Bespoke Shuffled Frog Leaping Algorithm and its Engineering Applications

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1 I.J. Intellient Systems and Applications, 5,, - Published Online March 5 in MECS ( DOI:.585/ijisa.5.. Bespoke Shuffled Fro Leapin Alorithm and its Enineerin Applications Anura Tripathi Maharishi Ved Vyas Enineerin Collee Jaadhari, Haryana, India anuratripathi5@mail.com Tarun K. Sharma, Vipul Sinh Amity University Rajasthan, Jaipur, India and JUET, Guna, MP, India {taruniitr, om.vipul5}@mail.com Abstract Shuffled Fro Leap Alorithm (SFLA, a metaheuristic alorithms inspired by PSO and DE has proved its efficacy in solvin discrete optimization problems. In this paper we have modified SFLA to solve constrained enineerin desin problems. The proposed modification interates a simple mechanism to update the position of fro in its memeple in order to accelerate the basic SFLA alorithm. The proposal is validated on four enineerin desin problems and the statistical results are compared with the state-of-art alorithms. The simulated statistical results indicate that our proposal is a promisin alternative to solve these types of optimization problems in terms of converence speed. Inde Terms Shuffled Leap Fro Alorithm, SFLA, Enineerin Desin Problems, Optimization, Constrained Handlin. information echanin of the shuffled comple evolution. Like other stochastic alorithms SFLA also bears some converence limitations. In this study we tried to improve the converence speed of SFLA by incorporatin a new search mechanism to update the position of fros. The proposal is named as Bespoke SFLA. Rest of the study is structured as follows: a brief introduction of SFLA is iven in Section. Section details the proposed Bespoke SFLA with constrained handlin procedure. Enineerin desin problems with mathematical models are briefed in Section. Parameter tunin is discussed in Section 5. Results are analyzed in Section. Finally, Conclusions drawn from the study are iven in Section 7. I. INTRODUCTION Optimization is a process of choosin the best or alternative solution that also satisfies all the constraints, from the iven set of solutions. Optimization plays a vital role in our real life too. Now a day s most of the researchers are workin and applyin various optimization techniques in optimizin real world enineerin desin problems, manaement problems, and ariculture problems and almost in every sphere. Stochastic techniques are ainin much attention in Optimization as they don t need any auiliary knowlede about problem domain as well as their computational cost and times is also comparatively less than traditional techniques. Stochastic techniques are either inspired by some evolution, natural or bioloical phenomenon. Genetic Alorithm (GA [], Differential Evolution (DE [], Particle Swarm Optimization (PSO [], Artificial Bee Colony (ABC [], Shuffled Fro Leapin Alorithm (SFLA [5] and a like falls into the cateory of stochastic alorithms. Their applications into various domains have already proved their efficacy to solve almost every type of optimization problems [] []. In this paper we are focusin on recently introduced Shuffled Fro Leapin Alorithm by Eusuff and Lansey, [5]. SFLA, mimics the social and natural behavior of species. SFLA is formulated on the concept of evolution of memeplees in Fros. SFLA combines the advantaes of local search process of PSO and II. INTRODUCTION TO SFLA SFLA is a population based metaheuristic for solvin discrete optimization problems. SFLA also, like others, is inspired by natural memetics. In SFLA, durin the local eploitation process, the fros use to share and inform the ideas to one another. The local searchin process in each memeplees is similar to that of PSO local search process. After eploitation process in each memeplees the information is shuffled amon different memeplees, and thus local search process moves towards lobal search process (eploration. Thus, the benefits of GA as well as PSO are combined in SFLA. SFLA consists of a population of fros partitioned into different Memeplees. In each memeple a local independent search is performed by the set of fros. This local searchin process is influenced by PSO. Later on eploration or diversification is performed by echanin information amon each memeplees. This phase is inspired by evolution process of GA. SFLA has fewer numbers of parameters to be set and is as: A. No. of fros called population size (P or initial solutions., B. number of Memeplees (m, and C. number of fros in each memeple (n. Fi. depicts the searchin process of SFLA in a search space. Copyriht 5 MECS I.J. Intellient Systems and Applications, 5,, -

