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Cross-dock job assignment problem in space-constrained industrial logistics distribution hubs with asingle docking zoneK.L. Choy a , H.K.H. Chow b , T.C. Poon a & G.T.S. Ho aa Department of Industrial and Systems Engineering , The Hong Kong PolytechnicUniversity , Hong Kongb Faculty of Management and Administration , Macau University of Science and Technology ,MacauPublished online: 28 Sep 2011.

To cite this article: K.L. Choy , H.K.H. Chow , T.C. Poon & G.T.S. Ho (2012) Cross-dock job assignment problem in space-constrained industrial logistics distribution hubs with a single docking zone, International Journal of Production Research,50:9, 2439-2450, DOI: 10.1080/00207543.2011.581006

To link to this article: http://dx.doi.org/10.1080/00207543.2011.581006

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International Journal of Production ResearchVol. 50, No. 9, 1 May 2012, 2439–2450

Cross-dock job assignment problem in space-constrained industrial logistics

distribution hubs with a single docking zone

K.L. Choya*, H.K.H. Chowb, T.C. Poona and G.T.S. Hoa

aDepartment of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hong Kong;bFaculty of Management and Administration, Macau University of Science and Technology, Macau

(Final version received April 2011)

This paper addresses a cross-dock operations problem in space-constrained industrial logistics distributionhubs. In these hubs, the number of incoming trucks exceeds the number of docks available, and inboundtrucks and orders arrive at random. The solution lies in minimising the waiting time of trucks by coordinatingthe pick up/delivery sequences of inbound and outbound orders in the storage zones. A mathematical modeland a meta-heuristics algorithm, which is based on a genetic algorithm, are developed to address the problem.This research is innovative because the proposed algorithm allows the insertion of inbound orders that arriveat random into the schedule, without causing any significant disturbance to the original outbound orderschedule. Computational experiments are conducted to examine the performance of the algorithm underheavy and normal cross-dock conditions. Results show that the algorithm reduces the total makespanof storage operations by 10% to 20% under heavy and normal conditions. The research study benefitsmanufacturers by increasing cross-docking efficiency in industrial logistics systems characterised by limitedtemporary storage capacity and the random arrival of inbound trucks.

Keywords: cross-dock; coordination of inbound and outbound orders; genetic algorithm; truck assignment

1. Introduction

The era of dynamic and changing markets with high-quality products and a high demand for fast services has forcedmanufacturing companies to maintain low operational costs and promote customer-oriented products with shortlife cycles. Manufacturing companies need to be more flexible in response to environmental changes in the market(Gunasekaran and Kobu 2007). A well-developed industrial logistics system has therefore become important amongmanufacturing companies, particularly in achieving a shorter transportation time, low inventory turns, improvedon-time delivery, and faster market access. E-commerce has also brought manufacturing companies to the level of aglobally connected and networked market. To cope with market changes, industrial logistics activities are beingdecentralised and dispersed in different countries. For example, companies establish their distribution hubs andtransportation channels in various countries to form a globalised industrial logistics system. The distributionhubs provide storage and cross-dock functions within such industrial logistics systems. The operational model ofdistribution hubs, unlike that of warehouses, aims to provide fast cross-dock processes, which are aided to minimisethe frequency of most labour-intensive storage and retrieval activities. According to Yu and Egbelu (2008), cross-docking is a material handling and distribution function in which items move directly from the receiving to theshipping dock, without being stored in warehouses or distribution centres. Therefore, the key performanceindicator for distribution hubs is their rate of efficiency in handling large volumes of items, complying withthe schedules of transportation networks, and responding quickly to customer demands in different markets. Bothhardware and software must be simultaneously considered in the design stage to ensure proper system integrationand sustain highly efficient cross-dock operations (Yu and Egbelu 2008). The hardware component refers to thelayout design and space requirement analysis for inbound and outbound flows and staging. The softwarecomponent refers to the coordination of inbound and outbound flows and scheduling. The majority of distributionhubs in Western countries have a one-way flow where the docking zones are built separately in the opposite sidesof the hubs. Expectedly, Western studies on cross-docking optimisation methods and models focus on such layoutconfiguration.

*Corresponding author. Email: mfklchoy@inet.polyu.edu.hk

ISSN 0020–7543 print/ISSN 1366–588X online

� 2012 Taylor & Francis

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Hong Kong, owing to its location as the gateway to and from China and the rest of Asia and the Pacific,has become the busiest hub in terms of ocean container throughput and international air-cargo tonnage, and oneof the largest container handling centres/distribution hubs in the world. Owing to spatial limitations, most of thedistribution hubs are located in five- to 10-floor buildings. Therefore, the hubs are designed with one docking zoneset up for cross-docking operations rather than for the one-way flow commonly seen in Western distributionsystems. Both inbound and outbound trucks in these systems are queued and processed in a single docking zone.Consequently, the changeover time for all inbound and outbound trucks does not follow the same queuingsequence. Therefore, research methods for one-way flow may not directly apply to the configuration of these hubs,and new and interesting research on the coordination of inbound and outbound truck sequencing to appropriatedocks is helpful.

