Graduate Program in
Business Information
Systems
Inventory Decisions with Uncertain Factors
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Uncertainties in real life
 Demand is usually uncertain.
 Probability distributions are used to represent
uncertain factors.
Ex: Demand is normally distributed OR
Demand is either 20,30,40 with respective
probabilities 0.2, 0.5, 0.3.
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Stochastic versus Deterministic
Models
 Mathematical models involving probability are
referred to as stochastic models.
 Deterministic models are limited in scope
since they do not involve uncertain factors.
But they are used to develop insight!
 Stochastic models are based on “expected
values”, i.e. the long run average of all
possible outcomes!
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Example: Drugstore

A drugstore stocks Fortunes.They sell each for $3 and unit cost
is $2.10. Unsold copies are returned for $.70 credit. There are
four levels of demand possible. How many copies of Fortune
should be stocked in October?
Payoff Table:
ACTS
Demand
Event
D = 20
Probability
.2
Q = 20
$18.00
Q = 21
$16.60
Q = 22
$15.20
Q = 23
$13.80
D = 21
.4
18.00
18.90
17.50
16.10
D = 22
.3
18.00
18.90
19.80
18.40
D = 23
.1
18.00
18.90
19.80
20.70
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Solution:
 The expected payoffs are computed for each
possible order quantity:
Q = 20
Q = 21
$18.00
$18.44
Q = 22
$17.90
Q = 23
$16.79
Optimal stocking level, Q*=21 at an optimal
expected profit of $18.44
 If the probabilities were long-run frequencies,
then doing so would maximize long-run profit.
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Example: Drugstore Payoff Table
(Figure 16-1)
A
B
C
D
E
PAYOFF TABLE EVALUATION
1
2
3
PROBLEM: Fortune Magazine
4
5
Problem Data
C
6
18 =SUMPRODUCT($B$9:$B$12,C9:C12)
7
Act 1
Act 2
Act 3
8
Events
Probability Q = 20
Q = 21
Q = 22
9
1 D = 20
0.2
$18.00
$16.60
$15.20
10
2 D = 21
0.4
$18.00
$18.90
$17.50
11
3 D = 22
0.3
$18.00
$18.90
$19.80
12
4 D = 23
0.1
$18.00
$18.90
$19.80
13
14
Act Summary
15
16
Act 1
Act 2
Act 3
17
Q = 20
Q = 21
Q = 22
18 Expected Payoff
$18.00
$18.44
$17.96
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F
Act 4
Q = 23
$13.80
$16.10
$18.40
$20.70
Act 4
Q = 23
$16.79
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Single-Period Inventory Decision:
The Newsvendor Problem




Single period problem (periodic review)
Demand is uncertain (stochastic)
No fixed ordering cost
Instead of h ($/$/period) we have hE ($/unit/period=ch)
 Instead of p ($/unit) we have pS and pR-c

Q: Order Quantity (decision variable)
D: Demand Quantity

Costs:
c = Unit procurement cost
hE = Additional cost of each item held at end of inventory cycle
= unit inventory holding cost-salvage value to the supplier
pS = Penalty for each item short (loss of customer goodwill)
pR = Selling price
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Modeling the Newsvendor Problem
 D if D  Q
Sales  
Q if D  Q
cQ  hE Q  d 
if d  Q

TC (Q)  
cQ   pS  pR  c d  Q  if d  Q
The objective is to minimize total expected cost, which can be
simplified as:
TEC(Q)  cm  hE  c Q  m  B(Q)   pS  pR  c  B(Q)
where m is the expected demand.
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Optimal order quantity of the
Newsboy Problem
Q* is the smallest possible demand such that
pS  p R  c
Pr[ D  Q*] 
 pS  pR  c   hE  c 
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Example: Newsboy Problem
 A newsvendor sells Wall Street Journals.
She loses pS = $.02 in future profits each time
a customer wants to buy a paper when out of
stock. They sell for pR = $.23 and cost c =
$.20. Unsold copies cost hE = $.01 to
dispose. Demands between 45 and 55 are
equally likely. How many should she stock?
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Example: Solution
 Discrete Uniform Distribution
Demand is either 45,46,47,..., 55 each with a
probability of 1/11.
pS  p R  c
.02  .23  .20

