The Truck Scheduling Problem at Crossdocking Terminals

Exclusive versus Mixed Mode

L. Berghman

, C. Briand

2,3

, R. Leus

and P. Lopez

Universit´e de Toulouse - Toulouse Business School, 20 BD Lascrosses BP 7010, 31068 Toulouse Cedex 7, France

CNRS, LAAS, 7 Avenue du Colonel Roche, F-31400 Toulouse, France

Universit´e de Toulouse, UPS, LAAS, F-31400 Toulouse, France

ORSTAT, KU Leuven, Naamsestraat 69, 3000 Leuven, Belgium

Keywords:

Crossdocking, Truck Scheduling, Parallel Machine Scheduling, Integer Programming.

Abstract:

In this paper we study the scheduling of the docking operations of trucks at a warehouse; each truck is either

empty and needs to be loaded, or full and has to be unloaded (but not both). We focus on crossdocking,

which is a recent warehouse concept that favors the transfers of as many incoming products as possible di-

rectly to outgoing trailers, without intermediate storage in the warehouse. We propose a time-indexed integer

programming formulation for scheduling the loading and unloading of the trucks at the docks, and we distin-

guish between a so-called “mixed mode”, in which some or all of the docks can be used both for loading as

well as unloading, and an “exclusive mode”, in which each dock is dedicated to only one of the two types of

operations. Computational experiments are provided to compare the efﬁciency of the two modes.

1 INTRODUCTION

Crossdocking is a warehouse management concept

in which items delivered to a warehouse by in-

bound trucks are immediately sorted out, reorganized

based on customer demands and loaded into outbound

trucks for deliveryto customers, without requiring ex-

cessive inventory at the warehouse ((van Belle et al.,

2012)). If any item is held in storage, it is usually

for a brief period of time that is generally less than

24 hours. Advantages of crossdocking can accrue

from faster deliveries, lower inventory costs, and a re-

duction of the warehouse space requirement ((Apte

and Viswanathan, 2000; Boysen, 2010)). Compared

to traditional warehousing, the storage as well as the

length of the stay of a product in the warehouse is lim-

ited, which requires an appropriate coordination of in-

bound and outbound trucks ((Boysen et al., 2010; Yu

and Egbelu, 2008)).

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. The problem studied concerns the

operational level: trucks are allocated to the differ-

ent docks so as to minimize the storage usage dur-

ing the product transfer. The internal organization of

the warehouse (scanning, sorting, transporting) is not

explicitly taken into consideration. We also do not

model the resources that may be needed to load or

unload the trucks, which implies the assumption that

these resources are available in sufﬁcient quantities to

ensure the correct execution of an arbitrary docking

schedule.

There exist two different service modes, which de-

pend on the degree of freedom in assigning inbound

and outbound trucks to docks. In the exclusive mode,

each dock is exclusively dedicated either to inbound

or to outbound operations. It is a widespread guide-

line in real-world terminals to ease product ﬂows and

supervision ((Boysen and Fliedner, 2010)). Typically,

one side of the terminal is dedicated to inbound and

the other to outbound operations. In the mixed mode,

on the other hand, an intermixed sequence of inbound

and outbound trucks to be processed per dock is al-

lowed. As stated in (Carlo and Bozer, 2011), in a

typical crossdock application, once a dock is classi-

ﬁed as an inbound (or an outbound) dock, it remains

that way until the docks are reclassiﬁed. Their ex-

periments show, however, that grouping the inbound

docks together and the outbound docks together is

generally not a good conﬁguration to use when the

decision maker wants to minimize the travel distance

of the forklifts, which follow a rectilinear travel path

between the doors inside the warehouse. Remark that

247

Berghman L., Briand C., Leus R. and Lopez P..

The Truck Scheduling Problem at Crossdocking Terminals - Exclusive versus Mixed Mode.

DOI: 10.5220/0005205102470253

In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 247-253

ISBN: 978-989-758-075-8

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

the exclusive mode resembles a hybrid ﬂow shop: the

inbound and outbound docks are the ﬁrst and the sec-

ond stage, respectively. The mixed mode, on the other

hand, resembles a parallel machine scheduling prob-

lem with precedence constraints: both the inbound

and the outbound trucks are scheduled on the same

set of identical machines.

