OPTIMAL INVENTORY MODELINGOF SYSTEMSMulti-Echelon Techniques

Second Edition

Ramík, J. & Vlach, M. / GENERALIZED CONCAVITY IN FUZZY OPTIMIZATIONAND DECISION ANALYSIS

Song, J. & Yao, D. / SUPPLY CHAIN STRUCTURES: Coordination, Information andOptimization

Kozan, E. & Ohuchi, A. / OPERATIONS RESEARCH/ MANAGEMENT SCIENCE AT WORKBouyssou et al. / AIDING DECISIONS WITH MULTIPLE CRITERIA: Essays in

Honor of Bernard RoyCox, Louis Anthony, Jr. / RISK ANALYSIS: Foundations, Models and MethodsDror, M., L’Ecuyer, P. & Szidarovszky, F. / MODELING UNCERTAINTY: An Examination

of Stochastic Theory, Methods, and ApplicationsDokuchaev, N. / DYNAMIC PORTFOLIO STRATEGIES: Quantitative Methods and Empirical Rules

for Incomplete InformationSarker, R., Mohammadian, M. & Yao, X. / EVOLUTIONARY OPTIMIZATIONDemeulemeester, R. & Herroelen, W. / PROJECT SCHEDULING: A Research HandbookGazis, D.C. / TRAFFIC THEORYZhu, J. / QUANTITATIVE MODELS FOR PERFORMANCE EVALUATION AND BENCHMARKINGEhrgott, M. & Gandibleux, X. /MULTIPLE CRITERIA OPTIMIZATION: State of the Art Annotated

Bibliographical SurveysBienstock, D. / Potential Function Methods for Approx. Solving Linear Programming ProblemsMatsatsinis, N.F. & Siskos, Y. / INTELLIGENT SUPPORT SYSTEMS FOR MARKETING

DECISIONSAlpern, S. & Gal, S. / THE THEORY OF SEARCH GAMES AND RENDEZVOUSHall, R.W./HANDBOOK OF TRANSPORTATION SCIENCE - Ed.Glover, F. & Kochenberger, G.A./HANDBOOK OF METAHEURISTICSGraves, S.B. & Ringuest, J.L. / MODELS AND METHODS FOR PROJECT SELECTION:

Concepts from Management Science, Finance and Information TechnologyHassin, R. & Haviv, M./ TO QUEUE OR NOT TO QUEUE: Equilibrium Behavior in Queueing

SystemsGershwin, S.B. et al/ ANALYSIS & MODELING OF MANUFACTURING SYSTEMSMaros, I./ COMPUTATIONAL TECHNIQUES OF THE SIMPLEX METHODHarrison, T., Lee, H. & Neale, J./ THE PRACTICE OF SUPPLY CHAIN MANAGEMENT: Where

Theory And Application ConvergeShanthikumar, J.G., Yao, D. & Zijm, W.H./STOCHASTIC MODELING AND OPTIMIZATION

OF MANUFACTURING SYSTEMS AND SUPPLY CHAINSNabrzyski, J., Schopf, J.M., J./ GRID RESOURCE MANAGEMENT: State of the Art

and Future TrendsThissen, W.A.H. & Herder, P.M./ CRITICAL INFRASTRUCTURES: State of the Art in Research

and ApplicationCarlsson, C., Fedrizzi, M., & Fullér, R./ FUZZY LOGIC IN MANAGEMENTSoyer, R., Mazzuchi, T.A., & Singpurwalla, N.D./ MATHEMATICAL RELIABILITY: An

Expository PerspectiveTalluri, K. & van Ryzin, G./ THE THEORY AND PRACTICE OF REVENUE MANAGEMENTKavadias, S. & Loch, C.H./PROJECT SELECTION UNDER UNCERTAINTY: Dynamically

Allocating Resources to Maximize ValueSainfort, F., Brandeau, M.L., Pierskalla, W.P./ HANDBOOK OF OPERATIONS RESEARCH AND

HEALTH CARE: Methods and ApplicationsCooper, W.W., Seiford, L.M., Zhu, J./ HANDBOOK OF DATA ENVELOPMENT ANALYSIS:

Models and Methods

* A list of the early publications in the series is at the end of the book *

Recent titles in theINTERNATIONAL SERIES INOPERATIONS RESEARCH & MANAGEMENT SCIENCE

Frederick S. Hillier, Series Editor, Stanford University

byCraig C. Sherbrooke, Ph.D.

