Scheduling trucks in cross docking systems with temporary storage and dock repeat truck holding pattern using GRASP method

Article history: Received 18 May 2012 Received in revised format 7 July 2012 Accepted July 31 2012 Available online 5 August 2012 Cross docking play an important role in management of supply chains where items delivered to a warehouse by inbound trucks are directly sorted out, reorganized based on customer demands, routed and loaded into outbound trucks for delivery to customers without virtually keeping them at the warehouse. If any item is held in storage, it is usually for a short time, which is normally less than 24 hours. The proposed model of this paper considers a special case of cross docking where there is temporary storage and uses GRASP technique to solve the resulted problem for some realistic test problems. In our method, we first use some heuristics as initial solutions and then improve the final solution using GRASP method. The preliminary test results indicate that the GRASP method performs better than alternative solution strategies. © 2012 Growing Science Ltd. All rights reserved


Introduction
Cross docking is one of the most important issues in supply chain management and there have been growing interests in this problem under different conditions (Barbarosoglu & Ozgur, 1999).Cross docking is a warehouse management idea where items delivered to a warehouse by inbound trucks are directly sorted out, reorganized based on customer demands, routed and loaded into outbound trucks for delivery to customers without virtually keeping them at the warehouse.If any item is held in storage, it is usually for a short time, which is normally less than 24 hours.This way, the turnaround times for customer orders, inventory management cost, and warehouse space requirements are reduced.Yu (2002) in his dissertation discussed cross docking problem under various assumptions.Yu and Egbelu (2008) determined the most suitable cross docking or scheduling sequence for both inbound and outbound trucks by minimizing total operation time when a temporary storage buffer is considered at shipping dock.The product assignment to trucks and the docking sequences of the inbound and outbound trucks were all determined, simultaneously.
Cross docking problem is normally formulated as a mixed integer problems and they are mathematically considered as NP-Hard problem (Feo & Resende, 1989;Mosheiov, 1998).Therefore, there is a need to use some metaheuristics to solve such problem.Rohrer (1995) discussed cross docking problem and the implementation of simulation to solve such problem.Vahdani and Zandieh (2010) presented the implementation of five meta-heuristic algorithms including genetic algorithm (GA), tabu search (TS), simulated annealing (SA), electromagnetism-like algorithm (EMA) and variable neighborhood search (VNS) to schedule the trucks in cross-dock systems to minimize total operation time when a temporary storage buffer to hold temporarily items is located at the shipping dock.They used response surface methodology (RSM) methodology to tune their problem parameters.They also considered two kinds of objective functions to develop multiple objective decision making model.Vahdani et al. (2009) considered another cross docking problem where it was a scheduling the truck holdover recurrent dock cross-dock problem using robust meta-heuristics.Soltani and Sadjadi (2010) proposed two hybrid meta-heuristics, hybrid simulated annealing and hybrid variable neighborhood search, to solve cross docking problem by achieving the best possible sequence of truck pairs.They used different sample problems to demonstrate the effectiveness of the proposed methods, especially for large-sized problems.Nascimento et al. (2010) discussed the independent multi-plant, multi-period, and multi-item capacitated lot-sizing problem where transfers among different plants were allowed.They developed a Greedy Randomized Adaptive Search Procedure (GRASP) heuristic as well as a path-relinking intensification procedure to detect cost-effective solutions for this problem.They also proposed some heuristics to solve some instances of the capacitated lot sizing problem with parallel machines.The results of the computational tests showed that the proposed heuristics outperform other heuristics previously described in the literature.
Boloori Arabani et al. (2011) developed another some meta-heuristics implementation for scheduling of trucks in a cross-docking system with temporary storage.Fig. 1 shows a sample of cross docking system.Yu (2002) proposed a model by assuming that there is a temporary storage in cross docking system and each two groups of trucks of receiving and shipping of loaded can alternatively enter into cross-dock.The temporary storage allows trucks to deliver more cargos, which are storage for future shipments.
The other trucks, which are responsible for shipping cargos to final destinations could also use this temporary storage to meet final customers' needs.

Continuous Variables:
Makespan, Time at which the variable transferring receiving truck to shipping truck starts to unload from receiving truck onto the receiving dock, Time at which the variable transferring from receiving truck to shipping truck finished loading from the shipping dock into shipping truck , Integer Variables: Number of units of product type which transfer from receiving truck to shipping truck , Total number of units of products which transfer from receiving truck to shipping truck , where ( ∑ ), Binary Variables: 1 if any products transfer from receiving truck to shipping truck 0 otherwise  (1) (18) All variables ≥ 0.
According to Eq. ( 1), makespan is greater equal to the time that the last product is loaded into the last scheduled shipping truck.Eq. ( 2) assures that total number of units of k th product from receiving truck i are shipped to all shipping trucks are the same as the number of products scheduled for receiving truck i.Similarly, Eq. (3) assures that, for shipping truck j, total number of outgoing products type k from all receiving trucks is the same as total number of incoming product type k.Eq. ( 10) assures that only one of receiving trucks' ij t  is scheduled in the beginning of the sequence and Eq. ( 11) assures that only one of receiving trucks' ij t should stay in the last schedule.Similarly, Eq. ( 12) and Eq. ( 13) assures that only one of the shipping trucks' ij t  comes at the beginning of the sequence and only one of ij t comes last.Eq. ( 14) and Eq. ( 15) guarantee that there is no consecutive sequence, which transfers products from the same receiving truck to the same shipping truck.
Eq. ( 16) and Eq. ( 17) provide an appropriate sequence for unloading times for ij t variables.If there is no change on receiving truck ( ii   ) we use Eq. ( 16) and when there is receiving truck ( ii   ) we need to compute delay time using Eq. ( 17).
Eq. ( 18) setup a good relationship between ij L and ij U and finally, Eq. ( 19) and Eq. ( 20) setup a valid loading time for ij t based on the orders received.If there is no changes between two consecutive shipments ( jj   ) Eq. ( 19) becomes active, otherwise Eq. ( 20) is used to calculate delay time.
As we can observe, there are literally considerable number of binary variables, which make it impossible to solve the resulted problem for real-world applications.For instance, there are 3 receiving truck and 3 shipping truck with 8 products, there will be 307 decision variables including 207 binary variables and 19 continuous variables.There are also 304 constraints, 189 inequality and 115 equality constraints.Yu (2002) proposed the following heuristic approach to solve the proposed model.

