integrated logistic

7/23/2019 Integrated Logistic

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Integrated Logistics

2

Contents

1 Introduction .............................................................................................. 4

1.1 Defining Supply Chain Management ........................................................... 4

1.2

Understanding Supply Chain Management .................................................... 4

1.3 Measuring Supply Chain Performance ......................................................... 4

1.4 What is Integrated Logistics? ................................................................... 4

2 IL & Finance .............................................................................................. 5

2.1 Strategy and Finance ............................................................................ 5

2.2 Time value of money ............................................................................. 6

2.2.1 Net Present Value for discrete interest rates ............................................. 6

2.2.2 Net Present Value for continuous interest rates ......................................... 8

2.2.3 Annuity Stream ............................................................................... 10

2.2.4 Linear approximations of NPV and AS ..................................................... 10

2.2.5

NPV and AS of a few useful cases .......................................................... 11

2.3 Economic Order Quantity (EOQ) Harris (1913) ............................................ 14

2.4 Economic Production Quantity (EPQ) Taft (1918) ........................................ 18

2.5 EOQ with batch demand Grubbstrm (1980) .............................................. 21

2.6 Lot-for-lot production at finite rate Monahan (1984) .................................... 24

2.7 EOQ for batch demand Goyal (1976) ....................................................... 26

2.8 EPQ for batch demand Joglekar (1988) .................................................... 28

2.9 Vending machine ................................................................................ 29

2.10 Payment structures ............................................................................. 30

2.11 Consignment arrangements .................................................................... 32

2.12

The profit function of the integrated supply chain ........................................ 34

2.13 Using the NPV framework to include other cost components ............................ 35

3 IL in Buyer-Supplier Supply Chains .................................................................. 38

3.1 One-to-one shipping ............................................................................. 38

3.1.1 Shortest Path Problem ....................................................................... 38

3.1.2 Economic Transport Quantity (ETQ) ....................................................... 41

3.1.3 Maximum Economic Haulage Radius (MEHR) ............................................. 45

3.1.4 Optimal policy to order N different items from a Cross-Docking Facility (CDF) .... 46

3.2 One-to-many shipping .......................................................................... 48

3.2.1 Length of an optimal TSP tour visiting many customers ............................... 48

3.2.2

Expected length of an optimal TSP tour visiting a few customers ................... 51

3.2.3 Continuous approximation of an optimal vehicle routing solution ................... 52

3.2.4 One-to-many ETQ ............................................................................ 54

4 Strategy in the supply chain: Alliances versus Leaders .......................................... 61

4.1 Alliances .......................................................................................... 61

4.1.1 Cooperative game theory ................................................................... 61

4.1.2 Alternative methods ......................................................................... 71

4.2 Two-party alliances in the supply chain ..................................................... 71

4.2.1 Two-party alliance: example ............................................................... 71

4.2.2 Two-party alliance: example 2 ............................................................. 79

4.3 Perfect coordination ............................................................................ 82


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3

4.3.1 Perfect coordination schemes: examples ................................................ 83

4.4 Leaders and followers .......................................................................... 88

4.4.1 Leaders and followers: Stackelberg games ............................................... 88

4.4.2 Stackelberg leader in the supply chain: examples ...................................... 90

5 Stochastic Models ...................................................................................... 96

5.1 When is the assumption of a constant demand rate valid? ............................... 96

5.2 (r, Q) reorder point models .................................................................... 97

5.2.1 Backorder case ................................................................................ 98

5.2.2 Lost sales case .............................................................................. 103

5.2.3 Service level approach .................................................................... 103

5.2.4 Standard Tables ............................................................................ 105

5.2.5 Exercises ..................................................................................... 110

5.3 News vendor problems ........................................................................ 111

5.3.1 Example ...................................................................................... 111

5.3.2 Cost minimisation .......................................................................... 111

5.3.3 Profit maximisation ........................................................................ 112

5.3.4 Regret minimisation ....................................................................... 113

5.3.5 Exercises ..................................................................................... 114

6 MRP, JIT, and Bottlenecks .......................................................................... 116

6.1 Materials Requirements Planning ........................................................... 116

6.1.1 MRP inputs ................................................................................... 116

6.1.2 MRP outputs ................................................................................. 118

6.1.3 Lot sizing policies .......................................................................... 122

6.1.4 Remarks on MRP ............................................................................ 125

6.2

Just-In-Time .................................................................................... 130

6.2.1 Motivation ................................................................................... 131

6.2.2 Push and pull systems ..................................................................... 131

6.2.3 The Kanban System ........................................................................ 131

6.2.4 How many cards? ........................................................................... 132

6.2.5 Subcontractors .............................................................................. 134

6.2.6 Fluctuations in demand ................................................................... 134

6.2.7 Comparison JIT and MRP .................................................................. 136

6.3 Bottleneck scheduling ........................................................................ 139

6.3.1 Optimised Production Technology (OPT) ............................................... 139

6.3.2

Theory Of Constraints (TOC).............................................................. 143


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4

1 Introduction

Foranyquestions,email:[email protected]

Seealsothematerialonblackboard.

Thistext follows the lecturesgiven,but Imayhavearranged thesequenceofa few topics inorder toarriveatamorelogicalflow.

1.1 DefiningSupply

Chain

Management

Seeslidesonblackboardandreadthefollowingarticle(onblackboard):

MENTZER J T, DE WITT W, KEEBLER J S, MIN S, NIX N W, SMITH C D, AND ZACHARIA Z G. 2001.

DEFININGSUPPLYCHAINMANAGEMENT.JOURNALOFBUSINESSLOGISTICS22(2),125.

1.2 UnderstandingSupplyChainManagement

Seeslidesonblackboardandreadthefollowingarticle(onblackboard):CHENIJANDPAULRAJA.2004.UNDERSTANDINGSUPPLYCHAINMANAGEMENT:CRITICALRESEARCH

ANDATHEORETICALFRAMEWORK. INTERNATIONAL JOURNALOFPRODUCTIONRESEARCH42 (1),131

163.

1.3 MeasuringSupplyChainPerformance

Seeslidesonblackboardandreadthefollowingarticle(handout):SHEPHERDCANDGNTERH.2006.MEASURINGSUPPLYCHAINPERFORMANCE:CURRENTRESEARCH

AND FUTURE DIRECTIONS. INTERNATIONAL JOURNAL OF PRODUCTIVITY AND PERFORMANCE

MANAGEMENT55(3/4),242258.

1.4 Whatis

Integrated

Logistics?

Seetheslidesonblackboard.


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5

2 IL Finance

2.1 StrategyandFinance

Source: SILVER A., PYKE D.F., AND PETERSON R. 1998. INVENTORY MANAGEMENT AND PRODUCTION

PLANNING AND SCHEDULING, THIRD ED. JOHN WILEY & SONS, NEW YORK, CHAPTER 2: STRATEGICISSUES,P.14.

Integrated Logistics (IL) should be linked to the corporate and business strategy. The most importantobjectiveofanyfirmisarguablylongtermprofitability.Inthiscontexttheoperatingprofitisdefinedas:

ILcanaffectbothtermsontherighthandside.Byreducinge.g.aggregateinventorylevelsinthefirm,

the investmentcostcanbe reduced.Byallocatingproper inventory levelsamongdifferent items inanimprovedway,salesrevenuemayincrease.

One common aggregate performance measure in IL and inventory management is the inventory

turnover:

An increase in sales without a corresponding increase in inventory will increase the inventory

turnover,aswilladecreaseininventorywithoutadeclineinsales.Turnovercanbeausefulmeasuretocomparedivisionsofafirmorfirmsinanindustry.

Ahigherturnoverratioforthesamelevelofsalesmeansamoreprofitablebusiness,aslessmoneyis

tiedupininventories.Thedangeristhatareductionofinventorylevelsmayalsonegativelyaffectsales.Whenitisnotknownwithcertaintyhowmuchdemandthereisperperiod,anamountofsafetystockis

needed to make sure enough products are in stock in case demand would be somewhat higher than

expected.Furthermore,theright figureof inventoryturnover foryour firmdependsonthe levelof in

houseproductionversusoutsourcing.Afirmthatdoeseverything inhousewillneeda lower inventory

turnoverthan

afirm

that

is

completely

based

on

outsourcing

of

the

production.

Indeed,

the

first

firm

will

alsohaveastockofrawmaterialsandworkinprocess,whilethesecondwillonlyhaveendproductsininventory.These considerationsare importantwhen comparing inventory turnover figures of different

firms.OtherusefulperformancemeasuresforILincludethosethatmeasureallsortsof:costs;averageand

variabilityofleadtimes(oftenseenasthetimebetweeninitiationofsomesalesorderandrealisationof

thesales,sometimesalsojust thetime istakes foraproduct tomovethroughaparticularpartofthe

supply chain); product and service quality; customer satisfaction; and innovativeness (see also Section

1.3).


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6

2.2 Timevalueofmoney

Source (on blackboard): GRUBBSTRM, R.W. 1980. A PRINCIPLE FOR DETERMINING THE CORRECT

CAPITAL COSTS OF WORKINPROGRESS AND INVENTORY. INT. J. OF PRODUCTION ECONOMICS 18(2),259271.

2.2.1 NetPresentValuefordiscreteinterestrates

IfIinvestV()intoaprojectnow,theprojecthavingarateofreturn =0.2afteroneyear,whatwillbetheamountofmoneya()thatIwillreceiveoneyearlater?

Theanswer: 1 1.2

If Ikeepthe investmentrunningnot foronebutforTyears intotal,whatwillbemyrewardattheend?

Theanswer: 1 1 1 1 Wecanturnthisaroundandaskthequestion:whatisthecurrentvalueV()ofreceivinga()within

Tyearstimefromnow?Algebraicmanipulationoftheaboveequationgives:

1

Time

Cashflows

0 1

V

a

Time

Cashflows

0 1

V

a

T...2


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7

ViscalledtheNetPresentValueofa. Itthusalldependsonthevalueof,calledthediscountfactor,interestrate,orinternalrateofreturn. Companiestypicallyusevaluesofwithintherange0.1to0.3.

