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The second part of the dissertation studies a two-echelon system a single central repair depot and multiple local inventory stocking centers for repairables with local-center-dependent d

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SERVICE PARTS LOGISTICS:

MODELING, ANALYSIS AND APPLICATION

Yunzeng Wang

A DISSERTATION

In Operations and Information Management For the Graduate Group in Management Science and Applied Economics

Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

UMI Number: 9830011

UMI Microform 9830011 Copyright 1998, by UMI Company All rights reserved

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copying under Title 17, United States Code

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DEDICATION

TO MY WIFE AND OUR DAUGHTER

Acknowledgements

I wish to express my sincere thanks and deep appreciation to my advisor, Professor Morris A Cohen, for his encouragement, guidance and financial support throughout the development of my research at the Wharton School He has constantly encouraged me to approach research problems in the direction of relevance, without sacrificing rigor

I am indebted to Professor Yu-Sheng Zheng for all that he taught me and for all

of his help during the last five years His sharp critiques and comments frequently led

me to clarify my ideas and to develop key results

I want thank my dissertation committee members, Professors Karen Donohue, Noah Gans and Enver Yucesan, for their suggestions and recommendations Their advice has significantly improved this dissertation [ am thankful to Professor David Pyke, of Dartmouth College, for his generous help and constant encouragement

This dissertation is based on a research project with Teradyne, Inc., a Boston- based electronic equipment manufacturer, and I owe special thanks to the following people there Jason Anton, Brian Amero, John Leutien, Ted Miller and Randy Stone They were extremely helpful in providing me with data and valuable managerial knowledge of service parts logistics systems My more than four years of experience working with them will surely be beneficial throughout my career

I would like to thank all the people in the Operations and Information Management Department for providing a friendly and supportive environment while I was pursuing my studies [ am especially appreciative of Marge Weiler’s assistance during that time I am also grateful to my peer doctoral students for their friendship, which will always recall good memories of those doctoral student days

Last, but not least, I am indebted to my wife, Sylvia, and our daughter, Diana, for their tremendous patience and fortitude in enduring my absence from home Their

ABSTRACT SERVICE PARTS LOGISTICS:

MODELING, ANALYSIS AND APPLICATION

Yunzeng Wang Morris A Cohen

Service parts logistics is concerned with providing support through service parts availability to the after-sales service operations of equipment manufacturers This dissertation studies strategies and methodologies for managing service parts logistics systems The approach is to combine extensive field study with rigorous modeling analysis The research was motivated

directly by a project with a major US electronic equipment manufacturer to

improve the operations of their world-wide service parts logistics system

The first part of the dissertation presents our preliminary study of the company’s service parts logistics system We build two simple models (one for consumables and the other for repairables) to approximate the company’s complex system and to show their effectiveness in providing improved policies Through this phase of study, we draw important insights regarding applications

of operational models to real-world systems and abstract research problems

These new problems are then rigorously analyzed in the next two parts of the

dissertation

The second part of the dissertation studies a two-echelon system (a single central repair depot and multiple local inventory stocking centers) for repairables with local-center-dependent depot replenishment lead times We characterize the system by deriving the various exact performance metrics and then analyze system optimization problems The following two features in the solution structure differentiate our model fundamentally from those in the literature: 1)

the random delays at the depot of local center replenishment orders are different

from center to center; 2) the lead time variances have a direct effect on system performance

The third part of the dissertation considers a two-echelon system for repairables with two classes of customer service: an emergency class, where a customer demand needs to be satisfied immediately upon its arrival, and a non- emergency class, where a customer demand needs not to be satisfied until a due date Using both transient and steady state analysis approaches, we derive exact system performance metrics and characterize various properties We then carry

out numerical studies, based on company data, to quantify and to compare

system improvement potential over company practice

Approximation

2.2.3 Axsater’s Approach

2.2.4 Svoronos' and Zipkin'sModel

2.2.5 The OPTIMIZER Model

2.2.6 Other Extensions 2.3 Concluding Remarks

3

4

Identifying Improvement Opportunities for a Service Parts Logistics

3.1 Introduction - eee eee eee ee 47 3.2 The Service Parts Logistics System and The Parts Flow Processes 50 3.3 The Diagnostic Process 52 3.3.1 Data Collection and a Pareto Analysis 52 3.3.2 Policy Recommendations for Controlling Consumables 54 3.3.3 Improvement Analysis for Repairables 58 3.4 Conclusions 00.00 eee eee cece e eens 70

