Enhanced Version of Multi-algorithm Genetically Adaptive for Multiobjective optimization

Multi-objective EAs (MOEAs) are well established population-based techniques for solving various search and optimization problems. MOEAs employ different evolutionary operators to evolve populations of solutions for approximating the set of optimal solutions of the problem at hand in a single simulation run. Different evolutionary operators suite different problems. The use of multiple operators with a selfadaptive capability can further improve the performance of existing MOEAs. This paper suggests an enhanced version of a genetically adaptive multi-algorithm for multi-objective (AMALGAM) optimisation which includes differential evolution (DE), particle swarm optimization (PSO), simulated binary crossover (SBX), Pareto archive evolution strategy (PAES) and simplex crossover (SPX) for population evolution during the course of optimization. We examine the performance of this enhanced version of AMALGAM experimentally over two different test suites, the ZDT test problems and the test instances designed recently for the special session on MOEA’s competition at the Congress of Evolutionary Computing of 2009 (CEC’09). The suggested algorithm has found better approximate solutions on most test problems in terms of inverted generational distance (IGD) as the metric indicator. Keywords—Multi-objective optimization, Multi-objective Evolutionary algorithms (MOEAs), Pareto Optimality, Multi-objective Memetic Algorithm (MOMAs).


I. INTRODUCTION
Multi-objective evolutionary optimization is a subject of intense interest in all fields of Science, Engineering, Economics, Logistics and others.Multi-objective optimization problems (MOPs) have more than one conflicting objective function and they have many real-world applications [6], [59].A general MOP can mathematically be formulated as follows.
minimize F (x) = (f 1 (x), . . ., f m (x)) T  (1) subject to x ∈ Ω where Ω is the decision variable space, x = (x 1 , x 2 , . . ., x n ) T is an individual or solution and x i , i = 1, . . ., n are their decision variables, F (x) : Ω → R m consists of m real valued objective functions and R m is called the objective space.If Ω is a closed and connected region in R n and all the objective functions in (1) are continuous for x, we call it a continuous MOP.Furthermore, if m ≥ 3, then problem (1) is said to be a many objectives problem.In single objective optimization, the main focus is on the decision space while in multi-objective optimization, the focus is mainly on the objective space because objective values are used in checking for optimality [43].In practical applications of optimization, it is very common that the objective functions of the MOP conflict with one another or are mostly incommensurable.One needs a set of optimal solutions to solve these problems.A solution u = (u 1 , u 2 , . . ., u n ) ∈ Ω is said to be Pareto optimal if there exist no another solution v = (v 1 , v 2 , . . ., v n ) ∈ Ω such that f j (u) ≤ f j (v) for all j = 1, . . ., m and also f j (u) < f j (v) for at least index k.An objective vector is said to be Pareto optimal if their corresponding decision vector is Pareto optimal.All Pareto optimal solutions in the decision space of MOP is called Pareto set (PS) and their corresponding image in their objective space is called Pareto front (PF).The idea Pareto optimality was first proposed by Francis Ysidro Edgeworth in 1881 and then later on generalized by Vilfredo Pareto in 1986 as discussed in [12], [10].
The Pareto dominance concept of MOEAs is very common for solving MOPs [28], [12].To promote diversity, most of these algorithms use different diversity techniques such as fitness sharing, niching, the kernel approach, the nearest neighbour approach, the histogram technique, crowding or clustering, a relaxed form of dominance and restricted mating [11].Among them, a fast non-dominated sorting algorithm (NSGA-II) [13], SPEA2 or improving the strength Pareto evolutionary algorithm [60], the Pareto archive evolution strategy (PAES) [27], multi-objective genetic algorithm (MOGA) [18], and niched Pareto genetic algorithm (NPGA) [19] are long-familiar and well known approaches.They have shown good behaviors in several comparative analysis.
Multi-objective memetic algorithms (MOMAs) form a new and attractive area of research in EC.They are inspired by models of adaptation found in nature.They are known as Baldwinian EAs, Lamarckian EAs, cultural algorithms, or genetic local search and hybrid MOEAs [45].Hybrid MOEAs have been developed with the aim to overcome the shortcomings of stand-alone MOEAs [34], [32], [39], [42], [40].
A genetically adaptive multi-algorithm for multi-objective (AMALGAM) optimisation is recently developed for solving both multi-objective optimization problems [52] and single optimization problems [53].It employs multiple search operators for its population evolution.The search operators used include the particle swarm optimizer (PSO) [15], differential evolution (DE) [47] and NSGA-II [13] and allocates resources dynamically to each search operators based on their individual performances.It does not involve any decomposition as in MOEA/D (multio-bjective evolutionary algorithm based on decomposition) [54].MOEA/D [54] decomposes the approximated PF of the given MOP into a number of different single objective optimization subproblems (SOPs).It then optimizes all SOPs simultaneously using generic evolutionary algorithm.MOEA/D paradigm have tackled diverse benchmark functions and it has several enhanced versions [31], [34], [32], [39], [36], [42].In this paper, our main objective is to further improve the algorithmic performance of ALMAGAM by employing by employing multiple search operators including the differential evolution (DE) [46], particle swarm optimization (PSO) [15], simulated binary crossover (SBX) [24], Pareto archive evolution strategy (PAES) [23] and simplex crossover (SPX) [50] with self-adaptive alternative procedures for dealing with both CEC'09 test instances [57] and ZDT test problems [62].
The rest of this paper is organized as follows.Section II outlines the framework of the enhanced version of the genetically adaptive multi-algorithm multi-objective (AMAL-GAM)method.Section III presents experimental results obtained with the enhanced AMALGAM on both CEC'09 [57] and five ZDT test problems [62].Section IV is devoted to a discussion on experimental results.Section V finally concludes this paper and suggest further areas of research on this topic an related ones.

