Elitist-Multi-objective Differential Evolution (E- MODE) Algorithm for Multi-objective Optimization

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1 Elitist-Multi-objective Differential Evolution (E- MODE) Algorithm for Multi-objective Optimization B. V. Babu, Ashish M. Gujarathi Dean Educational Hardware Division (EHD) and Professor, Department of Chemical Engineering, Birla Institute of Technology and Science (BITS), Pilani , Rajasthan, India, Phone: Ext. 59 / ; Fax: ; bvbabu@bits-pilani.ac.in, URL: Comment [S]: : Corresponding Author ashishg@bits-pilani.ac.in (Ashish Gujarathi) Abstract. Several problems in the engineering domain are multi-objective in nature. The solution to multi-objective optimization is a set of solutions rather than a single point solution. Such a set of non-dominated solutions are called Pareto optimal solutions or non-inferior solutions. In this paper, a new algorithm, Elitist-Multi-objective Differential Evolution (E-MODE) is proposed. The proposed algorithm is applied successfully on several test functions, and the results are discussed extensively. Results obtained from the proposed algorithm are compared with those obtained using Multi-objective Differential Evolution (MODE) algorithm. E-MODE is found to give better solutions in terms of wide range of solutions, spread, and diversity of Pareto front than those obtained using MODE. Keywords: Multi-objective optimization (MOO), Multi-objective Differential Evolution (MODE), Elitist-Multi-objective Differential Evolution (E-MODE), Evolutionary Algorithms, Evolutionary Multi-objective Optimization (EMO). Introduction Multi-objective optimization problems (MOOPs) deal with optimizing more than one objective simultaneously. In the presence of conflicting objectives MOOPs give rise to a set of non-dominated solutions (rather than a single point solution), referred as Pareto Optimal solutions or non-inferior solutions. As each of the solutions in the Pareto Front is equally good (depending upon the decision makers interest), it is better to have as many number of non-dominated solutions as possible.

2 While solving any MOOP, it is very important that the optimization algorithm gives the Pareto solutions which are as close as possible with a better spread in terms of all the objectives. Several algorithms are available to solve conflicting MOOPs. The problem with most of the classical (traditional) methods is that they converge to a single point solution. The detailed account of classical methods such as weighted sum methods, ε-constraint method, weighted metric methods etc. is available in the literature []. Unlike the classical methods, Evolutionary Algorithms (EAs) start with a set of solutions and finally result in a set of solutions rather than a single point solution. EAs have been gaining popularity since last decade due to this ability in giving an option to the decision maker of choosing from multiple solutions. Local Pareto front Objective Space Decision Space x f Global Pareto front x f Fig.. Decision space, objection space, local and global Pareto fronts involved in multiobjective optimization study. EAs are based on the law of nature of survival of the fittest. Fig. shows the decision variable space and objective space with global and local Pareto fronts. The dependency on decision variables is understood from decision space, and the performance of any solution is tested based on its fitness in the objective space. In this way, we can obtain different (local) Pareto optimal fronts using the concept of nondominance on the dominated outcome of each sorting operation. Popular EAs for single objective optimization are Genetic Algorithm, Differential Evolution, Simulated Annealing etc. [], [], [3], [4]. Several EAs were extended to solve multiobjective optimization problems. Some of the popular algorithms in this category are NSGA (Non-dominated sorting Genetic Algorithm) and its variants, SPEA (Strength Pareto Evolutionary Algorithm), MOSA (Multi-objective Simulated Annealing),

