This study proposes a genetic algorithm to solve the biobjective vehicle routing problem with time windows simultaneously considering total distance and distance balance of active vehicle fleet. A new complex chromosome is used to present the active vehicle route. Through tournament selection, onepoint crossover, and migrating mutation operator, the solution of the problem is solved. In experiment on Solomon’s benchmark problems, considering the total distance and distance balance, the results are improved in all classes of problems. According to the experimental results, the suggested approach is sufficient and the average GA performance is good.
The vehicle routing problem (VRP) is one of the most attractive topics in operation research, logistics, and supply chain management. VRP deals with minimizing the total cost of logistics systems. VRPs are wellknown combinatorial optimization problems arising in transportation logistic that usually involve scheduling in constrained environments. In transportation management, there is a requirement to provide services from a supply point (depot) to various geographically dispersed points (customers) with significant economic implications. Because of VRP’s important applications, many researchers have developed solution approaches for those problems.
Vehicle routing problem with time windows (VRPTW) is a variant of VRP with adding time windows constraints to that model. In VRPTW, a set of vehicles with limited capacity is to be routed from a central depot to a set of geographically dispersed customers with known demands and predefined time windows in order that fleet size of vehicles and total traveling distance are minimized and capacity and time windows constraints are satisfied. Due to its inherent complexities and usefulness in real life, the VRPTW continues to draw the attention of researchers and has become a wellknown problem in network optimization, so many authors developed different solution approaches based on exact and heuristics methods.
Many exact optimization approaches have been used to solve the VRPTW which is a wellknown NPhard problem [
In recent years, approximate approaches are used in VRPTW instead of exact methods considering latter’s intolerably high cost. Various heuristic methods may be found in literature in [
These above pieces of literature focus on the single objective problems of the VRPTW by far. In fact multiobjective problems attract many researchers’ attention since the multiobjective is closer to real environments in these years. Some multiobject VRPs are formulated as a single function using weight parameters determined only experientially. Paretobased approach is good to solve such problem since the managers can make their own decisions from the Paretooptimal output [
Different objectives were classified in [
This paper studies a biobjective VRPTW considering simultaneous minimization of the total traveling distance and workload imbalance of vehicles. Generally, the workload imbalance includes the distance imbalance and the load imbalance. However, in some real life environment, that is, fresh food delivery, the weight of good can be ignored because these orders are not heavy and make no influence on the workload cost. In other words, this paper will consider the multiobjective of the total travelling distance and the distance imbalance of active vehicle fleet. Section
The VRP problem was introduced by [
A nondirected complete graph
The VRPTW can be formulated as follows.
is the earliest time for customer
is the latest time for customer to allow the service.
is the cost for travelling from node
is the demand at customer
is the maximum number of vehicles that can be used.
is the number of customers plus the depot. The depot is denoted with number 1, and the customers are denoted as
is the loading capacity of each vehicle.
is the corresponding time at which vehicle
is a given large value.
is the decision variable. It is equal to 1 if vehicle
subject to
Equation (
The paper aims to solve the vehicle routing problem with hard time windows and route balance as a multiobject problem, where both the total travelling distance and route imbalance are minimized. The route balance often was related to the following factors:
balancing the number of customers visited by each active vehicle,
balancing the quantity or weight of the good delivered by each active vehicle, sometimes balancing the load rate (BLR), denoted as (
balancing the time required of the route,
balancing the waiting time required of the route,
balancing the delayed time of the route,
balancing the distance of the route travelled by active vehicles.
In this paper, we consider the imbalance of the distances of the route travelled, which is defined as (
Thus, from (
Minimize
Various heuristic and metaheuristic approaches have been proposed for solving the VRPTW. GA, compared with other heuristics [
This paper uses a complex twopart chromosome to represent the solution of VRPTW. The chromosome is separated into two parts by a zero number decorated by yellow. The first part of the chromosome is a chain of integers and each of the integers represents a customer. We also call this part customerpart. The customers on it are separated to several routes, each of them representing a sequence of delivers that must be covered by a vehicle. The second part of the chromosome contains vehicles information. We also can call this part vehiclepart. In the vehiclepart the quantity of genes equals the quantity of routes in the customerpart. The number on each of the genes represents the length of its corresponding route. The sum of these numbers in vehiclepart must be equal to the quantity of customers. For example, Figure
An example of a chromosome.
This design is different from the classical approach, in which the route information is mixed with the customer sequences in a single chromosome. Storing the route information and customer sequences separately can represent the solution more clearly and facilitate the implementation of the algorithm, but its effect on the computational efficiency would not be significant. Without loss of generality, we consider the following implementations of the three operators.