 .192
 pS  pR  c   hE  c  .02  .23  .20  .01  .20
P(D<=Q*)=0.2
Q*=47 units.
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Newsvendor Problem
(Figure 16-3)
A
B
C
D
E
F
G
1
SINGLE PERIOD INVENTORY MODEL -- NEWSVENDOR PROBLEM
2
3 PROBLEM:
Wall Street Journal
4
5
Parameter Values:
6
Cost per Item Procured: c =
0.20
Additional Cost for Each Leftover Item Held: hE =
7
0.01
Penalty for Each Item Short: pS =
8
0.02
Selling Price per Unit: pR =
9
0.23
10
Number of demands for probability distribution =
11
11
12
Optimal Values:
13
Optimal Order Quantity: Q* =
47
14
Expected Demand: mu =
49.5
15
Total Expected Cost: TEC(Q*) =
$10.07
16
Expected Shortages: B(Q*) =
2.66
17
Probability of Shortage: P[D>Q*] =
0.80
18
19
Cumulative Number of
20
Demand Probability Probability
shortages
21
45
0.05
0.05
0.0
22
46
0.06
0.11
0.0
23
47
0.09
0.20
0.0
24
48
0.12
0.32
1.0
25
49
0.17
0.49
2.0
26
50
0.20
0.69
3.0
27
51
0.12
0.81
4.0
28
52
0.08
0.89
5.0
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29
53
0.06
0.95
6.0
30
54
0.04
0.99
7.0
31
55
0.01
1.00
8.0
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Multiperiod Inventory Policies
 When demand is uncertain, multiperiod inventory
might look like this over time.
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Multiperiod Inventory Policies
 The multiperiod decisions involve two variables:
Order quantity Q
 Reorder point r
 The following parameters apply:
 A = mean annual demand rate
 k = ordering cost
 c = unit procurement cost
 ps = cost of short item (no matter how long)
 h = annual holding cost per dollar value
 m = mean lead-time demand

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Multiperiod Inventory Policies:
Discrete Lead-Time Demand
 The following is used to compute the expected
shortage per inventory cycle:
Br    d  r PrL D  d 
d r
 The following is used to compute the total annual
expected cost:
Q
 A
 A


TEC r ,Q    k  hc   r  m   pS   Br 
2

Q
Q
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Multiperiod Inventory Policies:
Discrete Lead-Time Demand
 Solution Algorithm.
2 Ak
Q1 
hc

Calculate the starting order quantity:

hcQ
Determine the reorder point r*: PrD  r *  1 
pS A

Determine optimal order quantity:

This procedure continues –using the last Q to obtain r and
r to obtain the next Q- until no values change.
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2 Ak  pS Br 
Q* 
hc
16
Example:
Annual demand for printer cartridges costing c = $1.50 is A = 1,500.
Ordering cost is k = $5 and holding cost is $.12 per dollar per year.
Shortage cost is pS = $.12, no matter how long. Lead-time demand
has the following distribution. Find the optimal inventory policy.
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Example: Solution
 The starting order quantity is:


 Pr D  r *  1 
21,5005
Q1 
 289
.121.50 
hcQ
(0.12)1.5289
 1
 .93
pS A
(0.5)1,500
r* = 7 cartridges.
 B(7) = (8–7)(.03) + (9–7)(.01) + (10–7)(.01)=.08 and
the optimal order quantity is:
21,5005  .50.08
Q* 
 290
.121.50 
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Example: Solution (cont’d.)
 Q=290 leads to r=7, so the solution is optimal.
The optimal inventory policy is:
r* = 7 Q* = 290

