CHAPTER 4 OPTIMIZATION TECHNIQUES

Transcription

1 48 CHAPTER 4 OPTIMIZATION TECHNIQUES 4.1 INTRODUCTION Unfortunately no sngle optmzaton algorthm exsts that can be appled effcently to all types of problems. The method chosen for any partcular case wll depend prmarly on the character of the objectve functon, the nature of the constrants and the number of ndependent and dependent varables. The general objectve n optmzaton s to choose a set of values of the varables subject to varous constrants that wll produce desred optmum response for the chosen objectve functon. In real world, many problems may be represented n terms of searchng for the values of some parameters to mnmze or maxmze a functon, often called the objectve functon. In ths thess, the tolerance optmzaton problem s formulated as a non-lnear multvarate problem. As non-tradtonal optmzaton technques overcome dffcultes and lmtatons encountered by tradtonal approaches, the tolerance optmzaton problem s solved by usng non-tradtonal optmzaton technques. 4.2 NON-TRADITONAL OPTIMIZATION TECHNIQUES Non-tradtonal optmzaton technques are mostly nspred from nature and apply nature le processes to solutons, n order to serve as very flexble and robust tools for complex combnatoral optmzaton problems. They are a class of approxmate methods to search for an optmal soluton out

2 49 of all possble solutons. Some of the examples are Genetc Algorthms (GA) (Goldberg 1989), and Dfferental Evoluton (DE) (Storn et al 1997). In ths thess, three non-tradtonal optmzaton technques namely Genetc Algorthm (GA), Partcle Swarm Optmzaton (PSO) and Dfferental Evoluton (DE) are appled for solvng tolerance optmzaton problem. In the followng sectons, GA, PSO and DE are explaned n detal. 4.3 GENETIC ALGORITHM Genetc Algorthm was ntroduced by Holland n the year It s a meta-heurstc search technque, whch wors wth the concept of Darwn s theory of natural evoluton (Goldberg 1989). GA s a drected random search method that reles on the mechancs of natural selecton and breedng to effcently explore a large space of canddate desgns and fnd optmum solutons (Goldberg, 1998). The power of ths algorthm stems from ts ablty to explot hstorcal nformaton structure from prevous soluton n an attempt to ncrease the performance of future solutons. Compared to tradtonal contnuous optmzaton methods, GA has the followng sgnfcant dfferences. GA manpulates coded versons of the problem parameters nstead of the parameters themselves. Whle almost all conventonal methods search from a sngle pont, GA always operates on a whole populaton of ponts (strngs). Ths contrbutes much to the robustness of genetc algorthm. It mproves the chance of reachng the global optmum and, vce versa, reduces the rs of becomng trapped n a local statonary pont. Normal genetc algorthms do not use any auxlary nformaton about the objectve functon value such as dervatves. Hence,

3 50 they can be appled to any nd of contnuous or dscrete optmzaton problem. GA uses probablstc transton operators whle conventonal methods for contnuous optmzaton apply determnstc transton operators. More specfcally, the way a new generaton s computed from the actual one has some random components Worng prncple of GA GA mantans a populaton of ndvduals that represent canddate solutons. Each ndvdual s evaluated to gve some measure of ts ftness to the problem from the objectve functon. In each generaton, a new populaton s formed by selectng the fttest ndvduals based on a partcular selecton strategy. Some members of the new populaton undergo genetc operatons to form new soluton. The two commonly used operatons are crossover and mutaton. Crossover s a mxng operator that combnes genetc materal from selected parents. Mutaton acts as a bacground operator and s used to search the unexplored search by randomly changng the values at one or more postons of the selected chromosome. After several generatons, the algorthm converges to the best chromosome, whch hopefully represents the optmum or near optmal soluton. GA has been proven to be robust, flexble and effcent n vast complex spaces, when searchng a problem space for the optmum soluton. GA uses the dea of randomness when performng a search. However, t must be clearly understood that GA s not smply a random search algorthm. It utlzes nowledge from prevous generatons of strngs n order to construct a new generaton that wll approach the optmal soluton. It also uses past nowledge to drect the search. GA has four components as descrbed by Davs (1991) whch are lsted below:

4 51 A means of encodng solutons to the problem as chromosome A means of obtanng an ntal populaton of solutons A functon that evaluates the ftness of a soluton Reproducton operators for the encoded solutons There are sx steps n GA and are () Problem representaton () Intalzaton of populaton () Evaluaton of ftness functon (v) Constrant handlng (v) Generaton of new populaton and (v) Stoppng crtera. () Problem representaton The frst and the foremost mportant step n applyng GA to a problem s the encodng scheme because t can severely lmt the wndow of nformaton that has been observed from the system. To enhance the performance of the algorthm, a chromosome representaton s desred. In general, the GA evolves a mult-set of chromosomes. The chromosome s usually expressed as a strng of varables, each element of whch s called a gene. The varable can be represented as bnary, real number, or other forms and ts range s usually defned by the problem specfed. The most commonly used representaton of chromosomes s that of the sngle-level bnary strng 0 s and 1 s. The length of the strng s usually determned, accordng to the desred soluton accuracy. () Intalzaton of populaton There are two parameters, whch have to be decded for ntalzaton, namely the populaton and the procedure to ntalze the populaton. Instead of relyng on a sngle pont, GA generates a populaton of ponts wth the predefned populaton sze. Ths gves the GA, the power to search through many dfferent possbltes of the problem space and results n

5 52 global optmal soluton. In general, the sze of 20 to 50 s preferable for a normal populaton. There are two ways to generate the ntal populaton namely the random ntalzaton and heurstc ntalzaton. The most common method s the random ntalzaton, whch randomly generates solutons for the entre populaton. For the problem under consderaton, each of the ndvdual s ntalzed randomly usng a unform random number dstrbuton wthn the feasble range of the control varables. Then the bnary coded strngs are decoded usng the followng lnear mappng rule: t j ut lt j j ut s (4.1) j l 2 1 where t j, s the decoded value, lt j and ut j are the lower and upper lmts of tolerance t j, l s the length of the strng and s s the bnary substrng. () Evaluaton of ftness functon The GA mmcs the survval of the fttest prncple of nature to mae the search process and uses the ftness functon value as pay off nformaton to gude them through the problem space. Once GA nows the current measure of goodness about a pont, t can use ths to contnue searchng for the optmum. GA s naturally sutable for solvng maxmzaton problems. Mnmzaton problems are usually transformed nto maxmzaton problem by some sutable transformaton. In general, a ftness functon F(x) s frst derved from the objectve functon and used n successve genetc operatons. Certan genetc operatons requre the ftness functon to be nonnegatve although, certan operators do not have ths requrement. For maxmzaton problems, the ftness functon F(x) can be consdered to be the same as the objectve functon f(x) and hence F(x) = f(x). For mnmzaton problems, the followng transformaton s often used.

6 53 1 F(x) (4.2) 1 f(x) Ths transformaton converts a mnmzaton problem to an equvalent maxmzaton problem. The ftness functon value of a strng s nown as the ftness of the strng. (v) Constrant handlng GA s deally suted for unconstraned optmzaton problems. However, most of the optmzaton problems are constraned n nature. Hence, t s necessary to transform t nto an unconstraned problem (Deb 2001). Transformaton methods acheve ths by addng a penalty term wth the objectve functon. There are two man approaches for penalty functon: ) based only on the number of constrants volated and ) based on some dstance from the feasble regon. In general, penalty functons should be chosen to satsfy the followng requrements: The penalty functon should be progressve, so that a more severe volaton of constrants attracts a hgher penalty value allowng the GA to be guded towards feasblty. The penalty factor for each volated constrant should be summed to form the overall penalty loss, so that a well-behaved penalty surface s produced. In the penalty functon, penalty term correspondng to the constraned volaton s added to the objectve functon. In a constraned mnmzaton problem, the objectve functon f(x) s replaced by the penalty functon j 2 K 2 P(x) f(x) u g j (x) y h (x) 1 j (4.3) 1

7 54 where u j and y are penalty coeffcents, whch are usually constant throughout GA smulaton and g j (x) and h (x) are nequalty constrants. (v) Generaton of new populaton Then the evaluaton concepts are translated nto the new populaton generaton to search for the best chromosome n a more natural way. It conssts of three genetc operators: (a) Selecton (b) Crossover and (c) Mutaton. (a) Selecton Selecton s a mechansm for selectng ndvduals (strngs) from a populaton accordng to ther ftness (objectve functon value). The ftness of an ndvdual s evaluated wth respect to a gven objectve functon. The hghest ran chromosome wll have more possblty of selecton and the worst wll be elmnated. There are number of selecton methods avalable. The methods nclude, roulette wheel selecton, tournament selecton, ran selecton, steady state selecton and so on. In general, Roulette wheel selecton method s used. In ths method, parents are selected accordng to ther ftness. The better the chromosomes they have, the more chances are there to be selected. Imagne a roulette wheel, where all the chromosomes n the populaton are placed. The sze of the secton n the roulette wheel s proportonal to the value of the ftness functon of every chromosome - the bgger the value, the larger the secton. A marble s thrown n the roulette wheel and the chromosome on whch the marble stop s selected. Clearly, the chromosomes wth hgher ftness value are selected more tmes.

