stochastic problem A co-evolutionary matheuristic for the car rental capacity-pricing European Journal of Operational Research

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European Journal of Operational Research

journalhomepage:www.elsevier.com/locate/ejor

Innovative Applications of O.R.

A co-evolutionary matheuristic for the car rental capacity-pricing stochastic problem

Beatriz B. Oliveira

^a,∗

, Maria Antónia Carravilla

^a

, José F. Oliveira

^a

, Alysson M. Costa

^b

aINESC TEC, Faculty of Engineering, University of Porto, Portugal

bSchool of Mathematics and Statistics, The University of Melbourne, Australia

a rt i c l e i n f o

Article history:

Received 2 March 2018 Accepted 7 January 2019 Available online 11 January 2019 Keywords:

Revenue management Pricing

Car rental fleet management Genetic algorithms Stochastic programming

a b s t r a c t

Whenplanningasellingseason,acarrentalcompanymustdecideonthenumberandtypeofvehicles inthefleettomeetdemand.Thedemandfortherentalproductsisuncertainandhighlyprice-sensitive, andthuscapacityandpricingdecisionsareinterconnected.Moreover,sincetheproductsarerentals,ca- pacity“returns”.Thiscreatesalinkbetweencapacitywithfleetdeploymentandothertoolsthatallow thecompanytomeetdemand,suchasupgrades,transferringvehiclesbetweenlocationsortemporarily leasingadditionalvehicles.

Weproposeamethodologythataimstosupportdecision-makerswithdifferentriskprofilesplana season,providinggoodsolutions andoutliningtheirability todealwithuncertainty whenlittleinfor- mation aboutit isavailable.Thismatheuristicis basedonaco-evolutionarygeneticalgorithm, where parallelpopulationsofsolutionsand scenariosco-evolve.Thefitnessofasolutiondependsontherisk profileofthedecision-makeranditsperformanceagainstthescenarios,whichisobtainedbysolvinga mathematicalprogrammingmodel.Thefitnessofascenarioisbasedonitscontributioninmakingthe scenariopopulationrepresentativeanddiverse.Thisismeasuredbytheimpactthescenarioshaveonthe solutions.

Computationalexperimentsshowthepotentialofthismethodologyregarding thequality oftheso- lutions obtained and the diversity and representativeness ofthe set ofscenarios generated. Its main advantages are that noinformation regarding probability distributions is required, it supports differ- ent decision-making risk profiles, and it provides a set of good solutions for an innovative complex application.

© 2019ElsevierB.V.Allrightsreserved.

1. Introduction

Whenplanningasellingseason,acarrentalcompanymustde- cideonthefleetsizeandmix,i.e.,thecapacityitwillhavetomeet demandthroughouttheseasonandrentallocations.Thedemandis uncertainandhighlyprice-sensitive.Therefore,thepricescharged byacompanyareconnectedwithandshouldinfluencethecapac- itydecisions.Capacitydecisionsarealsoconnectedwithotherin- struments that allow the company to “meet” its demand, which rangefromoffering upgradesto transferringvehicles betweenlo- cationsortemporarilyleasingadditionalvehicles.

Thegoalofthisworkistoprovidedecision-makerswithprof- itable solutions to capacity and pricing decisions, assessing and increasing their ability to deal with the different realizations of uncertainty, represented by scenarios, when little information

∗ Corresponding author.

E-mail addresses: beatriz.b.oliveira@inescporto.pt , beatriz.oliveira@fe.up.pt (B.B.

Oliveira).

regardingthoseis available.The methodologydevelopedis based onaco-evolutionarygeneticalgorithm,whereparallelpopulations ofsolutionsandscenarios co-evolve,dependingoneach otherfor the fitness evaluationof their individuals. Onthe one hand,this methodaims atobtaininga representativeanddiversepopulation ofscenarios, measuredaccordingto the impactthey haveon the population of solutions. On the other hand, the solutions evolve accordingtodifferentdecision-makingriskprofiles thatassessits performanceagainstthepopulationofscenarios.

1.1. Previousworks

Thisworkdealswiththeintegrationofcapacityandpricingde- cisionsunderuncertaintywithinthecontextofthecarrentalbusi- ness.Inthissection,therelevanceoftheapplicationandmethod- ologicalscope ofthe work will be discussed.Firstly, the recently growing body of research on car rental fleet management and pricingwill be briefly reviewed. Thisis an innovative and differ- ent application because the capacity is rented rather than sold.

https://doi.org/10.1016/j.ejor.2019.01.015

0377-2217/© 2019 Elsevier B.V. All rights reserved.

However, previous works that tackled the integration of pricing andcapacity, although not directlyapplicable, can bringrelevant insights to this problem. A stochastic approach to the problem is considered, where the uncertainty is represented by scenar- ios.Stochasticproblemswithsimilar characteristicsarebrieflyre- viewedregarding methodological approaches. Moreover, previous fundamentalworksthat laid thefoundationforthemethodologi- calideadevelopedinthispaperwillbepresented.

1.1.1. Carrentalfleetmanagementandpricing

The car rental fleet management problem is initially struc- tured in Pachon, Iakovou, Ip, and Aboudi (2003) and Pachon, Iakovou, and Ip (2006). Fink and Reiners (2006) extends the operational issues within fleet management and deployment, considering essential and realistic practical needs. In Oliveira, Carravilla,andOliveira(2017c),thelinkwithrevenuemanagement issues is introduced, andthe body of research developed in this fieldisreviewedandstructured.Existinggapsandrelevantfuture researchdirectionsarediscussed,includingtheintegrationofpric- ingand/or capacity allocation(revenue management issues) with operationaldecisionsrelatedtofleetsize/mixanddeployment.The needto consideruncertainty indemand in orderto approximate themodeltorealityisalsohighlighted.

