We are given the following linear programming problem: Mallory furniture buys 2

Question

We are given the following linear programming problem:
Mallory furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $200.

The linear programming formulation is
Max 300B + 200M
Subject to
500B + 300 M < 75000
100B + 90M < 18000
B, M > 0

I have solved the problem by using QM for Windows and the output is given below.

The Original Problem w/answers:
B M RHS Dual
Maximize 300 200
Cost Constraint 500 300 <= 75,000 .4667
Storage Space Constraint 100 90 <= 18,000 .6667
Solution-> 90 100 Optimal Z-> 47,000

Ranging Result:
Variable Value Reduced Cost Original Val Lower Bound Upper Bound
B 90. 0 300. 222.22 333.33
M 100. 0 200. 180. 270.

Constraint Dual Value Slack/Surplus Original Val Lower Bound Upper Bound
Cost Constraint 0.4667 0 75000 60000 90000
Storage Space Constraint 0.6667 0 18000 15000 22500




2. Find the range of optimality for the profit contribution of a big shelf from the output given above and interpret its meaning.

Explanation / Answer

To find the endpoints of the range of optimality, look under the ranging result: there is a lower bound (222.22) and an upper bound (333.33) for B, the big shelf variable. Then the range of optimality for profit contribution per big shelf is 222.22 to 333.33 (dollars). [Note that the profit per big shelf is currently $300, which is within the range of optimality]. Our interpretation: The range of optimality always refers to a range of values for an objective function coefficient; in this case, the objective function is the function we are trying to maximize [300B + 200M, the profit function], and the coefficient in question is the number in front of the 'B' variable (big shelves). The range of optimality for an objective function coefficient is the set of values for which the optimal solution stays the same. In this case, if we changed the profit contribution per big shelf to $220, or to $330, or to $310, or to any other value within the range (222.22, 333.33), we would get the same solution point for the problem, namely B = 90, M = 100. [Note: The total profit, of course, would change if we changed the profit contribution per big shelf]. Finally, if we change the profit contribution per big shelf to some number outside of our range of optimality, we will get a new solution point.

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