The cross-docking system is a new distribution strategy which can reduce inventories, lead times, and improve responding time to customers. This paper considers biobjective problem of truck scheduling in cross-docking systems with temporary storage. The objectives are minimizing both makespan and total tardiness. For this problem, it proposes a multiobjective iterated greedy algorithm employing advance features such as modified crowding selection, restart phase, and local search. To evaluate the proposed algorithm for performance, it is compared with two available algorithms, subpopulation particle swarm optimization-II and strength Pareto evolutionary algorithm-II. The comparison shows that the proposed multiobjective iterated greedy algorithm shows high performance and outperforms the other two algorithms.
Efficiently managing the flow of products is one of the most essentialsteps in supply chain management. How this flow is handled is basically affected by transportation networks and distribution structures. Hence, any action contributing to the improvement of these structures such as the execution of cross-docking systems is considered worthwhile. In cross-docking system, products are received by inbound trucks in the receiving dock; then, they are unloaded, sorted, and reorganized based on customer demands. Afterwards, these products are loaded into the outbound trucks for delivery to customers, without being actually held as inventory at warehouse. If any item is held in storage, it takes usually a short time, generally less than 24 h.
In comparison with traditional warehousing strategy, cross-docking systems can cut down or remove storing and retrieving functions, the two most expensive warehousing operations, by synchronizing the flows of inbound and outbound trucks. As a result, not only total operational costs are decreased as a result of reduction of a considerable level of inventory in the distribution system, but also the customers can be served by more precise and on-time shipment deliveries. The cross-docking system is the best way to handle high volume of items in a short time, reduce cost and space required for inventory (or eliminate storage), increase throughput, and improve efficiency by increasing level of customer satisfaction [
The problem of truck scheduling in the cross-docking systems is to determine the sequence of inbound/outbound trucks to unload/load their products. Besides, the assignment of product transshipment is determined as well. It is also assumed that there is temporary storage in front of the shipping dock. If a product arriving at the shipping dock is not intended for loading into the outbound truck currently at the dock, the product is stored in the temporary storage until the appropriate outbound truck comes into the shipping dock. The truck docking pattern employed requires that both inbound and outbound trucks must stay in docks until they finish their task once they come into docks. According to Yu and Egbelu [
Typical flows in a cross-docking system.
Most of the papers in the problem of scheduling trucks in a cross-docking system consider a single optimization criterion, although in practice the decision maker often faces several (usually conflicting) criteria. Therefore, this paper addresses a biobjective problem of truck scheduling in a cross-docking system. The first objective is to minimize makespan (i.e., the length of time from unloading the first product from the first inbound truck until loading the last product on the last outbound truck). The second objective is to minimize the total tardiness of outbound trucks. The tardiness of a truck is equal to the difference between departure time of the truck and its due date.
To solve the problem under consideration, we propose a novel multiobjective algorithm based on the iterated greedy algorithm. An efficient management of the Pareto front, a modified crowding selection operator, an effective local search, and other techniques are applied in order to attain high quality and well spread Pareto fronts. The performance of this algorithm is compared with the subpopulation particle swarm optimization-II (SPPSO-II) proposed by Boloori Arabani et al. [
The rest of the paper is organized as follows. Section
With the advent of cross-docking concept, many researchers have placed their emphasis on this field. After reviewing the literature, only few papers studied the problem of multiobjective truck scheduling in cross-docking systems. In the case of cross-docking systems, Yu [
Yu and Egbelu [
There are some papers considering multiobjective cross-docking systems. Boloori Arabani et al. [
Boloori Arabani et al. [
As we reviewed the literature of multiobjective cross-docking systems, two recent multiobjective algorithms of SPPSO II and SPEA II proposed by Boloori et al. [
This section proposes a multiobjective algorithm to solve the problem under consideration. This algorithm is an effective metaheuristic in form of iterated greedy algorithm (IGA). This algorithm has demonstrated to be very efficient for several combinatorial problems, including the flowshop scheduling problem with makespan objective (Ruiz and Stützle [
IGA starts from a single solution as its working solution and consists of iteratively removing some elements of the working solution randomly (called destruction phase) and then adding so that a new complete and hopefully better solution is generated (called reconstruction phase). Finally, this new solution probabilistically replaced the working solution even if it is worse than the working solution. This procedure iterates until a stopping criterion, commonly a time limit, is met.