Considerasequenceofnpaymentsofdifferentamounts , 1, , atequidistantpoints intimewithcycletimeT(seeFigurebelow).WhatistheNetPresentValueofthesepayments,if

istherateof

returnoveroneperiodT?

Theanswer:

1 Therateofreturn isafunctionofthetimeperiodoverwhich it isdefined.Forexample, if

isthe

rateofreturnoveraoneyearperiod,whatistheequivalentrateofreturndefinedoveraperiodofone

month(take

this

to

be

1/12th

of

ayear)

that

would

give

the

same

Net

Present

Value?

Answer:Letuscall theequivalentrateofreturnoveronemonth,thenwithT=12: 1 1 1 1

Mathsrefresher

ln ln1 ln1

1T ln1 ln1

Thus,iftheannualinterestrateis0.2,theequivalentmonthlyinterestratewouldbegivenby:

Time

Cashflows

0 1

Van

n...2

a2

a1

TT


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8

112 ln1 0 . 2 ln1

. 1 0.0153Notethatiftheinterestratewouldbedefinedoverasmalltimeperiod,thenln1 .

2.2.2 NetPresentValueforcontinuousinterestrates

In general, the conversion of interest rates corresponding to different lengths of period obeys the

formula:

1 ln1 ,

where isthe interestratepertainingtoaperiodof lengthand isthecontinuous interestratecorrespondingtothelimitlengthzero.ThedefinitionofNetPresentValueinthecontinuouscasebecomes:

,where

isthecashflowattime

plussomemultiple

ofDiracsDeltafunctionatpoints

at

which

there

are

finite

payments

.

Wewilltypicallyneedtosolveonlyspecialcases,givenbelow.

Example1Aoneofflumpsum()receivedatfuturetimeL

Thus the NPV ofacashflow received L time units in the future is the cashflowmultipliedwith its

delayfactor.Time

Cashflows

L

a

0

V


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9

Example2Afinitenumberofcashrevenues,eachreceivedtheirownfuturemoment

Example3Aninfinitenumberofequalcashrevenuesreceivedatequidistantmoments

Mathsrefresher

If|| 1,then For , || 1

Thus: 1

Time

Cashflows

1

an

n...2

a2

a1

T2

T1

0

V

Time

Cashflows

0 1 ...2

aa

TT

...

a


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10

Example4Acontinuouscashflowatarateofa(/year)receivedforeternity.

2.2.3 AnnuityStream

TheAnnuityStreamASofaseriesofcashflows istheNetPresentValueVofthesecashstreamstimestherateofreturn:

Fromexample4,itisclearthattheannuitystreamisthatcontinuousstreamofcashyieldingthesamenetpresentvalueastheoriginalseriesofcashflows.

2.2.4 LinearapproximationsofNPVandAS

Mathsrefresher

Maclaurinexpansionofanexponentialfunction ! 1 ! andconvergesfor

Usingthisresultfor ,itiseasytoseethat:

1

2

,

andit

can

also

be

proven

(after

lengthy

algebraic

manipulation):

1 1 2 12 This can be used to derive linear or quadratic approximations in of NPV or AS functions, as

illustratedinthenextsection.

Time

Cashflows

0

a ....


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11

2.2.5 NPVandASofafewusefulcases

Case1Aoneofflumpsuma()receivedatfuturemomentL

1 1 Foraoneofflumpsum,boththelinearapproximationoftheNPVandthequadraticapproximationin

of the AS are acceptable, and indeed would be of the same accuracy. However, the linear

approximationof

AS

would

be

insufficiently

accurate

as

indeed

the

above

shows,

cannotbetheASas a itself isnot itsNPV.The linear approximationof the AS would thus neglect the delayeffect altogether.Case2Aninfiniteseriesoflumpsumpaymentsa()receivedwithcycletimeT.

1 1 1 2 12

1 12 12

1 1 2Foraninfiniteseriesofcashflows,thelinearapproximationofASisacceptable.

Time

Cashflows

0 L

a

Time

Cashflows

0 1 ...2

aa

TT

...

a


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12

Case3Acontinuouscashflowatratea(/year),startingattimeL,andforeverlasting.

Itcanbeobservedthatthisresultcouldhavebeenobtainedbycombiningexample4andcase3fromabove.IndeedtheNPVattimeLofthecontinuouscashflowaisa/(example4),andthenaccountingforthedelaywithtimeL(case1)givestheaboveresult.

1 2 1

Case4 Acontinuouscashflowatratea(/year)receivedinfuturetimeperiodT,startingat

timeL.

1

Thisresultcanalsobeobtainedbyviewingthetemporarycontinuouscashflowaasthesumoftwo

infinitecontinuous

cash

flows

+a

and

a

(starting

atime

Tlater),

and

thus

by

applying

the

result

of

case

3

twice:

Time

Cashflows

0

a ....

L

Time

Cashflows

0

a

L T


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13

1 To find the linear approximation, it is safe to take the quadratic terms first into consideration and

approximatelateron:

1 2 1 1 2 2 1

Case5 Aninfiniteseriesofcontinuouscashflowsatratea(/year)withcycletimeLandeachtimereceivedforalengthoftimeT(T L).

The result of case 4 (without delay) can first be applied to find the NPV of the cashflow of every

periodL.Usingtheapproachasincase2thengives:

1 1 1 TofindthelinearapproximationofAS,itisagainsafetofirstconsiderthequadratictermsaswell:

1

1 1

2! 1 2

2 2

Time

Cashflows

0

a

L

a

T T

L

....


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14

2.3 EconomicOrderQuantity(EOQ)Harris(1913)

Aretailersellstocustomersatypeofproductatthepricepperproduct.Demandfortheproductcan

beassumedtooccurataconstantrateaccordingtoanannualdemandofproducts.Theretailerhastopurchasefromanexternalsupplieratprice

perproduct,andalsohastopaya

fixed

order

cost

for

each

order

placed

at

this

supplier

(this

could

be

the

fixed

cost

of

transport

plus

the

fixedcostofadministrationtoplaceanorder).

Figure 1. Lot-size model (EOQ).

When the retailerwouldplaceanorderorsize Q (products/order)withcycletime T,the inventory

levelovertimeattheretailerwillfollowtheclassicalsawtoothpatternasillustratedintheFigureabove.

Itcanbeobservedthat:

If there is a nonzero leadtime , and we want to make sure that the order arrives when the

inventory drops to zero, we must order in advance when the inventory level is . This is called thereorderpoint.

Thetraditionalapproachtoderivingtheoptimalpolicy

Weconsideradeterministicsysteminwhichallrelevantparametersareconstantandshortagesarenotallowed.Thepolicyusedis(r,Q).Althoughtheaimistofindoptimalvaluesforbothrandq,theoptimal

valuefor

iseasilydetermined.TheproblemthereforereducestofindingtheoptimallotsizeQ.Inthis

classiceconomiclotsizesystemthefollowingassumptionsaremade:

1. constantannualdemandratey(items/year);

w(/product)s(/order)

y(products/year)p(/product)

retailer

Q(products/order)?T(cycletime)?

Inventory

retailer

Time

...

T

Q

L

r


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15

2. constantinfinitereplenishmentrateR=;

3. constantunitholdingcosth(/item,year);

4. constantunitordercosts(/order);

5. noshortagesallowed;

6. constantleadtimeL=0,easilyextendedtoconstantL0(years).

7. constantreplenishmentquantityQ(items/order).

TheinventoryfluctuationsinthissystemareillustratedinFigure1.Itisclearthatweplacetheorderatexactlythatmomentsothatreplenishmentsarrivewhenonhandinventoryreacheszero:ifleadtime

L=0,weorderwhentheinventory levelI(t)=r*=0; ifL0(andL


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16

Figure 2. Global cost f unct ion of t he EOQ model.

yhsQ 2* 7

Andsince

y= Q/T, 8

theoptimalcycletimeis:

yh

s

y

QT 2

*

* 9

Graphically, the cost equations can be described as in Figure 2. At optimum, annual holding costs

equalannualreplenishmentcosts.

UsingtheNPVframeworktoderivetheoptimalpolicy

TheNPVframeworkcanalsobeusedtoderivetheoptimalvaluesofQandT.Therefore,thecashflows

for the retailer have to be determined. The following assumptions are adopted: Since demand isoccurring at a constant rate

, the customers pay a continuous cashflow of

to the retailer. The

retailerwill

pay

the

set

up

cost

uponreceivingeverybatch,aswellastheamount tothesupplier.ThisisillustratedinFigure3.

0 Q* Q

(Q)

Ordercost

Holdingcost

Totalcost


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17

Figure 3. Cash-f l ows repr esentat ion of t he EOQ model .

Theannuitystreamoftheprofitfunctionfortheretaileris:

1

Thelinearapproximation inisthus: 1 2

2

2

Since

:

2 2 The optimal value for Q can be obtained by taking the derivative of this profit function to Q, and

settingthisequalto0: 2 0

2 Thusthesameresult isobtained,and inaddition itbecomesveryclearhowtheholdingcost(per

productperyear),usedinthetraditionalinventoryframework,needstobecalculated: Thus,theholdingcostfortheretailerisrelatedtotheamountofmoneyinvestedperproduct,i.e..

Wewilllaterencounterexamplesweretheholdingcostsperproductperyearwillbedifferent.

Time

Cashflows

0

s+wyT

T

py ....

....


