Question

Sportswear company O'yeah designs and sells wetsuits to the U.S. market. The designs of the wetsuits are updated each year. The process for updating the design typically starts in January the year before the designs are to be released. At this time, the purchasing, design, and sales departments have a two-day meeting to discuss the product portfolio for the upcoming year, including design, functionality, and price. In March, the designs are finalized, and the purchasing department starts negotiating with suppliers across the world. Production usually starts in September or October the year before the designs are to be released, and lasts until December or January depending on the supplier. In February, retailers start placing their orders to O'yeah, with retail sales of the new designs typically starting in early April. The season stretches from April to September. As the season progresses, retailers place replenishment orders to O'yeah, who supplies the retailers from a central warehouse. Sales peak in early summer. Towards the end of the summer, sales decline dramatically. When the season ends in September, O'yeah pushes out any remaining inventory to the retailers by offering all models at only 25% of the original selling price. 1)Consider a specific SKU# 001237 (Model A200, size L). For the upcoming year, O'yeah has negotiated with a supplier to have SKU# 001237 produced and delivered to O'yeah at $175 per unit. SKU# 001237 is sold to retailers at $225, for a unit margin of $50. Suppose total demand for SKU# 001237 for the upcoming season (the time during which the product is sold at full price) is estimated to follow a Normal distribution with mean 2,500 and standard deviation 400. How many A200, size L, should O'yeah produce to maximize expected profit? Please round to closest integer. 2)Last year, SKU# 001237 turned out to be extremely popular and sold out quickly. Since O'yeah makes a good margin on the product, management has decided they do not want the same situation again. "We cannot have the probability for out-of-stock during the season to be more than 5% on this model", said the sales manager recently. How many SKU# 001237 should O'yeah produce to ensure that the stock-out probability is exactly 5% during the season? Assume the same demand distribution as in Part 1. Please round to closest integer. 3)Consider the policy from Part 2. With your suggested production quantity, how many units of SKU# 001237 can O'yeah expect to be short during the regular season? Please answer with two decimals. 4)Consider the policy from Part 2. What is the expected contribution to profit from SKU# 001237? Please round to closest integer dollars. Answer without the dollar symbol. 5)The policy suggested by the sales manager leads to a lower expected profit than the "optimal" policy calculated in Part 1. The sales manager motivates this by claiming that the policy in Part 1 is not taking into account future lost profits by being out of stock. "So", he argues, "the expected profit implied by the 'optimal policy' is biased upwards." Suppose there is an additional cost per unit out-of-stock, ? . Suppose further that the policy suggested in Part 2 is the policy that maximizes expected contribution when this additional cost is taken into account. What is the additional cost ? ? Please round to closest integer dollars. Answer without the dollar symbol Solution for Part 5

Answer 1

(1)

Cu = Cost of understocking = Selling price - Purchase cost = $225 - $175 = $50
Co = Cost of overstocking = Purchase cost - Salvage value = $175 - $225*25% = $118.75

Critical ratio = Cu / (Cu + Co) = 50 / (50+118.75) = 0.296

Z = NORMSINV(0.296) = -0.535

Production quantity (optimal) = 2500 + Z*400 = 2,286 units

(2)

Stock-out probability = 5%; so, in-stock probability = 95%

Z = NORMSINV(0.95) = 1.645

Production quantity = 2500 + Z*400 = 3,158 units

(3)

Normal loss function corresponding to Z=1.645 is L(Z) = 0.021

Expected lost sales = 0.021 x SD = 0.021 x 400 = 8 units

(4)

Q	2286
Z	-0.54
Loss function, L(Z) (use tables)	0.723
Expected lost sales, L(Q) = L(Z) x s	289.2
Expected Sales, S(Q) = D - L(Q)	2210.8
Expcted left over, V(Q) = Q - S(Q)	75.2
Expected profit = Cu*S(Q) - Co*V(Q)	$101,616