In this paper, we empirically study the performance of particle swarm optimization (PSO) using a nonlinear inertia weight compared with the performance using a constriction factor. Adjusting the parameters of nonlinear inertia weight directly affects the performance of PSO. Five benchmark functions are used for evaluation. Under all testing functions, the PSO with nonlinear inertia weights consistently converges faster than the PSO with constriction factors.

Particle swarm optimization (PSO) is a populationbased,selfadaptive search optimization technique first

introduced by Kennedy and Eberhart in 1995 [1]. The development of this method was based on the simulation of simplified animal social behaviors, e.g., fish schooling, bird flocking, etc.Shi and Eberhart have published several papers that describe research concerning a version of the particle swarm algorithm that incorporates what we call an inertia weight [2,4–6].Equations (1) and (2) describe the velocity and position update equations with the inertia weight included.Equation (1) calculates a new velocity for each particle (potential solution) based on its previous velocity, the locationwhere the best fitness so far has been achieved,and the population global (or local neighborhood, in the neighborhood version of the algorithm) location where the best fitness so far has been achieved. Equation (2)updates each particle’s position in the solution hyperspace.The two random numbers are independently generated. The use of the inertia weightw, which typ ically decreases linearly from about 0.9 to 0.4 during a run, has provided improved performance in a number of applications.

vid = w *vid + c1 * rand() * (pid − xid)

+ c2 *rand() *(pgd − xid) (1)

x id = xid + vid (2)

In this paper, we propose a nonlinear inertia weight for Eq. (1). Parameters of the nonlinear inertia weight

function directly affect PSO performance. Therefore,adjusting parameters should be considered before being used in real world application. The five functions were used to evaluate the performance in comparison with the constriction factor that was introduced by Clerc(1999) [3–6].The organization of this paper is as follows. In Section 2, we describe the nonlinear benchmark functions.Section 3 presents the background concept of the constriction factor. Section 4 introduces the background concept of the nonlinear inertia weight function that is to be applied to PSO. Then, in Section 5, the result of the experimentation is discussed. Finally, we summarize our work in Section 6.

** 2. Experimental approach**

For comparison, five nonlinear benchmark functions are used here. The first is Ackley’s function described by Eq. (3):

In all cases, the population size was set to 30, the maximum number of iterations was set to 800 and if the GBest was not changed at least 1e099 for 100 epochs,the program would be terminated.

In all cases for which the inertia weight was used, it was set to 0.9 at the beginning of the run, and made to decrease nonlinearly to 0.4 at the maximum number of iterations. Inertia weight cases used a Vmax set to the maximum range Xmax. Each of the two (p − x) terms was multiplied by an acceleration constant of 2.0 (times a random number between 0 and 1).

is thus 0.729. This resulted in the previous velocity being multiplied by 0.729, and each of the two (p − x) terms being multiplied by 0.729 * 2.05 = 1.49445 (times a random number between 0 and 1). Note that this is functionally equivalent to using Eq. (1)with w = 0.729 and c1 = c2 = 1.49445. These new parameter values are, however, related (see Eq. (8)).

**3. Constriction factor**

Since particle swarm optimization originated from efforts to model social systems, a thoroughmathematical foundation for the methodology has not been developed at the same time as the algorithm. Within the last few years, a few attempts have been made to begin to establish this foundation. Recent work done by [3] has indicated that the use of a constriction factor may be necessary to ensure convergence of the particle swarm algorithm. A detailed discussion of the constriction factor is beyond the scope of this paper, but a simplified method of incorporating it appears in Eq. (8), where is a function of c1 and c2 as reflected in Eq. (9) [3,7].

**4. Nonlinear inertia weight function**

Figure 1 shows the relation between a characteristic of inertia weight and adjusting parameters(F1 and F2). The Y axis is the inertia weight which starts 1 to 0 and the X axis is iteration numberswhich start 1 to 100. For example,F1 is 2 and F2 are 10(the solid line) and 25 (the dotted line) respectively. The inertia weight speed of the solid line is faster than the dotted line and these are shown in Fig. 1(a). Conversely, in Fig. 1(b),F1 are2(the solid line) and 4(the dotted line) respectively, and F2 is 10. The inertia weight speeds of the solid line and the dotted line are almost the same because F2 of both lines has been adjusted to the same value (F2 =10). Figure 1(c) and (d) also present the characteristic of inertia weight when F2 are 25 and 40 respectively,and F1 are still 2 and 4 respectively.

**5. Experiment and results**

The performance comparison between nonlinear inertiaweights and constriction factors is presented by the number of iterations. This evaluates a performance of convergence based on the average number of iterations and the range values of iteration. The experiment applied the Particle Swarm Optimization toolbox (PSOt) forMATLAB which is programmed by Brian Birge. In our experiment, each version of each approach (nonlinear inertia weight and constriction factor) was run 10 times for each test function.

Section 4 describes the main concept of setting the optimum parameter for preventing the particle position movement achieving an out of velocity rate. The nonlinear parameters F1 and F2 are 4 and 80 iterations(10% of maximum iterations) respectively, as defined

in this paper. We are seeking to define the velocity of PSO in pursuing the global objective within the first 10% of iterations (80 iterations), with F1 as 4. If F1 is a large value, the velocity will decrease too slowly. We set F2 as 80 iterations or 10% of iteration for pursuit of the local objective after 80 iterations.

