A new closed-loop supply chain logistics network of vehicle routing problem with simultaneous pickups and deliveries (VRPSPD) dominated by remanufacturer is constructed, in which the customers are originally divided into three types: distributors, recyclers, and suppliers. Furthermore, the fuel consumption is originally added to the optimization objectives of the proposed VRPSPD. In addition, a bee evolutionary algorithm guiding nondominated sorting genetic algorithm II (BEG-NSGA-II) with a two-stage optimization mechanism is originally designed to solve the proposed VRPSPD model with three optimization objectives: minimum fuel consumption, minimum waiting time, and the shortest delivery distance. The proposed BEG-NSGA-II could conquer the disadvantages of traditional nondominated sorting genetic algorithm II (NSGA-II) and algorithms with a two-stage optimization mechanism. Finally, the validity and feasibility of the proposed model and algorithm are verified by simulating an engineering machinery remanufacturing company’s reverse logistics and another three test examples.
Recycling and remanufacturing is an integral system by which the old or discarded products are recycled and then processed, recovered, and sold as like-new ones in the markets [
The VRPSPD problem was first introduced by Min [
Although there is a stream of research on the VRPSPD problem and related algorithms, we find that there are still some important problems which have been overlooked and needed to be solved. It is indicated that the traditional VRPSPD model only considers a single type of customer. However, in a real recycling and remanufacturing system, customers consist of various types, such as distributors, recyclers, and suppliers. In addition, although the optimization objective of fuel consumption has been studied in some models such as FCM-CVRP and CVRP, it has not been considered in the existing literatures with the traditional VRPSPD. However, the fuel consumption is actually a key factor affecting the resource consumption and environmental destruction in VRPSPD. With regard to the overlooked problems mentioned above, in this paper the customers are originally divided into three types which are distributors, recyclers, and suppliers, and a new closed-loop supply chain logistics network of VRPSPD dominated by remanufacturer is constructed, which is more consistent with the actual situation of recycling and remanufacturing. Furthermore, for the purpose of resource saving and environment protection, the fuel consumption is originally added to the optimization objectives of VRPSPD. Besides, inspired by the literature [
The rest of this paper is organized as follows. The proposed VRPSPD model is constructed in the next section. In Section
The proposed VRPSPD model in the paper is shown in Figure
The proposed logistics structure.
The following assumptions are made for the network structure:
(i) Due to the uncertainty of quantity and quality of recycled products, each service point (except waste treatment plan) in the model should establish an information-sharing platform at the beginning of each logistics cycle; then, according to the information, recycling and remanufacturing factory arranges distribution vehicles.
(ii) Distributor points are sale points of remanufacturing products as well as recycling points of waste products.
(iii) Recycling process center and recycling and remanufacturing plant are located in the same location.
The VRPSPD model proposed in this paper is based on the following assumptions:
(i) Every vehicle moves at a constant speed between any two customers; namely, the acceleration speed is zero.
(ii) Other energy consumption of distribution vehicles (such as air conditioning) is zero, which means
(iii) Every vehicle departs from the recycling company and returns to it in the end.
(iv) All vehicles’ maximum payload is identical and they have the same weight as well.
(v) The fuel consumption for each vehicle is only affected by three factors, namely, travel distance, payload, and travel speed.
(vi) The road slope (
By analysing the operating process of recycling and remanufacturing reverse logistics network, recycling points and suppliers can be regarded as special distribution points. So the vehicle routing problem of the proposed model can be changed into VRPSPD.
The parameters used in the definition of multiobjective VRPSPD are as follows:
If the real delivery time for point
Many scholars have carried out a lot of researches on the model of vehicle energy consumption and have made a series of research achievements. One of the classic models for the energy consumption of vehicle engine was based on the research of Barth and Boriboonsomsin [
The meanings of letter symbols in (
Letter symbols | Meanings |
---|---|
| Ratio of fuel to air weight |
| Engine speed |
| Engine friction factor |
| Engine exhaust volume |
| Acceleration of gravity |
| Transmission efficiency |
| Engine efficiency |
| Truck windward area |
| Air resistance coefficient |
| Air density |
| Travel distance |
| Other energy needs |
| Rolling resistance coefficient |
| Vehicle acceleration |
| Mass of vehicle and products |
| Vehicle travel speed |
| The road slope |
In (
According to the assumption, we can get the vehicle fuel consumption from service point
According to the assumption above, all vehicles’ travel distance can be calculated by
The optimization objectives and constraints in our multiobjective optimization model are defined as follows:
Equations (
Equation (
In this paper, a two-stage optimization mechanism is used to solve the proposed VRPSPD with strict constraints. In the first stage, we use an improved bee evolutionary genetic algorithm to optimize the initial population. The optimization efficiency of the first stage is improved by optimizing the selection operators, selecting the different crossover operators according to the similarity of individuals’ parent chromosomes, and selecting the different mutation operators according to the performance of individual’s parent chromosome. In the second stage, an improved NSGA-II is used to optimize the proposed VRPSPD model. Based on the improved crossover and mutation operators, we construct the methods of deleting duplicate individuals, introducing new individuals, and using elite population instead of the parent population to improve the population diversity. To deal with the strict constraints, we introduce an external auxiliary population. During the optimizing process, if the constraint violation degree of the infeasible solutions is smaller than a given value, these infeasible solutions are copied to the external auxiliary population and used to evolve in the next generation. By using this external auxiliary population, the infeasible solutions gradually evolve to the boundary of feasible solutions, and the convergence speed is accelerated. The framework of the proposed algorithm is shown in Figure
The framework of BEG-NSGA-II for VRPSPD.