3 PACO (Pareto ant colony optimization), PDE (Pareto Differential Evolution), MODE (Multi-objective Differential Evolution) etc. [], [], [3], [4], [5], [6]. It is observed that despite following an evolutionary approach, some of the algorithms get converged to a single point solution, whereas others get converged to local Pareto optimal front. Therefore a good multi-objective optimization algorithm should be able to give a better distribution of solutions with better convergence to Pareto optimal front. Complex and multi-dimensional decision variables may produce a non-linear and discontinuous objective space. There are chances that if the algorithm is not efficient, it may get converged to local Pareto front, as shown in Fig.. In our earlier study [7] it was observed that results obtained by using MODE algorithm were better when compared with results obtained when NSGA was used. Improved versions of NSGA such as NSGA-II and NSGA-II with jumping genes have found to give better results than NSGA [8]. In this study, we extended MODE algorithm to include the concept of elitism with crowding distance sorting in order to give maximum number of set of equally good solutions with diversification. The remaining section of this manuscript is arranged in the flowing order. Section- discusses the working principle of MODE and the proposed Elitist-MODE (E- MODE) algorithm. In Section-3, the results obtained for several test functions using MODE and E-MODE are discussed in detail and the comparative analysis on the performance of these two algorithms is made. Finally the conclusions of the study are included in Section-4. Multi-objective optimization using Differential Evolution. Differential Evolution (DE) Differential Evolution (DE) is simple, robust, and faster in optimization among the population based search algorithms that are stochastic in nature [Price & Storn,. In DE, after generating initial population randomly, a weighted difference of two randomly chosen individuals is added to a third randomly chosen individual to create a noisy random vector. Subsequently, crossover between the target and noisy random vector is carried out to give birth to offspring (trial vector). This newly generated trial vector then competes with the target vector, and the winner takes the position of target vector in next generation. Unlike Genetic Algorithms (GA), DE ensures that newly generated population is always better than the population in preceding generation. DE and its various improved and modified versions of strategies have been successfully applied to many complex and non-linear applications [6], [7], [8], [9], [0], [], [], [3], [4], [5], [6], [7], [8].. Multi-objective Differential Evolution (MODE) and its strategies Several studies on extension of differential evolution for solving MOOPs are reported in literature [9], [0], [], [], [3], [4], [5], [6] [7], [8]. Babu and Gujarathi

4 [] used MODE algorithm to solve multi objective optimization of supply chain planning and management. MODE algorithm is also applied successfully on multiobjective optimization of wiped film poly ethylene reactor [8]. The working principle of simple MODE algorithm consists of the following steps:. Generate NP number of initial population.. Calculate the cost of objective functions. 3. Carry out non-dominated sorting based on cost of objective functions. 4. Pass the non-dominated solutions to Differential Evolution algorithm. 5. Check for convergence. 6. Repeat steps -5 till convergence. 7. The obtained set of solutions is the required Pareto optimal front. Even though the Pareto optimal front obtained using MODE was better than those reported in literature [0], MODE algorithm was not able to give a good number of non-dominated solutions. If we explore the design structure of MODE, we observe that the non-dominated sorting of solutions is carried out before the solutions are passed to next generation of DE. The number of non-dominated solutions decreases as we move on from generation to generation. This results in a Pareto front consisting of a very few number of non-dominated solutions. Babu et al. [9] proposed three strategies of MODE for solving MOOPs to address the above problem. In MODE-II (one of the three strategies as mentioned above), after non-dominated sorting (as in simple MODE), new population points are generated to make the population size NP. These newly generated population points along with non-dominated solutions are then passed to DE algorithm. This operation is continued till convergence is reached. However, in MODE-II the population is generated randomly every time, instead of getting it from Parent population or from the offspring set. This deviates from the simple evolutionary approach of population based search algorithms. Considering this problem we are proposing a new algorithm, Elitist MODE (E-MODE) which gives maximum number of well diversified non-dominated solutions with smooth Pareto front..3 Elitist Multi-objective Differential Evolution (E-MODE) The schematic diagram of E-MODE is shown in Fig.. The working principle of E- MODE is explained in following three steps: () Processing of initial population of size NP using Differential Evolution, () Combining the solutions obtained from DE and those obtained from non-dominated sorting (NP+Q) ensuring elitism, and (3) Maintaining NP population points in the next generation by using crowding distance sorting. ) Processing of initial population using DE: In E-MODE, we start with an initial population of size NP (parent population). A target vector is chosen. Three vectors X a, X b and X c are also chosen randomly from the initial population. The weighted difference of X a and X b variables is then added into the third vector, X c. This vector is termed as noisy random vector. Cross over is carried out to generate Trial vector (X c ) from target and noisy random vectors.

5 Competition is then made between the trial and target vectors and the vector giving better cost values is replaced into the population for next generation. Population Size NP NP t Differential Evolution for generating offspring Size NP Non-Dominated Sorting Size Q Population Size NP+Q Non-Dominated Sorting Pareto front F i and dominated solutions Dominated Solutions If F i + NP t+ < NP t N Crowded Distance Sorting Y Pareto front, F i to NP t+ NP t - NP t+ Solutions NP t+ Rejected Solutions Population Size NP to next generation Fig.. Working Principle of E-MODE algorithm.

