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Unformatted text preview: Table of Contents Chapter 3 (Linear Programming: Formulation and Applications) Super Grain Corp. Advertising­Mix Problem (Section 3.1) Resource Allocation Problems (Section 3.2) Cost­Benefit­Trade­Off Problems (Section 3.3) Mixed Problems (Section 3.4) Transportation Problems (Section 3.5) Assignment Problems (Section 3.6) 3.2–3.5 3.6–3.16 3.17–3.22 3.23–3.28 3.29–3.33 3.34–3.37 Applications of Linear Programming with Spreadsheets (UW Lecture) 3.38–3.57 These slides are based upon lectures to first­year MBA students at the University of Washington that discuss the application and formulation of linear programming models (as taught by one of the authors). 1 Super Grain Corp. Advertising­Mix Problem Goal: Design the promotional campaign for Crunchy Start. The three most effective advertising media for this product are The limited resources in the problem are Television commercials on Saturday morning programs for children. Advertisements in food and family­oriented magazines. Advertisements in Sunday supplements of major newspapers. Advertising budget ($4 million). Planning budget ($1 million). TV commercial spots available (5). The objective will be measured in terms of the expected number of exposures. Question: At what level should they advertise Crunchy Start in each of the three media? 2 Cost and Exposure Data Costs Cost Category Ad Budget Planning budget Expected number of exposures Each TV Commercial $300,000 90,000 1,300,000 Each Magazine Ad $150,000 30,000 600,000 Each Sunday Ad $100,000 40,000 500,000 3 Spreadsheet Formulation 3 4 5 6 7 8 9 10 11 12 13 14 15 B Exposures per Ad (thousands) C TV Spots 1,300 D Magazine Ads 600 E SS Ads 500 F G H Ad Budget Planning Budget 300 90 Cost per Ad ($thousands) 150 30 100 40 Budget Spent 4,000 1,000 <= <= Budget Available 4,000 1,000 Total Exposures (thousands) 17,000 Number of Ads Max TV Spots TV Spots 0 <= 5 Magazine Ads 20 SS Ads 10 4 Algebraic Formulation Let TV = Number of commercials for separate spots on television M = Number of advertisements in magazines. SS = Number of advertisements in Sunday supplements. Maximize Exposure = 1,300TV + 600M + 500SS subject to Ad Spending: 300TV + 150M + 100SS ≤ 4,000 ($thousand) Planning Cost: 90TV + 30M + 30SS ≤ 1,000 ($thousand) Number of TV Spots: TV ≤ 5 and TV ≥ 0, M ≥ 0, SS ≥ 0. 5 The TBA Airlines Problem TBA Airlines is a small regional company that specializes in short flights in small airplanes. The company has been doing well and has decided to expand its operations. The basic issue facing management is whether to purchase more small airplanes to add some new short flights, or start moving into the national market by purchasing some large airplanes, or both. Question: How many airplanes of each type should be purchased to maximize their total net annual profit? 6 Data for the TBA Airlines Problem Small Airplane Net annual profit per airplane Purchase cost per airplane Maximum purchase quantity $1 million 5 million 2 Large Airplane $5 million 50 million — $100 million Capital Available 7 Violates Divisibility Assumption of LP Divisibility Assumption of Linear Programming: Decision variables in a linear programming model are allowed to have any values, including fractional values, that satisfy the functional and nonnegativity constraints. Thus, these variables are not restricted to just integer values. Since the number of airplanes purchased by TBA must have an integer value, the divisibility assumption is violated. 