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by Copyright Garrett Randall Lambert 2004 The Dissertation Committee for Garrett Randall Lambert Certifies that this is the approved version of the following dissertation: A TABU SEARCH APPROACH TO THE STRATEGIC AIRLIFT PROBLEM Committee: J. Wesley Barnes, Supervisor Leon S. Lasdon John J. Hasenbein Erhan Kutanoglu Christopher Bassham A TABU SEARCH APPROACH TO THE STRATEGIC AIRLIFT PROBLEM by Garrett Randall Lambert, B.S., M.S.I.E. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin December, 2004 Dedication To Bea, Connor and Gavin Acknowledgements I would like to thank my advisor Dr. J. Wesley Barnes for his encouragement, insight and, above all, his patience. His unwavering support and understanding helped make this dissertation possible. To the gang who made up the War Room at UT, my thanks for making the abstruse clear and the hard times fun. To my parents, Harry and Joan Lambert, thank you, as always for your support and belief in me. To my siblings, Heather and Priscilla, thanks for your words of encouragement, humor and commiseration on coding and dissertation woes. To my in-laws, Harry and Josephine Nieder, thanks for your support and prayers. Pop, I know you did not live to see this completed but I can now report Mission accomplished, Sergeant Major. To my two wonderful sons, Connor and Gavin, thanks for putting up with my absences from so many of the great things you do. I am so proud of you both. To my wife Bea, my rock and best friend, all my love and thanks for your patience and support. Your selfless efforts to hold things together at home while managing a day job in the Army still leaves me in awe. I will never forget your sacrifice. v A TABU SEARCH APPROACH TO THE STRATEGIC AIRLIFT PROBLEM Publication No._____________ Garrett Randall Lambert, Ph.D. The University of Texas at Austin, 2004 Supervisor: J. Wesley Barnes Air Mobility Command (AMC) planners currently use simulation systems or large-scale linear programming (LP) models in studying the strategic airlift problem. Simulations are descriptive in nature and therefore cannot prescribe optimal flight schedules. Aggregation is used in large-scale LP models to make the problem tractable and thus much operational level detail is lost. AMC planners need a tool which prescribes good solutions while maintaining the operational level detail necessary to produce flight schedules. This research outlines a robust algorithm that obtains excellent solutions to the strategic airlift problem that possess the operational level detail necessary for AMC planners to develop the detailed routing and scheduling of strategic airlift aircraft. algorithm utilizes the tabu search methodology. The vi Table of Contents List of Tables.......................................................................................................... xi List of Figures .......................................................................................................xii Chapter 1: 1.1 1.2 1.3 1.4 Introduction .................................................................................. 1 Historical Background........................................................................... 1 The Airlift System................................................................................. 2 The Strategic Airlift Network ............................................................... 4 Strategic Airlift Modeling Interest ........................................................ 8 1.4.1 Strategic Environment Impact...................................................... 8 1.4.2 Current Strategic Airlift Models ................................................ 10 1.4.3 Motivation .................................................................................. 12 1.5 Chapter 2: 2.1 2.2 2.3 Research Objective.............................................................................. 13 Literature Review....................................................................... 15 Strategic Airlift Literature ................................................................... 15 Strategic Airlift as a General Vehicle Routing Problem ..................... 19 Solution Approaches to General Vehicle Routing Problems .............. 23 2.3.1 Exact Algorithms........................................................................ 23 2.3.2 Classical Heuristics .................................................................... 25 2.3.2.1 2.3.2.2 2.3.2.3 Constructive Heuristics ............................................... 25 Improvement Heuristics .............................................. 26 Composite Heuristics .................................................. 27 2.3.3 Constraint Programming ............................................................ 28 2.3.4 Metaheuristics ............................................................................ 30 2.3.4.1 2.3.4.2 2.3.4.3 2.3.4.4 Simulated Annealing ................................................... 30 Genetic Algorithms ..................................................... 31 Tabu Search................................................................. 32 Tabu Search Variations ............................................... 33 vii 2.3.4.5 2.3.4.6 Chapter 3: 3.1 Tabu Search Applications to the GVRP...................... 36 Adaptive Tabu Search Approach to the SAP .............. 38 Detailed Problem Statement....................................................... 41 The Strategic Airlift Problem.............................................................. 41 3.1.1 SAP Inputs.................................................................................. 41 3.1.1.1 3.1.1.2 3.1.1.3 3.1.1.4 3.1.2.2 3.1.2.3 3.1.2.4 3.1.2.5 3.1.3.1 3.1.3.2 Requirements............................................................... 41 Aircraft ........................................................................ 43 Locations ..................................................................... 45 Routes.......................................................................... 47 A Standard Flight Leg ................................................. 50 Calculating Maximum Cargo for a Route ................... 51 Calculating Fuel Requirements for a given payload ... 59 Determining Aircraft Payloads.................................... 60 Determining Timings for a Mission ............................ 64 An Example Mission................................................... 66 3.1.2 Additional SAP Characteristics.................................................. 50 3.1.3 SAP Building Blocks: Missions................................................ 63 3.1.4 SAP Decisions............................................................................ 70 3.2 SAP Representation............................................................................. 71 3.2.2 Pure Node Based Approach ....................................................... 72 3.2.3.Arcs.. .......................................................................................... 74 3.3 Chapter 4: 4.1 4.2 4.3 4.4 Summary ............................................................................................. 76 A Tabu Search Approach to the Strategic Airlift Problem ........ 77 Tabu Search Architecture.................................................................... 77 4.1.1 JAVATM Software Programming Language and OpenTS ......... 77 SAP-TS Data Structures...................................................................... 79 SAP-TS Solution Structure ................................................................. 82 Initial Solution Construction ............................................................... 83 viii 4.5 4.6 Objective Function .............................................................................. 85 Move Neighborhoods .......................................................................... 87 4.6.1 SAP-TS Neighborhood Phases................................................... 88 4.6.2 New Mission Insert Move Neighborhood (NMI) ...................... 89 4.6.3 Between Pair Swap Move Neighborhood (BPS) ....................... 92 4.6.4 Within Pair Insert Move Neighborhood (WPI).......................... 94 4.6.5 Between Pair Insert Move Neighborhood (BPI) ........................ 97 4.6.6 Within Mission Arc Swap Move Neighborhood (WMAS) ....... 98 4.6.7 Load Reallocation Move Neighborhood (LR) ........................... 99 4.6.8 Recover to APOE Arc Insert Move Neighborhood (RAI) ....... 100 4.6.9 Impose Time Delay Move Neighborhood (ITD) ..................... 101 4.7 4.8 4.9 Tabu Structure ................................................................................... 103 SAP Tabu Search Algorithm............................................................. 106 Mission Assignment/Scheduler Heuristic ......................................... 110 4.10 Move Manager .................................................................................. 114 4.11 An Iteration in the SAP-TS Algorithm ............................................. 117 4.12 Summary ........................................................................................... 118 Chapter 5: 5.1 5.2 An Application of the SAP Tabu Search Algorithm................ 120 SAP Problem Instances ..................................................................... 120 An Example Strategic Airlift Problem .............................................. 121 5.2.1 Scenario 5 Details..................................................................... 122 5.2.2 SAP-TS Results for Example Problem .................................... 125 5.2.3 AMOS Results for Example Problem ...................................... 128 5.3 Other SAP Problem Instances ........................................................... 130 5.3.2 Scenario 1................................................................................. 131 5.3.3 Scenario 2................................................................................. 133 5.3.4 Scenario 3................................................................................. 135 5.3.5 Scenario 4................................................................................. 137 5.4 Extended Analysis of Scenario 2 ...................................................... 139 ix 5.5 Chapter 6: 6.1 6.2 6.3 6.4 Conclusion......................................................................................... 150 Conclusion................................................................................ 151 Major Contributions .......................................................................... 151 Further Enhancements to the SAP-TS Algorithm............................. 152 Extensions to other aspects of the SAP ............................................. 154 Summary ........................................................................................... 155 Appendix A: SAP-TS Best Solution Scenario Mission Summaries .................. 157 A.1 Scenario 1 Best Solution Mission Summary..................................... 157 A.2 Scenario 2 Best Solution Mission Summary..................................... 161 A.3 Scenario 3 Best Solution Mission Summary..................................... 170 A.4 Scenario 4 Best Solution Mission Summary..................................... 177 A.5 Scenario 5 Best Solution Mission Summary..................................... 181 References ........................................................................................................... 185 Vita .................................................................................................................. 202 x List of Tables Table 3.1 Table 3.2 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 4.7 Table 4.8 Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 5.5 Table 5.6 Table 5.7 Table 5.8 Table 5.9 Table 5.10 Table 5.11 Table 5.12 Table A.1 Table A.2 Table A.3 Table A.4 Table A.5 Route Segment Details for KGRK - OKBK Route.................... 52 C-5A Standard Weights and Fuel............................................... 53 SAP-TS Objects and their Attributes ......................................... 80 Aircraft and Missions Available ................................................ 90 APOE-APOD Pairs and Assigned Missions .............................. 91 APOE Arrival Times .................................................................. 95 Tabu Search Parameters ........................................................... 108 Problem Instance Parameters ................................................... 109 Arcs Connecting KDOV to OKBK .......................................... 112 Neighborhood Selection After Phase Change .......................... 117 Scenario Characteristics ........................................................... 121 Example Problem Available Aircraft ....................................... 122 Example Problem Requirements .............................................. 123 Example Problem Route Segments .......................................... 125 Example Problem Tabu Search Settings .................................. 126 Example Problem Initial Solution Objective ........................... 127 Example Problem Best Solution Objective.............................. 128 Example Problem AMOS & SAP-TS Objective...................... 129 AMOS and SAP-TS Results for Scenarios 1 through 4........... 131 Number of Required C17 Missions by Day for Scenario 2 ..... 142 Scenario 2 Aircraft Arrivals by Type ....................................... 145 Scenario 2 Working MOG Violation Bases (SAP-TS) ............ 147 Scenario 1 Mission Summary .................................................. 161 Scenario 2 Mission Summary .................................................. 169 Scenario 3 Mission Summary .................................................. 177 Scenario 4 Mission Summary .................................................. 181 Scenario 5 Mission Summary .................................................. 184 xi List of Figures Figure 1.1 Figure 1.2 Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 Figure 3.9 Figure 3.10 Figure 3.11 Figure 3.12 Figure 3.13 Figure 3.14 Figure 3.15 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.11 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15 Airlift Mission Classifications (AFDD 2-6.1 1999)..................... 3 Airlift Network Example.............................................................. 7 Extract from AMOS Payload Target File for C-5A ................... 42 Region Map ................................................................................ 46 Route Segment Extract from AMOS Plan File .......................... 48 Standard Flight Leg with Segments ........................................... 50 Climb Data Extract from fuelcalc.dat File for a C-5A............... 54 Cruise Data Extract from fuelcalc.dat file for a C-5A ............... 55 Example Maximum Cargo Calculation for a C-5A ................... 59 Extract from Requirements File ................................................. 60 Example Flight Leg Timings...................................................... 65 Route Segment Joining Region 21 to Region 20 ....................... 67 Extract from Payload Target File for C-5As .............................. 68 Route Segment for OKBK to LEMO ......................................... 69 Route Segment for LEMO to KDOV......................................... 70 SAP Representation using Pure Nodal Approach ...................... 72 SAP Representation using Arcs.................................................. 75 OpenTS Required Elements and Associated SAP-TS Classes .. 78 OpenTS Architecture (Harder 2000).......................................... 79 Pseudo Code for Initial Solution ................................................ 84 WPI Insert Points for Mission 0................................................. 97 RAI Move................................................................................. 100 Mission Assignment/Scheduler Heuristic Pseudo Code .......... 111 Determination of Neighborhood Phase and Search Context.... 115 Example Problem Requirements and Aircraft Available ......... 124 Example Problem PAX & Cargo Closure................................ 130 Scenario 1 Total Daily Requirements and Aircraft Available . 132 Scenario 1 PAX & Cargo Closure............................................ 133 Scenario 2 Total Daily Requirements and Aircraft Available . 134 Scenario 2 PAX & Cargo Closure............................................ 135 Scenario 3 Total Daily Requirements and Aircraft Available . 136 Scenario 3 PAX & Cargo Closure............................................ 137 Scenario 4 Total Daily Requirements and Aircraft Available . 138 Scenario 4 PAX & Cargo Closure............................................ 139 SAP-TS Scenario 2 Solution with Time Windows .................. 140 C17 Missions by Time Window and APOE-APOD Pair......... 143 SAP-TS Missions vs C17 Equivalent Missions ....................... 145 AMOS Missions vs C17 Equivalent Missions......................... 146 SAP-TS Working MOG for OKBK, LEMO and LICZ ........... 149 xii Chapter 1: Introduction Airlift is the transportation of personnel and materiel through the air and can be applied across the entire range of military operations in support of national objectives. A key function of the Air Force, airlift provides global reach for US military forces and the capability to quickly apply strategic global power to various crisis situations worldwide by delivering necessary forces. Air Force Doctrine Document 1: Air Force Basic Doctrine This dissertation describes an adaptive tabu search (ATS) approach to the Air Mobility Command s strategic airlift problem (SAP). To facilitate that description, the next section provides a brief historical background of the SAP followed by descriptions of the airlift system and the strategic airlift network. 1.1 HISTORICAL BACKGROUND Historically, strategic lift (airlift and sealift) has been seriously considered only in the crucible of war. Strategic airlift, the more responsive of the two, has fared no better than sealift. Because of the large, time-sensitive mobilization requirement placed upon the airlift network, the strategic airlift fleet often finds itself in the unenviable position of playing catch-up at the start of conflict. The U.S. Army championed initial interest in strategic airlift in the mid-1930 s with the purchase of airplanes designed specifically to transport cargo (Callander, 1998). At the start of World War II, a mere 254 of the 12,297 aircraft in the Air Corps were for cargo transport. By the end of the war, strategic airlift was firmly established as a key component of U.S. air doctrine. 1 Throughout the Cold War and in recent times, strategic airlift has continued to play a pivotal role in the support of national military strategy. The Berlin Airlift is perhaps one of the most striking examples of the efficacy of strategic airlift in the furtherance of U.S. objectives. By the end of the Korean and Vietnam wars, strategic airlift assets were organized under a single command, the Military Airlift Command (later transformed into Air Mobility Command), and included aircraft from the Civil Reserve Air Fleet (CRAF). The buildup of forces for Operation Desert Storm stretched air mobility capabilities to the limit and represented the most massive airlift in the history of airpower (Callander, 1998). Recent operations in Afghanistan mark the first time in US history in which the United States moved its war machine overseas exclusively by air (Loeb, 2002). Over its varied history, strategic airlift has become many things--simple cargo transport, airborne drops, glider support, medical evacuation, and sophisticated aerial resupply methods, to name only a few. Today, it is an indispensable component of national military power. It is imperative to carefully manage and efficiently utilize available strategic airlift resources so that US air mobility capabilities will be equal to the daunting tasks of the future. 1.2 THE AIRLIFT SYSTEM Air mobility is composed of airlift, air refueling, and air mobility support the air mobility triad (AFDD 2-6, 1999). Air mobility forces are drawn from active duty, Air Force Reserve Command (AFRC), Air National Guard (ANG), and Civil Reserve Air Fleet (CRAF) components. The active duty 2 component is the most responsive and flexible. AFRC and ANG, when mobilized, are equivalent to the active component. CRAF provides important surge capacity during contingency and wartime operations. In return for this service, CRAF participating carriers are contracted for movement of passengers and cargo during peacetime (AFDD 2-6.1, 1999). Together, these components provide the aircraft, crews and support forces that make up the airlift system. The airlift system pictured in Figure 1.1 is an integrated hierarchical system that consists of intertheater, intratheater, and dedicated Joint Task Force (JTF) airlift (AFDD 2-6.1, 1999). Figure 1.1 Airlift Mission Classifications (AFDD 2-6.1 1999) The airlift system delivers personnel and cargo when and where they are needed. Airlift operations can be characterized as intertheater airlift, intratheater airlift, or Operational Support Airlift (OSA). Intertheater airlift provides airlift 3 linking theaters to the continental United States (CONUS) and to other theaters as well as the airlift within CONUS. These airlift assets are assigned to the Commander in Chief, United States Transportation Command (AFDD 2-6.1 1999). Since typical intertheater distances are large, strategic airlift is normally comprised of the heavy, longer range, intercontinental airlift assets. Intratheater airlift is that airlift assigned or attached to a combatant commander other than Commander in Chief, US Transportation Command, which provides air movement and delivery of personnel and equipment directly into objective areas to meet specific theater objectives and requirements (AFDD 2-6.1 1999). OSA operations are time-sensitive movements of high-priority personnel and small amounts of cargo. The problem of scheduling intertheater airlift operations, the SAP, is the focus of this research. 1.3 THE STRATEGIC AIRLIFT NETWORK The SAP is primarily concerned with intertheater airlift of personnel and cargo from CONUS to the theater of operations or from one theater to a different theater. The mission of the US Transportation Command (USTRANSCOM) is to provide air, land, and sea transportation for the Department of Defense (DOD) in peace and in war. Air Mobility Command is the USTRANSCOM component responsible for conducting airlift operations (AMC). At the outset of a contingency operation or war, USTRANSCOM receives personnel and equipment requirements from the Time Phased Force Deployment Document (TPFDD) developed by the combatant forces. The TPFDD specifies 4 such details as the personnel and equipment to be moved, their origin and destination locations, when they are available for loading at origin and when they are required to be delivered at their destination. Once USTRANSCOM determines the mode (air or sea) of transport, requirements are then distributed to the appropriate subordinate commands in USTRANSCOM--AMC for airlift requirements and Military Sealift Command (MSC) for sealift requirements. Planners at AMC then establish an airlift network to meet the movement requirements defined by USTRANSCOM. The airlift network consists of aircraft, aircrews, airfields, and support equipment and personnel. The amount and type of aircraft employed and the airfields used are a function of the airlift requirements derived from the TPFDD. Sources for aircraft may be either military or CRAF. The aircraft employed can be cargo and/or passenger aircraft. The airfields employed in the network include the origin and destination airfields outlined in the TPFDD for each requirement. Origin airfields are aerial ports of embarkation (APOEs). At the APOEs, personnel and equipment are loaded onto aircraft. For the SAP, APOEs typically are located within CONUS, where the bulk of US forces are stationed. Personnel and equipment are unloaded at the destination airfields, aerial ports of debarkation (APODs). The APODs are generally located within or near the theater of operations. Because the large distances between APOEs and APODs generally preclude direct flights, AMC incorporates a series of en route bases into the network that service the various APOE-APOD pairs. En route bases can be selected from existing bases or established for a particular contingency using air 5 mobility support forces. These en route bases are used for refueling of aircraft, changing and staging of aircrews and performing required aircraft maintenance. The number and location of en route bases utilized are determined by the APOEAPOD pairs that need to be serviced and the number of aircraft available. There is a special virtual en route base called a waypoint that may also be established in defining the airlift network. Waypoints are points in space through which an aircraft must fly. Among other uses, waypoints serve as rendezvous points for aerial refueling and as additional control measures to further specify a route (e.g., route aircraft around restricted airspace). Crews are not assigned to a particular aircraft. Rather, aircraft move continuously through the airlift network changing crews at en route bases as necessary. A typical mission cycle for an aircraft starts at its home base (the aircraft s permanent base) with the initial aircrew. The aircraft is then flown to an APOE where some portion of the requirement (cargo and/or passengers) is loaded. The aircraft is then flown along a route leading to the APOD, passing through a series of en route bases along the way to refuel, make repairs or change crews as necessary. Ultimately, the aircraft reaches the APOD and the cargo and/or personnel are unloaded. The aircraft is then flown to a recovery base for any required maintenance and crew change. At this point, the aircraft may return to its home base or it may fly to another APOE for the start of a new mission. In either case, the aircraft will again pass through a series of en route bases for refueling, repair and crew changes as necessary. For the SAP, aircrews are, in general, not the bottleneck in the system. There is roughly a three to one ratio of 6 aircrews for military aircraft. Therefore, once aircraft and requirements are scheduled, the crew scheduling problem can be solved independently. A key constraint on the flow through the airlift network is the limited capacity of the various airfields to service arriving aircraft. Among the finite resources available for servicing aircraft are parking spaces, material handling equipment (MHE), available fuel, and support personnel. To simplify modeling this constraint, AMC planners use an aggregate number called maximum-on-theground (MOG) to capture the number of aircraft that can be simultaneously serviced (worked) or parked or refueled at the airfield. APOE 1 ER 1/ CS 1 RB 1 WP 1 HB 1 APOD 1 APOE 2 ER 2/ WP 1 CS 2 Key: RB WP HB ER CS Recovery Base Waypoint Home Base Enroute B ase Crew Staging Base Figure 1.2 Airlift Network Example Working MOG is the number of aircraft that can be simultaneously worked using MHE, servicing equipment, and support personnel so that all the 7 aircraft can leave the airfield within their allotted ground times. Parking MOG refers to the maximum number of aircraft that can be accommodated on the airfield. Finally, fuel MOG is the maximum number of aircraft that can be fueled simultaneously. Typically, working MOG is more restrictive than parking MOG and fuel MOG. An example of an airlift network is depicted in Figure 1.2. 1.4 STRATEGIC AIRLIFT MODELING INTEREST 1.4.1 Strategic Environment Impact History is replete with examples of the tremendous impact strategic airlift has had upon the accomplishment of national policy and objectives. Notable examples of the efficacy and importance of strategic airlift include the Berlin Airlift, support of Operations Desert Shield/Storm, and most recently the support of the war effort in Afghanistan and Iraq. In each of these cases, strategic airlift has played a significant role in the enhancement of US national objectives (Rower, 2001). Three primary factors influence strategic airlift today and create a need for the efficient use of air mobility assets. The first of these factors is the posture of US military forces. Over the last decade, the US military has de-emphasized forward deployment and has emphasized force projection. As a result, the ability to rapidly project large numbers of forces, strategic mobility, is even more critical. Air mobility is the most flexible and responsive means to rapidly project and sustain combat forces during peace and war. 8 A projected shortfall in military airlift capability is the second factor influencing the strategic airlift. In 1998, the Department of Defense initiated a comprehensive two-year study, the Mobility Requirements Study-2005 (MRS05), to determine strategic mobility requirements based on 2005 force structure. For strategic airlift, the accepted measure of capacity to deliver cargo (in tons) through a distance (in miles) in a period of time (in days) with a given fleet of vehicles is million ton-miles per day (MTM/D). MRS-05 predicts that US strategic airlift requirements will exceed the current airlift capability of 49.7 MTM/D by more than 5 MTM/D. Procurement of additional aircraft, modernization of existing aircraft and improvements to base infrastructure are among the many strategies established to deal with this shortfall. The third factor is the changing strategic environment. The US National Military Strategy (NMS) is evolving in response to the dynamic nature of global threats to US interests. US NMS mandates the ability to prosecute two nearly simultaneous major theater wars (MTWs). This presents a significant challenge for air mobility planners. Additionally, recent experience (e.g., Afghanistan) indicates that smaller scale contingencies (SSCs) are increasingly likely scenarios for military forces. SSCs are characterized by austere airfields and ports that limit the introduction of forces in theater. Thus, in this environment, even if the MTM/D requirement of above were met, infrastructure limitations could largely reduce the capacity of the airlift network. Thus SSCs, which generally have smaller requirements than an MTW, can still exert a major strain on the air mobility system. 9 The US military s transition from forward deployment to force projection, the projected airlift shortfalls, and a dynamic strategic environment together place a premium on the importance of effective employment of air mobility assets. In recent testimony to the US Senate Armed Services Committee on US operations in Afghanistan, General Thomas Franks, commander of US Central Command, stated, Strategic airlift remains key to current and future military operations. We are on a glide-path to expand our strategic airlift capabilities and must remain committed to the task. 1.4.2 Current Strategic Airlift Models Over the years, AMC has implemented a number of models to assist planners in assessing a particular airlift network s capacity, mission planning, and in developing aircraft schedules. The focus here will be on models that are used as aids in the SAP. Many of these models are legacy systems that have been retained and improved upon over the years. As detailed below, AMC uses ADANS, CAMPS, AFM, ACAS, JFAST and AMOS in various contexts to arrive at a plan for a given TPFDD requirement. A brief synopsis of each of these models follows. The Airlift Deployment Analysis System (ADANS), developed by Oak Ridge National Laboratory, was implemented in 1990 to support the scheduling of airlift missions for Operations Desert Shield/Storm. ADANS was the first major effort to automate scheduling of strategic airlift. ADANS incorporates a dynamic programming approach along with a greedy heuristic that sequentially builds missions. In 2002, the Consolidated Air Mobility Planning System (CAMPS) replaced ADANS adding the aerial refueling problem solution to the SAP by joining ADANS 10 with the Combined Mating and Ranging Planning System (CMARPS) for tanker scheduling. The Airlift Flow Model (AFM) is a legacy stochastic simulation model that incorporates stochastic elements such as winds and maintenance variability. AFM is a very complex, user-unfriendly model and is now obsolete. The Airlift Cycle Analysis Spreadsheet (ACAS) is a single cycle deterministic spreadsheet model used as a quick look tool. A major limitation of ACAS is that it is restricted to a single APOE/APOD pair. The Joint Flow Analysis System for Transportation (JFAST) is a dynamic programming based multimodal transportation analysis model designed for USTRANSCOM and implemented in 1991. It is used to determine transportation requirements, perform course of action analysis, and project delivery profiles of troops and equipment by air, land, and sea. AMC uses JFAST to analyze the airlift transportation requirements for deploying US forces and predict their arrival dates in-theater. JFAST can be used to determine the transportation feasibility of airlift deployment plans. Finally, JFAST may also be used as a TPFDD editor. The Air Mobility Operations Simulation (AMOS) is a developing stochastic simulation model that will eventually replace AFM. AMOS is being built to support in-depth analysis of global mobility requirements and deployment capabilities to include resource allocation, aircrew availability, route optimization, aerial refueling, and intra-theater airlift. AMOS is an ambitious effort that will attempt to tie all elements of AMC scheduling together airlift, air refueling, and air mobility support. While these models address some of AMC s planning needs, they are incapable of obtaining effective detailed solutions to realistic instances of the SAP. ADANS/CAMPS are used primarily for deliberate planning at AMC plans that are developed before a particular crisis or contingency occurs. Because ADANS/CAMPS and JFAST are based upon a dynamic programming approach, 11 they are particularly susceptible to the curse of dimensionality. As a result, realistic problem sizes are reduced through aggregation and the use of a greedy myopic heuristics (for pairing aircraft to requirements) to reduce the solution space. Solutions obtained are feasible but not necessarily good. Since ACAS is limited to a single APOE/APOD pair, it is not sufficient for obtaining solutions to realistic SAP instances. Thus AFM (and later AMOS) is the primary tool used to solve the SAP. A major drawback of simulation models like AFM and AMOS is that they do not describe what is best. Instead, these models enable analysts to investigate what if scenarios which, over a multitude of runs, can lead to insight into the airlift system. In the context of the SAP, AFM and AMOS proceed by loading cargo on the first available aircraft that is then routed according to a pre-selected prioritized list a greedy myopic measure of goodness. Solutions obtained in this manner, while feasible, are generally not very good. 1.4.3 Motivation Current models that incorporate simulations or traditional optimization methods do not provide, for a realistic problem size, a near-optimal combination of aircraft and routes for each requirement at a sufficient level of detail. AMC requires an AMOS compatible methodology that rapidly provides excellent solutions to realistic instances of the SAP with sufficient detail to build a mission schedule for each aircraft. The method must have the capability to address multicriteria objectives such as minimizing the time required for unit closure (delivery of a specified fraction of the unit to the APOD), minimizing aircraft 12 utilization and maximizing cargo throughput while still meeting TPFDD delivery requirements. Tabu search has been shown to be particularly effective in solving large scale combinatorial optimization problems related to the SAP. Recent work by Barnes, Wiley, Moore and Ryer (2004) on the Aerial Flight Refueling Problem (AFRP), Crino, Moore, Barnes and Nanry (2004) on the Theater Distribution Vehicle Routing and Scheduling Problem (TDVRSP), and Combs (2002, 2004) on the Crew Scheduling Problem (CSP) clearly demonstrate that a tabu search approach can provide robust, effective and efficient solutions to practical large scale combinatorial optimization problems. Thus an ATS approach to the SAP will provide an effective methodology to fill the gaps left by current models at AMC by providing timely, robust, detailed and excellent solutions to the SAP. 1.5 RESEARCH OBJECTIVE The primary objective of this research is to develop a robust and effective solution methodology that solves the Strategic Airlift Problem. The solution will dictate the detailed assignment of cargo to aircraft and the routing and scheduling of aircraft missions through the network. There are several supporting objectives for this research. The first supporting objective is to develop an effective and extensible representation of the SAP. Since the SAP instances are defined by the AMOS input files, the methodology must mirror many of the modeling aspects and problem structure imposed by AMOS. This representation captures essential aspects of the SAP yet remains tractable for application of the tabu search methodology. 