In a single-objective case, the quality of a solution is directly obtained by the objective function value. It can easily concluded that a solution
The main idea behind the proposed multiobjective IGA is handling a population of nondominated solutions as a working set instead of just a single solution. At each iteration, one solution from the working set is selected for further processing. The selection is done by a mechanism that accelerates the search and to maximize the spread of the final Pareto set. The selected solution has then undergone the greedy phase, in which some elements are eliminated. Next, the reconstruction procedure creates a whole set of nondominated solutions by inserting each removed element into a population of partial solutions. The working set is updated with the recently generated set of nondominated solutions from the reconstruction procedure. After the greedy phase, the working set is updated, possibly with new solutions. Thus, the solution selected previously for the greedy phase might not be exist in the working set anymore because of being dominated by other new solutions. As a consequence, the selection operator is employed again to select one solution that will go through local search. These two phases, namely, greedy and local search, are repeated until a termination criterion is satisfied. Algorithm
Before describing how to generate an initial solution, it is necessary to show the representation scheme of solutions in this algorithm. As mentioned earlier, there are two decisions in the problem under consideration, the sequence of inbound and outbound trucks as well as the assignment of product transshipment. Our strategy to make the decisions is to determine the sequences by the algorithm and the assignment by a rule.
Thus, an encoded solution consists of two parts such that the first part is related to the inbound trucks and the second part is related to the outbound trucks. Figure
The representation scheme.
To determine the assignment of product transshipment, products unloaded from early inbound trucks are transshipped to the first outbound truck available to load its products. To describe how to decode an encoded solution (i.e., calculation of the two objective functions), see Notations.
The departure time of the trucks determines makespan and total tardiness. The departure time of inbound trucks is calculated as such. Consider
The departure times of the first inbound truck in (
The departure time of
To select a quality Pareto set, it is necessary to consider both objective function and spread of solutions in the set. Hence, the fitness of each solution is defined according to these two factors. To determine the fitness of each solution in the set, we use modified crowding distance assignment (MCDA) procedure. This method is an extension of the well-known crowding distance operator initially presented by Deb et al. [
The original method divides the working set into dominance levels; that is, the set of nondominated solutions forms the first-level Pareto front. Once we eliminate these elements, we have another nondominated set of solutions, which correspond with the second-level Pareto front. This procedure is repeated until all solutions are assigned to a Pareto front. Subsequently, the crowding distance operator assigns a value to each solution of the working set according to the distance between it and its nearest neighbors belonging to the same Pareto front level. Such method favors the selection of the most solitary solutions of the first frontier. The idea is that isolated solutions need to be further explored in order to close gaps in the objective solution space. The main drawback of this technique is that it does not keep track of how many times a solution has been previously selected; because of that, it might keep selecting it again and again. After selection, IGA employs the greedy and local search phases. Thus, applying the standard crowding distance procedure results in an algorithm that gets easily stuck, as if no improvements are found after the greedy and local search phases, the Pareto fronts do not change, and the same solution is selected repeatedly.
To avoid this, Minella et al. [
//
// // Max Min
The greedy phase works in two steps: first, a block of
To overcome the size of the partial solutions that grows exponentially, each time a set of partial solutions is created, only the nondominated partial solutions are kept and the dominated ones are discarded. In the next step, the next removed element is only reinserted in the nondominated partial solutions. Algorithm
Afterwards, the solution set acquired from the greedy phase is appended to the working population and the dominated elements are eliminated. Finally, the MCDA is employed on this working set and a new solution is selected for the local search phase.
A simple local search phase is proposed to improve the search in the space close to the selected solution. The local search phase consists in randomly selecting each time, one element from inbound trucks sequence of the selected solution, and reinserting it into the
Similar to the greedy phase, instead of keeping one full solution in this local search phase, we keep a local working set of solutions. At each step, a removed element is reinserted, and we add the new solutions to the local working set. For each of the removed elements,
The last phase is the restart procedure and consists in archiving the current working set and then creating new one with randomly generated solutions. This is the simplest possible restart procedure that still allows the algorithm to escape from a situation in which the current working set is stalled. One of the difficulties of this phase is to understand when to implement the restart. A very simple method is to restart when the working population has not changed for a given number of iterations. The issue of determining when a working set has not changed is not trivial, as this can be calculated in a number of ways. Based on what proposed in Minella et al. [
IGA with additional restart phase.
This section assesses the performance of the proposed IGA. To this purpose, we bring the SPPSOII and SPEAII proposed by Boloori Arabani et al. [
To compare the approximation solution methods (ASM), we require some performance measures (PM). They should be designed to assess the performance of tested ASM with no bias or misleading results. In multi-objective cases, PMs are challenging since each ASM gives an approximation Pareto set and these sets need to be compared. For example, consider two ASMs that provide two approximation Pareto sets
After all, three PMs are known to give reliable analyses (Knowles et al. [
In this paper, we employ two major types of quality indicators that were shown to be Pareto-compliant in Zitzler et al. [
For a given instance, the reference Pareto set has been constructed from all nondominated solutions found from all tested methods and experiments.
