The optimal number of yard cranes in container terminal

Transcription

1 Journal of Industrial Engineering International January 2009, Vol. 5, o. 8, 7-76 Islamic Azad Universy, South Tehran Branch The optimal number of yard cranes in container terminal Gholam Reza Amin Assistant Professor, Department of Computer Engineering, Postgraduate Engineering Center, Islamic Azad Universy, South Tehran Branch, Tehran, Iran Hamed Golchubian Research Scholar, Department of Transportation Engineering, Islamic Azad Universy, Science and Research Branch, Tehran, Iran Abstract The aim of this study is to propose a ne mied integer linear programming (MILP) model to find the minimum required number of yard cranes, Rubber tyred gantry cranes (RTGCs), for completion the total amount of orks at the end of planning horizon. In other ord, e find the optimal number of yard cranes in a container terminal hich completes the total amount of orks. The paper supports the proposed method using a numerical eample. Keyords: Container terminal; Yard crane; Mied integer linear programming. Introduction A container terminal in a port is the place here container vessels dock on berths and unload inbound containers and load outbound containers. Three different types of containers are handled in a container terminal: inbound, outbound, and transshipment containers [4, 7, 8]. Inbound containers hich are usually import containers from abroad are discharged from a vessel in a short time hen the vessel arrives at a berth, and kept in temporary storage at a container yard for a fe days until they are transferred to outside trucks. Outbound containers hich are usually eport containers bound for other countries arrive at the terminal from several days before the arrival of the corresponding vessel and are loaded onto the vessel in a short amount of time hen she arrives at a berth. Transshipment containers are unloaded from a vessel, stay in the yard for several days, and loaded onto another vessel [4, 7, 8]. The storage yard in a terminal is usually divided into blocks. A typical block has seven ros of spaces, si of hich are used for storing containers and the seventh reserved for truck passing. The placing of a container in a yard is carried out by huge cranes called yard cranes. The most commonly used yard cranes are Rubber Tyred Gantry Cranes (RTGCs) [0]. The daily operations of a container terminal are comple and involve a variety of decisions to be made under condions varying h time. To perform a large number of container handling jobs a series of planning problems need to be solved and several epensive terminal resources should be allocated as ell [8, 0]. Therefore managing a container terminal is a challenging task. Researchers have paid attention to the operational viepoints of container terminals. Kim and Kim [2] suggested a method of determining the optimal amount of storage space and the number of transfer cranes for handling import containers as ell. Zhang et al. [0] assumed the given of orkload of each block in each period of a day and formulated a dynamic crane deployment problem. Their mathematical model [0] minimized the total delayed orkload in the yard and also proposed the optimal routes of crane movements among blocks. Also Kim and Park [4] discussed a dynamic spaceallocation method for outbound containers. Recently, Wang [] introduced container information tracing system. A comprehensive lerature revie on the application of operations research for container terminal is presented by Steenken et al. [9]. Recently, Kim and Kim [3] have offered a method of determining the optimal price schedule for storing inbound containers in a container yard. Corresponding author. g_amin@azad.ac.ir

2 72 G. R. Amin and H. Golchubian Also Lee et al. [5] introduced a ne simulated annealing based algorhm for scheduling of port container terminals. More recently, Canonaco et al. [] introduced a netork model for managing the berth crane operations. Further by introducing the noninterference constraints in port container terminals a quay crane scheduling scheme is proposed [6]. This paper borros the mied integer linear programming (MILP) model iniated by Zhang et al. [0] for dynamic crane deployment in a container storage yards and etends a ne model for finding the minimum required number of yard cranes for completion the total amount of orks at the end of planning horizon. Therefore the contribution of the current paper is that provides the optimal number of yard cranes in a container terminal for completion the total amount of orks. The rest of this paper is organized as follos: Section 2 shos a quick eplanation of dynamic crane deployment proposed by Zhang et al. [0]. The minimum required number of yard cranes need to be considered for completion the total amount of orks is given in Section 3. Section 4 illustrates the proposed model by a numerical eample. Section 5 ends the paper h conclusion. 2. Dynamic crane deployment In a container terminal for optimal deployment of yard cranes or minimizing the total orks delayed at the end of a planning horizon, Zhang et al. [0] proposed an integer linear programming model as ell as a Lagrangean relaation heuristic algorhm for finding the near-optimal solution. Their model [0] has some important practical assumptions given in Zhang et al. [0], pages 542. As they assumed in the container terminal a yard is divided into some blocks. Also there is a specified number of yard cranes and a planning horizon containing some periods. The folloing notations (including the model parameters and decision variables) are also introduced in their paper [0]. 2.. Parameters ) The number of cranes assigned for block i ( i = K,, ) at the beginning of the planning horizon denoted by ii0. Therefore the total number of available cranes is ii 0. i= 2) The capacy of each crane hin a planning period is shon by C = 240 minutes. 3) The total number of blocks under consideration is denoted by. 4) The total number of planning periods in a planning horizon is T = 6 periods. 5) The orkload in block i hin planning periodt is given byb, for each i = K,, and t = K,,T. 6) The traveling time of a crane from block i to blockj is indicated byt ij, i, j =, K,, i j Decision variables ) The number of cranes moving from block i to block j during a planning period t is denoted by. ote that heni = j, these cranes stay in the same block during periodt, for each i, j =, K, and t = K,,T. 2) The orkload fulfilled in block i by cranes that move from block i to block j during planning period t is indicated by z, i, j =, K,, t =, K, T. 3) The orkload fulfilled in blockj by cranes that move from block i to block j during planning period t is denoted by y, i, j =, K,, t =, K, T. 4) The orkload left in block i at the end of planning period t denoted by, i =, K,, t = K,,T. o e give the constraints of the dynamic crane deployment model used in Zhang et al. [0]. Constraints () ensure the crane flo or movement conservation in each block hen cranes are deployed from one period to the net. That is = i =, K,, t =, K,T ji( t ) () Constraints (2) arrant that only to cranes can serve a block in a planning period. These constraints assure us the number of eisting cranes in each block at the beginning of a planning period (and during a period) is at most to.

