rsearch agenda cross dock

ARTICLE IN PRESS

Omega ] (]]]]) ]]]–]]]

Contents lists available at ScienceDirect

Omega

0305-04

doi:10.1

� Corr

E-m

malte.fl

Pleas(200

journal homepage: www.elsevier.com/locate/omega

Review

Cross dock scheduling: Classification, literature review and research agenda

Nils Boysen �, Malte Fliedner

Friedrich-Schiller-Universitat Jena, Lehrstuhl fur ABWL/Operations Management, Carl-Zeiß-Str. 3, D-07743 Jena, Germany

a r t i c l e i n f o

Article history:

Received 16 March 2009

Accepted 28 October 2009

This manuscript was processed by

Associate Editor Pesch

next destination in the distribution chain. Accordingly, a cross dock is a consolidation point in a

distribution network, where multiple smaller shipments can be merged to full truck loads in order to

Keywords:

Logistics

Cross docking

Scheduling

Classification

83/$ - see front matter & 2009 Elsevier Ltd. A

016/j.omega.2009.10.008

esponding author.

ail addresses: [email protected] (N. Boy

[email protected] (M. Fliedner).

e cite this article as: Boysen N, Flied9), doi:10.1016/j.omega.2009.10.008

a b s t r a c t

At cross docking terminals incoming deliveries of inbound trucks are unloaded, sorted, moved across

the dock and finally loaded onto outbound trucks, which immediately leave the terminal towards their

realize economies in transportation. In this context, the truck scheduling problem, which decides on the

succession of truck processing at the dock doors, is especially important to ensure a rapid turnover and

on-time deliveries. Due to its high real-world significance, several truck scheduling procedures have

been introduced during recent years, which all treat specific cross dock settings. In order to structure

and promote scientific progress, this paper introduces a classification of deterministic truck scheduling.

With the help of this classification, existing literature is reviewed and future research needs are

identified. Moreover, we represent a yet unexplored class of truck scheduling problems which is highly

relevant in real-world distribution networks.

& 2009 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. Scope of review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3. Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1. Door environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2. Operational characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.3. Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4. Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

5. Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5.1. Research needs in relation to interdependent planning problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5.2. Implementation of real world truck scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

6. Cross docking with fixed outbound schedules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1. Introduction

A cross docking terminal is an intermediate node in adistribution network which is exclusively dedicated to thetransshipment of truck loads. In contrast to traditional ware-houses, a cross dock carries no or at least a considerably reducedamount of stock. Whenever an incoming truck arrives at the yard

ll rights reserved.

sen),

ner M. Cross dock scheduli

of a cross dock, it is assigned to a dock door, where inbound loadsare unloaded and scanned to determine their intended destina-tions. The loads are then sorted, moved across the dock andloaded onto outbound trucks for an immediate delivery elsewherein the distribution system. Fig. 1 gives a schematic representationof a cross docking terminal.

The primary purpose of a cross dock is to enable a consolida-tion of differently sized shipments with the same destination tofull truck loads, so that economies in transportation costs can berealized [1]. This advantage makes cross docking an importantlogistics strategy receiving increased attention in today’s globa-lized competition with its increasing volume of transported

ng: Classification, literature review and research agenda. Omega

www.elsevier.com/ome

dx.doi.org/10.1016/j.omega.2009.10.008

mailto:[email protected]

mailto:[email protected]

dx.doi.org/10.1016/j.omega.2009.10.008

ARTICLE IN PRESS

Fig. 1. Schematic representation of a cross docking terminal.

N. Boysen, M. Fliedner / Omega ] (]]]]) ]]]–]]]2

goods. Success stories on cross docking which resulted inconsiderable competitive advantages are reported for severalindustries with high proportions of distribution cost, such as retailchains (Wal Mart [2]), mailing companies (UPS [3]), automobilemanufacturers (Toyota [4]) and less-than-truckload logisticsproviders [5,6].

In contrast to traditional point-to-point deliveries, an addi-tional transshipment of goods at the cross docking terminal slowsdown the distribution process and generates a significant amountof double handling. Consequently, efficient transshipment pro-cesses are required where inbound and outbound truckloads aresynchronized, so that intermediate storage inside the terminal iskept low and on-time deliveries are ensured.

For this purpose, several scheduling procedures have beenintroduced in recent years, which aim at solving the so calledtruck scheduling problem. This problem decides on the successionof inbound and outbound trucks at a given set of dock doors of theterminal. On the basis of the truck schedule, each inbound andoutbound truck arriving at the yard is assigned to a specific dockdoor where shipments are processed. Obviously, this elementaryproblem repeatedly arises during the daily cross dock operationsand has vital influence on a rapid transshipment processes.

Different organizational and technical implementations lead toa large variety of possible truck scheduling problems in real-world settings. As cross docking is a comparatively new logisticsstrategy, there is not yet a massive body of academic literature onthis subject. In fact, no dedicated research on the short-term truckscheduling problem was published before 2005 (see [7]). Due tothe immense practical importance, there has been a considerableamount of follow-up research in the meantime, however, and westrongly assume that this trend continues in the future. On theone hand, this shows that we are in a formidable position tostructure the field in an early stage of exploration, so that futureresearch can be more easily coordinated. On the other hand, thismeans that the classification scheme needs to be easily adaptableand extendable, so that problem settings which have not yet beendiscussed can be readily considered. We therefore base ourclassification of deterministic truck scheduling problems on thevery successful and widely accepted tupel notation for machinescheduling [8], provide a concise review on existing solutionprocedures and use the insights to identify important fields ofinterest for future research.

The remainder of the paper is organized as follows. Section 2defines the scope of this review by characterizing the truckscheduling problem and establishing the relationship to inter-dependent decision problems. In Section 3 the truck scheduling

Please cite this article as: Boysen N, Fliedner M. Cross dock scheduli(2009), doi:10.1016/j.omega.2009.10.008

classification is presented which is employed to review existingoptimization models and solution procedures in Section 4. InSections 5 and 6, future research needs are specified. In particular,a yet unexplored class of truck scheduling problems is introduced,which is highly relevant in real-world cross docking applications,before Section 7 concludes the paper.

2. Scope of review

In general, scheduling problems deal with the allocation ofresources over time to perform a set of tasks being part of someprocess (e.g., [9, p. 1]). In the special case of truck scheduling, theprocess of transshipment can be subdivided into the tasks ofunloading inbound trucks and loading outbound trucks, which aretypically separated by a time lag for material handling inside theterminal, i.e., for scanning, sorting and moving shipments acrossthe dock. These two tasks are to be processed by the resources‘‘dock doors’’, which can process one truck at a time and areassumed to be sufficiently equipped with loading equipment(e.g., hand stackers or fork lifts) and workers. Typically, truckscheduling uses a time related objective function in order toevaluate a given solution. As with other (operational) schedulingproblems, cost arising as a consequence of task processing, e.g.,delayed deliveries influencing customer satisfaction, are hard toquantify accurately, so that a time related surrogate objectiveoften turns out to be the better (operational) choice. To conclude,a dispatcher of a cross docking terminal who seeks to solve a truckscheduling problem faces two interrelated decisions bound tosome (time related) objective function: where and when the trucksshould be processed at the dock doors of the terminal.

