a methodology to solve large-scale cooperative transportation planning problems

European Journal of Operational Research 223 (2012) 626–636

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Discrete Optimization

A methodology to solve large-scale cooperative transportation planning problems

Ralf Sprenger, Lars Mönch ⇑Department of Mathematics and Computer Science, University of Hagen, 58097 Hagen, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 April 2011Accepted 15 July 2012Available online 25 July 2012

Keywords:TransportationCooperative transportation planningRich VRPRolling horizonDiscrete event simulationPerformance assessment in a stochasticenvironment

0377-2217/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.ejor.2012.07.021

⇑ Corresponding author. Tel.: +49 0 23319874592;E-mail addresses: [email protected]

fernuni-hagen.de (L. Mönch).

In this paper, we suggest a methodology to solve a cooperative transportation planning problem and toassess its performance. The problem is motivated by a real-world scenario found in the German foodindustry. Several manufacturers with same customers but complementary food products share theirvehicle fleets to deliver their customers. After an appropriate decomposition of the entire problem intosub problems, we obtain a set of rich vehicle routing problems (VRPs) with time windows for the deliveryof the orders, capacity constraints, maximum operating times for the vehicles, and outsourcing options.Each of the resulting sub problems is solved by a greedy heuristic that takes the distance of the locationsof customers and the time window constraints into account. The greedy heuristic is improved by anappropriate Ant Colony System (ACS). The suggested heuristics to solve the problem are assessed withina dynamic and stochastic environment in a rolling horizon setting using discrete event simulation. Wedescribe the used simulation infrastructure. The results of extensive simulation experiments based onrandomly generated problem instances and scenarios are provided and discussed. We show that thecooperative setting outperforms the non-cooperative one.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Transportation planning problems are important for theGerman food industry because of an increasing competition amongthe manufacturers and smaller margins. In this paper, we considera real-world situation where several manufacturers with same oroverlapping customers but complementary food products collabo-rate by jointly using their vehicle fleets to reduce delivery costs.The different products are delivered to selected first-class hotels.Therefore, small delivery quantities are typical. Achieving highon-time delivery performance is an important goal in this context.

There are different types of deliveries in the researched real-world setting. In the simplest case, a manufacturer uses own vehi-cles at its manufacturing location and delivers the orders to thecustomers. Depending on the geographical location of the custom-ers and the capacity of the vehicles, it might be more promisingthat the manufacturer runs an intermediate distribution center ina far-away region, locates own vehicles there, and delivers ordersfrom the intermediate distribution center to the customers. An ex-press company delivers the orders from the manufacturing loca-tion to this intermediate distribution center. Using an expresscompany leads to additional costs and requires more time. Thethird and most expensive delivery type is to carry out a directtransport from the manufacturing location to a single customerby an express company.

ll rights reserved.

fax: +49 0 23319874519.(R. Sprenger), Lars.Moench@

This paper deals with cooperative transportation planning. Thebasic idea consists in offering the option that different manufactur-ers share their own vehicles either at the manufacturing locationsor at the distribution centers to reduce costs and to improve theon-time delivery performance. Problems of this type are rarelystudied in the literature so far. To the best of our knowledge, coop-erative problems similar to ours are studied only by Shang and Cuff(1996) and by Lin (2008). But there is some literature on transpor-tation planning with outsourcing options that covers some specif-ics of our problem (cf. Lee et al., 2003; Zäpfel and Bögl, 2008). Inthis paper, we propose a heuristic that decomposes the overallproblem into a set of problems containing a smaller number of or-ders. We will show that each of the smaller problems, so called subproblems, can be efficiently solved by ant colony optimization(ACO). Using discrete event simulation, we are able to show thatthe ACO approach outperforms a greedy heuristic and the cooper-ative approach the non-cooperative in the case of stochastic distur-bances when it is applied in a rolling horizon setting.

The paper is organized as follows. In Section 2, we describe theproblem. Related literature is discussed in Section 3. The problemis decomposed into different sub problems to deal with the prob-lem size that is found in real-world scenarios in Section 4. Theresulting sub problems form rich VRPs that are solved by a greedyheuristic and improved by an ACO type heuristic. In Section 5, westart by describing a simulation framework to assess the perfor-mance of the heuristic in a dynamic and stochastic setting. Resultsof simulation experiments based on randomly generated scenarios

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R. Sprenger, L. Mönch / European Journal of Operational Research 223 (2012) 626–636 627

are presented. Finally, we present conclusions and discuss some fu-ture research directions in Section 6.

2. Problem statement

In this section, we start by summarizing some basic notation.We then describe the problem.

2.1. Notation

Transportation network related:

m 2M
index of the main manufacturing location of asingle manufacturer, where M is the set of allmanufacturers
c 2 C
index of a customer, where C is the set of allcustomers
i 2 I
index of an intermediate distribution center,where I is the set of all intermediate distributioncenters
dl 2 DL
index of a single distribution location, where DL isthe set of all distribution locations, DL = M [ I
l 2 L
index of a single location, where L is the set of alllocations, L:¼DL [ C
e 2 E
index of a single express company, where E is theset of all express companies
rmax
maximum number of days (horizon)
cð1Þe ðd; qÞ
cost for carrying out the delivery to a distributionlocation by an express company e, where ddenotes the distance that is traveled by thevehicles of the express company, and q is therequested capacity of the express company
cð2Þe ðd; qÞ
cost for direct transportation of orders from amanufacturing location to a customer by anexpress company e, where d denotes the distancethat is traveled by the vehicles of the expresscompany and q is the requested capacity of theexpress company
qð1Þre
total volume at day r that is transported byexpress company e to one of the distributionlocations
dð1Þre
traveled distance of the express company e to oneof the distribution locations at day r
qð2Þre
volume that is directly transported by the expresscompany e to a customer at day r
dð2Þre
traveled distance of the express company e to acustomer at day r corrected by ve.
Vehicle related:
v 2 V index of a vehicle, where V is the set of all
vehicles
Cv capacity of vehicle v, i.e., maximum available
volume
otv maximum operating time per day hdv home depot of vehicle v dr
v
total distance traveled by vehicle v at dayr 6 rmax
d̂
average traveled distance of a single vehicle atone day
ve
multiplier that models the longer transportationtime that is needed by an express company due tohidden tours of the vehicles of the expresscompany.
Order related:
o 2 O index of an order, where O is the set of all orders co customer that orders order o
~ro
point of time when o is ordered ro point of time when the goods associated with o
are ready for delivery
wo:¼[lo,uo] time window associated with o, where lo and uo
are the earliest and latest time, respectively tocomplete the delivery of order o to co

so
delivery date of order o, i.e., the point of timewhen the order arrives at its customer location
qo
capacity demand of order o to lateness of order o with respect to uo.
2.2. Problem description

We study a transportation network that consists of manufactur-ers with one main manufacturing location m 2M, intermediatedistribution centers i 2 I, and customers c 2 C. In the following,the term distribution location dl 2 DL is used for manufacturinglocations as well as for intermediate distribution centers if a differ-entiation is not necessary. Both types of locations are able to deli-ver orders to customers.