A Two-Echelon System for Repairables with Local-Center-Dependent

4.1 Introduction 0.0 eee eee eens 87 4.2 The system, Assumptions and Notation .- 90 4.3 System Performance Evaluations — The Depot Problem 94 4.3.1 The Steady-State Depot Inventory Level Distribution 94 4.3.2 The Distributions of Random Delays at the Depot 97 4.4 System Performance Evaluation — The Local Center Problem 109

4.4.1 The Steady-State Outstanding Orders 109

4.4.2 Customer Service Level and Waiting Time lil

4.5.1 Minimizing Total Cost of Inventory Holding and Customer

Backlogging 113 4.5.2 Minimizing Total Cost of Inventory Holding and Customer

Backlogging Subject to Individual Service Constraints 115 4.5.2 Minimizing Total Inventory Value Subject to a Service Level

Constraint 117 4.6 Numerical Studies and Comparison 119 4.7 Concluding Remarks 126

A Two-Echelon System for Repairables with Two Classes of Customer

5.1 Introduction 2.2.0 ccc eee eee 130 5.2 Model Description, Assumption and Notation 133 5.3 System Performance Evaluation — The Depot Problem with a Single Local Center 140 5.3.1 The Depot Inventory Level Distribution 141 5.3.2 The Service Level for Local Center ES

Replenishment Orders 146 5.3.3 The Service Level for Local Center RR

Replenishment Orders 147

5.3.4 The Probability Distribution of Random Delays of Local

Center ES Replenishment Orders 150

5.3.5 The Probability Distribution of Random Delays of Local

Center RR Replenishment Orders 156

5.4 System Performance Evaluation — The Depot Problem with

N Local Centers 157 5.4.1 The Depot Inventory Level Distribution 157 5.4.2 The Service Level for Local Center ES

Replenishment Orders 160 5.4.3 The Service Level for Local Center RR

Replenishment Orders 161 5.4.4 The Probability Distribution of Random Delays of ES

Replenishment Orders from a Given Local Center 164 5.4.5 The Probability Distribution of Random Delays of RR

Replenishment Orders from a Given Local Center 169 5.5 System Performance Evaluations — The Local Center Problem 171 5.6 System Optimization 172 2.7 Numerical Results 173 5.8 Concluding Remarks 175

6 Major Contributions 178 6.2 Puture Research 184

List of Tables

3.1 Service Level Improvement Through Various Lead Time Reduction

4.1 The Expected Delays for a, =50,a@, =25, A, =A, =0.02

4.2 The Expected Delays for a =50, a =15, A) =A, =0.02

4.3 The Expected Delays for a, =50, ay =25, A, =A, =0.05

4.4 The Expected Delays for a, =50, @ =15, 4, =A, =0.05

4.5 Comparison Between Our Model and METRIC (A, =A, =0.1)

4.6 Comparison Between Our Model and METRIC (A, =A, =0.05)

4.7 Effect of Lead Time Variances on Performance (4, = 4, =0.05)

5.1 Inventory Values of Different Models ($1,000)

5.2 Improvement of Different Models Over Company Practice (%)

65

106

106

107

107

123

List of Figures

1.1

2.1

2.2

4.1

4.2

2.1

5.2

5.4

The Two-Echelon Service Parts Logistics System 6

A Two-Echelon System for Repairables 21

A Two-Echelon System for Consumables 23

Random Delay at the Depot_ 98

The Difference in Delay at the Depot 108

The Operational Process ofa RR Demand 135

The FDFS Rule at the Depot 137

The Depot Inventory Balance 142

The Random Delays of ES Replenishment Orders 150

Chapter 1

Introduction

1.1 Research Background

Service parts logistics systems provide support, through service parts

availability, to the after-sales service operations of equipment manufacturers

This type of logistics system exists widely, in both military (e.g., Navy and Air Force) and commercial industries (e.g., airlines, automobile, telecommunication computer and other high-tech industries) Inventory value tied to a company’s service parts logistics system often ranges from millions to billions of dollars (Cohen and Zhang 1997)

