A software approach to mathematical programming

A software approach to mathematical programming

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  A Software approachto Mathematical Programming Arian Razmi Farooji 4 March 2015
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  List of Contents •Introduction • Mathematical Programming • Mathematical Programming techniques • Solving a Mathematical Model • Mathematical Programming Software • GAMS Software • Heuristics and Metaheuristics • References A Software Approach to Mathematical Programming-Arian Razmi Farooji 1
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  Introduction A Software Approachto Mathematical Programming-Arian Razmi Farooji 2 Operations Research Mathematical Optimizations Simulations Markov Chains Data Analysis Statistics Neural Networks Queuing Theory Expert Systems Economic Methods Decision Analysis
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  Mathematical Programming A SoftwareApproach to Mathematical Programming-Arian Razmi Farooji 3 Problem Definition Model Construction Model Solution Model Validity Implementation Real World Assumed Real World Model
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  Mathematical Programming techniques 1.LinearProgramming 2.Integer Programming 3.Mixed Integer Programming 4.Dynamic Programming 5.Network Programming 6.Nonlinear programming A Software Approach to Mathematical Programming-Arian Razmi Farooji 4
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  Solving a MathematicalProgramming • Goal : “ To find an Optimum solution ” • Algorithms: − provides fixed computational rules − are applied repeatedly to the problem − each repetition (iteration) moving the solution closer to the optimum. A Software Approach to Mathematical Programming-Arian Razmi Farooji 5
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  Solving a MathematicalProgramming • Simplex Method − solves LP problems − tests adjacent vertices of the feasible sets − at each iteration Simplex chooses the variable that will produce the largest change towards optimum solution • Software − GAMS − AMPL − Lingo − … A Software Approach to Mathematical Programming-Arian Razmi Farooji 6
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  GAMS Software What isGAMS? • General Algebraic Modeling System • High-level modeling system for mathematical optimization • GAMS solves −Linear optimizations −Non-linear optimizations −Mixed-integer optimizations • Tailored for large scales optimization problems A Software Approach to Mathematical Programming-Arian Razmi Farooji 7
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  GAMS Software Transportation Problem ASoftware Approach to Mathematical Programming-Arian Razmi Farooji 8 1 2 m 1 n Sources (𝑖) Destinations (𝑗) . . . . . . 𝑐11 𝑐1𝑛 𝑐21 𝑐2𝑛 𝑐 𝑚1 𝑐 𝑚𝑛 𝑎1 𝑎2 𝑎 𝑚 𝑏1 𝑏 𝑛
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  GAMS Software Decision Variable 𝑥𝑖𝑗 𝑐𝑖𝑗 𝑎𝑖 𝑏𝑗 ASoftware Approach to Mathematical Programming-Arian Razmi Farooji 9 The amount shipped from i to j min 𝑧 = 𝑖=1 𝑚 𝑗=1 𝑛 𝑐𝑖𝑗 𝑥𝑖𝑗 𝑗=1 𝑛 𝑥𝑖𝑗 ≤ 𝑎𝑖 𝑓𝑜𝑟 𝑖 = 1, , , , , 𝑚 𝑖=1 𝑚 𝑥𝑖𝑗 ≥ 𝑏𝑗 𝑓𝑜𝑟𝑗 = 1, , , , , 𝑛 Mathematical Formulation Parameters Shipping Cost from i to j Supply Capacity of source i Demand of Destination j 𝑥𝑖𝑗 ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑖 𝑎𝑛𝑑 𝑗 𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:
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  GAMS Software Example (HamdyTaha, 2011) A Software Approach to Mathematical Programming-Arian Razmi Farooji 10 LA Detroit New Orleans Denver Miami Sources Destinations 1000 1500 1200 2300 1400
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  GAMS Software Example-Parameters A SoftwareApproach to Mathematical Programming-Arian Razmi Farooji 11 Table (1). Transportation Cost per Car Denver Miami Los Angeles $80 $215 Detroit $100 $108 New Orleans $102 $68
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  GAMS Software Mathematical Formulation min𝑧 = 80𝑥11 + 215𝑥12 + 100𝑥21 + 108𝑥22 + 102𝑥31 + 68𝑥32 𝑥11 + 𝑥12 = 1000 𝑥21 + 𝑥22 = 1500 𝑥31 + 𝑥32 = 1200 𝑥11 + 𝑥21 + 𝑥31 = 2300 𝑥12 + 𝑥22 + 𝑥32 = 1400 A Software Approach to Mathematical Programming-Arian Razmi Farooji 12 Los Angeles Detroit New Orleans Denver Miami 𝑥𝑖𝑗 ≥ 0 𝑓𝑜𝑟 𝑖 = 1,2,3 𝑎𝑛𝑑 𝑗 = 1,2
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  Heuristics and Metaheuristics •Both find “good and satisfactory” solutions in shorter time • The quality of algorithms is usually based on a tradeoff between: −Optimality −Completeness −Accuracy −Execution Time A Software Approach to Mathematical Programming-Arian Razmi Farooji 13
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  Assignment • Formulate thefollowing transportation problem and solve it in GAMS. • Download GAMS here A Software Approach to Mathematical Programming-Arian Razmi Farooji 14 Shipping Costs Warehouses Supply 1 2 3 4 Factories 1 470 520 654 890 75 2 350 416 690 750 100 3 995 670 350 685 125 Demand 85 70 65 80
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  References 1. Taha, H.,Operations Research an Introduction, Pearson, New Jersey, 2011 2. INFORMS: What is Operations Research? 3. GAMS Website 4. Richard E. Rosenthal’s GAMS Tutorial 5. Wikipedia: General Algebraic Modeling System 6. Education.com: Algorithms and Heuristics A Software Approach to Mathematical Programming-Arian Razmi Farooji 15
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  Thank You forYour Attention A Software Approach to Mathematical Programming-Arian Razmi Farooji
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  Question and Answer ASoftware Approach to Mathematical Programming-Arian Razmi Farooji
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  A Software approachto Mathematical Programming Arian Razmi Farooji 4 March 2015 arian.razmifarooji@student.oulu.fi