2 Bespoke Shuffled Fro Leapin Alorithm and its Enineerin Applications Group 5 Group Search Space Food Group Group Group Fi.. Searchin process in SFLA (initially locally then performs information echane The steps of SFL alorithm are as follows:. The population of fros (P solutions for D dimension is enerated randomly, a fro (solution i is represented as i = ( i, i,, is.. Fitness function value is calculated for each solution (fro.. Arrane the solutions (fros based on their of their fitness values in a descendin order. Then population of solutions (fros is divided into m number of memeplees, each containin n fros. Thus, P = m * n. 5. Assin the fros to the Memeplees in such a way that first oes to the first memeple, the second fro oes to the second memeple,, fro m oes to the m th memeple, and fro m+ oes back to the first memeple, etc.. Then in each memeple, the fros (solutions with worst and best positions are identified. 7. In each memeple the local search is performed usin the equation (: r ( i i b w ( r is a randomly enerated number between and. If fitness value of i+ is superior to i, reinstate current fro position i with new position i+. Else proceed to Step The fro with the lobal best fitness ( is identified and lobal search is done usin equation (: i i r ( w ( If i+ fitness value is better than the fitness value i, replace current fro position i with new position i+.else o to Step Generate new fro position i randomly to replace the worst fro. i min r ( min ( min ( min, min, min,..., min S and,,..., (, S. Does the termination criteria met? If yes then stop. Else o to Step. III. BESPOKE SFLA AND COSNTRAINT HANDLING In this study we tried to enhance the local search mechanism and converence rate of basic SFLA by incorporatin a scheme that is based on the best individual vector and Scalin factor at a particular eneration. The updatin scheme is as follows: F ( i b b w ( where F is scalin factor lies between and and rest of the notations are same as discussed in equation (. The modification is intended to keep the weihted difference vector between the best and the worst as it is in equation ( and the remainin base vector i and randomly selected vector are replaced with the best vector and a new parameter called scalin factor. The proposed scheme favors the eploitation since all the vectors are biased by the same direction as of best vector. As a result, the new position updatin scheme has better local search ability and faster converence rate. However, proper tunin of F is required. Here authors would like to mention that the proposed scheme is also inspired by the behavior of social animals called human beins. Everyone in his life style would like to follow the successful peoples or their paths in order to avoid the failures in achievin their oals. The new position update mechanism or scheme is embedded into basic SFLA as follows: i i b r ( F ( b w w if.5 else where α is a random real number lies between and. If the value of α is below.5 then the alorithm will (5 Copyriht 5 MECS I.J. Intellient Systems and Applications, 5,, -

3 Bespoke Shuffled Fro Leapin Alorithm and its Enineerin Applications behave as basic SFLA otherwise bespoke SFLA will be invoked. Constraint Handlin In our proposal we have adopted simple constrainthandlin methods. Fro positions are compared by pairs: if the position of two fros in a memeple are feasible, we choose the one with a better fitness function value; if the position of two fros in a memeple are infeasible, we choose the fro position with the lower infeasibility deree; if position of one fro is feasible and the other is infeasible, we choose the feasible one. where ( ' MR '' J M P( L R P z ( ' ' '' ( '' R, IV. ENGINEERING DESIGN PROBLEMS To evaluate the performance of the proposal a test bed of four enineerin desin optimization problems [] are considered and solved. A. Welded Beam Desin (WBD Minimizes the cost of the beam subject to constraints on shear stress, τ, bendin stress in the beam, σ, bucklin load on the bar, P c, end deflection of the beam, δ, and side constraints. This problem consists of a nonlinear objective function, five nonlinear and