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Rafael Diaz - Old Dominion University 1
A Simulation Optimization Approach to Solve Stochastic Inventory Problems with Autocorrelated Demand
Rafael Diaz
Old Dominion University Virginia Modeling, Analysis, & Simulation Center 1030 University Blvd. Suffolk, VA 23435
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Content
Introduction and Motivation Inventory Model Simulation Optimization Method Results Conclusions and Future Work
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Introduction A simulation-based optimization technique:
Simulating Annealing Pattern Search Ranking and Selection
used to approximate solutions to stochastic inventory models that consider autocorrelated demands.
Failing to capture the probabilistic properties of input processes that exhibit autocorrelation generates errors that can be characterized: Regression analysis.
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Motivation Stochastic Inventory System
Rewards and inconvenience Decisions and rules are designed to rationalize, coordinate,
and control Stochastic demand
Positively / Negatively autocorrelated. Examples:
Electronic retailing, consumer goods, and grocery shops i.e. Erkip 1990, Lee et al. 2001.
Other authors: Ray (1980, 1981), Lau and Wang (1987), Fotopoulos et al (1988), Marmorstein and Zinn (1993), Charnes et al. (1995), and Urban (2000, 2005).
The difficulty: Multivariate times series integration
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GoalThe purpose of this research:1. Design a simulation-based optimization method to
approximate solutions in terms of inventory policy. 2. Characterize the error generated by stochastic
modeling techniques that ignore serial correlation components in the demand
– lost-sale case inventory system, competitive markets.
3. Analyze the impact of ignoring autocorrelation components.
Average Total Costs Stockouts
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1. Inventory Model and Autocorrelated Demands
The lost sale case
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The Inventory Problem
Inventory System•Lost-sale
Stochastic demand
•Serially-correlated
Control system•Continuous review
Minimizing Costs•Ordering •Penalty•Holding
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Assumptions and Inventory sequence
Inventory X(i)S
i0
s
1 2
z1 z2
0Min ( ) ( , )ii ic y x L y D
If x
0 Elsewhere i i
i
S x sz
0 *( ) If ( ) 0( , )
*( ) (0) If ( ) 0 i i i i
ii i i i
h y D y DL y d
p D y C D y
500s 8,000S 1.10S s
i Order zi Demand ξi i+1
yixi
Hold
Objective Function Ordering Decision
(s, S) Constraints
; ;
Penalty and Holding Decision
9
Representing Discrete Markovian-modulated demand and Autoregressive AR(1)
1 2 30
P_01 P_12 P_23P_00 P_33
π 1 π2
1 2 30
π 0 π3
•Given Transition Probabilities
Autocorrelated Case
Correlation-free Case
Autoregressive - AR(1)
1( )i i i
Autocorrelation
Error
Stochastic Demand
Correlation boundaries
1 2, ,... i
1 1
1 2, ,... i
hh
Examples: electronics retail, grocery foodIndustry, and general Promotions
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2. Simulation-Based Optimization
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Simulation-Based Optimization
Optimization Procedure
Simulation Model
Input
Output
-Continuous:-Stochastic Approximation (gradient based methods)-RSM.
-Discrete:-Statistical selection-Random search-Metaheuristics
In this study:• Simulated Annealing• Pattern Search• R&S
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Simulation-Based Optimization
Simulated Annealing (SA) Large decision space. Mathematically proven. Generate, evaluate, and pre-select candidate solutions.
Pattern Search (PS) 80’s and 90’s Systematic exploring. Additional neighbors.
Ranking and Selection (R&S) Improve estimation MOP due to stochastic nature. Evaluate incumbent and neighbors.