13 The SAP representation can be extended to include additional aspects of the SAP not incorporated in this research such as transloading operations. A second supporting objective is to develop a tabu search algorithm that solves the SAP efficiently and effectively. The algorithm uses an adaptive tabu search approach incorporating effective neighborhoods and neighborhood selection schemes to provide the user with a suite of excellent solutions. The third supporting objective is to code the SAP algorithm in the JavaTM software programming language. The JavaTM language portability and re- useability facilitates use of the software on different platforms and expansion of the software. Using Harder s (2000) tabu search architecture as the framework, the SAP software implements the SAP tabu search algorithm providing the user with detailed solutions to SAP instances. 14 Chapter 2: 2.1 Literature Review STRATEGIC AIRLIFT LITERATURE Surprisingly, despite the premium placed on airlift capabilities, there is little in the literature on the SAP. As discussed in Section 1.4.2, most of the work on the SAP at AMC has centered on the use of simulation models. However, exact approaches such as linear programming (LP) and hybrid approaches, that combine exact methods with heuristics, have also appeared in the SAP literature. The following is a summary of the pertinent work on the SAP in the literature. The first major effort to automate the scheduling of strategic airlift, ADANS, was developed by Hilliard et al. (1992) of Oak Ridge National Laboratory just prior to Operation Desert Shield/Storm. ADANS, a hybrid method, incorporates a dynamic programming approach that uses a sequential procedure that schedules one mission at a time taking into consideration the constraints imposed by previously scheduled missions. Rule based heuristics are used to build and sort pairings of aircraft with requirements. A weighted objective function is used which considers minimization of such things as late deliveries, mission flying time, and the number of crew changes. There are several limitations to this model. Clearly, due to the scale of strategic airlift problems, a dynamic programming approach is particularly susceptible to the curse of dimensionality. Additionally, the aircraft-requirement pairing heuristic is a greedy (local) approach that restricts, a priori, the network configurations that can be considered. Although ADANS is still in use at AMC today, its use is 15 largely relegated to feasibility studies and force structure issues instead of actual scheduling of aircraft in a crisis. Rappoport et al. (1992) developed a myopic greedy airlift planning heuristic (APH) for incorporation into ADANS that first assigns requirements to aircraft and then routes and schedules the aircraft. Captured in a plane preference value, assignment of a requirement to an aircraft is based upon the idea of best fit in terms of the most constraining restriction associated with either weight, volume, or square feet. Rappoport et al. (1991) provide a detailed description of plane preference values. Once aircraft-requirement pairings are made, the heuristic attempts to obtain feasible routes (up to two for each aircraft) for up to five aircraft of the preferred type using a shortest path approach. Once feasible schedules are obtained, crew constraint feasibility is verified. Feasible schedules are assessed using a weighted objective function that incorporates total mission time, total on ground delays and time of delivery in excess of the RDD. The APH heuristic represented a significant improvement over the level of detail and the efficiency of methods in use by AMC at the time. However, it is limited by its greedy myopic underpinnings that significantly restrict the solution space. There are several linear programming (LP) based models described in the literature that were implemented to support studies at levels above AMC such as at the USTRANSCOM or Joint Staff level. As an alternative or in concert with detailed simulations, LP can be used to answer many questions concerning the Air Mobility system. However, they do not to provide operational level detail, rather 16 they provide insights into the airlift network. These models are generally built to assess more global considerations such as: (1) Where are the bottlenecks in the airlift network? (2) What is the best force structure (considering such things as fleet mix and airfield support team mix) given the requirements outlined in the National Military Strategy? (3) What is the force closure capability for a given fleet of aircraft over a specified network? These are broad level questions that are generally answered using broad models. As a consequence, key features of the problem are aggregated so that realistic instances can be addressed. A significant amount of the literature on the SAP originates from the Naval Post Graduate School. Yost (1994) developed a flow optimization model in the General Algebraic Modeling System (GAMS) called THRUPUT. THRUPUT is a time static model of the SAP on a general routing network. In Morton et al. (1996), several improvements to Yost s THRUPUT model are developed into a model called THRUPUT II. THRUPUT II formulates the SAP as a multi-period, multi-commodity network-based LP model that includes a number of side constraints such as aircraft & airfield capacity constraints, aircraft utilization constraints, and aircraft balance constraints. To capture the time dynamic nature of the SAP, THRUPUT II makes use of a time index to track aircraft locations in each discretized time period. To overcome the intractability issue common with realistic SAP instances, an aggressive preprocessing step is 17 used to reduce the model size through removal of unnecessary variables/constraints and data aggregation. Additionally, some key assumptions are made in THRUPUT II to reduce problem size and they include the aggregation of airfield capacity into working MOG and the use of deterministic ground times. THRUPUT II trades the high level detail and the inclusion of uncertainty of simulation models for the rapid answers to broad questions like those listed above. THRUPUT II was successfully used by the Air Force Studies and Analyses Agency (AFSAA) to analyze procurement issues for the C-17 Defense Acquisition Board (Rosenthal et al. 1997). Baker et al. (2002) further enhance the THRUPUT II model by melding the best of THRUPUT II and a RAND Corporation model called CONOP (CONcept of Operations). Like THRUPUT II, CONOP is a time dynamic model on a general airlift network. Unlike THRUPUT II, CONOP includes aerial refueling, crew constraints, transloading options, and the use of recovery bases as an option. A key drawback of CONOP is that it does not have sufficient resolution of the cargo. The result of the marriage of THRUPUT II and CONP was NRMO (NPS/RAND Mobility Optimizer). As in THRUPUT II, NRMO incorporates aggregation to reduce problem size. Baker et al. report several key insights gained while using NRMO. Chief among these is the impact of winds on throughput. Since NRMO incorporates many of the major features of the SAP and is computationally efficient, it has been successfully used in a variety of analyses to assist Air Force planners in answering questions concerning force structure and airfield resources issues. 18 2.2 STRATEGIC AIRLIFT AS A GENERAL VEHICLE ROUTING PROBLEM The General Vehicle Routing Problem (GVRP) is a useful paradigm for viewing the SAP. The GVRP is any problem that seeks to determine the best (as measured by some objective) routing of vehicles to satisfy a given set of customers. More fully, the GVRP can be expressed as follows: given a set of customers requiring a visit, and a fleet of vehicles based at a depot that can perform the visits, construct a set of routes for the vehicles that minimizes the costs of operation. The objective function is typically expressed as costs related to the number of vehicles and to distance traveled. Constraints include various capacity constraints on weight, volume, length; time constraints on when the customer will accept a visit; and the total length of routes. In practical problems there may be different kinds of additional constraints including legislative restrictions, established work practices and customer preferences, and a complex objective function reflecting complicated pay provisions. The Traveling Salesman Problem (TSP), the Vehicle Routing Problem (VRP) and the Pickup and Delivery Problem (PDP) all fall within the GVRP class. Distinctions between classes are due to the manner in which vehicles, customer service, and costs are defined. For example, in the TSP, the problem is to find a route for a single vehicle (salesman) with unlimited capacity that visits (services) each customer once at least cost (generally measured as distance traveled). For the PDP, the vehicle has a finite capacity and services to customers entail precedence constraints that ensure that an item is picked up before it is delivered. Variations of the TSP, VRP and PDP are numerous and difficult to 19 categorize. These variations include such considerations as multiple vehicles, multiple depots, time windows, and route length constraints that greatly increase the complexity of the GVRP. Carlton (1995) extends the classification of vehicle routing and scheduling problems by Bodin et al. (1983) by providing a multi-tiered framework to characterize the GVRP hierarchy. In this hierarchy the TSP, VRP and PDP represent succeeding floors where the stairs from the TSP to the VRP are formed in the presence of capacity constraints and the stairs from the VRP to the PDP reflect the existence of precedence constraints. includes other typical elements of GVRPs such as: 1. Number of vehicles a single vehicle (SV) or multiple vehicles (MV) 2. Type of vehicles homogeneous (same type) vehicles (H) or different types of vehicles ( H ) 3. 4. Number of depots a single depot (SD) or multiple depots (MD) Route Length (RL) constraints distance or amount of time vehicles may travel 5. Time Windows (TW) service for a customer must fall within a designated time window specified by an earliest and latest time to start a service Using Carlton s GVRP classification hierarchy, the SAP can be partly classified as a MV H , MD, PDP, with RL and TW constraints. Crino (2002) extends Carlton s hierarchy by adding the following characteristics: single trip Within each floor, Carlton 20 (ST) per vehicle versus multiple trip (MT) per vehicle; single service (SS) per customer versus multiple service (MS) per customer; single commodity (SC) versus multiple commodity (MC); and hubs (H) or transshipment points. Aircraft in the SAP usually make multiple trips. Aircraft move through the network performing multiple trips during the course of a typical planning horizon. Thus, we may add multiple trips as a characteristic of the SAP. The SAP also entails multiple services per customer (APOE) as defined by the TPFDD. Typically requirements arrive at an APOE over time and each requirement is normally much larger than a single plane load. As a result, multiple services for each APOE are required. SAP requirements are multiple commodity items that have different fill efficiencies based upon commodity type. For example, the ammunition commodity type will usually exceed the aircraft weight constraint before volume is exceeded. Conversely, the airmobile commodity type (army helicopters) normally exceeds available floor space before weight constraints. Finally, in the case where transloading is permitted, the SAP will include hubs that serve both as APOEs and APODs. In this instance, PDP precedence constraints may no longer hold. This is because the PDP assumes that the same vehicle that picks up a requirement delivers it to its ultimate destination. Transloading permits one vehicle to pickup and deliver a requirement to an intermediate point where it is subsequently picked up and delivered to its destination by at least one different vehicle. Thus, in essence, some customer nodes may be both source nodes (a pickup point) and sink nodes (a delivery point). Customers of this type are hubs (H). All of these additional characteristics with the exception of transloading 21 (hubs) are included in the SAP solved in this dissertation. The additional characteristics (and the notation) are consistent with the extensions to Carlton outlined in Crino (2002). Thus, the basic SAP is classified as: MD, MV H , PDP, RL, TW, MT, MS, MC. There is an additional characteristic of the SAP not captured in the classification above. There are other route constraints embedded within the SAP that are a significant departure from simple route length constraints that constrain the amount of time or distance that a vehicle (aircraft) may travel. Parking MOG, working MOG, and fuel MOG constrain the timing of aircraft flow along routes. For the SAP these additional route constraints impose temporal restrictions on when aircraft may move along routes that are not specifically customer driven. For classification purposes we call these characteristics route time windows (RTW) to discriminate them from customer driven time windows. A key feature of the SAP is that it can be broken into two components or phases: the assignment of aircraft to requirements and the subsequent routing of assigned aircraft to service missions. Unfortunately the assignment and routing components are interdependent. We need to know the routes to determine the cargo that can be loaded and we need to know the APOE-APOD assignments before the routes can be determined. This observation is pivotal in selecting a suitable SAP solution representation that captures the critical assignment and routing information while avoiding a representation that is computationally unwieldy. 22 2.3 SOLUTION APPROACHES TO GENERAL VEHICLE ROUTING PROBLEMS There is an extensive amount of work in the literature on the GVRP. Several survey papers have been done over the years such as Bodin et al. (1983), Bodin (1990), Laporte et al. (2000). The number of references in Bodin et al. alone totals 699. The following summary captures the main advances in the GVRP literature over the years and focuses on the work done on the GVRP classes that are germane to this research. 2.3.1 Exact Algorithms A good survey of exact algorithms for VRPs can be found in Laporte and Nobert (1987). Exact approaches to the GVRP center around dynamic programming (DP) and Integer Programming (IP) formulations (branch and bound). Much of the initial work on the VRP focused on solving the VRP as a set-partitioning or set-covering problem. While such formulations are mathematically correct for solving the VRP, they are not practical due to the difficult and time consuming column enumeration requirement involved in solving even modest-sized problems (Bodin 1990). Balas and Padberg (1970) discuss a method of solving a set covering problem (constraints restricted to be equations or less than or equal to inequalities) formulated as an IP using a sequence of primal pivots instead of a cutting plane method. Balas and Padberg (1976) provide an extensive survey of the theoretical results and solution methods for the set partitioning or the equality constrained set covering problems. Desrochers et al. (1992) formulate the VRPTW as a set covering problem. In this approach, the LP relaxation of the set covering formulation is solved using 23 column generation. A branch and bound strategy is then used to find an integer solution to the set covering problem. They solve to optimality several instances of 100 customer problems. Typically, as reported in Desrochers et al., the optimal solution to the set covering problem is close to the LP relaxation solution thus making the branch and bound procedure very efficient. Bramel and Simchi-Levi (1997) demonstrate for the VRPTW why this relative gap is so small by showing that the gap becomes arbitrarily small as the number of customers grows for any distribution of service times, time windows, customer demands and customer locations. Belenguer et al. (2000) present an interesting variant of the VRP in which the demand of a customer can be serviced by more than one vehicle. This is called the Split Delivery Vehicle Routing Problem (SDVRP). They formulate the SDVRP as an IP and solve it using a polyhedral approach. They identify some facets and develop some other valid inequalities for the SDVRP polyhedron. Unfortunately, their cutting plane algorithm was based upon a relaxed formulation of the SDVRP and only lower bounds for the SDVRP are obtained. Problem sizes solved were relatively small instances up to 50 customers. Further details on these exact methods can be found in Larsen (1999) and Cook and Rich (1999). Because the VRP class (and by extension the GVRP class) of problems have been shown to be NP-hard (Lenstra & Rinooy Kan 1981), there is little hope of the existence of a polynomial time algorithm to solve them. In seeking the optimal solution, exact methods fail to meet practical time and computing resources due to problem size and complexity. As a consequence, 24 exact approaches are impractical for problems of realistic size and therefore heuristic approaches are generally favored over exact methods for GVRPs. 2.3.2 Classical Heuristics Classical heuristic approaches to the GVRP can be roughly separated into two categories: constructive or improvement heuristics (Laporte et al. 2000). In constructive heuristics the emphasis is on building a feasible solution without a separate improvement scheme. Improvement heuristics work on improving an incumbent solution by using some type of edge exchange heuristic within and between routes. Classical heuristics attempt to get a good solution quickly using a simple heuristic. Thus a shallower search of the solution space is characteristic of these methods when compared with metaheuristic methods. 2.3.2.1 Constructive Heuristics Perhaps one of the most well known of the constructive heuristics for the VRP is the Clarke and Wright (1964) savings algorithm. This method attempts to construct a VRP solution by merging routes through the use of a savings criterion for a pair of customers. Laporte et al. (2000) experimented with the sequential and parallel implementations of the Clark and Wright savings algorithm and report that the parallel implementation dominates the sequential version. Gaskell (1967), Yellow (1970), Golden et al. (1977), Paessens (1988), and Nelson et al. (1985) report variations of the Clark and Wright savings algorithm which, in general, attempt to be more efficient. Laporte et al. (2000) report on other constructive heuristics such as the petal algorithm (an extension to the sweep algorithm) first proposed by Balinski 25 and Quandt (1964), the sweep algorithm attributed to Gillett and Miller (1974) and cluster-first, route-second algorithms (Fisher and Jaikumar 1981). The insertion heuristic of Solomon (1987) is a quick and effective method that is still used in constructing initial feasible tours in metaheuristics. Bramel and SimchiLevi (1995) describe a variant of the cluster-first, route-second algorithm in which the seeds used for clustering vertices are determined by solving a capacitated location problem with the remaining vertices being incorporated into routes in a second stage. Bramel and Simchi-Levi report this two-phase heuristic to be asymptotically optimal but not competitive in a practical sense. The converse, route-first, cluster-second, has also been a popular method for solving the VRP. In this method, a large (generally infeasible) route is constructed which includes all the customers. This route is then partitioned in some fashion into smaller, feasible routes. Golden et al. (1984) is an example of the route-first, cluster second method applied to the fleet size and mix VRP. Laporte et al. (2000) performed computational comparisons of these constructive heuristics using the fourteen Christofides et al. (1979) benchmark instances for the VRP. They report good solutions using the savings algorithm of Clark and Wright (1964) and the sweep algorithms of Gillett and Miller (1974). 2.3.2.2 Improvement Heuristics Improvement heuristics for the GVRP generally attempt to improve a particular route or set of routes through some type of edge exchange mechanism. Edge exchange mechanisms reorder some subset of customers within a route or between routes. Lin s (1965) -opt mechanism is typical of the edge-exchange 26 methods applied to the TSP (equivalent to within route improvement for the VRP). For the -opt mechanism, edges are removed from the tour and the remaining segments are reconnected in all possible ways. Other well-known edge-exchange methods are the Or-opt (Or 1976), 2-opt* (Potvin and Rousseau 1995), 4-opt* (Renaud, Boctor and Laporte 1996) and the CROSS exchange (Taillard et al. 1997). 2.3.2.3 Composite Heuristics Composite heuristics blend both route construction and route improvement procedures. Generally, the construction and improvement heuristics are applied in sequence when solving VRPs. Bodin (1990) reports that this approach may not work well for the more tightly constrained GVRPs. This is because the sequential nature of construction heuristics means that a bad decision in the construction phase can adversely affect subsequent steps. Improvement procedures simply may take too long or even be unable to overcome a poor decision in the construction phase. Nevertheless, composite heuristics represent a major portion of the literature on the GVRP. Examples of composite approaches for the VRPTW can be found in Derigs and Grabenbauer (1993), Kontoravdis and Bard (1995), and Russell (1995). Renaud, Boctor and Quenniche (2000) report on a composite heuristic for the PDTSP (pickup and delivery TSP) that first inserts each delivery customer simultaneously with its associated pickup customer and then invokes an improvement procedure that uses the 4-Opt* improvement heuristic of Renaud, Boctor, & Laporte (1996). 27 2.3.3 Constraint Programming Lustig and Puget (2001) and Gendreau (2002) provide excellent primers for the OR community on Constraint Programming (CP) and discuss how CP differs from traditional OR approaches. The ILOG Optimization Suite White Paper (ILOG 2001) provides an example of a comprehensive software implementation of CP. CP, also called constraint logic programming, is an emerging approach to solving large-scale combinatorial optimization problems such as the GVRP class. CP is the study of computational systems based on constraints (Bartak 1999). In CP, the idea is to solve problems by simply stating the constraints and then finding a solution that satisfies the constraints a constraint satisfaction problem (CSP). A CP algorithm has a two level architecture. The first level entails statements of constraints over the variables of the problem. In this level, a sort of handshaking between constraint propagation and domain reduction algorithms occur. Constrain propagation is the modification of all constraints that contain a variable whose domain has been modified. The domain reduction algorithm modifies, for each constraint, the domains of all variables in that constraint given the change in the domain of one of the variables in that constraint (Lustig and Puget 2001). A form of backtracking is used to retreat from domain reductions leading to unsatisfiable (infeasible) constraints. The handshaking between constraint propagation and variable domain reduction is carried out in an iterative manner so as to reduce variable domains as much as possible without removing potential solutions to the CSP. The second level pertains to the methodologies 28 used to program the search strategies. While the first level can determine if the CSP is infeasible, it does not necessarily find a solution if one exists. Search strategies (depth-first search, best-first search (Nilsson 1971), limited-discrepancy search (Harvey and Ginsberg 1995), depth-bounded-discrepancy search (Walsh 1997) and interleaved depth-first search (Meseguer 1997) are examples of some of the search strategies that have been implemented in commercial CP systems. Two major positive features of CP are expressivity and flexibility (Gendreau 2002). Expressivity implies that complex problems are very easy to describe (express) in a natural fashion. Flexibility means that difficult problems can be solved without having to create new algorithms. Numerous and complex side constraints, typical of the GVRP class, can be easily incorporated without requiring major revisions to the algorithmic machinery. A third positive feature of CP, called openness by Gendreau, means that CP can be fairly easily combined with other approaches, like local search heuristics, in a manner consistent with composite or hybrid approaches. Gendreau (2002) cautions that CP should not be used indiscriminately. In particular, CP can be very computationally expensive when applied to welldefined (especially loosely constrained) problems (Pesant, Gendreau, and Rousseau 1997). If one has to repeatedly solve similar instances of a problem for which other effective solution approaches exist or can be developed, I would not recommend using CP. (Gendreau 2002). In addition, mastering the technique and language of CP which is based on logical relationships rather than equations can be formidable. Lustig and Puget (2001) report that CP is better 29 than IP in applications that concern sequencing and scheduling as well as for problems which are in essence CSPs (tightly constrained). Current efforts in CP appear to be oriented towards combining CP with classical OR approaches and/or heuristics (hybrid approaches) in an attempt to capitalize on their respective strengths. Focacci, Lodi and Milano (2002) provide an overview of work done in combining CP with traditional OR methods. Examples of hybrid approaches to VRPs using CP combined with heuristics are reported in Rousseau, Gendreau and Pesant (2002), Backer et al. (2000), Caseau and Laburthe (1999) and Shaw (1998). 2.3.4 Metaheuristics According to Osman and Kelly (1996): A metaheuristic is an iterative master process that guides and modifies the operations of subordinate heuristics to efficiently produce high quality solutions. It may manipulate a complete (or incomplete) single solution or collection of solutions at each iteration. The subordinate heuristics may be high (or low) level procedures, or a simple local search, or just a construction method. Over the past decade, metaheuristics have been particularly popular and effective in obtaining timely and effective solutions to the GVRP class. The more popular metaheuristic approaches are simulated annealing (SA), genetic algorithms (GA) and tabu search (TS). A brief survey of the GVRP literature concerning these three approaches follows. 2.3.4.1 Simulated Annealing SA employs the cooling or annealing of solids as a paradigm for search in combinatorial optimization problems--different states of the solid correspond to 30 different solutions. Eglese (1990) provides a detailed theoretical treatment of SA. Koulmas, Antony and Jaen (1994) provide a survey of SA applications to a series of scheduling and routing problems among others. Chiang and Russell (1996) implement a SA approach to the VRPTW using the -interchange mechanism of Osman (1993) and a modified k-node interchange of Christofides and Beasley (1984). Osman s -interchange mechanism outperformed the modified k-node interchange on the larger problem instances. It can be proven that simulated annealing converges asymptotically to the optimal solution. Unfortunately, such convergence requires exponential time (Aarts and van Laarhoven 1985). Thus, in practice, simulated annealing is used with faster cooling schedules (not guaranteed to converge to the optimal) and thus it behaves like an approximation algorithm. To counter the faster cooling schedule, hybrid SA algorithms embed other methods to improve performance. Li and Lim (2001) is an example of an SA approach to the PDPTW that has an embedded TS procedure to restart the procedure after a series of non-improving iterations. SA relies heavily on a single control parameter, the temperature, and the cooling schedule the rate of descent. Most importantly, SA does not make use of or exploit memory a major drawback. For large scale problems like the SAP, knowledge gained during the course of a search can be invaluable in selecting appropriate neighborhoods and improving efficiency. 2.3.4.2 Genetic Algorithms Goldberg (1989) and Kolen and Pesch (1994) provide good introductory information on GAs. Thangiah et al. (1991) and Thangiah (1995) introduce a 31 successful GA approach to the VRPTW called GIDEON. Blanton and Wainwright (1993) introduce an implementation of a GA for the VRPTW using a set of problem specific crossover operators. A drawback of their method is that it often will not converge to a feasible solution. Moin (2002) overcomes this problem by using a hybrid GA that incorporates an insertion heuristic (Solomon 1987) and either a vertex sequencing or a parallel savings approach. Test results on a series of 30 customer problems indicate superior results for the vertex sequencing method. For a recent survey on genetic and evolutionary algorithms for the VRPTW, see Br ysy and Gendreau (2001b). GA algorithms, especially in terms of the genetic operators, can be difficult to construct. 2.3.4.3 Tabu Search Judging by volume alone, tabu search is the most popular metaheuristic for the GVRP class. Glover (1989) introduces the fundamental ideas behind TS and Glover (1990) follows up with an analysis of refinements and more advanced aspects of TS. A foundational presentation of TS can be found in Glover and Laguna (1997) who describe TS as a metaheuristic that guides a local heuristic search procedure to explore the solution space beyond local optimality. Key components of basic TS are the solution (S), objective function (f(S)), a neighborhood of the solution (N(S)), and a tabu memory structure. The search starts with an initial solution and defines a subset, N(S), of possible solutions that are neighbors of S under N. Each neighbor is evaluated against the objective function. In a strict local search paradigm, the best of these is selected and the process continues until it terminates at a local optimum. TS, however, employs a 32 short-term memory construct to force the search beyond local optima and to prevent cycling. Attributes of recently selected moves are recorded. Moves that lead back to recently visited solutions are forbidden, or tabu. Recently visited is defined by the tabu tenure. Moves with tabu attributes within tabu tenure iterations are tabu. The length of the tabu tenure must be large enough to avoid cycling, but small enough not to forbid too many moves (Glover & Laguna 1997). Given S, N(S) and f(S), the best non-tabu move in N(S) is selected. Thus, under TS, when the search arrives at a local optimum, it does not terminate but moves beyond the local optimum by choosing the best non-tabu move. In this way, the search moves from one solution to another, with the intent of improving the quality of the solutions visited until some stopping criteria (such as maximum iterations or maximum time) is met. There are times when excellent or influential moves (such as a move that improves the current best-known solution) may result in a tabu solution. To override such a move s tabu status, aspiration criteria are defined that may depend, for example, on the current solution and the best solution found. If the interesting move meets one of the aspiration criteria, the tabu move can be executed (Glover & Laguna 1997). An abundance of variations to the basic TS paradigm, described above, exist. The next section highlights some of the more important variations. 2.3.4.4 Tabu Search Variations Other than basic TS, there are many variations of the TS methodology that have shown great promise in solving large scale combinatorial optimization 33 problems. Some enhanced TS variations are vocabulary building (Kelly and Xu 1995), subpath ejection chains (Rego 1998) and path relinking (Glover, Laguna and Marti 2000). In particular, Reactive TS (RTS), Adaptive TS (ATS) and, more recently, Group Theoretic TS (GTTS) have been used successfully in a wide variety of contexts particularly routing and scheduling problems. RTS was first proposed by Battiti and Tecchiolli (1994). RTS consists of dynamically varying the tabu tenure during the search process so as to diversify or intensify the search depending upon the current context of the search. The tabu tenure is increased if identical solutions occur too frequently and reduced when solutions have not been frequently repeated. RTS requires that a history of visited solutions be maintained and effective RTS approaches efficiently identify previously visited solutions. Typically this is achieved using some type of hashing function (see Woodruff and Zemel (1993)). GTTS provides another way to efficiently identify solutions using group theoretic constructs. ATS changes selected tabu parameters to improve search quality by utilizing the search history. Typically the tabu tenure is perturbed dynamically to promote intensification of the search in promising regions and diversification in regions where improvements are small. If the current solution is the best solution found, the tabu tenure can be reset to the specified default value. If the current solution is better than the previous solution, but not the best solution found, the tabu tenure is decremented. If the current solution is not better than the previous solution, the tabu tenure is incremented. This dynamic updating of the tabu tenure promotes diversification and intensification (Glover & Laguna 1997). 34 There are many ATS schemes in the literature as exemplified by Chambers & Barnes (1996), Dell Amico & Trubian (1993), Wiley (2001), and Combs (2002). Group theory is one of the fundamental building blocks of abstract algebra. It is a powerful tool that has been applied to a host of disciplines such as physics, biology, cryptology and engineering (Gaglione 1992). It was also the basis for advances in exact approaches to solving integer programming problems (Gomory 1963, 1965, 1967, 1969 and Wolsey 1971a , 1971b). Until recently, however, group theory was rarely used for other problems in operations research. Colletti s (1999) landmark dissertation reveals the natural way in which combinatorial optimization problems, such as the SAP, can be couched in the unifying framework of group theory. Colletti (1999), Colletti et al. (1999), Barnes and Colletti (2000 & 2001), Colletti and Barnes (1999a and 1999b), and Barnes, Colletti and Neuway (2002) present strong evidence of the efficacy of group theory in the study and understanding of metaheuristic approaches to combinatorial problems. As mentioned in Section 1.4.3, there are several recent examples of the successful application of group theoretic tabu search (GTTS) to large-scale combinatorial optimization problems related to the SAP. Recent work by Wiley (2001) on the Aerial Flight Refueling Problem (AFRP), Crino (2002) on the Theater Distribution Vehicle Routing and Scheduling Problem (TDVRSP) and Combs (2002) on the Crew Scheduling Problem clearly demonstrate that GTTS can provide robust, effective and efficient solutions to practical large scale combinatorial optimization problems. 35 2.3.4.5 Tabu Search Applications to the GVRP TS applications to the GVRP and its variants are voluminous and a full survey is beyond the scope of this review. Instead key contributions in the literature are highlighted with emphasis on those applications that possess aspects of the SAP. Gendreau et al (1999) introduced a TS heuristic developed for the heterogeneous fleet vehicle routing problem. This is an extension of the classical VRP that includes a heterogeneous fleet with various vehicle capacities. The quite elaborate heuristic incorporates a generalized insertion heuristic, a route optimizer and TS embedded within an adaptive memory procedure. Gendreau et al (1999) employ Gendreau and Laporte s (1994) dynamic diversification mechanism that penalizes vehicle capacity constraints violations. This penalty structure is typical of TS applications and permits traversal of infeasible regions while encouraging a return to feasibility. Toth and Vigo (2003) introduce Granular TS (GTS) and its application to the capacitated and distance constrained VRP. GTS is based on significantly reduced neighborhoods called granular neighborhoods obtained by ignoring neighboring solutions with attributes unlikely to belong to high quality solutions. Granular neighborhoods can be viewed as a variant of the candidate list strategies described in Glover and Laguna (1997) and are best employed with problem structures whose solution cost is the sum of the included elements. TS implementations for the VRPTW are numerous as well. Rochat and Semet (1994) develop a TS approach to a VRPTW that has many elements of the 36 SAP not usually captured in the VRP literature. Some examples are vehicle accessibility issues (akin to airfield constraints), route time duration (crew constraints), and differing vehicle capacities. In Carlton (1995) and Barnes and Carlton (1995), a robust RTS approach to the VRPTW is implemented using simple insertion moves to define the neighborhood. Hashing structures, critical to efficiency in RTS, are also described. Potvin et al. (1996) implement a TS approach to the VRPTW that maintains feasibility throughout. They incorporate a dynamic neighborhood that alternates between 2-opt* and Or-opt neighborhoods to diversify the search. Interestingly, they conclude that better results may be obtained by relaxing vehicle capacity constraints thus allowing traversal of infeasible solutions. Taillard et al. (1997) present a TS approach to the VRPTW in which the time windows are considered to be soft. They introduce a new edge exchange heuristic called CROSS exchange that generalizes two edge exchange heuristics such as 2-opt* (Potvin and Rousseau, 1995) and Or-opt (Or 1976). A key feature of CROSS exchange is that it preserves the orientation of the routes a useful feature for problems with time windows. A drawback of the CROSS exchange is its complexity and therefore approximations are used to evaluate moves in constant time. Tests on Solomon s set of VRPTW (hard time windows) resulted in 17 improvements and 20 ties to the best known solution on the 56 test problems. Chiang and Russell (1997) develop a RTS in which the underlying local search is based on the -interchange mechanism of Osman (1993). More recent TS implementations for the VRPTW are Gehring and Homberger (1999 and 2001), Tan et al. (2000), and Cordeau et al. (2001). 