1. ' Contents nsrs available at SceQCeO~rectComputers & Industrial Engineering lournal homepage: www.elsevier.com/locate/ e:aieMulti-objective generic local search algorithm using Kohonen's neural map Mehrdad Hakimi-Asiabar . Seyyed Hassan Ghodsypour .... Reza Kerachian b ~Orrparrmt-nt11/ lndr.otrw~l rtt.Amuio0trthttv o/Ttdlf'IOitOloD. Vah-cml". T.-hron. rron (f'nrtt of .tdtiKl'/01 MhmnKUttt (IIJNtf'l"' oud Mono,ttoi'II'IU. 'aru/ty of CMJ ngmHnJ.,_ UnuttSlyARTICLEcf T~rtm. Tthtmt.1Jan:ABSTRACTINFOAnjcf~ IIISIOiy,Re.lck of the CAs is genetic drift. The CAs seo.rch process is a bl"dc box process .11 no one knows th.ll which d rfjlon Is being se.a.rched by the o~lgorithm .and it is poss1ble tl'wt GAs Stirch only.:. sm.aU re;ion in tht fu slble Sp.l. On the other h.lnd. GAs usu.ally do not use the existing infornl.ltlon .about th~ optimality rea,ons in p.ut lter.itions. In thiS p.aper . new method C.l.l~ SOM8.:1Sed Muhi..Ob,ttCIIVt CA (SBMOCA) I) proposed to improve the renet1 dtvemty. ln SBMOCA. a gnd of neurons use fht concept of leo1rn11'1 rule of Stlf Orlo1n1zinc c M.ll' (SOM) mppornng by Vanable Neighborhood Strc:h (VNS) le.am from genetiC .alaontlH1l lmprOVI"' bo~h loc41 nd &lob.ll S('.trch. SOMis n.tUr.al networlc wh1ch '' C:pblt of le.1mma .md '"" tmprove the tffic1ency ol d.at.a processm& la:onthms. The VNS .al,onthm 15 dt'wlopcten d~loptd bued on D.trw~n's pnnCiples of N~tur'1 st~tton .and survrv.11J of the finest individui11S m a popul~llon. EAs use comput,auon.al modtls of tvo1unonary processes as key clcmcncs in the destgn .and tmpltmtnr.uon of computer-~sed problem solvong systems (Cordon. Moy. & l.>rco. 2002; Goldlx'rg. 1989). Th(f( ,)J'( a V.lfl~ty ~VOIUiiOOlr'Y (Ompuurion.al models. There have been four well defined EAs. whoch hve served iS rho basis for che most of .1ct ivtties tn the field of evoluttOil.lf'Y computations: Genetic Algorothms (CAs)(Hollnd. 1975: Mlchalrwicz. 1996). Evolutoon Strtegies (SbJOCtwe EAs such as Vec1or Evlu.oted Ctnetic Algo nthm (VEGA)(Schffcr. 1984) nd Non-domonated Soned Ceneloc Algorirhms (NSCA) (Snnoas & Deb. 1994 ) hvo ~n proposed. These early EAs often J>t>rfonned poorly, cons1d~nns two key parameters: convergence ra.te c1nd d1verstty. Recent ,algonthms II let S1rcngth Par01o Evolutionary SPEA (Zotzlcr & Thoele. 1999) nd NSCA-11 (Deb. Pratp. Agarwal. & Mty11van. 2002) l)mnlygrmmmg (EP) (Fogel.1. Introductionahud:SV~~~~.c II (5 uChodsypour~ K~r.tiChiAnOut..lC.II' ( R.Ktr.Kh.l.ln)._ 0360 U5l/S $front mmtr e 2001 liHvttr Lid. All r,&tlrc ffS~rwd. doi 10 1016jJ(i~.l00110010ro-or

2. ' M. Hakimi -Asiabar rr ai)Compurers & l11dusrrial Engineering 56 (2009) 1566-1576from traditional GA by using specialized fitness functions and introducing methods to promote solution diversity. Many real -world problems do not satisfy necessary conditions such as continuity. differentiability. convexity, etc. Therefore. they can not be easily solved using traditional gradient-based optimization techniques. GAs have been considered as a practical optimization tool in many disciplines such as discontinuous multi-modal objective functions, combinatorial {together with discrete, continuous or integer design variables). dynamic. severely nonlinear, and non-d ifferen tiable, non-convex design spaces problems. Another advantage of MOEAs is definition of Pareto front set with an acceptable computational time. Traditional multi-objective algorithms define one solution in each run. The MOEAs usually attempt to generate (or closely approximate) the entire Pareto front in a single run and place emphasis on J.chieving solution diversity so as to avoid local optima (Rangarajan. Ravindran. & Reed, 2004).The advantages of GAs increasingly extend their applications. However. there are some drawbacks that limit their efficiency. The traditional GAs intensification process is not sufficiently accurate. GAs usually find the area of good fitness qu ite easily. However, finding the global optimal solution may be time-consuming and inaccurate. This is because the search strategy of GAs is probabilistic. In a probabilistic search process. when a chromosome is far from the local optima, there is a SO% chance that a random search direction will simultaneously improve .all the objectives. However. when a point is close to the Pareto set, the size of proper descent/ascent cone is extremely narrow and there is small probability that a random update improves the objective functions (Brown & Smith, 2005). Thus with a random search strategy, GAs generally require a great number of iterations and they converge slowly. especially in the neighborhood of the global opti mum. With a randomized reproduction strategy in which the crossover points are determined randomly, the resulting children are created without regard to the existing information about high fitness regions. Therefore. the fitness of a child can deviate quite widely from the fitness of its parents. Another drawback of the GAs is genetic drift. The GAs exploration process is a black