A Comparison of MOEA/D, NSGA II and SPEA2 Algorithms

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1 Iteratioal Joural of Egieerig Treds ad Applicatios (IJETA) Volume 4 Issue 6, Nov-Dec 7 RESEARCH ARTICLE A Comparo of MOEA/D, ad Algorithms Rajai [1], Dr. Aad Prakash [] Asstat Professor of Computer Sciece [1] GDC Memorial College, Behal, Bhiwai, Haryaa Asstat Professor of Computer Sciece [] Govt. College, Kawali, Rewaril, Haryaa Idia OPEN ACCESS ABSTRACT The optimizatio of multi-objective problems curretly a importat area of research ad developmet. The importace gaied by th type of problem has give re to the developmet of multi-objective metaheurtics to attai solutios for such problems. I th paper, a experimetal comparo of MOEA/D (Multiobjective Evolutioary Algorithm based o Decompositio), (Nodomiated Sortig Geetic Algorithm II), ad (Stregth Pareto Evolutioary Algorithm ) usig T bechmark, has bee doe to determie which multi-objective metaheurtic has the best performace with respect to a problem. The results thus obtaied are compared ad aalyzed based o three performace metrics amely Hyper volume, GD, ad IGD that evaluate the dpersio of the solutios ad its proximity to it. Keywords: Multiobjective optimizatio; Multiobjective evolutioary algorithm based o Decompositio, Nodomiated sortig geetic algorithm-ii; Stregth pareto evolutioary algorithm. I. INTRODUCTION May real-world decio makig problems eed to achieve several objectives such as miimized rks, maximized reliability, miimized deviatios from desired levels, ad miimized cost etc. The mai goal of sigle-objective (SO) optimizatio to fid the best solutio, which correspods to the miimum or maximum value of a sigle objective fuctio that lumps all differet objectives ito oe. Th type of optimizatio useful as a tool that provides decio makers the ights ito the ature of the problem, but usually lacks to provide a set of alterative solutios that have tradeoff amog differet objectives agat each other. O the cotrary, i a multi-objective optimizatio with coflictig objectives, there o sigle optimal solutio (Rajai et al 7). The iteractio amog differet objectives gives re to a set of compromed solutios, largely kow as the trade-off, o-domiated, o-iferior or Pareto-optimal solutios (Savic, 00). Optimizatio used to fid out oe or more feasible solutios that give(s) the best value(s) for oe or more objectives of some fuctio. Whe oly oe objective fuctio ivolved i the optimizatio problem, the task of fidig the best possible solutio called sigle objective optimizatio as show i figure 1(a). O the other had, whe more tha oe objective fuctio are ivolved i the optimizatio problem, the task of fidig oe or more best possible solutio(s) kow as multi-objective optimizatio as show i figure 1(b) (Badyopadhyay ad Saha, 3). Figure 1 (a) Sigle objective problem formulatio ad (b) Multi-objective problem formulatio feature feature feature 1 feature feature feature 3 feature 1 feature feature 3 Objective 1 Objective Objecti The task of simultaeously optimizig more tha oe coflictig objectives with respect to a set of certai G Go al ISSN: Page 58