The first stage includes the following.
Randomly generate
Calculate the constraint violation value of each chromosome with (
where
Calculate each chromosome’s fitness based on the degree of constraint violation using (
Select
Randomly generate
The new queen takes crossover and mutation with other individuals generated from
Delete the redundant repeating individuals and randomly generate
Return to
The second stage includes the following.
Take fast nondominated sorting and crowding degree calculation on population.
Carry out the following three kinds of operation for the population generated by
The individuals take mutation with the probability of 0.1 after crossover. If the offspring rank = 1, then select the conventional mutation; otherwise the two-binding mutation (TBM) or reverse mutation is selected randomly.
Randomly generate
where
Combine the individuals of
Use
If
According to the characteristics of logistics path planning, a string with different integers (the length equals
In this paper, in order to enhance the searching space and avoid prematurity of local optimal solutions, we use two types of crossover operators (single-point crossover and two-point crossover operator) in this proposed algorithm. The single-point crossover operator helps to play excellent high genetic characteristics and to improve the convergence speed. Nevertheless, when the two parent generations become identical or similar, the crossover offspring almost get no change especially in the middle and later periods of the evolution. Hence, a two-point crossover operator is proposed for its ability to improve the mentioned problems. Equation (
The procedure of single-point crossover is described as follows (
A random parameter
The elements from 1 to
The elements in
The elements in
The procedure of two-point crossover is described as follows (
Two random parameters
The elements from
The elements in
The elements in
The examples of singe-point crossover and two-point crossover are, respectively, shown in Figures
Singe-point crossover.
Two-point crossover.
In this paper, we have adopted three different mutation operators for the purpose of expending the solution space as well as maintaining the good solutions, which are conventional mutation operator, reverse mutation operator, and TBM operator. The examples of these mutation operators are shown in Figures
Conventional mutation.
Reverse mutation.
Two-binding mutation.
The main procedure of conventional mutation operator is described as follows
Randomly select two positions in
Swap the elements in the selected positions to generate
The main procedure of reverse mutation operator is described as follows (
Randomly select two positions in
Reverse the numbers between the two selected positions to generate
The main procedure of TBM operator is described as follows (
A random parameter
Exchange the elements
The proposed BEG-NSGA-II algorithm was coded in MATLAB R2014a and implemented on a computer configured with Intel Core i3 CPU with 2.67 GHz frequency and 8GB RAM. Three sets of examples are used to illustrate the performance of the proposed algorithm, in which three of the same trucks were used to deliver and pick up the goods. The first one is the VRPSPD problem of an actual engineering machinery remanufacturing company (with 12 customers). To further verify the effectiveness of the proposed algorithm, another two instances with 22 customers and 42 customers are used to simulate the problem of VRPSPD.
Because the problem of this paper has no benchmarks, we generate the test instances based on the SCA3-0 instance from Dethloff [
The values of parameters are shown in Table
The values of parameters used in the proposed VRPSPD model.
Parameter | value |
---|---|
| 1 |
| 25r/m |
| 0.2 |
| 8 |
| 9.8N/kg |
| |
| 0.6 |
| 0.4 |
| 0.01 |
| 1.2041[ |
| 4.5m2 |
| 0 |
| - |
| 0 |
| - |
| - |
| 0 |
In addition,
The adopted parameters of the proposed algorithm are set as follows:
Table
The basic information of test example 1 with 12 customers.
| | | | | | |
---|---|---|---|---|---|---|
| 0 | 35 | 35 | - | - | - |
| 1 | 49 | 69 | 8 | 2.5 | 8 am-10 am |
| 2 | 60 | 25 | 6 | 2 | 10 am-12 am |
| 3 | 0 | 36 | 12 | 6 | 8 am-10 am |
| 4 | 15 | 25 | 9 | 5 | 8.5 am-10.5 am |
| 5 | 35 | -6 | 9 | 4 | 9.5 am-11.5 am |
| 6 | 69 | 30 | 11 | 5 | 9 am-12 am |
| 7 | 50 | 9 | 0 | 8 | 10 am-3 pm |
| 8 | 36 | 70 | 0 | 8 | 10 am-3 pm |
| 9 | 4 | 60 | 0 | 10 | 10 am-3 pm |
| 10 | 15 | 69 | 0 | 6.3 | 10 am-3 pm |
| 11 | 63 | 55 | 0 | 6.3 | 10 am-3 pm |
| 12 | 5 | 5 | 0 | 6.3 | 10 am-3 pm |
(b) “-” means that these data do not need to be set in advance.
