Andreas Fink *, Torsten Reiners

on substantial real-world data, using a simulation model to assess optimization results for dierent scenar-

ios. The results indicate that the proposed approach can signicantly improve eciency.

as well as the general demand for an improved service quality in a competitive market. As carrental companies provide substitutional products, price and service quality are critical success

* Corresponding author. Tel.: +49 40 42838 4706; fax: +49 40 42838 5535.

E-mail address: nk@econ.uni-hamburg.de (A. Fink).

www.elsevier.com/locate/tre

Transportation Research Part E 42 (2006) 2722921366-5545/$ - see front matter 2005 Elsevier Ltd. All rights reserved. 2005 Elsevier Ltd. All rights reserved.

Keywords: Car rental logistics; Transportation in car rental networks; Minimum cost network ow model

1. Introduction

We consider the logistics processes in the short-term car rental industry. This industry faces cer-tain developments such as a disproportionate growth of car holding costs relative to pricing levelsInstitute of Information Systems, University of Hamburg, Von-Melle-Park 5, 20146 Hamburg, Germany

Received 19 March 2004; received in revised form 20 September 2004; accepted 29 October 2004

Abstract

Logistics management in the car rental business involves short-term decisions about the transportation

and deployment of cars with regard to optimizing eet utilization while maintaining a high service level. We

model and solve this problem by means of minimum cost network ow optimization under consideration ofessential practical needs such as multi-period planning, a country-wide network, customized transportation

relations, eeting and deeeting, and car groups with partial substitutability. Experiments were conductedModeling and solving the short-termcar rental logistics problemdoi:10.1016/j.tre.2004.10.003

is the description of an eective solution method that supports decision making in short-termlogistics management under consideration of essential practical needs. We disregard aecting de-

A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292 273mand from a revenue management perspective (by means of pricing policies depending on theshort-term relation between supply and demand); see Geraghty and Johnson (1997).The paper is organized as follows: First, we describe the major system components and core

processes of car rental operations and we introduce the resulting decision problem (Section 2).Section 3 focuses on modeling and solving the short-term car rental logistics problem. Thisincludes determining the supply of available cars, forecasting demand, balancing supply anddemand on the basis of a minimum cost network ow model, and eventually validating thegenerated plan by means of a simulation model. Computational results are presented in Section4. In Section 5, we describe the architecture of an integrated decision support system supportingcar rental logistics. Finally, in Section 6, we summarize the lessons learned and discuss require-ments for future research.

2. Problem description

In this section we present an overview of car rental operations (network, eet, rental and logis-tics processes) and introduce the core decision problems.

2.1. Network and eet

The major car rental companies operate cross national. However, logistics management ismainly split in accordance with national subsidiaries. Such organizations run a network of rentallocations (stations), where customers can pick up (check-out) and return (check-in) cars. Typi-cally, a national rental network exhibits some hierarchical structuree.g., by means of groupingstations in districts (pools), and districts in regions. Fig. 1 shows a possible structure of a carrental network in Germany with four regions, and, as an example, the Munich district, whichincludes all stations in and around this city. The map also shows some depots, marked by squares,factors. This underlines the importance of optimizing car rental logistics in terms of the utilizationof the eet of cars while maintaining a high degree of customer satisfaction. In this regard, sophis-ticated control systems have to be developed and used.According to the Economist Intelligence Unit (2000) the car rental industry is polarized be-

tween the major international companies providing services to both business and leisure custom-ers on the basis of international networks including outlets at all major airports and city centers,and small companies operating locally and primarily serving the leisure market. While the generalconcepts described in this paper may be applied to any car rental company that operates a sub-stantial integrated network of rental locations, we are inuenced by our work in an industry pro-ject with the German subsidiary of a major international car rental company. Our focus is on theshort-term deployment of passenger cars for a planning horizon from a few days up to about twoweeks. In this context, yield management basically involves optimizing car deployment with re-gard to the number and type of deployed cars as well as the incurred transportation costs dueto car movements between rental locations and/or depots. The main contribution of this paperwhich serve for the dispersal of new cars (eeting) and eventually collecting cars once the holding

Hamburg

Berlin

Cologne

NORTH

EASTWEST District Munich

274 A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292period expires (deeeting). Depots act as an intermediate layer between stations on one side andmanufacturers and resellers on the other.The car rental company is aliated with dierent kinds of stations. A corporate station oper-

ates by means of sta and cars that are both part of the car rental company. A station with auton-omous sta but without a separate eet of cars is called corporate agent. In addition, franchisepartner (licensee) stations as well as foreign stations generally operate a separate eet of carsand are for the main part autonomous regarding logistics management. Consequently, we focuson corporate stations and corporate agents, where the car rental company can centrally decideabout the deployment of their own cars.

Munich

Frankfurt

SOUTH

Airport Munich

Landsberg

Fig. 1. Possible structure of a car rental network.A car rental company usually operates up to about 15 car groups where each group containsdierent cars with comparable quality (e.g., concerning size and equipment). Each group repre-sents a homogeneous good with a base rental fee per day (rate). In case that a customer madea reservation for a certain group in advance and no corresponding car being available at the timeof check-out an upgrade to a superior car group can be granted by the station. In practice, singleupgrades and double upgrades correspond to one or two additional quality levels, respectively.There are some common rules that dene feasible upgrade relations (see Table 1). Double up-grades should only be granted if no car from a single upgrade car group is available. Note thatthe rules are partly treated as suggestions and the station sta may make exceptions to satisfyparticular customers.

Table 1

Car groups

Group Type Rate Single Double Holding costs per day

A Sub-compact r1 B, J C h1B Compact r2 C D h2C Economy r3 D, K E h3. . . . . . . . . . . . . . .

A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292 275Car rental companies generally enter into agreements with car manufacturers and resellers,which dene criteria for the car usage (in particular, in terms of a maximum holding periodand mileage). For our purposes holding costs per day (including interest, depreciation, mainte-nance, etc.) of a car of some car group are assumed to be constant during the standard holdingperiod (e.g., 6 months). If a car rental company continues to use some car for rental operations inspite of an applicable deeeting criterion, penalty costs may have to be taken into account.

