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# hw2-answers - IE 418 Integer Programming Problem Set#2...

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IE 418 – Integer Programming Problem Set #2 — Answers 1 Nonconvex Regions 1.1 Problem Consider the following polyhedra: P 1 = { x R 4 + | 2x 1 + 2x 2 - x 3 + 2x 4 10, 4x 1 + x 2 - x 3 + x 4 10 } P 2 = { x R 4 + | 8x 1 + 2x 2 - x 3 + 14x 4 70, x 1 + x 2 - x 3 + x 4 2 } P 3 = { x R 4 + | 4x 1 + x 2 + x 3 + 5x 4 20, 2x 1 + x 2 - 2x 3 + x 4 15 } P 4 = { x R 4 + | 0 x 10e } and objective function c def = ( 6, 3, - 1, 5 ) T . Formulate as a mixed integer linear program the prob- lem: max { c T x | x P 4 ( P 1 P 2 P 3 ) } . Answer: For each region P k , k = 1, 2, 3 , we need enforce that δ k = 1 x P k . We do this by enforcing δ k = 1 implies each constraint defining P k . The only trick for Felix’s bag that we need is δ = 1 j N a j x j b j N a j x j + M + b M is an upper bound on j N a j x j - b , and we should work to get as tight (small) an up- per bound on M as possible, which we can get my using the upper and lower bounds on the variables. Doing this gives us the following formulation: max 6x 1 + 3x 2 - x 3 + 5x 4 s.t. 2x 1 + 2x 2 - x 3 + 2x 4 + 50δ 1 60 4x 1 + x 2 - x 3 + x 4 + 50δ 1 60 8x 1 + 2x 2 - x 3 + 14x 4 + 170δ 2 240 x 1 + x 2 - x 3 + x 4 + 28δ 2 30 4x 1 + x 2 + x 3 + 5x 4 + 90δ 3 110 2x 1 + x 2 - 2x 3 + x 4 + 25δ 3 40 δ 1 + δ 2 + δ 3 1

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IE418 Problem Set #2 Prof Jeff Linderoth 1.2 Problem Solve your formulation from Problem 1.1 using the software tool of your choice. Answer: The Optimal solution is z * = 54 , x = ( 28/3, 8/3, 10, 0 ) , and the point x P 2 . 2 TSP Solving 2.1 Problem Find the optimal solution to the traveling salesperson problem where the distances between the cities are given in Figure 1. Please describe all the steps of your algorithm. Atlanta 0 587 1212 701 1936 604 748 2139 2182 543 Chicago 587 0 920 940 1745 1188 713 1858 1737 597 Denver 1212 920 0 879 831 1726 1631 949 1021 1494 Houston 701 940 879 0 1372 968 1420 1645 1891 1220 LosAngeles 1936 1745 831 1374 0 2339 2451 347 959 2300 Miami 604 1188 1726 968 2339 0 1092 2594 2734 923 NewYork 748 713 1631 1420 2451 1092 0 2571 2408 205 SanFrancisco 2139 1858 949 1645 347 2594 2571 0 678 2442 Seattle 2182 1737 1021 1891
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hw2-answers - IE 418 Integer Programming Problem Set#2...

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