Ina previouspaper–Oliveira,Carravilla,andOliveira(2018) – wetackledthe firstresearch direction.Amathematical modelfor thedeterministic integrationofdynamicpricingandcapacityde- cisions was proposed. Due to the complexity of the problem, a matheuristicwasproposed.Thismatheuristicisbasedonadecom- positionoftheproblem,wherethepricedecisionsaredirectlyen- codedin thechromosomes, andthe remaining decisionsandthe fitnessofthefullsolutionareobtainedbysolvingamathematical programmingmodel.Moreover,someperformance-boostinginitial populationgenerationprocedureswereproposed.

In this work, we propose to tackle the even more complex problemthat arises when uncertainty is incorporated. Moreover, additionalrealistic requirements (suchasprice hierarchy) are in- cluded,anddemand ismodeled considering its relationship with competitorprices.

1.1.2. Integrationofcapacity,inventoryandpricingdecisions Pricing decisionshaveoftenbeentackledindependentlyofca- pacityandinventory decisions.Arecentandgrowing bodyof re- searchontheintegrationofthesetopicshasbeenarising.

Den Boer (2015) presents an interesting and thorough litera- turereview onthetopicofdynamicpricing, primarilyfocusedon learningprocesses.Followingthestructureproposedbytheauthor, thecarrentalpricingproblemherein consideredcanbeseenasa dynamicpricingproblemwithinventoryeffects,morespecifically

“jointlydeterminingsellingpricesandinventory–procurement”.In Gallego and van Ryzin (1994), the dynamic pricing problem for inventories with price-sensitive and stochastic demand is tack- led, including an extension where the initial stock is considered as a decision variable. The rental facet of the problem at hand hindersthe directapplication ofthe insights drawn.Focusing on perishableassets, a dynamic pricing problem under competition is studied in Gallego and Hu (2014). Here, the dynamics of an oligopolyare considered, dealingwith substitutabilityamong as- sets.Thesecharacteristicsare moresimilar tothecarrental mar- ket,where vehicles that are available at a particular day(or the correspondingavailabledays-of-use)“expire” sincetheycannotbe used in a future time period. Relevant results are obtained re- gardingdynamicpricingstrategies.Asthis,otherimportantworks have dealt with similar environments with insightful outcomes.

AdidaandPerakis(2010)presentaninterestingwork,wherediffer- entjointdynamicpricingandinventorycontrol models thatdeal with demand uncertainty (which depends linearly on price) are

considered, within a make-to-stock manufacturing context. This workcomparesstochasticandrobustoptimizationapproaches,in- troducesdifferentformulationsandcomparestheir computational performance.

Nevertheless,thecarrentalbusinessischaracterizedbythere- turn of its “sold inventory” in a pre-determined future time pe- riod andlocation. This causessignificant changes to theproblem structureandrenderstheproblemevenmorecomplextosolve.In Oliveira, Carravilla, andOliveira (2017a),a dynamicprogramming approach isdeveloped fora deterministic andsimpler version of thisproblemandthisquestionisfurtherdiscussed.

In innovative transportation systems based on the sharing paradigm,thisissueisalsopresent.Bike-sharing,forexample,has been a key driverof research on managingcapacityto meet de- mand better, considering variations throughout time and space.

Relevant works have focused on capacity reallocation, such as Freund,Henderson,andShmoys(2017),yetonlyafewworkshave focusedontherole ofpricingininfluencingdemand,likeChemla, Meunier, Pradeau, Calvo, and Yahiaoui (2013). Nevertheless, this mobility system shows some differences to the car rental busi- ness, which hinder the direct application of the developed tech- niques, such as the homogeneity of the fleet, the design of the repositioningschemesandthemotivations(andconsequentdistri- bution)ofdemand,amongothers.Otherinnovativebusinessesare alsodrivingresearchinthisfield,suchasride-sharingore-hailing platforms such asUber. Forexample, inBimpikis, Candogan,and Saban(2016),pricingdecisionsforaride-sharingsystemareintro- ducedtomanagethesupply-demandbalanceconsideringnotonly variationsintimebutalsothegeographicaldistributionofdemand andsupply.Asthecarrentalbusiness,thevehiclesaresharedand thuscapacity“returns” tobeusedbyanotherclient.However,the capacitydecisionsarenotcentralizedinthesamedecision-maker, asthepricingitselfmayinducemoreorlesssupplyofdrivers,de- pendingonthebusinessformatofthee-hailingplatform.

Additionally,therelationshipbetweendemandandpriceinthe context of car rental is distinctive and challengingto determine duetotheeffectofcompetitionandtothemyriadofproductsof- fered (rental types)that sharethe sameresources (vehicle fleet).

Therefore,newapproachesareneededtotacklethisproblem.

1.1.3. Representinguncertaintybyscenarios

Scenarios can be important tools for companies dealingwith relevant uncertainties. Moreover, the process of scenario genera- tioniscriticalforthepracticalrelevanceoftheresultsobtained.