8 Spreadsheet Model 3 4 5 6 7 8 9 10 11 12 13 14 B Unit Profit ($millions) C Small Airplane 1 D Large Airplane 5 E F G Capital ($millions) Capital Per Unit Produced 5 50 Capital Spent 100 <= Capital Available 100 Total Profit ($millions) 10 Units Produced Maximum Small Airplanes Small Airplane 0 <= 2 Large Airplane 2 9 Integer Programming Formulation Let S = Number of small airplanes to purchase L = Number of large airplanes to purchase Maximize Profit = S + 5L ($millions) subject to Capital Available: and S ≥ 0, L ≥ 0 S, L are integers. 5S + 50L ≤ 100 ($millions) Max Small Planes: S ≤ 2 10 10 Think­Big Capital Budgeting Problem Think­Big Development Co. is a major investor in commercial real­estate development projects. They are considering three large construction projects Each project requires each partner to make four investments: a down payment now, and additional capital after one, two, and three years. Construct a high­rise office building. Construct a hotel. Construct a shopping center. Question: At what fraction should Think­Big invest in each of the three projects? 11 11 Financial Data for the Projects Investment Capital Requirements Year 0 1 2 3 Net present value Office Building $40 million 60 million 90 million 10 million $45 million Hotel $80 million 80 million 80 million 70 million $70 million Shopping Center $90 million 50 million 20 million 60 million $50 million 12 12 Spreadsheet Formulation 3 4 5 6 7 8 9 10 11 12 13 14 15 16 B C Office Building 45 D Hotel 70 E Shopping Center 50 F G H Net Present Value ($millions) Now End of Year 1 End of Year 2 End of Year 3 Cumulative Capital Required ($millions) 40 80 90 100 160 140 190 240 160 200 310 220 Office Building 0.00% Shopping Center 13.11% Cumulative Capital Spent 25 44.757 60.583 80 <= <= <= <= Cumulative Capital Available 25 45 65 80 Total NPV ($millions) 18.11 Participation Share Hotel 16.50% 13 13 Algebraic Formulation Let OB = Participation share in the office building, H = Participation share in the hotel, SC = Participation share in the shopping center. Maximize NPV = 45OB + 70H + 50SC subject to Total invested now: 40OB + 80H + 90SC ≤ 25 ($million) Total invested within 1 year: 100OB + 160H + 140SC ≤ 45 ($million) Total invested within 2 years: 190OB + 240H + 160SC ≤ 65 ($million) Total invested within 3 years: 200OB + 310H + 220SC ≤ 80 ($million) and OB ≥ 0, H ≥ 0, SC ≥ 0. 14 14 Template for Resource­Allocation Problems Activities Unit Profit profit per unit of activity Resources Used SUMPRODUCT (resource used per unit, changing cells) Resources Available Constraints resource used per unit of activity <= Level of Activity changing cells Total Profit SUMPRODUCT(profit per unit, changing cells) 15 15 Summary of Formulation Procedure for Resource­ Allocation Problems 1. 2. Identify the activities for the problem at hand. Identify an appropriate overall measure of performance (commonly profit). For each activity, estimate the contribution per unit of the activity to the overall measure of performance. Identify the resources that must be allocated. For each resource, identify the amount available and then the amount used per unit of each activity. Enter the data in steps 3 and 5 into data cells. Designate changing cells for displaying the decisions. In the row for each resource, use SUMPRODUCT to calculate the total amount used. Enter <= and the amount available in two adjacent cells. Designate a target cell. Use SUMPRODUCT to calculate this measure of performance. 16 16 3. 4. 5. 6. 7. 8. 