Optimization of Mechanical Design Problems Using Improved Differential Evolution Algorithm

Transcription

1 International Journal of Recent Trends in Enineerin Vol. No. 5 May 009 Optimization of Mechanical Desin Problems Usin Improved Differential Evolution Alorithm Millie Pant Radha Thanaraj and V. P. Sinh Department of Paper Technoloy Indian Institute of Technoloy Roorkee India. millifpt@iitr.ernet.in t.radha@ieee.or sinhfpt@iitr.ernet.in Abstract Differential Evolution (DE) is a novel evolutionary approach capable of handlin nondifferentiable non-linear and multi-modal objective functions. DE has been consistently ranked as one of the best search alorithm for solvin lobal optimization problems in several case studies. This paper presents an Improved Constraint Differential Evolution (ICDE) alorithm for solvin constrained optimization problems. The proposed ICDE alorithm differs from unconstrained DE alorithm only in the place of initialization selection of particles to the net eneration and sortin the final results. Also we implemented the new idea to five versions of DE alorithm. The performance of ICDE alorithm is validated on four mechanical enineerin problems. The eperimental results show that the performance of ICDE alorithm in terms of final objective function value number of function evaluations and converence time. Inde Terms Differential Evolution optimization Mechanical desin problems constraint optimization. I. INTRODUCTION Many real-world optimization problems are solved subject to sets of constraints. The search space in COPs consists of two kinds of solutions: feasible and infeasible. Feasible points satisfy all the constraints while infeasible points violate at least one of them. Therefore the final solution of an optimization problem must satisfy all constraints. A constrained optimization problem may be distinuished as a Linear Prorammin Problem (LPP) and Nonlinear Prorammin Problem (NLP). In this paper we have considered NLP problems where either the objective function or the constraints or both are nonlinear in nature. The eneral NLP is iven by nonlinear objective function f which is to be minimized/maimized with respect to the desin variables =... ) and ( n the nonlinear inequality and equality constraints. This can be formulated by Minimize / Maimize f () : j ( ) 0 j =... p () h k ( ) = 0 k =... q () i min i i ma ( i =... n). where p and q are the number of inequality and equality constraints respectively. There are many traditional methods in the literature for solvin NLP. However most of the traditional methods require certain auiliary properties (like conveity continuity etc.) of the problem and also most of the traditional techniques are suitable for only a particular type of problem (for eample Quadratic Prorammin Problems Geometric Prorammin Problems etc). Keepin in view the limitations of traditional techniques researchers have proposed the use of stochastic optimization methods and intellient alorithms for solvin NLP which may be constrained or unconstrained. Some eamples are: Genetic Alorithms [] [] Ant Colony Optimization [4] Chaos Optimization Alorithm [5] Particle Swarm Optimization [6] Differential Evolution [7] etcetera. Based on the research efforts in literature constraint handlin methods have been cateorized in a number of classes [8] - [0]: Reject infeasible solutions Penalty function methods Convert the constrained problem to an unconstrained problem Preservin feasibility methods Pareto rankin methods Repair methods In the present research paper we have concentrated our work to DE which is comparatively a newer addition to the class of population based search techniques. DE is a stochastic population based search stratey developed by Storn and Price [7] in 995. It is a novel evolutionary approach capable of handlin no-differentiable nonlinear and multimodal objective functions. DE has been desined as a stochastic parallel direct search method which utilizes concepts borrowed from the broad class of EAs. The method typically requires few easily chosen control parameters. This paper presents an Improved Constraint Differential Evolution (ICDE) alorithm for solvin constrained optimization problems. The structure of the paper is as follows: in section II we have briefly eplained the Differential Evolution Alorithm in section III; we have defined and eplained the proposed ICDE alorithm. Section IV deals with eperimental settins and test problems Section V ives the numerical results and discussion and finally the paper conclude with section VI. II. DIFFERENTIAL EVOLUTION ALGORITHM DE shares a common terminoloy of selection crossover and mutation operators with GA however it is the application of these operators that make DE different from GA. Whereas in GA crossover plays a sinificant 009 ACADEMY PUBLISHER

