A Hybrid Ant Colony and Genetic Algorithm to solve the Container Stacking Problemat Seaport Terminal

Ndeye Fatma NDIAYELaboratory of Applied mathematics of

Le Havre25 rue Philippe Lebon, B.P. 540,76063 Le Havre cedex, France

Email: farlou@live.fr

Adnan YASSINESuperior institute of logistic studies

Quai Frissard, B.P. 1137, 76063Le Havre cedex, France

Email: adnan.yassine@univ-lehavre.f

Ibrahima DIARRASSOUBAUniversity Institute of Technology,

Place Robert Schuman,76600 Le Havre

cedex, FranceEmail: diarrasi@univ-lehavre.fr

Abstract—This article addresses the container storage problem(CSP) in a port terminal. The good management of thestorage space is essential to ensure the productivity of a port.This explains the usefulness of studying this problem. In thispaper, we consider a modern container terminal which usesstraddle carriers instead of internal trucks. We propose alinear mathematical model which takes into account operationalconstraints and minimizes the total distance traveled by straddlecarriers between the quays and the container yard. For thenumerical resolution, we design an effective hybrid ant colonyand genetic algorithm (HAC/GA). Numerical simulations andcomparisons made with CPLEX show that this algorithm givesvery good results.

Keywords: Storage container, mathematical modeling, CPLEX,ant colony algorithm, genetic algorithm, hybrid algorithms.

I. INTRODUCTION

In a container terminal, several activities are performed.Among the most studied include: the berth allocation problem,the quay cranes scheduling problem, and the managementof the storage yard. In this paper, we focus on the containerstorage problem at a seaport terminal. Generally, thereare three kinds of containers: inbound, outbound, andtransshipment containers. All these containers are temporarilyplaced in the storage yard. Outbound containers leave theport onto ships, whereas they are brought by external trucks.Unlike them, inbound containers are unloaded from ships,and then after staying at the storage areas, they are claimedby internal trucks. Transshipment containers are special cases,they are unloaded from some ships, and thereafter, they areloaded onto other vessels after staying in the storage yard.

The operating modes of container terminals differ fromone port to another. But they all have the same goal whichconsist in satisfying customers and to be very attractive.There are different criteria to measure the service level of aport. In this study, we consider two of them: the waiting timeof external trucks, and the turnaround time of ships. Whenan external truck comes at port and claims a container, itwaits during all the time required to extract it. In some cases,the desired container is not directly accessible. Therefore, itis necessary to move in first the obstructive containers. Thefact of moving some containers in order to access to other

is named a reshuffle. It is very important to avoid this kindof movements, because they are unproductive and require somuch time. One of the reason which justify the importance ofreducing the turnaround time of ships is the fact that generallythe number of berths in a port is limited. Therefore it isbetter to quickly free them in order to assign them to otherincoming vessels. Another reason is the fact that shipownersprefer a quick service because this allows them to reduce thedowntime of ships and also the rental cost.

The kind of equipment used in a container terminal isvery important, because it must be compatible to the layoutof the storage yard. In this paper, we consider a moderncontainer terminal which use straddle carriers. These vehiclessubstitute the internal trucks and also the yard cranes, becauseeach of them is able to pick up itself one container and thengo stacking it in the storage yard. In order to allow goodcirculation of straddle carriers in the storage areas, thereare small spaces between rows of containers. Each row iscomposed of several stacks. The number of containers in astack must be inferior or equal to the limit fixed by the portauthorities. The container yard is divided into blocks, eachcontaining several rows.Figure 1 is a representation of a storage block.

In this paper, we propose an effective hybrid algorithm tosolve the container storage problem. This algorithm combinesan ant colony algorithm and a genetic algorithm. We also givea linear mathematical model which determines the optimalstorage plan and minimizes the total distance traveled by thestraddle carriers between the quays and the storage yard.

The remainder of the paper is organized as follows: aliterature review is given in the second part, the addressedproblem is detailed in the third part, the mathematical model isexposed in the fourth section, the hybrid algorithm is itemizedin the fifth section, the numerical results are presented in thesixth section, a conclusion is given in the seventh and lastsection.

