- Email: [email protected]
Applying an adaptive tabu search algorithm to optimize truck-dock assignment in the crossdock management system
Expert Systems with Applications 41 (2014) 16–22
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Applying an adaptive tabu search algorithm to optimize truck-dock assignment in the crossdock management system Zhaowei Miao, Shun Cai ⇑, Di Xu School of Management, Xiamen University, Xiamen 361005, PR China
a r t i c l e
i n f o
Keywords: Tabu search Dock assignment Artificial intelligence Crossdock management system
a b s t r a c t Different from warehouse, crossdock is considered as a ‘‘JIT’’ technique in logistics and supply chain management (LSCM), which is usually a short period of time to store cargos. Since cargos transported to a crossdock by inbound trucks are immediately sorted out, repackaged, routed and loaded into outbound trucks and then delivered to customers within one day, one of the key issues to operate crossdock successfully is to develop an efficient truck-dock assignment module in the crossdock management system so that all the cargos can be delivered to customers on time. In this paper, we propose an adaptive tabu search (ATS) algorithm as an artificial intelligence (AI) tool to optimize the truck-dock assignment problem within a crossdock in a very efficient way, and computational experiments are conducted, showing that our approach dominates the CPLEX Solver in both effectiveness and efficiency. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Crossdock is a material handling and distribution point in logistic network, in which items move directly from receiving dock to shipping dock without being stored for a long period, and has been highly emphasized by researchers and companies recently in warehouse management. The activities at crossdocks include assigning trucks to docks, sorting, scanning, repacking, routing and loading items shipped from inbound trucks to outbound trucks, as well as temporary inventory managing (Bartholdi & Gue, 2002; Boysen & Fliedner, 2010; Lim, Miao, Rodrigues, & Xu, 2005). One of the key decisions is the dock assignment for both inbound and outbound trucks. Dock assignment affects the performance of transshipment network, efficient dock assignment can be helpful to reduce shipment delays, operational times in crossdock and cost including operational cost for transferring, sorting and repacking (Acar, Yalcin, & Yankov, 2012; Alpan, Larbi, & Penz, 2011a, Alpan, Ladier, Larbi, & Penz, 2011b; Boloori Arabani, Fatemi Ghomi, & Zandieh, 2011; Boysen, 2010; Konur & Golias, 2013; Lee, Kim, & Joo, 2012; Liao, Egbelu, & Chang, 2013; Miao, Lim, & Ma, 2009; Vahdani & Zandieh, 2010; Yu & Egbelu, 2008). Up to now, more and more research papers deal with the shortterm scheduling problems which continuously arise during the daily operations of crossdock terminals. Lim et al. (2005) worked on various transshipment problems in a crossdock distribution network, and they found that well-scheduled trucks assignment can help to reduce the transportation time and inventory holding ⇑ Corresponding author. E-mail address: [email protected] (S. Cai). 0957-4174/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2013.07.007
time, correspondingly reduce the transportation costed as well as inventory cost by consolidation. Bartholdi and Gue (2002) examined minimizing labor costs in freight terminals by properly assigning incoming and outgoing trailers to doors with respect of types of congestion, however, they did not address actual dock assignments to arriving vehicles when considering the time window of trucks. Yu and Egbelu (2008) studied a crossdock system with unlimited temporary storage. They investigated to find the best truck docking or scheduling sequence for both inbound and outbound trucks to minimize total operation time when a temporary storage buffer to hold items temporarily is located at the shipping dock, but they left out various important issues in real application such as crossdock physical layout, warehouse capacity, and arrival/departure schedules of vehicles. Based on the strategies proposed by Yu and Egbelu (2008) and Vahdani and Zandieh (2010) applied other heuristic methods, including genetic algorithm, tabu search, population based simulated annealing with diversification, and variable neighborhood search algorithm, to solve the same problem in Yu and Egbelu (2008), and proved the robustness of each proposed heuristics by numerous computational experiments. Miao et al. (2009) provided scheduling procedures adopted from gate assignment in airports, where trucks are assumed to have given service time windows, and they proved the problem is NP-hard, therefore two kinds of meta-heuristics including genetic algorithm and tabu search were proposed to find an optimal solution. Boysen (2010) treated a special truck scheduling problem arising in the zero-inventory crossdocks of the food industry to minimize time makespan including the flow time, processing time and tardiness of outbound trucks. Boloori Arabani et al. (2011) proposed genetic algorithm, tabu search, particle
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
swarm optimization and so on to find the best sequence of inbound and outbound trucks in order to minimize the total operation time. Lee et al. (2012) aimed to maximize the amount of products that are able to ship within a given time horizon and developed intelligent genetic algorithms to optimize door-assigning and sequencing of trucks in crossdocks. Konur and Golias (2013) studied a crossdock truck scheduling problem by formulating a bi-level optimization problem and developed a genetic algorithm to solve it. Alpan et al. (2011a) proposed a bounded dynamic programming approach to schedule inbound and outbound trailers in a multiple receiving and shipping door cross dock environment. In their another paper, Alpan et al. (2011b) developed several heuristics to find the best schedule in order to minimize the sum of inventory holding and truck replacement costs. Liao et al. (2013) proposed six meta-heuristic algorithms, including simulated annealing, tabu search, ant colony optimization and so on, to minimize total weighted tardiness for the dock assignment and sequencing of inbound trucks in a crossdock. Acar et al. (2012) introduced a mixed integer quadratic model for the problem which considered variability of truck arrival and service time. We refer readers to Boysen and Fliedner (2010) for thorough and excellent literature reviews of crossdock scheduling. Artificial intelligence tools are widely applied to the dock assignment problem in crossdocks, because as the increasing of problem scale, traditional methods or softwares cannot find the best solution effectively. Besides those literatures mentioned above, there are a lot of papers which applied artificial intelligence tools such as genetic algorithm, tabu search, simulated annealing and so on, to the other optimization problems such as inventory routing problem, production planning problem, network design problem, vehicle routing problem, and so on. McWilliams, Stanfield, and Geiger (2005) treated a different truck scheduling problem arising from freight consolidation terminals in the parcel delivery industry, where the movement of goods inside the terminal is conducted by a convey or belt system, and proposed a simulation-based scheduling algorithm that used a genetic algorithm to drive the search for new solutions. Lin, Yu, and Lu (2011) studied the truck and trailer routing problem with time windows and proposed a simulated annealing heuristic which can obtain consistent quality solutions. In this paper, we design a new tabu search algorithm to resolve proposed truck-dock assignment problem in a crossdock since tabu search proposed by Glover and Laguna (1998) is known as one of the powerful artificial intelligence tools to solve NP-hard problems. Ramezanian, Rahmani, and Barzinpour (2012) considered general two-phase aggregate production planning problem which is NP-hard and implemented tabu search algorithm to produce good-quality solutions. Li, Leung, and Tian (2012) developed a multistart adaptive memory-based tabu search algorithm to solve the heterogeneous fixed fleet open vehicle routing problem (HFFOVRP), and computational experiments showed that this modified tabu search algorithm has efficiency and effectiveness. Tarantilis, Stavropoulou, and Repoussis (2012) studied another vehicle routing problem called consistent vehicle routing problem (ConVRP) and solved it by a template-based tabu search algorithm. Moreover, Liao, Lin, and Shih (2010) integrated crossdock into vehicle routing problem and applied tabu search algorithm to determine the number of vehicles and a set of vehicle schedules with a minimum total cost including operation cost and transportation cost. In this paper, we extend the problem proposed by Miao et al. (2009), and take consideration of time windows for both inbound and outbound trucks in a transshipment network through crossdocks where the number of trucks exceeds the number of docks available, of which the objective is to minimize the total costs including operational cost of the cargo shipment and penalty cost of unfulfilled shipments. And we formulate this problem into a
17
0 1 integer programming model. Since this kind of truck-dock assignment problem is NP-hard, meta-heuristic should be a good approach to solve it. Adaptive tabu search (ATS) is proposed and developed for conducting computational experiments on test sets, and the results compared with CPLEX Solver 11.0 show that our algorithm dominates the CPLEX Solver in most of all test cases. This paper is organized as follows: in the next section, we give an IP model of the problem. Adaptive tabu search is proposed and developed in Section 3. We provide computational results in Section 4. In Section 5, we summarize our findings and suggest future work.
