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7-LP-special cases

7-LP-special cases - QMT437 Pn Paezah Topic 3 LP Special...

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QMT437 Pn Paezah Topic 3: LP - Special Cases in Linear Programming Objective Learning how to detect identify special cases from the graphical solution and the simplex solution. Four special cases that may occur in solving LP problems: ± Multiple Optimal Solutions ± Infeasibility ± Unboundedness ± Degeneracy Case 1: Multiple Optimal Solutions LP problem has more than 1 optimal solution, or there are alternative optimal solutions. Hence, the optimal solution is not unique. Example : Consider the XYZ Co. problem. Suppose that the profit per batch of scooters is only RM2,000, instead of RM5,000. The problem is reformulated as follows: X 1 = number of batches of tricycles, X 2 = number of batches of scooters Maximize profit (Z) = 3X 1 + 2 X 2 (RM’000) subject to: X 1 4 ;Workstation 1 (hours) 2X 2 12 ;Workstation 2 (hours) 3X 1 + 2X 2 18 ;Workstation 3 (hours) X 1 ,X 2 0 The Graphical Solution Corner Point solution Corner points Z=3X 1 +2X 2 (RM’000) (0,0) 0 (4,0) 12 (0,6) 12 (4,3) 18 (2,6) 18 Two corner points have the same max value of RM18,000, indicating there are multiple optimal solutions . All points on the line segment from (2,6) to (4,3) also provide optimal solutions. Maximum profit = RM18,000 (4,0) X 1 ,0) (0 (0,6) 6 9 2 2X 2 =12 X 1 =4 3X 1 +2X 2 =18 Feasible region (2, 6) ( X 4,3) 1

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The Simplex Solution How to detect the case of multiple optimal solutions from the simplex procedure? Observe the final tableau. An LP problem has multiple optimal solutions if in the final tableau, a nonbasic variable has 0 coefficient in row-Z. The third iteration of simplex method gives the following optimal tableau.
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7-LP-special cases - QMT437 Pn Paezah Topic 3 LP Special...

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