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18

2.4 EconomicProductionQuantity(EPQ)Taft(1918)

Inthissectionwewillanalysealotsizesysteminwhichthereplenishmentrateisnotnecessarilyinfinite

ashasbeenassumedintheprevioussection.Specifically,thesystemhasauniformreplenishmentrateR

(items/year),where itisobviouslynecessarythatRy.Thistypeofreplenishinggenerallyoccurswhen

the

demand

has

to

be

met

by

a

manufacturing

department

inside

the

company.

The

inventory

fluctuationscanthenbedescribedgraphicallyasinFigure4.

Aproducersellstocustomersatypeofproductatthepricewperproduct.Demandfortheproduct

canbeassumedtooccurataconstantrateaccordingtoanannualdemandofproducts.Theproducerhastomaketheproductsatavariableproductioncostperproduct,andalsohasto

payafixedsetupcostforeachrunoftheseproducts. Productionoccursataconstantfiniteproductionrateequivalenttoanannualrateof(productsperyear).Thetraditionalapproachtoderivingtheoptimalpolicy

Figure 4. Manufact ur ing lot size syst em.

Theaverageamountofinventoryequals

E(I) = |bc|/2 10

Wehavethefollowingrelationships:

c(/product)s(/order)

R(products/yr)

y(products/year)w(/product)

producer

Q(products/order)?

T(cycletime)?

t

I(t)

q

0

T

Lr

R

y

b

a

c


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19

abacbc (geometrical relationship) 11

rL

aby (demand rate)

12

rr L

Q

L

acR (replenishment rate)

13

Hencetheaverageamountininventorycanberewrittenasafunctionoftheknownparametersyand

RandthevariableQ:

R

yQ

QR

y

QIE 122

1

)(

14

Thenumberofreplenishmentsperyearequalsy/Q.Thetotalsystemcostisthesumofholdingcosts

andreplenishmentcosts:

Q

ys

R

yhQ

Q

ysIhEQ

1

2)()(

15

ThevalueQ*whichminimisestotalcostscanbeobtainedasfollows:

012d

d2

Q

sy

R

yh

Q

(Q)

16

R

yQ

R

yh

syQ EOQ

1

1

1

2 ** 17

where

*

EOQQ

refers

to

the

economic

order

quantity

of

the

basic

EOQ

model.

The

corresponding

cost

isgivenbysubstitutionof:

R

yh

sy

ys

R

yh

sy

R

yhQ

1

21

21

2)( *

18

Rearranging:


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20

2

1

2

1

)( *

R

yshy

R

yshy

Q

19

R

y

R

yshyQ EOQ 112)(

** 20

where *EOQ referstothecostobtainedinbasicEOQmodel. ItiseasytoshowthattheEOQmodelis

aspecialcaseofthecontinuousrateEOQmodelbysimplysubstitutingR=.

Example.

Lety=1000items/year,R=2000units/year,h=1.6/year,ands=200/year.Then:

707500000

5.016.1

)2000)(1000(2

1

2*

R

yh

syQ units/order.

21


Thefollowingcashflowsareassumed:

Then,usingCase5ofSection2.2.5forthevariableproductioncosts:

1 1 2 2 2

2

1

2

Theoptimal

order

quantity

is

derived

in

the

usual

manner

from

the

first

order

conditions:

Time

Cashflows

0

s

T

wy ....

....

yT/R

cR


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21

21Thesameresultisthusobtainedasinthetraditionalderivationif:

2.5 EOQwithbatchdemandGrubbstrm(1980)

Weconsiderthesamesystemasintheprevioussectionbutwithtwomodifications:

1. Theproductionrateissetsuchthat ;2. Salesoccursinbatchesofsize

TheinventorylevelasafunctionoftimelookslikeamirrorimageoftheEOQsawtoothpattern:

Thetraditionalapproachtoderivingtheoptimalpolicy

ThiswouldbeexactlythesamemodelastheEOQofHarrisandwouldproduce:

Q

ys

hQ

Q

ysIhEQ

2)()(


FirstNPVsolution.ThisishowGrubbstrm(1980)derivedasolution:

c(/product)

s(/order)

R(products/yr)

y(products/year)

w(/product)

producer

Q(products/order)?T(cycletime)?

Time

Inventory

0

T

....


22/150

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Ther

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Noti

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Thiscan

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eansthatt

uctandnot

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estartofpr

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iveNPVsol

nisplaceda

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berewritte

pproximatio

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s:

obebasedo

eliveryofo

terpretthis

ysthebesta

othermodel

ustomerha

appen, bec

tofstockor

se circumst

nthesales

derstothe

asamodel

ssumption.

wherethis

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riceof

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herewe

oundary

red.This

ise if the

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need to


23/150

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Q,to

be

aditional;ingaso

that this

theNPV


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24

2.6 LotforlotproductionatfiniterateMonahan(1984)

Wecan extend the previousmodelof batchdemand by assuming that theproducer runs everybatch at some finite production rate . The producer would hence only start a production runsometime

/earlierrelativetothedeliveryofabatchtothecustomer.

Theclassiccostfunctionofthissystem(Monahan,1984)isgivenby:

2 A first solution using NPV is found from setting the Anchor Point at start of production at time 0,

assumingthecashflowsasinthefigurebelow.

Ifthestartofproductionhastooccurattime0,theannuitystreamis:

1 1 1 2 1 2 2 2

2 2 2 2 Comparingwiththeclassiccostfunction,wefindthat 2 andwehaveasanextraterm

thesuppliersrewardwith .Inthespecialcasethat weretrievethesolutionfromtheprevioussection:

2 2

Time

Inventory

0

T

....

yT/R

Time

Cashflows

0

s

T

wyT

....

....

yT/R

cR


25/150


25

andtheholdingcost is then .Note,however, thatwecanalwayswrite 2 ,soinsteadofadoptingaspecialinterpretation ,weseethatwecanequallyadoptthemoregeneralinterpretation(sincevalidforany ofhaving 2 andanextraterm,thesuppliersreward,with .

WhensettingtheAnchorPointatthestartofsales,wefind:

1 Thefunctioninbetweenthesquarebracketsisrewrittenas:

1 2 2 2 2 Thus we can take

and identify, again, the suppliers reward as an extra term with

. The special case of is as seen in the previous section and can use the sameinterpretationsforand.Usingthetraditionalinventorymodelingapproach,onewouldonlyfindthefirstpart.Thisisinfact

whathappenedintheliteraturethathasfollowedMonahans(1984)model,andhencetherearemodels

intheliteratureforwhichitmaynotbeeasytoseewhethertheywillleadtoinventorypolicieswhichwill

alsomaximisetheNPVofthefutureprofitsofthefirm.SeealsoBeullens(2014).

We finishthissectionbyprovidingsome intuitionbehindthesuppliersreward.Asshownabove, it

indicatesthatthereisapositiverevenuetermintheproducerslinearisedASprofitfunction:

2 Thesuppliersrewardarisesfromthefactthatthecustomerordersinbatchratherthanoneproductatatime.Anintuitiveexplanationisthefollowing:

CaseA.Supposeyouhavetwooptionstoreceiveincome:eitherreceiving1,200atthestartofevery

year, or 100 at the start of every month, what would you choose? The logical answer would be to

choosethefirstoption.

Case B. Suppose you have two options to pay expenses: either paying 1,200 at the start of every

year, or100 at thestartof every month,what wouldyou choose? The logical answer this time is to

choosethelatteroption.

The fact that a customer orders in batch will cause inventory costs for this customer. Thedisadvantageforthecustomertoorderinbatchisextrainventoryholdingcoststobevaluedatinvested

cost , but at the same time this creates an advantage for the supplier as he receives his revenues earlier. The suppliers reward term incorporates this advantage into the suppliers profitfunction.

Homework

Derivetheoptimalorderquantityusingtheclassiccostfunctioninwhich andthenderivetheoptimalorderquantityusingthelinearisedannuitystreamfunctionunderthetwoassumptionsoftheAnchorPoint.Comparethethreeresults.


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26

2.7 EOQforbatchdemandGoyal(1976)

Weconsideramodelofadistributorwhoneedstodeliverordersofbatchsizetocustomerswithanaverage demand rate . Customers pay per product to the distributor but the distributor incurs adeliverycost

perdelivery.Thedistributorcanplaceordersofsize

toitsownsupplierandhastopay

per

product

and

has

an

order

cost

of

per

order.

Itcan

be

proven

that

it

is

optimal

for

the

distributor

to

have

forsomepositiveinteger.Seealsotheabovefigure.Theclassicderivationofthecostfunctionisagainbasedontrigonometryandproducestheresult(Goyal,1976): 12

ToderivetheASprofitfunction,weusethefollowingcashflows:

NotethatplacingtheAnchorPointatstartofsalestothecustomersorplacingtheAnchorPointat

startofthefirstbatcharrivingfromthesupplierproducesthesameboundaryconditionattime0.TheASfunctionis:

c(/product) (/order) y(products/year)w(/product)(products/order)(/order)distributor

(products/order)? (cycletime)?

Time

Inventory

0

....

Time

Cashflows

0

s

wyT ........

wyT


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27

2 1 2 2Comparisonofthisresultwiththeclassiccostfunctionshowsthat

,butalsothatthere isan

extraterm,thesuppliersreward,with

.