II. LITERATURE REVIEW

One of the goals of papers dealing with inboundcontainers is the minimization of reshuffles. In [1], there arethree different strategies of storage, which are proposed by

International Conference on Advanced Logistics and Transport

978-1-4799-4839-0/14/$31.00 ©2014 IEEE 247

Figure 1. A storage block

Sauri et al. They aimed to find the best strategy to storeinbound containers which minimized re-handles. For this,they proposed a mathematical probabilistic model whichevaluated the number of reshuffles. A segregation strategyis exploited in [2], by Kim et al. In this method, emptystorage spaces are assigned to each vessel. Therefore, itis not possible to mix containers which are unloaded atdifferent periods. So, storage areas are allocated to ships in amanner which minimized the expected number of reshuffles.In [3], CAO Jinxi et al. addressed simultaneously the trucksscheduling and the storage of inbound containers. For thisthey proposed an integer programing model which minimizedthe number of congestions, the waiting time of trucks,and the unloading time of containers. For the numericalresolution, they proposed a genetic algorithm and a heuristicwhich procured them best results. An automatic containerterminal is considered in [4], by Yu et al. They minimizedreshuffles in two step. At first, they resolved the block spaceallocation problem. In the second step, they dealt with there-marshaling processes, which consisted to reorganize theblock space allocation. They attempted to minimize thenumber of reshuffles; and so, proposed three mathematicalmodels of storage: a non segregation model, a single-periodsegregation model, and a multiple-period segregation model.A convex cost network flow algorithm was applied to the twofirst models, while a dynamic programing was used for thethird. To numerically resolve the re-marshaling problem, theydesigned a heuristic algorithm. In a previous work [5], weproposed a hybrid algorithm including genetic algorithm andsimulated annealing to solve the container storage problem.We realized a new mathematical model which minimized thetotal distance traveled between quays and the storage areas.This model provided the optimal storage plan which assignedcontainers to locations without occasioning any reshuffles.

In [6], Kim et al. addressed the storage problem ofinbound containers without looking after reshuffles. In orderto invite customers to quickly collect their containers, theydetermined a limited free time storage beyond which therewill be storage costs. For this, they proposed a mathematicalmodel which gave them the optimal price schedule.

Unlike to the articles which deal with the storageof inbound containers, those interested in the storage ofoutbound containers have different goals. In [7], Preston et al.minimized the time service of container ships. They proposeda container location model (CLM), and a genetic algorithmto solve it. In [8], Kim et al. minimized the number ofrelocations expected during the ships loading. They assignedstorage spaces to outbound containers depending on theirweight. For this, they developed a dynamic programmingmodel and a decision tree to support real time decisions. In[9], Chen et al. presented a mixed integer programming modelwhich determines the number of necessary yard bays and the

number of locations to use in each of them. After that, eachcontainer is assigned to an exact location. In [10], Woo etal. assigned adjacent stacks per group of containers whichhave the same attributes. They also determined the size ofspace necessary to store all the outbound containers. In [11],Kim et al. considered two kinds of transfer: one is direct andthe other is indirect. For each of them they created a linearmathematical model of storage containers. They proposedtwo heuristic algorithms for the numerical resolution. That ofthe direct transfer method was based on the duration-of-stayof containers, while that of the indirect transfer method wasbased on the sub-gradient technical optimization.

III. CONTEXT

In this paper, we don’t only assign containers to storageblocks, but we specify the exact location of each inboundcontainer. Indeed, after the unloading of these containers fromships by the quay cranes, they are placed on the docks.Thereafter, they are picked up by straddle carriers whichtransfer and stack them in the storage yard following a planprovided by the port authorities. In our work, we determine theoptimal storage plan that satisfies the operational constraints.For this, we mainly consider the following assumptions:(1) Reshuffles are prohibited. This means some containers maystay at quay until suitable storage locations become vacant, ifthere is not enough available storage space.(2) In each stack, containers are disposed following the de-scending order of their departure times.(3) We consider that the number of containers in a stack mustnot exceed three.(4) We don’t mix in a stack containers which don’t have thesame dimensions.(5) We take into account containers which are already presentin the storage areas at the beginning of each period.