This work intends to develop an optimisation model that can tackle the coordination of cross-dock jobassignment problems in distribution hubs with limited storage space. Our research problem differs from the currentproblem on cross-docking in which the inbound and outbound docks are independent and distinct, the sequenceof the arrival of products at the outbound docks is the same as their unloading at the inbound docks, and nopreparation operations are needed in the terminal. An optimisation method is proposed to minimise the waitingtime of trucks by finding the optimum job sequence of pick up and delivery tasks in the storage zone. Furthermore,current optimisation methods usually seek to minimise the completion time of activities (makespan) or the total timespent by trucks in docks. Our proposed solution is innovative because it enables the operations staff to insertadditional tasks in the schedule without causing any significant disturbance to the original plan for outboundjob ordering. We believe that the research study will benefit manufacturers in terms of increasing cross-dockingefficiency, even under conditions of limited temporary storage capacity, one docking zone, and the random arrivalof inbound trucks, which are often observed in Hong Kong and in other countries.

2. Literature review

A supply chain is a network of facilities and distribution selections that facilitates the operation of materialprocurement, the production of intermediate and finished goods, and the distribution of finished products(Safaei et al. 2010). A holistic analysis and coordination of external partners in the supply chain is requiredto achieve lean, agile, flexible, and efficient operations, and improve internal efficiency (Hwarng et al. 2005).Cross-dock operations, a special function of the logistics operations of supply chains, are tailored to cope with thecharacteristics of industrial logistics systems that emphasise quick item deliveries to the market and a short storageperiod. Tsui and Chang (1990) presented one of the earliest technical papers analysing decision-support tools oncross-docking. They studied the assignment of dock doors to incoming and outgoing trucks; such assignmentdetermines the efficiency of operations. They also mentioned that decisions on dock assignment, which are oftenwith realistic constraints, are mostly made by individuals; hence, they recognised that studies on decision-supporttools are needed to improve operations in freight yards. Tsui and Chang (1992) later proposed an improved versionof the assignment heuristics. However, the proposed heuristics is only capable of handling small-scale consolidationproblems. Similarly, Masel (1998) developed a list-scheduling heuristics for adequate trailer-dock assignment.Gue (1995, 1999) used a greedy method in assigning trailers to doors, with the objective of minimising the totaltransfer time. The method reduces transfer time by 3–30% and labour costs by 2–5%. Bartholdi and Gue (2000)improved the solution methodology by replacing the greedy method with simulated annealing. Although the modelconsiders internal factors (e.g. the types of congestion), it fails to address the problem of actual dock assignment toarriving trucks, considering the time windows of trucks. To minimise the total cost of handling materials, Heraguet al. (2005) proposed a mathematical model and a heuristic algorithm for the determination of the size of functionalareas: areas for cross-docking operations, particularly for receiving, shipping, and staging; reserve and forwardareas; and warehouse areas. McWilliams et al. (2005) worked out a simulation-based scheduling algorithm (SSA) toassign incoming trucks to unloading docks in parcel consolidation terminals. The algorithm embeds an iterativeimprovement heuristics to search for good unloading schedules, and reduces the time required to complete transferoperations by 15% to 25%. Miao et al. (2009) recently adopted a two-meta-heuristics approach using Tabu searchand a genetic algorithm (GA) to consider a truck assignment problem with an operational time constraint in a cross-docking system. Rajkumar and Shahabudeen (2009) applied GA in flow-shop sequence scheduling and makespanminimisation. They proposed an improved genetic algorithm (IGA) framework, which can help avoid the prematureconvergence of solutions, multi-crossover, hypermutation, and re-assignment. Chien and Chen (2007) proposed an

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optimisation-based schedule generator (OptSG) that can solve generic scheduling problems in the manufacturingenvironment. The generator applies GA in searching for the optimal combinations of dispatching rules that canresolve conflicts; it acts as a scheduling strategy provider. Chiang et al. (2008) proposed a new paradigm thatconsiders multiple lots and equipment when evaluating matches, and that seeks for the best set of matches ratherthan a single best match in solving scheduling problems. They also developed a GA for generating appropriatedispatching rules. The proposed paradigm helps improve performance by 16–28%. Yu and Egbelu (2008) studieda cross-docking system with unlimited temporary storage. A mixed-integer program for defining the best inboundand outbound docking sequence minimises the total operations time, but ignores various important issues in realapplications, such as preparation processes, particularly the picking/storage of orders and the arrival/departureof trucks.