This (positive) definition of truck scheduling is now amendedwith a (negative) demarcation from related (and possiblyinterdependent) decision problems. The decision problems to besolved during the life cycle of a cross docking terminal orderedfrom strategic to operational are as follows:

(i)

ng: C

Location of cross docking terminal(s).
(ii) Layout of the terminal.
(iii)
Assignment of destinations to dock doors. (iv) Vehicle routing. (v) Truck Scheduling.
(vi)
Resource scheduling inside the terminal. (vii) (Un-)Packing loads into (from) trucks.
The strategic problem (i) of locating a single cross dock (or someother kind of intermediate warehouse) or a complete distributionnetwork consisting of multiple cross docks is vividly discussed inscientific literature. A good starting point into location theoryinvestigating intermediate nodes in a network of sites are thereviews on the hub location problem provided, e.g., by Campbell[10], Klose and Drexl [11] or Chen [12]. In relation to the locationproblem, truck scheduling merely considers an isolated terminalwith a given location.

The layout problem (ii) of a cross dock is investigated byBartholdi and Gue [13]. Here, the number of dock doors and theshape of the terminal building (e.g., I, T or X-shaped) are to bedetermined. For truck scheduling, we presuppose a giventerminal, so that the number of dock doors and their placementalong the perimeter of the terminal are known. Consequently, thedistance between any pair of doors is given, so that the time lagfor material handling (at least the load-independent part)between any pair of doors can be anticipated accurately.

Typically, the truck scheduling problem presupposes that theassignment decision of trucks (and the destinations they serve) todock doors is part of the short-term problem, so that each door

lassification, literature review and research agenda. Omega

dx.doi.org/10.1016/j.omega.2009.10.008

ARTICLE IN PRESS

N. Boysen, M. Fliedner / Omega ] (]]]]) ]]]–]]] 3

may serve multiple destinations in varying succession per daydepending on the actual truck schedule. However, the assignmentof destinations to dock doors (problem (iii)) can also be executedon a mid-term horizon, with the result of each dock doorexclusively serving a specific inbound or outbound destinationfor a longer period of time (e.g., a month). On the one hand, such afixed assignment eases the allocation of shipments to trucks, sinceworkers can ‘‘learn’’ the topology of the terminal and respectiveinformation systems become superfluous. On the other hand, afixed assignment of doors to destinations restricts the degrees offreedom for short-term truck scheduling, because peak loads forsingle destinations cannot be absorbed by additional dock doors.Consequently, such a fixed assignment seems especially suited forsteady commodity flows with a reliable distribution amonginbound and outbound destinations. Tsui and Chang [14,15] werethe first to tackle the mid-term problem of assigning doors todestinations. On the basis of a representative distribution ofshipments among related sites they solve the problem as aquadratic assignment problem, which minimizes the shipmentflows between doors. Other contributions for this problem arefrom Gue [5], Bartholdi and Gue [16], Bermudez and Cole [17], Ohet al. [18] as well as Bozer and Carlo [19]. If the decision ofassigning doors to destinations is solved at an early (mid-term)stage, the short-term truck scheduling problem reduces tosequencing all trucks of equal destination at the respective dockdoor. However, in either case there remains a short-term truckscheduling problem. Typically, we assume that inbound andoutbound destinations are not previously fixed, so that doorassignment is part of the truck scheduling problem.

Obviously, truck scheduling is also closely related to inboundand outbound vehicle routing (problem (iv)). On the inbound side,the vehicle routing schedule establishes the arrival times of trucksat the cross dock (see [20–22]). The estimated times can bedirectly taken up as inbound truck specific arrival times in truckscheduling. On the outbound side, succeeding vehicle routingspossibly set boundaries on the earliest and latest departure timeof outbound trucks. Obviously, there are some arguments againstplanning (inbound and outbound) vehicle routing and truckscheduling simultaneously. Diverging time frames and theresulting complexity of a monolithic optimization model questionsuch a simultaneous approach. We will follow these argumentsand investigate an isolated truck scheduling. However, vehiclerouting adds considerable degrees of freedom (by varying arrivaland departure times of trucks), so that considerable improve-ments of the overall planning task might occur [21]. Thus, itseems a promising field for future research to integrate bothplanning tasks (see Section 5).

For a given truck schedule, resource scheduling inside aterminal (problem (vi)), i.e., scanning, sorting and movingshipments across the dock, is a complex scheduling problem initself, since multiple resources need to be coordinated. Li et al. [23]as well as Alvarez-Perez et al. [24] model these tasks as a machinescheduling problem and present different meta-heuristics for itssolution. Truck scheduling is heavily interdependent with thisproblem, because the actual time lag between each inbound andoutbound task is the result of a detailed resource scheduling.Consequently, both planning tasks could be solved in a simulta-neous manner. However, existing research does not advocate suchan approach, because this would require to schedule each workerin detail, which would in turn necessitate a respective informationsystem and limits workers’ flexibility to react on unforeseenevents. Thus, average handling times, e.g., determined fromhistorical data, should capture this relation with sufficient precise-ness, so that in the subsequent discussion, we assume given fixedtime lags between inbound and outbound tasks, which onlydepend on the pair of doors between which the shipment is moved.


Finally, the packing of shipments inside trailers (problem (vii),see, e.g., [25]) also influences task times for truck processing andhandling times inside the dock. However, it does not seem usefulto interrelate packing decisions with truck scheduling, becausethe packing of inbound trucks is usually not known at the crossdock prior to opening the respective trailer. Furthermore,integrating packing aspects at the outbound side would alsorequire to integrate vehicle routing, which determines thesequence of customer visits and, thus, the needed arrangementof shipments inside a trailer. This would, however, result in a verycomplex centralized planning approach. Consequently, we as-sume that the influence of packing times is negligible and alreadyincluded in the transportation time lag.

Along with the (positive) definition of truck scheduling this(negative) separation from related decision problems definestruck scheduling and, thus, the scope of our review. The nextsection presents a classification, which characterizes this schedul-ing task.

3. Classification

Classifications of complex and versatile optimization problemshave proved very successful to concisely identify and describe aspecific optimization problem, so that the coordination ofscientific efforts is eased considerably. The most successful andwidely accepted classification schemes basing on a so-called tupelnotation are dedicated to machine scheduling [8] and queuingsystems [26]. Other tupel notations which successfully helpedstructuring complex research fields are, e.g., provided by Bruckeret al. [27] for project scheduling, Dyckhoff [28] for cutting andpacking, Boysen et al. [29,30] for assembly line balancing andsequencing, respectively. In a tupel notation respective objectivesand operational characteristics are referenced by a symbolicnotation, so that in spite of the multitude of possible propertiesof a planning task, a particular model can be briefly describedby a tupel.

Cross dock scheduling is closely related to traditional machinescheduling. Whenever possible we therefore take over theattributes of the Graham-notation. However, cross-dock schedul-ing bears some peculiarities, which cannot be directly denomi-nated with the classical machine scheduling notation. Forinstance, in a cross docking terminal incoming shipments arrivingon inbound trucks might contain multiple product units, whichare not preassigned to a specific truck but may satisfy the demandof multiple outbound trucks for the respective product, so that anassignment of product units to outbound trucks might be anadditional decision task. The classical Graham-notation has nocounterpart for these additional elements of cross docking. Thus,conventional machine scheduling attributes are to be augmentedby special cross docking attributes which in combination formthe truck scheduling classification. The classification scheme isstructured as follows:

�

ng:

Any truck scheduling problem will at least consist of threebasic elements: door (processor) environment, operationalcharacteristics and an objective to be followed. Accordingly,the presented classification will be based on those threeelements which are noted as tuple ½ajbjg�, where a is doorenvironment, b the operational characteristics and g theobjectives.
� One major advantage of the tuple-notation is that any default
value, represented by the symbol 3, can be skipped when atuple is specified. In the following notation, the symbol �always indicates that for the respective attribute the alter-native values (except for 3) do not exclude each other and can

Classification, literature review and research agenda. Omega

dx.doi.org/10.1016/j.omega.2009.10.008

ARTICLE IN PRESS


be combined arbitrarily. As all attribute values are chosen suchthat they are unique, it is not necessary to specify the attributedesignators within the tuples.

