[PDF]Classical Topics In Inventory Control

Classical Topics


In


Bazargan Inventory


Control





November 2021


Classical Topics in


Inventor
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Hamid Bazargan





Classical Topics in inventory Control





Co my parents
The late Mohammad Ali Bazargan (1905-1967)
&


The late Robabeh Eslampanah (1921-1999)





3 Classical Topics in inventory Control





CHAPTER ~~ TITLES
FOREWORD 10


Chapter 1 Introduction & Basic Concepts 17
Chapter 2 Deterministic Inventory Models 37
Chapter 3 Constrained Inventory Control Problems 130
Chapter 4 Dynamic Lot sing Techniques 165

Chapter 5 Inventory Control under Uncertainty 226


Chapter B prvi to Forecasting Methods 317





4 Classical Topics in inventory Control





CONTENTS


FO RE WO RD ic ienecicccsitseiiences dia catheteaktedteedscdasbddi sailors 10
Symbols and abbreviations .0.... 0... eee cee eeeeeeeeeeeeeeeeeeeeeeeeeeeseeeaeesaeesaes 12
Chapter 1 Introduction and Basic Concepts............eeeeeeeeeeeeeeees 17
Aims of the chapter .........ccceeccecsseseceeceeecesneeceaecesneecaeeeseeecsaeeeeneeenaeeenees 17
1.1 Definition of inventory control systems .......... eee eee eee eeeeereeees 17
1.2 The purpose of holding inventOry...........cceecceeeseceeneecseeceeeeeesaeeeenees 17
TiS: INVeEntOry ‘COSTS 3503555 Se tetees eel eegesen chi das sb dedesined get beneed aeeuecs betoassts 18
1.4 Calculation of inventory average... ee eee ceseceeceeeeseeeeeeeeeees 19
1. 5 Calculation of shortage average... eee ceseesecesecesecseecseeeeeeseeees 23
1.6 Some points on statistical distributions used in inventory control...25
1.7 Pareto Principle and ABC Analysis... cee cee cee eesecseecseeereeeeeeees 27
1.7.1 Steps in conduction ABC Analysis... eee eeeeeeeeeees 28
1.7.1 Control activities on different categories ........ 31
1-8 Inventory models classification... ee ee cee eeseceseceeeceeeceeeeereseeees 32
FEXCLCISES uss incisicpants ctesusieviebeces stbvaeayebtaacvaenyearusnvets sdoviecsedepmtssueeuicauniuneseacee 33
Chapter — Trivers bony MIOGGIS. ..-2..:0.ss256céscesazdescdeceeatenstee 37
Ads OF the Chapter is csscissiv.scctvssssesibcieeedbeeieoiess eceecaseisbocssobseteasedsctebeosts 37
2-1Economic Order Quantity(EOQ) model... eeeceeeeesseceeneeeseeeenees 37
2-1-1 Assumptions of Classic EOQ model ......... eee eee 37

2-1-2 The maximum and the average of Inventory in EOQ model 42
2-1-3 The reorder point(ROP) in EOQ model... eee 42
2-1-4 Sensitivity Analysis for EOQ Model... 45

2-1-4-1 Impact of Errors in Co « Ch and D on Q and total cost 46
2-1-4-2 Impact of Errors in Q,, on total variable cost....... 48
EOQ model for items with discrete order quantity(Q)............... 53
2-2-1Calculation of order quantity... eee ee eeeeeeeceeeeeeeeeeees 54
2-3 Safety stock Model]... ee ee eeeeeeeeeeeeeceseeeeeeseeeseeenees 56
2-4 Economic Order Interval(EOI) Model-Single item......... ee 58
2-5 EOQ Model -Back Order .0......cc.ccceccccecssececeessececseceecesseeeesssaeeeesseaees 60
2-5-1 Average inventory and stockout level 00... 61
2-5-2 Optimal order quantity(Q) and oo... eee eee ceee ener 63
maximum stockout (b) in EOQ model with backorder ............ 63
2-5-3 Reorder level in EOQ with backorder model ................. 63
2-5-4 Optimal (Q) and (b) when 7 #0 8&7 =0 2 oe 64
2-2-5 Some comments on backordering......... eee eee eee 67
2-6 On-hand inventory and on-order inventory ............. sees eee eeeeee 71


