Managing Flow Variability: Safety Inventory
The Magnitude of Shortages (Out of Stock)
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
1
Managing Flow Variability: Safety Inventory
What are the Reasons?
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
2
Managing Flow Variability: Safety Inventory
Consumer Reaction
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Optimal Service Level: The Newsvendor Problem
How do we choose what level of service a firm should offer?
Cost of ordering
too much: holding
cost, salvage
Cost of ordering too
little: lost of sale, low
service level
The decision maker balances the expected costs of ordering too much
with the expected costs of ordering too little to determine the optimal
order quantity.
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
News Vendor Model; Assimptions
 Demand is random
 Distribution of demand is known
 No initial inventory
 Set-up cost is zero
 Single period
 Zero lead time
 Linear costs




Purchasing (production)
Salvage value
Revenue
Goodwill
The Newsvendor Problem
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Managing Flow Variability: Safety Inventory
Optimal Service Level: The Newsvendor Problem
Cost =1800, Sales Price = 2500, Salvage Price = 1700
Underage Cost = 2500-1800 = 700, Overage Cost = 1800-1700 = 100
Demand
100
110
120
130
140
150
160
170
180
190
200
Probability of Demand
0.02
0.05
0.08
0.09
0.11
0.16
0.20
0.15
0.08
0.05
0.01
What is probability of demand to be equal to 130? 0.09
0.35
What is probability of demand to be less than or equal to 140? 0.02+0.05+0.08+0.09+0.11=
What is probability of demand to be greater than or equal to 140? 1-0.35+0.11= 0.76
What is probability of demand to be equal to 133? 0
P(R ≥ Q ) = 1-P(R ≤ Q)+P(R = Q)
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
R is quantity of demand
Q is the quantity ordered
6
Managing Flow Variability: Safety Inventory
Optimal Service Level: The Newsvendor Problem
Demand
100
101
102
103
104
105
106
107
108
109
Probability of Demand
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
0.002
Demand
110
111
112
113
114
115
116
117
118
119
Probability of Demand
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.005
What is probability of demand to be equal to 116? 0.005
What is probability of demand to be less than or equal to 116? 0.02+0.035 = 0.055
What is probability of demand to be greater than or equal to 116? 1-0.055+0.005 = 0.95
What is probability of demand to be equal to 113.3? 0
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Optimal Service Level: The Newsvendor Problem
Average Demand
100
110
120
130
140
150
160
170
180
190
200
The Newsvendor Problem
Probability of Demand
0.02
0.05
0.08
0.09
0.11
0.16
0.20
0.15
0.08
0.05
0.01
What is probability of demand to be equal to
130? 0
What is probability of demand to be less than
or equal to 145? 0.02+0.05+0.08+0.09+0.11 = 0.35
What is probability of demand to be greater
than or equal to 145? 1-0.35 = 0.65
P(R ≥ Q) = 1-P(R ≤ Q)
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Compute the Average Demand
N
Average Demand   Qi P( R  Qi )
i 1
Average Demand =
+100×0.02 +110×0.05+120×0.08
+130×0.09+140×0.11 +150×0.16
+160×0.20 +170×0.15 +180×0.08
+190×0.05+200×0.01
Average Demand = 151.6
Qi
P( R =Q i )
100
110
120
130
140
150
160
170
180
190
200
0.02
0.05
0.08
0.09
0.11
0.16
0.20
0.15
0.08
0.05
0.01
How many units should I have to sell 151.6 units (on average)?
How many units do I sell (on average) if I have 100 units?
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Deamand (Qi)
100
110
120
130
140
150
160
170
180
190
200
Porbability
Prob (R ≥ Qi)
0.02
1.00
0.05
0.98
0.08
0.93
0.09
0.85
0.11
0.76
0.16
0.65
0.20
0.49
0.15
0.29
0.08
0.14
0.05
0.06
0.01
0.01
Suppose I have ordered 140 Unities.
On average, how many of them are sold? In other words, what is the
expected value of the number of sold units?
When I can sell all 140 units?
I can sell all 140 units if  R ≥ 140
Prob(R ≥ 140) = 0.76
The expected number of units sold –for this part- is
(0.76)(140) = 106.4
