Question

Problem 14-27 (Algorithmic) A perishable dairy product is ordered daily at a particular supermarket. The product, which costs $1.2 per unit, sells for $1.64 per unit. If units are unsold at the end of the day, the supplier takes them back at a rebate of $1 per unit. Assume that daily demand is approximately normally distributed with µ = 145 and ? = 30. What is your recommended daily order quantity for the supermarket? If required, round your answer to two decimal places. Q* = What is the probability that the supermarket will sell all the units it orders? If required, round your answer to four decimal places. P(Stockout) = In problems such as these, why would the supplier offer a rebate as high as $$1? For example, why not offer a nominal rebate of, say, 25¢ per unit? What happens to the supermarket order quantity as the rebate is reduced? The higher rebate the quantity that the supermarket should order.

Answer 1

We use the concept of newspaper vendor problem, use the Co and Cu and critical ratio as shown below:

m	mean	145
s	Std Dev	30
C	Cost	1.2
P	Price	1.64
V	Salvage	1	(10+0.4)
			Formula used
Cu	Cost of under order	0.44	(P-C)
Co	Cost of over order	0.2	(C-V)
CR	Critical ratio	0.6875	(Cu/(Cu+Co)
	So actual order	m+Z*s
		159.6633	NORMINV(CR,m,s)
If	Optimal Order is	160	ANs
X	X value	160.00
	P value	0.6915	NORM.DIST(X,m,s,TRUE)
	Probability of sale	0.6915	Ans

If we take non rounded values

prob is

0.6875

The supplier would offer a rebale as high as $1 , so that the supermarket orders more and the terms are convincing to order higher volumes and the dairy is more preferred over other similar partners

If that is now $ 0.25

m	mean	145
s	Std Dev	30
C	Cost	1.2
P	Price	1.64
V	Salvage	0.25	(10+0.4)
			Formula used
Cu	Cost of under order	0.44	(P-C)
Co	Cost of over order	0.95	(C-V)
CR	Critical ratio	0.3165	(Cu/(Cu+Co)
	So actual order	m+Z*s
		130.6787	NORMINV(CR,m,s)
If	Optimal Order is	131

So the order is reduced from 160 to 131, hence a potential loss of 29* (1.2-1) = $ 5.8 per day.

Hence it is preferred to have a higher salvage value offered. as higher the rebate higher is the order and higher is the potential profit.