Consider the LP problem Maximize Z = 6x1 + x2 + 2x3 Let s1,s2, s3 denote the sla

Question

Consider the LP problem

Maximize Z = 6x1 + x2 + 2x3 Let s1,s2, s3 denote the slack variables for the respective constraints. After applying the simplex method, the final tableau reads, in part, as follows: Find the matrix D. Find the shadow price vector cbvB-1. Use the basic representation of the entries of the final tableau as matrix products to complete the final tableau. (Note: if you want to run the simplex algorithm to check your work, that's fine, but the point of this problem is to complete the table using the information you have in the problem statement.) Find the dual problem to this LP. and write out the optimal solution for the dual.

Explanation / Answer

a) B = [1 1 2 ; -2 0 4 ; 1 0 -1]

b) cBV = [2 0 2]

B^(-1) = [0 0.5 2 ; 1 -1.5 -4 ; 0 0.5 1 ]

cBV * B^(-1) = [0 2 6]

c) in final tableau = [-1 0 0 ; 2 4 0 ; 4 4 1 ] ---- corresponding to s2, x3 & x1

d) dual problem

minimize ( 2y1 + 3y2 + y3)

s.t.

2y1 - 4y2 - y3 >= 6

2y1 - 2y2 - 2y3 >= 1

y1 - 3y2 + y3 >= 4

y1, y2, y3 >= 0

optimal solution

y1= 10/3

y3= 2/3

y2 is unbounded

hence optimal value = 22/3

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