The purpose of this paper is to propose a mathe-

matical formulation of the truck scheduling problem

at crossdocking terminals operating in mixed mode

and to show its interest by comparison with the exclu-

sive mode. It is structured as follows. Section 2 gives

a brief overview of the relevant literature on the prob-

lem under study. A detailed problem description can

be found in Section 3. A time-indexed formulation is

presented in Section 4. The results of the computa-

tional experiments can be found in Section 5. Finally,

some conclusions round off the paper in Section 6.

2 LITERATURE REVIEW

The concept of crossdocking has received a lot of at-

tention in recent literature: cases with one receiving

and one shipping door are most frequently studied.

A comprehensive overview of different variations and

the available literature can be found in (Boysen and

Fliedner, 2010), (van Belle et al., 2012) and (Ma-

knoon, 2013).

(Alpan et al., 2011a) and (Alpan et al., 2011b),

on the one hand, consider a multiple-door crossdock

environment with exclusive mode of service, where

preemption of loading operations is allowed. Each

outbound truck serves a single destination; each in-

bound truck can contain products for several desti-

nations. There are several ways to treat the product

ﬂows: (i) products can be transshipped directly from

an inbound to an outbound truck if one is available;

(ii) they can be temporarily stored to be loaded later

on; or (iii) an outbound truck can be replaced to fa-

cilitate direct loading. Contrary to our model, the

objective here is not time-related; the sum of the in-

ventory holding cost (per unit product) and the truck

replacement cost is minimized. As the sequence of

the inbound trucks is known, the problem consists in

scheduling the outbound trucks. (Alpan et al., 2011b)

try to ﬁnd optimal or near-optimal scheduling poli-

cies using dynamic programming while (Alpan et al.,

2011a) present several heuristics based on constrain-

ing the solution space that is generated by the dy-

namic programming model of (Alpan et al., 2011b).

Numerical experiments show that the heuristics can

ﬁnd near-optimal solutions much faster.

(Miao et al., 2009), on the other hand, consider

the truck-dock assignment problem with operational

time constraint for the mixed mode. For each pair

of inbound and outbound trucks, the number of pal-

lets that has to be transferred from the inbound to the

outbound truck is deﬁned. Once more, the objective

is not time-related; the authors aim to minimize the

number of unfulﬁlled shipments and the total ship-

ment costs at the same time. The problem is for-

mulated as an integer programming model and two

metaheuristics are proposed: tabu search and a ge-

netic algorithm. It turns out that for medium-size and

large-size instances, the metaheuristic approaches are

preferred in order to get quick and good solutions.

Some articles in literature model the truck

scheduling problem at crossdocking terminals with

exclusive mode as a machine scheduling problem. In

(Chen and Lee, 2009) and (Chen and Song, 2009), the

crossdocking environment is treated as a two-stage

ﬂow shop, but only instances with a small number

of docks are considered. The number and types of

products to be loaded (unloaded) per outbound (in-

bound) truck are not deﬁned a priori, but each job

in the second stage (outbound; loading) can be pro-

cessed only after the processing of some jobs in the

ﬁrst stage (inbound; unloading). (Chen and Lee,

2009) show that the problem is NP-hard for the two-

machine case where the objective is to minimize the

makespan. Furthermore, they present a polynomial

approximation algorithm and a branch-and-bound al-

gorithm. (Chen and Song, 2009) consider the hybrid

case where at least one stage contains more than one

machine; they present a mixed integer programming

model for small-scale instances and different heuris-

tics for moderate and large-scale instances.

(Li et al., 2004) use JIT scheduling to solve the

problem of scheduling loading and unloading activi-

ties when the goal is to complete processing each con-

tainer exactly at its due date. Each incoming container

has a release time and a due date and each outgoing

container has a due date. The crossdock can be di-

vided into an import area and an export area. Products

have known destinations before they enter the cross-

dock, such that precedence relationships arise. They

present an integer programming model, as well as two

heuristics for this NP-hard problem. The ﬁrst uses

squeaky wheel optimization ((Joslin and Clements,

1999)) embedded in a genetic algorithm and the sec-

ond uses linear programming within a genetic algo-

rithm. (

Alvarez P´erez et al., 2009) consider the same

problem and present a solution approach based on a

combination of two metaheuristics, reactive GRASP

and tabu search. They conclude that their algorithm

is an excellent alternative to the approach of (Li et al.,

2004). Note that JIT scheduling as considered in the

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two above papers amounts to minimizing the differ-

ence in completion time between each pair of tasks

that is involved in a precedence constraint. The de-

tails of the problems studied in the latter references

are such that the proposed algorithms are not suitable

for our setting.