37 TS implementations have also been applied to the more difficult PDPTW class of problems that are generalizations of the VRPTW. Nanry (1998) and Nanry and Barnes (2000) extend the work of Carlton (1995) on the VRPTW with a RTS approach to solving the PDPTW. Solomon s (1987) insertion method is used to construct a feasible PDPTW solution. An RTS method is then invoked to improve the plan using three neighborhoods, namely, Single paired insertion (SPI), Swapping pairs between routes (SBR) and Within route insertion (WRI). Lau and Liang (2001) implement a RTS approach to the PDPTW that incorporates a unique construction heuristic to build a feasible initial solution. Their partitioned insertion heuristic combines an insertion heuristic with a sweep heuristic. The advantage of including the sweep heuristic is that near and far customers are included in the same route that has a balancing effect across routes. The neighborhood moves are those of Nanry and Barnes (2000). 2.3.4.6 Adaptive Tabu Search Approach to the SAP There are many approximation methods used to solve vehicle routing and scheduling problems and their variants (GVRP). This literature review indicates that the best of these is tabu search which has been used extensively over the past decade to solve GVRPs. In recent years, TS applications have provided the best solutions in the least amount of time for many instances of the GVRP. Because of this strong track record, a TS approach to the SAP was deemed appropriate for this research. Among the major variants of TS, which would be most appropriate? Both GTTS and RTS approaches were investigated as potential TS methodologies to 38 employ for the SAP. Due to the complex SAP problem structure, however, neither was deemed appropriate and therefore an ATS approach was adopted. As noted above, GTTS has proven very effective with such problems as VRPs and CSPs partitioning and ordering problems (P|O). A GTTS approach entails formatting the representation of the SAP using the Symmetric Group on nletters (see Crino (2002) for a detailed description). For a VRP, letters represent vehicles and customers. Letters must be unique therefore multiple letters are needed for customers requiring multiple services. Formatting the SAP using the symmetric group on n-letters is possible but inefficient. Unique letters would be needed to capture not only aircraft and customer information, but also the more complex routing information in the SAP. Since routes are used repeatedly by different aircraft, unique letters would need to be used to capture each use. Even for moderately sized problems, the number of letters required would be prohibitive. Routing also precludes the SAP from being cast completely as a P|O problem. For the SAP, the load an aircraft can carry is not just a function of aircraft (capacity) and customer (requirement). An allowable load in the SAP is also a function of the route selected. The assignment of aircraft to requirements by itself can be cast as a P|O problem, but the routing of aircraft cannot. For a RTS approach to be effective, a complete accounting of the solutions visited during the search must be maintained. This is difficult to achieve efficiently in the SAP. The SAP solution representation must capture the routing of each trip in each aircraft s mission. Two solutions that contain a mission that passes through the same routes may not be equivalent due to different timings and 39 loads. For practical sized SAP instances, any solution hashing scheme quickly becomes unwieldy and inefficient. For the above reasons, an ATS approach to the SAP was developed. An ATS approach still employs an adaptive procedure to intensify and diversify the search based upon the search history without requiring as detailed an accounting. ATS exploits the key facets of TS and, as documented in this dissertation, provides timely solutions to decision makers at AMC that are superior to those rendered by competing techniques. This chapter has provided a review of the literature associated with the SAP. Chapter 3 provides a detailed problem description of the SAP and the solution representation used. 40 Chapter 3: Detailed Problem Statement This chapter provides a detailed description of the SAP and the solution representation used in this research. 3.1 THE STRATEGIC AIRLIFT PROBLEM A stated objective of this research is to develop a decision support tool that provides excellent solutions to the SAP in reasonable time with the level of detail compatible with AMOS version 2.0. Incorporation of such a decision support system into AMOS should greatly improve the performance of AMOS. Decision support tools such as AFM and AMOS are the result of extensive experience and insights gained over time at AMC. Wherever possible, this knowledge was incorporated to properly characterize the framework for the SAP. 3.1.1 SAP Inputs The airlift network described in Chapter 1 forms the basic framework for the SAP. The following discussion details each of the components of the airlift network that are provided as input to the model. AMOS builds these files based upon information contained in the TPFDD and user input. components are requirements, aircraft, locations and routes. 3.1.1.1 Requirements Requirements are contained in the AMOS file reqts.dat. Data within this file is extracted from the TPFDD. The key attributes of each requirement are: Type and quantity of cargo to be delivered (outsize, oversize, and/or bulk) Commodity type for the requirement (Navy_Marines, CSS_Other, ...) 41 Their main Number of personnel to be moved Available to Load Date (ALD) Required Delivery Date (RDD) APOE and APOD locations Priority While a requirement can consist of any combination of cargo types or personnel, each requirement has a unique commodity type. The commodity type is derived from the requirement s unit type which, in turn, determines the fill efficiency achievable for a given aircraft. Fill efficiencies or payload targets are defined in the AMOS file payloadtgt.dat. A non-exhaustive listing of the payload targets for a C-5A aircraft is given in Figure 3.1. ID: "C-5A" { Begin Type_Commodity_List // Type Out>0 Over>0 Bulk>0 Over>0 Bulk Bulk Pax Pax // Commodity Out=0 Out=0 Both=0 Fill Only //-------------------------------------------------------------------------------"AF_Acft" 41.3 20.6 0.0 51.1 1.2 68.4 71 340 "AF_Supt" 42.6 17.0 0.0 64.6 0.2 68.4 71 340 "Type_3" 34.5 38.9 0.0 72.6 0.0 82.8 71 340 "Type_4" 34.5 38.9 0.0 72.6 0.0 82.8 71 340 "Army_Prepo" 34.5 38.9 0.0 72.6 0.0 82.8 71 340 "Airborne" 18.0 13.4 0.2 52.0 5.7 82.8 71 340 "Armor" 61.5 20.5 0.0 79.4 0.0 82.8 71 340 "Mech" 65.3 16.3 0.0 80.3 0.0 82.8 71 340 "Airmobile" 15.5 11.9 0.1 51.5 7.7 82.8 71 340 "Cbac" 18.5 10.0 1.1 53.2 9.5 82.8 71 340 "Infantry" 34.1 16.0 0.1 55.6 6.6 82.8 71 340 "Armor_Cav" 58.1 16.8 0.0 63.1 12.5 82.8 71 340 "Type_13" 0.0 0.0 0.0 0.0 0.0 78.3 71 340 "Navy_Marines" 29.0 13.9 0.0 62.4 0.1 78.3 71 340 "CS_Eng" 54.4 18.3 0.2 65.2 10.1 82.8 71 340 "CS_Artillery" 65.1 13.7 0.0 63.2 10.7 82.8 71 340 "CS_Other" 57.5 14.7 1.4 66.0 7.8 82.8 71 340 "CSS_Eng" 54.0 13.6 0.4 66.9 9.8 82.8 71 340 "CSS_Med" 37.9 20.5 0.0 56.6 8.2 82.8 71 340 "CSS_Signal" 46.1 19.0 0.4 54.8 6.4 82.8 71 340 "CSS_Supt_maint"52.1 17.3 0.4 62.1 9.0 82.8 71 340 "CSS_Trans" 61.7 14.1 0.1 70.3 3.3 82.8 71 340 "CSS_Other" 34.5 16.9 1.4 66.2 5.9 82.8 71 340 "Type_24" 34.5 17.9 0.4 62.1 9.0 78.3 71 340 "Resupply" 34.5 16.7 1.6 64.0 7.6 82.8 71 340 "Ammo" 0.0 0.0 0.0 0.0 0.0 90.0 0 340 Figure 3.1 Extract from AMOS Payload Target File for C-5A 42 Each number gives the maximum load (tons or number of personnel) for the associated commodity. The first three columns are related to available outsize cargo; columns four and five are used if no outsize cargo is available and oversize cargo is available; column six is used if only bulk is available; column seven gives the number of seats available for personnel if some cargo is loaded and column eight is the number of seats available if only personnel are loaded. Additional details regarding how payload targets are used are described below. The requirements in the reqts.dat file are ordered according to ALD, RDD and priority. The ALD and RDD define the time window for the requirement. Aircraft may arrive at an APOE prior to the ALD, but, in general, MOG restrictions make it preferable to arrive just in time. A requirement is deemed on time if all of the requirement arrives on or prior to its RDD. Finally, smaller integer requirement priority values indicate a higher priority when compared to other requirements. Priority is used to discern between requirements with identical ALD and RDD windows. 3.1.1.2 Aircraft Several files define aircraft attributes and availability for an instance of the SAP. The AMOS file typeac.dat defines the key characteristics of each type aircraft available. These attributes include the following: Type aircraft (C5-A, C-17, ) Cruising speed Cruising altitude Fuel configuration (capacity, holding fuel, approach & landing fuel, taxi to takeoff fuel, minimum landing fuel) Allowable cabin load (ACL) Trivial load 43 Standard times (ground times, onload time, offload time) Cargo compatibility (Preferred, Feasible, Incompatible) Body Type (Wide, Narrow, or Small) An aircraft s ACL is the maximum weight allowed for a given payload. The trivial load represents a threshold weight below which, ideally, cargo would not be carried. In this implementation all requirements are delivered even if delivery results in a mission with a trivial load. The cargo compatibility defines an aircraft s compatibility with outsize, oversize, and bulk cargo as well as personnel. Compatibility is partitioned into three categories: preferred, feasible, and incompatible. An aircraft s body type is used in determining the parking and working MOG assets an aircraft requires. Fuel consumption rates (climb and cruise) for each type aircraft are defined in the AMOS file fuelcalc.dat. The AMOS file airunit.dat defines the aircraft available for a given instance. This file defines each available aircraft s type, home base, and time available for the aircraft in a given SAP instance. Also defined is the maximum continuous number of hours an aircraft may remain away from its home base the return to base time (RTB). After this time limit, no further missions may be scheduled until the aircraft has returned to its home base and undergone maintenance. Thus, missions that are completed after the limit can be scheduled as long as the mission is commenced prior to the return to base limit. Once this mission is complete the aircraft must return to base. Finally, the uterate.dat file defines the utilization (UTE) target rate by aircraft type and stipulates the total, daily airlift capability for a particular fleet of aircraft, i.e., the 44 UTE rate is the total pool of daily flying hour capability for a fleet of aircraft distributed equally among each aircraft. 3.1.1.3 Locations The locations that make up the airlift system are defined in three AMOS files: airbase.dat, recover.dat, and waypoint.dat. All locations are assigned an identifier. The identifier for airbases and recovery bases are usually the fourletter International Civil Aviation Organization (ICAO) identifier. Waypoints are assigned alphanumeric identifiers by AMC. The airbase file defines the bases in the airlift system for a particular instance. Key attributes of airbases include: Latitude/Longitude Region Maximum on ground (MOG) o Working MOG (Wide & Narrow) o Parking MOG (Wide & Narrow) Daily fuel capacity Alternate landing base location (Civilian & Military) Compatible aircraft (Military or Civilian; Wide or Narrow) Permissions (operating hours, etc) Airbases may be home bases for aircraft units, APOEs, APODs, en route bases, or recovery bases. Some bases may satisfy more than one category. For example, a home base may also be an APOE and serve as an en route base for one or more routes. Home bases are specified in the airunit.dat file. APOEs and APODs are designated in the reqts.dat file. En route bases are specified in the plan.dat file and recovery bases are defined in the recover.dat file. These will be described in more detail below. 45 Each airbase is assigned to a region. Generally, several bases will share a region. This clustering of bases into regions reduces the number of possible routes in the airlift network by defining routes to connect regions rather than individual bases. Routing in the SAP will be discussed in more detail below. Figure 3.2 below shows the regions defined by AMC and used in constructing routes. Figure 3.2 Region Map Parking and working MOG at an airbase are an aggregation of the capacity of an airfield to park and service various aircraft at any given time. Daily fuel capacity is the number of gallons of fuel that an airbase apportions to airlift on a daily basis. Alternate airfields are where an aircraft is diverted if congestion or other circumstances preclude landing at the primary airbase. In this research, 46 alternate airfields are used only in determining fuel requirements for a particular mission. A portion of the fuel requirements for a mission must include enough fuel to fly the aircraft to an alternate airbase. Compatible aircraft for an airbase are partially defined by the wide-body and narrow-body MOG values if the wide-body working MOG is zero, widebody aircraft are not allowed at that airbase. More detail on compatible aircraft for an airbase are described in the permission.dat file which defines what activities can take place at an airbase based upon the type of aircraft. These activities include whether it can be used as an en route airbase, whether refueling is available, and whether onloading and/or offloading can occur. permissions can change over time. Recovery bases are defined in the AMOS recover.dat file. This file These defines where an aircraft will recover once it has offloaded at an APOD based upon the APOD region and the type of aircraft. If no recovery base is specified in this file, then the aircraft will recover at its APOD. The last type of location is the waypoint a point in space used as a routing control measure or as a rendezvous for aerial refueling. Waypoints are specified in the file waypt.dat in which the waypoint s location (latitude/longitude) and region are defined. 3.1.1.4 Routes All possible routes for a SAP instance are defined in the AMOS file plan.dat which is the most complicated of the AMOS input files. AMC planners build the plan.dat file based upon experience and the current situation. The first 47 of three parts defines the various aircraft groups. These groups are used to define specified routes based upon aircraft properties such as whether the aircraft is military or civilian or of a specific type such as C-5As. The second part of the plan.dat file, the planning group list, defines all the currently planned routes. Each planning group has a name, a source region, a sink region and an aircraft group. As an example, a plan which links source region 20 to sink region 33 for the aircraft group C-5s would be named 20_33_C-5S_1 where the one represents the fact that this is first such plan for the given source, sink and group. Begin Plan_Block Planning_Group { "17_33_C-5S_1" "20_33_C-5S_1" "17_80_C-5S_1" "20_80_C-5S_1" } "18_33_C-5S_1" "21_33_C-5S_1" "18_80_C-5S_1" "21_80_C-5S_1" "19_33_C-5S_1" "22_33_C-5S_1" "19_80_C-5S_1" "22_80_C-5S_1" Begin Plan_Sequence Begin Route_Segment_List // ----------- Next ----------// Location Type_Point Nav_Category Type_Maneuver //-------------------------------------------------------ORIGIN START 2 DEFAULT "KDOV" "ENROUTE" 1 CRUISE "ERLP" "ENTRY" 2 CRUISE "LEMO" "ENROUTE" 1 CRUISE "ERT2" "ENTRY" 1 CRUISE "ERT1" "ENTRY" 2 CRUISE "TLXR" "ENTRY" 2 CRUISE //-------------------------------------------------------End Route_Segment_List End Plan_Sequence End Plan_Block Figure 3.3 Route Segment Extract from AMOS Plan File The third and final part of the plan document defines the route segments that join the source and sink regions for the various planning groups. An extract from the 48 third part of the AMOS plan file for the route segment above is shown in Figure 3.3. Note that several planning groups may share the same route segment. This is how the reduction in number of possible routes in modeling the airlift system is realized. Essentially bases are clustered in regions and regions are joined together by route segments. Continuing the example, the planning group 20_33_C-5S_1 connects region 20 to region 33 according to the route segment in Figure 3.3. The aircraft would depart from its origin base in region 20 and fly to KDOV (KDOV is the ICAO symbol for Dover Airbase in Delaware) where it would land and undergo en route base activities such as refueling and routine maintenance. From KDOV it would fly via the waypoint ERLP (entry signifies a waypoint) to LEMO (Moron Airbase in Spain) where it would once again land and undergo en route base activities. From LEMO, the aircraft would pass in succession through the waypoints ERT2, ERT1 and TLXR and then finally land at its destination base in region 33. A route is the path defined by an origin, route segment and destination. The only differences between the routes 20_33_C5S_1 and 22_80_C-5S_1 are the starting and ending bases (all intermediate stops are identical). The navigation categories (NAVCAT) define whether additional fuel reserves are required due to flight over water. Category 1 requires a minimum of 10% and one hour of cruise fuel be added to fuel requirements. Category 2 requires no additional fuel reserves. Each plane s return trip is defined by a 49 different planning group and route segment that may or may not be the reverse of the outbound trip. 3.1.2 Additional SAP Characteristics There are several characteristics of the SAP that are critical to understanding later discussions. The next several sections describe: standard flight legs, how aircraft payloads are determined, how maximum cargo for a route is determined, how mission timings are determined, and how fuel requirements are calculated. Take Off Start Cruise End Cruise Land Cruise Ap pro pro ah ach Cl im b & & a nd La Taxi to Take Off Mission Complete Standard Flight Leg Figure 3.4 3.1.2.2 Standard Flight Leg with Segments A Standard Flight Leg A flight leg consists of several segments: climb, cruise, approach, landing and alternate base. The alternate base segment helps determine fuel requirements 50 in the event an aircraft must be diverted from its original destination. A graphical depiction of a flight leg is shown in Figure 3.4. The flight leg is the basis for calculating maximum cargo for an APOE to APOD route, determining mission timings, and calculating fuel requirements. 3.1.2.3 Calculating Maximum Cargo for a Route For each APOE-APOD route and aircraft type, there is a maximum payload weight that can be carried based upon the route critical leg and the aircraft characteristics. The route critical leg is the flight leg with the smallest maximum payload. This is calculated independent of volume issues associated with commodity and cargo types. Issues with fill efficiencies are considered during the loading process using the payload target data. Given the critical leg, distance to the alternate base, and aircraft type, the maximum payload weight for a given route and aircraft type is computed. In the first of two steps, the aircraft is hypothetically filled to its maximum fuel capacity and the maximum cargo possible is hypothetically loaded (subject to ACL or maximum take-off weight restrictions). Assuming this configuration, fuel consumption for the critical leg is calculated. In the second step, if unused fuel remains, its weight difference is added to the maximum cargo value. If the projected fuel consumed exceeds the fuel capacity, the maximum cargo is reduced by the weight of the excess consumed fuel. Next, an iterative non-linear binary search is performed to determine the projected maximum cargo because the reduction in payload-fuel consumption trade-off is not linear. This procedure is used to determine the maximum cargo for each route and aircraft type. 51 As an example maximum weight calculation, assume that a C-5A is carrying cargo from KGRK to OKBK and is using the route segment of Figure 3.3. The aircraft speed is 0.74 mach and flight level is 31,000 feet. The key features of the route are shown in Table 3.1. Airbases and waypoints are named according to the ICAO abbreviations. FROM TO TYPE POINT DIST ALT BASE KWRI NONE LERT NONE NONE NONE OKAJ DIST TO ALT 67 0 48 0 0 0 21 NAV CAT 2 1 2 1 1 2 2 KGRK KDOV ERLP LEMO ERT2 ERT1 TLXR Table 3.1 KDOV ERLP LEMO ERT2 ERT1 TLXR OKBK EN ROUTE ENTRY EN ROUTE ENTRY ENTRY ENTRY DESTINATION 1196 2229 1018 698 1232 264 836 Route Segment Details for KGRK - OKBK Route Distances are measured in nautical miles and calculated using the great circle distance method. The great circle distance method takes into account the curvature of the earth in calculating the distance between two points. Given the latitude and longitude of two points on the surface of the earth, the shortest distance between the two points is given by the length of an arc of a circle 52 concentric with the earth and with the same radius as the earth. The great circle distance is calculated as follows: sin (lat1 * 180 ) * sin (lat2 * 180 ) + cos (lat1 * 180 ) * 180 distance = 60 * arccos * cos (lat2 * 180 ) * cos ((lon2 lon1 ) * 180 ) where lat1 and lat2 are the latitudes and lon1 and lon2 are the longitudes for the first and second points respectively. In Table 3.1, the critical leg passes through KDOV-ERLP-LEMO for a distance of 3,247 nautical miles. characteristics of the C-5A. typeac.dat. Table 3.2 details the required flight These values are contained in the AMOS file Weights (lbs) Value ACL 185,822 Max Takeoff 769,000 Max In Flight 769,000 Empty 372,500 Trivial Load 52,000 Min Landing 0 Fuel (lbs) Capacity Holding Value Table 3.2 332,500 18,000 Approach & Land Taxi to Takeoff 5,200 3,000 C-5A Standard Weights and Fuel The fuelcalc.dat file contains the climb and cruise data for all aircraft types. Figure 3.5 is an extract of the climb data section for a C-5A which provides the quantity of fuel burned (in pounds), distance traveled during climb (nautical miles), and amount of time to reach altitude (minutes) based upon the type aircraft, its weight and cruising flight level (in feet). Figure 3.6 is an extract 53 ID: "C-5A" { Begin Fuel_Requirement_Data // Weight DataType -- Flight Levels (NM/1000 lbs fuel) -// 1 2 3 4 5 6 // -----------------------------------------------{ 27000 29000 31000 33000 35000 37000 } //-------------------------------------------------------------------360000 Fuel { 5313 5829 6207 6571 6949 7415 } Distance { 46 52 58 65 73 80 } Time { 8.0 9.2 10.3 11.4 12.4 13.6 } . . . . . . . . . . . . . . . . . . . . . . . . . . . 560000 Fuel { 9403 10384 11338 12235 13302 14915 } Distance { 84 100 116 130 147 178 } Time { 15.1 17.4 19.6 21.9 24.6 29.5 } 580000 Fuel { 9954 10998 12050 13012 14334 16406 } Distance { 89 107 123 139 159 202 } Time { 16.0 18.4 20.8 23.4 26.8 33.7 } 600000 Fuel { 10535 11665 12830 13900 15558 18274 } Distance { 94 114 131 149 175 238 } Time { 17.0 19.5 22.1 25.0 29.2 38.9 } 620000 Fuel { 11150 12393 13687 14937 17018 0 } Distance { 100 121 140 161 197 0 } Time { 18.0 20.9 23.8 27.2 33.7 0 } 640000 Fuel { 11806 13190 14633 16164 18768 0 } Distance { 106 129 150 176 228 0 } Time { 19.2 22.3 25.4 29.5 38.1 0 } 660000 Fuel { 12511 14065 15684 17459 21295 0 } Distance { 114 138 161 196 265 0 } Time { 20.5 23.8 27.3 32.6 43.4 0 } 680000 Fuel { 13278 15031 16861 19156 0 0 } Distance { 122 149 175 222 0 0 } Time { 21.8 25.4 29.7 36.5 0 0 } 700000 Fuel { 14122 16104 18198 21444 0 0 } Distance { 131 160 193 256 0 0 } Time { 23.1 27.1 32.2 40.5 0 0 } 720000 Fuel { 15062 17307 19743 24343 0 0 } Distance { 140 174 216 286 0 0 } Time { 24.8 29.6 36.5 48.1 0 0 } 740000 Fuel { 16120 18671 21578 0 0 0 } Distance { 152 190 245 0 0 0 } Time { 26.5 32.2 40.9 0 0 0 } 760000 Fuel { 17325 20243 23840 0 0 0 } Distance { 164 208 275 0 0 0 } Time { 28.1 35.6 45.8 0 0 0 } 780000 Fuel { 18708 22096 0 0 0 0 } Distance { 179 231 0 0 0 0 } Time { 30.7 39.7 0 0 0 0 } 800000 Fuel { 20307 24357 0 0 0 0 } Distance { 196 259 0 0 0 0 } Time { 33.9 44.2 0 0 0 0 }} Figure 3.5 Climb Data Extract from fuelcalc.dat File for a C-5A 54 of the cruise segment data from the fuelcalc.dat file. The cruise section provides the burn rate (expressed in nautical miles per 1,000 pounds of fuel) for a given aircraft type, cruising speed, weight and flight level. ID: "C-5A" { Delta_Tolerance: 1000 // +/- lbs Begin Fuel_Requirement_Data // Weight Speed -- Flight Levels (NM/1000 lbs fuel) -// (lbs) (Mach) 1 2 3 4 5 6 { 27000 29000 31000 33000 35000 37000 } //----------------------------------------------------------------. . . . . . . . . . . . . . . . . . 560000 {0.70 { 20.02 20.87 21.63 22.22 22.45 22.05 } 0.74 { 19.42 20.38 21.33 22.14 22.67 22.66 } 0.77 { 18.73 19.76 20.80 21.67 22.35 22.53 } 580000 {0.70 { 19.74 20.50 21.20 21.66 21.67 0.00 } 0.74 { 19.19 20.09 20.99 21.69 22.03 0.00 } 0.77 { 18.54 19.52 20.47 21.25 21.80 0.00 } } 600000 {0.70 { 19.45 20.13 20.79 21.06 20.85 0.00 } 0.74 { 18.96 19.80 20.65 21.20 21.33 0.00 } 0.77 { 18.35 19.27 20.14 20.81 21.21 0.00 } } 620000 {0.70 { 19.11 19.76 20.36 20.44 20.05 0.00 } 0.74 { 18.67 19.50 20.32 20.68 20.64 0.00 } 0.77 { 18.07 19.01 19.87 20.35 20.63 0.00 } } . . . . . . . . . . . . . . . . . . 700000 {0.70 { 17.84 18.26 18.26 0.00 0.00 0.00 } 0.74 { 17.66 18.26 18.55 0.00 0.00 0.00 } 0.77 { 17.22 17.90 18.36 0.00 0.00 0.00 } } 720000 {0.70 { 17.52 17.83 17.70 0.00 0.00 0.00 } 0.74 { 17.41 17.92 18.08 0.00 0.00 0.00 } 0.77 { 16.99 17.60 17.95 0.00 0.00 0.00 } } 740000 {0.70 { 17.21 17.39 17.16 0.00 0.00 0.00 } 0.74 { 17.15 17.55 17.64 0.00 0.00 0.00 } 0.77 { 16.75 17.28 17.58 0.00 0.00 0.00 } } 760000 {0.70 { 16.89 16.91 16.63 0.00 0.00 0.00 } 0.74 { 16.89 17.16 17.22 0.00 0.00 0.00 } 0.77 { 16.51 16.95 17.24 0.00 0.00 0.00 } } 780000 {0.70 { 16.58 16.40 0.00 0.00 0.00 0.00 } 0.74 { 16.64 16.71 0.00 0.00 0.00 0.00 } 0.77 { 16.33 16.49 0.00 0.00 0.00 0.00 } } 800000 {0.70 { 16.26 15.90 0.00 0.00 0.00 0.00 } 0.74 { 16.38 16.26 0.00 0.00 0.00 0.00 } 0.77 { 16.08 16.08 0.00 0.00 0.00 0.00 }} Figure 3.6 Cruise Data Extract from fuelcalc.dat file for a C-5A 55 Using the computation procedure described above, the maximum weight is determined as follows: Find the maximum weight for the aircraft with full fuel tanks: acWgt = emptyWgt + fuelCapWgt = 372,500 + 332,500 = 705,000 maxCargo = min(MaxTakeOff, maxTableWgt) acWgt = min(769,000;760,000) 705,000 = 55,000 The maxTableWgt is determined from the fuelcalc.dat file using the maximum feasible climb weight (760,000 in this case). (Zero entries in the climb and cruise tables indicate infeasible combinations.) acWgt = acWgt + maxCargo = 705,000 + 55,000 = 760,000 Determine the climb segment values climbWgt = acWgt taxiToTakeOff = 760,000 3,000 = 757,000 Since an aircraft weight of 757,000 is not one of the weight levels in the climb data table (Figure 3.5), linear interpolation is performed using the weight levels that bracket the climbWgt (740,000 & 760,000). The fuel burned during the climb is 23,500; the distance traveled is 271 nautical miles. Determine average cruise burn rate The average cruise burn rate is computed as the average of the burn rate at the start of the cruise segment and at the end of the cruise segment. startCruiseWgt = climbWgt climbFuel = 757,000 23,500 = 733,500 The interpolated value for start cruise burn rate is 17.78 nm/1000 lbs of fuel. cruiseDist = legDist climbDist = 3247 271 = 2976 56 cruiseFuel = (1/rate) * cruiseDist * 1000 = (1/17.78) * 2976* 1000 = 167,379 endCruiseWgt = startCruiseWgt cruiseFuel = 733,500 167,379 = 566,121 The interpolated value for the end of cruise burn rate is 21.23 nm/1000 lbs of fuel. avgCruiseBurnRate = (17.78 + 21.23)/2 = 19.51 The acWgt at end of cruise, using the average burn rate, is recalculated as cruiseFuel = (1/rate) * cruiseDist * 1000 = (1/19.51) * 2976 * 1000 = 152,606 endCruiseWgt = startCruiseWgt cruiseFuel = 733,500 152,606 = 580,894 Calculation of the alternate base fuel requirement The burn rate to the alternate base is calculated using the endCruiseWgt. The interpolated value is 20.98 nm/1000 lbs of fuel. altFuel = (1/rate) * altDist * 1000 = (1/20.98) * 48 * 1000 = 2288 Calculation of the required fuel reserves Required reserves include the alternate base fuel, minimum landing fuel, holding fuel, and, if applicable, a NAVCAT 1 reserve. The standard reserve is calculated as follows: stdReserve = max(minimum landing, holding + altFuel) = max(0, 18000 + 2288) = 20,288 If part of the cruise segment is navigation category 1, an additional reserve must be included. From Table 3.1, the flight from KDOV to ERLP is a navigation category 1 flight. The required NAVCAT 1 reserve is equal to the 57 minimum of 10% of the fuel for this portion of the cruise segment and one hour of cruise fuel. To compute one hour of cruise fuel, the distance traveled in one hour at the given speed is determined, and the avgCruiseBurnRate is used. NAVCAT1Fuel = min(10%*cruiseFuel, 1 hour cruise fuel) = min(10%*cruiseFuel, (1/rate)* oneHourDist*1000) = min(10%*100413, (1/19.51)*437*1000) = min(10041, 22399) = 10041 requiredFuelReserves = stdReserve + NAVCAT1Fuel = 20,288 + 10,041 = 30,329 Calculation of the total fuel required totFuelBurned = taxiToTakeOff + climbFuel + cruiseFuel + appLandFuel = 3000 + 23,500 + 152,606 + 5200 = 184,306 totFuelRequired = totFuelBurned + requiredFuelReserves = 184,306 + 30,329 = 214,635 Determination of whether there is surplus fuel at landing extraFuel = fuelCapWgt totFuelRequired = 332,500 214,635 = 117,865 Since the extraFuel value is positive, the excess is added to the current maxCargo to obtain the maximum cargo for this route. maxCargo = maxCargo + extraFuel = 55,000 + 117,865 = 172,865 The calculated values for the critical leg in the example are graphically summarized in Figure 3.7. 58 KDOV Start Cruise Burn Rate: 17.78 End Cruise ERLP Burn Rate: 21.23 LEMO Cruise AVG Burn Rate: 19.51 im b ach r a p ro Ap Ap Cl & nd La Taxi to Take Off Distances (nm) 0 Fuel burned (lbs) 3000 AC Weight (lbs) 760000 757000 733500 633087 NAVCAT 1: 10,041 580894 23500 100413 52193 271 1958 1018 Mission Complete Total Distance: 0 3247 Total Fuel Burned: 184306 Final AC Weight: 5200 575694 Fuel reserves (lbs) = NAVCAT1 + MAX( min landing, holding + alt fuel) = 10041 + MAX(0, 18000 + 2288) 575694 ALT Fuel: Total Fuel Reserves: 2288 30329 Figure 3.7 3.1.2.4 Example Maximum Cargo Calculation for a C-5A Calculating Fuel Requirements for a given payload Calculating the fuel requirements for a particular flight leg and a specific payload (rather than the maximum payload) is significantly more complex. An iterative procedure is used that varies fuel values until the minimum fuel required to fly the leg for the given payload is determined. The procedure employed to guess required fuel levels is a binary search. At the outset of the search the fuel required is set at the midpoint of zero fuel and the aircraft s fuel capacity. Calculations similar to those demonstrated above are then used at each step to determine if there is an excess or lack of fuel to complete the trip with the given 59 payload. For an excess amount of fuel, the upper half of the current range of fuel values is discarded and the binary search repeats with the lower half. For insufficient fuel, the lower half fuel range is discarded and the binary search continues with the upper half. This procedure yields the approximate minimum amount of fuel required to fly the leg for the given payload. 3.1.2.5 Determining Aircraft Payloads Payload determination for an aircraft is based on the AMOS liquid ton modeling method. Key factors are available cargo (cargo & commodity types), aircraft type, payload targets, and route maximum cargo constraints. The following example illustrates how a payload is determined. Figure 3.8 gives a set of available requirements for the KGRK/OKBK APOE-APOD pair. The assigned aircraft is a C-5A. Avail/Reqd Out Over Commodity -- Day -- Priority Size Size Bulk Pax APOE-APOD Description //-------------------------------------------------------------------------------"CSS_Other" "Infantry" "Ammo" "CSS_Other" 0 0 0 0 3 3 4 5 50 50 50 50 18.0 5.0 0.0 18.0 323.8 0.0 0.0 5.0 0.0 30.0 323.8 0.0 20 30 15 20 {"KGRK" {"KGRE" {"KGRK" {"KGRE" "OKBK"} "OKBK"} "OKBK"} "OKBK"} AFG AFG AFG AFG Reqt Reqt Reqt Reqt 1" 2" 3" 4" Figure 3.8 Extract from Requirements File When ALDs, RDDs and priorities are identical, the default ordering for a set of requirements is the TPFDD ordering. Here there are four different commodities, each with different payload targets. The Table 3.1 route is also used for this example. Thus, the maximum cargo for the route is 172,865 pounds (86.4 tons). The example requirements ALDs are identical with different RDDs. 60 Loading begins with Requirement 1. Because a C-5A is compatible with the available outsize cargo, the first three values for type commodity CSS_Other in the payloadtgt.dat file will be used. From Figure 3.1, the maximum values for type commodity CSS_Other are as follows: 34.5 tons outsize, 16.9 tons oversize, 1.4 tons bulk and 71 personnel (PAX). Based upon the available quantities and maximum table values, 18 tons of outsize, 16.9 tons of oversize and 20 PAX are loaded. Note that PAX are loaded first and their weight is determined by using a standard PAX weight of 400 pounds defined in the typeac.dat file. Additionally, cargo maximum values are predicated on loading to the limit of the PAX fill 71 PAX in this instance. Thus far then, the total cargo loaded is 34.9 tons and PAX weight is 4 tons for a total payload of 38.9 tons, well below the route maximum cargo value. The next step is to determine the percent payload capacity remaining for the aircraft. For PAX it is simply the number of seats remaining 51 seats in this instance. For cargo, the percent of capacity filled is the ratio of cargo loaded to the maximum cargo capacity defined by the table values. %Filled = 34.9/52.8 = 66% Thus 34% of cargo capacity remains to load items from the next requirement. Because there is remaining capacity and maximum cargo limitations have not been exceeded, loading continues with Requirement 2. Again because outsize cargo is available, the first three columns of the payload target table (Figure 3.1) for commodity Infantry define maximum cargo values: 34.1 tons outsize, 16 tons oversize and 0.1 tons of bulk. PAX are loaded first with all 30 PAX available 61 being loaded leaving 21 seats remaining. The maximum table values must be adjusted since only 34% of cargo capacity remains. Maximum table values are therefore reduced to 34% of their original value and the new limits become: 11.59 tons of outsize, 5.44 tons of oversize, and 0.03 tons of bulk. Thus, cargo loaded for Requirement 2 is 5 tons of outsize and 0.03 tons of bulk. The Requirement 2 payload weight total is 11.03 tons (includes PAX). Total payload is now 49.93 tons which is less than the route maximum cargo. The percent fill for Requirement 2 is determined as follows: %Filled = 5.03/17.06 = 29% Thus, assuming only Requirement 2 is loaded, the remaining capacity is 71%. However, Requirement 1 is already loaded, so the true capacity remaining is 71% of 34%. Thus 24% of cargo capacity remains. Requirement 3 is now considered for loading. All 15 PAX are loaded leaving 6 seats empty. Because there is only bulk cargo available, the sixth column of the payload target table (Figure 3.1) for the type commodity Ammo defines the maximum cargo: 90 tons of bulk. Again, this must be reduced to reflect the cargo capacity remaining. The maximum bulk that can be loaded is 24% of 90 tons of bulk or a maximum value of 21.6 tons of bulk can be loaded. Since there are 30 tons available, all 21.6 tons of bulk is loaded. The total weight of Requirement 3 items loaded is 24.6 tons. The total current payload is now 74.53 tons. The percent fill is 100% since the maximum bulk value was loaded. No cargo capacity remains. 62 Since PAX seats are available, loading proceeds with Requirement 4. No cargo can be loaded and only 6 of the 20 PAX available are loaded. The final total payload is 75.73 tons. The aircraft is fully loaded and proceeds from the APOE to the APOD at OKBK. It is possible that this process will load trivial amounts of requirements as the percent of remaining capacity becomes small. To prevent this, a user defined percent capacity threshold is enforced. (The default value is 5%.) 