A Branch-and-Cut Algorithm to Solve the Container Storage Problem

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1 A Branch-and-Cut Algorithm to Solve the Container Storage Problem dèye Fatma diaye University of Le Havre Le Havre, France Adnan Yassine Suerior institute of logistic studies Le Havre, France Ibrahima Diarrassouba University Institute of Technology Le Havre, France Abstract We study the Container Storage Problem in ort terminal (CSP), which consists to effectively manage the storage sace so as to increase the roductivity of ort. When a shi arrives, the inbound containers are unloaded by Quay Cranes (QC) and then laced on quays. So, they are collected by Straddle Carriers (SC). Each is able to carry one container at a time, and store it in its storage location. In order to reduce the waiting times of shis, we roose a mathematical model which minimizes the total distance traveled by SC between quays and container yards. In this aer, we tae into account additional constraints which are not considered reviously. We also roose an effective branch-and-cut algorithm (BC-CSP), which is an otimal resolution method. Performed simulations rove the effectiveness of our algorithm. Keywords-Container Storage Problem; comleity; branchand-bound; CPLEX. I. ITRODUCTIO In a seaort, the container terminal manages all actions concerning containers. Generally, three tyes of containers are distinguished: outbound, inbound, and transshiment containers. All these containers are temorarily staced in the container yard, before leaving the ort. Outbound containers are brought by Eternal Trucs (ET), also iced u by SC which store them in their storage location, and then loaded onto vessels. Inbound containers are unloaded from vessels by QC, transorted to their storage locations by SC, and then recuerated later by ET. owadays, the cometition between orts is very high. Therefore, each of them tries to imrove continuously the quality of its service in order to attract more customers. The most imortant criteria to measure service level include the waiting time of ET which collect inbound containers. In fact, when an ET arrives at ort and claims a secific container, it waits during all the time required to retrieve it. If the desired container is under others, it may be necessary to move these containers in first. This ind of movements, named reshuffles, are unroductive and require so much time. Therefore, it is very imortant to otimally store containers so as to avoid them. Another imortant criterion to measure the quality of service is the time required to unload shis. The imortance of this factor is justified by the fact that it is more beneficial for both the ort and the customers to shorten the stay of vessels. First, it is better for the ort to quicly free the berths in order to allocate them to others incoming vessels. Second, generally, shi-owners rent vessels; therefore, they tend to minimize the berthing durations in order to reduce rental costs. These two issues are addressed in this aer. We consider a modern container terminal, which uses SC instead of Internal Trucs (IT). The advantage of a SC is that it is able to lift and to store a container itself; therefore it is not necessary to use Yard Cranes (YC). A storage yard is comosed of several blocs. In order to allow good circulation of SC, each bloc is comosed of bays which are searated by small saces. In every bay, there are stacs wherein containers are stored. A stac must have a height inferior or equal to the limit fied by the ort authorities. Fig. shows an eamle of bloc wherein circulate straddle carriers. Fig. Straddle carriers circulating in a containers yard We roose in this aer an effective method to solve the container storage roblem. For this, we roose a new mathematical model which determines the otimum storage lan and minimizes the time required to transort containers between quays and storage areas. For the numerical solution 226

2 of the model, we roose an effective branch and cut algorithm (BC-CSP), which is an eact method. The remainder of the aer is organized as follows: a literature review is given in the second art. A detailed descrition of the addressed roblem is eosed in the third art. The comleity of the roblem is discussed in the fourth section, while the roosed mathematical model is elained in the fifth section. The branch-and-cut algorithm is itemized in the sith section; the numerical results are resented in the seventh section. Our conclusion is given in the eighth section. II. LITERATURE REVIEW There are many aers addressing the storage of outbound containers than inbound containers. However, there are some aers which consider both simultaneously. Zhang et