A Fuzzy Adaptive Programming Method of Particle Swarm Optimization

Qi Kang IEEE Student Member

Control Department of Tongji University

No.1239,Siping Road Shanghai,China,200092

E-mail: kangqi_kz@hotmail.com

Lei Wang IEEE Member

Control Department of Tongji University

No.1239,Siping Road Shanghai,China,200092

E-mail: wanglei@mail.tongji.edu.cn

Qidi Wu IEEE Senior Member

Ministry of Education of China No.37,Damucanghutong Beijing,China,100816

E-mail: wuqidi@moe.edu.cn

Abstract—This paper introduces a novel multi-optimum programming mode for the particle swarm optimization algorithm. Initially, to efficiently control the relationship between multi-optimum information and convergence to the global optimum solution, fuzzy logic is introduced into the process of multi-optimum distribution state dynamic programming. Then, the fuzzy adaptive programming method based on single-variable and two-dimension-input fuzzy control structure for optimum and sub-optimum distribution information is implemented and simulated with different function optimization problems.

Index Terms—Particle swarm optimization; Fuzzy adaptive programming; Multi-optimum information

I. INTRODUCTION ARTICLE swarm optimization ( PSO ) algorithm is a stochastic optimization technique first proposed by

Kennedy and Eberhart in 1995 [1]. PSO is a population-based, heuristic global optimization technology and is inspired by the emergent motion of a flock of birds searching for food. The PSO algorithm iteratively explores a multidimensional search space with a swarm of individuals, that are referred to as particles without mass and volume, looking for the global minimum (or maximum). Each particle flies through the search space by adjusting its velocity and position. In each iteration, the velocity vector of each particle is adjusted so that prior personal fittest position (cognitive aspect) and the best position found by the swarm (social aspect) act as attractors.

Since the introduction of the PSO algorithm, there has been a considerable amount of works done in developing the original version of PSO, through empirical simulations [2-8], in the aspects of parameter optimization, neighborhood topology and hybrid algorithms with other optimization algorithms, etc. In addition, the PSO algorithm has been effectively applied to many fields, such as power system optimization [9-10], industry product optimization [11], neural network training [12], system identification [13], and multiple objectives optimization [14], etc, for its concise conception and convenient realization.

This paper proposed a fuzzy adaptive programming

method for particle swarm optimization based on the static multi-optimum programming mode proposed by authors previously [15]. In the static multi-optimum programming mode, the knowledge of multi-modal distribution state analysis was introduced into general programming of the particle swarm movement. Although it can avoid falling into local optimums, the relationship of multi-optimum was obtained based on plenty of simulations, and the programming relationship cannot be adjusted dynamically. To overcome the problem, efficiently control the relationship between multi-optimum information and convergence to the global optimum solution, fuzzy logic is introduced into the process of multi-optimum distribution state dynamic programming. Then, the fuzzy adaptive programming method based on single-variable and two-dimension-input fuzzy control structure for optimum and sub-optimum distribution information is implemented and simulated with different functions optimization.

II. STATIC MULTI-OPTIMUM PROGRAMMING MODE IN PARTICLE SWARM OPTIMIZATION

In particle swarm optimization, the position vector and the velocity vector of particle i )~1( Ni = in D-dimensional

search space can be indicated as ),,,( 1 iDidiixxxx KK=

and )v,v,,v( 1v iDidiiKK= respectively. The trajectory of

each individual in the search space is adjusted by dynamically altering the velocity of each particle. The new flight velocity and the position of particle i in dth )~1( Dd = dimensional subspace are calculated using the following two equations:

idgd

idididid

xprandcxprandcvv

−⋅+−⋅+=

)(()))((

22

11ω (1)

ididid vxx += (2) where, idp represents the best position of the particle and

dpg is the fittest particle found so far by all particles.

Acceleration coefficients 1c and 2c are constants, which determine the relative influence of the social and cognition

P

components, and are often both set to the same value to give each component (the cognition and social learning rates) equal weight. )(1 ⋅rand and )(2 ⋅rand are two separately generated uniformly distributed random numbers in the range [0,1].The variableω is called the inertia weight, which can be changed linearly with the running time:

TtKKK /)( 121 −+=ω (3) where, T is the total cycle index, t is the cycle index of current computation and 21 , KK are constants which indicate the border of changing ω .

Initially, a population of particles is generated with random position, and then random velocities are assigned to each particle. The fitness of each particle is then evaluated according to the objective function. At each generation, the velocity of each particle is calculated according to equation (1) and the position for the next function evaluation is updated according to equation (2). Each time, if a particle finds a better position than the previously found best position, its location is stored in memory. Generally, a maximum velocity ( maxv ) for each modulus of the velocity vector ( idv ) of particles is defined in order to control excessive roaming of particles outside the search space. Whenever idv exceeds the defined limit, its velocity is set to maxv .