Based on the aforementioned literature review, heuristic approaches are promising methods for solving the truckdocking sequence problem, particularly in the case of distribution and consolidation terminals with two separatedocking zones located in opposite sides of a distribution hub. The sequencing of inbound and outbound trucksaffects the performance of cross-docking operations, such as put-away, sorting, picking, and storing operations. It isalso important to deal with uncertainties, including those on changes in transportation time (which, for instance,may be due to congestion), the arrival of rush orders during schedule execution, and order modifications during therandom arrival of inbound trucks, in the operational planning and control of many transportation networks(Mes et al. 2007). The above studies have mainly focused on distribution hubs with two large-scale separate dockingzones; very few of them have actually designed a truck dock assignment method that considers a spatiallyconstrained distribution hub. To address problems in the cross-dock operations environment, we intend to developan inbound/outbound cross-docking scheduling model for distribution hubs with limited storage space and whereinbound trucks randomly arrive.

This paper is organised according to topics. Section 3 describes the operation mode of distribution hubs in HongKong and its underlying scheduling problems. Section 4 introduces the notations used in the job-assignmentproblem, and Section 5 discusses the proposed meta-heuristics method. Section 6 discusses an experiment toevaluate the performance of the proposed method. Section 7 concludes the paper.

3. Operation mode of industrial logistics distribution hubs with limited storage space and size

Figure 1 shows the layout of an industrial logistics distribution hub with a storage zone and a docking bay, andwhere trucks are queuing. This sample setting is based on hubs located in Hong Kong International Terminals(HIT) and Modern Terminals (MTL) in Kwai Chung, Hong Kong. The proportion of the storage to the dockingsize is usually 80:20; the storage zone is used for different cross-dock operations, including sorting, labelling, qualityinspection, re-packing, and other similar tasks. Items are usually stored on the floor through a block-stack methodusing 1–2 pallets to facilitate the process of order picking and the loading/unloading of items by forklifts. The blockstack method provides quick access to stocks, thereby increasing put-away/picking productivity. However, it isimpossible to load all items on the floor when the overflow problem appears because the storage zone in such adistribution hub is limited. When a large amount of inbound items suddenly arrive, the hub does not have enoughspace and resources to accommodate the inbound and outbound orders simultaneously. Therefore, another kindof storage method, called pallet racking, is adopted in the hub. A certain number of racks are built in the hubfor temporary storage when using this storage method. Although racking storage improves space efficiency,it significantly decreases the efficiency of put-away and picking operations because extra time is required to movethe items from the rack to the floor by forklift. With this storage method, the order handling productivity of thedistribution hub decreases by at least 30%–40%. If the approach taken for inbound and outbound truck schedulingis inappropriate, the proportion of items stored in the racks is increased, such that the subsequent processing timeof put-away and picking tasks in the storage zone is also increased.

A logistics manager should bear in mind that an inefficient dock assignment approach in this situation triggersan overflow problem, low productivity, and serious congestion effects in the truck yard. The cross-dock operationsproposed in this study are described as follows:

(i) Inbound trucks arrive in the hub and unload items in the loading bay. Subsequently, the forklift drivermoves all items to the storage zone. Items are placed in the storage zone randomly because of thelimited storage space. Items are loaded on the racks for temporary storage if there is no extra space

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for block stacking. The inbound truck can leave the docking zone only after all the pallets are unloaded.The arrival time of inbound trucks is random and unknown.

(ii) Outbound trucks load pallets of items from the loading bay and leave the hub. The outbound truck canleave the docking zone only after all the pallets are loaded. The departure time of outbound trucks is knownand can be determined by the operations manager.

(iii) The movement of items from the loading zone to the loading bays and vice versa is carried out through theuse of forklifts.

(iv) All loading bays at the docking zone are used for handling both inbound and outbound orders. All inboundand outbound trucks queue at the parking zone.

4. Mathematical model formulation

This section presents the development of a mathematical model that can be used in handling the research problem.It involves planning for the inbound/outbound truck sequence, as well as for the subsequent order picking anddelivery job assignment of material-handling equipment (MHE) in the storage zone. This model is beneficial,particularly when it comes to saving costs by reducing the waiting time of trucks. The following sets of informationare known at the time of planning:

(i) Number of docks (nd);(ii) Number of MHE (NF);(iii) Number of trucks used for outbound orders (NOT) and the number of trucks used for inbound orders (NIT);(iv) For each (outbound) truck

. Arrival time in the docking zone (yA);

. Due date (or latest departure time) (yDue);

. Order picking list divided into pallets. Each pallet consists of a subset of items in the picking list to bepicked up by one MHE. A pallet of items that needs to be picked is defined as a task to be assigned toan MHE.

Figure 1. Configuration of a typical industrial logistics distribution hub and the process flow of outbound and inbound orders.

