A new adaptive differential evolution optimization algorithm based on fuzzy inference system

A new adaptive differential evolution optimization algorithm based on fuzzy inference system Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Contents lists available at[.]

Full Length Article

A new adaptive differential evolution optimization algorithm based on

fuzzy inference system

Department of Mechanical Engineering, Faculty of Engineering, University of Guilan, Rasht 3756, Iran

a r t i c l e i n f o

Article history:

Received 17 October 2016

Revised 8 January 2017

Accepted 16 January 2017

Available online xxxx

Keywords:

Differential evolution

Fuzzy logic

Mutation factor

Population diversity

Optimization

Vehicle vibration model

a b s t r a c t

In this paper, a new version of differential evolution (DE) with adaptive mutation factor has been pro-posed for solving complex optimization problems The propro-posed algorithm uses fuzzy logic inference sys-tem to dynamically tune the mutation factor of DE and improve its exploration and exploitation In this way, two factors, named, the number of generation and population diversity are considered as inputs and, one factor, named, the mutation factor as output of the fuzzy logic inference system The performance of the suggested approach has been tested firstly by using some popular single objective test functions It has been shown that the proposed method finds better solutions than the classical differential evolution and also the convergence rate of that is really fast Secondly, a five degree of freedom vehicle vibration model is chosen to be optimally designed by the aforesaid proposed approach Comparison of the obtained results with those in the literature demonstrates the superiority of the results of this work

Ó 2017 Karabuk University Publishing services by Elsevier B.V This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1 Introduction

Evolutionary algorithms (EAs), motivated by the natural

evolu-tion of species[1], are popular for their ability to handle nonlinear

and complex optimization problems[2] EAs are often called

meta-heuristic approaches because the structure of such optimization

process is based on the discovering issues from the experiences

of real life Most of EAs use random components during the search

process, therefore they belong to the category of the stochastic

optimization approaches [3–4] As long as meta-heuristic

algo-rithms are intrinsically non-deterministic and not sensitive to the

continuity and differentiability of the objective functions, use of

such methods contains broad range of complex optimization

prob-lems[4] In addition, the stochastic global optimizations can

dis-cover global minimum without trapping in the local minima[3]

One of the recently developed meta-heuristic methods is

differ-ential evolution (DE) presented by Storn and Price[5,6]is a fast and

robust[7,8]stochastic metaheuristic algorithm which needs not

any gradient-based data In addition, it is a population-based and

derivative-free method which can be applied for solving

non-convex, nonlinear, non-differentiable and multimodal problems

[7] Besides, real numbers are applied in DE as solution strings,

so no encoding and decoding is required[9] Empirical results have

shown that DE has good convergence characteristics and over-comes other popular EAs [10] DE uses three main operators, namely, mutation, crossover and selection, respectively[5,6] Due

to its simple structure, simple implementation, fast convergence and robustness, DE has been widely applied to the optimization problems arising in some fields of science and engineering, such

as robot control[11], controller design[12], data clustering[13], optimal design [14], microbiology [15], image processing [16]

and so forth

It is very important to notice that the behavior of DE largely depends on the two parameters named mutation and crossover

[9,17–19] As widely discussed in the literature, a larger mutation factor (F) can be effectual in global search; on the other hand, a smaller one can hasten the convergence rate In addition, the larger crossover probability (Cr) leads to the higher diversity of the pop-ulation but, a smaller one causes local exploitation [20] Conse-quently, it could be readily observed that selecting a proper control parameter is considerably an important issue The muta-tion factor is the most sensitive one F2 ½0; 2 is allowable in theory

[9,17,21]but F2 ð0; 1Þ is more effectual in reality As a matter of fact, F2 ½0:4; 0:95 seems a proper range while a good first choice can be F2 ½0:7; 0:9[9] The crossover probability as Cr2 ½0; 1 is acceptable in theory[17], but Cr2 ½0:1; 0:8 sounds a proper range, and the first choice which can be convenient to be used is Cr¼ 0:5

[9] Even though DE is a good and fast algorithm, but it has some deficiencies [22] Global exploration ability of DE seems proper

http://dx.doi.org/10.1016/j.jestch.2017.01.004

2215-0986/Ó 2017 Karabuk University Publishing services by Elsevier B.V.

Peer review under responsibility of Karabuk University.