In this kind of movement mode, the performance of algorithm is affected largely by the initial distribution character of the swarm, especially in multi-modal function optimization, the movement of the particle swarm will be over-affected by the optimum value during the early part of the search. So, the distributional characteristics of multi-optimum value of the whole swarm should be paid more attention. In other words, besides its former optimum position, the M optimum values of the swarm can be introduced into multi-variant programming of the movement of current particle and the velocity of the particle can be determined by following formula:

∑=

−+=M

kididkkkidid xprandcwvv

1, )()( (4)

where, idkp , is the kth optimum ranked in the whole swarm,

kc denotes the instruction factor (or called “programming

coefficient”) of M optimum values of the swarm, and ()krand indicate their matching random quantity

respectively. Although the comparison of multi-variant optimum value is added in velocity computation to some extent in such movement pattern, the ability to avoid falling into local optima is strengthened with such expense.

III. MULTI-OPTIMUM FUZZY ADAPTIVE PROGRAMMING MODE OF PARTICLE SWARM OPTIMIZATION

The searching process of particle swarm optimization is a nonlinear and dynamic process. Therefore, when the

environment itself is dynamically changed over the time, the algorithm should be able to adapt to the changing environment. Although the above programming strategies (static multi-optimum programming mode, SMOP-PSO) improves the general convergence performance of the algorithm compared with the basic PSO, the programming coefficient kc cannot be adjusted dynamically to the current

optimization ability, and getting better kc need plenty of experiments; therefore, its adaptive ability and general convergence performance were limited to some extent.

If we can introduce some intelligent method on the basis of multi-optimum programming to program the relationship between the multi-optimum information dynamically in computation process, in other words, the programming strategies can be adjusted adaptively according to the current optimization ability, which harmonizes the movement relation between itself and the swarm more flexibly, the general convergence performance of algorithm will be improved greatly. Consequently, the following idea of a multi-optimum fuzzy adaptive programming method of particle swarm optimization algorithm came into being.

A. Fuzzy control structure Shi and Eberhart[17] designed a kind of fuzzy system

successfully implemented to dynamically adapt the inertia weight of the particle swarm optimization algorithm. In this paper, fuzzy logic is introduced into the PSO algorithm again for the dynamic programming of multi-optimum distribution states, and a novel multi-optimum programming mode of particle swarm optimization is proposed.

In this paper, the optimum and the sub-optimum information are selected, and a kind of single-variable and two-dimension-input fuzzy control structure as Fig.1 is adopted to design the fuzzy instruction system, the output structure of which is single-variable increment mode.

single-variable andtwo-dimension-input

fuzzy controlstructured/dt

EE

EC

U

Figure 1. Fuzzy control structure based on single-variable and two-dimension-input

B. Design of the fuzzy programming system The current best performance evaluation Val_opt

measures the performance of the best candidate solution found so far by the algorithm. Different optimization problems have different ranges of performance measurement values. To design a fuzzy system with the Val_opt as one of the inputs to be applicable to a wide range of optimization problems, Val_opt has to be converted into a normalized format. Assume, the optimization problems to be solved are minimization problems, and the estimated minimum is denoted as Val_min , and the non-optimal Val_opt is denoted

as Val_max . The normalized Val_opt ( UVal_opt ) can be calculated as

Val_min-Val_maxVal_min-Val_optUVal_opt = (5)

where, UVal_opt ]1,1[−∈ . Then, we select UVal_opt as an input of the fuzzy

controller, that is UVal_opt=E , another input of the controller in Fig.1 is the differential coefficient about E to t , so we select the change rate of UVal_opt ( UVal_opt∆ ) ,that is

UVal_opt∆=EC . In this paper, we adopt the change value of UVal_opt in a definite period, and because the periods we define are same, so the normalized change value is adopted as the second input, that is ]1,1[−∈EC . The output structure is single-variable increment mode, and the output variable is the fuzzy change value of the ratio between the programming coefficients of binary optima information: 21 / cc∆ ( 21 / ccU ∆= ).