Scenariogenerationconsistsofdefiningdiscrete outcomes(re- alizations) for all random variables and time periods (Høyland

& Wallace, 2001), especially usefulforstochastic problems. Mitra andDi Domenica (2010) review thescenario generation methods applied in the literature for stochastic programming models, in- cludingsampling-basedgeneration(e.g.,MonteCarlo,bootstrapor conditional samplingmethods), statistical methods(e.g., property matching or regressions) and simulation-based generation (e.g., VectorAutoRegressivemethods),aswellasotherlessusedmeth- ods(e.g.,hybridmethods). Theauthorsdiscussrelevant,desirable characteristics that all scenariogeneration methods shouldincor- porate:includingavarietyoffactorsandexistingcorrelations,con- sideringthepurposeofthemodel(tounderstande.g.,ifitismore relevanttocapturevariance orhighermoments),beingconsistent with anytheory andwith empirical data observations. Kaut and Wallace(2003)evaluatedifferentscenariogenerationmethodsand propose twoproperties(andcorresponding methodologiesto test them)thatamethodshouldsatisfytobeapplicableandrelevantto a givenproblem.Mostofthesetechniques involveaconsiderable amountofknowledgeabouttheuncertaintyandrandomvariables, e.g.,theirprobabilitydistribution.

1.1.4. Methodologicalapproaches

Solvingintegerstochasticmathematicalprogrammingmodelsis becoming a promising approach to obtain good andaccurate so- lutionsforcomplexreal-worldsituations,suchashazardmanage- mentofpost-firedebrisflowsortransportationnetworkprotection againstextremeeventssuchasearth-quakes(Krasko&Rebennack, 2017; Lu, Gupte, & Huang, 2018). Often, solution approaches are requiredtodealwiththeinherentcomplexity,suchasdecomposi- tion or(meta)heuristics(Özcan, 2010;Puga &Tancrez, 2017;Yan, Tang,&Fu,2008).

Genetic algorithms have been proposed to tackle complex stochastic problems (Gu, Gu, Cao, & Gu, 2010; Wang, Makond,

& Liu, 2011). In these works, random variables are often associ- ated with probability distributions; thus scenarios are generated byrandomsamplingorsimulation.Furthermore,thehybridization of genetic algorithms and linearprogramming hasbeen success- fully used to develop alternative stochastic methodologies (Reis, Walters,Savic,&Chaudhry,2005).

In this field, scenario generation is heavily dependent on the knowledge of probability distributions for the random variables and consists on selecting a small set of scenarios that repre- sent it well,which ishighlycomplicatedinthe multivariate case (Löhndorf, 2016). The author presents an empirical analysis of popular scenario generation methods for stochastic optimization.

State-of-the-art methodsare comparedregardingsolution quality, usingaproblemwhereanalyticalsolutionsareavailable.Theirade- quacyisdependentontheproblemcharacteristicsandprobability distributions. Guastaroba, Mansini, and Speranza (2009)focus on optimal portfolioselection problemandcompare scenario gener- ation techniques forthisproblem. One ofthe conclusionsis that the adequacy of the method depends on the risk profile of the decision-maker.

1.1.5. Coremethodologicalpreviousworks

Forthisproblem,usingscenariostorepresentuncertaintyhasa practical interest concerningthe applicationofthe method,since scenarios can be useful to help decision-makers understand and actupontheoutputs.Nevertheless,theonlyinformationregarding theuncertainparametersavailable forthisproblemisthebounds onthevaluestheycantake.Therefore,amethodologythattackles thislackofinformationisneeded.

Herrmann(1999)proposesametaheuristicbasedongenetical- gorithmsthatisespeciallyadequateforproblemswherethesetof scenariosistoolargeforeachelementtobeevaluatedindividually, orevenknown.Inthiswork,theauthorproposestheco-evolution ofsolutionsandscenariosintwoparallelspaces,asfollows.

Considering that SO is the set of all solutions and SC the set of all possible scenarios, the value obtained by a solution i∈SO when scenario j∈SC occurs is given by F(i, j). In this work,thegoalwastofindthesolutionthat performsbestforthe worst-case, which is translated (in a minimization problem) to:

min_i∈SOmax_j∈SCF(ⁱ,j)^{.Theauthorthusproposesa two-spacege-} neticalgorithmwherescenarios(SC)andsolutions(SO)co-evolve in different populations (P_SO and P_SC) composed of individuals whose fitnessdependsnot only onits characteristics butalsoon thecharacteristics oftheother population.Thisgeneticalgorithm favors solutions withbetterworst-case performancesandscenar- ioswithworse“bestsolutions”.Thefitnessofasolutioni₀iseval- uated asmax_j∈PSCF(ⁱ0,j) ^{(worstscenario forthissolution),while} the fitness of a scenario j₀ is evaluated as min_i∈PSOF(ⁱ,j₀) ^(best solution forthis scenario). The groundbreakingidea in thiswork is thatusing efficientgeneticalgorithms to evolve populationsof scenarios requires only an initial sample that will evolve and is thus expectedto adequately represent the full set, which would otherwisetakesignificantlymoreefforttoexplore.Simultaneously, the solutions evolve to perform increasingly better. This work is

continuedbyotherauthors,namelyJensen(2001)whoproposesa ranking-basedevaluationforscenariofitnessthat performsbetter andfixessymmetryandbiasissuesoftheoriginalapproach.

