Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 940592 | 10 pages | https://doi.org/10.1155/2015/940592

Pareto-Ranking Based Quantum-Behaved Particle Swarm Optimization for Multiobjective Optimization

Academic Editor: Fabio Tramontana
Received27 Apr 2015
Accepted25 Jun 2015
Published10 Aug 2015

Abstract

A study on pareto-ranking based quantum-behaved 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 job-shop scheduling problems.

1. Introduction

Most real-world 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 trade-off solutions exists, which are referred to as the pareto-optimal 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. [1] gave rise to a fast and elitist multiobjective genetic algorithm (NSGA-II) to reduce the computational complexity and adopted an elitism strategy. In [2], Konak et al. gave a tutorial about several variants of genetic algorithms and compared their performance. A particle swarm optimization (PSO) with pareto dominance was proposed in [3], in which a secondary repository and a special mutation operator were incorporated. A survey of the state-of-the-art multiobjective PSOs is presented in [4]. From the above literatures, the great difference between genetic algorithm (GA) and PSO exists for solving the multiobjective optimization problems. In GA, a set of particles are needed to be selected into next generation, while, in QPSO, one particle has to be chosen from the nondominated optima as the global best position (). Therefore, different pareto-ranking strategies are required.

Inspired by the quantum mechanics, QPSO was proposed by Sun et al. in 2004 [5], which has been proved to outperform PSO in both convergence rate and global search ability [6, 7]. Similar to other population-based algorithms (GA, PSO), maintaining a diverse population is an important consideration in multiobjective optimization to ensure solutions uniformly distributed over the pareto front. Therefore, in this paper, QPSO with three pareto-ranking strategies is compared on some benchmark functions and two metrics are used to test the performance of each strategy.

The research on the multiobjective FJSP is not as widely as that on the monoobjective FJSP. Brandimarte [8] was first to apply the decomposition method to FJSP and solved the routing subproblem by using some existing dispatching rules and addressed the scheduling subproblem by using a tabu search algorithm. Davarzani et al. [9] used artificial immune system to solve the multiobjective FJSP and compared it with Approach by Localization [10], AL + CGA [11]. Considering the advantages and disadvantages of stochastic algorithms and local search methods, some hybrid approaches were also proposed [12, 13]. In this paper, QPSO with sigma value is applied to solve multiobjective flexible job-shop scheduling.

The remainder of this paper is organized as follows. In Section 2, some basic concepts for multiobjective optimization are provided. Section 3 describes the three variants of pareto-ranking based QPSO in detail. Numerical tests on benchmark functions are presented in Section 4. In Section 5, QPSO with sigma value is applied to solve multiobjective flexible job-shop scheduling. Finally, conclusion is given in Section 6.

2. Basic Concepts for Multiobjective Optimization

Without loss of generality, only minimization problem is assumed here: where is the vector of decision variables. , , are the objective functions, and , are the constraint functions of the problem, is the dimension size of the search space, and is the number of objectives.

Definition 1 (pareto dominance). A vector dominates (denoted by ) if and only if is partially less than ; that is, , , and , .

Definition 2 (pareto optimal set). One has   .

Definition 3 (pareto front). For a given pareto optimal set , the pareto front is defined as .

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.

3. Pareto-Ranking Based Quantum-Behaved Particle Swarm Optimization

3.1. Particle Swarm Optimization

PSO is a population-based optimization technique originally proposed by Kennedy and Eberhart in 1995 [14]. A PSO system simulates the knowledge evolvement of a social organism, in which particles represent candidate solutions of an optimization problem. The position of each particle is evaluated according to the objective function, and particles share memories of their “best” positions. These memories are used to adjust the particles’ own velocities and their subsequent positions. It has already been shown that PSO is comparable in performance with and may be considered as an alternative to GA [15].

Ina PSO system with particles in the -dimensional space, the position, and velocity vector of particle at the kth iteration is represented as and . The particle moves according to the following equations: where, , , and are acceleration coefficient. and are random numbers distributed uniformly in . is inertia weight. Vector is the previous best position of particle (), and vector is the position of the best particle in the swarm ().

3.2. Quantum-Behaved Particle Swarm Optimization

The main disadvantage of PSO is that it is not guaranteed to be global convergent [16]. Concepts of QPSO were developed to address this disadvantage and first reported by Sun et al. in 2004 [5]. Trajectory analysis in [17] demonstrated the fact that the convergence of PSO may be achieved if each particle converges to its local attractor, , defined as where . It can be seen that is a stochastic attractor of particle that lies in a hyperrectangle with and being two ends of its diagonal and moves following and .

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 [5]: where is a random number uniformly distributed in and , called mean best position, is defined as the mean value of personal best positions of the swarm: The parameter in (4) is named as contraction-expansion (CE) coefficient, which can be adjusted to control convergence rate. The most commonly used method to control CE is linearly decreasing from to : where is the current iteration step, is the predefined maximum iteration steps, and and are the maximum and minimum value of CE.

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 well-known benchmark functions show that QPSO has better performance than PSO [57].