Each manufacturer owns a given number of vehicles v 2 V thatare located at the distribution locations of this manufacturer. Thenumber of vehicles is small with respect to the expected transpor-tation demand. We assume that all the vehicles are identical andthat a single vehicle has a capacity of 15 orders. The first assump-tion is motivated by the real-world scenario where vehicles of typeMercedes Benz Sprinter are used. However, the assumption ofidentical vehicles is not crucial for the algorithms and the simula-tion approach proposed in the paper. The second assumption isbased on the fact that in the real-world scenario the majority of or-der quantities are small and therefore, a small number of E2 boxes,typically five, are enough for each order. Such boxes are most com-monly used in the food production industry in Europe. Therefore,we assume in this paper for the sake of simplicity that all ordershave the same volume. The vehicles can only operate within thedaily availability period of its distribution location, i.e., hdv. Thedaily availability period corresponds to the daily opening time ofhdv.

Orders o 2 O are ordered by customers co at time ~ro and areavailable at time ro at the corresponding manufacturer for delivery.The difference ro � ~ro is the time that an order o needs to get readyfor delivery. When the arrival time of the vehicle at the location ofcustomer co is smaller than lo, the vehicle has to wait until thebeginning of the time window before servicing the customer. Notethat the end points lo and uo of the time window wo have to bewithin the availability period of the distribution location.

We differentiate between three types of transportation. Thefirst type consists of transportations that can be performedusing own local vehicles at the main manufacturing location.The second type is formed by deliveries that take place usingown far-away vehicles at an intermediate distribution center.Vehicles of a different manufacturer at its distribution locationscan be used to deliver orders of the specific manufacturer in thecooperative scenario. In this situation, an express company e 2 Edelivers the orders from the manufacturing location of a manu-facturer to another distribution location. The express companycan also be used to directly transport orders from the mainmanufacturing location to the corresponding customer. This isthe third type. The different types of transportation in a trans-portation network that consists of four distribution centers areshown in Fig. 1.

The delivery by an express company takes place overnight. Weassume that the delivered orders arrive in the next morning shortly

Manufacturer 1

Manufacturer 2

Intermediate Distribution

Center (Manufacturer 1)

Intermediate Distribution

Center (Manufacturer 2)

Customer

Customer

Customer

Customer

Customer

Customer

Customer

Customer

Customer

Customer

Direct delivery with express

company

Delivery between distribution centers with express company (cooperative scenario)

Delivery with local vehicles

Delivery with far-away vehicles

Delivery between distribution centers with express company

Fig. 1. Different types of transportation in a transportation network.

628 R. Sprenger, L. Mönch / European Journal of Operational Research 223 (2012) 626–636

before the corresponding distribution location opens. Note that di-rect deliveries start also in the evening after the distribution loca-tion has closed and have a duration of ve d. If the arrival time is notwithin the opening times of the distribution location, the deliverywill be postponed to the beginning of the next opening period. Thequantities cðiÞe are express company dependent. We assume that anorder can only be transported with an express company if it hasnot been already on a vehicle of one of the manufacturers or ona vehicle of an express company.

For assessing the total costs over a certain horizon, we have totake into account the total distance traveled by the vehicles, thecosts for the express companies, and the costs caused by the viola-tion of time windows of the orders. We set t :¼

Po2Oto to normal-

ize the individual lateness values of the orders. The total volume isalso normalized by ~q :¼

Po2Oqo. The total cost function for a hori-

zon of rmax days is then given by the expression:

cos t :¼Xrmax

r¼1

Xv2V

drv þXe2E

x1cð1Þe dð1Þre ;qð1Þre =~q� �

þx2cð2Þe dð2Þre ;qð2Þre =~q� �n o !

þx3

Xo2O

d̂ffiffiffiffiffiffiffiffiffito=t

p; ð1Þ

where xi > 0 are weights expressing the importance of the differentcost types. We use cost functions of the form

cðiÞe ðd; qÞ :¼ffiffiffiqp

d; ð2Þ

i.e., the costs are linear in the traveled distance for a given transpor-tation volume, but for a given traveled distance the increase of thecost shows a square root behavior. Note that transportation costsof this form are often assumed (cf. Altiparmak and Karaoglan,2008). We are interested in minimizing total costs. The last termin cost function (1) is due to orders that violate the right end pointof the time window during carry out the transportation plans in arolling horizon setting. This term has to be included to take stochas-tic effects into account that occur during the execution of the deter-mined plans. These effects are modeled by discrete event simulation.

The problem researched in this paper is NP hard because it in-cludes traveling salesman type problems as special cases. There-fore, we have to look for efficient heuristics to tackle the problem.

3. Related research

In this section, we discuss related literature mainly with respect tocooperative transportation planning, outsourcing of transportation

operations, and with respect to simulation based performance assess-ment of transportation planning schemes.

Cooperative transportation planning problems are rarely dis-cussed in the literature. Shang and Cuff (1996) suggest a cooperativestrategy to solve a multi-criteria pickup and delivery problem that al-lows for transferring hospital documents between different vehicles.A second example is described in Lin (2008). Here, a multi vehiclepickup and delivery problem with a single depot is discussed. Thevehicles cooperate by meeting at some locations on their routesand put the already picked up orders onto a vehicle that has com-pleted its route and travels back to its home depot. But the problemresearched in this paper is quite different from the problem describedthere in various aspects. We have to deal with multiple depots andalso with different outsourcing options. Collaborative logistics prob-lems in a less than truckload transportation scenario are presented byDai and Chen (2009). But again, the researched problem in this paperincludes much more restrictions and is different.

Outsourcing options within transportation are discussed, forexample, by Lee et al. (2003) and by Zäpfel and Bögl (2008).Pankratz (2004) studies a pickup and delivery problem with out-sourcing options using a genetic algorithm to solve this problem.Extensions of VRP by different types of sub-contractions are dis-cussed by Kopfer and Wang (2009). However, our problem is dif-ferent from the problems researched in these papers.

Next, we discuss different metaheuristic approaches. Metaheu-ristics are popular for VRPs (cf. Bräysy and Gendreau, 2005a,b;Pankratz, 2005; Irnich, 2008; Paraskevopoulos et al., 2008). In ourcase, because of the large number of constraints, it makes senseto try to improve a simple construction heuristic instead of definingsophisticated neighborhood structures. Therefore, we look for ACOtype approaches. ACO is a nature inspired approach that is able tosignificantly improve construction heuristics for a large class ofcombinatorial optimization problems. ACO approaches are quitepopular for solving different types of VRPs (cf. Bullnheimer et al.,1997; Gambardella et al., 1999; Montemanni et al., 2003; Tanet al., 2005; Doerner et al., 2006; Favaretto et al., 2007; Pellegriniet al., 2007; Gajpal and Abad, 2009 amongst others). In (Lee et al.,2010), an initial solution provided by simulated annealing is im-proved by ACO approaches for the capacitated VRP.

The suggested ACO algorithms mainly differ in the includedheuristic information that is typically problem specific. Amongthe ACO type heuristics, ACS often leads quickly to high-qualitysolutions (cf. Gajpal and Abad, 2009). This feature is importantwhen the solution of real-world problems is addressed.