II. ENHANCED VERSION OF MULTI-ALGORITHM GENETICALLY ADAPTIVE FOR MULTIOBJECTIVE
OPTIMIZATION Algorithm 1 outlines the framework of the an enhanced version of the AMALGAM.In Sept 1, a population P with size N has been generated uniformly and randomly within the search space of the given MOPs.We then evaluate the fitness values of solution of population P .We calculate the crowding distance of each member of population after categorize them into different layers by using fast non-dominating sorting procedure adopted in NSGA-II [13] framework.After this, an Algorithm 1 divide the whole population according to k number of search operators in order to work each search operator on specified number of sub-populations N 1 ,N 2 ,N 3 to generate Q offspring population of sizes whose sum is equal to N .We have used five different search operators such as differential evolution (DE) [46], particle swarm optimization (PSO) [15], simulated binary crossover (SBX) [24] and Pareto archive evolution strategy (PAES) [23] and simplex crossover (SPX) [50] in the evolutionary process of the suggested algorithm.Each individual search operator is getting resources at population level according to their current individual performance based on self-adaptive procedure as explained in subsection II-A.

Algorithm 1 Enhanced Version of Multi-algorithm Genetically
Adaptive for Multiobjective optimization 1: Input: 2: MOP: the multiobjective optimization problem; N : the population size and other main parameters; F eval : maximum function evaluations; 3: Output:{x 1 , . . ., x N } and {F (x 1 ), . . ., F (x N )}; 4: Generate an initial population P of size N uniformly and randomly.5: Calculate the F-function values of each member of the P population.6: Assign rank to each member of P using fast nondominating procedure.7: Assign sub-populations P = {P 1 , P 2 , . . ., P k } to k operators for creating an offspring population

A. Alternative Adaptive Resources Allocation Scheme
• We calculate the number of solutions that successfully enter to the next generation in the evolutionary process of enhanced version of AMALGAM.A successful solution is rewarded by 1 and unsuccessful by 0. An efficient operator gets more resources in the form of subpopulation to be operate on them as compared to weaker one.
• Let δ k , k = 1, 2, . . ., q are total number of nondominated solutions produced by q search operators (i.e, differential evolution (DE) [46], particle swarm optimization (PSO) [15], simulated binary crossover (SBX) [24], Pareto archive evolution strategy (PAES) [23] and simplex crossover (SPX) [50]) that enter successfully to next generation are convert into normalized form to develop probability formula (3) , where Where P k is the current and P k−1 is the previous probability of successes of the k search operators.More importantly, the above mentioned dynamic resources allocation did not switch on at every generation of proposed algorithm.It can allocate resources at every multiple of 5 th generation to tackle ZDT test problems [62].The suggested enhanced AMALGAM allocates resources to each of its embedded search operator at every multiple of 10 th generation for dealing with CEC'09 test instances [57].

III. PARAMETERS SETTING AND EXPERIMENTAL RESULTS
We have carried out Experiments using benchmark functions with two and three objectives.The ZDT test [62] were tackled with parameter settings as explained in the subsection III-A while CEC'09 [57] were handled with parameter settings are explained in the subsection III-B, respectively.

A. Parameter Settings for ZDT Problems
• N = 100: population size for 2-objective test instances.

B. Parameter Settings for CEC'09 Test Instances
This subsection explains the parameters setting to validate enhanced AMALGAM on CEC'09 test instances [57].