3.1. Door environment

The jobs to be executed during cross docking are inbound(unloading) and outbound (loading) operations for which pro-cessors are required. These processors are the doors of the dock.The door environment a of a cross docking terminal can berepresented by the two attributes a1 and a2:

Service mode a1AfE;M;EM;Gg: The service mode of a crossdocking terminal influences the degrees of freedom in assigninginbound and outbound trucks to dock doors.

a1 ¼ E

Please cite(2009), do

Each dock door is either exclusively dedicated toinbound or outbound operations. Such an exclusive

mode of service is a widely spread guideline in real-world terminals. Typically, to ease product flows andsupervision one side of the terminal is dedicated toinbound and the other to outbound operations.

a1 ¼M
On the other hand, also an intermixed sequence ofinbound and outbound trucks to be processed per dockdoor can be allowed, because technical restrictions fora separation of inbound and outbound trucks do notexist. We label this service mode as the mixed mode.
a1 ¼ EM
Additionally, both service modes can be applied inparallel, which means that a subset of doors isoperated in exclusive service mode and the other ina mixed mode of service.
a1 ¼G
Finally, the assignment of trucks to dock doors canalso be solved in a mid-term horizon, so that fixedassignments between doors and destinations exist(see Section 2). In this case, the door assignment ofeach truck is given by each trucks’ destination.Consequently, truck scheduling reduces to a sequen-cing problem of a given truck set at each door.
Note that the case a1 ¼ E is closely related to a flow shopsystem, where inbound and outbound doors correspond to thefirst and second production stage, respectively. And further notethat the case a1 ¼M is related to a processor environment withidentical parallel processors. However, as was mentioned before,the special cross dock setting that products arriving on theinbound side might be variably split among multiple outboundtrucks cannot be directly covered by the Graham-notation. Thus,we prefer to highlight these peculiarities with novel attributes.

Number of dock doors a2Af3; kg: Typically, a cross dock consistsof multiple dock doors. Gue [5] reports on a terminal containingmore than 500 doors, whereas the typical number ranges between40 and 150. Consequently, it might be valuable to further specifythe number of dock doors.

a2 ¼ 3
In the real-world, the number of dock doors varies fromterminals to terminal. In the default case the number ofdoors may differ, too, so that algorithms dedicated to thiscase can solve truck scheduling problems having anarbitrary number of dock doors.
a2 ¼ k
On the other hand, the number of dock doors can berestricted to a given number k, where k can be any positiveinteger. Especially valuable, e.g., for bound computationand complexity issues, might be the minimum number ofdock doors. This minimum number amounts to a2 ¼ 1 and2 depending on whether a mixed mode of service (a1 ¼M)or an exclusive (a1 ¼ E) one is employed, respectively.
this article as: Boysen N, Fliedner M. Cross dock schedulii:10.1016/j.omega.2009.10.008

3.2. Operational characteristics

The operational characteristics influencing the structure oftruck scheduling can be classified by attribute set b, whichcontains nine different attributes (b1 to b9).

Preemption b1Af3; pmtng: Preemption in the context of crossdocking means that loading or unloading a truck is interrupted,the half-full trailer is removed from the dock, and replaced byanother one. Later on, the unfinished trailer has to revisit theterminal to finish processing.

b1 ¼ 3

ng: Classifi

No preemption is allowed, so that a once docked traileris completely processed.

b1 ¼ pmtn
Preemption of truck processing is allowed.
Arrival times b2Af3; rjg: Trucks are either already waiting onthe yard and, thus, readily available to be called up or arrive afterthe start of the schedule, so that they may only be processedafter their truck specific arrival time.

b2 ¼ 3
All arrival times are zero. b2 ¼ rj Arrival times differ per truck.
Processing time b3Af3; pj ¼ p; prpjrpg: The service time(or processing time) pj of a truck j comprises the whole timespan to (un-)load its products. Note that we (and existingresearch) only deal with deterministic scheduling. Thus, weassume certainty about product loads arriving at the terminal,so that service times can be estimated upfront and are givenparameters of a truck scheduling model. Furthermore, we alsoprescind from predetermined service intervals like they areproposed by Miao et al. [31]. Such an assumption seemssomewhat artificial, because it requires a rejection of trucks (lostshipments) whose given service windows are missed even if onlyby a few seconds. A real world justification for such strict timewindows is not apparent. However, our classification can be easilyextended at this point to account for both peculiarities. Hence, wedistinguish processing time in analogy to machine schedulingas follows:

b3 ¼ 3

cation

Service times may vary from truck to truck, sothat arbitrary processing times exist.

b3 ¼ ðpj ¼ pÞ
All trucks have a processing times equal to p. Theassumption of equidistant service slots can beseen as a reasonable approximation of reality,whenever vehicle capacities as well as thenumber and nature of products per vehicle donot strongly differ (see [32]). As trailers aretypically of a standardized size and cross dockingaims at moving only full truck loads, this premiseis fulfilled whenever all processed products areof comparable size (e.g., mail distribution cen-ters) or all truck loads resemble a representativeaverage truck load (e.g., rotational deliveries ofspecial promotional offers to all stores of a retailchain).
b3 ¼ ðprpjrpÞ
No processing time pj is less than p or greaterthan p.
Deadlines b4Af3;~dlg: Deadlines might restrict the departure

time of trucks and shipments.

b4 ¼ 3
No deadlines are assumed in the system (as hardconstraints). However, due dates may be defined as softconstraints which are taken up in the objective function.
, literature review and research agenda. Omega

dx.doi.org/10.1016/j.omega.2009.10.008

ARTICLE IN PRESS


b4 ¼~dl

Please(2009)

Deadlines are defined which are to be met by trucks orshipments, where subscript lAfi; o; j; sg specifies forwhich element of truck scheduling due dates exist.l¼ i: Only inbound trucks are bound to due dates, whichmeans that these trucks need to be unloaded on-time tomeet a later assignment.l¼ o: Deadlines for the departure of outbound trucksneed to be regarded to meet due dates negotiated withthe customers.l¼ j: Both inbound and outbound trucks might be boundby deadlines.l¼ s: Finally, each single shipment might have a specificdue date negotiated with the respective final recipient.