Exercise

Derivetheoptimalorderquantity usingtheclassiccost function inwhich andthenderivetheoptimalorderquantityusingthelinearisedannuitystreamfunction.Weillustratetheprocedurefortheclassicfunction.Weneedtominimise:

12 bychoosingandoptimalintegervalue,say

.Thisvaluemustsatisfytwoconditions:

1 1Fromthefirstconditionwecanderive:

12 1 1 12 Hence

1 1 1

2

Or

12.Thisinequalityisaquadraticfunctionin.Thenonnegativerootisgivenby:

12 1 1 8

But the solution has to be integer: (=the largest integer not larger than always satisfies thequadraticinequality.Fromthesecondconditionwederivesimilarlythat:

1 2.Andwecanproceedasforthefirstconditiontoderivethesameasalwaysfeasible.Bothconditionstogetherimply .Hence:

12 1 1 8


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Noteth

by subs

depend

2.8

E

We intr

producti

producti

Ifwe

thefollo

The

Setti

the timi

(Beullen

Henc

Setti

donot

d

tthederiva

itution of on.PQ

for

b

duce in th

oncostper

onlotsizef

assumetha

wingfigure(

lassiccostfu

gtheAncho

ng of cash

sandJansse

e, ,agtheAnch

erivethe

res

ionoftheo . The

tchdem

previous

roductand

rasinglepr

t thisisjustan

nctionisder

rPointatst

lows as in

s,2014):

dagainther

rPointatst

ulthere.

c(/produc (/order)R(products

timalfoeason is tha

ndJogl

odel a finitisthesetuductionrun.

s inthepre

example).

ivedinJogle

rtofsalesa

he previous

1 eistheextr

artofthefir

)

year)

(pr (cyc

Integrated Logi

rcanbet the suppli

ekar(19

production

pcostfora

.

iousmodel,

ar(1988):

sinthefigur

sections, p

2 2 termwith

stproductio

producer

ducts/orderletime)?

sticsderivedfrom

ers reward

8)

rate (witproductionr

the invento

1 eabove,an

roduces the

1 runwillpr

y(prodw(/pr

(pro

(/o

)?

thisresultf

is a constan

). Inun.Thedeci

y levelover

2

2

whenmaki

following li

2 .duceadiffe

cts/year)oduct)ucts/order)

der)

ortheclassic

t term that

this model,

sionvariable

timemay lo

gassumpti

nearised AS

2

rent

fun

28

function

does not

is theisthe

ok like in

nsabout

function

2

tion.We


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29

NotethatthemodelofJoglekarincorporatespreviouslyconsideredmodels:

For ,wegetGoyalsmodel; For 1 ,wegetMonahansmodel; For 1 and ,theEOQmodelwithbatchdemandofGrubbstrm.

Homework

Derivetheoptimalorderquantityusingtheclassiccostfunctioninwhich andthenderivetheoptimalorderquantityusingthelinearisedannuitystreamfunction.Comparethetworesults.2.9 Vendingmachine

WeillustratetheusefulnessoftheNPVframeworkforderivingtheprofitfunctionofavendingmachine

operator.Weconsiderasingleproductsoldatapriceinavendingmachineofcapacity.Theproducthasacostpriceforthevendor.Thevendorhasasetupcostfordeliveringabatchofproductstothevendingmachine.Uponthedeliveryofproductstothevendingmachine,theoperatorcollectsthecoins

of

the

customers.

Assume

a

constant

demand

rate

.

The inventory level over time follows the EOQ sawtooth pattern.The cashflows however differ from

thatintheEOQmodelandaregiveninthefigurebelow.

2 2

Theholdingcostforavendingmachineistobebasedon

,i.e.basedonthesumofcost

priceand

sales

price!

ThereasonwhythisresultdiffersfromtheEOQmodelisthatthecoinsputintoavendingmachinebycustomers iscashthat isnotyetaccessibletothevendoroperator.Onlyuponcollectionofthesecoinscanthevendorhaveaccesstothiscapitalforreinvestment. It isas ifthecustomersonlyexchangethe

cashwiththeoperatorthemomenttheoperatoremptiesthevendingmachinescoinsregister.

Itcanbeproventhattheoptimallotsizeis min , .ThisexampleillustratesthattheprofitfunctionintheNPVframeworkwilldependontheassumptions

wemake

about

when

cash

is

exchanged.

We

call

this

the

payment

structure

and

it

is

further

discussed

in

thenextsection.

Time

Cashflows

0

s+wyT

T

....

....

pyT


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2.10

P

Conside

product

thatthe

If th

Convent

buyerp

Ther

The

transact

We

ignored.

Itisnot

incases

EOQm

Assume

time.assume

ayments

thetransfe

thatthebu

totalamoun

paymento

ional (C). It

ysatdiffere

CashInAdv ,withCredit(CR): 0arefurther

Paymentsin

with0 atthesuppli

Transaction

instrument

theamount

igurebelow

ion costs a.

illhencefor

Allpayment

difficulttoa

wheretheti

delwithCR

thatallcust

Thecashfl

thatthefirst

tructure

rofabatch

erhastopa

t

ispaid

ccurs in full

isnot theo

tmoments

ance (CIA): 0 .thebuyerca

orethefoll

instalment1,asCIA(ter),andthe

costsandtr

sed,thesup

andtimethe

illustrates.T

d delays.

thassume t

sfurthercon

aptallprevi

eisspeciandCIApa

merspayw

wdiagram i

deliverywill

ofproducythesuppli

orthemom

at the time

ly reasonab

intime,inclu

hebuyerpa

paythesu

owingtwoc

.The

buyer

hiscouldbe

emainder1nsactiondel

pliermayre

buyerhasm

hepayment

e call the

at transacti

sideredare

ouslyconsid

ied.Anexa

mentstru

ithaCRofti

sgivenbelo

occuratsom

Integrated Logi

tsbetween

r. Inthepr

ent

thatth

ofdelivery

leassumpti

ding:

ysat some

plierlatert

nsideration

maypay,

fo

adepositpai wiays.Duetoi

ceiveadiffe

adeapaym

oftheamo

first two p

oncostsan

enceredmodels

plefollows.

tures

mebutt.Sincewe

efuturetim

sticsbuyerand

vioussectio

batcharriv

then it is s

nwecanm

timebefore

anthetime

:

example,a

dthemome

haCRarran

nefficiencies

entamount

nt.

nt isCIA,yments

delaysare.forsituation

atyouhave

havetopay

ewith

asupplier.L

ns,wehave

satthebuy

aid that the

ake. In reali

thedelivery

ofdelivery,s

fractionof

t

twhenthe

gement.

andcostsch

andatadiff

ofisC,a an

zeroorso s

swherepay

topaythe

thesupplier.

etbetheconsistently

er.

payment st

y, itmaybe

ismade, sa

ayattime

heamount

buyerplaces

argedbythe

erenttimer

dofisCd the third

all that th

entsoccur

upplierwith

inadvance,

30

priceper

assumed

ucture is

thatthe

at time

,with

ue,hisorder

financial

lativeto

andhas

payment

ycanbe

CIAorCR

CIAwith

wemust


31/150


31

Thisgives:

2 2

2 Wealsoget: 2 2 and

2 2

Note.ItiseasytoseethatifcustomerswouldpayCIAwecanusetheabovemodelbutconsidernegative

valuesfor.Likewise,wecanstudytheimpactofreceivingacreditperiodfromthesupplierbytakingintheabovemodelnegativevaluesfor.Numericalexample

The table below illustrates the impact for an example with 25/, 150/, 2000 /, 0 and 50/. We take 0.2.

The%gap isacommonmeasureforgetting insightinrelativedifferencesbetweenascenarioanda

base case scenario. In the table it is calculated as the percentage difference relative to the base case

scenarioofaconventionalpaymentofthesupplieri.e.for 0.The%gapformeasureis:% 100 00 ,where in the above table is, respectively, , , and. It can be observed that the increase in

logisticscosts ismuchsmaller thanthecorrespondingprofit loss.Forassessing the impactofdifferent

timingsofpaymentsitishencemuchsafertoconsidertheprofitfunction.

Time

Cashflows

0

s

T

py ....

....

L

wyT


32/150


32

(months)

(products/order)

% gap

(-)

(/year)

% gap

(-)

(/year)

% gap

(-)

0 346 0.00 1732 0.00 48253 0.00

1 344 0.83 1747 0.84 47398 1.77

2 341 1.65 1761 1.68 46529 3.57

3 338 2.47 1776 2.53 45646 5.40

4 335 3.28 1791 3.39 44747 7.27

5 332 4.08 1806 4.25 43834 9.16

6 330 4.88 1821 5.13 42906 11.08

2.11Consignment

arrangements

Consignment arrangements are popular payment structures in some industries between suppliers and

buyers.Thesebuyersarecompaniesthemselvesandmayberetailersorproductioncompanies.Thestockheld at a buyer under this arrangement is called the consignment stock. It is hence inventory that is

physicallyheldatthepremisesofthebuyer,butfinanciallyitisstillunderthe(partial)ownershipofthe

supplier.Onlywhenaproductisremovedfromthisconsignmentstock,willthebuyerhavetocompletethepaymentfortheproducttothesupplier.

Assumethatthepriceforaproductthatabuyerneedstopaytothesupplieris.Wecandistinguishbetweenthefollowingthreecommonconsignmentarrangements:

FullConsignment(FC):thesupplierretainsownershipofthe inventoryatthebuyerandthis isimplementedbylettingthebuyerpaysthesupplierthepriceforaproductonlyatthemomentthatthisproductisactuallytakenoutoftheconsignmentstockatthebuyer.Thisproductisthentobeused inthebuyersproductionprocess,orinthecaseofthebuyerbeingaretailer,when

thebuyeractuallysellstheproducttooneofitsowncustomers.

PartialConsignment(PC):thesupplierispaidanamountforeachproductwhentheproductisdelivered to the buyers consignment inventory, but the remainder is onlypaid out themoment the product is taken out of the consignment stock by the buyer (for reasons asexplained above for the case of full consignment). The price

can in principle be any agreed

amount;itdoesnotneedtobethesuppliersowncostprice.

Graceperiod(GP(z)):ThisissomewhatsimilartoPCinthatthebuyermayfirstpayanamountthemomentthatproducts aredelivered to itsconsignmentstock,but the remainder isthentobepaidbackaccordingtoacreditarrangementthatdependsontheaveragecycletimebetweendeliveries.Thiscreditperiodiscalledthegraceperiodanditsmomentofpaymentisingeneralasfollows: ,where

isasuitablychosenconstant.The idea isthatwhenabuyerorders in larger lotsizes,

thatthecreditperiodthesupplieriswillingtoofferalsobecomeslonger.

