Question

+38x4-8x5 -17x2 (b) Maximize subject to Z 2 107 + 16x5-7x6 -x1-13x2 + 45x3 + 30x7 S 81 3x3-18x =-13 -5x3 4x1 10 s x1 -2 16 S X3 x5 unrestricted 4. Write the duals of the following problems. problem. [15 pts for each] Verify that the dual of the dual is the original (a) Minimize Z = 3x1-7x2 + 6x4 + 5x5 -X6, subject to 3x1 1x2 4x3 1x4 5x5 + 9x6-265 5 s xj s 20 for all j

Answer 1

Maximize Z =-17x2 + 38x4-8x5 Subject to x -13% + 45% + 16x5-7x: 107 3x3 18x4 + 30x7 s 81 4X1-5Kg + X6 13 16 X3 xsunrestricted

To convert into nonnegative variables, we use y1-X1 + 10. УГ-X2 + 3.y3-X3-16, xiyi for i-4,5,6,7 Then the given problem becomes Maximize Z =-170"2-3) + 38V4-8y5 Subject to 30% + 16)-1814 + 30y7 81 40%,-10)-50's + 16) + y6--13 -10 S (1-10) s -2 -3 s 2-3) 13 16 S (y316) ysunrestricted 67 2 0

Further simplifying problem becomes Maximize Z-17y2 38y4 -8ys 51 Subject to 33-18V4 + 30% 33 (y2)s 16 0 0 уз ysunrestricted 6 20

By Excel solver we find the solution as below 4 75 0 0 0 0 662 533|<= 0 16 0 13 01<= 107 1071 = 4 16 51 The solution does not converge.