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[.]
Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Contents lists available at ScienceDirect Engineering Science and Technology, an International Journal journal homepage: www.elsevier.com/locate/jestch Full Length Article A new adaptive differential evolution optimization algorithm based on fuzzy inference system M Salehpour, A Jamali ⇑, A Bagheri, N Nariman-zadeh 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 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 proposed for solving complex optimization problems The proposed algorithm uses fuzzy logic inference system 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/) Introduction Evolutionary algorithms (EAs), motivated by the natural evolution of species [1], are popular for their ability to handle nonlinear and complex optimization problems [2] EAs are often called metaheuristic 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 algorithms 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 problems [4] In addition, the stochastic global optimizations can discover global minimum without trapping in the local minima [3] One of the recently developed meta-heuristic methods is differential 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 nonconvex, 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 ⇑ Corresponding author E-mail address: ali.jamali@guilan.ac.ir (A Jamali) Peer review under responsibility of Karabuk University shown that DE has good convergence characteristics and overcomes 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 (C r ) leads to the higher diversity of the population but, a smaller one causes local exploitation [20] Consequently, it could be readily observed that selecting a proper control parameter is considerably an important issue The mutation factor is the most sensitive one F ½0; 2 is allowable in theory [9,17,21] but F ð0; 1Þ is more effectual in reality As a matter of fact, F ½0:4; 0:95 seems a proper range while a good first choice can be F ½0:7; 0:9 [9] The crossover probability as Cr ½0; 1 is acceptable in theory [17], but Cr ½0:1; 0:8 sounds a proper range, and the first choice which can be convenient to be used is Cr ¼ 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 This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.004 M Salehpour et al / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx 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 adjustment of DE have been revealed Fuzzy logic plays a pivotal role in this category As a matter of fact, fuzzy logic is a knowledgebased 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 combination 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 evolution 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 evolution algorithm to adjust the mutation 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 dynamically adapt the mutation factor of conventional differential evolution 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 literature (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 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, xi can be shown as: xGi ¼ ðxG1;i ; xG2;i ; ; xGd;i ị; i ẳ 1; 2; ; n ð1Þ 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 solutions 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 xi at any generation, three distinct vectors xr1 , xr2 , and xr3 are randomly selected, and then a socalled mutant vector (perturbed or donor vector) is generated by applying the mutation scheme: v Gi ẳ xGr ỵ FxGr2 xGr3 Þ; r1 –r –r –i ð2Þ The constant F 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 F ½0; 2, but in practice, a scheme with F 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 ẳ Fxr2 xr3 ị is added to the base vector indicated by xr1 to generate a mutant vector v i , and as a result, such perturbation defines the direction and length of the search space [21] Secondly, the crossover operator amalgamates the mutant vector ðv Gi Þ with the parent vector (target vector) ðxGi Þ to create a socalled trial vector ðuGi Þ 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 probabilistic formula: uGj;i ¼ G > < v j;i ifri C r or j ¼ J r ; > : Otherwise: xGj;i j ¼ 1; 2; ; d ð3Þ in which ri is a random number extracted from the interval ½0; 1 [17], Jr is used to guarantee that uGi –xGi , which may