Weillustratethethreepaymentstructureswiththefollowingexample.


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33

Abuyersuppliermodelwithconsignmentarrangements

WeconsiderabuyerhavinganEOQproblemandthesupplierdeliveringtothisbuyerthentohavethe

EOQproblemwithbatchdemand.Bothmodelswerediscussedpreviously.TheASfunctionofthebuyer,

underconventionalpaymentstructures,was:

2 2,wherethesubscriptisusedtomakeclearthatitconcernsthebuyer.ThesuppliersASfunctionwas: 2 1 2 2.

Wenowintroduceageneralisedpaymentstructurewhichhastheabovediscussedthreevariationsof

consignmentinit:

Anamount

ispaidforaproductwhenitisdeliveredtothebuyer;

Anamountispaidforaproductwhenthebuyersellstheproducttoitsowncustomers; The remainder is paid for with a grace period time units after thedeliveryoftheproducttothebuyer.Hence, for 0 , 0 and this corresponds to FC; for 0 but 0 this

correspondstoPC;andwith 0but 0 toGP(z).Themoregeneralcasehasallthreeamountsnonzero.Notethat .

CustomersofthebuyerstillpayaccordingtotheconventionalstructureC,asintheEOQmodel.

ThebuyersASfunctionchangesto:

Welinearisetheexponentialtermsinthedecisionvariableandfind:

2 1 2 2 Note that the holding cost for the buyer is based on

1 2

. For

, the

holdingcostforthe

componentiszero.For

1,theholdingcostforthe

componentbecomesa

negativecost

i.e.

will

increase

the

buyers

profit

function.

ThesuppliersASfunctionchangesto:

2 1 2 1 2 2.

Thesuppliersrewardisaffectedandnowtobebasedon

1 2 .


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34

2.12Theprofitfunctionoftheintegratedsupplychain

If firms in a supply chain want to identify the optimal way of coordinating, then a commonly used

benchmarkisassumingthatthefirmswouldoperateasonevirtualorganisation.Thisvirtualorganisationrepresents the integratedsupplychain.Toderive itsoptimalpolicy,wearguablyneedtohaveaprofit

function

as

well.

We

illustrate

with

an

example

what

can

be

done,

and

where

there

are

still

some

open

problems.

Consider the buyersupplier model from the previous section, but assume conventional paymentstructures.Theprofitfunctionsofbothfirmsare,respectively:

2 2, 2 1 2 2.Itistemptingtopostulatethattheprofitfunctionoftheintegratedsupplychainisfoundastheirsum:

Thisgives:

2 1 2 2However,thisfunctionnowmakesuseoftwodifferentopportunitycostsofcapital,and.Would

an integrated firm not be in a position to use one common opportunity cost of capital instead? If wewouldassumethatbothcapitalcostratesarereplacedbyonecommonrate

wewouldfind:

2 1 2 2 2 2Sucharesultwouldalsobefoundifwestartedfromthecashflowdiagramsofbothfirms,andthen

recognisingthatcashexchangedbetweenthefirmswouldcanceleachotherout.

Thisconundrumhasnotyetbeenadequatelyaddressedintheliterature.Wewillassumehenceforththat

; in this case the problem does not present itself. Under this assumption, we can

indeedsumtheprofitfunctionsof individualfirmsto findtheprofitfunctionoftheir integratedsupply

chain.

Exercise1

Derivefortheabovebuyersuppliermodelwithconventionalpaymentstructurestheorderquantitiesandthatwillmaximise.

Wetakethepartialderivativeof, toforaconstant,andfromsettingthistozerowefind:

2

Substitutionofthisresultinto :


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35

2 2

2 2 Theoptimalpositiveintegervalueishence 1.Therefore,wealsoget2

and

2 2 .Exercise2Considerthebuyersuppliermodelwithconsignmentpaymentstructuresderivedpreviously.Repeattheaboveanalysisforthiscase,i.e:

a. Findtheprofitfunctionoftheintegratedsupplychain;

b. Findthevaluesofandthatmaximisethisfunction.Youwillfindthattheprofitfunctionoftheintegratedsupplychainisexactlythesame,andtherefore

also the optimal values derived above apply. This gives an important insight: the payment structuresbetweenthefirmsinasupplychaindonotaffecttheirintegratedprofitfunction!

Homework

Considerthesetofmodelswehave lookedatsofar,andmakeupyourownfeasiblecombinationofa

singlesuppliersinglebuyermodelandrepeattheaboveanalysis,i.e.determinetheintegratedsupply

chainprofitfunction,andderivetheoptimalpolicythatwillmaximisethisfunction.

2.13UsingtheNPVframeworktoincludeothercostcomponents

TheNPVframeworkalsoallowsfortheconsiderationofothercostcomponents.Weillustratewithafew

examples,usingtheEOQmodelasourbasecasescenario.

Materialhandlingandinsurance

A variable material handling cost e (/product) for each product placed in inventory, paid uponreceivingthebatch

Acostdirectlyproportionaltotheaverageinventorylevel(e.g.thecostofinsuranceagainstfirefortheaverageamountofinventoryheld),tobepaidase.g.acontinuousstreamattherateof /2(/year).

TheASfunctionfortheEOQmodelisadaptedto:

2 2 1


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36

2 1 2

2

2

Thusthe

optimal

lot

size

becomes:

2 andtheholdingcostsaretobecalculatedasfollows: Ingeneral,wecanhenceidentifytwodifferentcomponentsofholdingcosts:

a. The

financial

cost

of

keeping

stock,

arising

from

investments

made

into

products

stored,

here

the

term ;b. Outofpocketholdingcosts,arisingasrealexpensestobemadewhenplacingproductsinstock;here.Notethatthesecostsneedtobeafunctionofthestockposition i.e.thecorrespondingannualcostneedstovarywith.

It isgenerallyaccepted that inmanypracticalsituations the financialholdingcostsaremuch largerthan theoutofpocketholdingcosts.That isalsowhy inmanymodelsoutofpocketholdingcostsare

simplyignored.

Priceelasticdemandfunctions

Finaldemand

foraproductcouldbeafunctionofprice

.Forexample,onepossible function is (for

0): where and arepositiveconstants. If the retailer with an EOQ model could decideon theprice,thenthereisthepracticalconstraint: asotherwisetheretailerwouldnotmakeanyprofits.TheretailerthushastheASprofitfunction:

, 2 2

Onewaytosolvethisproblemistakingpartialderivativestoandandsolvethenonlinearsystemoftwoequations.Amorepragmaticapproachwouldbetoapplythefollowingalgorithm:1. Input:Valuesfor, ,,andafunctionwithmaximumprice2. Forasequenceofprices , , 2, ,

2.1.Determine2.2.Usethistocalculate 2

2.3.Calculate

,

3. Retainthatpricethatgivesthehighestvaluefor ,.


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37

Creditelasticdemandfunctions

Sometimes firms will offer a delay of payments in order to boost demand. In the simplest case, the

completeamounthastobepaidfortimeunitsafterthepurchase.Assumethatthefunctionisknown.TheASfunctionfortheretailernowbecomes:

1 , 1 2 , 2 2 TofindtheoptimalvaluesforLandQ,thefollowingpragmaticalgorithmcouldbeused:

1. Input:Valuesfor, ,,andafunctionwithmaximumcreditperiod.2. Forasequenceofdelayvalues

0, , 2, ,

2.1.Determine

2.2.Usethistocalculate 2 2.3.Calculate ,

3. RetainthatLthatgivesthehighestvaluefor ,.References

GRUBBSTRM,R.W.1980.APRINCIPLEFORDETERMININGTHECORRECTCAPITALCOSTSOFWORKINPROGRESSANDINVENTORY.INT.J.OFPRODUCTIONECONOMICS18(2),259271.

BEULLENS, P. AND JANSSENS, G.K. 2011. HOLDING COSTS UNDER PUSH OR PULL CONDITIONS THEIMPACTOFTHEANCHORPOINT.EUROPEANJOURNALOFOPERATIONALRESEARCH215,(1),115125.

BEULLENS, P. 2014. REVISITING FOUNDATIONS IN LOT SIZING CONNECTIONS BETWEEN HARRIS,

CROWTHER,MONAHAN,ANDCLARK.INT.J.OFPRODUCTIONECONOMICS155,6881.

BEULLENS, P. AND JANSSENS, G.K. 2014. ADAPTING INVENTORY MODELS FOR HANDLING VARIOUS

PAYMENT STRUCTURES USING NET PRESENT VALUE EQUIVALENCE ANALYSIS. INT. J. OF PRODUCTIONECONOMICS157,190200.


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38

3 IL in Buyer-Supplier Supply Chains

3.1 Onetooneshipping

ThisstrategyinvolvesshippingproductsfromalocationAtoalocationB,bylettingavehicle(truck,taxi,

train,ship,airplane)takingafeasibleandoptimalroute,typicallythecheapestorfastest.Let us assume that the vehicle is dedicated to the transaction, i.e. during its trip it will not make

detours or perform other transportation duties. It is then also called Direct Shipping (DS) or linehaul

shipping.

Figure 5: One-to-one shipping (Direct shipping, line-haul shipping, FTL shipping)

Thetransittimeisthetimethegoodsareunderwayfromthesourcetothedestination.Itisingeneralthesmallestrelativetoothershippingstrategies(discussedlater).

Becauseavehicleisdedicatedtothissingletransportationjob,itis ingeneralonlyjustifiablefroma

costperspectivetotransportsmallamountswhentheproducttransportedhasahighvaluedensity(/m3

or/kg)orwhenitneedstoarrivefast(emergencyshipment).Itisnotunusual,forexample,todispatcha

privatejetorhelicopterforthetransferofhumantransplantorgansinordertosavealife,ortodelivera

singlesparepartbytaxiinordertopreventexcessivedelaysforapassengerairflight.