box and the diversity information obtained from past generations is only implicitly and partially preserved in the current genome. This bears the risk of a regeneration of individuals that have already bee n see n in the search process. Even more problematic is the fact that the search can be negatively af~ fected by genetic drift. As a consequence, big parts of the search space, potentially containing the global optimum. will never be explored. Thus there is a need for consistent exploration techniques that do not repeat the same patterns in mutation process and also can improve the diversity when increasing gene tic generations. EAs are producing vast amounts of data during an optimization run without sufficient usage of them. In each of the numerous gen erations, a large number of chromosomes is generated and eval uated (Drobics, Bodenhofer, & Winiwarter. 2001 ). This data can be used to produce valuable insight to enhance EAs solution quality. GAs are complete probabilistic search-based optimization models because they do not use the knowledge aggregated about the optimality regions and searched areas in past iterations. Necessary requirements are for example, processing incoming data such that it creates some useful information that incrementally improves the next generation population and new chromosomes should not be created based on entire probabilistic processes. It is possible to extract and use previously computed knowledge in next generations. Then it can be concluded that there is an area of improvement in convergence rate and diversity of GAs. Thus there is a need to new GAs that their exploitations are knowledge orie nted to accelerate the intensification process and also have better diversification to avoid genetic drift.1567In this paper, a background for developing a new GA-based algorithm is provided in the next section. In Section 3. the new algorithm is presented in detail. SBMOGA is developed based on some well known ideas such as SOM learning rule, and VNS shaking process. The new algorithm can provide consistent diversity without repeated evaluations and a systematic local and variable neighborhood search. In Section 4, a complex real world problem namely multi -objective multi-resetvoir operation management problem is described and the optimization model formulation is presented. It is clear that this problem is non-convex nonlinear. In Section 5, the results of application of new algorithm to solve the multi-reservoir operation problem is shown. In the last section, conclusions and future research opportunities are presented.2. BackgTOund In previous section. the main advantages and disadvantages of the traditional GA-based optimization models were described in detail. In this section, literature regarding the models improving the traditional GAs is reviewed. A variety of techniques for incorporating local search methods with EAs have been reponed. These techniques include Genetic Local Search (Merz & Freisleben, 1999), Genetic Hybrids (Fleurent & Ferland, 1994), Random Multi-Start (Kernighan & Un, 1970) and GRASP (Feo & Resende, 1989). Local search schemes such as gradient-based methods are efficient algorithms for refi n ing arbitrary points in the search space into better solutions. Such algorithms are called local search algorithms because they define neighborhoods, typically based on initial "coarse" solutions. The tenn 'local search ' generally is applied for methods that cannot escape these minima. Some hybridization schemes that will be used to d evelop the proposed algorithm are discussed below. 2.1. Hybrid loco/ seorch GAsHybrid algorithms are a combination of two or more different techniques. Hybridization of local search and evolutionary algorithms has complementary advantages and combines the strengths of different approaches in order to overcome their weaknesses. Evolutionary algorithms have been successfully hybridized with other local searc h methods. Hybrid EAs have the local search power of traditional methods thus their accuracy is better than the ordinary EAs. Also the methods common ly take advantage of the good global search capabilities of evolutionary algorithms then they are robust against getting stuck at local optima. The hybridization of genetic algorithms and local search methods, called genetic local search, has been applied to a variety of singleobjective combinatorial problems. The role of the local search is to enhance the intensification process in the genetic search (Arroyo & Armenta no. 2005 ). Grosan and Abrah(n) ;'IIR-w ;u(1)w ;r1 whe rc.n represents a learning step and W1" and are the weight vecto rs of un it s before and after updati ng, respectively. ering 56 (2009) 1566- 1576Nou-Dominated Sotting Algo l ithm IITtainiug the Self-Otga uizingFig. 