The multiobjective vehicle routing problem considering customer satisfaction (MVRPCS) involves the distribution of orders from several depots to a set of customers over a time window. This paper presents a self-adaptive grid multi-objective quantum evolutionary algorithm (MOQEA) for the MVRPCS, which takes into account customer satisfaction as well as travel costs. The degree of customer satisfaction is represented by proposing an improved fuzzy due-time window, and the optimization problem is modeled as a mixed integer linear program. In the MOQEA, nondominated solution set is constructed by the Challenge Cup rules. Moreover, an adaptive grid is designed to achieve the diversity of solution sets; that is, the number of grids in each generation is not fixed but is automatically adjusted based on the distribution of the current generation of nondominated solution set. In the study, the MOQEA is evaluated by applying it to classical benchmark problems. Results of numerical simulation and comparison show that the established model is valid and the MOQEA is effective for MVRPCS.
The vehicle routing problem (VRP) is one of the most important and widely studied combinatorial optimization problems, with many real-world applications in logistic distribution and transportation [
The aim of VRP is to find optimal routes for a fleet of vehicles serving a set of customers with known demands. Each customer is serviced exactly once and must be assigned a satisfactory vehicle without exceeding vehicle capacities. A solution for this problem is to find out a set of minimum cost routes that are used to represent vehicles distribution and clients’ permutation. However, current studies on VRP [
Actually, to achieve competitive advantage, a service provider needs to consider not only service costs but also service quality that can determine customers’ satisfaction. Most of the research on multiobjective VRP (MOVRP) does not take into account this objective, only focusing on the traditional objectives of minimum costs and the length of the longest route [
The VRPTW is developed from VRP and has been widely studied in the last decade [
In practice, this time window actually does not well describe customers’ satisfaction. A major reason is that customers are asked to provide a fixed time window for service, but in reality they really hope to be served at a desired time. Cheng and Gen [
Cheng and Gen [
The above studies use the weighted sums of objectives to solve the multiobjective problem; the higher an objective’s importance, the larger its corresponding weight coefficient. In general, no single solution can attain the optimum of all objectives at the same time. Therefore, it is desirable to obtain a set of Pareto optimal solution, that is, the Pareto set. The points in the objective space that correspond to the results in the set are usually called the Pareto front.
In this paper, a self-adaptive grid multiobjective quantum evolutionary algorithm (MOQEA) is proposed to solve the MVRPCS problem. In particular, the quantum evolutionary algorithm (QEA) is used in the MOQEA due to its high efficiency, convergence speed, strong full-searching optimization ability [
The remainder of the paper is organized as follows. Section
In traditional VRP, customers’ time constraints are represented by time windows as shown in Figure
Traditional time windows.
Fuzzy due-time windows have been introduced to describe different degrees of satisfaction. Generally, the tolerable service time for customer
Triangular fuzzy number.
Trapezoidal fuzzy number.
In this paper, an improved fuzzy due-time window is proposed, as shown in Figure
Improved trapezoidal fuzzy number.
The MVRPCS can be described as follows: there are
Mark the demand of customer
Customer number is
Fixed cost of sending a vehicle is
Distribution cost from customer
Time window of customer
If vehicle
The service quality objective is to maximize average customers satisfaction:
This objective is equivalent to minimizing the average customer dissatisfaction:
The other objective of the service costs is to minimize travel cost, fixed cost and waiting cost. For this objective, the fixed cost of sending a vehicle is considered because vehicles in operation have depreciation and fuel consumption. Also the fixed cost is related to the number of vehicle, that is, the more vehicles, the higher fixed cost. To the best of our knowledge, no previous work has been done to take into account this fixed cost when solving multiobjective VRP with fuzzy due-time. Based on the above discussion, the mathematical model for the MVRPCS can be established as follows:
In this research, the multiobjective optimization method of Pareto optimal solution [ how to construct a Pareto optimal solution set, namely non-dominated solutions set, and make it close to the Pareto optimal front as much as possible? how to attain the diversity and variety of solutions?
To address these two issues, a self-adaptive grid multiobjective quantum evolutionary algorithm (MOQEA) to solve the MVRPCS problem is proposed. The method of constructing non-dominated solution set and attaining the diversity and variety of solutions is described in the following sections.
