Solve the following linear programs using the simplex method. If the problem is

Question

Solve the following linear programs using the simplex method. If the problem is two dimensional, graph the feasible region, and outline the progress of the algorithm. Apply the simplex method, but do not use the simplex tableau to solve this problem Minimize z=-5x1-7x2-12x3 + x4 subject to 2xl 3x2+2x3 +x4 38 3x1 + 2x2 4x3 x455 x1, x2, x3, x420

Explanation / Answer

X1 X2 X3 X4 RHS Equation form Minimize -5 -7 -12 1 Min - 5X1 - 7X2 - 12X3 + X4 1 2 3 2 1 = 0 5 0 0 1 0 >= 0 X3 >= 0 6 0 0 0 1 >= 0 X4 >= 0 X1 X2 X3 X4 RHS Solution 0 0 15.5 7 -179 Ans: x1,x2=0 x3=15.5, x4=7 and Z= -179 artfcl=artficial variable Iterations: Cj Basic Variables Quantity -5 X1 -7 X2 -12 X3 1 X4 0 slack 1 0 slack 2 0 artfcl 3 0 surplus 3 0 artfcl 4 0 surplus 4 0 artfcl 5 0 surplus 5 0 artfcl 6 0 surplus 6 Phase 1 - Iteration 1 0 slack 1 38 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 slack 2 55 3 2 4 -1 0 1 0 0 0 0 0 0 0 0 1 artfcl 3 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 artfcl 4 0 0 1 0 0 0 0 0 0 1 -1 0 0 0 0 1 artfcl 5 0 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 1 artfcl 6 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 zj 0 -1 -1 -1 -1 0 0 1 1 1 1 1 1 1 1 cj-zj 1 1 1 1 0 0 0 -1 0 -1 0 -1 0 -1 Iteration 2 0 slack 1 38 0 3 2 1 1 0 -2 2 0 0 0 0 0 0 0 slack 2 55 0 2 4 -1 0 1 -3 3 0 0 0 0 0 0 0 X1 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 artfcl 4 0 0 1 0 0 0 0 0 0 1 -1 0 0 0 0 1 artfcl 5 0 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 1 artfcl 6 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 zj 0 0 -1 -1 -1 0 0 2 0 1 1 1 1 1 1 cj-zj 0 1 1 1 0 0 -1 0 0 -1 0 -1 0 -1 Iteration 3 0 slack 1 38 0 0 2 1 1 0 -2 2 -3 3 0 0 0 0 0 slack 2 55 0 0 4 -1 0 1 -3 3 -2 2 0 0 0 0 0 X1 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 X2 0 0 1 0 0 0 0 0 0 1 -1 0 0 0 0 1 artfcl 5 0 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 1 artfcl 6 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 zj 0 0 0 -1 -1 0 0 2 0 2 0 1 1 1 1 cj-zj 0 0 1 1 0 0 -1 0 -1 0 0 -1 0 -1 Iteration 4 0 slack 1 38 0 0 0 1 1 0 -2 2 -3 3 -2 2 0 0 0 slack 2 55 0 0 0 -1 0 1 -3 3 -2 2 -4 4 0 0 0 X1 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 X2 0 0 1 0 0 0 0 0 0 1 -1 0 0 0 0 0 X3 0 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 1 artfcl 6 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 zj 0 0 0 0 -1 0 0 2 0 2 0 2 0 1 1 cj-zj 0 0 0 1 0 0 -1 0 -1 0 -1 0 0 -1 Iteration 5 0 slack 1 38 0 0 0 0 1 0 -2 2 -3 3 -2 2 -1 1 0 slack 2 55 0 0 0 0 0 1 -3 3 -2 2 -4 4 1 -1 0 X1 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 X2 0 0 1 0 0 0 0 0 0 1 -1 0 0 0 0 0 X3 0 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 0 X4 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 zj 0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 cj-zj 0 0 0 0 0 0 -1 0 -1 0 -1 0 -1 0 Phase 2 0 slack 1 38 0 0 0 0 1 0 -2 2 -3 3 -2 2 -1 1 0 slack 2 55 0 0 0 0 0 1 -3 3 -2 2 -4 4 1 -1 -5 X1 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 -7 X2 0 0 1 0 0 0 0 0 0 1 -1 0 0 0 0 -12 X3 0 0 0 1 0 0 0 0 0 0 0 1 -1 0 0 1 X4 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 zj 0 -5 -7 -12 1 0 0 5 -5 7 -7 12 -12 -1 1 cj-zj 0 0 0 0 0 0 -5 5 -7 7 -12 12 1 -1 Iteration 7 0 slack 1 10.5 0 0 0 0 1 -0.5 -0.5 0.5 -2 2 0 0 -1.5 1.5 0 surplus 5 13.75 0 0 0 0 0 0.25 -0.75 0.75 -0.5 0.5 -1 1 0.25 -0.25 -5 X1 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 -7 X2 0 0 1 0 0 0 0 0 0 1 -1 0 0 0 0 -12 X3 13.75 0 0 1 0 0 0.25 -0.75 0.75 -0.5 0.5 0 0 0.25 -0.25 1 X4 0 0 0 0 1 0 0 0 0 0 0 0 0 1 -1 zj 165 -5 -7 -12 1 0 3 -4 4 1 -1 0 0 2 -2 cj-zj 0 0 0 0 0 -3 4 -4 -1 1 0 0 -2 2 Iteration 8 0 surplus 6 7 0 0 0 0 0.6667 -0.3333 -0.3333 0.3333 -1.3333 1.3333 0 0 -1 1 0 surplus 5 15.5 0 0 0 0 0.1667 0.1667 -0.8333 0.8333 -0.8333 0.8333 -1 1 0 0 -5 X1 0 1 0 0 0 0 0 1 -1 0 0 0 0 0 0 -7 X2 0 0 1 0 0 0 0 0 0 1 -1 0 0 0 0 -12 X3 15.5 0 0 1 0 0.1667 0.1667 -0.8333 0.8333 -0.8333 0.8333 0 0 0 0 1 X4 7 0 0 0 1 0.6667 -0.3333 -0.3333 0.3333 -1.3333 1.3333 0 0 0 0 zj 179 -5 -7 -12 1 1.33 2.33 -4.67 4.67 -1.67 1.67 0 0 0 0 cj-zj 0 0 0 0 -1.3333 -2.3333 4.6667 -4.6667 1.6667 -1.6667 0 0 0 0

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