C. Performance Indicators
Two main goals for dealing with multiobjective evolutionary optimization are very important: 1) convergence towards the Pareto-optimal front, 2) to find uniform and welldistributive set of multiple solutions that cover the whole true PF of the problem at hand [12].Several performance metrics are found in the specialized literature of evolutionary computing (EC) [51], [13], [12], [64] which are using to judge which algorithm is better than others and in what aspects.Inverted generational distance (IGD) [64], [57], relative hypervolume [51], [12], Gamma Υ and delta ∆ [12], [13] which are commonly in several comparative analysis of different algorithms.The aforementioned performance indicators can www.ijacsa.thesai.orguse only if the reference set for the test problems are known in advance or available.In this paper, we have used the Inverted Generational Distance (IGD) as performance indicator to judge the quality of final approximated set of Pareto optimal solutions obtained by proposed algorithm in comparison with other MOEAs.Let P * be a set of uniformly distributed points along the PF.Let A be an approximate set to the PF, the average distance from P * to A is defined as [57]: where d(v, A) is the minimum Euclidean distance between v and the points in A. If P * is large enough to represent the PF very well, D(A, P ) could measure both the diversity and convergence of A in a sense.The closer the IGD metric values, better is the approximation set.We have used P * = 500 in our carried experiments to tackle 2-objectives test instances and P * = 1000 to solve 3-objectives problems.For two objectives case, the IGD metric is pictorially depicted in the figure 1.It measures both the convergence and the spread of the obtained solutions.Smaller IGD-metric values, better will be approximated set of Pareto optimal of solutions of corresponding MOEAs.

IV. DISCUSSION OF THE EXPERIMENTAL RESULTS OBTAINED ON ZDT AND CEC'09 TEST INSTANCES
The simulation related parameters are as follows.
• Operating system: Windows XP Professional • Programming language of the algorithms: Matlab

A. Discussion of IGD-metric Values
Table I and Table II record the IGD-metric values in terms of minimum (Best), Median, Mean, and standard deviation(std) and maximum (worst) which are found by enhanced AMALGAM and NSGA-II [13], respectively.These statistics have been collected by executing each algorithm 30 times independently with different random seeds on each ZDT test problem [62] and CEC'09 test instance [57].The average CPU time spent by each algorithm are also provided in the last columns of I and Table II.It is evident from these Tables, that enhanced AMALGAM has found better approximated solution set with reduced the IGD-metric values as compared to NSGA-II [13] for most test problems.In most test problems, global convergence has been got for both test suites of problems.However, the complete Pareto front for some CEC'09 test instances have not been attained by enhanced AMALGAM in multiobjective optimization context.The primary reasons of this weak performance could be reason of the complicated objective functions profile of some CEC'09 test instances [57] which are mostly multi-modal near the global Pareto-optimal frontier and a slight perturbation in their optimization variables causes their solutions to become dominated.

Table I and Table II record the IGD-metric values in terms
of minimum (Best), Median, Mean, and standard deviation (std) and maximum (worst) which are found by enhanced AMALGAM and NSGA-II [13], respectively.These statistics have been collected by executing each algorithm 30 times independently with different random seeds on each ZDT test problem [62] and CEC'09 test instance [57].The average CPU time spent by each algorithm are also provided in the last columns of I and Table II.It is evident from statistics gathered in these Tables, that enhanced AMALGAM has found a better approximate solution set with reduced IGDmetric values compared to those of NSGA-II [13] for most test problems.In most test problems, global convergence has been achieved for both test suites of problems.However, the complete Pareto front for some CEC'09 test instances has not been attained by the enhanced AMALGAM algorithm, in the multi-objective optimization context.The primary reasons for this weak performance could be the complicated objective functions profile in some CEC'09 test instances [57]; these are mostly multi-modal near the global Pareto-optimal frontier and a slight perturbation in their optimization variables causes their solutions to become dominated.Figure 2 and Figure 6 depict the approximated Pareto front (PF) against the real PF of ZDT test problems displayed by enhanced AMALGAM and NSGA-II [13], respectively.These figures indicate that both algorithms have found better approximated PF in their best run among 30 independent runs on each ZDT test problem.We have plotted 30 PFs together in Figures 3 and Figures 7 of the enhanced AMALGAM and NSGA-II [13], respectively.These figures indicate that enhanced AMALGAM has displayed all 30 in better distribution ranges in all 30 independent runs as compared to NSGA-II [13].within 30 times independent execution for dealing with each problem.The Figure 8 is of the NSGA-II [13] produced in its best run among 30 independent runs for CEC'09 test instances [57].The PFs displayed by enhanced AMALGAM are more promising than NSGA-II [13] in terms of diversity and proximity.
We have also plotted 30 PFs altogether estimated by enhanced AMALGAM in the figure 5 and the figure 9 exhibited by NSGA-II [13] in all 30times independent runs over the problems UF1-UF4 and UF7-UF10.These figures clearly indicate that enhanced AMALGAM has tackled the most CEC'09 test instances more effectively and spend less CPU time dealing with each test problem as compared to NSGA-II [13].
The problems UF5 and UF6 have not been tackled by both algorithms as per demand and genetic drift has been occurred in their respective population due to the presence of highly multi-modality in these problems like UF5-UF6.The search process of both algorithms are get stuck in the local basin of attraction of these problems and due to this both algorithms have not shown further improvement dealing with these problems.