We aim to extend the idea of a two-space genetic algorithm to evolve solutions and scenarios to other decision-making risk profiles beyond the limitation of the worst-case perspective in Herrmann (1999). Considering the expected value as the goal to evaluatesolutions(stochasticapproach)ratherthantheworst-case value significantly impacts the evolution ofthe scenario popula- tion.This focusestheevolutionary drivein obtaining a represen- tative population, rather than converging to the worst-case sce- nario.Toachievethis,recentdevelopmentsonthefieldofinstance generation were considered. In Gao, Nallaperuma, and Neumann (2016), an evolutionary algorithm is proposed for generating in- stances that are diversewith respect to different features of the problem.It aimsto“diversify” points inN-dimensionsby ranking candidatesbasedondistancetonearestneighborsineachaxis.Us- ingthistechniquewithelitismleadstonewchildrenbeingadded tothepopulationonlyifthey extendtheextremevaluesorlie in alargegapbetweenexistingpoints. AlsoinDeb,Agrawal, Pratap, andMeyarivan (2002), the concept of crowding distance is used toestimatethedensityofsolutionssurroundingaparticularpoint in a population. It compares to the largest cuboid enclosing the pointwithoutenclosinganyotherpoints, withasimilarreference tonearestneighborsineachaxis.

1.2.Contributions

Themaincontributionsofthispaperarerelatedtothemathe- maticalmodelandthesolutionmethodologyproposed.

• We propose a new two-stagestochastic model, extending the deterministicmodelproposedinOliveiraetal.(2018):

– Its main innovative feature is that the stochastic capacity- pricingproblemforcarrentalsismodeled.Fewpapersfocus onthe integrationofpricingwithcapacitydecisions,using tactical information and uncertaintyto deal with strategic decisions, especially in the complex rental context, where inventoryisnotdepletedbutonlytemporarilyunavailable.

– Theissueofvehiclegrouppricehierarchyisincluded,ona morerealisticapproachtotheproblem.

– Demanduncertaintyandprice-sensitivityaremodeledinan innovative andefficientway,withasignificant fitwiththe problemathandanditsstrategicscope.Themodelisadapt- abletodifferentshapesofthedemand-pricefunction,con- sideringtheeffectofcompetition.

• We proposeaninnovative solutionmethodto tackletheprob- lem, based on the decomposition of the stochastic model in first-stageandsecond-stagedecisions:

– Solutionstothefirst-stagedecisionsandscenariosaregen- eratedinparallelwithmutualimpactonfitnessevaluation, requiringlittleinformationonrandomvariablestodoso.

– Thefitnessdependsontheprofitobtainedbyeachpair(so- lution,scenario), which iscalculated using amathematical programmingmodel.

– The methodology is easily adaptable to different decision- makingriskprofiles.

– Specificproblemknow-howcanbeusedintheinitial pop- ulationstoboosttheevolutionaryprocedure(e.g.,providing extremescenarios).

– It can be implemented andrun in a reasonabletime in a decision-supportsystem.

Overall,thismethodologyhasaproperfitwiththeproblemat hand,makingitusefulinreal-worldapplications.Moreover,itisa

methodologythatcanbeeasilyextendedtootherproblemswhere informationregardinguncertaintyisscarce.

1.3.Paperstructure

This paperisstructured asfollow.Firstly, the problemwillbe statedand the mathematical model presented (Section 2). Then, Section3presentstheco-evolutionarymatheuristicdevelopedand inSection 4 theresults ofthe computational testsare discussed.

Finally,conclusionsaredrawn andfuturework andpromising re- searchdirectionsarediscussed(Section5).

2. Problemdefinition

Car rental companies preparing a season must decide on the sizeandmixoftheirfleet,i.e.,thecapacitytheywillhavetoface demandin that season. In order forthis capacityto be used ef- ficiently,some operational issues that will take place duringthe seasonmustbeconsidered,aswellastheuncertaindemand.

Acarrentalcompanyhasseveralrental stationsthatsharethe samefleet.Withinthescopeofthisproblem,thesestationsareof- tenaggregatedinregionsorlocationssuchthat transferringave- hiclebetweenstationswithinthesameregionisnegligibleregard- ingtimeorcost,unliketransfersfromoneregiontoanother.Also, theunitoftimeconsideredcanbeseenasanaggregatedmeasure withinthisscope,e.g.,oneweek.

Thefleetiscomposedofdistinctvehicles,aggregatedinvehicle groupsthat differ in several aspects, namely customer valuation.

Nevertheless,the“products” thatcarrentalcompanies“trade” are rentaltypes.Eachrental typeis characterizednot onlyby theve- hiclegroup requested by the customer butalsoby start andend timeperiodsandstartandendlocations(whichmaybedifferent).

Differentrental types(products)“compete” forthesamefleet(ca- pacity).Moreover, ifnotconflicting in time,two rental typescan usethe samevehicle. Thedemand foreach rental type is uncer- tain and highly price-sensitive. Since it is increasingly easier for customersto comparethe pricesof all companies offering acer- tain rental type, brand loyalty plays a relevant role, and if it is not a dominant effect for a certain company, demand is usually mostlyattractedby havingthelowestpriceinthemarket.Dueto consumervalue perception,thecompanymust alsoconsidercon- straintsonthehierarchyofpricesforrental typesthataresimilar inallcharacteristicsexceptforthevehiclegrouprequested.Thatis tosay, pricehierarchyfor rentalsthat start andendatthe same time and place must respect the hierarchy of vehicle value, i.e., allotherparametersbeingequal,dependingonthevehiclegroups considered,arentalpriceforamore-valuedvehiclecannotbeless thanthepriceofless-valuedone.