Research on service parts logistics started over 30 years ago Sherbrooke (1968) developed the well-known METRIC (Multi-Echelon Technique for Recoverable Item Control) model for the management of repairable items This

model was implemented by the US military Sherbrooke`s seminal work

generated a whole new research area, resulting in an extensive list of academic publications Most noticeably, among many others, are the works by Simon (1971), Muckstadt (1973), Deuermeyer and Schwarz (1981), Graves (1985), Cohen et al (1986, 1988, 1989 and 1992), Lee (1987), Lee and Moinzadeh (1987a,b), Axsater (1990a,b, 1991), Svoronos and Zipkin (1991), etc Multi- echelon models have huge potential to improve the operations of complex real world service parts logistics systems Muckstadt and Thomas (1980) showed that, in some cases, inventory costs can be reduced by over 50% and Cohen et al (1990) documented a savings of hundreds of millions of dollars in inventory investment by IBM

In practice, during the past two decades service parts logistics has become an increasingly important business function critical to the success of product manufacturers, especially in high technology companies Technology advancement has expedited the introduction of new products and reduced the length of product life cycles The installed base of specific product models has

decreased Marketing globalization has led to geographically dispersed

customers Demand for service parts as a result is low and unpredictable At the same time, the standards (or expectations) for customer service are increasing and becoming more complex (e.g., differentiated service, response time window, product vs part support goals, etc.) As a consequence of all these factors, the management of service parts logistics has become more complex, more expensive and more important

While there clearly is an increasing demand for methodologies and

models to improve the management of existing service parts logistics systems,

application of theoretical models to complex real world problems has not

occurred, with a few exceptions, e.g., the work by Sherbrooke (1968) for the military and those by Cohen et al (1986, 1988 and 1990) for computer industry

In practice, most companies still depend on ad hoc rules to control the operations

of their systems These rules are often far from optimal, and thus there is a huge potential for improvement

There are two principal reasons for this situation One is that most of the

theoretical models existing in the literature make stringent assumptions that

eliminate their relevance for complex systems Most noticeably, research on

systems for repairable items has not seen much progress since the METRIC

model which was developed for the special case of the military Yet, METRIC

and its derivative models cannot deal with many of the operational features observed in commercial service parts logistics systems, e.g., multi-class customer services, geographically dispersed customer sites, etc The second reason is that not much effort has been made by academic researchers to link models closely to real world problems and to address issues related to model implementation

The purpose of this dissertation is to study strategies and methodologies for managing service parts logistics systems Our research approach is to combine extensive field study with rigorous theoretical analysis We build model frameworks that are very closely related to systems existing in practice Through rigorous analysis of these models, we make significant contributions to the theory of service parts logistics Second, we try to link theory directly with

application That is, through applications, we address important issues

associated with model implementation and improvement of company practice

This research has been motivated directly by a project with a major US electronic equipment manufacturer to improve the operations of their worldwide service parts logistics system The company manufactures electronic tester machines, which are used widely by semiconductor and other high technology

companies as on-line equipment in capital intensive production lines Upon

failure, the testers can become a bottleneck in these lines (see, e.g., Ou and Wein

estimated that one hour of tester down-time can cost a typical user as much as

$50,000

The testers are complex electronic devices, composed of many different parts and circuit boards The boards, although relatively reliable, are subject to random failures Therefore, a very important part of this company’s business is

to provide their customers with a prompt and highly reliable parts service, which

is demanded by customers and is critical to the company’s long term success Furthermore, most of these parts are expensive items, with a unit cost of up to

$25,000 Thus, it is financially worthwhile for the company to collect those

defective parts from the customers, to repair them and to reuse them

The company runs their parts provision and repair function through a world-wide service parts logistics system, that is organized in a two-echelon

structure: a single central repair depot located in the US and multiple local inventory stocking centers located close to customers around the world The

operations of the logistics system is illustrated in Figure 1.1 A local center satisfies parts demand from customers, and replenishes its inventory from the central depot Defective parts are collected at the local center from customers and then shipped back to the repair center for repair and replenishment to the depot inventory There are over 10,000 different parts managed in the system

with a total inventory investment value of nearly 50 million dollars

The company provides two types of customer part service — emergency

service and non-emergency service For the emergency service, the local center ships out a good part immediately upon receiving notification from a customer

that a part is required For the non-emergency service, demands are not satisfied immediately Rather the customer is quoted a due date of, say, 20 days from his/her service request At the due date, the local center will ship out a part to the customer

