Question

A retail outlet sells a perishable product for $12 per unit. The cost of the product is $6 per unit. The product not sold has the salvage value of $1.

a) Compute the overage cost and underage cost per unit.

b) If the demand follows the normal distribution with the average of 400, and the standard deviation of 100. What is the optimal order quantity to maximize the expected profit?

c) What is the optimal ordering quantity if the retailer uses the following discrete distribution for demand?

quantity 200 250 300 350 400 450 500 550 600

probability 0.05 0.05 0.15 0.15 0.20 0.15 0.15 0.05 0.05

d) What is the profit or loss for the quantity obtained from c)?

Answer 1

a.
Overage cost = incremental cost per unit for items not sold = cost - salvage price = $6 - $1 = $5
Underage cost = incremental cost per unit for not meeting demand of items = price of item - cost of item = $12 - $6 = $6
b. Normal distribution of Demand

Average of demand ()= 400
Standard deviation () = 100
Optimal order quantity (EOQ) = + (Z*)() where Z* = Normsinv((Underage cost)/(Underage cost+overage cost))
Note, normsinv is an excel function to calculate Z*, alternatively you can use statistical tables to find value of Z*

So, here Z* = NORMSINV(5/11) = -0.1142
So, EOQ = 400 + (-0.1142)(100) = 388.581 i.e. 389 (as items have to be ordered in integer value)
c. Discrete distribution of Demand

Demand	Probability	Cumulative Probability of demand less than this value (F(x))	Cumulative Probability of demand less than this value(1-F(x))
200	0.05	0.05	0.95
250	0.05	0.1	0.9
300	0.15	0.25	0.75
350	0.15	0.4	0.6
400	0.2	0.6	0.4
450	0.15	0.75	0.25
500	0.15	0.9	0.1
550	0.05	0.95	0.05
600	0.05	1	0
Probability we are able to sell Q +1 unit is given by critical limit i.e. (Overage cost/(Overage cost + Underage cost))
Pr(Sell Q +1st unit) = 5/11 i.e. 0.45
We compare this value with the Cumulative probability of demand being less than this value
We see that .45 comes between .6 and .4 I.e. the optimal order quantity will be 400

d. Profit/ Loss for above quantity

Here we need to take cases for all the demands given and then calculate the expected profit for Q = 400

Demand	Probability	Overage (400 - D)	Revenue Price*D + Overage*Salvage value	Expected revenue Probability * Revenue
200	0.05	200	$                                                       2,600	$                                   130
250	0.05	150	$                                                       3,150	$                                   158
300	0.15	100	$                                                       3,700	$                                   555
350	0.15	50	$                                                       4,250	$                                   638
400	0.2	0	$                                                       4,800	$                                   960
450	0.15	0	$                                                       5,400	$                                   810
500	0.15	0	$                                                       6,000	$                                   900
550	0.05	0	$                                                       6,600	$                                   330
600	0.05	0	$                                                       7,200	$                                   360
			Total expected revenue	$                                4,840
			Total expected profit	Expected revenue - Cost
				= $4840 - $6*400
			Total expected profit	$                                2,440