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Unformatted text preview: 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 +  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 k = 1 x P k . We do this by enforcing k = 1 implies each constraint defining P k . The only trick for Felixs bag that we need is = 1 X j N a j x j b X j N a j x j + M 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 + 50 1 60 4x 1 + x 2 x 3 + x 4 + 50 1 60 8x 1 + 2x 2 x 3 + 14x 4 + 170 2 240 x 1 + x 2 x 3 + x 4 + 28 2 30 4x 1 + x 2 + x 3 + 5x 4 + 90 3 110 2x 1 + x 2 2x 3 + x 4 + 25 3 40 1 + 2 + 3 1 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....
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 Spring '08
 Ralphs

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