Intermediate storage b5Af3l;no�waitg: Although cross dockingaims at minimizing inventory, at least intermediate storage insidea terminal is often inevitable.

b5 ¼ 3

cite th, doi:1

Typically, products are stored in front of the doorthe respective outbound truck is (to be) docked atand, thus, remain in intermediate stock until loaded.The storage space inside the terminal might turnout a bottleneck, which is covered by lAf3; limitg:l¼ 3:Unlimited storage space exists inside the terminal.l¼ limit: Available storage space is limited by agiven capacity.

b5 ¼ no�wait
Some products must not be intermediately storedat all, so that the no-wait property must hold. Thisis a common constraint for refrigerated products,e.g., frozen food, pharmaceuticals or flowers, forwhich a defrost threatens inside the unchilledterminal. Instead, these products must be instanta-neously loaded on a cooled outbound truck oncethey are unloaded (see [33]). Furthermore, it canalso be an organizational guideline to instanta-neously load products on their respective outboundtrucks to avoid congestions inside a terminal. Notethat the no-wait property corresponds to anintermediate buffer of zero capacity.
Assignment restrictions b6Af3;doorsg: Assignment restrictionsconfine the degrees of freedom in assigning trucks to doors.

b6 ¼ 3
If no restrictions are to be considered any dock door isa possible choice for truck processing.
b6 ¼ doors
Some trucks might only be processed at a subset ofdoors, which fulfill specific requirements, e.g., a bus barto cool freezer trailers or a wider dock for loading largeproducts crosswise. Note that an exclusive mode ofservice (a1 ¼ E) is not considered as such an assign-ment restriction.
Transshipment time b7Aftio; 3; tj ¼ 0g: We define the time lagbetween the arrival of shipments inside the terminal after having
unloaded them from their respective inbound truck until theiravailability at an outbound door as the transshipment time. In thereal world, such a transshipment time depends on multiplefactors, i.e., the availability of resources (e.g., workers and forklifts) and congestions inside the terminal. However, to reducecomplexity of truck scheduling the transshipment time isapproximated as being a given constant (see Section 2), whichmight differ as follows:
b7 ¼ tio
In real world terminals, distances to be covered bymaterial handling and, thus, transshipment times t
depend on the dock doors between which a shipment

is article as: Boysen N, Fliedner M. Cross dock scheduling: Classifica0.1016/j.omega.2009.10.008

is moved. Typically, it takes considerably moretransshipment time to move items between fardistant doors than between neighboring ones. Tomodel this relationship, an individual transshipmenttime for each pair of dock doors is to be considered asinput data and the realized transshipment time forany shipment depends on the door assignment of therespective inbound truck i and outbound truck o.

b7 ¼ 3
To reduce complexity of truck scheduling, it might bereasonable to assume transshipment times being aunique constant independent of the door assignmentof trucks. In this case, a single constant greater zero
covers the unique time lag for material handling ofany shipment. This simplification is better suited forsmall terminals with only a few doors, where thedifferences in transshipment times caused by diver-ging distances are negligible compared to the servicetimes of trucks.

b7 ¼ ðtj ¼ 0Þ
The ultimate simplification of transshipment timesreduces the constant to zero for each truck j. This mightbe a suitable simplification to ease the extraction ofstructural properties in mathematical models. In realworld terminals, this assumption can, for instance, bejustified in the food industry, where only very smalldocks are utilized and products must be instanta-neously stored in outbound trucks once they areunloaded. Here, short transshipment times are inevi-table to ensure a continuous cooling chain (see [33]).
Outbound organization b8Af3; fixg: This tupel entry defines theorganizational guidelines which decide on the points in time atwhich outbound trucks leave the terminal. The following twopossibilities exist:

b8 ¼ 3
An outbound truck leaves the terminal as soon as itspredefined set of products is loaded.
b8 ¼ fix
An outbound truck departs at a predefined point in time,irrespective of the products loaded. Especially postalservices depend on reliable departures in their multi-stage distribution networks and therefore often applyfixed schedules.
Interchangeable products b9Af3; changeg: The interchangeabilityof products mainly depends on whether or not value addingservices (e.g., repacking) are fulfilled at a terminal.

b9 ¼ 3
Any product arriving at the terminal is dedicated to aspecific outbound truck. This might result fromproducts being indeed individual, e.g., shipmentscommissioned for individual retail stores, or is anorganizational policy to ease allocation of productsfor the workers at the terminal.
b9 ¼ change
On the other hand, merely the number and types ofproducts to be loaded per outbound truck might bedefined, so that product units of a respective type cansatisfy any outbound truck’s demand for this pro-duct. Consequently, the assignment of product unitsto outbound trucks becomes part of the decisionproblem. Such a policy allows for a more flexiblereaction on unforeseen events, like erring truck loads,and seems especially promising if a reduced numberof standardized products is to be transshipped. Onthe other hand, a repacking of products inside theterminal is required, which slows down the trans-shipment process.
tion, literature review and research agenda. Omega

dx.doi.org/10.1016/j.omega.2009.10.008

ARTICLE IN PRESS


Note that these attributes might need to be extended if furtherorganizational implementations become relevant. For instance,
split deliveries might be allowed, so that a single shipmentcomprising several products can be divided among multipleoutbound trucks, which serve the same destination. Furthermore,the capacity of outbound trucks might be relevant. However, asthese additional attributes have not been covered by truckscheduling research thus far, we abstain from including them tokeep the classification as concise as possible.
3.3. Objectives

Finally, the optimization will be guided by some objective,which evaluates solutions. In truck scheduling, the traditionalmachine scheduling objectives (see, e.g., [9]), such as minimiza-tion of makespan or tardiness are also reasonable. However, insome cases an additional distinction can be made as to whichelement of a truck scheduling problem is subject to theseobjectives, i.e., shipments s or outbound trucks o. Furthermore,in the case of multi-objective optimization more than a singleobjective can be selected out of the following set:

gAXðwlÞCl;Cmax; Lmax;

XðwlÞTl;

XðwlÞUl;

XðwpÞSp; Smax;�;%

n o�:

g¼PðwlÞCl

Please cite t(2009), doi:

The completion time Cl is the time an outboundtruck o or a shipment s is finally processed andready to leave the cross dock. Thus, to accelerate theturnover of goods and to reduce the probability oflate shipments minimizing the sum of completiontimes (g¼

PCl) might be a reasonable objective. If

shipments s (or the shipments contained in anoutbound truck o) are of diverging value also theweighted sum of completion times (g¼

PwlCl)

can be considered.
g¼ Cmax The schedule length or makespan is reached at the
point in time the last shipment is finally loaded.Because shipments leave the terminal on outboundtrucks, the makespan does not depend on thedistinction between outbound trucks o and ship-ments s, so that Cmax ¼maxoAOfCog ¼maxsASfCsg.This objective is especially suited to rapidly emptythe terminal, so that following trucks of adjacentplanning runs can be processed.

g¼ Lmax
The maximum lateness can be minimized for eachshipment s or outbound truck o. In the case ofshipments as the reference point, maximum late-ness is calculated as follows: Lmax ¼maxsA SfCs � dsg,where ds is the deadline of shipment s. Thisobjective is especially suited whenever small delaysare acceptable, e.g., lost time can be regained on theroad, and only bigger delays notably impactscustomer satisfaction.
g¼PðwlÞTl
If already smaller delays influence customer satis-
faction, it might be better to minimize the(weighted) tardiness, which can also be assignedto shipments and outbound trucks. For instance,each outbound truck’s tardiness amounts to:To ¼maxfCo � do;0g8oAO.

g¼PðwlÞUl
Furthermore, each delay can be harmful irrespec-
tive of its magnitude. In this case, the (weighted)number of tardy truck or shipments

Pl ðwlÞUl can

be minimized, where Ul ¼ 1, if Cl4dl.
g¼
PðwpÞSp
The cross docking concept relies on a rapid turnover
of shipments. Thus, the minimization of stocked

his article as: Boysen N, Fliedner M. Cross dock scheduling: Classificat10.1016/j.omega.2009.10.008

products p inside the terminal over the planninghorizon, possibly weighted (according to, e.g., sizeor value) with a product specific factor, can be avaluable objective. At least in tendency, also thedanger of delayed shipments is reduced becauseinventory of products once delivered can only bedecreased by loading them on outbound trucks toleave the terminal as early as possible. Moreover, areduced stock size also alleviates congestions ofvehicles for material handling inside the terminal.

g¼ Smax
Furthermore, it might be reasonable to minimizethe maximum inventory level during the planninghorizon, e.g., to not exceed available stock space orto reduce extraordinary congestions.
g¼�
No objective function is applied whenever testingfor feasibility, i.e., to meet deadlines, is considered.
g¼ %
Some other (surrogate) objective is considered notspecified in our classification.
4. Literature review

In the following, we review existing truck schedulingresearch (in chronological order) on the basis of our classificationscheme.