al. [2] considered in addition to these two categories, those which are in transition, that means containers which are unloaded from some vessels and are waiting for being loaded onto others. They used the rollinghorizon aroach [2] to solve the storage sace allocation roblem. For each lanning horizon, they solved the roblem in two stes formulated as mathematical rograms. In the first ste, they determined the total number of containers assigned to each bloc at a eriod so that the worloads of loading and unloading of each vessel are balanced. Then, in the second ste they determined the number of containers associated to every vessel in order to minimize the total distance traveled to transort these containers from quays to the storage blocs. Bazzazi et al. [3] roosed a genetic algorithm to solve an etended version of the Storage Sace Allocation Problem (SSAP). It consists to allocate temorarily locations to the inbound and outbound containers in the storage yard according to their tyes (regular, emty, and refrigerated). They aimed to balance the worloads between blocs with the goal to minimize the time required to store or to retrieve containers. Par and Seo [4] dealt the Planar Storage Location Assignment Problem (PSLAP), in which only lanar movements are allowed. The urose of PSLAP is to store inbound and outbound containers so as to minimize the number of moving obstructive objects. The authors made a mathematical formulation of PSLAP and roosed a genetic algorithm to solve it. Lee et al. [5] combined the truc scheduling and the storage allocation roblems. They considered inbound and outbound containers, and tend to minimize the weighted sum of total delay of requests and the total travel time of yard trucs. For numerical resolution, they roosed a hybrid insertion algorithm. Kozan and Preston [6] develoed an iterative search algorithm using a transfer model and an assignment model. At first, the algorithm determined cyclically the otimum storage locations for inbound and outbound containers, and secondly, it found the corresonding handling schedule. They solved the roblem using a genetic algorithm, a tabou search algorithm and a hybrid algorithm. Concerning inbound containers, most of aers deal with the management of reshuffles. Sauri and Martin [7] roosed three different strategies to store inbound containers. The urose of their wor is to determine the best strategy which minimizes re-handles in an imort container yard. For this, they develoed a mathematical model based on robabilistic distribution functions to evaluate the number of reshuffles. Kim and Kim [8] considered a segregation strategy to store inbound containers. This method does not allow lacing newly arriving containers over those which arrived earlier. Therefore, storage saces are allocated to each vessel so as to minimize the number of reshuffles eected during the loading oerations. Jinin et al. [9] roosed an integer rogramming model, which addressed the trucs scheduling and the storage of inbound containers. They minimized at the same time the number of congestions, the waiting time of trucs, and the unloading time of containers. The authors designed a genetic algorithm to solve the model, and another heuristic algorithm which gave them best results. Yu and Qi [0] treated the storage roblem of inbounds containers in a modern automatic container terminal. They aimed to minimize reshuffles in two stes. For this, they first resolved the bloc sace allocation roblem for newly arriving inbound containers, and then after the retrieving of some containers they tacled the re-marshaling rocesses in order to re-organize the bloc sace allocation. They suggested three mathematical models of storage containers, the first is a non-segregation model, the second is a single-eriod segregation model, and the last is a multile-eriod segregation model. They conceived a conve cost networ flow algorithm for the first and the second models, and a dynamic rogramming for the third. They found that the remarshaling roblem is P-hard and then designed a heuristic algorithm to solve it. We considered in [] a container terminal wherein reshuffles are not allowed. We roosed a new mathematical model to allocate storage saces to inbound containers in such a way that no reshuffles will be necessary to retrieve them later. We designed a hybrid algorithm including genetic algorithm and simulated annealing to solve it. In most container terminals, the dearture times of inbound containers are unnown. K. H. Kim and K. Y. Kim considered in [] a container terminal in which there is limited free time storage for inbound containers, beyond which customers have to ay storage costs er time unity. The authors roosed a mathematical