MODULE - 9 LECTURE NOTES 2 GENETIC ALGORITHMS

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1 1 MODULE - 9 LECTURE NOTES 2 GENETIC ALGORITHMS INTRODUCTION Mos real world opimizaion problems involve complexiies like discree, coninuous or mixed variables, muliple conflicing objecives, non-lineariy, disconinuiy and non-convex region. The search space (design space) may be so large ha global opimum canno be found in a reasonable ime. The exising linear or nonlinear mehods may no be efficien or compuaionally inexpensive for solving such problems. Various sochasic search mehods like simulaed annealing, evoluionary algorihms (EA) or hill climbing can be used in such siuaions. EAs have he advanage of being applicable o any combinaion of complexiies (muli-objecive, non-lineariy ec) and also can be combined wih any exising local search or oher mehods. Various echniques which make use of EA approach are Geneic Algorihms (GA), evoluionary programming, evoluion sraegy, learning classifier sysem ec. All hese EA echniques operae mainly on a populaion search basis. In his lecure Geneic Algorihms, he mos popular EA echnique, is explained. CONCEPT EAs sar from a populaion of possible soluions (called individuals) and move owards he opimal one by applying he principle of Darwinian evoluion heory i.e., survival of he fies. Objecs forming possible soluion ses o he original problem is called phenoype and he encoding (represenaion) of he individuals in he EA is called genoype. The mapping of phenoype o genoype differs in each EA echnique. In GA which is he mos popular EA, he variables are represened as srings of numbers (normally binary). If each design variable is given a sring of lengh l, and here are n such variables, hen he design vecor will have a oal sring lengh of nl. For example, le here are 3 design variables and he sring lengh be 4 for each variable. The variables are x, x 7and 1. Then he chromosome lengh is 12 as shown in he figure x 1 x 2 x x3 An individual consiss a genoype and a finess funcion. Finess represens he qualiy of he soluion (normally called finess funcion). I forms he basis for selecing he individuals and hereby faciliaes improvemens.

2 2 The pseudo code for a simple EA is given below i = 0 Iniialize populaion P 0 Evaluae iniial populaion while (! erminaion condiion) { i = i+1 Perform compeiive selecion Creae populaion P i from P i-1 by recombinaion and muaion } Evaluae populaion P i A flow char indicaing he seps of a simple geneic algorihm is shown in figure 1.

3 3 Sar Generae Iniial Populaion Encode Generaed Populaion Evaluae Finess Funcions R E G E N E R A T I O N Mees Opimizaion Crieria? No Selecion (selec parens) Crossover (seleced parens) Yes Bes Individuals Sop Muaion (muae offsprings) Fig. 1 The iniial populaion is usually generaed randomly in all EAs. The erminaion condiion may be a desired finess funcion, maximum number of generaions ec. In selecion, individuals wih beer finess funcions from generaion i' are aken o generae individuals of i+1 h generaion. New populaion (offspring) is creaed by applying recombinaion and muaion o he seleced individuals (parens). Recombinaion creaes one or wo new individuals by swaping (crossing over) he genome of a paren wih anoher. Recombined individual is hen muaed by changing a single elemen (genome) o creae a new individual.

