Question

1. TopGarment is a boutique store specializing in high-end female apparel. The store must decide on the quantity of silk scarfs to order from China for the upcoming holiday selling season, which will last 14 weeks. The unit cost of each scarf is c=$40, and the scarf is sold at p=$150. A local discount store agrees to buy any leftover scarfs at the end of the season for s=$30 each. The store manager forecasts the weekly demand to be normally distributed with a mean D=20 and a standard deviation of σD= 15.

a. What is the optimal ordering quantity for TopGarment if it has to stock all the inventory before the selling season starts? What is the optimal expected profit?

b. TopGarment worries that silk scarfs may not be appreciated by local customers. After negotiating with the supplier, the supplier commits on a lead time of 6 weeks. This allows TopGarment to place two orders, one at least 6 weeks before the season starts and another one at the end of week 1 after observing the sales of the first week. Thus, inventory ordered from the 2nd order will arrive right before week 8 and can fulfill the demand for week 8 to week 14.

(i) How much inventory should the store manager order for the 1st order? Note that the 1 st order only needs to cover the demand in weeks 1-7.

(ii) Due to the recent overwhelming workload, the store manager forgot to update her demand forecast by the end of week 1 and had to decide on the 2 nd order quantity based on her old forecast. How much inventory would she order? What is the total expected profit of the entire selling season? (You can assume that unmet demand in the first 7 weeks is lost, but leftover inventory from the 1st order can be carried over for sales in weeks 8-14.)

(iii) Fortunately, the product manager paid enough attention and did another forecast based on the sales data of week 1. She found that the demand for the product had less uncertainty than what the store manager initially thought. The new forecasted standard deviation of the weekly demand drops to 3, whereas the mean weekly demand stays at 20. The product manager reported the new forecasts to the store manager. Now, how much inventory should the store manager order for the 2nd order? What is the total expected profit of the entire selling season? (Again, unmet demand in the first 7 weeks is lost and leftover inventory from the 1st order will be carried over.)

Answer 1

Cost of under stocking= 150-40=110
Cost of overstocking= 40-30=10
Probablity = Cu/(Cu+Co) = 110/(110+10) =110/120 = 0.9166



z values comes out to be 1.38
So 1.38= (x-Mean)/Sigma = (x-20)/15

Hence = 1.38*15 +20 = 40.7
There fore order quantity =41