improve the efficiency of the searching ability of the algorithm In addition, Cr ½0; 1 [17] is the crossover probability (crossover rate) as mentioned 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 intervals f1; 2; ; ng, and the trial vector is created according to the formula below: G > < v j;i if jfk; < k ỵ 1>n ; ; < k ỵ L 1>n g ; uGj;i ¼ j ¼ f1; 2; ; ng > : G xj;i Otherwise: ð4Þ The main difference between binomial and exponential crossover 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 subsequences The influence of this difference on the performance of differential evolution is not fully understood yet Choosing Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.004 M Salehpour et al / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Fig Flowchart of proposed fuzzy differential evolution Table Fuzzy rule-based system used here Output of fuzzy system is Mutation Factor Number of Generation Diversity Low Medium High 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 relation to the other one [17] Therefore in this paper, binomial crossover has been applied to generate the so-called trial vector The selection operator performs based on the one-to-one competition between the parent vector (target vector) ðxGj;i Þ and the trial vector ðuGj;i Þ 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: Low Medium High Medium Low Very Low High Medium Low Very High High Medium ( xGỵ1 i ẳ uGi if f ðuGi Þ f ðxGi Þ xGi 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 crossover probability (C r ) 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 various schemes with the general convention as, DE=x=y=z in which x shows type of choosing the base vector in mutation scheme indicated 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 Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.004 M Salehpour et al / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx some of the popular variants of DE is presented below: DE=rand=1= [42]: v Gi ẳ xGr ỵ FxGr2 xGr3 Þ ð6Þ xGr2 Þ ð7Þ DE=best=2= [42]: v Gi ẳ xGbest ỵ FxGr v Gi ẳ xGr ỵ FxGr2 xGr3 ị ỵ FxGr4 xGr5 Þ ð9Þ DE=current to best=1= [44]: DE=rand=1= [42]: v Gi ẳ xGbest ỵ FxGr DE=rand=2= [43]: xGr2 ị ỵ FxGr3 xGr4 ị 8ị v Gi ẳ xGi ỵ FxGbest xGi ị þ FðxGr xGr2 Þ ð10Þ In which, indices r1 through r are distinctive integers randomly selected from f1; 2; ; ng and also are different from index i Besides, xGbest is the best vector at the G-th generation which has been found so far Fuzzy differential evolution Fig Membership functions of Number of Generation as one of the two inputs of the fuzzy system Fig Membership functions of Population Diversity as one of the two inputs of the fuzzy system 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 diversity 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 better to be low for fine-tuning the global optimal solutions and hastening 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 population are packed together and their relative distances are low, the low value of F can be effective and when those aforesaid distances 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 process by changing the number of generation and value of population diversity The flowchart of the proposed method is shown in Fig For assessing population diversity, the formula written below is used: Pn1 Pn Div ersirtyGị ẳ Fig Member functions of Mutation Factor as output of the fuzzy system i¼1 xi Gịxii Gị iiẳiỵ1 HL 11ị 2dn 1Þn In which, L and H indicate low and high boundary constraints of each chromosome that limit the feasible area The abovementioned 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 Table Membership functions of the Number of Generation and Value of mutation factor Number of Generation as one of the fuzzy inference system Value of mutation factor as output of the fuzzy system Maximum value Interval Membership function1 Membership function2 Membership function3 Membership function4 Membership function5 5000 [0 5000] [508.2 639.1] [543.6 2310] [513.8 4045] – – [0 1] [0.04459 0.1475] [0.04247 0.25] [0.06158 0.3525] [0.08281 0.5025] [0.1232 0.705] Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.004 M Salehpour et al / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Table Membership functions of the population diversity as one of the input of fuzzy system for each of the test functions