OPTIMAL INVENTORY MODELINGOF SYSTEMSMulti-Echelon Techniques

Second Edition

KLUWER ACADEMIC PUBLISHERSNEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: 1-4020-7865-XPrint ISBN: 1-4020-7849-8

©2004 Kluwer Academic PublishersNew York, Boston, Dordrecht, London, Moscow

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No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,mechanical, recording, or otherwise, without written consent from the Publisher

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Boston

Dedication

This book is dedicated toRosalie, the next generation of

mathematicians Andrew andEvan, and the following

generation Joshua and Michael

Contents

Dedication

List of Figures

List of Tables

List of Variables

Preface

Acknowledgements

1 INTRODUCTION1.11.21.31.41.51.61.71.81.9

CHAPTER OVERVIEWTHE SYSTEM APPROACHTHE ITEM APPROACH

REPAIRABLE VS. CONSUMABLE ITEMS

“PHYSICS” OF THE PROBLEM

MULTI-ITEM OPTIMIZATION

MULTI-ECHELON OPTIMIZATION

MULTI-INDENTURE OPTIMIZATION

FIELD TEST EXPERIENCE

THE ITEM APPROACH REVISITED

THE SYSTEM APPROACH REVISITED

SUMMARY

PROBLEMS

1.101.111.121.13

v

xv

xvii

xix

xxiii

xxix

112346789

1013141718

viii Optimal Inventory Modeling of Systems

2 SINGLE-SITE INVENTORY MODEL FOR REPAIRABLE ITEMS2.12.22.32.42.5

2.62.72.82.92.102.112.122.132.142.152.16

CHAPTER OVERVIEW

MEAN AND VARIANCE

POISSON DISTRIBUTION AND NOTATION

PALM’S THEOREM

JUSTIFICATION OF INDEPENDENT REPAIR TIMES AND

CONSTANT DEMAND

STOCK LEVEL

ITEM PERFORMANCE MEASURES

SYSTEM PERFORMANCE MEASURES

SINGLE-SITE MODEL

MARGINAL ANALYSIS

CONVEXITY

MATHEMATICAL SOLUTION OF MARGINAL ANALYSIS

SEPARABILITY

AVAILABILITY

SUMMARY

PROBLEMS

3 METRIC: A MULTI-ECHELON MODEL3.13.23.33.43.53.63.73.83.9

CHAPTER OVERVIEW

METRIC MODEL ASSUMPTIONS

METRIC THEORY

NUMERICAL EXAMPLE

CONVEXIFICATION

SUMMARY OF THE METRIC OPTIMIZATION PROCEDURE

AVAILABILITY

SUMMARY

PROBLEMS

4 DEMAND PROCESSES AND DEMAND PREDICTION4.14.24.34.44.54.64.74.84.94.10

CHAPTER OVERVIEW

POISSON PROCESS

NEGATIVE BINOMIAL DISTRIBUTION

MULTI-INDENTURE PROBLEM

MULTI-INDENTURE EXAMPLE

VARIANCE OF THE NUMBER OF UNITS IN THE PIPELINE

MULTI-INDENTURE EXAMPLE REVISITED

DEMAND RATES THAT VARY WITH TIME

BAYESIAN ANALYSIS

OBJECTIVE BAYES

1919202122

222425292930333437374142

45454648495354555656

5959616265676771727375

Contents ix

4.11

4.124.134.144.154.164.174.184.194.20

BAYESIAN ANALYSIS IN THE CASE OF INITIAL ESTIMATE

DATAJAMES-STEIN ESTIMATION

JAMES-STEIN ESTIMATION EXPERIMENT

COMPARISON OF BAYES AND JAMES-STEIN

DEMAND PREDICTION EXPERIMENT DESIGN

DEMAND PREDICTION EXPERIMENT RESULTS

RANDOM FAILURE VERSUS WEAR-OUT PROCESSES