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(v) For each pallet (task)

. Duration;

. Starting point;

. Ending point;

. Set of order picking/delivery preparation operations with corresponding duration.

Each pallet calls for two tasks during the cross-dock operations. The first task is picking items in the storagearea, and the second task is carrying these items to appropriate docks in the bay area (Figure 1). The tasks haveto be assigned to MHE. The actual tour for one pallet in the storage area is determined by a set of rules to befollowed by the drivers (as in most settings) or determined by the use of some routing algorithm (as implementedin some warehouses).

The formulation of an order picking and dock assignment plan for outbound orders is considered a staticproblem; hence, there is a need to formulate a mathematical programming problem. The following notations areused in the mathematical programming model:

yD�j departure time of truck � containing order j;yA�j arrival time of truck � containing order j;s1ij starting time of the processing of pallet i of order j at the storage area;s2ij starting time of the processing of pallet i of order j at the docking zone;p1ij processing time at the storage area of pallet i of order j;p2ij processing time at the docking area of pallet i of order j;qBij number of SKU handled at the block stack storage area;qRij number of SKU handled at the racking storage area;mB

ij service rate of MHE at the block stack storage area;mR

ij service rate of MHE at the racking storage area;� coefficient factor of mR

ij to p2ij.

The objective is to minimise the total makespan of trucks:

minX�

yD� � yA�

� �( ),

where yD� and yA� are the departure and arrival time of truck �.The following constraints are considered:

(i) MHE assignment. Each pallet is assigned to only one MHE.

XKk¼1

x1ijk ¼ 1 8i ¼ 1, 2, . . . , Ij 8j ¼ 1, 2, . . . , J,

where x1ijk ¼1 if pallet i of order j is assigned to MHE k at the storage area

0 otherwise:

�

(ii) Order sequencing.In the model, s1ij denotes the start of the processing time of pallet i of order j at the storage area. The startingtime for delivering a pallet from the storage area to the docking zone must be at least later than theprocessing time of order picking: s2ij � s1ij þ p1ij 8i, j, where p1ij is the processing time at the storage area ofpallet i of order j.

(iii) Beginning of the horizon. The starting time of each order must be greater than or equal to 0.

s1ij � 0 8i, j:

(iv) Arrival and departure of trucks:

(a) After the pallet of an order is ‘completely’ processed (i.e. ready to be loaded to the truck), thecorresponding truck should have already arrived.

yA�j � s2ij þ p2ij 8i ¼ 1, 2, . . . , Ij 8j ¼ 1, 2, . . . , J:

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(b) The truck can leave the docking zone only after all pallets are loaded.

yD�j � s2ij þ p2ij 8i ¼ 1, 2, . . . , Ij 8j ¼ 1, 2, . . . , J:

(c) The departure time of truck � for order j must be before the due date.

yD�j � yDuej 8j ¼ 1, 2, . . . , J:

(v) MHE disjunction. Each MHE can deliver only one pallet at a time.

s1qr � s1ij þ p1ij or s1ij � s1qr þ p1qr 8k ¼ 1, 2, . . . ,K 8qr 6¼ ij such that x1ijk ¼ x1qrk ¼ 1:

s2qr � s2ij þ p2ij or s2ij � s2iqr þ p2qr 8k ¼ 1, 2, . . . ,K 8qr 6¼ ij such that x2ijk ¼ x2qrk ¼ 1:

(vi) Dock assignment. Each truck is assigned to only one dock.

XMm¼1

z�m ¼ 1 8� ¼ 1, 2, . . . ,NOT

where z�m ¼1 if the truck � is assigned to dock m

0 otherwise:

�

(vii) Dock disjunction. Each dock can accommodate only one truck at a time.

yAj � yDk or yAk � yDj 8m ¼ 1, 2, . . . ,M 8j 6¼ k assigned to dock m:

(viii) The processing time of the outbound order at the storage zone is

p1ij ¼Xi¼1

ðmBij :q

Bij þmR

ij :qRij Þ:

(ix) The order chain is expressed as

o1 ¼ ð p1ij þ p2ijÞ, where p

2ij ¼ � �m

Rij � q

Rij :

The objective is to minimise the waiting time of trucks. A penalty function is added to the objective functionto ensure the proper arrangement of the block stack and racking storage area for inbound and outbound orders.To be more realistic, we impose the constraints of truck availability and truck capacity. We also considerunforeseeable block stack and racking storage problems because of the effects of unpredictable inbound trucks andlimited storage space. This requires extra operation time for the loading of items to the racks for temporary storage;hence, productivity in the distribution hub is significantly affected. We also consider the underloading andoverloading situations of trucks in the docking area to address the issue of truck capacity, which is very commonin logistics. An underloaded truck is required to wait at the docking zone until the excess capacity is taken upby another order (on the same route); the truck should deliver part of the order first if an order is likely to resultin a truck overload. The rest of the order is handled based on truck availability.