Also, there is 0.02 probability that I sell 100 units 2 units
Also, there is 0.05 probability that I sell 110 units5.5
Also, there is 0.08 probability that I sell 120 units 9.6
Also, there is 0.09 probability that I sell 130 units 11.7
106.4 + 2 + 5.5 + 9.6 + 11.7 = 135.2
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Deamand (Qi)
100
110
120
130
140
150
160
170
180
190
200
Porbability
Prob (R ≥ Qi)
0.02
1.00
0.05
0.98
0.08
0.93
0.09
0.85
0.11
0.76
0.16
0.65
0.20
0.49
0.15
0.29
0.08
0.14
0.05
0.06
0.01
0.01
Suppose I have ordered 140 Unities.
On average, how many of them are salvaged? In other words, what is
the expected value of the number of salvaged units?
0.02 probability that I sell 100 units.
In that case 40 units are salvaged  0.02(40) = .8
0.05 probability to sell 110  30 salvaged  0.05(30)= 1.5
0.08 probability to sell 120  20 salvaged  0.08(20) = 1.6
0.09 probability to sell 130  10 salvaged  0.09(10) =0.9
0.8 + 1.5 + 1.6 + 0.9 = 4.8
Total number Sold
135.2 @ 700 = 94640
Total number Salvaged 4.8 @ -100 = -480
Expected Profit = 94640 – 480 =
94,160
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Cumulative Probabilities
Qi
100
110
120
130
140
150
160
170
180
190
200
Probabilities
P(R =Qi) P(R <Qi) P(R ≥Qi)
0.02
0
1
0.05
0.02
0.98
0.08
0.07
0.93
0.09
0.15
0.85
0.11
0.24
0.76
0.16
0.35
0.65
0.2
0.51
0.49
0.15
0.71
0.29
0.08
0.86
0.14
0.05
0.94
0.06
0.01
0.99
0.01
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
12
Managing Flow Variability: Safety Inventory
Number of Units Sold, Salvages
Qi
100
110
120
130
140
150
160
170
180
190
200
Probabilities
P(R =Qi) P(R <Qi) P(R ≥Qi)
0.02
0
1
0.05
0.02
0.98
0.08
0.07
0.93
0.09
0.15
0.85
0.11
0.24
0.76
0.16
0.35
0.65
0.20
0.51
0.49
0.15
0.71
0.29
0.08
0.86
0.14
0.05
0.94
0.06
0.01
0.99
0.01
The Newsvendor Problem
Units
Sold
Salvage
100
0
109.8
0.2
119.1
0.9
127.6
2.4
135.2
4.8
141.7
8.3
146.6
13.4
149.5
20.5
150.9
29.1
151.5
38.5
151.6
48.4
Ardavan Asef-Vaziri, Oct 2011
Sold@700
Salvaged@-100
13
Managing Flow Variability: Safety Inventory
Total Revenue for Different Ordering Policies
Qi
100
110
120
130
140
150
160
170
180
190
200
Probabilities
P(R =Qi) P(R <Qi) P(R ≥Qi)
0.02
0
1
0.05
0.02
0.98
0.08
0.07
0.93
0.09
0.15
0.85
0.11
0.24
0.76
0.16
0.35
0.65
0.2
0.51
0.49
0.15
0.71
0.29
0.08
0.86
0.14
0.05
0.94
0.06
0.01
0.99
0.01
The Newsvendor Problem
Units
Sold
Salvaged
100
0
109.8
0.2
119.1
0.9
127.6
2.4
135.2
4.8
141.7
8.3
146.6
13.4
149.5
20.5
150.9
29.1
151.5
38.5
151.6
48.4
Ardavan Asef-Vaziri, Oct 2011
Sold
70000
76860
83370
89320
94640
99190
102620
104650
105630
106050
106120
Revenue
Salvaged
0
20
90
240
480
830
1340
2050
2910
3850
4840
Total
70000
76840
83280
89080
94160
98360
101280
102600
102720
102200
101280
14
Managing Flow Variability: Safety Inventory
Denim Wholesaler; Marginal Analysis
The demand for denim is:
– 1000 with probability 0.10
– 2000 with probability 0.15
– 3000 with probability 0.15
– 4000 with probability 0.20
Unit Revenue (p ) = 30
Unit purchase cost (c )= 10
Salvage value (v )= 5
Goodwill cost (g )= 0
– 5000 with probability 0.15
– 6000 with probability 0.15
– 7000 with probability 0.10
How much should we order?
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Marginal Analysis
Marginal analysis: What is the value of an additional unit ordered?
Suppose the wholesaler purchases 1000 units
What is the value of having the 1001st unit?
Marginal Cost: The retailer must salvage the
additional unit and losses $5 (10 – 5).
P(R ≤ 1000) = 0.1
Expected Marginal Cost = 0.1(5) = 0.5
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Marginal Analysis
Marginal Profit: The retailer makes and extra profit of $20 (30 – 10)
P(R > 1000) = 0.9
Expected Marginal Profit= 0.9(20) = 18
MP ≥ MC
Expected Value = 18-0.5 = 17.5
By purchasing an additional unit, the expected profit increases by
$17.5
The retailer should purchase at least 1,001 units.
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Marginal Analysis
Should he purchase 1,002 units?
Marginal Cost: $5 salvage  P(R ≤ 1001) = 0.1
Expected Marginal Cost = 0.5