Before the season starts, the company must decide on how many vehicles of each group to purchase to meet the (uncer- tain)demand andwhere to make them available at the start of theseason. Since thepricing strategyheavily influences demand, thecompany mustalsodecide previously thepriceit willcharge for each rental type. After the season starts and demand is re- vealed,the company hasother tools to meet demand that must be considered since they impact the capacity decisions. On the onehand, since two rentalscan usethe samevehicle aslong as they do not overlap in time, it is critical to decide on fleet de- ployment throughout the season and network of locations. This deployment is achieved either by actual rentals that start and end in different locations (whose number, limited by demand andcapacity, is decided by the company) or by empty transfer- ring vehicles by truck or driver. Pricing is a relevant tool to in- fluence demand and, consequently, fleet deployment and utiliza- tion.Also, thecompanyhasthepossibilitytoupgraderentals:of- fering a more-valued vehicle than requested for the same price.

Upgrades are a common practice in this business. Nevertheless, they are used sporadically as a “lastresource” to avoid thesitu- ationwherecustomersrequestaless-valuedvehicle becausethey areexpectingan upgrade.Finally,tomeettemporarypeaksinde- mand, the company may lease more vehicles for a significantly highercost.

Thisproblemisheremodeledasatwo-stagestochasticmodel, wheretheuncertain parametersare relatedtodemand andcom- petitors’prices.Thedecisionsmadebeforetheseasonstartsdefine the first-stageand, after uncertaintyis revealed,therecourse ac- tions orsecond-stagedecisions includethe deploymentdecisions, upgrading, and leasing. The objective of this work is to provide decision-makers with profitable solutions to the first stage deci- sions,describingtheirabilitytodealwiththedifferentrealizations ofuncertainty,representedbyscenarios.Themaingoalistoobtain good capacitydecisions, whichwill last fora full season. Due to thismorestrategicsetting,aggregatedlevelsofdemandandprices areconsidered.Infact,inthisproblem,operationaldecisionssuch aspricinganditsimpacton demandareusedtogetbetterinfor- mation forthe strategic decisions of capacity, such asfleet size, which cannot be changed throughout the season. Therefore, the problemofupdatingthe pricingpolicy astheseasonunrolls – in an“online” mannerandconsideringafixedcapacity– isnotcon- sideredinthescopeofthiswork.

2.1. Problemmodeling

In this section, the uncertain integrated pricing and capacity problemin car rental is fully definedusing a mathematical pro- gramming model. This model is extended from the determinis- ticmodelpresentedinOliveiraetal.(2018).However,thismodel differs not only because it considers some parameters to be un- certain but also because it models more accurately the relation- shipsbetweendemand,thepricedecided andtheminimumprice in the market. Moreover, it considers that the price charged for a rental requiring a vehicle of a certain group may be limited by the pricechargedfora rental that only differs onthe vehicle group requested(price hierarchy).The notationused ispresented inTable1.

2.1.1. Demandmodeling

Ageneralizedrelationshipbetweendemand,thepricedecision andthecompetitors’pricesforeachrentaltypeisproposed,based onthefollowingassumptions,foreachrentaltype:

1. Thedemandforarentaltypedependsonitsprice,followinga givenrelationship (e.g.,acommondemandfunctioncould fol- lowalinearlyinverserelationshipwithprice).

2. Ifacompanyhasthelowestpriceinthemarket,comparingto competitors,itwillattractmorecustomersthanintheopposite case.

3. Foreach situation(priceaboveorbelowminimumcompetitor price),thereisarelationshipdescribedbyafunctiondependent onprice(seeitem1).

4. Foreach possible price, thedemand ishigher ifthe company hasthelowestpriceinthemarket.

5. The minimumcompetitor pricein themarket is an uncertain parameter,withinalimitedrange.

6. Theparametersthatdefinethedemandfunctionsareunknown (althoughtheshapeoftherelationshipisknown).

These assumptions aim to capture andadapt to the scope of thisworktheprice-sensitiveanduncertainnatureofcarrentalde- mand. Fig. 1 graphically representsthese assumptions, when the shapeofthedemand functionsis linear.Ainscough,Trocchia,and Gum(2009)presentaninteresting,althoughlimited,surveyoncar rental consumers where the effects of rental agency brand and

Table 1 Notation.

Indices, parameters and other notation

θ⁼ {^{1 , . . . ,} } ^Index for the set of scenarios t , t = {0 , . . . , T } Indices for the set T of time periods ¹ g, g1 , g2 = {1 , . . . , G } Indices for the set Gof vehicle groups s, s 1 , s 2 , c = {1 , . . . , S} Indices for the set Sof rental stations

r, r = {1 , . . . , R } Indices for the set R of rental types (characterized by check-out station and time period, check-in station and time period, and group requested)

so r Check-out station of rental type r do r Check-out time period of rental type r si r Check-in station of rental type r di r Check-in time period of rental type r gr r Vehicle group requested by rental type r

p r = {1 , . . . , P r} Index for the set P rof price levels allowed for rental type r LBP r Lower bound on prices for rentals of type r

UPB r Upper bound on prices for rentals of type r

PRI rp Pecuniary value charged for rental type r related with price level p . The value for p = 1 corresponds to the lower bound on price ( LBP r) and for p = P rcorresponds to the upper bound on price ( UBP r). The intermediate levels are a discretization of this range.