The key issues regarding the management of this service parts logistics system are:

e At the strategic level, where and how many stocking locations should be set up?

e At the operational level, what is the proper inventory stocking level for each part and at each stocking location in the logistics network? The effective management of this system is further complicated by the following factors: low and stochastic parts demand - as low as a few units per year system-wide; expensive items; high customer service level requirements -

same day demand fill rate over 90%; complex managerial processes - inventory

planning and procurement, transportation, defective repair, etc.; stochastic lead times - procurement, repair, transportation, etc

The goals of this research are two-fold First, by studying extensively this company’s service parts logistics system, we abstract various problems which are both common to the management of this type of systems and of theoretical interest Through modeling and analysis of these problems, we

develop frameworks and methodologies for the effective operation of these

systems These analyses will lead to significant extensions to the body of knowledge associated with multi-echelon systems for repairables Second, we apply these research results directly back to the company in order to dramatically improve their management practices

1.2 Organization of the Dissertation and Contributions

The dissertation includes three major parts Part I (Chapter 3) presents our preliminary study of the company’s service parts logistics system During

this initial phase of study, we performed a detailed logistics process analysis —

material flow process, inventory planning and control process, etc We collected and analyzed huge amounts of data Applying our knowledge of inventory theory, we then developed two simple models (one for consumable items and the other for repairable items) for inventory planing and control of the company's multi-echelon service parts logistics system The models allocate inventory for each part and at each stocking location in the system in a manner which considers the effects of parts usage rates, unit costs and various lead times

These models are rich enough to capture key characteristics of a complex

modeling literature (e.g., the two classes of customer service) They are,

however, simple enough to be readily understandable, which makes it easier to

communicate our findings to company managers Furthermore, according to our test results, these models are very effective in providing good inventory planning

and control policies In particular, compared with the company's current

practice, for consumables, customer lateness can be reduced by over 90% with limited increase in inventory investment (less than 3%); and, for repairables, inventory investment can be reduced by 35% and, at the same time, customer

service level can be improved by 4%

Through this phase of the research, we abstracted key problems to be rigorously analyzed in the next two parts of the dissertation Furthermore, we

draw the following important conclusions regarding application of operations

models: 1) For complex logistic systems, basic models can be very useful for both operational control and strategic analysis purposes, i.e., the policies recommended by these models can dominate the decision rules used in practice 2) A successful application of a basic model is often not an easy task It must

build on a thorough understanding of both the theory and the operation of the

underlying logistics processes Thus, attention must be paid to the choice of the

model, the definition of its parameters, the interpretation of its assumptions, the

data collected, the estimation of parameter values, etc 4) The simplicity of basic models and policies greatly enhances their potential applicability Both quantitative and qualitative insights can be communicated effectively to managers As a consequence, the models and/or policies generated by the models can be understood and actually implemented

The next two parts of the dissertation each study a different logistics system for repairable items (that were motivated by the field study) and makes significant theoretical contributions to the literature

Specifically, Part II (Chapter 4) considers a two-echelon system (a single central repair depot and multiple local inventory stocking centers) with local- center-dependent depot replenishment lead times In this system, customers arrive at the local centers for good items and return their defective items A local center replenishes its inventory from the central depot Its replenishment

lead time includes a random delay/waiting time at the depot (due to random

stocking-out) and a transportation time The central depot replenishes its inventory from defective return and repair process Its replenishment lead time will thus include a transportation time from a local center plus a repair time at the depot

The local-center-dependent depot replenishment lead time is due, for example, to the fact that the transportation of defective items to the depot takes different amounts of time from different local centers In our company’s system, the transportation to the depot from a US local center takes, on the average, 2 days, while, from an international local center, it takes 2 weeks

Our model assumptions for the system are: Poisson customer arrivals; one-for-one or (S-1, S) inventory replenishment policies; independent and

identically distributed depot replenishment lead times (from a given local center); and full back-logging unsatisfied demand

