1 Consider a perishable product whose daily demand can range from a low of 3 1 t
ID: 3206721 • Letter: 1
Question
1
Consider a perishable product whose daily
demand can range from a low of 3
1 to a high of 5
0.
Demand is uniformly distributed over this range, so that sales of the pro
duct are equa
lly likely to
be anywhere from 3
1 to 5
0. Unsold units at the end of the day must be discarded.
a)
The retailer selling the product wishes to maintain an 85% service level (i.e., probability of
fulfilling demand should be at least 0.85).
How
many units should
the retailer order each day?
b
)
What is the minimum number of additional units that the retailer should order each day if the
desired servi
ce level is increased to 95
%?
c
) If the service level is 85
%, what is the average numbers of un
its
that are unsold each day?
d
)
If the
retailer sells each unit for $10
, what is the average daily revenue
if the
service level is
85
%?
e)
The retailer purchases each unit at a cost of $7. If the retailer is interested in maximizing
expected profit, what is
the optimal order quantity?
f
)
If the retailer were able to lower the unit purchase cost by 10%, what would be the
corresponding percentage increase in profit?
g
) The retailer sells a second products with similar price and purchase cost, but whose demand
is
uniformly distributed over the range of 36 to 45 (i.e., the mean of the two products are the same
but their variances are different). If $10 is the selling price and $7 is the purchase cost, which
product is more pr
ofitable (has a higher expected profit
)? Compare the optimal order quantities
for the two products.
h
)
Using Excel,
generate the following plots for the two products discussed in
g
)
:
i) Expected profit
versus order quantity (
i.e.,
vary
Q
from 31 to 50 and plot the corresponding
expected cost)
for both products on the same figure
,
ii) Expected revenue versus order quantity (i.e., vary
Q
from 31 to 50 and plot the corresponding
expected cost) for both products on the same figure,
iii
)
Optimal order quantity versus the purchase cost (
i.e.,
v
ary the purchase costs from 0 to 10
and plot the corresponding
optimal order quantity
).
Discuss
the results observed.
1) Consider a perishable product whose daily demand can range from a low of 31 to a high of 50 Demand is uniformly distributed over this range, so that sales of the product are equally likely to be anywhere from 31 to 50. Unsold units at the end of the day must be discarded aO The retailer selling the product wishes to maintain an 85% service level (i.e., probability of fulfilling demand should be at least 0.85). How many units should the retailer order each day? b) What is the minimum number of additional units that the retailer should order each day if the desired service level is increased to 95%? c) the service level is 85%, what is the average numbers of units that are unsold each day? d) If the retailer sells each unit for $10, what is the average daily revenue if the service level is 85%? e) The retailer purchases each unit at a cost of $7. If the retailer is interested in maximizing expected profit, what is the optimal order quantity? If the retailer were able to lower the unit purchase cost by 10%, what would be the corresponding percentage increase in profit? g) The retailer sells a second products with similar price and purchase cost, but whose demand is uniformly distributed over the range of 36 to 45 (i e., the mean of the two products are the same but their variances are different). $10 is the selling price and $7 is the purchase cost, which product is more profitable (has a higher expected profit)? Compare the optimal order quantities for the two products h Using Excel, generate the following plots forthe two products discussed in g) i) Expected profit versus order quantity (i.e., vary from 31 to 50 and plot the corresponding expected cost for both products on the same figure ii) Expected revenue versus order quantity (i.e., vary from 31 to 50 and plot the corresponding expected cost) for both products on the same figure iii) Optimal order quantity versus the purchase cost (i.e., vary the purchase costs from 0 to 10 and plot the corresponding optimal order quantity) Discuss the results observedExplanation / Answer
Solution
Demand, say D, is equally likely between 31 and 50 =>
P(D = 31) = P(D = 32) = ….. = P(D = 50) = 1/(50 - 30) = 1/20……………. (1)
Also, P(D N) = (N - 30)/20 ……………. …………………………………(2)
Part (a)
Service level = 85% => P(Demand is fulfilled) = 0.85.
If order quantity per day is Q, then the above condition => P(D Q) = 0.85 = 17/20 ….. (3)
From (2) and (3), (Q - 30) = 17 or Q = 47 ANSWER
Part (b)
Service level = 95% => P(Demand is fulfilled) = 0.95.
If order quantity per day is Q, then the above condition => P(D Q) = 0.95 = 19/20 ….. (4)
From (2) and (4), (Q - 30) = 19 or Q = 49 ANSWER
Part (c)
At service level = 85%, Q = 47. Daily sales can be any number between 30 and 47 each with the same probability of 1/20. Average number sold = sum(number sold x probability)
= (31)(1/20) + (32)(1/20) + (33)(1/20) + …… + (47)(1/20) = (1/20)(31+ 32 + 33 + …. + 47)
= (1/20)663 = 663/20 = 33.15 ~ 33 ANSWER
Part (d)
Average daily sales revenue at 85% service level = (Average daily sales quantity) x price per unit
= 33.15[from (c) above] x 10 [given] = $331.5 ANSWER
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