2.2. Rentals

A rental starts with a check-out at some station where the customer signs the contract and endswith a check-in at the same or a dierent station where the car is returned. Check-out data in-cludes the planned check-in station and rental length. The revenue due to a rental is composedby the base rate per day (multiplied by the rental days) as well as additional services, e.g., feesfor insurance, gasoline, or extra equipment. In case of an upgrade, the revenue is based on therate of the car group originally reserved. In general, the rate may depend on factors such asthe season, day of the week, or special contracts with certain groups or companies.Customers make reservations specifying at least the check-out and check-in station and time as

well as the requested car group. The typical policy of car rental companies is to accept reserva-tions for passenger cars without examinationhowever, these reservations are usually not bind-ing on either side. Achieving a high service level, in particular providing all customers that hold areservation with a car of the requested group (or an upgrade), is extremely important. Althoughthis may be unprotable from a short-term perspective on some cases, a high service level is cru-cial to build long-term customer relationships in competitive markets.Achieving a high utilization of cars is a main goal of the planning concepts discussed in Section

3, but nonetheless requires the ecient execution of operations processes. In particular, car rentalcompanies aim at a short turnaround with regard to the time needed from a check-in until acheck-out of the car is possible again (e.g., due to refueling and cleaning). For standard casesthe turnaround time should be shorter than 1 h. In general, there is a high degree of uncertaintyin the processes throughout the day. For example, there may be delayed check-ins, returned carsmay be in need of repair, reservations may expire when no customer turns up (no-show), or alot of walk-in customers may unexpectedly arrive. Furthermore, the current status of cars is ofteninaccurately represented in the information system (e.g., shortly after check-in or during turn-around). Balancing supply and demand throughout the day is complicated by these uncertainties.One consequence is that car rental companies usually do not operate by xed and automated pre-assignments of specic cars to (forecasted) customers, but by exibly handling the allocation whenthe customer arriveswith some degree of manual forward planning by a rough matching of res-ervations with the pool of available cars for dierent groups.

2.3. Logistics processes

The typical life cycle of a car is illustrated in Fig. 2. New cars are delivered from the manufac-turers to feasible depots where the cars are prepared for rental operations including supplement-ing special equipment and registering a vehicle license. Cars are brought into the active eet

(eeting) using trucks (with a capacity of up to eight cars) from the depots to designatedwith

276 A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292respect to direct eetingstations within the districts. This applies analogously for deeeting(transporting a car at the end of the holding period from some station to some depot).Ideally, foreseeable car shortages at some stations are balanced by arranging eeting and

deeeting appropriately. Nonetheless, there is generally also the need to schedule car transfersbetween stations. We distinguish transfers via truckkeeping the mileage of the caror by drivingthe car itself (by axle, carried out by station sta or sta from service providers). While combiningseveral car transfers on one truck usually leads to lower costs, it may also stand for inexibilitybecause of longer transfer times, mandatory advance planning procedures, and delays resultingfrom the combination of more than one car transfer. Transportation by axle is typically fasterand more exible but also more expensive than transportation via trucks. Transportation by axlemay be combined with transportation via truck as some designated stations may act as collectionpoints where trucks drop and pick-up cars (in particular in connection with eeting and deeet-ing). Note that transportation services are for the most part assigned to external shipping compa-nies on the basis of basic agreements, which include certain service guarantees and cost structures.We generally found that shipping companies charge xed costs per car depending upon the trans-portation relation, the distance as well as the mode of transport.The question marks shown in Fig. 2 indicate the logistics processes that are the subject of this

paper. That is, we aim to optimize car deployment by means of eeting, transfers between sta-tions, and deeeting, focusing on short-term logistics decisions for a planning horizon from afew days up to one or two weeks. We assume tactical and strategic decisions as given. In partic-ular, there are generally xed arrangements for the range of aggregate eeting and deeeting con-

Fleeting

Defleeting

Transfer

Rental

Car Manufacturer

Reseller

Cars on yard

Station

DepotNew carsCars for saleand/or

Fig. 2. Life cycle of cars.tingents with xed charges per car provided. Agreements with car manufacturers and wholesalersdetermine to what extent the holding period of cars in the active eet can be exibly adjusted toresolve shortages or to reduce over-capacity. The strategic management of the purchase, dispersal,and disposal of cars is an important issue of the car rental industry; see Economist IntelligenceUnit (1997). More detailed plans are generally determined through some hierarchical eet plan-ning process. This mainly includes forecasting rental demand as well as controlling the availabilityof cars at dierent aggregate levels (in particular, with respect to the planning horizon, regions,and car groups).The management of car deployment is highly complex due to the connection of car availability

across time, the station network, and dierent car groups. Furthermore, the detailed planning ofshort-term car logistics involves thousands of potential rental cases each day. For these reasons,eective decision support by appropriate information systems seems indispensable. The literature

ply and demand from the perspective of short-term logistics management. (As discussed in Section2.3, we assume strategic and tactical decisions such as the station network or the available car

A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292 277types as given. In Section 5, we outline the integration of the proposed optimization model withina decision support system, which also includes mid- and long-term planning functionality as wellas aecting demand by exible rate adaptations in the sense of revenue management.) On the onehand, revenue may be increased by serving rental requests. On the other hand, we strive for anecient deployment of cars with regard to transportation costs as well as holding costs accordingto the number and type of used cars. These conicting goals are integrated by the objective ofmaximizing prots, as dened by subtracting the incurred variable costs from the obtainedrevenue.We model and solve the car logistics problem on the basis of a rolling planning horizon of one