two linear inequality constraints. The solution is located on the boundaries of the feasible reion. The ratio of feasible reion to entire search space is quite small for welded beam problem. More details can be found in []. There are four desin parameters,, and correspond to h, l, t and b variables, respectively, shown in Fi.. J PL ( PL ( E.E Pc L L and P = lb., L = in., E G =.5 in., E = psi, G = psi, =, psi, =, psi, = (,,, T,.,.,.,. B. Pressure Vessel Problem (PVP The objective of the PVP is to minimize the total cost of material, formin and weldin of a cylindrical vessel shown in Fi.. The nature of the problem is nonlinear objective function with a nonlinear and three linear inequality constraints. The details of the problem can be found in []. The mathematical model of PVP is iven as: Fi.. The welded beam problem min f (.7.8 (. Subject to ( : ( ( : ( ( : ( :.7 ( :.5 5 ( : ( ( : P P ( 7 c.8 (. 5. min f (. Subject to Fi.. The pressure vessel problem ( :.9.95 ( : ( : ( : 9 where = (,,, T. The ranes of the desin parameters are, 99,,. Copyriht 5 MECS I.J. Intellient Systems and Applications, 5,, -

4 Bespoke Shuffled Fro Leapin Alorithm and its Enineerin Applications C. Speed Reducer Desin (SRD The objective of SRD is to minimize the weihts of the speed reducer. Fi. presents the outline raph of the SRD. This problem consists of constraints out of which are linear and 7 are nonlinear. The mathematical model is iven below and the details of the SRD problem can be consulted in [] min f min f ( ( N Dd Subject to D N ( : 7785d D dd ( : 5( Dd d 58d.5d ( : D N D d ( :.5 = (d, D, N T,.5 d.,.5 D.,. N 5. Subject to Fi.. The speed reducer problem 7 ( : 97.5 ( :.9 ( :.9 ( : 75 ( : 5 75 ( : 7 ( : 5 8( : 9 ( : ( : 5 where..,.7.8, 7 8, 7. 8., ,.9.9, ( : D. Tension/Compression Problem (TP The objective of TP is to minimize of the weiht of the tension/compression sprin revealed in Fi. 5. This problem has one linear and non-linear inequality constraints with non linear objective function. Details about TP can be referred in []. Mathematically the minimization TP problem is formulated as: / / Fi. 5. The tension/compression sprin problem V. PARAMETER SETTINGS The parameter settins of SFLA and Bespoke-SFLA, for the fair comparison are stated in Table. The proposed CM-SFLA is eecuted in Dev C++. The population of fros is enerated usin inbuilt rand( function. Table. Parameterizations for test systems Population Size of Fros 5 Memeplees (m 5 Local Eplorations iterations in each Memeplees Number of Function Evaluations (NFE 5 D % of variable rane F (Scalin Factor.5 VI. STATISTICAL RESULTS The simulated statistical results of four enineerin desin problems are compared with the results reported in the literature. The best values with the values of decision variables are for each enineerin desin problem are iven in the Table Table 5. We have performed independent runs; where in each run, objective functions were evaluated. We have compared the simulated results with that of available in the literature [] [5]. [5] reached the best solution after, function evaluations, which is comparatively larer than taken by our proposal. The statistical results in terms of best values achieved are illustrated in Table Table 5. Comparative results in terms of mean, standard deviation (Std. Dev. and best are presented in Table. Copyriht 5 MECS I.J. Intellient Systems and Applications, 5,, -

5 Bespoke Shuffled Fro Leapin Alorithm and its Enineerin Applications 5 Table. Results of WBD Table. Results of SRD Best Values (Solutions Best Values (Solutions ( ( ( ( 5 ( ( 7 ( f(.57e-.75e+ 9.E+.57E- -.89E- -.7E-.E+ -.E E- -.55E-.E+.79E+ 5 7 ( ( ( ( 5 ( (.5E+ 7.E-.7E+ 7.E+ 7.8E+.5E+ 5.87E E- -.98E E- -9.7E-.E+ -5.E- Table. Results of PVP Best Values (Solutions 8.5E-.75E-.98E+.7E+ 7 ( 8 ( 9 ( ( ( f( -7.5E- -.E- -5.8E- -5.5E- -.85E-.99E+ ( -.5E-5 Table 5. Results of TP ( ( ( f( -.588E- -.E- -.E+.597E+ ( ( ( ( f( Best Values (Solutions 5.58E-.59E-.9E+ -.E- -.E- -.88E E-.5E- Table. Comparative Statistical Results Problems Optimal SFLA Bespoke SFLA COPSO Mezura WBD Std. Dev..785.E-.E-.E-5 8.8E- Best PVP Std. Dev E+.8E+.5E+.E+ Best SRD Std. Dev. NA.E-.E-.8E-.E+ Best TP Std. Dev..5.78E-.9E-.E-.9E- Best VII. CNCLUSIONS AND FUTURE SCOPE In this study we have presented a simple modification in the local searchin process of Fros in each memeplees. The searchin process is updated usin new search mechanism that is based and biased on best search and secondly embeddin scalin factor enhances the converence rate of the proposal. The proposal is named Copyriht 5 MECS I.J. Intellient Systems and Applications, 5,, -