V. CONCLUSION AND FURTHER WORK
Different operators suite different optimization and search problems.The dynamic use of multiple operators in the EA framework has exhibited good performance on complicated MOPs, [31], [39], [34], [36], [32], [41], [40].A multialgorithm genetically adaptive multi-objective (AMALGAM) has recently been developed for solving both single objective [53] and multi-objective optimization problems [52].In this paper, we have suggested new adaptive resource allocation procedure and developed a enhanced version of AMALGAM to cope with CEC'09, [57], and ZDT test problems [53].The suggested algorithm has shown promising results on most test problems compared to NSGA-II [13] in terms of proximity and diversity.Furthermore, the suggested algorithm is more efficient which is desirable when solving real-word problems where time can be an issue.In the future, we intend to examine the performance of enhanced AMALGAM over real-world problems such as: • Tubular permanent magnet linear synchronous motor      6. Plots of the final non-dominated solutions in the objective space displayed by NSGA-II [13] in its best run among 30 independent runs over ZDT1-ZDT4 and ZDT6 problems. .

9 :
Assign rank to each member of Q using fast nondominating procedure.10: Combine the new and old population P and Q, R = P ∪Q.11: Select population P of size N from population R of size 2N based on their ranks and crowding distances for next generation.12: Update N best individuals among C population with high ranks and crowding density.13: Update P = {P 1 , P 2 , . . ., P k } (Explanation can be found in subsection II-A) based on the individual performances of each search operator.

Fig. 1 .
Fig. 1.Inverted Generation Distance (IGD) has been used as a Performance Indicator.The above solutions (blue fill circle) are approximated solution of the algorithm and (black fill circle) are the Pareto solutions uniformly distributive along the Pareto Front (PF).

Figures 4
Figures 4 display the best approximated PF of the CEC'09 test instances as demonstrated by enhanced AMALGAM

Fig. 2 .
Fig.2.Plots of the final non-dominated solutions in the objective space displayed by enhanced AMALGAM in its best run among 30 independent runs over ZDT1-ZDT4 and ZDT6 problems.

Fig. 3 .
Fig. 3. Plots of the 30 PFs altogether in the objective space displayed by enhanced AMALGAM for ZDT1-ZDT4 and ZD6 problems.

Fig. 4 .
Fig.4.Plots of the final non-dominated solutions in the objective space displayed by enhanced AMALGAM in its best run among 30 independent runs over UF1-UF10 problems.

Fig. 5 .
Fig.5.Plots of the 30 PFs altogether in the objective space displayed by enhanced AMALGAM for UF1-UF4 and UF7-UF10 problems.
Fig.6.Plots of the final non-dominated solutions in the objective space displayed by NSGA-II[13] in its best run among 30 independent runs over ZDT1-ZDT4 and ZDT6 problems.

Fig. 8 .Fig. 9 .
Fig.8.Plots of the final non-dominated solutions in the objective space display by NSGA-II[13] in its best run among 30 independent runs over UF1-UF10 problems.

TABLE I .
THE IGD-METRIC VALUES OF THE ENHANCED AMALGAM FOR ZDT1-ZDT4 AND ZDT6.AVG-T MEANS AVERAGE CPU TIME IN

TABLE III .
THE IGD-METRIC VALUES OBTAINED BY ENHANCED AMALGAM OVER CEC'09 TEST INSTANCES.AVG-T MEANS AVERAGE CPU TIME

TABLE IV .
THE IGD-METRIC VALUES GENERATE BY NSGA-II [13] IN 30 INDEPENDENT RUNS FOR CEC'09 TEST INSTANCES.AVG-T MEANS AVERAGE CPU TIME IN SECONDS.