model to find the otimal rice schedule. Paers dealing with the storage roblem of outbound containers have generally different goals. Preston and Kozan [2] roosed a Container Location Model (CLM) to store outbound containers in such a way that the time service of container shis is minimal. They designed a genetic algorithm for the numerical resolution. Kim et al. [3] develoed a dynamic rogramming model to determine the storage locations for outbound containers according to their weight. They minimized the number of relocations eected during the shis loading. They also made a decision tree using the set of otimal solutions to suort real-time decisions. Chen and Lu [4] addressed in two stes the storage sace allocation roblem of outbound containers. In the first ste, they used a mied integer rogramming model to calculate the number of yard bays and the number of 227

3 locations in each of them. So, in the second ste, they determined for each container the eact location where it must be stored. Woo and Kim [5] roosed a method to allocate storage saces to grous of outbound containers. They reserved for each grou of containers having the same attributes, a collection of adjacent stacs. At the end, the authors roosed a method to determine the necessary size of the storage sace eected for all the outbound containers. Kim and Par [6] gave two linear mathematical models to store outbound containers. In the first, they considered a direct transfer, and so, in the second, they dealt with an indirect transfer system. They designed two heuristic algorithms to solve these models. The one is based on the duration-of-stay of containers, and then, they used the subgradient otimization technique in the other. Among the few aers dealing only transshiment containers include that of ishimura et al. [7]. They develoed an otimization model to store temorarily transshiment containers in the storage yard, and roosed a heuristic based on lagrangian relaation method for the numerical resolution. III. COTEXT When a container shi arrives at ort, QCs unload containers and lace them on quays. So, they are iced u by SCs, which carry and store them in the container yard. The firsts containers which are laced on quays are the first iced u. In order to avoid congestion on quays, which could increase the time required to unload shis, we minimize the total distance traveled by SCs between quays and the container yard. In this study, we consider the following five main hyotheses: () Reshuffles are not allowed, (2) In each stac, containers are arranged according to: (2.a) the same order that they are unloaded from shis, (2.b) and the descending order of their dearture time, (3) In a stac, all containers have same dimensions, (4)We tae into account containers which are already resent in the storage areas before the start of the new storage eriod, (5) We resect the maimum caacity of each stac. Eceted (2.a), these hyotheses are considered in []. IV. COMPLEXITY OF THE PROBLEM In this section, we study the comleity of the CSP. In articular, we show that it is equivalent to the Bounded Coloring Problem (BCP); therefore, is P-hard in the general case. A. Some reminders about the BCP Let us begin by recalling some concets and definitions that will be useful for the following. ) Preliminary notions: Let G(V,E) be an undirected grah, V is the set of vertices and E is the set of edges. G is a comarability grah if and only if there are a sequence of vertices v,...,vn of V such that for each (, q, r) checing < < q < r < n, if ( v, vq ) E and ( v, v ) E then ( v, v ) E. q r r A co-comarability grah is the comlement of a comarability grah. An undirected grah G = (V,E) is a ermutation grah if and only if there are a sequence of vertices v,...,vn of V and a ermutation σ of the vertices such that for all i and j satisfying i j n, ( v, v ) E if and only if σ(i) σ(j). Theorem : A grah G is a ermutation grah if and only if G and its comlement are comarability grahs [8]. 2) The bounded coloring roblem: Given an undirected grah G = (V,E), a set of s colors l,...,l s, an integer H and a vector that gives the weight of assigning a color l i to a verte of the grah. The bounded coloring roblem with minimum weight consists to determine a minimum weight coloring of G using at most s colors in such a way that a color is assigned to at most H vertices. Theorem 2: [9] The bounded coloring roblem with minimum weight is P-hard for the class of ermutation grahs for all H 6. B. Equivalence between the CSP and the BCP We show that the CSP is P-hard. For this, we introduce an undirected grah G(,T) = (V,E) constructed from an instance of the CSP, where is the set of containers and O are vectors, which give