Forgoods

with

lower

value

density,

direct

shipping

is

only

cost

effective

when

avehicle

can

transport

a large enough amount. Therefore this strategy is sometimes also referred to as FullTruckLoad (FTL)shipping,althoughitisnotrestrictedtoroadtransport,norisitalwaysoptimaltoshipinquantitiesthatfillupthevehiclescapacity.

3.1.1 ShortestPathProblem

FindingtheoptimalroutefromAtoBcanbemodelledasaShortestPathProblem(SPP).TheSPPcallsfor

findingtheshortestpathfromanoriginnodetoadestinationnodeinaconnecteddirectedgraphG=(N,

A)withnodesetNandarcsetAandwhereeveryarc aAhasanonnegative length.ShortestpathproblemscanbeefficientlysolvedusingDijkstrasalgorithm.

Figure 6: A shortest path problem

1

2

6

4

53

4

33

3

2

2

2

AB


39/150


39

Figure 6 showsan exampleof ashortest path problem. There are sixnodes in thegraph numbered

from1to6,andsevenarcs,whereeacharcsdirectionandlengthisalsoindicated. Theheadofanarcis

thenodeadjacenttothearcsarrowhead,andtheotheradjacentnodeofanarciscalledthearcstail.

DijkstrasAlgorithm

Tofindtheshortestpathfrom insuchgraphfromsomeoriginnodetosomedestinationnode,wecan

useDijkstras

algorithm:

1. Associatewithallnodesatemporarylabelwithvalue2. Startattheoriginnodebychangingitslabelto0andmakethelabelpermanent.Callit

thecurrentnode.3. Foreveryarcgoingoutofthecurrentnodethathasaheadnodewithatemporarylabel,

replacethevalueofthetemporarylabelofthisheadnodewiththevalue:

min ,

4. Amongalltemporarylabellednodes,selectonewiththesmallestlabelvalueandmakethe labelpermanent. If thisnode is thedestinationnode,stop,elsecall thisnode thecurrentnodeandgobacktoStep3(iterate).

Theoptimaltotallengthisnowequaltothelabelvalueofthedestinationnode.Todetermine

theoptimalpath,startatthedestinationnodeandworkbackwards inthegraphbyfinding

thearcwhichcostcorrespondstothedifferenceofthelabelsofitsheadandtailnodes.There

maybemorethanoneoptimalpath.

AppliedtotheprobleminFigure6,thealgorithmgivesthefollowingsequenceoflabelvaluesforeach

node

(note

that

permanent

label

values

are

indicated

with

a*,

and

the

position

in

the

sequence

correspondstothenodenumber):

[ ][0* ][0* 4 3 ][0* 4 3* ][0* 4 3* 6 ][0* 4* 3* 6 ][0* 4* 3* 7 6 ][0* 4* 3* 7 6* ]

[0*

4*

3*

7

6*

8][0* 4* 3* 7* 6* 8]

[0* 4* 3* 7* 6* 8*]

Thelengthoftheoptimalpathisthus8.Theoptimalpathisderivedstartingfromnode6andgoing

backtonode5sincethelength2ofthatarcisequaltothedifferenceofitsheadlabelandtaillabels,2=

86.Itistheneitherpossibletogotonode2,sincethearcslength2=64,ortonode3sinceitsarcslength3=63.Fromnode2wegobacktonode1.Fromnode3wewouldalsogobacktonode1.There

arethustwooptimalpaths:eitherthesequenceofnodes1256orthesequenceofnodes1356.

Note that all nodes in this examples have permanent labels at the end. This is not always so; ingeneraltherecouldbenodesthatstillcarrytemporarylabels.Notealsothatthealgorithmstillworksif

theretherearemultiplearcshavingthesametailandheadnodes,aslongasweselectinStep3ofthe

algorithmthe

shortest

arc

in

the

formula.

Dijkstras

Algorithm


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40

Thealgorithmfindstheoptimalsolutionsinceeverytimewemakealabelofsomenodejpermanent,

wewillhavefoundtheshortestpathfromtheoriginnodetothatnodejanditspermanentlabelvalueis

the length of this optimal path. The optimality of this path from origin toj is not depending on any

decisionsweneedtomake lateron inthealgorithm,andviceversa: iftheoptimalpathfromoriginto

destinationwouldpassnodej,thentheoptimalpathfromtheorigintojwillbecompletelypartofthe

optimalsolutionindependentofthedecisionmadeinthepathfromjtothedestination.Wecallthisthe

principleofoptimality.(Wewillfurtherdiscussconditionsunderwhichthisprincipleisnolongertrueandthereforethealgorithmwouldnotbeapplicable.)

Differentobjectivefunctions

Differentobjectivefunctionscanbeusedminimisingtotallength,timeorcostbylettingthelengthofeacharcintheSPPcorrespondtoitsdistance,expectedtransporttimetocrossthearc,ortotalcostto

useit,respectively.Itiseasytoincorporateanyfixedcostsorfixeddelaysencounteredonarcs,suchas

ontollroads,attollbridges,andattolltunnels,oratbordercrossings(administration,bordercontrol).

Tachographlegislation

Fortrucktransportoverlongerdistances,attentionhastobepaidtothesocalledtachographlegislation

which requires drivers to take breaks and limits the total number of driving hours per day. Typicalconstraints

may

include

the

following:

1. nomorethan2hoursofconsecutivedrivingisallowed;2. 45minutesofresttimeneedstobetakeneitherafter2hoursofconstantdrivingorduringthe2

hoursofdriving(inbreaksofminimum15minuteseach);

3. totalnumberofdrivinghoursperdriveranddayisrestrictedto8hours.

Companiesmayhavethechoicebetweenusingtwodriversreducingtotalroutetimeorusingasingledriver with longer total travel time. To minimise to total costs, one can run the SPP algorithm on the

graphusingonlytraveltimerelatedcostsforonedriverandthen,alsoknowingthetotaltraveltime,add

thecostsoftheextradriver,requiredbreaks,andresttimestofindthetotalcostandrealtotaltime.The

optionthat

is

the

cheapest

can

then

be

retained.

Timedependenttraveltimes

Theproblemofminimisingtotaltraveltimeingraphswithtimedependentexpectedtraveltimesonarcs

can be adequately modelled as a SPP if the start time at the origin node is given. The algorithm now

needstouseinStep3theexpectedtraveltimeofanarcbasedonthecalculatedexpectedarrivaltimeat

itstailnode.

Undertheassumptionthatvehiclesthatarrivelateratthetailnodeofanarccanneverarriveearlierattheheadofthearcthenvehiclesofthesametypethatarrivedearlieratthetailnode,thealgorithmfindstheoptimalpath.Theassumption is ingeneralrealistic forqueue induced traveldelaysonroadssincevehiclesofsometypeattheendofaqueuewillfind itverydifficulttobeatvehiclesofthesame

typeatthetopofthequeue.Theassumption isknownasthenoovertakingproperty.Notethatthis

propertyimplies

that

waiting

at

any

node

in

the

graph

is

never

optimal.

This

approach

also

works

for

when a desired arrival time at the destination node is given by letting the algorithm start at the

destinationnode,workingbacktotheoriginnode,andsubtractingtimesduringthesearch.Tofindminimumcostsolutionsfortimedependenttraveltimesforwhichthenoovertakingproperty

holds, the travel cost must be monotone increasing with travel time and a given units of driving timemustcostthesameasthesameunitsofwaitingtimeonnodes.Inthatcasethere isalwaysanoptimalsolutioninwhichnowaitingoccurs,andthusthealgorithmwillfindanoptimalsolution.

If waiting (i.e. resting) is less costly than driving, it may be optimal to wait for times with lesscongestion on roads in order to minimise costs. However, waiting at some tail node means that

opportunities for faster driving may be lost on arcs closer to the destination. Thus, the principle of

optimalitynolongerholdsandotheralgorithmsaretobeused.


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41

Multimodeltransportation

MultimodaltransportationproblemscanbemodelledasSPPsbyassociatingwitheacharcinthegraph

thedatarelevantforaspecificmodeandvehicletype;multimodeltransfernodesareintroducedinthe

graphwhichconnect thegraphsof thedifferent transportationmodes.Each transfernode willadd its

transfercostorexpectedtimeoftransfertotherelevantarcleavingthenode.ThenormalSPPalgorithmwillthenbeapplicable.

However,theremaybeotherconstraintsinpracticemakingthisapproachlessrealistic.Ships,trains,andairplanestypicallytravelaccordingtoprespecifiedschedulesandroutesandsomeshipsortrainsare

cheaperthanothers. Itmaythusbecheapertowaitatatransfernodeforthecheapestvehicle(train,

ship)orroute.Thishowever,violatestheprincipleofoptimality.

Note. For calculating SPP on road networks, various internetbased resources can now be used. For

example,GooglemapshasafunctiontoallowyoutoseekforthequickestroutefromAtoB,whereyoucanspecifyyourstartingtimeandwhichtakesintoaccounttimedependenttraveltimes.

3.1.2 EconomicTransportQuantity(ETQ)

Considerthesituationthatabuyerneedsregularsupplyofsomegood.Weconstructacostmodelforthe

situationof

direct

shipping

of

the

product

from

asupplier

to

this

buyer.

We

assume

that

the

optimal

travelroutehasbeendetermined(ase.g.anSPP)andthatitscostandtotaltraveltimeareindependent

ofthetimeoftheyearatwhichitisundertaken.Wedenotebythetotaltransittime.WeassumeabuyerwithanEOQmodelandasupplierproducinglotforlotatafiniteproductionrate.

Wenowalsoconsidertheintermediatestagewheretheproductsareonavehicleintransit.WeplacetheAnchorPointatthedeliveryofthefirstbatchtothebuyeratarbitrarytime.