2-7 EOQ Model —lost sale Case ..........cccssccesssceceessececeeceeeesseeeesssaeeeesneeees 72





5 Classical Topics in inventory Control





2-8 Total Discount Model... eee eceeceeecesecesecesecseesaecssecsaereeeaeeeaes 77
2-8-1 Quantity discount model —Ch variable ........ ee 719
2-8-1-1 The algorithm for finding optimal Q -..... 719
Case 1: Cy variable........cccccccescccccecsessssscececesessaseeeeeseeeseanee 719
2-8-2 Quantity discount model —case II:C, Fixed... 81
2-9Converse of Discount Model (rate increase with quantity increase) .83
2-10 Incremental Discount Modelo... eee ee cee esse cseeseeesecnsecneeeeeenes 86
2-10-1 The algorithm for finding optimal Q - incremental model 87
2-11 EOQ Model with sale price(temporary discount)... ee 91
2-11-1 Summary : EOQ Model with sale... ee 94
2-12EOQ Model -permanent reduction price ...... eee eee este tee teeeeees 96
2-13 EOQ Model —known increase price ....... eects eeseeseeseeesseteeeteeeeees 96
2-14 Economic Production Quantity—single item... eee 100
2-14-1 EPQ-single item,stockout unpermitted 0.0.0... 100
2-14-1-1 The reorder point in EPQ model —single item ...... 102
2-14-2 Single-item EPQ model with backorders...........0..00... 104
2-14-2-1 EPQ model with backorder- m2 =0&m #0....105
2-15 Make or Buy Decision... eee eceeceeeceeeceeceaecseesaecssecsaecseeeaeeeaes 106
2-16 Economic Production Quantity:Multiple-item ..0.... eee 106
2-16-1 Multiple-item EPQ model: n machines for n products with no
COMSILAINIS « 3 3c8s.

2-16-1 Multiple-item EPQ model: 1 machine for n products... 108
2-16-2-1 Multiple-item EPQ model: 1 machine &S, 0 oe. 109


2-16-2-2 Multiple-item EPQ model: 1 machine &S, #()....113
2-17 Multiple-item EOQ model... ee eee cesecsecseeesecsseceeeceeeseeeneeees 116
2-17-1 Unconstrianed multiple-item EOQ model... 116


2-17-2 Multiple-item EOQ model- annual number of orders the same for all 116
2-17-2-1 Multiple-item EOQ Model : order cost independent of


number and quantity Of 1teMS ..0...... eee eeeeeeeeeeeeeeeeeceteeeeneetees 117
2-17-2-1 Multiple-item EOQ Model : separate order cost for items 119
2-18 Deterministic continuous & periodic review Models................04+ 119
2-18-1 Deterministic continuous review=deterministic (r,Q) Model=

Deterministic ( FOS)Model ............::cccccsccecesssececeeecessseeeeessneeeeseaaes 119
2-18-2 Deterministic periodic review=deterministic (R,T) Model=
Deterministic ( FOT)Model ..............ccccccessecesssececenseeesseeeecessseeeeseaaes 120

2-19 Inventory Models for Deteriorating Items... eee 121
EXGrCIS@S ccs iicsiecstbleskkedcesacneterlaviees elo aia saeialia halides 121
Chapter 3 Constrained Inventory Control Problems..........00. ee 130
Aims'of the: chapter .sii.caihseivuisn sian ail netinad ets 130


3-1 Lagrange multiplies technique and Karush-Kuhn-Tacker conditions 130





6 Classical Topics in inventory Control





3-1-1 Nonlinear optimization problems with equality constraints


131


3-1-2 optimization of nonlinear problems with in-equality constraints 133


3-1-3 Nonlinear optimization problems with equality and in-equality


COM SULAINS 25 soncces sv pasensc ves costs vosh en suseses ccstbtaenavasvoncvuasvancebereevinteat nace 135
3-1-4 Nonlinear optimization problems inequality constraints and