GOODNESS-OF-FIT TESTS

SUMMARY

PROBLEMS

5 VARI-METRIC: A MULTI-ECHELON, MULTI-INDENTURE MODEL

5.15.25.35.45.5

5.6

5.7

5.85.95.105.115.125.135.145.155.165.175.185.19

CHAPTER OVERVIEW

MATHEMATICAL PRELIMINARY: MULTI-ECHELON THEORY

DEFINITIONS

DEMAND RATES

MEAN AND VARIANCE FOR THE NUMBER OF LRUS IN

DEPOT REPAIR

MEAN AND VARIANCE FOR THE NUMBER OF SRUS IN

BASE REPAIR OR RESUPPLY

MEAN AND VARIANCE FOR THE NUMBER OF LRUS IN

BASE REPAIR OR RESUPPLY

AVAILABILITY

OPTIMIZATION

GENERALIZATION OF THE RESUPPLY TIME ASSUMPTIONS

GENERALIZATION OF THE POISSON DEMAND ASSUMPTION

COMMON ITEMS

CONSUMABLE AND PARTIALLY REPAIRABLE ITEMS

NUMERICAL EXAMPLE

ITEM CRITICALITY DIFFERENCES

AVAILABILITY DEGRADATION DUE TO MAINTENANCE

AVAILABILITY FORMULA UNDERESTIMATES FOR AIRCRAFT

SUMMARY

PROBLEMS

6 MULTI-ECHELON, MULTI-INDENTURE MODELS WITH PERIODICSUPPLY AND REDUNDANCY6.16.26.36.4

SPACE STATION DESCRIPTION

CHAPTER OVERVIEW

MAINTENANCE CONCEPT

AVAILABILITY AS A FUNCTION OF TIME DURING THE CYCLE

80818385858789929596

101101103106107

108

109

110111112112113114114120122123124125125

129129130131132

x Optimal Inventory Modeling of Systems

6.56.6

6.7

6.86.96.106.116.126.136.14

6.156.166.17

PROBABILITY DISTRIBUTION OF BACKORDERS FOR AN ORUPROBABILITY DISTRIBUTION FOR NUMBER OF

SYSTEMS DOWN FOR AN ORUPROBABILITY DISTRIBUTION FOR NUMBER OF

SYSTEMS DOWN

AVAILABILITY

NUMERICAL EXAMPLE FOR ONE ORUOPTIMIZATION

MULTIPLE RESOURCE CONSTRAINTS

REDUNDANCY BLOCK DIAGRAMSNUMERICAL EXAMPLES

OTHER REDUNDANCY CONFIGURATIONS WITH 50%ORUS OPERATING

SUMMARY OF THE THEORY

APPLICATION OF THE THEORY

PROBLEMS

7 SPECIAL TOPICS IN PERIODIC SUPPLY7.17.27.3

7.47.57.6

7.77.87.97.107.11

CHAPTER OVERVIEW

AVAILABILITY OVER DIFFERENT CYCLE LENGTHS

AVAILABILITY DEGRADATION DUE TO REMOVE/REPLACE

IN ORBIT

FAILURES DUE TO WEAR OUT

NUMERICAL EXAMPLE

MULTIPLE WEAR OUT FAILURES AT ONE LOCATION DURING

A CYCLE

COMMON ITEMS

CONDEMNATIONS

DYNAMIC CALCULATIONS

SUMMARY

PROBLEMS

8 MODELING OF CANNIBALIZATION8.18.28.38.48.5

8.68.7

CHAPTER OVERVIEW

SINGLE SITE MODEL

MULTI-INDENTURE MODEL

OPTIMIZATION OF AVAILABILITY

COMPARISON OF OBJECTIVE FUNCTIONS FOR

CANNIBALIZATION

GENERALIZATIONS

DYNA-METRIC AND THE AIRCRAFT SUSTAINABILITY

MODEL

133

136

139140141142143145147

153156158159

163163164

165167170

172177178179179180

181181183186188

190193

194

Contents xi

8.8

8.98.108.118.128.138.14

8.158.168.17

DRIVE - DISTRIBUTION AND REPAIR IN VARIABLE

ENVIRONMENTS

PURPOSE OF DRIVEMODEL ASSUMPTIONS WITH DRIVEIMPLEMENTATION PROBLEMS WITH DRIVEDISTRIBUTION ALGORITHM FOR DRIVEFIELD TEST RESULTS FOR DRIVEOVERDRIVE - SEPARATE DISTRIBUTION & REPAIR MODELS