3 DETAILED PROBLEM

STATEMENT

We examine a crossdocking warehouse where incom-

ing trucks i ∈ I need to be unloaded and outgoing

trucks o ∈ O need to be loaded (where I is the set

containing all inbound trucks while O is the set con-

taining all outbound trucks). The warehouse features

n docks that can be used both for loading and un-

loading (mixed mode). The processing time of truck

j ∈ I ∪ O equals p

. This processing time includes

the loading or unloading but also the transportation of

goods inside the crossdock and other handling opera-

tions between dock doors. It is assumed that there is

sufﬁcient workforce to load/unload all docked trucks

at the same time. Hence, a truck assigned to a dock

does not wait for the availability of a material handler.

The products on the trucks are packed on unit-size

pallets, which move collectively as a unit: re-packing

inside the terminal is to be avoided. Each pallet on

an inbound truck i needs to be loaded on an outbound

truck o, which gives rise to a precedence constraint

(i,o) ∈ P ⊂ I×O, with P the set containing all couples

of inbound trucks i and outbound trucks o that share a

precedence constraint. Each truck j has a release time

(planned arrival time) and a deadline

(its latest

departure time).

Products can be transshipped directly from an in-

bound to an outbound truck if the outbound truck is

placed at a dock. Otherwise, the products are tem-

porarily stored and will be loaded later on. Each

couple (i,o) ∈ P has a weight w

, representing the

number of pallets that go from inbound truck i to

outbound truck o. The objective is to minimize the

weighted sum of sojourn times of the pallets stocked

in the warehouse. According to (Boysen and Fliedner,

2010), this is a valuable objective because the cross-

docking concept relies on a rapid turnover of ship-

ments. It also reduces the danger of late shipments:

the number of products in the storage area can only

be decreased by loading them on outbound trucks to

leave the terminal as early as possible. Moreover, a

lower stock size also reduces the material handling

effort inside the terminal. Remark that the time spent

by a pallet in the storage area is equal to the ﬂow time

of the pallet: the difference between the start of load-

ing the outbound trailer and the start of unloading the

inbound trailer.

Our problem can be modeled as a parallel machine

scheduling problem with release dates, deadlines, and

precedence constraints, denoted by Pm|r

, prec|−.

As this problem is a generalization of the 1|r

|−

problem which is NP-complete ((Lenstra et al.,

1977)), even ﬁnding a feasible solution for the prob-

lem is NP-complete.

For all trucks j ∈ I ∪O, let s

be the starting time

of the handling of truck j. A conceptual problem

statement with these variables is the following:

min z =

∑

(i,o)∈P

− s

) (1)

subject to

≥ r

∀ j ∈ I ∪ O (2)

+ p

≤

∀ j ∈ I ∪ O (3)

− s

≥ 0 ∀(i,o) ∈ P (4)

| ≤ n ∀τ ∈ T (5)

with A

= { j ∈ I ∪ O| s

< τ ≤ s

+ p

} the set con-

taining all tasks being executed during time period τ

and T the set containing all time periods considered

(time horizon). The objective function (1) minimizes

the total weighted usage of the storage area. Con-

straints (2) and (3) impose the time windows for all

trucks. Constraints (4) ensure that, if there exists a

precedence constraint between inbound truck i and

outbound truck o, then o cannot be processed be-

fore i. Finally, constraints (5) enforce the capacity

of the docks.

Remark that if we replace constraint (5) by two

capacity constraints (one for the inbound docks with

right-hand side n

equal to the number of inbound

docks and one for the outbound docks with right-hand

side n

equal to the number of outbound docks), we

obtain a formulation for the exclusive mode. This can

be easily done for the time-indexed formulation pre-

sented in the next section as well.

4 TIME-INDEXED

FORMULATION

A time-indexed formulation discretizes the continu-

ous time space into periods τ ∈ T of a ﬁxed length.

Let period τ be the interval [t − 1,t[. It is well

known that time-indexed formulations perform well

for scheduling problems because the linear program-

ming relaxations provide strong lower bounds ((Dyer

and Wolsey, 1990)). For this reason, we will test

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249

the integer programming formulation below, which is

called F1 in the sequel.