For a given instance, the best (worst) reference point is the best (worst) values available in the approximation Pareto sets for each objective function.
The first quality indicator is the hypervolume indicator (
The second quality indicator is the so-called Unary Epsilon Indicator
This section studies the influence of parameter’s values on the performance of the IGA. We employ the design of experiments (DOE) technique for the experiment, where the parameters affecting the performance of the IGA are tested in a full factorial experimental design which is later analyzed by means of the analysis of variance (ANOVA) technique. The proposed multiobjective IGA has two parameters:
Parameters and their level of IGA algorithm.
Parameters | Level’s number | Levels of parameters |
---|---|---|
|
1 | 4 |
2 | 6 | |
3 | 8 | |
4 | 10 | |
5 | 12 | |
|
||
|
1 | 1 |
2 | 2 | |
3 | 3 | |
4 | 4 | |
5 | 5 |
As a result, each parameter is measured at five levels. This algorithm is run using different combinations of parameters. Therefore, this gives a total of 25 algorithm configurations. Each configuration is tested with the 2 instances and 5 different problem sets (as shown in Table
The size characteristics of five problem sets.
Problem set | Problem size | Total number of products | ||
---|---|---|---|---|
Number of inbound trucks | Number of outbound trucks | Number of product types | ||
1 | 8 | 11 | 12 | 5293 |
2 | 15 | 15 | 15 | 5326 |
3 | 19 | 16 | 18 | 7964 |
4 | 17 | 16 | 19 | 10852 |
5 | 20 | 20 | 12 | 5314 |
The experiments are implemented and the obtained nondominated Pareto sets are transformed into hypervolume and epsilon indicators. The average of hypervolume indicator that was obtained in each level of mentioned parameters is shown in Table
The average of hypervolume indicator that was obtained in each level of parameters.
Problem set |
|
| ||||||||
---|---|---|---|---|---|---|---|---|---|---|
4 | 6 | 8 | 10 | 12 | 1 | 2 | 3 | 4 | 5 | |
1 | 0.788 | 0.826 | 0.781 | 0.746 | 0.785 | 0.749 | 0.771 | 0.824 | 0.788 | 0.772 |
2 | 0.843 | 0.844 | 0.823 | 0.878 | 0.778 | 0.701 | 0.902 | 0.874 | 0.856 | 0.831 |
3 | 0.690 | 0.690 | 0.693 | 0.655 | 0.679 | 0.647 | 0.673 | 0.717 | 0.714 | 0.665 |
4 | 0.814 | 0.825 | 0.794 | 0.713 | 0.705 | 0.751 | 0.747 | 0.832 | 0.799 | 0.722 |
5 | 0.706 | 0.739 | 0.741 | 0.660 | 0.674 | 0.737 | 0.685 | 0.766 | 0.699 | 0.632 |
|
||||||||||
Average | 0.768 | 0.785 | 0.766 | 0.732 | 0.724 | 0.717 | 0.755 | 0.803 | 0.771 | 0.724 |
The average of epsilon indicator that was obtained in each level of parameters.
Problem set |
|
| ||||||||
---|---|---|---|---|---|---|---|---|---|---|
4 | 6 | 8 | 10 | 12 | 1 | 2 | 3 | 4 | 5 | |
1 | 1.089 | 1.111 | 1.120 | 1.129 | 1.113 | 1.116 | 1.113 | 1.102 | 1.126 | 1.107 |
2 | 1.072 | 1.067 | 1.073 | 1.066 | 1.076 | 1.088 | 1.060 | 1.065 | 1.071 | 1.071 |
3 | 1.083 | 1.080 | 1.079 | 1.082 | 1.076 | 1.080 | 1.081 | 1.076 | 1.078 | 1.085 |
4 | 1.042 | 1.045 | 1.042 | 1.050 | 1.050 | 1.046 | 1.045 | 1.040 | 1.046 | 1.053 |
5 | 1.066 | 1.059 | 1.060 | 1.066 | 1.064 | 1.060 | 1.066 | 1.051 | 1.062 | 1.077 |
|
||||||||||
Average | 1.071 | 1.072 | 1.075 | 1.079 | 1.076 | 1.078 | 1.073 | 1.067 | 1.076 | 1.078 |
For statistical significance test of the parameters, the analysis of variance (ANOVA) is carried out. The response variables of the ANOVA experiment are the hypervolume (Table
Analysis of variance (ANOVA) for a statistical significance test of the parameters on the hypervolume indicator values.