CURRENT STATUS OF DRIVESUMMARY

PROBLEMS

9 APPLICATIONS9.19.29.39.49.59.69.79.89.99.10

9.119.12

9.139.149.15

CHAPTER OVERVIEW

AIRLINE APPLICATIONS

REDISTRIBUTION AND SALE OF ASSETS

PERIODIC RESUPPLY

NO RESUPPLY: FLYAWAY KITS

ITEMS THAT ARE SOMETIMES REPAIRED-IN-PLACE

CONTRACTOR REPAIR

PROBABILITY DISTRIBUTION OF DELAY TIME

SITES THAT ARE BOTH OPERATING AND SUPPORT

LARGE SYSTEMS WHERE INDENTURE INFORMATION MAY

BE LACKING

SYSTEMS COMPOSED OF MULTIPLE SUB-SYSTEMS

ITEMS WITH LIMITED INTERCHANGEABILITY AND

SUBSTITUTABILITY

REDUNDANCY

UNFILLED DEMAND MAY NOT BE A BACKORDERSUMMARY

10 IMPLEMENTATION ISSUES10.110.2

10.310.410.510.610.710.810.9

CHAPTER OVERVIEW

COMPARISON OF VARI-METRIC WITH OTHER

STOCKAGE POLICIES

USE OF STANDARDS VERSUS MEASURED QUANTITIES

ROBUST ESTIMATION

ASSESSMENT OF ALTERNATIVE SUPPORT POLICIES

MODEL IMPLEMENTATION – AIR FORCE

MODEL IMPLEMENTATION - ARMY

MODEL IMPLEMENTATION - NAVY

MODEL IMPLEMENTATION – COAST GUARD

195195197199200201

202206207208

211211212213213214215216216218

218219

220220221221

223223

225225226227228230231231

xii Optimal Inventory Modeling of Systems

10.1010.1110.1210.13

MODEL IMPLEMENTATION - WORLDWIDE

MODEL HIERARCHIES

SYSTEM APPROACH REVISITED ONE MORE TIME

PROBLEMS

Appendix A PALM’S THEOREMA.1A.2A.3A.4A.5A.6

APPENDIX OVERVIEW

PRELIMINARY MATHEMATICS

PROOF OF PALM’S THEOREM

EXTENSION OF PALM’S THEOREM TO FINITE POPULATIONS

DYNAMIC FORM OF PALM’S THEOREM

PROBLEMS

Appendix B MULTI-ECHELON SYSTEMS WITH LATERAL SUPPLY

B.1B.2B.3B.4B.5B.6B.7B.8

APPENDIX OVERVIEW

BACKGROUND

SIMULATION DESCRIPTION

PARAMETER VALUES

DEPOT-REPAIRABLE-ONLY ITEMS

BASE-REPAIRABLE ITEMS

NUMBER OF LATERAL SHIPMENTS

SUMMARY

Appendix C DEMAND PREDICTION STUDIESC.1C.2C.3C.4

C.5

C.6C.7C.8C.9

C.10

BACKGROUND

APPENDIX OVERVIEW

DESCRIPTION OF THE DEMAND PREDICTION EXPERIMENT

RESULTS OF THE DEMAND PREDICTION EXPERIMENT FOR

C-5 AIRFRAME

RESULTS OF THE DEMAND PREDICTION EXPERIMENT FOR

A-10 AIRFRAME

RESULTS OF THE F-16 DEMAND PREDICTION EXPERIMENT

DEMAND PREDICTION FOR F-16 USING FLYING HOUR DATA

CORRELATIONS

SMALLER SMOOTHING CONSTANT FOR LOW-DEMAND

ITEMS

SUMMARY

Appendix D PREDICTING WARTIME DEMAND FOR AIRCRAFTSPARESD.1D.2

APPENDIX OVERVIEW

DESERT STORM EXPERIENCE

232232234235

237237238239241241242

245245246247249250257258258

261261263264

269

274275276281

285286

291291292

Contents xiii

D.3D.4D.5D.6D.7

LITERATURE REVIEW

PROPOSAL FOR A CONTROLLED EXPERIMENT

DATA ANALYSIS – F-15 C/D AIRCRAFT

ANALYSIS OF OTHER DATA SETS

SUMMARY

Appendix E VMETRIC MODEL IMPLEMENTATIONE.1E.2

CHAPTER OVERVIEW

VMETRIC SCREENS

Appendix F DEMAND ANALYSIS SYSTEM

References

Index

292293294296298

301301302

315

321

327

List of Figures

1-1.1-2.1-3.2-1.2-2.2-3.2-4.2-5.4-1.4-2.4-3.4-4.5-1.5-2.6-1.6-2.6-3.6-4.

6-5.6-6.6-7.6-8.6-9.7-1.7-2.7-3.

Availability vs. Cost CurveDeterministic DemandArborescent tree with ragged echelonsExample of fill rate and backorders over one year.Optimal system backorders vs. costNonconvex exampleOptimality conditions: for any item iOptimal system availability vs. costVBO(s)/EBO(s) for various mean values of the PoissonBayes’ procedureExperimental procedure for demand prediction experimentGamma and Weibull comparisonBase-depot demand and backorder calculation sequencesNormal and Laplace distributions comparedAvailability on the space station: different measures.Combinations of demand that result in y broken units at time 0Constant availability curvesRedundancy Block Diagram, communications and transmission system.

Power generation system, comparison with the optimal policyComputer-generated availability-cost curve for no cannibalizationPower generation system, optimal and 95% POS policy comparedAlternative 50% power configurationsDiagram of redundancy designComparison of optimal and 95% POS policy.Failure rate for a wear-out item.Probability distribution of time to failure for a wear-out item

459

273235374168798891

109118134136146

148151154155157162167170172

xvi Optimal Inventory Modeling of Systems

7-4.7-5.8-1.10-1.B-1.D-1.E-1.E-2.E-3.E-4.E-5.E-6.E-7.E-8.F-1.F-2.F-3.F-4.F-5.F-6.

Comparison of random failure and wear out.Cost-availability of having separate or a common ORUF-16 Tradeoffs of Aircraft Down vs.LRU EBOs

Cost of storage per cubic foot as a function of warehouse capacityComparison of estimated and actual backorders for Cases 3a-3c.Demands vs. Sortie Length for A-10 aircraftVMetric Welcome ScreenVMetric Parts LibraryVMetric Structure ManagerVMetric DeploymentVMetric Parts at SiteVMetric Run ScreenAvailability vs. Cost Progress ScreenVMetric Output Report Screen for .90 Site AvailabilitiesTypes of Analysis in DASDAS Stability AnalysisAutocorrelations for various lagsComparison of 3 ProceduresResults of Comparing 3 ProceduresQuarterly Details for 3 Predictions

179181197240260301307308310312313315316317319320322323324324

List of Tables

1-1.1-2.1-3.1-4.1-5.2-1.2-2.2-3.3-1.3-2.3-4.3-3.4-1.4-2.4-3.4-4.4-5.4-6.4-7.4-8.4-9.4-10.5-1.6-1.6-2.6-3.