2. Problem description For operations in crossdocks, inbound trucks transport cargos to it, which are immediately sorted out, repackaged, routed and loaded into outbound trucks and then delivered to customers within one day (Alpan et al., 2011a; Boloori Arabani et al., 2011; Boysen, 2010; Konur & Golias, 2013; Miao et al., 2009; Vahdani & Zandieh, 2010). As a result, one of the key issues to implement crossdock as a ‘‘JIT’’ technique in logistics and supply chain management successfully is to develop an efficient truck-dock assignment module in the crossdock management system so that all the cargos can be delivered to customers on time (Acar et al., 2012; Alpan et al., 2011b; Konur & Golias, 2013; Lee et al., 2012; Yu & Egbelu, 2008). As mentioned before, our proposed truck-dock assignment problem is NP-hard, the artificial intelligence tool, such as genetic algorithm, tabu search, simulated annealing and so on, should be a good approach to obtain good-quality solutions efficiently (Li et al., 2012; Liao et al., 2013; Lin et al., 2011; Miao et al., 2009). In order to develop ATS to solve this problem, we need to formulate it into a mathematical model first. Before introducing the model, we should explain some common assumptions which are usually adopted in related literatures (Alpan et al., 2011a, 2011b; Liao et al., 2013; Miao et al., 2009). For this problem, we do not address secondary constraints which account for the loading/unloading times or other buffers between arrival and departure times, since they can be easily dealt with by extending the truck arrival and departure durations. The important constraint is the time window constraint for both inbound trucks and outbound trucks, i.e. each inbound/outbound truck can occupy an inbound/outbound dock to unload/load only within its pre-allocated arrival time and departure time. Also we take operational time of transferring, sorting and repacking within the crossdock into consideration. For example, if outbound truck A at outbound dock a is going to receive cargos from inbound truck B at inbound dock b, then inbound truck B needs to ship the cargos to dock a before truck A leaves when taking time window and the operational time for cargos between inbound dock b to outbound dock a into consideration, while the operational time is usually decided by the physical distance between the two docks. As we know, the crossdock management system should schedule the trucks well in order to deliver cargos to customers on time. If it is found that some trucks cannot be assigned due to the limited number of docks, those trucks need to be rescheduled to complete their tasks. The trucks can be delayed, or rescheduled to other crossdocks where a vacancy is available. In such situations, a penalty cost is incurred for any unfulfilled cargo shipment. Fig. 1 illustrates an outline of the crossdock and major elements of our problem, where the notations will be introduced in details next. We formulate our proposed problem into a 0 1 IP model, for which the following notations are used: NI: set of inbound trucks arriving at the crossdock, and jNIj = nI, where j j represents the cardinality of a set;
18
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
Fig. 1. Truck dock assignment problem.
MI: set of inbound docks available in the crossdock, and jMIj = mI, where nI > mI; NO: set of outbound trucks departing from the crossdock, and jNOj = nO; MO: set of outbound docks available in the crossdock, and jMOj = mO, where nO > mO; aIi : arrival time of inbound truck i (1 6 i 6 nI); I di : departure time of inbound truck i (1 6 i 6 nI); O aj : arrival time of outbound truck j (1 6 j 6 nO); O dj : departure time of outbound truck j (1 6 j 6 nO); tk,l: operational time for pallets from inbound dock k to outbound dock l (1 6 k 6 mI, 1 6 l 6 mO); fi,j: number of pallets transferring from inbound truck i to outbound truck j (1 6 i 6 nI, 1 6 j 6 nO); ck,l: operational cost per pallet from inbound dock k to outbound dock l (1 6 k 6 mI, 1 6 l 6 mO); pi,j: penalty cost per pallet cargo from inbound truck i to outbound truck j (1 6 i 6 nI, 1 6 j 6 nO). We also define dIi;j 2 f0; 1g: 1 iff inbound truck i departs no later than inbound truck j arrives; 0 otherwise. And dOi;j 2 f0; 1g: 1 iff outbound truck i departs no later than outbound truck j arrives; 0 otherwise. Then we can set dIi;j and dOi;j according to the truck time n o n o I I O O and aOi ; di ; aOj ; dj . window data aIi ; di ; aIj ; dj The decision variables are as follows:
yIi;k
¼
yOj;l ¼
1; if inbound truck i is assigned to inbound dock k 0;
min COST Operation þ COST Penalty where
COST Operation ¼
mI X mO X nI X nO X ck;l fi;j zi;j;k;l k¼1 l¼1 i¼1 j¼1
and
COST Penalty
nI X nO mI X mO X X ¼ pi;j fi;j 1 zi;j;k;l i¼1 j¼1
!
k¼1 l¼1
s.t. mI X yIi;k 6 1 ð1 6 i 6 nI Þ
ð1Þ
k¼1 mO X yOj;l 6 1 ð1 6 j 6 nO Þ l¼1
ð2Þ
yIi;k þ yIj;k 1 6 2 dIi;j þ dIj;i ð1 6 i;j 6 nI ; i – j;1 6 k 6 mI Þ yOi;l þ yOj;l 1 6 2 dOi;j þ dOj;i ð1 6 i;j 6 nO ;i – j;1 6 l 6 mO Þ yIi;k þ yOj;l P 2zi;j;k;l ð1 6 i 6 nI ;1 6 j 6 nO ; 1 6 k 6 mI ; 1 6 l 6 mO Þ O dj aIi t k;l fi;j zi;j;k;l P 0 ð1 6 i 6 nI ;1 6 j 6 nO ; 1 6 k 6 mI ; 1 6 l 6 mO Þ
ð3Þ ð4Þ ð5Þ ð6Þ
yIi;k 2 f0; 1g;yOj;l 2 f0;1g; zi;j;k;l 2 f0;1g ð1 6 i 6 nI ;1 6 j 6 nO ;1 6 k 6 mI ;1 6 l 6 mO Þ ð7Þ
1; if outbound truck j is assigned to outbound dock l 0; otherwise:
zi;j;k;l ¼
otherwise
The objective is to minimize the total cost: the sum of the operational cost and the penalty cost. The 0 1 IP model is as follows:
1; if f i;j is transfered from inbound dock k to outbound dock l 0; otherwise: I
I
Note that yIi;k ¼ yIj;k ¼ 1 implies that aIi > dj or aIj > di ð1 6 i; j 6 nI Þ, O and similarly, yOi;l ¼ yOj;l ¼ 1 implies that aOi > dj or O O O aj > di ð1 6 i; j 6 n Þ.
The first term in the objective function is total operational cost and the second term is the total penalty cost. Constraints (1) and (2) ensure that each truck cannot be assigned to more than one dock. Constraints (3) and (4) specify that one dock cannot be occupied by two different trucks simultaneously. Constraints (5) represent the logic relationship among yIi;k ; yOj;l and zi,j,k,l. Finally, constraints (6) say that the transfer process of cargoes from inbound truck i at inbound dock k to outbound truck j at outbound dock l should take place within the specified time window, i.e. from the arrival time of inbound truck i to the departure time of outbound truck j.
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
19
Algorithm 1. Initial Solution Generation Procedure. Fig. 2. The solution representation.
3. Adaptive tabu search Since this problem is NP-hard, it is necessary to adopt artificial intelligence tool like tabu search to handle large real world cases effectively and efficiently. Tabu search proposed by Glover and Laguna (1998) has been proved powerful in optimization for NPhard problems. It is a meta-heuristic search procedure that proceeds iteratively from one solution to another by moves in a neighborhood space with the assistance of adaptive memory, which plays an important role in the search process. Therefore, neighborhood structures are rather key in the whole procedure. In our proposed tabu search algorithm, we adopt four kinds of moves to explore neighborhood space. What’s more, we apply adaptive rule to control the exploring procedure alternately rather than use all moves simultaneously, that is why the propose metaheuristic is called adaptive tabu search (ATS) in this paper. In the following section, we will show the details of ATS.
Initialize num_list for all trucks according to their total penalty costs; for i 1 to nI do Randomly select a feasible inbound dock for the inbound truck according to num_list until all inbound trucks have been selected; Otherwise, set the value of the inbound dock for this inbound truck to 0; end for for j 1 to nO do Randomly select a feasible outbound dock for the outbound truck according to num_list until all outbound trucks have been selected; Otherwise, set the value of the outbound dock for this outbound truck to 0; end for Calculate the value of this solution.
3.3. Neighborhood search methods 3.1. Solution representation We first define solution representation that is the key element in meta-heuristics design. The solution is the dock assignment represented by a sequence, which has length nI + nO, and involves twopart decisions: the first part is dock assignment for inbound trucks (the sequence number is from 1 to nI), and the second part is for outbound trucks (the sequence number is from nI + 1 to nI + nO). A sequence sI1 ; sI2 ; . . . sInI ; sOnI þ1 ; . . . ; sOnI þnO , which means that inbound truck 1 is assigned to inbound dock sI1 , inbound truck 2 is assigned to inbound dock sI2 , and so on and so forth ð1 6 i 6 nI þ nO ; 0 6 sIi 6 mI ; 0 6 sOi 6 mO Þ. If the inbound/outbound truck i is unassigned to any of the inbound/outbound docks, which is possible when all the inbound/outbound docks are occupied, we give sIi =sOi the value of 0. If the solution is feasible, the dock assignment is then uniquely determined by the sequence of sI1 ; sI2 ; . . . ; sOnI þnO . The representation is depicted in Fig. 2. In the tabu search memory we implement, only the assignment information is captured so that only the move that has the identical assignment to the ones in tabu search memory will be forbidden. 3.2. Initial solution generation Initial solution is rather important in meta-heuristic design, a good initial solution generation strategy will provide an effective input for algorithms. As we know, a good strategy used to generation initial solution depends on attributes of the proposed problem. Therefore, we design a greedy strategy based on penalty cost for inbound/outbound trucks. First, we construct a list as num_list, to store the number of inbound and outbound trucks. We define the total penalty cost for PO inbound truck i as PIi ¼ nj¼1 pi;j and that for outbound truck j as PnI O P j ¼ i¼1 pi;j . During the construction process, we first calculate the total penalty cost for each truck respectively, then each truck number is stored in descending order according the corresponding total penalty cost in the list. After initializing such a list, we assign a dock to each truck according to the total penalty cost, i.e. the truck with higher total penalty cost has a larger probability to be assigned a dock. Algorithm 1 displays the procedure of generating initial solutions.