resectively the unloading order and the dearture times of each container. The grah G is constructed as follows. A verte of the grah corresonds to a container. To simlify notation, the inde is used to denote both a container and the verte of the grah which corresonds to it. There is an edge between two vertices and if and only if O < O ' < T '. We have the following lemma. Lemma : The grah G(,T) obtained from a instance of CSP is a ermutation grah. Proof: To rove that the grah G(,T) is a ermutation grah, it suffices to show that it is a comarability grah as well as its comlement (see Theorem ). First, we show that G(,T) is a comarability grah. The vertices are ordered according to the same order that the unloading of the corresonding containers from shis. If two containers and are unloaded from shis at the same time (that is to say if O = ), then the vertices and are O ' ordered in the ascending order of their dearture times. If i j 228

4 O = O ' = T ' then the vertices are ordered in the leicograhical order. Without loss of generality, we consider that the vertices are numbered in the order that is reviously determined. ow, consider any three vertices, and of the grah such that < ' < ",(, ' ) E and ( ', ' ') E. We will rove that necessarily (, '') E. As (, ') E and ( ', '') E, we have O < O ' < T ', and we have also O ' < O '' ' < T ''. We thus obtain that O < O ' < O '' and T < T < T, which imlies that the grah G(,T) has ' '' an edge between vertices and. So G(,T) is a comarability grah. ow, we will rove that the comlement of G(,T), denoted G (, T ) is also a comarability grah. First, note that there is an edge between two vertices and of G (, T ) if and only if there is no edge in G(,T) between and in other words O < O ' > T '. The vertices of G are ordered in the same order as those of G. As before, for any three vertices,, and of the grah G(, T ) such that < ' < '',(, ' ) E and ( ', ' ') E, we have O < O ' < O '' and T > T ' > T ''. So, O < O ' ' > T ' ', and then there is an edge between and in G (, T ). Therefore, G (, T ) is also a comarability grah. ow, it is easy to see that a solution of the container storage roblem is a solution of the corresonding bounded coloring roblem. In fact, a similar result is given in [20]. Consider an instance ICSP = (,T,,H,r,R,d) of the CSP and the grah G(,T) associated. ow, consider an H-coloring of G(,T) that has s colors. Each color of the bounded coloring roblem is matched to a stac of the CSP. Indeed, as all vertices having the same color form a stable set, in other words they are not connected by any edge, therefore any two containers corresonding to two vertices of this stable set satisfy these two inequalities O < and O ' T T '. The unloading order as well as the dearture times of containers corresonding to the vertices of a stable set are comatible; thereby, they can be stored in a same stac if it has enough emty slots. In addition, there are at most H vertices in this stable set. So, the number of containers assigned to the corresonding stac is inferior or equal to H. Therefore, an H-coloring corresonds to a valid assignment for the CSP. Similarly, it is easy to see that a solution of the CSP is a solution of the H-bounded coloring roblem in the grah G(,T). We have the following lemma. Lemma 2: Let ICSP = (,T,,H,r,R,d) an instance of the container storage roblem. The CSP has a solution for this instance if and only if the bounded H-coloring roblem on the grah G(,T) has a solution. We now give the main result of this section. Theorem 3: The container storage roblem is equivalent to the bounded coloring roblem with minimum weight. Proof: To establish this result, we rove that an instance of the CSP is equivalent to an instance of the BCP and vice versa. Let ICSP = (,T,,H,r,R,d) an instance of the storage container roblem and G(,T) the ermutation grah associated. Consider IBCP = (G(,T),H,,d) an instance defined on the grah G(,T), where is the number of colors, H is the bound, and d a matri containing the weights. According to the Lemma 2 a solution of the CSP is a solution of the BCP, and similarly a stac of CSP corresonds to a color of BCP and vice versa. It follows then that the cost stac d of assigning a container to the, is the same as the assignment of the verte to the color corresonding to the stac. So, the cost of H- coloring in the grah G(,T) is the same as the cost of the solution of the corresonding CSP and vice versa. Therefore, we can find the otimal solution of the CSP if and only if we find the otimal solution of BCP. According to the Theorem 2 the bounded coloring roblem is P-hard for the class of ermutation grahs if H 6. It therefore follows from Theorem 3 that the CSP is P-hard if H 6. Corollary : The container storage roblem is P-hard if the maimum caacity of every stac is suerior or equal to si. V. MATHEMATICAL MODELIG In the mathematical model, we use the following indices: : stac, : container. The data of the roblem are: : number of containers, : number of stacs, c : number of emty slots in the stac, r : tye of container which can be laced in the stac, t : dearture time of the container which was on the to of the stac at the begin of the new storage eriod. 229