9. Union Airways Personnel Scheduling Union Airways is adding more flights to and from its hub airport and so needs to hire additional customer service agents. The five authorized eight­hour shifts are Question: How many agents should be assigned to each shift? Shift 1: Shift 2: Shift 3: Shift 4: Shift 5: 6:00 AM to 2:00 PM 8:00 AM to 4:00 PM Noon to 8:00 PM 4:00 PM to midnight 10:00 PM to 6:00 AM 17 17 Schedule Data Time Periods Covered by Shift Time Period 1 2 3 4 5 Minimum Number of Agents Needed 48 √ √ √ √ √ √ √ √ √ √ √ 79 65 87 64 73 82 43 18 18 6 AM to 8 AM 8 AM to 10 AM 10 AM to noon Noon to 2 PM 2 PM to 4 PM 4 PM to 6 PM 6 PM to 8 PM 8 PM to 10 PM √ √ √ √ Spreadsheet Formulation 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 B C 6am-2pm Shift $170 D 8am-4pm Shift $160 E Noon-8pm Shift $175 F 4pm-midnight Shift $180 G 10pm-6am Shift $195 H I J Cost per Shift Time Period 6am-8am 8am-10am 10am- 12pm 12pm-2pm 2pm-4pm 4pm-6pm 6pm-8pm 8pm-10pm 10pm-12am 12am-6am 1 1 1 1 0 0 0 0 0 0 6am-2pm Shift 48 Shift Works Time Period? (1=yes, 0=no) 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 0 0 0 8am-4pm Shift 31 Noon-8pm Shift 39 4pm-midnight Shift 43 0 0 0 0 0 0 0 0 1 1 10pm-6am Shift 15 Total Working 48 79 79 118 70 82 82 43 58 15 >= >= >= >= >= >= >= >= >= >= Minimum Needed 48 79 65 87 64 73 82 43 52 15 Number Working Total Cost $30,610 19 19 Algebraic Formulation Let Si = Number working shift i (for i = 1 to 5), Minimize Cost = $170S1 + $160S2 + $175S3 + $180S4 + $195S5 subject to Total agents 6AM–8AM: S1 ≥ 48 Total agents 8AM–10AM: S1 + S2 ≥ 79 Total agents 10AM–12PM: S1 + S2 ≥ 65 Total agents 12PM–2PM: S1 + S2 + S3 ≥ 87 Total agents 2PM–4PM: S2 + S3 ≥ 64 Total agents 4PM–6PM: S3 + S4 ≥ 73 Total agents 6PM–8PM: S3 + S4 ≥ 82 Total agents 8PM–10PM: S4 ≥ 43 Total agents 10PM–12AM: S4 + S5 ≥ 52 Total agents 12AM–6AM: S5 ≥ 15 and Si ≥ 0 (for i = 1 to 5) 20 20 Template for Cost­Benefit Tradoff Problems Activities Unit Cost cost per unit of activity Benefit Achieved SUMPRODUCT (benefit per unit, changing cells) Benefit Needed Constraints benefit achieved per unit of activity >= Level of Activity changing cells Total Cost SUMPRODUCT(cost per unit, changing cells) 21 21 Summary of Formulation Procedure for Cost­Benefit­Tradeoff Problems 1. 2. Identify the activities for the problem at hand. Identify an appropriate overall measure of performance (commonly cost). For each activity, estimate the contribution per unit of the activity to the overall measure of performance. Identify the benefits that must be achieved. For each benefit, identify the minimum acceptable level and then the contribution of each activity to that benefit. Enter the data in steps 3 and 5 into data cells. Designate changing cells for displaying the decisions. In the row for each benefit, use SUMPRODUCT to calculate the level achieved. Enter >= and the minimum acceptable level in two adjacent cells. Designate a target cell. Use SUMPRODUCT to calculate this measure of performance. 22 22 3. 4. 5. 6. 7. 8. 9. Types of Functional Constraints Type Resource constraint Form* LHS ≤ RHS Typical Interpretation For some resource, Amount used ≤ Amount available For some benefit, Level achieved ≥ Minimum Acceptable For some quantity, Amount provided = Required amount Main Usage Resource­allocation problems and mixed problems Cost­benefit­trade­off problems and mixed problems Transportation problems and mixed problems Benefit constraint LHS ≥ RHS Fixed­requirement constraint LHS = RHS * LHS = Left­hand side (a SUMPRODUCT function). RHS = Right­hand side (a constant). 23 23 Continuing the Super Grain Case Study David and Claire conclude that the spreadsheet model needs to be expanded to incorporate some additional considerations. In particular, they feel that two audiences should be targeted — young children and parents of young children. Two new goals Furthermore, exactly $1,490,000 should be allocated for cents­off coupons. The advertising should be seen by at least five million young children. The advertising should be seen by at least five million parents of young children. 