8 55 (b) Crossover Once the selecton process s over, the crossover operator s appled next. Crossover s a recombnaton operator that combnes subparts of two parent chromosomes to produce offsprng that contans some parts of both parents genetc materal. In the crossover, hghly ft ndvduals are gven opportuntes to reproduce by exchangng peces of ther genetc nformaton wth other hghly ft ndvduals. Ths produces new "offsprng" solutons, whch share some good characterstcs taen from both parents. The commonly used sngle pont crossover operator s performed by randomly selectng a ste along the strng and by exchangng all bts on the rght sde of the crossover ste. Fgure 4.1 shows the crossover operaton between the two parent strngs and the creaton of off sprngs. Parent strng Offsprng Parent strng Offsprng Fgure 4.1 Crossover operaton In sngle pont crossover, two ndvdual strngs are selected at random from the matng pool. Next, crossover ste s selected randomly along the strng length and bnary dgts are swapped between the two strngs at the crossover ste. The crossover operator bascally combnes substructures of two parent chromosomes to produce new structures wth the chosen crossover probablty P c. It ndcates how often crossover s performed. A probablty of 0% means that the offsprng wll be the exact replca of ther parents and a probablty of 100% means that each generaton s composed of entrely new sprng.

9 56 (c) Mutaton The selecton and crossover operators wll generate a large amount of dfferent off sprngs. However, there are two man problems wth ths. They are () Dependng upon the ntal populaton chosen, there may not be enough dversty n the ntal strngs to ensure that the GA searches the entre problem space and () The GA may converge on sub-optmum strngs due to a bad choce of ntal populaton. These problems may be overcome by the ntroducton of mutaton operator nto GA. The mutaton operator s used to nject new genetc materal nto the genetc populaton. Mutaton can be realzed as a random deformaton of the strngs wth certan probablty. The postve effect s preservaton of genetc dversty and, as an effect, that local maxma can be avoded. In the mutaton, the offsprng can ether replace the whole populaton or replace less ft ndvduals. Mutaton operator changes 1 as 0 and vce versa by bt wse. Btwse mutaton s done bt by bt by flppng a con wth low probablty. If the outcome s true, then the bt s altered; otherwse the bt s not altered. Hgh mutaton rate would destroy the ft strngs and degenerate the GA nto a random search. Mutaton probablty P m of 0.01 to s common and these values represent the probablty that a certan strng wll be selected for mutaton,.e, for a probablty of 0.01, one strng n one thousand, wll be selected for mutaton. Fgure 4.2 llustrates the btwse operaton. As shown n Fgure 4.2, btwse mutaton operaton randomly selects a strng and swtches the randomly chosen bt from 0 to 1 or 1 to Fgure 4.2 Btwse mutaton

10 57 (d) Termnaton crtera Durng the GA run, ftness value ncreases gradually and at one partcular generaton, there s no further mprovement n the ftness value whch hopefully represents the optmal or near optmal soluton. At ths stage, executon of GA s to be termnated. 4.4 PARTICLE SWARM OPTIMIZATION Partcle Swarm Optmzaton was nvented by Kennedy et al (1995, 2001), whle attemptng to smulate the choreographed, graceful moton of swarms of brds as part of a soco cogntve study nvestgatng the noton of collectve ntellgence n bologcal populatons. PSO s a populaton-based evolutonary technque that has many ey advantages over other optmzaton technques as follows: It s a dervatve-free algorthm unle many conventonal technques. It has the flexblty of ntegraton wth other optmzaton technques to form hybrd tools. It has less parameter to adjust unle many other competng evolutonary technques. It has the ablty to escape local mnma. It s easy to mplement and program wth basc mathematcal and logc operatons. It can handle objectve functons wth stochastc nature, le n the case of representng one of the optmzaton varables as random. It does not requre a good ntal soluton to start ts teraton process.

