HYBRID FUZZY MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM: A NOVEL PARETO-OPTIMIZATION TECHNIQUE

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1 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February 22 HYBRID FUZZY MULTI-OBJECTIVE EVOLUTIONARY ALGORITHM: A NOVEL PARETO-OPTIMIZATION TECHNIQUE Amt Saraswat and Ashsh San 2 Department of Electrcal Engneerng, Faculty of Engneerng, Dayalbagh Educatonal Insttute, Agra-282, Uttar Pradesh, Inda amtsaras@gmal.com, 2 ashsh_7@redffmal.com ABSTRACT A novel pareto-optmzaton technque based on newly developed hybrd fuzzy mult-objectve evolutonary algorthm (HFMOEA) s presented n ths paper. In HFMOEA, two sgnfcant parameters such as crossover probablty (P C ) and mutaton probablty (P M ) are dynamcally vared durng optmzaton based on the output of a fuzzy controller for mprovng ts convergence performance by gudng the drecton of stochastc search to reach near the true pareto-optmal soluton effectvely. The performance of HFMOEA s tested on three benchmark test problems such as ZDT, ZDT2 and ZDT3 and compared wth NSGA-II. KEYWORDS Mult-objectve evolutonary algorthms, Fuzzy logc controller, Global optmal soluton, Pareto-optmal front.. INTRODUCTION Almost all real world optmzaton problems nvolve optmzng (.e. whether mnmzaton or maxmzaton or combnatons of both) the multple objectve functons smultaneously. In fact, these objectve functons are non-commensurable and often conflctng objectves. Multobjectve optmzaton wth such conflctng objectve functons gves rse to a set of optmal solutons, nstead of one optmal soluton []. In general, a common mult-objectve optmzaton problem may be formulated [] as follows: Mnmze f () x =,..., N () g j () x = j =,..., M, subject to : hk () x k =,..., K, obj Where f () x s the th objectve functon, x s a decson vector that represents a soluton, and N obj s the number of objectve functons, M and N are number of system equalty and nequalty constrants respectvely. For a mult-objectve optmzaton problem, any two solutons x and x 2 can have one of two possbltes- one domnates the other or none domnates the other. In a mnmzaton problem, wthout loss of generalty, a soluton x domnates x 2, f and only f, the followng two condtons are satsfed: {,..., obj} :()() 2 {,..., obj} :()() j j 2 = N f x f x (3) j = N f x < f x (2) (4) DOI :.52/jfls

2 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February 22 If any of the above condtons s volated, the soluton x does not domnate the soluton x 2. If x domnates the soluton x 2, then x s called the non-domnated soluton wthn the set {x, x 2 }. The solutons that are non-domnated wthn the entre search space are denoted as Pareto-optmal and consttute the Pareto-optmal set or Pareto-optmal front []. In last couple of decades, a number of mult-objectve evolutonary algorthms (MOEA s) have been suggested for solvng such a complex mult-objectve problems []-[6]. The man purpose behnd the development of the MOEA approach s that t has ablty to fnd multple Paretooptmal solutons n one sngle smulaton run. The non-domnated sortng genetc algorthm (NSGA) proposed n [4] was one of the frst such EAs. Over the years, NSGA was crtczed n [5] on the bass of some aspects such as hgh computatonal complexty of non-domnated sortng, lack of eltsm and need for specfyng the sharng parameter. In reference [6], an mproved verson of NSGA called as NSGA-II. Two other contemporary MOEAs: Pareto-archved evoluton strategy (PAES) [7] and strength Pareto EA (SPEA) [8] were also reported n the lterature. A detaled survey of all mult-objectve evolutonary and real coded genetc algorthms s gven n reference [9]. In present paper, a new Hybrd Fuzzy Mult-Objectve Evolutonary Algorthm (HFMOEA) has been proposed for solvng complex mult-objectve problems. In proposed HFMOEA, a fuzzy logc controller ( FLC_HMOEA) has been developed, whch would cause varaton n two HFMOEA parameters such as crossover probablty ( P C ) and mutaton probablty ( P M ) dynamcally durng optmzaton process after each k number of teratons. These parameter varatons provde HFMOEA a knd of adaptablty to the nature of targeted optmzaton problem and help to reach the near global optmal solutons and hence arrve near to true pareto-optmal front. The performance of HFMOEA s examned on three benchmark test problems such as ZDT, ZDT2 and ZDT3. Ths paper progresses wth desgnng a fuzzy logc controller for HFMOEA presented n secton 2. Secton 3 dscusses the complete procedural steps nvolved n the mplementaton of proposed HFMOEA. Smulaton results of HFMOEA and ts comparson wth NSGA-II on three benchmark test problems (ZDT, ZDT2 and ZDT3) are presented n secton 4. Fnally, concluson s drawn n secton Desgn of Fuzzy Logc Controller for HFMOEA In present paper, a HFMOEA approach s developed for solvng complex mult-objectve problems. In ths HFMOEA, ts two parameters such as crossover probablty (P C ) and mutaton probablty (P M ) are vared dynamcally after k teratons perodcally durng the executon of the program. These varatons n parameters would be taken place accordng to a fuzzy knowledge base whch has been developed from experence to maxmze the effcency of HFMOEA. Therefore, a fuzzy logc controller ( FLC_HMOEA) s desgned n ths papper. The schematc dagram of Fuzzy Rule Base and the workng of proposed fuzzy logc controller (.e. FLC_HMOEA) are shown n Fg.(a) and (b) respectvely. Fg.(a). Fuzzy Rule Base n FLC_HMOEA 3