Our major analytical contribution is due to the following important

observation for this system While all replenishment orders from different local centers share the same good parts shelf at the depot and are satisfied according

to the first-come-first-served rule, the random delay at the depot of orders from different local centers will be different This is because each depot replenishment order is trigged by a local center order to the depot As a

consequence, when the depot orders associated with a specific local center have

short lead times, they will arrive quickly and, hence, help to reduce the random delay at the depot of orders from this local center This key insight allows us to derive the probability distribution functions of random delays of orders from different local centers

All the related models (e.g., the well-known METRIC) in the literature have ignored the existence of local-center-dependent depot replenishment lead times Thus, orders from different local centers are assumed to have the same

random delay at the depot — one of the major differences in solution structure

from our model

A second major difference in solution structure between our model and

the METRIC is the effect of lead time variances on system performance While METRIC predicts that change of lead time variances has no effect, our model

demonstrates that an increase in variances will cause system performance to deteriorate

We formulate various system optimization problems and _ provide

numerical studies to show: 1) ignoring depot lead time differences across local

centers can degrade system performance dramatically; 2) the effect of change in lead time variances on system performance can be significant

Part III of the dissertation (Chapter 5) studies a two-echelon system for repairables with two classes of customer service The two-echelon system we consider has the same structure and assumptions as that in Part II, except for the following important feature There are two classes of customer service: emergency class, where a customer demand needs to be satisfied immediately upon its arrival, and a non-emergency class, where a customer demand need not

be satisfied until a due date At the due date, a non-emergency customer has to

be satisfied immediately if there is available inventory So, in that sense, a non- emergency customer becomes an emergency customer at the due date The operations corresponding to the two classes of customer demands interact with each other in the system, since they share the same good item inventory at each location and are satisfied according to First-Due-First-Served rule

To our knowledge, no previous work in the literature has been carried out to analyze systems with features similar to those in this system Our major technical contributions through analyzing this system include: 1) We characterize the system by deriving the following performance measures for each of the two service classes and at each location: inventory level distribution; customer service level; customer waiting time distribution 2) We show analytically that the service level of non-emergency customers is always higher than that of emergency ones in the system Due to the existence of the two

evaluations becomes an even more challenging problem For example, the well-

known Palm’s theorem (Palm 1938), which is used widely for analyzing inventory systems for repairables (e.g., Sherbrooke 1968), cannot be applied directly to the analysis of this system Our approach is to analyze the inventory stochastic processes in the system directly and to draw heavily on M/G/o queuing system results

We formulate an optimization problem to minimize the total inventory investment subject to a system-wide customer service level constraint, where the

decision variables are the inventory stocking levels for each item and at each

location We then provide numerical studies based on company data to compare the performances of this accurate model and those models developed in Chapter

3 and Chapter 4 as approximations to this system

Chapter 2

Literature Review

In this chapter we review various inventory models related to service

parts logistics operations Since service parts, especially for repairable items,

are characterized by high unit costs and low usage rates, the so-called one-for- one or (S—1, S) policies are often used for the inventory control in this type of systems Thus, our focus will be on models with one-for-one policies, while we will also discuss briefly other models when they are relevant We first review models for single location systems and then those for multi-echelon systems,

2.1 Single Location System and One-For-One Policy

An important building block for analyzing multi-echelon systems is a single location system with the following parameters: Customer demands arrive according to a Poisson process with rate 4 Demands are satisfied according to first-come-first-served rule and all unsatisfied demands are fully backlogged The inventory replenishment lead times are independent and identically

distributed with a mean value of L and any general distribution function The

system operates according to the so-called one-for-one or (S—1, S) policy That is, each time one or more units are demanded, the inventory position drops

below S and an order of an equal number of units is placed immediately Thus, the system will keep the inventory position at the fixed level of S constantly

Note that the net inventory (on-hand minus backorders) becomes negative when

a backorder condition exists

The one-for-one policy, instead of a batch ordering policy, is proper for

controlling of items which are characterized by high unit costs and low usage

rates, since the economies of scale in ordering will then be insignificant compared with holding and shortage costs