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enough which it can recognize the feasible region of the global

optimum, but its local exploitation one is considered slow at

fine-tuning the solution[1,18,23–25] In addition, DE suffers from

loss of diversity which happens while the population stagnation or

premature convergence occurs [22] Besides, DE is a parameter

dependent algorithm, and, therefore it is a difficult task to adapt

its control parameter for various problems[23,24] Furthermore,

by increasing the dimensionality of the optimization problem,

the efficacy of the algorithm debases [23,24,26] Consequently,

the aforementioned drawbacks make the scholars find methods

to improve performance and increase the effectiveness of DE Such

modifications, not only for DE but also for other EAs, can generally

classified into two main categories The first one are based on the

tuning or controlling the control parameters[27] of DE, and the

second one concentrates on the hybridization of DE with other

optimization methods such as particle swarm optimization[28]

or so forth

In terms of tuning the control parameters of DE (as done in this

work), recently, some methods, based on the dynamical

adjust-ment of DE have been revealed Fuzzy logic plays a pivotal role

in this category As a matter of fact, fuzzy logic is a

knowledge-based system considering a set of fuzzy rules proposed by Zadeh

[29]that shows the relationship between the input(s) and output

(s) of the system Some hybridization of differential evolution

and fuzzy logic is reviewed here

Patricia Ochoa et al.[30]proposed a method based on the

com-bination of fuzzy logic and DE to dynamical adjustment of the

mutation parameter In this case, fuzzy logic provides optimal

parameters for improving the efficiency of DE It has been shown

that the differential evolution algorithm with Fuzzy F (mutation

factor) Decrease performs better than the differential evolution

algorithm with F increase Liu and Lampinen [31] presented an

approach based on the hybrid application of the differential

evolu-tion algorithm and fuzzy logic The aim of this methodology is to

dynamically adapt the population size of the search process The

obtained results have shown that the adaptive population size

might lead to the higher convergence velocity and, of course,

decrease the number of the model assessments After that, Liu

and Lampinen[32], suggested a fuzzy adaptive differential

evolu-tion algorithm to adjust the mutaevolu-tion and crossover parameters

using a set of standard test functions It has been shown that the

proposed method works better than the original DE when the

dimensionality of the problem is high Furthermore, this method

was applied to hasten the convergence rate of DE by the use of

adaptive parameters

More description of the hybrid usage of EAs and fuzzy logic can

be seen in[33–38]

In this paper, fuzzy logic inference system is used to

dynami-cally adapt the mutation factor of conventional differential

evolu-tion In this way, two main factors, namely, number of

generation and population diversity of each generation which

may affect the exploration and exploitation ability of the algorithm

are selected as inputs and mutation factor as output of the fuzzy

logic inference system The ability of the proposed algorithm for

resolving single optimization problems is appraised by using six

well-known benchmark functions Afterwards, the proposed

method has been used for the single optimization of the five degree

of freedom vehicle vibration model for analyzing the performance

of the proposed method on the engineering problems The

obtained results show the very good behavior of the proposed

method, and also, comparison with the ones reported in the

liter-ature (two categories of previous works used here which contains

one work related to benchmark functions [30] and two works

related to the vehicle vibration model[39,40]) demonstrates the

superiority of the suggested method of this work

2 Differential evolution Like all other evolutionary algorithms, DE uses a population of potential solutions and genetic operators to seek for the optimums through feasible search space For each solution vector indicated by

xi, at any generation G, xican be shown as:

xG

i ¼ ðxG

;i; xG

;i; ; xG

in which, n indicates the number of population which is composed

of d-elements This vector is called chromosome or genome Differential evolution comprises three major operators, namely, mutation, crossover and selection Initially a population of n solu-tions is randomly generated using uniform distribution, and then the aforesaid operators are applied to the population to produce next generation In this way, for each vector xi, mutation scheme

is carried out firstly For each vector xiat any generation, three dis-tinct vectors xr 1, xr 2, and xr 3are randomly selected, and then a so-called mutant vector (perturbed or donor vector) is generated by applying the mutation scheme:

vG

i ¼ xG

r 1þ FðxG

r 2 xG

r 3Þ; r1–r2–r3–i ð2Þ

The constant F2 ½0; 2 [9,17,21] in the previously mentioned equation, is a mutation factor (scale factor or differential weight) which affects the diversity of the set of mutant vectors and helps

to manage the trade-off between exploration and exploitation of the search process[21] Essentially, in theory F2 ½0; 2, but in prac-tice, a scheme with F2 ½0; 1 is more efficient and stable, and it seems that it is used by almost all the studies in the literature