2 International Journal of Recent Trends in Enineerin Vol. No. 5 May 009 role it is the mutation operator which effects the workin of DE []. The workin of DE may be described as follows: Mutation: For a D-dimensional search space for each taret vector X i at the eneration its associated mutant vector is enerated via certain mutation stratey. The most frequently used mutation strateies implemented in the DE codes are listed below. DE/rand/(S): Vi = X r *( ) + F X r X r () DE/rand/ (S): Vi = X r *( ) *( ) + F X r X r + F X r4 X r5 (4) DE/best/ (S): Vi = Xbest + F *( X r ) X r (5) DE/best/ (S4): Vi = X best + F *( X r ) *( ) X r + F X r X r4 (6) DE/rand-to-best/ (S5): Vi = X r *( ) *( ) + F X best X r + F X r X r4 (7) where r r r r4 r5 {... NP} are randomly chosen inteers must be different from each other and also different from the runnin inde i. F (>0) is a scalin factor which controls the amplification of the difference vector. X best is the best individual vector with the best fitness value in the population at eneration. Crossover: In order to increase the diversity of the perturbed parameter vectors crossover is introduced []. The parent vector is mied with the mutated vector to produce a trial vector u ji + v ji + if ( rand j CR) or ( j = j u ji + = rand ) (8) ji if ( rand j > CR) and ( j jrand ) where j = D; rand [0] ; CR is the crossover constant takes values in the rane [0 ] and j rand (... D) is the randomly chosen inde. Selection: Selection is the step to choose the vector between the taret vector and the trial vector with the aim of creatin an individual for the net eneration. The simple flow of DE alorithm is iven in Fi. Initialize the population Calculate the fitness value for each particle Do For i = to number of particles Do mutation Crossover and Selection. Until stoppin criteria is reached. Fi Flow of DE alorithm III. PROPOSED ICDE ALGORITHM The proposed alorithm ICDE is a simple alorithm for solvin constraint optimization problems it is easy to implement. It differs from unconstrained optimization j alorithm only in the place of initialization selection of particles to the net eneration and sortin the final results. The proposed ICDE alorithm uses the mean zero standard deviation one normal distribution for initializin the population and uses the followin three selection criteria: After calculatin the trial vector (i) If the trial vector and the taret vector are feasible then select the best one (ii) If both the particles are infeasible then select the one havin smaller constraint violation (iii) If one is feasible and the other one is infeasible then select the feasible one. Also at the end of every iteration the particles are sorted by usin the three criteria: (a) Sort feasible solutions in front of infeasible solutions (b) Sort feasible solutions accordin to their fitness function values (c) Sort infeasible solutions accordin to their constraint violations. The computational steps of ICDE alorithm is iven below: Step Initialize the population usin normal distribution with mean zero and standard deviation one. Step Evaluate the objective function Calculate the constraint violation Step While stoppin criterion is not satisfied Do Step. Mutation Generate a mutated vector V i correspondin to the taret vector X i via one of the equations () to (7) Step. Crossover //Generate trial vector U i Select j rand { D} For j = to D If (rand(0) CR or j= j rand ) Then U i = V i Else U i = X i End if Step. Selection Set X i+ accordin to the three selection criteria Step.4 Sort the particles usin the three sortin rules Step.5 Go to net eneration Step 4 End while IV. EXPERIMENTAL SETTINGS AND TEST PROBLEMS In order to make a fair comparison of all versions of DE alorithms we fied the same seed for random number eneration so that the initial population is same for both the alorithms. The population size is taken as 009 ACADEMY PUBLISHER