3 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February 22 Fg.(b). Block dagram for Fuzzy Logc Controller (.e. FLC_HMOEA) for HFMOEA parameters tunng HFMOEA parameters (P C and P M ) are vared based on the ftness functon values as per followng logc as n reference []: I. The value of best compromzed ftness for each teraton (B CF) s expected to change over a number of teratons, but f t does not change sgnfcantly over a number of teratons (UN) then ths nformaton s consdered to cause changes n both P C and P M. II. The dversty of a populaton s one of the factors, whch nfluences the search for a true optmum. The varance of the ftness values of objectve functon (VF) of a populaton s a measure of ts dversty. Hence, t s consdered as another factor on whch both P c and P m may be changed. Thus the ranges of three nput parameters such as best compromzed ftness (BCF), number of teraton durng whch best ftness does not changed (UN) and varance n ftnesses n current populaton (VF) and two output parameters such as crossover probablty ( P C ) and mutaon probabltes ( P M ) have been dvded nto three lngustc terms as LOW, MEDIUM and HIGH. The detals of membershp functons for nput and output varables of FLC_HMOEA are shown n Fg.2 and Fg.3 respectvely. Low Medum Hgh Low Medum Hgh Low Medum Hgh Degree of membershp Degree of membershp Degree of membershp BCF UN VF Fg.2. Input parameters for Fuzzy Logc Controller 3

4 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February 22 (a) (b) D e g r e e o f m e m b e r s h p L o w M e d u m H g h D e g r e e o f m e m b e r s h p L o w M e d u m H g h P c P m Fg.3. Output parameters for Fuzzy Logc Controller Fuzzy Rule Base for HFMOEA. If (BCF s Low) then (Pc s Hgh)(Pm s Low) () 2. If (BCF s Medum) and (UN s Low) then (Pc s Hgh)(Pm s Low) () 3. If (BCF s Hgh) and (UN s Low) then (Pc s Hgh)(Pm s Low) () 4. If (BCF s Medum) and (UN s Medum) then (Pc s Medum)(Pm s Medum) () 5. If (BCF s Hgh) and (UN s Medum) then (Pc s Medum) () 6. If (UN s Hgh) and (VF s Low) then (Pc s Low)(Pm s Hgh) () 7. If (UN s Hgh) and (VF s Medum) then (Pc s Low) () 8. If (UN s Hgh) and (VF s Hgh) then (Pc s Medum) () 9. If (BCF s Hgh) and (VF s Medum) then (Pm s Low) (). If (BCF s Hgh) and (VF s Hgh) then (Pm s Low) (). If (VF s Hgh) then (Pc s Hgh)(Pm s Low) () 2. If (VF s Medum) then (Pc s Hgh)(Pm s Low) () 3. If (BCF s Hgh) and (VF s Low) then (Pc s Hgh)(Pm s Low) () 4. If (BCF s Medum) and (VF s Medum) then (Pc s Low)(Pm s Hgh) () 5. If (BCF s Low) and (UN s Low) and (VF s Low) then (Pc s Hgh)(Pm s Low) 3. Implementaton of HFMOEA The flowchart of proposed HFMOEA for soluton of complex mult-objectve problems s outlned n Fg.4. Detals of proposed algorthm are dscussed as below: Intalzaton: HFMOEA based optmzaton starts wth ntalzaton of varous nput parameters of HFMOEA such as populaton sze (popsze), maxmum numbers of teratons (max_teraton), number of control varables, system constrants lmts, crossover probablty (P C ), mutaton probablty (P M ) etc. Generaton of Intal populaton: t s generated randomly accordng to followng procedural steps: Step : Generate a strng of real valued random numbers wthn ther gven varable lmts to form a sngle ndvdual; Step 2: Place the ndvdual as vald ndvdual n ntal populaton; Step 3: Evaluate ftness value for vald ndvdual; Step 4: Check f the ntal populaton has not completed then go to step ; Non-Domnaton Sortng: The generated ntal populaton s sorted on the bass on nondomnaton sortng algorthm proposed by Deb [] and [5]. 32