6 ) Combining the solutions obtained from DE and those obtained from nondominated sorting (NP+Q) ensuring elitism: In parallel operation, the initial population (parent population) is also sorted for getting the non-dominated solutions. Say, the number of these non-dominated solutions is Q. This Q number of non-dominated population points is then mixed with NP number of offspring population points (as obtained in step- above) to generate a total mixed population of size NP+Q. 3) Maintaining NP population points in the next generation by using crowding distance sorting: The NP+Q number of population points is sorted for non-dominance to classify the solutions in different fronts. The first non-dominated solution is referred as front.this front is copied into NP t+ population of next generation. The non-dominated sorting is repeatedly carried out on remaining number of population points and fronts, front 3, front 4, upto front N are copied into the population. Since the overall size of intermediate population is NP+Q, not all the fronts may be accommodated into NP t+ (of size NP). Out of NP+Q population points (as generated in steps and above), only NP number of solutions are to be passed through to the next generation. For ensuring this, a crowded tournament selection operator is used. According to the crowded tournament selection operator, a solution i wins tournament with another solution j if any of the following conditions are true []:. If solution i has a better rank, that is r i <r j.. If they have the same rank but solution i has a better crowding distance than solution j, that is r i =r j and d i >d j. This procedure is illustrated in Fig.. When last considered front is not allowed into NP t+, it may be containing more number of solutions than the vacant space in NP t+. Crowded distance sorting is then carried out, and the solution containing higher crowding distance value is copied into NP t+ in vacant space. This operation is continued till the convergence criterion is met. 3 Results and Discussions 3. Application of MODE and E-MODE on Test Functions The performance of E-MODE is compared with that of MODE by applying both the algorithms on several test functions. Several trial runs were carried out to obtain the Pareto optimal front for each of these test functions. 3.. SCH Test Problem Minimize f( x) = x SCH: Minimize f ( x) = ( x ) A x A

7 The search space for Schaffer s function, SCH is shown in Fig. 3. The Pareto optimal region in the search space lies between x [0,]. The Pareto optimal set lies in the range of 0 f 4 as shown in Fig. 3. Different values of bound of (A) were chosen and experimental runs were carried out. It is observed that as the value of A is increased, the complexity of problem is increased. The Pareto optimal set shown in Fig. 3 corresponds to the bound value of A=0. The Pareto front is convex in nature. Even though MODE is found to cover entire range in the feasible search space, the density of solutions as compared to E-MODE is very less SCH Search (Objective) Space Pareto front using E-MODE Pareto front using MODE f Pareto optimal front f Fig. 3. Search space and Pareto front using MODE and E-MODE. 3.. SCH Test Problem Schaffer [30] introduced the two objective test problem as given below. x if x x if < x 3 SCH : Minimize f( x) = 4 x if 3 < x 4 x 4 if x > 4 Minimize f ( x) = ( x 5) 5 x 0

8 The Search space for Schaffer s function (SCH) is shown in Fig. 4. The search space is made up of four discontinuous regions. The Pareto front lies in two discontinuous regions in the search space. The Pareto front lies in the variable range of x * { [,] [4,5] }. The feasible search space, near the Pareto front is shown by points P, Q, R, S and T. Schaffer s function SCH is constructed in such a way that as the value of first objective function f increases, that of f decreases and vice versa. This is true as we move from point P to Q, in increasing order of f. But the region QR does not fall into Pareto optimal region, as this region is dominated by region ST. The dominated region QR corresponds to the f value in the range of [0,] and f value in the range of [,9]. Fig. 5 shows the Pareto fronts obtained by using both the MODE and E-MODE algorithms. MODE algorithm is found to be spread uniformly in the feasible search space of Pareto region but with lesser density. E-MODE gives 00 % convergence of solutions in the Pareto region. E-MODE algorithm may be useful in getting maximum number of solutions in highly complex and non-linear search space compared to other algorithms where elitism is not considered. Unlike MODE, E- MODE is found to give a stable population on each of the two disconnected Paretooptimal regions (which is difficult to maintain) SCH Objective Space Pareto front using E-MODE Pareto front using MODE 60 f P Q R S T f Fig. 4. Search space and Pareto front using MODE and E-MODE.

9 8 6 4 Pareto front using E-MODE Pareto front using MODE 0 f f Fig. 5. Pareto optimal solutions using MODE and E-MODE algorithms POL Test Problem Poloni et al. [3] introduced the two objective test function having two variables. Minimize Minimize where POL : A B B f ( x) = f ( x) = A = 0.5sin x =.5sin x [ + ( A B ) + ( A B ) ] [( x + 3) + ( x + ) ] = 0.5sin cos+ sin.5cos, =.5sin cos+ sin 0.5cos cos x cos x + sin x π ( x, x ) π = sin x.5 cos x 0.5 cos x

