MATH 251 ACTIVITY 10:
WHY:
Inventory Management – Perishable Goods, Uncertain Demand
The single-period, uncertain demand case extends many of the basic principles of inventory control and of operations
research generally into a different situation. It also provides an opportunity to recall the use of expected value in
decision-making.
LEARNING OBJECTIVES:
1.
Work as a team, using the team roles.
2.
Understand and be able to apply the single – period, uncertain demand inventory model.
CRITERIA:
1.
Success in completing the exercises.
2.
Success in working as a team
RESOURCES:
1.
Your class notes from Tuesday and today
2.
Your text - Section 8.8
3.
Microsoft Excel and the “inventory.xls” template from Public>Courses>Math>Math251 or Blackboard
4
40 minutes
PLAN:
1.
Select roles, if you have not already done so, and decide how you will carry out steps 2 - 4
2.
Work Read the discussion and work through the models given here
3.
Work through the exercises given here - be sure everyone understands all results
4.
Assess the team's work and roles performances and prepare the Reflector's and Recorder's reports including team grade.
DISCUSSION:
The formula for service level given in the text applies to any probability distribution for X = demand level, not only the
normal and the discrete uniform. All that is necessary to use it is knowledge of the data of the situation, including the
probability distribution of the demand function. The main components of the decision process are
1.) Bayes’ decision rule: Under conditions of uncertainty, select the alternative that gives the best expected value when the
whole range of possible situations is considered. [Yes – this is the Bayes of Bayes’ theorem on conditional probabilities]
2.) Marginal analysis: The process of making decisions of “what level to work/order/spend “ based on the marginal cost (or
profit – but here we’ll use cost) – the change in cost for increasing effort/order size/spending. The general rule – we obtain the
lowest cost by looking at the point at which our marginal cost changes from negative to positive [Calculus students note
“marginal cost” is a mask worn by the derivative (rate of change) when it wishes to sneak into economic discussions].
Thus we look at the Per-unit cost of over-ordering Co = c – s and the per-unit cost of under-ordering Cu = p – c + g [These
are marginal costs – only one applies at a time, since we can’t under-order and over-order on the same day]. If the demand
level is D (text uses x), then the cost of ordering Q units is (Q-D) Co (if D ≤ Q - we order correctly or over-order) or (D-Q)
Cu (if D > Q – we under-order). Bayes’ rule says we should set q at the level Q that makes the expected cost E(Q) lowest.
Focusing on the discrete case (both D & Q are integers) we can calculate the marginal value of the expected value:

E(Q  1)  E(Q)  P(D  Q)Co  P(D  Q)Cu  P(D  Q)Co  1 P(D  Q)Cu  (Co  Cu )P(D  Q)  Cu
Cu
This will be negative (so increasing Q decreases cost) whenever P(D  Q) 
and positive whenever
C u  Co
Cu
Cu
, so we get the least cost when Q is the largest value for which P(D  Q) 
P(D  Q) 
C u  Co
C u  Co

Since P(D≤Q) is the probability that all demand is satisfied (called the service level ), the fraction is referred to as the optimal
service level. Replacing C0 and Cu with their respective formulas, we obtain the formula on p. 483 of the text:
The optimal service level is L 
pc  g
p s g
You can automate this calculation by using either the “single period (uniformº” or “single period(normal) sheets in the
Inventory.xls template. The calculation of Q* applies only if demand fits the appropriate distribution – but the “Optimal
service level” output is correct regardless of the distribution.

MODEL:
Harvey’s’ Newsstand sells “What’s New at ND”. It costs Harvey’s $1.50 for each copy, and they sell the paper for $2.50.
Unsold papers can be returned to the publisher for a credit of $.50, and the manager believes that customer goodwill (if
customers who want the paper can’t get it) costs about $25 for each disappointed customer. The demand each day is 9, 10, or
11 papers, with probability .3, .4, .3, respectively. We would like to decide the best quantity to order.
From the data given, we see that the unit cost of underordering Cu is the cost of the lost sale ($2.50 - !.50 = 1.00} plus the
loss of goodwill ($.25) for a total $1.25. The unit cost of overordering Co is the difference between the purchase cost and the
salvage cost – so Co = $1.50 - .50 = $1.00.
The expected value from ordering 10 papers per day is P(D=9)(10-9)($1.00) + P(D = 10)(0) + P(D=11)(11-10)($1.25) =
$.675 and we can calculate expected values for other plans similarly – with so few options, we could use Bayes’ decision rule
directly.
On the other hand, we can immediately calculate the optimal service level
L
2.50 1.50  .25
 .555 - Harvey’s
2.50  .50  .25
optimal service level is 55.5% (order enough papers that demand is satisfied 55.5% of the time – use of either of the single –
period templates confirms this). This is achieved by ordering 10 papers a day (demand is 10 or fewer papers per day 70% of
the time, but for 9 or fewer only 30% of the time).
EXERCISES:
1.

Jennifer’s Donut House serves a large variety of doughnuts, including a specialty – a large-size, blueberry-filled chocolatecovered doughnut with sprinkles (intended to be shared by a family). Preparation must begin by 4:00 am, so a decision must
be made ahead of time on the number of these doughnuts to make. The cost for materials and labor is $1 per doughnut, and
the sale price is $3. Any doughnuts not sold during the day are sold to a local discount grocery store for $.25 each. Over time,
Jennifer’s has developed the following distribution of number of doughnuts requested per day (at the shop):
# doughnuts
0
1
2
3
4
5
% of days
10
15
20
30
15
10
a.) What are the unit cost of underordering (Cu) and the unit cost of overordering (Co)?
b.) what is the expected cost (it might be negative, indicating a profit) of a policy of making 5 of these doughnuts each day?
c.) What is the optimal service level?
d.) What is the optimal number of doughnuts to make per day?
e.) Some families make a special trip to the shop just for this special doughnut, so the manager believes there is a goodwill
cost, which he estimates at $.75 per unmet request. Does including this cost change the answers for c.) and d.)?
2.
Text – exercise #12 for chapter 8
CRITICAL THINKING QUESTIONS:(answer individually in your journal)
1.
Which of the important values in the model (Co, Cu, optimal service level, optimal order quantity) will increase if the selling
price of the item increases? What if the salvage value increases? Do these changes seem reasonable, given the situation we are
modeling?
2.
How difficult might it be to combine this model with a quantity-discount pricing structure (for the goods being purchased –
affecting the value c]
SKILL EXERCISES:(This is the assignment due next class)
Text: p.492 # 6, 25, 26 (goodwill cost is negative), 34, 44 (note the salvage value is negative)