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IEOR E4007 G. Iyengar September 26, 2018 Homework #3 1. Continuation of the toy LP from Homework 2 Consider the following linear program max 2x1 + x2 , subject to 12x1 + 3x2 ≤ 60, 3x1 − x2 ≥ −7, x2 ≤ 10, x1 ≥ 0, x2 free. The standard form LP is given by max 2y1 + (y2 − y3 ), subject to 12y1 + 3(y2 − y3 ) + y4 = 60, 3y1 − (y2 − y3 ) − y5 = −7, (y2 − y3 ) + y6 = 10, y ≥ 0. (a) Show that the constraints imply that y2 = 7 + 3y1 + y3 − y5 y4 = 39 − 21y1 + 3y5 y6 = 3 − 3y1 + y5 From this set of equations it follows that B = {y2 , y4 , y6 } is a basis. Using only the information in the above set of equalities, and no matrix inversions compute the reduced cost with respect to each of the non-basic directions N = {y1 , y3 , y5 }. Recall that the reduced cost is the impact on the objective when any one of the non-basic variables is increased. (b) Choose the non-basic direction that corresponds to the largest reduced cost, and compute the next corner, i.e. the new basis. You are only allowed use the information in the set of equations in part (a). (c) Suppose the objective function is given by max 3y1 − y2 + y3 . Start from the basis B = {y2 , y4 , y6 } to conclude the problem is unbounded. You are only allowed to use the information in part (a). (d) Show that there does not exist any vector c such that the basis B = {y2 , y4 , y6 } is optimal for the linear program max c1 y1 + c2 (y2 − y3 ). 1 (e) The basis B ∗ = {y1 , y2 , y5 } is optimal for the objective max 2y1 +y2 −y3 . Suppose the RHS 60 1 −7 bθ = + θ 2 10 1 What is the range of θ for which this basis remains optimal. −1 Note that since the reduced costs c̄> = c> − c> B AB A does not change as θ is changed, the current basis will remain optimal provided xB = A−1 B bθ ≥ 0. 2. Exercise 3.14 in in Optimization Methods in Finance Sensitivity table Time -----1 2 3 4 5 6 7 8 9 Shadow price --------------1.0829 -1.0775 -1.0657 -1.0397 -1.0345 -1.0294 -1.0243 -1.0050 -1.0000 RHS Low ---------299.8099 198.6910 0.0000 -912.2537 -793.0497 -94.5149 304.0125 -inf -inf 2 RHS --------100.0000 500.0000 100.0000 -600.0000 -500.0000 200.0000 600.0000 -900.0000 0.0000 RHS Up --------535.4807 937.6581 300.8861 -394.0917 -293.0622 407.9725 809.0124 -430.7829 471.5631