week. Resulting plans are re-optimized each night on the basis of new data, assuming that allcheck-outs and check-ins are generally keyed into the information system not later than by theend of the day. The assumption of time periods of half days results in partitioning a week into14 periods. (Dierent planning horizons or denitions of time buckets are of course possible.)We aim at a robust car logistics plan, which provides the stations in each period with enough carsof the dierent car groups, without fully automating the detailed decisions in the stations duringthe course of the day. The station sta should be enabled to handle customer requests by exiblydeploying locally available cars. That is, we do not propose an automatic assignment of speciccars to customers by the decision support system. This is due to the uncertainties of the actualthat considers the car rental business mainly encompasses revenue management and pool controlsystems as descriptive tools. Edelstein and Melnyk (1977) describe a pool control system with thepurpose to clarify and evaluate alternatives for, e.g., assigning cars within a pool of cities oraccepting reservations. The system is a descriptive interactive tool that leaves the actual deploy-ment decisions to the manager. A yield management system was designed at Hertz by Carroll andGrimes (1995) to combine various stand-alone decision support systems addressing mainly tacti-cal and strategic questions concerning the eet size and deployment as well as product design. Thisyield management system supports decision making by gathering information from the corre-sponding subsystems as well as presenting alternatives for mid- and short-term planning. Gera-ghty and Johnson (1997) focus on revenue management, especially capacity management,pricing, and reservations control. Pachon et al. (2003) describe a prescriptive model for daily eetplanning within a pool of neighboring rental locations. All papers mentioned provide valuable in-sight into the car rental business. However, the literature generally lacks prescriptive optimizationmethods for short-term logistics management under consideration of important requirementsfrom practice such as multi-period planning, a country-wide network, customized transportationrelations, eeting and deeeting, and car groups with partial substitutability. Our aim is to pro-vide an ecient solution method that takes these aspects into account and eectively supportsdecision making in car rental logistics management.

3. Modeling and solving the car logistics problem

The car logistics problem is modeled with the objective of maximizing prot by balancing sup-processes in the stations in conjunction with the often delayed availability of events (in particular,

278 A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292check-ins) in the information system. In the following sections, we describe the determination ofsupply and demand and dene the decision model and a corresponding minimum cost networkow model.

3.1. Supply

The determination of the supply of available cars throughout the planning horizon is compli-cated due to the connection of car availability across time and space. The option of car transfersbetween stations and the partial substitutability of car groups due to the possibility of upgradesmean that in theory almost any particular car may serve any future rental request at some arbi-trary station. This is related to the problem of deploying empty freight cars in rail system net-works; see, e.g., Spieckermann and Vo (1995).It is reasonable to assume that the information system of a car rental company provides at the

end of each day accurate data concerning the number of cars of dierent groups being available ata station, the cars currently on rent, which will become available at some station in the future, andthe number of new cars of dierent groups being available at a depot (with a eeting option). Thisprovides accurate data for the initial period of the planning horizon. The availability of cars insubsequent periods is inuenced by possible check-outs due to future rental requests, whichmay or may not be served, as well as check-ins due to cars currently on rent. Future rental re-quests have to be estimated on the basis of demand forecasts (see the next section), and one alsohas to rely on tentative check-in plans which are requested at check-out and updated during therental time. In combination, car availability may depend on uncertain check-in data of forecastedrental requests. Therefore, the accuracy of data about car availability decreases for future periods,which sets limits on an adequate planning horizon for detailed car logistics planning.We assume pre-dened eeting and deeeting contingents at the depots, which dene upper and

lower bounds for how many new cars of dierent groups are available and how many cars have tobe taken out of the active eet, respectively. Specic cars may eventually become unavailable forrental at the end of the contracted holding period or as a result of a repair necessity. While onecan estimate the former events, the latter ones are uncertain.

3.2. Demand

The planning of car logistics crucially depends on a sensible demand forecast. The main prob-lem is that a substantial proportion of rental requests are not due to a reservation. Therefore, wegenerally need to establish detailed short-term forecasts of rental requests over the planninghorizon.First of all, a forecast of a rental request must include information about the check-out station

and time as well as the requested car group. For rental requests that are linked to a reservation wealso know, from the corresponding reservation data, about the planned check-in station and ren-tal length (and so we usually have an accurate estimate of the revenue). However, to forecastwalk-in customers one has to rely on past data to guess the check-in station and the rental length.Consequently, forecast data becomes increasingly inaccurate after the average rental length of vedays. In practice, the demand forecast might be adapted by the local station sta, which may have

additional information available that aect demand (e.g., some new local event or weather

Thstatiosome location at some period) as well as short-term demand forecasts (i.e., potential number of

A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292 279check-outs of some car group at some station in some period), as elaborated in Sections 3.1and 3.2, respectively. Moreover, we rely on cost parameters concerning holding costs per dayand car group as well as transportation costs depending on the dierent modes of transport.The objective is to maximize short-term prots as measured by the revenue of the satised rentalrequests minus the sum of the variable costs, considering additional or saved car holding costs andthe incurred transportation costs. Within the planning horizon of about one week, decision vari-ables represent the number of cars of dierent car groups that are to be moved between dierentlocations at dierent time periods. This includes transfers between stations as well as eeting anddeeeting (moving cars from depots to stations and vice versa) under consideration of dierenttransportation options. A solution must meet the constraints set by the availability of cars (initialand inferred supply), the forecasted rental requests (demand), and the allowed upgrade relations.Because of the size of problem instances from practice it is crucial to devise an ecient solution

method. Our solution approach rests on modeling the problem as a specic minimum cost net-

worke decision model is based on detailed data about the car supply at the stations (corporatens and corporate agents) and depots (i.e., number of available cars of some car group atconditions). Nevertheless, detailed demand forecasts can at best provide a reasonable basis for ashort-term planning horizon, while mid- or long-term decision problemswhich are not subjectof this papergenerally rely on aggregate data. Furthermore, inuencing factors such as season-ality or local events are ignored in the current model due to a restricted access to historic data.Within the car rental logistics decision support system, as introduced in Section 5, the forecastmodule may exploit aggregate data over a long time period together with information about rel-evant (external) events.We assume that linear regression functions can map reservations to a forecast of rental re-

quests, i.e., to an estimate of the requested number of check-outs of a car of some car group atsome station in some period. We have identied four main factors that aect such a regressionfunction: the station, the period (for which we forecast) within a week, the lead time (betweenthe current period and the period for which we forecast), and the car group. This results in thefollowing general regression function (the subscripts/indices represent factor combinations):

#check-outsstation;period;leadtime;group astation;period;leadtime;group bstation;period;leadtime;group#reservationsstation;currentperiodleadtime;group

The parameters a and b of these regression functions are estimated on the basis of past data. Fur-thermore, we estimate a walk-in factor that represents the average proportion of walk-in customerrequests on the total demand (for each station, period within a week, and car group). The lengthand the check-in station of forecasted walk-in rental requests have to be guessed by sampling frompast rentals with the same characteristic (for each station, period within a week, and car group).The revenue of forecasted walk-in rentals is estimated on the basis of the average revenue per daythat has been obtained in the past for the relevant car group.