It is often beneficial to combine decomposition approaches withACO type solution approaches. For example, Bouhafs et al. (2006)decompose the capacitated-location routing problem into subproblems using simulated annealing and then use ACO to solve eachof the sub problems individually. A decomposition approach cou-pled with ACO along with some preliminary computational resultsis discussed by the present authors in (Sprenger and Mönch, 2009).However, in this paper we suggest several modifications of this ap-proach and provide results of extensive simulation experiments.

We have to embed the transportation planning approaches intoa rolling horizon setting because of the consideration of a real-world scenario. Therefore, we have to solve a sequence of smallertransportation planning problems. Feedback from the logisticsbase system can be taken into account for the construction of thesetransportation planning problems. Discrete event simulation canbe used to represent the logistics system. The approach of simula-tion-based performance assessment of transportation planning ap-proaches is rarely discussed in literature. We refer to Ebben et al.(2004) where simulation is applied to assess the performance oftransportation planning approaches. However, there is no clearseparation between the transportation planning approaches andthe simulation engine. Simroth and Baumbach (2007) suggest aconceptual framework to assess the performance of approachesfor dynamic transportation planning problems. However, no con-crete implementation strategies for a simulation environment aredescribed. In this paper, we apply the simulation framework sug-gested by the present authors (cf. Sprenger and Mönch, 2008) toassess the performance of our solution schemes in a dynamicand stochastic environment using a rolling horizon setting.

4. Cooperative transportation planning approach

In this section, we start by presenting a methodology to solvelarge-scaled cooperative transportation planning problems. Wediscuss a decomposition heuristic that results in a set of rich VRPproblems. We continue with discussing a simple greedy heuristicto solve the sub problems. Finally, we present an ACS type heuristicthat improves the greedy heuristic.

4.1. Overall methodology for large-scale cooperative transportationproblems

Our methodological approach consists of the following twosolution steps and a third performance assessment step:

1. Decomposition of the overall transportation planning probleminto rich VRP sub problems that are related to the differentdelivery locations taking the transportation alternatives forthe cooperative setting into account.

2. Improvement of the solution obtained in Step 1 by applying alocal search approach to the solution for the overall approach.

3. Usage of discrete event simulation to allow for the performanceassessment of different operational strategies when theapproach proposed in Steps 1 and 2 is executed in a rolling hori-zon setting.

Next, we have to motivate and justify the three steps. A largenumber of orders, i.e., some hundred or even one thousand, haveto be transported every day in the transportation network foundin the real-world scenario. Because of the NP hardness of the re-searched problem, we apply a decomposition approach by dividingthe transportation network into different zones in the first step. Thedecomposition approach allows us to compute feasible solutions ofthe overall approach using a relatively short amount of time. Thisbehavior is because of the reduced size for each sub problem.

Therefore, the search space for heuristics is smaller. The relativelyshort resultant computing time is an advantage when replanningis often required in a real-world setting to deal with disturbances.Decomposition-based heuristics are popular for multi-depot typeVRPs (cf. Lian and Castelain, 2010). After the decomposition, a richVRP is associated with each zone. In addition, this approach allowsfor using zone-specific algorithms in a decentralized manner.

The second step is desirable because the assignment decisionstaken in Step 2 may lead to high costs for transportation with ex-press companies. These assignments can be corrected by the localsearch approach.

The algorithms used in Steps 1 and 2 are based on deterministicdata, i.e., the traveling times and the number of available vehiclesare deterministic. However, because of traffic jams or vehiclebreakdowns the planning results are executed in a dynamic andstochastic environment. Consequently, a rolling horizon approachis required that takes appropriate feedback from the base systemand base process into account. We apply discrete event simulationto represent the dynamic and stochastic base system and base pro-cess. This allows us especially to test different operational strate-gies for a longer time horizon. Because there is an overlap oforders across the different periods, static problem instances donot have the potential to allow for similar insights. Therefore, Step3 of the proposed methodology is highly desirable.

4.2. Decomposition heuristic and route construction

After all orders have been arrived in the evening and no furthertraveling of vehicles takes place, we assign each order that is at amanufacturer location and that has not already been transportedto its nearest distribution location and determine inter zone trans-portation plans for transporting the orders to the correspondingdistribution location. The nearest distribution location is deter-mined with respect to the location of the customer who is assignedto the order.

We assume that this inter zone transportation plan can be exe-cuted and each order will be transported to their nearest distribu-tion location overnight by an express company when the order iscurrently not at this location. As a consequence, each order is atits nearest distribution location at the beginning of the nextmorning.

A VRP is solved for each zone using the orders and the vehiclesthat are assumed to be available at the next morning. Some orderseventually cannot be included into the plan because the utilizationof the zone is too large or because of the violation of the end pointsof their time windows. Then these orders are removed from the in-ter zone transportation plan.

The updated inter zone transportation plans are executed. Theorders that are not part of the inter zone transportation plans be-cause of the violation of some constraints wait for another daywhen this is possible or have to send directly to their customersby an express company.

When the distribution locations open in the morning, we solvethe corresponding VRP for each distribution location again andexecute the corresponding transportation plans using the vehiclesin the zones. This step is necessary to take the stochastic behaviorof the logistics base system like the breakdowns of some vehicle,new orders that are arrived in the morning, or the lateness of interzone transports appropriately into account.

A VRP has to be solved for the orders in each distribution loca-tion as a result of the decomposition step. Note that the resultantsub problems still are NP hard because they include traveling sales-man type problems as special cases. Orders are assigned to vehiclesand then a route is computed for each vehicle to deliver these or-ders to the customers. While computing a solution for the VRP,the maximum traveling time, the capacity of the vehicles, and the


opening times of the distribution location have to be taken into ac-count. A tour is called a feasible assignment of orders to a vehicleand the selection of a route for the vehicle to travel to the corre-sponding customers. A tour plan TP consists of the set of tours T thatare formed to deliver the orders to customers. We use the notationOT :¼ {ojo 2 T} for the set of all orders of a single tour T.

A seed vehicle has to be selected to find a first order for anempty route for an available vehicle. Here, we choose the vehiclewith the earliest availability time as seed vehicle. Ties are brokenrandomly. In the next step, orders are selected from the set X offeasible orders based on the index described in Subsection 4.3,and they are iteratively assigned to the vehicle. An order is in Xif the constraints with respect to capacity, maximum travelingtime of the vehicle, and the right end point of the time windoware not violated when this order is scheduled next within the routefor the current period. The waiting time of an order at a customerlocation has to be added to the traveling time of the vehicle whenthe left end point of the time window is violated, i.e., when the or-der is too early at the customer location. Note that we obtain a tourplan and a set of orders that are directly sent to customers by anexpress company as a result for each zone.

The sequence in which the orders are inserted into the routedetermines the sequence in which the customers are visited bythe vehicle. If X = £, then the current vehicle returns to its homedepot and its availability time is updated appropriately. A newroute is started for the next available vehicle unless there are nomore unrouted orders left.

