An active-set trust-region algorithm for solving warehouse location problem

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JournalofTaibahUniversityforSciencexxx(2016)xxx–xxx

ScienceDirect

An

active-set

trust-region

algorithm

for

solving

warehouse

location

problem

Y. Abo-Elnaga

b,1

,

B. El-Sobky

a,∗

,

L. Al-Naser

b

a_Department_of_Mathematics,_Faculty_of_Science,_Alexandria_University,_Alexandria,_Egypt b_Department_of_Mathematics,_Faculty_of_Science,_Taibah_University,_Saudi_Arabia Received6March2016;receivedinrevisedform7April2016;accepted10April2016

Abstract

Inthispaper,anactive-setstrategyisusedtogetherwithapenaltymethodandatrust-regiontechniquetosolveawarehouses locationproblem.Thetrustregionisusedtomodifythelocalmethodinsuchawaythatitisguaranteedtoconvergeatallevenifthe startingpointisfarawayfromthesolution.Thetrust-regionmethodisawell-acceptedtechniqueinnonlinearoptimizationtoassure globalconvergenceandismorerobustwhentheydealwithroundingerrors.Oneoftheadvantagesoftrust-regionmethodisthatit doesnotrequiretheobjectivefunctionofthemodeltobeconvex.Thewarehouseslocationprobleminvolvesthedeterminationof thenumberandsizeofservicecenter(warehouses)tosupplyasetofdemandcenterssoastominimizetotaldistributioncost.

Theproposedapproachistestedontwoproblemstoconfirmtheeffectivenessofthealgorithm.Ourresultswiththeproposed approachhavebeencomparedtothosereportedintheliterature.

Keywords: Warehouseslocation;Activeset;Penaltymethod;Trustregion

1. Introduction

Thewarehouselocationproblem(WLP)commonly faced in management of distribution systems is that todetermineasetof geographicalwarehouselocation

∗_{Corresponding}_author._Tel.:₊₂₀_1156646045.

E-mailaddress:[email protected](B.El-Sobky). 1 _Permanent_address:_Department_of_Basic_Science,_Ten_of_Ramadan City,HigherTechnologicalInstitute,Egypt.

PeerreviewunderresponsibilityofTaibahUniversity.

http://dx.doi.org/10.1016/j.jtusci.2016.04.003

whichsatisfy andsatisfactorythe demand of the cus-tomerservicewithminimumtotaldistributioncostover arelativelylongplanningperiod.

Thewarehouselocationproblem(WLP)consistsof theordinarytransportationproblemwiththeadditional featureofafixedcostassociatedwitheachsupplier.A suppliercanbe used towardsmeetingthe demandsof the customers only if the corresponding fixed cost is incurred.Theproblemistodeterminewhichsuppliers touseandhowthecustomerdemandsshouldbemet,so thattotalcostisminimized.

Mostoftherecentlypublishedalgorithmsfor(WLP) usebranchandboundbasedonaLagrangianrelaxation ofdemandconstraintsThewarehouselocationproblem (WLP)consistsofthefollowing:

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There are n potential supply points or warehouse andmdemand points(stores).Ifwarehouseiis avail-able,providing a fixed cost zi is incurred first. If the

warehouse i is not use at all, the fixed cost zi is not

incurred.Inadditiontofixedcosts,portedeachunitofthe commoditytransformwarehouseitodemandjincurs atransportationcost Aij.The problemis todetermine

whichwarehousestouseandhowmanyunitsofthe com-moditytotransportfromthewarehousetoeachdemand pointsoastomeetallthedemands(stores)atminimum totalcost.

The solution of the warehouse location problem shouldmeetsomeobjectives.Itmustbeableto evalu-atepossiblenumberofwarehouseconfiguration,capable withnonlinearitiesduetoboththefixedcostassociated withalternativeconfigurationsandvariablecost associ-atedwiththesystemthroughout,andthesolutionmust befeasibleandefficient.

Manyauthorshavebeenuseddifferentapproachesto meetsomeobjectives.Authorsin[1]prefertodescribe heuristicprogrammingapproachtosolvethisproblem, wheretheyemphasisonworkingtowardsoptimum solu-tionproceduresratherthanoptimumsolution.Authorsin [2]attemptedtotreatthenon-linearityofthewarehouse costfunction.

Morerecently,authorin[3]proposedextensionsof theworkdoneby [1,4]andothers.Manyof the tech-niquescurrentlyavailablearebasedonthebranchand boundwhichisdescribedindetailintheliterature[5,6]. Also thereare many other treatment for solving such problem, all of these treatments success in meeting someoftheobjectivesseealso,[5,7–12].Weconclude fromtheliteraturethat,thewarehouselocationproblem (WLP)isagreatinteresttomanyresearchesandthere areseverallocalmethodshavebeenproposedtosolve it.Alocalmethodisthemethodwhichisdesignedto convergetooptimalsolutionfromcloseststartingpoint. Thatis,thereisnoguaranteethatitconvergesifitstarts fromremote.

