PROBABILISTIC MODELS

Outline

• Probabilistic inventory models

• Single- and multi- period models
• A single-period model with uniform distribution of
demand
• A single-period model with normal distribution of
demand

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 Probabilistic Inventory Models

• The demand is not known. Demand characteristics

such as mean, standard deviation and the distribution
of demand may be known.
• Stockout cost: The cost associated with a loss of
sales when demand cannot be met.
– For example, if an item is purchased at $1.80 and sold at
$3.00, the loss of profit is $3.00-1.80 = $1.20 for each unit of
demand not fulfilled.

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 Single- and Multi- Period Models

• The classification applies to the probabilistic demand

case
• In a single-period model, the items unsold at the end
of the period is not carried over to the next period.
The unsold items, however, may have some salvage
values.
• In a multi-period model, all the items unsold at the
end of one period are available in the next period.
• In the single-period model and in some of the multi-
period models, there remains only one question to
answer: how much to order.

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 SINGLE-PERIOD (NEWS VENDOR) MODEL

• Imagine a vendor, selling newspapers on the street.

• Each morning, they have one chance to buy
newspapers in bulk from the printer.
• How many copies of today’s paper should the vendor
stock, knowing that unsold copies end up worthless?
• This problem can also occur in:
– Computer that will be obsolete before the next order
– Perishable product
– Seasonal products such as bathing suits, winter coats, etc.

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 Trade-offs in a Single-Period Models

Loss resulting from the items unsold

ML= Purchase price - Salvage value

Profit resulting from the items sold

MP= Selling price - Purchase price

Trade-off
Given costs of overestimating/underestimating demand
and the probabilities of various demand sizes
how many units will be ordered?

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Consider an order quantity Q
Let P = probability of selling all the Q units
= probability (demandQ)

Then, (1-P) = probability of not selling all the Q units

We continue to increase the order size so long as

P( MP)  (1  P ) ML
ML
or , P 
MP  ML

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Decision Rule:

Order maximum quantity Q such that

ML
P
MP  ML

where P = probability (demandQ)

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Text Problem 21, Chapter 15: Demand for cookies:
Demand Probability of Demand
1,800 dozen 0.05
2,000 0.10
2,200 0.20
2,400 0.30
2,600 0.20
2,800 0.10
3,000 0,05
Selling price=$0.69, cost=$0.49, salvage value=$0.29
a. Construct a table showing the profits or losses for each
possible quantity
b. What is the optimal number of cookies to make?
c. Solve the problem by marginal analysis.

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Demand Prob Prob Expected Revenue Revenue Total Cost Profit
(dozen) (Demand) (Selling Number From From Revenue
all the units) Sold Sold Unsold
Items Items
1800 0.05 1 1800 1242.0 0.0 1242 882 360
2000 0.1 0.95 1990 1373.1 2.9 1376 980 396
2200 0.2 0.85 2160 1490.4 11.6 1502 1078 424
2400 0.3 0.65 2290 1580.1 31.9 1612 1176 436
2600 0.2 0.35 2360 1628.4 69.6 1698 1274 424

highest expected profit

2800 0.1 0.15 2390 1649.1 118.9 1768 1372 396
3000 0.05 0.05 2400 1656.0 174.0 1830 1470 360

Sample computation for order quantity = 2200:

Expected number sold=1800(0.05)+2000(0.10)+2200(0.85)
=2160
Revenue from sold items=2160(0.69)=$1490.4
Revenue from unsold items=(2200-2160)(0.29)=$11.6
Total revenue=1490.4+11.6=$1502
Cost=2200(0.49)=$1078
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Profit=1502-1078=$424
Solution by marginal analysis:
MP  .69  .49  $0.20, ML  .49  .29  $0.20
Order maximum quantity, Q such that
ML 0.20
P  Probabilitydemand  Q     0.50
MP  ML 0.20  0.20
Demand, Q Probability(demand) Probability(demandQ), P
1800 0.05 1
2000 0.1 0.95
2200 0.2 0.85
2400 0.3 0.65*
2600 0.2 0.35