We design four kinds of neighborhood search methods to make deepest search to the best solution found at each iteration. The four move strategies are implemented to both inbound and outbound truck-dock assignments. The following will give the detail procedures: Move I-Insert unfilled truck: The move is randomly insert an unfilled truck into current truck dock assignment scheme. In this process, constraints (3) and (4) may be violated so that infeasible solutions will be generated. Move II-Exchange unfilled truck with filled truck: This move is implemented after Move I, that is, any unfilled truck cannot be inserted into assignment sequence. In this move, we first randomly select an unfilled truck and a filled truck and then assign the unfilled truck to the dock the filled truck assigned, meanwhile, the sequence value of the selected filled truck is set to 0. It should be noticed that the time window of the unfilled truck may conflict with that of the trucks assigned to the selected dock. Move III-Exchange two filled trucks: In this move, two filled trucks are randomly selected, then we exchange their assigned docks in order to get a better solution. Similar to Move II, infeasible solutions can be produced because the time window constraints. Move IV-Exchange truck subsequences assigned to two docks: This move is different from Move III, rather than randomly select two trucks, we randomly select two docks and swap the truck subsequences assigned to them. The procedure is terminated until find a superior solution. Noticed, Move I to III may produce infeasible solutions, while, in Move IV, exchanging trucks subsequences between two docks will not violate the time constraints, therefore, Move IV always can get a feasible solution. For any infeasible solution, we repair it by setting the sequence values of those conflicting filled trucks to 0 one by one until it becomes feasible. 3.4. Adaption rule Extensively computational experiments show Move IV can help to explore neighborhood explicitly and consume less time to attain
20
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
a feasible neighborhood, and Move I to III can help to move out of the local optima and explore other regions of the solution due to their large neighborhood but are more time-consuming. Based on this consideration of the four types of moves, we apply adaption scheme to use the four moves to explore neighborhood according to a rule set which includes two criteria as follows (Miao, Yang, Fu, & Xu, 2012). The iteration to be used. This one defines the iteration of each move used to explore its neighborhood. The iteration for Move IV initiates as 20, and the iterations for Move I to III are 10 respectively, because Move IV is less time-consuming, and does well in exploring the neighborhood of solutions, the iteration for it is relatively large. The iteration to be terminated. It defines the iteration that a certain move cannot make improvement. The iteration is set to 10 for Move IV and 5 for other moves. Additionally, if all moves can not improve the best current solution within 50 iterations, the current best solution will be replaced by the current solution with aim to reach deepest search of the current best solution. All moves are controlled by adaptive rules to explore different neighborhood space alternately. The principle of adaption is to take deepest search to the current best solution meanwhile avoid the local optimum. Let #term_iter denote the maximum number of iterations for which the best solution can’t be improved. Note that, if the current best solution can not be improved within #term_iter iterations, then terminate the algorithm and the current best solution is considered as the goal optimum. A neighbor x0 of current solution xcurr to be selected must meet one of the following conditions: (a) It is not forbidden (i.e. the assignment is not identical to any assignments of recent tabu tenure moves); (b) The cost of x0 is better than the current best cost (aspiration criterion). 3.5. ATS framework In this section, we outline the framework of ATS algorithm. Notations are defined as following: xcurr denotes the current best solution, xinit denotes the initial solution, #max_iter denotes the maximum number of iterations, and #term_iter denotes the maximum number of iterations for which the best solution can’t be improved. Algorithm 2. ATS Framework. Initialize the parameters of ATS; Generate an initial solution xinit by Algorithm 1, set xcurr xinit; Empty tabu list for iter 1 to ]max_iter do if Move I meets its adaptive exploring rules then Generate a set of neighborhood solutions N(xcurr) of xcurr by Move I; Repair the infeasible solutions. end if if Move II meets its adaptive exploring rules then Generate a set of neighborhood solutions N(xcurr) of xcurr by Move II; Repair the infeasible solutions. end if if Move III meets its adaptive exploring rules then Generate a set of neighborhood solutions N(xcurr) of xcurr by Move III; Repair the infeasible solutions.
⇑ (continued)
Algorithm 2. ATS Framework. end if if Move IV meets its adaptive exploring rules then Generate a set of neighborhood solutions N(xcurr) of xcurr by Move IV. end if if The solution x0 2 N(xcurr) meet either condition (a) or (b) then Set xcurr x0 ; Update the TS memory. end if if xcurr can’t be improved within #term_iter then Consider the solution as the goal optimal solution and terminate the procedure. end if end for Output the best solution and escaped time
4. Experimental results and analysis All the algorithms were coded in Java. As comparison, we use ILOG CPLEX 11.0 to solve the formulation presented in Section 2, all the experiments are conducted on Intel (R) Core (TM)2 Duo CPU E4500 @ 2.2 GHz, 2.19 GHz machine with 1.99 GB memory. In the following, we first demonstrate the parameter setting, and then detailed computational results are shown. 4.1. Test data generation and parameter setting The topology of crossdock is referred to the layout of crossdock designed by Miao et al. (2009). But we modify it to make its topology more reasonable according to the real-world situation. We revise their representative layout of a crossdock to have two parallel sets of terminals where docks are symmetrically located as shown in Fig. 3; meanwhile, the above sets are considered as inbound docks, and the below terminals as outbound docks. The distance between two adjacent inbound/outbound docks within one terminal (e.g., inbound dock 1 and inbound dock 2) is 1 unit and the distance between two parallel docks in opposite terminals (e.g., inbound dock 1 and outbound dock 1) to be 3 units. To simplify the problem, we assumed that forklift can only walk ‘horizontally’ or ‘vertically’, i.e., if one forklift wants to transfer one pallet from inbound dock 1 to outbound dock 4, the walking distance is 3 + 1 3 = 6 units (thus the distances are rectilinear). This is similar to the so-called Manhattan metric. We note that it is easy to use other metrics for the problem to compare performance. The test data generation program requires two parameters: the number of inbound trucks nI and the number of inbound docks mI. Simply, the number of outbound trucks nO and the number of outbound docks mO are equal to nI and mI respectively. The arrival time of inbound truck aIi ð1 6 i 6 nI Þ is uniformly generated in h i I , and the interval of the outbound trucks is the interval 1; n m60 I h i I I 15; n m60 þ 15 . The departure time of inbound truck di is generated I I
O
as di ¼ aIi þ ½25; 40, for the outbound truck is di ¼ aOi þ ½25; 40. The rectilinear walking distances between inbound dock k and outbound dock l is calculated according to the topology and then proportionally converted to the corresponding operational cost ck,l and operational time tk,l. The number of shipping pallets fi,j representing the quantity shipped from inbound truck i to outbound truck j is generated randomly in the interval [6, 60]. The penalty cost
21
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
Fig. 3. Crossdock topology.
are calculated by averaging values of all the instances in each group.
Table 1 Results of CPLEX and ATS of instances with small scales. Problem size
3-6-3-6
3-7-3-7
3-9-3-9
3-12-3-12
CPLEX
Obj
7033 6.35 7036
11563 16.40
20085 94.46
ATS
Time(s) Obj
4747 3.28
11563 0.15
20085 0.26
Time(s)
CPLEX
Obj
ATS
Time(s) Obj Time(s)
4747 0.06
0.06
4-10-4-10
4-12-4-12
5-12-5-12
5-15-5-15
14431 90.06
21242
14431 0.21
20827 0.56
20706 545.02 20720
32939 >1800 32948
0.62
1.55
260.50
per unit cargo from truck i to truck j(pi,j) is generated in the interval [1, 4] + maxk,l{ck,l}. We conducted a great number of experiments to test the performance of our heuristic approach, and the following parameter setting of ATS algorithm has been proved efficient: maximum number of iterations is 105 and each time 30 neighbors are generated with a tabu tenure equal to 10. The algorithm is to terminate if the best solution was not improved within 500 iterations.