It is easily seen that the perturbation term indicated by

d ¼ Fðxr 2 xr 3Þ is added to the base vector indicated by xr1to gen-erate a mutant vectorvi, and as a result, such perturbation defines the direction and length of the search space[21]

Secondly, the crossover operator amalgamates the mutant vec-torðvG

iÞ with the parent vector (target vector) ðxG

iÞ to create a so-called trial vector ðuG

iÞ The crossover scheme is classified into two forms, namely, binomial and exponential In the binomial scheme, the trial vector is generated according to the next proba-bilistic formula:

uG

j ;i¼

vG j;i ifri6 Cror j¼ Jr;

j¼ 1; 2; ; d

xG

j ;i Otherwise:

8

>

in which ri is a random number extracted from the interval½0; 1

[17], Jris used to guarantee that uG

i–xG

i, which may improve the effi-ciency of the searching ability of the algorithm In addition,

Cr2 ½0; 1[17]is the crossover probability (crossover rate) as men-tioned earlier

In the exponential scheme, a section of the mutant vector is chosen, and this section commences with an integer k and length

L randomly selected from the intervalsf1; 2; ; ng, and the trial vector is created according to the formula below:

uG

j ;i¼

vG

j ;i if jfk; < k þ 1>n; ; < k þ L  1>ng ;

j¼ f1; 2; ; ng

xG

j ;i Otherwise:

8

>

>

ð4Þ

The main difference between binomial and exponential cross-over is the fact that while in the binomial case the components inherited from the mutant vector are arbitrarily selected, in the case of exponential crossover they form one or two compact sub-sequences The influence of this difference on the performance of differential evolution is not fully understood yet Choosing

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between one of the two crossover schemes mentioned earlier is a

difficult task due to the fact that the superiority of one variant

for a given class of problems over another is not clearly uncovered

[17] But, the binomial scheme is the one which is frequently used

in the literature[17,41]due to its simple implementation in

rela-tion to the other one[17] Therefore in this paper, binomial

cross-over has been applied to generate the so-called trial vector

The selection operator performs based on the one-to-one

com-petition between the parent vector (target vector)ðxG

j;iÞ and the trial vectorðuG

j;iÞ In this way, it favors the better one between those two

aforementioned vectors with respect to their fitness value of the

assuming objective function If such value of the trial vector is less

than or equal to that of the parent vector, the trial vector will enter

to the next generation Otherwise, the parent vector will survive to

go to the next generation[20] This operator can be described by

the formula below:

xGþ1i ¼ uGi if fðuG

iÞ 6 f ðxG

iÞ

xG

i Otherwise:

(

ð5Þ

Consequently, the effectiveness of DE is largely depended on both mutation and crossover schemes and the values of their two associated parameters namely, mutation factor (F) and cross-over probability (Cr) as mentioned earlier[9,17–19]

In fact, different strategies can be adopted in DE based on the way of using mutation and crossover schemes which results in var-ious schemes with the general convention as, DE=x=y=z in which x shows type of choosing the base vector in mutation scheme indi-cated by rand (stands for random) or best Furthermore, y indicates the number of difference vectors, and z is the type of crossover scheme indicated by exp and bin which stand for exponential and binomial, respectively Hence, DE=rand=1= presents the basic

DE variant using random mutation and one difference vector with either a binomial or exponential crossover scheme A quick look on

Fig 1 Flowchart of proposed fuzzy differential evolution.

Table 1

Fuzzy rule-based system used here.

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some of the popular variants of DE is presented below:

DE=rand=1=[42]:

vG

i ¼ xG

r 1þ FðxG

r 2 xG

 DE=rand=1=[42]:

vG

i ¼ xG

bestþ FðxG

r 1 xG

 DE=best=2=[42]:

vG

i ¼ xG

bestþ FðxG

r 1 xG

r 2Þ þ FðxG

r 3 xG

 DE=rand=2=[43]:

vG

i ¼ xG

r1þ FðxG

r2 xG

r3Þ þ FðxG

r4 xG

 DE=current  to  best=1=[44]:

vG

i ¼ xG

i þ FðxG best xG

iÞ þ FðxG

r 1 xG

In which, indices r1through r5are distinctive integers randomly selected from f1; 2; ; ng and also are different from index i Besides, xG

best is the best vector at the G-th generation which has been found so far

3 Fuzzy differential evolution

In order to overcome the drawbacks of DE discussed earlier, a method based on the hybrid usage of fuzzy logic and differential evolution called Fuzzy Differential Evolution (FDE) is proposed here In this way, a fuzzy system considering the variation of two parameters, namely, number of generation and population diver-sity is applied to improve the performance of DE algorithm It could

be readily observed that in low number of generations, there is a considerable necessity to explore through the search space to find the approximate zone of the global optima, therefore the value of F

is better to be high But, on the other hand, in high number of that when approaching toward global optimal solution, its value is bet-ter to be low for fine-tuning the global optimal solutions and has-tening the convergence rate Consequently, by considering the mentioned facts, it is obvious that the value of F could not be fixed during the searching process Consequently, depending upon changing the number of generations, it seems that the value of F

is better to be changed for the purpose of seeking better feasible solutions Another factor which seems to be effective on the value

of F is population diversity In fact, when the individuals of popu-lation are packed together and their relative distances are low, the low value of F can be effective and when those aforesaid dis-tances are high, the high one of that may be practical As a result, for the low values of diversity, the low one of F sounds proper and for high ones of that, the high one of F may be effective Therefore, the mutation factor is adapted dynamically during the search pro-cess by changing the number of generation and value of population diversity The flowchart of the proposed method is shown inFig 1 For assessing population diversity, the formula written below is used:

DiversirtyðGÞ ¼

Pn1 i¼1

Pn

¼iþ1xiðGÞxHLiiðGÞ

In which, L and H indicate low and high boundary constraints of each chromosome that limit the feasible area The above-mentioned formula evaluates an average normalized distance between the individuals of the population of each generation

[21] which could be a good criterion for measuring population diversity

The fuzzy system used here is of Mamdani type containing two inputs namely, the number of generation and value of population diversity and one output which is the value of F The inputs and output of one selected test function (named Ackley described in

Fig 2 Membership functions of Number of Generation as one of the two inputs of

the fuzzy system.

Fig 3 Membership functions of Population Diversity as one of the two inputs of the

fuzzy system.

Fig 4 Member functions of Mutation Factor as output of the fuzzy system.

Table 2

Membership functions of the Number of Generation and Value of mutation factor.

Maximum value

function1

Membership function2

Membership function3

Membership function4

Membership function5 Number of Generation as one of the fuzzy

inference system

Value of mutation factor as output of the

fuzzy system

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next section) will be depicted in section 5 Furthermore, fuzzy rules

of the aforementioned system are given inTable 1

4 Benchmark functions

In order to analyze the capability and flexibility of the proposed

method, six standard and popular benchmark functions used here

which are concisely clarified as follows[30]:

 Sphere function

Formula:

fðxÞ ¼Xd

j¼1

x2

As discussed earlier, d represents the dimension of the opti-mization problem

Search domain:The hypercube xj2 ½500; 500 is utilized for the analysis.Global minimum:

 Griewank function Formula:

fðxÞ ¼ 1 4000

Xd j¼1

x2

j Yd j¼1

cos xjffiffi j p

!

Search domain:The hypercube xj2 ½500; 500 is utilized for the analysis

Global minimum:

Table 3

Membership functions of the population diversity as one of the input of fuzzy system for each of the test functions.

Fig 5 Optimal solutions obtained by the proposed method versus the ones

obtained by the classical differential evolution for Sphere, Griewank and Schwefel

function.

Fig 6 Optimal solutions obtained by the proposed method versus the ones obtained by the classical differential evolution for Rastringin, Ackley and Rosen-brock function.

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fðxÞ ¼ 0 at x¼ ð0; ; 0Þ ð15Þ

 Schwefel function

Formula:

fðxÞ ¼ 418:9829d X

d

j¼1

xjsin

ffiffiffiffiffiffiffi

jxjj q

ð16Þ

Search domain:The hypercube xj2 ½500; 500 is utilized for the

analysis.Global minimum:

 Rastringin function

Formula:

fðxÞ ¼ 10d þXd

j¼1

½x2

j  10 cos 2 pxjÞ

Search domain:The hypercube xj2 ½500; 500 is utilized for the analysis

Global minimum:

 Ackley function Formula:

Table 4

The values of objective functions for the mean value of thirty runs of each generation for both DE and FDE algorithms for several generations.