IV. MATHEMATICAL MODEL

A. Notations

We use the following notations.

Indices:

p: stack,i: empty slot in a stack,k: container.

Data:

N : number of containers,Np: number of stacks,cp: number of empty slots in the stack p,rp: size of container which can be placed in the stack p,tp: departure time of the container which was on the top

of the stack p at the beginning of the new storage period,Rk: size of the container k,Tk: departure time of the container k,dkp: traveled distance to transport the container k from

quay to stack p.

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Decision variables:

xki,p =

{1 If container k is assigned to the

empty location i in the stack p0 Otherwise

B. The proposed model

We model the problem as follows :

MinimizeN∑k=1

Np∑p=1

cp∑i=1

xki,pdkp (1)

Np∑p=1

cp∑i=1

xki,p = 1, ∀k = 1, ..., N (2)

N∑k=1

xki,p ≤ 1, ∀p = 1, ..., Np, ∀ i = 1, ..., cp (3)

cp∑i=1

xki,p = 0, ∀ p = 1, ..., Np, ∀ k = 1, ..., N,

rp 6= Rk or tp < Tk

(4)

N∑k=1

Tkxki,p ≥

N∑k=1

Tkxki+1,p,

∀ p = 1, ..., Np, ∀ i = 1, ..., cp − 1

(5)

xki,p ∈ {0, 1}, ∀ k = 1, ..., N, ∀ p = 1, ..., Np,

∀ i = 1, ..., cp(6)

The objective function (1) minimizes the total distancetraveled by straddle carriers between quays and the containeryard. Constraints (2) require that each container is assigned toa single location. Constraints (3) ensure that several containersaren’t assigned to a same empty slot. Constraints (4) securethe compatibility between containers and stacks. Constraints(5) impose that containers are stored following the descendingorder of their departure times in every stack.

V. RESOLUTION METHODS

To solve the container storage problem, we propose anew hybrid algorithm (HAC/GA) by combining a geneticalgorithm (GA) and an ant colony algorithm (AC). Theseare evolutionary algorithms which are inspired by naturalphenomena.

In all these three algorithms, we use the samerepresentation of solution, which is an array having tworows and N columns (where N is the number of containers tostore). In the first row there are containers, and in the secondthere are the assigned stacks. For example, suppose that wehave five containers and night stacks. A solution can be as

follows.

1 2 5 3 49 5 6 3 5

This means the following assignment:• Container 1 to stack 9,• Containers 2 and 4 to stack 5,• Container 3 to stack 3,• Container 5 to stack 6.

The order of containers in the solution is very importantbecause if several containers are assigned to a same stack, itspecifies the exact location of each of them. For example, inthe stack 5, the container 2 will be stored before the container4.

A. Genetics Algorithms (GA)

Genetics algorithms are initiated by John Henry Holland.In 1975, he published in [14], the canonical genetic algorithmwhich is the first to be published.

To perform a genetic algorithm, it is necessary to definesix elements: a solution representation, a method to generatethe initial population, a function to be optimized, a selectionmethod, the operators of crossover and mutation, and alsothe sizing parameters (the maximum number of individualsin a population “NIMax”, and the maximum number ofgenerations “NGMax”).

Generally, a genetic algorithm progresses as follows :

1: Create the initial population.2: Evaluate the initial population.3: Initialize the number of generations, NG = 1.4: While NG < NGMax do:

4.a: Select a part of the population.4.b: Initialize the New generation by adding to it the

selected population.4.c: While the size of the New generation is inferior

to NIMax, do:• Choose randomly two parents (P1 and P2) of

the selected population.• Do a crossover between P1 and P2 in order

to create two children (C1 and C2).• Add the best child to the New generation.

End while.4.d: Do a mutation on an individual selected randomly

in the New generation.4.e: Evaluate the New population.4.f: NG = NG+ 1.

End while.