6 Bespoke Shuffled Fro Leapin Alorithm and its Enineerin Applications as Bespoke SFLA. Bespoke SFLA has shown a ood performance on four constrained enineerin desin problems. We have compared the statistical results of our proposal with the basic SFLA and two alorithms results found in the literature. Our alorithm found to be capable in obtainin near optimal solutions with fewer number of function evaluations. This proposal could be an alternative for solvin such kind of problems. In future we will try to study the theoretical concept of the proposal and applications in multi-objective optimization problems. REFERENCES [] Goldber D., Genetic Alorithms in Search Optimization and Machine Learnin, Addison-Wesley, (989. [] Storn R. and Price K. V., Differential evolution a simple and efficient heuristic for lobal optimization over continuous spaces, J. GlobalOptimization, vol., pp. 59, (997. [] J. Kennedy and R. C. Eberhart, Particle Swarm Optimization, Proceedin of IEEE International Conference on Neural Networks,Perth, Australia, IEEE Service Center, Piscataway, NJ (995, pp [] D. Karaboa, An Idea based on Bee Swarm for Numerical Optimization, Technical Report, TR-, Erciyes University Enineerin Faculty, Computer Enineerin Department (5. [5] Eusuff, M., Lansey, K.E.: Optimization of water distribution network desin usin the shuffled fro leapin alorithm, Water Resources Plannin and Manaement 9(, 5 (. [] TK Sharma, M Pant. Redundancy Level Optimization in Modular Software System Models usin ABC, International Journal of Intellient Systems and Applications (IJISA (,,. [7] TK Sharma, M Pant. Improvised Scout Bee Movements in Artificial Bee Colony, I.J. Modern Education and Computer Science,,, -. [8] TK Sharma, M Pant. Swarm Intellience in Pulp and Paper Process Optimization. Applications of Metaheuristics in Process Enineerin, -5,. [9] M Ali, CW Ahn, P Siarry. Differential evolution alorithm for the selection of optimal scalin factors in imae watermarkin, Enineerin Applications of Artificial Intellience,, 5-,. [] Panchi Li, Hon iao An improved quantum-behaved particle swarm optimization alorithm, Applied Intellience, April, Volume, Issue, pp 79-9 [] Sunita Panda, Archana Sarani, Siba Prasada Panirahi. A new trainin stratey for neural network usin shuffled fro-leapin alorithm and application to channel equalization, AEU - International Journal of Electronics and Communications. DOI:./j.aeue..5.5 [] Tarun K. Sharma, Millie Pant and Deepshikha Bharava, Impact of Modification Rate in Artificial Bee Colony for Enineerin Desin Problems, J. Information Enineerin and Electronic Business,,, 55- [] Leticia C. Canina and Susana C. Esquivel, Carlos A. Coello Coello. Solvin Enineerin Optimization Problems with the Simple Constrained Particle Swarm Optimizer, Informatica, (8 9. [] A. Hernandez Auirre, A. Muñoz Zavala, E. Villa Diharce and S. Botello Rionda. COPSO: Constrained Optimization via PSO Alorithm, Technical report No. I- 7-/--7, Center for Research in Mathematics (CIMAT, 7. [5] E. Mezura and C. Coello. Useful Infeasible Solutions in Enineerin Optimization with Evolutionary Alorithms, Lect. Notes Comput. Sc., 789:5, 5. Authors Profiles into his credit. Anura Tripathi, B.Tech and pursuin his M.Tech in Electeronics and Communication Enineerin from Maharishi Ved Vyas Enineerin Collee Jaadhari, Haryana, India. His area of interest is evolutionary alorithms and their enineerin applications. He has several national and international publications Tarun Kumar Sharma: PhD from IIT Roorkee and presently alined with Amity Institute of Information Technoloy, Amity University Rajasthan, India. His research areas are evolutionary & Swarm intellience alorithms and their applications in Software Enineerin. He is in Editorial Board and reviewer of many refereed Journals. He has published about 5 research papers in Journal of repute and in refereed international Journals. Vipul Sinh is an Enineerin raduate in Computer Science discipline from. Jaypee University of Enineerin & Technoloy, Guna, MP, India. He is very keen into research in the field of enineerin optimization. usin memetic alorithms. Copyriht 5 MECS I.J. Intellient Systems and Applications, 5,, -