Generations Sphere Function Griewank Function Schwefel Function Rastringin

Function

Ackley Function Rosenbrock

Function

100 64,225 227,337 149 53.3 11,125 11,660 600,786 231,166 17.3 14.7 2177 1030

500 19,013 7.3 6.29 0.19 6272 5799 22,855 279.5 7.02 0.16 428.6 67.9

1000 347 2.2  10 5 1.08 1.1  10 6 2549 671 824 144.3 2.19 1.8  10 4 213 45.1

2000 0.093 4.3  10 17 7.49  10 3 0 0.83 6.36379  10 4 139 85.3 0.013 2.5  10 10 100 43.3

3000 2.9  10 5 4.6  10 31 2.6  10 6 0 8.8  10 4 6.36378  10 4 61.7 51.3 2.02  10 4 7.8  10 15 57.7 42.9

4000 6.7  10 9 2.4  10 46 5.9  10 10 0 6.3644  10 4 6.36378  10 4 27.3 0.91 3.14  10 6 7.99  10 15 46.22 42.70

5000 2.002  10 12 1.4  10 62 1.3  10 13 0 6.36378  10 4 6.36378  10 4 0.202 0 5.31  10 8 4.44  10 15 43.11 42.18

Fig 7 Curve of variation of mutation factor values versus the number of generation

obtained by the proposed method for Sphere, Griewank and Schwefel function.

Fig 8 Curve of variation of mutation factor values versus the number of generation obtained by the proposed method for Rastringin, Ackley and Rosenbrock function. Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng

fðxÞ ¼ a:exp b:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 d

Xd j¼1

x2 j

v u

0

@

1

d

Xd j¼1

cosðcxjÞ

!

þ a þ expð1Þ

ð20Þ

Search domain:The hypercube xj2 ½32:768; 32:768 is utilized

for the analysis

Global minimum:

 Rosenbrock function

Formula:

fðxÞ ¼Xd1

j¼1

½100ðxjþ1 x2

jÞ2þ 1  xj

Search domain:The hypercube xj2 ½2:048; 2:048 is utilized for

the analysis

Global minimum:

5 Results of the optimization of the benchmark functions

In case of conventional differential evolution, a population of

250 individuals with a crossover probability of 0.1 and mutation

probability of 0.9 has been used in 5000 generations Further, the dimension of the optimization problem is chosen to be 50 and lower and upper bound assigned to the optimization problem are

500 and 500, respectively Since, the selected search space is vast, the magnitude of the lower and upper bound of the search space decreases to32.768 and 32.768, respectively, for the case of Ack-ley Function, and, also,2.048 and 2.048, respectively, for the case

of Rosenbrock Function In case of fuzzy differential evolution (FDE), all of the above-mentioned conditions are used except the value of mutation factor which is adapted dynamically during the searching process It must be noted that both classical DE and proposed algorithm have been executed 30 times for each test function and the mean value for each test instance is given in the manuscript

A desktop system using an IntelÒCoreTM2 CPU 6420 @ 2.13 GHz 2.13 GHz as processor and a 4.00 GB RAM as installed memory is used to run both procedures in the MATLAB R2015a software Run-time results show that the elapsed time for executing classical

DE and FDE for Sphere, Griewank, Schwefel, Rastringin, Ackley and Rosenbrock function are about 258.4 and 2137.3, 270.8 and 2169.4, 278.7 and 2368.3, 248.3 and 2144.2, 265.1 and 2142.1, and 208.5 and 2139.8 s, respectively

As discussed earlier, the inputs and output of Ackley function as

a selected benchmark are shown inFigs 2–4 Based on the proper-ties ofTable 1, three Gaussian membership functions, named, Low,

Fig 9 Curve of variation of population diversity versus the number of generation

obtained by the proposed method for Sphere, Griewank and Schwefel function.