COM rθ Minimum price charged by the competitors for rental type r in scenario θ

DEM ˜ ^A_rθ(p) Demand function for rental type r in scenario θ, dependent on price level p ; valid when price is a bove the minimum price in the market for a similar product COM rθ

DEM ˜ ^Brθ(p) Demand function for rental type r in scenario θ, dependent on price level p ; valid when price is b elow the minimum price in the market for a similar product COM rθ, with DEM ˜ ^Arθ(p)≤DEM ˜ ^Brθ(p), ∀^p

MGP Marginal price difference

PLM g1g2 Whether the price charged for a vehicle of group g 1 should be lesser than or equal to the price charged for a vehicle of group g 2, considering the same check-out and check-in locations and time periods, ( = 1 ) or not ( = 0 )

COS g Buying cost of a vehicle of group g . The value considered is the net cost: purchase gross cost minus salvage value derived from its sale after one year

OWN g Ownership cost per time unit of a vehicle of group g LEA g Leasing cost (per time unit) of a vehicle of group g LP g Leasing period for a a vehicle of group g

PYL g Penalty charged for each day that a leasing return of group g is late PYU rg Penalty charged for each upgrade to vehicle group g applied to rental type r

UPG g1g2 Whether a vehicle of group g 1 can be upgraded to a vehicle of group g 2 ( = 1 ) or not ( = 0 ) TT s1s2 Transfer time from station s 1 to station s 2

TC gs1s2 Transfer cost of a vehicle of group g from station s 1 to station s 2 BUD Total budget for the purchase of vehicles

M Big-M large enough coefficient

E _θ Mathematical expectation with respect to scenarios θ Inputs from previous periods

Assumption : For all periods, T , G, S, R are the same, as well as rental types r ∈ R .

INX gs^O Initial number of owned ( O ) vehicles of group g located at station s , at the beginning of the time period ( t = 0 )

ONY _gts^L/O Number of owned ( O ) or leased ( L ) vehicles of group g on on-going transportation (previously decided), being transferred to station s , arriving at time t

ONU _gts^L/O Number of owned ( O ) or leased ( L ) vehicles of group g on on-going rentals (previously decided), being returned to station s at time t Other sets

R ^g− Rental types that do not require group g

R ⁱⁿ_st Rental types whose check-in is at station s at time t R ^out_st Rental types whose check-out is at station s at time t R ^use_t Rental types that require a vehicle to be in use at time t

1t = 0 represents the initial conditions of the time period and “overlaps” with t = T for the previous period.

price are studied. It concluded that the rental agency brand has a positive impact on willingnessto rent.Theseassumptions may thusbesuitableforcompanieswithdifferentlevelsofbrandrecog- nitionandloyalty, byallowingfordifferentshapesinthedemand functions. Furthermore,in thestudy, theconclusions supportthe hypothesis that higher prices lead to lower willingness to rent, with nosupport that they lead to a higherperception of service quality. These conclusions sustain the relatively simple assump- tions made within the strategic scope of this model. Lastly, one should noticethatdifferentfunctionsmaymodelthedemand for each differentrental type;nevertheless, themodel proposed en- surestheinterconnectionbetweendifferentvehiclegroupsarising frompossibleupgradesandpricehierarchyrules.

These assumptions also allow establishing a level of potential exogenous demand. For specific rental types (or all),it might be usefultoconsiderashareofdemandthatisnotinfluencedbythe priceschargedby thecompany. Functionswherethevalue ofde- mandishigherthanzerofortheentirepricedomainallowmod- elingthistypeofcustomers.

2.1.2. Mathematicalmodel Decisionvariables:

w^O_gs Numberofvehiclesofgroupgacquiredfortheownedfleet availableattimet=0instations

qr p =1ifthepricechargedforrentaltyperisassociatedwith pricelevelp;=0otherwise

w^L_gtsθ Numberofvehiclesofgroupgacquiredbyleasingattimetto beavailableatstationsinscenarioθ

y^L/O_s1s2gtθ Numberofleased(L)orowned(O)vehiclesofgroupg transferredfromstations1tostations2inscenarioθ^;the transferbeginsatt

u^L/O_rgpθ Numberofrentalsoftyperthatareservedbyaleased(L)or owned(O)vehicleofgroupgwithacorrespondingpriceof levelpinscenarioθ

x^L/O_gtsθ Numberofleased(L)orowned(O)vehiclesofgroupglocated atstationsatthestartoftimeperiodtinscenarioθ f_gt^L_θ Auxiliaryvariable:totalleasedfleetofgroupgattimetin

scenarioθ

zrθ Auxiliaryvariable:=1ifthepricechargedforrentaltyperis abovetheminimumvalueinthemarketinscenarioθ^;=0 otherwise

Fig. 1. Relationship between the price decision for a specific rental type r and the minimum competitor price and demand functions, which are dependent on scenario θ.

Optimizationmodel:

max −

G g=1

^S

s=1

w^O_gs

COSg+T×OWNg

+E_θ

R r=1

Pr

p=1

PRIr p

G g=1

u^L_rgpθ+u^O_rgpθ

− G g=1

^T

t=1

f_gt^L_θ

LEAg

− S

s1=1

S

s2=1

G g=1

^T

t=1

y^L_s1s2gtθ+y^O_s1s2gtθ

TCgs1s2

− G

g=1

r∈R^g− Pr

p=1

u^L_rgpθ+u^O_rgpθ

PYUrg

(1)

s.t.