We continue by proposing two heuristics to deal with the subproblems. These heuristics are clearly influenced by standard tech-niques to solve rich VRP. However, existing techniques cannot beused directly because of the possibility to transport orders with ex-press companies. The first heuristic is a simple greedy heuristicthat includes a specific look-ahead index that allows for combingthe least distance heuristic and a slack-based heuristic to deal withthe time windows. By selecting an appropriate problem instance-based scaling parameter, the influence of the two parts can bechanged. The second heuristic is a standard ACO approach, how-ever, its heuristic information is based on the look-ahead indexused in the greedy heuristic (cf. Subsection 4.3 for details) andthe embedded local search heuristic exploits the possible transpor-tation by express companies. Note that we do not aim to propose anew algorithm for a rich VRP, but we are interested in combiningand extending existing approaches.

4.3. Greedy heuristic for the sub problems

We describe how we select the next order o from X. Orders areinserted into the route taking the distance from the location of thepreviously inserted order i to the destination of co and the timeslack between uo and the arrival time of o at its destination into ac-count. The following index is used to select the next order o:

IioðtÞ :¼ 1maxðtio; lo � tÞ e

�maxðuo�tio�t;0Þ=j�t; ð3Þ

where we introduce the following notation:

tio
time to travel from the customer of order i or from thedistribution location when o is the first order on the routeto the customer of order o,
t
current time, �t average time to travel from one customer to its
consecutive customer,
j scaling parameter, lo earliest delivery time.
The order o with the largest value for the index (3) is selectednext.

Only the traveled distance and the number of orders that arenot part of the tour plan are important for evaluating the costs atthis point. Orders that are not part of the tour plan are directlytransported by an expensive express transport to the correspond-ing customer. These orders cause the largest costs. The traveleddistance is only secondary. Therefore, we determine the costs asso-ciated with the tour plan tp by

cos ttp :¼ dtp þ 1ðjototalj � jotpjÞ; ð4Þ

where dtp is the traveled distance according to the tour plan, jototaljis the total number of orders that have to be delivered, jotpj is thenumber of orders that are scheduled to be delivered by tp, and final-ly 1 = 1000 kilometers is a penalty term. Each order that will not bedelivered by a vehicle is penalized by 1000 kilometers. Conse-quently, a tour plan with one more delivered order results in lowercosts than a tour plan that does not include this order but has a low-er traveled distance.

Note that index (3) is influenced by the Apparent Tardiness Cost(ATC) dispatching rule from manufacturing. It is well-known thatATC type rules lead to schedules with high on-time delivery perfor-mance (cf. Pinedo, 2008). Similar greedy look-ahead heuristics areproposed by Atkinson (1994) and by Ioannou et al. (2001). Thenotation GH is used as an abbreviation for the greedy heuristic inthe remainder of this paper.

4.4. ACO type approaches for the sub problems

The main idea of ACS consists in a set of artificial ants that con-struct solutions to a VRP. Each single ant starts by an empty routeand constructs routes according to the description in Section 4.2.The search for high-quality solutions of a VRP is coupled betweenthe different ants by artificial pheromone trails. The ants indirectlycommunicate by modifying the pheromone trails after the con-struction of a new solution. The solutions found by the ants canbe improved by local search techniques. The overall scheme of anACS type algorithm consists of the following steps (cf. Dorigo andSocha, 2007):

1. Set parameters and initialize pheromone trails.2. Construct new solutions of the VRP by creating new ants. The

ants take pheromone trail information into account during theirconstruction of a solution.

3. Improve the solutions found in Step 2 by local search.4. Update pheromone trails using the solutions obtained by the

ants from Steps 2 and 3.5. When the termination criterion is reached then stop, otherwise

go to Step 2 and start a new iteration.

In the remainder of this section, we describe Steps 2 and 4 inmore detail. Note that Steps 2–4 forms a single iteration of ACS.Several ants are created within a single iteration. We denote bygio the heuristic desirability to deliver order o immediately afterorder i. The gio values are typically derived by using appropriateproblem specific construction heuristics. In our experiments, weuse the index (3) to provide the heuristic information within theACS approach. We denote by sio(t) the pheromone intensity thatis associated with the selection of order o immediately after orderi. The parameter t is used to denote the current iteration of the ACSscheme.

We continue with describing a single iteration of the ACSscheme. We assume that order i is the last selected order and wewant to choose the successor order o. We create a sample of a


U[0,1] distributed random variable and denote the obtained valueby ~q. When ~q 6 q then the order o 2 X is selected that maximizesthe value of siogio. Note that in contrast to the literature, a moregeneral expression of the form sk

iogcio with k > 0 and c > 0 is not

necessary for the researched problem. Here, q 2 (0,1) is a givenparameter of the ACS scheme. When ~q > q then order o 2X is se-lected according to the following discrete distribution with theprobabilities pio given by

pio :¼siogioPh2Xsihgih

; if o 2 X

0; otherwise

(: ð5Þ

A local update of the pheromone intensities is performed by theexpression

sio :¼ ð1� qÞsio þ qs0; ð6Þ

when a new order o is inserted into a route. Here, the quantity s0 isthe initial pheromone value. Furthermore, the quantity q 2 (0,1] is aparameter of the ACS scheme that influences the balance betweenexploration and evaporation.

When all ants have computed a tour plan within a single itera-tion, a local search procedure can be applied to improve the solu-tion obtained by one or more ants. We use the following two-phaselocal search scheme:

Step 1: Determine the costs for inserting each single order that isassigned to an express company at each possible positionof the tour plan. An insertion that leads to an infeasibletour plan is not carried out. Insert the order at the positionthat leads to the largest improvements of the tour planwith respect to cost function (4).

Step 2: Calculate the costs of moving each single order from itscurrent to an alternative position in the same tour as wellas in the other tours. An alternative position is chosenbased on the following two neighborhood structures foreach tour T and a given positive integer n:
� MoveDist(o,n,T): It consists of the min (n,jOTj) orders for
each tour T where the locations of the customers ofthese orders have the smallest distance to the locationof the customer of o among all orders OT.

� MoveTime(o,n,T): It consists of the min (n,jOTj) orderswhere the difference between delivery dates jso � s~ojis the smallest among all orders ~o 2 OT with o – ~o.
Only feasible insertions are considered. Move the orderto the position that leads to the largest improvement ofthe tour plan.
Step 3: Steps 1 and 2 are repeated until they do not lead toimprovements of the tour plan or a prescribed maximumnumber of iterations imax is reached.

The computational effort of the two-phase local search ap-proach can be estimated by O(jOjnjTPjimax), where we denote byjTPj the number of tours included in TP. Because the number oftours is usually a multiple of the number of vehicles, we approxi-mately obtain an O(nimaxjOjjVj) effort. The proposed local searchheuristic is faster and leads to smaller costs for the problem re-searched in this paper than k-opt type moves (see Subsection4.5). Especially the insertion of orders that are currently not partof the tour plan leads to improved results if the number of avail-able vehicles is limited. However, the local search procedure istime-consuming compared to computing a solution by a singleant. Therefore, we apply this local search scheme only for the iter-ation-best ant to reduce the computational burden.

After the local search step, a global update of the pheromonevalues is performed. We apply the update equation

sioðtþ1Þ :¼ð1�aÞsioðtÞþa=cost�; if o is chosen directly after i

in the global�best solutionð1�aÞsioðtÞ; otherwise

8><>: :

ð7Þ

The quantity a 2 (0,1] is a parameter of the ACS scheme. Here, cost⁄

denotes the cost associated with the global-best solution found sofar by an ant. Alternatively, it is allowed for the iteration-best antto deposit additional pheromone. In our experiments, a mixed strat-egy is applied by using the iteration-best ant for global update afterfive consecutive iterations.