George AFB Field Test ResultsGeorge AFB Simulation ResultsExample Data, Section 1.11Optimal Policies for the ExampleOptimal Policies for Negative Binomial DemandNumerical Example for Single-Site ModelTrial-and-Error SolutionMarginal AnalysisExpected backorders at any Base (Depot Stock Level = 0)Optimal Expected Backorders for Depot Stock Level = 0Optimal Expected BackordersOptimal Expected Backorders for any Depot Stock LevelMulti-Indenture ExamplePoisson and Negative Binomial Distributions with Mean =1Variance/Mean Ratio as a Function of m, MJames-Stein Simulation ExampleJames-Stein Simulation Example - More YearsDemand Prediction on C-5 aircraftBinomial Distributions with Mean = 1Goodness-of-Fit TestBinomially Distributed ObservationsValues of

% Reduction in Aircraft DownHypergeometric ExamplePV Module Input DataTranslation of Redundancy Block Diagram

111215151730303151525253677073848488929394

100124139144145

xviii Optimal Inventory Modeling of Systems

6-4.6-5.6-6.7-1.7-2.8-1.8-2.8-3.8-4.8-5.9-1.9-2.B-1.B-2.B-3.B-4.C-1.C-2.C-3.C-4.C-5.C-6.C-7.C-8.C-9.C-10.C-11.C-12.C-13.C-14.C-15.C-16.C-17.C-18.C-19.D-1.D-2.D-3.D-4.D-5.D-6.D-7.

System ResultsStockage PoliciesAlternative 50% Power ConfigurationsVariability of Demand: Single Failure, Tracking CaseVariance-to-Mean Ratio of Cycle Demand - No TrackingExample of Nonoptimal Solution Generated by Marginal AnalysisMaximum Availability vs. Probability of y or Fewer Aircraft DownAvailabilities when Bases have Equal EssentialitiesAvailabilities when Base Essentialities ChangeAvailabilities when Base Essentialities Change - Different TargetsIllustration of Repair-in-PlaceProbabilities of DelayRange of Parameter Values for U.S. Air ForceDepot-Repairable ParametersExpected Backorders under Lateral Supply (Depot Repairable)Three Simulated Backorder Solutions for T = 1,2, and 4Procedures for Predicting Mean DemandProcedures for Predicting Variance-to-Mean RatioEvaluation of Demand Prediction TechniquesDemand Prediction ProcessEvaluation ProcessList of Demand Prediction TechniquesAvailability of C-5 Airframe: $80 million budgetAvailability of C-5 Airframe: $100 Million BudgetVariance-to-Mean Ratio Over Repair Time

Availability of A-10 Airframe: $80 Million BudgetAvailability of F-16 Engine/Airframe: $80 million budgetEstimators A and B for F-16Average Demand/Item by Quarter for F-16Availability (%) Group A: Demand per QuarterAvailability (%) Group B: Demand per Flying HourCorrelations between Demand per Program Element: F-16Correlations of Demand per Program Element: A-10Correlations of Demand per 2-week Period: A-10Availabilities With Different Smoothing Constants

Desert Storm Spares DemandRegressions of Maintenance Removals on Sortie DurationRandom assignment of aircraft to treatment and control groupsData of Table D-3 Broken into Older and Newer Aircraft GroupsImpact of Sortie Number on Langley F-15C/D DemandImpact of Mission Type on F-15C/D DemandSlope % of Demand vs. Sortie Length by Aircraft Type

150151153171175186191203204205215217250251252255265266267268268270271272273274275277278279280282283284286292292293294295295296

List of Variables

The variables below will sometimes carry subscripts as defined in the text.We have used three letter mnemonics for probability density functions,abbreviated pdf below, except that p is used for the Poisson. Random variablesare abbreviated r.v. The symbol indicates an estimated value.

ROMAN LETTERSaAbBObin(x)cCdDDIeE[X]EBEBO(s)EFR(s)erl(x)exp(t)Ex(x)fFg(x)

Parameter of a pdf or type of eventAvailabilityParameter of a pdf or type of eventr.v. for backordersBinomial pdf of xItem cost ($)Cost of system spares ($)Demand/quarterSum of daily demand rates at bases/number basesr.v. for stock due-in2.718..(Euler’s constant)Expected value of the random variable XEstimated backorders from regression (Appendix B)Expected backorders with a stock level sExpected fill rate with a stock level sErlang pdf of xExponential pdf of tExpected number of periods with x demandsFraction or probabilityFlying hours/quarterProbability of x aircraft (end-items) grounded/down

xx Optimal Inventory Modeling of Systems

G(x)gam(t)h(x)H(x)hyp(x)iI

jJkKLlap(x)logLBmnNneg(x)NRTSOObs(x)OHp(x)P(x)Pr{X = x}qQrRsS(K)tTuUBvVVar[X]VBO(s)wW

Cumulative probability of x or fewer aircraft downGamma pdf of tProbability of xCumulative probability of x or lessHypergeometric pdf of xIndex for item number