Fig 10 Curve of variation of population diversity versus the number of generation obtained by the proposed method for Rastringin, Ackley and Rosenbrock function. Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng

Medium and High are considered as inputs of the fuzzy system

(depicted inFigs 2 and 3) and similarly, five Gaussian membership

functions, named Very Low, Low, Medium, High, and Very High as

the output of that (depicted inFig 4) Besides, more descriptions of

the characteristics of Membership functions of the inputs and

out-put of the fuzzy system used here are given inTables 2 and 3

Figs 5 and 6show the obtained optimal solutions from the

pro-posed method versus the classical differential evolution for all test

functions As can be readily observed in the aforesaid figures, for

the case of Sphere Function, in generation 848 the value of

objec-tive function by using FDE reaches to 8.16 104, but by using

DE, in generation 2538 reaches to the value of 9.99 104 By

con-sidering that aforementioned curve again, it is figured out that the

values of the function by using FDE and DE in the generations 2538

and 848 are about 3.05 1024 and 1073, respectively Hence,

these obtained values represent the really good behavior of the

proposed algorithm

As can be evidently discerned inFigs 5 and 6, in generations

such as, 740 for Griewank Function, 1540 for Schwefel Function,

4186 for Rastringin Function, 870 for Ackley Function and 1200

for Rosenbrock Function, the values of aforesaid objective

func-tions resulted by FDE and DE algorithms are about, 9.21 104

and 1.76, 9.99 104 and 41.20, 9.64 104 and 23.90

9.56 104and 2.91, and 44.4 and 173.7, respectively The values

of the objective functions for the best solutions of each generation

for both DE and FDE algorithms for several generations such as

100, 500, 1000, 2000, 3000, 4000 and 5000 are shown inTable 4

Comparison of the results of those two aforementioned

meth-ods shows the very good performance and promising results of

the proposed approach of this work Such collation demonstrates

that in most cases the obtained results of the proposed method

of this work are considerably better than the ones obtained by

clas-sical DE Further, the convergence rate of fuzzy DE significantly

improves in relation to the classical one These important issues

confirm the fact that the mutation factor appraised by fuzzy

method strengthens the DE algorithm performance and makes up

the deficiencies of that which are mentioned before Furthermore,

it can be concluded that the proposed algorithm acts substantially

in terms of exploration and exploitation of the search space

Figs 7 and 8exhibit the variation curve of obtained mutation

factor values by using fuzzy differential evolution versus the

num-ber of generation for different test functions It can be easily seen

through these figures that by increasing the number of generation,

the mutation factor decreases Also,Figs 9 and 10show the

popu-lation diversity variation versus the number of generation for

dif-ferent test functions

Table 5represents the comparison between the results of the

proposed method of this work and the results of the method based

on the combination of differential evolution algorithm with Fuzzy

F Decrease adopted from Ref.[30] It can be obviously perceived

from that the results of this work in many cases, vibrantly better

than the ones of Ref [30], and this matter brilliantly uncovers

the superiority of the methodology suggested here As a result, this

method can be used to resolve complex engineering optimization

problem

It is important to notice that all of the aforesaid figures and

tables are depicted based on the mean values of thirty runs of both

procedures

In the remainder of this work, the proposed method is applied

to optimize the popular 5-degree of freedom vehicle vibration

model which is used in the earlier works reported by the Ref

[39,40]in which the methods utilized for the purpose of the

opti-mization are somehow powerful and fast But, this evaluation

rep-resents the very good behavior of the proposed method of this

work in relation with the ones reported in the literature

Reference [30]

Method proposed

Reference [30]

Method proposed

Reference [30]

Method proposed

Reference [30]

Method propo

Reference [30]

Method proposed

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6 Optimization of the vehicle vibration model using fuzzy

differential evolution

A 5-degree of freedom vehicle vibration model with passive

suspension adopted from Ref [39–40]is shown in Fig 11 This

model contains one sprung mass that connects to three unsprung

masses (display the tires and seat) The values of vehicle

fixed parameters indicated by, m1;, m2;, mc;, Is; kp 1;,kp 2;, l1 and

l2which represented as forward tire mass, rear tire mass, seat

mass, sprung mass, momentum inertia of sprung mass, forward

tire stiffness coefficient, rear tire stiffness coefficient, forward

and rear suspension position in relation to the center of

mass, respectively, are given in Table 6 according to the Ref

[39–40] Design variables indicated by 50000< Kss Nm

 

< 150000,

10000< Ks1; Ks2 mN

 

< 20000, 1000 < Css Nsm

 

< 4000, 500 < Cs1;

Cs2 Nsm

 