Related work Simulated Tampering SA and R&S (Ahmed & Alkhamis, 2002) PS and R&S (Sriver & Chrissis, 2004)
1[ ( ) ( )]/
1
if
Otherwise
jH y H x T
j
j
y U ex
x
2
0 *max , i
i
hSN n
d
s s s s s s s s sC
S S S S S S S S S
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Simulation Optimization ProcedureStart
Generate stochastic
demand
Inventory model
Continuous DemandMultivariate input parameters
Inventory input parameters:holding, ordering, and penalty cost
Evaluation and
selection
Simulating Annealing parameters
Pattern Search parameters
R&S parameters
Termination
End
Arbitrary Inventory Policy
Inventory policy constrains
True
False
Stopping criteria
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Combining SA and PS &RS
s
S
Cost
T1
T2
T3
T4
Decision Space
Sample Path
Obtained by SA
Obtained by PS
Obtained by R&S
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Numerical Experimentation
2.3.1 Indifference zone value 5%.2.3.2 h based on the indifference value and the number of neighbor to explore 3.6192.3.3 Initial number of replications
2.2.1 Step Size2.2.2 Number of neighbors to explore per iteration = 3^2
2.1.1 Maximum temperature (based on acceptation 98%)2.1.2 Temperature Gradient 2.1.3 Length of the stage (20,000 periods)2.1.4 Stopping criteria
2.SAPSR&S algorithm
1.3 Maximum / minimum inventory level allowed in the system (s = 500; S = 8,000).
1.2.1 Ordering = 11.2.2 Holding = 2.51.2.3 Penalty = 19
1.1.2. Continuous demand modeled as AR(1) process
1. Inventory model
DescriptionInput Type
10.85*i i
Demand distribution
Costs
Constraints
Simulated Annealing
Pattern Search
R&S
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3. Results
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Results – Ignoring Autocorrelation
φ Cost s S D
IID 3,789.09 2,202 2,854 651
φ Cost
0.1 3,795.28
0.2 3,815.09
0.3 3,845.03
0.4 3,901.62
0.5 3,979.33
0.6 4,096.55
0.7 4,280.87
0.8 4,591.31
0.9 5,098.00
0.95 8,228.63
0.99 18,832.72
•Assuming IID demands
•Ignoring autocorrelated demands
Cost Ignoring Autocorrelation
0.00
2,000.00
4,000.00
6,000.00
8,000.00
10,000.00
12,000.00
14,000.00
16,000.00
18,000.00
20,000.00
0 0.2 0.4 0.6 0.8 1
Autocorrelation
$
Cost Ignoring Autocorrelation
Figure 1
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Results – Acknowledging Autocorrelation
φ Cost s S D
IID 3,789.09 2,202 2,854 651
φ Cost s S D
0.1 3,795.28 2,220 2,856 636
0.2 3,815.09 1,970 2,866 896
0.3 3,845.03 2,030 2,871 841
0.4 3,901.62 2,390 2,888 498
0.5 3,979.33 2,221 2,907 686
0.6 4,096.55 2,169 2,949 780
0.7 4,280.87 1,968 2,992 1,024
0.8 4,591.31 1,626 3,104 1,477
0.9 5,098.00 1,807 3,290 1,483
0.95 6,059.10 1,846 3,805 1,959
0.99 9,486.20 2,593 5,511 2,918
Order Quantity
0
500
1,000
1,500
2,000
2,500
3,000
3,500
IID 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.99
Autocorrelation
# Ite
ms
D
(s, S) - Considering Autocorrelation
0
1,000
2,000
3,000
4,000
5,000
6,000
IID 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.99
Autcorrelation
# o
f u
nit
s
s
S
•Assuming IID demands
•Considering autocorrelated demands
Figure 2
Figure 3
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Total Costs and Stockouts
Service Level
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.99
Autocorrelation
Sto
cko
uts
Stockout Ignoring Autocorrelation Stockout Considering Autocorrelation
Average Total Costs
0.00
2,000.00
4,000.00
6,000.00
8,000.00
10,000.00
12,000.00
14,000.00
16,000.00
18,000.00
20,000.00
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.99
Autcorrelation
$
Cost Ignoring Autocorrelation Cost Considering Autocorrelation
Ignoring Autocorrelation Considering Autocorrelationφ Costs Stockout Service Costs Stockout Service