S

s=1

G

g=1

w^O_gsCOSg≤BUD (2)

Pr

p=1

qr p=1

∀

^{r (3)}

Pr

p=1

qr pPRIr p≤

Pr

p=1

q_rpPRI_rp

∀

^r,r:

sor=so_r∧sir=si_r

∧dor=dor∧dir=dir

∧PLMgrr,gr_r=1

(4)

G

g=1

u^L_rgpθ+u^O_rgpθ

≤qr pDEM˜ ^B_rθ

(

^p

) ∀

^r,p,

θ

⁽⁵⁾

G

g=1

u^L_rgpθ+u^O_rgpθ

≤DEM˜ ^A_rθ

(

^p

)

+

DEM˜ ^B_rθ

(

^p

)

^{−DEM˜ A}rθ

(

^p

)

(

^1−zrθ

) ∀

^r,p,

θ

⁽⁶⁾

COM_rθ≥

Pr

p=1

qr pPRIr p−Mz_rθ

∀

^r,

θ

⁽⁷⁾

r∈R^outst Pr

p=1

u^L_rgp^/O_θ+ S

c=1

y^L_scgt^/O_θ≤x^L_gts^/O_θ

∀

^g,t,s,

θ

⁽⁸⁾

u^L_rgpθ+u^O_rgpθ≤UPGgrr,g×M

∀

^r,g,p,

θ

⁽⁹⁾

x^O_g0sθ=INX_gs^O+w^O_gs

∀

^g,s,

θ

⁽¹⁰⁾

x^L_g0sθ=0

∀

^g,s,

θ

⁽¹¹⁾

x^O_gtsθ =x^O_g,t−1,s,θ+ONY_gts^O +ONU_gts^O

+

r∈Rⁱⁿs,t−1 Pr

p=1

u^O_rgpθ−

r∈R^outs,t−1 Pr

p=1

u^O_rgpθ

+ S

c=1

y^O_c,s,g,t−TT

cs−1,θ− S

c=1

y^O_{s,c,g,t−1,θ}

∀

^g,t>0,s,

θ

⁽¹²⁾

x^L_gtsθ =x^L_g,t−1,s,θ+ONY_gts^L +ONU_gts^L

+

r∈Rⁱⁿ_s,t−1 Pr

p=1

u^L_rgpθ−

r∈R^outs,t−1 Pr

p=1

u^L_rgpθ

+ S c=1

y^L_c,s,g,t−TT

cs−1,θ− S c=1

y^L_{s,c,g,t−1,θ}

+w^L_gtsθ

∀

^g,0<t<LPg,s,

θ

⁽¹³⁾

x^L_gtsθ =x^L_g,t−1,s,θ+ONY_gts^L +ONU_gts^L

+

r∈Rⁱⁿs,t−1 Pr

p=1

u^L_rgpθ−

r∈R^out_s,t−1 Pr

p=1

u^L_rgpθ

+ S

c=1

y^L_{c,s,g,t−TTcs−1,θ}− S

c=1

y^L_{s,c,g,t−1,θ}

+w^L_gtsθ−w^L_{g,t−LPg,s,θ}

∀

g,t≥LPg,s,

θ

⁽¹⁴⁾

f_gt^L_θ = S

s=1

x^L_gtsθ+

r∈R^uset Pr

p=1

u^L_rgpθ

+ S

s1=1

S

s2=1 t−1

t=max{⁰,t−TTs1s2}

y^L_s1,s2,g,t,θ

∀

^g,t,

θ

⁽¹⁵⁾

w^O_gs∈Z⁺₀

∀

g,s qr p∈

{

^0,1

} ∀

^r

w^L_gtsθ∈Z⁺₀

∀

^g,t,s,

θ

y^L_s1^/O_s2gtθ∈Z⁺₀

∀

^s1,s2,g,t,

θ

x^L/O_gtsθ∈Z⁺₀

∀

g,t,s,

θ

u^L_rgp^/O_θ∈Z⁺₀

∀

^r,g,p,

θ

f_gt^L_θ∈Z⁺₀

∀

^g,t,

θ

z_rθ∈

{

^0,1

} ∀

^r,

θ

⁽¹⁶⁾

The objective function (Eq.(1)) represents the profitobtained by the fulfilled rentals. It considers: the one-time cost of pur- chasingvehicles andthe costper timeperiodofmaintainingthis owned fleet, the revenue earned with the rentals fulfilled, and other costs such leasingvehicles, performing empty transfers be- tween stations and an artificial cost to penalize upgrades. The rental revenueiscalculated bymultiplying thenumberofrentals fulfilledfor a givenpricelevel by thepecuniary value associated with that price level. Constraints (5) ensure that the number of

Price Demand

DEM^B_rθ

DEM^A_rθ

COMrθ

Fig. 2. Relationship between the price decision for a specific rental type r and the minimum competitor price and demand values, which are dependent on scenario θ.

rentalsisonlypositiveifthecorresponding pricelevelisselected.

Withthis,non-linearityintheobjectivefunctionisavoided.

Constraint(2)establishesthepurchasingbudget.Constraints(3) limit theselectionofpricelevels toa singlelevel perrental type (lowerandupperboundsonpriceareguaranteedbythedefinition ofthePRIprparameters,asexplainedinthenotationsection).

Anovelissueintroducedinthismodelisthehierarchyamong vehiclegroupsconcerningprice.Besidesbeinganessentialrequire- mentfromthebusinessperspective,itintroducessomechangesto the structureoftheproblem, relevantforthe methodology.More specifically, itis requiredthat thepricesforrental typesthat are similarineverythingexceptvehiclegrouprequiredfollowsomehi- erarchical rules.The goalisto avoidthat aluxury vehicleissold forasmallerpricethanacompactvehicle,forthesamedatesand locations. TheunitarymatrixPLM_rr describestherelationshipbe- tween groups, indicating whether theprice ofa group is limited bythepriceofother.Constraints(4)translatethisrequirement.