Inthispaper,wewilluseatrust-regionglobalization strategytofindthesolutionofwarehouseslocation prob-lem.Globalizationstrategymeansmodifyingthelocal methodinsuchawaythatitisguaranteedtoconverge atallevenifthestartingpointisfarawayfromthe solu-tion.Asweknowtrust-regionmethodisawell-accepted technique in unconstrainedand constrained optimiza-tionproblemstoassureglobalconvergenceandismore robustwhentheydealwithroundingerrors.Oneofthe advantages of trust-region method is that it does not requirethe objectivefunctionof themodel tobe con-vex.However,intraditionaltrust-regionmethod,after solvingatrust-regionsubproblem,weneedtousesome

criteriontocheckifthetrialstepisacceptable.Ifnot,the subproblemmustberesolvedwithareducedtrust-region radius.Formoredetailssee[13–17].

In this paper, we use an active-setstrategy in[18] toconvertthe warehouselocationproblemtoequality constrainedoptimizationproblem.Theheadfeature of thesuggestedactivesetisthatitisidentifiedandupdated naturallybythestep.See[13,19].

Apenaltymethodisusedinthispaper,totransform the equality constrained optimization problem which obtained from the above step to unconstrained opti-mization problem.Some penalty functions have been suggestedandmanycontributionsaddressingthe con-vergenceofthesemethodshavebeenmade,see[20,21]. Thispaperisorganizedasfollows.Amathematical formulationofthewarehouselocationproblemis pre-sentedinSection2.Adetaileddescriptionofthemain stepsofthealgorithmwhichisusedtosolvewarehouse locationproblemarepresentedinSection3.InSection4, numerical results for two test problems are reported. Finally,Section5containsconcludingremarks.

Thefollowingnotationsareusedthroughouttherest of thepaper. Thesequenceof pointsgenerated bythe algorithmisdenoted{xk}.Asubscriptedfunctionmeans

thevalueofthefunctionevaluatedataparticularpoint. Forexample,fk=f(xk),∇fk=∇xf(xk),∇2f_k =∇_x2f(x_k),

and so on. We use the notation x_k(i) to denote the ith component ofthevectorxk,andsoon.Finally,allthe

normsusedinthispaperare2-norms.

2. MathematicalformulationofWLP

Themathematicalformulationofthewarehouse loca-tionproblemhasthefollowingform:

minimize f = n i=1 m j=1 Aijxij+ n i=1 ziyi subjectto n i=1 xij =1, ∀j={1,...,m}, xij≤yi, ∀i={1,...,n}, j ={1,...,m} xij≥0, ∀i={1,...,n}, j={1,...,m} yi ∈ {0,1}, ∀i={1,...,n}, (2.1) whereiandjarethenumbersofwarehousesandstores respectively,Aijrepresentamatrixcontainingthecosts

associatedtothesupplying(supplyCost),xijisamatrix

indicatingifthewarehouseiissuppliedbythestorej,zis thefixedcost,andyisavectorindicatingwhatwarehouse

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are opened. The above problemcan bewritten in the followingconstrainedoptimizationproblemform

minimize f(x)

subject to cp(x)=0, p ∈ E, cp(x)≤0, p ∈ I,

(2.2)

where f :Rn(m+1) →R, cp:Rn(m+1) →R2n(m+1),

EI={1,...,2n(m+1)},andEI=∅.Thefunctions

f andcp,p={1,...,2n(m+1)}are presumedtobeat

leasttwicecontinuouslydifferentiable.

Following the active-set strategy in [18], we define a 0-1 diagonal indicator matrix W(x) ∈ R2n(m+1)×2n(m+1)_,_whose_diagonal_entries_are

wp(x)= ⎧ ⎪ ⎨ ⎪ ⎩ 1, ifp ∈ E, 1 ifp ∈ Iandcp(x)≥0, 0 ifp ∈ Iandcp(x)<0 (2.3)

Usingtheabovematrix,wetransformproblem(2.2)to thefollowingequalityconstrainedoptimizationproblem

minimize f(x)

subjectto C(x)TW(x)C(x)=0, (2.4)

where C(x)=(c1(x), ..., c2n(m+1)(x))T is continuously

differentiablefunction.

Usingapenaltymethod,theequalityconstrained opti-mization problem (2.4) transformed to the following unconstrainedoptimizationproblem

minimize (x;ρ)=f(x)+ρ 2W(x)C(x) 2 2, subjectto x ∈ Rn(m+1), (2.5) where ρ>0 is a parameter usually called the penalty parameter.