Marginal Profit: $20 profit  P(R >1002) = 0.9  18
Expected Marginal Profit = 18
Expected Value = 18-0.5 = 17.5
Assuming that the initial purchasing quantity is between
1000 and 2000, then by purchasing an additional unit
exactly the same savings will be achieved.
Conclusion:
Wholesaler should purchase at least 2000 units.
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
18
Managing Flow Variability: Safety Inventory
Marginal Analysis
Marginal analysis: What is the value of an additional unit ordered?
Suppose the retailer purchases 2000 units
What is the value of having the 2001st unit?
Marginal Cost: The retailer must salvage the
additional unit and losses $5 (10 – 5).
P(R ≤ 2000) = 0.25
Expected Marginal Cost = 0.25(5) = 1.25
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
19
Managing Flow Variability: Safety Inventory
Marginal Analysis
Marginal Profit: The retailer makes and extra profit of $20 (30 – 10)
P(R > 2000) = 0.75
Expected Marginal Profit= 0.75(20) = 15
MP ≥ MC
Expected Value = 15-1.25 = 13.75
By purchasing an additional unit, the expected profit increases by
$13.75
The retailer should purchase at least 2,001 units.
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
20
Managing Flow Variability: Safety Inventory
Marginal Analysis
Should he purchase 2,002 units?
Marginal Cost: $5 salvage  P(R ≤ 2001) = 0.25
Expected Marginal Cost = 1.25
Marginal Profit: $20 profit  P(R >2002) = 0.75
Expected Marginal Profit = 15
Expected Value = 18-0.5 = 13.75
Assuming that the initial purchasing quantity is between
2000 and 3000, then by purchasing an additional unit
exactly the same savings will be achieved.
Conclusion:
Wholesaler should purchase at least 3000 units.
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
21
Managing Flow Variability: Safety Inventory
Marginal Analysis
Why does the marginal value of an additional unit decrease, as
the purchasing quantity increases?
– Expected cost of an additional unit increases
– Expected savings of an additional unit decreases
Cumulative Expected
Expected
Expected
Demand Probability Probability Marginal Cost Marginal Profit Marginal Value
1000
0.10
0.1
0.50
18
17.50
2000
0.15
0.25
1.25
15
13.75
3000
0.15
0.40
2.00
12
10.00
4000
0.20
0.60
3.00
8
5.00
5000
0.15
0.75
3.75
5
1.25
6000
0.15
0.90
4.50
2
-2.50
7000
0.10
1.00
5.00
0
-5.00
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
22
Managing Flow Variability: Safety Inventory
Marginal Analysis
What is the optimal purchasing quantity?
– Answer: Choose the quantity that makes marginal value: zero
Marginal value
17.5
13.75
10
5
1.3
Quantity
-2.5
1000
2000
3000 4000
5000
6000
7000 8000
-5
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Analytical Solution for the Optimal Service Level
Marginal Profit:
MP = p – c
MP = 30 - 10 = 20
Marginal Cost:
MC = c - v
MC = 10-5 = 5
Suppose I have ordered Q units.
What is the expected cost of ordering one more units?
What is the expected benefit of ordering one more units?
If I have ordered one unit more than Q units, the probability of not
selling that extra unit is the probability demand to be less than or equal
to Q.
Since we have P( R ≤ Q).
The expected marginal cost =MC× P( R ≤ Q)
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
24
Managing Flow Variability: Safety Inventory
Analytical Solution for the Optimal Service Level
If I have ordered one unit more than Q units, the probability of
selling that extra unit is the probability of demand to be greater than Q.
We know that P(R > Q) = 1- P(R ≤ Q).
The expected marginal benefit = MB× [1-Prob.( r ≤ Q)]
As long as expected marginal cost is less than expected marginal
profit we buy the next unit.
We stop as soon as: Expected marginal cost ≥ Expected marginal
profit.
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
25
Managing Flow Variability: Safety Inventory
Analytical Solution for the Optimal Service Level
MC×Prob(R ≤ Q*) ≥ MP× [1 – Prob( R ≤ Q*)]
Prob(R ≤ Q*) ≥
MB
MB  MC
MP = p – c = Underage Cost = Cu
MC = c – v = Overage Cost = Co
P( R  Q* ) 
MP
MP  MC
The Newsvendor Problem
cu