In this paper, the Challenge Cup rule [
To attain the variety of the set, the individual space is divided into several small areas each of which is a called a grid, as shown in Figure
Individual space divided by grid.
A grid is used in many different ways to maintain the diversity and variety of solutions. Knowles and Crone [
When a grid contains more than one individual, these individuals are treated as the same solution. As such, the size of grid is very important. When the grid is too large, multiple individuals will exist in the same grid, and the resultant solution distribution is not accurate. When the grid is too small, it is likely that there are no individuals in some grids, and so it takes longer computation time though the resultant solution distribution is accurate. Therefore, computation time and accuracy must be traded off when determining the grid size.
There are two objective functions in this optimization problem. The range of the customers’ satisfaction is
In this paper, the number of grids is not fixed in each generation but automatically adjusted based on the distribution of the current generation of non-dominated solution set. The grid boundary is a fixed value. In the process of each evolutionary generation, the number of grid is adjusted by the
The two objections can be described by generation 1: generation The number of each dimension grid in generation
To keep the diversity and variety of the non-dominated solution set, choose the individual with the maximum extrusion coefficient and delete it form the non-dominated solution set.
Quantum evolutionary algorithm (QEA) [
The smallest unit of information stored in two-state quantum computer is called a Q-bit, which may be in the “1” state, or in the “0” state, or in any superposition of the two. The state of a Q-bit can be represented as follows
So a Q-bit individual with a string of m Q-bits can be expressed as follows
The main advantage of the representation is that any linear superposition of solutions can be represented. For example, a three-Q-bit system can contain the information of eight states. QEA with Q-bit representation has a better characteristic of population diversity than other representations, as it can represent linear superposition of state’s probabilities.
In this paper, a method of converting integer representation to Q-bit representation is designed. For the MVRPCS with
The “Customers permutation Route First, Vehicles distribution Cluster Second” rule is adopted for decoding. Firstly, get the customers permutation route. The solution of MVRPCS is a permutation of all customers and Q-bit representation cannot be evaluated directly. So it should be converted to permutation for evaluation. The Q-bit string is firstly converted to binary string Secondly, distribute the vehicles and get the subroute. A vehicle is dispatched to service customers according to the customers’ permutation route, if the vehicle cannot serve the next customer when it cannot meet the time window or loading capacity constraints, a new vehicle will be dispatched. For example, the customers’ permutation route is [
In the MOQEA, a Q-gate is an evolution operator which is the same as the QEA in [
The lookup table of
Lookup table of rotation angle.
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0 | 0 | False | 0 | 0 | 0 | 0 | 0 |
0 | 0 | True |
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+1 | −1 | 0 |
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0 | 1 | False | 0 | 0 | 0 | 0 | 0 |
0 | 1 | True |
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−1 | +1 |
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0 |
1 | 0 | False |
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−1 | +1 |
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0 |
1 | 0 | True |
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−1 | +1 | 0 |
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1 | 1 | False | 0 | 0 | 0 | 0 | 0 |
1 | 1 | True |
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−1 | +1 | 0 |
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In the above table,
The flow chart of the MOQEA for this problem is illustrated in Figure
Flow chart of MOQEA.
The detailed procedure of the MOQEA is as follows.
Let
Convert
According to the decoding method to get the subroute, evaluate the objectives to get the MVRPCS solution set
Use the formulas (
When
Adjust the size of
If the stopping condition is satisfied, then output the Pareto set; otherwise, go on to the following steps.
