A study on paretoranking based quantumbehaved particle swarm optimization (QPSO) for multiobjective optimization problems is presented in this paper. During the iteration, an external repository is maintained to remember the nondominated solutions, from which the global best position is chosen. The comparison between different elitist selection strategies (preference order, sigma value, and random selection) is performed on four benchmark functions and two metrics. The results demonstrate that QPSO with preference order has comparative performance with sigma value according to different number of objectives. Finally, QPSO with sigma value is applied to solve multiobjective flexible jobshop scheduling problems.
Most realworld optimization problems have more than one objective, with at least two objectives conflicting with each other. The conflicting objectives lead to a problem where a single solution does not exist. Instead, a set of optimal tradeoff solutions exists, which are referred to as the paretooptimal front or pareto front. This kind of optimization problems is referred to as multiobjective optimization problems.
In the past ten years, a wide variety of algorithms have been proposed to address such problems. Deb et al. [
Inspired by the quantum mechanics, QPSO was proposed by Sun et al. in 2004 [
The research on the multiobjective FJSP is not as widely as that on the monoobjective FJSP. Brandimarte [
The remainder of this paper is organized as follows. In Section
Without loss of generality, only minimization problem is assumed here:
A vector
One has
For a given pareto optimal set
Generally, the analytical expression of the line or surface which contain those points does not exist. It is only possible for us to determine the nondominated points and to produce the pareto front.
PSO is a populationbased optimization technique originally proposed by Kennedy and Eberhart in 1995 [
Ina PSO system with
The main disadvantage of PSO is that it is not guaranteed to be global convergent [
In quantum world, the velocity of the particle is meaningless, so, in QPSO system, position is the only state to depict the particles, which moves according to the following equation [
QPSO does not require velocity vectors for the particles and has fewer parameters to control, making the algorithm easier to implement. Experimental results performed on some wellknown benchmark functions show that QPSO has better performance than PSO [
A great deal of efforts has been made to PSO for solving multiobjective optimization problems. A survey of stateoftheart works is presented in [
As we all know, when the optimization problem has two objectives, the pareto optimal solutions can be plotted on a curve. Though the tradeoff surface can also be visualized for three objectives, it is not easy to pick the final point. For problems with more than three objectives, it becomes extremely difficult to find the optimized solution through visualization.
The idea of preference order was firstly proposed in 1999 [
Consider a minimization example with three objectives in Table
Nondominated set.
Points 




Point 1  0.5  2.1  11.2 
Point 2  6  1.1  1.3 
Point 3  3.14  0  3.76 
Point 4  1.1  0.5  4.32 
Point 5  3.89  0.1  3.23 
The dominant relation of all points in 2element subsets is shown in Table
Dominant relation.
Points  Point 1  Point 2  Point 3  Point 4  Point 5 

Point 1 





Point 2 





Point 3 





Point 4 





Point 5 





About efficiency of order, [
In a space with
The pseudocode of QPSO with preference order is shown below.
Initialization: initialize the swarm size
Update the following:
Identify global best position using preferenceorder method.
Get combination of all subsets for
Compute the efficiency order for each nondominated solution.
Sort nondominated solutions in descending order according to their efficiency order.
Get
Update position of each particle according to (
Update
Update external archive and remove solution with lower efficiency order if archive size
Update
Repeat
The sigma method was first proposed in [
Sigma value for a twoobjective space.
The pseudocode of QPSO with sigma value is shown below.
Initialization: initialize the swarm size
Update the following:
Identify local best position using sigma method.
Assign the value
Calculate the value
Calculate the distance between
The particle
Update position of each particle according to (
Update
Update
Seven common test functions [
The relatively small implementation effort.
The ability to be scaled to any number of objectives and decision variable.
The global pareto front being known analytically.
Convergence and diversity difficulties that can be easily controlled.
Description of the tested functions.
Name  Expression  Optimal solution 

DTLZ1 




DTLZ2 




DTLZ3 




DTLZ4 


Difficulty factor, number of runs, iterations, particles, and external archive.
Name 

Number of runs  Number of iterations 

ND 

DTLZ1  5  10  600  80  80 
DTLZ2  10  10  600  80  80 
DTLZ3  10  10  600  80  80 
DTLZ4  10  10  600  80  80 
Usually, paretobased multiobjective algorithms consider two aspects: closeness to the global pareto front (
Tables
Mean value of generational distance.
Name 