4.5. Parameter setting for GH and ACS and preliminary computationalresults

We start with describing how we select the various parametersin GH and ACS. The j value in index (3) is selected by a grid searchapproach, i.e., we determine tour plans using GH for allj = 1, . . . , 100 and then choose the tour plan with the smallest cost.We use the value of the scaling parameter j that we obtain fromGH for the heuristic desirability gio. Furthermore, we choose theparameter settings a = 0.1, q = 0.9, and q = 0.95 in our ACS ap-proach based on computational experiments on a small numberof problem instances in combination with a trial and error strategy.Finally, we initialize the pheromone intensities sio by choosing thereciprocal value of the costs obtained by GH. We use 50 ants for asingle iteration.

We perform some preliminary computational experiments toassess the performance of ACS using single depot problem in-stances. It turns out that ACS outperforms GH by 26% on averagein terms of the traveled distance. The proposed two-phase localsearch scheme clearly outperforms the 3-opt-based local searchstrategy. Furthermore, ACS is assessed using the Solomon bench-mark instances (cf. Solomon, 1987). It turns out that the traveleddistance by ACS is on average 4.5% higher and needs 0.5 more vehi-cles on average as the best known solution after a computing timeof 15 minutes. Because of the relative small computing time usedby ACS and the relatively small difference to the best known solu-tions for the Solomon instances we conclude that our ACS schemeperforms well. The Solomon instances are considered to get animpression of the performance of our heuristics.

Note that the proposed decomposition heuristic is appropriatefor the described problem because the distance between most ofthe distribution locations is large (see the description of the scenar-ios in Subsection 5.2). Therefore, an assignment to the nearest zoneis reasonable for most of the orders and Step 2 of the overall ap-proach described in Subsection 4.1 can be avoided for the situationthat is given by our problem data. An exchange of orders acrosszones does not improve the solution in many situations. We usedthe local search heuristic described in Subsection 4.4 for improvingthe entire solution obtained by the decomposition approach andACS for the researched transportation problem. However, it turnedout that the objective value calculated based on expression (1) canbe improved in this way by only 1–3% compared to the solutionwithout additional improvement phase using 50% more computa-tion time. In addition, a more sophisticated decomposition heuristicis assessed. In contrast to the previous decomposition strategy, or-ders are not assigned to their nearest delivery location if the corre-sponding customer is nearby the zone of the manufacturinglocation that belongs to the order. Consequently, this decomposi-tion heuristic has the potential to avoid transportations across dif-ferent zones. As a result of some computational experimentation,we found that a few transportations across zones are eliminated,but we did not observe that a full transport across different zonesis saved in any situation. Therefore, according to the form of the cost

Intermediate Distribution Center


function (2) the cost savings are small and the additional traveltime of own vehicles and the costs caused by late orders exceedthese savings.

Manufacturer

Customer

600km

Hamburg

Cologne Frankfurt

Berlin

Munich

Fig. 2. Logistics network in Germany.

5. Simulation-based performance assessment of the heuristics

In this section, we assess the performance of ACS in a rollinghorizon setting. Therefore, we describe an appropriate simulationframework in Subsection 5.1. The design of experiments used ispresented in Subsection 5.2. The results of simulation experimentsare shown and discussed in Subsection 5.3.

5.1. Framework for experimentation

A simulation framework is developed to assess the performanceof the algorithms for the researched problem in a dynamic and sto-chastic environment. This corresponds to Step 3 of the proposedmethodology presented in Subsection 4.1. Note that our algorithmsare based on deterministic input data. We apply the proposed heu-ristics in a rolling horizon setting. Stochastic events like traffic jamsor breakdowns of vehicles might occur that influence the perfor-mance of the tour plans when they are executed. When the num-ber of orders is large compared to the number of available vehiclesand the time windows are tight, an efficient heuristic leads often totour plans where the delivery dates of the orders are near to theright end point of the time window. This can lead to a significantdeterioration in performance of high-quality solutions when theyare executed in a stochastic environment.

The simulation framework consists of the simulation engineAutoSched AP 9 for modeling the transportation network and thevehicles and a blackboard-type data layer between tour planningcomponent and simulator (cf. Sprenger and Mönch, 2008). The datalayer stores current status related information about the transpor-tation network, like locations and distances between the locations,the set of vehicles, and the orders in the memory of the computer.The transportation network and the transportation process itselfare emulated by the simulation model and the simulation engine.The simulation framework takes the following stochastic behaviorof the transportation network and process into account:

� Vehicles might break down randomly for an entire day. The cur-rent status of all vehicles is known at the beginning of each day.� The traveling, loading, and unloading times are stochastic rather

than deterministic as assumed by the planning heuristics.

The transportation network and process treat the consequencesof the described uncertainty as follows:

� The vehicles travel back to their home depot when there is adanger that their maximum daily operating time will be vio-lated or the return time of the vehicles will be outside the dailyopening hours of the home depot. Orders that are not served tocustomers due to these earlier returns will be unloaded at thedelivery location.� When the arrival time of the vehicle is smaller than lo, then the

vehicles have to wait until the beginning of the time windowbefore servicing the customer.

5.2. Design of experiments

In our simulation experimentation, we consider scenarios thatare based on a real-world data set obtained from food manufactur-ers in Germany. Note that in contrast to the standard problems ofthe transportation planning domain (cf. Toth and Vigo, 2002)benchmark problem instances are not available in the literature.

In our scenarios, two manufacturers produce food. In addition,there are three intermediate distribution centers. Each manufac-turer owns vehicles at its distribution centers. The transportationnetwork is shown in Fig. 2.

The different number of vehicles per location that is assumed inthe scenarios is summarized in Table 1. Totally, we consider 20vehicles.

We generate problem instances according to the followingdescription. Each single problem instance is described by a set oforders with corresponding attributes and by a set of vehicles. Notethat we have to specify the arrival pattern of the orders over time.We start by describing the order generation scheme.

The ready date ro of an order o is set according to the expression

r0 � U½24k;10aþ 24k�; k 2 IN; ð8Þ

where we denote by a 2 (0,1] a parameter that controls how earlyin the morning the transportation orders are ready for transporta-tion. Incoming orders are concentrated in the morning. In ourexperiments, we simply set a = 0.5. Because we run the simulationfor 120 days, we use k = 0, . . . , 119 in expression (8).

The earliest delivery date lo and the latest possible delivery dateuo are chosen in a similar way. We obtain for order o:

lo � ro � U½dtmin þ dmoco ;dtmax � tsmin�; ð9Þ

and

uo � ro � tsmin � U½l0 � ro;dtmax � tsmin�; ð10Þ

where we denote by dtmin and dtmax the minimum and maximumtime relative to the order release date when a delivery has to takeplace. We denote by dmoco the time to travel between the distribution

Table 1Number of vehicles at the different locations.