1. Computational results of small scale instances. The size for eight groups of small scale instances are generated from 3-6-3-6 to 5-15-5-15. The performance of CPLEX solver and ATS algorithm of this category are shown in Table 1. We highlight the lower average objective attained by both of them with underline in the table. As the results shown, CPLEX and our algorithm attain the same average objective in four groups, and CPLEX performs a little bit better than our algorithm, but they are very close; however, ATS can get near optimal solutions within much shorter time than CPLEX. 2. Computational results of large scale instances. Table 2 presents the results of eight large scale instance groups with size ranging from 8-20-8-20 to 14-28-14-28. Judged from value of Obj, ATS has a rather better performance than CPLEX in all instances. What’s more, values in row Time(s) show that, CPLEX has to take more than 7200s to attain a near optimal solution. And ATS has a great advantage in speed, of which average time is ranging from 5s to 90s in each group. From the results of the experiments, the superiority of our heuristic method is obvious, especially for large scale instances. Our algorithm not only can get better solutions than CPLEX, but also consumes much less time. The advantage of ATS in time dimension is quite important, which is one of our major contributions of this paper, because as a ‘‘JIT’’ technique in logistics and supply chain management, crossdocks need to tranship cargos to customers quickly, which requires an efficient truck-dock assignment module in the crossdock management system. In conclusion, our algorithm can greatly improve the performance of truck-dock assignment module in term of effectiveness and efficiency, which is a good approach to tackle this kind of problem in real-world application.
4.2. The results We design two categories of random instances, including small and large scales, to test the algorithm. In each category, we conduct 32 test cases and sorted them into 8 groups, and record the detailed results in Tables 1 and 2. The first row of each table specifies the instance size. mI nI mO nO denotes that there are nI inbound trucks and mI inbound docks, likewise, nO outbound trucks and mO outbound docks for this instance group. The effectiveness and efficiency of CPLEX and ATS are evaluated by objective and time, which are shown in rows Obj and Time(s). Both of values
Table 2 Results of CPLEX and ATS of instances with large scales. Problem size CPLEX ATS
Obj Time(s) Obj Time(s)
CPLEX ATS
Obj Time(s) Obj Time(s)
8-20-8-20
8-24-8-24
10-20-10-20
10-24-10-24
63020 >7200
90412 >7200
64228 >7200
90939 >7200
61733 5.02
89202 17.27
61819 15.23
88935 14.06
12-24-12-24
12-28-12-28
13-26-13-26
14-28-14-28
93694 >7200
127331 >7200
110085 >7200
128584 >7200
91555 47.73
124713 71.50
107651 58.63
125552 90.75
22
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
5. Conclusions and future work Compared with the traditional warehouse, the crossdock is considered as a ‘‘JIT’’ technique in logistics and supply chain management, which is usually a short period of time to store cargos. Since cargos transported to a crossdock by inbound trucks are immediately sorted out, repackaged, routed and loaded into outbound trucks and then delivered to customers within one day, one of the key issues to operate crossdock successfully is to develop an efficient truck-dock assignment module in the crossdock management system so that all the cargos can be delivered to customers on time. In this paper, we consider an over-constrained truck-dock assignment problem in a crossdock, in which the number of trucks are much larger than that of docks. The objective of the study is to find the best truck-dock assignment scheme for both inbound and outbound trucks to minimize total cost including the penalty cost of unfulfilled shipments and the operational cost of shipments within the crossdock. The paper contributes to the extant literatures in the following two ways. First, we provide a 0 1 Integer Programming model for such a complex problem, which can be input to those optimization softwares like ILOG CPLEX solver to get optimal or near optimal solutions so as to help crossdock managers make a decision if they don’t have other better methods to solve it. Second, we propose an adaptive tabu search algorithm as an artificial intelligence tool to optimize this truck-dock assignment problem which is known to be NP-hard in a very efficient way. For more details, our algorithm not only can get better solutions than CPLEX, but also consumes much less time. The advantage of ATS in time dimension is the key contribution because our algorithm can be integrated into truck-dock assignment module in the crossdock management system, which can help crossdocks to tranship cargos to customers within a short time with a lower cost so that not only the customer’s satisfactory level can be improved by reducing responsive time, but also inventory cost can be reduced for both customers and crossdocks. With the current problem, the topology of crossdock is quite regular. One future study is to work on other kinds of crossdock with different topology as mentioned in Bartholdi and Gue (2004). In our problem, we haven’t consider the truck capacity constraint, i.e. the total loading of a truck cannot excess its capacity. Moreover, the maximum loading capacity of a truck also depends on how to pack cargoes with different volumes. Hence another possible direction could take the capacity constraint and the packing issue into consideration in the future. Acknowledgments The authors thank the Editors and anonymous referees for their valuable comments and suggestions that have greatly improved this article. This research was supported in part by National Natural Science Foundation of China (70802052, 71171171, 71202059), the Fundamental Research Funds for the Central Universities (2012221011, 2011221016, 2010221025), MOE (NCET-10-0712),
the Academic Outstanding Youthful Research Talent Plan of Fujian Province (JA10001S), Soft Science Projects of Fujian Province (2011R0081), and Research Fund for the Doctoral Program of Higher Eduction of China (20100121120013). References Acar, K., Yalcin, A., & Yankov, D. (2012). Robust door assignment in less-thantruckload terminals. Computers and Industrial Engineering, 63, 729–738. Alpan, G., Ladier, A. L., Larbi, R., & Penz, B. (2011b). Heuristic solutions for transshipment problems in a multiple door cross docking warehouse. Computers and Industrial Engineering, 61, 402–408. Alpan, G., Larbi, R., & Penz, B. (2011a). A bounded dynamic programming approach to schedule operations in a cross docking platform. Computers and Industrial Engineering, 60, 385–396. Bartholdi, J. J., & Gue, K. R. (2002). Reducing labor costs in an LTL crossdocking terminal. Operations Research, 48(6), 823–832. Bartholdi, J. J., & Gue, K. R. (2004). The best shape for a crossdock. Transportation Science, 38(2), 235–244. Boloori Arabani, A. R., Fatemi Ghomi, S. M. T., & Zandieh, M. (2011). Meta-heuristics implementation for scheduling of trucks in a cross-docking system with temporary storage. Expert Systems with Applications, 38, 1964–1979. Boysen, N. (2010). Truck scheduling at zero-inventory cross docking terminals. Computers and Operations Research, 37(1), 32–41. Boysen, N., & Fliedner, M. (2010). Cross dock scheduling: Classification, literature review and research agenda. Omega, 38, 413–422. Glover, F., & Laguna, M. (1998). Tabu search. Dordrecht, The Netherlands: Kluwer Academic Publishers. Konur, D., & Golias, M. M. (2013). Cost-stable truck scheduling at a cross-dock facility with unknown truck arrivals: A meta-heuristic approach. Transportation Research Part E, 49, 71–91. Lee, K., Kim, B. S., & Joo, C. M. (2012). Genetic algorithms for door-assigning and sequencing of trucks at distribution centers for the improvement of operational performance. Expert Systems with Applications, 39, 12975–12983. Liao, T. W., Egbelu, P. J., & Chang, P. C. (2013). Simultaneous dock assignment and sequencing of inbound trucks under a fixed outbound truck schedule in multidoor crossdocking operations. International Journal of Production Economics, 141, 212–229. Liao, C. J., Lin, Y., & Shih, S. C. (2010). Vehicle routing with cross-docking in the supply chain. Expert Systems with Applications, 37, 6868–6874. Li, X., Leung, S. C. H., & Tian, P. (2012). A multistart adaptive memory-based tabu search algorithm for the heterogeneous fixed fleet open vehicle routing problem. Expert Systems with Applications, 39, 365–374. Lim, A., Miao, Z., Rodrigues, B., & Xu, Z. (2005). Transshipment through Crossdocks with inventory and time windows. Naval Research Logistics, 52(8), 724–733. Lin, S. W., Yu, V. F., & Lu, C. C. (2011). A simulated annealing heuristic for the truck and trailer routing problem with time windows. Expert Systems with Applications, 38, 15244–15252. McWilliams, D. L., Stanfield, P. M., & Geiger, C. D. (2005). The parcel hub scheduling problem: A simulation-based solution approach. Computers and Industrial Engineering, 49(3), 393–412. Miao, Z., Lim, A., & Ma, H. (2009). Truck dock assignment problem with operational time constraint within crossdocks. European Journal of Operational Research, 192(1), 105–115. Miao, Z., Yang, F., Fu, K., & Xu, D. (2012). Transshipment service through crossdocks with both soft and hard time windows constraints. Annals of Operations Research, 192, 21–47. Ramezanian, R., Rahmani, D., & Barzinpour, F. (2012). An aggregate production planning model for two phase production systems: Solving with genetic algorithm and tabu search. Expert Systems with Applications, 39, 1256–1263. Tarantilis, C. D., Stavropoulou, F., & Repoussis, P. P. (2012). A template-based tabu search algorithm for the consistent vehicle routing problem. Expert Systems with Applications, 39, 4233–4239. Vahdani, B., & Zandieh, M. (2010). Scheduling trucks in cross-docking systems: Robust meta-heuristics. Computers and Industrial Engineering, 58(1), 12–24. Yu, W., & Egbelu, P. J. (2008). Scheduling of inbound and outbound trucks in cross docking systems with temporary storage. European Journal of Operational Research, 184(1), 377–396.