NONME SALVE 6)'S sesh i cces bes ceedee Hh canes cescsassetbsands iacarenececheteeresesiheneeeeere 138
3-1-5 Interpretation of Lagrange multiplies ....... ee 138

3-2 Constraint in INVENtOTY SYSTEMS 00... ee eee eeeeceeeeeeeteeeeeeeaeenaees 139
3-2-1 Constraint on the space or surface of the warehouse...... 140

3-2-2 Constraint on the budget... eee eeeeeeeeeeeeeeeeeeeeees 144

3-2-2-1 The budget for ordering is exactly C dollars........... 145


3-2-2-2 The budget for ordering is less than or equal to C.146
3-2-3-1 Constraint on annual number of orders-Co negligible


148


3-2-4 Constraint on the number of orders of multiple items having the same


MUMIDEL- OL. OLGELS sv. i.s...c.sacceneceshsaxasstevseessesbansennsveeessayvenssvaedeabenncvease 154
3-2-5 constraint on the cycle time of classic EOQ model-single item
3-2-7 Multiple-constraint inventory models........0.... eee 161
PXCLCISES isi p2ii2s shgiscs eels uses cteel igo cas gieckcbatselecas cuaptaeestay ab gheebohatscabeteastaees 162
Chapter Zh sami ILOt ane Tee WAGES: ssscntecnn extasastereretatnawcses 165
Adms:OF the: chapter secs: cccctiecisvices hsusteecli adie bes eieci ease ns eserkeecrpeeteeees 165
A= 1-IntrOduction si:acsiictdins ted ie sheaincs Piles Aion rien: 165
4-2 Dynamic Lot Sizing Problem ou... eee eee eee eeeeeeeeeeeeeeeeeeeeenenes 166
4-2-1 Assumptions of Dynamic Lot Sizing Algorisms........... 166


4-3-2-1 Economic order Quantity (EOQ) lot sizing policy .175


4-3-3 Fixed Order Period (FOP ) or Periods of Supply (POS) policy
4-3-3-1 Economic Order Interval (EOD) method or Period Order


Quantity (POQ) or Fixed Order Interval(FOID)........... ce eeeeeeeeee 179

4-3-4 Least Unit Cost (LUC( Algorithm... 183

4-3-5 Least total Cost (LTC) method... eee eneeeees 187

or Part Period Algorithm(PPA) .0...... ee eceeeseeeeneceeeeeeneeeees 187

4-3-6 Part Period Balancing(PPB) algorithm... 190

4-3-7 Incremental Part- Period Algorithm(IPPA).................. 192

4-3-8 Silver —Meal algorithm... eee eee eeeeeeeceeeeeeeeneeeeees 195

4-4 Wagner and Whitin's Exact Algorithm... ee eeeeeeeeeeeereees 199
4-4-1 The steps of Wagner-Whitin Algorithm... 200
EEX CT CISES ei sei cess ease pe Aisa cacei gue chived cons cuttecl svhcaacubchpaelinndcessbiiecdebbiasibesepeciees 221
Chapter 5 Inventory Control under Uncertainty... eeeeeeeeeeeetees 226
Aams:Of the: chapter vsccie:.sesdscligscusoiees chides viens avs hecsbeicceesdiebateastaeees 226
Sell NtOGUCHON ceichAseiGie hihi ie AMIE eek idinenis 226
5.2 Single Period Inventory Model with Probabilistic demand.............. 226


5.2.1 Single Period Inventory Model —order/setup cost ignorable


156


176


229





7 Classical Topics in inventory Control





5.2.1.1 Single Period Inventory Model ,Co = 0 and continuous demand


weed shee beeseysoeues Giecoahe ithe ey veh cancer eaeast ia tiendiatubael Ee ebeesuny covecebesteetts 229
5-2-1-1-1 Optimal value of maximum inventory(R *)......... 235
5-2-1-1-2 Optimal strategy in single period model............. 236


5-2-1-1-3 average shortage cost in the single period model 237
5.2.1.2 Single period Inventory model :Co = 0 & discrete demand 239
5.2.2 Single Period Model —order/setup cost (Co( considerable 241


