lecture3_ip_intro

lecture3_ip_intro - INTEGER PROGRAMMING 1.224J/ESD.204J...

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INTEGER PROGRAMMING 1.224J/ESD.204J TRANSPORTATION OPERATIONS, PLANNING AND CONTROL: CARRIER SYSTEMS Professor Cynthia Barnhart Professor Nigel H.M. Wilson Fall 2003
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IP OVERVIEW Sources: -Introduction to linear optimization (Bertsimas, Tsitsiklis)- Chap 1 - Slides 1.224 Fall 2000
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12/31/2003 Barnhart 1.224J 3 Outline • When to use Integer Programming (IP) • Binary Choices – Example: Warehouse Location – Example: Warehouse Location 2 • Restricted range of values • Guidelines for strong formulation • Set Partitioning models • Solving the IP – Linear Programming relaxation – Branch-and bound – Example
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12/31/2003 Barnhart 1.224J 4 When to use IP Formulation? • IP (Integer Programming) vs. MIP (Mixed Integer Programming) – Binary integer program • Greater modeling power than LP • Allows to model: – Binary choices – Forcing constraints – Restricted range of values – Piecewise linear cost functions
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12/31/2003 Barnhart 1.224J 5 Example: Warehouse Location A company is considering opening warehouses in four cities: New York, Los Angeles, Chicago, and Atlanta. Each warehouse can ship 100 units per week . The weekly fixed cost of keeping each warehouse open is $400 for New York, $500 for LA, $300 for Chicago, and $150 for Atlanta. Region 1 requires 80 units per week, region 2 requires 70 units per week, and Region 3 requires 40 units per week. The shipping costs are shown below. Formulate the problem to meet weekly demand at minimum cost . From/To Region 1 Region 2 Region 3 New York 20 40 50 Los Angeles 48 15 26 Chicago 35 18 Atlanta 24
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12/31/2003 Barnhart 1.224J 6 Warehouse Location- Approach • What are the decision variables? – Variables to represent whether or not to open a given warehouse (y i =0 or 1) – Variables to track flows between warehouses and regions: x ij • What is the objective function? – Minimize (fixed costs+shipping costs) • What are the constraints? – Constraint on flow out of each warehouse – Constraint on demand at each region – Constraint ensuring that flow out of a closed warehouse is 0.
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12/31/2003 Barnhart 1.224J 7 ¦¦ ¦ ¡¡ ¡ . W iW iR j ij ij i i x t y c MIN . . ( ) s.t. ¦ ¡ % d j i ij W i y x , . 100 R j b x i j ij ¡ % ¦ , ^` 1 , 0 , ¡ ¡ . i ij y Z x Forcing constraint Warehouse Location- Formulation •L e t y i be the binary variable representing whether we open a warehouse i (y i =1) or not (y i =0).
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lecture3_ip_intro - INTEGER PROGRAMMING 1.224J/ESD.204J...

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