5. (10 points) The following linear programming problem has no feasible solution

Question

5. (10 points) The following linear programming problem has no feasible solution. min 0x 0x2 s.t. 3x1 + 2x2 2 30 (1) 2x1 + x2 2 40 (2) 5x1 3x2 s 50 (3) 2 20 Assume that we have the penalty cost $100 for failing to satisfy 1 unit of 30 in the 1st constraint; the penalty cost $50 for failing to satisfy 1 unit of 40 in the 2nd constraint; also $10 penalty is assessed for each unit exceeding 50 in the 3rd constraint. . Please add some deviational variables on these three constraints to formulate this problem as a goal programming problem to minimize the penalty cost (just write a linear programming model with explanations on new variables) SIE340 Final Exam- Summer 1 2017 Page 8 of 12

Explanation / Answer

Formulation as Goal Programming model is following

Deviation variables:

d1- be the negative deviation variable (i.e. shortage of 1 unit of 30) in the 1st constraint

d1+ be the positive deviation variable (i.e. excess of 1 unit of 30) in the 1st constraint

d2- be the negative deviation variable (i.e. shortage of 1 unit of 40) in the 2nd constraint

d2+ be the positive deviation variable (i.e. excess of 1 unit of 40) in the 2nd constraint

d3- be the negative deviation variable (i.e. shortage of 1 unit of 50) in the 3rd constraint

d3+ be the positive deviation variable (i.e. excess of 1 unit of 50) in the 3rd constraint

Min 100d1- + 50d2- + 10d3+

s.t.

3x1 + 2x2 + d1- - d1+ = 30

2x1 + x2 + d2- - d2+ = 40

5x1 + 3x2 + d3- - d3+ = 50

x1, x2, d1-, d1+, d2-, d2+, d3-, d3+ >= 0

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