EXOLCISES .:c5ccscacscsessncek ies tages heed lvecs pobeke cs sedses cocebons chevbete la cpbods seb cagsboescenenede 249
5.3 Probabilistic Continuous and Periodic review models- introduction 252
D23= L/Salety StOCK. .3.20/:swdeescbds ooehesenietiee ee bho tetheeancauteereee tes 253
923-2. Service Level ives ini cessesesthchiweeel vehi hgeeede eek ober 253
5.4Continuous Review Inventory Model... eee cece ese eecseeeneeenes 255
or (1, Q ) policy or FOS system 0... ee eecesecesecesecsseceeceeeseceeeaeeenes 255
5.3.1 Order quantity in (7,Q) SysteM oo... eee eee eects eete eee 256
5-3.2 Safety stock in (1,Q) SYSteM ...... eee eeceeeeceeseeceeeeeneeeeeeeees 256
5-4-4 Reorder point and safety stock for normally distributed DL in FOS
POVIC Yc sssceecceocess ees penenesteta See, Seesepas ete pete nant cub sda oeksepseentovbecesbeegees 262
5.4.5 Determining safety stock and reorder point in (r,Q) system when
demand and/or lead time is probabilistic .....0.. eee eeeeeee 262
5-4-5-1: Case 1: Demand and lead time (D &L=T{) probabilistic and
INGEPENCENE s -exsssss5. ease sshsvesshs savages Maastae suseestishtites BEE eee esos 263
5-4-5-1-1 Some points on the unit conversion of demand's variance and
Standard deviation... eee cece cseeeeeeeeeeeeeeeeceeceseseseeseeseeaees 264
5-4-5-2 Case 2: Demand(D) Deterministic but lead time (L=T_,)
Probabilistic: sc.si..20, secsscaetcastes savsseete aeheen sesh maachabissed sevaae. stare ees 266
TABS CASE Fs obscas gi cariaty sta sces ess et vekvaetan shneeaaSe seh oe taeaeaten sees 266
5-4-5-3 Case 3: Demand(D) probabilistic but lead time deterministic
POU ha Sh atthe atovelel ete ee ei ek ee 267
5-4-5-4 Case 4: Both demand and lead time deterministic .268
5-4-6 On Lost sale and stockout in FOS systems..........0..00.. 268
5-4-6-1 Calculation of average shortage in FOS systems when Dy is
normally distributed using normal loss integral .......00... 273
5-4-7 Average inventory in FOS system... eee eee 274
5-4-8 Other ways for determining reorder point in ............... 276
FOS SYStCMS oo. .sscics ceedpessaescneseeestius esses ents cdsbaneebechistesee ssessnes toes 276
Determining reorder point given the service level and lead time
CONSUMPTION istribUtiON ..... eee ee eee cese cee ctseceecaecsaecnaeeaee 276
Determining reorder point given the average consumption and the
maximum Of lead time... eee ee cess cese esse cnsecaeecaeeeaecseeeneeenes 277
Determining reorder point given the demand maximum and the lead
{IME AVELABE goss sesh vociesapevcodsaieosvendpeesegratssttesp ye seloeedhsues pueraueestes 278
5-STwo-bin Or MaX-MiN POLICY eee eee eee ese cesecesecseeceecseeesecseecaeeenes 280
5.6Back ordering in FOS system... ee eeceeceseceecnsecseecseeesecseeeneeenes 282
5-6-1 Backordered (r Q) - Stockout cost/ unit (77)known......... 282
5-6-2 Backordered (r Q) - Stockout cost/ outage (g)known.....285


5-7Lost sale case in FOS system oo... ee eeeesecesecssecseecseeeeesseeeneeenes 289





8 Classical Topics in inventory Control





5-7-1 Lost sale (r Q) - Stockout cost/ unit (71)known............... 289

5-7-1-1 Safety Stock in (r Q) - Lost Sale case... 289

5-7-2 Lost sale (r Q) - Stockout cost/ outage (g) known ......... 292

5.8 Periodic Review Inventory Model...........ceeeeeeeesecesecseeetssetseeeaes 295
or (R, T ) policy or FOT system oo... eee cee eeee cee cneeceecaeeaeceeeaeeeaes 295