3.3. Network ow modelow problem, which makes available polynomial time algorithms from network ow theory;

Figrouare awithdue teeti

g = g , . . . ,g , which dene eeting and deeeting options, respectively. The corresponding

280 A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 2722921 G

parameter values qi represent eeting and deeeting contingents.Arcs (i, j) represent ow variables xij with regard to dierent options for car deployment. If not

dened otherwise, we assume the parameter values as lij = 0, uij =1, and eij = 0. For cars that are

neithrst of all, for each combination of some station s = s1, . . . , sS, period t = t1, . . . , tT, and carp g = g1, . . . ,gG, a stock node is dened that represents the number of cars of group g thatvailable at station s at the beginning of period t. Some of the stock nodes i are source nodesa positive supply qi which results from the initial car stock as well as anticipated check-inso cars on rent at the beginning of the planning horizon. In the same manner, we introduceng nodes and deeeting nodes for each depot c = c1, . . . ,cC, period t = t1, . . . , tT, and car groupsee, e.g., Ahuja et al. (1993) or Kennington and Helgason (1980). On the basis of a directed graphthat consists of a set of nodes N and a set A of arcs (i, j) that connect nodes, a general minimumcost network ow model can be formalized as follows:

Minimize zx X

i;j2Aeijxij

subject toX

j:i;j2Axij

X

j:j;i2Axji qi 8i 2 N ;

lij 6 xij 6 uij 8i; j 2 AA node i 2 N comes with a parameter qi that represents, if dierent from null, a positive (source

node) or a negative (sink node) amount of goods (in this case, cars) in the sense of a given supplyor demand, respectively. In total, the supply and demand of all network nodes must equal eachother (i.e.,

Pi2Nqi 0. Arcs represent a potential ow of goods. The variable ow quantity

xij on an arc (i, j) 2 A is restricted by a lower bound lij and an upper bound uij. Each arc (i, j)has a cost parameter eij. Multiplying this parameter by the ow quantity results in the costs thatare incurred by the ow on a particular arc. Solving a minimum cost network ow problem meansdetermining feasible (integer) values for the arc ow variables xij in such a way that at each node ithe quantity of in-ow minus out-ow equals qi and the sum of incurred costs is minimized. Suchminimum cost network ow models can be solved to optimality by ecient algorithms.Modeling the short-term car rental logistics problem by means of a network ow model re-

quires on original transformation of the characteristics of the described decision problem. Ourapproach is based on a time expanded network (timespace network), where specic points in time(periods) and space (stations and depots) are represented as corresponding nodes. Arcs that con-nect these nodes are related to temporal and spatial movements in the sense of dierent deploy-ment options. Time expanded networks have been applied in application elds such as railroadsystems (see, e.g., Kwon et al., 1998), air trac systems (see, e.g., Gu et al., 1994), or freight ship-ping (see, e.g., Chardaire et al., in press). We must also take into account dierent car groups andupgrade relations. Modeling this additional dimension by means of multiple commodity typeswould lead to a multi-commodity ow problem, which presumably cannot be eciently solvedto optimality for realistic (i.e., large size) problem instances (see the discussion in Section 6).Therefore, we integrate the possibility of upgrades into a single-commodity minimum cost net-work ow model as described below (note the example in Fig. 3).er deployed for a check-out nor transported to a dierent location we dene, for all locations

A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292 281(stations and depots), periods, and car groups, carryover arcs that allow keeping cars on the yardin the sense of movement in time. That is, such arcs connect (stock and eeting) nodes of the sub-sequent period. Furthermore, for each station, period, and car group, each stock node is con-nected to a corresponding rental node (see the discussion below) that is traversed by cars thatare actually deployed for rental requests with a preceding reservation. Such a rental request is rep-resented by a reservation rental arc between the applicable rental node (with respect to the check-out) and the applicable stock node (with respect to the check-in). Forecasted walk-in customersare represented by walk-in rental arcs that connect two corresponding stock nodes. The costparameter eij of a reservation rental arc is set to the negative value of the expected revenue accord-ing to the reservation data, while the costs of walk-in rentals are estimated on the basis of historicaverage revenues. In each case, the destination node of a rental arc is determined as follows: If theexpected check-in is within the considered planning horizon at a station inside the considered net-work, the arc leads to the corresponding stock node of the period that follows the check-in period.(Our experiments have shown that the followingnonetheless rather conservativeapproachmay also be reasonable: A check-in before 8 a.m. or 1 p.m. still leads to assigning the arc tothe morning or afternoon period, respectively, while later arrivals lead to the subsequent period.)If the check-in is after the considered planning horizon or at a station outside the considered net-work, the arc ends at a virtual super-sink node, which means that such a car (ow) cannot be usedagain within the planning horizon. Each rental arc has an upper bound uij = 1. Eventually, theresulting ow on such arcs, one or zero, indicates whether the rental request is served (xij = 1)or turned-down (xij = 0). (Rental requests with identical parameters can also be represented bya common arc with a resulting upper bound larger than one.)For each combination of a station and a period, upgrade arcs are introduced that start from the

superior car groups stock node and lead to the inferior car groups rental node in accordance withthe feasible upgrade relations. To prevent from unnecessary upgrades, we dene the cost param-eter of upgrade arcs as a small value d > 0. (By increasing this penalty factor, one can decrease theprobability for granting upgradesin general at the cost of more transportation or unfullledrental requests.) There are two reasons for distinguishing between stock nodes and rental nodes(instead of directly connecting stock nodes by upgrade arcs). First, this prevents a concatenationof upgrades which would unintentionally result in the transitive closure of the upgrade relations.Second, this restricts upgrades to be applied in direct connection with a rental request (but notbefore a transportation or while waiting at a station). Note that this modeling of upgrades leads,for each car that is used in connection with an upgrade, to irrevocable group degradation until theend of the planning horizon. As a result, the model is slightly more restrictive than the underlyingapplication. However, this does not severely aect the soundness of the model as we assume anaverage rental length of ve days in connection with a planning horizon of about one week.Fig. 3 illustrates the basic structure of the network ow model by means of an example for two

stations, one depot, and two car groups, where car group B is a feasible upgrade option for cargroup A. At station 1 in period 2, there are two rental requests for a car of group A, both of whichwith an estimated check-in at station 2 in some future period k. While the reservation rental arcstarts at the rental node and thus can be served by the superior car group B, a walk-in customermust not receive an upgrade. Therefore, the walk-in rental arc starts at the stock node.Every reasonable transportation option between stations is represented by a transportationarc. Such arcs are connected to stock nodes. Essentially in conformance with industry practices

