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math_prog - Analysis of Air Transportation Systems...

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1 NEXTOR - National Center of Excellence for Aviation Research Analysis of Air Transportation Systems Mathematical Programming (LP) Dr. Antonio A. Trani Associate Professor of Civil and Environmental Engineering Virginia Polytechnic Institute and State University Falls Church, Virginia June 9-12, 2003
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2 NEXTOR - National Center of Excellence for Aviation Research Resource Allocation Principles of Mathematical Programming Mathematical programming is a general technique to solve resource allocation problems using optimization. Types of problems: Linear programming Integer programming Dynamic programming Decision analysis Network analysis and CPM
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3 NEXTOR - National Center of Excellence for Aviation Research Mathematical Programming Operations research was born with the increasing need to solve optimal resource allocation during WWII. Air Battle of Britain North Atlantic supply routing problems Optimal allocation of military convoys in Europe Dantzig (1947) is credited with the first solutions to linear programming problems using the Simplex Method
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4 NEXTOR - National Center of Excellence for Aviation Research Resource Allocation Linear Programming Applications Allocation of products in the market Mixing problems Allocation of mobile resources in infrastructure construction (e.g., trucks, loaders, etc.) Crew scheduling problems Network flow models Pollution control and removal Estimation techniques
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5 NEXTOR - National Center of Excellence for Aviation Research Linear Programming General Formulation Maximize subject to: for for c j j 1 = n x j a ij j 1 = n x j b i i 1 2 m , , , = x j 0 j 1 2 n , , , =
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6 NEXTOR - National Center of Excellence for Aviation Research Linear Programming Maximize Subject to: ... and Z c 1 x 1 c 2 x 2 c n x n + + + = a 11 x 1 a 12 x 2 a 1 n x n + + + b 1 a 21 x 1 a 22 x 2 a 2 n x n + + + b 2 a m 1 x 1 a m 2 x 2 a mn x n + + + b m x 1 0 x 2 0 x n 0 , , ,
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7 NEXTOR - National Center of Excellence for Aviation Research Linear Programming Objective Function (OF) Functional Constraints ( m of them) Nonnegativity Conditions ( n of these) are decision variables to be optimized (min or max) are costs associated with each decision variable c j j 1 = n x j a ij j 1 = n x j b i x j 0 x j c j
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8 NEXTOR - National Center of Excellence for Aviation Research Linear Programming are the coefficients of the functional constraints are the amounts of the resources available (RHS) Some definitions Feasible Solution (FS) - A solution that satisfies all functional constraints of the problem Basic Feasible Solution (BFS)- A solution that needs to be further investigated to determine if optimal Initial Basic Feasible Solution - a BFS used as starting point to solve the problem a ij b i
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9 NEXTOR - National Center of Excellence for Aviation Research LP Example (Construction) During the construction of an off-shore airport in Japan the main contractor used two types of cargo barges to transport materials
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