3 The optimal number of yard cranes in container terminal i =, K,, t =, K,T j (2) j i Constraints (3) imply the balance beteen the orkload that should be finished ( + b ) and i(t ) the orkload that can be finished ( ( z + = y j ) ) j in each block by using slack variables. That is i( t ) + b + = z y j i =, K,, t =, K,T (3) Constraints (4) assure that the total orkload fulfilled by a crane in each planning period cannot eceed s total net crane capacy. z + y ij ( C t ) 0 i =, K,, j =, K,, t =, K, T (4) Constraints (5) inialize orkload at the beginning of a planning horizon; any delayed orkload from the previous planning horizon is counted as the orkload in the first planning period. i = 0 i =,, (5) Constraints (6), together h the parameters ii0, define the inial locations of the cranes. ij = 0 i =, K,, j =,,, i j (6) Constraints (7) and (8) are non-negative and integer constraints. Constraints (8) also restrict that at most one RTGC (crane) can be moved from one block to another in a period. 0, z 0, y 0 j i =, K,, j =, K,, t =, K,T (7) { 0,, 2}, { 0,} (8) i i =, K,, j =, K,, i j, t =, K,T Therefore Zhang et al. [0] introduced the folloing mathematical model for the optimal crane deployment in a container storage yard. 0 min T t= i= subject to Constrains (),(2), K,(8) (9) The net section of this paper proposes a ne MILP model to find the minimum required number of yard cranes for completion the total amount of orks at the end of planning horizon. 3. The optimal number of yard cranes As can be seen Zhang et al. [0] proposed a novel MILP model and a heuristic based solution procedure as ell to solve the dynamic crane deployment problem in a container terminal. One important issue related to model (9) is assigning the inial values of yard cranes for each blocki = K,,. As they have assigned ii0 { 0,, 2} crane(s) to the i th block subjectively ( i = K,, ). For eample those authors [0] assigned the available yard cranes as follos: Assuming = 20 blocks if the proportion of the available number of yard cranes and the number of blocks is equal then one crane assigns to each block. When the proportion is.5, that is the number of yard cranes is30, then the order of assignment of cranes to the blocks is as 2,,, 2,, respectively. That is they assigned 2 cranes to the first block, one crane for the second block, and so on. In order to find the optimal number of cranes or the minimum number of required yard cranes for completion the total amount of delayed orks at the end of planning horizon, this paper considers the inial posion of yard cranes as the ne decision variables. Therefore Constraints () of model (9) is converted to the folloing constraints: i= ji t = 0 i =, K,, t =, K,T (-) ( ) 0 k (-2) ii = { 0,,2 } i,, ii = (-3) here, k is the number of inial yard cranes that e consider as unknon. Here Constraints (2) of model (9) remain unchanged. Regarding the objective

4 74 G. R. Amin and H. Golchubian of this paper Constraints (3) of model (9) is also converted to the folloing constraints: i( t ) + b + = z y j j = j = i =, K,, t =, K,T - (3-) i( T ) + bit + = 0 zijt y jit i = K,, (3-2) Because to find the minimum number of required cranes for completion the total amount of orks at the end of planning horizon e have it = 0 for each block i = K,,. In fact Constraints (3) disjoined into to constraints. The first type for t =, K, T is denoted as (3-) remain unchanged and the second type for t = T is shon as (3-2). ote that for t = T the left hand side of (3-2) is equal to and therefore should be vanished. To it find the optimal number of required yard cranes the remaining constraints of model (9) remain unchanged ecept for constraints (5) and (8) hich are modified as follos: i = it = 0 i =,, (5-) { 0,, 2}, { 0,} (8-) i i =, K,, j =, K,, i j, t = 0,K,T Because is needed to finish the amount of orks at the end of planning horizon for each of the blocks, in constraints (5), it = 0 and also ii0 is considered as a ne decision variable, in constraints (8). Therefore e propose the folloing model (0): k = min k Subject to ji( t ) = 0 0 i =, K,, t =, K,T i= ii 0 = k { 0,,2} i,, ii = + j i j 2 i =, K,, t =, K,T - i =, K,, t =, K,T i( t ) + b + = z y j i ( T ) + bit + = 0 =, K, zijt y jit i z + y ( C tij) 0 i =, K,, j =, K,, t =, K,T (0) i 0 = it = 0 i =, K, = 0 i =, K,, j =,,, i ij 0, z 0, y 0 i =, K,, j =, K,, t =, K,T i j { 0,,2}, { 0,} i =, K,, j =, K,, i j, t = 0,K,T Clearly the objective of the above model is finding the minimum required number of yard cranes in such a ay that at the end of planning horizon the amount of orkloads in each of the blocks is zero. According to the constraints of the above model, Constraints (2) and (8-), at most to yard cranes can ork in each of the blocks at every time period, so if the amount of orkloads is very large, model (0) may be infeasible. In such case changing the parameter C rectifies this problem. The net section gives a numerical illustration for the ne model (0). j 0