2 Iteratioal Joural of Egieerig Treds ad Applicatios (IJETA) Volume 4 Issue 6, Nov-Dec 7 costraits dealt with the help of multi-objective optimizatio. I multi-objective optimizatio problem, if optimizatio of oe objective automatically leads to optimizatio of aother objective, the th kid of problem ot cosidered as a multi-objective optimizatio problem. However, i may real-life situatios we come across problems where a attempt to improve oe objective leads to degradatio of the other objective. Such problems belog to the class of multi-objective optimizatio problems (Deb, 4). Mathematically, the multi-objective problem could be writte as follows: Maximize/miimize (1) Subject costraits: (3) (4) where parameter a dimesioal vector with desig or decio variables,. The costraits i equatio (4) are called variables boud, that restrict each decio variable to take a value withi a lower ad a upper boud. These bouds make up a decio space,. A solutio that satfies all the costraits ad all the variable bouds called a feasible solutio otherwe it kow as a ifeasible solutio. The set of all feasible solutios called the feasible regio or search space, (Deb, 4). Give two vectors (where the decio space), the vector said to domiate,, iff ot worse tha i ay objective fuctio ad it strictly better i at least oe objective fuctio (Deb, 4). If either domiates, or domiates, ad are said to be o-comparable, deoted as. For a give MOP, the Pareto optimal set the set cotaiig all the solutios that are o-domiated with to () respect to. It ca be deoted as (Veldhuize ad Veldhuize, 1999): The, for a give MOP ad its correspodig Pareto optimal set ), the Pareto optimal frot the result of mappig to (where the objective space). defied as (Veldhuize ad Veldhuize, 1999): where the vector fuctio cotaiig objective fuctio. A approximatio set defied by Zitzler et al. as follows (Zitzler et al., 003): let be a set of objective vectors. called a approximatio set if ay elemet of does ot domiate or ot equal to ay other objective vector i. The set of all approximatio sets deoted as. The rest of the paper orgaized as follows: Sectio itroduces brief review of multi-objective optimizatio algorithms. Sectio 3 talks about the basics of three multiobjective optimizatio techiques i.e. MOEA/D,, ad. Sectio 4 covers experimetatio, results ad dcussio. Sectio 5 cocludes the work. II. REVIEW OF ALGORITHMIC CONCEPTS OF MOEA/D,, AND Optimimizig multiple objectives i a problem curretly a importat area of research ad developmet. Recetly, researchers focused o developmet of multiple metaheurtics for solvig multi-objective problems. The followig sectio gives the review of algorithmic cocepts of MOEA/D,, ad..1 MOEA/D (Multiobjective Evolutioary Algorithm based o Decompositio) MOEA/D (Zhag ad Li 007) explicitly decomposes the MOP ito scalar optimizatio sub problems. It solves these sub problems simultaeously by evolvig a populatio of solutios. At each geeratio, the populatio composed of the best solutio foud so far (i.e. sice the start of the ru of the algorithm) for each subproblem. The eighborhood relatios amog these subproblems are defied based o the dtaces betwee their aggregatio coefficiet vectors. The optimal solutios to two eighborig subproblems should be very similar. Each subproblem (i.e., scalar aggregatio fuctio) optimized i MOEA/D by usig iformatio oly from its eighborig subproblems. ISSN: Page 59

3 Iteratioal Joural of Egieerig Treds ad Applicatios (IJETA) Volume 4 Issue 6, Nov-Dec 7. NSGA-II (Nodomiated Sortig Geetic Algorithm II) The multi-objective evolutioary algorithms (MOEA) suffered lots of drawbacks such as: 1. Computatioal complexity where the umber of objective ad the populatio size to.. No-elitm approach ad 3. Need for specifyig a sharig parameter. To overcome these difficulties, K. Deb et al. (00) proposed a elitt o-domiated Sortig GA (NSGA-II). I NSGA-II, the paret populatio used to create the offsprig (child) populatio. To esure a better spread amog the solutio, a ichig strategy used to choose the diverse set of solutio from the set (i.e. paret populatio + offsprig populatio) followed by crowded touramet selectio, crossover ad mutatio operator (Rajai et al 7)..3 (Stregth Pareto Evolutioary Algorithm ) (Zitzler, Laumas, ad Thiele 0) a extesio of SPEA (Stregth Pareto Evolutioary Algorithm) (Zitzler ad Thiele 1999) that resolves the followig weakess of SPEA i.e. 1) Fitess assigmet, ) Desity estimatio, 3) Archive trucatio. SPEA uses a regular populatio ad a archive (exteral set). Startig with a iitial populatio ad a empty archive the followig steps are performed per iteratio. First, all odomiated populatio members are copied to the archive; ay domiated idividuals or duplicates (regardig the objective values) are removed from the archive durig th update operatio. If the size of the updated archive exceeds a predefied limit, further archive members are deleted by a clusterig techique which preserves the charactertics of the odomiated frot. Afterwards, fitess values are assiged to both archive ad populatio members. III. EXPERIMENTATION ad Result Dcussio 3.1 Bechmark Fuctios Used The defiitio of the bechmark fuctios used for experimetatio as follows (Table 1): Table 1 Costrait Bechmark Fuctios 3. Parameter Settig for Ivolved Algorithms The followig table idicates the parameter settigs for the three algorithms. Table MOEA/D Parameter Settigs for, MOEA/D, ad Populatio Mutatio Crossover Performace Metrics ISSN: Page 60

