The LP formulation given below represents a problem whose solution will tell a s
ID: 352184 • Letter: T
Question
The LP formulation given below represents a problem whose solution will tell a specialty retailer how many of four different styles of umbrellas to stock for the next quarter in order to maximize profit. It is assumed that every umbrella stocked will be sold. The variables represent the number of women's (x1), golf (x2), men's (x3), and folding (x4) umbrellas to be stocked. There exist constraints for storage space availability, budget limitation, demand restriction, and marketing requirement. Max 25x1+ 22x2 + 16x3 18x4 Subject to (storage space availability) (budget limitation) (demand restriction) (marketing requirement) xi 2x2 + x3 + 0.5x4 400 8x 12x2 + 8x3 + 15x4 š 10000 x3 + x4 200 xlt x3 2 100 xl, x2, x3, x4 20 The model was solved with Excel and the following Sensitivity Report was generated. Adjustable Cells Final Reduced Objective Allowable Allowable Cell $B$8Women's x1 $C$8Golf x2 $D$8 Men's x3 $ES8 Folding x4 Name Value Cost Coefficient Increase Decrease 25 200 100 1E+30 5.99999999 -9 16 1E+30 1.5 18 1E+30 Constraints Final Shadow Constraint Allowable Allowable Name Value Price R.H. Side Increase Decrease Cell $B$13 Storage $B$14 Budget $B$15 Demand 400 2800 200 200 400 10000 200 100 1200 1E+30 200 100 200 7200 100 1E+30 B$16 Marketin Using the information in the Sensitivity Report, answer the following questions a. The retailer was disappointed to see that the computer solution did not recommend keeping any of men's umbrellas and believes this would not be a good strategy in the long run. If the retailer decides to stock men's umbrellas, what unit profit would justify it? b. If the retailer can manage to make an additional $1 profit for cach falding umbrella, how would the optimal solution be affected? Explain why c. If 100 additional storage units were available at a cost ofS12/unit, how many additional storage units would you recommend purchasing? Explain why d. Would you recommend the retailer to increase its budget by getting a loan? ExplainExplanation / Answer
a. The reduced cost of men's umbrella is -9 . Which means the unit profit is short of 9, before it beacomes feasible to stock men's umbrellas. Therefore, unit profit that would justify it to stock men's umbrellas is 16+9 = 25
b. The allowable increase of Folding umbrellas is 1.5, which means the optimal solution would remain unchanged, as long as increase in unit profit is less than 1.5 . Therefore, with increase of $ 1 in profit of folding umbrellas, the optimal solution would remain unchanged.
c. Shadow price of storage units is 11, which means each additional storage unit will increase the total profit by $ 11. Hence I would NOT recommend purchasing 100 additional storage units at a cost of $ 12/unit, as this will result in net loss of 11-12 = -1 per unit x 100 units = $ -100
d. Shadow price of budget is 0, which means each additional dollar of budget increase will lead to 0 increase in profit. Hence I would NOT recommend increasing budget by getting a loan.
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