Total number of itemsIndex for base number or system numberTotal number of bases (sites)Protection level, degrees of freedom for the chi-squareNumber of systems that must operateInventory position (on-hand + due-in – backorders)Laplace pdf of xNatural logarithm (base e)Lower bound on backorders (Appendix B)Average annual demandNumber of time periods, number of trialsTotal number of systems or aircraft (end-items)Negative binomial pdf of xNot Repairable This Site (1- r)Average order and ship timeNumber of periods with x demands observedr.v. for stock on-handPoisson pdf of xCumulative Poisson pdf of x or lessProbability that r.v. X equals the value xProbability that failure of an item is due to this childOrder quantityProbability of base repairReorder pointStock levelProbability that systems 1, 2 . . K operate; K + 1, . . N downTimeAverage repair timeUnits on handUpper bound on backorders (Appendix B)VolumeVariance/mean ratio of demandVariance of the random variable XVariance in backorders with a stock level sWeightSet of probabilities defined in Equation 6.6

List of Variables xxi

wei(x)xXyYzz

Weibull pdf of xNumber of demands, number in pipelineRandom variable for number of demands, number in pipelineNumber of demands, number in pipelineRandom variable for number of demands, number in pipelineMinimum number of locations of an item for parent operationTotal number of locations for an item in its parent item,

LOWER-CASE GREEK LETTERS[a] , the integer aBackorder target for an itemDemands/flying hourChi square probability distributionInterest rateShrinking constantLagrange multiplierAverage demand over the lead time, average pipelineAnnual cost of a backorder ($)Cumulative probability of x or fewer backorders due to LRUs

or SRUsEquation 5.34Probability of demand, correlation of demandStandard deviation (square root of the variance)TimeExponential smoothing constant

UPPER CASE GREEK LETTERSFirst difference h(x + 1) - h(x)

The gamma function, defined as x! for integral x.Order cost ($)

Preface

This book is written for the logistician who is concerned with one ormore systems or end equipments and with the percent of time that theyare operational. We develop the mathematical modeling techniques todetermine the optimal inventory levels by item and location for any specifiedsystem availability or total spares investment. The optimizations considertrade-offs between stock at the operating locations and the supportingdepots, known as the multi-echelon problem; between stock for an item andits sub-items, known as the multi-indenture problem.

In addition, this book is written for the graduate student in operationsresearch who is interested in the mathematics of inventory theory and itsapplication to real problems. The theoretical foundations of the requisiteinventory theory are covered in detail. As the sub-title indicates, multi-echelon (and multi-indenture) techniques are an important part of the book.We believe this is the first text to consider these topics in depth.

However, this is not primarily a book on multi-echelon inventory theory.We restrict our attention in the optimization theory to the case where thestock level is s and a reorder or repair of one unit is initiated whenever thelevel falls to s - 1. This is the only policy that we consider, because it is theoptimal policy for the high-cost, low-demand repairable items of whichsystems are composed. We do calculate order quantities that can be largerthan one for low cost, high demand items. However, because theseitems appear at lower indentures in the parts hierarchy, we are content to use

xxiv Optimal Inventory Modeling of Systems

approximations to the optimal policy, knowing that the impact on systemavailability and system cost will be slight.

The reader who is primarily interested in the mathematics ofthe general multi-echelon problem should refer to other sources. The classicreference is Clark and Scarf (1960), and there is an excellent anthology bySchwarz (1981). Several of the papers in the Schwarz anthology deal withthe multi-echelon problem, including over 200 references. Other more recentworks of note include Federgruen and Zipkin (1984) and Svoronos andZipkin (1988). Due to the complex iterative nature of the solutiontechniques for these optimal, multi-echelon policies, there have beenfew applications to date. An important exception, Cohen et al (1990), isdiscussed in Chapter 10.

In the past twenty years there have been two important, conflictingdevelopments in the management of inventories. The manufacturing sectorhas tended to place more emphasis on better planning and “just-in-time”methods to reduce investment in in-process inventories. At the otherextreme, logisticians who are responsible for the support of complexequipments such as ships, telecommunications networks, electric utilities,computer systems, space shuttles and orbiting vehicles are making use ofever more sophisticated inventory models. This is due in part to theincreasing complexity of these equipments, and the need to meet specifiedavailability targets. Central to both developments has been the tremendousincrease in computing power, computer literacy and widespread user access.

Demand forecasting and inventory modeling are becoming less importantto the former group, while they are becoming more critical to the latter.Between the extremes there are many other applications, such as those forretailers in the commercial world. In some cases retailers have been ableto shorten lead times, and depend on greater responsiveness from theirsuppliers; in others, the variability of lead times and the number ofwholesale suppliers has been increasing. Inventory theory and forecastingmay still be important for them, but there is less of a need for new andbetter techniques.