5 R : tye of the container, T : dearture time of the container, O : unloading order of the container from shis. d : traveled distance to transort the container from quay to stac, G(V,E): a grah, where V is the set of vertices and E the set of edges. Every verte reresents a container, and V =. There is an edge between two vertices v and v if and T < and O < ' only if T ' O, this means that container and can t be assigned to a same stac. The decision variables are defined as follows: = if container is assigned to stac 0 otherwise ' Proosition : For an integer solution, the inequalities (3) and (7) are equivalents. Proof: It suffices to rove that (7) imlies (3), because the reverse is highlighted by the definition of (7). Since or. ' is a binary variable, then it can be equal to either 0 If ' = then ' ' = 0 ' ( ). Therefore, for all ' neighbor of, we have 0. ' = ' Thereby ' + ' =, ' ( ). If ' ' = 0 then ' ( ) which means ' ( ) that for all belonging to (), ' ' ' or. Thus, +, ' ( ). ' ' can be equal to either 0 We roose the following mathematical model: VI. BRACH-AD-CUT ALGORITHM Minimize = = d = =, =,..., ' +, (, ') E, =,..., () (2) (3) c, = = 0, =,...,, T > t orr r =,..., (4) (5) 0, =,...,, =,..., (6) The objective function () minimizes the total distance traveled between shis and the container yard. Constraints (2) require that each container is assigned to a single stac. Constraints (3) ensure that the containers of each stac are arranged following the same order that they were unloaded from shis, and the decreasing order of their dearture times. Constraints (4) enforce the stac caacity. Constraints (5) secure the comatibility between containers and stacs. Let a verte of the grah ( ), () the set of its neighbors, a stac ( ' ). Constraints (3) lead to the following neighborhood inequality as in [2]. ' ' ' ( ) + ( ) ' ( ) (7) The branch-and-cut is an imrovement of the branchand-bound, which is an eact resolution method. Each of these two methods uses a search tree to elore the solution sace. To do this, the search sace is divided into smaller subsets, each reresenting a node of the search tree. So, the roblem is solved by considering one by one all subsets. This strategy is called divide and conquer. To build the search tree, we first create the root node; it corresonds to the released roblem. Other nodes of the tree are obtained by maing connections. In the branch-and-cut, unlie the branch-and-bound, at each node of the search tree, some constraints called valid inequalities are added to the released roblem so as to imrove the solution. A. Relaation of the roblem After the relaation of integrity constraints (6), we find that the total number of constraints of the mathematical model remains great. Therefore, since the adjacency constraints (3) are equivalent to the neighborhood inequalities (7), so we delete them from the model nowing that the admissibility of solutions will be ensured by the gradual addition of valid inequalities along the branch-andcut algorithm. B. Prerocessing The number of variables increases deending on the number of stacs ) and containers (). In the case ( where all stacs were emty at the beginning of the storage eriod, we can reduce the number of variables. We consider that all containers are equidistant to the stacs. Knowing 230

6 that, generally,, we only use the stacs which are more near to quays. This allows to significantly reduce the number of variables and to seed u the comutation. C. Uer bound To find an uer bound, we solve the bounded verte coloring roblem on the grah defined in section V. Each color corresonds to a stac. We use a heuristic algorithm which colors vertices one by one following the descending order of their number of uncolored neighbors. For each verte, it chooses among the admissible colors the one that fits to the nearest stac. The eligible colors are those not assigned to a verte which is a neighbor of the considering verte, and corresond to the stacs which are not full. Whenever a verte is colored, the number of emty slot of the stac corresonding to the used color is reduced. D. Branchings rules We use the classical branching rule. At each node of the search tree, we create two branches by rounding the largest