Fashionables is a franchisee of The UnLimited, the well-known retailer of fashio

Question

Fashionables is a franchisee of The UnLimited, the well-known retailer of fashionable clothing. Prior to the winter season, The UnLimited 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 UnLimited, Fashionables will also 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 500 and a standard deviation of 225. Further, you may assume that the demand for each sweater is independent of the demand for any other color. The UnLimited offers the sweaters to Fashionables at the wholesale price of $45 per sweater, and Fashionables plans to sell each sweater at the retail price of $74 per unit. The UnLimited 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 $25 each If a part of the question specifies whether to use Table 13.4, or to use Excel, then credit for a correct answer wll depend on using the specified method a How many units of each sweater-type should Fashionables order to maximize its expected profit? Use Table 13.4 and round to nearest integer. If Fashionables wishes to ensure a 97.5% in-stock probability, what should its integer b. order quantity be for each type of sweater? Use Table 13.4 and round to nearest Say Fashionables orders 650 of each sweater. What is Fashionables' expected profit? Use Table 13.4 Say Fashionables orders 650 of each sweater. What is the stockout probability for each sweater? Use Excel. d. Round your answer to 4 decimal places.)

Explanation / Answer

Cu = Cost of underage = Selling price - Cost = $74 - $45 = $29
Co = Cost of overage = Cost - Salvage value = $45 - $25 = $20

Critical ratio = Cu / (Co+Cu) = 29 / 49 = 0.5918

The nearest value from the table is 0.5793 and the corresponding Z is 0.2

So, the order size should be 500 + 0.2 x 225 = 545 -------------------(a)

For in-stock probability of 0.975, the corresponding nearest value from the table is 0.9772 and the corresponding Z is 2.0.

So, the order size should be 500 + 2.0 x 225 = 950 -------------------(b)

For order size = 650,

650 = 500 + Z x 225 or, Z = 0.667

The nearest value of Z from the table is Z=0.7 for which F(Z) = 0.758 and L(Z) = 0.8429

Lost sales = L(Q) = Std. deviation of demand x (L(Z) - Z) = 225 x 0.1429 = 32.15
Expected sales = S(Q) = Mean Demand - L(Q) = 467.85
Expected left over inventory = V(Q) = Q - S(Q) = 650 - 467.85 = 182.15

Expected profit = Cu x S(Q) - Co x V(Q) = 29*467.85 - 20*182.15 = 9924.65 (or, 9924.5275 if no intermediate rouding off is done) -----------------(c)

With excel, Z = 2/3

L(Q) = (NORMDIST(2/3,0,1,0)-(2/3)*(1-NORMDIST(2/3,0,1,1)))*225 = 34.00

S(Q) = 466

V(Q) = 184

Expected profit = Cu x S(Q) - Co x V(Q) = 29*466 - 20*184 = 9834 (or, 9833.9059 if no intermediate rouding off is done) -----------------(d)

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