24 24 Benefit and Fixed­Requirement Data Number Reached in Target Category (millions) Each TV Commercial Young children Parents of young children 1.2 0.5 Each Magazine Ad 0.1 0.2 Each Sunday Ad 0 0.2 Minimum Acceptable Level 5 5 Contribution Toward Required Amount Each TV Commercial Coupon redemption 0 Each Magazine Ad $40,000 Each Sunday Ad $120,000 Required Amount $1,490,000 25 25 Spreadsheet Formulation 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 B Exposures per Ad (thousands) Ad Budget Planning Budget C TV Spots 1,300 D Magazine Ads 600 Cost per Ad ($thousands) 150 30 E SS Ads 500 F G H 300 90 100 40 Budget Spent 3,775 1,000 Total Reached 5 5.85 Total Redeemed 1,490 <= <= Budget Available 4,000 1,000 Minimum Acceptable 5 5 Required Amount 1,490 Total Exposures (thousands) 16,175 Young Children Parents of Young Children 1.2 0.5 Number Reached per Ad (millions) 0.1 0 0.2 0.2 Magazine Ads 40 SS Ads 120 >= >= Coupon Redemption per Ad ($thousands) TV Spots 0 = Number of Ads Maximum TV Spots TV Spots 3 <= 5 Magazine Ads 14 SS Ads 7.75 26 26 Algebraic Formulation Let TV = Number of commercials for separate spots on television M = Number of advertisements in magazines. SS = Number of advertisements in Sunday supplements. Maximize Exposure = 1,300TV + 600M + 500SS subject to Ad Spending: 300TV + 150M + 100SS ≤ 4,000 ($thousand) Planning Cost: 90TV + 30M + 30SS ≤ 1,000 ($thousand) Number of TV Spots: TV ≤ 5 Young children: Parents: Coupons: and TV ≥ 0, M ≥ 0, SS ≥ 0. 27 27 1.2TV + 0.1M ≥ 5 (millions) 0.5TV + 0.2M + 0.2SS ≥ 5 (millions) 40M + 120SS = 1,490 ($thousand) Template for Mixed Problems Activities Unit Profit or Cost profit/cost per unit of activity Resources Used SUMPRODUCT (resource used per unit, changing cells) Resources Available resource used per unit of activity <= Benefit Needed Constraints Benefit Achieved benefit achieved per unit of activity SUMPRODUCT (benefit per unit, changing cells) >= = Level of Activity changing cells Total Profit or Cost SUMPRODUCT(profit/cost per unit, changing cells) 28 28 The Big M Transportation Problem The Big M Company produces a variety of heavy duty machinery at two factories. One of its products is a large turret lathe. Orders have been received from three customers for the turret lathe. Question: How many lathes should be shipped from each factory to each customer? 29 29 Some Data Shipping Cost for Each Lathe To From Factory 1 Factory 2 Order Size $700 800 10 lathes $900 900 8 lathes $800 700 9 lathes Customer 1 Customer 2 Customer 3 Output 12 lathes 15 lathes 30 30 The Distribution Network C1 10 lathes needed $700/lathe 12 lathe produced F1 $900/lathe $800/lathe C2 $800/lathe 15 lathes produced $900/lathe 8 lathes needed F2 $700/lathe C3 9 lathes needed 31 31 Spreadsheet Formulation B Shipping Cost (per Lathe) Factory 1 Factory 2 C Customer 1 $700 $800 D Customer 2 $900 $900 E Customer 3 $800 $700 Total Shipped Out 12 15 F G H 3 4 5 6 7 8 9 10 11 12 13 14 15 Units Shipped Factory 1 Factory 2 Total To Customer Order Size Customer 1 10 0 10 = 10 Customer 2 2 6 8 = 8 Customer 3 0 9 9 = 9 = = Output 12 15 Total Cost $20,500 32 32 Algebraic Formulation Let Sij = Number of lathes to ship from i to j (i = F1, F2; j = C1, C2, C3). Minimize Cost = $700SF1-C1 + $900SF1-C2 + $800SF1-C3 + $800SF2-C1 + $900SF2-C2 + $700SF2-C3 subject to Factory 1: SF1-C1 + SF1-C2 + SF1-C3 = 12 Factory 2: SF2-C1 + SF2-C2 + SF2-C3 = 15 Customer 1: SF1-C1 + SF2-C1 = 10 Customer 2: SF1-C2 + SF2-C2 = 8 Customer 3: SF1-C3 + SF2-C3 = 9 and Sij ≥ 0 (i = F1, F2; j = C1, C2, C3). 