The following constraintshave beenadded orsignificantly al- tered compared to the model in Oliveira et al. (2018), based on the assumptionspresented andFig. 1.Asmentioned before,Con- straints (5) support the modeling of a linear objective function.

Constraints (6) limit the numberof rentals fulfilledto the exist- ing demand and Constraints (7) relate the price charged forthe rental type,theminimumpricethatthecompetitorsarecharging andthedemandlevels,usingbinaryvariablesz_rθ.

Fortheremainderofthepaper,itwasdecidedtouseconstant- shaped demandfunctions, inordertodescribe acompany thatis not amarket leaderoperatingona significantlycompetitive mar- ket. That is to say, it is assumed that if the company prices a rentalmarginallylowerthantheminimumpriceinthemarketfor agivenscenario(z_rθ=0),itwillattractamajorsliceofthemarket.

Ifnot(z_rθ=1),itwillsecureonlyaresidual shareofthemarket, as demonstratedin Fig.2, withthe following adaptation ofCon- straints(5)and(6),whereDEM^A_rθ andDEM^B_rθ areparameters:

G g=1

u^L_rgpθ+u^O_rgpθ

≤qr pDEM^B_rθ

∀

r,p,

θ

⁽¹⁷⁾

G

g=1

u^L_rgpθ+u^O_rgpθ

≤DEM^A_rθ+

DEM_r^B_θ−DEM^A_rθ

(

^1−zrθ

) ∀

^r,p,

θ

(18) Sincethe rentaltypessharethe sameresources(vehicles) and there arepricehierarchyandsubstitution issues betweengroups, such asupgrades, these demand functions still involve a pricing decisionandnot onlya“sell/no-sell” decision,i.e., beingaboveor belowthethresholdprice.Nevertheless,anyrelationshiporshape

can be considered for these demand functions. In Section 4.5, a differentshapeandfunctionwillbeusedtovalidatethisfeature.

Constraints(8)limitthevehiclesthatexitacertainstationina time periodby thestockavailable. Constraints(9) reflectthe up- grading policies. Following what is set by the binary parameter UPGgrr,g,theydefine whichvehiclegroupsrequested intherental type(gr_r)canbeupgradedtowhichvehiclegroups(g).

Constraints(10)–(14)definetheevolutionofthestockvariables, asinthe previouswork. Atthebeginningofthetime period,the stock of owned fleet is given by the initial purchases (10), and thereisnostockofleasingfleet(11).Forlatertimeperiods,ineach stationandforeachrentalgroup,thestockofownedfleetineach scenario is given by the previous stock, increasedby the arrival oftransfersandrentalsfromthepreviousseason(parameters)and fromprevioustimeperiodsofthecurrentseasonanddecreasedby othersthatstartonthistimeperiod(12).Fortheleasingfleet,the stockis also changed by leasing acquisitions (13)and, when the leasingperiodexpires,removalsfromthefleet(14).Itisassumed thattheremovaltakesplaceinthesamestationastheacquisition.

Finally,theauxiliaryvariablesthatsummarizetotalleasedfleet pergroupandtimeperiodarecalculated(Constraints(15))andthe domainofalldecisionsvariablesisestablished(Constraints(16)).

3. Solutionmethod

Two main issues underline the need for a specific solution methodto solvethe mathematicalmodelpresented inthe previ- oussection.First,thenumberofdecisionvariablesinareal-world- sizedinstanceissignificantlylargeinthisproblem.Eveninasmall casestudy,a significant amountof rental types(different combi- nations of starting/ending times and locations) leads to an even highernumberof(binary)decisionvariables, whichmakes itdif- ficult to solve thismodel to optimality. However, even ifthis is- sue could be overcome for some specific instances with signifi- cantcomputational power, a second issue is related to the need to define and generatevalid scenarios for thisproblem properly.

Inthiswork,weproposeamethodologythatsimultaneouslygen- eratesscenariosandachievesgoodsolutionsfortheproblem.This methodologyisbased ongenetic algorithms,since itis grounded onthepreviousworkbyHerrmann(1999)presentedinSection1. Aspresented,themethodologyproposedinthisworkisbased ontheideaofco-evolutionbetweenapopulationofsolutionsand apopulationofscenariostoachievesolutionsthathaveagoodper- formanceacrossadiversesetofscenarios.Theevolutionarycom- ponents of the algorithm are based on a biased random-key ge- neticalgorithm (BRKGA)framework (Gonçalves &Resende, 2011), since it is a widely used andwell-performing genetic algorithm, whichisstructured sothattheevolutionaryprocedures areinde- pendentoftheproblemandisavailableonanAPI(Toso&Resende, 2015). Appendix A.1, in the Supplementary materials, details the adaptationsmadetothe“problem-independent” partoftheorigi- nalBRKGAframeworkinorderto establishtwoparallel spacesof evolution.

Inthissection,thebasicco-evolutionaryprocedureswillbedis- cussed,andtheproblem-dependentpartsofthegeneticalgorithm – decodingchromosomesandcalculatingfitness– willbedetailed forbothtypesofpopulations.

3.1. Co-evolutionofsolutionsandscenarios

Theprimarygoalofthissolutionmethodistoobtaingoodsolu- tionsforthestochasticproblemdefinedinSection2,forwhichthe onlyinformationregardinguncertainparametersarethelowerand upperboundsthattheirvaluescantake.Inordertoobtainscenar- iosthat havea diverseimpactonthesolutions,a setofscenarios