Our objective in this book is to address the problem of supporting high-technology equipments. Though many of the most natural applications arein the military sector, the techniques that we develop are appropriate forcomplex civilian programs, too. Rather than talk in abstract terms abouthigh-technology equipments and retail sites, it will be convenient in ourdiscussions to adopt military examples and refer to aircraft, operatingbases, supporting depots, etc. We hope this will make the context clearerand less academic without causing the reader to ignore other applications.

The stimulus for writing this book was a four-day (now three day) courseon spares management and modeling that I first presented in April 1989.

Preface xxv

The course has now been presented over fifty times in various locations inthe United States, Europe, and the Far East. Before each subsequentcourse, the material was revised to reflect students’ comments and theauthor’s experience. The current form owes much to the feedback fromhundreds of students.

The attendees have ranged from logisticians and engineers with extensiveexperience and doctoral degrees to managers with limited mathematicaltraining. The book is intended to appeal to a similar audience with a rangeof interests and ability. Many of the mathematical proofs are placed in theProblems and Appendices to make the text easier for the reader who has lessmathematical facility. (Calculus is unavoidable in a few sections ofChapters 4, 5, and 7, however).

I have taught inventory theory courses in graduate schools of operationsresearch - usually using Analysis of Inventory Systems by Hadley and Whitin(1963) as the principal text. That book contains some excellentmaterial, though it is out of date and out of print. However, studentscomplained that Hadley and Whitin and other texts had few real examples,and they wanted to know more about whether the models had beenimplemented. Consequently, this book includes actual data from field testsof the techniques, demand prediction studies, and from work for SpaceStation Freedom wherever possible. Furthermore, every model discussed inthe book has been programmed on personal computers, and most are beingused today.

It is important to emphasize that the models developed in this bookare all analytic. Simulation is used to verify the accuracy of the analyticmodels, but the models themselves consist of mathematical equations thatcan be solved for optimal stockage policies in an efficient manner. Theanalytic nature of the models is essential for practical application on personalcomputers or even mainframes.

The book includes a careful development of the mathematicalfoundations of the theory, appropriate for a one-semester graduatecourse. It is the author’s hope that the book will be used both by practicinglogisticians who want to keep up with the state of the art in inventorymodeling, and by graduate students of operations research who areinterested both in theory and practice.

A large part of the material in the book is based on my research. Much ofit has been published in journals such as Operations Research, ManagementScience, and the Naval Research Logistics Quarterly. However, themodeling of periodic resupply for Space Station Freedom, where there isredundancy at both the system and item levels, is too recent to haveappeared in print. Much of the demand prediction work has been describedonly in Logistics Management Institute publications.

xxvi Optimal Inventory Modeling of Systems

The material in this book begins with research performed in the early1960s. The research showed that it was possible to operate an Air Forcebase and achieve higher performance at significantly less cost for spares.Subsequently the research findings were validated in a field test in whichthe recommended stocks were actually implemented at the base, resultingin the same performance at about half the inventory cost. Thephilosophical basis for this new approach is given in Chapter 1,and the mathematical techniques in Chapter 2. It is shown that minimizingthe sum of base backorders is equivalent to maximizing availability.

In Chapter 3 the mathematics is extended to the joint optimization ofstock levels at bases and at the supporting depot. Chapter 4 treatsdemand rate estimation, and suggests techniques to model demand rates thatdo not stay constant. We show that this results in larger variance-to-meanratios than the value of 1 that characterizes the Poisson distribution.The negative binomial distribution is used to model this effect as well asthe larger variance-to-mean ratios that occur because the pipeline delaysbetween echelons and indentures are not independent. This isillustrated with a two-indenture example. We describe demand predictionstudies using actual Air Force data, and present methods for dealingwith items whose failures are dominated by wear out. In Chapter 5 wedevelop the mathematics for the combination multi-echelon, multi-indentureoptimization problem.

Chapter 6 and 7 are concerned with the periodic resupply problem forrepairable items, and its application to Space Station Freedom. One of thenew results presented in this book for the first time is an optimizationtechnique where redundancy is modeled at both the system and item level.This has important implications in the design of systems. The same modelcan be used for long-term procurement problems and for short-termresupply manifesting of the space shuttle; in the latter case the age ofinstalled units subject to wear out can be used to improve the set of itemsresupplied in a given shuttle flight.

In some applications, maintenance performs cannibalization: consolid-ation of item shortages on the smallest number of end-items. Themathematics for cannibalization is different; this is the subject of Chapter 8.We show that it is possible to use the same objective function, expectedavailability, though the results are only quasi-optimal. We note thatregardless of the procurement model used, it is possible to achievebetter short-term performance if information on the location and condition ofassets at each point in time is used in decision-making. The DRIVE(Distribution and Repair in Variable Environments) model for distributionof serviceable assets from depot to bases and for prioritization of repair atdepot is such a model. We describe some of the benefits and problems of

Preface xxvii

implementing such a technique. Chapter 9 is new in the second edition,describing a dozen problems that can be modeled with the same theory,including modifications for commercial airlines, variations in the resupplyand repair assumptions, treating sites that operate aircraft and support othersites.