In order to demonstrate the performance of the proposed algorithm and research the effect of the duplicate individuals, test example 1 was tested by the traditional NSGA-II (T-NSGA-II), BEG-NSGA-II without eliminating the duplicate individuals (W-BEG-NSGA-II), and BEG-NSGA-II, respectively. Table
Comparisons of the optimization results of test example 1 with 12 customers.
| | | | | |
---|---|---|---|---|---|
| 0-3-1-8-10-0 0-4-12-5-7-0 0-2-11-6-9-0 | 546.39 | 0.74 | 489.70 | 1.2 |
0-3-1-8-10-0 0-4-12-5-7-0 0-11-2-6-9-0 | 548.06 | 0.74 | 481.77 | ||
0-3-8-1-10-0 0-4-12-5-7-0 0-2-11-6-9-0 | 546.33 | 31.28 | 493.11 | ||
0-3-8-1-10-0 0-4-12-5-7-0 0-11-2-6-9-0 | 548.00 | 31.28 | 485.18 | ||
0-3-10-1-8-0 0-4-12-5-7-0 0-2-11-6-9-0 | 539.77 | 50.13 | 475.41 | ||
| 0-4-5-7-10-0 0-2-6-11-1-8-0 0-3-9-12-0 | 512.69 | 95.5 | 476.94 | 0.9 |
0-4-5-7-10-0 0-6-2-11-1-8-0 0-9-3-12-0 | 506.21 | 147.4 | 458.14 | ||
0-4-5-7-10-0 0-6-2-11-1-8-0 0-3-9-12-0 | 525.85 | 73.8 | 488.82 | ||
0-4-5-7-10-0 0-6-2-11-1-8-0 0-9-3-12-0 | 519.36 | 125.6 | 470.02 | ||
0-5-7-4-10-0 0-2-6-11-1-8-0 0-9-3-12-0 | 519.20 | 180.9 | 452.91 | ||
| 0-2-6-11-1-8-0 0-4-12-5-7-0 0-3-9-10-0 | 418.99 | 95.5 | 371.69 | 0.5 |
| | | | ||
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| | | |
(b) “
(c) “
(i) The algorithms of T-NSGA-II, W-BEG-NSGA-II, and BEG-NSGA-II could find 5, 6, and 6 optimal Pareto solutions with average computing time of 1.2, 0.9, and 0.5 minutes correspondingly during 10 run times.
(ii) The algorithm of BEG-NSGA-II could find 2 Pareto solutions (bold and marked as “
From the information mentioned above, we can draw the following conclusions:
(i) The algorithms of T-NSGA-II and W-BEG-NSGA-II may generate amount of duplicate individuals, especially after 100 iterations. These duplicate individuals may cause the problems of inefficiency, slow convergence, and converging to local Pareto optimal solutions.
(ii) The proposed algorithm of BEG-NSGA-II is more effective than the algorithms of T-NSGA-II and W-BEG-NSGA-II to solve the VRPSPD problem with rigor constraints.
Figures
Solution 1 of test example 1 (fuel consumption=418.99kg, waiting time = 95.5 min, and distance =371.69km).
Solution 2 of test example 1 (fuel consumption=432.14kg, waiting time = 73.7 min, and distance = 383.57km).
The number of duplicate individuals in test example 1 with 12 customers.
Table
The basic information of test example 1 with 22 customers.