6. A s ketch of NSGA-11 .md its relrttion to SOM's neurons le.nning process.4. Case studyThe optimal operation of multi-purpose multi-reseJVoir systems is a real world complex problem. There are many advances in operation of reservoirs which are cited in the literature. L:lbadie (2004 ) presented a state of the art review on mathematical programming and heuristic methods in optimal operation of multireservoir systems. He concluded that although there are a few areas of application of optimization models with a richer or more diverse history than in reservoir systems optimization and opportuni ties for real-world applications are enormous, actual implementations remain limited or have not been sustained. In this section, the applicability of the proposed algorithm will be examined in developing operating policies for the Karoon-Dez multi-purpose multi -reservoir system. The Dez and Karoon reservoirs. with a tota l storage capacity of more than 6.4 billion cubic meters (BCM ). form the most important reservoir system in south western Iran close to the Persian Gulf (see Fig. S ). The system carry more than one-fifth of the Iran's surface water supply ( Karamouz & Mousavi. 2003). The reservoirs have been constructed on the Karoon and Dez Rivers. The two rivers join together at a location called Band-e-Gh ir, north ofthe City of Ahwaz. to form the Great Karoon River. The average annual inflows of the Dez and Karoon reservoirs are 8.5 and 13.1 (BCM ). respectively. The water downstream of the Karoon and Dez dams, supply domestic. industrial, agricultural and agro-indu strial demands. Total water de mand downstream of the Dez and Karoon dams is estimated as 1.95 (BCM), from which 42% is a llocated to downstream of the Dez dam (d 11 ) : 35% is allocated to downstream of the Karoon dam between the Karoon reservoir and Band-e-Ghir (d 2 , ) and the rest goes downstream of Band-e-Ghir to the Persian Gulf (d 3 , ). There is also an environmental water demand equal to 0.62 (BCM ) as in-stream flow in the Great Karoon River {d 41 ) (see Fig. 9). The reservoirs have a hydropower generation capacity of 1.15 million megawart hours (MWh ) per month.Other model variables are:,.in"Maximum storage volume of the ith reservoir Minimum storage volume of the irh reservo ir Storage volume of ith reservoir in time period t Water head elevation in irh reservoir in time period r Inflow to the ith river in time period t Release from outlet of hydropower plant of ith reservoir in time period t Maximum capacity out let of hydropower plant of ith reservoir Minimum water required to activate the hydropower plant of ilh reservoir Water release from spillway of ith reservoir Maximum spillway capacity for ith reservoir Minimum spillway capacity for ith reservoir Maximum release for ith reservoir Minimum release for ith reservoir jth water demand in time period t4.2. Objecrive funccionsFirst objective function: minimizing unsatisfied water demand 3MinZ1 =T2::: ~ (dJr - d11 j: l1=1).j1)23=Tf" l1= 1L L dJr (1 - /.j2(8)1)where i.1, dit -Xtit Ru = 0 0 :; ,; : i.jl:;;,;;: 1(9) ( 10)4.1 . Model fonnularionIn this study, a mathematical model for monthly operation of the Karoon and Dez reservoirs is developed considering the objectives of water supply to downstream demands and power generation. The decision variables of the optimization model are as follows:R1r Releases for reservoir i in time period r Xii1 Percentage of outflow from reservoir i allocated to water demand j in time period t (0 :;;; Xijr :;;; 1 ). Ajr Satisfied portion of jth demand location in period r Second objective function: maximizing power generationMaxl2 =tf=lt K;.e;. rl lr Htr (Su, Stt- t . Ra )( 11 )i= lwhere K1 Energy transfer coefficient of ith hydropower plant e1 Efficiency index of ith hydropower plant H11 Mean value of water head behind the ith reservoir whose storage is equal to S1r 7. ' M. Hokimi-Asiobar t.>t ol. / Compurm & lndusl'riaf Engineering 56 (2009) 1566-15761572Randomly generate fr~t genetiC popu!atrcn and we.ght ;ec:orsofS01.1 ulltts E 3IUJle !l'lt Cl11111: XIOUI,ncn JM l'.'f !Qtll,edcrs ol SOU uml:St:ase~en Ct jeat.e.. lfld>COSRecordnoo-K:tr_j_H ,H r.l1HPP FiJ. 10. The reservorr p.umeters.Fig. 7.Th~The effective water level height for producing hydropowe r energy can be calculated as follow (see Fig. 10 for more details ):flowchart of SBMOGA.(13) flu is usually a non -linear function of reservoir srorage volume, which can be presented as:The tail water height of released water from reselVoir i in time period c can be est imated as:(12)( 14) 8. M. Hakimi -Asiabar l't al.f(omputl'fS & Industrial Enginttring 56 (2009)Therefore. the second objective function can be wrinen as follows : lT:LK, e,. r., . (H(S., )- H(R., ))=LM~z,' ""]The hydropower energy production is a nonlinear and non-convex function.1. Water storage capacity constraims~' 1 " ~ 511 ~ ~11.11 ~fin ~ S21 :;;t = 1, 2. 3 .... T l = I. 2. 3 .... T1i11!12. Water demand constraints R tr X11r = ).J, dlr0~ Xn 1 ~1R2r X22r = /.21 d 2ro ~ X22r~1Rl r" XIlr + R2r Xnr = /.Jr d2r X1r = i. lr d lrR1,-0~ ).11 :!i;1X2r = i.2r d 2rR2r-0~h,~1lnst ream flows d41620:;;l!:3. Continuity equations S1 ,1= S1.o52,= 52 .0S, r-1= 5 tr -S2r .. 1 =52t-R1r:s::;Rlt (ft.fa~0 ~ R2,