Rafael Diaz

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


Content

Introduction and Motivation Inventory Model Simulation Optimization Method Results Conclusions and Future Work


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.


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


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


1. Inventory Model and Autocorrelated Demands

The lost sale case


The Inventory Problem

Inventory System•Lost-sale

Stochastic demand

•Serially-correlated

Control system•Continuous review

Minimizing Costs•Ordering •Penalty•Holding


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


2. Simulation-Based Optimization


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


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


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


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


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


3. Results


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


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


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


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


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










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


Conclusions and Future Work


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.


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.


Thank you!!!!


Questions?????


Appendices


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.


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


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

Rafael Diaz

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