Contents lists available at ScienceDirect
Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa
Applying an adaptive tabu search algorithm to optimize truck-dock assignment in the crossdock management system Zhaowei Miao, Shun Cai ⇑, Di Xu School of Management, Xiamen University, Xiamen 361005, PR China
a r t i c l e
i n f o
Keywords: Tabu search Dock assignment Artificial intelligence Crossdock management system
a b s t r a c t Different from warehouse, crossdock is considered as a ‘‘JIT’’ technique in logistics and supply chain management (LSCM), which is usually a short period of time to store cargos. Since cargos transported to a crossdock by inbound trucks are immediately sorted out, repackaged, routed and loaded into outbound trucks and then delivered to customers within one day, one of the key issues to operate crossdock successfully is to develop an efficient truck-dock assignment module in the crossdock management system so that all the cargos can be delivered to customers on time. In this paper, we propose an adaptive tabu search (ATS) algorithm as an artificial intelligence (AI) tool to optimize the truck-dock assignment problem within a crossdock in a very efficient way, and computational experiments are conducted, showing that our approach dominates the CPLEX Solver in both effectiveness and efficiency. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Crossdock is a material handling and distribution point in logistic network, in which items move directly from receiving dock to shipping dock without being stored for a long period, and has been highly emphasized by researchers and companies recently in warehouse management. The activities at crossdocks include assigning trucks to docks, sorting, scanning, repacking, routing and loading items shipped from inbound trucks to outbound trucks, as well as temporary inventory managing (Bartholdi & Gue, 2002; Boysen & Fliedner, 2010; Lim, Miao, Rodrigues, & Xu, 2005). One of the key decisions is the dock assignment for both inbound and outbound trucks. Dock assignment affects the performance of transshipment network, efficient dock assignment can be helpful to reduce shipment delays, operational times in crossdock and cost including operational cost for transferring, sorting and repacking (Acar, Yalcin, & Yankov, 2012; Alpan, Larbi, & Penz, 2011a, Alpan, Ladier, Larbi, & Penz, 2011b; Boloori Arabani, Fatemi Ghomi, & Zandieh, 2011; Boysen, 2010; Konur & Golias, 2013; Lee, Kim, & Joo, 2012; Liao, Egbelu, & Chang, 2013; Miao, Lim, & Ma, 2009; Vahdani & Zandieh, 2010; Yu & Egbelu, 2008). Up to now, more and more research papers deal with the shortterm scheduling problems which continuously arise during the daily operations of crossdock terminals. Lim et al. (2005) worked on various transshipment problems in a crossdock distribution network, and they found that well-scheduled trucks assignment can help to reduce the transportation time and inventory holding ⇑ Corresponding author. E-mail address: [email protected] (S. Cai). 0957-4174/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2013.07.007
time, correspondingly reduce the transportation costed as well as inventory cost by consolidation. Bartholdi and Gue (2002) examined minimizing labor costs in freight terminals by properly assigning incoming and outgoing trailers to doors with respect of types of congestion, however, they did not address actual dock assignments to arriving vehicles when considering the time window of trucks. Yu and Egbelu (2008) studied a crossdock system with unlimited temporary storage. They investigated to find the best truck docking or scheduling sequence for both inbound and outbound trucks to minimize total operation time when a temporary storage buffer to hold items temporarily is located at the shipping dock, but they left out various important issues in real application such as crossdock physical layout, warehouse capacity, and arrival/departure schedules of vehicles. Based on the strategies proposed by Yu and Egbelu (2008) and Vahdani and Zandieh (2010) applied other heuristic methods, including genetic algorithm, tabu search, population based simulated annealing with diversification, and variable neighborhood search algorithm, to solve the same problem in Yu and Egbelu (2008), and proved the robustness of each proposed heuristics by numerous computational experiments. Miao et al. (2009) provided scheduling procedures adopted from gate assignment in airports, where trucks are assumed to have given service time windows, and they proved the problem is NP-hard, therefore two kinds of meta-heuristics including genetic algorithm and tabu search were proposed to find an optimal solution. Boysen (2010) treated a special truck scheduling problem arising in the zero-inventory crossdocks of the food industry to minimize time makespan including the flow time, processing time and tardiness of outbound trucks. Boloori Arabani et al. (2011) proposed genetic algorithm, tabu search, particle
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
swarm optimization and so on to find the best sequence of inbound and outbound trucks in order to minimize the total operation time. Lee et al. (2012) aimed to maximize the amount of products that are able to ship within a given time horizon and developed intelligent genetic algorithms to optimize door-assigning and sequencing of trucks in crossdocks. Konur and Golias (2013) studied a crossdock truck scheduling problem by formulating a bi-level optimization problem and developed a genetic algorithm to solve it. Alpan et al. (2011a) proposed a bounded dynamic programming approach to schedule inbound and outbound trailers in a multiple receiving and shipping door cross dock environment. In their another paper, Alpan et al. (2011b) developed several heuristics to find the best schedule in order to minimize the sum of inventory holding and truck replacement costs. Liao et al. (2013) proposed six meta-heuristic algorithms, including simulated annealing, tabu search, ant colony optimization and so on, to minimize total weighted tardiness for the dock assignment and sequencing of inbound trucks in a crossdock. Acar et al. (2012) introduced a mixed integer quadratic model for the problem which considered variability of truck arrival and service time. We refer readers to Boysen and Fliedner (2010) for thorough and excellent literature reviews of crossdock scheduling. Artificial intelligence tools are widely applied to the dock assignment problem in crossdocks, because as the increasing of problem scale, traditional methods or softwares cannot find the best solution effectively. Besides those literatures mentioned above, there are a lot of papers which applied artificial intelligence tools such as genetic algorithm, tabu search, simulated annealing and so on, to the other optimization problems such as inventory routing problem, production planning problem, network design problem, vehicle routing problem, and so on. McWilliams, Stanfield, and Geiger (2005) treated a different truck scheduling problem arising from freight consolidation terminals in the parcel delivery industry, where the movement of goods inside the terminal is conducted by a convey or belt system, and proposed a simulation-based scheduling algorithm that used a genetic algorithm to drive the search for new solutions. Lin, Yu, and Lu (2011) studied the truck and trailer routing problem with time windows and proposed a simulated annealing heuristic which can obtain consistent quality solutions. In this paper, we design a new tabu search algorithm to resolve proposed truck-dock assignment problem in a crossdock since tabu search proposed by Glover and Laguna (1998) is known as one of the powerful artificial intelligence tools to solve NP-hard problems. Ramezanian, Rahmani, and Barzinpour (2012) considered general two-phase aggregate production planning problem which is NP-hard and implemented tabu search algorithm to produce good-quality solutions. Li, Leung, and Tian (2012) developed a multistart adaptive memory-based tabu search algorithm to solve the heterogeneous fixed fleet open vehicle routing problem (HFFOVRP), and computational experiments showed that this modified tabu search algorithm has efficiency and effectiveness. Tarantilis, Stavropoulou, and Repoussis (2012) studied another vehicle routing problem called consistent vehicle routing problem (ConVRP) and solved it by a template-based tabu search algorithm. Moreover, Liao, Lin, and Shih (2010) integrated crossdock into vehicle routing problem and applied tabu search algorithm to determine the number of vehicles and a set of vehicle schedules with a minimum total cost including operation cost and transportation cost. In this paper, we extend the problem proposed by Miao et al. (2009), and take consideration of time windows for both inbound and outbound trucks in a transshipment network through crossdocks where the number of trucks exceeds the number of docks available, of which the objective is to minimize the total costs including operational cost of the cargo shipment and penalty cost of unfulfilled shipments. And we formulate this problem into a
17
0 1 integer programming model. Since this kind of truck-dock assignment problem is NP-hard, meta-heuristic should be a good approach to solve it. Adaptive tabu search (ATS) is proposed and developed for conducting computational experiments on test sets, and the results compared with CPLEX Solver 11.0 show that our algorithm dominates the CPLEX Solver in most of all test cases. This paper is organized as follows: in the next section, we give an IP model of the problem. Adaptive tabu search is proposed and developed in Section 3. We provide computational results in Section 4. In Section 5, we summarize our findings and suggest future work.
