Optimizing Empty Containers Distribution among Ports

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1 Journal of Mathematics and tatistics 7 (3): 26-22, 20 IN cience ublications Otimizing Emty Containers Distribution among orts Mohammed A. Hajeeh and Weam Behbehani Techno-Economics Division, Kuwait Institute for cientific Research, Kuwait Abstract: roblem statement: Emty containers are an essential art of the logistics of the movement, reositioning and distribution of containers. Otimizing the allocation and transortation of emty containers contributes significantly in reducing cost. Aroach: A mathematical model is formulated with the objective of finding an otimal sequence of shis movement among orts in order to satisfy demands at the orts with minimum total cost. Results: everal numerical examles are used to comare the erformance of the develoed otimization model with well known standard models available in the literature. Conclusion/Recommendation: It was observed that the solution obtained by the develoed method to be either as good as or better than the solution rovided by other standard models. Future work should consider contribution of full containers in meeting demand for emty containers. Key words: his, otimal sequence, minimum cost, mathematical model, containers movement, caacitated lant location roblem, time eriod, transortation roblem INTRODUCTION Containers were introduced in maritime shiing in the 960 s of the last century as an imortant means of transorting goods. Containers had contributed significantly in making transortation easier and faster and in reducing encountered handling costs. However, it had brought about some roblems such as reloading, owning or leasing and reositioning in addition to the roblem of handling, acquisition and maintenance which are very costly. Today aroximately 60% of maritime cargo is transorted in containers. Literature related to maritime containerizing and otimization is extensive and detailed. Francesco et al. (2009) studied the maritime roblem dealing with the non-deterministic nature of the historical data that deals with emty containers reositioning. The existence of uncertainty in the arameters makes it very difficult for the decision makers to take the roer actions. In order to overcome such roblem, a time-extended multi-scenario otimization model was develoed. The different scenarios were based on exert oinion obtained from the different shiing comanies. In their attemt to study resource allocation in shiing lines, Qingcheng et al. (200) built a deterministic model based on equilibrium rincile for shiing liners. The model exanded to an otimization model that takes into account the different uncertain arameters and factors. This robust model treats the shi size and slot allocation in the ort as the design and the control variables, resectively. Hajeeh (20) formulated two nonlinear mathematical models with the objective of finding the otimal failure and reair rates for a system that is deteriorating over time. The system was assumed to be imerfectly reaired after each failure. In the first model, the failed system was relaced by a new one after several imerfect reairs, while in the second model and uon each failure the system was either relaced or imerfectly reaired with different robabilities. Lei and Church (20) develoed models for locating emty containers yards, the objective was to select a convenient location away from the orts such that the travel time involved in reositioning emty containers is minimized. The ort comlexes at Los Angeles and Long Beach were taken as an examle in order to show that the savings associated with storing emty containers are usually overestimated when neglecting the trucking comany s viewoint. Chou et al. (200) studied containers transortation roblems using fuzzy logic aroach. In this regards, a two stage fuzzy multile criteria otimization model was develoed with the objective of minimizing the transortation cost in addition to considering the volume of containers and ort facility conditions. In the first stage, containers demand slit rate was comuted, while in the second case network otimization model was develoed to determine the inland origin destination of imort/exort containers. Corresonding Author: Mohammed A. Hajeeh, Techno-Economics Division, Kuwait Institute for cientific Research,.O. Box 24885; afat-309; Kuwait Tel: Fax:

2 J. Math. & tat., 7 (3): 26-22, 20 ong and Dong (20) used simulation and mathematical modeling to find an aroriate olicy for the roblem of reositioning emty containers in the resence of flexible destination orts. In this regards, no secific orts were chosen in advance, however, emty containers are unloaded in orts based on demand. Onut et al. (20) examined a real life roblem which was faced by a firm in Turkey. This firm deals with around thirty-arty logistics comanies and uses different orts each time. The objective was to select among many orts, the most convenient one. The selection rocess was based on several quantitative and qualitative factors that are comlex and conflicting. Hence, the fuzzy analytic network rocess aroach was utilized in order to overcome the vagueness associated with the roblem. In this aroach, six main criteria were used along with twenty sub-criteria. Fan et al. (200) develoed an otimization model for studying the transortation network of containers shied to the United tates of America. ensitivity analysis was used for evaluating the imacts of congestion on caacity constraints and on the alternative routes in the shiment transort. Bandeira et al. (2009) resented decision suort systems integrating both full and emty containers. The roblem was modeled as a network; a management aroach was develoed for the allocation and the distribution of the containers. Here, the distribution lanning was integrated in a user friendly manner. Wong et al. (2009) develoed a Hybrid Artificial Immune ystems (HAI) algorithm for solving multiobjective otimization roblems. This algorithm has been successfully alied to the roblem of global reositioning containers. It was claimed that the develoed HAI will assist shiing liners too otimally and cost effectively reosition containers globally. Investigated the relationshi between quay cranes, yard machines and container locations in a multi-berth and multi-shi environment. The main objective was to build a model for imroving the oeration efficiency of the seaorts and to develo an analytical tool for yard oeration lanning. MATERIAL AND METHOD and have to be otimally allocated in the different orts. There are many costs associated with containers; these include storage cost, reair cost, maintenance cost and insection cost, transortation cost, handling cost, shi berthing cost and cleaning cost among others. The main objective of this research work is to find the otimal movement of emty containers among orts to meet demand at minimum cost. There are many constraints that are considered in this roblem, these are: Demand for shiments to be transorted from one ort to another at a given time, demand for emty containers to carry such shiments among orts at a given time eriod, demand for sace on board of a shi that is journeying among orts in order to carry the demanded shiments (full containers) and emty containers, balance equations for emty containers in each ort at any given time eriod, a balance equation for the sace available in a shi after leaving a ort in a given time eriod. Note that sace in a shi is counted in terms of number of containers. The assumtions are: All containers are of the same size and tye, emty containers stored in a given ort should not exceed its storage caacity, containers loaded on board of a shi should not exceed the shi s caacity at any time and number of shis berthed in a ort in any time eriod should not exceed the maximum allowable number (ort caacity). The roblem is resented schematically in Fig.. The general model for transorting emty containers by s shis to satisfy demands at orts is resented by the following mathematical model (): minimize Z = (C X + K Y ) ubject to: s = X = a, =,2,..., s s s s Xs A s, =,2,..., () Containerization has become the most significant mode of oerations nowadays. Containers come into different shaes and sizes and move between orts either as full carrying goods demanded in other orts, or as emty to be filled in another ort. Although full containers have allocation riority, emty containers are on the move and are necessary for new shiments. Emty container are in high demand, their allocation roblem is comlex. They are either owned or leased 27 Xs a Y s, =,2,...,; s =,2,..., Fig. : A schematic resentation of the shiing roblem

3 J. Math. & tat., 7 (3): 26-22, 20 Xs are ositive integers, =,2,...,; s =,2,..., Ys = 0 or, =,2,...,; s =,2,..., Where: X s = Number of emty containers carried by shi s to meet demand at ort. Y s = Number of stoage made by shi s at ort. C s = Cost of transorting an emty container by shi s to ort. K s = toage cost imosed on shi s at ort. a = Number of emty containers demanded at ort. A s = ace availability on board of shi s, measured in number of emty containers. The above roblem () cannot be easily solved analytically by existing techniques. However, when examined closely, the structure of the roblem is found to ossess some features of the well known Caacitated lant Allocation roblem (CL). The CL roblem could be simly stated as follows: Given there are n otential lants locations and m set of customers to be sulied from selected lants with the objective of satisfying customers demand from the different lants at minimum cost. The mathematical model for the CL is resented in several forms in the literature. The most oular structure is as follows: m n m minimize Z = (C X ) + K Y ubject to: n j= ij i i ij ij i i i= j= i= m Xij = b j, j =,2,...,n (2) i= ij ij X a Y, i =,2,...,m X 0, i =,2,...,m; j =,2,...,n Y = 0 or, i =,2,...,m; j =,2,...,n The inut arameters are: K i = Fixed cost associated with lant i a i = Maximal caacity (outut) at lant i C ij = Cost of sulying one unit of the roduct from the ith lant to the jth customer b j = Number of units demanded by customer j m = Total number of customers Y i = Whether a lant is oen in location i ( Y i = ) or is closed at the same location., (Y i = 0) X ij = Number of roducts shied from the lant in location i to customer j The aarent difference between the main roblem and CL lies mainly in the decision variable Y. In order to show that the solution of the original roblem () is close to that of some standard models in the literature; the original roblem has been modified and it was roved that its solution is close to that of the standard models. In this regards, the original roblem solution was comared to that of the Transortation roblem and the Caacitated lant Location roblem as exlained by the two aroaches below. Aroach : Transortation roblem (T): This aroach is based on solving the transortation version of the general roblem and adding the aroriate stoage costs to the otimal solution. The main stes of this aroach are follows: Transform the original roblem () into the well known Transortation roblem (T) by excluding the stoage costs. olve the new T using existing techniques in the literature. Add the ort stoage costs to the costs value obtained by the T model. The roblem is mathematically resented as follows: = = minimize Z (C X ) ubject to: s s X = a, =,2,..., s Xs A s, =,2,..., (3) X areositive integers, =,2,...,; s =,2,..., s olving the above roblem gives the otimal assignment of shis to orts. The added stoage cost is as follows: K s = 0 if Y s = 0 The decision variables are: K s > 0 if Y s > 0 28