5 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February 22 Fg.4. Flowchart for Hybrd Fuzzy Mult-Objectve Evolutonary Algorthm (HFMOEA) For producng the new populaton for next teraton, the followng operators are appled to parent populaton: Selecton: The Bnary Tournament selecton as proposed n reference [5] s used as a selecton operator for reproducng the matng pool of parent ndvduals for crossover and mutaton operatons. Crossover: The BLX- crossover as proposed n reference s appled on randomly selected pars (,)(2,) t t, P whch s a combnaton of an of parent ndvduals ( ) extrapolaton/nterpolaton method. x x wth a crossover probablty ( ) Mutaton: The PCA based Mutaton as proposed n reference [] wth mutaton probablty ( ) P m s appled to generate the offsprng populaton. C 33

6 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February 22 Crteron to prepare populaton for next teraton: After the executon of above genetc operators, offsprng populaton s checked to prepare new populaton for next teraton by gong through followng procedural step: Step : Evaluate the ftness values for each ndvdual n offsprng populaton; Step 2: Combne the parent and offsprng populaton to obtan the ntermedate populaton; Step 3: Perform the non-domnaton sortng algorthm on ntermedate populaton; Step 4: Remove the worse ndvduals to mantan the new populaton sze constant. Here the new populaton for next teraton s prepared; Step 5: Check f k th teratons (let k = ) has completed go to next step 6 otherwse go to step 7. Step 6: Update HFMOEA parameters (.e. Pc and Pm) by usng fuzzy logc controller (FLC_HMOEA). Step 7: Check the termnaton condton of HFMOEA..e. f the current teraton number s equal to max_teraton, termnate the teraton process otherwse go to next teraton. Step 8: Select the best compromse soluton usng fuzzy set theory. FLC_HMOEA: fuzzy logc controller (FLC_HMOEA) s desgned n secton 2. The membershp functons and membershp values for these three varables (BF, UN and VF) are selected after k th teratons (let k = ) to get optmum results. A dagrammatc representato n of an approach to ncorporate fuzzy logc controller n HFMOEA s shown n Fg.4. Best compromse soluton: Upon havng the Pareto-optmal set of non-domnated soluton usng proposed HFMOEA approach, an approach proposed n [4] selects one soluton to the decson maker as the best compromse soluton as used n [5]. Ths approach suggests that due to mprecse nature of the decson maker s judgment, the th objectve functon F s represented by a membershp functon defned as n reference [4]: F F F F mn max mn max = F < < max mn F F F F mn F F (5) mn max Where F and F are the mnmum and maxmum values of the th objectve functon among all non-domnated solutons, respectvely. For each j th non-domnated soluton, the normalzed j membershp functon s calculated as: j = Nobj j = N N dom obj j j= = (6) Where N dom s the number of non-domnated solutons. The best compromse soluton s that j havng the maxmum value of. 34