4 Iteratioal Joural of Egieerig Treds ad Applicatios (IJETA) Volume 4 Issue 6, Nov-Dec 7 For the performace measure, as may as three differet performace metrics amely, Hypervolume, Geeratioal Dtace (GD), ad Iverted Geeratioal Dtace (IGD) have bee used (Rajai et al 7). These are defied as: The Hypervolume (HV) (Auger et al, 009), also kow as S metric, hyper-area or Lebesgue measure, a uary metric that measures the size of the objective space covered by a approximatio set. HV cosiders all three aspects: accuracy, diversity ad cardiality, beig the oly uary metric with th capability. A algorithm with a large value of HV desirable. Hypervolume calculated as: Geeratioal Dtace (GD) (Riquelme et al, 5) takes as referece a approximatio set ad calculates how far it from the Pareto optimal frot (or referece set ). Th uary measure cosiders the average Euclidea dtace betwee the members of ad the earest member of. It ca be oticed that GD cosiders oly oe aspect of : the accuracy. A algorithm havig a small value of GD better. It calculated as: Iverted Geeratioal Dtace (IGD) (Riquelme et al, 5) a iverted variatio of GD but it sigificatly differet from GD: i) it calculates the miimum Euclidea dtace (tead of the average dtace) betwee a approximatio set ad the Pareto optimal frot, ii) IGD uses as referece the solutios i solutios i (ad ot the ) to calculate the dtace betwee the two sets ad iii) if sufficiet members of are kow, IGD could measure both the diversity ad the covergece of calculated as:. It where the umber of true pareto optimal solutios ad idicates the Euclidea dtace betwee the ith true pareto optimal solutio ad the closest obtaied pareto optimal solutios i the referece set. 3.4 Results ad Dcussio The above performace metrics allow us to quatitatively compare MOEA/D,, ad. All the algorithms are ru 0 times o the test problems ad the stattics results of these 0 rus are provided i Tables The comparo of the results over five multi-objective test fuctios i.e.,,, ad for three algorithms amely MOEA/D,, ad give i table below: Populatio 80: Table 3 provides stattical results of the algorithms for Hypervolume (HV) o populatio 80. Sice HV the metric that cosiders accuracy ad diversity of a algorithm. Th table shows that MOEA/D algorithm outperforms others o stattical metric mea for,, ad. For, outperforms MOEA/D ad SPEA i terms of mea. MOEA/D also outperforms others i terms of stattical metric stadard deviatio. Table 3 Stattical results for Hypervolume o Populatio 80 o T bechmark T 1 T T 3 T 4 T 6 MOEA/ D Table 4 Stattical results for GD o Populatio 80 o T bechmark MOEA /D E E E E E E E E E E ISSN: Page 61

5 Iteratioal Joural of Egieerig Treds ad Applicatios (IJETA) Volume 4 Issue 6, Nov-Dec 7 Table 5 Stattical results for IGD o Populatio 80 o T bechmark MOE A/D SPEA The stattical results of the algorithms o T bechmark for GD are show i Table 4. It has bee observed from the table that for stattical metric mea, has the better results tha others except o. For, has maximum value for mea. GD the performace metric that shows the accuracy of a algorithm. So it ca be stated that algorithm superior except for. For stattical metric stadard deviatio, MOEA/D outperform better tha others. Table 5 shows the stattical results of the algorithms o T bechmark fuctios for IGD. It has bee observed from the table that has better results for,, ad. For, has good results for mea. It has also bee observed that for stattical metric stadard deviatio, MOEA/D agai outperform better tha others. Populatio 100: The stattical results of MOEA/D,, ad o T bechmark o populatio 100 for GD, IGD ad Hypervolume are show i Tables 6-8. The stattical results of, Abyss ad OMOPSO o T bechmark o populatio 100 for hypervolume metric show i Table 6. Accordig to th, MOEA/D outperforms others for stattical metrics mea except for. For, has greater value for mea. For stattical metric stadard deviatio, MOEA/D domiates others. It demostrates the supremacy of MOEA/D algorithm. The stattical results of MOEA/D,, ad o T bechmark o populatio 100 for GD metric show i Table 7. Accordig to th, outperforms others o,,, ad. For, outperforms others. Agai for stattical metric stadard deviatio, MOEA/D has best values. Table 6 Stattical results for Hypervolume o Populatio 100 o T bechmark MOE A/D E 3.30E E 5.81E E.18E E 8.61E E.8E Table 7 Stattical results for GD o Populatio 100 o T bechmark MOE A/D E.07E E 4.36E E 1.3E E 3.97E E 1.68E Table 8 Stattical results for IGD o Populatio 100 o T bechmark MOEA ISSN: Page 6