For all inbound trucks i ∈ I and for all time periods

τ ∈ T

, we have

iτ











1 if the unloading of inbound truck i is

started during time period τ,

0 otherwise,

with T

= {r

+ 1,r

+ 2,.. . ,

− p

+ 1}, the relevant

time window for inbound truck i. Additionally, for all

outbound trucks o∈ O and for all time periods τ ∈ T

we have

oτ











1 if the loading of outbound truck o is

started during time period τ,

0 otherwise,

with T

= {r

+1,r

+2,...,

− p

+1}, the relevant

time window for outbound truck o.

A time-indexed formulation for the considered

truck scheduling problem is the following:

min z =

∑

(i,o)∈P

∑

τ∈T

τ(y

oτ

− x

iτ

) (6)

subject to

∑

τ∈T

iτ

= 1 ∀i ∈ I (7)

∑

τ∈T

oτ

= 1 ∀o ∈ O (8)

∑

τ∈T

τ(x

iτ

− y

oτ

) ≤ 0 ∀(i,o) ∈ P (9)

∑

i∈I

∑

u=τ−p

∑

o∈O

∑

u=τ−p

≤ n ∀τ ∈ T (10)

iτ

∈ {0,1} ∀i ∈ I;∀τ ∈ T

(11)

oτ

∈ {0,1} ∀o ∈ O;∀τ ∈ T

(12)

The objective function (6) minimizes the total

weighted usage of the storage area. Constraints (7)

and (8) demand each truck to be assigned to exactly

one gate. Constraints (9) ensure that if there exists

a precedence constraint between inbound truck i and

outbound truck o, then o cannot be processed before i.

Constraints (10) enforce the capacity of the docks.

An alternative precedence constraint is the follow-

ing:

∑

u=1

− y

oτ

≥ 0 ∀(i,o) ∈ P;∀τ ∈ T (13)

Informally, this constraint states that in fractional so-

lutions, the loading task can only be started up to the

fraction to which the unloading task has been started.

(Christoﬁdes et al., 1987) call this constraint disag-

gregated. The formulation obtained by replacing con-

straint (9) in formulation F1 by (13) will be referred

to as F2. When we take a look at the polyhedron that

contains all feasible solutions for the LP-relaxations,

F2 is theoretically stronger since each feasible solu-

tion for the LP relaxation of formulation F2 is also

a feasible solution to the LP relaxation of formula-

tion F1. (Laborie and Nuijten, 2008) observe, how-

ever, that the additional CPU time needed to solve the

larger linear program can counterbalance the signif-

icant improvement of the bound. Both formulations

will be tested empirically.

Although in mixed mode all gates can serve both

to unload incoming trailers and to load outgoing trail-

ers, it might not be needed that every gate has this

double purpose. It is possible that in an optimal

schedule, at some gates only incoming trailers are un-

loaded and at other gates only outgoing trailers are

loaded. Indeed, since switching completely to mixed

mode might impact signiﬁcantly the company orga-

nization, both because of the placing of the docks

and because of the internal transportation within the

warehouse, it is worth determining the gain obtained

when switching only a small number of docks from

exclusive to mixed mode. We remark that when a

warehouse is expanding, the additional docks can be

mixed docks, or changing only a limited number of

docks does not necessarily change the internal orga-

nization of the warehouse in a drastic way. To de-

termine the minimal number of gates that has to be

double purpose so that the optimal objective value is

kept, we can work as follows. We refer to n

(respec-

tively n

) as the number of gates that are used for un-

loading (loading) purposes. Moreover, let δ

(δ

) be

the number of gates that we allow to unload (load) in-

coming (outgoing) trailers, on top of n

), during

certain time periods and deﬁne δ

∗

, δ

∗

as the optimal

values for these variables. A schematic representa-

tion is given in Figure 1. In a ﬁrst stage, we solve

the formulation F1, which gives the optimal objective

value z

∗

. In a second stage, we solve a second time-

indexed formulation: we minimize δ = δ

+δ

subject

to (7)–(12) and we add the following constraints:

∑

(i,o)∈P

∑

τ∈T

τ(y

oτ

− x

iτ

) ≤ z

∗

(14)

∑

i∈I

∑

u=τ−p

≤ n

+ δ

∀τ ∈ T (15)

∑

o∈O

∑

u=τ−p

≤ n

+ δ

∀τ ∈ T (16)

+ n

= n (17)

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Figure 1: Schematic representation.

The values of δ

and δ

are calculated in constraints

(15) and (16), respectively. Constraints 10 ensuring

in any case that the sum of used gates never exceeds

the total gate capacity. By minimizing their sum, we

minimize the number of gates that are used both for

loading as well as for unloading.

5 COMPUTATIONAL RESULTS

To the best of our knowledge, the problem studied in

this paper was never studied as such. Thus, we create

new instances in line with (Chen and Song, 2009) and

(Li et al., 2004) in the following way.

Remark that we rounded all fractional values to

obtain integer data. The number of gates is n ∈

{10, 20,30}. The number of inbound trucks is |I| ∈

{3n, 4n,5n} and the number of outbound trucks is

|O| ∈ {0.8|I|,|I|,1.2|I|}. Since the time needed to

unload one pallet equals one time unit, the time

needed to unload a trailer i, equals the number of pal-

lets to be unloaded. The processing time p

is uni-

formly distributed in [a,30] with a ∈ {10,20,30}. For

each inbound truck i, the number of outbound trucks

in which goods of truck i will be loaded is nb

∈

{1,.. . ,

}. The number of precedence constraints is

determined by γ: the larger γ, the more precedence

constraints. These outbound trucks are chosen ran-

domly. The number of pallets that will be charged

from this inbound truck i to one of the correspond-

ing outbound trucks o is nb

∈ {0.8

,1.2

There exists a precedence constraint (i,o) ∈ P be-

tween an inbound trailer i and an outbound trailer o

when at least one pallet is associated with both in-

bound trailer i and outbound trailer o. The weight

of a precedence constraint (i,o) ∈ P is equal to

the number of pallets that need to be unloaded from

inbound trailer i and loaded to outbound trailer o

afterwards. The time needed to load an outbound

trailer o is equal to the number of pallets that will be

loaded in this trailer. The ready times r

for the in-

bound trailers are uniformly distributed in [0, α

∑

]

with α ∈ {0.3, 0.6,0.9}. The deadlines for the out-

bound trailers are in

∈ [φΩ

,βΩ

] with Ω

max

(i,o)∈P

+ p

}. The length of the time horizon

is |T| = max

o∈O

{

}. The ready times r

for the out-

bound trailers are uniformly distributed in [r

max

−

] with r

max

= max

(i,o)∈P

}. The deadlines for

the inbound trailers

are uniformly distributed in

[1.5(r

+ p

max

] with

max

= min{min

(i,o)∈P

{

−

} + p

,max

(i,o)∈P

{

}}.

Remark that γ, φ and β are parameters. The tight-

ness of the time windows is determined by φ and β:

the smaller the difference between β and φ, the tighter

the time windows. We will assign different values to

both γ and β to obtain different datasets.

All models are encoded in C using the Microsoft

Visual Studio programming environment, and exe-

cuted on a PC computer with an Intel Core i3-2350M

CPU 2.30-GHz processor and 2 GB RAM, equipped

with Windows 7. ILOG CPLEX 12.4 is used to solve

the models.

As preliminary experiments, we created 567 in-

stances : we ﬁxed the parameters γ = 3, φ = 1.5 and

β = 5 and we created three instances for each combi-

nation of the other parameters. An instance is named

after its parameters : “n

|I| |O| a α index”. We have

implemented both constraints (9) and (13) as prece-

dence constraints and we have remarked that the for-

mulation performs better with constraint (9). The fol-

lowing results refer to this formulation. Detailed re-

sults can be found in Table 1: for each set of in-

stances, the average number of instances that were

proved to be infeasible, the average number of in-

stances for which a feasible solution was found that

was not proven to be optimal and the average num-

ber of instances for which an optimal solution was

found are mentioned both for exclusive and for mixed

mode. Overall, for the exclusive mode, 22% of the

instances were proved to be infeasible, while none of

them were proved infeasible for the mixed mode. This

is a ﬁrst evidence that companies can expect more

ﬂexibility when switching from exclusive to mixed

mode. For the exclusive mode, a feasible solution

was found within a time limit of ﬁve minutes for 69%

of the instances; for 2% of all instances, this solu-

tion was proven to be an optimal solution. For the

mixed mode, a feasible solution was found for 93%

of the instances, and for 3% of all instances this solu-

tion was optimal. When we only look at the instances

for which an optimal solution was found, the aver-

age computation time to ﬁnd these optimal solutions

is 70 seconds for the exclusive mode and 23 seconds

for the mixed mode. To give an idea about the qual-

ity of the solutions founds, we mention that the av-

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251

Table 1: Computational results.

exclusive mode mixed mode

n |I| infeasible feasible optimal infeasible feasible optimal

10 30 23.81% 65.08% 11.11% 0.00% 84.13% 15.87%

10 40 23.81% 74.60% 1.59% 0.00% 96.83% 3.17%

10 50 26.98% 63.49% 0.00% 0.00% 98.41% 0.00%

20 60 12.70% 82.54% 3.17% 0.00% 96.83% 3.17%

20 80 22.22% 73.02% 0.00% 0.00% 98.41% 0.00%

20 100 23.81% 61.90% 0.00% 0.00% 90.48% 0.00%

30 90 15.87% 77.78% 0.00% 0.00% 95.24% 4.76%

30 120 19.05% 65.08% 0.00% 0.00% 95.24% 0.00%

30 150 28.57% 53.97% 0.00% 0.00% 79.69% 0.00%

total 21.87% 68.61% 1.76% 0.00% 92.95% 3.00%

Figure 2

erage GAP between the best solution found within 5

minutes and the optimal solution of the linear relax-

ation of the formulation is 13,28%. The average GAP

with respect to a Lagrangian relaxation that we imple-

mented is 6,73%. When we compare the instances for

which a feasible solution was found both for the ex-

clusive and mixed modes, we calculated an improve-

ment of 8% of the objective value with a mixed mode.

This is a second evidence that companies might take

proﬁts from switching to a mixed organization.

We minimized the number of double purpose

gates δ for the 17 instances for which we found an

optimal solution for the mixed mode. For 6 of these

instances, we found a feasible solution that was not

guaranteed to be optimal; the average number of dou-

ble purpose gates is 49%. For the other instances, we

found an optimal solution. The average number of

double purpose gates is 27% and the average compu-

tation time is 50 seconds.

In Figures 2 and 3, we illustrate the results ob-

tained with three instances having 10 docks. On the

horizontal axis, we display the number of docks δ =

Figure 3

+ δ

(out of 10) switched in mixed mode. On the

vertical axis, we indicate the GAP = 100∗ (z− z

∗

)/z

∗

between the obtained solution z and the optimal solu-

tion z

∗

when all docks are in mixed mode. In Figure 2,

we can see that for instances with a rather small differ-

ence between exclusive and mixed mode (some 2%),

only changing one, two, or three docks (out of 10) to

mixed mode is enough to obtain the optimal solution

obtained when all docks are in mixed mode. Figure 3

shows instances for which the exclusive mode is in-

feasible. Having only one dock in mixed mode allows

ﬁnding a feasible solution, and better solutions are ob-

tained when the number of mixed gates increases.

6 CONCLUSIONS

We have presented a time-indexed (integer program-

ming) formulation for the truck scheduling problem

at crossdocking terminals. We have experimentally

compared the mixed mode strategy with the exclu-

sive one. As might be expected, the results conﬁrm

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252

that it is easier to ﬁnd a feasible solution, or even an

optimal one, when handling terminals operating with

a mixed mode. Moreover, our experiments provide

insight into the number of gates to be changed from

exclusive to mixed in order to guarantee the best per-

formance.

For future research, it may be interesting to inves-

tigate the special case of the problem with p

= p.

The complexity of Pm|r

, p

= p|

∑

is open

((Kravchenko and Werner, 2011)) and this problem is

a special case of our problem with p

= p : take |I| = 1

with i ∈ I, deﬁne

= r

+ p

such that s

= r

and de-

ﬁne for all o ∈ O, r

′

= max{r

+ p

} and w

= w

Another interesting problem is an extension in

which trailers are allowed to remain at the gate longer

than strictly needed for loading or unloading. In this

way, the number of direct transfers from inbound to

outbound trailers can be augmented and consequently,

the usage of the storage area can be decreased.

ACKNOWLEDGEMENTS

This research was part of ROCKS project supported

by CNRS/INS2I (Centre National de la Recherche

Scientiﬁque/Institut des Sciences de l’Information et

de leurs Interactions), PICS action n 6421.

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