5. Meta-heuristic model

The same set of stated assumptions and constraints that appear in Section 4 are adopted in developing the heuristicsalgorithm. The heuristics algorithm is constructed to optimise scheduling and dock assignments for trucks withperturbations (e.g. randomly arriving inbound orders), carry out the scheduling of order picking/storing and dockassignment within the shortest possible time, and minimise the total makespan of both inbound and outboundtrucks. The configurations and routines for both outbound and inbound orders, as shown in Figure 1, represent thesystematic algorithm for the distribution hub. There are two operations in the hub, namely, outbound and inboundoperations. An order chain is generated based on information obtained from the Enterprise Resource Planning(ERP) system, and this represents the sequence of truck dock assignment and corresponding order tasks.In determining the optimal dock assignment sequence, order tasks, including order pick ups and the storingsequence of orders, should be well defined. Figure 2 shows the five-step mechanism of the proposed model.

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Step 1: Initialisation of the population

It is assumed that outbound truck assignment for outbound orders is assigned by the ERP system. We also assume

that each outbound order has its predefined due time based on the request of the customer. A fundamental

two-dimensional matrix is then constructed to represent outbound truck assignment based on the ERP system

(Figure 3).In cross-dock situations, inbound trucks randomly arrive and are immediately handled. The original

fundamental two-dimensional matrix is modified into perturbed matrices to represent both inbound and outbound

truck dock assignment and precedence sequence constraints. The dimension of a perturbed matrix is defined as

nd � nt, where nd is the number of docks in the distribution hub, and nt is the number of trucks. Each element of a

matrix (MQR) represents a unique truck identity or null index. Therefore, nd rows are constructed, representing

the nd number of parallel constraints on truck sequences, in which the sequence of trucks is read from left to right.

The perturbed matrices are later modified in real time to represent a new truck docking sequence that is based on the

first-come first-served (FCFS) strategy. By doing this, perturbed matrices with population size np, which represent

different combinations of inbound and outbound truck assignments, are generated.

Step 2: Initialisation of the order chain

An order chain, representing a sequence of incomplete orders with the first precedence rating of the aisles within the

perturbed matrices, is generated based on the population generated in Step 1 (Figure 4). For each order chain, there

is an MHE origin time records set (T�

0), representing the time when each MHE can start to process the top-ranking

order in the given order chain. All elements of the origin time records set at the initial status are assumed to be the

current system time. The objective value of a chromosome (orders chain) is defined as �i ¼ ð p1ij þ p2ijÞ.

Figure 3. Fundamental two-dimensional matrix.

Figure 2. Mechanism of the proposed meta-heuristics model.

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Step 3: Evaluation of the makespan

The completion time of each inbound/outbound order job process is obtained by (i) reporting by staff members

and (ii) recording through a real-time data capturing device, such as an Radio Frequency Identification (RFID)

technology. The elements of the order chain, which follow the precedence constraints of the proposed orders and the

time records set, are also determined after all order completion time is collected. It is essential to minimise the wrong

arrangement of job order sequences/schedule (mRij ) to minimise the waiting time of trucks. Therefore, the swap

process in this proposed algorithm is performed. In the swap process, crossover and mutation operations are

conducted by re-allocating and changing the genes in the chromosomes to determine new offsprings with enhanced

makespan.

Step 4: Execution of swap processes

A two-point swapping crossover is applied to the swapping of two equal-length non-zeros sub-segments from

two rows of the selected matrix. An illustrative example of a two-point swapping crossover operator is shown

in Figure 5. Suppose the element of the selected matrix is represented asMQR, the Rth truck queues at the Qth dock,

and " and � are the first and last genes of the selected element, respectively. The sum of the selected elements cannot

be zero, such that the swapping process is not interpreted, that is,P�1

R¼"1MQ0R þ

P�2R¼"2

MQ00R 6¼ 0.The transition process of the order chain is created through a swap mutation operator (Figure 6). The mutation

operation is conducted by swapping nm pairs of the non-zero elements of the selected matrix through the swapping

mutation operators. The number of mutation pairs (nm), which is related to the variance of the objective values at

the parent jth iteration, is derived as shown in the following mathematical expression. When the variance increases,

we must select more pairs of non-zero elements for the mutation process.

nm ¼ exp �var �0j

� �var �0j�Dj

� � !

, where j� Dj ¼0 if j5Dj

j� Dj elsewhere

�:

A new order chain, called a transition order chain, is formed after a swapping process is completed. If the

objective value of the created transition order chain (�trans) is less than or equal to the objective value of the jth

parent iteration �0 ¼ minð�0jj j Þ or within the acceptance value (paccept), where paccept ¼ e��trans��0

��0 and � ¼ � ln 2, then

Selected matrix for transition

Q„th row

Created transition matrix after crossover

Q„th row

1ε 1κ 2ε2κ

2ε 2κ 1ε1κ

Q„„th row Q„„th row

Figure 5. Two-point swapping crossover operator.