4 4 Finally, he new populaion is evaluaed and he process is repeaed. Each sep is described in more deail below. PARENT SELECTION Afer finess funcion evaluaion, individuals are disinguished based on heir qualiy. According o Darwin's evoluion heory he bes ones should survive and creae new offspring for he nex generaion. There are many mehods o selec he bes chromosomes, for example roulee wheel selecion, Bolzmann selecion, ournamen selecion, rank selecion, seady sae selecion and ohers. Two of hese are briefly described, namely, roulee wheel selecion and rank selecion: Roulee Wheel Selecion: Parens are seleced according o heir finess i.e., each individual is seleced wih a probabiliy proporional o is finess value. In oher words, depending on he percenage conribuion o he oal populaion finess, sring is seleced for maing o form he nex generaion. This way, weak soluions are eliminaed and srong soluions survive o form he nex generaion. For example, consider a populaion conaining four srings shown in he Table 1. Each sring is formed by concaenaing four subsrings which represens variables a,b,c and d. Lengh of each sring is aken as four bis. The firs column represens he possible soluion in binary form. The second column gives he finess values of he decoded srings. The hird column gives he percenage conribuion of each sring o he oal finess of he populaion. Then by "Roulee Wheel" mehod, he probabiliy of candidae 1 being seleced as a paren of he nex generaion is 28.09%. Similarly, he probabiliy ha he candidaes 2, 3, 4 will be chosen for he nex generaion are 19.59, and respecively. These probabiliies are represened on a pie char, and hen four numbers are randomly generaed beween 1 and 100. Then, he likeliness ha he numbers generaed would fall in he region of candidae 2 migh be once, whereas for candidae 4 i migh be wice and candidae 1 more han once and for candidae 3 i may no fall a all. Thus, he srings are chosen o form he parens of he nex generaion.

5 5 Table 1 Candidae Finess value Percenage of oal finess Toal Rank Selecion: The previous ype of selecion may have problems when he finesses differ very much. For example, if he bes chromosome finess is 90% of he enire roulee wheel hen he oher chromosomes will have very few chances o be seleced. Rank selecion firs ranks he populaion and hen every chromosome receives finess from his ranking. The wors will have finess 1, second wors 2 ec. and he bes will have finess N (number of chromosomes in populaion). By his, all he chromosomes will have a chance o be seleced. Bu his mehod can lead o slower convergence, because he bes chromosomes may no differ much from he ohers. CROSSOVER Selecion alone canno inroduce any new individuals ino he populaion, i.e., i canno find new poins in he search space. These are generaed by geneically-inspired operaors, of which he mos well known are crossover and muaion. Crossover can be of eiher one-poin or wo-poin scheme. In one poin crossover, seleced pair of srings is cu a some random posiion and heir segmens are swapped o form new pair of srings. In wo-poin scheme, here will be wo break poins in he srings ha are randomly chosen. A he break-poin, he segmens of he wo srings are swapped so ha new se of srings are formed. For example, le us consider wo 8-bi srings given by ' ' and ' '. Then according o one-poin crossover, if a random crossover poin is chosen afer 3 bis from lef and segmens are cu as shown below: and he segmens are swapped o form

6 According o wo-poin crossover, if wo crossover poins are seleced as Then afer swapping boh he exreme segmens, he resuling srings formed are Crossover is no usually applied o all pairs of individuals seleced for maing. A random choice is made, where he probabiliy of crossover being applied is ypically beween 0.6 and 0.9. MUTATION Muaion is applied o each child individually afer crossover. I randomly alers each gene wih a small probabiliy (generally no greaer han 0.01). I injecs a new geneic characer ino he chromosome by changing a random a bi in a sring depending on he probabiliy of muaion. Example: is muaed as I is seen in he above example ha he sixh bi '0' is changed o '1'. Thus, in muaion process, bis are changed from '1' o '0' or '0' o '1' a he randomly chosen posiion of randomly seleced srings. REAL-CODED GAs As explained earlier, GAs work wih a coding of variables i.e., wih a discree search space. GAs have also been developed o work direcly wih coninuous variables. In hese cases, binary srings are no used. Insead, he variables are direcly used. Afer he creaion of populaion of random variables, a reproducion operaor can be used o selec good srings in he populaion. AREAS OF APPLICATION IN WATER RESOURCES Waer disribuion sysems Hydrological modeling Waershed Managemen