The first contribution to short-term truck scheduling comesfrom McWilliams et al. [7]. They investigate a real world terminalof a postal service provider where delivered parcels are forwardedto outbound trucks by a system of interconnected conveyor belts.They tackle the resulting problem case ½EjtiojCmax� with a geneticalgorithm, which is coupled with a simulation model (simulationbased optimization). The simulation model is applied to evaluatethe congestions on the conveyor belt system (and their impact onthe makespan) caused by different inbound schedules.

Another real world setting from the food industry is presentedby Boysen [33]. The peculiarity of frozen foods and otherrefrigerated products, e.g., pharmaceuticals or flowers, is thatthe cooling chain must be intact. Consequently, a shipment onceunloaded must instantaneously be stored in its respective cooledoutbound trailer. No intermediate storage inside the uncooledterminal is allowed, so that the no-wait property (b5 ¼ no�wait)must hold. For the case ½Ejpj ¼ p;no�wait; tj ¼ 0j

PTo� Boysen [33]

presents an exact Dynamic Programming approach extended withlower and upper bounds (so-called Bounded Dynamic Program-ming) and a heuristic simulated annealing approach.

A more stylized model with only a single inbound and a singleoutbound door operated in an exclusive mode of service (a¼ E2)is investigated by Boysen et al. [32]. For such a terminal setting,they aim at minimizing the makespan of the schedule (g¼ Cmax)with the peculiarity of products being interchangeable betweenoutbound trucks (b9 ¼ change), so that the following constellationis considered: ½E2jpj ¼ p; changejCmax�. Boysen et al. [32] decom-pose the overall problem into an inbound and an outboundproblem, which are proven to have identical structure. Iteratively,they solve inbound and outbound problems with differentalgorithms. This procedure is tested in a comprehensive compu-tational study.

Chen and Lee [34] extend the traditional two-machine flow-shop problem by precedence constraints between inbound tasksat the first stage and outbound tasks at the second stage. Anoutbound truck receives multiple dedicated shipments fromdifferent inbound trucks so that respective precedence constraintsbetween inbound and outbound tasks enforce that an outboundtruck on the second stage cannot be processed before all itspredecessor tasks have been completed on the first stage.Although, the resulting cross docking case with only one inbound

ion, literature review and research agenda. Omega

dx.doi.org/10.1016/j.omega.2009.10.008

ARTICLE IN PRESS


and one outbound door is a stylized problem not directlyapplicable in the real world, it is a good starting point foranalyzing the structure of cross docking problems. Especially, theNP-hardness proof for the very basic case ½E2jtj ¼ 0; pj ¼ pjCmax� isvery useful since many real world problems are generalizations ofthis problem. Furthermore, they present a branch and boundprocedure for the case ½E2jtj ¼ 0jCmax�, which is able to solveproblems with up to 60 trucks to optimality.

The development of a decision support system called ‘‘Load-Dock’’ for real world cross docking in the less-than-truckloadindustry is documented by Chmielewski [35]. Such a decisionsupport tool has to cover multiple real world settings, so thatmany attributes of our classification are treated by Chmielewski:½EMjrj;

~dj; limit;doors; tioj%�. The solution approach bases on anetwork flow formulation where inbound and outbound doorsalong with different storage areas inside the terminal are definedas nodes with given capacity (deduced from processing speed perperiod). A heuristic solution approach for the extended networkflow model based on Column-Generation is presented and testedwith real world data.

Miao et al. [31] investigate the following cross dock setting:½Mjlimit; tioj%�, which has important characteristics often relevantin the real world, i.e., multiple doors (a2 ¼ 3) operated in a mixedservice mode (a1 ¼M), a limited storage space inside the terminal(b5 ¼ limit) and a transshipment time depending on the doorassignment of the respective inbound and outbound trucks of ashipment (b7 ¼ tio). They also presuppose that each truck has apredefined and fixed service window during which a dock door isfully occupied. Whenever no door can be found to be reserved forthe complete time span the truck cannot be processed at all and,thus, becomes a lost shipment. Consequently, one term of theirobjective function is to minimize penalty cost for lost shipments.An additional term covers operational costs which are mainlyinfluenced by the (door assignment dependent b7 ¼ tio) distancesto be covered by material handling devices. However, the modelof Miao et al. [31] suffers from the fact that trucks are counted aslost shipments even if their service window is only violated for afew seconds. Such a strict service window without the slightestvariability seems hard to imagine in a real world cross docksetting. The resulting optimization model is solved by a tabusearch approach and a genetic algorithm. Especially the tabusearch approach is shown to be very efficient in a computationalstudy with differently sized test instances when compared withstandard solver CPLEX.

Another stylized model with only a single door operatedin a mixed service mode (a¼M1) is considered by Boysen [36].After proving NP-hardness for the investigated case: ½M1jpj ¼

p; tj ¼ 0; changejP

Sp� an exact Dynamic Programming approachand a Beam Search heuristic are presented and tested.

Yu and Egbelu [37] treat the case ½E2jchangejCmax�, which isalso a very elementary problem when products delivered byinbound trucks may serve product demands of multiple outboundtrucks (b9 ¼ change). However, with merely a single inbound anda single outbound door it is only a stylized model to investigatethe structure of related (more complex) cross docking problems.For the solution of their problem Yu and Egbelu [37] introduced apriority rule based heuristic. This heuristic is evaluated against acomplete enumeration of all inbound and outbound sequenceswith test instances up to 12 trucks (6 inbound and 6 outbound)and 9 products.

Chen and Song [38] extend the work of Chen and Lee [34]by considering multiple parallel processors (multiple doors)per (inbound and outbound) stage. For the case ½Ejtj ¼ 0jCmax�

heuristic procedures which are adaptations of the famousJohnson-rule for two machine flow-shop scheduling are presentedand tested.


Table 1 summarizes the literature review, where thecontributions of each paper are stated with the help of thefollowing notation:

M mathematical modelHI heuristic improvement procedureHM meta-heuristicS simulation approachB bound computationHS start heuristic for initial solutionE exact solution procedureP properties (e.g., complexity) of problem

Additionally, complexity of the problems is reported. If nocomplexity proof is provided by the authors, the label ‘‘open’’highlights that the complexity status of the problem is unknown.

5. Future research

Open research can be divided into three categories: (i) theunexplored cases of our classification, (ii) research needs inrelation to interdependent planning problems, and (iii) imple-mentation of real world truck scheduling. With the help of ourclassification scheme yet unexplored cases of truck scheduling(category (i)) can be easily identified. Preemption of truckprocessing (b1 ¼ pmtn), door-dependent transshipment times(b7 ¼ tio), an outbound organization with fixed outbound sche-dules (b8 ¼ fix) and objectives other than minimizing the make-span (g¼ Cmax) should especially be considered. Research needsfalling into the latter two categories are described in more detailin the following two sections.