3.1.3 SAP Building Blocks: Missions In solving the SAP, we want to find the best allocation of aircraft to requirements and the best routing and scheduling of aircraft so that late deliveries and non-deliveries are minimized and essential constraints are not violated. Since a SAP solution is a set of missions flown by the various aircraft, a mission is the basic solution building block. Each mission has an associated APOE/APOD pair. A typical mission consists of flight segments sequencing through home base, APOE, APOD, recovery base and then back to home base. Each flight segment traverses an appropriate route (defined in the plan file). Upon return to home base, the aircraft undergoes maintenance and refueling, and then is available for subsequent missions. There are variations on this basic mission pattern. For example, subject to the return to base constraint, an aircraft could move from a recovery base directly to an APOE and start a new mission instead of returning to home base. 63 3.1.3.1 Determining Timings for a Mission Each mission consists of a set of route segments, each with a defined origin and destination. Within a route segment there are one or more flight legs. The timings for a mission are determined by combining the timings for all the flight legs that make up the mission, i.e., the flight leg is the fundamental component of a mission. As an example of computing the timings of a flight leg, consider the critical leg used in the maximum cargo calculation example (KDOV-ERLPLEMO) for a C-5A. Suppose the payload is set to the maximum cargo found in the example, 86.4 tons and takeoff from KDOV is at time zero. A graphical depiction of the flight leg, with segment times, is presented in Figure 3.9. Upon takeoff, the aircraft climbs to its cruising altitude, the climb segment. The time to reach altitude (31,000 feet in this example) based upon a given aircraft weight is defined in the climb section of the fuelcalc.dat file (see Figure 3.5 above). In this example, the climb weight of the aircraft is 757,000 pounds which yields a climb time of 45 minutes. 64 KDOV Start Cruise Burn Rate: 17.78 End Cruise ERLP Burn Rate: 21.23 LEMO Cruise AVG Burn Rate: 19.51 Cl im b a p pro Ap & h& ch n nd L La Taxi to Take Off Distances (nm) 0 Fuel burned (lbs) 3000 AC Weight (lbs) 760000 Timings (minutes) 0 45 269 140 757000 733500 633087 580894 23500 100413 52193 271 1958 1018 Mission Complete Total Distance: 0 3247 Total Fuel Burned: 184306 Final AC Weight: 5200 575694 575694 Total Flight Time: 15 469 Figure 3.9 Example Flight Leg Timings The aircraft now enters the cruise segment of the flight leg. The standard cruising speed for a C-5A is 0.74 mach. Computation of the cruise time, requires conversion of the true air speed into knots according to accepted formulae (Williams, 2002). Required inputs are the Mach number (MachNum), the temperature at zero altitude (zeroTemp) in degrees Celcius and the altitude (alt) at the Mach number (hundreds of feet). For our example, the Mach number is 0.74, the zero altitude temperature is set at 18.33 C (about 65 F), and the altitude is 310 hundred feet. The many factors used to obtain the constants used in the formulae below are defined by Williams (2002). The computation proceeds as follows: temp = zeroTemp - (0.0019812 * (alt*100)) = 18.33 (0.0019812 * (310*100)) = -43.08 C 65 TAS = MachNum * (38.967854 * = .74 * (38.967854 * = 437.38 knots temp + 273.15 ) - 43.08 + 273.15 ) Once the true air speed (TAS) is known, cruise times are calculated as follows: cruiseTime = 60 * cruiseDist /TAS For the first cruise portion to ERLP, the cruise time is cruiseTime = 60 * 1958/437.38 = 268.60 minutes. In a similar way, cruise time from ERLP to LEMO is calculated to be 139.65 minutes for a total cruise time of 408.25 minutes. As in AMOS, the cruise distance used places the aircraft directly above the destination. A standard approach and landing time is then added (currently set at 15 minutes) to obtain the total flight time from KDOV to LEMO. Since LEMO is an en route base, the aircraft will undergo refueling and routine maintenance checks before embarking on the next leg. En route times are defined in the typeac.dat file. For a C-5A, the en route time is 3.25 hours or 195 minutes. In summary, the total flight time for this leg is 469 minutes and the total ground time at LEMO is 195 minutes. Standard en route times are also used to model delays at recovery bases and home bases. There are separate standard times set for onloading and offloading. Timing for an entire mission is determined by summing the timings for the various legs that make up the mission. 3.1.3.2 An Example Mission To illustrate how a mission is built using the various input files, suppose that there is a requirement of commodity type CSS_Other consisting of 50 tons of 66 outsize, 12 tons of oversize and 15 tons of bulk to be moved from Robert Gray Army Airfield in Texas (KGRK, Region 20) to Kuwait International Airport in Kuwait (OKBK, Region 33). Since there is outsize cargo available, a C-5A would be a suitable choice. Assume that the C-5A home base is at Dover Air Force Base in Delaware (KDOV, Region 21). The recovery base for a C-5A in Region 33 is at Moron Airbase, Spain (LEMO, Region 24). For this mission, the aircraft will go from KDOV to KGRK to OKBK to LEMO and back to KDOV. Begin Plan_Block Planning_Group { "33_49_CIV_1" "80_49_CIV_1" "8_21_CIV_1" "8_22_CIV_1" "10_26_MIL_1" "10_26_CIV_1" "17_36_CIV_1" "18_36_CIV_1" "18_8_CIV_1" "18_9_CIV_1" "19_8_CIV_1" "19_9_CIV_1" "20_36_MIL_1" "20_36_CIV_1" "20_8_CIV_1" "20_9_CIV_1" "21_36_MIL_1" "21_36_CIV_1" "21_8_CIV_1" "21_9_CIV_1" "22_36_MIL_1" "22_36_CIV_1" "22_8_MIL_1" "22_8_CIV_1" "22_9_MIL_1" "22_9_CIV_1" "36_17_CIV_1" "36_18_CIV_1" "36_19_ALL_1" "36_19_CIV_1" "36_20_ALL_1" "36_20_CIV_1" "36_21_ALL_1" "36_21_CIV_1" "36_22_ALL_1" "36_22_CIV_1" "37_39_CIV_1" "37_41_CIV_1" "37_51_CIV_1" "39_17_CIV_1" "39_18_CIV_1" "39_19_CIV_1" "39_20_CIV_1" "39_37_CIV_1" "41_39_CIV_1" "51_37_CIV_1" "18_17_ALL_1" "19_17_ALL_1" "20_17_ALL_1" "21_17_ALL_1" "22_17_ALL_1" "17_18_ALL_1" "19_18_ALL_1" "20_18_ALL_1" "21_18_ALL_1" "22_18_ALL_1" "17_19_ALL_1" "18_19_ALL_1" "20_19_ALL_1" "21_19_ALL_1" "22_19_ALL_1" "17_20_ALL_1" "18_20_ALL_1" "19_20_ALL_1" "21_20_ALL_1" "22_20_ALL_1" "17_21_ALL_1" "18_21_ALL_1" "19_21_ALL_1" "20_21_ALL_1" "22_21_ALL_1" "17_22_ALL_1" "18_22_ALL_1" "19_22_ALL_1" "20_22_ALL_1" "21_22_ALL_1" "19_36_MIL_1" "19_36_CIV_1" "51_41_ALL"} Begin Plan_Sequence Begin Route_Segment_List // ----------- Next ----------// Location Type_Point Nav_Category Type_Maneuver //-------------------------------------------------------ORIGIN START 2 DEFAULT //-------------------------------------------------------End Route_Segment_List End Plan_Sequence End Plan_Block Figure 3.10 Route Segment Joining Region 21 to Region 20 67 For the first leg, the regions are 21 and 20. The plan document route segment joining these regions is shown in Figure 3.10. This route segment indicates a direct flight from KDOV to KGRK (no intermediate bases or waypoints). At KGRK, the amount of cargo that can be loaded is a function of the requirements available, the payload targets defined in the payloadtgt.dat file, and the limitations imposed by the route. The payload targets for commodity type CSS_Other for a C-5A are given in Figure 3.11. ID: "C-5A" { Begin Type_Commodity_List // Type Out>0 Over>0 Bulk>0 Over>0 Bulk Bulk Pax Pax // Commodity Out=0 Out=0 Both=0 Fill Only //-----------------------------------------------------------------------------"AF_Acft" 41.3 20.6 0.0 51.1 1.2 68.4 71 340 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . "CSS_Other" 34.5 16.9 1.4 66.2 5.9 82.8 71 340 "Type_24" 34.5 17.9 0.4 62.1 9.0 78.3 71 340 "Resupply" 34.5 16.7 1.6 64.0 7.6 82.8 71 340 "Ammo" 0.0 0.0 0.0 0.0 0.0 90.0 0 340 //-------------------------------------------------------------------------------End Type_Commodity_List } Figure 3.11 Extract from Payload Target File for C-5As The first three columns define what can be loaded for this commodity and aircraft type 34.5 tons of outsize, 16.9 tons of oversize and 1.4 tons of bulk. Thus, the payload should be 34.5 tons of outsize, all 12 tons of the oversize, and 1.4 tons of bulk for a total payload of 47.9 tons. Next, we must confirm whether the chosen route segment s maximum cargo can support 47.9 tons. Determining a route maximum cargo begins with identifying the route between the APOE and APOD (KGRK and OKBK). 68 Since planning group 20_33_C-5S_1 joins KGRK (Region 20) to OKBK (Region 33), we could use the route segment in Figure 3.3. As derived earlier, the route s maximum cargo of 86.43 tons exceeds the payload target total of 47.9 tons. Following offloading at OKBK, the aircraft proceeds to recovery base LEMO. Figure 3.12 presents one of the route segments joining OKBK to LEMO. Begin Plan_Block Planning_Group { "33_10_CIV_1" "33_24_MIL_1" "33_24_CIV_1" "80_8_CIV_1" "80_10_CIV_1" "80_24_MIL_1" "80_24_CIV_1" "80_8_CIV_2" } Begin Plan_Sequence Begin Route_Segment_List // ----------- Next ----------// Location Type_Point Nav_Category Type_Maneuver //-------------------------------------------------------ORIGIN START 2 DEFAULT "TLXR" "ENTRY" 2 CRUISE "ERT1" "ENTRY" 2 CRUISE "ERT2" "ENTRY" 2 CRUISE //-------------------------------------------------------End Route_Segment_List End Plan_Sequence End Plan_Block Figure 3.12 Route Segment for OKBK to LEMO If this route is selected, the aircraft proceeds to its recovery base via OKBK-TLXR-ERT1-ERT2-LEMO. Following required maintenance and refueling at LEMO, the aircraft returns to its home base. One route segment joining LEMO to KDOV is shown in Figure 3.13. Note that the destination, KDOV, is a part of the route segment. If this route is selected, the aircraft flies directly from LEMO to KDOV. 69 Begin Plan_Block Planning_Group { "24_17_C-5S_1" "24_20_C-5S_1" } "24_18_C-5S_1" "24_21_C-5S_1" "24_19_C-5S_1" "24_22_C-5S_1" Begin Plan_Sequence Begin Route_Segment_List // ----------- Next ----------// Location Type_Point Nav_Category Type_Maneuver //-------------------------------------------------------ORIGIN START 1 DEFAULT "KDOV" "ENROUTE" 2 CRUISE //-------------------------------------------------------End Route_Segment_List End Plan_Sequence End Plan_Block Figure 3.13 Route Segment for LEMO to KDOV In summary, the example mission traverses the following points: (KDOV, KGRK, KDOV, ERLP, LEMO, ERT2, ERT1, TLXR, OKBK, TLXR, ERT1, ERT2, LEMO, KDOV). 3.1.4 SAP Decisions For a given SAP instance, a solution requires the following decisions: The detailed asset level visibility of aircraft and TPFDD level visibility of requirements (aircraft/requirement pairings). The selected detailed routings flown by each aircraft in servicing an APOE/APOD pair (aircraft missions). Time phasing of aircraft along routes, both to the APODs and on return flights, to account for the dynamically changing network characteristics (for example, changing available MOG due to the arrival and departure of aircraft). This includes the takeoff and landing times of each aircraft at the APOEs, APODs and at recovery and en route bases. 70 To facilitate time phasing decisions, it is assumed that AMC can dictate aircraft departure times and vary flight speeds. The SAP objective is multicriteria and hierarchical in nature. Depending upon mission priorities, pertinent criteria and their ordering may change. Key objectives include minimizing the lateness of all requirements at their APOD, minimizing congestion at airfields in the airlift network, minimizing the number of missions flown, and minimizing the number of aircraft used. The above set of inputs and decisions represent the basic SAP model. Aircraft are assigned a series of missions to cover the given requirements. Aircraft may return to home base at the end of each mission, or within the defined return to base time, an aircraft may move from its recovery base to an APOE. The following section discusses the SAP representation in the context of the chosen solution methodology. 3.2 SAP REPRESENTATION One goal in formulating our representation is to compactly synthesize the critical SAP aspects so that construction and implementation of moves in a TS framework are facilitated. The inherent ordering already present for each SAP mission strongly effects this consideration. 71 (4,5) REQ A REQ B APOE 1 ER 1/ RB 2 (17,18) (8,9,10,11,12) RB 1 (22,23,24,25,26,27) HB 1 A/C X (1,2) A/C Y (3) WP 1 (13,14,15) APOD 1 (19,20,21) (6,7) APOE 2 REQ C Key: RB WP HB ER CS (16) WP 1 ER 2/ CS 2 Recovery Base Waypoint Home Base Enroute Base Crew Staging Base A/C Aircraft REQ Requirement A/C X trip (1) A/C X trip (2) A/C Y trip (3) Figure 3.14 SAP Representation using Pure Nodal Approach 3.2.2 Pure Node Based Approach Representing the SAP using a pure nodal approach is the most atomic representation. In this representation, every point visited by an aircraft in a mission would be assigned a node. Figure 3.14 depicts a simple instance of a SAP airlift network. There are three requirements (A, B & C) located at two different APOEs. Arrows indicate a possible solution using feasible paths through the network. The numbers at each node represent a separate visit to that particular node. Thus, using the pure nodal approach, the solution in the figure may be represented in the following manner: 72 (1,22,4,8,13,19,17,9,23)(2,24,5,10,14,20,18,11,25)(3,26,6,16,21,15,12,27). Each subset of numbers (within the parentheses) represents a mission. The succeeding numbers represent points visited during the mission. In the first mission, aircraft 1 is used. It departs HB1(represented by 22) and proceeds to APOE1(4). There, some portion of requirements A and/or B are loaded. Payload attributes would be stored with the aircraft trip number. Using the route segment represented by ER1(8) and WP1(13), the aircraft proceeds to its destination APOD1(19) where the cargo is offloaded. The aircraft then flies to its recovery base, RB1(17). Finally, the aircraft flies via the route segment ER1(9) to its home base HB1(23) where it prepares for subsequent missions. For realistic instances of the SAP, the cardinality of the pure nodal representation will be very large. Because locations are often visited multiple times by many aircraft, a unique node for every visit would have to be created to capture such attributes as when an aircraft passed through a node. Additionally, there is an inherent structure and ordering for a SAP mission. For example, route segment sequences must be followed and APOEs must be visited prior to APODs. Thus, in an ATS sense, a move that swaps nodes within a route segment or puts an APOD prior to an APOE are inadmissible. To change a route segment under this representation, a string of nodes would have to be removed and a new string inserted. Most importantly, this representation does not exploit the tremendous reduction in possible routes due to the region concept. Thus the large cardinality 73 and complexity of dealing with the SAP using a pure nodal approach outweighs the potential gains due to its atomic nature. 3.2.3 Arcs A fundamental fact concerning the manner in which the airlift system is built and modeled at AMC is that there are a restricted number of ways to get from one region to another. The plan file lists all possible route segments joining two regions for defined aircraft groups (Military, Civilian, C-5s, etc.). The use of regions dramatically reduces the number of possible routes. Thus, instead of using a purely nodal approach, consider the use of a supernode or arc as an aggregate representation of a set of nodes. For our purposes, arcs will have an origin node, a route segment (comprised of a series of bases and waypoints) and a destination node. In every mission, there is an origin location for the aircraft (normally the home base), an APOE, an APOD, a recovery base, and an ultimate destination (normally home base). Joining these locations are the route segments. Thus, for a typical mission, there is an arc for the home base to APOE node set, an arc for APOE to APOD, an arc for APOD to recovery base, and an arc for recovery base back to home base. Consider the first mission in the pure nodal approach example presented in the previous section. That nine node mission would now be represented by five objects: a node for the aircraft trip, and an arc each for HB1 to APOE1, APOE1 to APOD1, APOD1 to RB1 and RB1 to HB1. Figure 3.15 shows the missions pictured in Figure 3.14 using the arc representation. 74 (5) REQ A REQ B (4) APOE 1 ER 1/ RB 2 (7) (10) (7) RB 1 (6) (5) (5) HB 1 A/C X (1,2) A/C Y (3) (8) (11) WP 1 (10) APOD 1 (9) APOE 2 REQ C Key: RB WP HB ER CS WP 1 ER 2/ CS 2 Recovery Base Waypoint Home Base Enroute Base (9) Crew Staging Base A/C Aircraft REQ Requirement A/C X trip (1) A/C X trip (2) A/C Y trip (3) Figure 3.15 SAP Representation using Arcs Numbers next to directed arrows represent the Arc object that contains the nodes connected by the arrow. The three mission solution representation now is: (1,4,5,6,7)(2,4,5,6,7)(3,8,9,10,11). This representation exploits the structure of the airlift network and the structure of typical airlift missions, providing a much more compact representation than the pure nodal approach. This representation is used throughout this dissertation. Arc attributes and other details concerning the SAP representation and data structures are described in Chapter 4. 75 3.3 SUMMARY This chapter addressed the AMOS input file structure, key SAP decisions and the SAP representation. The key decisions for the SAP are the assignment of aircraft missions to requirements and the routing and scheduling of the assigned missions. The arc-based SAP representation compactly captures mission assignment information (mission node) and routing and scheduling information (arcs). The structure of a typical mission using this representation facilitates the implementation of a tabu search methodology. Chapter 4 provides a detailed description of the tabu search algorithm developed based upon this representation. 76 Chapter 4: A Tabu Search Approach to the Strategic Airlift Problem This chapter describes the Strategic Airlift Problem Tabu Search algorithm (SAP-TS). 4.1 TABU SEARCH ARCHITECTURE Basic tabu search was described in Chapter 2. The ATS approach developed for the SAP incorporates several features to enhance the search. These features are: an adaptive memory feature that varies the tabu tenure to encourage intensification in promising areas and diversification in less promising regions; the use of dynamic neighborhood selection to guide the choice of neighborhood based upon the search trajectory; and, once the neighborhood is selected, the aggressive use of candidate list strategies to target moves that are most likely to improve solution quality. These features along with the TS framework employed are described in detail below. 4.1.1 JAVATM Software Programming Language and OpenTS The SAP tabu search algorithm was developed using the JavaTM software programming language. The portability and object-oriented nature of the JavaTM programming language were key factors. Additionally, prior efforts by Harder (2000), Wiley (2001) and Crino (2002) illustrate the efficacy of JavaTM in rapidly developing effective tabu search algorithms. In particular, Harder s JavaTM implementation of a generalized tabu search engine, OpenTS, provides the framework within which specialized tabu search algorithms can be developed. In 77 OpenTS, the user defines the basic components of tabu search in the context of the problem. The OpenTS required components along with the associated SAPTS classes are shown below (Figure 4.1). OpenTS Solution structure Move manager Objective function Tabu list Move Figure 4.1 SAP sapSolution sapMoveManager sapObjFunction sapTabuStructure sapMove OpenTS Required Elements and Associated SAP-TS Classes Given a An iteration in OpenTS starts with an incumbent solution. starting or current solution, the move manager is then called and generates a list of moves for consideration. These moves are then invoked in turn on the incumbent solution and sent to the objective function object for evaluation. Using the tabu list, the best non-tabu move is selected. High quality solutions that are tabu may still be selected using some aspiration criteria such as best solution found so far , to override the tabu status. Once a move is selected, the move is performed on the incumbent solution and the new solution is passed to the move 78 manager to generate the next set of moves. graphically. Figure 4.2 shows this process Figure 4.2 4.2 OpenTS Architecture (Harder 2000) SAP-TS DATA STRUCTURES During preprocessing, AMOS data input files are read and a series of objects created. These objects and their key attributes are summarized in the table below. 79 Object AC Key Attributes Type, weights, times, fuel data, payload targets, cargo compatibility Aircraft Tail number, Home base, unit, missions, RTB time, time available, last time at HB Locations ID number, Lat-Lon, region, narrow working MOG, wide working MOG, MOG logic, daily fuel capacity, fuel deliveries Requirements Requirement number, ALD, RDD, APOE, APOD, commodity type, outsize, oversize, bulk, PAX Routes Arc Source Region, Sink region, planGroupName ID number, Start base, stop base, Route segment, legs, leg timings, leg fuelRequired, leg fuelBurned Node Mission number, Tail number, % capacity left, payload, APOE-APOD timings array, APOEAPOD fuel arrays Table 4.1 SAP-TS Objects and their Attributes Available aircraft units define the number, type, home base (HB), and available time for the available aircraft. An AC object is created for each type of aircraft available. An Aircraft object is created for each physical aircraft available for the instance. Aircraft objects inherit the attributes of the associated AC object. A Locations object is created for each base and waypoint available in the airlift 80 system. A Requirements object is created for each requirement listed in the requirements file. The set of available APOEs and APODs and their unique pairings are defined by the list of available requirements. The set of possible recovery bases (REC) is based upon the set of APODs and aircraft types available one or more REC bases for each APOD and aircraft type available. The set of HBs, APOE to APOD pairs, and REC bases generated above are used to construct the Arc objects. The set of possible Arc types is comprised of HBtoAPOE Arcs, APOEtoAPOD Arcs, APODtoREC Arcs, RECtoHB Arcs, and RECtoAPOE Arcs. An Arc object is generated for each of these types based upon all available bases (HB, APOE, APOD, REC) and the different route segments (defined in the Routes object) connecting the endpoint bases. Arcs feasible for more than one aircraft type store unique timings and fuel requirements for respective aircraft types. The number of Arc objects generated for a particular Arc type, say HBtoAPOE, is: i HBs j APOEs } k Segments } { { I (}HB , APOE { i j , Segment k ) , where I (HBi , APOE j , Segmentk ) is the indicator function that has value one if Segmentk joins HBi to APOEj and value zero otherwise. The total number of Arcs generated is the sum over all the Arc types. For an instance of moderate size containing 10 HBs, 48 APOEs, 30 APODs, with 56 unique APOE-APOD pairs, and 18 RECs, over 11,000 Arcs were generated. To keep the number of Arcs generated at a reasonable level, RECtoAPOE Arcs are 81 generated only as needed during the course of the search since return to base (RTB) times limit the number of missions that can employ RECtoAPOE Arcs. The number of aircraft mission Node objects generated is a balance between ensuring sufficient missions are available for each aircraft while not having so many that the search is degraded. The default number of mission Node objects generated for each aircraft allows the aircraft to be active for about one third of the planning horizon. This is based on a typical mission cycle time of just over two days. 4.3 SAP-TS SOLUTION STRUCTURE As described in Section 3.2.2, a solution to an instance of the strategic airlift problem is comprised of a set of missions. The sapSolution object holds the set of assigned missions in an array. Each mission is an array containing the identifiers that point to the Node and Arc objects that hold assigned payload, timings and fuel consumption as well as the routing information. Missions are ordered in the solution by aircraft number and then temporally for each mission assigned to the aircraft. Recall that a typical mission starts from a HB, travels to an APOE and loads cargo, flies to an APOD and offloads the cargo, moves to a regional REC base and then returns to the HB. A mission Node contains information about the departure and arrival times for the HB, APOE, APOD, REC and HB bases along the route, a detailed breakdown of the payload (requirement number and quantity of each cargo category), and the percent capacity remaining for the aircraft. Four separate Arc objects capture the routing information and standard times and fuel 82 requirements for arcs with no payload. The mission Node holds the timings and fuel requirements for the APOE-APOD arc based upon the payload. standard mission then, the mission array structure is: (MissionNode#, HBtoAPOE#, APOEtoAPOD#, APODtoREC#, RECtoHB#) A collection of these missions constitutes a solution to the SAP. 4.4 INITIAL SOLUTION CONSTRUCTION The initial solution is constructed incrementally using a greedy method to assign aircraft missions to APOE-APOD pairs. Available missions are sorted according to time available. The first mission for each aircraft is assigned an available time equal to the original time available for the aircraft. Subsequent missions for the aircraft are assigned later available times (artificially) by adding an additional day to the original available time. Requirements are sorted and grouped by APOE-APOD pair and ALD. The pseudocode for this procedure is presented in Figure 4.3. For a 83 Pseudo code for initial solution Sort aircraft missions by time available Initialize requirement array for each day i { if no aircraft missions left, break for each APOE-APOD pair j { while cargo remaining for day i and APOE-APOD pair j { while mission not assigned { get next aircraft mission if not compatible with available cargo, continue else { assign the mission remove mission from mission candidate list, set mission assigned = true }//end else }//end while mission not assigned //build the mission select APOE-APOD Arc id while cargo left space left on day i at APOE-APOD pair j { load cargo available }//end while build mission route update aircraft time available add mission to solution array if no aircraft missions left, break }end while cargo remaining }//end for j }//end for i Figure 4.3 Pseudo Code for Initial Solution The initial solution construction method quickly yields a solution to the SAP that is typically an infeasible but suitable starting point for SAP-TS. 84 4.5 OBJECTIVE FUNCTION The SAP-TS objective function object calculates the current solution value based upon the following criteria: Undelivered cargo Late cargo Working MOG violations Fuel MOG violations UTE Rate violation Trivial load violations Number of trivial loads Number of missions Number of aerial refueling missions Total objective value Undelivered and late cargo measure how well a solution meets RDDs for a given SAP instance. Undelivered cargo is the total weight (tons) of all cargo not delivered to their respective APODs. Late cargo is the total weight (tons) of all cargo not delivered to their respective RDDs. Both undelivered cargo and late cargo are weighted according to the amount of time (in days) that the cargo is late. The amount late for the undelivered cargo is dependent on the user specified planning horizon. The weight for late cargo is the difference between the arrival time at the APOD and the cargo s RDD plus one--delivery anytime on the RDD is considered on time . 85 Working MOG, fuel MOG and UTE rate violations measure the feasibility of a solution. Working MOG violations are the sum of the total number of MOG violations at each base at each time interval. The default time interval is 15 minutes. Fuel MOG violations are the sum of the daily fuel shortages (in tons) at all bases under the current solution. A solution with a feasible UTE rate has no more aircraft in the air at any given time than the respective aircraft UTE rates. The UTE rate violation is the sum of the difference between of the number of aircraft of each type in the air at any given time and the particular aircraft type s UTE rate. Working MOG and fuel MOG violations indicate overly optimistic throughput of aircraft in the airlift network for a given solution. violations indicate overly optimistic aircraft operational rates. feasible settings for these measures are 0. UTE rate The default Experience with feasible AMOS solutions to SAP instances show that these measures are rarely 0 especially for MOG and UTE measures. Thus, user defined threshold values for these measures are permitted in SAP-TS to preclude overly constraining the airlift system. Trivial loads are loads whose weight falls below an aircraft s trivial load threshold. A mission with a trivial load is not permitted in AMOS. Depending on rules invoked by the user, trivial loads are either aggregated at a central base and then delivered or are simply not delivered. In the SAP-TS implementation however, trivial loads are permitted all requirements are delivered. Thus the presence of trivial load missions does not violate feasibility of a solution. Missions with payloads less than the trivial load are counted and the trivial load 86 violation, the sum of the difference between the trivial load threshold and the payload, is computed. The remaining criteria incorporated into the objective function are simple counts. The number of trivial loads is a count of the number of missions whose payload is less than the aircraft s trivial load threshold. The number of missions is the current total number of assigned missions. The number of aerial fueling missions is the sum of all aerial refueling legs used in the current solution. Air Mobility Command desires to limit the number of Aerial Refueling missions in support of strategic airlift because refueling assets are dedicated to supporting deployment of fighters and bombers. This is especially true in the surge phase of a major deployment. These criteria are combined using a weighted sum (default weights are 1 for each of the criteria) and the goal is to minimize this weighted sum. In Harder s OpenTS, objective function comparison is conducted by comparing each component of the objective in a hierarchical fashion until a winner is determined or all objective components have been checked. The order in which the SAP-TS objective components are compared is the objective total, undelivered cargo, late cargo, working MOG violation, UTE rate violation, fuel MOG violation, trivial load violation, number of missions, number of trivial loads, and the number of aerial refueling missions. 4.6 MOVE NEIGHBORHOODS Critical decisions for the Strategic Airlift Problem are assignment of aircraft to requirements, routing of aircraft in assigned missions and the detailed 87 timings of aircraft movement through the airlift network. The moves developed target aspects of the SAP that work to obtain feasibility and improve upon assignment, routing and scheduling decisions. This section describes the various move neighborhoods developed for SAP-TS. 4.6.1 SAP-TS Neighborhood Phases The Strategic Airlift Problem can be broken into two phases a mission assignment phase and a mission improvement phase. In the assignment phase, aircraft missions are allocated to service requirements at APOE-APOD pairs. A mission assignment/scheduler heuristic (described in section 4.9) is then called to build and schedule missions based upon the given assignment. Once an assignment of an aircraft mission to an APOE-APOD pair is made, the mission improvement phase is invoked to obtain the best routing and timings for the given mission assignment scheme. Thus, given a set of APOE-APOD pairs and a set of available missions, we seek a good assignment of these missions to the pairs that minimizes the objective function. Then, based upon the mission assignments, local changes to missions are made seeking further improvement. The New Mission Insert Move Neighborhood, Within Pair Insert Move Neighborhood, Between Pair Swap Move Neighborhood, and Between Pair Insert Move Neighborhood traverse the mission assignment space. The Within Mission Arc Swap Move Neighborhood, Load Reallocation Move Neighborhood, APOE to Recover Arc Insert Move Neighborhood and the Impose Time Delay Move Neighborhood work to improve 88 attributes of individual missions. These neighborhoods are described in detail below. 4.6.2 New Mission Insert Move Neighborhood (NMI) NMI moves target APOE-APOD pairs that have either undelivered or late requirements. Undelivered requirements (if any) are targeted first and are sorted according to largest undelivered quantity. Late requirements are then considered and are sorted according to the product of the quantity of cargo late (tons) and the amount late (days). Insert missions are selected from the set of current unused missions and inserted into the set of missions currently assigned to the respective APOE-APOD pairs. The number of candidate new missions is restricted to no more than one per available aircraft inserting more than one mission for the same aircraft is redundant. Additionally, candidate new missions are checked for load feasibility for each undelivered or late requirement considered. For example, if an undelivered requirement under consideration consists only of outsize cargo and the current candidate new mission is a PAX only aircraft, the new mission is not considered for insert. The number of possible insertion points for a given unused mission and APOE-APOD pair is one more than the current number of missions assigned. Even for moderate sized problems, the number of possible inserts can be prohibitively large. Thus a candidate list structure is imposed on the NMI neighborhood to focus on insert locations most likely to reduce undelivered or late requirements. Candidate insert locations for undelivered requirements are the locations of those current missions whose last requirement ALD is either close 89 to the undelivered requirement ALD or whose payload contains some portion of the undelivered requirement. A mission s last ALD is close if it is within the user specified move distance used in the LR neighborhood (default is 3 days). Similarly, candidate insert locations for late requirements are the locations of those missions whose last ALD is within the LR move distance or whose payload contains some portion of the late requirement. As an example, consider the mission number assignments presented in Table 4.2. In this example there are five aircraft available, each with five possible missions. Tail Number 1 2 3 4 5 Table 4.2 Aircraft Type C-5A C-5A WBC WBP C-17 Available Mission Numbers 1 through 5 6 through 10 11 through 15 16 through 20 21 through 25 Aircraft and Missions Available An example of an APOE-APOD pair to mission assignment is given in Table 4.3. 90 Pair Number 0 1 2 3 Table 4.3 APOE-APOD Pair [KDOV, LTAG] [KDOV, OKBK] [KDMA, ETAR] [KBLV, RJBB] Assigned Missions 1,11,7,16,24 2,6,21,12,3 13,8,17,23 22,18, APOE-APOD Pairs and Assigned Missions This arrangement can be represented as follows: (0, (1, 11, 7, 16, 24), 1, (2, 6, 21, 12, 3), 2, (13, 8, 17, 23), 3, (22, 18)), where APOE-APOD pair numbers are followed by an array of the ordered mission assignments to the APOE-APOD pair. In this example, there is undelivered cargo for the pair KDOV to OKBK (pair 1). Additionally, there are late deliveries of requirements for the pair KDMA-ETAR (pair 2). The NMI neighborhood first determines the missions available for insert. The unused missions are: (4, 5, 9, 10, 14, 15, 19, 20, 25). From the set of unused missions, at most one mission number for each aircraft is added to the insert list. Thus, in this example, the set of possible insert missions are (4, 9, 14, 19, 25). Unused missions are then sorted according to earliest time available under each aircraft s current assignment scheme. The set of APOE-APOD pairs considered are restricted to those pairs with undelivered cargo and/or late cargo. Undelivered cargo is targeted first and pairs are sorted for insert based on the total quantity of undelivered cargo. Pairs with 91 late cargo are considered next and ordered according to the product of the tonnage and amount of lateness. Once the insert missions and pairs are identified, the set of insert locations are determined. Returning to the example, there are only 6 possible insert points for unused mission 19 into pair 1 (KDOV to OKBK). Using the candidate list structure described above, the possible insert points are identified and added to the candidate list. Unless all missions for the APOE-APOD pair under consideration possess a last ALD close to the late/undelivered requirement ALD or carry some portion of the requirement, the number of insert points will be less than the 6 possible insert points. Typically requirement ALDs are spread over time for realistic TPFDDs and the number of filtered insert points will be significantly less. 4.6.3 Between Pair Swap Move Neighborhood (BPS) BPS moves swap missions between two different APOE-APOD pairs. Allowable swaps are those that swap two different types of aircraft or two different aircraft. Swapping missions assigned to the same aircraft would be a null move, i.e., illogical and wasteful of computational effort. In an effort to target poor mission assignments, a dynamic functional candidate list is created to sort the APOE-APOD pairs and the missions within those pairs for consideration. Swaps are then investigated based upon the sorted pairs and missions within the pairs. Under the functional candidate list scheme, mission quality is assessed using the mission attributes of late requirements and unused aircraft capacity. 