QPSO + PO  QPSO + Rand  QPSO + Sigma 

DTLZ1  2 



3 




4 




5 




6 


 
7 


 
8 


 


DTLZ2  2 



3 




4 




5 




6 




7 


 
8 


 


DTLZ3  2 



3 




4 




5 




6 




7 




8 






DTLZ4  2 



3 


 
4 


 
5 




6 




7 


 
8 



Mean value of spacing.
Name 

QPSO + PO  QPSO + Rand  QPSO + Sigma 

DTLZ1  2 



3 




4 


 
5 




6 


 
7 




8 


 


DTLZ2  2 



3 




4 




5 




6 




7 


 
8 


 


DTLZ3  2 



3 




4 


 
5 




6 




7 


 
8 


 


DTLZ4  2 



3 


 
4 


 
5 


 
6 


 
7 


 
8 



When
Furthermore, in QPSO with sigma value, each particle flies towards its nearest nondominated particle in external archive, which will easily result in a premature convergence and the particles will get trapped in a local optimum.
The FJSP could be formulated as follows. There is a set of
In this paper, the following criteria are to be minimized:
The maximal completion time of machines, that is, the makespan.
The maximal machine workload, that is, the maximum working time spent on any machine.
The total workload of machines, which represents the total working time over all machines.
Each operation cannot be interrupted during its performance (nonpreemptive condition).
Each machine can perform at most one operation at any time (resource constraint).
The precedence constraints of the operations in a job can be defined for any pair of operations.
Setting up time of machines and move time between operations are negligible.
Machines are independent from each other.
Jobs are independent from each other.
All machines are available at time 0.
Each job has its own release date.
The order of operations for each job is predefined and invariant.
Particle’s position representation is the most important task for successful application of PSO or QPSO to solve the FJSP. In this paper, the particle’s position is represented by a vector which consists of two parts (take Table
Example of 3 jobs in 4 machines.
Job  Operation 







1  3  4  1 

3  8  2  1  

3  5  4  7  




4  1  1  4 

2  3  9  3  

9  1  2  2  




8  6  3  5 

4  5  8  1 
Encoding of a particle.
The initial population has great influence on the performance of the algorithm. Here in this paper, a guided initialization method is used. In order to ensure the diversity of the population, 50% of the particles are initialized randomly, and the others are initialized according to the following rules.
Create an array for a random job order J (e.g., J4 J2 J3 J1).
Assign the first operation
Identify the machine with earliest stop time Ms and the last operation
Find a suitable machine to process
Identify all the machines that can process
Calculate the waiting time of each machine
The machine with shortest waiting time is chosen to process
For combinational optimization problems, effective information exchanging is able to help find the better solution. The crossover operator on the operation part is shown in Figure
Crossover on operation part.
The crossover operator on the machine part is shown in Figure
Crossover on machine part.
The mutation of a particle is also divided into two steps; one is on operation part (Figure
Mutation on operation part.
Another mutation is on machine part, but the operation to be mutated cannot be randomly chosen. Because the makespan and workload are determined by the critical path [
The Gantt graph of moving operation.
As the benchmark results analysis in Section
To illustrate the performance of QPSO with sigma value, four representative examples based on practical data have been selected [
The parameters are set as population size
From Table
Comparison of results in four problems.
Problem 
Objective  AL + CGA  PSO + SA  GA  QPSO + Sigma 



16  11  12 


10  10  11 



34  32  33 






16  16  15 


15  13  14 



75  73  73 






7  7  7 


5  6  6 



45  44  43 






24  12  12 


11  11  11 



91  91  92 

Figures
Gantt graph of problem
Gantt graph of problem
Gantt graph of problem
Gantt graph of problem
In this paper, QPSO with three elitist selection strategies are used to solve multiobjective optimization problems. The results show that QPSO with preference order has comparative performance with sigma value. When the number of objectives is small (e.g., 2, 3, 4, and 5), QPSO with sigma value performs better. However, when the number of objectives becomes larger (e.g., 6, 7, and 8), QPSO with preference order obtains solutions closer to the true pareto front but is very timeconsuming. Therefore, in the future work, we will try to combine these two methods to achieve a balance between global search and local search.
Moreover, we apply QPSO with sigma value to solve multiobjective flexible jobshop scheduling problems, the results of which demonstrate the competitive performance of the algorithm.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by Jiangsu Postdoctoral Funding (Project no. 1401004B), by National HighTechnology Research Development Plan Project (Project no. 2013AA040405), and by National Natural Science Funding (Project no. 61300152).