5.1. Research needs in relation to interdependent planning problems

An important research question within this category refers tothe problem whether or not destinations should be fixed over amid-term horizon. Alternatively, the assignment of trucks todoors can also remain part of the truck scheduling problem. Toanswer this question the trade-off between a clear arrangement ofmaterial handling inside the terminal and the degrees of freedomfor truck scheduling needs to be observed. On the one hand adurable assignment of destinations eases shipment allocation todoors for the workforce. On the other hand, it complicates findinggood truck schedules, because peak loads for single destinationscan not be absorbed by additional doors. It would be a valuablecontribution if the disadvantage of a durable assignment withregard to efficient truck schedules could be evaluated withdifferent data settings. This way, decision support could beprovided under which real world circumstances a durableassignment between doors and destinations is less disadvanta-geous for truck scheduling than in others.

Even if the assignment of doors to destinations is executed ona mid-term horizon there remains a truck scheduling problem,which is to sequence the trucks of a specific destinations at theirrespective door. For some cases of our classification the problemdecomposes into single door problems. However, for otherproblems the relation of inbound trucks with regard to outbounddepartures hinders a decomposition. It would be a valuablecontribution to investigate the structure of the remaining truckscheduling problems. This would answer the question whetherthese problems are indeed easier (e.g., with regard to complexity)to solve compared to their counterpart truck scheduling problemsincluding the door assignment of trucks.


dx.doi.org/10.1016/j.omega.2009.10.008

ARTICLE IN PRESS

Table 1Overview on truck scheduling research.

Publication Notation Complexity Contribution

McWilliams et al. [7] ½EjtiojCmax� Open HM, S

Boysen [33] ½Ejpj ¼ p;no�wait; tj ¼ 0jP

To� Open M, HM, E

Boysen et al. [32] ½E2jpj ¼ p; changejCmax� NP-hard M, HS, HI, E, P

Chen and Lee [34] ½E2jtj ¼ 0jCmax� NP-hard B, E, P

Chen and Lee [34] ½E2jtj ¼ 0;pj ¼ pjCmax� NP-hard P

Chmielewski [35] ½EMjrj ;~dj ; limit;doors; tioj%� Open M, E

Miao et al. [31] ½Mjlimit; tioj%� NP-hard M, HM, P

Boysen [36] ½M1jpj ¼ p; tj ¼ 0; changejP

Sp� NP-hard M, HM, E, P

Yu and Egbelu [37] ½E2jchangejCmax� Open M, HS

Chen and Song [38] ½Ejtj ¼ 0jCmax� NP-hard M, HS, B

This paper ½Ejtio; fixjP

wsUs� NP-hard M, P

This paper ½Ejti ¼ 0; fixjP

wsUs� NP-hard P


Furthermore, the relationship to (inbound and outbound) vehiclerouting should be closely investigated. First, the sensitivity of truckscheduling with regard to diverging arrival times of inbound trucksand departure times of outbound trucks should be considered. Aconsiderable sensitivity would be a hint that a simultaneous truckscheduling and vehicle routing is a promising planning approach.Then, detailed models in different levels of aggregation and suitedsolution procedures should be developed. Especially, decompositionapproaches solving both problems separately but iteratively coupledby some adjustment procedure, e.g., an auction mechanism, seempromising. This way, existing solution procedures out of both fields’extensive body of research could be reused.

Finally, in relation to material handling inside the terminal thefollowing research question seems especially relevant: How toanticipate congestions of material handling devices resulting fromdifferent truck schedules? If such congestions can be easilyestimated, a mechanism could be integrated into truck schedulingwithout requiring integration of a detailed material handling.

5.2. Implementation of real world truck scheduling

Finally, implementing truck scheduling in real world crossdocks seems an especially challenging field. The most straightfor-ward implementation would be to consider all trucks (i.e., alltrucks already waiting on the yard plus all those whichpresumably arrive during the planing horizon) in a singleplanning run. Then, the resulting schedule would fix truckprocessing over the complete planning horizon. However, arrivaltimes of trucks are typically bound to heavy inaccuracies, becausetraffic congestions or engine failures delay inbound trucks withthe utmost probability. Thus, the following research questionsneed to be answered in this context: up to which ‘‘level ofuncertainty’’ are expected arrival times of trucks useful informa-tion to be considered in truck scheduling? How to derive robustplans, i.e., plans which remain feasible in spite of (shorter) delays?

To attenuate the impact of uncertain arrivals, truck schedulingis often applied in a rolling horizon setting. Scientific advice onhow to dimension the planning horizon and how to link adjacentplanning runs is still missing. In the extreme case, truckscheduling is executed once a door is released to merelydetermine the truck to be called-up taking over the empty door(see [33]). Such an online procedure has the advantage thatuncertain truck arrivals become irrelevant because only trucksalready waiting on the yard would need to be considered. In thiscontext the question whether a complete planning run each timea door is released is actually better than a (simple) selection ruleneeds to be investigated.


Testing all planning scenarios (static planning vs. rollinghorizon vs. online selection of the next truck) with real worlddata seems a fruitful research task, as advice on the suitability ofthose alternatives under specific real world circumstances couldbe gained.

Furthermore, organizational policies should be challenged. Forinstance, an exclusive mode of service (a1 ¼ E), which is oftenapplied in many real world terminals, eases material handlinginside the terminal. Nevertheless, a mixed mode of service(a1 ¼M) leaves more degrees of freedom for truck schedulingand, thus, promises better plans. Quantifying the advantage of amixed mode of service within different truck scheduling instancescould provide valuable information for the practitioner toreasonably decide on this organizational guideline. Anotherpolicy to question is the widespread exclusion of preemption(b1 ¼ pmtn).

Finally, cross docks in different branches of industry should beinvestigated. Especially, material handling considerably deviatesbetween branches. On the one hand, in retail or less-than-truckload industries material handling is mostly a manual tasksupported by fork lifts or pallet jacks [5]. On the other hand, inpostal services material handling is automated by conveyor beltsystems [7]. In the automobile industry, even highly automatedrobots, which sort material into the sequence they are required atthe final assembly (Just-in-Sequence), can be found. These andfurther peculiarities of different branches applying cross dockingcould be an important step towards learning the needs ofdifferent branches. At least, different cost structures (e.g., withregard to the products shipped and the penalty of delays) could beconsidered, so that the choice of an appropriate objective functionfor different cost structures would be facilitated. This way existingresearch, which mainly deals with formulating and solvingisolated truck scheduling models, could be enhanced to service-able decision support in real world cross docking.

6. Cross docking with fixed outbound schedules

As the literature review revealed, present research ontruck scheduling is restricted to a single kind of outboundorganization. Up to now, all studies presuppose that any outboundtruck leaves the terminal not before all dedicated products orshipments are loaded (b8 ¼ 3). However, such an outboundorganization is only possible if all shipments to arrive are knownin advance at the respective cross docking terminal. Moreover, froman economic point of view it seems especially sensible if fewshipments of high value are transported. Consequently, this out-bound pattern is, for instance, applied in cross docks of the


dx.doi.org/10.1016/j.omega.2009.10.008

ARTICLE IN PRESS

Table 2Notation.