| | | | | | |
---|---|---|---|---|---|---|
| 0 | 50 | 50 | - | - | - |
| 1 | 41 | 60 | 3 | 1.5 | 8 am-10 am |
| 2 | 39 | 40 | 2 | 1 | 10 am-12 am |
| 3 | 60 | 58 | 4 | 2 | 8 am-10 am |
| 4 | 58 | 42 | 3.5 | 2 | 8.5 am-10.5 am |
| 5 | 18 | 86 | 3 | 0.5 | 9.5 am-11.5 am |
| 6 | 4 | 12 | 3 | 1.5 | 9 am-12 am |
| 7 | 90 | 6 | 2 | 2 | 10 am-3 pm |
| 8 | 92 | 86 | 4 | 2 | 10 am-3 pm |
| 9 | 25 | 76 | 2.5 | 2 | 10 am-3 pm |
| 10 | 30 | 40 | 3 | 2 | 10 am-3 pm |
| 11 | 72 | 36 | 2.5 | 1 | 10 am-3 pm |
| 12 | 68 | 80 | 3 | 1.5 | 10 am-3 pm |
| 13 | 22 | 69 | 3.5 | 2.5 | 10 am-3 pm |
| 14 | 28 | 44 | 2 | 1.5 | 8.5 am-10.5 am |
| 15 | 56 | 36 | 4 | 1.5 | 9.5 am-11.5 am |
| 16 | 64 | 62 | 0 | 6 | 9 am-12 am |
| 17 | 32 | 80 | 0 | 7 | 10 am-3 pm |
| 18 | 28 | 36 | 0 | 7 | 10 am-3 pm |
| 19 | 62 | 40 | 0 | 6 | 10 am-3 pm |
| 20 | 63 | 55 | 0 | 6.3 | 10 am-3 pm |
| 21 | 14 | 70 | 0 | 6.3 | 10 am-3 pm |
| 22 | 20 | 30 | 0 | 6.3 | 10 am-3 pm |
(b) “-” means that these data do not need to be set in advance.
In order to demonstrate the performance of the proposed algorithm and research the effect of the duplicate individuals, test example 2 was also tested by T-NSGA-II, W-BEG-NSGA-II, and BEG-NSGA-II, respectively. Table
Comparisons of the optimization results of test example 2 with 22 customers.
| | | | | |
---|---|---|---|---|---|
| 0-1-4-7-11-19-13-2-6-22-0 0-18-14-21-5-9-20-0 0-15-3-16-12-8-10-17-0 | 702.56 | 109.37 | 708.43 | 2.5 |
0-4-15-7-11-19-2-13-6-22-0 0-18-14-21-5-9-16-0 0-1-3-20-12-8-10-17-0 | 654.17 | 125.09 | 676.71 | ||
0-4-15-7-11-19-2-13-6-22-0 0-18-14-21-5-9-16-0 0-1-20-3-12-8-10-17-0 | 652.72 | 141.55 | 678.08 | ||
0-4-15-11-7-19-2-13-6-22-0 0-18-14-21-5-9-16-0 0-1-3-20-12-8-10-17-0 | 650.68 | 150.43 | 680.65 | ||
0-4-15-11-19-1-13-2-6-22-0 0-18-14-21-5-9-20-0 0-3-7-16-12-8-10-17-0 | 698.33 | 96.99 | 712.65 | ||
0-4-15-19-11-1-13-2-6-22-0 0-18-14-21-5-9-20-0 0-3-7-16-12-8-10-17-0 | 695.59 | 122.56 | 714.06 | ||
0-4-15-19-11-7-2-13-6-22-0 0-18-14-21-5-9-16-0 0-1-3-20-12-8-10-17-0 | 644.90 | 161.72 | 676.88 | ||
0-4-15-19-11-7-2-13-6-22-0 0-18-14-21-5-9-16-0 0-1-20-3-12-8-10-17-0 | 643.45 | 178.18 | 678.25 | ||
0-4-15-19-11-7-2-13-6-22-0 0-18-14-21-9-5-16-0 0-1-3-20-12-8-10-17-0 | 644.72 | 168.08 | 683.36 | ||
0-4-15-19-11-7-2-13-6-22-0 0-18-14-21-9-5-16-0 0-1-20-3-12-8-10-17-0 | 643.26 | 184.54 | 684.73 | ||
| | | | | 1.6 |
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| 0-4-15-2-10-6-22-18-14-0 0-1-17-16-3-11-19-0 0-9-13-21-5-12-8-7-20-0 | 544.83 | 147.38 | 569.64 | 0.9 |
0-4-15-2-10-6-22-18-14-0 0-9-17-3-16-11-19-0 0-1-13-21-5-12-8-7-20-0 | 543.80 | 147.15 | 570.21 | ||
0-4-15-2-10-6-22-18-14-0 0-9-17-16-3-11-19-0 0-1-13-21-5-12-8-7-20-0 | 540.38 | 159.78 | 569.18 | ||
0-4-15-2-10-6-22-18-14-0 0-17-9-16-3-11-19-0 0-1-13-21-5-12-8-7-20-0 | 537.94 | 164.41 | 572.82 | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| | | | ||
| | | |
(b) “
(c) “
(i) The algorithms of T-NSGA-II, W-BEG-NSGA-II, and BEG-NSGA-II could find 10, 12, and 10 optimal Pareto solutions with average computing time of 2.5, 1.6, and 0.9 minutes correspondingly during 10 run times.