Location Manufacturer 1 Manufacturer 2

Manufacturer 1 (Frankfurt) 5 /Manufacturer 2 (Hamburg) / 3Intermediate distribution center 1 (Cologne) 2 2Intermediate distribution center 2 (Berlin) 2 2Intermediate distribution center 3 (Munich) 2 2


location and the location of the customer of order o. A minimumlength of the time window is given by tsmin. We use tsmin = 2 hoursand dtmin = 24 hours. Notice that because of expression (9) and(10) the inequality

uo � lo P tsmin; ð11Þ

is valid. We consider only orders that have a unit volume.We continue with the description of the non-order related part

of the generation scheme. The maximum daily operating time otvof the vehicles is set to 10 hours. We assume the all the distribu-tion locations are open between 8 am and 6 pm, i.e., 10 hours perday. The loading time of orders consists of a fixed part that takes10 minutes and an additional variable part that is one minute perorder. The corresponding unloading times of orders are five min-utes as fixed component and one additional minute per order.The traveling time multiplier for direct transportation ve is set to 4.

In order to mimic the stochastic behavior of the transportationnetwork and process we consider random instances of the travel-ing time of the vehicles including loading and unloading, the dailynumber of arriving orders jbOj, and the number of non-availablevehicles per day jbV j. Furthermore, we study the cooperative andthe non-cooperative case. The used design of experiments is shownin Table 2.

Totally, we have to consider 32 scenarios for the cooperativeand 32 scenarios for the non-cooperative transportation planningapproach, respectively. Note that in case of d = 0.1 and p = 0.1 thetour planes are based on values for the traveling times and thenumber of available vehicles that are different from those thatare used during the execution of the tour planes. In contrast,c = 0.2 leads to orders that are taken into account during the deter-mination of the tour plans.

The simulation is performed for 120 days per scenario wherethe first ten days are ignored due to a warm-up phase. We takeall orders o into account where ro is between day 10 and 110 forgathering the statistics for the performance measures. We areinterested in the percentage of late orders, the traveled distanceof own vehicles, and the traveled distance and transported volumeof express company transports according to cost function (1). Weset x1 = 15 and x2 = 250 in expression (1) to model that the sec-ond and third type of transportation is much more expensive thanthe first type. Furthermore, we use x3 = 1. Because of the rollinghorizon approach, we have to run the transportation planning ap-

Table 2Design of experiments.

Factor Level

Transportation planning approach CooperatTraveling time of the vehicles including loading and unloading time d̂ � U½ð1�Daily number of arriving orders jbOj � U½ð1

number oDeterministic daily number of arriving orders jeOj ¼ 100Daily number of non-available vehicles jbV j � U½0Maximum time relative to the order release date where a delivery has to

take placedtmax 2 {4

Total number of combinations

proach in the morning and in the evening of each day. A maximumtime of five minutes is allowed to solve the corresponding trans-portation planning problem. One minute out of the five minutesis spent to obtain a initial solution using GH. GH terminates afterone minute or when the grid search for appropriate j values iscompleted within a smaller period of time. Due to the longer deliv-ery horizon it turns out that the best j is between 0 and 1000. As aconsequence, we set j = 1, . . . , 1000. The remaining four minutesare used to run ACS including the two-phase local search heuristicwith n = imax = 3. Note that usually small values of j lead to high-quality solutions. All simulations runs are performed on a 2 Giga-hertz Intel Core Duo computer using one core and 1.5 Gigabyteof its RAM. We use three independent replications for each simula-tion run and take average values for the performance measures.

5.3. Results of computational experiments

An overview about the results can be found in Table 3. Insteadof comparing all scenarios individually, the scenarios are groupedaccording to factor levels such as daily number of arriving orders,daily number of non-available vehicles, etc. For example, c = 0.2implies that all other factors have been varied, but the noise inthe daily arriving number of orders has been kept constant atc = 0.2. For a particular factor in Table 3 the value provided is theaverage value of the performance measure for three independentruns.

The cooperative scenario turns out to be more beneficial. Thenumber of orders where the delivery time violates the right endpoint of the time window is between 1% and 3%, whereas up to7% of the orders are too late in the non-cooperative scenario. Inaddition, the own vehicles have to travel around 20% more andthe vehicles for direct express transports 100% more. This resultsin additional costs and demonstrates the benefits for the manufac-turers if they cooperate and share their distribution fleets.

The longer time period for delivery, i.e., dtmax = 72 hour, savesabout 25% of direct transportation kilometers paying the price of10% more traveled kilometers by own vehicles. Due to the maingoal of saving expensive direct transports, the increasing numberof transports by own vehicles is acceptable and leads to lowercosts.

The number of direct transportation kilometers increasesquickly if the number of vehicles is not sufficient. This can be seen

Count

ive vs. non-cooperative 2

dÞd; ð1þ dÞd�with d 2 {0.0,0.1} and d is the deterministic traveling time 2

� cÞjeOj; ð1þ cÞjeOj� with c 2 {0.0,0.2} and jeOj is the deterministicf orders

2

and jeOj ¼ 200, equally distributed over the two manufacturer 2

;pjV j� where p 2 {0.0,0.1} 2

8 hours, 72 hour} 2

64

Table 3Computational results for different factor settings.

Compare Factor Traveled distance of ownvehicles in km

Traveled distance of theexpress companies in km(direct transport)

Percentage oflate orders (%)

Cost perorder

Cooperative dtmax = 48 627,953 712,109 2.57 164.75dtmax = 72 657,737 421,307 1.55 121.31c = 0.0 649,983 581,776 2.08 142.90c = 0.2 635,707 551,640 2.04 143.17

jbOj ¼ 100 549,996 173,032 0.96 121.58

jbOj ¼ 200 735,694 960,384 3.17 164.48

d = 0.0 642,938 554,501 2.02 141.52d = 0.1 642,751 578,914 2.10 144.54p = 0.0 661,757 549,352 2.17 142.81p = 0.1 623,933 584,063 1.96 143.25

Non-cooperative dtmax = 48 871,965 1,529,501 5.61 354.33dtmax = 72 960,031 1,187,711 5.03 312.86c = 0.0 919,087 1,356,443 5.21 329.99c = 0.2 912,909 1,360,768 5.43 337.20

jbOj ¼ 100 810,112 597,531 3.98 316.68

jbOj ¼ 200 958,012 1,985,930 6.21 326.51

d = 0.0 918,909 1,362,669 5.47 337.60d = 0.1 913,088 1,354,543 5.17 329.59p = 0.0 949,443 1,153,271 3.80 280.46p = 0.1 882,553 1,563,941 6.84 386.73


by increasing the number of orders per day from 100 to 200. As aconsequence, it is beneficial to run a sufficient number of ownvehicles at the distribution locations. In addition, it turns out thatthe number of necessary vehicles is lower in the cooperativescenario.

The number of orders that violate the right end point of the timewindow increases if vehicle breakdowns are considered. Increasedcosts of about 20% are the result in the non-cooperative scenario.Deviations in the traveling time between locations have only a sig-nificant effect if the number of orders is considerably large with re-spect the number of available vehicles.