7 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February 22 Ftness functon: The ftness functon correspondng to each ndvdual n the populaton s assgned based on ther respectve generalzed augmented functons as evaluated n equaton (7). Thus the ftness functon ( H ) for m th objectve s evaluated as: Km H m = ; m = : N + f obj, m Where N obj s the total number of objectves and m th objectve, n ths work. m obj (7) Km s the approprate constant correspondng to 4. Smulaton Results The proposed HFMOEA s mplemented accordng to the procedure explaned n prevous sectons and all the smulatons are carred out n MATLAB 7. programmng envronment on Pentum IV 2.27 GHz, 2. GB RAM computer system. In present case study, the proposed HFMOEA s examned and compared wth a popular multobjectve evolutonary algorthm.e. NSGA-II presented n reference [5]. The detaled specfcatons of both NSGA-II and HFMOEA are summarzed n Table. The NSGA-II comprses a smulated bnary crossover (SBX) operator and a polynomal mutaton [3] lke real coded GAs. For real-coded NSGA-II, dstrbuton ndexes [3] as c = 2, and m = 2 are used for crossover and mutaton operators respectvely (see Table ). Whereas n HFMOEA, a BLX-α crossover and PCA-mutaton [] operators are used wth dynamcally varyng after each teratons wth probabltes ( P C and P M ) based on fuzzy logc controller ( FLC_HMOEA) as descrbed n secton 2. Table. Specfcatons of optmzaton algorthms Algorthm Parameters NSGA-II HFMOEA Selecton operator Tournament Tournament Crossover operator Smulated Bnary (SBX) BLX-α crossover Mutaton operator polynomal mutaton PCA mutaton Crossover probablty (P C ).9 varyng based on FLC output Mutaton probablty (P M ) = 2, and = 2 varyng based on FLC output c m Three benchmark test problems such as ZDT, ZDT2 and ZDT3 out of sx as suggested by Ztzler, Deb and Thele [6] are taken for testng and comparson of proposed HFMOEA. All three test problems have two objectve functons as descrbed n Table 2. None of these problems have any constrant. Table 2 also shows the number of varables, ther bounds, the Pareto-optmal solutons, and the nature of the Pareto-optmal front for each problem.e. whether t s convex or non-convex, contnuous or dscontnuous. In ths paper, the whole smulaton s dvded nto four cases such that n each case, both the algorthms (NSGA -II and HFMOEA) are evaluated for deferent populaton szes and number of maxmum teratons. Thus, the Populaton sze and number of maxmum teratons are taken as ( and 3), ( and 5), (2 and 5) and (3 and 5) n Case:, Case: 2, Case: 3 and Case: 4 respectvely (see Table 3). For all three test problems, the best compromsed solutons obtaned after optmzaton usng NSGA-II and HFMOEA are summarzed n Table 3. The best compromsed soluton s calculated accordng to () and (2) descrbed n prevous secton. 35

8 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February 22 Table 2. Detals of Test Problems used n case study Test Problem Varable Bounds Objectve Functon Remark ZDT [,] n = 3 ( ) f x = x and ( ) = () ( /() ) f2 x g x f g x n 9 g() x = + x ( ) f x = x and n = 2 Optmal Solutons x [,] x = = 2 : n Convex paretooptmal Front ZDT2 [,] n = 3 2 ( ) ( ) = () ( /() ) f x g x f g x 2 9 g() x = + n n = 2 x Optmal Solutons x [,] x = = 2: n Non convex paretooptmal Front ZDT 3 [,] n = 3 ( ) f x = x and f f f ( ) 2 x = g() x sn ( f ) g()() x g x 9 g() x = + n n = 2 x Optmal Solutons x [,] x = = 2: n Convex and dsconnected paretooptmal Front Table 3. Best Compromsed solutons obtaned by NSGA-II and HFMOEA Test Case Pop. Sze Case: 3 Case:2 5 Case:3 2 5 Case:4 3 5 Max. Iteratons Best Compromsed Soluton after optmzaton optmzaton Algorthm Test Problem : ZDT Test Problem : ZDT2 Test Problem : ZDT3 f(x) f2(x) f(x) f2(x) f(x) f2(x) NSGA-II HFMOEA NSGA-II HFMOEA NSGA-II HFMOEA NSGA-II HFMOEA