fractional variable. Let this variable. We ut = 0 in a branch; it means that container will not be assigned to stac in this branch. Then, in the other branch, we ut =, which means that container will inevitably be assigned to stac in this branch. E. Searation method At each node of the search tree, before creating branches, we use a simle heuristic algorithm to loo for neighborhood inequalities which are violated. To do this, we treat one by one all variable which is suerior to 0.5 in the otimal solution of the current node. Let one of these variables and S an integer initialized to zero. We calculate the number () of neighbors of the verte. Then, we add to S the value of multilied by (). And we see all variables ' ' such that and are neighbors and =, and we add the sum of their values to S. If S > () then there is a violated inequality therefore we add to the sub-roblem a constraint to avoid this. F. Descrition of the algorithm : We begin by solving the roblem using a heuristic algorithm to find an uer bound named BS. 2: Then, we create the root node of the search tree which reresents the released roblem. 3: We solve the sub-roblem using the CPLEX solver. 4: Then, we see all neighborhood inequalities that are not satisfied by the solution of the current node, and then we add them to the released roblem. 5: We solve the roblem again using the CPLEX solver. 6: If the solution is integer and inferior to BS then we udate BS. 7: If the solution is fractional and inferior to BS then we do: 7.a: Perform connections, 7.b: Choose an uneloited node, 7.c: Go bac to 3. VII. UMERICAL RESULTS In this section, we resent the numerical results of our branch-and-cut algorithm. For the imlementation, we use SCIP, which is a framewor allowing a total control of the solution rocess. The eeriments were erformed using a comuter DELL PRECISIO T3500 with an Intel Xeon 5 GHz rocessor. To test the effectiveness of our algorithm, we naturally comare it to CPLEX version 2.5. Performed tests on several instances rove that BC-CSP is very fast and it is able to solve large instances which can not be solved by CPLEX because requiring a lot of memory. In Table I, we note the eecution times of BC-CSP and of CPLEX for various instances. means that the eecution is interruted because it lasted more than 3 hours. --- means that the comuter memory is insufficient to resolve this instance. TABLE I. UMERICAL RESULTS BC-CSP CPLEX sec 2 min 58 sec sec 5 min 45 sec sec 4 min sec sec min 6 sec sec 2 sec sec 4 min sec min 5 sec sec min 29 sec sec 2 min sec sec 53 min 50 sec sec sec sec sec sec min 36 sec min 3 sec min 57 sec min 49 sec min 35 sec h 4 min 26 sec h 43 min 47 sec h 5 min 4 sec h 2 min 20 sec

7 In Table I, we remar that, in the most cases, the resolution of a great instance requires more time than the resolution of a small instance. However, in some cases, we observe the reverse. This henomenon can be justified by the influence of the values of arameters lie the dearture times, the unloading order, etc. In fact, in some cases the search tree can have too lot of nodes; therefore its eloration may require more time. But, even with these instances, our branch-and-cut algorithm is faster than CPLEX. The mathematical model of our container storage roblem has too many variables, esecially when there are a lot of emty stacs in the terminal. Therefore, the elimination of the farthest stacs reduces the size of the roblem and imroves the resolution. Fig. 2 shows that rerocessing reduces the eecution times. Fig. 2 Comarisons of eecution times As can be seen in Fig. 2, the rerocessing is more efficient when the number of stacs is suerior to the number of containers. VIII. COCLUSIO AD FUTURE WORK In this aer, we studied the container storage roblem. We widely imrove the wor that we did in [] by considering additional constrains in order to avoid reshuffles at quays. We tae into account the order in which containers are unloaded from vessels, and we minimize the total distance traveled by SC between quays and container yards in order to shorten the berthing times of shis. The major contribution of this aer is the effective branch-and-cut algorithm, which is very fast and is able to solve great instances. This is an eact resolution method, unlie the hybrid algorithm roosed in [], which has an average ercentage deviation equal to 0.22%. It may be ossible to imrove our branch-and-cut algorithm; therefore we rosect to design more effective branching rules and searation methods. We also lan to adat our aroach to container terminals which use modern equiments, such as automatic guided vehicles. REFERECES [] R. Moussi,. F. diaye, and A. 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