For the fuzzification of input EC and output 21 / cc∆ , we select the same fuzzy sets: ( NB ， NM ， NS ，O ， PS ，

MP ， PB ), in which, NB denotes “Negative Big”, NM is “Negative Medium”, NS is “Negative Small”, O is ”Zero”, PS is “Positive Small”, PM is ”Positive Medium” and PB is “Positive Big”. The sketch map of the membership functions is Figure 2(a), in which, ),( ba cc ++ , ),( ba cc −− and ),( ibia cc

4,,1,0 L=i denote the boundary values of the corresponding membership function. For input UVal_opt=E , we select the fuzzy sets: （ L ， SL ， SL ， B ）,and the sketch map of the membership functions is Figure 2(b).

(a) Input EC and output U

1

0

)( xµ

x

Lf SLf SBf Bf

1 (b) Input E

Figure2. Sketch map of fuzzy membership function for fuzzy variables

The authors established 28 dynamic fuzzy adjustment rules of multi-optimum information based on experience (Table I).

TABLE I FUZZY RULES OF THE FUZZY SYSTEM (OUTPUT IS 21 / cc∆ )

EC E

NB NM NS O PS PM PB

L PB PM PM PB PM PS O

SL PM PS PM PM PM O PS

SB PB PM PM PM O PS NS

B PB PB PM PB PS O NS

The dynamic adjustment value of the ratio between the

programming coefficients of binary optima information is obtained by the defuzzification of the fuzzy output.

C. Flow chart of multi-optimum fuzzy adaptive programming method of PSO algorithm

We describe the multi-optimum fuzzy adaptive programming method of PSO algorithm (called MFPSO-2D, because the fuzzy system has a two dimensional structure) as follows: Step.1 Initialize the population (position vector and velocity

vector) and set the corresponding parameters in the way adopted in SMOP-PSO [15];

Step.2 Evaluate each of particles in the swarm, compute the best position for each particle and store it as bestp ;

Step.3 Compare and sort all the current best positions( bestp ), find the anterior M optimums, locate them for the subarea number according the space grids setting in the initialization period and get their centers of gravity(used to multi-optimum programming [15] );

Step.4 Detect the current best performance Val_opt and calculate its normalized value UVal_opt according to equation (5) as one input of the fuzzy programming controller UVal_opt=E ; compute the change value of UVal_opt in a definite period as the other input: UVal_opt∆=EC ;

Step.5 Output the change value of the ratio between the programming coefficients of binary optima information: 21 / cc∆ after fuzzy computation and defuzzification (fuzzy programming system), and update the current 1c 、 2c ;

Step.6 Particle enters the optimal neighbor area? If it is true, then update the velocity and position of particle according to multi-optimum programming rules and update multi-optimum, else update the velocity and position of particle according to traditional PSO algorithm mode;

Step.7 Test the terminate conditions (the maximal generation or finding an idea optimum). If the terminate conditions is met, end the algorithm; otherwise, continue the iteration.

Figure. 3 displays the flow char of MFPSO-2D algorithm.

Figure3. The flow chart of MFPSO-2D algorithm

IV. SIMULATION AND COMPUTATION The simulation of this paper is running in Windows XP,

the emluator is written in VB6. In the simulation comparison work of this paper, two well-known Benchmarks: Rosenbrock function and Griewank function were used to evaluate the performance. The first function is simple unimodal function whereas the second function is multi-modal function designed with a considerable amount of local minima. Simulations were carried out to find the global minimum of each function. Both benchmarks used are given in Table II.

We use the asymmetric initialization method to observe the performance of the new development introduced in this paper. Table III shows the range of population initialization and the maximum velocity with the limitation of

maxmax XV = for the benchmarks considered in this paper.

TABLE II

FUNCTIONS FOR SIMULATION Name of function Mathematical representation

Rosenbrock ∑=

− −+−=n

iiii xxxxf

1

22211 ])1()(100[)(

Griewank ∑ ∏= =

+−=n

ii

in

ii x

ixxxf

1 1

22 1)(cos

40001)(

TABLE. III INITIALIZATION RANGE AND MAXIMUM VELOCITY FOR FUNCTIONS

Function Range of search Range of initialization maxV

)(f1 x n)100,100(−

n)30,15( 100

)(f2 x n)600,600(−

n)600,300( 600

In the simulation work of this paper, both functions were

tested with dimensions 10, 20, and 30. A different number of maximum generations (Gmax) are used according to the complexity of the problem under consideration. For each function, 100 trials were carried out and the average optimal values are presented. In addition, the basic parameters setting are: 80=N ; 4.01 =k , 9.02 =k .

For two functions, we attempt the same boundary settings of each fuzzy membership function for input fuzzy variables E and EC as follows:

UVal_opt=E : L （0, 0.06）, SL （0.05, 0.4）, SB （

0.3, 0.6）, B （0.5, 1）; UVal_opt∆=EC : NB（-1, -0.50）, NM （-0.60, -0.20

）, NS （-0.30, 0.05）, O （-0.05, 0.05）, PS （-0.05, 0.30）, PM （0.20, 0.60）, PB （0.50, 1）.