11 Worng prncple of PSO PSO s a nd of algorthm, searchng for the best answer by smulatng the movement and flocng of buds. The algorthm ntalzed the floc of buds randomly over the searchng space, n whch every brd s called as a partcle. For each partcle, the poston and velocty vectors wll be randomly ntalzed wth the same sze as the problem dmenson. In each teraton, the ftness of each partcle (pbest) s measured and the partcle wth the best ftness (gbest) value s stored. Then the velocty and poston vectors are updated for each partcle. The above steps 2 3 are repeated untl a termnaton crteron s satsfed. The varous components of PSO are explaned below. () Generaton of the partcles postons and veloctes PSO s ntalzed wth a group of random partcles (solutons) wth ntal poston of s and velocty, V usng the followng Equatons (4.4) and (4.5) s V s mn rand s s (4.4) max mn s mn rand s s (4.5) max mn where s max and s mn are the upper and lower bounds on the desgn varables values, rand s a unformly dstrbuted random varable that can tae any value between 0 and 1.

12 59 () Fndng the gbest and pbest Then, PSO searches for optmal soluton by updatng generatons. Each partcle s updated by means of two best values namely pbest and gbest n successve teraton. The pbest s the best soluton (ftness) that has been acheved so far. The gbest s the best value obtaned so far by any partcle n the populaton. () Updatng the partcles poston After fndng the two best values, the partcle updates ts velocty and poston by the followng Equaton (4.6) and (4.7) V 1 w V c rand (pbest s ) 1 1 c rand (gbest s ) (4.6) 2 2 s 1 s V 1 (4.7) where V 1 s the velocty of (+1) th teraton of th ndvdual, V s the velocty of th teraton of th ndvdual, w s the nertal weght, c 1 and c 2 are the self and swarm confdence factors. rand and rand are the random numbers selected 1 2 between 0 and 1. pbest s the best poston of th ndvdual, gbest s the best poston among the ndvdual. (v) Termnaton crtera The above procedure s repeated, untl a crteron s met, usually a suffcently good ftness value or a maxmum number of generatons.

13 DIFFERENTIAL EVOLUTION ALGORITHM Storn and Prce (1995) proposed a new floatng pont encoded evolutonary algorthm for global optmzaton and named t Dfferental Evoluton owng to a specal nd of dfferental operator, whch they nvoed to create new offsprng from parent chromosomes nstead of classcal crossover or mutaton. The DE dffers from GA wth respect to the mechancs of mutaton, crossover and selecton performed. GA reles on crossover whle DE reles on mutaton operaton. In GA, the mutaton taes place randomly, whereas n DE, t taes place by some rule. The choce of dfferental evoluton algorthm for numercal optmzaton s based on the followng useful features. It s a stochastc search algorthm that s orgnally motvated by the mechansms of natural selecton. It s less lely to become trapped n a local optmum because t searches for the global optmal soluton by manpulatng a populaton of canddate solutons, or equvalently, searchng a number of dfferent areas smultaneously n search space. It s very effectve for solvng the optmzaton problems wth non smooth objectve functons as t does not requre the dervatve nformaton. It allows for the parameters to be nput, manpulated, and output as ordnary floatng-pont numbers wthout extra processng and, therefore, utlzes computer resources effcently. It uses arthmetc addton rather than random bt flppng to search for contnuum.

14 61 It wors well as a local optmzer because the dfferentals generated by a convergng populaton eventually become nfntesmal. It does not need to mantan a large sze of populaton Worng prncple of DE In DE algorthm, solutons are represented as chromosomes based on floatng-pont numbers. In the mutaton process of ths algorthm, the weghted dfference between two randomly selected populaton members s added to a thrd member to generate a mutated soluton followed by a crossover operator to combne the mutated soluton wth the target soluton so as to generate a tral soluton. Then a selecton operator s appled to compare the ftness functon value of both competng solutons, namely, target and tral solutons to determne who can survve for the next generaton. The basc DE algorthm conssts of four steps, namely, ntalzaton of populaton, mutaton, crossover and selecton. () Intalze the populaton Smlar to other evolutonary algorthm, DE also starts wth a populaton of NP D-dmensonal search varable vectors. Let t = 0, 1, 2... t, t + 1, etc. where t represents generaton ndex. Snce, the vectors are lely to be changed over dfferent generatons, the followng notaton s adopted for representng the th vector of the populaton at the current generaton. X (t) [x,1 (t),x,2 (t),x,3 (t)...x,d (t)] (4.8) These vectors are referred as chromosomes or target vector. For example, f the lower lmt x l and upper lmt x u of the varable x are nown, j j j