Cu  C o
MP
MP  MC
pc
pc


pccv pv
Ardavan Asef-Vaziri, Oct 2011
26
Managing Flow Variability: Safety Inventory
Marginal Value: The General Formula
P(R ≤ Q*) ≥ Cu / (Co+Cu)
Cu / (Co+Cu) = (30-10)/[(10-5)+(30-10)] = 20/25 = 0.8
Order until P(R ≤ Q*) ≥ 0.8
P(R ≤ 5000) ≥ = 0.75 not > 0.8 still order
P(R ≤ 6000) ≥ = 0.9 > 0.8 Stop
Demand
1000
2000
3000
4000
5000
6000
7000
Probability
0.1
0.15
0.15
0.2
0.15
0.15
0.1
In Continuous Model where demand for example has Uniform or
Normal distribution
MP
cu
P( R  Q ) 

MP  MC
Cu  C o
*
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
pc

pv
27
Managing Flow Variability: Safety Inventory
Type-1 Service Level
What is the meaning of the number 0.80?
80% of the time all the demand is satisfied.
–
Probability {demand is smaller than Q} =
–
Probability {No shortage} =
–
Probability {All the demand is satisfied from stock} = 0.80
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
28
Managing Flow Variability: Safety Inventory
Marginal Value: Uniform distribution
Suppose instead of a discreet demand of
Demand
1000
2000
3000
4000
5000
6000
7000
Probability
0.1
0.15
0.15
0.2
0.15
0.15
0.1
Cumlat
Probab
0.1
0.25
0.4
0.6
0.75
0.9
1
We have a continuous demand uniformly distributed between
1000 and 7000
1000
7000
Pr{r ≤ Q*} = 0.80
How do you find Q?
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Marginal Value: Uniform distribution
Q-l = Q-1000
?
0.80
l=1000
1/6000
u=7000
u-l=6000
(Q-1000)/6000=0.80
Q = 5800
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
30
Managing Flow Variability: Safety Inventory
Marginal Value: Normal Distribution
Suppose the demand is normally distributed with a mean of 4000
and a standard deviation of 1000.
What is the optimal order quantity?
Notice: F(Q) = 0.80 is correct for all distributions.
We only need to find the right value of Q assuming the normal
distribution.
P(z ≤ Z) = 0.8  Z= 0.842
Q = mean + z Standard Deviation  4000+841 = 4841
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
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Managing Flow Variability: Safety Inventory
Marginal Value: Normal Distribution
0.00045
0.0004
Probability of
excess inventory
0.00035
0.0003
Probability of
shortage
4841
0.00025
0.0002
0.00015
0.0001
0.80
0.00005
0
0
2000
4000
0.20
6000
8000
Given a service level, how do we calculate z?
From our normal table or
From Excel Normsinv(service level)
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
32
Managing Flow Variability: Safety Inventory
Additional Example
Your store is selling calendars, which cost you $6.00 and sell for $12.00
Data from previous years suggest that demand is well described by a
normal distribution with mean value 60 and standard deviation 10.
Calendars which remain unsold after January are returned to the
publisher for a $2.00 "salvage" credit. There is only one opportunity to
order the calendars. What is the right number of calendars to order?
MC= Overage Cost = Co = Unit Cost – Salvage = 6 – 2 = 4
MB= Underage Cost = Cu = Selling Price – Unit Cost = 12 – 6 = 6
Cu
6
P( R  Q ) 

 0. 6
Cu  C o 6  4
*
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
33
Managing Flow Variability: Safety Inventory
Additional Example - Solution
Look for P(x ≤ Z) = 0.6 in Standard Normal table or
for NORMSINV(0.6) in excel  0.2533
Q*  
Q*  
P( Z 
)  0.6 
 0.2533


Q*    0.2533  60  10(0.2533)  62.533  63
By convention, for the continuous demand distributions, the
results are rounded to the closest integer.
Suppose the supplier would like to decrease the unit cost in order to
have you increase your order quantity by 20%. What is the minimum
decrease (in $) that the supplier has to offer.
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
34
Managing Flow Variability: Safety Inventory
Additional Example - Solution
Qnew = 1.2 * 63 = 75.6 ~ 76 units
76  60
P( R  Q )  P( R  76)  P( Z 
)  P( Z  1.6)
10
*
Look for P(Z ≤ 1.6) = 0.6 in Standard Normal table
or for NORMSDIST(1.6) in excel  0.9452
Cu
pc
12  c 12  c
P( R  Q )  0.9452 