(ii) All of the results (bold and marked as “
(iii) The algorithm of BEG-NSGA-II could find 6 Pareto (bold and marked as “
From the information mentioned above, we can draw the following conclusions:
(i) The algorithms of T-NSGA-II and W-BEG-NSGA-II may generate amount of duplicate individuals, especially after 100 iterations. These duplicate individuals may cause the problems of inefficiency, slow convergence, and converging to local Pareto optimal solutions.
(ii) The proposed algorithm of BEG-NSGA-II is more effective than the algorithms of T-NSGA-II and W-BEG-NSGA-II to solve the VRPSPD problem with rigor constraints.
Figures
Solution 1 of test example 2 (fuel consumption=544.83kg, waiting time = 147.38 min, and distance = 569.64km).
Solution 2 of test example 2 (fuel consumption=543.80kg, waiting time = 147.15 min, and distance = 570.21km).
The number of duplicate individuals in test example 2 with 22 customers.
Table
The basic information of test example 3 with 42 customers.
| | | | | | |
---|---|---|---|---|---|---|
| 0 | 50 | 50 | - | - | - |
| 1 | 41 | 60 | 2 | 1 | 8 am-10 am |
| 2 | 39 | 40 | 1.5 | 0.8 | 10 am-12 am |
| 3 | 60 | 58 | 3 | 1.2 | 8 am-10 am |
| 4 | 58 | 42 | 2.5 | 1 | 8.5 am-10.5 am |
| 5 | 18 | 86 | 2 | 0.5 | 9.5 am-11.5 am |
| 6 | 4 | 12 | 2 | 0.8 | 9 am-12 am |
| 7 | 90 | 6 | 1.5 | 0.6 | 10 am-3 pm |
| 8 | 92 | 86 | 3 | 1.5 | 10 am-3 pm |
| 9 | 25 | 76 | 2 | 1 | 10 am-3 pm |
| 10 | 30 | 40 | 2 | 1 | 10 am-3 pm |
| 11 | 72 | 36 | 2 | 0.7 | 10 am-3 pm |
| 12 | 68 | 80 | 2 | 1.2 | 10 am-3 pm |
| 13 | 22 | 69 | 2 | 1.8 | 10 am-12 am |
| 14 | 28 | 44 | 1.5 | 0.9 | 8.5 am-10.5 am |
| 15 | 56 | 36 | 2.5 | 1.1 | 9.5 am-11.5 am |
| 16 | 9 | 56 | 1 | 0.8 | 9 am-12 am |
| 17 | 8 | 40 | 1.7 | 0.8 | 10 am-3 pm |
| 18 | 85 | 12 | 1 | 0.7 | 10 am-3 pm |
| 19 | 88 | 84 | 0.9 | 0.6 | 10 am-3 pm |
| 20 | 40 | 80 | 1 | 0.8 | 10 am-3 pm |
| 21 | 45 | 13 | 1.5 | 1 | 10 am-3 pm |
| 22 | 70 | 25 | 1.1 | 1 | 10 am-3 pm |
| 23 | 69 | 77 | 1.2 | 0.6 | 10 am-12 am |
| 24 | 24 | 81 | 1 | 0.7 | 8.5 am-10.5 am |
| 25 | 50 | 30 | 1.2 | 0.6 | 9.5 am-11.5 am |
| 26 | 48 | 56 | 0.9 | 0.5 | 9 am-12 am |
| 27 | 22 | 60 | 1.4 | 0.6 | 10 am-3 pm |
| 28 | 6 | 50 | 0.5 | 0.2 | 10 am-3 pm |
| 29 | 48 | 10 | 0.6 | 0.4 | 10 am-3 pm |
| 30 | 92 | 50 | 0.8 | 0.4 | 10 am-3 pm |
| 31 | 64 | 62 | 0 | 2.5 | 10 am-3 pm |
| 32 | 32 | 80 | 0 | 2 | 10 am-3 pm |
| 33 | 28 | 36 | 0 | 3 | 10 am-12 am |
| 34 | 62 | 40 | 0 | 1.8 | 8.5 am-10.5 am |
| 35 | 76 | 70 | 0 | 3.5 | 9.5 am-11.5 am |
| 36 | 72 | 41 | 0 | 5 | 9 am-12 am |
| 37 | 12 | 63 | 0 | 4 | 10 am-3 pm |
| 38 | 21 | 19 | 0 | 4.2 | 10 am-3 pm |
| 39 | 63 | 55 | 0 | 4 | 10 am-3 pm |
| 40 | 14 | 70 | 0 | 3 | 10 am-3 pm |
| 41 | 20 | 30 | 0 | 7 | 10 am-3 pm |
| 42 | 80 | 53 | 0 | 4.9 | 10 am-3 pm |
(b) “-” means that these data do not need to be set in advance.