Inthefollowingsection,wepresentmainstepsofour trust-regionalgorithmforsolving(WLP).

3. Trust-regionalgorithmoutline

This section is devoted to the description of our method.

3.1. Computingastep

Inthissection,atrialstepdkiscomputedbysolving

thefollowingtrust-regionsubproblem minimize fk+∇fkTd+ 1 2d T_H kd+ρk 2Wk(Ck+∇C T kd) 2 subjectto d≤δk, (3.1)

whereHkistheHessianmatrixoftheobjectivefunction

f(xk)or anapproximationtoit.Sinceourconvergence

theoryisbasedonthefractionofCauchydecrease condi-tion,thereforeageneralizeddoglegalgorithmintroduced by[22]and[11]isusedtocomputedk.

3.2. Testingthestepandupdatingδk

Oncedk iscomputed,itneedstobetestedto

deter-minewhetheritwillbeaccepted.Soweusethefunction

(x;ρ)asameritfunction.

Theactualreductioninthemeritfunctioninmoving from(xk)to(xk+1)isdefinedas

Aredk =Φ(xk;ρk)−Φ(xk+1;ρk).

Aredkcanbewrittenasfollows

Aredk =f(xk)−f(xk+1)

+ρk

2 [WkCk 2₋

Wk+1Ck+12]. (3.2) Thepredictedreductioninthemeritfunctionisdefined as Predk =−∇fkTdk− 1 2d T kHkdk +ρk 2 [WkCk 2₋ Wk(Ck+∇CTkdk)2]. (3.3) Totestdktoknowwhetheritisaccepted.Thisisdone

bycomparingPredkagainstAredk.Ourwayoftesting

dkandupdatingthetrust-regionradiusδkispresentedin

Step4ofAlgorithm3.1below.

Afteracceptingdk,weupdatetheparameterρkusing

aschemesuggestedbyYuan[23].Toupdateρk,weset

ρk+1=ρkif

Predk ≥σ∇CkWkCkmin{∇CkWkCk,δk}, (3.4)

whereσ ispre-specifiedfixedconstant.Otherwise,we setρk+1=2ρk.

Finally, the algorithm is terminated when either ∇fk+∇CkWkCk≤1 or dk≤2 for some

1>0and2>0.

3.3. Themasteralgorithm

Masterstepsofouralgorithmispresentedinthe fol-lowingalgorithm.

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Algorithm3.1. Trust-regionalgorithm Step0.

Givenx0 ∈ Rn(m+1).ComputeW0.Setρ0=1. Choose1,2,α1,α2,η1,andη2suchthat1>0,

2>0,0<α1<1<α2,and0<η1<η2≤1.Chooseδmin,

δmax,andδ0suchthatδmin≤δ0≤δmax.

Setk=0.

Step1.If∇fk+∇CkWkCk≤1,thenstop.

Step2.Computethestepdkbysolvingsubproblem

(3.1).

Ifdk≤2,thenstop.

Endif.

Setxk+1=xk+dk.

Step3.ComputeWk+1.

Step4.While Aredk

Predk <η1,orPredk≤0.

Donotacceptthestep.

Reduce the trust-region radius by setting δk=

α1dk.

Computeanewtrialstepdk.

If η1≤ Ared_Predk

k <η2, then accept the step: xk+1= xk+dk.

Setthetrust-regionradius:δk+1=max(δk,δmin).

Endif. IfAredk

Predk ≥η2,thenacceptthestep:xk+1=xk+dk.

Set the trust-region radius: δk+1=min{δmax,

max{δmin,α2δk}}. Endif. Step5. (a)Setρk+1=ρk. (b)IfPredk≤σ∇CkWkCkmin{∇CkWkCk, δk},thensetρk+1=2ρk. Endif.

Step6.Setk=k+1andgotoStep1.

A global convergence theorem of the trust-region Algorithm3.1ispresentedin[15].

Inthefollowingsection,we introduceawarehouse locationtwo testproblems toobviousthe goalof our paper.

4. Warehouselocationtestproblems

Inthis section,we present thenumericalresults of twotestproblemsforthewarehouselocation optimiza-tionproblem.Theproposedtrust-regionAlgorithm3.1 hasbeenperformedonalaptopwithIntelCore (TM)i7-2670QMCPU2.2GHzand8GBRAM.Algorithm3.1 was implementedas aMATLAB code and rununder MATLABversion7.10.0.499(R2010a).

In our algorithm we begin from any starting point

x0.Wechosethe initialtrust-regionradiustobe δ0=

max(scp0 ,δmin),whereδmin=10−3.Wechosethe

max-imumtrust-regionradiustobeδmax=105δ0.