2. Problem description For operations in crossdocks, inbound trucks transport cargos to it, which are immediately sorted out, repackaged, routed and loaded into outbound trucks and then delivered to customers within one day (Alpan et al., 2011a; Boloori Arabani et al., 2011; Boysen, 2010; Konur & Golias, 2013; Miao et al., 2009; Vahdani & Zandieh, 2010). As a result, one of the key issues to implement crossdock as a ‘‘JIT’’ technique in logistics and supply chain management successfully is to develop an efficient truck-dock assignment module in the crossdock management system so that all the cargos can be delivered to customers on time (Acar et al., 2012; Alpan et al., 2011b; Konur & Golias, 2013; Lee et al., 2012; Yu & Egbelu, 2008). As mentioned before, our proposed truck-dock assignment problem is NP-hard, the artificial intelligence tool, such as genetic algorithm, tabu search, simulated annealing and so on, should be a good approach to obtain good-quality solutions efficiently (Li et al., 2012; Liao et al., 2013; Lin et al., 2011; Miao et al., 2009). In order to develop ATS to solve this problem, we need to formulate it into a mathematical model first. Before introducing the model, we should explain some common assumptions which are usually adopted in related literatures (Alpan et al., 2011a, 2011b; Liao et al., 2013; Miao et al., 2009). For this problem, we do not address secondary constraints which account for the loading/unloading times or other buffers between arrival and departure times, since they can be easily dealt with by extending the truck arrival and departure durations. The important constraint is the time window constraint for both inbound trucks and outbound trucks, i.e. each inbound/outbound truck can occupy an inbound/outbound dock to unload/load only within its pre-allocated arrival time and departure time. Also we take operational time of transferring, sorting and repacking within the crossdock into consideration. For example, if outbound truck A at outbound dock a is going to receive cargos from inbound truck B at inbound dock b, then inbound truck B needs to ship the cargos to dock a before truck A leaves when taking time window and the operational time for cargos between inbound dock b to outbound dock a into consideration, while the operational time is usually decided by the physical distance between the two docks. As we know, the crossdock management system should schedule the trucks well in order to deliver cargos to customers on time. If it is found that some trucks cannot be assigned due to the limited number of docks, those trucks need to be rescheduled to complete their tasks. The trucks can be delayed, or rescheduled to other crossdocks where a vacancy is available. In such situations, a penalty cost is incurred for any unfulfilled cargo shipment. Fig. 1 illustrates an outline of the crossdock and major elements of our problem, where the notations will be introduced in details next. We formulate our proposed problem into a 0 1 IP model, for which the following notations are used: NI: set of inbound trucks arriving at the crossdock, and jNIj = nI, where j j represents the cardinality of a set;
18
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
Fig. 1. Truck dock assignment problem.
MI: set of inbound docks available in the crossdock, and jMIj = mI, where nI > mI; NO: set of outbound trucks departing from the crossdock, and jNOj = nO; MO: set of outbound docks available in the crossdock, and jMOj = mO, where nO > mO; aIi : arrival time of inbound truck i (1 6 i 6 nI); I di : departure time of inbound truck i (1 6 i 6 nI); O aj : arrival time of outbound truck j (1 6 j 6 nO); O dj : departure time of outbound truck j (1 6 j 6 nO); tk,l: operational time for pallets from inbound dock k to outbound dock l (1 6 k 6 mI, 1 6 l 6 mO); fi,j: number of pallets transferring from inbound truck i to outbound truck j (1 6 i 6 nI, 1 6 j 6 nO); ck,l: operational cost per pallet from inbound dock k to outbound dock l (1 6 k 6 mI, 1 6 l 6 mO); pi,j: penalty cost per pallet cargo from inbound truck i to outbound truck j (1 6 i 6 nI, 1 6 j 6 nO). We also define dIi;j 2 f0; 1g: 1 iff inbound truck i departs no later than inbound truck j arrives; 0 otherwise. And dOi;j 2 f0; 1g: 1 iff outbound truck i departs no later than outbound truck j arrives; 0 otherwise. Then we can set dIi;j and dOi;j according to the truck time n o n o I I O O and aOi ; di ; aOj ; dj . window data aIi ; di ; aIj ; dj The decision variables are as follows:
yIi;k
¼
yOj;l ¼
1; if inbound truck i is assigned to inbound dock k 0;
min COST Operation þ COST Penalty where
COST Operation ¼
mI X mO X nI X nO X ck;l fi;j zi;j;k;l k¼1 l¼1 i¼1 j¼1
and
COST Penalty
nI X nO mI X mO X X ¼ pi;j fi;j 1 zi;j;k;l i¼1 j¼1
!
k¼1 l¼1
s.t. mI X yIi;k 6 1 ð1 6 i 6 nI Þ
ð1Þ
k¼1 mO X yOj;l 6 1 ð1 6 j 6 nO Þ l¼1
ð2Þ
yIi;k þ yIj;k 1 6 2 dIi;j þ dIj;i ð1 6 i;j 6 nI ; i – j;1 6 k 6 mI Þ yOi;l þ yOj;l 1 6 2 dOi;j þ dOj;i ð1 6 i;j 6 nO ;i – j;1 6 l 6 mO Þ yIi;k þ yOj;l P 2zi;j;k;l ð1 6 i 6 nI ;1 6 j 6 nO ; 1 6 k 6 mI ; 1 6 l 6 mO Þ O dj aIi t k;l fi;j zi;j;k;l P 0 ð1 6 i 6 nI ;1 6 j 6 nO ; 1 6 k 6 mI ; 1 6 l 6 mO Þ
ð3Þ ð4Þ ð5Þ ð6Þ
yIi;k 2 f0; 1g;yOj;l 2 f0;1g; zi;j;k;l 2 f0;1g ð1 6 i 6 nI ;1 6 j 6 nO ;1 6 k 6 mI ;1 6 l 6 mO Þ ð7Þ
1; if outbound truck j is assigned to outbound dock l 0; otherwise:
zi;j;k;l ¼
otherwise
The objective is to minimize the total cost: the sum of the operational cost and the penalty cost. The 0 1 IP model is as follows:
1; if f i;j is transfered from inbound dock k to outbound dock l 0; otherwise: I
I
Note that yIi;k ¼ yIj;k ¼ 1 implies that aIi > dj or aIj > di ð1 6 i; j 6 nI Þ, O and similarly, yOi;l ¼ yOj;l ¼ 1 implies that aOi > dj or O O O aj > di ð1 6 i; j 6 n Þ.
The first term in the objective function is total operational cost and the second term is the total penalty cost. Constraints (1) and (2) ensure that each truck cannot be assigned to more than one dock. Constraints (3) and (4) specify that one dock cannot be occupied by two different trucks simultaneously. Constraints (5) represent the logic relationship among yIi;k ; yOj;l and zi,j,k,l. Finally, constraints (6) say that the transfer process of cargoes from inbound truck i at inbound dock k to outbound truck j at outbound dock l should take place within the specified time window, i.e. from the arrival time of inbound truck i to the departure time of outbound truck j.
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
19
Algorithm 1. Initial Solution Generation Procedure. Fig. 2. The solution representation.
3. Adaptive tabu search Since this problem is NP-hard, it is necessary to adopt artificial intelligence tool like tabu search to handle large real world cases effectively and efficiently. Tabu search proposed by Glover and Laguna (1998) has been proved powerful in optimization for NPhard problems. It is a meta-heuristic search procedure that proceeds iteratively from one solution to another by moves in a neighborhood space with the assistance of adaptive memory, which plays an important role in the search process. Therefore, neighborhood structures are rather key in the whole procedure. In our proposed tabu search algorithm, we adopt four kinds of moves to explore neighborhood space. What’s more, we apply adaptive rule to control the exploring procedure alternately rather than use all moves simultaneously, that is why the propose metaheuristic is called adaptive tabu search (ATS) in this paper. In the following section, we will show the details of ATS.
Initialize num_list for all trucks according to their total penalty costs; for i 1 to nI do Randomly select a feasible inbound dock for the inbound truck according to num_list until all inbound trucks have been selected; Otherwise, set the value of the inbound dock for this inbound truck to 0; end for for j 1 to nO do Randomly select a feasible outbound dock for the outbound truck according to num_list until all outbound trucks have been selected; Otherwise, set the value of the outbound dock for this outbound truck to 0; end for Calculate the value of this solution.
3.3. Neighborhood search methods 3.1. Solution representation We first define solution representation that is the key element in meta-heuristics design. The solution is the dock assignment represented by a sequence, which has length nI + nO, and involves twopart decisions: the first part is dock assignment for inbound trucks (the sequence number is from 1 to nI), and the second part is for outbound trucks (the sequence number is from nI + 1 to nI + nO). A sequence sI1 ; sI2 ; . . . sInI ; sOnI þ1 ; . . . ; sOnI þnO , which means that inbound truck 1 is assigned to inbound dock sI1 , inbound truck 2 is assigned to inbound dock sI2 , and so on and so forth ð1 6 i 6 nI þ nO ; 0 6 sIi 6 mI ; 0 6 sOi 6 mO Þ. If the inbound/outbound truck i is unassigned to any of the inbound/outbound docks, which is possible when all the inbound/outbound docks are occupied, we give sIi =sOi the value of 0. If the solution is feasible, the dock assignment is then uniquely determined by the sequence of sI1 ; sI2 ; . . . ; sOnI þnO . The representation is depicted in Fig. 2. In the tabu search memory we implement, only the assignment information is captured so that only the move that has the identical assignment to the ones in tabu search memory will be forbidden. 3.2. Initial solution generation Initial solution is rather important in meta-heuristic design, a good initial solution generation strategy will provide an effective input for algorithms. As we know, a good strategy used to generation initial solution depends on attributes of the proposed problem. Therefore, we design a greedy strategy based on penalty cost for inbound/outbound trucks. First, we construct a list as num_list, to store the number of inbound and outbound trucks. We define the total penalty cost for PO inbound truck i as PIi ¼ nj¼1 pi;j and that for outbound truck j as PnI O P j ¼ i¼1 pi;j . During the construction process, we first calculate the total penalty cost for each truck respectively, then each truck number is stored in descending order according the corresponding total penalty cost in the list. After initializing such a list, we assign a dock to each truck according to the total penalty cost, i.e. the truck with higher total penalty cost has a larger probability to be assigned a dock. Algorithm 1 displays the procedure of generating initial solutions.