92 The late requirement value is the product of the quantity late (tons) and the amount late (days). Unused aircraft capacity is the difference between the aircraft ACL and the current payload. A mission s quality value is the sum of these two measures. Missions are sorted in descending order within the APOE-APOD pair according to this measure. APOE-APOD pair quality is the sum of the assigned mission values and any undelivered cargo. These three measures are readily obtained and provide a sufficient means to evaluate current mission assignments. Late requirements and undelivered requirements provide a quick assessment of aircraft assignments. Unused aircraft capacity measures not only how well aircraft are loaded but also (indirectly) the current missions assignment s impact on MOG and UTE constraints small loads mean more aircraft missions and hence a greater strain upon the airlift network. Continuing with the example, for the current pair to mission assignment, (0, (1, 11, 7, 16, 24), 1, (2, 6, 21, 12, 3), 2, (13, 8, 17, 23), 3, (22, 18)), there are 8 possible swap moves between APOE-APOD pairs 2 and 3. Under the BPS move neighborhood however, only 6 are allowed swaps (22, 23) and (17, 18) are not allowed since they are missions for the same aircraft. The order in which these swaps are considered is based upon the candidate list structure described above. Subsequent iterations begin at the last BPS swap location to ensure that other regions of the solution space are investigated. This creates a rotating list of swaps that moves through the APOE-APOD pairs sequentially, starting over whenever all swaps for the last two APOE-APOD pairs are exhausted. 93 4.6.4 Within Pair Insert Move Neighborhood (WPI) WPI moves seek improvements in mission assignments within APOE- APOD pairs. Thus WPI moves are intensifying moves. Under the WPI neighborhood, a mission assigned to a given APOE-APOD pair can be inserted in a different position within the current set of missions assigned to the pair. Candidate APOE-APOD pairs for WPI moves are sorted by contribution to the objective function. In this way, APOE-APOD pairs that contribute most to the objective are investigated first. The mission assignment/scheduler heuristic attempts to assign missions to requirements based upon the requirement RDD, mission ordering, and aircraft compatibility with the requirement. Changing the order of missions within an APOE-APOD pair can lead to better load efficiencies and improvements in temporally based objective values (working MOG, fuel MOG, cargo lateness, and UTE rate). As an example, consider the following mission assignment for an APOEAPOD pair: (252, 140, 147, 154, 161, 168, 0, 163, 170, 289, 296, 303, 310, 387). The current APOE arrival times for the missions are summarized in Table 4.4. 94 Mission Number 252 0 140 147 154 161 168 387 163 170 289 296 303 310 Table 4.4 APOE Arrival Time (days) 0.17708 0.18167 0.22324 0.22324 0.22324 0.22324 0.22324 11.09375 11.12500 11.12500 11.17708 11.17708 11.17708 11.17780 APOE Arrival Times Missions are sorted in the table by arrival time at the APOE. Observe the position of Mission 0 in the mission assignment array and in Table 4.4. The temporal ordering of missions is not (necessarily) the same as the mission assignment ordering. Under the mission assignment/scheduler heuristic, aircraft missions are assigned based upon several factors that may preclude missions from 95 being assigned strictly according to mission assignment order. These factors are: aircraft availability time (due to prior missions), aircraft compatibility with the current controlling requirement, and aircraft compatibility with other requirements available at the APOE on the current controlling requirement s ALD. Because of this phenomenon, it is difficult to determine where good insert points lie. Ideally all possible inserts are considered, however, for typical instances of the SAP, the number of inserts can be prohibitively large. As a compromise, the insert points for each mission within an APOE-APOD pair are restricted to be within an insert move distance that is based upon instance size. The number of missions per pair in the initial solution is used as a rough measure of instance size. Two measures are used to determine the insert move distance the average number of missions per pair (AM/P) and the median number of missions per pair (MM/P). These measures provide an estimate of the typical number of missions per APOE-APOD pair. The insert move distance is set to the maximum of the default move distance (set to 5 in this implementation), AM/P and MM/P. In this way the insert move distance is tied to instance size and a reasonable portion of the possible inserts are investigated. WMI inserts are all possible inserts within insert move distance to the left and right of the insert mission. Returning to the example, assume that the insert move distance is 5. The possible WPI insert points for Mission 0 are shown in Figure 4.4 below: 96 (252, 140, 147, 154, 161, 168, 0, 163, 170, 289, 296, 303, 310, 387) Figure 4.4 4.6.5 WPI Insert Points for Mission 0 Between Pair Insert Move Neighborhood (BPI) The BPI move neighborhood removes an assigned mission from one APOE-APOD pair and inserts it into a different APOE-APOD pair. BPI moves change the cardinality of the current mission assignment structure and therefore is a diversifying move. Because BPI reduces the number of missions assigned to an APOE-APOD pair, it likely results in a large undelivered penalty. Thus, BPI is used sparingly as a temporary infeasible diversifier that may also add heretofore unavailable trajectories to the search. Its use is curtailed to conditions where diversification is appropriate. Diversification conditions are described in the Move Manager section below (Section 4.10). The dynamic functional candidate list described for BPS moves in Section 4.6.3 is used to determine the ordering of APOE-APOD pairs and their mission assignments. However, for the BPI neighborhood, ordering of candidate pairs and missions for consideration is in ascending order. This targets poor pair assignments and identifies inserts from other missions that may improve the objective. BPI moves that render an APOE-APOD pair devoid of missions are not allowed. 97 4.6.6 Within Mission Arc Swap Move Neighborhood (WMAS) WMAS moves target improvements in working MOG and fuel MOG at the mission level. WMAS candidates are those missions in the current solution that contribute to working MOG and/or fuel MOG violations. Recall each Arc has a start base and end base that are connected by a route segment. The route segment consists of the waypoints and bases the aircraft passes through when traveling from the start base to the end base. In the case of a direct flight, no intermediate bases are traversed and swapping the associated Arc would have no effect. Thus swapping an Arc in the current routing scheme can change the set of bases that the aircraft passes through and thus reduce congestion at intermediate bases. For the special case of APOE-APOD Arcs, a new Arc can have a smaller maximum cargo than the incumbent Arc. In this case cargo must be removed to meet the new maximum cargo limit. For WMAS, late requirements are first considered for ejection then requirements are dropped based upon compatibility and RDD order until the maximum cargo limit is met. Ejected cargo is then added to the undelivered cargo list. In the case where cargo is ejected as a result of a WMAS move, there is an opportunity to employ powerful compound moves called ejection chains (Glover and Laguna, 1997). A route change that forces cargo to be ejected causes a change in the assignment of cargo as well as the routing of the mission. Ejected cargo as part of a WMAS move immediately invokes the LR move neighborhood to determine the best mission to load some or all of the undelivered cargo. The 98 value of the WMAS move then is the result of this compound move. This ejection chain feature is incorporated into SAP-TS and can be invoked throughout the search or as a result of conditions encountered during the search. 4.6.7 Load Reallocation Move Neighborhood (LR) The LR move neighborhood strives to increase fill efficiencies and reduce the total number of missions required. LR moves target undelivered cargo and those missions with payloads below the aircraft s trivial load threshold. If there is undelivered cargo, LR moves attempt to load undelivered cargo on existing missions. The LR move neighborhood assigns cargo from aircraft with trivial loads to aircraft with excess cargo capacity (percent capacity remaining > 0). Spatial and temporal restrictions limit candidate missions for the LR move neighborhood. Candidate onto missions must service the same APOE-APOD pair as that of the undelivered requirement or the trivial load mission. Additionally, onto missions must fall within the LR move distance. The LR move distance is a temporal restriction. An onto mission is a candidate if the difference between the requirement new ALD and the onto mission s APOE arrive time is less than the LR move distance. A default move distance of 3 days was selected because typical mission cycle times are less than three days and delaying a mission more than this amount is usually more costly than just creating a new mission. Undelivered cargo moves are considered first and then moves for missions with less than trivial loads. If all requirements of a trivial load mission are removed, the mission is deleted from the solution. 99 4.6.8 Recover to APOE Arc Insert Move Neighborhood (RAI) RAI moves are specialized moves designed to reduce requirement lateness by altering an aircraft s routing. Specifically, RAI moves delete a REC to HB Arc and insert a REC to APOE Arc based upon the APOE visited by the aircraft s next mission. In this way the time from REC base to HB and from HB to APOE is replaced with the (shorter) REC to APOE time. Figure 4.5 illustrates the RAI move. REC-REC (0, HB-APOE,APOE-APOD,APOD-REC,REC-HB) REC-APOE (1, HB-APOE,APOE-APOD,APOD-REC,REC-HB) (2, HB-APOE,APOE-APOD,APOD-REC,REC-HB) (3, HB-APOE,APOE-APOD,APOD-REC,REC-HB) Figure 4.5 RAI Move RAI moves are constrained by the return to base (RTB) time. The RTB constraint is an aircraft unit attribute. Aircraft assigned to the same unit share the same RTB time. In practice, a new mission cannot be planned if the time since the aircraft last visited its HB exceeds the RTB time. If the RTB is exceeded, the aircraft must return to its HB before a new mission can be planned. In SAP-TS, RTB time violations are determined as follows: the last aircraft HB departure 100 time is determined. Then, for the mission in which a REC to HB arc is replaced with a REC to REC arc, the time available at the REC base (arrival time at the REC base plus en route time) is determined. If the difference between time available at the recovery base and the last HB departure time is less than the RTB time, the RAI move is permissible. As an example, suppose that an aircraft (a C5A) has the following set of missions assigned under the current solution: (0, 2, 53, 170, 309) (1, 2, 72, 176, 541) (2, 2, 43, 162, 179). Further, suppose that we desire to invoke an RAI move that involves missions 0 and 1. Let the RTB time for a C-5A be 3 days and the en route time 3.25 hours (0.14 days). Let the depart time from the HB for mission 0 be at time 0 the last time the aircraft was at its HB. Arc object 170 is the APOD to REC arc for mission 0. Suppose that arrival at the REC base is at 1.61 days. Thus the aircraft is available at the REC base at 1.61 + 0.14 or 1.75 days. Since the difference between the time available at the REC base (1.75 days) and the time last at its HB (0.00 days) is less than the RTB time of 3 days, the RAI mission may be planned. A candidate list of RAI missions is constructed as follows. Aircraft with missions containing late requirements are identified and sorted by their lateness contribution. Then, for each of these aircraft, missions are identified that may feasibly move directly to another APOE. 4.6.9 Impose Time Delay Move Neighborhood (ITD) WMAS moves may not completely remove working MOG and fuel MOG violations. Violations that occur at endpoint bases are usually unaffected by 101 WMAS moves since only intermediate bases are exchanged. Also, route segments frequently have some bases in common thus swapping an arc may not remove MOG violations for some intermediate bases. ITD moves are invoked to temporally stagger aircraft through the network to alleviate congestion (working MOG and fuel MOG violations) and reduce the number of aircraft operating at any given time (UTE rate). Generally, working MOG violations accompany fuel MOG and UTE rate violations since working MOG violations indicate aircraft clustering. Thus, ITD moves that target working MOG violations often have the secondary effect of reducing fuel MOG and UTE rate violations as well. For this reason, ITD moves target in sequence working MOG, then fuel MOG and finally UTE rate violations. ITD move attributes consist of the mission number, the arc location at which the delay is imposed, and the amount of the delay (in days). A specialized candidate list strategy is imposed to target missions that contribute to MOG violations (working and fuel) and UTE rate violations. For working and fuel MOG violations, bases are ordered by the respective total violation. Then, for each base, the missions contributing to the violation are identified and sorted by slack time. Slack time is defined as the difference between earliest RDD time for the requirements on the mission and mission arrival time at the APOD. The intent is to target missions that can be delayed without increasing lateness first. Sorted missions are then investigated in turn and the delay is computed. The delay for a mission violating either a working MOG or fuel MOG constraint is the maximum ground time (in days) divided by 102 the base MOG. The maximum ground time is a user setting. The default setting of 5 hours exceeds the maximum ground time for all aircraft types (en route, onloading or offloading) in the current AMOS implementation. This delay has the effect of staggering aircraft through the constraining base sufficiently to reduce MOG violations without imposing too great a delay. For UTE rate violations, the focus is on aircraft and not bases. Aircraft types are sorted for consideration using the aircraft type s UTE rate violation. Missions are then sorted by aircraft type and then, once again by slack (as defined above) within aircraft type. The amount of delay imposed is computed in the same manner as described above. 4.7 TABU STRUCTURE Tabu restrictions are constraints imposed on allowable moves based upon move attributes. Tabu restrictions are used to prevent the reversal of recent moves and to allow the search to escape from local optima. The duration of tabu restrictions is the tabu tenure. Moves whose attributes are restricted under the current tabu list are considered tabu and are not allowed for the duration of tabu tenure iterations. There are two tabu lists implemented in SAP-TS to impose tabu restrictions. They mirror the dichotomy in the SAP-TS move neighborhoods a tabu list for the mission assignment phase (assignmentTabuList) and a tabu list for the mission improvement phase (missionTabuList). The tabu list for the mission assignment phase, assignmentTabuList, is invoked during the mission assignment phase of the search. The assignmentTabuList attributes consist of the APOE-APOD pair, mission number 103 and the position of the mission in the current APOE-APOD pair mission assignments. For insert moves (NMI, BPI, and WMI), the position is the location from which the inserted mission came from. For swap moves (BPS), both missions are added to the tabu list along with their original position in the current APOE-APOD pair mission assignments. As an example, suppose that the current tabu tenure is 7, the current iteration is 18 and the current APOE-APOD pair to mission assignment is: (0,(1,11,7,16,24),1,(2,6,21,12,3),2,(13,8,17,23),3,(22,18)). If a BPS move is performed swapping mission 16 with 21, any move that either moves mission 16 back to APOE-APOD pair 0 at position 3 OR moves mission 21 back to APOE-APOD pair 1 at position 2 before iteration 25 is considered tabu. For an insert move, if mission 16 is inserted into APOE-APOD pair 2 at position 1 under a BPI move, a move that returns mission 16 to position 3 in APOE-APOD pair 0 before iteration 25 is considered tabu. The tabu structure for the mission improvement phase, missionTabuList, is a list consisting of the move mission number and location within the mission array where the change occurred. Tabu active status duration (current iteration plus tabu tenure) is stored in an array of size 5 at the position where the change occurred. This array mirrors the mission array structure in a SAP-TS solution (mission node#, arc#, arc#, arc#, arc#). Thus the first position corresponds to the mission node, the second to the first arc traversed in the route, the third to the second arc traversed in the route and so on until the last arc at the fifth and final 104 position. The position in the array at which the tabu active status is stored is based upon the move neighborhood and where the change in the mission array occurs. An LR move changes the payload for two missions from mission and to mission. Payload changes affect the mission node (position 0 in a mission array) therefore tabu active status for an LR move is stored in position 0. WMAS and RAI moves represent changes in the routing of a mission. Tabu active status for these moves is stored at the location corresponding to the arc(s) that was changed. ITD moves store tabu active status at the arc position at which the time delay was imposed. In this way, mission improvement phase move attributes that affect specific portions of the mission, payload or routing, are recorded to prevent reversion to recent solutions. Note that mission improvement phase tabu attributes are based upon the solution array and not the APOE-APOD pair to mission assignment used in the mission assignment phase above. As an example, suppose that the following missions represent a portion of the current solution at iteration 10 (and that the tabu tenure is still 7): (0, 2, 53, 170, 309) (1, 2, 72, 176, 541) (2, 2, 43, 162, 179). Suppose that the current move is a WMAS move for mission 1 that swaps arc 72 with arc 73 resulting in the new mission (1, 2, 73, 176, 541). The tabu array corresponding to mission 1 would be set to (0, 0, 17, 0, 0). Mission improvement moves that affect the arc at position 2 are tabu if the current iteration is not greater than 17. Suppose an LR move is applied to the current solution that removes some portion of the payload from mission 0 (from mission) and places it on mission 1 (to mission). Only the to mission (mission 2) tabu 105 status is updated to preclude removal of cargo from a mission that just received additional cargo. This also permits consideration of the from mission payload in the next iteration if payload still remains. In this way trivial load missions can be quickly removed from the solution. Under this move, the tabu array for mission two would be set to (17, 0, 0, 0, 0). Tabu active status for an ITD move is set at the arc location where the time delay is imposed. RAI moves affect the tabu status of more than one mission. Continuing with the solution excerpt from above, (0, 2, 53, 170, 309) (1, 2, 72, 176, 541) (2, 2, 43, 162, 179) , suppose that an RAI move is imposed on missions 0 and 1 such that the last arc of mission 1 becomes a REC to REC arc and the first arc of mission 1 becomes a REC to APOE arc. For this move the tabu arrays of mission 0 and 1 become (0, 0, 0, 0, 17) and (0, 17, 0, 0, 0) respectively. A tabu active move may be accepted if it meets some aspiration criterion. In SAP-TS, the aspiration criterion is met if the tabu move value is better than the global current best solution. In this case, the tabu status is overridden and the move is allowed. 4.8 SAP TABU SEARCH ALGORITHM SAP-TS proceeds in turn through the following phases: Preprocessing o Read input files o Build objects/Initialize data o Construct arcs and aircraft nodes 106 o Construct Initial Solution o Evaluate Initial Solution Conduct Tabu Search o Initialize Tabu Search Objects o Start search o Select Move Neighborhood Phase o Select Move Neighborhood o Evaluate Moves o Select Best Non-Tabu Move o Perform selected move o Update Solution/Solution Value Elite List Iterations (Optional) Postprocessing o Save solution and solution objects to file o Output solution Mission Summary Output Detailed Itinerary Output Key tabu search parameters and default settings are summarized in Table 4.5. 107 Parameter NoChangeMovesTolerance UnimprovingMovesTolerance SuperDiversifyTolerance SolutionHistoryLength ObjectiveTolerance ChooseFirstImprovingMove MaxCompTime NumAircraftTrips NumRoutesToConsider MsnAssignmentPhase MsnImprovementPhase MsnAssignmentNBHSizeLimit msnImprovementNBHSizeLimit WorkingMOGTolerance FuelMOGTolerance UteRateTolerance NumberOfIterations Tenure EliteListSize ObjFnWeights loadReallocationMoveDistance InsertDistance Default Setting 3 3 10 20 0.01 TRUE 40 12 2 TRUE FALSE 100 800 0 0 0 400 7 3 {1,1,1,1,1,1,1,1,1,1 } 3 5 Description # no change value moves allowed before diversifying # unimproving moves allowed before diversifying # small change moves allowed before superdiversifying # previous objectives to consider for superdiverisfication tolerance for objective in superdiversification count boolean for selecting the first improving move maximum search time (hours) alotted for the search # aircraft trips per aircraft maximum number of routes to create for each arc boolean that sets mission assignment phase status boolean that sets mission improvement phase status mission Assignment Phase maximum NBH size mission Improvement Phase maximum NBH size threshold below which working MOG feasible threshold below which fuel MOG feasible threshold below which UTE rate feasible # tabu search iterations duration (iterations) of tabu attribute number of elite solutions to retain weights for changing priority of objective values temporal distance (days) for LR moves maximum insert distance (left & right) for WPI moves Table 4.5 Tabu Search Parameters 108 Tabu search parameters control the course of the search through a series of counters, tabu search listeners and thresholds. Details on how these parameters are employed are described in the Move Manager section (Section 4.10). There are several settings related to calculations in an instance of the SAP. These parameters affect calculations concerning timings, loading, and fuel calculations. Additionally, the search space size is controlled by airlift network size (numRoutesToConsider) and the allocation of aircraft missions (numAircraftTrips). These problem instance parameters are summarized in Table 4.6 below. Parameter payLoadThreshold Precision appLandTime PaxWeight ZeroTemp mogInterval Horizon poundsPerGallonOfFuel maxGroundTime numAircraftTrips numRoutesToConsider Table 4.6 Default Setting 0.05 0.001 15 400 18.3333 0.25 40 6.8 5 12 2 Description percentage below which payload calculations will stop precision level used in calculations standard approach & land time (minutes) standard passenger weight (lbs) temperature (0C) at zero alttitude interval (hours) for working MOG calculations planning horizon (days) conversion factor for fuel calculations maximum ground time (hours) across aircraft types # aircraft trips per aircraft maximum number of routes to create for each arc Problem Instance Parameters 109 The next several sections describe details of major components of the SAP-TS algorithm. 4.9 MISSION ASSIGNMENT/SCHEDULER HEURISTIC The mission assignment/scheduler heuristic is similar to the method used to construct the initial solution. The major difference is that the mission assignment/scheduler heuristic obtains a solution using a given ordered assignment of aircraft missions to the APOE-APOD pairs. Requirements are sorted by earliest ALD and then by RDD. The heuristic attempts to sequentially load the next available requirement on the next available aircraft assigned to the APOE-APOD pair. If the current aircraft is not compatible with the current requirement or not completely filled by the current requirement, other requirements available at the same APOE-APOD pair and on the same day are considered for loading. In this way, a solution is constructed based upon the ordered assignment of available missions at APOE-APOD pairs and prioritized requirements. Pseudocode for the mission assignment/scheduler heuristic is presented in Figure 4.6. Two additional measures drive the mission assignment/scheduler heuristic to obtain solutions that are closer to feasibility than the initial solution. The first is associated with how arcs are selected during mission construction and the second is associated with mission timings along the arcs. 110 createSolutionFromAssignedMissions(){ get sorted requirement list for each requirement i {//loop through & load each requirement get current requirement object and its APOE-APOD pair get current missions assigned to this APOE-APOD pair if( no cargo available) continue otherwise find a mission to assign while(cargo remains for requirement i){ while(no mission assigned){ if( no missions available) break get the next mission from current missions list if(aircraft not compatible with current requirement){ check if other cargo available on the same day/location is compatible }//end if else{ continue;} if(no mission assigned) assigned=false else{//mission was assigned remove assigned mission from current missions list }//end else }//end while no mission assigned if (no mission assigned) {//no feasible assigned aircraft left to take this reqt break; }//end if no feasible ac else create the new mission add new mission to newPairMsn list find an applicable APOE to APOD arc set time of arrival at apoe to time cargo is available to load (ald) load the aircraft with as much of the requirement as possible if(requirement cargo all delivered) remove requirement from reqt list if (aircraft not full){ try to load other reqts available for this APOE-APOD pair and ALD for each remaining requirement{ load until this aircraft is full or reqts are "exhausted" if(requirement cargo all delivered) remove requirement from reqt list }//end for each remaining requirement }//end if aircraft not full Create path for mission given aircraft start point & APOE-APOD arc Add mission to aircraft's current missions Add mission array to current solution }//end while cargo remains for this reqt }//end for each reqt i }//end createSolutionFromAssignedMissions Figure 4.6 Mission Assignment/Scheduler Heuristic Pseudo Code 111 Arcs are selected using a rotating list for each set of route segments connecting two endpoints. For example, suppose that the current mission for a C5A (aircraft type MIL) requires a route from KDOV to OKBK the APOE to APOD arc. The set of arcs connecting KDOV to OKBK under the current route plan are shown in Table 4.7. C-5A AcTypes Max Cargo (tons) # En route Bases CIV N/A 2 CIV N/A 2 MIL 83.98 1 MIL 84.40 1 MIL 77.31 1 MIL 79.42 1 MIL 88.07 1 MIL 87.06 1 MIL 85.31 2 MIL 85.72 2 MIL 78.82 2 MIL 80.93 2 MIL 88.59 2 MIL 88.07 2 MIL 109.94 3 Arc# 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 Table 4.7 Seg# 747 748 734 735 736 737 738 739 740 741 742 743 744 745 746 Arcs Connecting KDOV to OKBK Since the aircraft is a military aircraft, only arcs 54 to 66 apply. As this is an APOE-APOD route, we desire the arc with the largest maximum cargo so that payload is not constrained by the route. The maximum cargo range is [77.31, 109.94] and the average is 85.97 tons. Note that the arc with the greatest maximum cargo (arc 66) also uses the most en route bases three in all. Thus the increase in possible payload comes at the cost of additional strain on the airlift 112 network. Thus if APOE to APOD arcs were selected solely on the basis of arc maximum cargo, bottlenecks would develop reducing throughput. For this reason arcs are selected from the possible set on a rotating basis. If the current mission is assigned arc 66 the next mission (with a KDOV to OKBK arc) will be assigned arc 54 and so on. The second feature employed in the mission assignment/scheduler heuristic is a mechanism to stagger aircraft flow through the airlift network. As arcs are selected, a running tally is kept of the depart times from the arc s starting base. As an arc is employed in subsequent missions, a check is made to see if too many aircraft are starting down the arc at the same time. Too many aircraft is defined as more aircraft moving down the arc than the minimum wide body aircraft working MOG (WBWMOG) for all the bases in the arc. WBWMOG is selected because wide body aircraft slots at airfields are generally more limited than narrow body slots. Additionally, narrow body aircraft can often use narrow and wide body aircraft slots interchangeably. Aircraft are considered to be moving down the arc at the same time if they depart from the arc start base within the user defined maximum ground time (default is 5 hours) of each other. Aircraft missions meeting this temporal criterion are counted and the total is compared to the WBWMOG. If the WBWMOG is exceeded, the current mission s departure time from the arc start base is delayed by the ratio maximum ground time/WBWMOG. 113 4.10 MOVE MANAGER The move manager is the heart of SAP-TS and determines the appropriate phase, search context and move neighborhoods to apply to the current solution. Move neighborhoods are selected dynamically based upon current solution objective values and the search history. SAP-TS alternates between neighborhood phases missionAssignmentPhase (MAP) and the missionImprovementPhase (MIP). Performing moves in the MAP are more expensive than MIP moves. For each MAP move considered, the mission assignment/scheduler heuristic is called. As problem size increases so does the time required to execute a call to the mission assignment/scheduler heuristic. MIP moves employ change calculations updating only the affected missions and are therefore much faster to evaluate. This is why the MAP maximum neighborhood size is set to a smaller value than the MIP maximum neighborhood size. Several mechanisms are employed to ensure that the neighborhood phases alternate periodically. Figure 4.7 illustrates how the search phase and search context is determined. The search phase will remain the same if a new best or improving move is made. If either the superdiversification, unimproving move or no change move thresholds are exceeded, the search phase is changed. Once the search phase is determined, the context of search (superdiversify, intensify or diversify) is determined. 114 New Current Solution Yes Add solution to feasible elite list Feasible? No In superDiversify? No Yes Continue superDiversify? Yes superDiversify = true Phase = MAP superDiversifyCtr++ No Superdiversify Now? Yes Select Move NBH Evaluate NBH Perform Move No Current Phase? MAP MIP Yes New best or improving move? No No Iters since change phases > tolerance? Yes No unimproving Ctr OR no change Ctr > tolerance? Yes Iters since last best solution > tolerance? Change Phase & Intensify Yes No Diversify Intensify Superdiversify Figure 4.7 Determination of Neighborhood Phase and Search Context 115 Superdiversification conditions are met if, within the last solutionHistoryLength iterations, the number of objective function values that are within the objectiveTolerance of the current objective exceeds the superDiversifyTolerance. If superdiversification is called for, the phase is set to MAP and the neighborhood to BPI. Once superdiversification conditions are met, a total of numberSuperDiversifyMoves superdiversification iterations are executed. Default settings for each of these parameters are listed in Table 4.5. If superdiversification criteria are not met, additional checks are performed to determine if the search context should be intensify or diversify. As shown in Figure 4.7, new best or improving moves invoke intensification as the search context. Diversification is called for if either the threshold for the number of iterations since phases were changed or the threshold for the number of iterations since the best solution is exceeded. If both the unimproving move and no change move thresholds are not exceeded, diversification becomes the search context. Otherwise the phase is changed and the search context becomes intensification. Once the search phase and context are specified, the appropriate move neighborhood is selected. A given neighborhood phase restricts the possible move neighborhoods to the four neighborhoods associated with the phase. The search context also impacts upon the neighborhood selected. If the context is superdiversify, BPI becomes the active neighborhood. Intensification implies that the current neighborhood is finding good solutions and therefore the current neighborhood is maintained. In the special case where the phase changed and 116 intensify is the context, the neighborhood is selected based upon the current objective. Neighborhood selections under this situation are summarized in Table 4.8. Phase Mission Assignment Condition Undelivered/Late Cargo Infeasible Neighborhood NMI or BPS BPS or WMI LR or RAI WMAS or ITD Mission Improvement Undelivered/Late Cargo Infeasible Table 4.8 Neighborhood Selection After Phase Change Diversification implies that the current neighborhood should be changed. The new neighborhood is determined based upon the phase, the previous neighborhood and the current objective. For the MAP, if the current objective has undelivered or late cargo the selected neighborhood alternates between NMI & BPS. Otherwise, the selected neighborhood alternates between BPS and WMI. For the MIP, the selected neighborhood alternates between LR and RAI if there is undelivered or late cargo. between WMAS and ITD. 4.11 AN ITERATION IN THE SAP-TS ALGORITHM At the start of the search, the phase is set to MAP, search context to intensify and the neighborhood to BPS. An iteration in SAP-TS begins with a new current solution. As described in the previous section, search phase, context 117 Otherwise, the selected neighborhood alternates and neighborhood are chosen based upon a series of criteria. Once the neighborhood is identified, the set of possible moves given the neighborhood and the current solution are evaluated. The best non-tabu move is selected. The tabu active status of tabu moves that meet the aspiration criterion (move value is a new best solution value) are overridden. The selected move is performed and the solution is updated to reflect the change. If the move is feasible, it is inserted into the feasible elite list. If the move is an unimproving move, the tabu tenure is incremented by one and the unimproving move counter is incremented. If the move yields no change in the objective function value, the no change in value counter is incremented. If the move is a new best or improving move, the unimproving and no change in value counters are reset to zero. A new best solution causes the tabu tenure to be reset to the default tenure. Additionally, the new best solution is added to the elite list. The move manager is then invoked to determine the appropriate phase, context and neighborhood to employ based upon the new solution and the search history and the process repeats until either the maximum number of iterations have been performed or the maximum search time has elapsed. 4.12 SUMMARY This chapter described the SAP-TS algorithm. SAP-TS partitions the search into two alternating phases: mission improvement phase and the mission assignment phase. Within each phase a set of specialized neighborhoods is employed. For the mission assignment phase, the NMI, BPS, WPI, and BPI move neighborhoods seek to improve the assignment of missions to APOE-APOD 118 pairs. Moves are sent to the mission assignment/scheduler heuristic to obtain a new assignment. The mission improvement phase neighborhoods, LR, WMAS, ITD and RAI, are specialized neighborhoods that attempt to improve missions under the current mission assignment. The SAP-TS move manager uses a series of counters and search history to dynamically determine the appropriate phase, context and neighborhood at each iteration. The next chapter describes the SAP problem instances employed to validate the efficacy of the SAP-TS algorithm. 119 Chapter 5: An Application of the SAP Tabu Search Algorithm This chapter provides an example application of the SAP Tabu Search algorithm (SAP-TS) to the Strategic Airlift Problem. Results are reported for application of SAP-TS to five SAP scenarios. SAP-TS results are then compared to AMOS results. 