Scarf (1958) appears to be the first one to consider this system He

identifies the analogy between this model and an M/G/o queuing system At

each placement of an order, one may think of a customer entering service in the queuing system The replenishment order lead times in the inventory system corresponds to the customer service time in the queuing system; the number of customers (i.e., the number of busy servers) in the queuing system corresponds

to the number of outstanding orders The well-known Palm’s Theorem (Palm 1938) states that if customers arrive according to a stationary Poisson process and service times are independent and identically distributed random variables with finite mean, then the steady-state probability distribution of the number of busy servers is a Poisson random variable with the mean value of customer arrival rate multiplied by the mean service time, independent of the form of the

service time distribution In our inventory problem this implies that the steady-

state number of outstanding orders, denoted by X, is a random variable having

a Poisson distribution That is,

Given the distribution of the number of outstanding orders, the expected back orders, E[B], and the expected net inventory level, E[/], can be computed respectively as

per unit and unit time The steady-state expected total cost per time unit is

C(S) = E|l[S~ X]" + of x-s]*} (24)

It is easy to show that C(S) is a convex function As a consequence, the optimal inventory stocking level can be easily determined

Although traditional inventory models often involve cost minimization,

in the context of service parts system it is more common to use one of two types

of service level criteria (Nahmias 1981) The first service level criterion is

commonly known as fill rate which is the probability of non-stocking-out at a

random point in time, that is, the probability for a customer demand to be

satisfied without delay Given that the number of outstanding orders is a Poisson random variable, the fill rate service level can be computed as

In steady state, each demand has the same probability of being delayed

A(1—) gives us the expected number of late demands per unit time in the

system The second service criterion is based on looking at the number of backorders in the system at a random point of time, which is the expected number of backorders in steady-state, i.e E[B]

Each of the two service criteria has its drawbacks in measuring the system performance or effectiveness The fill rate is only concerned with the number of delayed demands; it completely ignores the length of delay experienced by a demand The expected backorder criterion considers the average unit-years of demand delay per year; a delay of one demand for ten

hours is considered to be equivalent to a delay of ten demands each for one hour

Another dimension of interest for the single location system is to determine the delay or waiting time required to fill a demand arriving randomly

to the system The expected waiting time can be computed by direct application

of Little’s Law (Little 1961) from queuing theory, which says that the expected

queue length is the product of the arrival rate and the expected waiting time of

an entering customer, independent of the form of the inter-arrival or service time

is the expected number of backorders E[B] (the expected queue length of backorders) divided by the arrival rate 2 Kruse (1980) and Berg and Posner (1990) derive the customer delay/waiting time distribution

Several extensions have been made to the above basic single location system Galliher et al (1959) and Hadley and Whitin (1963) consider the system with lost sales The steady-state number of outstanding orders then become a truncated Poisson random variable That is,

eA LE (x), x=0, 1, 2, .,

the compound Poisson probability of x demands is

x 4y

A(x|A) = yf exp(-a): f” (x), x=0, 1, 2, , (2.7)

y=0 7°

where, f € (x) is the k fold convolution of ⁄ i} Feeney and Sherbrooke

consider both of the cases of full backlogging and lost sales They show that, for

the full backlogging case, the distribution of the steady-state number of outstanding orders X is given by

with state dependent arrival rates He also mentions that the results hold for arbitrarily distributed lead times as well

complemented Higa’s results by deriving an expression for the exact waiting time distribution when replenishment lead times are constant

2.2 Multi-Echelon Systems

When items are delivered to customers over a large geographical region,

it is often practical to have multiple inventory stocking locations The overall system often has a hierarchical structure: the lower echelon locations are located closer to customers and the upper echelon locations supply inventory to the lower echelon For repairable items, the repair facilities are often centrally located at the upper echelon

Figure 2.1 A Two-Echelon System for Repairables

Figure 2.1 is a typical two-echelon system for repairables In this

system, there is a single central repair depot and multiple local stocking centers

A local center stocks good/repaired items to satisfy customer demands directly

21

and it replenishes its inventory from the central depot All defective items

returned to the local centers from customers are shipped back to the central

depot for repair Depot inventory are replenished from those repaired items

Inventory systems of this type are usually managed in practice using adaptations of single location models But Muckstadt and Thomas (1980) have shown that special multi-echelon methods may reduce the total costs substantially (in some cases by about 50 per cent) compared with an application

of single location techniques

The fundamental problem in connection with a multi-echelon system is

to find the best balance between central stock (at the upper echelon) and local stock (at the lower echelon) It is clear that inventories at different levels will support each other Local inventory is advantageous in the sense that it is closer

to customers The central inventory, on the other hand, can take the advantage

of risk-pooling to provide service to customers at all locations Also, with a large central inventory, orders from the local centers will rarely be delayed and,

as a consequence, local centers’ safety stocks can be reduced The optimal solution of a multi-echelon system problem will depend on system structure, demand processes, lead times, cost parameters, etc

