Multi-objective optimization problem under fuzzy rule constraints using particle swarm optimization

Abstract

In this paper, a fuzzy multi-objective programming problem is considered where functional relationships between decision variables and objective functions are not completely known to us. Due to uncertainty in real decision situations sometimes it is difficult to find the exact functional relationship between objectives and decision variables. It is assumed that information source from where some knowledge may be obtained about the objective functions consists of a block of fuzzy if-then rules. In such situations, the decision making is difficult and the presence of multiple objectives gives rise to multi-objective optimization problem under fuzzy rule constraints. In order to tackle the problem, appropriate fuzzy reasoning schemes are used to determine crisp functional relationship between the objective functions and the decision variables. Thus a multi-objective optimization problem is formulated from the original fuzzy rule-based multi-objective optimization model. In order to solve the resultant problem, a deterministic single-objective non-linear optimization problem is reformulated with the help of fuzzy optimization technique. Finally, PSO (Particle Swarm Optimization) algorithm is employed to solve the resultant single-objective non-linear optimization model and the computation procedure is illustrated by means of numerical examples.

Appendix

Particle swarm optimization (PSO), introduced by (Kennedy and Eberhart 1995; Shi and Eberhart 1998a), is a nature-inspired heuristic global optimization technique. It simulates the social behavior of bird flocking or fish schooling to configure the heuristic learning mechanism. PSO normally starts with a set of initial solution (called swarm) of the decision making problems under consideration. Individual solutions are called particles and food is analogous to optimal solution. The particles are flown through a multi-dimensional search space, where the position of each particle is adjusted according to its own experience and that of its neighbors. Let us assume that the dimension of the searching space is $D$ and the number of particle present in initial solution set is $l$. We further assume that at $s$th generation the position of the $r$th particle is $x_r (s)=\left[ {x_{r1} (s),\,\,x_{r2} (s),\ldots ,\,x_{rD} (s)\,} \right]$, the velocity of the $r$th particle is denoted as $v_r (s)=\left[ {v_{r1} (s),\,\,v_{r2} (s),\ldots ,\,v_{rD} (s)\,} \right]$, the position vector of the $r$th particle at which best fitness encountered so far is denoted as $pbest_r (s)=\left[ {pbest_{r1} (s),\,\,pbest_{r2} (s),\ldots ,\,pbest_{rD} (s)\,} \right]$ and the best position of all the particle is denoted as $gbest(s)\,{=}\,\left[ {gbest_1 (s),\,\,gbest_2 (s),\ldots ,\,gbest_D (s)\,} \right]$. In $(s+1)^\mathrm{th}$ generation the position and velocity of each particle are updated using the following rules:

$$\begin{aligned}&v_r (s+1)=w.v_r (s)+c_1 .uv_1 .\nonumber \\&\quad \times (pbest_r (s)-x_r (s))+c_2 .uv_2 .(gbest_r (s)-x_r (s)) \end{aligned}$$

(28)

$$\begin{aligned}&x_r (s+1)=x_r (s)+v_r (s+1), \end{aligned}$$

(29)

where the parameters $c_1$ and $c_2$are constants, $uv_1$ and $uv_2$are two random variables with uniform distribution in $[0,1]$ and $w(0<w<1)$ is called the inertia weight which controls the influence of previous velocity on new velocity. The pseudo-code of the PSO algorithm is given below.

Note For solving the optimization models given in Sect. 4, different parameters settings of the PSO algorithm and system environments are used. The optimization method is implemented in MATLAB and program is run on a Intel(R) Core(TM) i5-2500 CPU @ 3.30 GHz processor with 4 GB RAM under windows environment. To remove stochastic dependency 30 independent runs are made. In each run PSO parameters are set as follows: initial population size I $=$ 20, maximum number of generation M $=$ 100, the acceleration parameters $c_1 =c_2 =1.5$, inertia weight $w$ is obtained by putting $w_1 =0.9$ and $w_2 =0.4$ in the formula $w=(w_1 -w_2 )( {\frac{M-s}{M}})+w_1$ where $M$ is the maximum number of generation and $s$denotes the current generation number (Shi and Eberhart 1998b; Clerc and Kennedy 2002). The termination criteria have been set as either limited to maximum number of 100 generations or the order of relative error $10^{-5}$, whichever is achieved first. For solving each optimization problem program has been run 30 times and the best values are chosen.