3 International Journal of Recent Trends in Enineerin Vol. No. 5 May The crossover constant CR is set as 0.95 and the scalin factor F is set as 0.7. For each alorithm the stoppin criteria is to terminate the search process when one of the followin conditions is satisfied: () the maimum number of enerations is reached (assumed 0000 enerations) () f ma - f min < 0-4 where f is the value of objective function. A total of 0 runs for each eperimental settin were conducted. If the run satisfies the second stoppin condition then that run is called successful run. Also we implemented the new idea to five versions of DE alorithm. To check the efficiency of the proposed ICDE alorithm we have tested it on four optimization problems arisin commonly in the field of Mechanical enineerin. All the problems considered here are hihly non linear in nature and are subject to various constraints. The mathematical models of these problems may be iven as: A. Weiht Minimization of a Speed Reducer (WMSR) [] The mathematical model of this problem is Min f ) = ( ) (.508 ( ) ( ) ( ) A B Where A = [( ) ] B = 0. 6 A B Where A = [( ) ] B = B. Heat Echaner Network Desin (HEND) [4] The mathematical model is Minimize f ( ) = ( 4 + 6) ( ) ( 8 5 ) i 0000 ( i = ) 0 i 000 ( i = 4...8) Fi Heat Echaner Network Desin Problem C. Gas Transmission Compressor Desin (GTCD) [5] The mathematical model is 5 / / / 4 f ( ) = D. Optimal Desin of Industrial refrieration System (ODIRS) [6] The mathematical model is.664 f ( ) = i 5 i =... 4 V. EXPERIMENTAL RESULTS AND DISCUSSION Tables I IV ives the numerical results iven four real life constrained optimization problems. These problems occur frequently in the field of mechanical desins. The comparison criteria for the alorithms is done in terms of best averae and worst fitness function values NFE std SR and time. For all the alorithms NFE represents the number of function evaluations which helps in determinin the converence of the alorithm. Lesser value of NFE implies faster ACADEMY PUBLISHER

4 International Journal of Recent Trends in Enineerin Vol. No. 5 May 009 converence. std represents the standard deviation which tells the stability of the alorithms. Smaller std implies that the alorithm is more stable. SR represents the success rate which sinifies the efficiency of an alorithm. It tells us how many times the alorithm was able to convere successfully within % of the true lobal optima. For all the problems we compared our results with those available in literature. Form Tables I to IV the results obtained by the different DE versions and the ones available in literature are iven. From the numerical results it is quite visible that the versions of DE ave better results than the ones available in literature. In terms of best averae and worst fitness function values all the alorithms ave more or less similar results. However in terms of NFE SR and time taken the alorithms showed different behavior. TABLE I COMPARISON RESULTS OF WMSR TABLE II COMPARISON RESULTS OF HEND TABLE III COMPARISON RESULTS OF GTCD TABLE IV COMPARISON RESULTS OF ODIRS Item DE/rand/ DE/rand/ DE/best/ DE/best/ DE/randto-best/ Result in [] Best Averae NA- Worst NA- Std.56e-05.84e NA- NFE NA- SR 00% 00% 00% 00% 00% -NA- Time (sec) NA- Item DE/rand/ DE/rand/ DE/best/ DE/best/ DE/randto-best/ Result in [4] Best Averae NA- Worst NA- Std 6.7e-05.e e-05.45e-05 -NA- NFE SR 00% 00% 96% 00% 00% 88% Time (sec) Item DE/rand/ DE/rand/ DE/best/ DE/best/ DE/rand-tobest/ Result in [5] Best.96e+06.96e+06.96e+06.96e+06.96e+06.99e+06 Averae.96e+06.96e+06.96e+06.96e+06.96e+06 -NA- Worst.96e+06.96e+06.96e+06.96e+06.96e+06 -NA- Std 8.79e e-06.7 e e e-06 -NA- NFE NA- SR 00% 00% 00% 00% 00% -NA- Time (sec) NA- Item DE/rand/ DE/rand/ DE/best/ DE/best/ DE/randto-best/ Result in [6] Best Averae NA- Worst NA- Std 7.8e NA- NFE NA- SR 00% 0% 96% 0% 0% -NA- Time (sec) NA ACADEMY PUBLISHER

