Segregating space allocation models for container inventories in port container terminals

Transcription

1 Int. J. Production Economics 59 (1999) Segregating space allocation models for container inventories in port container terminals Kap Hwan Kim*, Hong Bae Kim Department of Industrial Engineering, College of Engineering, Pusan National University, Changjeon-dong, Kumjeong-ku, Pusan , South Korea Department of Industrial Engineering, Kyungsung University, Daeyeon-dong, Nam-ku, Pusan , South Korea Abstract This paper considers how to allocate storage space for import containers. In the segregation strategy, stacking newly arrived containers on the top of containers that arrived earlier is not allowed. We analyze cases where the arrival rate of import containers is constant, cyclic, and dynamic. Spaces are allocated for each arriving vessel so as to minimize the expected total number of rehandles. Mathematical models and solution procedures are suggested for obtaining the optimal solution. Numerical examples are provided in order to illustrate the solution procedure Elsevier Science B.V. All rights reserved. Keywords: Space allocation; Container; Terminal 1. Introduction The level of customer service is an important issue in the management of container terminals because of the severe competition between port terminals. One of the most important factors in customer service is the turn-around time for an outside truck to pick up an import container. This consists of the travel time in the yard, the waiting time for service, and the container transfer time by a transfer crane (TC). The waiting time and the transfer time are very dependent on the number of rehandles by the TC during the transfer operation. * Corresponding author. Tel.: # ; fax: # ; kapkim@hyowon.cc.pusan.ac.kr. This paper focuses on methodology for increasing the level of customer service by reducing the number of rehandles through efficient space allocation. In order to unload import containers from a vessel onto the container yard in port container terminals, the planner has to pre-assign the yard space. Once the yard space is allocated, the gantry crane unloads the containers and yard tractors deliver them to the TC. Then the TC stacks the delivered containers in the allocated yard space. When an outside truck arrives in a random manner and requests the TC to pick up an arbitrary container, the TC transfers it from the yard to the outside truck. One of the difficult problems in the yard operation is that it takes too much handling effort for the TC to rehandle containers on the top of the requested container /99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S ( 9 8 )

2 416 K.H. Kim, H.B. Kim/Int. J. Production Economics 59 (1999) Since it is very difficult to secure enough storage space in most port container terminals, the space allocation strategy affects significantly the efficiency of container handling operations. One of the key issues in space allocation is the height (tier) of container stacking. The higher the container stack in the yard is, the less is the ground space requirement for the same number of import containers. But, as the container stack becomes higher, the expected number of rehandles increases when the TC transfers a requested container to an outside truck. There are two strategies in space allocation for import containers: the segregation policy and the non-segregation policy. In the segregation policy, an empty space is allocated whenever import containers are unloaded from each vessel. In the nonsegregation policy, we select bays with empty slots, considering the duration of stay (DOS) of containers located at the bottom, and stack the unloaded containers onto the bays until they are filled up to a prescribed level [1]. The advantages of the segregation policy are the easy traffic control for the external trucks during the retrieval operation and the reduced number of rehandling operations, by preventing the hot containers which are more likely to be picked up in the near future from being buried under the cold ones. When using the segregation policy for allocating spaces for import containers we have to consider: (1) the arrival time of the containerships, (2) the number of containers to be unloaded from each vessel, and (3) the DOS of containers in the yard. We can get information on the arrival time of each containership from the berth schedule. For the information on the number of containers to be unloaded, shipping agents usually provide a booking prospect which contains the number of loading and unloading containers several days before the arrival of the containership. Most of container terminal operators establish the maximum limit on the DOS (called the free time limit ) and impose charges on containers whose DOS exceed the limit. Thus, it is safe to assume that few containers remain in the yard after the free time limit. Little attention has been paid by academic researchers on the operational aspects of container terminals until recently. Castilho and Daganzo [1] developed general expressions for the expected number of handlings to retrieve one container from a storage stack under segregation and non-segregation storage policies. However, they did not consider the dynamic nature of the problem: that the number of unloaded containers changes from time to time. McDowell et al. [2] analyzed the container handling operation and developed a mathematical model for determining the stacking configuration. Watanabe [3] suggested a simple method for estimating the number of rehandles, called the index of selectivity. Kim [4] developed a more accurate formula for estimating the number of rehandles than the one by Watanabe. Kim and Kim [5] and Taleb-Ibrahimi [6,7] examined the space allocation problem for export containers. Van Hee et al. [8,9] developed models for evaluating the performance of the key elements of a port terminal and introduced a decision support system for capacity planning of container terminals. Evers and Koppers [10] suggested a distributed traffic control strategy for automated guided vehicles at a container terminal. The load planning problem of export containers is dealt with in some studies [11,12]. Little research has been done on space allocation for import containers due to its dynamic characteristics. In this paper we deal with a storage space allocation problem for import containers with a dynamic space requirement with the objective of minimizing the number of rehandles. In the next section, we derive a formula which describes the relationship between the height of container stacks and the number of rehandles. We propose a mathematical model for the space allocation problem with a constant arrival rate of import containers. In Section 3, we analyze the case of a cyclic arrival rate of containers. In Section 4, we propose a mathematical model to schedule the space allocation for the case of a dynamic space requirement. We provide a conclusion in the final section. 2. The case of a constant arrival rate of import containers We begin with the most simple case in which the arrival rate of the import containers is constant. As