Parent order chain before swap mutation Transition order chain after swap mutation

1ε 1κ 2ε2κ 1

'ε '1κ '

2ε '2

κ

Figure 6. Swapping mutation for an order chain.

O3O1 O2... … … … ON

Figure 4. Structure of an order chain.

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such a created transition order chain automatically replaces the original one. The looping steps are repeated until

two situations occur: (i) the minimum value of the average makespan is determined, and (ii) the number of iterations

for that transition order chain meets the termination condition nStep ¼ nmaxStep. The maximum number of iterations

in the algorithm is defined as nmaxStep ¼ n� ln n�, where n� is the number of trucks, and n� is the average number of

incomplete orders for the given order chain.

6. Experimental results and analysis

Three tests are conducted to evaluate the performance of the proposed algorithm under two different cross-dock

conditions: (i) the estimated arrival time of inbound and outbound orders is nearly the same, and (ii) the expectedarrival time interval between inbound and outbound orders is scaled at 0.3 of the operation period of the

distribution hub. Tables 1–3 present the results of the proposed algorithm under the two given conditions. There are

five problem sets within each cross-dock condition. The number of outbound orders in the problem set is the same

for all 10 instances, and each problem set runs 500 times. The number of inbound orders differs across instances.

The arrival of the trucks follows the Poisson distribution. Meanwhile, the waiting time for the arrival of trucks

follows an exponential distribution, and converges in a probability distribution to a statistically normal distribution.

(i) The estimated arrival time of inbound order �in and outbound order �out is the same, and the variance

collapses when compared with the operation period of the distribution hub (Tperiod) for the first case in

which a large number of inbound trucks arrive in the hub. The mathematical model is expressed as follows:

�in � �outj j

Tperiod¼ 0 and

�inTperiod

¼�outTperiod

¼ 1� 10�3:

(ii) The expected arrival time interval between inbound order �in and outbound order �out is scaled at 0.3 Tperiod

and is widely distributed for the second case in which a normal number of inbound trucks arrive at the hub.

The following mathematical model is used:

�in � �outj j

Tperiod¼ 0:3 and

�inTperiod

¼�outTperiod

¼ 0:1:

The heuristics algorithms are coded in Matlab and run on a personal computer (Intel Dual 2.0GHz 2.0GB

DDRAM). Within each table and using two algorithms, each problem set is simulated 10 times to determine

Table 1. Results of the FCFS dock assignment and the heuristics method on random instances with small inbound order sizes.

ConditionProblem

set

No. ofinboundorders

No. ofoutboundorders

Totalmakespan

(FCFS) (min)

Total makespan(Heuristics)(min)

Comparison of thetwo cases in termsof total makespan

Computationaltime of theheuristicscase (min)Worst Average Best (min) (%)

1st 1 10 6 709 641 593 541 116 16.36 9.112 30 6 3541 3152 2813 2687 728 20.56 5.323 40 6 5610 4652 4257 4094 1353 24.12 9.204 60 6 8416 6851 6230 5961 2186 25.97 7.305 70 6 11,263 10541 8269 8074 2994 26.58 8.02

Average 5907.8 5167.4 4432.4 4271.4 1475.4 22.72 7.79

2nd 1 10 6 261 234 223 216 38 14.56 1.412 30 6 449 431 426 417 23 5.12 1.713 40 6 698 670 656 638 42 6.02 1.684 60 6 1201 1093 1021 967 180 14.99 1.435 70 6 1520 1432 1338 1309 182 11.97 1.96

Average 825.8 772 732.8 709.4 93 10.53 1.64

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Table 2. Results of the FCFS dock assignment and heuristics method on random instances with medium size inbound orders.

ConditionProblem

set

No. ofinboundorders

No. ofoutboundorders

Totalmakespan(FCFS)(min)

Total makespan(Heuristics)(min)

Comparison of thetwo cases in termsof total makespan Computation time

of the heuristicscase (min)Worst Average Best (min) (%)

1st 1 110 6 18,412 16,652 15,198 14,854 3214 17.46 10.562 130 6 34,851 29,645 27,451 27,098 7400 21.23 9.963 140 6 36,211 30,744 28,544 28,161 7667 21.17 8.544 160 6 38,444 37,512 36,497 35,926 1947 5.06 7.945 170 6 41,219 40,851 39,844 38,653 1375 3.34 9.62

Average 33,827.4 31,080.8 29,506.8 28,938.4 4320.6 13.65 9.32

2nd 1 110 6 4311 4265 4132 4104 179 4.15 2.412 130 6 5895 5792 5731 5713 164 2.78 1.893 140 6 9143 8912 8516 8397 627 6.86 2.434 160 6 12,389 12,067 11,622 10,931 767 6.19 2.855 170 6 22,174 21,321 20,414 18,876 1760 7.94 3.49

Average 10,782.4 10,471.4 10,083 9604.2 699.4 5.58 2.61

Table 3. Results of the FCFS dock assignment and heuristics method on random instances with large size inbound orders.