5.1 SAP PROBLEM INSTANCES SAP-TS was applied to a set of five different scenarios of the Strategic Airlift Problem. These scenarios represent operationally realistic problem instances of the SAP and provide a sound basis to investigate the efficacy of SAPTS. A few of the scenarios were derived from AMC studies loosely based upon the deployment of an Army Stryker Brigade. Each scenario was adjusted to emphasize different aspects typical of SAP instances. In particular, feasibility (working MOG, fuel MOG and UTE rate constraints) and problem size (number of requirements, number of unique APOEAPOD pairs and number of aircraft) elements were varied to provide a robust assessment of SAP-TS. To discern between different scenarios, we employ the following elements to capture key problem characteristics: the number of aircraft, number of unique APOE-APOD pairs, total cargo (tons), total number of personnel and the feasibility constraints emphasized. summarized in Table 5.1. Scenarios and their key features are 120 Scenario Number of Aircraft Problem Parameters Number of Unique APOE-APOD Pairs Total Cargo (tons) Total PAX Scenario Emphasis Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 Table 5.1 297 124 297 165 70 56 45 95 12 12 7647 15482 13777 7647 3786 14769 10576 16683 14769 9769 (UTE/LATE) (LATE/MOG/FUEL) (MOG) (LATE) (Typical) Scenario Characteristics Several measures were used to compare SAP-TS with AMOS. First, the SAP-TS objective function was used to provide a wholistic measure of SAP-TS results against AMOS results. Second, the makespan or closure time for the respective solutions was computed to measure the time required for all requirements to reach the respective APODs under the two schemes. Finally, a comparison of the daily cumulative tonnage closed was tabulated to determine how well required delivery times were met throughout the deployment. 5.2 AN EXAMPLE STRATEGIC AIRLIFT PROBLEM The smallest problem instance, Scenario 5, was selected to illustrate SAPTS. Scenario 5 details and the results associated with SAP-TS and AMOS are now presented. 121 5.2.1 Scenario 5 Details This particular deployment entails movement of equipment and personnel from five different APOEs in CONUS to four APODs in three different theaters Southwest Asia, Europe and the Pacific. The aircraft, home base and time available for each unit involved in the deployment are shown in Table 5.2. Multiple entries for a particular unit indicate that additional aircraft become available at later times in the deployment horizon. A total of 70 aircraft are available during the course of the deployment. Type Aircraft WBP NBP Home Base KBOS KBOS RTB Time Day (days) Available Qty 0 5 10 20 5 0 5 10 20 5 0 5 6 5 10 20 5 0 10 8 5 10 20 5 10 0 5 10 0 5 10 0 5 Unit Name KBOS WBP Unit KBOS NBP Unit KBOS WBC Unit KDOV C-5A Unit ETAR C-17 Unit KCHS C-17 Unit PHIK C-17 Unit Table 5.2 WBC C-5A C-17 C-17 C-17 KBOS KDOV ETAR KCHS PHIK Example Problem Available Aircraft Requirements for this problem are shown in Table 5.3. 122 REQT ID COMMODITY ALD RDD OUT OVER BULK PAX APOE APOD 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 CSS_Other AF_Acft AF_Supt Type_3 Type_4 Army_Prepo Airborne Armor Mech AF_Supt CSS_Other CSS_Other AF_Acft AF_Supt Armor Mech Airmobile Cbac Infantry Armor_Cav Type_13 NEW_1 CSS_Med CSS_Signal CSS_Supt_maint CSS_Trans CSS_Other AF_Acft AF_Supt Type_3 Type_4 Army_Prepo Airborne Armor Mech AF_Acft AF_Supt Type_3 Type_4 Army_Prepo Airborne Armor 0 0 0 1 0 0 0 1 1 1 4 4 7 7 5 5 5 7 7 9 7 7 7 9 9 1 9 11 11 11 11 14 9 11 22 4 12 24 4 15 24 11 1 1 2 3 3 3 3 3 3 3 7 7 9 9 9 9 9 9 9 10 11 12 12 12 12 13 16 16 16 16 16 21 22 23 32 9 21 27 9 21 25 28 0 33.2 0 0 13 0 0 0 37.2 0.5 0 16.5 0 22.3 0 0 5.6 0 0 50.6 0 0 0 32 0 0 90.3 0 0 0 1.2 0 0 98.8 0 50.6 0 0 23.3 6.5 64 0 33.5 0 18.1 0 6 30.9 0 6.6 0 11.2 0 98.6 0 0 0 88.8 6.7 75.9 0 0 0 90.4 0 0 10.5 14.5 0 0 6.7 0 33.6 33.6 0 0 14.2 0 0 0 0 6.7 0 110.4 0 23.5 0 155.4 180 0 200.6 0 0 45.5 10.2 0 155.3 0 34.5 0 0 0 20.3 0 90.4 0 0 0 8.9 0 1.2 287.8 0 155.4 80 122.3 0 7.9 0 0 90.4 155.4 180 5.6 0 0 300 0 0 604 233 5 0 1700 0 28 341 3 643 0 0 442 41 0 2 322 4 0 398 0 0 120 0 0 7 604 233 0 12 0 133 322 4 604 233 123 89 0 KBLV KBLV KDOV KDMA KDOV KDOV KDOV KDSM KDYS KBLV KDMA KBLV KBLV KDMA KDOV KBLV KBLV KDMA KBLV KDOV KDOV KBLV KDMA KDMA KDOV KDMA KDOV KDOV KDMA KDOV KBLV KBLV KDMA KDOV KDMA KDOV KDOV KDMA KDMA KBLV KDOV KBLV LTAG LTAG LTAG OKBK OKBK OKBK OKBK LTAG LTAG LTAG LTAG LTAG LTAG LTAG LTAG LTAG LTAG OKBK LTAG LTAG LTAG LTAG ETAR ETAR ETAR ETAR ETAR OKBK LTAG LTAG LTAG LTAG OKBK OKBK OKBK RJBB RJBB RJBB RJBB RJBB RJBB RJBB Table 5.3 Example Problem Requirements 123 REQT ID COMMODITY ALD RDD OUT OVER BULK PAX APOE APOD 43 44 45 46 47 48 49 Mech Airmobile Cbac Infantry Armor_Cav Type_13 Navy_Marines 14 11 20 15 16 24 21 29 23 23 23 23 28 23 0 32 0 0 90.3 0 18 0 0 10.5 44.5 0 0 0 0 0 8.9 0 6.2 0 0 678 0 0 120 0 309 0 KDOV KDMA KBLV KDMA KDOV KDOV KBLV RJBB RJBB RJBB RJBB RJBB RJBB RJBB Table 5.3 (continued) Example Problem Requirements Figure 5.1 graphically depicts the example problem requirements and aircraft available over time. Scenario 5 Total Daily Requirements & Aircraft Available 3000 70 2500 2452 70 80 2000 50 50 45 1500 40 1462 40 1047 1000 899 539 538 271 379 136 0 0 1 00 00 2 3 4 5 00 6 7 814 852 1012 30 20 483 455 334 194 00 8 9 00 90 4 00 293 289 140 219 141 133 123 970 79 300 0 0 0 019 0 180 14 0 10 Cargo (tons) #PAX Aircraft Arrivals 500 0 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Day Figure 5.1 Example Problem Requirements and Aircraft Available 124 #Aircraft Available t 60 Quantity The deployment horizon begins with the first requirement ALD at day zero and ends with the last RDD at day 32. Forty-nine requirements composed of 20 different commodities yield a total requirement of 3786 tons of cargo and 9769 PAX. The airlift network is summarized in Table 5.4. Type Arc HB-APOE APOE-APOD APOD-REC REC-HB REC-APOE Table 5.4 Number of Route Segments 8 40 5 107 64 Example Problem Route Segments Based upon the number of home bases, APOEs, APODs, recovery bases, aircraft types and route segments, a total of 1,228 arcs were generated to define the airlift network for Scenario 5. 5.2.2 SAP-TS Results for Example Problem Tabu search settings for Scenario 5 are summarized in Table 5.5. These settings are the same as the default settings listed in Chapter 4 and are provided here for convenience. Table 5.6 presents the initial solution objective function values. The initial solution is infeasible due to a working MOG violation of 451 and a UTE rate violation of 1. The makespan for the initial solution is 27.95 days. 125 Parameter NoChangeMovesTolerance UnimprovingMovesTolerance SuperDiversifyTolerance SolutionHistoryLength ObjectiveTolerance ChooseFirstImprovingMove MaxCompTime NumAircraftTrips NumRoutesToConsider MsnAssignmentPhase MsnImprovementPhase MsnAssignmentNBHSizeLimit msnImprovementNBHSizeLimit WorkingMOGTolerance FuelMOGTolerance UteRateTolerance NumberOfIterations Tenure EliteListSize ObjFnWeights loadReallocationMoveDistance InsertDistance Default Setting 3 3 10 20 0.01 TRUE 40 12 50 TRUE FALSE 100 800 0 0 0 800 7 3 {1,1,1,1,1,1,1,1,1,1} 3 5 Description # no change value moves allowed before diversifying # unimproving moves allowed before diversifying # small change moves allowed before superdiversifying # previous objectives to consider for superdiverisfication tolerance for objective in superdiversification count boolean for selecting the first improving move maximum search time (hours) alotted for the search # aircraft trips per aircraft maximum number of routes to create for each arc boolean that sets mission assignment phase status boolean that sets mission improvement phase status mission Assignment Phase maximum NBH size mission Improvement Phase maximum NBH size threshold below which working MOG feasible threshold below which fuel MOG feasible threshold below which UTE rate feasible # tabu search iterations duration (iterations) of tabu attribute number of elite solutions to retain weights for changing priority of objective values temporal distance (days) for LR moves maximum insert distance (left & right) for WPI moves Table 5.5 Example Problem Tabu Search Settings 126 Criterion Undelivered cargo penalty Late Delivery penalty workingMOGViolation Fuel MOG violation Total Number of missions Total Number of trivial loads Trivial load penalty Total Number of Aerial refueling missions UTE Rate violation Total Objective value Table 5.6 Value 0.00 9.72 451.00 0.00 159.00 17.00 64.21 0.00 1.00 701.93 Example Problem Initial Solution Objective A total of 800 tabu search iterations were performed. The best solution mission summary is presented in Appendix A. The best solution objective function values, shown in Table 5.7, were obtained at iteration 384 in 3 hours and 33 minutes. The best solution, with a makespan of 26.3 days, achieves a 74% reduction in the initial solution objective value and is feasible with respect to UTE rate and fuel MOG constraints. It is near feasible with a small working MOG violation of 16. 127 Criterion Undelivered cargo penalty Late Delivery penalty workingMOGViolation Fuel MOG violation Total Number of missions Total Number of trivial loads Trivial load penalty Total Number of Aerial refueling missions UTE Rate violation Total Objective value Table 5.7 5.2.3 Value 0.00 16.24 16.00 0.00 138.00 3.00 9.50 0.00 0.00 182.74 Example Problem Best Solution Objective AMOS Results for Example Problem For comparison the AMOS results were evaluated using the SAP-TS objective function. AMOS and SAP-TS results are given in Table 5.8. The SAP-TS best solution achieves an overall 83% reduction in the AMOS objective value greatly reducing late cargo (a critical issue for combatant commanders) and achieving better results for feasibility constraints. The AMOS makespan of 30.85 days is 4.55 days greater than the SAP-TS best solution. Figure 5.2 depicts the daily cumulative required total tonnage and the associated totals obtained with AMOS and SAP-TS. 128 Criterion Undelivered cargo penalty Late Delivery penalty workingMOGViolation Fuel MOG violation Total Number of missions Total Number of trivial loads Trivial load penalty Total Number of Aerial refueling missions UTE Rate violation Total Objective value Table 5.8 AMOS Value 0.00 762.97 58.00 0.00 116.00 19.00 136.92 0.00 0.00 1092.89 SAP-TS Value 0.00 16.24 16.00 0.00 138.00 3.00 9.50 0.00 0.00 182.74 Example Problem AMOS & SAP-TS Objective 129 Daily PAX & Cargo Closure 7000 6000 5000 Tonnage 4000 3000 2000 Required 1000 SAP Delivered AMOS Delivered 0 0 5 10 15 20 25 30 35 40 45 Day Figure 5.2 Example Problem PAX & Cargo Closure SAP-TS dominates AMOS throughout the time span of the deployment and incurs a single instance of delivery time shortfall. 5.3 OTHER SAP PROBLEM INSTANCES Scenarios 1 through 4 were run using the SAP-TS settings indicated in Table 5.5 with the following exceptions. Each scenario was run for a total of 15 hours. The standard tabu search phase was capped at 12 hours. After the standard tabu search phase, a series of elite iterations were performed starting with the best solution found during the 12 hour phase. neighborhoods were restricted to the mission Elite iteration phase improvement neighborhoods LR, WMAS, ITD, and RAI to intensify the search around the 130 elite solution. The number of elite iterations performed is the smaller of 800 iterations and the number of iterations performed in 3 hours. AMOS and SAP-TS results for the Scenarios 1 through 4 are summarized in Table 5.9. Criterion Undelivered cargo Late Cargo Working MOG Fuel MOG # Missions # Trivial Loads Trivial Load penalty # Aerial Refueling missions UTE Rate violation Total Objective value Makespan Scenario 1 AMOS 144.40 7005.83 163.00 0.00 240.00 56.00 507.89 0.00 4237.00 12354.12 31.52 TS 0.00 69.61 114.00 0.00 248.00 12.00 27.00 0.00 141.00 611.61 26.99 Scenario 2 AMOS 0.80 9992.05 1038.00 0.00 425.00 88.00 805.28 0.00 30096.00 42445.13 18.40 TS 0.00 8978.00 1033.00 0.00 389.00 34.00 348.32 0.00 5069.00 15851.32 18.82 Scenario 3 AMOS 4.95 20974.74 578.00 0.00 577.00 239.00 1837.79 0.00 12531.00 36742.48 40.47 TS 0.00 749.55 314.00 0.00 403.00 22.00 71.70 0.00 2718.00 4278.25 32.35 Scenario 4 AMOS 0.00 1595.08 51.00 0.00 116.00 19.00 136.92 0.00 1623.00 3541.00 29.02 TS 0.00 135.20 41.00 0.00 228.00 2.00 1.51 0.00 11.00 418.71 28.62 Table 5.9 5.3.2 AMOS and SAP-TS Results for Scenarios 1 through 4 Scenario 1 Scenario 1 consists of 59 requirements totaling 7,647 tons of cargo and 14,769 PAX that become available for delivery over 25 days. A total of 166 aircraft are available at the start building to a final total of 297 aircraft available at day 20. Requirement and aircraft arrival details are shown in Figure 5.3. 131 Scenario 1 Total Daily Requirements & Aircraft Available 5000 4500 4000 3500 3000 3452 4462 350 297 300 250 Quantity 2500 2000 1500 166 173 179 2114 1852 181 200 150 100 1047 899 739 538 571 679 483 136 1012 859 594 334 0 0 0 0 0 0 0 0 0 0 90 4 0 0 493 289 140 97 230 0 0 0 0 133 123 0 0 19 0 18 0 14 0 141 79 775 1000 500 0 50 Cargo (tons) #PAX Aircraft Arrivals 0 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Day Figure 5.3 Scenario 1 Total Daily Requirements and Aircraft Available The Scenario 1 best solution was found at iteration 509 after 12 hours and 43 minutes. The Scenario 1 SAP-TS best solution achieves an overall 95% reduction in the AMOS objective value, a reduction of 99% in late cargo penalty, and no undelivered cargo. SAP-TS delivers (closes) all requirements 4.53 days earlier than AMOS. Figure 5.4 provides a comparison of the closure rates for AMOS and SAP-TS for Scenario 1. 132 #Aircraft Available t Daily PAX & Cargo Closure 12000 10000 Cumulative Tonnage 8000 6000 4000 Required 2000 SAP Delivered AMOS Delivered 0 0 5 10 15 20 25 30 35 40 45 Day Figure 5.4 5.3.3 Scenario 1 PAX & Cargo Closure Scenario 2 Scenario 2 consists of 132 requirements totaling 15,482 tons of cargo and 10,576 PAX that become available for delivery over 10 days. Aircraft available start at 60 aircraft on day 0 and build to a final total of 124 aircraft by day 8. Scenario 2 is the most challenging scenario in that requirements become available in less than half the time (10 days vice 25) of the other scenarios and the available aircraft are not commensurate with the total requirement. aircraft arrival details for Scenario 2 are shown in Figure 5.5. Requirement and 133 Scenario 2 Total Daily Requirements & Aircraft Available 3500 3032 3000 114 2675.2 2500 95 100 124 120 140 Quantity 2000 1757.7 60 1500 1293 1074.8 1000 822 645 500 492 70 1501.7 1923.9 1904 1906.5 80 60 1157.6 807.8 585 1209.1 40 736 528 509 300 645.2 552 20 #Aircraft Available t Cargo (tons) #PAX Aircraft Arrivals 0 0 1 2 3 4 5 6 7 8 9 10 0 Day Figure 5.5 Scenario 2 Total Daily Requirements and Aircraft Available The Scenario 2 best solution was found at iteration 134 after 9 hours and 31 minutes. The Scenario 2 SAP-TS best solution achieves an overall 63% reduction in the AMOS objective value and a reduction of 10% in late cargo penalty. The SAP-TS solution closed 10 hours later than the AMOS solution. Figure 5.6 provides a comparison of the closure rates for AMOS and SAP-TS for Scenario 2. Further analysis of Scenario 2 is provided in Section 5.4. 134 Daily PAX & Cargo Closure 20000 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 0 5 10 15 20 25 30 35 40 45 Cumulative Tonnage Required SAP Delivered AMOS Delivered Day Figure 5.6 5.3.4 Scenario 2 PAX & Cargo Closure Scenario 3 Scenario 3 consists of 96 requirements totaling 13,777 tons of cargo and 16,683 PAX that become available for delivery over 25 days. Aircraft available start at 166 aircraft on day 0 and build to a final total of 297 aircraft by day 20. Requirement and aircraft arrival details for Scenario 3 are shown in Figure 5.7. 135 Scenario 3 Total Daily Requirements & Aircraft Available 6000 350 5021 297 5000 300 250 4000 Quantity 3000 3086 166 173 179 2719 181 200 2996 150 2000 1904 1839 1440 1044 1700 100 1000 673 305 129 821 850 625 89 00 6 7 639 635 651 00 8 9 00 10 11 90 4 00 13 14 335 230 33 92 0 0 15 16 17 709 400 28 7 18 19 20 876 50 133 123 18 0 14 0 21 22 23 24 131 72 0 25 Cargo (tons) #PAX Aircraft Arrivals 0 0 1 00 00 2 3 4 5 12 Day Figure 5.7 Scenario 3 Total Daily Requirements and Aircraft Available The Scenario 3 best solution was found at iteration 908 after 15 hours. The Scenario 3 SAP-TS best solution achieves an overall 88% reduction in the AMOS objective value and a reduction of 96% in late cargo penalty with no undelivered cargo. The SAP-TS solution closed 9.6 hours earlier than the AMOS solution. Figure 5.8 provides a comparison of the closure rates for AMOS and SAP-TS for Scenario 3. 136 #Aircraft Available t Daily PAX & Cargo Closure 18000 16000 14000 Cumulative Tonnage 12000 10000 8000 6000 4000 Required SAP Delivered 2000 AMOS Delivered 0 5 10 15 20 25 30 35 40 45 0 Day Figure 5.8 5.3.5 Scenario 3 PAX & Cargo Closure Scenario 4 Scenario 4 consists of 59 requirements totaling 7,647 tons of cargo and 14,769 PAX that become available for delivery over 25 days. Aircraft available start at 91 aircraft on day 0 and build to a final total of 165 aircraft by day 20. Requirement and aircraft arrival details for Scenario 4 are shown in Figure 5.9. 137 Scenario 4 Total Daily Requirements & Aircraft Available 5000 165 4500 4000 3500 3000 3452 120 4462 160 180 140 Quantity 98 91 104 100 2114 2500 2000 1500 1000 500 0 80 1852 60 1047 775 483 00 00 0 1 2 3 4 136 5 00 6 7 594 334 00 8 9 00 10 11 90 4 00 14 493 289 230 140 97 133 123 0 0 0 0 0 0 19 0 18 014 0 15 16 17 18 19 20 21 22 23 24 141 79 25 1012 739 538 571 899 679 859 40 20 Cargo (tons) #PAX Aircraft Arrivals 0 12 13 Day Figure 5.9 Scenario 4 Total Daily Requirements and Aircraft Available The Scenario 4 best solution was found at iteration 270 after 4 hours and 38 minutes. The Scenario 4 SAP-TS best solution achieves an overall 88% reduction in the AMOS objective value and a reduction of 92% in late cargo penalty with no undelivered cargo. The SAP-TS solution closed 8.4 days earlier than the AMOS solution. Figure 5.10 provides a comparison of the closure rates for AMOS and SAP-TS for Scenario 4. 138 #Aircraft Available t Daily Cargo Closure 9000 8000 7000 Cumulative Tonnage 6000 5000 4000 3000 2000 Required SAP Delivered AMOS Delivered 1000 0 0 5 10 15 20 25 30 35 40 45 Day Figure 5.10 Scenario 4 PAX & Cargo Closure 5.4 EXTENDED ANALYSIS OF SCENARIO 2 Scenario 2 represented the most challenging deployment amongst the 5 different scenarios. The requirement time windows spanned a minimum ALD of day 0 to a max RDD of 10 a 10 day horizon for the deployment. Along with this compressed horizon, the number of aircraft allocated is insufficient to meet the requirement demand within the given time windows. As a result, both AMOS and SAP-TS solutions incurred a substantial late delivery penalty. A snapshot of the SAP-TS solution deliveries juxtaposed with requirement time windows is 139 shown in Figure 5.11. Significant delays in requirement delivery begin to occur at about day 8 however the origin of these delays starts at around day 4. SAP-TS Requirement Deliveries 140 135 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Requirement# ALD RDD Late On Time 18 19 20 Day Figure 5.11 SAP-TS Scenario 2 Solution with Time windows The following is an extended analysis of the SAP-TS best solution for Scenario2. This analysis is performed to illustrate how SAP-TS results can be used to gain additional insight into a particular SAP instance. What additional resources are needed to better meet deployment demands? Which bases are restricting the flow of aircraft through the network? These are some of the questions analysts are interested in when developing a strategic airlift plan. For the purposes of this discussion, we confine our study to determining what changes 140 in available aircraft and working MOG at critical bases will improve closure times of the requirements. The number of aircraft required to move all requirements within their respective time windows is a function of the requirements themselves (commodity type; amount of outsize, oversize, bulk and PAX), aircraft capabilities (ACL, cargo compatibility, UTE rates), and the strategic airlift network (critical legs, MOG, number of routes). We use the C17 as the base aircraft to determine approximately how many C17 missions are required to deliver all requirements. The planning payloads for the C17 are described in Table 3 of Air Mobility Planning Factors (Air Force Pamphlet 10-1403, 1998). These are: 45 tons of cargo with 90 passengers and 102 passengers if there is no cargo on board. These payloads represent historical averages from Desert Shield/Desert Storm. Since all Scenario 2 APOE-APOD pairs are from CONUS to Southwest Asia, these payloads will be adequate for obtaining a rough estimate of the number of C17 missions required. The daily cargo and PAX requirements for Scenario 2 are shown in Figure 5.5. Using the C17 planning factors outlined above, the calculation for the number of C17 missions required to deliver all cargo and PAX is defined by the following function: cargo 90 pax 45 102 cargo + f ( cargo, pax ) = 45 0, , pax cargo 90 > 0 45 pax cargo 90 0 45 where cargo is the amount of cargo in tons and pax is the number of personnel. 141 As an example, we calculate the number of C17 missions required to move the total requirements available on day 0 (ignoring the APOE-APOD pairs): 1,757.7 tons of cargo and 1,293 PAX. First calculate the number of cargo missions: 1757.7/45 = 39.06 cargo missions or 40 missions total. These cargo missions may carry 40*90 or 3600 total PAX. Since this exceeds the 1,293 PAX on hand, the PAX are delivered using the cargo missions and no additional PAX only aircraft are necessary. C17 mission requirements for the remaining days (again, ignoring APOE-APOD pairs) are shown in Table 5.10. DAY 0 1 2 3 4 5 6 7 8 9 10 TOTALS CARGO (tons) 1757.7 1074.8 822.0 1501.7 1923.9 807.8 1157.6 1209.1 2675.2 1906.5 645.2 15482 #PAX 1293 3032 492 645 1904 585 736 528 509 300 552 10576 # C17 Missions 40 33 19 34 43 18 26 27 60 43 15 358 Table 5.10 Number of Required C17 Missions by Day for Scenario 2 142 KSZL OKBK KDOV OEDR KDOV OBBI KSVN OBBS KPOB OBBS KSUU OKBK KTIK OKBK KNGU OKBK KRIV OBBS KAEX OBBS PHIK OKBK KRIV OKBK KDOV OEKJ KDOV OKAJ KDOV OTBD KBIF OTBD KAEX OKBK KLFI OKBK KMUO OTBD ETAD OKBK KHMN OKBK KPOB OKBK KWRB OEKJ KDMA OEKJ KMUO OOTH KTIK OEKJ KBIF OBBS KDMA OBBS KPOB OEKJ ETAD OOTH KDOV OOTH KHOP OEKJ KBIF OEKJ KCHS OBBI KHOP OKBK KAGS OKBK KNGP OBBI KDOV OBBS KSKF OKBK KWRI OKBK KDMA OKBK 50 KDOV OKBK KGRK OKBK KBIF OKBK KMGE OKBK Scenario 2 C17 Equivalents By Time Windows 441 1 425 1 1 376 1 2 7 2 9 1 1 1 311 7 1 2 276 11 1 9 245 1 2 6 4 1 3 6 4 1 2 1 2 5 3 1 1 6 3 1 2 2 2 4 1 2 2 2 1 1 1 5 218 6 166 9 3 122 2 1 100 5 7 4 1 6 1 1 2 8 2 1 1 8 12 8 8 2 9 1 2 3 2 6 1 1 1 17 8 3 8 2 4 5 6 1 5 1 1 3 2 1 2 3 2 2 4 2 7 1 8 9 10 11 1 1 2 9 3 4 1 12 1 1 1 16 2 11 2 4 3 11 21 18 Day 7 Figure 5.12 C17 Missions by Time Window and APOE-APOD Pair 143 The total number of C17 missions required is 358. Determining the number of required C17 missions based solely upon daily cargo available is admittedly a gross simplification. The results in Table 5.10 do not take into account the impact of different APOE-APOD pairs (nor loading issues due to commodity type and base constraints which we ignore here). Figure 5.12 depicts the impact of multiple APOE-APOD pairs on the number of missions required. Requirements with the same time window and APOE-APOD pair are grouped together. The red number on the time window represents the number of C17 missions needed to deliver the requirements for the given time window and APOE-APOD pair. The blue line (and numbers) indicates the cumulative number of C17 missions required over time. The total number of C17 missions required has risen from 358 to 441 when the different APOE-APOD pairs are included. Scenario 2 aircraft available by type are shown in Table 5.11. The actual number of missions performed by SAP-TS and AMOS by type are shown in Figure 5.13 and Figure 5.14 respectively. 144 Day 0 0 0 0 0 0 1 3 3 5 5 8 Quantity 5 10 10 10 10 15 10 10 15 9 10 10 Type WBC WBP C-5A C-17 C-5A WBP WBC C-17 WBC C-5A C-17 C-5A Table 5.11 Scenario 2 Aircraft Arrivals by Type SAP-TS Missions vs C17 Equivalent Missions 70 124 60 50 40 30 20 10 0 0 2 4 6 8 Day 10 12 14 16 18 95 114 SAP-TS Total SAP-TS C17s SAP-TS C5s SAP-TS WBC SAP-TS WBP C17 Equivalents #Aircraft Available 120 100 80 60 40 20 0 Number of Aircraft Available 140 Number of Missions 70 60 Figure 5.13 SAP-TS Missions vs C17 Equivalent Missions 145 AMOS Missions vs C17 Equivalent Missions 70 124 60 50 40 30 20 10 0 0 2 4 6 8 Day 10 12 14 16 18 95 114 AMOS Total AMOS C17s AMOS C5s AMOS WBC AMOS WBP C17 Equivalents #Aircraft Available 120 100 80 60 40 20 0 Number of Aircraft Available 140 Number of Missions 70 60 Figure 5.14 AMOS Missions vs C17 Equivalent Missions A clear shortfall or mission gap, measured as the difference between the C17 mission equivalents and the total missions performed in the respective solutions, is apparent in both solutions. The mission gap is most prominent between days 8 and 10 achieving a maximum shortfall of about 30 C17 missions on day 8 for SAP-TS and 40 C17 missions for AMOS on day 8. Since the requirements during the period from day 8 to day 10 are cargo intensive (5227 tons of cargo, 1361 PAX, see Figure 5.5), an increase in the available number of C5s and/or C17s prior to or during this window would improve closure times of the requirements. 146 A determination of the quantity and mix of additional aircraft must include the impact of airlift network constraints on aircraft throughput. Specifically, we investigate those bases with working MOG violations in the SAP-TS solution. These represent the bottlenecks or locations where aircraft flow is constrained due to working MOG limitations. For the SAP-TS solution, 10 bases have working MOG violations totaling 1033. These bases along with the respective total working MOG violation and number of aircraft visits are summarized in Table 5.12. Base Name ETAD OKBK OBBS OBBI OEKJ OTBD LEMO LICZ EGVA HECA Working MOG (Wide/Logic/Narrow) 1/OR/1 5/OR/5 2/OR/2 2/OR/2 2/OR/2 2/OR/2 3/OR/4 2/OR/4 1/OR/1 1/OR/2 MOG Violation 46 283 42 14 49 14 492 7 30 56 #Visits 42 254 52 15 40 20 136 57 40 34 Base Role(s) APOE/ENROUTE APOD APOD APOD APOD APOD REC/ENROUTE REC/ENROUTE ENROUTE ENROUTE Table 5.12 Scenario 2 Working MOG Violation Bases (SAP-TS) 147 Half of the MOG violation bases for this solution are APOD bases, contributing about 39% of the total working MOG violation. APOD bases typically do not have large wide or narrow working MOG. Located in the theater of operations, APOD facilities are often limited when compared to CONUS bases. Of course a large number of APOD visits for requirement delivery are expected and thus these bases are likely to be the sources of bottlenecks in the airlift system. If the requirement time windows are tight as in this scenario, aircraft begin to compete for working MOG space. LEMO accounts for 48% of the MOG violation the largest single contributor among the bases. The 126 aircraft visits to LEMO are bested only by the 256 visits to OKBK. The high aircraft visit frequency for LEMO is because this base serves as the sole recovery base for C-5s in the theater as well as an en route base for 23 of the 49 arcs servicing CONUS to theater and back again. LICZ is one of the two recovery bases for C17s the other is ETAR. LICZ has a small working MOG violation of 7 and ETAR has no working MOG violation in the SAP-TS solution for Scenario 2. 0 depicts the total number of aircraft that visit OKBK, LEMO and LICZ over time. Major spikes in the number of aircraft occur during days 1 to 3 and days 9 to 11. Referring to Figure 5.13, we see that these time periods coincide with the mission gaps. Increasing the number of C5s to bridge the mission gap during these periods would only increase working MOG violations at OKBK and LEMO. An increase in available C17s during these intervals would be better from a working MOG standpoint since LICZ and 148 ETAR (recovery bases for C17s) are less constrained and the C17s do not recover at LEMO. SAP-TS Working MOG 16 14 12 # Aircraft 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 OKBK LEMO LICZ DAY Figure 5.15 SAP-TS Working MOG for OKBK, LEMO and LICZ Based upon this simple analysis of the SAP-TS solution, several broad conclusions can be made about the current strategic airlift plan for Scenario 2. First, additional aircraft are required to successfully meet requirement RDDs. Because the late requirements are predominantly cargo, the type of aircraft should be C5s or C17s. Ideally, the additional aircraft should be made available no later than day 4 the time when late cargo begins to manifest itself. C17s are a better choice than C5s based upon the characteristics of the airlift system. Current C5 missions are straining base capacity at several key bases (LEMO, OKBK). Additional C5 missions will increase congestion in the network. 149 Secondly, changes to the strategic airlift network to improve throughput are necessary to meet requirement delivery windows for this deployment. A quick fix would be to identify an alternate recovery base for C5s. ETAR, one of the C17 recovery bases, has a working MOG that can support 20 wide aircraft (C5s) and 20 narrow body aircraft (C17s,) simultaneously. If deemed suitable, ETAR would be an ideal alternate recovery base for C5s and would also help divert some missions away from LEMO. A longer term fix would be to identify bases in the airlift network that are likely to be bottlenecks, such as LEMO and the APOD bases and to make long term investments in facility improvements to increase the MOG capacity. 5.5 CONCLUSION This chapter discussed the application of SAP-TS to 5 separate scenarios. SAP-TS results dominate AMOS results in virtually all aspects. SAP-TS offers high quality solutions to the SAP in a reasonable amount of computation time. SAP-TS results provide a level of detail that facilitates analysis of SAP instances as well as overall performance of the strategic airlift network. Insights into network bottlenecks, airlift shortfalls, and potential improvements in the airlift network infrastructure may be obtained by analyzing SAP-TS results. Chapter 6 contains concluding remarks describing key contributions of this research and extensions of SAP-TS for future research. 150 Chapter 6: Conclusion This chapter outlines the major contributions of this dissertation and outlines improvements to the SAP-TS and extensions to the Strategic Airlift Problem. 6.1 MAJOR CONTRIBUTIONS This research has made several contributions. These contributions are: the development of a flexible representation of the SAP for a TS approach the creation of an effective solution methodology for the SAP a proof of concept of a TS approach to the SAP The SAP-TS solution representation captures key elements of the strategic airlift network using super arcs to represent routing of aircraft. This representation maintains sufficient fidelity of the SAP to create detailed schedules while avoiding the more cumbersome nodal approach. The SAP-TS solution representation also lends itself to the TS methodology. Key aspects of the SAP mission structure are captured and this structure is exploited in the development of SAP-TS neighborhoods. The SAP-TS algorithm provides excellent solutions to the SAP. A series of neighborhoods specifically tailored to the SAP are dynamically selected during the conduct of the search based upon the phase of the search (mission assignment phase or mission improvement phase) and the search context (superdiversify, 151 diversify or intensify). A flexible objective function was developed that can be easily adapted to target key aspects of the SAP. Weighting of feasibility (working MOG, fuel MOG, or UTE rate) or operational (undelivered cargo, late cargo) objective components enables aspects of SAP to be emphasized during the course of the search. The efficacy of the SAP-TS algorithm was demonstrated using a series of operationally realistic scenarios. The SAP-TS results for these scenarios represented a significant improvement over AMOS results in a reasonable amount of time. These results also illustrate how the SAP-TS can be used to bridge the gap between lower quality (yet detailed) solutions obtained using simulation and the less detailed (yet higher quality) solutions obtained using large scale optimization such as integer or linear programming. 6.2 FURTHER ENHANCEMENTS TO THE SAP-TS ALGORITHM Although SAP-TS obtains excellent results for the SAP, further research into several aspects of SAP-TS would lead to improved efficiency and effectiveness of the algorithm. Specifically, enhancements in the initial solution construction, mission assignment heuristic and partitioning of the mission assignment space would improve algorithm performance. The SAP-TS initial solution construction heuristic ignores both MOG infeasibilities and UTE rate violations. The SAP-TS then strives to achieve feasibility during the conduct of the search. Starting the search with a near feasible solution may improve overall SAP-TS results. The additional time 152 required to generate a near feasible initial solution may be well spent in achieving an improved starting solution for the search. Several enhancements to the mission assignment heuristic warrant further study. Currently aircraft are assigned to loads based upon temporal constraints (aircraft availability) and load feasibility. Incorporating more sophisticated aircraft selection rules may improve aircraft mission loads. For example, it may be beneficial to incorporate metrics or rules that measure the goodness of a particular type of aircraft to load assignment based upon aircraft cargo preferences, percent fill of the aircraft and requirement priority. These and other rules could be adjusted to favor the assignment of particular aircraft types under different circumstances such as favoring the use of C17s over the older C5 fleet. Another enhancement to the mission assignment heuristic centers around improving feasibility as the solution is constructed. Currently, as missions are created, arcs are selected out of set of possible arcs on a rotating basis. An improvement to this strategy would be to maintain a running tally of the fuel and working MOG status as missions are built. Arcs would then be selected based upon the MOG status along the arc for the time period of the current mission. In this way, the flow through the network would be dynamically adjusted as the solution is constructed yielding near feasible solutions. A third enhancement to the mission assignment heuristic concerns the use of portions of the previous solution instead of a complete reconstruction of the solution. For example, for a BPS move, identify and update only the missions affected by swapping the two missions. The difficulty in this improvement lies in 153 determining which missions are affected. Network constraints (MOG), temporal constraints (aircraft and requirement availability) and spatial constraints (route constraints) can cause missions other than the ones associated with the BPS move to be affected. The final and perhaps most promising enhancement entails the use of a Group Theoretic TS (GTTS) approach to partition the search space during the mission assignment phase. As discussed in Section 2.3.4.4, GTTS has been shown by Wiley (2001) and Crino (2001) to be particularly effective in efficiently searching the solution space of portioning and ordering problems. During the mission assignment phase of SAP-TS, we could seek a good partitioning of the set of possible missions among the set of APOE-APOD pairs and a good ordering of the missions within APOE-APOD pairs. In this way, GTTS could be used to efficiently search the solution space during the mission assignment phase. The structure of neighborhoods used in the mission improvement phase neighborhoods does not readily lend itself to the use of GTTS. 6.3 EXTENSIONS TO OTHER ASPECTS OF THE SAP Several extensions to the basic SAP investigated in this research would add to realism and encompass other aspects important to AMC analysts. Chief among these are the inclusion of transloading, theater constraints and more realistic modeling of the airlift network. SAP-TS requirements have a known destination a priori. For the SAP, the destination is the APOD. However, during strategic airlift operations, transloading is frequently employed to stage requirements at or near the APODs 154 so that requirements can be moved by other aircraft to their final destination in theater (which may or may not be the APOD). Typically this entails modeling the intra-theater airlift. Strategic assets deliver cargo to APODs and then intra-theater aircraft (usually smaller aircraft such as C130s and C141s) move requirements to their final destinations (usually to more austere bases) in theater. Modeling intratheater airlift is beyond the scope of this research. However, even in a strictly strategic sense, efficiencies may be gained by using some aircraft for the major legs from CONUS to near theater bases and other aircraft to shuttle from transload points to the APODs. Operational plans often allocate aircraft to support a particular theater. An air unit s mission may support one or more theaters during the course of a deployment. Incorporating theater constraints imposes a partitioning of the aircraft which may be artificial. However, operational plans may require such a partitioning and thus research into the impact of this restriction would be beneficial. Incorporating improvements that enhance realism in modeling the airlift network would provide additional insight into the SAP. In particular, adding the impact of weather, winds and seasonal temperatures can have a large impact on fuel considerations, timings along the route and aircraft loads. 