Time

Inventorysupplier

0

T

....

yT/R

Time

Inventoryintransit

0

T

....

L

Time

Inventorybuyer

0

T

....

L


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42

Figure 7: Inventory positions and cash-flows in the direct shipping model

Weassumethatthetransport isundertakenbya3PLandthatthebuyerpaysthiscompanyafixed

transportcostpershipmentandavariabletransportcostthatdependsonthelotsizebeingshipped.The

buyerpaysthesupplierfortheproductsthemomentthatabatch isdelivered.Weassumethatnexttosetupcostsforloadingandunloadingthevehicle,the3PLalsoincursatransportcostatarateforthedurationofthejourney.SeealsoFigure7.Thecapacityofthevehicleis

.

Weproceed

by

deriving

the

AS

profit

functions

of

the

three

firms

involved

from

their

cash

flow

functions.Forthebuyer,wehave:

2 2

Forthe3PL,wefind:

1

Time

Cashflowssupplier

0

s

T

wyT

....

yT/R

cR

L

Time

Cashflows

3PL

0

T

....

L

Time

Cashflowsbuyer

0

T

....

L


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43

Define: 1Then:

2 2 Forthesupplier,finally,wederive:

1 Thisleadsto,aftersomealgebraicmanipulation:

1 2 2 2 2 Optimalthreefirmsolution

Ifthethreefirmsareinterestedindeterminingtheoptimalshippingstrategyfortheirintegratedsupply

chain, we can derive this from their supply chain profit function. This function is found from the

summationoftheirprofitfunctions.(Asalways,assuming

).Thisproduces:

Define: 1 1 1 2

Then:

2 1 2 Thisproducesanoptimalunrestrictedlotsize:

2 1 However,sincethevehiclecapacityis

,theoptimalfeasibleEconomicTransportQuantity(ETQ)is:

min,


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44

DeterminingthevalueofTheparameterisanannuitystreamcostrateandhenceneedstobeexpressedin(/year).Weshowhowtoincorporatetworelevantcomponentsincasethatthevehicleisaroadvehicle:driverwagesand

fuel costs. Driver wages are typically given in (/hr), so if a driver costs (/hr), then it has to beconvertedtoacostrate

(/year)asfollows:

8760 ,since there are approximately 24365 8760 hours in one year (i.e. not counting years with anextradayinFebruary).Fuelcostsaretypicallyexpressedin(/km).Therefore,afuelcost(/km)hastobeconvertedintoarate(/year)byassuminganaveragespeedofthevehicleof(km/hr): 8760 .

Ifthesearetheonlyrelevantcosts,itwouldhenceproduce .Exercise

Determinethe

optimal

lot

size

when

the

buyer

would

independently

be

able

to

determine

.Inthatcase,thebuyerwouldaimtomaximise.Thisproduces min , where:2

Note.Observethattheunrestrictedoptimallotsizefortheintegratedthreefirmsolutionisafunctionof

thetransitleadtime.However,fortheabovesolutionforthebuyeritisindependentofthisleadtime.

Optimalsolutionwhentransportisoutsourced

If

the

3PL

works

independently

and

the

supplier

and

buyer

want

to

determine

the

optimal

shipping

strategyforthemselvesastwofirms,wecanderivethisfrom: Redefine:

1 2

Then:

2 1 2 Thisproducesanoptimalunrestrictedlotsize:

2 1

However,

since

the

vehicle

capacity

is

,theoptimalfeasibleresultis min , .


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45

Note

Ininternationaltransportbyseaitisoftenthatcasethatbuyerandsupplierneedtorelyona3PLforthe

shipping.Here, couldbe themaximum load of theproduct intoonecontainer.The 3PLmay chargedifferentratesbetweenshippingafullcontainerversusshippingafractionofacontainerload.

Homework

Derivethe

optimal

lot

size

when:

(1)

buyer

and

3PL

would

seek

to

find

their

integrated

optimal

solution;

(2)supplierand3PLwouldseektofindtheirintegratedoptimalsolution.

Note

Thesituationoflotforlotataninfiniteratecanberetrievedfromtheabovefunctionsbyconsideringthe

case .ThecasethatthesupplierproducesaccordingtoanEOQwithbatchdemandisretrievedbysetting .3.1.3 MaximumEconomicHaulageRadius(MEHR)

Thereisalimitastohowfaronecanreasonablytransportacertaintypeofproductusingdirectshipping.Ingeneral,thehigherthevaluedensityofaproductthefurtheritcanbetransportedinsmallquantities.

This

makes

it

reasonable

to

use

dedicated

small

transport

vehicles

such

as

small

vans,

taxis,

or

expensive

vehiclessuchasaeroplanes.Thelowerthevaluedensityofaproduct,theshorterthedistanceoverwhichitcanbeeconomically

transported.Tocoverlongerdistanceswouldneedshipmentsinlargequantitiessuchaslargetruckswithasecondtrailerorbulktransportinabarge(inlandwaterways),orcontainertransportontrains(rail)orships (sea and ocean transport) where transport costs can be shared with other goods from other

companies.

UsingtheprofitfunctiontoderivetheMEHR

KnowledgeoftheASprofitfunctionallowsustocalculatethemaximumdistanceoverwhichaproduct

can be transported. Since

, only strictly positive value for

will also produce strictly

positivevaluesforNPV.Asanyproject isnotconsideredworthwhilebyacompany ifitsrespectiveNPV

wouldbecome

negative,

the

boundary

condition

would

be

that:

, 0Note that when, 0, this doesnot mean that the company would not produce a positive

profitinaccountingterms.Itsimplymeansthatitwouldnotproducemoreprofitthanfromthenextbest

available alternative! If 0.20 then the firm would still gain a respective 20% of profits from thisactivity.

Wesketch theapproachwith the followingexample.Take the threefirm integratedprofit functionderivedpreviously:

, 2 1 2 It is sufficient to focus on the function in between the square brackets. Furthermore, we can

substituteintothisfunction,producingtheboundarycondition: 2 1 2 0,

where

1


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Wen

CaseThelot

types of

function

wher

The

express

CaseSubstitu

It m

practicalreduces

3.1.4

Figur

owneedto

.izeisinoth

terms in th

of.

eworkingo

righthand s

din(years).

tionofther

y be difficu

approach

tozero.(Sim

Optimalpo

e 8: Cross-D

Inbou

onsidertwo

rwordsac

e boundary

orkingoutt

ttheactual

ide of this

Toconvertt

.sultfor lt to derive

ouldbe

to

ilartotheal

licytoorde

cking Facilit

ndtransport

cases.

nstant.Itca

condition: t

hisboundary

aluesfor

inequality wistoadista

derivedp 2an analytic

sean

algori

orithmspre

rNdiffere

Cross

Integrated Logi

nbeobserv

erms that a

condition,

andisleft

1 ould be th

ce,younee

reviouslywo

2 result for th

thmwhere

sentedatth

titemsfro

ockingFacili

1

stics1 1

dfromthe

re not a fun

ewillhence

,asanexercis

maximum

dtoconsider

uldproduce

e maximum

ouwould

in

endofSecti

aCrossD

Outbound

ty

2

bovefuncti

ction of ,findaformu

totheread

economic h

thetypeof

heconditio

1 possible val

creaseunon2.13).

ockingFaci

transport

onsthatwe

and terms t

laofthesor

er.Thismea

aulage radi

ehicleused.

:

0lue of

. Ho

ilthe

right

lity(CDF)

2

46

havetwo

at are a

:

sthat:

s

wever, a

andside


47/150


47

A CrossDocking Facility (CDF) is typically located near the boundaries of a populated area. It receives

goods in (large) vehicles from various suppliers; these incoming goods are separated and mixed as

required at the CDF, and subsequently sent out in vehicles without being held in storage to different

destinationsinthelocalarea.

Thecrossdockingoperationsmayrequirelargeareasinthewarehousewhereinboundmaterialsare

sorted,consolidated,andstoreduntiltheoutboundshipmentiscompleteandreadytoship.IfthistakesseveraldaysorevenweeksitisnotconsideredaCDFbutawarehouse.InmostCDFs,goodsdonotstaylongerthan48hours.

OptimalorderpolicyforNitemsreceivedfromaCDF

Considertheproblemwhereyourfirmrepresentsstockingpoint2inFigure8.Yourfirmhasademandfor

N different types of items (i = 1, ..., N). For each item the uniform annual demand rate of yourcustomers is yi and your cost price is . You order each of these items from the CDF. Each time avehiclefromtheCDFvisitsyourfirm,however,youhavetopayafixedtransportcost.Whatwouldbetheoptimallotsize fororderingeachitem?

Since

you

have

constant

annual

demand

for

each

item,

you

have

an

EOQ

type

problem

for

each

item.

YourASprofitfunctionforitem,wheneachitemisorderedseparately,isthenarguablyofthefollowingform: 2 2 For each delivery, the CDF charges a setup cost , and that is why in the above function wehave .Yourtotalprofitfunctionwouldthenbethesumoverallitems:

2

2

Couldtherebeabetterwayofordering?Isitperhapsworthwhiletoordersomeitemstogetherintoonetrip?Thiswouldsafeontransportationcharges.

Property.AnoptimalpolicycontainsschedulingperiodsTiofequallength,i.e.Ti=TforalliN.

Proof. Suppose the theorem does not hold. Let T designate the smallest scheduling period which

happenstobe for item k, i.e. T=TkTand letthe

averageinventorycarriedofthisitembeE(Ij) .Nowsupposewedecrease itsschedulingperiodtoTj =T.Theaverage inventorywilldecrease from

E(Ij) to E(Ij ). Since the replenishment cost is independent of the quantity ordered, no additional

replenishmentcost

is

incurred

for

replenishing

item

type

jevery

Tj

=Tunits

of

time.

Hence

the

global

costwilldecrease.