There are several commonly used selection operators used in GA selection process. Roulette wheel selection (RWS) is to stochastically select from one generation to create the basis of the next generation. RWS enables the fittest individuals to have a greater chance of survival than weaker ones. This replicates nature in that fitter individuals will tend to have a better probability of survival and will go forward to form the mating pool for the next generation. Weaker individuals are not without a chance. In nature such individuals may have genetic coding that may be proven to be useful for future generations. Unlike RWS, uniform selection (US) assigns the same probability to each chromosome of the population. The US operation proceeds at random and is easy to implement. However, it has been criticized for lacking the spirit of natural evolution compared with RWS. Tournament selection (TS) is the most commonly used operation besides RWS. The TS operator involves running several “tournaments” among a set of chromosomes chosen at random. The one with the largest fitness is selected for crossover in a pair of chromosomes. The tournament size can be used to adjust the selection pressure. If the tournament size is larger, weaker individuals will have a smaller chance to be selected. This process is repeated until the mating pool is full. Since some experiments indicate that the TS operator outperforms the RWS and US, we choose TS as the selection operator. A possible explanation is that TW always selects the best set of individuals to crossover, whereas RWS and TS are probabilistic and hence some good individuals may be lost in the evolutionary process [
Onepoint crossover operator evolves selecting one point randomly which divides a parent into two parts. Each of these points is selected with equal probability. For example, the crossover point is selected at the third gene of parent 1 randomly. The offspring inherits the left side from parent 1 and other genes are inherited from parent 2. Another offspring is produced by exchanging the roles of two parents. Figure
Onepoint crossover operator.
In mutation process, there are also several mutation operators in different literatures. Some of them are very complex. However, some different mutation operators were experimented that they did not make significant difference in GA efficiency. A possible explanation to that maybe the mutation rate is always very small, typically between 0.01 and 0.1. A migrating mutation is adopted to produce heterogeneous chromosomes in the pool to avoid early convergence of the algorithm. This mutation method is to randomly select a chromosome from the pool and then randomly choose a customer from one route. Then the selected customer is tried to be inserted into a new route. If the insertion results produce a feasible route, this mutation operator succeeds. Otherwise, this process is repeated until a feasible solution is achieved. Figure
Migrating mutation process.
The Solomon’s problems consist of 56 data sets, which have been extensively used for benchmarking different algorithms for VRPTW in literature over the years, since they represent relatively well different kinds of routing scenarios [
This section describes computational experiments carried out to investigate the efficiency of the proposed GA. The algorithm is coded in JAVA and run on a PC with 2.53 GHz CPU and 2000 MB memory. The standard Solomon’s VRPTW benchmark problem instance is used as experimental data [
Population size = 100,
Generation number = 500,
Crossover rate = 0.9,
Mutation rate = 0.2.
To illustrate the influence of different models with routing balance, we have employed the best solution of the benchmark problems in Table
in the first column, the benchmark problem instance according to [
in the second column, the best known solution for that problem in literatures,
in the third column, the best solution found by the algorithm of this paper,
in the fourth column, the difference by percent between the best known and the best found,
in the third column from last, the balance rate of single objective search,
the second column from last and the final column, the new distance value and the balance rate of biobjective algorithm of this paper.
Results of C instances.
No.  Instance  Best known  Suggestive algorithm  

Single objective  Biobjective  
Total distance  Distance  Difference (%)  Balance  Distance  Balance  
1  C101  828.94  828.94  0.00  42.2%  839  10.3% 
2  C102  828.94  828.94  0.00  42.2%  838  4.2% 
3  C103  828.06  828.06  0.00  42.3%  1116  1.8% 
4  C104  824.78  824.78  0.00  41.2%  1253  2.9% 
5  C105  828.94  828.95  0.01  42.2%  969  4.2% 
6  C106  828.94  828.95  0.01  42.2%  1098  6.0% 
7  C107  828.94  828.95  0.01  42.2%  842  3.9% 
8  C108  828.94  828.95  0.01  42.2%  1169  5.8% 
9  C109  828.94  828.95  0.01  42.2%  1198  8.9% 
10  C201  591.56  591.58  0.01  11.2%  565  2.7% 
11  C202  591.56  591.58  0.01  11.2%  792  0.8% 
12  C203  591.17  591.18  0.01  10.7%  716  0.8% 
13  C204  590.60  590.62  0.01  11.2%  759  2.4% 
14  C205  588.16  588.18  0.01  10.7%  784  1.7% 
15  C206  588.49  588.51  0.01  10.7%  815  2.2% 
16  C207  588.29  588.30  0.01  10.7%  693  3.9% 
17  C208  588.32  588.32  0.00  10.7%  839  4.5% 
Results of R instances.