6.4 SUMMARY This research presented an effective solution methodology, the SAP-TS, which solves the SAP and obtains excellent solutions. The SAP-TS incorporates an efficient representation of the SAP that captures sufficient mission details for 155 the construction of a detailed schedule yet remains flexible enough for application of the TS methodology. 156 Appendix A SAP-TS Best Solution Scenario Mission Summaries A.1 Tail# 0 1 2 2 3 3 4 4 5 5 6 6 7 8 8 9 9 10 10 21 21 22 22 23 24 25 25 27 28 29 30 41 41 42 43 44 45 46 47 48 49 50 51 52 SCENARIO 1 SAP-TS MISSION SUMMARY Msn# 1 12 24 25 36 37 48 49 60 61 72 73 84 96 97 108 109 120 121 252 253 264 265 276 289 300 301 324 336 348 360 492 495 504 516 528 540 552 564 576 588 600 612 624 REQT# 47 4 0 54 7 54 7 54 7 54 7 54 7 7 37 3 37 3 37 3 37 3 37 3 55 9 58 10 10 38 35 35 40 16 15 15 15 12 12 12 12 22 22 22 APOE~APOD KNZY~RJTY KDNL~OJ1X KBLV~LTAG KNYL~RJFU KDSM~LTAG KNYL~RJFU KDSM~LTAG KNYL~RJFU KDSM~LTAG KNYL~RJFU KDSM~LTAG KNYL~RJFU KDSM~LTAG KDSM~LTAG KNLC~RJCC KDMA~OKBK KNLC~RJCC KDMA~OKBK KNLC~RJCC KDMA~OKBK KNLC~RJCC KDMA~OKBK KNLC~RJCC KDMA~OKBK KNZC~RJNN ETAR~LTAG KDOV~RJTY KFSI~LTAG KFSI~LTAG KNTD~RJCJ KNKT~RJAA KNKT~RJAA KNUQ~RKTC KBLV~LTBF KGTB~LTBA KGTB~LTBA KGTB~LTBA KFWA~LTAJ KFWA~LTAJ KFWA~LTAJ KFWA~LTAJ KNGZ~ETAR KNGZ~ETAR KNGZ~ETAR ARR APOD 25.3723 1.43128 0.86795 24.65088 2.17085 25.07701 1.87217 24.73459 2.47444 24.94374 3.95039 25.67685 2.31668 2.05439 25.84286 2.36032 25.00862 1.92282 25.85186 3.08948 25.90129 2.06787 24.70233 1.92281 26.99249 1.5091 26.0375 6.65251 6.52556 5.53394 5.08209 4.87706 25.27149 5.66889 8.22802 8.23754 5.89444 7.84795 8.11845 7.8353 9.43053 8.06274 7.62524 8.06247 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 0 309 61.8 0 0 0 233 46.6 0 0 0 300 60 0 0 0 286 57.2 0 0 0 335 67 0 0 0 252 50.4 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 22 4.4 0 0 0 335 67 0 0 0 25 5 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 260 52 0 0 0 335 67 0 0 0 325 65 0 0 0 264 52.8 0 0 0 18 3.6 0 0 0 28 5.6 0 0 0 123 24.6 0 0 0 335 67 0 0 0 6 1.2 0 0 0 233 46.6 0 0 0 260 52 0 0 0 62 12.4 0 0 0 89 17.8 0 0 0 41 8.2 0 0 0 195 39 0 0 0 195 39 0 0 0 52 10.4 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 58 11.6 0 0 0 195 39 0 0 0 195 39 0 0 0 8 1.6 157 Tail# 53 54 55 57 58 59 60 61 62 63 64 85 86 87 88 89 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 135 135 136 136 137 137 138 138 139 139 140 140 141 141 142 143 143 144 144 145 145 146 146 Msn# 636 648 660 684 696 708 720 732 744 756 768 1020 1032 1044 1056 1068 1092 1104 1116 1128 1140 1152 1164 1176 1188 1200 1212 1224 1236 1248 1260 1272 1284 1296 1308 1620 1621 1632 1633 1644 1645 1656 1657 1668 1669 1680 1681 1692 1693 1705 1716 1717 1728 1729 1740 1741 1752 1753 REQT# 20 18 19 32 50 30 30 29 29 29 29 29 29 29 29 29 36 42 42 42 42 53 53 57 45 39 34 51 49 49 49 49 49 49 49 6 30 6 30 6 29 6 29 4 29 4 29 4 29 29 1 27 3 27 3 27 3 27 APOE~APOD KNGU~LTCC KNFW~LTBU KNGP~LTBU KSPS~OKBK KPDX~RJNN KSLC~LTAN KSLC~LTAN KSKF~LTAG KSKF~LTAG KSKF~LTAG KSKF~LTAG KSKF~LTAG KSKF~LTAG KSKF~LTAG KSKF~LTAG KSKF~LTAG KNKX~RJBB KNXX~RJNN KNXX~RJNN KNXX~RJNN KNXX~RJNN KRCA~RJBB KRCA~RJBB KNZW~RJSM KNZJ~RJTA KNTU~RJSM KSTJ~OMAD KPOB~RJTY KPBG~RJFU KPBG~RJFU KPBG~RJFU KPBG~RJFU KPBG~RJFU KPBG~RJFU KPBG~RJFU KDPG~OJHF KSLC~LTAN KDPG~OJHF KSLC~LTAN KDPG~OJHF KSKF~LTAG KDPG~OJHF KSKF~LTAG KDNL~OJ1X KSKF~LTAG KDNL~OJ1X KSKF~LTAG KDNL~OJ1X KSKF~LTAG KSKF~LTAG KDLF~LTAG KSGH~OJHF KDMA~OKBK KSGH~OJHF KDMA~OKBK KSGH~OJHF KDMA~OKBK KSGH~OJHF ARR APOD 8.10298 7.84703 9.90857 9.85513 11.60866 11.9553 12.34963 14.233 13.08586 14.67066 13.21942 14.08717 12.06503 14.67066 13.69337 15.1078 12.74322 15.2444 14.69913 14.73923 15.36442 15.08596 14.96691 15.72142 15.84363 15.75544 22.859 23.71979 26.11948 25.60014 25.21156 26.26698 26.53825 26.98131 25.97322 1.54012 11.94681 1.55101 12.03769 1.68563 12.15363 1.68797 11.8504 1.34813 11.85732 0.91101 14.87635 1.06376 12.15363 14.14718 1.33576 11.93983 2.11025 11.75546 2.10945 12.53198 2.25529 13.39577 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 0 4 0.8 0 0 0 2 0.4 0 0 0 195 39 0 0 0 12 2.4 0 0 0 8 1.6 0 0 0 195 39 0 0 0 38 7.6 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 44 8.8 0 0 0 4 0.8 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 93 18.6 0 0 0 195 39 0 0 0 139 27.8 0 0 0 46 9.2 0 0 0 120 24 0 0 0 123 24.6 0 0 0 133 26.6 0 0 0 123 24.6 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 60 12 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 38.4 0 38.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 18.4 0 18.4 0 0 23.5 0 23.5 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 72.2 0 72.2 158 Tail# 147 147 148 148 149 149 150 150 151 151 152 152 153 153 154 154 182 182 183 183 184 184 185 185 186 186 187 187 188 188 189 189 190 190 191 191 192 192 193 193 194 194 195 196 196 212 212 213 213 214 214 215 215 216 216 217 217 218 Msn# 1764 1765 1776 1777 1788 1789 1800 1801 1812 1813 1824 1825 1836 1837 1848 1849 2184 2185 2196 2197 2208 2209 2220 2221 2232 2233 2244 2245 2256 2257 2268 2269 2280 2281 2292 2293 2304 2305 2316 2317 2328 2329 2341 2352 2353 2544 2545 2556 2557 2568 2569 2580 2581 2592 2593 2604 2605 2616 REQT# 3 27 3 27 3 27 9 27 10 27 38 27 38 27 38 36 38 36 14 31 12 31 12 31 12 31 12 53 12 57 12 39 12 46 20 44 20 37 18 37 24 37 37 33 37 5 28 4 31 2 40 1 53 0 57 8 45 7 APOE~APOD KDMA~OKBK KSGH~OJHF KDMA~OKBK KSGH~OJHF KDMA~OKBK KSGH~OJHF ETAR~LTAG KSGH~OJHF KFSI~LTAG KSGH~OJHF KNTD~RJCJ KSGH~OJHF KNTD~RJCJ KSGH~OJHF KNTD~RJCJ KNKX~RJBB KNTD~RJCJ KNKX~RJBB KGSP~LTAZ KSPI~LTAG KFWA~LTAJ KSPI~LTAG KFWA~LTAJ KSPI~LTAG KFWA~LTAJ KSPI~LTAG KFWA~LTAJ KRCA~RJBB KFWA~LTAJ KNZW~RJSM KFWA~LTAJ KNTU~RJSM KFWA~LTAJ KNZW~RJTT KNGU~LTCC KNZC~RJSM KNGU~LTCC KNLC~RJCC KNFW~LTBU KNLC~RJCC KBLV~ETAR KNLC~RJCC KNLC~RJCC KSSC~OMAA KNLC~RJCC KDOV~OJ2X KSKA~LTAG KDNL~OJ1X KSPI~LTAG KDLH~LTAG KNUQ~RKTC KDLF~LTAG KRCA~RJBB KBLV~LTAG KNZW~RJSM KDYS~LTAG KNZJ~RJTA KDSM~LTAG ARR APOD 2.98525 12.19504 2.83862 11.90129 1.96431 12.82364 1.55073 13.24993 6.54819 12.19504 6.22758 11.75546 6.16659 12.53197 6.6839 12.97403 6.35995 12.90567 5.75363 15.05318 9.3673 22.19201 9.63838 22.06512 8.37063 22.04623 8.79964 15.19035 8.63813 15.81964 8.32588 15.93689 8.07888 17.14444 8.03112 20.8154 7.72749 25.44674 7.78595 26.07103 9.4791 25.66004 26.67198 12.68375 25.29804 0.98825 13.78139 1.23157 22.17242 1.88993 26.18085 1.84806 14.84693 1.71172 16.57491 4.01011 15.83203 4.23208 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 18.4 0 18.4 0 0 72.2 0 72.2 0 0 45.5 0 45.5 0 0 72.2 0 72.2 0 0 10.2 0 10.2 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 65.8 0 65.8 0 0 65.53 0 65.53 0 0 72.2 0 72.2 0 0 39.67 0 39.67 0 0 18.2 0 18.2 0 0 34.5 0 34.5 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 60.1 0 60.1 0 0 72.2 0 72.2 0 0 56.9 0 56.9 0 0 72.2 0 72.2 0 0 56 0 56 0 0 72.2 0 72.2 0 0 5.6 0 5.6 0 0 22.1 0 22.1 0 0 6.2 0 6.2 0 0 82.7 0 82.7 0 0 8.9 0 8.9 0 0 7.7 0 7.7 0 0 87.4 0 87.4 0 0 20.3 0 20.3 0 0 87.4 0 87.4 0 0 8.9 0 8.9 0 0 87.4 0 87.4 0 0 65.53 0 65.53 0 0 7.9 0 7.9 0 0 78.66 0 78.66 0 30.9 0 5 31.9 0 6.7 0 7 8.1 13 6 17.8 0 36.8 0 33.6 0 0 33.6 0 18.1 0 0 18.1 18 0 0 0 18 33.2 0 0 0 33.2 19.2 23.1 0 0 42.3 0 33.5 0 0 33.5 0 20.9 0 0 20.9 37.2 0 0 0 37.2 0 44.5 0 0 44.5 0 6.6 0 0 6.6 159 Tail# 218 219 219 220 220 221 221 227 227 233 233 234 234 235 235 236 236 237 237 238 238 239 239 240 240 241 241 242 242 250 250 251 251 252 252 253 253 254 254 255 255 256 256 257 257 258 258 259 259 260 261 261 262 262 263 263 264 264 Msn# 2617 2628 2629 2641 2640 2652 2653 2724 2725 2796 2797 2808 2809 2820 2821 2832 2833 2844 2845 2856 2857 2868 2869 2880 2881 2892 2893 2904 2905 3000 3001 3012 3013 3024 3025 3036 3037 3048 3049 3060 3061 3072 3073 3084 3085 3096 3097 3108 3109 3121 3132 3133 3144 3145 3156 3157 3168 3169 REQT# 39 25 46 46 40 11 52 25 19 11 52 38 52 35 52 35 52 35 52 35 40 35 44 35 48 35 34 16 40 15 40 15 40 21 40 21 40 21 40 17 40 17 40 17 40 13 40 23 52 40 19 40 19 53 19 40 19 40 APOE~APOD KNTU~RJSM KWRI~ETAR KNZW~RJTT KNZW~RJTT KNUQ~RKTC KFSM~LTAI KPSM~RJAA KWRI~ETAR KNGP~LTBU KFSM~LTAI KPSM~RJAA KNTD~RJCJ KPSM~RJAA KNKT~RJAA KPSM~RJAA KNKT~RJAA KPSM~RJAA KNKT~RJAA KPSM~RJAA KNKT~RJAA KNUQ~RKTC KNKT~RJAA KNZC~RJSM KNKT~RJAA KOAK~RODN KNKT~RJAA KSTJ~OMAD KBLV~LTBF KNUQ~RKTC KGTB~LTBA KNUQ~RKTC KGTB~LTBA KNUQ~RKTC KNGU~LTAN KNUQ~RKTC KNGU~LTAN KNUQ~RKTC KNGU~LTAN KNUQ~RKTC KBLV~OKBK KNUQ~RKTC KBLV~OKBK KNUQ~RKTC KBLV~OKBK KNUQ~RKTC ETAR~LTAG KNUQ~RKTC KNHK~ETAR KPSM~RJAA KNUQ~RKTC KNGP~LTBU KNUQ~RKTC KNGP~LTBU KRCA~RJBB KNGP~LTBU KNUQ~RKTC KNGP~LTBU KNUQ~RKTC ARR APOD 16.56 1.52107 16.96273 17.52428 26.18302 5.06488 18.24575 8.83547 11.19433 5.03734 18.55087 5.9667 18.11437 5.53619 17.9582 5.18676 18.52429 5.1764 18.24575 5.50964 25.8384 5.304 20.93269 5.53619 22.28878 5.22315 23.22109 6.02388 25.83634 7.99828 26.07979 7.9592 24.79895 8.26226 25.70268 8.37494 26.05527 8.40497 25.71999 7.98359 25.27782 7.98391 26.12376 7.98274 25.56334 7.24411 25.78812 9.55477 18.33849 26.19576 11.07868 25.99033 10.13311 14.83351 11.04319 25.60885 9.91228 25.12201 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 6.5 6.7 0 0 13.2 0 0 0 106 21.2 58.1 0 0 0 58.1 32.2 0 0 0 32.2 18 0 0 0 18 16.5 53.97 0 0 70.47 34.5 0 0 0 34.5 0 14.5 0 14 17.3 51.39 0 0 62 63.79 0 44.63 0 3 45.23 33.71 0 0 0 33.71 23.3 0 0 0 23.3 34.5 0 0 0 34.5 35.53 0 0 0 35.53 34.5 0 0 0 34.5 41.3 0 0 0 41.3 34.5 0 0 0 34.5 41.3 0 0 0 41.3 34.5 0 0 0 34.5 41.3 0 0 0 41.3 18 0 0 0 18 41.3 0 0 0 41.3 0 10.5 0 0 10.5 35.53 0 0 0 35.53 18 0 0 0 18 14.33 0 0 0 14.33 0 14.2 0 0 14.2 5.6 6.7 0 0 12.3 18 0 0 0 18 0 62 0 0 62 23.7 0 0 0 23.7 0 26.8 0 0 26.8 23.9 0 0 0 23.9 0 47.9 0 0 47.9 23.7 0 0 0 23.7 0 47.9 0 0 47.9 23.7 0 0 0 23.7 0 14.6 0 0 14.6 23.7 0 0 0 23.7 0 31.4 0 0 31.4 23.7 0 0 0 23.7 0 31.4 0 0 31.4 23.7 0 0 0 23.7 0 13.1 0 0 13.1 23.7 0 0 0 23.7 22.3 0 0 0 22.3 23.7 0 0 0 23.7 32 0 0 0 32 23.99 0 0 0 23.99 23.7 0 0 0 23.7 50.5 0 0 17 53.9 23.7 0 0 0 23.7 50.5 0 0 17 53.9 26.2 11.5 0 0 37.7 50.5 0 0 17 53.9 23.7 0 0 0 23.7 50.5 0 0 14 53.3 23.7 0 0 0 23.7 160 Tail# 265 265 266 266 267 267 268 268 269 270 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 Msn# 3181 3180 3192 3193 3204 3205 3216 3217 3229 3240 3241 3252 3264 3276 3288 3300 3312 3324 3336 3348 3360 3372 3384 3396 3408 3420 3432 3444 3456 3468 REQT# 9 19 19 40 19 37 19 37 56 24 58 26 26 26 26 43 43 41 41 41 41 41 41 33 33 33 33 30 30 30 APOE~APOD ETAR~LTAG KNGP~LTBU KNGP~LTBU KNUQ~RKTC KNGP~LTBU KNLC~RJCC KNGP~LTBU KNLC~RJCC KNZJ~RJOI KBLV~ETAR KDOV~RJTY KWRI~ETAR KWRI~ETAR KWRI~ETAR KWRI~ETAR KNYL~RJOI KNYL~RJOI KNUW~RKTD KNUW~RKTD KNUW~RKTD KNUW~RKTD KNUW~RKTD KNUW~RKTD KSSC~OMAA KSSC~OMAA KSSC~OMAA KSSC~OMAA KSLC~LTAN KSLC~LTAN KSLC~LTAN ARR APOD 1.53577 10.75323 10.13121 24.9749 10.787 25.19839 11.00809 24.85889 26.85406 9.61163 26.31932 9.5546 9.43654 9.43657 9.55455 12.13939 11.93732 12.0323 12.73392 11.88505 12.55906 12.76434 11.86462 12.15142 12.28297 13.01233 13.01276 12.81735 12.99006 12.99246 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0.5 11.2 0 0 11.7 50.5 0 0 0 50.5 50.5 0 0 0 50.5 23.7 0 0 0 23.7 50.5 0 0 0 50.5 0 0 41.4 0 41.4 45.71 0 0 0 45.71 0 0 7.62 14 10.42 0 24.5 20 0 44.5 0 10.5 0 0 10.5 34 0 0 0 34 26.2 0 1.2 0 27.4 26.2 0 0 0 26.2 26.2 0 0 0 26.2 11.7 0 0 0 11.7 16.8 0 0 0 16.8 15.2 0 0 0 15.2 0 59.6 0 0 59.6 0 59.6 0 0 59.6 0 59.6 0 0 59.6 0 47.93 0 0 47.93 0 47.93 0 0 47.93 0 35.75 0 0 35.75 51.4 0 0 0 51.4 51.4 0 0 0 51.4 51.4 0 0 0 51.4 44.6 0 0 0 44.6 1.2 33.6 28.02 0 62.82 0 0 41.4 0 41.4 0 0 35.78 0 35.78 Table A.1 A.2 Tail# 0 0 0 1 1 2 2 2 3 3 4 5 5 5 5 5 5 6 Scenario 1 Mission Summary SCENARIO 2 SAP-TS MISSION SUMMARY Msn# 0 2 2 12 13 24 25 27 36 39 48 60 60 61 61 62 62 74 REQT# 2 54 33 2 41 2 34 60 1 110 22 3 42 31 76 56 40 71 APOE~APOD KGRK~OKBK KBIF~OBBS KBIF~OBBS KGRK~OKBK KMUO~OOTH KGRK~OKBK KGRK~OKBK KBIF~OBBS KBIF~OKBK KGRK~OKBK KMGE~OKBK KDOV~OKBK KDOV~OKBK KPOB~OEKJ KPOB~OEKJ KTIK~OEKJ KTIK~OEKJ KGRK~OKBK ARR APOD 1.07281 5.74413 5.74413 1.06135 3.97048 1.0561 3.87702 8.55218 1.10818 9.87685 2.82176 3.7705 3.7705 6.79366 6.79366 9.11462 9.11462 6.87178 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 0 335 67 0 0 0 70 14 0 0 0 70 14 0 0 0 335 67 0 0 0 28 5.6 0 0 0 81 16.2 0 0 0 335 67 0 0 0 124 24.8 0 0 0 229 45.8 0 0 0 146 29.2 0 0 0 176 35.2 0 0 0 63 12.6 0 0 0 45 9 0 0 0 28 5.6 0 0 0 54 10.8 0 0 0 108 21.6 0 0 0 50 10 0 0 0 276 55.2 161 Tail# 6 7 7 8 8 8 9 9 9 10 10 10 10 10 10 11 11 11 11 12 12 12 12 13 13 13 13 13 14 14 14 14 14 14 14 14 15 15 15 15 15 15 15 16 16 17 17 17 17 17 17 18 19 19 19 19 20 20 Msn# 74 86 87 97 98 98 108 108 108 121 122 123 124 124 124 132 133 134 136 145 145 146 148 156 156 156 157 159 171 171 171 169 169 169 172 172 180 181 182 183 183 183 184 192 193 205 204 206 207 207 208 220 228 229 230 232 240 240 REQT# 79 131 131 29 128 72 10 11 12 24 48 86 112 120 124 6 24 48 112 23 26 48 126 10 11 12 26 35 96 99 106 110 71 79 125 97 14 26 52 83 94 96 119 13 41 34 35 44 93 66 121 116 20 35 43 117 10 12 APOE~APOD KGRK~OKBK KGRK~OKBK KGRK~OKBK KDOV~OOTH KMGE~OKBK KMGE~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KBIF~OEKJ KMUO~OTBD KRIV~OBBS KDOV~OKBK KDOV~OKBK KDOV~OKBK KSKF~OKBK KBIF~OEKJ KMUO~OTBD KDOV~OKBK KDOV~OKBK KDOV~OKBK KMUO~OTBD KDOV~OBBS KGRK~OKBK KGRK~OKBK KGRK~OKBK KDOV~OKBK KWRI~OKBK KDOV~OKBK KDOV~OKBK KDOV~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KDOV~OKAJ KDOV~OKAJ KBIF~OKBK KDOV~OKBK KAEX~OKBK KDOV~OKBK KDOV~OKBK KDOV~OKBK KDOV~OEKJ KAGS~OKBK KMUO~OOTH KGRK~OKBK KWRI~OKBK KWRB~OEKJ KDOV~OTBD KDOV~OTBD KDOV~OEDR KTIK~OKBK KDOV~OBBS KWRI~OKBK KDMA~OEKJ KSUU~OKBK KGRK~OKBK KGRK~OKBK ARR APOD 6.87178 10.87178 13.22322 4.33611 10.82171 10.82171 1.99356 1.99356 1.99356 4.6022 7.03701 9.50734 11.95085 11.95085 11.95085 2.11152 4.55727 6.98475 9.8755 4.51399 4.51399 6.96796 9.83534 2.15586 2.15586 2.15586 4.57607 7.00967 8.81202 8.81202 8.81202 11.26696 11.26696 11.26696 13.7021 13.7021 2.2079 4.6448 7.1011 9.53238 9.53238 9.53238 11.96293 2.17256 4.63236 3.91328 6.34269 8.79009 11.23951 11.23951 13.69153 13.24948 2.10262 4.54281 6.96994 11.81884 2.05202 2.05202 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 0 16 3.2 0 0 0 335 67 0 0 0 196 39.2 0 0 0 25 5 0 0 0 15 3 0 0 0 33 6.6 0 0 0 335 67 0 0 0 18 3.6 0 0 0 6 1.2 0 0 46.3 0 46.3 0 0 54.2 0 54.2 0 0 81.1 0 81.1 0 0 4.7 0 4.7 0 0 21.7 0 21.7 0 0 23.8 0 23.8 0 0 44.6 0 44.6 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 70.4 0 70.4 0 0 25.45 0 25.45 0 0 72.2 0 72.2 0 0 18.9 0 18.9 0 0 59.7 0 59.7 0 0 8.5 0 8.5 0 0 5.6 0 5.6 0 0 87.4 0 87.4 0 0 65.41 0 65.41 0 0 9.75 0 9.75 0 0 4.5 0 4.5 0 0 2.2 0 2.2 0 0 2.9 0 2.9 0 0 9.3 0 9.3 0 0 3.6 0 3.6 0 0 18.5 0 18.5 0 0 1.5 0 1.5 0 0 22.5 0 22.5 0 0 87.4 0 87.4 0 0 14.9 0 14.9 0 0 59.9 0 59.9 0 0 22.7 0 22.7 0 0 13.25 0 13.25 0 0 23 0 23 0 0 55 0 55 0 0 11.1 0 11.1 0 0 47.7 0 47.7 0 0 87.4 0 87.4 0 0 60.2 0 60.2 0 0 3.6 0 3.6 0 0 3.6 0 3.6 0 0 4.4 0 4.4 0 0 35.9 0 35.9 0 0 18.9 0 18.9 0 0 87.4 0 87.4 0 0 37.38 0 37.38 0 0 25.9 0 25.9 0 0 0 335 67 0 0 0 24 4.8 162 Tail# 20 21 21 21 22 22 22 23 23 23 24 24 24 24 25 26 26 26 27 27 27 27 28 29 29 29 29 29 29 29 30 30 31 31 31 32 32 32 32 32 33 33 33 33 33 33 34 34 34 35 35 35 35 35 35 36 36 36 Msn# 242 252 252 253 264 265 266 276 277 278 288 288 289 290 300 312 312 313 324 325 325 325 338 348 348 349 349 350 350 350 361 362 373 374 374 384 385 385 386 386 396 397 398 398 398 398 408 409 409 420 422 423 421 425 425 432 437 433 REQT# 77 10 12 46 21 48 83 21 48 75 21 27 53 81 27 27 34 55 14 45 58 59 85 9 28 44 74 70 73 78 51 102 47 101 52 25 49 50 100 84 24 43 112 130 99 106 57 62 82 1 35 82 108 115 88 5 16 17 APOE~APOD KAGS~OKBK KGRK~OKBK KGRK~OKBK KHMN~OKBK KGRK~OKBK KMUO~OTBD KDOV~OKBK KGRK~OKBK KMUO~OTBD KDOV~OEKJ KGRK~OKBK KGRK~OKBK KGRK~OKBK PHIK~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KBIF~OTBD KBIF~OKBK KPOB~OKBK KPOB~OKBK KPOB~OKBK KDMA~OKBK KDOV~OKBK KDOV~OKBK KWRB~OEKJ KWRB~OEKJ KDOV~OKBK KDOV~OKBK KDOV~OKBK KLFI~OKBK KPOB~OBBS ETAD~OKBK KAEX~OKBK KAEX~OKBK KHOP~OEKJ KDOV~OKBK KDOV~OKBK KAEX~OBBS KAEX~OBBS KBIF~OEKJ KDMA~OEKJ KDOV~OKBK KDOV~OKBK KDOV~OKBK KDOV~OKBK KMGE~OKBK KBIF~OKBK KBIF~OKBK KBIF~OKBK KWRI~OKBK KBIF~OKBK KCHS~OBBI KNGU~OKBK KNGU~OKBK KWRI~OKBK KCHS~OBBI KNGP~OBBI ARR APOD 6.8162 1.88847 1.88847 4.91441 1.87178 4.93636 7.77573 1.94263 4.93642 6.83167 1.87689 1.87689 4.877 7.63356 1.87702 3.88846 3.88846 6.04799 1.97641 5.80078 5.80078 5.80078 7.9177 1.82887 1.82887 6.82904 6.82904 8.7064 8.7064 8.7064 4.79773 8.80658 4.3351 8.85741 8.85741 2.96754 4.83965 4.83965 8.87555 8.87555 3.04517 5.22174 10.78298 10.78298 10.78298 10.78298 5.82185 7.96511 7.96511 1.24288 3.81144 7.94815 12.28922 14.29997 14.29997 0.96239 3.02069 5.2416 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 0 30 6 0 0 0 322 64.4 0 0 0 9 1.8 0 0 0 312 62.4 0 0 0 335 67 0 0 0 335 67 0 0 0 211 42.2 0 0 0 335 67 0 0 0 96 19.2 0 0 0 4 0.8 0 0 0 49 9.8 0 0 0 286 57.2 0 0 0 273 54.6 0 0 0 16 3.2 0 0 0 335 67 0 0 0 119 23.8 0 0 0 56 11.2 0 0 0 75 15 0 0 0 170 34 0 0 0 39 7.8 0 0 0 18 3.6 0 0 0 30 6 0 0 0 220 44 0 0 0 17 3.4 0 0 0 288 57.6 0 0 0 209 41.8 0 0 0 27 5.4 0 0 0 70 14 0 0 0 59 11.8 0 0 0 7 1.4 0 0 0 148 29.6 0 0 0 27 5.4 0 0 0 2 0.4 0 0 0 33 6.6 0 0 0 8 1.6 0 0 0 144 28.8 0 0 0 26 5.2 0 0 0 64 12.8 0 0 0 33 6.6 0 0 0 57 11.4 0 0 0 143 28.6 0 0 0 160 32 0 0 0 88 17.6 0 0 0 4 0.8 0 0 0 35 7 0 0 0 21 4.2 0 0 0 98 19.6 0 0 0 127 25.4 0 0 0 40 8 0 0 56.2 0 56.2 0 0 87.4 0 87.4 0 0 58.8 0 58.8 0 0 22.8 0 22.8 0 0 16.9 0 16.9 0 0 11.4 0 11.4 0 0 87.4 0 87.4 0 0 9.9 0 9.9 0 0 57 0 57 163 Tail# 36 36 37 37 37 37 38 38 38 38 38 38 39 39 39 39 39 39 39 39 40 40 40 41 41 41 41 42 42 42 43 43 44 45 45 46 46 46 47 48 49 49 49 50 51 52 52 53 55 55 55 55 55 55 55 56 56 56 Msn# 434 436 445 446 448 449 456 456 457 458 459 461 468 469 470 471 471 472 472 473 481 481 481 493 492 492 492 504 504 505 517 516 529 540 541 552 553 553 565 577 588 588 589 601 613 624 625 637 660 661 663 664 665 666 662 672 673 674 REQT# 35 89 19 35 90 118 3 9 25 35 59 131 7 25 35 87 95 92 64 128 104 127 53 33 61 65 67 67 70 84 18 39 80 68 114 37 113 123 123 123 57 72 123 123 123 44 123 123 2 7 48 80 85 101 111 2 7 33 APOE~APOD KWRI~OKBK KTIK~OKBK KSKF~OKBK KWRI~OKBK KSUU~OKBK KSKF~OKBK KDOV~OKBK KDOV~OKBK KHOP~OEKJ KWRI~OKBK KPOB~OKBK KGRK~OKBK KDOV~OBBS KHOP~OEKJ KWRI~OKBK KWRI~OKBK KWRI~OKBK KSKF~OKBK KSKF~OKBK KMGE~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KBIF~OBBS KDOV~OKBK KDOV~OKBK KDOV~OKBK KDOV~OKBK KDOV~OKBK KAEX~OBBS KWRI~OKBK KDOV~OBBS KRIV~OKBK KDOV~OKAJ KRIV~OKBK KSKF~OKBK KWRI~OKBK KWRI~OKBK KWRI~OKBK KWRI~OKBK KMGE~OKBK KMGE~OKBK KWRI~OKBK KWRI~OKBK KWRI~OKBK KWRB~OEKJ KWRI~OKBK KWRI~OKBK KGRK~OKBK KDOV~OBBS KMUO~OTBD KRIV~OKBK KDMA~OKBK KAEX~OKBK KHOP~OKBK KGRK~OKBK KDOV~OBBS KBIF~OBBS ARR APOD 7.22651 9.37787 3.17781 5.13921 9.48716 11.686 1.82868 1.82868 3.91994 5.89223 7.92781 12.11096 0.92115 3.14519 5.12172 7.80635 7.80635 10.00299 10.00299 12.06817 11.01687 11.01687 11.01687 4.18807 6.16684 6.16684 6.16684 5.81716 5.81716 7.97584 3.90389 5.88775 8.1667 5.81602 9.99202 4.13362 9.80635 9.80635 9.86977 9.92821 6.87478 6.87478 9.88138 9.80636 9.87715 4.86313 9.86977 9.86982 1.43354 3.66387 6.60399 9.73223 12.63489 15.38405 18.06055 1.4431 4.16457 7.19957 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 87.4 0 87.4 0 0 23.2 0 23.2 0 0 22.3 0 22.3 0 0 87.4 0 87.4 0 0 17.4 0 17.4 0 0 28.8 0 28.8 0 0 4.8 0 4.8 0 0 3.4 0 3.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 1.4 0 1.4 0 0 32.8 0 32.8 0 0 43.8 0 43.8 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 69.7 0 69.7 0 0 11.1 0 11.1 0 0 25.4 0 25.4 0 0 22.3 0 22.3 0 0 21 0 21 0 0 27.6 0 27.6 0 0 16.9 0 16.9 0 0 12 0 12 0 0 79.2 0 79.2 0 0 36.6 0 36.6 0 0 2.7 0 2.7 0 0 50.58 0 50.58 0 0 24.42 0 24.42 0 0 58.28 0 58.28 0 0 3.3 0 3.3 0 0 48.2 0 48.2 0 0 17.9 0 17.9 0 0 64.2 0 64.2 0 0 21.5 0 21.5 0 0 74 0 74 0 0 22.3 0 22.3 0 0 11.11 0 11.11 0 0 78.77 0 78.77 0 0 82.7 0 82.7 0 0 82.7 0 82.7 0 0 19.5 0 19.5 0 0 5.6 0 5.6 0 0 82.7 0 82.7 0 0 82.7 0 82.7 0 0 82.7 0 82.7 0 0 72.2 0 72.2 0 0 82.7 0 82.7 0 0 36.73 0 36.73 34.5 2 0 0 36.5 29 0 0 0 29 0 39.22 0 0 39.22 0 62.4 0 0 62.4 0 55.91 0 0 55.91 31.5 36.5 0 0 68 0 63.79 5.69 51 79.68 34.5 0 0 0 34.5 29 0 0 0 29 0 51.2 0 0 51.2 164 Tail# 56 56 56 56 56 57 57 57 57 57 57 57 57 58 58 58 58 58 58 59 59 59 59 59 59 59 59 59 60 60 60 60 60 61 61 61 61 61 61 61 62 62 62 62 62 62 62 63 63 63 63 64 64 64 64 64 64 64 Msn# 674 675 676 678 677 684 684 684 686 687 688 689 690 696 697 698 699 701 702 708 709 710 710 711 712 714 714 713 720 721 723 724 726 732 733 735 736 738 734 737 744 745 747 746 748 749 750 756 757 758 761 768 770 771 772 769 773 773 REQT# 60 48 80 84 89 2 21 27 54 54 80 90 100 1 4 54 84 92 130 1 8 31 76 54 104 105 7 131 1 8 54 80 107 1 17 60 72 103 113 131 1 8 54 63 76 110 111 1 8 35 131 0 30 55 76 84 110 131 APOE~APOD KBIF~OBBS KMUO~OTBD KRIV~OKBK KAEX~OBBS KTIK~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KBIF~OBBS KBIF~OBBS KRIV~OKBK KSUU~OKBK KAEX~OBBS KBIF~OKBK KDMA~OKBK KBIF~OBBS KAEX~OBBS KSKF~OKBK KDOV~OKBK KBIF~OKBK KNGP~OBBI KPOB~OEKJ KPOB~OEKJ KBIF~OBBS KGRK~OKBK KDOV~OBBS KDOV~OBBS KGRK~OKBK KBIF~OKBK KNGP~OBBI KBIF~OBBS KRIV~OKBK KDOV~OBBI KBIF~OKBK KNGP~OBBI KBIF~OBBS KMGE~OKBK KSVN~OBBS KWRI~OKBK KGRK~OKBK KBIF~OKBK KNGP~OBBI KBIF~OBBS KWRI~OKBK KPOB~OEKJ KGRK~OKBK KHOP~OKBK KBIF~OKBK KNGP~OBBI KWRI~OKBK KGRK~OKBK KMGE~OKBK ETAD~OOTH KBIF~OTBD KPOB~OEKJ KAEX~OBBS KGRK~OKBK KGRK~OKBK ARR APOD 7.19957 10.3012 13.26815 16.03399 18.81674 2.16848 2.16848 2.16848 6.4588 9.34664 12.45978 15.44378 18.35698 1.51554 4.41366 7.29428 10.06809 12.88776 15.16028 1.50585 4.77514 7.40807 7.40807 10.29079 13.10362 15.52342 15.52342 18.33173 1.51554 7.05223 12.06589 15.17839 17.31153 1.50585 4.32214 7.21982 9.94717 12.62031 15.20147 18.00184 1.56418 4.38512 7.27314 9.85935 12.4702 15.25424 18.09182 1.61266 4.68762 7.42578 15.7677 1.36234 3.81297 6.83022 9.46868 12.2226 15.02108 15.02108 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 4.2 0 4.2 0 64.6 0 0 64.6 0.6 62.16 0 71 76.96 2.1 28.1 0 0 30.2 3 0 0 0 3 30.8 0 0 0 30.8 0 0 11.6 0 11.6 0 0 0.4 0 0.4 34.5 16.9 0 0 51.4 34.5 5.9 0 0 40.4 0 62.4 0 71 76.6 20 0 0 0 20 31.5 39.2 0 0 70.7 34.5 16.9 0 0 51.4 0 47.6 0 12 50 34.5 16.9 0 0 51.4 34.5 16.9 0 0 51.4 0 6 0 0 6 0 0 60.4 0 60.4 34.5 16.9 0 0 51.4 29 13.9 0 0 42.9 0 18.9 0 0 18.9 0 0 10.2 0 10.2 34.5 0 0 0 34.5 34.5 16.9 0 0 51.4 0.7 1.3 0 0 2 7 0 0 0 7 34.5 16.9 0 0 51.4 34.5 16.9 0 0 51.4 24 43.47 0 0 67.47 34.5 0 0 0 34.5 0 1.54 0 0 1.54 7.9 0 0 6 9.1 34.5 16.9 0 0 51.4 29 13.9 0 0 42.9 34.5 16.9 0 0 51.4 12 25.1 0 0 37.1 12.6 12 0 17 28 0 0 70.99 7 72.39 34.5 16.9 0 0 51.4 34.2 0 0 0 34.2 0 62.4 0 0 62.4 50.7 0 0 0 50.7 0 0 53 0 53 37 11.2 0 0 48.2 0 63.79 0 0 63.79 0 6.96 0 0 6.96 34.5 4.1 0 0 38.6 0 62.4 0 0 62.4 0 0 70.99 0 70.99 34.5 16.9 0 0 51.4 0 59.23 0 55 70.23 0 0 51.7 3 52.3 0 40.35 13.2 0 53.55 34.5 16.9 0 0 51.4 34.5 16.9 0 0 51.4 0 12.61 0 0 12.61 0 14.89 0 0 14.89 165 Tail# 65 65 65 66 66 66 67 67 67 68 68 69 69 70 70 71 71 71 72 72 72 73 73 74 74 74 74 74 74 74 74 74 75 75 75 75 75 75 75 75 75 76 76 76 76 76 76 76 76 77 77 77 77 77 77 77 77 78 Msn# 780 780 781 792 792 793 804 804 805 816 817 829 828 840 841 852 852 852 864 864 865 876 877 888 889 889 891 890 890 892 893 894 900 901 901 901 903 902 904 905 906 912 913 913 914 915 916 917 918 924 925 925 926 927 928 929 930 936 REQT# 26 70 111 71 79 111 71 79 109 104 127 99 104 104 122 104 109 127 104 127 119 104 119 0 12 21 55 71 79 77 127 117 0 10 11 12 55 69 74 127 117 0 10 11 32 55 74 127 117 0 10 11 32 55 74 127 115 0 APOE~APOD KDOV~OKBK KDOV~OKBK KHOP~OKBK KGRK~OKBK KGRK~OKBK KHOP~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KDOV~OKBK KGRK~OKBK KGRK~OKBK KDOV~OTBD KGRK~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KDOV~OEKJ KGRK~OKBK KDOV~OEKJ KMGE~OKBK KGRK~OKBK KGRK~OKBK KBIF~OTBD KGRK~OKBK KGRK~OKBK KAGS~OKBK KGRK~OKBK KSUU~OKBK KMGE~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KBIF~OTBD KDOV~OBBS KWRB~OEKJ KGRK~OKBK KSUU~OKBK KMGE~OKBK KGRK~OKBK KGRK~OKBK KDMA~OBBS KBIF~OTBD KWRB~OEKJ KGRK~OKBK KSUU~OKBK KMGE~OKBK KGRK~OKBK KGRK~OKBK KDMA~OBBS KBIF~OTBD KWRB~OEKJ KGRK~OKBK KNGU~OKBK KMGE~OKBK ARR APOD 5.90345 5.90345 10.17684 7.16862 7.16862 10.16716 7.17819 7.17819 9.97418 9.17823 12.11617 8.91305 11.70392 9.17823 11.31709 10.84825 10.84825 10.84825 9.16847 9.16847 11.30118 9.16846 11.28692 1.21665 3.70666 3.70666 6.41079 8.91879 8.91879 11.63572 14.2202 16.93428 1.13654 3.65977 3.65977 3.65977 6.25052 8.77158 11.3026 13.96494 16.7124 1.36256 3.94705 3.94705 6.75077 9.44843 11.8437 14.2512 17.13996 1.42089 4.0243 4.0243 6.74784 9.35104 11.74628 14.32844 16.81233 1.36252 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 54.65 0 54.65 0 0 9.02 0 9.02 0 63.79 0 0 63.79 0 59.23 0 0 59.23 2.97 13.11 0 0 16.07 0 59.23 0 0 59.23 0 56.86 0 0 56.86 0 2.22 0 0 2.22 29.4 38.3 0 0 67.7 34.5 16.9 0 71 65.6 34.5 16.9 0 0 51.4 0.3 45 0 0 45.3 34.5 16.9 0 71 65.6 34.5 16.9 0 71 65.6 0 33 31.6 0 64.6 34.5 16.9 0 46 60.6 0 0 0 15 3 0 0 0 10 2 34.5 16.9 0 0 51.4 0 0 0 5 1 0 3.21 63.21 0 66.42 34.5 16.9 0 0 51.4 0 63.79 5.69 0 69.48 26.2 11.5 0 28 43.3 0 9.56 0 0 9.56 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 8 51.41 0 0 59.41 12.43 9.77 0 0 22.21 0 14.1 2 0 16.1 0 42.19 0 0 42.19 12.6 0 0 0 12.6 11.8 30.89 0 27 48.09 16.6 0 0 0 16.6 0 16.84 0 0 16.84 0.6 15.84 0 0 16.44 26.2 11.5 0 0 37.7 0 0 39.2 10 41.2 26.2 11.5 0 0 37.7 0 47.9 0 0 47.9 26.2 0 0 0 26.2 0 47.9 0 28 53.5 26.2 11.5 0 0 37.7 6.17 4.85 0 0 11.03 0 47.9 0 0 47.9 26.2 11.5 0 0 37.7 26.2 4.7 0 0 30.9 26.2 11.5 0 0 37.7 26.2 0 0 0 26.2 0 47.9 0 28 53.5 26.2 0 0 0 26.2 0 52.07 0 0 52.07 0 42.19 0 0 42.19 26.2 11.5 0 0 37.7 26.2 0 0 0 26.2 0 39.61 0 0 39.61 26.2 0 0 0 26.2 0 47.9 0 0 47.9 166 Tail# 78 78 78 78 78 78 78 79 79 79 79 79 79 79 80 80 80 80 80 80 80 80 80 81 81 81 81 81 81 81 81 81 81 81 82 82 82 82 82 82 82 83 83 83 83 83 83 83 83 84 84 84 84 85 85 85 85 85 Msn# 937 937 938 939 940 941 942 948 949 949 950 951 952 953 960 961 962 963 963 964 965 966 966 972 973 973 974 975 975 976 976 976 977 978 984 985 986 987 988 989 990 996 997 998 999 1000 1000 1001 1002 1008 1009 1010 1011 1020 1020 1020 1021 1022 REQT# 21 27 32 55 74 110 115 0 10 11 32 55 74 127 0 21 40 45 59 74 127 108 16 5 10 11 40 44 74 73 49 50 127 131 5 13 40 44 70 127 131 5 13 40 44 75 91 127 131 51 81 109 127 36 38 23 81 109 APOE~APOD KGRK~OKBK KGRK~OKBK KDMA~OBBS KBIF~OTBD KWRB~OEKJ KGRK~OKBK KNGU~OKBK KMGE~OKBK KGRK~OKBK KGRK~OKBK KDMA~OBBS KBIF~OTBD KWRB~OEKJ KGRK~OKBK KMGE~OKBK KGRK~OKBK KTIK~OEKJ KPOB~OKBK KPOB~OKBK KWRB~OEKJ KGRK~OKBK KCHS~OBBI KCHS~OBBI KWRI~OKBK KGRK~OKBK KGRK~OKBK KTIK~OEKJ KWRB~OEKJ KWRB~OEKJ KDOV~OKBK KDOV~OKBK KDOV~OKBK KGRK~OKBK KGRK~OKBK KWRI~OKBK KAGS~OKBK KTIK~OEKJ KWRB~OEKJ KDOV~OKBK KGRK~OKBK KGRK~OKBK KWRI~OKBK KAGS~OKBK KTIK~OEKJ KWRB~OEKJ KDOV~OEKJ KDOV~OEKJ KGRK~OKBK KGRK~OKBK KLFI~OKBK PHIK~OKBK KGRK~OKBK KGRK~OKBK KDOV~OKBK KDOV~OKBK KDOV~OKBK PHIK~OKBK KGRK~OKBK ARR APOD 3.8525 3.8525 6.66061 9.35828 12.47805 14.8856 17.28946 1.13654 3.62656 3.62656 6.40148 9.03797 11.54571 14.14676 1.19482 3.71805 6.34386 8.77545 8.77545 11.50571 14.02656 16.31437 16.31437 1.32761 3.85085 3.85085 6.30596 8.68091 8.68091 11.07471 11.07471 11.07471 13.59791 16.00772 1.18924 3.69461 6.34941 8.6912 11.25972 13.78295 16.44748 1.32761 3.57827 6.09169 8.52801 10.95289 10.95289 13.44047 15.93037 5.01605 8.24425 10.84596 13.65632 4.17615 4.17615 4.17615 7.5176 10.10206 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 4.47 0 0 4.47 0 39.1 0 0 39.1 0 33.51 0 0 33.51 14.7 30.65 0 0 45.35 23.4 0 0 0 23.4 17.7 28.3 0 0 46 17.8 0 0 0 17.8 0 42.19 0 24 46.99 26.2 11.5 0 0 37.7 6.53 35.86 0 0 42.39 0 47.9 0 5 48.9 26.2 11.5 0 0 37.7 26.2 0 0 0 26.2 0 47.9 0 0 47.9 0 36.29 0 0 36.29 5.3 38.03 0 0 43.33 0 0 1.06 0 1.06 0 33 0 0 33 0 13.29 0 0 13.29 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 10 5.5 0 0 15.5 0 0 0 16 3.2 49.6 0 0 0 49.6 26.2 1.7 0 0 27.9 0 20.37 0 0 20.37 0 37.7 10.44 0 48.14 0 28.86 0 0 28.86 0 0 1.7 0 1.7 0 21.81 0 0 21.81 0 0 0.3 0 0.3 0 5.4 4.4 0 9.8 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 26.2 0 9 0 35.2 26.2 11.5 0 0 37.7 0 41.9 0 0 41.9 0 41.9 0 0 41.9 0 32.7 0 0 32.7 0 42.19 0 0 42.19 26.2 11.5 0 0 37.7 26.2 0 0 0 26.2 26.2 11.5 0 0 37.7 0 41.9 0 0 41.9 11.9 35.44 0 0 47.34 0 2 4 0 6 0 0 9 0 9 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 10.7 16.3 0 0 27 4 41.38 0 0 45.38 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 0 0 2.4 0 2.4 0 0 15.2 0 15.2 0 0 0 29 5.8 0 4.72 5.8 0 10.52 26.2 11.5 0 0 37.7 167 Tail# 85 86 86 86 86 87 87 87 88 88 88 88 89 89 89 89 89 89 90 90 90 90 91 91 91 91 92 92 92 92 93 93 93 94 94 95 95 96 96 97 97 97 98 99 99 100 101 102 103 104 104 104 104 105 105 105 105 105 Msn# 1023 1032 1033 1034 1035 1044 1046 1047 1056 1057 1058 1059 1068 1069 1069 1069 1070 1071 1080 1081 1082 1083 1093 1092 1094 1095 1104 1105 1106 1107 1116 1117 1118 1128 1129 1140 1141 1152 1152 1165 1164 1164 1176 1188 1188 1200 1212 1224 1236 1248 1249 1251 1252 1260 1261 1262 1263 1264 REQT# 127 43 82 109 127 43 109 127 43 82 109 127 62 70 83 94 109 127 62 80 109 127 15 33 109 127 33 84 109 127 33 60 109 80 129 104 129 104 109 80 104 109 109 73 78 109 109 109 109 5 13 60 84 5 13 46 60 84 APOE~APOD KGRK~OKBK KDMA~OEKJ KBIF~OKBK KGRK~OKBK KGRK~OKBK KDMA~OEKJ KGRK~OKBK KGRK~OKBK KDMA~OEKJ KBIF~OKBK KGRK~OKBK KGRK~OKBK KBIF~OKBK KDOV~OKBK KDOV~OKBK KDOV~OKBK KGRK~OKBK KGRK~OKBK KBIF~OKBK KRIV~OKBK KGRK~OKBK KGRK~OKBK KHOP~OKBK KBIF~OBBS KGRK~OKBK KGRK~OKBK KBIF~OBBS KAEX~OBBS KGRK~OKBK KGRK~OKBK KBIF~OBBS KBIF~OBBS KGRK~OKBK KRIV~OKBK KSZL~OKBK KGRK~OKBK KSZL~OKBK KGRK~OKBK KGRK~OKBK KRIV~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KDOV~OKBK KDOV~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KGRK~OKBK KWRI~OKBK KAGS~OKBK KBIF~OBBS KAEX~OBBS KWRI~OKBK KAGS~OKBK KHMN~OKBK KBIF~OBBS KAEX~OBBS ARR APOD 12.59196 5.13639 8.3427 11.00723 13.61051 5.21654 10.86574 13.44727 5.13638 8.11675 10.52657 13.11102 6.3427 8.77225 8.77225 8.77225 11.18207 13.59192 6.17513 9.02557 12.19335 14.79674 4.1756 6.76082 9.58355 12.33266 4.3652 8.14136 10.677 13.48737 4.44537 7.07635 9.75324 7.1682 11.05884 9.07529 11.5322 9.15537 9.15537 7.24841 9.65822 9.65822 9.15537 7.02849 7.02849 9.15537 9.07529 9.15537 9.07529 1.4777 4.62579 7.80031 14.08466 1.4777 5.08821 8.23094 11.39576 14.58199 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 26.2 11.5 0 0 37.7 11.9 33.67 0 29 51.37 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 0 41.9 0 28 47.5 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 0 13.23 27.82 30 47.05 33.9 5.2 0 0 39.1 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 26.2 11.5 0.1 0 37.8 0 40.1 0 0 40.1 5 0 0 0 5 0 0.8 0 0 0.8 26.2 11.5 0 0 37.7 21.5 25.32 0 0 46.82 12 19 9.2 0 40.2 36.3 11.6 0 25 52.9 26.2 11.5 0 0 37.7 0 47.9 0 0 47.9 0 6.1 4.9 15 14 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 16.8 29 0 0 45.8 26.2 11.5 0.1 0 37.8 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 0 47.9 0 0 47.9 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 0 40.1 0 22 44.5 26.2 10.5 0 0 36.7 26.2 11.5 0 0 37.7 41.3 0 0 2 41.7 26.2 9.1 0 0 35.3 0 2.4 0 0 2.4 36.3 11.6 0 25 52.9 22.1 0 0 0 22.1 4.1 11.5 0 0 15.6 26.2 11.5 0 0 37.7 0 42.19 0 0 42.19 0 8.4 0 0 8.4 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 26.2 11.5 0 0 37.7 34.5 0 0 0 34.5 34.5 16.9 0 0 51.4 34.5 16.9 0 0 51.4 34.5 16.9 0 0 51.4 34.5 0 0 0 34.5 7.6 2.2 0 10 11.8 42.6 17 0 0 59.6 18.6 30.5 0 0 49.1 34.5 16.9 0 0 51.4 168 Tail# 106 106 106 106 106 106 107 107 107 107 107 107 107 107 108 108 108 108 108 108 109 109 109 109 109 109 110 110 110 110 110 110 111 111 111 111 111 111 112 112 112 112 112 112 113 113 113 113 113 113 114 114 114 114 Msn# 1272 1274 1273 1276 1276 1277 1285 1286 1287 1287 1284 1288 1288 1289 1296 1297 1297 1298 1300 1301 1308 1309 1310 1312 1313 1311 1320 1321 1322 1324 1325 1323 1332 1333 1334 1336 1337 1335 1344 1345 1346 1348 1349 1347 1356 1357 1358 1360 1361 1359 1368 1368 1368 1369 REQT# 5 46 108 120 124 131 16 46 58 59 111 125 97 131 5 17 8 46 85 131 7 8 46 85 102 109 7 17 46 85 102 109 7 25 48 85 102 109 7 25 48 85 102 109 7 25 48 85 102 109 69 98 105 71 APOE~APOD KWRI~OKBK KHMN~OKBK KCHS~OBBI KDOV~OKBK KDOV~OKBK KGRK~OKBK KCHS~OBBI KHMN~OKBK KPOB~OKBK KPOB~OKBK KHOP~OKBK KDOV~OKAJ KDOV~OKAJ KGRK~OKBK KWRI~OKBK KNGP~OBBI KNGP~OBBI KHMN~OKBK KDMA~OKBK KGRK~OKBK KDOV~OBBS KNGP~OBBI KHMN~OKBK KDMA~OKBK KPOB~OBBS KGRK~OKBK KDOV~OBBS KNGP~OBBI KHMN~OKBK KDMA~OKBK KPOB~OBBS KGRK~OKBK KDOV~OBBS KHOP~OEKJ KMUO~OTBD KDMA~OKBK KPOB~OBBS KGRK~OKBK KDOV~OBBS KHOP~OEKJ KMUO~OTBD KDMA~OKBK KPOB~OBBS KGRK~OKBK KDOV~OBBS KHOP~OEKJ KMUO~OTBD KDMA~OKBK KPOB~OBBS KGRK~OKBK KDOV~OBBS KDOV~OBBS KDOV~OBBS KGRK~OKBK ARR APOD 1.4777 5.21074 11.519 14.51051 14.51051 17.72522 2.11599 5.28542 8.57833 8.57833 11.7212 14.85555 14.85555 18.16742 1.41937 4.76019 4.76019 8.07673 14.36487 17.52116 1.27498 4.48696 7.64224 10.79027 13.95336 17.12206 1.27498 4.67687 7.84175 11.14159 14.45637 17.72747 1.27498 4.42335 7.56801 10.74825 13.91583 17.2317 1.42081 5.01269 8.30908 11.64596 14.79938 17.97326 1.42081 4.57379 7.72294 10.89857 14.20369 17.38208 9.27498 9.27498 9.27498 12.45329 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 34.5 0 0 0 34.5 7.8 62.19 0.18 0 70.17 29 13.9 0 8 44.5 8 0 0 0 8 21 0 0 0 21 25.3 42.71 0 0 68.01 22.6 44.9 0 0 67.5 42.6 17 0 0 59.6 0 19.7 0 0 19.7 19.4 36.01 0 0 55.41 0 59.23 3.11 0 62.35 0 22 0 0 22 0 0.9 0 0 0.9 34.5 16.9 0 0 51.4 34.5 0 0 0 34.5 15 0 0 8 16.6 0 0 0 48 9.6 0 0 48.43 0 48.43 34.5 16.9 0 0 51.4 34.5 16.9 0 0 51.4 29 0 0 0 29 0 30.13 7 0 37.13 42.6 17 0 0 59.6 34.5 16.9 0 0 51.4 34.5 16.9 1.4 0 52.8 34.5 16.9 0 0 51.4 29 0 0 0 29 29 10.4 0 0 39.4 0 32.21 37.29 0 69.5 29.6 39.9 0 0 69.5 0.8 47.6 2.2 0 50.6 34.5 16.9 0 0 51.4 29 0 0 0 29 0 0 45.42 0 45.42 0 63.14 0 0 63.14 0 63.79 3.7 0 67.49 34.5 16.9 1.4 0 52.8 34.5 16.9 0 0 51.4 29 0 0 0 29 0 36.9 39.97 0 76.87 0 64.6 0 0 64.6 34.5 16.9 0 0 51.4 34.5 16.9 1.4 0 52.8 34.5 16.9 0 0 51.4 29 0 0 0 29 0 0 65.91 0 65.91 0 63.14 0 0 63.14 34.5 16.9 0 0 51.4 34.5 16.9 1.4 0 52.8 34.5 16.9 0 0 51.4 0 0 32.7 0 32.7 0 0 3.4 0 3.4 0 0 0 48 9.6 34.5 16.9 1.4 0 52.8 Table A.2 Scenario 2 Mission Summary 169 A.3 Tail# 0 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 21 21 22 22 23 23 24 25 25 26 26 27 28 28 29 29 30 30 41 41 42 42 43 43 44 44 45 45 46 46 47 47 48 SCENARIO 3 BEST SOLUTION MISSION SUMMARY Msn# 1 12 24 25 36 37 48 49 60 61 72 73 84 85 96 97 108 109 120 121 252 253 264 265 276 277 289 300 301 312 313 325 336 337 348 349 360 361 492 493 504 505 516 517 528 529 540 541 552 553 564 565 577 REQT# 84 0 11 79 10 92 10 67 10 61 10 88 9 73 7 73 7 73 7 73 7 73 7 73 7 71 71 57 56 25 82 80 21 76 21 76 20 76 20 76 20 76 19 69 19 69 16 93 14 89 13 89 89 APOE~APOD KRCA~RJBB KBLV~LTAG KELP~OKBK KPHX~RJSM KEGI~LTAG KNZW~RJSM KEGI~LTAG KNZJ~RJTA KEGI~LTAG KNTU~RJFF KEGI~LTAG KNXX~RJFK KEDW~LTAG KOQU~RJCJ KDSM~LTAG KOQU~RJCJ KDSM~LTAG KOQU~RJCJ KDSM~LTAG KOQU~RJCJ KDSM~LTAG KOQU~RJCJ KDSM~LTAG KOQU~RJCJ KDSM~LTAG KOFF~RJBB KOFF~RJBB KNKT~RJAA KSTJ~OMAD KGFA~LTAG KPOB~RJTY KPIA~RJTA KFSI~LTAG KPBG~RJFU KFSI~LTAG KPBG~RJFU KFSD~LTAG KPBG~RJFU KFSD~LTAG KPBG~RJFU KFSD~LTAG KPBG~RJFU KFOK~LTAG KNZY~RJTY KFOK~LTAG KNZY~RJTY KFLL~LTAG KNZY~RJTA KFFO~LTAG KNYL~RJFU KFAR~OMAD KNYL~RJFU KNYL~RJFU ARR APOD 15.395 0.86795 1.90573 16.46436 1.88039 16.46786 3.91505 15.92786 3.05899 16.07507 4.20677 20.54683 2.85132 20.02224 3.3305 21.81772 3.05778 24.02677 3.62226 20.72105 3.92085 22.92615 3.47634 19.68552 2.03687 23.48611 22.09189 5.08951 22.87553 6.09981 24.33188 24.65623 4.9026 27.65861 6.07618 26.53951 5.47274 26.72147 7.19636 27.55908 8.52832 27.56455 4.64694 26.0851 6.37736 25.14247 8.36564 25.31842 4.67095 26.46359 4.86241 24.77759 25.05778 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 0 334 66.8 0 0 0 300 60 0 0 0 8 1.6 0 0 0 46 9.2 0 0 0 335 67 0 0 0 46 9.2 0 0 0 335 67 0 0 0 120 24 0 0 0 335 67 0 0 0 123 24.6 0 0 0 225 45 0 0 0 28 5.6 0 0 0 28 5.6 0 0 0 291 58.2 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 260 52 0 0 0 335 67 0 0 0 144 28.8 0 0 0 25 5 0 0 0 335 67 0 0 0 65 13 0 0 0 322 64.4 0 0 0 133 26.6 0 0 0 65 13 0 0 0 123 24.6 0 0 0 65 13 0 0 0 335 67 0 0 0 286 57.2 0 0 0 6 1.2 0 0 0 252 50.4 0 0 0 335 67 0 0 0 335 67 0 0 0 195 39 0 0 0 195 39 0 0 0 165 33 0 0 0 162 32.4 0 0 0 195 39 0 0 0 195 39 0 0 0 49 9.8 0 0 0 114 22.8 0 0 0 123 24.6 0 0 0 65 13 0 0 0 65 13 0 0 0 195 39 0 0 0 46 9.2 0 0 0 195 39 0 0 0 195 39 170 Tail# 49 49 50 51 51 52 52 53 53 54 55 55 56 56 57 59 60 61 62 63 64 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 101 102 103 104 105 106 107 108 109 135 135 135 136 136 136 136 137 137 137 137 138 138 Msn# 588 589 601 612 613 624 625 637 636 648 660 661 672 673 684 708 720 732 744 756 768 1020 1032 1044 1056 1068 1080 1092 1104 1116 1128 1140 1152 1164 1176 1188 1212 1224 1236 1248 1260 1272 1284 1296 1308 1620 1622 1623 1632 1633 1634 1635 1644 1645 1646 1647 1656 1657 REQT# 28 89 89 41 89 41 89 44 35 39 33 90 38 95 23 34 34 46 46 54 77 86 86 86 86 86 86 86 86 86 51 51 51 51 51 51 58 75 64 64 64 64 48 48 48 6 72 94 6 55 72 94 6 51 72 94 6 51 APOE~APOD KGSB~LTAY KNYL~RJFU KNYL~RJFU KNGZ~LTAG KNYL~RJFU KNGZ~LTAG KNYL~RJFU KWRI~ETAR KNGU~LTCC KNGP~LTAC KNFW~LTBL KNZC~RJNN KNFW~LTAG KDOV~RJTY KFWA~LTAG KNGP~LTBU KNGP~LTBU KDOV~LTAG KDOV~LTAG KSPS~OKBK KPDX~RJNN KBLV~RJCJ KBLV~RJCJ KBLV~RJCJ KBLV~RJCJ KBLV~RJCJ KBLV~RJCJ KBLV~RJCJ KBLV~RJCJ KBLV~RJCJ KSKF~LTAG KSKF~LTAG KSKF~LTAG KSKF~LTAG KSKF~LTAG KSKF~LTAG KNKX~RJBB KPAM~RJFK KNXX~RJNN