Question

The Fair Lady Hat Company produces three distinct styles of hat. The operations manager has modeled the operations using linear programming. She is using x1, x2, and x3 to represent each hat style. Below is her formulation:

Maximize Z = 285 x1 + 245 x2 + 295 x3

Subject to: Machine A: 2 x1 + 2 x2 + 3 x3 < 240 hours

Machine B: 4 x1 + 1 x2 + 1 x3 < 320 hours

Machine C: 3 x1 + 2 x2 + 4 x3 < 260 hours

x1 > 0; x2 > 0; x3 > 0

a.(6 points) Use Excel and Solver to find the optimal solution for this problem. Attach the printouts of worksheets for the initial and final solution along with the Sensitivity Report (No other worksheets or reports are needed). Write down the answers to the following questions:

1. What is the value of the objective function?

2. What are the optimal values of the product mix?

b. Consider the statements below independently and explain each of your answers, based only on the Sensitivity Report (DO NOT RE-Solve the problem using Solver).

1.(3 points) If the marginal price of Hat 3 were to increase by $85, would she have to make changes to the values of the product mix? Explain.

2. (3 points) Suppose the capacity of Machine B were to be reduced to 260 hours , would that have any impact on total profits? Explain.

3.(3 points) If machine C had a capacity of 230 hours instead of its current value of 260, how would the Total Profit be impacted? Explain.

Answer 1

The screenshots are shown below

Value of objective function = 30077.5

Optimal values of product mix are x1 = 20, x2 = 99.5, x3 = 0

Hat 3 (x3) has an allowable increase of 112.5. This means an increase of 85 will not change the optimal solution or the product mix.

Machine B is not completely utilized. Also the allowable decrease on machine B is 139.5. Thus if the time was reduced by 60 to 260, the optimal solution will not get impacted.

Machine C has a shadow price of 40. This means if the hours were reduced by 30, the profit would reduce by 30*40 = 120.