Maximum value of population diversity Interval Membership function1 Membership function2 Membership function3 Sphere Function Griewank Function Schwefel Function Rastringin Function Ackley Function Rosenbrock Function 41.3420 [0 41.3420] [2.926 6.89] [5.56 14.13] [5.852 27.56] 41.6127 [0 41.6127] [2.945 6.935] [5.596 14.22] [5.89 27.74] 45.0907 [0 45.0907] [3.191 7.515] [6.064 15.41] [6.383 30.06] 41.3420 [0 41.3420] [2.926 6.89] [5.56 14.13] [5.852 27.56] 2.7546 [0 2.7546] [0.195 0.4591] [0.3704 0.9412] [0.3899 1.836] 0.1681 [0 0.1681] [0.0119 0.02801] [0.0226 0.05742] [0.02379 0.112] next section) will be depicted in section Furthermore, fuzzy rules of the aforementioned system are given in Table 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]: f x ị ẳ at x ẳ 0; ; 0ị Formula: 12ị jẳ1 Fig Optimal solutions obtained by the proposed method versus the ones obtained by the classical differential evolution for Sphere, Griewank and Schwefel function ð13Þ Griewank function Formula: ! d d Y X xj f xị ẳ x cos p ỵ 4000 jẳ1 j j jẳ1 Sphere function d X f xị ẳ x2j As discussed earlier, d represents the dimension of the optimization problem Search domain:The hypercube xj ½500; 500 is utilized for the analysis.Global minimum: 14ị Search domain:The hypercube xj ẵ500; 500 is utilized for the analysis Global minimum: Fig Optimal solutions obtained by the proposed method versus the ones obtained by the classical differential evolution 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 Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.004 M Salehpour et al / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Table 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 100 500 1000 2000 3000 4000 5000 Sphere Function Griewank Function Schwefel Function Rastringin Function Ackley Function Rosenbrock Function DE FDE DE FDE DE FDE DE FDE DE FDE DE FDE 64,225 19,013 347 0.093 2.9 105 6.7 109 2.002 1012 227,337 7.3 2.2 105 4.3 1017 4.6 1031 2.4 1046 1.4 1062 149 6.29 1.08 7.49 103 2.6 106 5.9 1010 1.3 1013 53.3 0.19 1.1 106 0 0 11,125 6272 2549 0.83 8.8 104 6.3644 104 6.36378 104 11,660 5799 671 6.36379 104 6.36378 104 6.36378 104 6.36378 104 600,786 22,855 824 139 61.7 27.3 0.202 231,166 279.5 144.3 85.3 51.3 0.91 17.3 7.02 2.19 0.013 2.02 104 3.14 106 5.31 108 14.7 0.16 1.8 104 2.5 1010 7.8 1015 7.99 1015 4.44 1015 2177 428.6 213 100 57.7 46.22 43.11 1030 67.9 45.1 43.3 42.9 42.70 42.18 f x ị ẳ at x ẳ 0; ; 0ị 15ị Schwefel function Formula: f xị ẳ 418:9829d d q X xj sin jxj j f xị ẳ 10d ỵ d X ẵx2j 10 cos 2pxj ị 18ị jẳ1 16ị jẳ1 Search domain:The hypercube xj ẵ500; 500 is utilized for the analysis.Global minimum: f x ị ẳ at x ẳ 420:9687; ; 420:9687ị Formula: ð17Þ Rastringin function Fig Curve of variation of mutation factor values versus the number of generation obtained by the proposed method for Sphere, Griewank and Schwefel function Search domain:The hypercube xj ½500; 500 is utilized for the analysis Global minimum: f x ị ẳ at x ẳ 0; ; 0ị 19ị Ackley function Formula: Fig 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 Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.004 M Salehpour et al / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ! u d d u1 X 1X 2A t @ exp f xị ẳ a:exp b: x coscxj ị ỵ a ỵ exp1ị d jẳ1 j d jẳ1 20ị Search domain:The hypercube xj ½32:768; 32:768 is utilized for the analysis Global minimum: f x ị ẳ at x ẳ 0; ; 0Þ ð21Þ Rosenbrock function Formula: f ðxÞ ẳ d1 X 2 ẵ100xjỵ1 x2j ị ỵ xj 22ị jẳ1 Search domain:The hypercube xj ½2:048; 2:048 is utilized for the analysis Global minimum: f x ị ẳ at x ẳ 1; ; 1Þ ð23Þ 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 Fig Curve of variation of population diversity versus the number of generation obtained by the proposed method for Sphere, Griewank and Schwefel function 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 to 32.768 and 32.768, respectively, for the case of Ackley 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 IntelÒ CoreTM 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 in Figs 2–4 Based on the properties of Table 1, three Gaussian membership functions, named, Low, 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 Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.004 1030 67.9 45.1 43.3 42.9 42.70 42.18 141.7 34.4 1.55 2.2 1010 2.0 1020 0 14.7 0.16 1.8 104 2.5 1010 7.8 1015 7.99 1015 