Note that there is a difference between the cooperative and thenon-cooperative scenarios concerning the traveled distance of ex-press companies between distribution locations. This differenceis not explicitly shown in Table 3. Due to two additional deliverylinks in the cooperative scenarios, the traveled distance of the ex-press companies increase from around 258,000 to 365,000 kilome-ters. This information is not shown in Table 3, because there is only

Table 4Computational results for the deterministic and the stochastic setting.

Deterministic vs.stochastic

jbOj dtmax Traveled disown vehicle

Cooperative Stochastic 100 48 523,604100 72 542,947200 48 695,157200 72 730,882Average 623,147

Deterministic 100 48 558,515100 72 573,669200 48 750,475200 72 794,889Average 669,387

Non-cooperative Stochastic 100 48 744,133100 72 831,332200 48 916,692200 72 1,006,016Average 874,543

Deterministic 100 48 800,027100 72 865,898200 48 1,019,180200 72 1,131,542Average 954,162

a difference between the cooperative and the non-cooperative sce-narios, but not among the scenarios in the cooperative and thenon-cooperative case itself. However, this information is implicitlycontained in the average cost per order. This value is calculatedbased on expression (1).

In Table 4, we compare purely deterministic scenarios and sce-narios where stochastic events happen. The traveled distance ofthe express companies increase for direct transports, and more or-ders are late in the stochastic setting. The traveled distance of di-rect transports increase by 20–30% especially in scenarios with asmall number of orders, i.e., jbOj ¼ 100. This is another justificationfor the need to carry out a simulation-based performance assess-ment of the algorithms in a rolling horizon setting, i.e. for Step 3of our overall methodology. Moreover, we see from the results thatthe impact of the stochastic base system is larger in the non-coop-erative setting.

Next, we are interested in assessing the performance in a sce-nario where the heuristics can react faster to newly arrived orders

tance fors in km



Cost perorder

256,327 1.32 145.12129,315 0.88 111.211,159,094 3.07 175.68801,772 2.45 140.76586,627 1.93 143.19222,868 1.09 134.2798,020 0.78 101.491,153,354 3.60 182.93683,990 2.21 131.58539,558 1.92 137.57

816,243 5.28 385.29577,630 5.00 344.002,620,213 8.15 408.152,166,710 7.84 380.601,545,199 6.57 379.51604,873 2.97 291.11378,133 2.71 246.381,978,039 4.39 293.351,620,587 4.71 279.181,145,408 3.69 277.51

Table 5Computational results for tight time windows and different planning frequencies.

Compare Traveled distance ofown vehicles in km



Cost perorder

Two calculations Cooperative Deterministic twmax = 24 hour 728,506 644,030 3.53 184.67twmax = 3 hour 731,587 675,763 3.84 190.19

Stochastic twmax = 24 hour 655,605 795,322 4.43 208.61twmax = 3 hour 663,837 842,000 4.78 217.20

Non-cooperative Deterministic twmax = 24 hour 961,185 1,627,235 8.06 424.46twmax = 3 hour 950,787 1,761,885 9.54 449.40

Stochastic twmax = 24 hour 864,137 1,982,579 10.83 492.23twmax = 3 hour 855,861 2,091,172 12.44 524.26

Three calculations Cooperative Deterministic twmax = 24 hour 727,804 634,673 3.81 185.57twmax = 3 hour 732,874 707,110 4.02 194.74

Stochastic twmax = 24 hour 667,985 767,957 4.47 208.13twmax = 3 hour 668,756 857,721 5.06 220.88

Non-cooperative Deterministic twmax = 24 hour 971,674 1,599,180 7.73 401.58twmax = 3 hour 962,345 1,764,835 9.02 427.96

Stochastic twmax = 24 hour 875,718 1,957,849 10.42 480.49twmax = 3 hour 864,670 2,039,869 11.54 503.34


and stochastic events. Therefore, we compute new tour plans at1 pm when half of the possible delivery duration of the currentday has been reached. We can compute a new tour plan for theremaining delivery period at 1 pm, taking all orders that are avail-able at this point of time and the estimated completion date of thecurrently executed tours as possible start time for new tours intoaccount. We refer to this setting as scenarios with three computa-tions, i.e., in the evening, morning, and at noon. We reduce themaximum number of orders per vehicle from 15 to six to increasethe potential number of daily tours per vehicle. A smaller numberof orders per tour lead often to shorter tours. Consequently, a morefrequent replanning has the potential to offer some advantage inthis situation.

In addition, we choose the size of the time windows by the set-ting dtmin = 24 hour to determine the left end points of the timewindow according to expression (9). In order to set the right endpoint uo, we replace expression (10) by

uo � ro � tsmin � U½l0 � ro; lo � ro þ twmax � tsmin�: ð12Þ

Therefore, we have

tsmin 6 uo � lo 6 twmax; ð13Þ

i.e., twmax is the maximum length of the time window. In the sce-narios with tight time windows, we set tsmin = 2 hour andtwmax = 3 hour, i.e., the time windows have a length between twoand three hours. We set jbOj ¼ 150 to avoid increasing the numberof vehicles. Because the new scenario is different from that scenariodescribed in Table 2, we present for comparison purposes also re-sults for twmax = 24 hour. As a result, we are able to study the effectof reducing the size of the time windows. The levels of the factorsthat influence the stochastic behavior of the transportation networkare selected according to Table 2. The computational results of thenew scenarios are shown in Table 5. The best results in comparisonof two and three computations are bold-marked.

The results show again that the cooperative scenario outper-forms the non-cooperative one. Time windows of reduced size leadto increased lateness of the orders and the influence of stochasticevents is higher and results in increased travel time and largercosts. The additional computations at the mid of the day have a po-sitive impact in the non-cooperative setting. In the cooperative set-ting, we observe slightly larger costs. This coincides with ourobservation that the effect of a stochastic base system is muchstronger in case of the non-cooperative setting. It is clear that amore frequent feedback from the base system leads to a better per-formance of the executed tour plans in this situation.

6. Conclusions and future research

In this paper, we discussed a methodology for a cooperativetransportation planning problem that is motivated by a real-worldproblem found in the German food industry. Because the researchedproblem is NP hard, we looked for appropriate heuristics. Wedecomposed the overall problem into a set of smaller rich VRPs thatare solved by ACS. ACS type heuristics are appropriate because theyare able to deal appropriately with the large number of differentconstraints. We applied the proposed heuristics in a rolling horizonsetting using discrete event simulation to represent the dynamicand stochastic logistics system. It turned out that the cooperativestrategy clearly outperforms the non-cooperative algorithms.

There are several directions for future research. The first direc-tion is related to the decomposition scheme. So far, we used in ourexperimentation a simple decomposition. However, we believethat we can improve the performance of our heuristics when weuse a more dynamic decomposition scheme (cf. Bouhafs et al.,2006 for such a heuristic for a capacitated location routing prob-lem) that allows for an iterative improvement of the decomposi-tion after the resulting sub problems are solved. This is especiallytrue when many orders have a similar distance to different distri-bution locations. Note that this situation is not covered by theproblem instances used so far. Finally, in the real-world setting,backhauls are quite important. It seems to be interesting to con-sider approaches that treat backhauls within a VRP in the situationresearched in this paper. However, carrying out all the necessarydetails is part of future research.