Optimal annual expected cost is:
TEC(7,290) 
1500
1500
 290

5  (0.12)(1.5) 
 7  4  0.5
0.08  $52.71
290
290
 2

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Multiperiod Discrete Backordering
Iteration 1
G
14 =SQRT((2*G7*G6)/(G9*G8))
=INDEX(C23:C42,MATCH(LOOKUP(1((G9*G8*G14)/(G10*G7)),E23:E42,C23:
15 C42),C23:C42,0)+1)
16 =SUMPRODUCT(C23:C42,D23:D42)
=(G7/G14)*G6+G9*G8*(G14/2+G1517 G16)+G10*(G7/G14)*G18
18 =SUMPRODUCT(D23:D42,F23:F42)
19 =1-VLOOKUP(G15,C23:E42,3)
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM:
Printer Cartridges
4
5
Parameter Values
6
Fixed Cost per Order: k =
5
7
Annual Demand Rate: A =
1500
8
Unit cost of Procuring an Item: c =
1.5
9
Annual Holding Cost per Dollar Value: h =
0.12
Shortage Cost per Unit: pS =
10
0.5
11
Number of demands for probability distribution =
11
12
13
Optimal Values:
14
Optimal Order Quantity: Q* =
288.68
15
Optimal Reorder Point: r* =
7
16
Expected Lead-Time Demand: mu =
4
17
Total Expected Cost: TEC(Q*) =
$ 52.7094
18
Expected Shortage: B(r*) =
0.08
19
Probability of Shortage: P[D>r*] =
0.05
20
21
Cumulative
Number of
22
Demand Probability
Probability
Shortages
23
0
0.01
0.01
0
24
1
0.07
0.08
0
25
2
0.16
0.24
0
26
3
0.20
0.44
0
27
4
0.19
0.63
0
28
5
0.16
0.79
0
29
6
0.10
0.89
0
30
7
0.06
0.95
0
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20
31
8
0.03
0.98
1
32
9
0.01
0.99
2
33
10
0.01
1.00
3
Multiperiod Discrete Backordering
Iteration 10
14
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM:
Printer Cartridges
4
5
Parameter Values
6
Fixed Cost per Order: k =
5
7
Annual Demand Rate: A =
1500
8
Unit cost of Procuring an Item: c =
1.5
9
Annual Holding Cost per Dollar Value: h =
0.12
Shortage Cost per Unit: pS =
10
0.5
11
Number of demands for probability distribution =
11
12
13
Optimal Values:
14 =SQRT((
14
Optimal Order Quantity: Q* =
290
=INDEX(
15
Optimal Reorder Point: r* =
7
G
((G9*G8*
16
Expected Lead-Time Demand: mu =
4 15 ),C23:C4
=SQRT((2*G7*(G6+'9'!G18*G10))/(G9*G8))
17
Total Expected Cost: TEC(Q*) =
$ 52.71 16 =SUMPR
18
Expected Shortage: B(r*) =
0.08
=(G7/G1
19
Probability of Shortage: P[D>r*] =
0.05
G16)+G1
17
20
18
=SUMPR
21
Cumulative
Number of
19
=1-VLOO
22
Demand Probability
Probability
Shortages
23
0
0.01
0.01
0
24
1
0.07
0.08
0
25
2
0.16
0.24
0
26
3
0.20
0.44
0
27
4
0.19
0.63
0
28
5
0.16
0.79
0
29
6
0.10
0.89
0
30
7
0.06
0.95
0
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21
31
8
0.03
0.98
1
32
9
0.01
0.99
2
33
10
0.01
1.00
3
Multiperiod Discrete Backordering
Summary
A
B
C
D
E
F
G
1 MULTI-PERIOD EOQ MODEL (Backordering) - DISCRETE LEAD-TIME DEMAND
2
3 PROBLEM: Printer Cartridges
4
5
Parameter Values
6
Fixed Cost per Order: k =
5
7
Annual Demand Rate: A =
1500
8
Unit cost of Procuring an Item: c =
1.5
9
Annual Holding Cost per Dollar Value: h =
0.12
Shortage Cost per Unit: pS =
10
0.5
11
Number of demands for probability distribution =
11
12
Qi
ri
B(ri) TEC(Qi,ri)
13
Iteration, i
14
15
16
17
18
19
20
21
22
23
1
2
3
4
5
6
7
8
9
10
289
290
290
290
290
290
290
290
290
290
7
7
7
7
7
7
7
7
7
7
0.08 $ 52.71
0.08 $ 52.71
0.08 $ 52.71
0.08 $ 52.71
0.08 $ 52.71
0.08 $ 52.71
0.08 $ 52.71
0.08 $ 52.71
0.08 $ 52.71
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0.08
$ Aslı52.71
22

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