4.44 1015 13.3 0.26 5.90 104 7.5 109 1.02 1013 7.52 1015 6.8 1015 231,166 279.5 144.3 85.3 51.3 0.91 184,622 342.5 150.8 84.7 57.9 37.6 1.13 104 11,660 5799 671 6.36379 104 6.36378 104 6.36378 104 6.36378 104 10,932 4842 3.95 6.36378 104 6.36378 104 6.36378 104 6.36378 104 53.3 0.19 1.1 106 0 0 48.1 0.51 1.8 105 2.3 1015 0 227,337 7.3 2.2 105 4.3 1017 4.6 1031 2.4 1046 1.4 1062 191,810 19.9 2.4 104 3.7 1014 5.6 1024 9.4 1034 1.4 1043 100 500 1000 2000 3000 4000 5000 Rosenbrock Function Reference [30] Method proposed by this work Ackley Function Reference [30] Method proposed by this work Rastringin Function Reference [30] Method proposed by this work Method proposed by this work FDE proposed by this work Schwefel Function Reference [30] Griewank Function Reference [30] Sphere Function Reference [30] Generations Medium and High are considered as inputs of the fuzzy system (depicted in Figs and 3) and similarly, five Gaussian membership functions, named Very Low, Low, Medium, High, and Very High as the output of that (depicted in Fig 4) Besides, more descriptions of the characteristics of Membership functions of the inputs and output of the fuzzy system used here are given in Tables and Figs and show the obtained optimal solutions from the proposed 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 objective 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 considering 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 in Figs 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 functions 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 9.56 104 and 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 in Table Comparison of the results of those two aforementioned methods 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 classical 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 and exhibit the variation curve of obtained mutation factor values by using fuzzy differential evolution versus the number 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 and 10 show the population diversity variation versus the number of generation for different test functions Table represents 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 optimization are somehow powerful and fast But, this evaluation represents the very good behavior of the proposed method of this work in relation with the ones reported in the literature Method proposed by this work M Salehpour et al / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Table The values of objective functions for the mean value of thirty runs of each generation for the method of this work and the one of [30] Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.004 M Salehpour et al / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Table The input values of fixed parameters of this paper [39–40] 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 ; kp1 ;,kp2 ;, l1 and l2 which 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 according to the Ref N [39–40] Design variables indicated by 50000 < K ss m < 150000, N Ns 10000 < K s1 ; K s2 m < 20000, 1000 < C ss m < 4000, 500 < C s1 ; < 2000, and < rðmÞ < 0:5 denote seat stiffness coefficient, C s2 Ns m stiffness coefficients for vehicle suspension, seat damping coefficient, damping coefficients for vehicle suspension and seat position in relation to the center of mass, respectively Further, subscripts and 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 in Fig 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 sm2 ), vertical velocity of form ward tire (z_ s ), vertical velocity of rear tire (z_ ms ), relative displacement between sprung mass and forward tire ðd1 Þ and relative displacement between sprung mass and rear tire ðd2 Þ The aforementioned objective function is formulated as follows: f ¼ Z T a1 jzc j ỵ a2 jd1 j ỵ a3 jd2 j ỵ a4 jz_ j ỵ a5 jz_ jÞdt T ð24Þ l1 l2 m1 m2 mc ms Is kp1 kp2 1.011 m 1.803 m 40 kg 35.5 kg 75 kg 730 kg 1230 kg m2 175,500 N/m 175,500 N /m Fig 12 Double-bump excitation adopted from references [39,40] where a1 ¼ 10, a2 ¼ a3 ¼ 200 and a4 ¼ a5 ¼ 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 Fig 11 Vehicle vibration five-degree of freedom model with passive suspension adopted from references [39,40] Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.004 0.4336 18.796 20.350 0.4328 0.4486 20.348 26.258 0.4470 0.4335 19.001 