* the highest Q with P > 0.50

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 Demand Characteristics

Suppose that the historical sales data shows:

Quantity No. Days sold Quantity No. Days sold

14 1 21 11
15 2 22 9
16 3 23 6
17 6 24 3
18 9 25 2
19 11 26 1
20 12

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Demand Characteristics
Mean = 20
Standard deviation = 2.49

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Demand Characteristics

Normal distribution

Uniform distribution

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Example 1: The J&B Card Shop sells calendars. The once-
a-year order for each year’s calendar arrives in
September. The calendars cost $1.50 and J&B sells them
for $3 each. At the end of July, J&B reduces the calendar
price to $1 and can sell all the surplus calendars at this
price. How many calendars should J&B order if the
September-to-July demand can be approximated by
a. uniform distribution between 150 and 850

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Solution to Example 1:

Loss resulting from the items unsold

ML= Purchase price - Salvage value = $1.5 - $1 = $0.5

Profit resulting from the items sold

MP= Selling price - Purchase price = $3 - $1.5 = $1.5

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 ML
P = 0.5 / (1.5+0.5) = 0.25
MP  ML

Now, find the Q so that P(demandQ) = 0.25

Q* = 675

0.25*(850-150)

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Q*
Example 2: The J&B Card Shop sells calendars. The once-
a-year order for each year’s calendar arrives in
September. The calendars cost $1.50 and J&B sells them
for $3 each. At the end of July, J&B reduces the calendar
price to $1 and can sell all the surplus calendars at this
price. How many calendars should J&B order if the
September-to-July demand can be approximated by
b. normal distribution with  = 500 and =120.

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Solution to Example 2: ML= $?, MP=$? (see example 1)

ML
P = 0.25
MP  ML

Now, find the Q so that P =

Q
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We need z corresponding to area =
From Normal Table (see next slide)

z = 0.675
Hence, Q* =  + z = 500 + 0.675(120) = 581

Q*

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[Untuk dipakai di ujian]

.75

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Normal distribution online calculator
[NB: tidak boleh dipakai di ujian]
https://stattrek.com/online-calculator/normal

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Example 3: A retail outlet sells a seasonal product for $10
per unit. The cost of the product is $8 per unit. All units not
sold during the regular season are sold for half the retail
price in an end-of-season clearance sale. Assume that the
demand for the product is normally distributed with  =
500 and  = 100.
a. What is the recommended order quantity?
b. What is the probability of a stockout?
c. To keep customers happy and returning to the store
later, the owner feels that stockouts should be avoided if
at all possible. What is your recommended quantity if the
owner is willing to tolerate a 0.15 probability of stockout?
d. Using your answer to part c, what is the goodwill cost
you are assigning to a stockout?
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Solution to Example 3:
a. Selling price=$10,
Purchase price=$8
Salvage value=10/2=$5
MP =10 - 8 = $2, ML = 8-10/2 = $3
Order maximum quantity, Q such that
ML 3
P   0.6
ML  MP 2  3
Now, find the Q so that
P = 0.6
or, area (2)+area (3) = 0.6
or, area (1) = 1-0.6=0.40

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Find z for area = 0.40 from the standard normal table
So, z = -0.255
So, Q*=+z =500+(-0.255)(100)=474.5 units.

b. P(stockout) = P(demandQ) = P = 0.6

c. P(stockout)=Area(3)=0.15
find z for Area (1+2) = 0.85

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z = 1.03 for area = 0.3485
z = 1.04 for area = 0.3508
So, z = 1.035 for area = 0.35
So, Q*=+z =500+(1.035)(100)=603.5 units.

d. P=P(demandQ)=P(stockout)=0.15
For a goodwill cost of g
MP =10 - 8+g = 2+g, ML = 8-10/2 = 3

ML 3
Now, solve g in p =   0.15
ML  MP 2  g  3

Hence, g=$15.

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Exercise

Cost = Rp 4250
Sell = Rp 5000
Salvage = Rp 125

a. What is the
recommended order
quantity?
b. What is the
probability of unsold
item?

Hint: find  and  !

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