33 33 Sellmore Company Assignment Problem The marketing manager of Sellmore Company will be holding the company’s annual sales conference soon. He is hiring four temporary employees: Each will handle one of the following four tasks: Ann Ian Joan Sean Question: Which person should be assigned to which task? Word processing of written presentations Computer graphics for both oral and written presentations Preparation of conference packets, including copying and organizing materials Handling of advance and on­site registration for the conference 34 34 Data for the Sellmore Problem Required Time per Task (Hours) Temporary Employee Ann Ian Joan Sean Word Processing 35 47 39 32 Graphics 41 45 56 51 Packets 27 32 36 25 Registrations 40 51 43 46 Hourly Wage $14 12 13 15 35 35 Spreadsheet Formulation 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 B Required Time (Hours) Assignee Ann Ian Joan Sean C D Word Processing 35 47 39 32 E Task Graphics 41 45 56 51 Packets 27 32 36 25 Registrations 40 51 43 46 Hourly Wage $14 $12 $13 $15 F G H I J Task Cost Assignee Ann Ian Joan Sean Word Processing $490 $564 $507 $480 Graphics $574 $540 $728 $765 Packets $378 $384 $468 $375 Registrations $560 $612 $559 $690 Task Assignment Assignee Ann Ian Joan Sean Total Assigned Demand Word Processing 0 0 0 1 1 = 1 Graphics 0 1 0 0 1 = 1 Packets 1 0 0 0 1 = 1 Registrations 0 0 1 0 1 = 1 Total Assignments 1 1 1 1 Supply 1 1 1 1 Total Cost $1,957 = = = = 36 36 The Model for Assignment Problems Given a set of tasks to be performed and a set of assignees who are available to perform these tasks, the problem is to determine which assignee should be assigned to each task. To fit the model for an assignment problem, the following assumptions need to be satisfied: 1. 2. 3. 4. 5. The number of assignees and the number of tasks are the same. Each assignee is to be assigned to exactly one task. Each task is to be performed by exactly one assignee. There is a cost associated with each combination of an assignee performing a task. The objective is to determine how all the assignments should be made to minimize the total cost. 37 37 Formulating an LP Spreadsheet Model Enter all of the data into the spreadsheet. Color code (blue). What decisions need to be made? Set aside a cell in the spreadsheet for each decision variable (changing cell). Color code (yellow with border). Write an equation for the objective in a cell. Color code (orange with heavy border). Put all three components (LHS, ≤/=/≥, RHS) of each constraint into three cells on the spreadsheet. Some Examples: Production Planning Diet / Blending Workforce Scheduling Transportation / Distribution Assignment 38 38 LP Example #1 (Product Mix) The Quality Furniture Corporation produces benches and picnic tables. The firm has a limited supply of two resources: labor and wood. 1,600 labor hours are available during the next production period. The firm also has a stock of 9,000 pounds of wood available. Each bench requires 3 labor hours and 12 pounds of wood. Each table requires 6 labor hours and 38 pounds of wood. The profit margin on each bench is $8 and on each table is $18. Question: What product mix will maximize their total profit? 39 39 Algebraic Formulation Let B = Number of benches to produce, T = Number of tables to produce. Maximize Profit = $8B + $18T subject to Labor: 3B + 6T ≤ 1,600 hours Wood: 12B + 38T ≤ 9,000 pounds and B ≥ 0, T ≥ 0. 40 40 Spreadsheet Formulation 3 4 5 6 7 8 9 10 11 12 B C D E F G Profit Resources Labor Wood Benches $8 Tables $18 Total 1,600 9,000 Available 1,600 9,000 Total Cost $4,638.10 Used per Unit Produced 3 6 12 38 <= <= Units Produced 161.90 185.71 41 41 LP Example #2 (Diet Problem) A prison is trying to decide what to feed its prisoners. They would like to offer some combination of milk, beans, and oranges. Their goal is to minimize cost, subject to meeting the minimum nutritional requirements imposed by law. The cost and nutritional contents of each food, along with the minimum nutritional requirements are shown below. Navy Oranges Minimum Milk (gallons) 3.2 1.12 32 2.00 Beans (cups) 4.9 1.3 0 0.20 (large Calif. Valencia) 0.8 0.19 93 0.25 Daily Requirement 13.0 1.5 45 Niacin (mg) Thiamin (mg) Vitamin C (mg) Cost ($) Question: What should the diet for each prisoner be? 42 42 Algebraic Formulation Let x1 = gallons of milk per prisoner, x2 = cups of beans per prisoner, x3 = number of oranges per prisoner. Minimize Cost = $2.00x1 + $0.20x2 + $0.25x3 subject to Niacin: 3.2x1 + 4.9x2 + 0.8x3 ≥ 13 mg Thiamin: 1.12x1 + 1.3x2 + 0.19x3 ≥ 1.5 mg Vitamin C: 32x1 + 93x3 ≥ 45 mg and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0. 