I Set of inbound trucks with I¼ f1;2; . . . ;ng

O Set of outbound trucks

D Set of inbound doors available for processing inbound trucks

pi Processing time for unloading inbound truck i

do Departure time of outbound truck o

tko Transshipment time from inbound dock k to the dock outbound truck o

is processed

wio Weight, e.g., the number of products, of a shipment delivered by

inbound truck i dedicated to outbound truck o

M Big integer

Ci Continuous variable: completion time of inbound truck i

xkij

Binary variable: 1, if inbound truck j is processed directly after inbound

truck i at dock door k; 0, otherwise

xk0i

Binary variable: 1, if inbound truck i is processed first at door k; 0,

otherwise

xki;nþ1

Binary variable: 1, if inbound truck i is processed last at door k; 0,

otherwise

yio Binary variable: 1, if shipments delivered by inbound truck i are too late

to reach outbound truck o; 0, otherwise


automobile industry, where large transport boxes of Just-in-Timematerials are transshipped. Here, an early departure of an outboundtruck ahead of an only slightly delayed inbound truck wouldjeopardize on time deliveries of Just-in-Time materials at the finalassembly with the hazard of material stock outs and, thus, linestoppages.

Nevertheless there are several other industries where amultitude of smaller and low valued shipments are transported.In such a setting, delays of shipments are still undesired and, thus,to be reduced to a minimum; but they are by far not as harmful.Especially in larger hub-and-spoke networks where a multi-stagecross docking is applied, a reliable and steady flow of trucks seemsmuch more essential. Consequently, especially postal services andless-than-truck load service providers typically rely on fixedoutbound schedules (b8 ¼ fix). Consequently, trucks are supposedto leave a terminal exactly at a predefined point in time over afixed route to a specific destination [35]. All shipments for therespective destination which arrive before the truck’s departureare loaded and, thus, shipped the same day. Any other shipment isdelayed until the next day when the next truck serves thedestination. In such a setting, truck scheduling should aim atminimizing the (weighted) number of shipments delayed untilthe next day (g¼

PðwsÞUs). Unfortunately, present research has

not yet considered this highly relevant truck scheduling setting.To stimulate this important field of truck scheduling research, we

present an optimization model for the case: ½Ejtio; fixjP

wsUs�

according to our classification. We presuppose that the outboundschedule is already planned over a mid-term horizon, so that alloutbound trucks are previously fixed concerning the destinationthey serve, the point in time they leave the terminal and the dockdoors they are served at. Thus, short-term truck scheduling has todetermine the inbound schedule at a separated set of inbound doors(exclusive service mode: a1 ¼ E). Each inbound truck deliversshipments dedicated to multiple destinations any of which servedby a specific outbound truck over a predetermined dock at a specificpoint in time (b8 ¼ fix). Consequently, some shipments delivered byan inbound truck might reach their dedicated outbound trucks to beshipped the same day whereas others arrive late. Thus, in such asetting each inbound truck is bound to multiple different dead lines.Note that, typically, in postal services the shipments arriving on aninbound truck are not previously announced to the cross dock, thus,average flows determined from historical data might need to beapplied. A reasonable objective in such a setting is to minimize theweighted number of shipments delayed up to the next day(g¼

PwsUs). Furthermore, in real world cross docks (especially

larger ones) the diverging transshipment time for material handlingbetween the inbound and outbound door needs to be considered(b7 ¼ tio). In our model, we capture this context with a transship-ment time tko, which measures the time lag between dock door k therespective inbound truck i is processed at and the dock doorassigned to outbound truck o.

With the help of the notation summarized in Table 2 truckscheduling for the case ½Ejtio; fixj

PwsUs� consists of objective

function (1) and constraints (2)–(8):

Minimize ZðC;X;YÞ ¼XiA I

XoAO

wio � yio ð1Þ

subject toXkAD

Xi A I[f0g

i a j

xkij ¼ 1 8jA I ð2Þ

XiA I

xk0ir1 8kAD ð3Þ

Xi A I[f0g

i a j

xkij ¼

Xi A I[fnþ 1g

i a j

xkji 8jA I; kAD ð4Þ


Ci ¼XkAD

pi � xk0iþ

XjA I

ðCjþpiÞxkji

0@

1A 8iA I ð5Þ

yio �MZCi � doþXkAD

tko

Xj A I[f0g

i a j

xkji

0B@

1CA 8iA I; oAO ð6Þ

xkijAf0; 1g 8iA I [ f0g; jA I [ fnþ1g; kAD ð7Þ

yioAf0; 1g 8iA I; oAO ð8Þ

The objective function (1) seeks to minimize the weightednumber of delayed shipments, i.e., the number of ship-ments which remain in the terminal until the next outboundtruck (e.g., of the next day) serving the respective destinationleaves. Constraints (2) ensure that each inbound truck isprocessed exactly once. Inequalities (3) guarantee thateach dock door is utilized at most once by restricting thenumber of startup trucks to at most one per door. Constraints(4) ensure that the succession of inbound trucks at the dock doorsis well-defined. These constraints play the same role as flowconservation constraints in many network flow problems.Constraints (5) define completion time Ci for each inbound trucki. Inequalities (6) ensure that shipments of late inbound trucks i

cannot reach a respective outbound truck o whenever completiontime Ci exceeds departure time do plus movement time tko

required to move a shipment processed at inbound door k to theoutbound door truck o is processed at, where big integer M can bedimensioned as follows: M¼

PiA I pi �minoAO fdoþminkAD ftkogg.

Finally, constraints (7) and (8) represent binary integralityrequirement of 0–1 variables.

This model is NP-hard in the strong sense, which is proven inthe appendix. Thus, future research should develop efficient exactand especially heuristic solution procedures for this and relatedoptimization models. Especially, the close relationship of ourmodel with parallel machines scheduling (e.g., [39,40]) should bea good starting point in this direction.

7. Conclusions

This paper introduces a classification of truck scheduling problemsbasing on a tupel notation. With the help of this classification schemeexisting research is briefly summarized and future research needs areidentified. Especially, a yet unexplored truck scheduling problem forfixed outbound schedules is formalized by an optimization model


dx.doi.org/10.1016/j.omega.2009.10.008

ARTICLE IN PRESS


along with a complexity proof. In addition to the academic effortspent on describing the mathematical properties of alternativemodels and deriving suitable solution procedures, there is anapparent lack of empirical research evaluating the goodness of fit ofalternative truck scheduling approaches to real world applications.Therefore, contributions which provide insight into this complexmatter are to be seen as especially valuable.

Appendix

We will prove NP-hardness for the case ½Ejtj ¼ 0; fixjP

wsUs� by atransformation from the 3-Partition Problem, which is well knownto be NP-hard in the strong sense (see [41]). To ease representation,we will refer to the truck scheduling problem as CD. Note that themodel for the case ½Ejtio; fixj

PwsUs� presented in Section 6 is a

generalization of ½Ejtj ¼ 0; fixjP

wsUs� and the reduction can be usedto show NP-hardness for the former case as well.

3-Partition problem: Given 3q positive integers ai (i¼ 1; . . . ;3q)and a positive integer B with B=4oaioB=2 and

P3qi ¼ 1 ai ¼ qB,

does there exist a partition of the set f1;2; . . . ;3qg into q setsfA1;A2; . . . ;Aqg such that

PiAAj

ai ¼ B 8j¼ 1; . . . ; q?Transformation of 3-Partition to CD: Consider an instance of CD

with 3q inbound trucks, whose processing times pi equal theinteger values ai of 3-Partition. Furthermore, we have a singleoutbound truck o with a departure time do ¼ B which receivesshipments from all inbound trucks. Finally, we assume jDj ¼ q

dock doors for processing inbound trucks. As there is a one-to-onemapping between integer values of 3-Partition and inboundtrucks of CD, this transformation is polynomial. The question weask is whether we can find a solution for CD with objective valueZ ¼ 0, i.e., no delay of shipments.