KNXX~RJNN KNXX~RJNN KNXX~RJNN KSFO~OJ2X KSFO~OJ2X KSFO~OJ2X KDPG~OJHF KOKC~RJCC KBLV~RJTT KDPG~OJHF KSSC~OMAA KOKC~RJCC KBLV~RJTT KDPG~OJHF KSKF~LTAG KOKC~RJCC KBLV~RJTT KDPG~OJHF KSKF~LTAG ARR APOD 5.64383 25.73609 25.50948 11.32473 26.68171 11.18595 26.0257 1.80037 7.66526 7.73625 8.72817 27.23256 9.36275 27.29539 9.27807 10.44629 9.74103 11.84901 11.68367 10.14593 11.52648 11.64207 13.15766 14.29984 12.34679 13.83495 13.50464 12.66291 12.86599 13.27887 14.37899 12.19858 14.08717 11.9192 14.08733 12.78171 13.43142 16.26572 15.22421 15.2444 14.69913 14.88481 14.85944 14.99448 15.14198 1.68804 21.00367 29.59636 1.24845 12.06317 20.81976 30.28236 1.84268 12.98052 21.53155 30.17782 1.53974 12.45259 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 0 76 15.2 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 183 36.6 0 0 0 60 12 0 0 0 106 21.2 0 0 0 4 0.8 0 0 0 89 17.8 0 0 0 2 0.4 0 0 0 8 1.6 0 0 0 123 24.6 0 0 0 123 24.6 0 0 0 43 8.6 0 0 0 195 39 0 0 0 127 25.4 0 0 0 195 39 0 0 0 114 22.8 0 0 0 12 2.4 0 0 0 8 1.6 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 140 28 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 29 5.8 0 0 0 4 0.8 0 0 0 28 5.6 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 93 18.6 0 0 0 195 39 0 0 0 195 39 0 0 0 10 2 0 0 87.4 0 87.4 0 0 78.66 0 78.66 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 7.9 0 7.9 0 0 71.76 0 71.76 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 38.4 0 38.4 0 0 87.4 0 87.4 171 Tail# 138 138 139 139 139 139 140 140 140 141 141 141 142 142 143 143 143 144 144 144 145 145 145 146 146 146 147 147 147 148 148 148 149 149 149 150 150 150 151 151 151 152 152 152 153 153 153 154 154 154 182 182 182 183 183 183 184 184 Msn# 1658 1659 1668 1669 1670 1671 1680 1681 1682 1692 1693 1694 1705 1706 1716 1717 1718 1728 1729 1730 1740 1741 1742 1752 1753 1754 1764 1765 1766 1776 1777 1778 1788 1789 1790 1800 1801 1802 1812 1813 1814 1824 1825 1826 1836 1837 1838 1848 1849 1850 2184 2185 2186 2196 2197 2198 2208 2209 REQT# 72 94 4 51 72 94 4 51 72 4 51 72 51 72 1 49 72 9 49 72 3 49 72 60 49 72 21 49 72 15 49 72 15 49 72 15 49 72 15 49 72 29 49 72 27 49 72 26 58 66 26 58 85 26 75 85 26 75 APOE~APOD KOKC~RJCC KBLV~RJTT KDNL~OJ1X KSKF~LTAG KOKC~RJCC KBLV~RJTT KDNL~OJ1X KSKF~LTAG KOKC~RJCC KDNL~OJ1X KSKF~LTAG KOKC~RJCC KSKF~LTAG KOKC~RJCC KDLF~LTAG KSGH~OJHF KOKC~RJCC KEDW~LTAG KSGH~OJHF KOKC~RJCC KDMA~LTAG KSGH~OJHF KOKC~RJCC KNTD~RJCJ KSGH~OJHF KOKC~RJCC KFSI~LTAG KSGH~OJHF KOKC~RJCC KFHU~LTAG KSGH~OJHF KOKC~RJCC KFHU~LTAG KSGH~OJHF KOKC~RJCC KFHU~LTAG KSGH~OJHF KOKC~RJCC KFHU~LTAG KSGH~OJHF KOKC~RJCC KGSP~LTAZ KSGH~OJHF KOKC~RJCC KGRK~LTAG KSGH~OJHF KOKC~RJCC KGPT~LTAG KNKX~RJBB KNZC~RJSM KGPT~LTAG KNKX~RJBB KHIF~RJCC KGPT~LTAG KPAM~RJFK KHIF~RJCC KGPT~LTAG KPAM~RJFK ARR APOD 24.06594 32.10397 1.49362 11.99624 24.13377 26.93349 1.2023 12.00315 22.99011 1.20268 13.70969 21.20784 12.30664 20.81976 1.0441 13.99031 22.11488 4.3702 15.10931 24.15344 2.73378 13.79921 22.50043 4.60994 12.19296 23.16511 5.23555 14.13614 21.20784 7.62224 14.23431 20.81976 8.06666 13.79921 22.11488 6.74715 11.90129 23.86177 6.76609 15.01114 23.63793 5.75363 14.81764 23.22344 10.91694 12.68436 21.20784 10.29882 13.53935 24.02373 8.99324 13.66761 21.26461 10.00706 17.72846 20.78976 8.7135 18.6031 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 87.4 0 87.4 0 0 65.53 0 65.53 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 15.07 0 15.07 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 65.53 0 65.53 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 78.66 0 78.66 0 0 18.4 0 18.4 0 0 71.76 0 71.76 0 0 23.5 0 23.5 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 45.5 0 45.5 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 13 0 13 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 23.2 0 23.2 0 0 72.2 0 72.2 0 0 65.53 0 65.53 0 0 10.2 0 10.2 0 0 72.2 0 72.2 0 0 78.66 0 78.66 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 71.76 0 71.76 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 80.6 0 80.6 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 34.5 0 34.5 0 0 72.2 0 72.2 0 0 65.53 0 65.53 0 0 17.8 0 17.8 0 0 65.8 0 65.8 0 0 78.66 0 78.66 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 8.9 0 8.9 0 0 87.4 0 87.4 0 0 18.2 0 18.2 0 0 71.76 0 71.76 0 0 87.4 0 87.4 0 0 85.03 0 85.03 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 172 Tail# 184 185 185 185 186 186 186 187 187 187 188 188 188 189 189 189 190 190 190 191 191 191 192 192 192 193 193 193 194 194 195 195 195 196 196 196 212 212 212 212 213 213 213 213 214 214 214 214 215 215 215 215 216 216 216 217 217 217 Msn# 2210 2220 2221 2222 2232 2233 2234 2244 2245 2246 2256 2257 2258 2268 2269 2270 2280 2281 2282 2292 2293 2294 2304 2305 2306 2316 2317 2318 2328 2330 2340 2341 2342 2352 2353 2354 2544 2545 2547 2546 2556 2557 2558 2559 2568 2570 2571 2569 2580 2582 2583 2581 2592 2593 2594 2604 2605 2606 REQT# 85 26 75 85 37 75 85 35 75 85 35 53 85 33 53 85 38 53 85 23 53 85 23 84 85 23 61 85 23 85 23 81 85 23 81 85 5 45 62 62 4 65 85 62 2 85 62 62 1 85 62 62 0 63 85 9 63 85 APOE~APOD KHIF~RJCC KGPT~LTAG KPAM~RJFK KHIF~RJCC KNHK~LTFJ KPAM~RJFK KHIF~RJCC KNGU~LTCC KPAM~RJFK KHIF~RJCC KNGU~LTCC KSPI~LTAG KHIF~RJCC KNFW~LTBL KSPI~LTAG KHIF~RJCC KNFW~LTAG KSPI~LTAG KHIF~RJCC KFWA~LTAG KSPI~LTAG KHIF~RJCC KFWA~LTAG KRCA~RJBB KHIF~RJCC KFWA~LTAG KNTU~RJFF KHIF~RJCC KFWA~LTAG KHIF~RJCC KFWA~LTAG KPIT~RJTT KHIF~RJCC KFWA~LTAG KPIT~RJTT KHIF~RJCC KDOV~OJ2X KWRI~ETAR KNUQ~RJFK KNUQ~RJFK KDNL~OJ1X KNYL~RJOI KHIF~RJCC KNUQ~RJFK KDLH~LTAG KHIF~RJCC KNUQ~RJFK KNUQ~RJFK KDLF~LTAG KHIF~RJCC KNUQ~RJFK KNUQ~RJFK KBLV~LTAG KNUW~RJFU KHIF~RJCC KEDW~LTAG KNUW~RJFU KHIF~RJCC ARR APOD 20.94516 8.98626 16.60288 22.22584 7.6879 14.78355 21.86917 7.75271 18.44862 20.73585 8.74857 15.04617 21.61461 7.77132 14.77345 21.1033 11.44538 18.10876 20.94516 10.51364 18.26144 22.22584 9.92332 15.31523 21.86917 10.23393 16.01875 20.73585 10.79842 21.26461 10.07614 19.10431 24.30202 10.94415 19.21485 22.64503 0.98792 9.52122 26.1763 29.12746 1.81475 12.25285 22.36077 26.53157 1.45243 21.76609 26.7386 29.98135 1.55629 21.79944 25.00877 28.55904 1.71165 12.13073 23.92325 4.38146 12.33911 22.67362 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 87.4 0 87.4 0 0 50.8 0 50.8 0 0 65.53 0 65.53 0 0 87.4 0 87.4 0 0 23.2 0 23.2 0 0 69.05 0 69.05 0 0 65.53 0 65.53 0 0 82.7 0 82.7 0 0 38.49 0 38.49 0 0 78.66 0 78.66 0 0 7.7 0 7.7 0 0 87.4 0 87.4 0 0 71.76 0 71.76 0 0 20.3 0 20.3 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 5.6 0 5.6 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 60.1 0 60.1 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 56.9 0 56.9 0 0 65.53 0 65.53 0 0 72.2 0 72.2 0 0 5.6 0 5.6 0 0 78.66 0 78.66 0 0 72.2 0 72.2 0 0 71.76 0 71.76 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 30.9 0 5 31.9 37.9 0 0 0 37.9 18 0 0 0 18 18 0 0 0 18 13 0 17.8 0 30.8 15.5 0 0 0 15.5 0 0 41.4 0 41.4 18 0 0 0 18 0 18.1 0 0 18.1 0 0 41.4 0 41.4 18 0 0 0 18 18 0 0 0 18 3.2 0 0 0 3.2 0 0 41.4 0 41.4 18 0 0 0 18 18 0 0 0 18 0 3.5 0 0 3.5 0 58.28 0 0 58.28 0 0 41.4 0 41.4 0.5 11.2 0 0 11.7 0 55.54 0 0 55.54 0 0 41.4 0 41.4 173 Tail# 217 218 218 218 218 219 219 219 219 220 220 220 220 221 221 221 221 227 228 233 233 233 233 234 234 234 234 235 235 235 235 236 236 236 236 237 237 237 237 238 238 238 238 239 239 239 239 240 240 240 241 241 241 242 242 242 250 250 Msn# 2607 2616 2617 2619 2618 2628 2629 2630 2631 2640 2641 2642 2643 2652 2654 2655 2653 2724 2736 2796 2797 2798 2799 2809 2810 2811 2808 2820 2821 2822 2823 2832 2834 2833 2835 2844 2845 2846 2847 2856 2857 2858 2859 2868 2869 2870 2871 2880 2881 2882 2892 2893 2894 2904 2905 2906 3000 3001 REQT# 62 8 63 83 85 7 63 85 62 44 63 85 62 60 85 62 62 62 62 57 55 85 62 55 85 62 62 57 55 85 59 57 85 62 94 57 52 85 74 57 50 85 74 57 75 85 91 22 53 85 22 84 85 17 84 85 15 67 APOE~APOD KNUQ~RJFK KDYS~LTAG KNUW~RJFU KPSM~RJAA KHIF~RJCC KDSM~LTAG KNUW~RJFU KHIF~RJCC KNUQ~RJFK KWRI~ETAR KNUW~RJFU KHIF~RJCC KNUQ~RJFK KNTD~RJCJ KHIF~RJCC KNUQ~RJFK KNUQ~RJFK KNUQ~RJFK KNUQ~RJFK KNKT~RJAA KSSC~OMAA KHIF~RJCC KNUQ~RJFK KSSC~OMAA KHIF~RJCC KNUQ~RJFK KNUQ~RJFK KNKT~RJAA KSSC~OMAA KHIF~RJCC KNLC~RJCC KNKT~RJAA KHIF~RJCC KNUQ~RJFK KBLV~RJTT KNKT~RJAA KSLC~LTAG KHIF~RJCC KORD~RJFF KNKT~RJAA KSKA~LTAG KHIF~RJCC KORD~RJFF KNKT~RJAA KPAM~RJFK KHIF~RJCC KNZJ~RJOI KFSM~LTAG KSPI~LTAG KHIF~RJCC KFSM~LTAG KRCA~RJBB KHIF~RJCC KFMH~LTAG KRCA~RJBB KHIF~RJCC KFHU~LTAG KNZJ~RJTA ARR APOD 25.99958 3.28333 11.95741 18.67012 23.36537 3.67613 11.80072 22.02129 25.8636 1.68157 12.36338 21.55882 25.30044 5.36534 22.89513 25.80903 29.03074 24.8525 25.88379 5.53619 12.30616 22.14133 25.59297 12.61152 24.15452 27.1149 29.59947 4.95765 12.46394 21.54734 25.4087 5.50964 21.29632 26.2399 32.34795 5.23108 13.31986 23.0045 26.09286 5.53619 12.73017 23.21321 25.93661 5.11377 16.93717 24.08889 27.25913 7.23905 17.97592 22.67025 6.97938 15.00323 21.35465 8.66709 14.847 22.89513 5.69164 15.70454 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 18 0 0 0 18 37.2 0 0 0 37.2 0 65.53 0 0 65.53 34.5 0 0 0 34.5 0 0 41.4 0 41.4 0 6.6 0 0 6.6 0 72.71 0 0 72.71 0 0 41.4 0 41.4 18 0 0 0 18 0 14.5 0 14 17.3 0 58.35 0 0 58.35 0 0 41.4 0 41.4 18 0 0 0 18 23.3 0 0 0 23.3 0 0 41.4 0 41.4 18 0 0 0 18 18 0 0 0 18 18 0 0 18 21.6 18 0 0 71 32.2 35.53 0 0 0 35.53 50.42 0 0 0 50.42 0 0 41.4 0 41.4 18 0 0 0 18 61.5 0 0 0 61.5 0 0 41.4 0 41.4 18 0 0 0 18 18 0 0 0 18 41.3 0 0 0 41.3 50.92 0 0 0 50.92 0 0 41.4 0 41.4 23.9 0 0 8 25.5 41.3 0 0 0 41.3 0 0 41.4 0 41.4 18 0 0 0 18 9 0 0 0 9 41.3 0 0 0 41.3 0 33.6 0 0 33.6 0 0 41.4 0 41.4 51.18 0 0 0 51.18 35.53 0 0 0 35.53 0 6.7 0 7 8.1 0 0 41.4 0 41.4 16.02 0 0 0 16.02 13.63 0 0 0 13.63 0.5 11.2 0 0 11.7 0 0 41.4 0 41.4 0 4.5 0 0 4.5 16.5 36.71 0 0 53.21 0 33.6 0 0 33.6 0 0 41.4 0 41.4 0 61.89 0 3 62.49 34.5 16.9 0 0 51.4 0 0 41.4 0 41.4 30.2 0 0 0 30.2 10.9 17.7 0 0 28.6 0 0 41.4 0 41.4 9 0 0 0 9 0 14.5 0 0 14.5 174 Tail# 250 251 251 251 252 252 252 253 253 253 254 254 254 255 255 255 256 256 256 257 257 257 258 258 258 259 259 259 260 260 260 261 261 261 262 262 262 263 263 263 264 264 264 265 265 265 266 266 266 267 267 267 268 268 268 269 269 269 Msn# 3002 3012 3013 3014 3024 3025 3026 3036 3037 3038 3048 3049 3050 3060 3061 3062 3072 3073 3074 3084 3085 3086 3096 3097 3098 3108 3109 3110 3120 3121 3122 3132 3133 3134 3144 3145 3146 3156 3157 3158 3168 3169 3170 3180 3181 3182 3192 3193 3194 3204 3205 3206 3216 3217 3218 3228 3229 3230 REQT# 85 14 61 85 12 68 85 30 68 85 18 55 85 18 83 85 18 83 85 31 83 85 37 83 85 40 83 85 40 83 85 40 83 85 39 83 85 39 81 85 39 81 85 39 81 85 39 81 85 39 81 85 38 88 85 24 88 85 APOE~APOD KHIF~RJCC KFFO~LTAG KNTU~RJFF KHIF~RJCC KEND~OMAA KNZW~RJTT KHIF~RJCC KGTB~LTBA KNZW~RJTT KHIF~RJCC KFOE~LTAC KSSC~OMAA KHIF~RJCC KFOE~LTAC KPSM~RJAA KHIF~RJCC KFOE~LTAC KPSM~RJAA KHIF~RJCC KBLV~LTBF KPSM~RJAA KHIF~RJCC KNHK~LTFJ KPSM~RJAA KHIF~RJCC KNGU~LTAF KPSM~RJAA KHIF~RJCC KNGU~LTAF KPSM~RJAA KHIF~RJCC KNGU~LTAF KPSM~RJAA KHIF~RJCC KNGP~LTAC KPSM~RJAA KHIF~RJCC KNGP~LTAC KPIT~RJTT KHIF~RJCC KNGP~LTAC KPIT~RJTT KHIF~RJCC KNGP~LTAC KPIT~RJTT KHIF~RJCC KNGP~LTAC KPIT~RJTT KHIF~RJCC KNGP~LTAC KPIT~RJTT KHIF~RJCC KNFW~LTAG KNXX~RJFK KHIF~RJCC KFWH~LTAG KNXX~RJFK KHIF~RJCC ARR APOD 20.95076 6.46584 16.28466 21.25168 5.07023 17.24112 20.96952 5.81859 16.83531 20.79713 5.90084 12.00451 22.03399 6.04901 18.11236 22.56212 6.15592 18.33849 23.63187 5.82181 17.94472 22.50252 7.79938 17.83067 20.93426 8.5344 18.34848 22.38494 8.711 18.11236 21.25168 8.89561 18.41141 23.07921 8.22629 17.9447 20.95076 7.93506 19.04081 23.41099 8.22913 19.88065 23.81696 8.07993 19.20842 23.22875 7.97125 19.09429 24.15292 8.81204 19.56488 23.48556 10.2668 19.42022 22.90882 9.79769 20.15135 24.17168 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 41.4 0 41.4 0 12.5 0 0 12.5 6.5 6.7 0 0 13.2 0 0 41.4 0 41.4 0 4.5 0 0 4.5 50.5 0 0 0 50.5 0 0 41.4 0 41.4 0 8.8 0 12 11.2 39.8 0 1.2 0 41 0 0 41.4 0 41.4 6.7 50.84 0 0 57.54 35.96 0 0 0 35.96 0 0 41.4 0 41.4 0 47.9 0 0 47.9 26.2 0 0 0 26.2 0 0 41.4 0 41.4 0 45.46 0 0 45.46 26.2 0 0 0 26.2 0 0 41.4 0 41.4 5.6 6.7 0 1 12.5 26.2 0 0 0 26.2 0 0 41.4 0 41.4 23.3 0 0 0 23.3 26.2 0 0 0 26.2 0 0 41.4 0 41.4 0 47.9 0 0 47.9 26.2 0 0 0 26.2 0 0 41.4 0 41.4 0 47.9 0 0 47.9 26.2 0 0 0 26.2 0 0 41.4 0 41.4 0 14.6 0 0 14.6 26.2 0 0 0 26.2 0 0 41.4 0 41.4 38.1 0 0 0 38.1 12.3 0 0 0 12.3 0 0 41.4 0 41.4 38.1 0 0 0 38.1 9 0 41.4 0 50.4 0 0 41.4 0 41.4 38.1 0 0 0 38.1 0 0 41.4 0 41.4 0 0 41.4 0 41.4 38.1 0 0 0 38.1 0 0 41.4 0 41.4 0 0 17.54 0 17.54 38.1 0 0 0 38.1 0 0 41.4 0 41.4 0 0 41.4 0 41.4 13.5 0 0 0 13.5 0 0 2.4 0 2.4 0 0 41.4 0 41.4 6.5 6.7 0 0 13.2 0.5 11.2 34.57 0 46.27 0 0 41.4 0 41.4 22.3 0 0 0 22.3 0 0 41.4 0 41.4 0 0 41.4 0 41.4 175 Tail# 270 270 270 271 271 271 272 272 272 273 273 273 274 274 274 275 275 276 276 276 277 277 277 278 278 278 279 279 279 280 280 280 281 281 281 282 282 282 283 283 283 284 284 284 285 285 285 286 286 286 287 287 287 288 288 289 289 289 Msn# 3240 3241 3242 3252 3253 3254 3264 3265 3266 3276 3277 3278 3288 3289 3290 3301 3302 3312 3313 3314 3324 3325 3326 3336 3337 3338 3348 3349 3350 3360 3361 3362 3372 3373 3374 3384 3385 3386 3396 3398 3397 3408 3409 3410 3420 3421 3422 3432 3434 3433 3446 3444 3445 3456 3457 3468 3470 3469 REQT# 23 88 85 32 88 85 32 88 85 32 88 85 42 88 85 88 70 36 88 56 34 73 80 34 72 78 34 72 93 34 72 87 34 72 87 34 72 87 34 86 72 34 72 62 34 72 62 47 65 72 57 43 72 45 72 45 66 72 APOE~APOD KFWA~LTAG KNXX~RJFK KHIF~RJCC KBLV~LTBG KNXX~RJFK KHIF~RJCC KBLV~LTBG KNXX~RJFK KHIF~RJCC KBLV~LTBG KNXX~RJFK KHIF~RJCC KNHK~ENBO KNXX~RJFK KHIF~RJCC KNXX~RJFK KOAK~RJAA KNGZ~LTCJ KNXX~RJFK KSTJ~OMAD KNGP~LTBU KOQU~RJCJ KPIA~RJTA KNGP~LTBU KOKC~RJCC KPHL~RJOI KNGP~LTBU KOKC~RJCC KNZY~RJTA KNGP~LTBU KOKC~RJCC KNUW~RJFF KNGP~LTBU KOKC~RJCC KNUW~RJFF KNGP~LTBU KOKC~RJCC KNUW~RJFF KNGP~LTBU KBLV~RJCJ KOKC~RJCC KNGP~LTBU KOKC~RJCC KNUQ~RJFK KNGP~LTBU KOKC~RJCC KNUQ~RJFK KBNA~OJ1X KNYL~RJOI KOKC~RJCC KNKT~RJAA KBLV~ENTC KOKC~RJCC KWRI~ETAR KOKC~RJCC KWRI~ETAR KNZC~RJSM KOKC~RJCC ARR APOD 9.76296 18.8705 21.12867 7.85176 19.36871 21.94582 7.99994 19.18291 22.65549 7.81916 19.31084 23.2926 9.72644 20.15135 23.53626 19.7455 22.03675 10.06368 19.47798 23.04018 11.07895 20.37114 24.76563 10.20635 20.95992 26.45675 10.16767 21.56709 24.89339 10.64117 20.8379 25.22579 10.78935 20.72385 25.06704 11.33484 21.54731 25.20457 11.07895 14.1185 20.95992 11.22719 21.56709 25.71412 10.89678 20.8379 25.01668 9.8768 13.39867 20.72385 4.94426 9.79835 21.54731 9.5546 20.95992 9.43654 20.80529 23.83555 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 22.1 0 22.1 0 0 41.4 0 41.4 0 0 41.4 0 41.4 0 31.4 0 0 31.4 0 0 41.4 0 41.4 0 0 41.4 0 41.4 0 31.4 0 0 31.4 0 0 41.4 0 41.4 0 0 41.4 0 41.4 0 13.1 0 0 13.1 0 0 41.4 0 41.4 0 0 41.4 0 41.4 32 0 0 0 32 0 0 41.4 0 41.4 0 0 41.4 0 41.4 0 0 41.4 0 41.4 18 0 0 0 18 23.9 0 0 8 25.5 0 0 21.13 0 21.13 0 14.2 0 0 14.2 50.5 0 0 0 50.5 0 6.6 0 0 6.6 0 12.5 0 0 12.5 50.5 0 0 0 50.5 0 0 41.4 0 41.4 0 4.5 0 0 4.5 50.5 0 0 0 50.5 0 0 41.4 0 41.4 0 12.5 0 0 12.5 50.5 0 0 0 50.5 0 0 41.4 0 41.4 26.2 0 0 0 26.2 50.5 0 0 0 50.5 0 0 41.4 0 41.4 26.2 0 0 0 26.2 50.5 0 0 0 50.5 0 0 41.4 0 41.4 14.8 0 0 0 14.8 50.5 0 0 0 50.5 0 6.6 0 0 6.6 0 0 41.4 0 41.4 50.5 0 0 0 50.5 0 0 41.4 0 41.4 23.7 0 0 0 23.7 46.6 0 0 0 46.6 0 0 41.4 0 41.4 23.7 0 0 0 23.7 18 0 0 0 18 16.5 0 0 0 16.5 0 0 41.4 0 41.4 42 0 0 0 42 0 10.5 8.9 0 19.4 0 0 41.4 0 41.4 26.2 0 0 0 26.2 0 0 41.4 0 41.4 26.2 0 1.2 0 27.4 0 10.5 0 0 10.5 0 0 32.1 0 32.1 176 Tail# 290 290 Msn# 3480 3482 REQT# 62 62 APOE~APOD KNUQ~RJFK KNUQ~RJFK ARR APOD 25.60001 27.71175 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 23.7 0 0 0 23.7 8.9 0 0 0 8.9 Table A.3 A.4 Tail# 0 0 0 0 0 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 8 8 9 9 10 21 22 22 23 23 24 24 24 25 25 26 26 26 27 27 28 28 28 Scenario 3 Mission Summary SCENARIO 4 BEST SOLUTION MISSION SUMMARY Msn# 0 0 1 2 2 12 14 24 25 26 36 37 38 48 49 50 60 61 62 72 73 74 84 96 96 108 109 120 253 264 265 276 277 288 289 289 300 301 313 313 313 324 325 336 337 337 REQT# 4 5 19 37 49 0 37 3 29 37 3 29 54 10 29 54 3 29 54 3 29 54 25 9 11 7 30 7 42 7 42 7 42 7 42 53 7 53 53 36 50 35 45 35 39 57 APOE~APOD KDOV~OKBK KDOV~OKBK KDOV~LTAG KDMA~RJBB KDMA~RJBB KBLV~LTAG KDMA~RJBB KDMA~OKBK KDOV~LTAG KDMA~RJBB KDMA~OKBK KDOV~LTAG KBLV~RJBB KDMA~LTAG KDOV~LTAG KBLV~RJBB KDMA~OKBK KDOV~LTAG KBLV~RJBB KDMA~OKBK KDOV~LTAG KBLV~RJBB KDMA~ETAR KBLV~LTAG KBLV~LTAG KDSM~LTAG KBLV~LTAG KDSM~LTAG KDOV~RJBB KDSM~LTAG KDOV~RJBB KDSM~LTAG KDOV~RJBB KDSM~LTAG KDOV~RJBB KDOV~RJBB KDSM~LTAG KDOV~RJBB KDOV~RJBB KDOV~RJBB KDOV~RJBB KDOV~RJBB KDMA~RJBB KDOV~RJBB KBLV~RJBB KBLV~RJBB ARR APOD 0.92256 0.92256 10.82988 25.39351 25.39351 1.15861 24.63572 1.92282 12.84874 24.62856 2.21371 12.24656 25.62282 5.41254 11.81598 25.25137 1.92282 12.10064 25.01248 2.21371 12.11955 26.31969 1.79609 4.72281 4.72281 3.18476 11.86169 1.87918 14.78855 2.89294 14.90332 2.15149 15.51355 1.70674 15.31665 15.31665 2.45526 14.70644 14.81736 14.81736 14.81736 5.51355 15.69815 5.04355 15.65626 15.65626 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 0 161 32.2 0 0 0 5 1 0 0 0 305 61 0 0 0 335 67 0 0 0 24 4.8 0 0 0 300 60 0 0 0 310 62 0 0 0 335 67 0 0 0 335 67 0 0 0 280 56 0 0 0 335 67 0 0 0 335 67 0 0 0 310 62 0 0 0 146 29.2 0 0 0 335 67 0 0 0 280 56 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 335 67 0 0 0 264 52.8 0 0 0 305 61 0 0 0 120 24 0 0 0 28 5.6 0 0 0 3 0.6 0 0 0 335 67 0 0 0 118 23.6 0 0 0 335 67 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 93 18.6 0 0 0 101 20.2 0 0 0 105 21 0 0 0 195 39 0 0 0 38 7.6 0 0 0 4 0.8 0 0 0 8 1.6 0 0 0 195 39 0 0 0 120 24 0 0 0 59 11.8 0 0 0 123 24.6 0 0 0 46 9.2 177 Tail# 29 29 29 30 30 31 32 34 34 35 35 36 36 36 36 37 38 38 39 39 40 41 41 42 42 43 44 44 44 44 65 65 65 66 66 66 67 67 67 67 68 68 68 69 69 69 69 70 70 70 71 71 71 72 72 72 73 73 Msn# 348 349 349 360 361 373 384 408 409 420 421 432 432 433 433 445 456 457 468 469 481 492 493 504 505 516 528 528 529 529 780 781 782 792 793 794 804 804 805 806 817 816 818 829 829 828 830 840 841 842 852 853 854 864 866 865 876 877 REQT# 10 34 32 3 51 58 38 15 37 15 37 15 16 37 49 37 22 49 22 49 49 12 49 12 49 12 12 18 49 55 4 12 36 4 12 36 4 6 12 53 12 27 37 24 26 27 31 27 29 31 27 29 31 1 29 29 20 29 APOE~APOD KDMA~LTAG KDMA~OKBK KDMA~OKBK KDMA~OKBK KBLV~RJBB KDOV~RJBB KDMA~RJBB KBLV~LTAG KDMA~RJBB KBLV~LTAG KDMA~RJBB KBLV~LTAG KBLV~LTAG KDMA~RJBB KDMA~RJBB KDMA~RJBB KDMA~ETAR KDMA~RJBB KDMA~ETAR KDMA~RJBB KDMA~RJBB KBLV~LTAG KDMA~RJBB KBLV~LTAG KDMA~RJBB KBLV~LTAG KBLV~LTAG KBLV~LTAG KDMA~RJBB KDMA~RJBB KDOV~OKBK KBLV~LTAG KDOV~RJBB KDOV~OKBK KBLV~LTAG KDOV~RJBB KDOV~OKBK KDOV~OKBK KBLV~LTAG KDOV~RJBB KBLV~LTAG KDOV~OKBK KDMA~RJBB KDOV~ETAR KDOV~ETAR KDOV~OKBK KBLV~LTAG KDOV~OKBK KDOV~LTAG KBLV~LTAG KDOV~OKBK KDOV~LTAG KBLV~LTAG KBLV~LTAG KDOV~LTAG KDOV~LTAG KDOV~LTAG KDOV~LTAG ARR APOD 4.93866 22.90827 22.90827 2.34504 23.84507 25.95178 5.37763 5.70322 25.02637 6.42514 25.42368 5.71532 5.71532 25.52348 25.52348 25.73677 7.59349 25.18989 7.59349 24.79842 24.75477 7.69597 25.09929 8.00716 25.13202 8.86279 7.99471 7.99471 26.29905 26.29905 1.06045 8.48643 12.83923 0.91541 9.38038 12.88397 1.06047 1.06047 8.48653 15.21214 8.78509 12.30325 25.58073 9.50815 9.50815 11.86496 15.1948 12.49769 14.37095 16.18218 11.81635 13.68926 15.4816 1.21873 12.14215 13.82912 9.16304 12.45276 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 0 195 39 0 0 0 133 26.6 0 0 0 12 2.4 0 0 0 195 39 0 0 0 123 24.6 0 0 0 123 24.6 0 0 0 184 36.8 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 52 10.4 0 0 0 41 8.2 0 0 0 94 18.8 0 0 0 100 20 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 195 39 0 0 0 58 11.6 0 0 0 2 0.4 0 0 0 131 26.2 0 0 0 18 3.6 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 69.51 0 69.51 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 20.89 0 20.89 0 0 51.55 0 51.55 0 0 45.95 0 45.95 0 0 72.2 0 72.2 0 0 56.9 0 56.9 0 0 72.2 0 72.2 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 8.9 0 8.9 0 0 1.2 0 1.2 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 72.2 0 72.2 0 0 87.4 0 87.4 0 0 72.1 0 72.1 0 0 23.5 0 23.5 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 51.2 0 51.2 0 0 87.4 0 87.4 178 Tail# 73 74 74 74 74 75 75 76 76 76 77 77 77 78 78 78 79 79 79 80 80 81 81 82 82 83 83 84 84 112 112 112 112 113 113 113 113 114 114 114 114 114 114 115 115 115 115 115 116 116 116 116 117 117 117 117 118 118 Msn# 878 889 888 888 890 900 902 913 912 914 925 924 926 936 937 938 949 948 950 960 961 972 973 984 985 996 997 1008 1009 1344 1346 1347 1345 1356 1358 1357 1359 1368 1368 1370 1369 1371 1371 1380 1380 1381 1382 1383 1392 1394 1393 1395 1405 1405 1406 1407 1416 1417 REQT# 46 29 39 57 44 3 37 6 3 37 6 3 37 3 27 30 6 9 37 10 27 38 27 38 27 38 27 38 27 2 8 40 40 4 41 46 40 0 1 3 12 40 47 1 9 19 41 40 25 41 41 40 23 22 53 40 26 43 APOE~APOD KDOV~RJBB KDOV~LTAG KBLV~RJBB KBLV~RJBB KBLV~RJBB KDMA~OKBK KDMA~RJBB KDOV~OKBK KDMA~OKBK KDMA~RJBB KDOV~OKBK KDMA~OKBK KDMA~RJBB KDMA~OKBK KDOV~OKBK KBLV~LTAG KDOV~OKBK KBLV~LTAG KDMA~RJBB KDMA~LTAG KDOV~OKBK KDMA~RJBB KDOV~OKBK KDMA~RJBB KDOV~OKBK KDMA~RJBB KDOV~OKBK KDMA~RJBB KDOV~OKBK KDOV~LTAG KDYS~LTAG KDOV~RJBB KDOV~RJBB KDOV~OKBK KBLV~RJBB KDOV~RJBB KDOV~RJBB KBLV~LTAG KBLV~LTAG KDMA~OKBK KBLV~LTAG KDOV~RJBB KDOV~RJBB KBLV~LTAG KBLV~LTAG KDOV~LTAG KBLV~RJBB KDOV~RJBB KDMA~ETAR KBLV~RJBB KBLV~RJBB KDOV~RJBB KDMA~ETAR KDMA~ETAR KDOV~RJBB KDOV~RJBB KDOV~ETAR KDMA~RJBB ARR APOD 17.12994 13.34532 15.95788 15.95788 21.07324 1.96362 24.85955 1.35212 3.64887 24.85239 1.40153 3.67396 25.41572 1.9636 12.25464 14.24431 1.20627 3.47633 26.16402 5.43526 11.99864 5.50196 12.54631 4.67726 12.39968 4.96177 12.09665 5.19685 12.39968 1.43289 4.16798 25.76336 28.62361 0.94073 12.18793 17.12625 25.50717 1.24154 1.24154 4.19003 9.77556 25.65415 25.65415 2.78575 2.78575 11.01632 13.64869 25.73385 1.84848 12.45053 15.3434 25.27197 9.84855 9.84855 15.04274 25.00637 9.52755 12.36064 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 0 0 6.2 0 6.2 0 0 18.4 0 18.4 0 0 5.6 0 5.6 0 0 56 0 56 0 0 8.9 0 8.9 0 0 87.4 0 87.4 0 0 75.67 0 75.67 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 69.51 0 69.51 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 87.4 0 87.4 0 0 78.63 0 78.63 0 0 72.2 0 72.2 0 0 83.74 0 83.74 0 0 79.85 0 79.85 0 0 45.5 0 45.5 0 0 72.86 0 72.86 0 0 10.2 0 10.2 0 0 72.2 0 72.2 0 0 65.53 0 65.53 0 0 72.2 0 72.2 0 0 75.67 0 75.67 0 0 72.2 0 72.2 0 0 69.51 0 69.51 0 0 72.2 0 72.2 0 0 31.45 0 31.45 0 0 72.2 0 72.2 0 18.1 0 0 18.1 37.2 0 0 0 37.2 18 0 0 0 18 18 0 0 0 18 13 6 53.65 72 87.05 0 62.17 0 0 62.17 50.2 0 0 0 50.2 18 0 0 0 18 0 33.5 0 0 33.5 23.78 0 0 0 23.78 0 0 73.17 41 81.37 0 0 65.27 0 65.27 18 0 0 67 31.4 0 0 0 221 44.2 9.42 0 0 0 9.42 0.5 11.2 0 0 11.7 55.9 0 0 0 55.9 0 69.05 0 0 69.05 18 0 0 0 18 0 14.5 0 0 14.5 0 55.54 0 0 55.54 0 64.04 0 0 64.04 18 0 0 0 18 32 0 0 0 32 0 0 0 8 1.6 34.5 16.9 0 0 51.4 18 0 0 0 18 34.5 0 0 0 34.5 16.5 0 0 0 16.5 179 Tail# 118 118 119 119 119 120 120 120 121 121 121 121 127 133 133 133 133 134 134 134 135 135 135 135 136 136 136 137 137 137 138 138 139 139 139 140 140 141 141 142 142 142 143 143 143 143 144 144 144 144 145 145 145 146 146 146 147 147 Msn# 1419 1418 1428 1429 1431 1440 1441 1442 1452 1453 1454 1454 1524 1596 1597 1597 1598 1608 1609 1610 1620 1621 1621 1622 1632 1633 1634 1644 1645 1646 1656 1658 1668 1669 1670 1680 1682 1693 1694 1704 1705 1706 1717 1717 1718 1716 1728 1728 1729 1730 1740 1741 1742 1752 1753 1754 1766 1764 REQT# 43 53 7 15 58 35 30 45 35 19 39 57 37 35 19 20 46 35 19 41 35 24 26 52 35 26 52 35 12 52 19 52 11 19 52 11 52 5 52 14 27 52 27 33 52 40 15 16 33 44 20 33 48 3 33 34 12 19 APOE~APOD KDMA~RJBB KDOV~RJBB KDSM~LTAG KBLV~LTAG KDOV~RJBB KDOV~RJBB KBLV~LTAG KDMA~RJBB KDOV~RJBB KDOV~LTAG KBLV~RJBB KBLV~RJBB KDMA~RJBB KDOV~RJBB KDOV~LTAG KDOV~LTAG KDOV~RJBB KDOV~RJBB KDOV~LTAG KBLV~RJBB KDOV~RJBB KDOV~ETAR KDOV~ETAR KDMA~RJBB KDOV~RJBB KDOV~ETAR KDMA~RJBB KDOV~RJBB KBLV~LTAG KDMA~RJBB KDOV~LTAG KDMA~RJBB KBLV~LTAG KDOV~LTAG KDMA~RJBB KBLV~LTAG KDMA~RJBB KDOV~OKBK KDMA~RJBB KDOV~LTAG KDOV~OKBK KDMA~RJBB KDOV~OKBK KDOV~OKBK KDMA~RJBB KDOV~RJBB KBLV~LTAG KBLV~LTAG KDOV~OKBK KBLV~RJBB KDOV~LTAG KDOV~OKBK KBLV~RJBB KDMA~OKBK KDOV~OKBK KDMA~OKBK KBLV~LTAG KDOV~LTAG ARR APOD 15.4322 18.52309 3.49924 6.23277 25.96998 5.25207 12.61129 16.36079 5.00644 11.1286 16.07014 16.07014 25.17754 5.33249 10.73825 10.73825 17.30252 5.08217 10.01142 12.30845 5.33898 9.5482 9.5482 17.84969 4.99295 9.44297 18.14858 4.9517 9.91389 17.97122 10.99965 18.19162 4.89086 10.08404 17.9638 5.29575 17.88614 1.26658 18.18504 6.44895 11.86112 17.97122 11.86152 11.86152 18.19162 25.3324 6.03438 6.03438 12.09925 20.78642 8.44661 12.18031 22.23186 2.1009 12.05465 23.0896 8.76197 11.00444 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 15.5 0 0 0 15.5 10.9 17.7 0 0 28.6 0 0 0 340 68 0 66.97 0 0 66.97 34 0 0 0 34 39.78 0 0 68 53.38 1.2 33.6 30.66 59 77.26 0 44.5 0 0 44.5 41.3 0 0 0 41.3 41.2 0 0 0 41.2 6.5 6.7 0 0 13.2 0 20.9 0 0 20.9 0 0 62.56 0 62.56 42 0 0 0 42 50.5 0 0 0 50.5 0 0 0 4 0.8 40.1 0 0 0 40.1 42 0 0 0 42 50.5 0 0 0 50.5 0 59.6 0 0 59.6 35.63 0 0 0 35.63 0 10.5 0 0 10.5 26.2 0 0 0 26.2 26.2 0 0 0 26.2 42 0 0 0 42 29.6 0 0 0 29.6 26.2 0 0 0 26.2 7.89 0 0 0 7.89 0 0 34.2 0 34.2 26.2 0 0 0 26.2 50.5 0 0 0 50.5 26.2 0 0 0 26.2 16.5 46.27 0 0 62.77 50.5 0 0 0 50.5 26.2 0 0 0 26.2 0 47.9 0 0 47.9 26.2 0 0 0 26.2 0 30.9 0 0 30.9 26.2 0 0 0 26.2 0 0 34.5 0 34.5 0 0 34.2 0 34.2 26.2 0 0 0 26.2 0 0 31.6 0 31.6 23.52 0 0 0 23.52 20.6 0 0 0 20.6 23.7 0 0 0 23.7 0 21.83 0 0 21.83 5.6 6.7 0 0 12.3 51.4 0 0 0 51.4 0 10.5 0 0 10.5 0 0 39.2 0 39.2 51.4 0 0 0 51.4 18 0 0 0 18 0 0 41.4 28 47 51.4 0 0 0 51.4 0 14.2 0 0 14.2 0 0 34.2 0 34.2 50.5 0 0 17 53.9 180 Tail# 147 148 148 149 149 149 150 150 150 151 151 152 152 152 152 152 153 153 153 154 154 154 154 155 155 155 155 156 156 156 157 157 157 158 158 158 Msn# 1765 1776 1778 1788 1789 1790 1800 1801 1802 1812 1814 1824 1824 1824 1825 1826 1836 1837 1838 1848 1849 1849 1850 1861 1860 1860 1862 1872 1873 1874 1884 1885 1886 1897 1896 1898 REQT# 33 17 40 17 28 40 17 19 40 13 40 12 18 21 30 40 21 30 40 21 40 47 40 38 21 11 40 19 31 40 30 31 40 7 30 56 APOE~APOD KDOV~OKBK KDMA~OKBK KDOV~RJBB KDMA~OKBK KDMA~LTAG KDOV~RJBB KDMA~OKBK KDOV~LTAG KDOV~RJBB KDMA~LTAG KDOV~RJBB KBLV~LTAG KBLV~LTAG KBLV~LTAG KBLV~LTAG KDOV~RJBB KBLV~LTAG KBLV~LTAG KDOV~RJBB KBLV~LTAG KDOV~RJBB KDOV~RJBB KDOV~RJBB KDMA~RJBB KBLV~LTAG KBLV~LTAG KDOV~RJBB KDOV~LTAG KBLV~LTAG KDOV~RJBB KBLV~LTAG KBLV~LTAG KDOV~RJBB KDSM~LTAG KBLV~LTAG KDOV~RJBB ARR APOD 12.98718 8.09098 25.59421 8.0913 13.13221 24.99291 8.09011 11.14871 24.84233 9.45659 25.40531 9.201 9.201 9.201 13.09166 25.22797 9.02492 12.9117 25.55775 9.05363 24.9565 24.9565 27.55366 5.14862 8.9101 8.9101 24.84233 10.84667 15.18258 25.08213 12.76197 15.69874 25.2279 2.63139 12.47269 25.95649 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX TOTAL 21.08 0 7.9 0 28.98 0 31.4 0 0 31.4 23.7 0 0 0 23.7 0 31.4 0 0 31.4 0 6.7 0 7 8.1 23.7 0 0 0 23.7 0 13.1 0 0 13.1 50.5 0 0 0 50.5 23.7 0 0 0 23.7 22.3 0 0 0 22.3 23.7 0 0 0 23.7 0 0 32.83 0 32.83 0 0 20.3 0 20.3 0 10.24 0 0 10.24 0 0 41.4 0 41.4 23.7 0 0 0 23.7 0 47.9 0 0 47.9 0 0 41.4 0 41.4 23.7 0 0 0 23.7 0 47.9 0 0 47.9 23.7 0 0 22 28.1 0 0 0 88 17.6 23.7 0 0 0 23.7 23.3 0 37.84 49 70.94 0 4.36 0 0 4.36 0 4.43 0 0 4.43 23.7 0 0 0 23.7 50.5 0 0 0 50.5 0 33.6 34 0 67.6 23.7 0 0 0 23.7 0 0 41.4 28 47 0 0 41.4 0 41.4 17.3 0 0 0 17.3 0 6.6 0 0 6.6 0 0 41.4 28 47 0 24.5 20 0 44.5 Table A.4 Scenario 4 Mission Summary A.5 Tail# 0 0 0 0 0 0 1 1 SCENARIO 5 BEST SOLUTION MISSION SUMMARY Msn# 0 0 1 3 3 2 12 13 REQT# 4 5 35 15 16 19 0 34 APOE~APOD KDOV~OKBK KDOV~OKBK KDOV~RJBB KBLV~LTAG KBLV~LTAG KDOV~LTAG KBLV~LTAG KDMA~OKBK ARR APOD 0.87393 0.87393 5.17778 7.12879 7.12879 9.67014 0.98996 22.92198 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX 0 0 0 133 0 0 0 5 0 0 0 280 0 0 0 335 0 0 0 24 0 0 0 322 0 0 0 300 0 0 0 133 TOTAL 26.6 1 56 67 4.8 64.4 60 26.6 181 Tail# 1 1 2 2 2 2 3 3 3 3 3 4 4 4 4 10 10 10 10 10 11 11 11 12 12 13 13 13 13 14 14 14 14 14 20 20 20 20 20 21 21 21 21 22 22 22 22 23 23 23 23 23 24 24 24 35 35 35 Msn# 13 15 24 25 26 27 37 38 36 39 39 50 50 48 51 120 120 121 121 123 133 132 134 145 144 156 159 157 157 168 171 169 170 170 241 242 242 240 243 252 252 253 254 264 265 266 266 276 277 278 279 279 289 290 288 422 420 420 REQT# 32 37 3 10 29 51 38 29 45 37 49 42 53 53 54 9 11 15 16 58 3 7 30 12 12 7 12 39 57 7 10 22 42 53 10 24 26 36 53 4 6 38 29 6 38 39 57 6 38 27 31 30 14 27 29 27 27 33 APOE~APOD KDMA~OKBK KDMA~RJBB KDMA~OKBK KDMA~LTAG KDOV~LTAG KBLV~RJBB KDMA~RJBB KDOV~LTAG KDMA~RJBB KDMA~RJBB KDMA~RJBB KDOV~RJBB KDOV~RJBB KDOV~RJBB KBLV~RJBB KBLV~LTAG KBLV~LTAG KBLV~LTAG KBLV~LTAG KDOV~RJBB KDMA~OKBK KDSM~LTAG KBLV~LTAG KBLV~LTAG KBLV~LTAG KDSM~LTAG KBLV~LTAG KBLV~RJBB KBLV~RJBB KDSM~LTAG KDMA~LTAG KDMA~ETAR KDOV~RJBB KDOV~RJBB KDMA~LTAG KDOV~ETAR KDOV~ETAR KDOV~RJBB KDOV~RJBB KDOV~OKBK KDOV~OKBK KDMA~RJBB KDOV~LTAG KDOV~OKBK KDMA~RJBB KBLV~RJBB KBLV~RJBB KDOV~OKBK KDMA~RJBB KDOV~OKBK KBLV~LTAG KBLV~LTAG KDOV~LTAG KDOV~OKBK KDOV~LTAG KDOV~OKBK KDOV~OKBK KDOV~OKBK ARR APOD 22.92198 24.79628 1.92282 5.55837 11.87848 23.91638 4.87719 11.76729 15.71802 24.71809 24.71809 14.91798 14.91798 17.26863 25.03191 4.69576 4.69576 6.39537 6.39537 25.74461 1.90754 3.81757 11.69588 7.69613 9.3834 1.92232 7.69597 15.90185 15.90185 1.70674 4.89699 7.59349 14.70644 14.70644 5.47693 9.50815 9.50815 13.13003 15.25706 0.91463 0.91463 4.67726 12.03607 0.95014 5.19094 15.71651 15.71651 1.01878 5.28019 11.81714 14.7573 14.7573 5.71156 11.81635 13.70698 11.9392 14.34324 14.34324 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX 0 0 0 12 0 0 0 280 0 0 0 335 0 0 0 146 0 0 0 335 0 0 0 123 0 0 0 233 0 0 0 269 0 0 0 120 0 0 0 296 0 0 0 63 0 0 0 335 0 0 0 24 0 0 0 63 0 0 0 230 0 0 0 28 0 0 0 3 0 0 0 107 0 0 0 17 0 0 0 123 0 0 0 195 0 0 0 195 0 0 0 65 0 0 0 195 0 0 0 43 0 0 0 195 0 0 0 195 0 0 0 123 0 0 0 46 0 0 0 195 0 0 0 195 0 0 0 195 0 0 0 37 0 0 0 158 0 0 10.2 0 0 0 8.9 0 0 0 1.2 0 0 0 72.2 0 0 0 56.9 0 0 0 84.08 0 0 0 13.42 0 0 0 75.67 0 0 0 68 0 0 0 87.4 0 0 0 69.51 0 0 0 5.6 0 0 0 56 0 0 0 87.4 0 0 0 34.82 0 0 0 72.2 0 0 0 87.38 0 0 0 8.2 0 0 0 34.5 0 0 0 72.2 0 0 0 87.4 0 0 0 68.4 0 0 0 6.6 0 46.17 0 0 0 TOTAL 2.4 56 67 29.2 67 24.6 46.6 53.8 24 59.2 12.6 67 4.8 12.6 46 5.6 0.6 21.4 3.4 24.6 39 39 13 39 8.6 39 39 24.6 9.2 39 39 39 7.4 31.6 10.2 8.9 1.2 72.2 56.9 84.08 13.42 75.67 68 87.4 69.51 5.6 56 87.4 34.82 72.2 87.38 8.2 34.5 72.2 87.4 68.4 6.6 46.17 182 Tail# 35 35 36 36 36 37 37 37 38 38 38 38 39 39 39 39 40 40 40 40 40 41 41 41 41 41 42 42 42 42 43 43 43 43 44 44 44 44 44 44 44 55 55 55 55 56 56 56 56 57 57 57 57 58 58 58 58 59 Msn# 423 421 432 434 433 444 446 445 456 457 458 459 468 469 470 471 481 482 480 483 483 492 495 495 493 494 506 504 505 507 516 517 518 519 528 528 528 530 531 531 529 663 660 662 661 672 675 674 673 684 685 685 686 698 698 696 697 708 REQT# 53 58 4 7 11 3 17 41 1 11 26 43 8 35 23 48 13 26 34 40 47 3 15 16 20 27 25 25 17 45 9 19 33 41 12 18 21 28 39 57 52 3 7 12 22 7 30 43 42 7 24 26 31 5 6 7 56 7 APOE~APOD KDOV~RJBB KDOV~RJBB KDOV~OKBK KDSM~LTAG KBLV~LTAG KDMA~OKBK KDMA~OKBK KBLV~RJBB KBLV~LTAG KBLV~LTAG KDOV~ETAR KDMA~RJBB KDYS~LTAG KDOV~RJBB KDMA~ETAR KBLV~RJBB KDMA~LTAG KDOV~ETAR KDMA~OKBK KDOV~RJBB KDOV~RJBB KDMA~OKBK KBLV~LTAG KBLV~LTAG KDOV~LTAG KDOV~OKBK KDMA~ETAR KDMA~ETAR KDMA~OKBK KDMA~RJBB KBLV~LTAG KDOV~LTAG KDOV~OKBK KBLV~RJBB KBLV~LTAG KBLV~LTAG KBLV~LTAG KDMA~LTAG KBLV~RJBB KBLV~RJBB KDMA~RJBB KDMA~OKBK KDSM~LTAG KBLV~LTAG KDMA~ETAR KDSM~LTAG KBLV~LTAG KDMA~RJBB KDOV~RJBB KDSM~LTAG KDOV~ETAR KDOV~ETAR KBLV~LTAG KDOV~OKBK KDOV~OKBK KDSM~LTAG KDOV~RJBB KDSM~LTAG ARR APOD 16.78814 25.96998 0.93905 3.51543 6.14988 2.27469 8.27208 11.91356 1.24155 5.47573 9.67517 12.36759 2.32322 5.12614 9.84855 22.18779 8.47674 10.50588 23.25778 25.56914 25.56914 2.26092 6.05824 6.05824 8.27762 12.10211 1.84632 4.05531 8.4176 16.10455 2.13205 9.99306 12.46969 15.57839 8.68336 8.68336 8.68336 12.37489 15.91395 15.91395 18.71123 2.10225 4.37672 8.48319 10.45128 1.90229 12.09769 14.27905 18.18421 2.00884 9.44297 9.44297 14.88935 1.36148 1.36148 3.62845 26.30258 2.11362 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX 34.5 16.9 0 71 34 0 0 0 13 6 54.52 72 0 0 0 196 16.5 54.66 0 0 0 0 72.05 46 0 22.7 0 0 0 76.34 0 0 6.06 0 23.5 0 0 43.94 0 0 29.6 0 0 0 15.2 0 0 0 37.2 0 0 0 14.97 0 0 27 32 0 0 0 18 0 0 0 22.3 0 0 0 34.5 0 0 0 0 14.2 0 0 16.6 0 0 45 0 0 0 190 0 0 41.95 0 0 26.8 0 0 5.6 6.7 0 0 0 0 51.2 4 0 0 68.4 0 0 14.5 0 14 0 0 0 106 0 53.2 0 0 0 44.5 0 0 0.5 11.2 45.5 0 50.6 0 0 0 52.63 0 7.9 0 0 34.06 0 0 0 0 52.7 54 0 0 17.48 2 0 0 0 0 0 6.7 0 7 6.5 6.7 0 0 0 20.9 0 0 30.2 0 0 0 0 0 41.4 28 0 0 0 102 0 0 34.2 18 0 0 0 102 0 0 0 102 0 0 41.4 28 16.8 0 0 0 0 0 0 102 0 0 0 102 0 10.5 0 0 26.2 0 0 0 0 33.6 34.92 0 0 30.9 0 0 0 0 12.38 0 0 0 0 102 0 24.5 20 0 0 0 0 102 TOTAL 65.6 34 87.92 39.2 71.16 81.25 22.7 76.34 29.56 43.94 29.6 15.2 37.2 20.37 32 18 22.3 34.5 14.2 25.6 38 41.95 26.8 12.3 52 68.4 17.3 21.2 53.2 44.5 57.2 50.6 60.53 34.06 63.5 17.88 0 8.1 13.2 20.9 30.2 47 20.4 37.8 20.4 20.4 47 16.8 20.4 20.4 10.5 26.2 68.52 30.9 12.38 20.4 44.5 20.4 183 Tail# 59 59 60 60 60 60 61 61 61 61 61 62 62 62 62 62 63 63 63 63 63 64 64 64 64 64 65 65 66 66 66 67 67 67 67 68 68 68 69 69 69 Msn# 709 711 722 720 721 723 733 733 732 734 735 746 744 745 747 747 757 756 758 758 759 769 769 768 770 771 782 781 792 793 794 805 804 806 807 817 818 818 829 830 830 REQT# 12 37 2 7 12 37 0 1 7 42 37 4 7 12 37 49 7 7 36 50 49 18 21 22 42 55 20 21 35 30 46 38 15 46 44 53 40 47 30 40 47 APOE~APOD KBLV~LTAG KDMA~RJBB KDOV~LTAG KDSM~LTAG KBLV~LTAG KDMA~RJBB KBLV~LTAG KBLV~LTAG KDSM~LTAG KDOV~RJBB KDMA~RJBB KDOV~OKBK KDSM~LTAG KBLV~LTAG KDMA~RJBB KDMA~RJBB KDSM~LTAG KDSM~LTAG KDOV~RJBB KDOV~RJBB KDMA~RJBB KBLV~LTAG KBLV~LTAG KDMA~ETAR KDOV~RJBB KDMA~RJBB KDOV~LTAG KBLV~LTAG KDOV~RJBB KBLV~LTAG KDOV~RJBB KDMA~RJBB KBLV~LTAG KDOV~RJBB KBLV~RJBB KDOV~RJBB KDOV~RJBB KDOV~RJBB KBLV~LTAG KDOV~RJBB KDOV~RJBB ARR APOD 8.20187 25.14863 1.0753 3.22431 8.06416 24.85445 1.15761 1.15761 3.38106 15.08209 24.74032 0.96105 3.29059 7.89087 25.14858 25.14858 1.97533 4.22835 13.2959 13.2959 24.7403 8.72185 8.72185 10.81496 14.95645 25.93468 7.89544 10.97587 5.30251 12.19945 17.30252 4.7403 7.02831 16.95658 21.23186 15.296 25.08217 25.08217 12.3052 25.3025 25.3025 --------DELIVERED QUANTITIES--------OUT OVER BULK PAX 0 0 34.2 18 0 0 41.4 28 0 18.1 0 0 0 0 0 102 0 0 34.2 18 0 0 41.4 0 0 33.5 0 0 27.14 0 0 0 0 0 0 102 0 0 0 102 0 0 41.4 0 0 0 41.4 28 0 0 0 102 0 0 0 102 0 0 31.2 0 0 0 0 90 0 6.6 0 1 0 0 0 102 0 0 18.2 4 0 0 0 8 0 0 0 77 0 0 2.82 0 0 47.9 0 0 0 0 0 101 0 0 0 102 0 0 0 18 0 0 39.2 0 0 42.5 0 0 35.63 0 0 15 1.2 33.6 30.4 38 40.1 0 0.16 0 23.3 0 0 0 0 62 0 0 50.2 0 6.04 0 0 10.5 8.9 0 10.9 17.7 0 18 23.7 0 0 22 0 0 0 74 0 0 0 102 23.7 0 0 22 0 0 0 45 TOTAL 37.8 47 18.1 20.4 37.8 41.4 33.5 27.14 20.4 20.4 41.4 47 20.4 20.4 31.2 18 6.8 20.4 19 1.6 15.4 2.82 47.9 20.2 20.4 3.6 39.2 42.5 38.63 72.8 40.26 23.3 62 56.24 19.4 32.2 28.1 14.8 20.4 28.1 9 Table A.5 Scenario 5 Mission Summary 184 References Aarts, E.H. & van Laarhoven, P.J. (1985). Statistical Cooling: A General Approach to Combinatorial Optimization Problems, Philips Journal of Research 40: 193-226. AFDD 2-6 (1999). Air Mobility: Air Force Doctrine Document 2-6, Part of Joint Publication 3-17, Joint Tactics, Techniques, and Procedures for Air Mobility Operations. AFDD 2-6.1 (1999). Airlift Operations: Air Force Doctrine Document 2-6.1, Part of Joint Publication 3-17, Joint Tactics, Techniques, and Procedures for Air Mobility Operations. Baker, S.F., Morton, D.P., Rosenthal, R.E. & Williams, L.M. (2002). Optimizing Military Airlift, Operations Research 50(4): 582-602. Balas, E. & Padberg, M. W. (1970). On the Set Covering Problem, Operations Research 20(6): 1152 1161. Balas, E. & Padberg, M. W. (1976). Set Partitioning: A Survey, SIAM Review 18, (4): 710 760. Balinski, M.L. & Quandt, R.E. (1964). On an Integer Program for a Delivery Problem, Operations Research 12, 300 304. 185 Barnes, J. W. & Carlton, W. B. (1995). 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Yost, K. A. (1994). The THRUPUT Strategic Airlift Flow Optimization Model, Air Force Studies and Analyses Agency, The Pentagon, Washington, D.C. 201 Vita Garrett Randall Lambert was born in Topeka, Kansas on April 12th, 1962, the son of Henry Dillon and Joan Randall Lambert. A 1980 graduate of Iona Preparatory High School, New Rochelle, New York, he entered the United States Military Academy in West Point, New York. He received a B.S. from the United States Military Academy in May 1984 and was commissioned as a second lieutenant in the Infantry. His military service has been continuous to the present day and he has held positions as Rifle Platoon Leader, Company Executive Officer, Battalion Logistics Officer (S4), Brigade and Battalion Assistant Operations Officer (Asst. S3), Company Commander, Instructor and Assistant Professor in the Department of Mathematical Sciences at West Point, Battalion Executive Officer, and Corps Assistant Operations Officer for Training (Asst. G3 Training). He currently holds the rank of Lieutenant Colonel. He received a M.S. in Industrial Engineering from Texas A&M University, College Station, Texas in May 1995. He received a M.A. in National Security and Strategic Studies from the Naval War College, Newport, Rhode Island in June 1998. In June 2000 he began doctoral studies in the Operations Research, Industrial Engineering Program at the University of Texas, Austin, Texas. Permanent address: 211E Barry Road, West Point, NY 10996 This dissertation was typed by the author. 202

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