7 7 Groundwaer modeling Reservoir Operaion ADVANTAGES AND DISADVANTAGES OF EA: EA can be efficienly used for highly complex problems wih muli-objeciviy, non-lineariy ec. I provides no only a single bes soluion, bu he 2 nd bes, 3 rd bes and so on as required. I gives quick approximae soluions. EA mehods can very well incorporae wih oher local search algorihms. There are some drawbacks also in using EA echniques. An opimal soluion canno be ensured on using EA mehods, which are usually known as heurisic search mehods. Convergence of EA echniques are problem oriened. Sensiiviy analysis should be carried ou o find ou he range in which he model is efficien. Also, he implemenaion of hese echniques requires good programming skill. MULTI-OBJECTIVE EVOLUTIONARY ALGORITHMS Geneic algorihms are efficien in solving muliobjecive problems. Consideraions in Muli- Objecive Evoluionary Algorihms (MOEAs) implemenaion are 1. Preserve non-dominaed poins eliism 2. Progress owards poins on Pareo fron 3. Mainain diversiy of poins on Pareo Fron (phenoype) and/or Pareo Opimal soluions (genoype) 4. Provide decision maker a limied number of Pareo Fron (PF) poins. Non-dominaed soluions are always beer han 1s-level dominaed soluions, which are always beer han 2nd-level dominaed soluions, ec. Wihin he same level of dominance, soluions which are isolaed are beer han soluions ha are clumped ogeher.

8 8 Flow char of Muli-objecive Geneic Algorihm wih Eliism Fig. 2 Muli Objecive Geneic Algorihms for Opimal Reservoir Operaion Case Sudy Bhadra Reservoir A muli-purpose reservoir locaed in he disric of Chickmangalur, Karnaaka sae, India; 75 o E longiude and 13 o 42 N laiude.

9 9 Fig. 3 Schemaic diagram of Bhadra reservoir Projec Muli Objecive Reservoir Operaion Model Objecive funcions: 1. Minimize irrigaion defici (f 1 ) SQDV Dl, IRl, Dr, IRr, Maximize hydropower producion (f 2 ) 2 Subjec o consrains in 12 E k1rl, Hl, k2rr, H r, k3rb, Hb, 1 (i) Reservoir sorage coninuiy consrain (ii) Sorage bounds S 1 S I R1, R2, R3, Smin S S max (iii)turbine capaciy limis pr pr pr 1, 2, 3, H H (iv) Canal capaciy limis H 1, 2, 3, E 1,max E E 2,max 3,max E O

10 10 R R 1, 2, C C 1,max 2,max (v) Irrigaion demands D D 1min, 2min, R 1, R 2, (vi) Waer qualiy requiremens D 1max, D 2max, R 3, MDT Pareo opimal soluion for reservoir operaion f gen=50 gen=200 gen= f 1 x 10 4 Fig. 4 Improvemen in Pareo opimal fron over he ieraions. f 1 is annual squared irrigaion defici; f 2 is hydropower generaed MkWh Model Applicaion MOGA model is solved for hree differen inflow scenarios ino he reservoir Scenario 1: Mean monhly inflows 0.5 * SD Scenario 2: Mean monhly inflows Scenario 3: Mean monhly inflows * SD where SD is he sandard deviaion of monhly flows

11 11 Fig. 5 Pareo opimal fron, showing he rade-off beween irrigaion ( f 1 ) and hydropower ( f 2 ) for differen inflow scenarios. f 1 = sum of squared irrigaion deficis, (Mm 3 ) 2 ; f 2 = hydropower generaed, (MkWh)

12 12 Fig. 6 Reservoir operaing policies for differen inflow scenarios, showing he iniial sorages for differen siuaions, viz., equal prioriy case, irrigaion only prioriy case and hydropower only prioriy case.

13 13 Fig. 7 Opimal release policy obained for equal prioriy case, showing releases in Mm 3 for Lef bank canal (R1), Righ bank canal (R2) and River bed (R3) for differen inflow scenarios. Advanages of MOEAs: MOEAs are easy o adop and can provide efficien soluions for muli-objecive problems. They are capable of handling nonlinear objecives/ consrains, disconneced Pareo-frons,

14 14 non-convex decision space. They can find soluions o exremely complex and high dimensional real-world applicaions in reasonable compuaion ime. They have high poenial for muli-objecive opimizaion of hydrological and waer resources problems.