6 Iteratioal Joural of Egieerig Treds ad Applicatios (IJETA) Volume 4 Issue 6, Nov-Dec 7 /D T E T E E 6.69E 6.19E T E E E E E 1.67E 4.18E Table 10 Stattical results for GD o Populatio 10 o T bechmark 1.68E.19E 6.86E MOEA/ 1.40E 4.58E 5.83E D The stattical results of MOEA/D,, ad T o T bechmark o populatio 100 for IGD metric show 1 i Table 8. Accordig to th, agai outperforms T others except for mea. For, has maximum value for mea. Agai for stattical metric T Estadard deviatio, MOEA/D domiates others. 3 T Populatio 10: 4 The stattical results of MOEA/D,, ad T o T bechmark o populatio 10 for GD, IGD 6 ad Hypervolume metric are show i table The stattical results of MOEA/D,, ad o T bechmark o populatio 10 for Hypervolume Table 11 Stattical results for IGD o Populatio 10 o metric show i Table 9. Accordig to th, MOEA/D T bechmark outperforms others i terms of mea ad stadard deviatio except. For, has the maximum value for mea. MOEA /D SPEA Table 10 shows the stattical results of MOEA/D, NSGA II, ad o T bechmark o populatio 10 for GD metric. Accordig to th, outperforms others o,,, ad. For, outperforms others. For stadard deviatio, MOEA/D has best values T 1 T Table 9 Stattical results for Hypervolume o Populatio 10 o T bechmark MOEA/ D E E ISSN: Page 63

7 Iteratioal Joural of Egieerig Treds ad Applicatios (IJETA) Volume 4 Issue 6, Nov-Dec 7 Table 11 shows the stattical results of MOEA/D,, ad o T bechmark o populatio 10 for IGD metric. Accordig to th, outperforms others o, ad. For ad, outperforms others. Agai for stadard deviatio, MOEA/D has best values. IV. CONCLUSIONS I th paper, we have performed a experimetal comparo amog three algorithms for multi-objective optimizatio. To evaluate the performace of the algorithms we have used five well-kow bechmark problems (T) by three uary metrics i.e. Hypervolume, GD, IGD. Accordig to experimets with test fuctios examied ad the parameter settigs used, we ca coclude that for Hypervolume metric MOEA/D yields the best performace i terms of mea o populatio 80,100 ad 10. For GD ad IGD metric, MOEA/D yields best results for stadard deviatio ad NSGA II yields best results for mea o populatio 80,100 ad 10. REFERENCES [1] Rajai, Kumar D., Kumar V.(7), A Coceptual Comparo of, OMOPSO ad AbYss algorithms i: Proceedigs of the 3 rd Iteratioal Coferece o Iovatios i Egieerig ad Techology (ICET). [] Savic, D. (00) Sigle-objective vs. Multiobjective Optimatio for Itegrated Decio Support, I: Itegrated Assessmet ad Decio, i: Proceedigs of the First Bieial Meetig of the Iteratioal Evirometal Modellig ad Software Society, pp [3] Deb K. (4) Multi-objective Optimizatio Usig Evolutioary Algorithms, Studet Wiley Editio, pp.1-1. [4] Badyopadhyay, S., Saha, S. (3) Some Sigle- ad Multiobjective Optimizatio Techiques, i: Usuperved Classificatio. Spriger Berli Heidelberg, pp [5] Veldhuize, D.A.V., Veldhuize, D.A.V. (1999) Multiobjective Evolutioary Algorithms: Classificatios, Aalyses, ad New Iovatios. Evolutioary Computatio. [6] Zitzler, E., Thiele, L., Laumas, M., Foseca, C.M., Foseca, V.G. da (003) Performace assessmet of multiobjective optimizers: a aalys ad review. IEEE Trasactios o Evolutioary Computatio 7, [7] Zhag, Q., Li, H., (007) MOEA/D: A Multiobjective Evolutioary Algorithm Based o Decompositio. IEEE Trasactios o Evolutioary Computatio 11(6), [8] Deb, K., Pratap, A., Agarwal, S., Meyariva, T. (00) A fast ad elitt multiobjective geetic algorithm: NSGA-II. IEEE Trasactios o Evolutioary Computatio 6, [9] Eckart Zitzler, Marco Laumas, ad Lothar Thiele (0) : Improvig the Stregth Pareto Evolutioary Algorithm. [10] Zitzler, E. ad L. Thiele (1999). Multiobjective evolutioary algorithms: A comparative case study ad the stregth pareto approach. IEEE Trasactios o Evolutioary Computatio 3(4), [11] Auger, A., Bader, J., Brockhoff, D., Zitzler, E., (009). Theory of the Hypervolume Idicator: Optimal - dtributios ad the Choice of the Referece Poit, i: Proceedigs of the Teth ACM SIGEVO Workshop o Foudatios of Geetic Algorithms, FOGA 09. ACM, New York, NY, USA, pp [1] Riquelme, N., Lücke, C.V., Bara, B. (5) Performace metrics i multi-objective optimizatio, i: 5 Lati America Computig Coferece (CLEI). pp ISSN: Page 64