282 A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292A A

stat

ion

1st

atio

n 2

A

period 1 period 2 period k

...+1

+10

+10

A A

A

A

B B

B B

B

B

upgrad

e

+5

A A+3

A A

A

A

B B

B B

B

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keep on yard

+4

transportation (axle)

A A

B+20 B B

...

depo

t 1

fleet

ing +10

transportation (truck)

fleeting

defleeting

reservation rental arc

walk-in rental arc

...

...

stock node

rental nodetransportation costs are assumed to be proportional to the number of cars according to the par-ticular cost parameter value depending upon the transportation relation, the distance as well asthe mode of transport. Possible upper bounds (transportation capacities) and destination nodes(in accordance with the transportation times) are dened depending on each specic transporta-tion relation. Consequently, one can exibly map dierent modes of transport such as by axle(expensive, fast, exible) or via truck (cheap, not so fast, some minimum lead time due to advanceplanning) with regard to dierent distances and network structures.For each depot and car group, eeting arcs and deeeting arcs from depot nodes to stock nodes

of specic stations and vice versa represent eeting and deeeting options, respectively, which alsoincludes the actual transportation (eeting is usually done via truck). Car holding costs are takeninto account by means of the cost parameters of eeting and deeeting arcs. That is, the costparameters of eeting and deeeting arcs represent the transportation costs plus the additionalor minus the saved holding costs. Minimum and maximum eeting and deeeting contingentsprimarily result from the supply and demand at the depot nodes, but can also be constrainedby suitable lower and upper bounds on specic arcs from depots to stations and vice versa. Someminimum lead times may have to be taken into account, which restricts the periods where eetingarcs may start and lead to. If deeeting takes longer than the planning horizon, such arcs are con-nected to corresponding depot nodes at period T (to allow for deeeting even at the end of theplanning horizon). Fleeting and deeeting contingents that are not pre-assigned to specic depots

depo

t 1

defle

etin

g A-

10A A

B B B

-

15......

Fig. 3. Basic structure of the network ow model.

A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292 283are represented, for each car group, by virtual super-eeting nodes and super-deeeting nodes.These nodes are connected to corresponding depot eeting and deeeting nodes, respectively.In order to balance supply and demand, we introduce a general super-sink node that eventuallyabsorbs all remaining cars at the end of the planning horizon in accordance with the resulting neg-ative supply value.Solving the network ow model generates ow values for each arc (variable), which essentially

results in a short-term car logistics plan with regard to proposed transfers between stations as wellas eeting and deeeting decisions. Reservation rental arcs with no ow indicate the possible needfor special action to enable serving such rental requests. Detailed examinations of the resultingsolution are possible by means of sensitivity analysis. For example, one can evaluate the eectof modifying the initial availability of cars at some station, or determine the additional revenuethat is a necessary to serve an unserved rental request.The introduced network ow model does not comprise all aspects of the short-term car logistics

problem and thus should be embedded in a more general decision support system, which providesthe eet manager with a exible interactive planning environment. In particular, a solution of thenetwork ow model does not determine specic cars for deeeting. Moreover, detailed restrictionswith regard to the transportation options (e.g., xed track routes throughout a day or a minimumtruck loading) cannot be represented. Therefore, the core optimization model must be comple-mented by appropriate pre- and post-optimization steps depending on the specics of the practicalsituation. First, particular cars with a deeeting status may be scheduled separately before solvingthe network ow model. Second, the results of the network ow optimization should be adaptedregarding detailed transportation procedures. In general, the planning process may involve dier-ent scenarios (e.g., dierent demand estimations or dierent eeting and deeeting contingents),which are evaluated by a combined application of the network ow optimization, pre- andpost-optimization steps, and a simulation system.

3.4. Simulation model

For a given scenario the described network ow model generates a schedule how cars should bedeployed. Due to the uncertainties of the rental operations throughout the day it is reasonable toevaluate the generated car logistics plan by means of simulation experiments before being imple-mented in practice. Fig. 4 shows the combined iterative application of demand forecast, networkow optimization, and simulation experiments as elements of a general logistics planning process.Simulation experiments are based on initial data in a global database (in particular, information

about the state of the car eet and existing reservations at the day of planning), which is developedthrough a simulation of seven days. Based upon a rolling planning horizon of one week, the se-quence of forecast, optimization, and simulation is executed each day until a whole week has beensimulated. The results of the detailed simulation of car rental operations for 24 h, considering thescheduled car transfers as well as eeting and deeeting decisions due to the network ow optimi-zation, lead to modied data, which serves as input for the next iteration. Depending on the out-come the whole process can either be repeated using a variation of the input parameters (e.g.,adapted eeting and deeeting contingents), or the deployment plan can be put into practice.To enable the assessment of the application of an optimized car rental logistics planning incomparison to historic processes, we distinguish between two modes of simulation. In replay

mode, historic processes (rentals, transfers, eeting, and deeeting) are simulated in accordancewith historic data. This mode is mainly used to obtain reference values such as the incurred costs

Forecast

Anticipated rentals

OptimizationDatabase Simulation

Logistics PlanReservations, fleet

7 days7 daysResults: car fleet, pending transports and rental s

24 hoursInput from stations Yes: Apply

Repeat until a week is simulated

Assessment

Logistics Planning Process

Accepted?

No: Repeat with changed input parameters

Fig. 4. Logistics planning process.

