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 selfadaptive grid multiobjective 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 duetime 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 realworld 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 selfadaptive 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 fullsearching 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 duetime 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 duetime 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 duetime. 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 nondominated solutions set, and make it close to the Pareto optimal front as much as possible?
how to attain the diversity and variety of solutions?
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 nondominated 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
Quantum evolutionary algorithm (QEA) [
The smallest unit of information stored in twostate quantum computer is called a Qbit, which may be in the “1” state, or in the “0” state, or in any superposition of the two. The state of a Qbit can be represented as follows
So a Qbit individual with a string of m Qbits can be expressed as follows
The main advantage of the representation is that any linear superposition of solutions can be represented. For example, a threeQbit system can contain the information of eight states. QEA with Qbit 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 Qbit 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 Qbit representation cannot be evaluated directly. So it should be converted to permutation for evaluation.
The Qbit 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 Qgate is an evolution operator which is the same as the QEA in [
The lookup table of
Lookup table of rotation angle.












0  0  False  0  0  0  0  0 
0  0  True 

+1  −1  0 

0  1  False  0  0  0  0  0 
0  1  True 

−1  +1 

0 
1  0  False 

−1  +1 

0 
1  0  True 

−1  +1  0 

1  1  False  0  0  0  0  0 
1  1  True 

−1  +1  0 

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 Qbit
Use (
Let
There are few studies on multiVRP 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 nondominated solution set and a method for attaining keeping the variety of the solution set, is designed, based on selfadaptive 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.