SinceanoptimalpolicycontainsperiodsTofequallength,wehave:

Andweget:

2 2


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48

Wefind:

2

1

And:

2 2

3.2 Onetomanyshipping

3.2.1 LengthofanoptimalTSPtourvisitingmanycustomers

Drawonapieceofpaperasquare.Callthisyourserviceregion.Indicatethexandyaxis(see

Figure9).Wearbitrarilytakethelengthofthesidesequalto1m,sothattheserviceregioncoversanarea

of1x1=1m2.

Figure 9: Randomly distr ibut ed points in a rectangle.

Nowdrawarandomnumberfromtheinterval[0,1]andcallthisx1.Drawasecondrandomnumberfrom the interval [0, 1] and call this y1. Use these coordinates to draw a point (x1, y1) in your serviceregion. Repeat this procedure.Nowyouwill haveasecond point locatedatsomecoordinates (x2, y2).

Continue until you have generated n points in your service region. Figure 9 shows the exercise at four

differentstages,i.e.forn=5,n=10,n=15andn=20.WecallthesetofnpointobtainedXn:

)},(),....,,(),,{( 2211 nnn

yxyxyxX

Drawatourthroughyournpoints,visitingeachpointonlyonceandreturningtothefirstcitywhere

youstartedasinFigure10.Notethateachtimeyouleaveapointyouhavetodecidewhichpointtogoto

next.Youcanthusconstructseveraldifferenttours,allhavingadifferenttotaldistance.Infact,thereare(n1)!/2differentpossibletoursthroughnpoints.

x

100

1

(x1,

y1) (x1,y1) (x1,

y1)(x1,

y1)


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49

Figure 10: A TSP tour t hrough 9 point s.

Suppose we are interested in that tour of which the total distance travelled is minimal. If n grows

large,the

number

of

possible

tours

becomes

excessively

large

and

we

cant

just

find

the

best

tour

by

tryingallpossibletoursandretainthebestone.

Wecallthisproblemof findingtheshortesttourthroughnpointsthe TravellingSalesmanProblem

(TSP).TheTSPisoneoftheclassicproblemsinOperationalResearch.

Withoutknowingtheoptimaltour itself, letuscallthe lengthoftheoptimalTSPtour T*(Xn) inourexampleofrandomlydistributedpointsinasquareofarea1.ThenBeardwoodetal.(1959)provedthat

whenyoumakenverylarge,theratioT*(Xn)/ nbecomesconstant.Inotherwords,forverylargen:

nXT n )(*

wheretheconstantisbelievedtobe0.7124.

Infact,thisalsoholdsforaserviceregionofmore irregularshapes(as inFigure11). IfA isthearea

(m2)ofaserviceregionofanyfiniteshape,then

AnXT n )(*

Figure 11: Locat ions randomly di st r i buted in a serv i ce region.


50/150


50

Finally,theresultisevenmoregeneralthanthat.Thepointsdonotneedtobedrawnfromauniform

distributionacrosstheservicearea,anydistributionwilldo.TheTheoremofBeardwoodetal.(1959) is

giveninFigure12.

Theorem BBH. (Bear dwood et al . , 1959) . I f T*( Xn) i s

t he l engt h of t he opt i mal t r avel l i ng sal esman t ourt hrough n poi nt s whi ch ar e i ndependent l y dr awn f r om

an i dent i cal di str i but i on over a bounded regi on aof t he Eucl i dean pl ane, t hen t here exi st s a const ant

such t hat wi t h pr obabi l i t y one

dn

XT

a

n

n

)(lim

*,

wher e the i nt egr ati on i s wi t h r espect t o theLebesgue ( area) measure. For t he uni f ormdi str i but i on over [ 0, 1] 2, the i ntegrat i on term i sequal t o one.

Note:0.7124(Johnsonetal.,1996,PercusandMartin,1996).

Figure 12:Theorem of Beardw ood et al . (1959).

Example1

Aprintedcircuitboardof20cmby10cmneeds1000littleholesdrilled in it.Drilling isperformedbya

moving pin. What is the expected distance that the pen needs to travel? Assume that the holes areuniformlydistributedacrosstheboard.

UsingtheTheorem,wegetforA=200cm2,n=1000

)1000(20071.0)(* AnXT n 318cm

Example2

In his small van, an express courier has small packages destined for 250 customers located acrossHampshire.AssumethatHampshirecovers22500km

2andcustomersareuniformlydistributed.

a)Estimatethedistancetobetravelledtodeliverallmail.b)Giveareasonableestimateofhowlongitisgoingtotakethecouriertodeliverthemail.

c)Whatwillbetheresultofdividingtheworkupinfivevans,eachcoveringanequalpartofHampshire?

a)UsingtheTheorem,wegetforA=22500km2,n=250

)250(2250071.0)(* AnXT n 1690km

b)Takinganaveragedrivingspeedof50km/hour,totaltraveltime=1690km/(50km/hr)=33.8hours.Inaddition, itmaytakeourdriverat leastoneminute foreverycustomertomakethedelivery,which

addsanother250min=4.2hourstothetotaldeliverytime.

c)Thedistancetravelledbyeachdriver,sincenowA=22500/5km2andn=250/5

)5/250)(5/22500(71.0)(* AnXT n 1690/5=338km

Takinganaveragedrivingspeedof50km/hour,totaltraveltime=338km/(50km/hr)=6.76hours.A

drivernowvisitsabout50customers,whichshouldkeeptotalworkingtimeperdriverwithin8hours.


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51

3.2.2 ExpectedlengthofanoptimalTSPtourvisitingafewcustomers

ComputationaltestsforuniformlygeneratedrandomcustomersinasquareareafindthattheformulaofBeardwoodetal.alsoservesasagoodpredictionfortheexpectedoraverageoptimalTSPtour lengthevenwhennisrelativelysmall(Eilonetal.,1971,seealsoHaimovichandRinnooyKan,1985).

Figure 13: Few customers uniformly distributed in a square.

Figure13showsthreeexamples.Thefirstrowshowsfourdifferentinstancesforn=5.Thelengthof

theoptimalTSPtourinallfourwillbedifferent,say*iT (i =1,,4).Theaveragelength,however,canbe

expectedtobe

571.054

1 4

1

*

AAnTi

i

Example

A pizza takeaway also makes home deliveries using scooters. On average there are 10 home delivery

ordersper

hour,

randomly

located

in

a50

km2

service

region

located

around

the

pizza

restaurant.

a) Estimate the average total distance a scooter will travel to deliver 3 orders and return to therestaurant.

b) Estimate the average total time needed for a scooter to deliver 3 orders and return to the pizza

restaurant.c) Estimate the minimum number of scooters needed when a scooter will deliver on average to 3

customerspertrip.d)Estimatethetimewhenthescooterarrivesatthethirdcustomeronitstrip(relativetoitsstartingtimeattherestaurant).

a)With threecustomersplusthe restaurant, n=3+1,andA=50,the formulagivesusanestimated

averagetimeofvisitingthreecustomersinthebestsequenceastominimisetraveldistance:


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52

10)4(5071.0 An km

b)Assumingtheaveragespeedofascooterisaround35km/hour,totaltraveltimeis10/35=0.28hour=

17min.Assumingittakes4minutestodeliveratacustomerlocation,itwilltakeintotalonaverage17+

12=29minutesorabouthalfanhourtomakethedeliverytrip.

c)It

takes

one

scooter

half

an

hour

to

deliver

to

3customers

and

in

the

same

time

10/2

=5new

orders

arrive.Thereforeoneneedsminimum2scooters.(Inordertobeabletoperformalldeliverieswithonescooter,theaveragenumberofdeliveryordersneedstodropto6perhour,orsmaller.)d)Thetotaltraveltimeof17minpertrip isdividedover4legs,andthethirdcustomer isreachedat

theendofthethirdleg.Thescooter isthereforeexpectedtoarriveaftertravellingatotaltimeof

(17)=12.75min.However,wehavetoaddtothisthetime istakesfordeliveringthe firstandsecond

order,intotal8min.Thereforethetimeofarrivalatthethirdcustomerisestimatedtobe12.75+821min.afterleavingtherestaurant.

3.2.3 Continuousapproximationofanoptimalvehicleroutingsolution

Inphysicaldistribution,goodsneedtobedeliveredusingvehiclesoflimitedcapacity.Intheonetomany

shipping

mode,

a

vehicle

will

deliver

to

more

than

one

customer

during

one

trip.

In

the

socalled

Capacitated Vehicle Routing Problem (CVRP) (Dantzig and Ramser, 1959), all vehicles have the samecapacity.TheCVRPconsistsoffindingasetofvehicleroutesofminimumcostsuchthat:everycustomerisservicedexactlybyonevehicle,eachroutestartsandendsatthedepotandthetotaldemandservicedby a route does not exceed vehicle capacity. The CVRP, like TSP, is a classic problem in Operational

Research.

To find an optimal set of routes for the CVRP is a difficult problem that can require a lot of

computationaltimeforlargeproblems.Oneoftenmakesuseofheuristicsi.e.methodsthatcanfindina

relativelyshorttimeagood,butnotnecessarilyoptimal,solution.Good(nearoptimal)solutionstypicallylook like in Figure 14 a: the total service region is divided into a number of districts; and customersbelongingtoonedistrictareservedbyoneandthesamevehicle,asindicatedinFigure14b.

Agoodapproximationforthedistancetravelledbyavehicleservingadistrictiis(Daganzo,1984):

iiiii nAnrD )(2*

whereAiisthesizeofthedistrict, ir istheaveragedistancefromacustomerindistrictitothedepot,

ni isthenumberofcustomers inthedistrict,andthedimensionless factor(ni)dependsonthemetric

andthenumberofcustomersinthedistrict. Forni6andtheEuclideanmetric,(ni)isaconstant=0.57,andforsmallervalue

integrated logistic

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