We design four kinds of neighborhood search methods to make deepest search to the best solution found at each iteration. The four move strategies are implemented to both inbound and outbound truck-dock assignments. The following will give the detail procedures: Move I-Insert unfilled truck: The move is randomly insert an unfilled truck into current truck dock assignment scheme. In this process, constraints (3) and (4) may be violated so that infeasible solutions will be generated. Move II-Exchange unfilled truck with filled truck: This move is implemented after Move I, that is, any unfilled truck cannot be inserted into assignment sequence. In this move, we first randomly select an unfilled truck and a filled truck and then assign the unfilled truck to the dock the filled truck assigned, meanwhile, the sequence value of the selected filled truck is set to 0. It should be noticed that the time window of the unfilled truck may conflict with that of the trucks assigned to the selected dock. Move III-Exchange two filled trucks: In this move, two filled trucks are randomly selected, then we exchange their assigned docks in order to get a better solution. Similar to Move II, infeasible solutions can be produced because the time window constraints. Move IV-Exchange truck subsequences assigned to two docks: This move is different from Move III, rather than randomly select two trucks, we randomly select two docks and swap the truck subsequences assigned to them. The procedure is terminated until find a superior solution. Noticed, Move I to III may produce infeasible solutions, while, in Move IV, exchanging trucks subsequences between two docks will not violate the time constraints, therefore, Move IV always can get a feasible solution. For any infeasible solution, we repair it by setting the sequence values of those conflicting filled trucks to 0 one by one until it becomes feasible. 3.4. Adaption rule Extensively computational experiments show Move IV can help to explore neighborhood explicitly and consume less time to attain
20
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
a feasible neighborhood, and Move I to III can help to move out of the local optima and explore other regions of the solution due to their large neighborhood but are more time-consuming. Based on this consideration of the four types of moves, we apply adaption scheme to use the four moves to explore neighborhood according to a rule set which includes two criteria as follows (Miao, Yang, Fu, & Xu, 2012). The iteration to be used. This one defines the iteration of each move used to explore its neighborhood. The iteration for Move IV initiates as 20, and the iterations for Move I to III are 10 respectively, because Move IV is less time-consuming, and does well in exploring the neighborhood of solutions, the iteration for it is relatively large. The iteration to be terminated. It defines the iteration that a certain move cannot make improvement. The iteration is set to 10 for Move IV and 5 for other moves. Additionally, if all moves can not improve the best current solution within 50 iterations, the current best solution will be replaced by the current solution with aim to reach deepest search of the current best solution. All moves are controlled by adaptive rules to explore different neighborhood space alternately. The principle of adaption is to take deepest search to the current best solution meanwhile avoid the local optimum. Let #term_iter denote the maximum number of iterations for which the best solution can’t be improved. Note that, if the current best solution can not be improved within #term_iter iterations, then terminate the algorithm and the current best solution is considered as the goal optimum. A neighbor x0 of current solution xcurr to be selected must meet one of the following conditions: (a) It is not forbidden (i.e. the assignment is not identical to any assignments of recent tabu tenure moves); (b) The cost of x0 is better than the current best cost (aspiration criterion). 3.5. ATS framework In this section, we outline the framework of ATS algorithm. Notations are defined as following: xcurr denotes the current best solution, xinit denotes the initial solution, #max_iter denotes the maximum number of iterations, and #term_iter denotes the maximum number of iterations for which the best solution can’t be improved. Algorithm 2. ATS Framework. Initialize the parameters of ATS; Generate an initial solution xinit by Algorithm 1, set xcurr xinit; Empty tabu list for iter 1 to ]max_iter do if Move I meets its adaptive exploring rules then Generate a set of neighborhood solutions N(xcurr) of xcurr by Move I; Repair the infeasible solutions. end if if Move II meets its adaptive exploring rules then Generate a set of neighborhood solutions N(xcurr) of xcurr by Move II; Repair the infeasible solutions. end if if Move III meets its adaptive exploring rules then Generate a set of neighborhood solutions N(xcurr) of xcurr by Move III; Repair the infeasible solutions.
⇑ (continued)
Algorithm 2. ATS Framework. end if if Move IV meets its adaptive exploring rules then Generate a set of neighborhood solutions N(xcurr) of xcurr by Move IV. end if if The solution x0 2 N(xcurr) meet either condition (a) or (b) then Set xcurr x0 ; Update the TS memory. end if if xcurr can’t be improved within #term_iter then Consider the solution as the goal optimal solution and terminate the procedure. end if end for Output the best solution and escaped time
4. Experimental results and analysis All the algorithms were coded in Java. As comparison, we use ILOG CPLEX 11.0 to solve the formulation presented in Section 2, all the experiments are conducted on Intel (R) Core (TM)2 Duo CPU E4500 @ 2.2 GHz, 2.19 GHz machine with 1.99 GB memory. In the following, we first demonstrate the parameter setting, and then detailed computational results are shown. 4.1. Test data generation and parameter setting The topology of crossdock is referred to the layout of crossdock designed by Miao et al. (2009). But we modify it to make its topology more reasonable according to the real-world situation. We revise their representative layout of a crossdock to have two parallel sets of terminals where docks are symmetrically located as shown in Fig. 3; meanwhile, the above sets are considered as inbound docks, and the below terminals as outbound docks. The distance between two adjacent inbound/outbound docks within one terminal (e.g., inbound dock 1 and inbound dock 2) is 1 unit and the distance between two parallel docks in opposite terminals (e.g., inbound dock 1 and outbound dock 1) to be 3 units. To simplify the problem, we assumed that forklift can only walk ‘horizontally’ or ‘vertically’, i.e., if one forklift wants to transfer one pallet from inbound dock 1 to outbound dock 4, the walking distance is 3 + 1 3 = 6 units (thus the distances are rectilinear). This is similar to the so-called Manhattan metric. We note that it is easy to use other metrics for the problem to compare performance. The test data generation program requires two parameters: the number of inbound trucks nI and the number of inbound docks mI. Simply, the number of outbound trucks nO and the number of outbound docks mO are equal to nI and mI respectively. The arrival time of inbound truck aIi ð1 6 i 6 nI Þ is uniformly generated in h i I , and the interval of the outbound trucks is the interval 1; n m60 I h i I I 15; n m60 þ 15 . The departure time of inbound truck di is generated I I
O
as di ¼ aIi þ ½25; 40, for the outbound truck is di ¼ aOi þ ½25; 40. The rectilinear walking distances between inbound dock k and outbound dock l is calculated according to the topology and then proportionally converted to the corresponding operational cost ck,l and operational time tk,l. The number of shipping pallets fi,j representing the quantity shipped from inbound truck i to outbound truck j is generated randomly in the interval [6, 60]. The penalty cost
21
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
Fig. 3. Crossdock topology.
are calculated by averaging values of all the instances in each group.
Table 1 Results of CPLEX and ATS of instances with small scales. Problem size
3-6-3-6
3-7-3-7
3-9-3-9
3-12-3-12
CPLEX
Obj
7033 6.35 7036
11563 16.40
20085 94.46
ATS
Time(s) Obj
4747 3.28
11563 0.15
20085 0.26
Time(s)
CPLEX
Obj
ATS
Time(s) Obj Time(s)
4747 0.06
0.06
4-10-4-10
4-12-4-12
5-12-5-12
5-15-5-15
14431 90.06
21242
14431 0.21
20827 0.56
20706 545.02 20720
32939 >1800 32948
0.62
1.55
260.50
per unit cargo from truck i to truck j(pi,j) is generated in the interval [1, 4] + maxk,l{ck,l}. We conducted a great number of experiments to test the performance of our heuristic approach, and the following parameter setting of ATS algorithm has been proved efficient: maximum number of iterations is 105 and each time 30 neighbors are generated with a tabu tenure equal to 10. The algorithm is to terminate if the best solution was not improved within 500 iterations.
