Problem 3: The Outdoor Furniture Corporation manufactures two products, benches

Question

Problem 3: The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: labor and redwood. A (defined as x1) and how many picnic tables (defined as x2) to produce in the next production cycle so as to obtain the largest possible profit. The printout below shows the optimal solution and sensitivity reports for this LP obtained from Excel/Solver linear program has been formulated to solve the problem of how many benches Microsoft ExcelAnswer Report Target Cell (Max Cell Name Original Value Final Value SD$7 Total Profit 2575 Adjustable Cells Cell Name Original Value Final Value SB$5 SC$5 175 50 Constraints Name Cell Value Formu Status SD$10 SD$11 Labor Used (hours Redwood Used (feet) 1000 SDS18-sFS10 3500 SD$11-$F$11 Bindin Binding 0 Microsoft Excel Sensitivity Report Adjustable Cells Final Value Reduced Cost Objective Coefficient Allowable Allowable Name Decrease 3.285714286 Cell Increase 4.333333333 175 50 9 20 SC$5 0 11.5 6.5 Constraints Final Value Shadow Price Constraint R.H. Side Allowable Increase Allowable Decrease Cell D$10 Labor U Name 1000 3500 1.4375 0.325 1000 3500 400 2333.333333 400 1000 urs SD$11 Redwood Used (feet) Use the information from these printouts to answer questions below. Be sure to explain/justify your answers to receive full credit a). How many benches and how many picnic tables should be produced to maximize profit? What is the maximum profit? Explain/justify your response. [5 pts] Note: Consider the changes indicated in parts b, c, and d independently b). Suppose that it is possible to increase the amount of labor available by 150 hours. By how much would this increase the total profit? Explain/justify your response. [5 pts] c). If the profit for a bench increases to $10 per unit, what will the optimal value of x1 and x2 be? What will the objective function value be? Explain/justify your response. [5 pts] d). The production manager has just learned that the supply of redwood is only 2400 feet for the next production cycle, due to some quality problems. What is the effect of this change? What will the new optimal solution and objective function value be? Explain/justify your response. [5 pts]

Explanation / Answer

a) Refer final value in the sensitivity report

Number of benches (x1) = 175

Number of picnic table (x2) = 50

Maximum profit = 175*9+50*20 = 2575

b) Allowable increase for labor used is 400

Shadow price of labor used is 1.4375  

Therefore, 150 hours of increase in labor capacity is within the allowable increase limit.

Increase in total profit = 150*1.4375 = 215.63

c) Allowable increase for bench(x1) is 4.333.

So increase from 9 to 10 is 1, which is within allowable increase limit.

Therefore there will be no change in optimal solution

New objective value = 175*10+50*20 = 2750

d) This is reduction from 3500 to 2400 = 1100, it is more than allowable decrease (which is 1000) for Redwood used. Therefore, current basis will not remain optimal.

Shadow price is not valid for this change. Therefore, new optimal solution and objective value cannot be determined from the sensitivity report. It requires the problem to be solved again. For that we need data about how much labor and redwood each items required.

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