IID 3,789.09 0.11 0.89 3,789.09 0.11 0.890.1 3,799.22 0.12 0.88 3,795.28 0.11 0.890.2 3,819.20 0.12 0.88 3,815.09 0.11 0.890.3 3,855.06 0.13 0.87 3,845.03 0.11 0.890.4 3,914.24 0.14 0.86 3,901.62 0.10 0.900.5 4,006.39 0.16 0.84 3,979.33 0.11 0.890.6 4,154.06 0.18 0.82 4,096.55 0.11 0.890.7 4,406.62 0.20 0.80 4,280.87 0.11 0.890.8 4,896.30 0.25 0.75 4,591.31 0.11 0.890.9 6,187.01 0.31 0.69 5,098.00 0.11 0.890.95 8,228.63 0.36 0.64 6,059.10 0.10 0.900.99 18,832.72 0.50 0.50 9,486.20 0.11 0.89
Figure 5 Figure 6
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Error Characterization
β( C )
y = 6.6683e6.1759x
R2 = 0.9629
0.00
500.00
1,000.00
1,500.00
2,000.00
0 0.2 0.4 0.6 0.8 1
Autocorrelation
Figure 6
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Analyzing the difference - ANOVA
H0 = The autocorrelated demand does not change the average total cost of the inventory system.
HA = The autocorrelated demand does change the average total cost of the inventory system
Significance level - 0.95
P-value Hypothesis
φ Cost Ho Ha
0.1 1.43E-10 Reject Accept
0.2 1.90E-15 Reject Accept
0.3 8.89E-19 Reject Accept
0.4 1.78E-21 Reject Accept
0.5 6.36E-24 Reject Accept
0.6 5.87E-21 Reject Accept
0.7 1.09E-24 Reject Accept
0.8 3.38E-23 Reject Accept
0.9 3.71E-24 Reject Accept
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Improving candidate solutions Using PS combined with R&S improves candidate solutions. The enhancement was defined and measured by:
The number of times that a candidate solution improved upon the evaluation process.
It enhances between 40-60% candidate solution proposal
SAPSRS for AR(1) Case
61%
39%
EnhancedRegular
Figure 7
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Conclusions and Future Work
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Summary
Ignoring φ leads to errors. Substantial and significant.
It shows a better performance. Autocorrelation is high,
Stockouts are controlled. Lower costs
It demonstrated 40%-60% improvement evaluating and selecting candidate solutions.
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Future work
1. Study other types of stochastic dependent demand.
2. Compare with other inventory models.3. Study more complex supply chain
problems.4. Study the effect of autocorrelation demand
in other settings, i.e. scheduling.
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Thank you!!!!
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Questions?????
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Appendices
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Autocorrelation It refers to the correlation of a time series with its
own past and future values. +: positive/negative departures from the mean tend
to be followed by positive/negative departures from the mean.
-: a tendency for positive departures to follow negative departures, and vice versa.
Tools: 1. the time series plot, 2. the lagged scatterplot, and 3. the autocorrelation function.
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Mathematical formulation
•Stochastic formulation
y x0
( ) Min ( ) ( ( ) (0)) ( ) ( ) ( )i
y
i i i i C i i C
y
f x c y x p y c d h y d
y x0
( ) Min ( ) ( ( ) (0)) ( ) ( ) ( )i
i
y
i i i i D i iy
f x c y x p y C h y
•Continuous case
•Discrete case
*
(1 )
h CRp
CR
•Service Level
( , ) ( ) max 0, max 0,i i i i i i i iC D y c y x p D y h y D
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1. Given values of the parameters , , and
Autoregressive - AR(1)
1( )i i i
Autocorrelation
Error
Stochastic Demand
Correlation boundaries1 2, ,... i
1 1
1 2, ,... i
hh
.
Generating AR(1) demands
2i
2
1( )i i i
1i i
2. Generate from the normal distribution with a given mean and variance
3. Set
4. Set and go to 2
Examples: electronics retail, grocery foodindustry