ConditionProblem

set

No. ofinboundorders

No. ofoutboundorders

Totalmakespan

(FCFS) (min)

Total makespan(Heuristics)(min)

Comparison of thetwo cases in termsof total makespan Computation time

of the heuristicscase (min)Worst Average Best (min) (%)

1st 1 210 6 63,218 59,521 57,511 55,121 5707 9.03 12.392 230 6 78,513 72,152 69,461 67,845 9052 11.53 14.963 240 6 83,114 78,516 75,415 74,116 7699 9.26 15.324 260 6 92,845 90,127 89,313 88,695 3532 3.80 11.125 270 6 109,841 97,128 93,251 91,976 16,590 15.10 13.67

Average 85,506.2 79,488.8 76,990.2 75,550.6 8516 9.75 13.49

2nd 1 210 6 29,744 28,120 27,854 26,941 1890 6.35 3.022 230 6 32,954 32,511 31,454 30,894 1500 4.55 2.673 240 6 35,891 35,741 35,341 35,309 550 1.53 2.794 260 6 47,414 46,129 45,765 43,120 1649 3.48 4.15 270 6 50,259 49,125 48,123 47,652 2136 4.25 1.64

Average 39,252.4 38,325.2 37,707.4 36,783.2 1545 4.03 2.84

Table 4. Optimal strategies for real scenarios.

Condition

No. ofinboundorders

No. ofoutboundorders

Normal case oftotal makespan

(min)

Heuristics caseof total

makespan (min)

Comparisonof the two cases in

terms of total makespan Computation timeof the heuristics

case (min)Worst Average Best (min) (%)

1st 30 6 3541 3152 2813 2687 728 20.56 5.3240 6 5610 4652 4257 4094 1353 24.12 9.20

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the worst, average, and best total makespan of the heuristics algorithm. Total makespan comparisons between theFCFS dock assignment strategy and the heuristics algorithm are presented in Tables 1–4.

(1) Small inbound order size testFive problem sets under two conditions, with the number of inbound orders ranging from 10 to 70, aregenerated. The results are shown in Table 1. With an increasing number of inbound orders, long truckmakespans are observed under both FCFS dock assignment method and the proposed heuristics method.However, the proposed heuristics method provides better results on the total makespan than the FCFSmethod. The average makespan generated by the heuristics method is 22.72% shorter than that of the FCFSstrategy under the first condition (10.53% shorter under the second condition). The results prove that theproposed heuristics method is capable of substantially decreasing the total makespan of trucks—at areasonable computational runtime—when handling small sizes of inbound and outbound orders.

(2) Medium inbound order size testFive medium inbound order problem sets ranging from 110 to 170 inbound orders are used, and the resultsare presented in Table 2. The results are similar to those generated in the small inbound order size test.This means that the total makespan generated by the proposed heuristics method outperforms that of theFCFS dock assignment method. However, the average percentage deviations in the total makespan of thetwo approaches differ by nearly 40%; this is not observed in the small inbound order size test.

(3) Large inbound order size testFive large inbound order problem sets ranging from 210 to 270 inbound orders are tested. The results arepresented in Table 3. The average percentage deviations in the makespan generated by the two approachesare 9.75% and 4.03% in the first and second conditions, respectively.

These test results indicate that the proposed heuristics method is capable of better decreasing the total makespanin the distribution hub compared with the FCFS method. The proposed method provides a good way, with areasonable computational runtime, of addressing the random arrival of inbound orders, under both the firstand second conditions. The algorithm provides an optimal solution for coping with the limited storage spaceand single docking zone in Hong Kong’s distribution hubs. Considering that a hub with six docks operates for12 h a day, the total feasible makespan for handling inbound and outbound truck orders is 4320min(6 docks� 12 h� 60min¼ 4320min). Table 1 shows that the total feasible makespan for a truck in a distributionhub in Problem Sets 2 and 3 is 3541 and 5610min, respectively. Therefore, as illustrated in Table 4, the best solutionprovided by the proposed heuristics method for the distribution hub is between 30 and 40 inbound orders under thefirst condition, in which the estimated arrival time of the inbound and outbound orders is nearly the same.