284 A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292and revenues as well as the resulting service level. In evaluation mode, initial data is given as be-fore, but rental requests are stochastically generated on the basis of demand forecasts. Further-more, the transfers between stations as well as the transports between stations and depots andvice versa due to eeting and deeeting decisions, respectively, are generated using plans due tothe network ow optimization.Since the simulation is more detailed than the network ow model (in particular by estimating

and using the exact time when a customer appears at the station counter instead of using timebuckets of half days) and depends on stochastic inuences, the implementation of the generatedcar logistics plan is not always completely feasible. For example, if a customer unexpectedly ex-tends the rental length a previously planned transfer of this particular car may become impossible.In such cases we abandon the respective element of the plan, while in practice there may be thepossibility of some exible short-term action to improve matters. Such deviating developmentsof the rental and logistics operations are eventually taken into account at the next iteration ofthe planning process (i.e., at the end of each day).Fig. 5 demonstrates the basic simulation process in more detail. Following the model initializa-

tion, a generator calls a procedure every specied time unit to handle the queued events in theevent calendar. Such events are primarily due to the start and the end of rentals and transports. Generator event_checker

Loo

p

Check_eventMethod rental_end (rental_nr)

Method transport_start (transport_nr)

Get planned rental agreement Assign carUpdate fleet, region, stationCreate real agreement

Reading Data

Create Model

Start Simulation

Call event

Event Calendar

Method rental_start (rental_nr)

Update fleet, region, station

Get planned transportCheck if transport can be performedUpdate fleet, region, station

transport_end (transport_nr)Update fleet, region, station

Method

Fleeting, defleetingTransfer between stations (by truck, per axle)

Fig. 5. Simulation process.

A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292 285For each event, a corresponding procedure has to be executed, which adapts the state of the sim-ulation model. During the simulation every event is recorded and eventually stored in the data-base together with accumulative data (e.g., revenues, costs). Report generators produce variousreports for dierent purposes of logistics management. The simulation model has been imple-mented using the simulation system eM-Plant from Tecnomatix. By means of an ODBC interfaceto a relational database, the simulation system obtains all relevant data and nally writes back thenew state of the car eet after simulation.

4. Results

Our results are based on data of the German subsidiary of one of the major international carrental companies. We generalize some of our ndings from this case study. The problem data ischaracterized by the following parameters: There are a few hundred stations and about ten de-pots. The active eet typically includes up to 18,000 cars from 15 car groups. There are up to3000 new rentals (check-outs) each day. Therefore, we have to consider about 20,000 rental re-quests simultaneously, for a planning horizon of one week.

4.1. Forecast quality

The parameters of the regression functions introduced in Section 3.2 were estimated on real-world data that cover the past rental operations for a period of 4 months. Each combinationof some station, some period (for which we forecast) within a week, the lead time (between thecurrent period and the period for which we forecast), and the car group leads to a specic regres-sion function, which maps the number of reservations to an estimate of the number of potentialcheck-outs. Despite processing several gigabytes of data, each least-squares estimate results fromconsidering only 16 data points (one for each week).The quality of the demand forecast was evaluated taking into account the coecient of deter-

mination (r2) and the expected forecast error (s2

p). As an example, we consider the forecast of

rental requests for car group B for a Monday at some large airport station. Carrying out thedemand forecast on Friday evening means applying the following regression function: #check-outs = 6.26 + 0.97 * #reservations. This regression function comes with a coecient of determina-tion of r2 = 0.84 and thus a rather high correlation between the number of reservations and thenumber of check-outs. If we need to carry out the forecast already on Wednesday evening oneexpects that the parameter b is larger than one taking into account the reservations yet to be re-ceived. This is conrmed by the following regression function for the described scenario: #check-outs = 14.06 + 1.21 * #reservations. For this regression function the coecient of determinationdrops to r2 = 0.49. In the considered scenario, the t-statistics values for the slope regression coef-cients are 8.5 and 5.0 when forecasting with a look-ahead of two (Friday) or four (Wednesday)periods, respectively, which means statistical signicance (assuming a level of signicance of 99%and a degree of freedom of 14 which corresponds to a theoretical t-value of 3.0).Fig. 6 illustrates the eects of the look-ahead period on the forecast quality for the same

scenario as discussed before (with an average number of check-outs per day of about 45). Theexpected forecast error (

s2

p) approximately doubles throughout the planning horizon of oneweek. That is, the expected forecast error is critical, at least for a look-ahead of more than very

286 A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292few days. (For a look-ahead of more than ve days the regression becomes statistically insignif-icant.) The situation worsens for small stations with only a minor number of check-outs per day.However, such forecast errors partly oset each other with regard to dierent car groups that mayserve as substitutes due to upgrades. The same eect is to be expected when aggregating over dif-ferent stations (in the same district or region). Therefore, the demand forecast is useful to estimatesuch aggregate demand values for a planning horizon of one week (which is important to decideabout eeting and deeeting contingents), while detailed short-term forecasts for the next fewdays allow balancing supply and demand at the stations on the basis of planning individual cartransfers between stations. This is in agreement of using a rolling planning horizon of one week

0

2

4

6

8

10

12

1 2 3 4 5 6 7Look-ahead [days]

Fore

cast

err

or

Fig. 6. Expected forecast error depending on the look-ahead period.with a re-optimization each night on the basis of new data.There are two main options to improve the forecast quality. First, one may estimate the param-

eters of the regression functions on the basis of a larger data set (covering complete data aboutoperations of some years), which should lead to an improved forecast quality. This option includesapplying medium range forecast models, which would also allow taking into account eects of sea-sonality, local events, and specic incentive plans. Second, we observed that information systems ofcar rental companies often do not include complete information about all relevant events. Forexample, the sta in small stations sometimes misses to key-in local reservations into the global res-ervation system (partly handling them in a paper and pencil manner). Moreover, not every check-out that is due to a reservation is actually labelled accordingly, and not all rental requests that areturned-down at the counter (usually in connection with a walk-in customer) are recorded. Car ren-tal companies should generally take action to completely record all relevant business processes intheir information system, which might substantially improve the general forecast quality.