In this sub-section, we review various models dealing with multi-echelon systems Our focus will be on systems with the following features: a two- echelon system structure, continuous review and one-for-one inventory control

policies and Poisson demand arrivals For review on general multi-echelon

A very important and simplifying consequence of a continuous review and one-for-one policy is that Poisson demand arrivals at the local centers will give a demand process at the depot which is also Poisson, since the ordering process will replicate the demand process at each location and the demand at the depot is simply the superimposition of orders from all local centers

In our review, we will try to emphasize the main characteristics of different technical approaches and thus will sometimes not give a complete

description of the models considered in the original papers We will also discuss

other various extensions, e.g., compound Poisson arrivals, batch-ordering, lateral transshipment, etc

Customer Demands

Figure 2.2 A Two-Echelon System for Consumables

We note that although models in the literature are often labeled either for

a repairable system (e.g., Figure 2.1) or for a consumable system/distribution system (e.g., Figure 2.2), it is easy to realize, as pointed out by Axsater (1993), that the two type of systems are almost identical from the operational point

23

view All the assumptions are the same, except the following: while the central

depot in Figure | replenishes its inventory through the defective's return and

repair process, the depot in Figure 2 through the ordering process from an outside supplier The defective's return and repair lead time in Figure 1 corresponds to the replenishment order lead time in Figure 2 Thus, models for a

system for repairables can be applied to a system for consumables and vice

versa But, note that one can only draw this analogy when the lead times of

defectives’ return and repair in Figure | are nor local center dependent (i.e., the defectives’ return and repair lead times are the same, in a stochastic sense, across

all local centers) Indeed, all the models in the literature conform to this scenario, though, by assumption sometimes Our model in Chapter 4 will relax this assumption and address related modeling and analysis issues

2.2.1 The METRIC Model

Sherbrooke (1968) develops the well-known METRIC model (Multi- Echelon Technique for Recoverable Item Control) It is important in that it appears to capture many of the significant features of the problem of determining suitable stocking levels in a large-scale repairable item inventory

system It provides a sound approach/approximation to the analysis of the

problem

Consider the two-echelon system as in Figure 2.1 We introduce the following notation and assumptions regarding the operations of the system:

A, = the Poisson customer demand arrival rate at local center 7,

Lạ = the expected replenishment lead time for the depot Physically, this lead time includes the defective item’s transportation time from a local center and the repair time at the repair center It is assumed that the depot has

the same replenishment lead time from different local centers Second,

the lead times corresponding to different demands are assumed to be independent and identically distributed

S, = the order-up-to inventory level at local center ,

So = the order-up-to inventory level at the depot,

[,, = the random on-hand inventory level at local center 7 in steady state,

[y = the random on-hand inventory level at the depot in steady state,

B, = the random backorders at local center n in steady state,

By = the random backorders at the depot in steady state,

W,, = the random delay/waiting time of customer demand due to stock-out at local center 1 in steady state,

Wo = the random delay/waiting time of local center replenishment orders due to stock-out at the depot in steady state,

O,, = the random outstanding orders at local center n in steady state,

Op = the random outstanding orders at the depot in steady state

A key issue in designing a multi-echelon inventory system of this type is

to determine the proper inventory stocking levels at various locations The common approach to this problem takes two steps (Nahmias 1981) The first step is to characterize, exactly or approximately, the system performance (e.g., service level, waiting time distribution, expected inventory and backlog levels, etc.) for a given set of inventory stocking levels The second step is to optimize the system in terms of certain given criteria (objective function and constraint)

by searching systematically over the possible stocking level combinations

The following is the approach adopted in METRIC for characterizing the system First, consider the depot Due to the assumptions of i.i.d replenishment lead times and Poisson demand arrivals, the depot can be described as a single location system with parameters of Lo, Sg and 4) As a consequence, one can apply Palm’s Theorem (Palm 1938) to specify the outstanding orders as a simple Poisson random variable Thus, the expected backorders, expected on-hand