5 The optimal number of yard cranes in container terminal 75 0,, Table. The optimal inial posions of yard cranes ,0, ,2,0 3,3,0 4,4,0 5,5,0 6,6,0 7,7,0 8,8,0 9,9, ,20,0 4. A numerical illustration As a numerical demonstration for the proposed model, e use the numerical eample regarding the container terminal consisting 20 blocks given in Zhang et al. [0]. The problem contains to tables, table as appeared in Zhang et al. [0] gives the crane traveling time beteen blocks, t, for i, j =, K, = 20 and Table shos the orkload in each block i hin every planning period t, b, for i =, K, = 20, t =, K, T = 6, here T = 6 indicates the number of planning periods. Also Zhang et al. [0] assigned an inial posion for the available 30 cranes into 20 blocks. Furthermore the capacy of one crane hin a planning period is denoted as C. To importance issues corresponding to comparison of the proposed model in this paper, model (0), and the model introduced by Zhang et al. [0], model (9), are, the proposed model (0) takes the inial posion of yard cranes as decision variables, ii, i =, K, 20, and the number of yard cranes 0 = is also considered as unknon, k, in our ne model. Whout loss of generaly e consider C=600. Also to obtain the eact optimal solution of the proposed model e used LIGO softare. Solving model (0) for the stated numerical data gives the optimal number of yard cranes k = 6. Table shos the optimal inial posion of yard cranes, ii, i =,, 20, ij hen C=600 minutes and the optimal number of cranes is also identified as k = 6. As the above table shos assigning the inial posion of yard cranes based on a subjective method, as Zhang et al. [0] assigned, may not produce an optimum posion for yard cranes. For eample if one uses the scheme used in Zhang et al. [0] assigned the inial posion of yard cranes for the given eample as for ii0 = i 6 = i = K,,6 and ii0 = 0, i = 7,8,9, 20. Furthermore, the optimal solution shos 0 for every block i = K,, 20. This indicates the completion amount of eisting orkloads at the end of planning horizon for each block is achieved. 5. Conclusion This paper has proposed a modified mathematical model to obtain the optimal number of yard cranes for completion the amount of orkloads at the end of planning horizon in a container terminal. We have formulated the problem as a mied integer programming model and using a numerical eample the results of the proposed model have been illustrated. Finding the optimal capacy of every crane hin a planning period and relationship beteen C and k needs further investigation. References [] Canonaco, P., Legato, P., Mazza, R. M. and Musmanno, R., 2008, A queuing netork model for the management of berth crane operations. Computers & Operations Research, 35(8), [2] Kim, K. H. and Kim, H. B., 2002, The optimal sizing of the storage space and handling facilies for import containers. Transportation Research, Part B, 36, [3] Kim, K. H. and Kim, K. Y., 2007, Optimal price schedules for storage of inbound containers. Transportation Research Part B: Methodological, 4(8), [4] Kim, K. H. and Park, K. T., 2003, A note on a dynamic space-allocation method for outbound containers. European Journal of Operational Research, 48, [5] Lee, D. H., Cao, Z. and Meng, Q., 2007, Scheduling of to-transtainer systems for loading outbound containers in port container terminals

6 76 G. R. Amin and H. Golchubian h simulated annealing algorhm. International Journal of Production Economics, 07(), [6] Lee, D. H., Wang, H. Q. and Miao, L., 2008, Quay crane scheduling h non-interference constraints in port container terminals. Transportation Research Part E: Logistics and Transportation Revie, 44(), [7] Murtya, K. G., Liu, J., Wan, Y. W. and Linn, R., 2005, A decision support system for operations in a container terminal. Decision Support Systems, 39, [8] g, W. C., 2005, Crane scheduling in container yards h inter-crane interference. European Journal of Operational Research, 64, [9] Steenken, D., Voß, S. and Stahlbock, R., 2004, Container terminal operation and operations research: A classification and lerature revie. OR Spectrum, 26(), [0] Zhang, C., Wan, Y. W., Liu, J. and Linn, R., 2002, Dynamic crane deployment in container storage yards. Transportation Research, Part B, 36, [] Wang, L. 2004, Container information tracing system. Journal of Traffic and Transportation Engineering, 4(),