5-8-3-1 Demand(D) and the lead time(L =T,) independent random


B21 F216) (oh eee nO en a Oe ne ee ee Pe eee 298


5-8-3-2 Demand(D) random variables and the lead time(L =T_) constant


330


debiseve dba wes ooeuiuasvas eetpencesoctndeossdeaseeaacaiecdeseuivegebeudedescasscctacedecsantoetes 300
5-7-3-3 Demand(D) constant and the lead time(L =T,) random variables
sabe cag aaeceahe pence eras caueo ea cee obs anasl saves teeshs tyes uneiateveasebaatetdedeacenseevacs 300
5-7-4 Average Shortage 00... eee eeeeeeeeeeeeeeseeeeeeseceeeeereeseenaees 301
5-7-4-1 Average shotage, maximum inventory, safety stock when
DEFT AS MOTMA . seicc.sscéscevsaccesersticeseosccersacesgecdvetensnedavsasveesensees 302
5-9 Back ordering in FOI system... ee ee cece cese esse cneecaeeseecseeeneeenes 307
5-9-1 Backordered (R T) - Stockout cost/ unit (72)known....307
OO ciaesacet arvesecetseneepcsitntiesangaita sats aveedanaes corse anicapuentunvucesintantecaseses Lost sale
CASE! IN FOS SYStOM susesst da cegseencsdseveses sasencaisevigedaseveneiaseveevsasbuts fecdeesenctsys 309
5-10-1 Lost sale (R T) - Stockout cost/ unit (7)known........... 309
5-10-2 Lost sale (R T) - Stockout cost/ outage (g)known....... 309
5-1 1Inventory control under complete uncertainty... eee 310
5-11-1 Decision criteria in minimization problems.................. 312
The minimax decision criterion(Tule) ............cc:cccesseeeeeeeees 312
The minimin decision rule oo... eee eeeceeseeceeeeeeeeeceeeeeneeeenees 312
The expected value criterion (Bayes method)... 313
PX CTCISCS ets sca cers exstecatecs san cuneea taaut evoeststenaseenteuneceaneeniceeteceteterssareverectmnecs 314
Chapter B prvi to Forecasting WICHIOS ay caniesapsitcenapsenagsanaossys SF
AIMS OF ThE CHAPLEM % vsscavescdsccusecessucdecsacadetuctactdeescesencecsaqacececadetivaessecsuniers 317
G1, TMTPOMUCHON si sesin csecasestesucczecsssuadesdetduscciagecessocedncececeueescdcadetecivessocedusnes 318
6-2 Classification of Forecasting Methods ............cceeeceessceeneeeeeeeeneeeeees 319
6-3 Subjective or qualitative Methods ...........ceceeessceenceesseceeeeeceeeeeneeeenees 321
6-3-1 Delphi Technique 000.0... eee eeeeesceeeceeceseeseeeeennees 321
6-4 Objective or quantitative Methods .00.... eee eee ceseeseessecseecseeeneeenes 322
O=4= 1 RE BTESSION, s25sssecsceesa ser sebetscese ss ieee ieessctbasencessideaeesssaceeesthcs 322
6-4-1-1Simple Linear Regression Model... eee 323
6-4-1-1-1Estimation of model parameters with the method of Least
SQUATES «05 cies sens cbSesscechses side dvestesesvancsSecoqasesorteussdecoutsssensebncowsscdent ches 324
6-4-1-1-2 Correlation coefficient... eeeeseeeereeeeeeeetees 327
6-5 Measures of Model Effectiveness 0.0.0.0... eeeceeeseceeneecseeceeeecseeeeeneees 328
5-6-1 Application of "t-test for paired data" to model effici study
6-6 Multiple Linear Regression... eee eeceeecesecesecssecseeceesaeceeeneeenes 331
6-7 Simple Moving Average(SMA) .0.... cece eecesecesecsseceeeceenseceeeneeenes 336
6-8 Modified Moving Average... eeeceeceseceseceecssecseecseessecsseeaeeenes 338