4.2. Optimization by means of the network ow model

Beside transportation costs, car holding costs due to the active eet constitute the main costfactor in car rental operations. Consequently, the logistics management in car rental companies

A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292 287generally aims at reducing the eet size as far as possible and keeping transportation costs to areasonable amount without inducing too many turned-down rental requests. Even if loweringthe service level might be protable in the short-term in certain situations, maintaining a high ser-vice level (e.g., above 99%) is extremely important from a long-term perspective (with regard tobuilding stable customer relationships and a resulting strong market share). Under considerationof these essential demands of practical logistics management we are primarily interested in meansfor reducing the eet size without signicant negative side eects. Therefore, we analyze, rst, theeect of applying the network ow model in comparison to the present manual practices, and,second, modifying upgrade restrictions in order to increase the overall eet utilization.A typical problem instance leads to a network ow model with more than 100,000 nodes and

3,000,000 arcs (variables). The resulting problem instances were solved by using the network sol-ver of ILOG Cplex (version 6.6). The data structure of the network model was constructed on thebasis of real-world data stored in a relational database, which was accessed through an ODBCinterface, using the programming language C++ and accessing Cplex in the form of a dynamiclink library (dll). We used a standard personal computer with an 1.8 GHz CPU and 512 MB mainmemory. All problem instances considered were solved to optimality in about 1 min, requiringabout 300 MB of main memory. That is, computation time is no critical factor for the scenariosexamined, which allows an integration of the network optimizer in an interactive decision supportsystem.Assuming demand forecast data as deterministic data and solving the network ow model to

optimality for dierent eet sizes should result in an upper bound for the potential prot increase.However, our analysis of real-world data has shown that a signicant proportion of check-outs inpractice are not handled in accordance with the allowed upgrade relations, which complicatescomparing real-world conditions with results from the network ow optimization. For example,the comparative assessment is problematic if in practice up to 10% of rental requests have notbeen served by a car from a feasible group, while the solution of the network ow model includesup to 3% of rental requests that are turned-down fully conforming to the feasible upgrade options.A deployment plan that conforms to strict rules but turns-down a few rental requests may be per-fectly adequate if it enables the station sta to serve these requests by disregarding some of theserules.We examine the eects of eet size reductions with regard to the percentage of rental requests

that are turned down as well as the resulting prot change. Hypothetical eet size reductions aregenerated by applying rules such as if there are initially three cars of some group at some station,remove one of these. By using dierent rule sets we created various reduced eet size scenariosthat match the data points shown in Fig. 7. The depicted results are typical for various periodsconsidered in dierent experiments. Starting with an initial eet size from practice of 100%(e.g., 15,500 cars), according to the upper diagram one might save more than 20% of the cars(e.g., more than three thousand cars) before the service level drops below 99%. That is, in theorywe are able to signicantly reduce the active eet size without losing a high service level. The lowerdiagram illustrates the eects of analogue eet size reductions on the revenue as well as the rev-enue minus the additional transportation costs (in each case normalized with reference to the rev-enue obtained for the initial eet size). As may be expected, lost revenues and additionaltransportation costs are only moderately aected at the critical point where the service level begins

to drop sharply, since the optimized deployment plan generally gives preference to the most

288 A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 27229296.00%

96.50%

97.00%

97.50%

98.00%

98.50%

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100.00%

10000 11000 12000 13000 14000 15000 16000

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ice

leve

l

98.00%

98.50%

99.00%

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100.00%

ison

to in

itial

reve

nueprotable rental requests. Without disclosing tangible and detailed revenue and cost gures in thispaper, in our experiments reductions of car holding costs oset lost revenues and additional trans-portation costs by a factor of more than ten. Assuming car holding costs of 10 Euro per day, thisresults in a prot increase of more than a million Euro per year by means of reducing the averageeet size by a few hundred cars. The general protability of the proposed approach even holds ifthe currently suboptimal forecast quality is taken into account.In order to assess the impact of upgrades we examine the eect of modifying upgrade restric-

tions. In comparison to the existing upgrade rules from practice, we consider two alternative sce-narios. On the one hand, we prohibit all kinds of upgrades. On the other hand, we use thetransitive closure of upgrade options in the sense that a rental request may be served by anycar with a quality that is not lower than the requested car group. Fig. 8 shows that the possibilityof granting upgrades is crucial to achieve a high service level. In case that one enables all conceiv-able upgrade options, the eects on the service level and resulting prot gures are slightly betterin comparison with present practices. Consequently, more upgrade exibility constitutes an op-tion for increasing protability in connection with an improved eet utilization. However, notethe possible long-term aspects of loose upgrade rules. Namely, if the probability of being granteda valuable upgrade is high enough, customers might be tempted to make reservations for cargroups of lower quality than actually wanted, which results in overall revenue degradation.

96.00%

96.50%

97.00%

97.50%

10000 11000 12000 13000 14000 15000 16000Fleet size

Red

uctio

n in

com

par

Revenue Revenue - transportation costs

Fig. 7. Inuence of eet size reductions.

90.00%

ice

lev

A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292 28975.00%

80.00%

85.00%

10000 11000 12000 13000 14000 15000 16000

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venu

e m

inus

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tatio

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Normal upgrades All upgrades95.00%

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Normal upgrades All upgrades No upgrades4.3. Validation per simulation

We analyzed the eects of implementing the car logistics plan from the network ow model bysimulation experiments. As a basis for comparison we used the simulation model to replay rentaland logistics processes of periods of one week based on historic data. With regard to this referencemeasurement, we validated optimized car logistics plans by assessing the results in comparison toboth the historic processes as well as the network ow scenario.We found that the general potential for improvement due to the network ow optimization is

conrmed by results from simulation experiments. As an example, some simulation experimentfor a given week may result in an approximate revenue of 3 million Euro and transportation costsof less than 100,000 Euro. Removing one thousand cars from the active eet, and simulating thesame seven day period on the basis of the car logistics plan generated by the network ow model,essentially led to a rental service level of 99.9% with about the same transportation costs and anacceptable upgrade ratio of about 16%.

5. Decision support system

In Fig. 9 we propose the system architecture of a car rental logistics decision support system.We distinguish between a core system database and an events database. Core data are xed over a

98.00%10000 11000 12000 13000 14000 15000 16000

Fleet size

Re

Fig. 8. Inuence of upgrade rules.