Randomly select some individuals from the Q-bit
Use (
Let
There are few studies on multi-VRP taking into account customers’ satisfaction. Among the studies that have taken into account customer satisfaction, most of them are evaluated using randomly generated test cases. Therefore, there is no standard test cases library. The tests data used in this research is from the benchmark problems in the standard example library of MDVRPTW (multiple depot vehicle routing problem with time windows), and all examples can be downloaded from
The parameters involved in the MOQEA include coefficient
Optimization results of coefficient
Problem | pr02 (96, 4) | pr07 (72, 6) | ||||
---|---|---|---|---|---|---|
|
VN | CS | Cost | VN | CS | Cost |
0.08 | 16 | 0.402 | 4539 | 15 | 0.492 | 4116 |
0.06 | 15 | 0.378 | 4045 | 14 | 0.424 | 3904 |
0.05 | 14 | 0.355 | 3906 | 13 | 0.399 | 3331 |
0.02 | 15 | 0.364 | 3998 | 14 | 0.431 | 3969 |
Optimization results of constant
Problem | pr02 (96, 4) | pr07 (72, 6) | ||||
---|---|---|---|---|---|---|
|
VN | CS | Cost | VN | CS | Cost |
0.1 | 15 | 0.381 | 4013 | 14 | 0.435 | 3391 |
0.2 | 14 | 0.357 | 3902 | 13 | 0.389 | 3367 |
0.3 | 15 | 0.361 | 3926 | 15 | 0.486 | 4124 |
0.4 | 16 | 0.411 | 4625 | 15 | 0.498 | 4102 |
All the programs in this research are developed using the JAVA language and run on a PC with Dual 2.8 GHz CPU and 1.0 GB of memory. A manufacturing company has 4 warehouses and provides goods to 48 vendors. The actual distribution process can be attributed to the open, capacity constraints, and multidepot VRP. The capacity is 20 t. The proposed MOQEA is used to solve this problem, and the distance and demand of each client and depot are shown in Tables
The distance and demand of each client.
CN | Coordinate/km | ST | De/ |
TW | |
---|---|---|---|---|---|
1 | −29.730 | 64.136 | 2 | 12 | [ |
2 | −30.664 | 5.463 | 7 | 8 | [ |
3 | 51.642 | 5.469 | 21 | 16 | [ |
4 | −13.171 | 69.336 | 24 | 5 | [ |
5 | −67.413 | 68.323 | 1 | 12 | [ |
6 | 48.907 | 6.274 | 17 | 5 | [ |
7 | 5.243 | 22.260 | 6 | 13 | [ |
8 | −65.002 | 77.234 | 5 | 20 | [ |
9 | −4.175 | −1.569 | 7 | 13 | [ |
10 | 23.029 | 11.639 | 1 | 18 | [ |
11 | 25.482 | 6.287 | 4 | 7 | [ |
12 | −42.615 | −26.392 | 10 | 6 | [271 420] |
13 | −76.672 | 99.341 | 2 | 9 | [108 266] |
14 | −20.673 | 57.892 | 16 | 9 | [340 462] |
15 | −52.039 | 6.567 | 23 | 4 | [226 377] |
16 | −41.376 | 50.824 | 18 | 25 | [446 604] |
17 | −91.943 | 27.588 | 3 | 5 | [444 566] |
18 | −65.118 | 30.212 | 15 | 17 | [434 557] |
19 | 18.597 | 96.716 | 13 | 3 | [319 460] |
20 | −40.942 | 83.209 | 10 | 16 | [192 312] |
21 | −37.756 | −33.325 | 4 | 25 | [414 572] |
22 | 23.767 | 29.083 | 23 | 21 | [371 462] |
23 | −43.030 | 20.453 | 20 | 14 | [378 472] |
24 | −35.297 | −24.896 | 10 | 19 | [308 477] |
25 | −54.755 | 14.368 | 4 | 14 | [329 444] |
26 | −49.329 | 33.374 | 2 | 6 | [269 377] |
27 | 57.404 | 23.822 | 23 | 16 | [398 494] |
28 | −22.754 | 55.408 | 6 | 9 | [257 416] |
29 | −56.622 | 73.340 | 8 | 20 | [198 294] |
30 | −38.562 | −3.705 | 10 | 13 | [375 467] |
31 | −16.779 | 19.537 | 7 | 10 | [200 338] |
32 | −11.560 | 11.615 | 1 | 16 | [456 632] |
33 | −46.545 | 97.974 | 21 | 19 | [72 179] |
34 | 16.229 | 9.320 | 6 | 22 | [182 282] |
35 | 1.294 | 7.349 | 4 | 14 | [159 306] |
36 | −26.404 | 29.529 | 13 | 10 | [321 500] |
37 | 4.352 | 14.685 | 9 | 11 | [322 430] |
38 | −50.665 | −23.126 | 22 | 15 | [443 564] |
39 | −22.833 | −9.814 | 22 | 13 | [207 348] |
40 | −71.100 | −18.616 | 18 | 15 | [457 588] |
41 | −7.849 | 32.074 | 10 | 8 | [203 382] |
42 | 11.877 | −24.933 | 25 | 22 | [75 167] |
43 | −18.927 | −23.730 | 23 | 24 | [459 598] |
44 | −11.920 | 11.755 | 4 | 3 | [174 332] |
45 | 29.840 | 11.633 | 9 | 25 | [130 225] |
46 | 12.268 | −55.811 | 17 | 19 | [169 283] |
47 | −37.933 | −21.613 | 10 | 21 | [115 232] |
48 | 42.883 | −2.966 | 17 | 10 | [414 531] |
The distance of each depot.