The benefit of parameters setting is that the global objective is explored rapidlywhile the local objective is sought slowly, and convergence to the target is achieved fast and correctly. The nonlinear inertia weight parameter setting is used in a real world application as [9] which defines F1 is 4 and F2 is 10% of maximum iterations.As a result, the PSO algorithm converges to the target faster than the other parameter as presented in Section 4.

**5.1. Ackley’s function**

The dimensions and generation of Ackley’s function are 30 and 800, respectively. The program would be terminated when an error of less than 1e099 for 100 epochs was achieved. Table 1 shows the number of iterations which converge to the target by the nonlinear inertia weight method; Table 2 shows iterations needed using the constriction factor method.

The average number of iterations using the nonlinear inertia weight is thus 158; the range of values is 145 (from 105 to 250), about 91.7 percent of the average,and standard deviation is 50.2.

The average number of iterations using the constriction factor is thus 288.3; the range of values is 217(from 103 to 320), about 75.3 percent of the average,and standard deviation is 65.32.

As a result, the nonlinear inertia weight method achieves convergence on the target faster than the constriction factor method, evaluated by the average number of iterations and the range of values. If testing of the performance is based on the processing time, the results of both methods are not different.

Fig. 1. The characteristic of inertia weight will be changed when the F1 and F2 parameters are adjusted.

**5.2. Alpine function**

The dimensions and generation of the Alpine function are 30 and 800, respectively. The program would be terminatedwhen an error of less than 1e099 for 100 epochs was achieved. Table 3 shows the number of iterations which converge to the target by the nonlinear inertia weight method; Table 4 shows iterations needed using the constriction factor method.

The average number of iterations using the nonlinear inertia weight is thus 131.7; the range of values is 187(from 101 to 288), about 142 percent of the average,and standard deviation is 58.07.

The target error was not achieved within 800 iterations for all runs.

As a result, the constriction factor method is not able to converge to the target of the function within 800 iterations but the nonlinear inertia weight method converges to the target quickly. If testing of the performance is based on the processing time, the results of both methods are not different.

**5.3. Tripod function**

The dimensions and generation of the Tripod function are 30 and 800, respectively. The program would be terminatedwhen an error of less than 1e099 for 100 epochs was achieved. Table 5 shows the number of iterations which converge to the target by the nonlinear

inertia weight method; Table 6 shows iterations needed using the constriction factor method.

The average number of iterations using the nonlinear inertia weight is thus 161.5; the range of values is 160(from 105 to 265), about 99 percent of the average, and standard deviation is 57.56.

The average number of iterations using the constriction factor is thus 214.6; the range of values is 28 (from 293 to 321), about 8.9 percent of the average, and standard deviation is 8.74.

As a result, the performance of the nonlinear inertia weightmethod achieves convergenceto the target faster than the constriction factor method, evaluated by the average number of iterations and the range of values.If testing of the performance is based on the processing time, the results of both methods are not different.

**5.4. Ratrigrin function**

The dimensions and generation of the Ratrigrin function are 30 and 800, respectively. The program would be terminatedwhen an error of less than 1e099 for 100 epochs was achieved. Table 7 shows the number of iterations which converge to the target by the nonlinear inertia weight method; Table 8 shows iterations needed using the constriction factor method.

The average number of iterations using the nonlinear inertia weight is thus 142.2; the range of values is 180(from 104 to 284), about 126.6 percent of the average,and standard deviation is 57.51.

The average number of iterations using the constriction factor is thus 214.1; the range of values is 232(from 105 to 337), about 102 percent of the average,and standard deviation is 58.76.

As a result, the performance of the nonlinear inertia weightmethod achieves convergenceto the target faster than the constriction factor method, evaluated by the average number of iterations and the range of values.If testing of the performance is based on the processing time, the results of both methods are not different.

**5.5. Schaffer’s f6 function**

The dimensions and generation of the Schaffer’s f6 function are 30 and 800, respectively. The program would be terminated when an error of less than 1e099 for 100 epochs was achieved. Table 9 shows the number of iterations which converge to the target by the nonlinear inertia weight method; Table 10 shows iterations needed using the constriction factor method.

The average number of iterations using the nonlinear inertia weight is thus 191.8; the range of values is 187(from 139 to 326), about 97.5 percent of the average,and standard deviation is 55.85.

The average number of iterations using the constriction factor is thus 251.4; the range of values is 99 (from 205 to 304), about 39.4 percent of the average, and standard deviation is 35.34.

As a result, the performance of the nonlinear inertia weightmethod achieves convergenceto the target faster than the constriction factor method, evaluated by the average number of iterations and the range of values.If testing of the performance is based on the processing time, the results of both methods are not different.

**6. Conclusion**

In this paper, the performances of the PSO algorithm with nonlinear decreasing inertia weight and constriction factor are evaluated by experimental studies of five nonlinear functions. The experiment results show that the PSO algorithm with nonlinear decreasing inertia weight has the ability to converge faster than the PSO algorithm with constriction factor.

The results also show not only that the average number of iterations in each function that applies constriction factor were more than with decreasing inertia weight, but also that the target error was not achieved within 800 iterations for all runs in the Alpine function.Applying the nonlinear inertia weight function to PSO improves its performance. Adjusting the parameters directly affects, and can benefit, evaluation in the search space. Therefore, before real world application of the PSO with nonlinear decreasing inertia weight,nonlinear inertia weight parameters should be selected with care.

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