# Tutorial 11 - 4 2 6[3 8-4-2(a Write the primal and dual...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA3252 Linear and Network Optimization Tutorial 11 1. In the following two maximum ﬂow problems with source s = 1 and sink t = 5, determine all s - t cuts. Find the minimum s - t cut in each case. (8,0) (4,0) (2,0) (6,0) (4,6) 2 4 5 3 (4,4) 1 30 25 15 20 12 2 3 5 4 18 22 12 1 (0,2) (4,0) u ji u ij i j (u ij, u ji ) i j i j For i < j u ij 2. Consider the maximum ﬂow problem between source 1 and sink 4 in the network. u ij i j 1 2 4 3 2 2 4 1 6 (a) Write the linear program for the maximum ﬂow problem. (b) Find the optimal arc ﬂows by intuition. Is this solution a basic solution? (c) Write the dual linear program. Use complementary slackness to determine a dual optimal solution and identify the minimum s - t cut based on this. 1

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3. Consider the minimum cost uncapacitated network ﬂow problem where the number along each arc represents the unit cost and the number beside the node denotes the supply/demand. [5] [-2] 1 2 4 3 3 2 5 3
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Unformatted text preview: 4 2 6 [3] 8 [-4] [-2] (a) Write the primal and dual linear programs for the MCF problem. (b) Based on complementary slackness conditions, verify the optimality of each of the following primal solutions: i. x 12 = 3 ,x 23 = 4 ,x 24 = 4 ,x 45 = 2 ii. x 13 = 3 ,x 24 = 5 ,x 43 = 1 ,x 45 = 2 iii. x 13 = 2 ,x 12 = 1 ,x 43 = 2 ,x 24 = 6 ,x 45 = 2 4. Consider the uncapacitated MCF problem where the number along each arc rep-resents the unit cost and the value beside the vertices denote the supply/demand. 1 2 4 3 2 4 4 1 2 [-2] [5] [-3] (a) Start with the spanning tree solution consisting of arcs T = { (1 , 2) , (2 , 4) , (1 , 3) } and use the network simplex method to solve the problem to optimality. (b) By how much can we increase the cost of arc (2 , 3) and still have the same optimal basic feasible solution? 2...
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## This note was uploaded on 12/02/2011 for the course MATH 3252 taught by Professor Tanbanpin during the Spring '10 term at National University of Singapore.

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Tutorial 11 - 4 2 6[3 8-4-2(a Write the primal and dual...

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