1. Computational results of small scale instances. The size for eight groups of small scale instances are generated from 3-6-3-6 to 5-15-5-15. The performance of CPLEX solver and ATS algorithm of this category are shown in Table 1. We highlight the lower average objective attained by both of them with underline in the table. As the results shown, CPLEX and our algorithm attain the same average objective in four groups, and CPLEX performs a little bit better than our algorithm, but they are very close; however, ATS can get near optimal solutions within much shorter time than CPLEX. 2. Computational results of large scale instances. Table 2 presents the results of eight large scale instance groups with size ranging from 8-20-8-20 to 14-28-14-28. Judged from value of Obj, ATS has a rather better performance than CPLEX in all instances. What’s more, values in row Time(s) show that, CPLEX has to take more than 7200s to attain a near optimal solution. And ATS has a great advantage in speed, of which average time is ranging from 5s to 90s in each group. From the results of the experiments, the superiority of our heuristic method is obvious, especially for large scale instances. Our algorithm not only can get better solutions than CPLEX, but also consumes much less time. The advantage of ATS in time dimension is quite important, which is one of our major contributions of this paper, because as a ‘‘JIT’’ technique in logistics and supply chain management, crossdocks need to tranship cargos to customers quickly, which requires an efficient truck-dock assignment module in the crossdock management system. In conclusion, our algorithm can greatly improve the performance of truck-dock assignment module in term of effectiveness and efficiency, which is a good approach to tackle this kind of problem in real-world application.
4.2. The results We design two categories of random instances, including small and large scales, to test the algorithm. In each category, we conduct 32 test cases and sorted them into 8 groups, and record the detailed results in Tables 1 and 2. The first row of each table specifies the instance size. mI nI mO nO denotes that there are nI inbound trucks and mI inbound docks, likewise, nO outbound trucks and mO outbound docks for this instance group. The effectiveness and efficiency of CPLEX and ATS are evaluated by objective and time, which are shown in rows Obj and Time(s). Both of values
Table 2 Results of CPLEX and ATS of instances with large scales. Problem size CPLEX ATS
Obj Time(s) Obj Time(s)
CPLEX ATS
Obj Time(s) Obj Time(s)
8-20-8-20
8-24-8-24
10-20-10-20
10-24-10-24
63020 >7200
90412 >7200
64228 >7200
90939 >7200
61733 5.02
89202 17.27
61819 15.23
88935 14.06
12-24-12-24
12-28-12-28
13-26-13-26
14-28-14-28
93694 >7200
127331 >7200
110085 >7200
128584 >7200
91555 47.73
124713 71.50
107651 58.63
125552 90.75
22
Z. Miao et al. / Expert Systems with Applications 41 (2014) 16–22
5. Conclusions and future work Compared with the traditional warehouse, the crossdock is considered as a ‘‘JIT’’ technique in logistics and supply chain management, which is usually a short period of time to store cargos. Since cargos transported to a crossdock by inbound trucks are immediately sorted out, repackaged, routed and loaded into outbound trucks and then delivered to customers within one day, one of the key issues to operate crossdock successfully is to develop an efficient truck-dock assignment module in the crossdock management system so that all the cargos can be delivered to customers on time. In this paper, we consider an over-constrained truck-dock assignment problem in a crossdock, in which the number of trucks are much larger than that of docks. The objective of the study is to find the best truck-dock assignment scheme for both inbound and outbound trucks to minimize total cost including the penalty cost of unfulfilled shipments and the operational cost of shipments within the crossdock. The paper contributes to the extant literatures in the following two ways. First, we provide a 0 1 Integer Programming model for such a complex problem, which can be input to those optimization softwares like ILOG CPLEX solver to get optimal or near optimal solutions so as to help crossdock managers make a decision if they don’t have other better methods to solve it. Second, we propose an adaptive tabu search algorithm as an artificial intelligence tool to optimize this truck-dock assignment problem which is known to be NP-hard in a very efficient way. For more details, our algorithm not only can get better solutions than CPLEX, but also consumes much less time. The advantage of ATS in time dimension is the key contribution because our algorithm can be integrated into truck-dock assignment module in the crossdock management system, which can help crossdocks to tranship cargos to customers within a short time with a lower cost so that not only the customer’s satisfactory level can be improved by reducing responsive time, but also inventory cost can be reduced for both customers and crossdocks. With the current problem, the topology of crossdock is quite regular. One future study is to work on other kinds of crossdock with different topology as mentioned in Bartholdi and Gue (2004). In our problem, we haven’t consider the truck capacity constraint, i.e. the total loading of a truck cannot excess its capacity. Moreover, the maximum loading capacity of a truck also depends on how to pack cargoes with different volumes. Hence another possible direction could take the capacity constraint and the packing issue into consideration in the future. Acknowledgments The authors thank the Editors and anonymous referees for their valuable comments and suggestions that have greatly improved this article. This research was supported in part by National Natural Science Foundation of China (70802052, 71171171, 71202059), the Fundamental Research Funds for the Central Universities (2012221011, 2011221016, 2010221025), MOE (NCET-10-0712),
the Academic Outstanding Youthful Research Talent Plan of Fujian Province (JA10001S), Soft Science Projects of Fujian Province (2011R0081), and Research Fund for the Doctoral Program of Higher Eduction of China (20100121120013). References Acar, K., Yalcin, A., & Yankov, D. (2012). Robust door assignment in less-thantruckload terminals. Computers and Industrial Engineering, 63, 729–738. Alpan, G., Ladier, A. L., Larbi, R., & Penz, B. (2011b). Heuristic solutions for transshipment problems in a multiple door cross docking warehouse. Computers and Industrial Engineering, 61, 402–408. Alpan, G., Larbi, R., & Penz, B. (2011a). A bounded dynamic programming approach to schedule operations in a cross docking platform. Computers and Industrial Engineering, 60, 385–396. Bartholdi, J. J., & Gue, K. R. (2002). Reducing labor costs in an LTL crossdocking terminal. Operations Research, 48(6), 823–832. Bartholdi, J. J., & Gue, K. R. (2004). The best shape for a crossdock. Transportation Science, 38(2), 235–244. Boloori Arabani, A. R., Fatemi Ghomi, S. M. T., & Zandieh, M. (2011). Meta-heuristics implementation for scheduling of trucks in a cross-docking system with temporary storage. Expert Systems with Applications, 38, 1964–1979. Boysen, N. (2010). Truck scheduling at zero-inventory cross docking terminals. Computers and Operations Research, 37(1), 32–41. Boysen, N., & Fliedner, M. (2010). Cross dock scheduling: Classification, literature review and research agenda. Omega, 38, 413–422. Glover, F., & Laguna, M. (1998). Tabu search. Dordrecht, The Netherlands: Kluwer Academic Publishers. Konur, D., & Golias, M. M. (2013). Cost-stable truck scheduling at a cross-dock facility with unknown truck arrivals: A meta-heuristic approach. Transportation Research Part E, 49, 71–91. Lee, K., Kim, B. S., & Joo, C. M. (2012). Genetic algorithms for door-assigning and sequencing of trucks at distribution centers for the improvement of operational performance. Expert Systems with Applications, 39, 12975–12983. Liao, T. W., Egbelu, P. J., & Chang, P. C. (2013). Simultaneous dock assignment and sequencing of inbound trucks under a fixed outbound truck schedule in multidoor crossdocking operations. International Journal of Production Economics, 141, 212–229. Liao, C. J., Lin, Y., & Shih, S. C. (2010). Vehicle routing with cross-docking in the supply chain. Expert Systems with Applications, 37, 6868–6874. Li, X., Leung, S. C. H., & Tian, P. (2012). A multistart adaptive memory-based tabu search algorithm for the heterogeneous fixed fleet open vehicle routing problem. Expert Systems with Applications, 39, 365–374. Lim, A., Miao, Z., Rodrigues, B., & Xu, Z. (2005). Transshipment through Crossdocks with inventory and time windows. Naval Research Logistics, 52(8), 724–733. Lin, S. W., Yu, V. F., & Lu, C. C. (2011). A simulated annealing heuristic for the truck and trailer routing problem with time windows. Expert Systems with Applications, 38, 15244–15252. McWilliams, D. L., Stanfield, P. M., & Geiger, C. D. (2005). The parcel hub scheduling problem: A simulation-based solution approach. Computers and Industrial Engineering, 49(3), 393–412. Miao, Z., Lim, A., & Ma, H. (2009). Truck dock assignment problem with operational time constraint within crossdocks. European Journal of Operational Research, 192(1), 105–115. Miao, Z., Yang, F., Fu, K., & Xu, D. (2012). Transshipment service through crossdocks with both soft and hard time windows constraints. Annals of Operations Research, 192, 21–47. Ramezanian, R., Rahmani, D., & Barzinpour, F. (2012). An aggregate production planning model for two phase production systems: Solving with genetic algorithm and tabu search. Expert Systems with Applications, 39, 1256–1263. Tarantilis, C. D., Stavropoulou, F., & Repoussis, P. P. (2012). A template-based tabu search algorithm for the consistent vehicle routing problem. Expert Systems with Applications, 39, 4233–4239. Vahdani, B., & Zandieh, M. (2010). Scheduling trucks in cross-docking systems: Robust meta-heuristics. Computers and Industrial Engineering, 58(1), 12–24. Yu, W., & Egbelu, P. J. (2008). Scheduling of inbound and outbound trucks in cross docking systems with temporary storage. European Journal of Operational Research, 184(1), 377–396.