Acknowledgements

The authors gratefully acknowledge the support from the Ger-man Federal Ministry of Economics (BMWi) within the project iCo-Trans (Grant 19G7022A). Furthermore, they would like to thankCan Dündar Karaduman for his valuable programming efforts.

References

Altiparmak, F., Karaoglan, I., 2008. An adaptive tabu-simulated annealing forconcave cost transportation problems. Journal of the Operational ResearchSociety 59, 331–341.

Atkinson, J.B., 1994. A greedy look-ahead heuristic for combinatorial optimization:an application to vehicle scheduling with time windows. Journal of theOperational Research Society 45 (6), 673–684.

Bouhafs, L., Hajjam, A., Koudam, A., 2006. A combination of simulated annealing andant colony system for the capacitated location-routing problem. In: ProceedingsInternational Conference on Knowledge-Based Intelligent Information andEngineering Systems, LNCS, vol. 4251, pp. 409–416.


Bräysy, O., Gendreau, M., 2005a. Vehicle routing problem with time windows, partI: route construction and local search algorithms. Transportation Science 39 (1),104–118.

Bräysy, O., Gendreau, M., 2005b. Vehicle routing problem with time windows, partII: metaheuristics. Transportation Science 39 (1), 119–139.

Bullnheimer, B., Hartl, R.F., Strauss, C., 1997. Applying the ant system to the vehiclerouting problem. In: Proceedings 2nd International Conference onMetaheuristics, pp. 285–296.

Dai, B., Chen, H., 2009. Mathematical model and solution approach for collaborativelogistics in less than truckload (LTL) transportation. Proceedings ComputersIndustrial Engineering 2009 (CIE 2009), 767–772.

Doerner, K.F., Hartl, R.F., Benker, S., Lucka, M., 2006. Ant colony system for a VRPwith multiple time windows and multiple visits. Parallel Processing Letters 16(3), 351–369.

Dorigo, M., Socha, K., 2007. Ant Colony Optimization. Handbook on ApproximationAlgorithms and Metaheuristics. Chapman & Hall, New York, pp. 1–14 (Chapter26).

Ebben, M.J.R., van der Heijden, M.C., van Harten, A., 2004. Dynamic transportscheduling under multiple resource constraints. European Journal ofOperational Research 167 (2), 320–335.

Favaretto, D., Moretti, E., Pellegrini, P., 2007. Ant colony system for a VRP withmultiple time windows and multiple visits. Journal of InterdisciplinaryMathematics 10, 263–284.

Gajpal, Y., Abad, P., 2009. An ant colony system (ACS) for vehicle routing problemwith simultaneous delivery and pickup. Computers & Operations Research 36(12), 3215–3223.

Gambardella, L.M., Taillard, E., Agazzi, G., 1999. MACS-VRPTW: a multiple antcolony system for vehicle routing problems with time windows. In: Corne, D.,Dorigo, M., Glover, F. (Eds.), New Ideas in Optimization. McGraw-Hill, London,pp. 63–76.

Ioannou, G., Kritikos, M., Prastacos, G., 2001. A greedy look-ahead heuristic for thevehicle routing problem with time windows. Journal of the OperationalResearch Society 52 (5), 523–537.

Irnich, S., 2008. A unified modeling and solution framework for vehicle routing andlocal-search-based heuristics. INFORMS Journal of Computing 20 (2), 270–287.

Kopfer, H., Wang, X., 2009. Combining vehicle routing with forwarding: extension ofthe vehicle routing problem by different types of sub-contraction. Journal of theKorean Institute of Industrial Engineers 35 (1), 1–14.

Lee, L.H., Tan, K.C., Ou, K., Chew, Y.H., 2003. Vehicle capacity planning system: a casestudy on vehicle routing problem with time windows. IEEE Transactions onSystems, Man and Cybernetics, Part A: Systems and Humans 33 (2), 169–178.

Lee, C.-Y., Lee, Z.-Y., Lin, S.-W., Ying, K.-C., 2010. An enhanced ant colonyoptimization (EACO) applied to capacitated vehicle routing problem. AppliedIntelligence 32, 88–95.

Lin, C.K.Y., 2008. A cooperative strategy for a vehicle routing problem with pickupand delivery time windows. Computers & Industrial Engineering 55, 766–782.

Lian, L., Castelain, E., 2010. A decomposition approach to solve a general deliveryproblem. Engineering Letters 18 (1).

Montemanni, R., Gambardella, L.M., Rizzola, A.E., Donati, A.V., 2003. A newalgorithm for a dynamic vehicle routing problem based on ant colony system.Second International Workshop on Freight Transportation and Logistics, 27–30.

Pankratz, G., 2004. Dynamische speditionelle Transportdisposition unterbesonderer Berücksichtigung der Fremdvergabe. Logistik Management 6 (1),42–57.

Pankratz, G., 2005. A grouping genetic algorithm for the pickup and deliveryproblem with time windows. OR Spectrum 27, 21–41.

Paraskevopoulos, D.C., Repoussis, P.P., Tarantilis, C.D., Ioannou, G., Prastacos, G.P.,2008. A reactive variable neighborhood tabu search for the heterogeneous fleetvehicle routing problem with time windows. Journal of Heuristics 14, 425–455.

Pellegrini, P., Favaretto, D., Moretti, E., 2007. Multiple ant colony optimization for arich vehicle routing problem: a case study. In: Proceedings InternationalConference on Knowledge-Based Intelligent Information and EngineeringSystems, LNCS, vol. 4693, pp. 627–634.

Pinedo, M., 2008. Scheduling: Theory, Algorithms, and Systems, Third Edition.Prentice Hall.

Shang, J.S., Cuff, C.K., 1996. Multicriteria pickup and delivery problem with transferopportunity. Computers & Industrial Engineering 30 (4), 631–645.

Simroth, A., Baumbach, B., 2007. Eine Planungs- und Experimentierumgebung fürdie Tourenplanung unter Unsicherheit. Technical Report. Fraunhofer Institut fürVerkehrs- und Infrastruktursysteme Dresden.

Solomon, M.M., 1987. Algorithms for the vehicle routing and scheduling problemswith time window constraints. Operations Research 35 (2), 254–265.

Sprenger, R., Mönch, L., 2008. A simulation framework for assessing theperformance of cooperative transportation planning algorithms. In:Proceedings of the 2008 Winter Simulation Conference, pp. 2769–2776.

Sprenger, R., Mönch, L., 2009. An ant colony optimization approach to solvecooperative transportation planning problems. In: Proceedings of the 2009Winter Simulation Conference, pp. 2488-2495.

Tan, X., Luo, X., Chen, W.N., Zhang, J., 2005. Ant colony system for optimizing vehiclerouting problem with time windows. In: Proceedings of the InternationalConference on Computational Intelligence for Modelling, Control andAutomation, pp. 209–214.

Toth, P., Vigo D., (Eds.), 2002. The Vehicle Routing Problem. SIAM Monographs onDiscrete Mathematics and Applications. Philadelphia.

Zäpfel, G., Bögl, M., 2008. Multi-period vehicle routing and crew scheduling withoutsourcing options. International Journal of Production Economics 113 (2),980–996.

a methodology to solve large-scale cooperative transportation planning problems

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