7. Conclusion

A cross-dock job assignment problem in an industrial logistics distribution hub with a limited storage spaceis analysed. In contrast to current cross-docking studies, which deal with a situation where the arrival sequence ofinbound/outbound trucks in the shipping docks is the same as the sequence of their unloading in the receivingdocks, this research aims to address the job assignment problem in an industrial logistics hub operation environmentwith limited storage space and a single docking zone. This research proposes a method that can minimise the waitingtime of in-/outbound trucks by coordinating schedules for order picking/deliveries in the storage area. We formulatea mathematical programming model to represent the operational constraints and the time window of each palletfor both inbound and outbound orders in the distribution hub. A GA-based meta-heuristics algorithm is alsodeveloped to decrease computational time in solving medium- and large-scale problems.

This research is innovative because the proposed meta-heuristics algorithm allows manufacturers to insertadditional tasks into the schedule even with inbound orders arriving randomly, without significantly disturbingthe outbound order schedule. This is contrary to the usual practice of minimising the completion time of activities(makespan) or the total time spent by trucks in the docks.

Experiments are performed with a number of data sets, ranging from small to large, on inbound and outboundorders to examine the performance of the method under heavy and normal operational congestion scenarios in thedistribution hub. The overall results show that the heuristics method improves the operations performance of cross-dock operations by 10% to 20% in both heavy and normal operational congestion scenarios. The best solutionprovided by the proposed heuristics method for the distribution hub is between 30 and 40 inbound orders under the

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heavy-traffic situation. The above results highlight the contribution of this research in the industrial logistics systemsof manufacturers who operate cross-dock operations. This research also provides insights into possible researchdirections, including the optimal allocation of truck docking assignment and the coordination of job schedulesin a distribution hub characterised by limited storage space and a single docking zone.

Acknowledgements

The authors wish to thank the Research Office of the Hong Kong Polytechnic University and Mr Fermi Cheung, the office’sresearch assistant, for supporting the project (Project Code: A-SA07).

References

Bartholdi, J. and Gue, K., 2000. Reducing labor cost in an LTL cross-docking terminal. Operations Research, 48 (6), 823–832.

Chiang, T.C., Shen, Y.S., and Fu, L.C., 2008. A new paradigm for rule-based scheduling in the wafer probe centre. InternationalJournal of Production Research, 46 (15), 4111–4133.

Chien, C.F. and Chen, C.H., 2007. Using genetic algorithms (GA) and a coloured timed Petri net (CTPN) for modelling the

optimization-based schedule generator of a generic production scheduling system. International Journal of ProductionResearch, 45 (8), 1763–1789.

Heragu, S.S., et al., 2005. Mathematical model for warehouse design and product allocation. International Journal of Production

Research, 43 (2), 327–338.Hwarng, H.B., et al., 2005. Modelling a complex supply chain: understanding the effect of simplified assumptions. International

Journal of Production Research, 43 (13), 2829–2872.

Gue, K., 1995, Freight terminal layout and operations. Dissertation. Georgia Institute of Technology.Gue, K., 1999. Effects of trailer scheduling on the layout of freight terminals. Transportation Science, 33 (4), 419–428.Gunasekaran, A. and Kobu, B., 2007. Performance measures and metrics in logistics and supply chain management: a review

of recent literature (1995–2004) for research and applications. International Journal of Production Research, 45 (12),

2819–2840.Masel, D., 1998, Adapting the longest processing time heuristics for output station assignment in a transfer facility.

In Proceedings of the 7th Annual Industrial Engineering Research Conference, Phoenix, AZ.

McWilliams, D.L., Stanfield, P.M., and Geiger, C.D., 2005. Simulation optimization applied to the hub scheduling problem.Computers and Industrial Engineering, 49 (3), 393–412.

Mes, M., Heijden, M., and Harten, A., 2007. Comparison of agent-based scheduling to look-ahead heuristics for real-time

transportation problems. European Journal of Operational Research, 181 (1), 59–75.Miao, Z., Lim, A., and Ma, H., 2009. Truck dock assignment problem with operational time constraint within cross-docks.

European Journal of Operational Research, 192 (1), 105–115.Rajkumar, R. and Shahabudeen, P., 2009. An improved genetic algorithm for the flowshop scheduling problem. International

Journal of Production Research, 47 (1), 233–249.Safaei, A.S., et al., 2010. Integrated multi-site production-distribution planning in supply chain by hybrid modeling. International

Journal of Production Research, 48 (14), 4043–4069.

Tsui, L. and Chang, C., 1992. An optimal solution to dock assignment problem. Computers and Industrial Engineering, 23 (1–4),283–286.

Tsui, L.Y. and Chang, C.H., 1990. A microcomputer based decision support tool for assigning dock door in freight yard.

Computer and Industrial Engineering, 19 (1–4), 309–312.Yu, W. and Egbelu, P.J., 2008. Scheduling of inbound and outbound trucks in cross docking systems with temporary storage.

European Journal of Operational Research, 184 (1), 377–396.

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