9 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February (a) Case : NSGA-II HFMOEA.4.2 (b) Case : 2 NSGA-II HFMOEA Objectve 2 : FQ Objectve 2 : FQ Objectve : FQ Objectve : FQ.4.2 (c) Case :3 NSGA-II HFMOEA.4.2 (d) Case : 4 NSGA-II HFMOEA Objectve 2 : FQ Objectve 2 : FQ Objectve : FQ Objectve : FQ Fg.5. Comparson of Pareto-optmal fronts obtaned usng NSGA-II and HFMOEA for ZDT test problem for dfferent populaton szes after varous teratons O b j e c t v e 2 : F Q ( a) C a s e : N S G A - II O b j e c t v e 2 : F Q ( b) C a s e : 2 N S G A - II O b j e c t v e 2 : F Q O b j e c t ve : F Q ( c) C a s e : 3 N S G A - II O b j e c t v e 2 : F Q O b j e c t ve : F Q ( d) C a s e : 4 N S G A - II O b j e c t ve : F Q O b j e c t ve : F Q Fg.6. Comparson of Pareto-optmal fronts obtaned usng NSGA-II and HFMOEA for ZDT2 test problem for dfferent populaton szes after varous teratons 37

10 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February 22 O b je c t v e 2 : F Q (a) C a se : N S G A -II O b je c t v e 2 : F Q (b) C a se : 2 N S G A -II O b je c t v e 2 : F Q O b je c t ve : F Q (c) C a se : 3 N S G A -II O b je c t v e 2 : F Q O b je c t ve : F Q (d) C a se : 4 N S G A -II O b je c t ve : F Q O b je c t ve : F Q Fg.7. Comparson of Pareto-optmal fronts obtaned usng NSGA-II and HFMOEA for ZDT3 test problem for dfferent populaton szes after varous teratons The pareto-optmal fronts obtaned by NSGA-II and HFMOEA for ZDT test problem n all four cases are depcted n Fg.5. It has been observed that NSGA-II could not be fully converged n case and Case: 2 when the populaton sze s and maxmum numbers of teratons are 3 and 5 respectvely. Whle, proposed HFMOEA has been converged and able to acheve near global pareto-optmal front even n case: (see Fg.5). Smlar nvestgatons are conducted on another to benchmark test problems such as ZDT2 and ZDT3 for Case:, Case: 2 and Case: 3, the pareto-optmal fronts are obtaned shown n Fg.6 and Fg.7, respectvely. Durng the executon of optmzaton based on proposed HFMOEA, t s two parameters such as crossover probablty (P C ) and mutaton probablty (P M ) are vared dynamcally after each ten teratons. These varatons are taken place based on the output of fuzzy controller (FLC_HMOEA) as descrbed n secton 2. The varatons n P C and P M for all three test problems (ZDT, ZDT2 and ZDT3) n Case: 3 are shown n Fg. 8. It has been observed that the varatons n crossover and mutaton probabltes are such that f P C s gong to reduce, P M wll ncrease (see Fg.8.). These varatons n parameters are helpng the HFMOEA n searchng the global optmal solutons. Therefore, ths property wll enhance the capablty of HFMOEA to acheve the near global pareto-optmal front. 5. Concluson A fuzzy logc controller called as FLC_HMOEA has been developed and successfully appled n a proposed mult-objectve optmzaton algorthm.e. HFMOEA. Ths mplementaton returns the advantage n terms of mprovement n the performance of HFMOEA.e. good convergence wth better qualty of the pareto-optmal solutons and consequently arrves to a near paretooptmal front. Bascally, FLC_HMOEA helps n gudng the drecton of stochastc search to reach the near global optmal soluton effectvely. HFMOEA has been tested on three benchmark test problems such as ZDT, ZDT2 and ZDT3 and compared wth NSGA-II. The smulaton results revealed the effectveness of HFMOEA for solvng mult-objectve problems. 38

11 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February 22 M u t a t o n P r o b a b l t y ( P m) C r o s s o v e r P r o b a b M u t a t o n P r o b a b l t y ( C P r m) o s s o v e r P r o b a b M u t a t o n P r o b a b l t y ( P m) C r o s s o v e r P r o b a I t e r a t o n N o I t e r a t o n N o. (a) Case 3: For ZDT problem I t e r a t o n N o I t e r a t o n N o (b) Case 3: For ZDT2 problem I t e r a t o n N o I t e r a t o n N o. (c) Case 3: For ZDT3 problem Fg.8. Varatons P C and P M durng optmzaton based on HFMOEA for three test problems REFERENCES. Deb, K.,: Mult-objectve Optmzaton Usng Evolutonary Algorthms. Chchester, U.K.: Wley, Fonseca, C. M. and Flemng, P. J.: Genetc algorthms for mult-objectve optmzaton: Formulaton, dscusson and generalzaton, n Proceedngs of the Ffth Internatonal Conference on Genetc Algorthms, S. Forrest, Ed. San Mateo, CA: Morgan Kauffman, pp , (993). 3. Horn, J., Nafplots, N. and Goldberg, D. E.: A nched Pareto genetc algorthm for mult-objectve optmzaton, n Proceedngs of the Frst IEEE Conference on Evolutonary Computaton, Z. Mchalewcz, Ed. Pscataway, NJ: IEEE Press, pp , (994). 39