Cu  Co p  c  c  v 12  2
10
*
12  c  9.452  c  2.55
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
35
Managing Flow Variability: Safety Inventory
Additional Example
On consecutive Sundays, Mac, the owner of your local newsstand,
purchases a number of copies of “The Computer Journal”. He
pays 25 cents for each copy and sells each for 75 cents. Copies he
has not sold during the week can be returned to his supplier for 10
cents each. The supplier is able to salvage the paper for printing
future issues. Mac has kept careful records of the demand each
week for the journal. The observed demand during the past weeks
has the following distribution:
Qi
4
5
6
7
8
P(R=Qi) 0.04 0.06 0.16 0.18 0.2
The Newsvendor Problem
9
0.1
Ardavan Asef-Vaziri, Oct 2011
10
0.1
11 12 13
0.08 0.04 0.04
36
Managing Flow Variability: Safety Inventory
Additional Example
Qi
4
5
6
7
8
P(R=Qi) 0.04 0.06 0.16 0.18 0.2
9
0.1
10
0.1
11 12 13
0.08 0.04 0.04
a) How many units are sold if we have ordered 7 units
There is 0.18 + 0.20 + 0.10 + 0.10 + 0.08 + 0.04 + 0.04 = 0.74
There is 0.74 probability that demand is greater than or equal to 7.
There is 0.16 probability that demand is equal to 6.
There is 0.06 probability that demand is equal to 5.
There is 0.04 probability that demand is equal to 4.
The expected number of units sold is
0.74(7) + 0.16 (6) + 0.06 (5) + 0.04 (4) = 6.6
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
37
Managing Flow Variability: Safety Inventory
Additional Example
Qi
4
5
6
7
8
P(R=Qi) 0.04 0.06 0.16 0.18 0.2
9
0.1
10
0.1
11 12 13
0.08 0.04 0.04
b) How many units are salvaged?
7-6.6 = 0.4. Alternatively, we can compute it directly
There is 0.74 probability that we salvage 7 – 7 = 0 units
There is 0.16 probability that we salvage 7- 6 = 1 units
There is 0.06 probability that we salvage 7- 5 = 2 units
There is 0.04 probability that we salvage 7-4 = 3 units
The expected number of units salvaged is
0.74(0) + 0.16 (1) + 0.06 (2) + 0.04 (3) = 0.4 and 7-0.4 = 6.6 sold
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
38
Managing Flow Variability: Safety Inventory
Additional Example
c) Compute the total profit if we order 7 units.
Out of 7 units, 6.6 sold, 0.4 salvaged.
P = 75, c= 25, v=10.
Profit per unit sold = 75-25 = 50
Cost per unit salvaged = 25-10 = 15
Total Profit = 6.6(50) + 0.4(15) = 333 - 9 = 324
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
39
Managing Flow Variability: Safety Inventory
Additional Example
Qi
4
5
6
7
8
P(R=Qi) 0.04 0.06 0.16 0.18 0.2
9
0.1
10
0.1
11 12 13
0.08 0.04 0.04
d) Compute the expected Marginal profit of ordering the 8th unit.
MP = 75-25 = 50
P(R ≥ 8) = 0.2 + 0.1 + 0.1 + 0.08 + 0.04 + 0.04 = 0.56
Expected Marginal profit = 0.56(50) = 28
d) Compute the expected Marginal cost of ordering the 8th unit.
MC = 25 – 10 = 15
P(R ≤ 7) = 1-0.56 = 0.44
Expected Marginal cost = 0.44(15) = 6.6
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
40
Managing Flow Variability: Safety Inventory
Additional Example - Solution
e) What is the optimum order quantity for Mac to minimize his cost?
Overage Cost = Co = Unit Cost – Salvage = 0.25 – 0.1 = 0.15
Underage Cost = Cu = Selling Price – Unit Cost = 0.75 – 0.25 = 0.50
Cu
0.50
P( R  Q*) 

 0.77
Cu  Co 0.50  0.15
P( R  Q* )  0.77
Qi
4
5
6
7
8
9
10
11
12
13
Probability
P(R=Qi)
0.04
0.06
0.16
0.18
0.20
0.10
0.10
0.08
0.04
0.04
Cumulative
Probability
F(Qi)
0.04
0.10
0.26
0.44
0.64
0.74
0.84
0.92
0.96
1.00
Q* = 10
The Newsvendor Problem
Ardavan Asef-Vaziri, Oct 2011
41