3 K.H. Kim, H.B. Kim/Int. J. Production Economics 59 (1999) mentioned above, the height of stack is directly related to the ground space requirements. On the other hand, the height is a variable which determines the number of rehandles. The height of stacks and the amount of space allocation are decision variables in the container inventory problem and are closely related to each other. The analysis in this paper is based on the following assumptions and notations: Assumptions 1. No container remains in the yard after the free time limit. 2. The rehandled container is moved to another slot within the same bay of the container stack. 3. The segregation space allocation policy is adopted. That is, containers unloaded during different periods are not mixed with each other in a same bay. 4. A transfer crane is the container handling equipment in the yard. 5. There is no prior information on the sequence of retrievals for import containers. Notation b the number of bays in the yard, r the number of stacks in a bay, t period index, h the average height of container stacks for containers unloaded during period t (a decision variable), A the total number of containers (TEU: Twenty-feet Equivalent Unit) unloaded during period t, N n c R(x) the planning horizon, the free time limit, the length of the cyclic space requirement (In this paper, we assume c"7), the expected total number of rehandles to pick up all the containers in a bay when the initial number of containers in the bay is x. The container yard consists of several blocks depending on the size of the yard and each block has usually bays. Each bay has several stacks (usually six) in which several (usually 3 4) containers can be stacked one over another. Fig. 1 shows the configuration of the block. Fig. 2 illustrates how the inventory level of import containers changes. When all the containers from a vessel are unloaded the inventory level reaches a maximum. After that, the shipper sends trucks to pick up its own containers in a random way. Most of the containers are picked up no later than the free time limit because the storage penalty is charged to every container which remains after then. However, since we assume the segregating strategy, the allocated space for a period can be released only after the free time limit. Although there may remain some containers even after the free time limit, they are usually cleared to make empty spaces for the next vessel. In order to estimate R(x), we assume that the rehandled containers are moved to other slots within the bay, which is a valid assumption if the TC has a significant setup time for moving between bays. Since the average initial height of container stacks is denoted by h, the average initial number of containers in a bay is h r. Then R(h r) can be Fig. 1. The configuration of the block.