Finally, Chapter 10 is concerned with many of the real-world problems ofusing models. What are the advantages and what are the limitations?Implementation experiences by several different user groups are presented.The appendices provide mathematical proofs of Palm’s theorem, anddiscussions of special topics such as lateral supply between bases anddemand prediction studies. Appendices D-F are new in the second edition.Appendix D is concerned with predicting spares demand in a wartimeenvironment, based on observations from Desert Storm. Appendices E and Fdescribe implementations of the optimization theory (VMetric) and thedemand prediction theory (Demand Analysis System).

This book differs from other books on inventory theory in severalimportant ways. We use the system approach, whereby we focus on theavailability of the end-items such as aircraft, and then determine theappropriate inventory policies. We believe that logisticians should providemanagement with cost-availability curves, from which an optimal systemtarget can be chosen. In fact, the system approach is used in several ways -not just in the determination of stockage levels but in demand prediction andin the evaluation of alternative policies.

Repairable items are the central focus here, because they most directlyrelate to aircraft availability, whereas consumable items are the focus inmost books. We devote a lot of time to multi-echelon, multi-indentureinventory theory, though these are only given a couple of summary pages inmost texts

Although only four chapters and appendices are totally new in this secondedition, I have made extensive revisions in all chapters, adding numerousworked-out examples. The first edition was published twelve years ago, andmany things have changed since that time as reflected in the new edition. Forexample, the personal computer models in 1992 did not use Windows, nowthe standard; the original book was done in WordStar, not Word, requiringan archaelogical project on the part of my son, Evan, to reconstruct theoriginal manuscript.

I can be reached by email at csherbro@alum.mit.edu.

CRAIG SHERBROOKECamarillo, CA.

Acknowledgements

The materials in this book that were developed by me were performedunder the sponsorship of several organizations. As a graduate student at theMassachusetts Institute of Technology during 1958-1960, I worked on Armyinventory problems. From 1962-1969 at the RAND Corporation, much ofthe material in Chapters 1-4 was developed for the Air Force. From 1981-1993 I was a consultant to the Logistics Management Institute where thematerial in Chapters 5 and 8 as well as Appendices B-D was done for theAir Force; Chapters 6 and 7 for NASA and Space Station Freedom. Atother times the author has worked on Navy and Defense LogisticsAgency studies, and consulted with private companies on inventoryproblems.

It is impossible to thank everyone who has influenced and helped me,because of the large number of such people. My earliest productive workwas largely spurred by a collaboration with George Feeney in my firstdays at the RAND Corporation, under the supportive management of thelate Murray Geisler. Later at the Logistics Management Institute (LMI) Iwas fortunate to work with Mike Slay, one of the most creative logisticsmodelers I have known. Rob Kline worked with me on the SpaceStation application, and T. J. O’Malley supervised the research for the AirForce and NASA. The DRIVE model was developed jointly with Jack Abelland Lou Miller of RAND. Several others deserve thanks for encouraging meto write the book including Saul Gass, Jack Muckstadt, Ben Blanchard, andRod Stewart. Bob Butler should be included in this category, because he firsturged me to teach the course on which the book is based. My mathematiciansons, Andrew and Evan, suggested several changes to the notation andexposition, all of which were incorporated. The notation would have been farmore confusing but for the patience of the editor, Isabel Stein, among whosemany contributions was the insistence that a given symbol have the samemeaning from one chapter to the next.

Finally I want to thank LMI and its former President, Bob Pursley,for providing me with some time to write the first edition; thanks also go toseveral colleagues who critiqued individual chapters including ChrisHanks, Rob Kline, and Sal Culosi. Mike Slay spent many hourspatiently looking for errors and suggesting improvements throughout thebook. Though I am responsible for all remaining errors of commission andomission, I am most grateful that so many have were eliminated by theirefforts.

xxx Optimal Inventory Modeling of Systems

The second edition was motivated by the large number of things that havehappened in spares logistics over the past eleven years. I have added nearly ahundred pages including a new chapter, Chapter 9, three new appendices (D-F), and substantial revision and expansion of several chapters including moreworked-out examples. Unfortunately, the first edition had a large number oftypos and some substantive errors in Chapter 6 concerning finitepopulations. I apologize because it is hard enough to read an advanced textwithout encountering errors.

The new edition would not have happened without the strong support ofFred Hillier, whose distinguished career in operations research is wellknown. I appreciate the help of many people in updating the book includingRich Moore, Bob McCormick, Norm Scurria, Jim Russell, Meyer Kotkin,Sal Culosi, Randy King, and Mike Slay who brought me up-to-date onimplementation by the services; to Ken Woodward, the architect of theVMetric interface and much more, who assisted in getting the latestinformation on VMetric; and to my wife Rosalie who has become a wizardat downloading pictures. Deborah Doherty and others at Kluwer helped meto overcome the sometimes mystifying ways of Microsoft Word and theKluwer templates where objects can appear and disappear capriciously.

C.C.S.