Question

Given the following output, what would happen if the coefficient of X2 increased by 1? LINEAR PROGRAMMING PROBLEM MAX 31X1+35X2+32X3 S.T. 1) 3X1+5X2+2X3>90 2) 6X1+7X2+8X3<150 3) 5X1+3X2+3X3<120 OPTIMAL SOLUTION Objective Function Value = 763.333 Variable Value Reduced Cost X1 13.333 0.000 X2 10.000 0.000 X3 0.000 10.889 Constraint Slack/Surplus Dual Price 1 0.000 −0.778 2 0.000 5.556 3 23.333 0.000 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit X1 30.000 31.000 No Upper Limit X2 No Lower Limit 35.000 36.167 X3 No Lower Limit 32.000 42.889 RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 1 77.647 90.000 107.143 2 126.000 150.000 163.125 3 96.667 120.000 No Upper Limit

Answer 1

Increasing the coefficient of X2 by 1 while remaining within the allowable increase will likely raise the objective function value without changing the optimal solution set, given X2's positive contribution.

Step-by-step explanation:

If the coefficient of X2 in the linear programming problem were to increase by 1, the objective function would change from MAX 31X1+35X2+32X3 to MAX 31X1+36X2+32X3.

Since the optimal solution indicates that the upper limit for the coefficient of X2 can go up to 36.167 without altering the current basis (solution set), increasing the coefficient of X2 by 1 would still fit within this range, and it would likely result in an increase in the objective function value, assuming that the variable X2 remains positive in the optimal solution.

This is because each unit of X2 would contribute more to the objective function value.

However, we must note that the change in the coefficient could potentially alter the optimal solution if the new solution set becomes more profitable, leading to a new optimal mix of X1, X2, and X3.