5 International Journal of Recent Trends in Enineerin Vol. No. 5 May 009 The first problem is involves the desin of a speed reducer for small aircraft enine. It has a nonlinear objective function and it consists of eleven inequality constraints and seven unknown variables. For this problem all the DE versions ave same results in terms of best worst and averae fitness function values. If we compare the NFC and converence time then DE/rand-tobest/ is better than all other versions. The second problem addresses the desin of a heat echaner network as shown in Fi. It has three equality constraints three inequality constraints and eiht decision variables. For this problem also all the alorithms ave same result in comparison best fitness function value. In comparison of averae fitness value DE/best/ ave slihtly worse value than other alorithms. But in terms of converence time it is better than all other versions. The third problem is a as transmission compressor desin problem. For this problem DE/best/ ave better result in terms of standard deviation NFE and converence time. DE/rand/ ave better result than other alorithms in terms of best fitness function value. VI CONCLUSION In this paper we proposed an Improved DE alorithm called ICDE for solvin constrained optimization problems. ICDE differs from the basic DE in the initialization selection and sortin phases. These modifications are embedded on various versions of DE and their efficiency is validated on a set of four real life enineerin desin problems occurrin frequently in the field of mechanical enineerin. We would like to add that in the present article thouh we have considered only four problems the preliminary numerical results obtained show that the proposed modifications are beneficial for solvin constrained optimization problems effectively. Moreover this is a eneral technique/ modification and can be applied to any population based search technique like Genetic Alorithms Particle Swarm Optimization etc. REFERENCES [] Y. Li M.Gen Non-linear mied inteer prorammin problems usin enetic alorithm and penalty function In Proc. of IEEE Int. Conf. on SMC pp [] Y. Takao M. Gen T. Takeaki Y. Li A method for interval 0- number non-linear prorammin problems usin enetic alorithm Computers and Industrial Enineerin Vol. 9 pp [] J. F. Tan D.Wan et al. A hybrid enetic alorithm for a type of nonlinear prorammin problem Computer Math. Applic Vol. 6(5) pp [4] M. Dorio V. Maniezzo A. Colori Ant system optimization by a colony of cooperatin aents IEEE Trans. on system Man and Cybernetics Vol. 6() pp [5] Z. L. Wan L. Qiu L. Function C. Lian Application of chaos optimization alorithm to nonlinear constrained prorammin Journal of North China Institute of Water Conservancy and Hydroelectric Power Vol. () pp [6] J. Kennedy and R. Eberhart Particle Swarm Optimization IEEE International Conference on Neural Networks (Perth Australia) IEEE Service Center Piscataway NJ IV pp [7] R. Storn and K. Price Differential Evolution a simple and efficient adaptive scheme for lobal optimization over continuous spaces Technical Report International Computer Science Institute Berkley 995. [8] G. Coath S. K. Halamue A Comparison of Constraint- Handlin Methods for the Application of Particle Swarm Optimization to Constrained Nonlinear Optimization Problems In Proc. of the IEEE Conress on Evolutionary Computation Vol. 4 pp [9] S. Koziel Z. Michalewicz Evolutionary Alorithms Homomorphus Mappins and Constrained Optimization Evolutionary Computation Vol. 7() pp [0] Z. Michalewicz A Survey of Constraint Handlin Techniques in Evolutionary Computation Methods In Proc. of the Fourth Annual Conf. on Evolutionary Prorammin pp [] D. Karaboa and S. Okdem A simple and Global Optimization Alorithm for Enineerin Problems: Differential Evolution Alorithm Turk J. Elec. Enin. () 004 pp [] R. Storn and K. Price Differential Evolution a simple and efficient Heuristic for lobal optimization over continuous spaces Journal Global Optimization. 997 pp [] Floudas C.A. Pardalos P.M. A collection of test problems for constrained lobal optimization alorithms LNCS. Spriner Germany 990. [4] B. V. Babu and R. Anira Optimization of Industrial Processes Usin Improved and Modified Differential Evolution Studies in Fuzziness and Soft Computin Vol pp.. [5] Beihtler C. S and Phillips D. T Applied Geometric Prorammin Jhon Wiley & Sons New York 976. [6] Paul H and Tay Optimal Desin of an Industrial Refrieration System In Proc. of Int. Conf. on Optimization Techniques and Applications 987 pp ACADEMY PUBLISHER