Question

Fashionables is a franchisee of The Limited, the well-known retailer of fashionable clothing. Prior to the winter season, The Limited offers Fashionables the choice of five different colors of a particular sweater design. The sweaters are knit overseas by hand, and because of the lead times involved, Fashionables will need to order its assortment in advance of the selling season. As per the contracting terms offered by The Limited, Fashionables also will not be able to cancel, modify, or reorder sweaters during the selling season. Demand for each color during the season is normally distributed with a mean of 620 and a standard deviation of 250. Further, you may assume that the demands for each sweater are independent of those for a different color.

The Limited offers the sweaters to Fashionables at the wholesale price of $40 per sweater and Fashionables plans to sell each sweater at the retail price of $70 per unit. The Limited delivers orders placed by Fashionables in truckloads at a cost of $2,000 per truckload. The transportation cost of $2,000 is borne by Fashionables. Assume unless otherwise specified that all the sweaters ordered by Fashionables will fit into one truckload. Also assume that all other associated costs, such as unpacking and handling, are negligible.

The Limited does not accept any returns of unsold inventory. However, Fashionables can sell all of the unsold sweaters at the end of the season at the fire-sale price of $20 each.

a. How many units of each sweater type should Fashionables order to maximize its expected profit? (Round your answer to the nearest whole number.)

For parts c, assume Fashionables orders 820 of each sweater.

c. What is Fashionables’ expected profit? (Round your answer to one decimal place.)

e. Now suppose The Limited announces a truckload capacity of 3,100 total units of sweaters. If Fashionables orders more than 3,100 sweaters in total, it will have to pay for two truckloads. What now is Fashionables’ optimal order quantity for each type of sweater? (Round your answer to the nearest whole number.)

Answer 1

Answer:

Where and are the cost per unit of demand underestimated and that of demand overestimated respectively.

a) Cu = selling price - purchase cost = 70-40 =30

Cv = purchase cost - salvage value = 40-20 =20.

So, Critical ratio (CR) =Cu/(Cv+Cu) = 30/50 = 0.6

For optimal order quantity, Q, the distribution function, F(Q) must be equal to the critical ratio.

S0, F(Q) = 0.60.

Lets calculate Q through Excel Formula:

Q= NORMINV(CR,Mean,Standard Deviation) =NORMINV(0.6,620.250) =683.34

After roundoff 683.34 will be 683.

Hence,Optimum order quantity to maximize its profit will be 683.

c)

With the given production quantity,Lerts calculate the left-over units suitable for discount.

Order quantity, Q = 820

Standard normal Variable (z) = Q-D/σ = 820- 620/250 = 0.8

From the chart ,Loss function of Z (0.8) will be 0.120.

L(0.8) = 0.120

So, Expected loss sales = 0.120 * 250 = 30.

So,Expected loss S(Q) = D- Expected loss sales = 620-30 = 590

Expected leftover V(Q) = 820-590 = 230

Expected profit = Cu*S(Q)-Cv*V(Q) = 30*590 - 20*230 = 17700-4600 =$13100

Fashionables’ expected profit will be $13100.

e)Lets compare both the scenario :single truckload for order quantity of 620 and the optimal order quantity for two truckload

Q = 683.(as calculated in question a)

Standard normal Variable (z) = Q-D/σ = 683- 620/250 = 0.252

From the chart ,Loss function of Z (0.252) will be 0.286.

L(0.8) = 0.286

So, Expected loss sales = 0.286 * 250 = 71.5 = 71(Round off)

So,Expected loss S(Q) = D- Expected loss sales = 620-71 = 549

Expected leftover V(Q) = 683-549 = 134

Expected profit = Cu*S(Q)-Cv*V(Q) = 30*549 - 20*134 = $13790

Cost of two truckloads (as the order quantity is more than 500) is 2 x 2000 = $4000.

Hence, the expected net profit =   5 *13790 - 4000 = 64950 -(A)

For the order quantity, Q = 620

After , Calculating expected net profit like above procedures ,

tandard normal Variable (z) = Q-D/σ = 620- 620/250 = 0

From the chart ,Loss function of Z (0.252) will be 0.286.

L(0) = 0.40

So, Expected loss sales = 0.40* 250 = 100

So,Expected loss S(Q) = D- Expected loss sales = 620-100 = 520

Expected leftover V(Q) = 620-520 = 100

Expected profit = Cu*S(Q)-Cv*V(Q) = 30*520 - 20*100 = $13600

Expected net profit on one truckload = 5*13600-2000 = 66000 ---(B)

After Comparing equation (A) and (B), it is efficient to apply 620 order size and single truckload instead of the optimal order size and two truckloads.

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