19.761 0.4330 0.4380 25.6495 18.971 0.4334 0.4316 21.127 35.587 0.4279 0.4335 19.111 0.5087 26.220 19.123 f 0.4333 0.4324 FDE Comparison of the obtained result of this paper with the ones of the literature [43,44] has proven the effectiveness of the proposed 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 at Table As can be easily seen in Table 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 readily 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 C , C , C , and C (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, C , C , C , and C in terms of €zc , d1 ; d2 ; z_ and z_ 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 C , thirdly between the ones of G and C , fourthly between the ones of G and C , and finally between the ones of G and C 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 d1 the variation are about 25% decrease, 27% decrease, 26% decrease, approximately without change and z_ 25% 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 Further, it can be found that the variation of the values of z_ are about 2.1% increase, 2.2% increase, 2.1% increase, approximately without change and 2% decrease, respectively, which those afore C2 mentioned 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 z_ ; 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, C , C , C3 , and C4 are shown in Fig 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 20.558 0.0895 0.0588 0.4176 0.1194 0.0579 0.4092 0.1192 0.0515 0.4092 0.1977 0.0500 0.4157 0.0947 0.0586 0.4157 0.0890 0.0815 0.4175 0.0865 0.1512 0.4182 0.0895 0.0625 0.4176 0.1210 0.0705 0.4090 0.2352 0.1011 0.4227 0.0926 0.0608 0.4165 0.1230 0.0575 0.4085 0.1484 0.1282 0.4059 0.0960 0.0601 0.4151 d1 ðmÞ d2 ðmÞ z_ ms z_ ms 1999.99 0.5000 2.7889 0.4960 2.8929 0.4991 2.950 0.4996 3.6546 0.4997 2.7913 0.4929 2.8094 0.0000 3.6054 0.4978 2.7956 0.4711 2.9455 0.0028 5.1074 0.4776 2.8181 0.4963 2.9131 0.4992 3.3315 0.4981 2.7947 rðmÞ €zc sm2 10000.0 10196.1 1982.35 1999.61 19999.9 19998.5 1999.908 1999.742 10020.0 10003.8 1432.873 785.6314 10000.1 10037.3 1879.508 1613.628 10036.3 10000.4 1112.495 1935.256 10107.8 548.9686 1947.486 1989.187 19907.7 10020.7 10029.6 10000.0 1999.9 1294.1 10000.0 10000.3 1336.8 500.2 10001.8 10001.4 1843.6 1999.9 10001.6 10000.1 1999.9 1997.3 10012.4 10000.4 1259.9 502.1 11852.1 10028.0 1906.3 940.1 1807.0 10003.3 10003.2 10006.9 1232.8 149999.0 3999.3 2788.2 3482.9 3959.2 3941.8 3768.222 3999.4 3926.0 3484.0 3737.8 3292.9 3119.2 1796.6 3964.8 mN K s1 m Ns C s1 m N K s2 m Ns C s2 m m Ns C ss F 144,902 14 6265.7 C4 B4 149232.8 147107.3 A4 C3 147210.7 50006.2 B3 A3 147769.6 139269.5 C2 B2 50504.3 131574.7 A2 C1 144263.6 149700.2 B1 A1 146728.6 N K ss G [This work] M Salehpour et al / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx Table The values of objective function, its components and the associated design variables of the optimum points of this work and the design points A1 , B1 , C , A2 , B2 , C ,A3 , B3 , C , A4 , B4 and C of [40], design point F of [39] and point G of this work 10 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 Sci Tech., Int J (2017), http://dx.doi.org/10.1016/j.jestch.2017.01.004 M Salehpour et al / Engineering Science and Technology, an International Journal xxx (2017) xxx–xxx The significant matters mentioned above confirm the superiority of the proposed method of this work; especially with regarding the considerable fact that the optimum point suggested by this work is obtained by single objective optimization but the optimum points reported by [39,40] are resulted by multi-objective optimization As a result, this crucial fact discloses the ability, efficacy and ascendancy of the proposed method of this work Conclusion A method based on the hybrid usage of fuzzy logic and differential evolution proposed here The ability of the proposed algorithm for 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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. .. Tasgetiren, Differential evolution algorithm with ensemble of parameters and mutation strategies, Appl Soft Comput 11 (2011) 1679–1696 [3] S Kitayama, M Arakawa, K Yamazaki, Differential evolution as... of this work and the one of [30] Please cite this article in press as: M Salehpour et al., A new adaptive differential evolution optimization algorithm based on fuzzy inference system, Eng Sci