1) Creating the initial population: The individuals in theinitial population are created individually. Each of them is avalid solution of the storage problem, therefore is representedas an array having two rows and N columns.

To build an element of the initial population, we treat thecontainers singly in a randomly order. Whenever a container isselected, it is assigned to the nearest stack among the available

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stacks which are compatible to it (this means the stacks whichare not full, have the same size as the container, and haven’ta container whose departure time is inferior to that of theselected container).

2) Selection method: We use the wheel of lucky method ateach iteration to select the half of the population in order tocreate the next generation. With this method, the luck of eachindividual to be selected is proportional to its performance.Let G the set of individuals belonging to the population. Weconsider a circular wheel, and allocate to each individual i ansector measuring 360◦×Oi∑

j∈G Oj(where Oi is the performance of

i). To select an individual, the wheel is rotated. When it stopsspinning, the pointed individual is selected.

For example, suppose that we have three solutions: O1 =123, O2 = 201, O3 = 84. The associated wheel is

3) Crossover method: After selecting the half of apopulation, we do several crossovers in order to create newssolutions. The selected population and the news individualsform together a new generation. To perform a crossover, wetake arbitrarily two parents in the selected population anda real number rc between 0 and 1. If rc is inferior to thecrossover probability pc, then we choose randomly a sectionposition and invert the parts which are to the right. So weget two children, and we keep the best. For example, supposethese two following parents:

After performing a crossover, we get these two followingchildren:

Before evaluating these new solutions, we check if theysatisfy all constraints, and we make corrections if necessary.For example, if a container is assigned to two stacks, thenwe delete the assignment corresponding to the farthest stack.We also check if there are containers which are not in thesolution, and we assign them to the nearest stacks among theavailable, if so. After correcting these two kinds of defects,we verify if the capacity constraints are satisfied and assignsome containers to another stacks if necessary. The last stepof the correction is to reorganize containers assigned to thesame stacks, in order to satisfy the departure time constraints.

After correcting the children, we get:

4) Mutation method: After making crossovers, we do amutation on a single individual. For this, we choose randomlyan individual and a real number rm between 0 and 1. If rm isinferior to the mutation probability pm, then we take arbitrarilya container and assign it to another available stack which iscompatible to it.

In the following example, the container 5 was initiallyassigned to stack 6; but it is transfered to the stack 2 after themutation.

B. Ant colony algorithm (AC)

The first ant colony algorithms are created by Dorigo etal [15]. They are inspired by the manner that natural antsproceed during the research of food.

Due to a natural substance called pheromone, antscommunicate with each other in order to find the shortestpath between their anthill and a location where there isfood. At the beginning of the collection, the shortest pathwhich leads to food is unknown. Therefore each ant followsarbitrarily a road and puts over pheromone. Throughout thecollection, ants secrete continuously this substance. Thus allpaths traveled contain it. Thereafter, when an ant detectsthe presence of pheromone onto several paths, it choosesthe shortest among them. Due to the fact that pheromoneevaporates over time, it will remain only onto the shortest path.

In the ant colony algorithm, we use essentially four parame-ters: the number of ants (NA), the maximum number of travelsper ant (NTMax), the minimum threshold of pheromone(τMin) and the maximum threshold (τMax) of pheromone. Theant colony algorithm is generally as follows:

1: Initialize pheromone.2: Each ant builds a solution.3: Evaluate the solutions.4: Initialize the number of travels per ant, NT = 1.5: While NT < NTMax do:

5.a: Update pheromone.5.b: Each ant builds a new solution.5.c: Evaluate the solutions.5.d: NT = NT + 1.

End while.

1) Method to build a solution: Before starting theconstruction of solutions, we first seek all pairs (stack,container) which are compatible. This means all couples (p,k)satisfying thee conditions:• cp > 0,• rp = Rk,

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• tp ≥ Tk.

Let E be the set of these pairs. Each element (p, k) of Ehas a pheromone rate τ(p,k) which is initialized to τMax. Theconstruction of a solution by an ant is performed using thefollowing algorithm.

1: Initialization of the solution S = ∅.2: Add to S an arbitrary element of E.3: While E is not empty do:

3.a: Update E in order to delete all pairs whichcan occasion the violation of a constraint ifthey are added in S.

3.b Calculate the probability of each element (p, k)remaining in E by using the following formula

P(p,k) =(τ(p,k))

α × ( 1dkp)β∑

(p,k)∈E (τ(p,k))α × ( 1dkp)β

where α and β are real positive numbers.3.c Add to S the element of E which has

the highest probability.End while.

4: If |S| < N then the solution is not valid, so4.a: Go back to 1, and restart the construction of another

solution.

2) Method to update pheromone: At the end of an iteration,the pheromone rate of each couple is updated. To do this, wefirst do an evaporation:

∀(p, k) ∈ E, τ(p,k) = (1− ρ)τ(p,k)

where ρ is the evaporation rate, and 0 < ρ < 1.After this, we only increase the pheromone rate of couplesbelonging to the best solution Sbc of the current iteration:

∀(p, k) ∈ Sbc, τ(p,k) = τ(p,k) +1

|Obc −Ob + 1|

Where Ob is the value of the best solution found since thebeginning until the current iteration, and Obc is the value ofSbc.

C. Hybrid Ant Colony and Genetic Algorithm (HAC/GA)

Our hybrid algorithm is an ant colony algorithm whereina genetic algorithm is inserted. We begin it by initializing thepheromone. After that, each ant builds a solution. We thenobtain a population on which we apply a genetic algorithm.Thereafter, we update the pheromone and the ants build othernew solutions. These steps are repetitive and succeed eachother, as can be seen in Figure 2.

Figure 2. The HAC/GA algorithm

VI. NUMERICAL RESULTS

All algorithms are coded in C++ language. To performsimulations, we use a computer DELL PRECISION T3500with an Intel Xeon 5 GHz processor.

At first, we look for the best parameters. So we get thefollowing values mentioned in table I.

TABLE I. VALUES OF THE PARAMETERS

Parameters Values

Ant colony

Number of iterations NTMax = 50Number of ants NA = 100Exponent of pheromone α = 0.3Exponent of the visibility β = 0.2Evaporation rate of pheromone ρ = 0.2Minimum threshold of pheromone τMin = 1Maximum threshold of pheromone τMax = 10

Genetic algorithmNumber of generations NGMax = 50Size of the population NIMax = 100probability of crossover pc = 0.75probability of mutation pm = 0.025

To prove the effectiveness of our algorithms, we comparethem to CPLEX version 12.5, which is an software giving inthe most cases optimal results. To do this, we calculate thepercentage deviations using the following formula :

gap =Obj(algorithm) −Obj(CPLEX)

Obj(CPLEX)× 100

where Obj(CPLEX) is the value of the optimal solution foundby CPLEX, while Obj(algorithm) can be the value of solutionfound by the ant colony algorithm (in this case it is namedObj(AC)) or that of the solution of the genetic algorithm(Obj(GA)), or that of the hybrid algorithm (and then it isnominated by Obj(HAC/GA)).

The percentages deviation of the three algorithms arereported in Table II.— means that the computer memory is insufficient to resolvethis instance.

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TABLE II. PERCENTAGE DEVIATIONS

Instances CPLEX AC GAW AC/GAWObj(CPLEX) Obj(AC) gap Obj(GA) gap Obj(HAC/GA) gap

1 2651 2762 4.187% 2651 0% 2651 0%2 5077 5200 2.423% 5088 0.217% 5077 0%3 4292 4292 0% 4292 0% 4292 0%4 5137 5161 0.467% 5151 0.273% 5137 0%5 5922 6054 2.229% 5930 0.135% 5922 0%6 6743 6921 2.640% 6758 0.222% 6743 0%7 5809 5826 0.293% 5821 0.207% 5809 0%8 6647 6755 1.625% 6660 0.196% 6647 0%9 7452 7645 2.590% 7514 0.832% 7455 0.040%10 7344 7344 0% 7344 0% 7344 0%11 7445 7603 2.122% 7467 0.296% 7445 0%12 8279 8442 1.96% 8319 0.483% 8292 0.157%13 7380 7417 0.501% 7392 0.163% 7380 0%14 8028 8095 0.835% 8034 0.075% 8028 0%15 8895 8948 0.596% 8916 0.236% 8911 0.180%16 9714 9770 0.576% 9752 0.391% 9730 0.165%17 10576 10672 0.908% 10628 0.492% 10592 0.151%18 10527 10836 2.935% 10602 0.712% 10602 0.712%19 10443 10483 0.383% 10454 0.105% 10450 0.067%20 11315 11435 1.061% 11340 0.221% 11315 0%21 10545 10699 1.460% 10570 0.237% 10566 0.199%22 11339 11427 0.776% 11381 0.370% 11351 0.106%23 12163 12259 0.789% 12227 0.526% 12174 0.090%24 12030 12243 1.771% 12147 0.973% 12085 0.457%25 12858 13151 2.279% 12983 0.972% 12954 0.747%26 13660 13987 2.394% 13763 0.754% 13763 0.754%27 13633 13862 1.680% 13803 1.247% 13736 0.756 %28 14474 14726 1.741% 14629 1.071% 14609 0.933%29 16080 16393 1.947% 16305 1.399% 16243 1.014%30 15979 16270 1.821% 16058 0.494% 16051 0.451%31 15184 15412 1.502% 15262 0.514% 15245 0.402%32 17254 17297 0.249% 17258 0.023% 17254 0%33 31475 31583 0.343% 31520 0.143% 31475 0%34 51222 51754 1.039% 51547 0.634% 51547 0.634%35 15959 15977 0.113% 15959 0% 15959 0%36 31762 31810 0.151% 31789 0.085% 31789 0.085%37 45793 46024 0.504% 45894 0.221% 45793 0%38 63737 64076 0.532% 63962 0.353% 63960 0.350%39 84492 85203 0.841% 85190 0.826% 85111 0.733%40 18901 18908 0.037% 18902 0.005% 18901 0%41 30748 30796 0.156% 30758 0.033% 30756 0.026%42 46437 46516 0.170% 46482 0.097% 46472 0.075%43 61967 62086 0.192% 62061 0.152% 62042 0.121%44 17807 17827 0.112% 17807 0% 17807 0%45 31902 31970 0.213% 31909 0.022% 31906 0.013%46 48941 48973 0.065% 48953 0.025% 48940 0%47 60819 60923 0.171% 60874 0.090% 60869 0.083%48 93062 93149 0.093% 93118 0.060% 93111 0.053%49 — 270591 — 270551 — 270540 —50 — 285386 — 285290 — 285288 —51 — 315203 — 315137 — 315127 —52 — 360163 — 360144 — 360130 —

Average percentage deviation 1.072% 0.345% 0.199%

As can be seen in table II, the HCA/GA is very efficientsince its average percentage deviation is equal to 0.199% whilethose of AC and GA are respectively equal to 1.072% and0.345%. In addition to this, it has 20 optimal results on 48instances, unlike to AC and GA which have respectively 2and 5 optimal results on 48 instances.The effectiveness of HAC/GA can be also seen in Figure 3.

Figure 3. Numerical results

VII. CONCLUSION

In this paper, we propose an efficient hybrid ant colony andgenetic algorithm (HAC/GA) to solve the container storageproblem in port terminal. This hybridization improves theperformance of the ant colony algorithm and the geneticalgorithm, which are very suitable to solve the containerstorage problem as shown in the numerical results. HAC/GA isalso able to solve more efficiently large instances which can not

be solved by CPLEX because requiring too much memory. Thecontainer storage problem is known to be NP-hard, thereforeit is useful to create efficient algorithms to solve it.

Another contribution of this paper is the linear mathemati-cal model which determines the exact location assigned to eachcontainer without causing reshuffle. This paper also contains arecent state of the art regarding the storage container problem.

In future reacherch, we plan to take into account otheroperational constraints. We also envisage to study the storagecontainer problem in an automated port and propose otherefficient resolution methods.

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