Factor |
|
|
|
|
|
---|---|---|---|---|---|
|
4 | 0.1625 | 0.0406 | 0.80 | 0.524 |
|
4 | 0.4758 | 0.1189 | 2.35 | 0.054 |
Interaction | 16 | 1.1571 | 0.0723 | 1.43 | 0.125 |
Error | 375 | 18.9766 | 0.0506 | ||
|
|||||
Total | 399 | 20.7720 |
Analysis of variance (ANOVA) for a statistical significance test of the parameters on the epsilon indicator values.
Factor |
|
|
|
|
|
---|---|---|---|---|---|
|
4 | 0.0048 | 0.0012 | 0.59 | 0.668 |
|
4 | 0.0067 | 0.0016 | 0.82 | 0.512 |
Interaction | 16 | 0.0385 | 0.0024 | 1.18 | 0.280 |
Error | 375 | 0.7656 | 0.0020 | ||
|
|||||
Total | 399 | 0.8158 |
After the calibration, we can see from the results, apart from some minor exceptions, the best size for the destruction block resulting in being
After adjustment parameters, in order to implement IGA and compare the performance of this algorithm with the existent meta-heuristic algorithms from the literature (SPPSOII and SPEAII) that are proposed by Boloori Arabani et al. [
The size characteristics of twenty problem sets.
Problem set | Problem size | Total number of products | ||
---|---|---|---|---|
Number of inbound trucks | Number of outbound trucks | Number of product types | ||
1 | 8 | 11 | 12 | 5293 |
2 | 9 | 14 | 8 | 5600 |
3 | 10 | 8 | 11 | 4923 |
4 | 11 | 10 | 10 | 3115 |
5 | 11 | 15 | 8 | 4786 |
6 | 12 | 9 | 11 | 6007 |
7 | 12 | 12 | 12 | 3060 |
8 | 13 | 11 | 13 | 2614 |
9 | 14 | 14 | 13 | 5040 |
10 | 15 | 15 | 15 | 5326 |
11 | 15 | 19 | 16 | 13374 |
12 | 16 | 17 | 16 | 5676 |
13 | 17 | 15 | 17 | 9944 |
14 | 17 | 16 | 19 | 10852 |
15 | 17 | 18 | 11 | 5377 |
16 | 18 | 16 | 14 | 5905 |
17 | 19 | 16 | 18 | 7964 |
18 | 19 | 19 | 13 | 6384 |
19 | 20 | 17 | 14 | 5488 |
20 | 20 | 20 | 12 | 5314 |
The characteristics of five instances.
Parameters | Instances | ||||
---|---|---|---|---|---|
Instance 1 | Instance 2 | Instance 3 | Instance 4 | Instance 5 | |
|
1 |
|
3 |
|
|
|
1 |
|
5 |
|
|
|
75 | 90 | 80 | 60 | 110 |
|
100 | 120 | 115 | 85 | 150 |
In this section, we compare our proposed algorithm with the best performing algorithms proposed by Boloori Arabani et al. [
After running each of the applied multiobjective algorithms under their optimal parameters, we will have the final nondominated Pareto set of solutions for each problem set. As stated before, these Pareto sets must be evaluated, analyzed, and compared with each other by the means of hypervolume and epsilon indicators so that the effectiveness of each multiobjective algorithm can be clarified.
As mentioned before, each algorithm is run for 5 instances and 20 problem sets and the results are shown in Table
The average values of hypervolume and the epsilon indicator of all problem sets for each algorithm.