4 418 K.H. Kim, H.B. Kim/Int. J. Production Economics 59 (1999) to the fraction of a bay occupied by the requirement of the specific period. We will delete the constant term in the following formulation. Because 1#1/(4r)(A/4r) is also constant, the objective function of the problem can be expressed as Fig. 2. Inventory level of import containers. estimated by the following equation: R(h r)"h r(h!1)/4#h (h #2)/16. (1) The validity of above Eq. (1) was numerically demonstrated in Ref. [4] and is also discussed in Appendix A. The minimization of the expected number of rehandles is the objective of the problem. By reducing the number of rehandles, we can expect to reduce the service time for outside trucks by the transfer cranes, which results in a higher customer service and a lower operating cost of transfer cranes. In this section, we assume that A "A for all t. We formulate mathematically the problem of minimizing the expected total number of rehandles under the condition that the space requirements are satisfied. The expected total number of rehandles during period t becomes h r(h!1)/4#h (h #2)/16A/(h r). (2) Note that A/(h r) denotes the number of bays required in order to accommodate the unloaded con- tainers during period t. If we denote A/(h r)byx, then Eq. (2) may be rewritten as 1#1/(4r)(A/4r)/x #constant term. (3) We treat x and h as continuous variables. Note that h represents the average height of stacks. A value of 11.5 for x implies that containers of period t are stacked on 11 empty bays and a half of another bay. We assume that, when a partial bay is allocated, the number of rehandles is proportional Min B, (4) x where B"1#1/(4r)(A/4r). The constraints are expressed as follows: x )b, (5) x *0 for all t. The optimal solution is obtained when x* "b/n. Thus, the optimal height of stacks becomes h* " A " NA rx* rb. (6) 3. The case of a cyclic arrival rate of import containers The berthing time of each container ship is determined by the long-term schedule. Since container ships are usually scheduled on the weekly basis the number of unloaded containers varies from day to day, but the same pattern is usually repeated each week. In this section we assume that the arrival rate of import containers follows a cyclic pattern with the period of one week. That is, we assume A "A. If we set B "1#1/(4r)(A /4r), it also holds that B "B. Since the parameters for decision making are repeated on a weekly cycle, it is also valid to assume x "x. Thus, the problem of space allocation may be formulated as follows: (P) Minimize B, (7) x x )b, t"1, 2, 2, 7, (8) x *A /(hm r), t"1, 2, 2, 7, (9)

5 K.H. Kim, H.B. Kim/Int. J. Production Economics 59 (1999) where α(i) is one plus the remainder of i divided by 7 and hm is the maximum allowed height restricted by the height of the transfer crane. Constraint (8) implies that the total available space is limited. When n"4, it becomes x #x #x #x )b for t"1, x #x #x # x )b for t"2 and so on. The maximum height of the stack may not exceed hm by constraint (9). Constraint (8) can be relaxed to get the following formulation: (R) Minimize B # x λ x!b (10) x *A /(hm r), t"1, 2, 2, 7. (11) Thus, the solution of (P) may be obtained by solving the following dual problem and the decomposed seven primal problems using subgradient optimization. The dual problem is as follows: (D) Maximize λ x!b #B x. (12) The decomposed primal problems are as follows: (DP ) Minimize B #x λ, (13) x x *A /(hm r), t"1, 2, 2, 7. (14) Note that for n"4 and t"1, λ becomes λ #λ #λ #λ which corresponds to the constraints associated with x. We can get the optimal solution of (DP ) for a given set of values for λ as follows: x* " B λ A /(hm r) if B λ *A /(hm r), otherwise. The subgradient optimization procedure is summarized as follows: 1. Set the initial upper bound, Z " (B /x ), where x "A /(hm r). And let π"2 and λ "1 for all i. 2. Evaluate x* "max B λ, A /(hm r). Let G(t)" x!b. If G(t))0, for t"1, 2, 7, then stop and the x* give the optimal solution. Otherwise go to step Calculate Z (λ ) Z (λ )" λ x*!b #B x. 4. Modify x* to satisfy constraints (8) and (9) using the procedure in Appendix B. Calculate the upper bound Z " (B /x*) using the modified values of x* s. 5. If Z!Z /Z (ε then stop. Otherwise ¹"π(Z!Z )/ (G(i)) and λ "max0, #¹*G(i), i"1, 2, 7. Go to step The case of a dynamic arrival rate of import containers In this section, we deal with the case in which the arrival rate of containers varies in an irregular way. In this case, we allocate spaces on a rolling horizon basis. In period 1 we plan the space allocation for periods 1 N, which is the length of the planning horizon. Based on the solution obtained, we allocate the space for period 1 only. At the beginning of period 2, we plan the space allocation for periods 2 to(n#1). We implement the space allocation only for period 2, and so on. In this manner the same process is repeated every period. At any period, we may formulate the following problem for space allocation: B Minimize, (15) x where B "(1#1/(4r))A/(4r), x )b, t"1, 2, N!1 (16)

