IEEM 517
Inventory Control –
Newsvendor Model
LEARNING OBJECTIVES
1.
Understand how demand uncertainty can affect inventory decision
2.
Understand modeling assumptions, formulation, and optimal solution of the
newsvendor model
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1
CONTENTS
2
• Newsvendor Model
• Estimation of Demand Distribution
• Summary
EXAMPLE (1)
3
A newsvendor needs to decide on the quantity of newspaper to order for a day.
The demand during the day is stochastic. Before ordering, the newsvendor has
some knowledge about the demand and estimates that the demand follows a
Normal distribution with mean 3000 and variance 10002.
The newsvendor orders the newspaper from a publisher. After ordering, the
plushier delivers the ordered quantity immediately. No replenishment is
conducted later on. The unit purchase cost from the publisher is $0.60 per
piece. The selling price is $1.00 per piece. At the end of the day, excessive
newspaper can be returned to publisher at $0.20/piece.
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4
PROBLEM ANALYSIS
A newsvendor must decide at beginning of day the inventory level Q of a newspaper:
• If the inventory level Q is higher than the realized demand x, excess inventory can be
only partially salvaged, and thus the newsvendor suffers some overage cost
• If the inventory level Q is lower than the realized demand x, excess demand is lost,
and thus the newsvendor suffers some shortage cost
Case 2: shortage
Q<x
Knowledge
on demand
distribution X
Inventory
level Q
Realized
demand x
Case 1: overage
Q>x
MODELING ASSUMPTIONS
1.
Consider the case of a single product
2.
One-period planning horizon (e.g., one day, one week, or one selling
season)
3.
Demand is random, and we can characterize our knowledge of the
demand by a probability distribution
4.
Inventory stock is made and becomes available before demand realization
5.
Costs of overage and shortage are both linear in quantity
6.
The decision-maker is an expected-value decision-maker (i.e., risk neutral)
5
3
MODEL PARAMETERS AND DECISION VARIABLE
X
Demand (in units), a random variable
x
Realized demand
6
G(x) Cumulative density function of X, i.e., G(x) = P(X ≤ x)
g(x) Probability density function of X, i.e., g(x) = dG(x)/dx
µ
Mean demand (in units)
σ
Standard deviation of demand (in units)
co
Unit overage cost (in dollars)
cs
Unit shortage cost (in dollars)
Q
Inventory level, i.e., production/order quantity (in units) Å decision variable
7
RELEVANT COSTS
1. Expected overage cost = c o ∫0 (Q − x )g(x)dx
Q
Units over = (Q – X)+ = max {Q – X, 0}
Expected number of units over = E(Q – X)+ = E(max {Q – X, 0})
= ∫0 max{Q − x,0}g(x)dx = ∫0 (Q − x )g(x)dx + ∫Q 0g(x)dx = ∫0 (Q − x )g(x)dx
∞
∞
Q
Q
2. Expected shortage cost = c s ∫Q (x - Q )g(x)dx
∞
Units short = (X – Q)+ = max {X – Q, 0}
Expected number of units short = E(X – Q)+ = E(max {X – Q, 0})
=
∫
∞
0
max{x - Q,0}g(x)dx = ∫ 0g(x)dx + ∫ (x - Q )g(x)dx = ∫ (x - Q )g(x)dx
Q
∞
∞
0
Q
Q
Total cost Y(Q) = c o ∫0 (Q − x )g(x)dx + c s ∫Q (x - Q )g(x)dx
Q
∞
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8
OPTIMAL SOLUTION (1)
a (Q)
a (Q)
2
d 2
da (Q)
∂
f(x, Q)dx = ∫
f(x, Q)dx + f(a 2 (Q), Q) 2
∫
dQ a1 (Q)
Q
dQ
∂
a1 (Q)
Total cost function
Y(Q) = c o ∫ (Q − x )g(x)dx + c s ∫ (x - Q)g(x)dx
Q
∞
0
Q
− f(a1(Q), Q)
Leibniz’s rule
da1(Q)
dQ
First-order condition
9
OPTIMAL SOLUTION (2)
Second-order condition
G(x)
Optimal solution
G(Q*) =
1
cs
co + cs
cs
co + cs
Q*
x
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10
SPECIAL CASE: NORMAL DEMAND DISTRIBUTION
Demand X is Normally distributed with mean µ and variance σ2
Æ
X −µ
follows a standard Normal distribution (with mean 0 and variance 12)
σ
Æ
cs
X − µ Q * −µ
= G(Q*) = P(X ≤ Q*) = P(
≤
) = Φ (z * )
co + cs
σ
σ
where z* =
Q * −µ
and Φ is the cumulative density function of standard Normal
σ
Æ Q* = µ + z*σ, where Φ(z*) =
cs
co + cs
Φ(z)
cs
co + cs
z*
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EXAMPLE (2)
demand ~ Normal (3000, 10002)
selling price = $1.00/piece
z
purchase cost = $0.60/piece
salvage value = $0.20/piece
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12
EXAMPLE (3)
demand ~ Normal (3000, 10002)
selling price = $1.00/piece
purchase cost = $0.40/piece
salvage value = $0.20/piece
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EXAMPLE (4)
demand ~ Normal (3000, 10002)
selling price = $1.00/piece
purchase cost = $0.60/piece
salvage value = $0.00/piece
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CONTENTS
14
• Newsvendor Model
• Estimation of Demand Distribution
• Summary
ESTIMATING DEMAND DISTRIBUTION
15
We have assumed so far that the demand distribution is known.
However, we need to estimate this demand distribution in practice.
There are two main approaches for estimating the distribution
1. Fit a demand distribution to historical demands
2. Use a forecasting approach to determine the expected daily
demand and the forecast-error distribution
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16
FITTING DISTRIBUTION TO DEMAND
Period
Demand
Period
Demand
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
32
21
39
32
29
26
43
39
37
21
41
36
30
25
8
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
17
27
20
20
53
29
18
4
6
27
31
32
33
33
28
Demand Distribution
Probability
0.3
0.2
~Normal(27.9, 10.9)
0.1
0.0
0
20
40
60 Demand
17
FITTING DISTRIBUTION TO FORECAST
One-step forecast by exponential smoothing with α = 0.2
Period Demand Forecast
At
ft
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
32
21
39
32
29
26
43
39
37
21
41
36
30
25
8
32
30
32
32
31
30
33
34
35
32
34
34
33
31
Error2
2
εt
121
81
0
9
25
169
36
9
196
81
4
16
64
529
Period Demand Forecast
At
ft
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
17
27
20
20
53
29
18
4
6
27
31
32
33
33
28
26
24
25
24
23
29
29
27
22
19
21
23
25
27
28
Error2
2
εt
81
9
25
16
900
0
121
529
256
64
100
81
64
36
0
9
DISTRIBUTION OF FORECAST ERROR
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Probability
0.4
εt ~ Normal (0, 11.2)
0.2
Demand of Period 31
~ Normal (28, 11.2)
0.0
-35 -25 -15 -5
5
15 25 35
Error εt
CONTENTS
19
• Newsvendor Model
• Estimation of Demand Distribution
• Summary
10
SUMMARY
20
• Newsvendor problem is a fundamental problem in stochastic inventory
control. It is used as a building block for many advanced inventory models
• The tradeoff decision in the newsvendor model is made between the
overage cost and the shortage cost
• Demand distribution can be fit to historical demand or estimated with a
forecasting approach
ANNOUNCEMENTS
21
• For the Newsvendor model, read Section 2.4.1
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