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INEN 420 EXTRA REVIEW PROBLEMS Chapter 4 Simplex Method Page 213 Problem 7 (Also use the graphical method to verify your optimal solutions) One optimal solution is z = 2, x1 = 1/2, x2 = 0. An alternative optimal solution z = 2, x1 = 1/3, x2 = 2/3. This LP has an infinite number of optimal solutions (see your graph!) The Big-M and the 2-Phase Simplex Methods Page 213 Problem 3 The optimal solution is z = 15, x1 = 3, x2 = 0. Chapter 6 Sensitivity Analysis and Duality Page 349 Problem 13 a. The optimal dual solution is w = 16, y1 = 0, y2 = 2. b. The current basis remains optimal for c3 ≥ 1. c. c1 ≤ 12. Complementary Slackness Page 328 Problem 3 The optimal primal solution z = 49/3, x1 = 5/3, x2 = 8/3, x3 =0. Chapter 7 Transportation Problem Page 408 problem 5: In part (b) use the Northwest corner method. Let xij = number of type j files stored on medium i. The cost of 40, for example, is derived from the fact that if one word processing file is stored on a hard disk it will be accessed eight times per month and require five minutes per access. Thus 8(5) = 40 minutes per month will be spent accessing a word processing The optimal tableau is as follows: WP PP 16 8 DUMMY 0 16 4 2 0 80 32 12 0 40 HD CM TAPE 200 DATA 0 200 100 100 100 100 100 1 300 300 100 100 100 Page 408 Problem 10: The optimal solution has z = 10(4) + 5(2) + 5(8) + 10(4) = 130. Assignment Problem Page 411 problem 28: Multiplying by (-1) converts problem to a minimization yielding P1 P1’ P2 P2’ P3 P3’ 9 AM -8 -8 -9 -9 -7 -7 10 AM -7 -7 -9 -9 -6 -6 11 AM 1 PM -6 -5 -6 -5 -8 -8 -8 -8 -9 -6 -9 -6 2 PM -7 -7 -4 -4 -9 -9 3 PM -6 -6 -4 -4 -9 -9 ROW MIN -8 -8 -9 -9 -9 -9 Optimal Tableau and Assignment: P1 P1’ P2 P2’ P3 P3’ 9 AM 0 0 0 0 3 3 10 AM 1 1 0 0 4 4 11 AM 1 1 0 0 0 0 1 PM 2 2 0 0 3 3 2PM 0 0 4 4 0 0 3 PM 1 1 4 4 0 0 Six lines are now needed to cover the zeroes. The optimal solution x66 = 1, x15 = 1, x21 = 1, x32 = 1, x44 = 1, x53 = 1 is now available. Thus Professor 1 teaches 9 AM and 2 PM class, Professor 2 teaches 10 AM and 1 PM class, and Professor 3 teaches 11 AM and 3 PM class. Chapter 8 Shortest Path Problem Page 418 Problem 1: 1-3-6 is the shortest path (of length 31) from node 1 to node 6. Maximum Flow Problems Page 430 Problem 4: Maximum flow is 45. Min Cut Set = {1, 3, and si}. Capacity of Cut Set = 20 + 15 + 10 = 45. 2 Page 430 Problem 5: Maximum flow = 9. Min Cut Set = {2, 4, si}. Capacity of Cut = 3 + 1 + 3 + 2 = 9. Minimum Spanning Tree Page 473 Problem 5: Begin at NY and include NY-Cleveland arc (length 400). Next include NY-Nashville arc (length 800). Next include Nashville-Dallas(length 600). Next include Dallas-St. Louis(length 600). Next include Dallas-Phoenix(length 900). Next include Phoenix-LA (length 400. Finally include Salt Lake City-Los Angeles (length 600). Total length of minimum spanning tree is 4300 miles. 3