A furniture store chain with 25 stores is stocking a fashionable chair in each of their stores with a normal demand distribution with a mean of 250 and a standard deviation of 125.  The cost of the chair is $750, and it is sold for $1,500. Chairs that are not sold by the end of the season are sold for $250 per chair. If you would like partial credit, please show your work and the assumptions you made.

a)     What are the values of cu and co?                                     

b)    What is the optimal service level for an individual store?                  

c)     What is the Z value?                                                                                      

d)    What is the optimal purchasing quantity for an individual store?          

e)     What is the probability of having some shortage at an individual store?        

f.)     If the chain is purchasing and keeping the chairs centrally (for all 25 stores), what is the demand distribution at the center?              

g)     If the chain is purchasing and keeping the chairs centrally (for all 25 stores), what is the optimal order quantity?                          

h)    What is the probability of having some shortage at the center??    

a) Cost of Understocking = Cost of lost profit on additional unit sold

Cost of Understocking = Selling price - cost per unit

Cost of Understocking = 1500 - 750

Cost of Understocking C_u = $750

Cost of Overstocking = Cost of selling the unsold unit at a lower price

Cost of Overstocking = Cost per unit - reduced selling price

Cost of Overstocking = 750 - 250

Cost of Overstocking C_o = $500

Critical Ratio = Cost of Understocking /(Cost of Understocking + Cost of Overstocking)

Critical Ratio = 750 / (750 + 500)

Critical Ratio = 0.60

b) Optimal service level = Critical Ratio = 60%

c) z = NORM.S.INV(Service Level)

z = NORM.S.INV(0.60)

z = 0.253

d) Optimal Purchasing quantity for individual store = Mean demand + z * Standard deviation of demand

Optimal Purchasing quantity for individual store = 250 + 0.253 * 125

Optimal Purchasing quantity for individual store = 281.63 =~ 282

Optimal Purchasing quantity for individual store = 282 units

e) In-stock probability = NORM.DIST[(Q - Mean)/Standard deviation, 0, 1, 1]

Here, Q = Optimal Purchasing quantity for individual store = 281.63

Stock-out probability = 1 - In-stock probability

Stock-out probability = 1 - NORM.DIST[(Q - Mean)/Standard deviation, 0, 1, 1]

Stock-out probability = 1 - NORM.DIST[(281.63 - 250)/125, 0, 1, 1]

Stock-out probability for individual store = 0.40

f) For central purchasing and holding of chairs for all the 25 stores,

Mean Demand for the central location = Mean demand for each store * Number of stores

Mean Demand for the central location = 250 * 25

Mean Demand for the central location = 6250 units

Standard deviation of demand for the central location = Standard deviation of demand for each store * sqrt(Number of stores)

Standard deviation of demand for the central location = 125 * sqrt(25)

Standard deviation of demand for the central location = 625

g) Optimal Order quantity for individual store = Mean demand for the central location + z * Standard deviation of demand for the central location

Optimal Order quantity for individual store = 6250 + 0.253 * 625

Optimal Order quantity for individual store = 6408.13 =~ 6409

Optimal Order quantity for individual store = 6409 units

(Rounding off to the next higher whole number)

h) In-stock probability = NORM.DIST[(Q - Mean)/Standard deviation, 0, 1, 1]

Here, Q = Optimal ordering quantity for central location = 6408.13

Stock-out probability = 1 - In-stock probability

Stock-out probability = 1 - NORM.DIST[(Q - Mean)/Standard deviation, 0, 1, 1]

Stock-out probability = 1 - NORM.DIST[(6408.13 - 6250)/625, 0, 1, 1]

Stock-out probability for central location = 0.40