Depot no. | Coordinate/km | Service time | Demand/ |
Time windows | |
---|---|---|---|---|---|
49 | 4.163 | 13.559 | 0 | 0 | [0 1000] |
50 | 21.387 | 17.105 | 0 | 0 | [0 1000] |
51 | −36.118 | 49.097 | 0 | 0 | [0 1000] |
52 | −31.201 | 0.235 | 0 | 0 | [0 1000] |
The results obtained are shown in Table
Optimization results of pr01.
Pareto set | The solution of maximum satisfaction | The solution of minimizing travel cost, fixed cost, and waiting cost |
---|---|---|
(472, 4532) | (630, 3523) | |
VN | 9 | 11 |
| ||
Route | 52 21 36 31 27 6 24 | 52 43 21 14 13 |
51 28 13 44 25 | 52 30 45 44 38 | |
51 5 39 17 | 50 6 40 48 47 17 | |
49 32 19 20 18 42 | 49 7 37 11 32 26 16 36 42 | |
51 4 11 41 26 1 29 22 | 52 12 41 15 2 4 | |
51 14 12 43 16 33 | 51 33 18 3 | |
50 3 45 8 40 10 37 | 51 1 35 20 39 | |
52 23 34 46 38 30 15 47 | 51 8 29 9 22 | |
49 7 9 35 48 2 | 51 5 34 24 23 | |
51 19 27 28 10 31 25 | ||
49 46 |
Pareto optimal solution set.
In order to evaluate the performance of the algorithm, the proposed MOQEA is compared with the hybrid multiobjective evolutionary algorithm (HMOEA) developed in [
Table
Comparisons of the MOQEA to the HMOEA in [
Algorithm | Project 1 | Project 2 | ||||
---|---|---|---|---|---|---|
MOQEA | HMOEA in [ | |||||
Problems | VN | CS | Cost | VN | CS | Cost |
pr02 (96, 4) | 14 | 0.355 | 3906 | 15 | 0.391 | 4005 |
pr07 (72, 6) | 13 | 0.399 | 3331 | 13 | 0.402 | 3397 |
This paper presents the modeling of vehicle scheduling problem that takes into account customer satisfaction and the development of the MVRPCS. Specifically, an improved trapezoidal fuzzy number is proposed to represent customers’ satisfaction and the MOQEA for this problem is developed. The MOQEA can get multiple solutions, namely, the Pareto optimal solution set, according to his own expectations. These solutions will be used by the decision maker to choose the best one on the basis of different preferences on satisfaction maximization and travel costs minimization. In the MOQEA, the Challenge Cup rule is constructed for non-dominated solution set and a method for attaining keeping the variety of the solution set, is designed, based on self-adaptive grid. Simulation results and comparisons show that the MOQEA is an effective method. In our future work, we will focus on improving the algorithm and test it on other datasets.
This paper is supported by the National Natural Science Foundation of China (Grant no. 60970021), the Postdoctoral Science Foundation of Zhejiang Province, and the Department of Education Foundation of Zhejiang Province (No. Y201225032). The authors are also most grateful for the constructive suggestions from anonymous reviewers which led to the making of several corrections and suggestions that have greatly aided in the presentation of this paper.