15 62 then the j th component of the th populaton s ntalzed by the followng Equaton (4.9). x l u l,j (0) x j rand(0,1) x x (4.9) j j where rand (0,1) s a unformly dstrbuted random number lyng between 0 and 1. () Mutaton After the populaton s ntalzed, mutaton s carred out for each target ndvdual X (t). Mutaton s an operaton that adds a vector dfferental to a populaton vector. Based on mutaton rule, the DE s classfed nto fve dfferent strateges (Storn 1997). They are as follows. Strategy 1: Rand /1 In ths strategy, mutant vector or donor vector V (t) s found by addng the weghted dfference between the defned numbers of ndvdual randomly selected from the prevous populaton to another ndvdual by the followng Equaton (4.10): V, j(t) xr1, j(t) F(xr2, j(t) xr3,j(t)) (4.10) where r1, r2 and r3 are randomly chosen vectors from the current populaton, F s a scalng factor (0,2), whch controls the amplfcaton of the dfferental varaton (x (t) x (t)). r2,j r3,j

16 63 Strategy 2: Rand-to-best/1 Rand to best/1 follows the same procedure as that of the Rand/1. The only dfference s that, the donor vector, whch s used to perturb each populaton member, s created usng any two randomly selected members of the populaton as well as the best vector of the current generaton. Ths can be expressed as V t 1 X(t) Xbest (t) X(t) FXr2(t) Xr3(t) (4.11) where s another control parameter of DE n [0, 2], X (t) s the target vector and X (t) s the best member of the populaton best regardng ftness at t th generaton. Strategy 3: Best/1 In ths scheme everythng s dentcal to DE/rand/1 except the fact that the tral vector s formed as V t 1 Xbest (t) FXr1(t) Xr2(t) (4.12) where the vector to be perturbed s the best vector of the current populaton and the perturbaton s caused by usng a sngle dfference vector. Strategy 4: Best/2 Under ths method, the donor vector s formed by usng two dfferent vectors as shown below: V t 1 Xbest (t) FXr1(t) Xr2(t) Xr3(t) Xr4(t) (4.13)

17 64 Owng to the central lmt theorem, the random varatons n the parameter vector seem to shft slghtly nto the Gaussan drecton, whch s benefcal for many functons. Strategy 5: Rand/2 Here, the vector to be perturbed s selected randomly and two weghted dfference vectors are added to the same to produce the donor vector. Thus for each target vector, a totalty of fve other dstnct vectors s selected from the rest of the populaton. The process can be expressed n the form of an equaton as V t 1 Xr1(t) F1 Xr2(t) Xr3(t) F2Xr4(t) Xr5(t) (4.14) where F1 and F2 are two weghng factors selected n the range from 0 to 1. To reduce the number of parameters, F1 = F2 = F s chosen. () Crossover Followng the mutaton operator, crossover operator s appled to ncrease the dversty of the populaton. The crossover operator s appled on each target ndvdual x (t) and tral ndvdual u (t) s formed usng,1,j the followng equaton, u,j (t) v,j (t) x,j (t) f rand(0,1) CR, otherwse (4.15) where v (t) s the donor ndvdual and CR [0,1] s the crossover,j constant that controls the dversty of the populaton.

18 65 (v) Selecton After the crossover phase, the selecton phase arses to decde, whether the tral vector (off sprng) U (t) would be a member of the populaton of the next generaton. For that, U (t) s compared to the ntal target ndvdual X (t 1) by the followng one-to-one based greedy selecton crteron. X (t 1) U (t) X (t) f f (U otherwse (t)) f (X (t)), (4.16) If the new tral vector yelds a better value of the ftness functon, t replaces ts target n the next generaton; otherwse the target vector s retaned n the populaton. Hence, the populaton ether gets better wth respect to the ftness functon or remans constant. The above procedure s repeated for all other target vectors and new populaton s generated. (v) Termnaton crtera The above procedure s repeated, untl a crteron s met, usually a suffcently good ftness value or a maxmum number of generatons. 4.6 CONCLUSION The non-tradtonal optmzaton technques GA, PSO and DE have been appled to optmzaton problems n all felds of engneerng and attaned astonshng success. The applcaton of these technques for the optmzaton of desgn and manufacturng tolerance by consderng the qualty loss of the assembly, when t s manufactured by alternate sequence of manufacturng

19 66 processes, alternate machnes and fxed sequence of manufacturng process are dealt n the subsequent chapters.