- holding cost/period- fleeting/defleeting date

290 A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292Events Database

Reservation/Rental/Transport: - check-out station/date - check-in station/date - car group/extras

Car Status:- on yard- on rent- on transport

Data Warehouse

Strategic Planning

EvaluationAnalysis Tools

Logistics Planning Process

Tactical-operational Planning

Forecast

OptimizationSimulation

Decision Support System

DatabaseManagementSystemFleetInformation:

- id- group/type

NetworkStation/Depot/Region:

- name/id- location

Core System Databasecertain period (such as the station network, car groups with holding costs, and basic informationabout the cars in the active eet). Data about incoming reservations, check-outs and check-ins,and the start and end of transports are stored in the events database.In addition to the described logistics planning process, we propose the implementation of a data

warehouse that stores aggregate data for further processing by appropriate analytical tools. Onthe one hand, resulting insights (e.g., with regard to the protability of specic stations or the con-tribution of dierent customer segments) will be a valuable input for strategic planning. On theother hand, aggregate data (e.g., concerning empirical distributions for the length and reliabilityof transportation processes or probabilities for the unexpected extension of the rental length) mayalso be used in the logistics planning process. In particular, the data warehouse may provide long-term data for enhancing the demand forecast model.

6. Conclusion

In this paper we considered logistics management in the car rental business. After giving anoverview of car rental operations, we presented a novel quantitative decision model to ecientlysolve short-term car rental logistics problems by means of network ow optimization. Our deci-sion model includes essential practical aspects such as multi-period planning, a country-widenetwork, specic transportation relations, eeting and deeeting, and dierent car groups.Experiments were conducted on substantial real-world data, using a simulation model to assess

Fig. 9. Architecture of a car rental logistics decision support system.

A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292 291optimization results for dierent scenarios. Our experience from an industry research project indi-cates that the approach can signicantly improve prots by reducing the costs for the eet of carsand limiting transportation costs.From a practical point of view, the crucial requirement for the implementation of our approach

is the availability of an integrated information system with data about the current state of the ren-tal system as well as a sensible short-term demand forecast model. From a methodological pointof view, modeling and solving the problem by means of a multi-commodity network ow formu-lation may pose an interesting research subject. In such a model, dierent commodity types rep-resent the various car groups. As rental requests can be served by dierent car groups according tothe allowed upgrade relations, the sum of respective ow variables is linked to corresponding de-mand data. Because of the large network size, special techniques such as DantzigWolfe decom-position and Lagrangian relaxation may have to be exploited to solve the problem to optimality ina reasonable time horizon; cf. Kennington and Helgason (1980) or Minoux (1986).A crucial question of car rental logistics is the degree of central vs. local planning. Implement-

ing a nation-wide car logistics plan restricts the degrees of freedom for local actions. On the onehand, our proposal intentionally does not prescribe the operations in the stations during thecourse of the dayin particular, we do not recommend centrally assigning specic cars to custom-ers. However, our empirical experience is that local planning of car transfers often leads to anuncoordinated and inecient use of the rental network resources. The local cause for car transferscan only be avoided by means of an eective car logistics planning and implementation, whichprovides the stations with adequate availability of cars throughout each day. Central planning de-pends on the availability of rather complete and accurate data in information systems. The qualityof data can be improved by technical and organizational measures. From a technical point ofview, the use of transponder systems and the global positioning system may contribute to theautomatic representation of the status of each car at each time in the information system. More-over, the planning and execution of car transportation via truck in collaboration with externalshipping companies requires online interfaces between relevant information systems. From anorganizational point of view, there are two main sources of information that we have to deal with:the customers and the companys sta in rental stations. For both we need to introduce reasonableincentives for making available their mental knowledge as data in the information system.To increase the information about short-term demand, one must think about incentives for cus-

tomers to make serious reservations in advance. Today, reservations are usually not binding oneither side, which means that customers often make reservations at dierent car rental companiesand eventually select only one (e.g., by entering the rst station with an empty queue at the coun-ter upon arrival at the airport). For a car rental company it is dicult to introduce penaltycharges for no-shows. First, non-binding reservations are the norm in the car rental business. Sec-ond, if a customer expects that enough cars are available at the station even without a reservationthe customer might be tempted to refrain from making reservations. Therefore, car rental compa-nies generally strive to make it as easy as possible to make reservations (e.g., by phone or by web),in combination with incentives such as the possibility of being granted an upgrade (which is notpossible for walk-in customers).Considering the station sta, a strong negative incentive to key-in information about a check-in

may arise in case that the station sta must reckon with a decision from central planning that a

particular car (e.g., an attractive sports car which might attract additional walk-in customers) will

soon be transferred to a dierent station. However, the check-in must eventually be keyed-in withthe correct check-in time. Thus, if station performance is measured by taking into account thecosts of unutilized cars the station sta may indeed strive to properly enter data about the avail-ability of this car for the general network as soon as possible. In this respect we draw attention tothe importance of a deliberate measurement of station performance by an incentive-compatibleassignment/sharing of costs (in particular, holding and transportation costs) and revenues inthe context of intertwined central and local decisions. This poses a rather complex problem, which

Pachon, J.E., Iakovou, E., Ip, C., Aboudi, R., 2003. A synthesis of tactical eet planning models for the car rental

industry. IIE Transactions 35, 907916.

292 A. Fink, T. Reiners / Transportation Research Part E 42 (2006) 272292Spieckermann, S., Vo, S., 1995. A case study in empty railcar distribution. European Journal of Operational Research

87, 586598.is even more true in the case of licensees, which might over-exploit resources of the franchise sys-temin this case by stockpiling cars of otherswhile under-investing in own resources (see, e.g.,Dnes, 1996 for a general economic analysis of franchise systems).

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Modeling and solving the short-term car rental logistics problemIntroductionProblem descriptionNetwork and fleetRentalsLogistics processes

Modeling and solving the car logistics problemSupplyDemandNetwork flow modelSimulation model

ResultsForecast qualityOptimization by means of the network flow modelValidation per simulation

Decision support systemConclusionReferences