A feasible solution for an instance of 3-Partition can betransformed to a feasible solution of the corresponding CD-instancein polynomial time by assigning each set of integers to a separateinbound door. As the sum of integer values of each set amounts to B,all three inbound trucks can be processed up to period B infacultative succession at their respective door, so that no delay atneither door occurs and the objective value of Z ¼ 0 is realized.

On the other hand, each feasible solution for any CD instance isalso a feasible solution for 3-Partition. This holds true because anysolution of CD with Z ¼ 0 must have exactly three inbound trucksassigned per door because of the restriction on the processing timevalues B=4opi ¼ aioB=2. Any solution with more or less thanthree trucks at a door must result in a makespan higher than B, sothat a delay would be inevitable. Thus, any CD-solution with Z ¼ 0must have a makespan of B at any door, so that a direct mappingbetween the trucks per door and the sets of integers exist.

Note that the problem remains NP-hard even for a givennumber of doors with jDj41. For jDj ¼ 2 this can be proven by ananalogous transformation from the Partition problem.

References

[1] Apte UM, Viswanathan S. Effective cross docking for improving distributionefficiencies. International Journal of Logistics: Research and Applications2000;3:291–302.

[2] Stalk G, Evans P, Shulman LE. Competing on capabilities: the new role ofcorporate strategy. Harvard Business Review 1992;70(2):57–69.

[3] Forger G. UPS starts world’s premiere cross-docking operation. ModernMaterial Handling 1995;36–38.

[4] Witt CE. Crossdocking: concepts demand choice. Material Handling Engineer-ing 1998;53(7):44–9.

[5] Gue KR. The effect of trailer scheduling on the layout of freight terminals.Transportation Science 1999;33:419–28.

[6] Kim C, Yang KH, Kim J. A strategy for third-party logistics systems: a caseanalysis using the blue ocean strategy. Omega 2008;36:522–34.


[7] McWilliams DL, Stanfield PM, Geiger CD. The parcel hub scheduling problem:a simulation-based solution approach. Computers & Industrial Engineering2005;49:393–412.

[8] Graham RL, Lawler EL, Lenstra JK, Rinnooy Kan AHG. Optimization andapproximation in deterministic sequencing and scheduling: a survey. Annalsof Discrete Mathematics 1979;5:287–326.

[9] B"azewicz J, Ecker K, Pesch E, Schmidt G, Weglarz J. Handbook on scheduling:from theory to application. Berlin: Springer; 2007.

[10] Campbell JF. A survey of network hub location. Studies in Locational Analysis1994;6:31–49.

[11] Klose A, Drexl A. Facility location models for distribution system design.European Journal of Operational Research 2005;162:4–29.

[12] Chen J-F. A hybrid heuristic for the uncapacited single allocation hub locationproblem. Omega 2007;35:211–20.

[13] Bartholdi JJ, Gue KR. The best shape for a crossdock. Transportation Science2004;38:235–44.

[14] Tsui LY, Chang C-H. A microcomputer based decision support tool forassigning dock doors in freight yards. Computers & Industrial Engineering1990;19:309–12.

[15] Tsui LY, Chang C-H. An optimal solution to dock door assignment problem.Computers & Industrial Engineering 1992;23:283–6.

[16] Bartholdi JJ, Gue KR. Reducing labor costs in an LTL crossdocking terminal.Operations Research 2000;48:823–32.

[17] Bermudez R, Cole MH. A genetic algorithm approach to door assignments inbreakbulk terminals. Technical Report MBTC-1102, Mack Blackwell Trans-portation Center, University of Arkansas, Fayetteville, Arkansas; 2001.

[18] Oh Y, Hwang H, Cha CN, Lee S. A dock-door assignment problem for theKorean mail distribution center. Computers & Industrial Engineering2006;51:288–96.

[19] Bozer Y, Carlo H. Optimizing inbound and outbound door assignments inless-than-truckload crossdocks. IIE Transactions 2008;40:1007–18.

[20] Chen P, Guo Y, Lim A, Rodrigues B. Multiple crossdocks with inventory andtime windows. Computers & Operations Research 2006;33:43–63.

[21] Lee YH, Jung JW, Lee KM. Vehicle routing scheduling for cross-docking insupply chain. Computers & Industrial Engineering 2006;51:247–56.

[22] Lim A, Miao Z, Rodrigues B, Xu Z. Transshipment through crossdocks withinventory and time windows. Naval Research Logistics 2005;52:724–33.

[23] Li Y, Lim A, Rodrigues B. Crossdocking—JIT scheduling with time windows.Journal of the Operational Research Society 2004;55:1342–51.

[24] Alvarez-Perez GA, Gonzalez-Velarde JL, Fowler JW. Crossdocking—just intime scheduling: an alternative solution approach. Journal of the OperationalResearch Society 2009;60:554–64.

[25] Lim A, Rodrigues B, Wang Y. A multi-faced buildup algorithm for three-dimensional packing problems. Omega 2003;31:471–81.

[26] Kendall DG. Stochastic processes occurring the theory of queues and theiranalysis by the method of imbedded Markov chains. Annals of MathematicalStatistics 1953;24:338–54.

[27] Brucker P, Drexl A, Mohring RH, Neumann K, Pesch E. Resource-constrainedproject scheduling: notation, classification, models and methods. EuropeanJournal of Operational Research 1999;112:3–41.

[28] Dyckhoff H. A typology of cutting and packing problems. European Journal ofOperational Research 1990;44:145–59.

[29] Boysen N, Fliedner M, Scholl A. A classification of assembly line balancingproblems. European Journal of Operational Research 2006;183:674–93.

[30] Boysen N, Fliedner M, Scholl A. Sequencing mixed-model assembly lines,classification and model critique: survey. European Journal of OperationalResearch 2009;192:349–73.

[31] Miao Z, Lim A, Ma H. Truck dock assignment with operational time con-straint within crossdocks. European Journal of Operational Research2009;192:105–15.

[32] Boysen N, Fliedner M, Scholl A. Sequencing inbound and outbound trucks atcross docking terminals. OR Spectrum 2007 (to appear).

[33] Boysen N. Truck scheduling at zero-inventory cross docking terminals.Computers & Operations Research 2010;37:32–41.

[34] Chen F, Lee C-Y. Minimizing the makespan in a two-machine cross-docking flow shop problem. European Journal of Operational Research2009;193:59–72.

[35] Chmielewski A. Entwicklung optimaler Torbelegungsplane in Stuckgutspe-ditionsanlagen. Dissertation, University of Dortmund, Germany; 2007.

[36] Boysen N. Uber die Synchronisierung von Guterstromen in der Umschlaglo-gistik. Zeitschrift fur Betriebswirtschaft 2008;78:1285–315.

[37] Yu W, Egbelu PJ. Scheduling of inbound and outbound trucks in cross dockingsystems with temporary storage. European Journal of Operational Research2008;184:377–96.

[38] Chen F, Song K. Minimizing makespan in two-stage hybrid crossdocking scheduling problem. Computers & Operations Research 2009;36:2066–73.

[39] Chen ZL, Powell WB. Solving parallel machine scheduling problems bycolumn generation. Informs Journal on Computing 1999;11:78–94.

[40] M’Hallah R, Bulfin RL. Minimizing the weighted number of trady jobs onparallel processors. European Journal of Operational Research 2005;160:471–84.

[41] Garey MR, Johnson DS. Computers and intractability: a guide to the theory ofNP-completeness. New York: Freeman; 1979.


dx.doi.org/10.1016/j.omega.2009.10.008

rsearch agenda cross dock

Documents