In order to demonstrate the performance of the proposed algorithm and research the effect of the duplicate individuals, test example 3 was also tested by T-NSGA-II, W-BEG-NSGA-II, and BEG-NSGA-II, respectively. Table
Comparisons of the optimization results of test example 3 with 42 customers.
| | | | | |
---|---|---|---|---|---|
| 0-16-13-27-34-35-36-11-18-7-21-40-37-0 | 1035.36 | 1006.49 | 1120.63 | 3 |
0-16-13-27-34-35-36-11-18-7-21-40-37-0 | 1089.13 | 943.67 | 1156.32 | ||
0-16-13-27-34-35-36-11-18-7-21-40-37-0 | 1037.93 | 1003.33 | 1119.99 | ||
0-16-13-27-34-35-36-11-18-7-21-40-37-0 | 1091.70 | 940.52 | 1155.68 | ||
0-16-13-27-34-35-36-11-18-7-21-40-37-0 | 1099.02 | 926.88 | 1162.02 | ||
0-16-13-34-35-36-11-18-7-21-27-40-37-0 | 1050.16 | 987.16 | 1116.56 | ||
0-16-13-34-35-36-11-18-7-21-27-40-37-0 | 1106.49 | 921.19 | 1151.61 | ||
0-27-40-13-37-16-34-36-11-18-7-21-35-0 | 983.25 | 1239.32 | 1065.36 | ||
| 0-26-39-11-22-25-2-27-16-28-17-6-41-10-31-19-32-0 | 1031.87 | 508.12 | 1185.27 | 2.6 |
0-26-39-11-22-25-2-27-16-28-17-6-41-10-31-19-32-0 | 1040.27 | 369.66 | 1191.60 | ||
0-26-39-11-22-25-2-27-16-28-17-6-41-10-32-19-31-0 | 1022.91 | 525.40 | 1168.28 | ||
0-26-39-11-22-25-2-27-16-28-17-6-41-10-32-19-31-0 | 1034.94 | 386.94 | 1176.93 | ||
0-26-39-11-22-25-2-27-16-28-17-6-41-10-32-19-31-0 | 1031.05 | 614.06 | 1157.60 | ||
0-26-39-11-22-25-2-27-16-28-17-6-41-10-32-19-31-0 | 1030.05 | 628.25 | 1161.22 | ||
| 0-6-29-21-25-15-2-33-10-14-27-13-24-5-37-16-28-17-22-39-0 | 983.70 | 463.29 | 992.51 | 2.1 |
| | | | ||
0-6-29-21-25-15-2-33-10-14-27-13-24-5-37-16-28-17-22-39-0 | 984.27 | 463.29 | 990.59 | ||
| | | | ||
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| | | | ||
| | | | ||
| | | | ||
| | | |
(i) The algorithms of T-NSGA-II, W-BEG-NSGA-II, and BEG-NSGA-II could find 8, 6, and 11 optimal Pareto solutions with average computing time of 3, 2.6, and 2.1 minutes correspondingly during 10 run times.
(ii) The algorithm of BEG-NSGA-II could find 9 Pareto solutions (bold and marked as “
From the information mentioned above, we can draw the following conclusions:
(i) The algorithms of T-NSGA-II and W-BEG-NSGA-II may generate amount of duplicate individuals, especially after 100 iterations. These duplicate individuals may cause the problems of inefficiency, slow convergence, and converging to local Pareto optimal solutions.
(ii) The proposed algorithm of BEG-NSGA-II is more effective than the algorithms of T-NSGA-II and W-BEG-NSGA-II to solve the VRPSPD problem with rigor constraints.
Figures
Solution 1 of test example 3 (fuel consumption=945.59kg, waiting time = 466.37min, and distance = 959.02km).
Solution 2 of test example 3 (fuel consumption=945.59kg, waiting time = 463.29 min, and distance = 992.51km).
The number of duplicate individuals in test example 3 with 42 customers.
Dethloff’s benchmark [ PILS: parallel iterated local search [ VLBR: variable length bone route [ HPSO: hybrid particle swarm optimization [ ACS: ant colony system [ ACSEVNS: ant colony system empowered variable neighborhood search algorithm [
Therefore, we compare the performance of the proposed BEG-NSGA-II against these algorithms reported in the literature. Table
Computational results for Dethloff’s benchmark.