ThevaluesoftheconstantsthatareneededinStep0of Algorithm3.1weresettobeη1=10−4,η2=0.5,α1=0.5,

α2=2,1=10−8,2=10−7,σ=10−2andβ=0.1. Thetwowarehouselocationoptimizationtest prob-lemsarepresentedinthefollowingtwosubsections.

4.1. Warehouselocationtestproblem1

Letusconsiderthefollowingnumericalexample,to illustratetheapplicationoftheproposedalgorithm.The problemhasthefollowingcharacteristics:

Thenumberofstoresi=6andthenumberof ware-housesj=6.

Thefixedcostz={33,47,27,39,41,26}.

ThematrixAijwhichcontainingthecostsassociated

tothesupplycostis:

Aij = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 5 19 8 2 23 12 1 7 15 8 7 27 15 6 4 5 19 1 4 27 33 24 6 9 11 8 16 11 5 2 21 14 4 26 18 6 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (4.1)

Byusingourproposedalgorithmtosolvetestproblem1 we foundthat theobjective functionis520.6467.The same test problem 1 was solved by many researches suchas[2]whoseintroduceapartialdualalgorithmand obtained the objective function 549.0. Also,the same problemwassolvedbyresearches[24]withtheir algo-rithm andthe objective function whose obtained it is 551.2.Itisseemthatouralgorithmhasperformedvery welloverallcomparedwiththosepreviouslyreported. Distribution of units transported from warehouse j to demandpointiaresummarizedinTable1.Table2show the distributionof the optimal value function at each cell.

Table1

Distributionofunitstransportedfromwarehousetodemandpoint.

xi y1=1 y2=0 y3=0 y4=0 y5=0 y6=1 x1 1 0 0 0 0 0 x2 0.60527 0 0 0 0 0.39473 x3 0 0 0 0 0 1 x4 1 0 0 0 0 0 x5 0 0 0 0 0 1 x6 0 0 0 0 0 1

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Table2

Distributionoftheoptimalvaluefunction.

38 33 33 33 33 33 6.0527 0 0 0 0 106.5906 0 0 0 0 0 1 39.999 0 0 0 0 0 0 0 0 0 0 2 26 26 26 26 26 32 Table3

Comparison objective function value of method in [2] with Algorithm3.1.

Problem Methodin[2] Algorithm(3.1)

16×50 97.4 94.2 16×50 98.7 94.1 16×50 97.3 92.7 16×50 99.2 91.7 16×50 96.5 89.4 16×50 96.7 88.2 16×50 95.5 93.2 16×50 93.8 94.7 16×50 98.2 92.9

4.2. Warehouselocationtestproblem2

TodemonstratetheeffectivenessoftheAlgorithm3.1, we compareourresultswiththepartialdualapproach introducedin[2].Weuseasetrandomlygenerated prob-lems,withuniformlydistributionof thecoefficienton the following ranges Aij=[10, 50]and z=[300, 600].

Author in [2] generatednine problemswith16 ware-houseand50demandpointsandwesolvetheseproblems byAlgorithm3.1.Theresultsofthesetestsareshownin Table3.FromthereportedresultsinTable3itisclear thattheproposedtrustregionalgorithmproducedresults betterthanthoseintroducedin[2].

5. Conclusions

Inthispaper,weintroduceatrustregionalgorithmfor solving the warehouselocation problem.In this algo-rithm, an active set strategy is used together with a penaltymethodtoconvertthe computationof thetrial steptoeasytrust-regionsubproblemsimilartothisforthe unconstrainedcase.Thecomputationalresultsshowthat the solution of warehouselocation problemusing our algorithm hasconsistentlyperformed well,and gener-allyproducedresultswhichareagoodas,orbetterthan, thoseofthepreviouslyacceptedapproach.Wetestour algorithmusingrandomlygeneratedproblemsoflarge sizetoclarifytheeffectivenessofouralgorithm.

Forfuturework,therearemanyquestionsthatshould beanswered.Althoughwehaveimplementedthe algo-rithmandtestedit,webelievethattheimplementation of the algorithm should be refined with efficiency in mined.In particular,abetterwayforsolvingthe trust-regionsubproblemsthatcanhandlelarge-scaleproblems should be used. Updating the penalty parameter is anotherpointthatneedstoberefined.Thisindeedwill reducethecostofthecomputationofthesteps.

Acknowledgements

Firstandforemost,wegivethankstoGodforthegrace andmercyhehasshownus.WedependonHimandwe could not have accomplished thiswork withoutHim. WordscannotexpressourdeepgratitudetoThe Dean-shipofScientificResearch,TaibahUniversity,KSAfor supportingthisprojectbygrantNo.1437/1416.

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