Problem set | Performance indicators | |||||
---|---|---|---|---|---|---|
|
| |||||
Algorithms | ||||||
IGA | SPPSOII | SPEAII | IGA | SPPSOII | SPEAII | |
1 | 1.409 | 0.488 | 0.317 | 1 | 1.172 | 1.171 |
2 | 1.423 | 0.721 | 0.371 | 1 | 1.159 | 1.376 |
3 | 1.387 | 0.718 | 0.423 | 1 | 1.144 | 1.290 |
4 | 1.423 | 0.745 | 0.300 | 1 | 1.355 | 1.505 |
5 | 1.438 | 0.912 | 0.465 | 1 | 1.143 | 1.354 |
6 | 1.399 | 0.792 | 0.274 | 1 | 1.245 | 1.332 |
7 | 1.435 | 0.564 | 0.281 | 1 | 1.146 | 1.178 |
8 | 1.400 | 0.734 | 0.217 | 1 | 1.086 | 1.152 |
9 | 1.437 | 0.568 | 0.226 | 1 | 1.107 | 1.151 |
10 | 1.435 | 0.570 | 0.308 | 1 | 1.105 | 1.164 |
11 | 1.433 | 0.758 | 0.180 | 1 | 1.071 | 1.149 |
12 | 1.438 | 0.546 | 0.313 | 1 | 1.088 | 1.121 |
13 | 1.434 | 0.654 | 0.280 | 1 | 1.100 | 1.136 |
14 | 1.430 | 0.651 | 0.141 | 1 | 1.069 | 1.144 |
15 | 1.431 | 0.656 | 0.253 | 1 | 1.085 | 1.166 |
16 | 1.436 | 0.712 | 0.389 | 1 | 1.112 | 1.185 |
17 | 1.406 | 0.681 | 0.130 | 1 | 1.057 | 1.127 |
18 | 1.435 | 0.752 | 0.274 | 1 | 1.124 | 1.156 |
19 | 1.436 | 0.811 | 0.322 | 1 | 1.092 | 1.164 |
20 | 1.437 | 0.590 | 0.278 | 1 | 1.153 | 1.149 |
|
||||||
Average | 1.425 | 0.681 | 0.287 | 1 | 1.131 | 1.208 |
Based on the obtained average of two quality indicators values in each algorithm, we can conclude that IGA favorably outperforms the other two algorithms with
It can be noticed that according to the both quality indicators, IGA algorithm is the best performer, the second one in the ranking is SPPSO II while the worst is SPEAII. However, for small instances, we can see how the IGA performs much better. For the further analysis, we carry out the ANOVA. The related results show that there is statistically significant difference between the performances of the algorithms with
Analysis of variance (ANOVA) for a statistical significance test of the algorithms on the hypervolume indicator values.
Factor |
|
|
|
|
|
---|---|---|---|---|---|
Algorithms | 2 | 13.35867 | 6.67933 | 1078.93 | 0.000 |
Error | 57 | 0.35287 | 0.00619 | ||
|
|||||
Total | 59 | 13.71154 |
Analysis of variance (ANOVA) for a statistical significance test of the algorithms on the epsilon indicator values.
Factor |
|
|
|
|
|
---|---|---|---|---|---|
Algorithms | 2 | 0.44602 | 0.22301 | 42.68 | 0.000 |
Error | 57 | 0.29783 | 0.00523 | ||
|
|||||
Total | 59 | 0.74386 |
Figure
Means and 95% confidence interval for each algorithm in terms of the quality indicators.
As a conclusion and according to the results of the two analyzed measures, it can be stated that the IGA algorithm with average hypervolume and epsilon values of 1.425 and 1.00, respectively, can surmount the other two algorithms.
In this paper, we have proposed the iterated greedy algorithm for a biobjective scheduling problem in the cross-docking system with the objectives of minimizing the makespan and the tardiness in order to fill the current research gap in the case of multiobjective optimization problems. In the cross-docking system, inbound trucks are coming into the receiving dock while their products are unloaded. Then, the product items are categorized and sorted out in the temporary storage. Afterwards, they are loaded into the outbound trucks leaving the shipping dock. In fact, the scheduling problem is how to schedule and assign the inbound and outbound trucks in order to minimize the two objective functions. The effectiveness of the approach is established by comparing it with the best existing algorithms for the problem under consideration (SPPSOII and SPEAII) presented by Boloori Arabani et al. [
The performance of each of these multiobjective algorithms was analyzed and compared by means of two Pareto-compliant performance measures which demonstrate that the proposed IGA can relatively overwhelm other two algorithms.
Regarding future research trends in the case of cross-docking systems, some of the assumptions of this paper can be modified. For example, the arrival time of trucks can be changed to variable times. Moreover, instead of single receiving and shipping docks, multiple receiving and shipping docks can be implemented so that several inbound and outbound trucks can be dealt with concurrently. The capacity of temporary storage which in this paper is supposed to be infinite can be limited. Additionally, other conditions can be taken into account in which a particular set of inbound and outbound trucks commute between the specific set of cross-dock terminals and transport products in a distribution network. Furthermore, we can suggest more complicated and applicable versions of multiobjective algorithms in which hybrid metaheuristics are employed. Moreover, to obtain new areas of cross-docking systems, new constraints can be added to the problem such as time windows and maximum acceptable due dates for product delivery.
Makespan
Total tardiness
The number of inbound trucks
The number of outbound trucks
The number of product types
Truck changeover time
Moving time of product from the receiving dock to the shipping dock per product unit
The
Due date for the
The number of units of product type
The number of units of product type
The total number of products type
1 if any product is transshipped for the
The loading time per product unit type
The unloading time per product unit type
The time at which the
The time at which the
The authors declare that there is no conflict of interests regarding the publication of this paper.