No.  Instance  Best known  Suggestive algorithm  

Single objective  Biobjective  
Total distance  Distance  Difference (%)  Balance  Distance  Balance  
1  R101  1645.79  1650.8  0.30  64.6%  1673  8.1% 
2  R102  1486.12  1486.12  0.00  62.9%  1746  4.5% 
3  R103  1292.68  1292.68  0.00  49.3%  1471  5.1% 
4  R104  1007.24  1007.24  0.00  4.5%  1141  3.2% 
5  R105  1377.11  1377.11  0.00  50.8%  1332  4.7% 
6  R106  1251.98  1251.98  0.00  51.8%  1505  4.5% 
7  R107  1104.66  1104.66  0.00  4.5%  1303  1.5% 
8  R108  960.88  960.88  0.00  0.9%  1164  0.5% 
9  R109  1194.73  1194.73  0.00  30.4%  1518  3.9% 
10  R110  1118.59  1118.59  0.00  5.4%  1090  3.7% 
11  R111  1096.72  1096.72  0.00  7.3%  1335  1.7% 
12  R112  982.14  987.24  0.52  3.7%  992  2.9% 
13  R201  1252.37  1252.37  0.00  22.0%  1282  2.5% 
14  R202  1191.70  1191.70  0.00  14.9%  1146  2.4% 
15  R203  939.54  939.54  0.00  29.1%  1041  0.8% 
16  R204  825.52  832.14  0.80  0.7%  847  3.2% 
17  R205  994.42  994.42  0.00  4.2%  1138  1.1% 
18  R206  906.14  906.14  0.00  7.6%  1054  4.7% 
19  R207  890.61  896.88  0.70  0.4%  895  3.4% 
20  R208  726.75  726.75  0.00  8.0%  891  0.1% 
21  R209  909.16  909.16  0.00  20.8%  1171  5.6% 
22  R210  939.34  939.37  0.00  7.7%  1046  2.4% 
23  R211  892.71  904.78  1.33  0.7%  925  9.0% 
Results of RC instances.
No.  Instance  Best known  Suggestive algorithm  

Single objective  Biobjective  
Total distance  Distance  Difference (%)  Balance  Distance  Balance  
1  RC101  1696.94  1696.95  0.00  51.2%  1963  4.5% 
2  RC102  1554.75  1554.75  0.00  21.6%  1894  2.6% 
3  RC103  1261.67  1261.67  0.00  28.8%  1669  5.3% 
4  RC104  1135.48  1135.48  0.00  21.1%  1447  1.0% 
5  RC105  1629.44  1629.44  0.00  30.3%  1956  3.5% 
6  RC106  1424.73  1424.73  0.00  17.8%  1858  1.7% 
7  RC107  1230.48  1230.48  0.00  22.3%  1764  2.9% 
8  RC108  1139.82  1139.82  0.00  11.4%  1545  1.6% 
9  RC201  1406.91  1406.94  0.00  16.8%  1646  4.3% 
10  RC202  1365.65  1365.65  0.00  8.8%  1439  1.9% 
11  RC203  1049.62  1058.33  0.82  10.0%  1243  0.4% 
12  RC204  789.41  798.46  1.13  16.3%  936  9.6% 
13  RC205  1297.19  1297.65  0.04  14.8%  1485  3.4% 
14  RC206  1146.32  1146.32  0.00  17.3%  1187  5.1% 
15  RC207  1061.14  1061.14  0.00  15.0%  1327  3.9% 
16  RC208  828.14  828.71  0.07  25.0%  1046  8.1% 
From the results of Tables
After considering the distance balance, the biobjective solution data illustrated much improvement on the balance rate without much influence on the distance cost. For example, in C101 when the balance decreases from 42.2% to 10.3%, the distance only reduces by about 10 (839 minus 828.94). Not only the instances with wide time windows get a great improvement, but also the ones with the narrow time windows reduce the balance rates.
The comparative data shows that suggestive algorithm of this paper reaches better route balance without significant deterioration of the VRPTW solution, in terms of the active vehicle fleet. From Table
Compared results of all instances.
No.  Instance class  Best known  Suggestive approach  Improved  







 
1  C1  42.3%  41.2%  42.1%  1.8%  8.9%  5.3%  87.4% 
2  C2  11.2%  10.7%  10.9%  0.8%  5.5%  2.4%  78.3% 
3  R1  64.6%  0.9%  28.0%  0.5%  8.1%  3.7%  86.8% 
4  R2  29.1%  0.4%  10.6%  0.1%  9.1%  3.2%  69.7% 
5  RC1  51.2%  11.4%  25.6%  1.0%  5.3%  2.9%  88.7% 
6  RC2  25.0%  8.0%  15.5%  0.42%  8.08%  4.6%  73.9% 
The problem discussed in this paper is of significant practical importance in cases where employee’s labor balance is key motivation of vehicle routing. In some situation, the weight is not very important compared with the distances of the active vehicle fleet when considering labor balance.
This paper proposed a genetic algorithm to solve the biobjective vehicle routing problem with time windows simultaneously considering total distance and distance balance of active vehicle fleet. We used a new complex chromosome to present the active vehicle route. We choose Tournament selection, onepoint crossover, and migrating mutation operator to solve this GA. After iterator operation, the solution of the problem was solved. In experiment on Solomon’s benchmark problems, we found that this objective is close to best known value in literatures, even though it was not designed for the single objective problems. Considering the distance balance, those instances are imbalanced and have much space to improve. From the results, the distance balance was improved in all classes of problems in the biobjective problem. According to the produced results, the suggested approach was sufficient and the average GA performance was adequate.
The authors thank the anonymous reviewers for their useful suggestions and comments. This work was supported by the National Natural Science Funds of China no. 51105157.