12 Internatonal Journal of Fuzzy Logc Systems (IJFLS) Vol.2, No., February Srnvas, N., and Deb, K.: Mult-objectve Optmzaton Usng Non-domnated Sortng n Genetc Algorthms. Evolutonary Computaton, Vol.2, No.3, pp , (994). 5. Deb, K., Pratap, A., Agarwal, S., and Meyarvan, T.: A Fast Eltst Mult-objectve Genetc Algorthm: NSGA-II. IEEE Transactons on Evolutonary Computaton, Vol. 6, No. 2, pp , (22). 6. Ztzler, E. and Thele, L.: Multobjectve optmzaton usng evolutonary algorthms A comparatve case study, n Parallel Problem Solvng From Nature, V, A. E. Eben, T. Bäck, M. Schoenauer, and H.- P. Schwefel, Eds. Berln, Germany: Sprnger-Verlag, pp , (998). 7. Knowles, J. and Corne, D.: The Pareto archved evoluton strategy: A new baselne algorthm for mult-objectve optmzaton, n Proceedngs of the 999 Congress on Evolutonary Computaton. Pscataway, NJ: IEEE Press, pp (999). 8. Ztzler, E.: Evolutonary algorthms for multobjectve optmzaton: Methods and applcatons, Doctoral dssertaton ETH 3398, Swss Federal Insttute of Technology (ETH), Zurch, Swtzerland, (999). 9. Raghuwansh, M.M., and Kakde, O.G.: Survey on mult-objectve evolutonary and real coded genetc algorthms. In Proceedngs of the 8th Asa Pacfc Symposum on Intellgent and Evolutonary Systems, pp. 5-6, (24).. San, A, Chaturved, D.K. and Saxena, A.K.: Optmal power flow soluton: a GA-Fuzzy system approach, Internatonal Journal of Emergng Electrc Power Systems, Vol. 5, Issue 2, (26).. Saraswat A. and San A,: Optmal reactve power dspatch by an mproved real coded genetc algorthm wth PCA mutaton, n proceedngs of second nternatonal conference on Sustanable Energy and Intellgent System (IET SEISCON 2), Vol. 2, pp (2). 2. Beyer, H.G., and Deb, K.: On Self-Adaptve Features n Real-Parameter Evolutonary Algorthm. IEEE Transactons on Evolutonary Computaton, Vol. 5, No. 3, pp , (2). () 3. Deb, K., and Agarwal, R. B.: Smulated Bnary Crossover for Contnuous Search Space. Complex Systems, Vol. 9, pp. 5-48, (995). 4. Dhllon, J.S., Part, S.C. and Kothar, D.P.: Stochastc economc emsson load dspatch. Electrc Power Systems Research, Vol. 26, pp (993). 5. Abdo, M.A. and Bakhashwan, J.M.: Optmal VAR dspatch usng a mult-objectve evolutonary algorthm. Electrc Power and Energy Systems, Vol. 27, No., pp. 3-2 (25). 6. Ztzler, E., Deb, K., and Thele, L.: Comparson of mult-objectve evolutonary algorthms: Emprcal results. Evolutonary Computng, Vol. 8, No. 2, pp , Summer 2. Authors Amt Saraswat receved hs B.Sc.Engg. (Electrcal Engg.) and M.Tech. (Engneerng Systems) from Faculty of Engneerng, Dayalbagh Educatonal Insttute, Agra, Inda n 23 and 26 respectvely. Presently, he s pursung hs Ph.D from the same nsttuton. Hs research area ncludes compettve electrcty markets, reactve power management, applcatons of soft computng technques n power systems etc. Ashsh San, Ph.D., s presently workng as an Assocate Professor n the Dept. of Electrcal Engg., Faculty of Engneerng, Dayalbagh Educatonal Insttute, Agra, Inda. Hs research nterest ncludes the applcatons of artfcal ntellgence technques n power system optmzaton, plannng and operaton of power systems, power system deregulaton, transmsson prcng. 4