6 420 K.H. Kim, H.B. Kim/Int. J. Production Economics 59 (1999) x )b!s, (17) x *A /(hm r), t"1, 2, 2, N. (18) When S is a given value for the boundary condition at the final stage and x (t)0) has also a predetermined value which corresponds to the amount of allocated space in a past stage, the above problem is converted into the following two subproblems which may be solved using the subgradient optimization technique: (D) Maximize λ ( x!b)# B x # λ ( x!b#s )# B x. (19) B (DP ) Minimize #x λ, x (20) x *A /(hm r), t"1, 2, N, where λ, 2, λ "0. The optimal solution of (DP ) becomes " B if B *A /(hm r), λ λ x* A /(hm r) otherwise. (21) The same procedure for subgradient optimization can be used to solve problem (D) and (DP ) iteratively. 5. Numerical experimentation In Table 1 an example for the case of cyclic arrival rate is illustrated. Data for the example are as follows: b"200, r"6, hm "5, n"4. An example for the case of dynamic arrival rate is solved in Table 2. Data for the example are as follows: b"200, S "60, hm "5, r"6, n"4, x "0.0, x "73.0, x "47.0. Table 1 A numerical example of the case of cyclic arrival rate t A x* h* R Note: R denotes the expected number of rehandles. In order to evaluate the performance of the algorithm we solved 20 problems for the case of cyclic arrival rate and 60 problems for the case of dynamic arrival rate. For the case of cyclic arrival rate the free time limit (n) is varied from 2 to 5. For the case of dynamic arrival rate the planning horizon (N) is varied from 10 to 25 and the free time limit is assumed to be between 3 and 6. For both cases the generation of the arrival rate is controlled so that the average amount of occupied space becomes 60 70% or 20 30% of the maximum capacity of the yard. The results are summarized in Table 3. When we set ε"0.01, the ratio of the duality gap (Z!Z /Z ) was less than for all the cases shown in Table 3. The solution procedure was programmed using Delphi 2.0 and run on an IBM PC (Pentium). The run time was less than 1 s for all cases. 6. Conclusions In this paper we present a formula which represents the relationship between the stack height and the number of rehandles. We formulate the problem of minimizing the expected total number of rehandles under the condition that the space requirements should be satisfied. We consider the cases of a constant arrival rate, a cyclic arrival rate and a dynamic arrival rate of import containers. For each case we suggest a methodology, based on the Lagrangian relaxation technique, for finding the optimal solution. We provide numerical examples and also the results of numerical experimentation for evaluating the performance of the algorithm developed.