Instance | Best so far know solution in | ACSEVNS | BEG-NSGA-II | ||||||
---|---|---|---|---|---|---|---|---|---|
the literature | |||||||||
Ref. | BKS | Best | Avg. | T | Best | Avg. | T | | |
SCA3-0 | PILS, VLBR, HPSO, ACS | | | 635.62 | 4.77 | 732.12 | 745.36 | 10.21 | 684.23 |
SCA3-1 | PILS, VLBR, HPSO, ACS | | | 697.84 | 5.24 | 821.13 | 838.39 | 11.32 | 761.96 |
SCA3-2 | PILS, VLBR, HPSO, ACS | | | 659.34 | 7.47 | | 659.34 | 12.65 | 612.62 |
SCA3-3 | PILS, VLBR, HPSO, ACS | | | 680.34 | 5.20 | 762.36 | 776.61 | 9.69 | 715.37 |
SCA3-4 | PILS, VLBR, HPSO, ACS | | | 690.50 | 4.96 | 765.58 | 770.67 | 13.52 | 726.27 |
SCA3-5 | PILS, VLBR, HPSO, ACS | | | 659.91 | 5.18 | 675.69 | 680.21 | 12.38 | 606.39 |
SCA3-6 | PILS, VLBR, HPSO, ACS | | | 651.11 | 4.68 | | 651.09 | 14.32 | 610.92 |
SCA3-7 | PILS, VLBR, HPSO, ACS | | | 659.17 | 6.06 | 763.47 | 773.6 | 12.21 | 723.26 |
SCA3-8 | PILS, VLBR, HPSO, ACS | | | 719.56 | 4.51 | 768.36 | 782.65 | 10.36 | 716.28 |
SCA3-9 | PILS, VLBR, HPSO, ACS | | | 681.00 | 7.08 | | 681.00 | 9.36 | 621.23 |
SCA8-0 | PILS, VLBR, HPSO, ACS | | | 961.50 | 5.33 | 1012.38 | 1026.54 | 15.63 | 973.62 |
SCA8-1 | PILS, VLBR, HPSO, ACS | | | 1049.65 | 5.62 | 1099.63 | 1129.78 | 16.22 | 1038.53 |
SCA8-2 | PILS, VLBR, HPSO, ACS | | | 1041.62 | 6.05 | 1126.53 | 1139.25 | 16.37 | 1079.92 |
SCA8-3 | PILS, VLBR, HPSO, ACS | | | 983.34 | 8.39 | 1016.29 | 1035.69 | 13.23 | 968.37 |
SCA8-4 | PILS, VLBR, HPSO, ACS | | | 1065.49 | 6.07 | | 1065.49 | 11.32 | 1016.27 |
SCA8-5 | PILS, VLBR, HPSO, ACS | | | 1027.14 | 6.96 | 1089.36 | 1096.75 | 10.63 | 1023.63 |
SCA8-6 | PILS, VLBR, HPSO, ACS | | | 971.82 | 7.76 | 985.78 | 996.34 | 9.89 | 928.56 |
SCA8-7 | PILS, VLBR, HPSO, ACS | | | 1051.28 | 8.14 | 1156.36 | 1173.95 | 12.35 | 1103.93 |
SCA8-8 | PILS, VLBR, HPSO, ACS | | | 1071.22 | 7.06 | 1210.35 | 1242.25 | 18.69 | 1129.46 |
SCA8-9 | PILS, VLBR, HPSO, ACS | | | 1060.50 | 5.29 | 1126.52 | 1153.34 | 19.99 | 1086.95 |
CON3-0 | PILS, VLBR, HPSO, ACS | | | 616.52 | 6.80 | 712.45 | 732.32 | 15.36 | 672.97 |
CON3-1 | PILS, VLBR, HPSO, ACS | | | 554.47 | 5.01 | 589.76 | 593.67 | 11.25 | 521.36 |
CON3-2 | PILS, VLBR, HPSO, ACS | | | 518.00 | 7.55 | 562.98 | 583.53 | 16.87 | 512.39 |
CON3-3 | PILS, VLBR, HPSO, ACS | | | 591.19 | 5.75 | | 591.19 | 13.63 | 538.13 |
CON3-4 | PILS, VLBR, HPSO, ACS | | | 588.79 | 3.90 | 623.82 | 643.78 | 9.36 | 579.24 |
CON3-5 | PILS, VLBR, HPSO, ACS | | | 563.70 | 6.86 | | 563.70 | 12.36 | 513.21 |
CON3-6 | PILS, VLBR, HPSO, ACS | | | 499.05 | 8.54 | | 499.05 | 12.21 | 432.63 |
CON3-7 | PILS, VLBR, HPSO, ACS | | | 576.48 | 4.26 | 596.35 | 601.28 | 10.86 | 536.52 |
CON3-8 | PILS, VLBR, HPSO, ACS | | | 523.05 | 3.89 | 556.31 | 571.36 | 10.35 | 506.37 |
CON3-9 | PILS, VLBR, HPSO, ACS | | | 578.25 | 6.33 | 623.67 | 635.73 | 13.62 | 576.23 |
CON8-0 | PILS, VLBR, HPSO, ACS | | | 857.17 | 5.40 | 926.46 | 942.39 | 15.36 | 876.56 |
CON8-1 | PILS, VLBR, HPSO, ACS | | | 740.85 | 8.46 | | 740.85 | 15.78 | 695.35 |
CON8-2 | PILS, VLBR, HPSO, ACS | | | 712.89 | 4.79 | 763.56 | 781.22 | 10.36 | 713.69 |
CON8-3 | PILS, VLBR, HPSO, ACS | | | 811.07 | 7.21 | 862.35 | 873.68 | 14.36 | 812.38 |
CON8-4 | PILS, VLBR, HPSO, ACS | | | 772.25 | 6.70 | 812.65 | 832.47 | 15.69 | 756.24 |
CON8-5 | PILS, VLBR, HPSO, ACS | | | 754.88 | 5.74 | | 754.88 | 12.63 | 703.63 |
CON8-6 | PILS, VLBR, HPSO, ACS | | | 678.92 | 4.36 | 723.52 | 736.61 | 12.53 | 676.29 |
CON8-7 | PILS, VLBR, HPSO, ACS | | | 813.46 | 8.38 | 866.76 | 873.69 | 20.32 | 813.53 |
CON8-8 | PILS, VLBR, HPSO, ACS | | | 767.53 | 6.16 | 813.65 | 826.63 | 12.36 | 763.96 |
CON8-9 | PILS, VLBR, HPSO, ACS | | | 809.00 | 7.19 | 901.23 | 923.57 | 14.23 | 863.24 |