43 43 Spreadsheet Formulation 3 4 5 6 7 8 9 10 11 12 13 B C D E F G H Cost Niacin (mg) Thiamin (mg) Vitamin C (mg) Quantity (per prisoner) Milk (gal.) $2.00 Beans (cups) $0.20 Oranges $0.25 Total 13 3.438 45 >= >= >= Minimum Requirement 13 1.5 45 Total Cost $0.64 Nutritional Contents (mg) 3.2 4.9 0.8 1.12 1.3 0.19 32 0 93 0 2.574 0.484 44 44 George Dantzig’s Diet Stigler (1945) “The Cost of Subsistence” Dantzig invents the simplex method (1947) heuristic solution. Cost = $39.93. Dantzig goes on a diet (early 1950’s), applies diet model: Stigler’s problem “solved” in 120 man days. Cost = $39.69. ≤ 1,500 calories objective: maximize (weight minus water content) 500 food types 500 gallons of vinegar 200 bouillon cubes Initial solutions had problems For more details, see July­Aug 1990 Interfaces article “The Diet Problem” 45 45 Least­Cost Menu Planning Models in Food Systems Management For more details, see Sept­Oct 1992 Interfaces article “The Evolution of the Diet Model in Managing Food Systems” Used in many institutions with feeding programs: hospitals, nursing homes, schools, prisons, etc. Menu planning often extends to a sequence of meals or a cycle. Variety important (separation constraints). Preference ratings (related to service frequency). Side constraints (color, categories, etc.) Generally models have reduced cost about 10%, met nutritional requirements better, and increased customer satisfaction compared to traditional methods. USDA uses these models to plan food stamp allotment. 46 46 LP Example #3 (Scheduling Problem) An airline reservations office is open to take reservations by telephone 24 hours per day, Monday through Friday. The number of reservation agents needed for each time period is shown below. A union contract requires that all employees work 8 consecutive hours. Time Period 12am – 4am 4am – 8am 8am – 12pm 12pm – 4pm 4pm – 8pm 8pm – 12am Number of Agents Needed 11 15 31 17 25 19 Question: How many reservation agents should work each 8-hour shift? 47 47 Algebraic Formulation Let x1 = agents who work 12am – 8am, x2 = agents who work 4am – 12pm, x3 = agents who work 8am – 4pm, x4 = agents who work 12pm – 8pm, x5 = agents who work 4pm – 12am, x6 = agents who work 8pm – 4am. Minimize Number of agents = x1 + x2 + x3 + x4 + x5 + x6 subject to 12am–4am: x1 + x6 ≥ 11 4am–8am: x1 + x2 ≥ 15 8am–12pm: x2 + x3 ≥ 31 12pm–4pm: x3 + x4 ≥ 17 4pm–8pm: x4 + x5 ≥ 25 8pm–12am: x5 + x6 ≥ 19 and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0, x4 ≥ 0, x5 ≥ 0, x6 ≥ 0. 48 48 Spreadsheet Formulation A 1 2 3 4 5 6 7 8 9 10 11 B C D E F G H Reservation Agents Scheduling Problem Time Period 12am Š 4am 4am Š 8am 8am Š 12pm 12pm Š 4pm 4pm Š 8pm 8pm ŠŹ12am Number Working 11 15 31 33 25 19 Minimum Required 11 15 31 17 25 19 Shift 12am - 8am 4am - 12pm 8am - 4pm 12pm - 8pm 4pm - 12am 8pm - 4am Total Number Working 0 15 16 17 8 11 67 >= >= >= >= >= >= 49 49 Workforce Scheduling at United Airlines United employs 5,000 reservation and customer service agents. Some part­time (2­8 hour shifts), some full­time (8­10 hour shifts). Workload varies greatly over day. Modeled problem as LP: For more details, see Jan­Feb 1986 Interfaces article “United Airlines Station Manpower Planning System” Saved United about $6 million annually, improved customer service, still in use today. Decision variables: how