10 The feasible search space for POL test problem is non-convex as shown in Fig. 6. For any inefficient algorithm it would be difficult to get a uniformly distributed spread of solutions in the Pareto front. E-MODE is found to give a smooth Pareto front with uniform spread in the Pareto region. POL problem consists of two disconnected regions of Pareto front as shown in Fig. 7. The search space of POL is so complex that MODE algorithm is able to give only 9 non-dominated solutions out of initial 500 number of population points. A greater number of solutions is observed in lower Pareto front (out of two disconnected regions). This is because, the lower limit of x is -π. While calculating the crowded distance, solutions having higher value of x get discarded, thus increasing the number of solutions in lower Pareto front out of two disconnected regions Objective Space Pareto front using E-MODE Pareto front using MODE 30 f f Fig. 6. Feasible Search Space and Pareto optimal solutions using MODE and E-MODE algorithms.

11 30 Pareto front using E-MODE Pareto front using MODE 5 0 f 5 Pareto Optimal fronts f Fig. 7. Pareto optimal solutions using MODE and E-MODE algorithms BNH Test Problem Bi-objective optimization problem with two constraints and two variables was introduced by Binh and Korn [3] Minimize Minimize BNH : Subjectto c ( x) ( x c ( x) ( x f( x) = 4x f ( x) = ( x 5) 5) 8) + 4x + x + ( x + ( x 5) 5 + 3) 7.7 The feasible search space and Pareto optimal solutions obtained using MODE and E-MODE algorithm for this problem is shown in Fig. 8. Both MODE and E-MODE algorithms are found to give smooth and wide spread of solutions in the Pareto front. E-MODE algorithm is found to give 00 % convergence of solutions on the Pareto optimal front. The Pareto optimal solutions are due to variables in the range of x * = x* [0,3] and x* [3.5], x* = 3 as shown in Fig. 9. Fig. 9 also shows the decision variable space and the variables corresponding to Pareto optimal set shown in Fig. 8 using E-MODE algorithm. Both x and x are found to increase from 0 to 3 linearly till x =3. After x =3, with further in crease in x value, value of variable x remains almost constant as shown in Fig. 9.

12 Fig. 0 shows the effect of constraints on the Pareto optimal front. It is observed that the Pareto optimal front remains the same, even after removing the constraints. However, some portion (shown by box in Fig. 0) of Pareto front is eliminated because of the constraint C. The comparison of the results obtained using MODE and E-MODE are shown in Table. The number of population points those converged to a set of Pareto optimal solutions after specified number generation for different test problems are shown in Table. It is observed that MODE algorithm gives good number of Pareto solutions, if the search space is not complex. As the search space become more and more complex (e.g. POL Test problem Search Space), MODE is found to give 09 solution with initial NP size of 500.However, irrespective of the search space, EMODE is found to give a well spread and 500 number of Pareto solution for all the test problems considered in this study. Table shows the values of different control parameters used in simulation study in the present study. Several trial runs well carried out with varied set of control parameters, and those used in the results discussed are reported in Table. 50 BNH Search Space Pareto front using E-MODE Pareto front using MODE f f Fig. 8. Feasible Search Space and Pareto optimal solutions using MODE and E-MODE algorithms.

13 Generation 0 (E-MODE) Generation Variables corresponding to Pareto Optimal set x x Fig. 9. Decision variable space and Decision variables corresponding to points on Pareto front. Table. MODE and E-MODE initial population and number of converged solutions. MODE EMODE Problem Number of solutions Number of solutions Initial NP Initial NP in Pareto front in Pareto front SCH SCH POL BNH Table. Key parameters of MODE and E-MODE used in the present study. Key MODE/ E-MODE Example parameters used in study NP 500 Generation 00 CR 0.9 F Random

14 50 Without constraints With Constraints Less dense solutions region f 0 0 Pareto front f Fig. 0. Effect of constraints on Pareto optimal front 4 Conclusions Elitist-Multi-objective Differential Evolution algorithm is proposed in this study. The proposed algorithm is successfully applied on well known test problem. The results obtained are compared with the results of MODE algorithm. E-MODE is found to give better spread and diversity of solutions. Such an algorithm may prove to be highly useful, especially when the search space is complex and nonlinear. E-MODE is found to give 00 % convergence of initial population to Pareto optimal front. References. Deb, K.: Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons Limited, New York (00). Babu, B. V.: Process Plant Simulation, New York: Oxford University Press (004). 3. Onwubolu G. C. and Babu, B. V.: New Optimization Techniques in Engineering, Springer- Verlag, Heidelberg, Germany (004) 4. Goldberg, D. E.: Genetic Algorithms for Search, Optimization, and Machine Learning. Kluwer Academic Publishers, Boston, MA (989)

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