4 J. Math. & tat., 7 (3): 26-22, 20 Thus the otimal solution of the roblem (Z) is as follows: Z() = Z() + K Y for Y > 0 = s s s However, this aroach does not always roduce the otimal solution to the general roblem (). Y = 0 or, =,2,..., The above roblem is the caacitated lant location roblem with one shi and several orts. It also could be looked at as an assignment roblem. olving the roblem, the following otimal solutions are obtained: Z,X,a Aroach 2: Caacitated lant Location roblem (CL): In this aroach, the roblem is transformed into several one shi roblems. Hence, a secific shi is chosen and orts are assigned to it otimally by solving the one shi roblem. Next, another shi is selected and orts are assigned to it and so on until either the demand for emty at all orts are met, or all available shis have already been assigned. The detailed stes are as follows:. Take the one shi model of roblem and by selecting any shi s, the structure becomes similar to the CL. 2. Find the otimal solution of the revious structure by the available CL algorithm, or develo a new algorithm. 3. Remove those orts that their demand is comletely met by the shi in the system. 4. In orts where the demand is artially satisfied; the demand is modified to be equal to original demand minus the allocated number of emty container by the assigned shi. 5. If all demands are met go to 7, otherwise continue. 6. Assign a non-assigned shi and go to. If all shis are already assigned, go to Find the otimal solution and sto. The solution of the general model is the sum of the solutions of the one shi models. The mathematical model for the one shi is as follows: minimize Z = (C X + K Y ) ubject to: = X a Y, =,2,..., (4) = X A X are ositiveintegers, =, 2,..., X Can take one of several categories: = = X, a a some X, a a = < some different than ort X = 0, a = 0 some different than orts and 2 It should be emhasized that any ort in the system can be in one and only one of the above categories. Next a different shi is selected and a new mathematical model is formulated. This model has a similar structure as before, but with the following modifications: All orts with fully satisfied demand, i.e., as in the first category above are omitted from the first shi orts where their demands are artially met by the first shi are included, but with modified demand, i.e., the new demand is equal to the original demand minus the demand met by the first shi orts that are not sulied with any emty containers by the first shi. The new demand for these orts is the same as their original demand Next, another non-assigned shi is selected with similar modification until either the demand for all orts are met or no shi is available in the system or both. Therefore the otimal solution of the comlete roblem is the sum of the solutions of the one shi roblems. The following examle is resented for illustration. Examles: The following examle is used to illustrate how the different models are used and to comare the results obtained from each. Inut data: Number of shis () = 3, Number of orts () = 5. In Table, the transortation costs encountered by moving one emty container by the various shis to the different orts are given. Other relevant arameters are also shown. In Table 2, the stoage cost imosed by each ort s authority on each shi is rovided. 29

5 J. Math. & tat., 7 (3): 26-22, 20 Table : Transortation costs to the orts by the different shis orts his Caacity Demand Table 2: hi stoage cost at each ort orts toage cost REULT olving the roblem using the above data in the different mathematical models is discussed below: The general model (): X 64 = X = 0, X = 2,X = 7, X = 6,X = 3,X = Y =, Y =, Y =, Y =,Y = The transortation model: X = 25 X = 0, X = 2, X = 7,X = 6, X = 3,X = Z = Z + Z = = 64 T j The Caacitated lant Location (CL) Model: equence-: hi, hi 2, hi 3. Z = Z + Z2 + Z3 = = 64 X = 0, X = 2, X = 7, X = 6, X = 3,X = 3,X = Y =, Y =,Y =,Y =,Y = equence-2: hi 3, hi, hi 2. Z = Z3 + Z + Z2 = = 88 X = 5, X = 3, X = 0, X = 3, X = 7, X = 9,X = Y =, Y = 2, Y =,Y = 2, Y = equence-3: hi 3, hi 2, hi. = + + = + + = Z Z Z2 Z X = 0, X = 7, X = 3,X = 2,X = 6,X = 2,X = Y =,Y =, Y =,Y = 2, Y = DICUION It can be deduced from the results roduced by the examle that transforming the original roblem into the well known transortation roblem version is the best otion. uch aroach results in roducing an otimal solution with minimum cost. While, using that of the caacitated lant location roblem version requires solving several combinations of the roblem in order to come u with the sequence that roduces the otimal solution. CONCLUION trategically and otimally moving and storing containers is essential in the transortation establishments. Emty containers movement and distribution is an integral art of maritime business. In this research work, attemts were made to solve relaxed versions of the comlex roblem of emty containers shiment. ince solving the comlete roblem analytically is tedious and challenging, the strategy was to transform the original into well known structured standard methods in the literature and to develo new algorithms to solve the resulted roblems. Future work should address the movement of both emty and full containers, since full containers contribute to satisfying the demand for emty containers in subsequent orts uon unloading their goods. Another otion is to introduce nonlinear cost functions associated with transorting emty containers among orts which are more realistic and thus solving a nonlinear mathematical rogramming roblem. REFERENCE Bandeira, D.L., J.L. Becker and D. Borenstein A D for integrated distribution of emty and full containers. Decision uort yst., 47: DOI: 0.06/j.dss Chou, C.-C., F.-T. Kuo, R.-H. Gou and C.-L. Tsai Alication of a combined fuzzy multile criteria decision making and otimization rogramming model to the container transortation demand slit. Alied oft Comut., 0: DOI: 0.06/j.asoc

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