many employees of each shift length should begin at each potential start time (half­hour intervals). Constraints: minimum required employees for each half­hour. Objective: minimize cost. 50 50 LP Example #4 (Transportation Problem) A company has two plants producing a certain product that is to be shipped to three distribution centers. The unit production costs are the same at the two plants, and the shipping cost per unit is shown below. Shipments are made once per week. During each week, each plant produces at most 60 units and each distribution center needs at least 40 units. Distribution Center 1 Plant A B $4 $6 2 $6 $5 3 $4 $2 Question: How many units should be shipped from each plant to each distribution center? 51 51 Algebraic Formulation Let xij = units to ship from plant i to distribution center j (i = A, B; j = 1, 2, 3), Minimize Cost = $4xA1 + $6xA2 + $4xA3 + $6xB1 + $5xB2 + $2xB3 subject to Plant A: xA1 + xA2 + xA3 ≤ 60 Plant B: xB1 + xB2 + xB3 ≤ 60 Distribution Center 1: xA1 + xB1 ≥ 40 Distribution Center 2: xA2 + xB2 ≥ 40 Distribution Center 3: xA3 + xB3 ≥ 40 and xij ≥ 0 (i = A, B; j = 1, 2, 3). 52 52 Spreadsheet Formulation B C D E F G H 3 4 5 6 7 8 9 10 11 12 13 14 15 Cost Plant A Plant B Distribution Center 1 $4 $6 Distribution Center 2 $6 $5 Distribution Center 3 $4 $2 Shipment Quantities Plant A Plant B Shipped Needed Distribution Center 1 40 0 40 >= 40 Distribution Center 2 20 20 40 >= 40 Distribution Center 3 0 40 40 >= 40 Shipped 60 60 Cost Available <= 60 <= 60 = $460 53 53 Distribution System at Proctor and Gamble Proctor and Gamble needed to consolidate and re­design their North American distribution system in the early 1990’s. For more details, see 1997 Jan­Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain” Solved many transportation problems (one for each product category). Goal: find best distribution plan, which plants to keep open, etc. Closed many plants and distribution centers, and optimized their product sourcing and distribution location. Implemented in 1996. Saved $200 million per year. 50 product categories 60 plants 15 distribution centers 1000 customer zones 54 54 LP Example #5 (Assignment Problem) The coach of a swim team needs to assign swimmers to a 200­yard medley relay team (four swimmers, each swims 50 yards of one of the four strokes). Since most of the best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and their best times (in seconds) they have achieved in each of the strokes (for 50 yards) are shown below. Backstroke Carl Chris David Tony Ken 37.7 32.9 33.8 37.0 35.4 Breaststroke 43.4 33.1 42.2 34.7 41.8 Butterfly 33.3 28.5 38.9 30.4 33.6 Freestyle 29.2 26.4 29.6 28.5 31.1 Question: How should the swimmers be assigned to make the fastest relay team? 55 55 Algebraic Formulation Let xij = 1 if swimmer i swims stroke j; 0 otherwise tij = best time of swimmer i in stroke j Minimize Time = ∑ i ∑ j tij xij subject to each stroke swum: ∑ i xij = 1 for each stroke j each swimmer swims 1: ∑ j xij ≤ 1 for each swimmer i and xij ≥ 0 for all i and j. 56 56 Spreadsheet Formulation B 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 C D E F G H I Best Times Carl Chris David Tony Ken Backstroke 37.7 32.9 33.8 37.0 35.4 Breastroke 43.4 33.1 42.2 34.7 41.8 Butterfly 33.3 28.5 38.9 30.4 33.6 Freestyle 29.2 26.4 29.6 28.5 31.1 Assignment Carl Chris David Tony Ken Backstroke 0 0 1 0 0 1 = 1 Breastroke 0 0 0 1 0 1 = 1 Butterfly 0 1 0 0 0 1 = 1 Freestyle 1 0 0 0 0 1 = 1 1 1 1 1 0 Time <= 1 <= 1 <= 1 <= 1 <= 1 = 126.2 57 57 ...
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This note was uploaded on 04/21/2009 for the course BUS 338W-1 taught by Professor White during the Winter '09 term at University of Michigan-Dearborn.

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