GRASP algorithm
Greedy randomized adaptive search procedure(GRASP) (Feo, 1989(Feo, , 1994(Feo, , 1995;;Resende & Ribeiro, 2002;Pitsoulis & Resende, 2002) starts with an empty solution and as the search algorithm continues, new solutions are added to solution set until there is no further solution found.There are many applications of GRASP to solve engineering problems (Zapfel et al., 2010;Javanshir & Haghighi, 2011).There are some similarities between GRASP and Greedy construction method but there are some important changes between these two methods.GRASP uses restricted candidate list (RCL) and allows the algorithm diversified solution sets.The primary objective is to simplify the process of adding new solutions.Fig. 7 shows details of GRASP algorithm.

Fig. 2. GRASP-solution Processing
The solution procedure starts with an empty set and evaluates all elements based on performance function and their effects on quality of solution.These elements are sorted based on their performances in candidate list.Fig. 3 shows three levels of the operations.

Set of remaining solution elements
Restricted candidate solution Partially constructed solution

Fig. 3. Solution construction of the GRASP metaheuristic
Let g be objective function, to build construction list, we need to have and with { } and { } .We use these two limits for the implementation of our GRASP method.For the proposed model of this paper, there are two RCL For instance, if k = 5, there are five best elements in RCL and from this list, one sample is selected randomly.The value based method uses a parameter α to build RCL.Suppose we deal with an optimization problem and all elements must maintain a value less than their limits to become eligible for RCL list, we use When 1   the method is completely random and when 0   the algorithm is greedy.The following shows the summary of the proposed GRASP algorithm, Step 1: Execute three heuristic algorithms 1, 2 and 3 Step 2: Compute j c Step 3: Compute min g and max g Step 4: Add generated solution to RCL if min max min () Step 5: Select one of shipping truck and update the information Step 6: If the number of solutions obtained is equal to S (number of shipping trucks) stop, otherwise goto step 1 The proposed GRASP algorithm first generates three heuristic solutions proposed by Yu (2002) and in each stage, we choose and dispatch one truck.Based on the calculated j c we calculate min g and max g and based on the values of j c we add any solution with min max min () . One advantage of GRASP method is that the proposed GRASP method has only two parameters including the value of  and termination criteria and the implementation of this algorithm chooses 0.2.

The results
The proposed GRASP method has implemented using three strategies.All programs were coded using personal computer with 4GB RAM and Intel ® Core™ DUE CPU processor.Table 1 shows details of our implementations on some test problems.As we can observe from the results, in most cases, the proposed model of this paper provides better objective values.Fig. 4 shows the difference between the objective functions of the proposed GRASP and Yu's heuristics results.As we can observe, except two cases, there are 16 cases where GRASP performs better than best solution strategies provided by Yu's heuristic method, in two case there is no difference and only in two cases, Yu's method beats GRASP method.

Fig. 5. GRASP performance for the test problem
Fig. 5 shows the performance of the proposed GRASP CPU time in seconds for 20 test problem.Note that the CPU time does not increase significantly as inputs of changes.In other words, the proposed GRASP can be easily implemented for some real-world case studies in reasonable amount of time.

Conclusion
In this paper, we have presented an extended cross docking problem by considering temporary storage and repeat holding pattern in the system.The proposed model of this paper was formulated as mixed integer programming and GRASP method was developed to solve the resulted problem.The performance of the proposed model has been compared with Yu's method using some randomly generated test problems.The preliminary results indicate that the proposed GRASP provides better objective values compared with alternative method.Besides, the proposed GRASP seems to be applicable for real-world application in reasonable amount of time.The proposed model of this paper can be used for cross docking problems with more than one single objective function.Such a problem can be solved using multi objective GRASP methodologies to generate efficient Pareto solutions and we leave it for interested researchers as future research.

Fig . 1 .
Fig .1.Cross docking distribution center placed at the first position in the shipping sequence of receiving trucks in the set, S = Number of shipping trucks in the set, N =Number of product types in the set, Number of units of product type k, which is initially loaded in receiving truck i, Number of units of product type k, which is initially loaded for shipping truck j, D = Delay time for truck change, V = Moving or travel time of products from the receiving dock to the shipping dock, based and value Based.The cardinality-based method includes k best alternative solutions.

Table 1
Makespan obtained by the GRASP and Heuristics for the test problem