G. Avg. | 6.13 | 13.25 | |||||||
BKS found | 40 | 40 | 9 |
(i) The algorithm of BEG-NSGA-II could find 9 BKSs in 40 instance.
(ii) The objective of vehicle distance and computation time obtained from algorithm of BEG-NSGA-II is little worse than the ones obtained from ACSEVNS.
(iii) In addition to the objective of vehicle distance and computation time, the optimization objective of fuel consumption that is not considered in the compared algorithm is also obtained in the proposed algorithm of BEG-NSGA-II.
From the information mentioned above, we can draw the following conclusions:
(i) The proposed algorithm of BEG-NSGA-II can obtain the BKS or near BKS in reasonable computation time considering the objectives of vehicle distance and fuel consumption simultaneously.
(ii) The reason why some results gained from proposed BEG-NSGA-II are little worse than the compared algorithms is that we consider the optimization objectives of vehicle distance and computation simultaneously, while the compared ones only consider the single objective of vehicle distance.
(iii) From the simulations of examples 1-4, we can see that our proposed algorithm of BEG-NSGA-II has high performance in multiobjective optimization as well as meeting the single objective optimization.
In this paper, we construct a new closed-loop supply chain logistics network of VRPSPD dominated by remanufacturer, in which the customers are originally divided into three types: distributors, recyclers, and suppliers. The proposed model is more consistent with the actual situation of recycling and remanufacturing. For the purpose of resource saving and environment protection, we originally add the fuel consumption as an optimization objective into VRPSPD. In order to solve the proposed model and conquer the disadvantages of the traditional NSGA-II and the existing algorithms with two-stage optimization mechanism, a bee evolutionary algorithm guiding nondominated sorting genetic algorithm II (BEG-NSGA-II) is proposed with the optimization objectives of minimum fuel consumption, minimum waiting time, and the shortest delivery distance while meeting the travel constraints and load constraints at the same time. To verify the effectiveness of the proposed algorithm, test examples are used to simulate the proposed VRPSPD and the traditional VRPSPD, respectively. From the simulation results of test examples 1-3, we can see that our proposed algorithm BEG-NSGA-II is superior to the traditional NSGA-II (T-NSGA-II) and the proposed BEG-NSGA-II without eliminating the duplicate individuals (W-BEG-NSGA-II). From the simulation results of test example 4, we can see that the proposed algorithm of BEG-NSGA-II can obtain the best known solution (BKS) or near BKS in reasonable computation time considering the objectives of vehicle distance and fuel consumption simultaneously. That is to say, our proposed algorithm of BEG-NSGA-II works better in multiobjective optimization as well as meeting the single objective optimization. It would be prosperous to apply our algorithm to other multiobjective optimization problems, especially in the logistics distribution field.
The authors confirm that this work does not have any conflicts of interest with other work.
This work was supported by the National Natural Science Foundation of China (Grant no. 71473077), the National Science-Technology Support Plan Projects of China (2015BAF01B00), the National High Technology Research and Development Program of China (863 Program) (Grant no. 2013AA040206), and Project of State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University (Grant no. 71775004).