7 K.H. Kim, H.B. Kim/Int. J. Production Economics 59 (1999) Table 2 A numerical example of the case of dynamic arrival rate t A x* h* R Table 3 Summary of the performance of the algorithm The case of cyclic arrival rate The case of dynamic arrival rate Number of problems solved The ratio of duality gap Appendix A. Estimation of the number of rehandles We estimate the expected total number of rehandles in picking up all the containers in a bay, when the sequence of the pick-up orders for containers is completely unknown. Let us assume that there are h r containers in a bay. And let n(h, r) be the expected number of rehandles in picking up the next arbitrary container, when there are h r containers in the bay. Then n(h, r) depends on not only the value of h r but also on the configuration of the bay. Let the configuration of the bay be as in Fig. 3. The expected number of rehandles for containers in the first tier is (h!1)r/(h r), and the one in the second tier is (h!2)r/(h r), and so on. Thus, the expected number of rehandles for the next pickup becomes n(h, r)"(1/h r)r#2r#3r#2#(h!1)r "(h!1)/2. (A.1) But formula (A.1) underestimates the true value of n(h, r) since the assumption of the even height among stacks in a bay is too optimistic. Since the rehandled containers are removed to slots as low as possible the height of all the stacks except the one Fig. 3. The configuration of the bay of an even height (the number inside the box represents the number of rehandles). where the pick-up occurred will differ by no more than one tier. For the stack where the last pick-up occurred the average height of the stack will be approximately h /2. Thus, we can estimate the number of rehandles, using the configuration of the bay as in Fig. 4. Comparing Fig. 4 with Fig. 3, one container is put on the top of each of the h /2 stacks that have h containers. Thus, h (h /2)/(h r) should be added to the expected number of rehandles of (A.1). And in the last stack, the expected number of rehandles reduced to 1#2#3#2#(h /2!1)/(h r). It has been reduced to 1#2#3#2#(h / 2!1)/(h r). The difference between two term is (3h!2h )/(8h r). The change in the expected number of rehandles resulting from the modification of the configuration becomes h (h /2)/(h r)!(3h!2h )/(8h r)"(h #2)/(8r). Thus, n(h, r)"(h!1)/2#(h #2)/(8r). (A.2)

8 422 K.H. Kim, H.B. Kim/Int. J. Production Economics 59 (1999) Table 4 Comparison of the total expected number of rehandles (percentage error) Number of rows Number of tiers True value of exact method Approximated formula Fig. 4. The modified configuration of the bay (h "6, r"6). If we assume that n(h, r) also decreases linearly as h r decreases linearly, then the expected total number of rehandles in picking up h r containers becomes R(h r)" n(h, r)/(h r)x dx "h r(h!1)/4#h (h #2)/16. (A.3) The numerical comparison between the results of Eq. (A.3) and the results of the exact method from Ref. [4] are given in Table 4. The exact method is based on the assumption that the probability of a container being picked up next is the same for all the containers in a bay and rehandled containers are positioned at the lowest possible slot in a bay. The comparison is done for various configurations of container stacks. The average percentage absolute error of the approximate method compared with exact method is only 1.88%. The advantage of Eq. (A.3) is its simplicity in estimating the expected total number of rehandles. Appendix B. The procedure for producing a feasible solution 1. Compute G(t), t"1, 2, Let»"tG(t)'0. Note that» is the set of indices of violated constraints. 3. If»"H, stop (!2.7) (!4.9) (!4.9) (!5.0) (!2.8) (!2.0) (!1.6) (!1.7) (!2.4) (!0.4) (0.2) (0.1) (!2.1) (1.0) (1.4) (1.2) 4. Let t*"arg min G(t) and x be the variable which is included in the t*th constraint and has the least value of B among variables included in the constraint. Let z(j)"min[x!a /(hm r), G(t*)]. 5. x "x!z(j). Go to step (1). References [1] B.D. Castilho, C.F. Daganzo, Handling strategies for import containers at marine terminal, Transportation Research 27B (1993) [2] E. McDowell, G. Martin, D. Cho, T. West, A Study of Maritime Container Handling, Oregon State University, Sea Grant College Program, Ads. Corvallis, Oregon, 97331, 1985, pp [3] Watanabe, Characteristics and analysis method of efficiencies of container terminal an approach to the optimal loading/unloading method, Container Age (1991) [4] K.H. Kim, Evaluation of the number of rehandles in container yards, Computer and Industrial Engineering 32 (4) (1997) [5] K.H. Kim, D.Y. Kim, Group storage methods at container port terminals, MH-vol. 2, The Material Handling Engineering Division 75th Anniversary Commemorative Volume ASME 1994.

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