ABC Corporation has a total of 120 stores. All stores are served by a single dis
ID: 369746 • Letter: A
Question
ABC Corporation has a total of 120 stores. All stores are served by a single distribution center. ABC offers a free gift with every purchase. All of the 120 stores offer this deal. When the order must be placed with the factories that produce the gifts, demand (in units of gifts) for each store is forecasted by a Normal distribution (assume that demands are independent at every store location). Right now ABC differentiates the stores in terms of their sales in 3 categories: Small, Large, Largest. The number of stores in each category and the Average and Standard Deviation of demand for gifts are given as follows:
30
Suppose 6 months before, ABC must make an order for each single store. In addition, ABC wants to have an 85% probability of a store being in stock at the end of the season.
Find -
a) Find the order quantity for each store in the each category.
b) What is the implied unit cost of underage if the unit cost of a gift is $0.5?'
c) Suppose ABC decides to make a single order for all 120 stores. The order will be delivered to and stored at the distribution center. It will then be dispatched to each store during the period as requirement. Find the order quantity in this case if ABC wants an 85% probability of no stock out at the warehouse . What is the average number of leftover (unsold) gifts in this case?
d) If ABC ordered the total quantity ordered in (a) but stored the entire order at the distribution center. It then delivered to each store only as needed, then what is the probability of no stock out at the distribution center at the end of the period? What is the average number of leftover (unsold) gifts in this case? Also comment on the policies found in (c) and (d) regarding the tradeoff between service level and average leftover units and cost of leftovers?
Category Number of Stores Average Demand per Store (Gifts) Standard Deviation of Demand at every store (Gifts) Small30
250 75 Large 40 1000 200 Largest 50 2000 600Explanation / Answer
a) z value for 85% probability = NORMSINV(0.85) = 1.0364
Order qantity for each store in Small category = + z =250 + 1.0364*75 = 328
Order qantity for each store in Large category = + z =1000 + 1.0364*200 = 1207
Order qantity for each store in Largest category = + z =2000 + 1.0364*600 = 2622
b) Critical value = Cu/(Cu+Co) = 0.85
Overage cost, Co = 0.5 (cost of a gift)
Solving the critical value equation for Cu
Cu*(1-0.85) = 0.85*Co
Cu = 0.85*0.5/(1-0.85) = 2.83
Implied cost of underage = $ 2.83
c) Total average demand of 120 stores = 30*250+40*1000+50*2000 = 147,500 units
Standard deviation of demand of 120 stores = (30*752 + 40*2002 + 50*6002) = 4446 units
Optimal order quantity for 85% in-stock probability = 147500 + 1.0364*4446 (1.04 is the z value as determined earlier)
Optimal order quantity = 152,108 units
d) Total quantity ordered in part (a), Q = 328*30+1207*40+2622*50 = 189,220 units
z value = (189200 - 147500)/4446 = 9.38
Corresponding to z value as calculated above, in-stock probability (service level) - NORMSDIST(9.38) = 1 or 100%
Therefore, probability of no stockout at the distribution center at the end of the period = 0% (approx)
From standard normal table, corresonding to z=1, Lost sales, L(z) = 0
Expected lost sales, L = *L(z) = 0*4446 = 20
Expected sales, S = µ - L = 147500 - 0 = 147500
Expected leftover (unsold) inventory = Q - S = 189220 - 147500 = 41,720
Cost of leftover inventory = 41720*0.5 = $ 20,860