6-9 Weighted Moving Average ......... cc eceeseecesecesecesecesecseecseesaecseecaeeenes 339





9 Classical Topics in inventory Control





6-10 Exponential Smoothing .0.0... eee eeceeeceeeceeceseesecesecssecaecsaeeaes 340
6-10-1 Relation between simple moving average and simple exponential

SIMO OLIN Gh ss seases sbitass oes cestsssebatassed sxbeseeeentd Lip es nedovseenseeae deh sou Seed 345

6-11 Double Exponential Smoothing 0.0... eee ceseceecneeeneeeaee 345

6-12 Forecasting techniques for time series having seasonal variations.350

6-12-1 Ratio-to-trend technique for seasonal adjustment......... 351

6-13 Verifying and controlling forecasters using control charts............. 354

6-13-1 A control chart for forecast error... eeeeeeeeeeteceeeeees 354

Definition of Moving Range(MR) ........ eee eeeceeeeseeeneeeeees 355

6-13-1-1 Upper and lower limits of the control chart for forecast error356

6-13-1-2 Some criteria for out-of- control status ................. 357

6-13-2 Tlustrations .00... eee eeeeeeceeneeceseeeseeceeeceaeceeeeecsaeeeeeeees 359

EXOT CISGS vis is selaetSoveceectciabvececuss su chesanece snsseeetuiaees cosas Daneeusncoeseetemeegscnss ties 373

Reference ..i sevens cet esceaseens ches bs diwaneegs cesornudbtaeeye chasis Sbuneverscnelu tates aedtees 379

SPADES: sc seecbes we tvszeews cat vatisn cons ensues od usavapheswsselteeneaveas ants Seeverncoueeteinents chee toey 383

Table A Unit Loss Normal Integrals oes ceseceeceseeneeenes 384

Table B Cumulative Poisson Probabilities ..0.....0 cee eeeeeeseeseeeeeneees 385

Table C Area under normal curve from—©00 to Z =X — UO... eee 391

Table D Area under normal curve from Za to :PrZ > Za = a395

Table E MATLAB commands related to some distributions............... 397

Table F Some characteristics of 6 distributions ........... cc eeeeeeeeseeeeeeee 398

Table G Some useful formulas for Inventory Models ........0........ 399


ABOGT THE AGTHORB 2c cccsscscecseseeeseeeeeeeseeeeeaeeeees 405





10 Classical Topics in inventory Control





FO REWORD


This book’ is the outcome of teaching a course titled "Inventory
planning and control" for several years to B.S. students using many
books especially the book written by Dr Tersine.


Thanks God for making me successful to present this work which I
hope to be useful in both academic and industrial environments.


The book covers the classic topics in inventory control as well as
some demand forecasting methods. The Persian version of the book
has a chapter on MRP. But the author did not translate the chapter
because there are many works available in the internet and in books.


Mr Masoud Hajghani gets the credit for the last part of Chapter 6
i.e verifying the forecasts. I would like to thank Mr Ali Bazargan
who helped the author in some phases of editing.


The author would be pleased if the readers write him about any
kind of deficiencies of the book.


Hamid Bazargan

College of Engineering,

Shahid Bahonar University of Kerman , Iran
bazargan @uk.ac.ir


November 2021





' The Persian version of book wriiten by the same author has been published by
Shahid Bahonar Univerisity of Kerman, Iran in October 2021





Classical Topics in inventory Control








Symbols and abbreviations


Symbols and abbreviations















































































































































A D
The increase in price froma : a temporary special d
future dats reduction of price d per unit.
The current level of inventory., A Amount of demand or D
the level of inventory before requirement
ordering at_the period The average of deviation D
Accumulated Part-Period APP between observed and predicted
Artificial Neural Networks ANN values
B
Maximum back-ordered demand | B Demand for period t_(t=1,2,...T) Dy
Average shortage per unit time b Estimated amount of demand D'
Optimal value of b b* Annual demand for i" product Di
>>>