The boundary settings of each fuzzy membership function for output fuzzy variable 21 / cc∆ as follows:

NB （-0.1, -0.04）, NM （-0.045, -0.015）, NS （-0.025, 0.005） , O （ -0.005, 0.005） , PS （ -0.005, 0.025）, PM （0.015, 0.045）, PB （0.04, 0.1）

Under the same parameter setting except 21 ,cc (the initialization values are different, that is )0(/ 21 cc are different), the simulations were proceeded for 100 trial runs separately, and compared with the static multi-optimum programming algorithm (SMOP-PSO) in [15], SPSO algorithm in [16], and FPSO algorithm in [17]. Table IV shows the simulation results.

In addition, we observed the real-time change current of the programming coefficient 21, cc with different initialization settings, to verify the adaptive performance of the new method. Figure 4(a-c) display the typical dynamic curves of

21 / cc from different initialization values (The initialization value is 1)0(/ 21 =cc , 0.2)0(/ 21 =cc , or 2)0(/ 21 =cc separately ).

c 1/c

2(2/div)

(a) 1)0(/ 21 =cc

(b) 0.2)0(/ 21 =cc

(c) 2)0(/ 21 =cc

Figure 4. The typical dynamic of 21 / cc from different initialization values

V. SIMULATION RESULTS ANALYSIS From the results presented in Table IV, it has been

understood that the new MFPSO-2D method is superior to all the other methods for both benchmarks considered in this investigation. Therefore, we can draw such a conclusion that

TABLE.ІV. COMPARISON OF THE AVERAGE OPTIMIZATION RESULTS

Function Dimension Gmax SPSO [16]

FPSO [17]

SMOP -PSO[15]

MFPSO-2D 2.0)0(/ 21 =cc

MFPSO-2D 5.2)0(/ 21 =cc

MFPSO-2D 1)0(/ 21 =cc

10 1000 36.2945 15.8165 8.02339 7.78574 7.01125 7.90700

20 1500 87.2802 45.9999 32.8245 23.46281 20.43854 19.77443 )(1 xf

30 2000 205.559 124.418 53.8489 32.26450 35.99866 30.74123

10 1000 0.07600 0.06832 0.08759 0.08794 0.08429 0.07952

20 1500 0.02880 0.02596 0.02410 0.02471 0.02346 0.02287 )(2 xf

30 2000 0.01280 0.01495 0.01037 0.01053 0.01209 0.000981

the general optimization performance of the algorithm is improved greatly when the programming coefficient ratio had been dynamic adjusted through fuzzy rules. In addition, from the average optimization results, we can see that the performance of the algorithm was hardly influenced by the initialization value of 1c and 2c (or 21 / cc ), that is to say, even when the sub-optimum is setting to have more attraction to particle than the optimum at initialization ( 1)0(/ 21 <cc ), MFPSO-2D can also find good optimization result.

In the multi-optimum programming mode, the proportion between the programming coefficients for multi-optima affects the performance of algorithm, and need amount of analysis and tests in advance. This paper presents an intelligent fuzzy programming method to implement the adaptive instruction for multi-optimum programming coefficients of particle swarm optimization, and the general optimization performance of the PSO algorithm was improved. All these are validated in the resolving process of both benchmarks in this paper. Certainly, when the algorithm is used for other kinds of function optimizations, the fuzzy rules of course should be adjusted according to specific characteristics of the given function in order to achieve the best effects of speed and convergence.

Ⅵ. CONCLUSIONS This paper introduces a novel multi-optimum intelligent

programming mode for the particle swarm optimization algorithm. To efficiently control the relationship between multi-optimum information and convergence to the global optimum solution, fuzzy logic is introduced into the process of multi-optimum distribution state dynamic programming, and the fuzzy adaptive programming method based on single-variable and two-dimension-input fuzzy control structure for optimum and sub-optimum distribution information is implemented. The MFPSO-2D method was tested with two benchmarks, and the results prove its effectiveness in this investigation.. In the view of the authors, the applications of fuzzy logic in particle swarm optimization algorithm with intelligent characteristics can be discussed further in future and the convergence pattern, dynamic and steady-state performances of the algorithm can be improved further to specific complex optimization functions.

ACKNOWLEDGMENT The work of this paper is supported by National Science

Foundation of China (60104004, 70271035), and Shanghai Riding-star Foundation for Young Researchers (03QG14053).

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