A great deal of efforts has been made to PSO for solving multiobjective optimization problems. A survey of state-of-the-art works is presented in [4]. The most contributive work is proposed by Ceollo et al. [3], in which the objective space is divided into hypercubes using an adaptive grid. The nondominated solutions are distributed into grids and then the grid is chosen according to a roulette wheel selection strategy. However, the grid is problems dependent and computation complicated. While in [18] a new density measure strategy named as sigma method was introduced to multiobjective PSO, so as to choose a from the nondominated archive, the sigma method can guide particles to the pareto optimal front. However, if the initialized solutions in nondominated archive are bad, the algorithm will fall into local optimum due to the fast convergence. Yang et al. [19] proposed a hybrid method which combined the density information and sigma method so as to achieve a balance between global search and local search. Considering the calculation of the density value and crowding distance is limited by two objectives. When the number of objectives becomes large, the measure of density and crowding distance will become invalid. Therefore, the idea of preference order is introduced [20].

3.3. Preference Order

As we all know, when the optimization problem has two objectives, the pareto optimal solutions can be plotted on a curve. Though the trade-off 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 [21]. A point is said to be efficiency of order if is not dominated by any of the -element subsets of .

Consider a minimization example with three objectives in Table 1; combination of all subsets of the objectives is .


Points

Point 10.52.111.2
Point 261.11.3
Point 33.1403.76
Point 41.10.54.32
Point 53.890.13.23

The dominant relation of all points in 2-element subsets is shown in Table 2, from which only Point 3 is nondominated by any points on all 2-element subsets. So Point 3 is said to be efficiency of order 2.


PointsPoint 1Point 2Point 3Point 4Point 5

Point 1
Point 2
Point 3
Point 4
Point 5

About efficiency of order, [22] gave three claims.

Claim 1. In a space with objectives, there is no more than one point with efficiency of order .

Claim 2. is efficient of order , if it is efficient of order .

Claim 3. is not efficient of order , if it is not efficient of order .

The pseudocode of QPSO with preference order is shown below.

Preference-Order Based QPSO(1)Initialization: initialize the swarm size , the external archive size , and the maximum number of iterations , CE . Position of each particle , , is randomly initialized. Personal best position of each particle is equal to . Construct external nondominated set .(2) Update the following:(a) Identify global best position using preference-order method.(i) Get combination of all subsets for objectives.(ii) Compute the efficiency order for each nondominated solution.(iii) Sort nondominated solutions in descending order according to their efficiency order.(iv) Get .(b) Update position of each particle according to (3), (4), and (5).(c) Update according to dominance rule.(d) Update external archive and remove solution with lower efficiency order if archive size is exceeded.(e) Update according to (6).(3) Repeat until the maximum number of iterations is reached.

3.4. Sigma Method

The sigma method was first proposed in [18] for finding the best local guide for each particle. Each nondominated solution in the external repository is assigned a value , , which is defined as (for two objectives shown in Figure 1.) In the general case, is a vector of elements, and each element is the combination of two coordinates. Take a problem of three objectives; for example,

The pseudocode of QPSO with sigma value is shown below.

QPSO with Sigma Method(1)Initialization: initialize the swarm size , the external archive size , and the maximum number of iterations , CE . Position of each particle , , is randomly initialized. Personal best position of each particle is equal to . Construct external nondominated set .(2)Update the following:(a)Identify local best position using sigma method.(i)Assign the value to each particle in the external archive.(ii)Calculate the value for each particle in the swarm.(iii)Calculate the distance between and .(iv)The particle in the archive with has the minimum distance to selected as the best local guide for the particle .(b)Update position of each particle according to (3), (4), and (5).(c)Update according to dominance rule.(d)Update according to (6).(3)Repeat until the maximum number of iterations is reached.

4. Numerical Experiments

4.1. Test Functions

Seven common test functions [2023] are used to test the proposed algorithm (DTLZ1, DTLZ2, DTLZ3, and DTLZ4), as listed in Table 3. We call difficulty factor. Table 4 gives the difficulty factor, number of runs, number of iterations, number of particles, and size of nondominated archive used in our test. The reason why these functions are chosen is due to the following features:(1)The relatively small implementation effort.(2)The ability to be scaled to any number of objectives and decision variable.(3)The global pareto front being known analytically.(4)Convergence and diversity difficulties that can be easily controlled.


NameExpressionOptimal solution

DTLZ1,





,

DTLZ2,



  

,

DTLZ3,





,

DTLZ4,





,


NameNumber of runsNumber of iterationsND

DTLZ15106008080
DTLZ210106008080
DTLZ310106008080
DTLZ410106008080

4.2. Performance Metrics

Usually, pareto-based multiobjective algorithms consider two aspects: closeness to the global pareto front (distance) and spread along the pareto front (diversity) [2], which are defined as follows.

Generational Distance (GD). It is a way of estimating how far the elements in the nondominated set are found so far from those in the pareto optimal set, which is defined aswhere is the number of points in nondominated set and is the Euclidean distance between a point and the nearest member of Pareto optimal set.

Spacing (SP). It is a metric measuring the distance variance of neighboring points in the nondominated set, which is defined as where ,   , is the mean of all , and is the number of objectives.

4.3. Experiments Results

Tables 5 and 6 show the mean values of generational distance and spacing for three methods (QPSO + PO, QPSO + Rand, and QPSO + Sigma) with different number of objectives (2, 3, 4, 5, 6, 7, and 8). It is easy to note 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. 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 it is very time-consuming.


NameQPSO + POQPSO + RandQPSO + Sigma

DTLZ12
3
4
5
6
7
8

DTLZ22
3
4
5
6
7
8

DTLZ32
3
4
5
6
7
8

DTLZ42
3
4
5
6
7
8


NameQPSO + POQPSO + RandQPSO + Sigma

DTLZ12
3
4
5
6
7
8

DTLZ22
3
4
5