< 2000, and 0 < rðmÞ < 0:5 denote seat stiffness coefficient,

stiffness coefficients for vehicle suspension, seat damping

coeffi-cient, damping coefficients for vehicle suspension and seat position

in relation to the center of mass, respectively Further, subscripts 1

and 2 show tire axes, respectively It should be noted that in this

case study, seat type is composed of a linear spring and damper

This model is excited by a double-bump shown inFig 12

The aforementioned design variables are optimally obtained

based on the single optimization utilizing proposed method of this

work by using an objective function composing of some important

components The components used in the objective function are,

namely, vertical seat acceleration (€zc ms2

  ), vertical velocity of for-ward tire (_z1 ms

 

), vertical velocity of rear tire (_z2 ms

  ), relative dis-placement between sprung mass and forward tire ðd1Þ and

relative displacement between sprung mass and rear tireðd2Þ

The aforementioned objective function is formulated as follows:

T

 ZT

0

ða1j€zcj þa2jd1j þa3jd2j þa4j_z1j þa5j_z2jÞdt ð24Þ

where a1¼ 10, a2¼a3¼ 200 and a4¼a5¼ 10 which are called weighting coefficients and used for compromising between road holding capability and comfort In fact, the trade-off between road holding capability and comfort is a difficult task to achieve[45,46]

A population of 80 individuals with a crossover probability of 0.9 and dynamically adaptable mutation factor has been used in

240 generations for the purpose of the single optimization using

Table 6

Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng

FDE Comparison of the obtained result of this paper with the ones

of the literature[43,44]has proven the effectiveness of the pro-posed method of the present work Output of elapsed time of run

of the proposed algorithm applied on the above-mentioned vehicle model using the desktop PC with the characteristics described in the last section has been about 6156.8 s

The obtained optimum point of this work, the trade-off point suggested by Ref.[39]and the optimum points suggested by Ref

[40]are presented atTable 7 As can be easily seen inTable 7, point

G (the optimum point suggested by the method of this work) has the least value of the objective function (f) shown in Eq.(24)) amongst all of the points presented here In fact, with a thorough consideration on the previously mentioned table, it could be read-ily observed that the comparison of the results of using point G, point F (the trade-off points resulted by 5-objective optimization suggested by[39], points C1, C2, C3, and C4(the trade-off points resulted by bi-objective optimization suggested by[40]) reveals the fact that the value of the objective function f improves about 7.6%, 8.6%, 11%, 5% and 7.6%, respectively Further, it can be seen through the table that the values of the elements of the objective function f resulted by point G in comparison with those of resulted

by the points F, C1, C2, C3, and C4in terms of€zc, d1; d2; _z1and_z2

improve in many cases which prove the really good behavior of the proposed method of this paper Such comparison is described elaborately as follows by the order of firstly, comparison between the results of G and F, secondly, between the ones of G and C1, thirdly between the ones of G and C2, fourthly between the ones

of G and C3, and finally between the ones of G and C4 In fact, the collation of the values of €zc shows the variation about 3.6% decrease, 4.3% decrease, 5.3% decrease, approximately without change and 5.5% decrease, respectively For the case of d1the vari-ation are about 25% decrease, 27% decrease, 26% decrease, approx-imately without change and_z125% decrease, respectively For the case of d2 the variation are about 1.6% increase, 2.2% increase, 16.5% decrease, 28% decrease and 14% increase, respectively Fur-ther, it can be found that the variation of the values of_z1are about 2.1% increase, 2.2% increase, 2.1% increase, approximately without change and 2% decrease, respectively, which those afore C2 men-tioned increase are diminutive and not considerable In fact, with

a deep observation, it could be found that the values of are not changed and remained the same, approximately Finally, in case

of_z2; the comparison of the results reveals that all of the variations are approximately without change except the last one which shows 3.3% improvement

Time response behaviors of vertical seat acceleration resulted

by points G, F, C1, C2, C3, and C4 are shown inFig 13 As can be readily seen through the figure, time response obtained by point

G is superior to those of the other trade-off points

A1

,A3

C4

A1

B1

C1

A2

B2

C2

A3

B3

C3

A4

B4

C4

Kss

Css

Ks1

Cs1

Ks2

Cs2

€z c

m 2

d1

d2

_z 1

_z 2

Fig 13 Time response behaviors of vertical seat acceleration resulted by points G,

F, C1, C2, C3, and C4.

Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng