A Genetic Algorithm Approach to the Selection of Engineering Controls for Optimal Noise Reduction

Transcription

1 doi: 0.306/scienceasia ScienceAsia 33 (007): 89-0 A Geneic Algorihm Approach o he Selecion of Engineering Conrols for Opimal Noise Reducion Krisada Asawarungsaengkul a and Suebsak Nanhavanij b* a Indusrial Engineering Program, Sirindhorn Inernaional Insiue of Technology, Thammasa Universiy Pahum Thani, Thailand. b Managemen Technology Program, Sirindhorn Inernaional Insiue of Technology, Thammasa Universiy Pahum Thani, Thailand. *Corresponding auhor, suebsak@sii.u.ac.h Received 0 May 006 Acceped 6 Aug 006 ABSTRACT: The selecion of engineering conrols for workplace noise reducion can be formulaed as a zero-one nonlinear programming problem which is NP hard. Given a noise conrol budge and a se of worker locaions, he problem objecive is o find a combinaion of feasible engineering conrols o minimize a maximum daily noise load. A geneic algorihm (GA) is developed o find an opimal (or near-opimal) noise conrol soluion. Suiable GA parameers and operaions are deermined from a compuaional experimen. GA is found o be efficien in solving engineering noise conrol problems irrespecive of he problem size. KEYWORDS: Geneic algorihm, noise reducion, opimizaion, knapsack problem, capial budgeing. INTRODUCTION Noise-induced hearing loss is one of he mos common occupaional diseases and he second mos self-repored occupaional illness or injury. In Briish Columbia, Canada, he workers compensaion board paid $8 million in permanen disabiliy awards during o 3,07 workers suffering hearing loss. An addiional $36 million was paid ou for hearing aids. According o he Naional Insiue for Occupaional Safey and Healh (NIOSH), USA; approximaely 30 million U.S. workers are currenly exposed o noise hazard on he job and an addiional 9 million U.S. workers risk geing hearing loss. A major cause of occupaional hearing loss is a repeiive exposure o excessive noise levels. Since hearing loss is hardly cured, he prevenion of such exposure is he bes approach. When workers are exposed o a noise condiion exceeding a permissible level, hey are a high risk of hearing loss and a noise conrol program mus be conduced o reduce heir noise exposures. To preven workplace noise hazard, safey praciioners are srongly recommended o apply appropriae engineering conrols as he firs resor. Implemenaion of engineering conrols is he mos effecive approach for he noise conrol program. This approach involves he noise reducion a noise sources and he blocking of noise ransmission beween noise sources and workers. Examples of engineering conrols include proper acousical design of machinery, enclosure of noisy machine (if pracical), modificaion of he exising machine design, and use of sound barrier and sound absorpion along he ransmission pah. Deails of engineering conrols can be found in a number of publicaions. 3-8 More specifically, opics such as a developmen of quieer machines, noise reducion mehods, noise absorpion maerials, and process change for noise reducion have been discussed in he lieraure. 9-8 Engineering approach is he bes approach for noise conrol problems because i solves he problem a is roo cause. Neverheless, i usually is he mos expensive approach. Engineering conrols differ in he cos and noise reducion capabiliy. Suon 9 presened a procedure o idenify possible engineering mehods of noise reducion and o selec he bes mehod using a cos/benefi analysis. The cos/benefi approach is a simple approach bu i does no guaranee an opimal soluion. Briefly, a cos/benefi raio, which is a raio of he noise reducion cos o he amoun of noise ha is reduced a he noise source, is developed for each noise conrol echnique. The noise conrol echnique having a lowes cos/benefi raio will hen be seleced. This selecion procedure, however, has limied applicaion since i considers only engineering conrols a he noise sources. Nor does i consider he noise conrol budge. The selecion of engineering conrols o reduce noise levels a worker locaions can be formulaed as a zero-one nonlinear programming problem. This problem is a varian of he knapsack problem which is NP-hard. Pisinger 0 provided an overview of recen exac soluion approaches and showed he difficuly in solving he knapsack problem. A review on he

2 90 ScienceAsia 33 (007) nonlinear knapsack problem was given by Brehauer and Shey. They discussed algorihms and applicaions of he nonlinear knapsack problem. Among various mea-heurisic echniques, geneic algorihm (GA) has been well adoped by researchers o find good soluions for global and hard-solving opimizaion problems. GA is widely used o solve linear/nonlinear zero-one programming problems as well as linear/nonlinear ineger programming problems. Yokaa e al., 3 formulaed an opimal design problem of sysems reliabiliy as he zero-one nonlinear programming problem wih inerval coefficiens and solved i using GA. GA is also used o deermine soluions for various opimizaion problems in recen research sudies GA has been applied o solve manufacuring problems such as scheduling problems in flexible manufacuring sysems 34, sequencing problems in mixed model assembly lines 35 and in nonmanufacuring problems such as fair bandwidh allocaion 36 and muli-objecive land use planning problems. 37 This paper discusses a GA approach o deermine a se of feasible engineering conrols for opimal noise reducion. The problem objecive is o minimize a maximum daily noise load a any worker locaion wihou exceeding he given noise conrol budge. The paper is organized as follows. Iniially, we presen relevan mahemaical equaions for compuing a daily noise load and an opimizaion model for selecing feasible engineering conrols. Nex, we develop GA for he engineering noise conrol problem (ENCP) and explain how he suiable seings of GA parameers are deermined. We also discuss he effeciveness of GA by comparing GA soluions wih hose from an opimizaion approach. Lasly, we demonsrae how GA is applied o solve he ENCP. Engineering Noise Conrol Problem Problem descripion Generally, an indusrial faciliy has several primary noise sources (manufacuring machines) and secondary noise sources (air compressors, indusrial fans, indusrial pumps, and cooling owers). The noise generaed from hese sources is ransmied o workers who are presen in ha faciliy. In mos counries, a safey law requires ha workers do no receive a daily noise exposure beyond he permissible level. For example, he permissible noise exposure level in he U.S. is se a 90 dba for an 8-hour workday. If here is any worker whose daily noise exposure exceeds his limi, an effecive noise conrol program needs o be implemened. The engineering approach for noise conrol is recommended as he firs line of defense owing o is high effeciveness. Typical engineering conrols include reducing noise levels a he noise sources (or conrolling a he source) and blocking he noise ransmission pah (or conrolling along he pah). To reduce he generaed noise a any noise source, here usually are several echniques ha can be applied. An imporan resul of his noise conrol is ha noise levels a all worker locaions are aenuaed, bu aenuaion levels differ depending on disances beween he noise source and individual worker locaions. Similarly, here usually are several echniques for conrolling noise along is pah (e.g., puing up physical barriers or curains), and wih varying noise aenuaion capabiliies. Only noise levels a he worker locaions in which he direc pahs beween he noise source and he locaions are blocked will be reduced. Comparing beween conrolling a he source and conrolling along he ransmission pah, he former is more effecive han he laer, bu i is also more expensive o implemen. Wih he given noise conrol budge, safey praciioners mus decide on a combinaion of engineering noise conrol(s) o yield a maximum noise aenuaion. More specifically, if conrolling a he source is being considered, i is necessary o deermine which noise source(s) is/are o be conrolled and wih which noise conrol echnique(s). In case of blocking along he pah, he ype of barrier/ curain and is locaion will depend on he inended worker locaions. Esimaion of Combined Noise Level and Daily Noise Exposure When here are muliple noise sources in he faciliy, he combined noise level a worker locaion j, L j (dba), can be compued. Leing L ab be ambien noise level (dba), L be noise level generaed by noise source (dba, measured a -m disance), q be number of noise sources, n be number of worker locaions, and d j be Euclidean disance beween worker locaion j and noise source, we have L 0 Lab 0 q log 0 0 L j = + + = d j j =,, n () If a worker is o be assigned o worker locaion j hroughou an enire workday, his/her daily noise exposure (or 8-hour ime-weighed average noise level, 8-hr TWA) will be equal o L j. In several counries, he permissible daily noise exposure is se a 90 dba. For he sake of mahemaical modeling, we define a uniless variable called daily noise load l o represen he daily noise exposure. The daily noise load l can be compued

3 ScienceAsia 33 (007) 9 from L j 90 l = 5 () Noe ha a permissible daily noise load l p is equal o one. From Eq. (), i is seen ha if an amoun of generaed noise a noise source, L, is reduced, all L j s will be reduced as well. However, noise aenuaion a any worker locaion is nonlinear and dependen of he amoun of noise reducion and he disance beween he concerned worker locaion and noise source. As such, he noise reducion a noise sources having high noise levels may no yield a good soluion if hose sources are locaed far from he worker locaion under consideraion. MATHEMATICAL FORMULATION The definiion of he binary knapsack problem can be given as follows. 38 Suppose ha a hiker has o fill up his/her knapsack by selecing from among various possible objecs hose of which will give he maximum comfor. Leing c be size of a knapsack, n be number of objecs, w j be size of objec j, p j be measure of comfor given by objec j, x j be (0, ) binary variable (where x j = if objec j is seleced, and x j = 0 oherwise), he knapsack problem can be mahemaically formulaed as shown below. Maximize subjec o n px j j (3) j= n wx j j c (4) j= x j = {0, } j =,, n (5) The knapsack problem is one of he mos inensively sudied discree programming problems. The reason for such ineres basically derives from hree facs: () i can be viewed as he simples ineger linear programming problem; () i appears as a sub-problem in many more complex problems; and (3) i may represen a grea many pracical siuaions. 38 The engineering noise conrol problem (ENCP) is a varian of he binary knapsack problem. Given he limied budge, a se of engineering conrols are o be seleced for implemenaion o achieve he maximum noise aenuaion wihou exceeding he budge. Objecive Funcion The objecive of he ENCP is o minimize he maximum daily noise load a any worker locaion, l max. Minimize l max (6) Consrains The ENCP requires hree ses of consrain: () budge consrain; () noise load consrain; and (3) binary variable consrain. A oal noise conrol cos consiss of cos of conrolling noise a he source and cos of blocking he noise ransmission by a barrier. Leing ys u be (0, ) binary variable such ha ys u = if conrolling a noise source using engineering conrol mehod u is applied, and ys u = 0 oherwise; cs u be cos of conrolling a noise source using engineering conrol mehod u; q be number of noise sources; r be number of engineering conrol mehods of conrolling a noise source, we have Cos of conrolling noise a he source = (7) Cos of blocking he noise ransmission by a barrier can be deermined in a similar fashion. Leing yb v be (0, ) binary variable such ha yb v = if blocking he noise ransmission pah using barrier v is applied, and yb v = 0 oherwise; cb v be cos of insalling barrier v; s be number of barriers, we obain Cos of blocking he noise ransmission pah = (8) Since he sum of boh coss mus no exceed he given noise conrol budge EB, he budge consrain can be formulaed as EB (9) Afer applying he seleced noise conrol a he source, he reduced noise level a noise source,, can be compued from = =,, q (0) where NRs u is he amoun of noise reducion (dba) a noise source afer applying noise conrol mehod u. As a resul of noise conrol, he combined noise levels a all (if conrolling a he noise source has been applied) or some (if blocking he noise ransmission pah has been applied) worker locaions will be reduced. Leing NRb jv be amoun of noise reducion (dba) a worker locaion j afer insalling barrier v, he combined

4 9 ScienceAsia 33 (007) noise level a ha locaion hen becomes L j = L 0 Lab 0 q log = d j s j =,, n () v= NRb jv ybv ( ) From Eq. (), he daily noise load consrain can be wrien as L j 90 5 l max () Finally, he binary variable consrain is defined for ys u and yb v. ys u, yb v = (0, ) =,, q; u =,, r ; v =,, s (3) Thus, he ENCP model will have he objecive funcion (Eq. (6)) and consrains ((9) (3)) as summarized below. L j Minimize subjec o l max = =,, q EB L 0 Lab 0 q log 0 0 = + + = d j s NRb yb j =,, n v= jv v solved o opimaliy when he problem size is large. To yield he opimal or near-opimal soluion, we develop he geneic algorihm (GA) for he ENCP. GENETIC ALGORITHM APPROACH TO ENCP The geneic algorihm (GA) approach is employed o opimally selec feasible engineering conrols for he maximum noise aenuaion wihou exceeding he noise conrol budge. Deailed discussion on GAs can be found in Holland 39, Michalewicz 40, and Gen and Cheng. 4, 4 In his secion, we explain GA ha is specifically developed for he ENCP. Topics covered include: () GA procedure, () chromosome coding and iniial populaion, (3) crossover, (4) muaion, (5) finess and evaluaion funcion definiions, (6) repairing procedures (7) selecion echniques, and (8) erminaion rules. GA Procedure GA procedure is illusraed in Fig.. Parameers required for he proposed GA include crossover probabiliy Pc, muaion probabiliy Pm, populaion size Popsize, and maximum generaion Max gen. Firsly, se an iniial generaion as gen = 0. If a repair procedure is required, a repair rae mus also be specified. Nex, binary sring v k (k =,,, Popsize) is creaed. Each sring (chromosome) represens a feasible soluion for he ENCP. Essenial GA operaions including crossover, muaion, and selecion are par of he evoluion process. According o he survival-of-he-fies rule, an evaluaion funcion (o deermine a finess value) mus be evaluaed prior o he selecion. The bes chromosome is regisered afer he selecion process. Then, updae he gen value (gen = gen +). Repea GA procedure unil gen = Max_gen. In addiion, if he repairing procedure is employed, i will be execued afer he crossover and muaion operaions. Iniial parameer Iniial populaion L j 90 5 l max Repair Procedure Crossover No Terminaion? Yes Sop ys u, yb v = {0, } =,, q; u =,, r ; v =,, s Repair Procedure Muaion Regiser he bes chromosome As one can see, he ENCP is a minimax opimizaion problem wih nonlinear consrains. Also, since i is he zero-one nonlinear programming problem, i canno be Evaluaion Fig. The geneic algorihm procedure. Selecion

5 ScienceAsia 33 (007) 93 Chromosome Coding and Iniial Populaion Binary encoding is employed in he proposed GA o creae chromosomes because he decision variables of he ENCP are zero or one. The lengh of chromosome is equal o he number of engineering conrols ha are feasible for he problem being considered, q s+ r =. An iniial populaion is randomly generaed. Noe ha he number of chromosomes in he (iniial and subsequen) populaions is consan and is denoed by Popsize. Crossover Crossover is a GA operaion which aemps o generae wo new chromosomes ha may be sronger han heir parens. Two paren chromosomes are randomly seleced from he curren populaion for maing. Two new chromosomes, called offspring, will be creaed by swapping some pars of he paren chromosomes. Crossover probabiliy Pc indicaes he number of chromosome pairs ha will be involved in he crossover operaion. For our GA procedure, wo crossover echniques are considered: () single-poin crossover, and () wo-poin crossover. Single-poin Crossover Single-poin crossover is a simple echnique ha combines wo paren chromosomes o generae wo offspring. To achieve his, a random cu-poin is chose and wo new offspring are generaed by swapping he lef-hand-side segmens afer he cu-poin of he seleced parens. Fig. (a) illusraes he single-poin crossover echnique. he number of muaed bis. A single-poin muaion, which is used in his paper, alers a value o 0, and vice versa. Leing mu_no and chro_l denoe number of muaed bis and lengh of chromosome, respecively, hen mu_ no can be compued as follows. mu _ no = Pm chro _ l popsize (4) The muaion operaion is illusraed in Fig Fig 3. Muaion operaion Parens Offspring Finess and Evaluaion Funcion Definiions An evaluaion funcion is used o evaluae he finess of chromosomes in each generaion. The chromosomes having high evaluaion values will poenially be seleced for he nex generaion. To obain he evaluaion funcion, a finess funcion and a penaly coefficien have o be defined. Deails of hese opics can be found in Michalewicz e al. 43 4, 4 and Gen and Cheng. Finess Funcion The finess funcion is problem specific. For he ECNP, a finess value is defined as he maximum daily noise load l max. When comparing beween wo chromosomes, since he problem objecive is o minimize l max, a sronger chromosome is he chromosome ha has a lower l max han he oher one. The finess funcion f k (v k ) can be wrien as Parens Parens f k (v k ) = l max (5) Offspring Offspring (a) Fig. (a) Single-poin crossover, and (b) Two-poin crossover Two-poin Crossover This crossover echnique combines wo paren chromosomes by choosing wo random cu-poins. Unlike he single-poin crossover, he middle segmens beween boh cu-poins of he wo parens are swapped o creae wo offspring as illusraed in Fig. (b). Muaion Muaion is a GA operaion which makes random aleraions o various chromosomes. Random muaion changes a small number of bis in chromosomes depending on muaion probabiliy Pm ha indicaes (b) Penaly Funcion Since he ENCP has an upper bounded consrain which is he engineering conrol budge EB, a penaly erm is added o he finess funcion so ha he chromosome ha falls in infeasible space will have a lesser chance o be seleced for he nex generaion. A penaly coefficien p k, where k =,, Popsize, is proporional o he amoun of exra budge ha can be deermined from he following funcion. p k = 0, if budge consrain is saisfied q r s θ ( csu ysu ) ( cbv yb + v ) = u= v= - EB, oherwise (6) where q is a large posiive value.

6 94 ScienceAsia 33 (007) Evaluaion Funcion From Eqs. (5) and (6), he evaluaion funcion eval(v k ), where k =,, Popsize, can be expressed as eval(v k ) = f ( v ) + p k k k k =,,, Popsize (7) Repair Procedures Afer performing he crossover and muaion operaions, new chromosomes may be infeasible since he oal cos exceeds he noise conrol budge. They have o be repaired before hey can be considered for he nex generaion. Furher discussion on he chromosome repairing issue can be found in Michalewicz e al. 43 The number of infeasible offspring o be repaired mus no be greaer han he value compued from [repair rae Popsize]. Here, we consider wo repair procedures, each of which can be employed o repair any infeasible chromosomes. Random Repair Procedure This echnique randomly changes bis ha have a value o 0. This random change is repeaed unil he budge consrain is saisfied. Ordered Repair Procedure The amoun of noise generaed from a noise source reaching a worker locaion depends on how far he locaion is from he noise source. Le us define a noise impac of he noise source as a sum of inensiies (in W) of noise from ha noise source measured a all worker locaions. Thus, he noise impac of noise source, T, can be compued from L 0 n 0 0 T = j= d j (8) There are seps required o complee he ordered repair procedure. Given bn i = value of bi number i i* = seleced bi number j* = seleced worker locaion number rank_no = ranking number sum_bi = sum of bi values of bi number ranging beween L and U * = seleced noise source number Sep : Rank T in ascending order. Sep : Deermine rank_no of all noise sources from he order of T obained in Sep. Sep 3: Se e =. Sep 4: Selec * having rank_no = e. Sep 5 : Se L = * r + and U = r. = * = Sep 6 : Calculae sum_bi = U bni. If sum_bi is i= L greaer han or equal o, hen go o Sep 7. Oherwise, go o Sep 0. Sep 7 : Randomly selec i* where L i* U. Sep 8 : If bn i* =, hen change i o 0 and go o Sep 9. Oherwise, reurn o Sep 7. Sep 9 : If, hen go o Sep 0. Oherwise, sop repairing. Sep 0: If e < q, hen se e = e + and go o Sep 4. Oherwise, go o Sep. Sep : Calculae L j when no engineering conrol is implemened and rank L j in ascending order. Sep : Deermine rank_no of all worker locaions from he order of L j in Sep. Sep 3: Se e =. q Sep 4: Se L = r + and U = r +s. = q = Sep 5: Selec j* having rank_no = e. Sep 6: Calculae sum_bi = U bni. If sum_bi is i= L greaer han or equal o, hen go o Sep 7. Oherwise, sop repairing. Sep 7: Se k =. Sep 8: If NRb j*,v=k > 0 and bn (where i = L + k ) i =, hen le bn i = 0. Oherwise, go o Sep 9. Sep 9: If, hen go o Sep 0. Oherwise, sop repairing. Sep 0: If k < s, hen se k = k + and reurn o Sep 8. Oherwise, go o Sep. Sep : If e < n, hen se e = e + and reurn o Sep 5. Oherwise, sop repairing. Selecion Techniques For he selecion procedure, wo basic opics are discussed: () sampling space, and () sampling mechanism. Various mehods for selecing chromosomes are laer examined in he compuaional experimen.

7 ScienceAsia 33 (007) 95 Sampling Space Two ypes of sampling space are invesigaed in his paper. They are: () regular sampling space, and () enlarged sampling space. Regular Sampling Space The size of regular sampling space is always equal o Popsize. This is because newly generaed offspring will replace heir parens afer heir birh. Originally, his procedure is called generaional replacemen. Enlarged Sampling Space Boh parens and offspring have been reained in he sampling space, called enlarged sampling space. Therefore, he size of sampling space is equal o Popsize + (cross_pair ) + mu_no. In his mehod, parens and offspring have heir chances o be seleced for he new generaion depending upon heir finess values. Sampling Mechanism The sampling mechanism involves how o selec chromosomes from he sampling space for he new generaion. Two sampling mechanism echniques are considered. Roulee Wheel Selecion wih Eliis Selecion The roulee wheel selecion echnique is an eliis approach in which he bes chromosome has a highes probabiliy o be seleced for he new generaion. The basic roulee wheel is a sochasic sampling wih replacemen. The higher he evaluaion funcion value a chromosome has, he greaer poenial i will be seleced as a member of he new generaion. The new generaion has he same populaion size as he previous one. Wih he eliis selecion, he bes chromosome is firsly seleced for inclusion in he new generaion. Ranking Selecion The evaluaion funcion values of all chromosomes Table. ANOVA able for he 8 x 8 problem size in he sampling space are firsly calculaed. Then, hey are sored and lised in descending order (i.e., from he bes o he wors). The number of chromosomes o be seleced for inclusion in he new generaion is Popsize. This approach prohibis duplicae chromosomes from passing ono he new generaion. Ter erminaion Rules Since GA is an ieraive approach, GA procedure is erminaed when he number of ieraions has reached he maximum generaion denoed by Max_gen. ANALYSIS OF GA PARAMETERS When applying GA, i is known ha he qualiy of he soluion and he effeciveness of GA are likely o be influenced by he parameer seings. A compuaional experimen is conduced o invesigae effecs of he crossover probabiliy Pc, muaion probabiliy Pm, populaion size Popsize, and maximum generaion Max_gen on l max. The experimen is designed as a full-facorial experimen wih four facors (i.e., Pc, Pm, Popsize, and Max_gen) and hree replicaes. A dependen variable in his experimen is he maximum daily noise load l max. The number of levels (reamens) and he seings of each facor are shown in Table. There are 360 runs in he experimen. Two problem sizes (deermined by he numbers of noise sources and worker locaions) are invesigaed: () 8 noise sources (q = 8) and 8 worker locaions (n = 8), and () 0 noise sources (q = 0) and 0 worker locaions (n = 0). The resuls of he Table. Facors and levels of he full-facorial experimen Facors Number Seings of Levels Pc 5 0., 0., 0.3, 0.4, 0.5 Pm , 0.0, 0.5, 0.0, 0.5, 0.30 Popsize 50, 00 Max_gen 00, 0000 Source of Variaion Degrees of Freedom Sum of Mean F 0 P-value Squares Square Pc Pm Popsize Max_gen Pc x Pm Pc x Popsize Pc x Max_gen Pm x Popsize Pm Max_gen Popsize Max_gen Error Toal

8 96 ScienceAsia 33 (007) Table 3. ANOVA able for he 0 x 0 problem size Source of Variaion Degrees of Freedom Sum of Mean F 0 P-value Squares Square Pc Pm Popsize Max_gen Pc x Pm Pc x Popsize Pc x Max_gen Pm x Popsize Pm x Max_gen Popsize Max_gen Error Toal analysis of variance (ANOVA) of he 8 x 8 and 0 x 0 problem sizes are shown in Tables and 3, respecively. From Tables and 3, i is found ha Pc, Popsize and Max_gen have significan effecs on l max in boh problem sizes. Pm only has a significan effec on l max in he 0 0 problem. Based on he resuls from he saisical analysis, we se Pc = 0.5, Pm = 0.05, and Popsize = 50. Max_gen, however, will vary wih he problem size. ANALYSIS OF GA OPERATIONS In his secion, he effecs of sampling space, selecion mehod, crossover and muaion echniques, and repair procedure are invesigaed in anoher compuaional experimen. As shown in Table 4, six reamens (P, P,, P6) wih differen combinaions of sampling space, selecion mehod, crossover and muaion echniques, and repair procedure are described. For each reamen, nine problem sizes as indicaed by he numbers of noise sources and worker locaions (4 x 4, 6 x 6, 8 x 8, 0 x 0, 5 x 5, 0 x 0, 30 x 30, 40 x 40, and 50 x 50) are examined. For each problem size, five sub-problems (S- o S-5) are esed. All sub-problems are randomly generaed using L, r, and s ranging beween dba, 0-3 mehods, and - 6 mehods, respecively. Table 5 shows he maximum generaions for he nine problem sizes in he experimen. The experimen is repeaed wih 0 replicaes for each sub-problem. Therefore, he experimen consiss of 450 experimenal runs (9 problem sizes 5 sub-problems 0 replicaes). GA procedure is implemened in VBA (Microsof Excel) and is run on Penium IV,.80 GHz, and 5 MB RAM personal compuers. An average l max from he 0 replicaes is used as a quaniaive measure o represen GA soluion and o compare among differen soluions. A plo of he average l max versus he number of generaions of he 0 0 problem size (sub-problem: S-) is illusraed in Fig. 4. I is seen ha he average l max converges quickly o he bes soluion wihin he firs wo hundred generaions, afer which i levels off. Fig. 5(a) and Fig. 5(b) show changes in he average l max and average CPU ime, respecively, wih respec o he problem size for Table 4. GA operaions for he six reamens GA Operaion P P Treamen P3 P4 P5 P6 Sampling Space Regular Enlarged Enlarged Enlarged Enlarged Enlarged Selecion Mehod Roulee Wheel Roulee Wheel Roulee Wheel Ranking Roulee Wheel Roulee Wheel Crossover Single-poin Crossover Muaion Single-poin Muaion Repair Procedure None None None None Random a Ordered a a Repair rae = 0.0 Table 5. Maximum number of generaions for he nine problem sizes Problem size 4 x 4 6 x 6 8 x 8 0 x 0 5 x 5 0 x 0 30 x x x 50 Max_gen,000,000,000 4,000 4,000 7,000 8,000 8,000 8,000

9 ScienceAsia 33 (007) 97 all six reamens and for he opimizaion approach. An opimizaion sofware ool called LINGO is used o solve he ENCP o opimaliy. Is compuaion ime limi is se a 50,000 seconds. From Fig. 5(a), i is seen ha combinaions P, P3, P5, and P6 are superior o he ohers due o heir lower average l max s. For he wo larges problem sizes (40 x 40 and 50 x 50), he average l max from combinaion P6 is found o be he lowes. In erms of compuaion ime, he average CPU ime increases wih he problem size in all six reamens (see Fig. 5(b)). Furhermore, he increases are found o be progressive when he problem size is 5 x 5 or larger. When LINGO is uilized, he opimal soluion could be obained only when he problem size is small (no larger han 5 x 5). Among he six reamens, combinaion P requires he leas amoun of CPU ime o yield he bes soluion, while combinaions P, P3, P4, and P6 require relaively equal compuaion imes. Since our emphasis is on he qualiy of he soluion (as measured by how low he average l max is), GA operaions employed in combinaion P6 are chosen as hose o be used in GA approach o he ENCP. Specifically, enlarged sampling space, roulee wheel selecion, single-poin crossover, single-poin muaion, and ordered repair procedure are employed, wih P c = 0.5, P m = 0.5, and Popsize = 50. Nex, we perform a saisical analysis o sudy differences in he average l max beween soluions from he GA approach (i.e., combinaion P6) and he opimizaion approach (i.e., LINGO). The resuls (% deviaion) are shown in Table 6. When he problem sizes are small (e.g., 4 x 4, 6 x 6, and 8 x 8), GA is able o yield he opimal soluions in all sub-problems. For he nex wo larger problem sizes (0 x 0 and 5 x 5), GA is effecive in abou 50% of he sub-problems solved. Neverheless, a is wors performance, he soluion from GA is sill only 0.9% greaer han he opimal soluion. Table 6. % deviaion of average l max [(l max (P6) l max (LINGO))/ l max (LINGO)] Problem Size Sub-problem S- S- S-3 S-4 S-5 4 x x x x x x a a a -3.7 a 30 x a a a -9.0 a -.50 a 40 x 40.4 * * 50 x a * * * * *LINGO canno find any feasible soluion wihin 50,000 seconds. a LINGO can find only he feasible soluion. Average l max Number of Generaions Fig 4. Plo of number of generaions vs. average l max for he 0 0 problem size (sub-problem: S-) Average l max Average CPU Time x4 4x4 a) 6x6 b) 6x6 8x8 8x8 0x0 5x5 0x0 Problem Size P P P3 P4 P5 P6 Opimizaion 0x0 5x5 0x0 Problem Size 30x30 30x30 P P P3 P4 P5 P6 Opimizaion 40x40 40x40 50x50 50x50 Fig 5. Plos of (a) problem size vs. average l max, and (b) problem size vs. average CPU ime For he problem sizes greaer han 5 x 5, LINGO is able o solve four (ou of 0) problems o opimaliy. In some problems, LINGO can find only feasible soluions wihin 50,000 seconds. There are six problems (wih he problem sizes 40 x 40 and 50 x 50) for which LINGO canno obain feasible soluions wihin 50,000 seconds. When GA is used, a maximum % deviaion from he opimal soluions is found o be.4% (a he 40 x 40 problem). In hose problems for which LINGO can find feasible soluions, he soluions from GA are superior o hose from LINGO.

10 98 ScienceAsia 33 (007) Thus, i is eviden ha GA approach is an effecive means for solving he ENCP. GA soluion is opimal when he problem size is small. For larger problems for which he opimizaion approach fails o find he opimal soluions, GA can yield he soluions wih small deviaions from he bes soluions obained by he opimizaion sofware ool used. Addiionally, he compuaion ime when using GA is also shor, making i a very pracical means for solving he ENCP. NUMERICAL EXAMPLE [NRs u ] =, [cs u ] = ,000 3,500 3,500 5,000 4,500 5,500 4,500 5,500 4,500 5,500 4,500 5,500 Le us consider an indusrial faciliy wih eigh machines (q = 8) and eigh worker locaions (n = 8). Locaion coordinaes of he eigh machines and heir noise levels (measured a -m disance) are shown in Table 7. A presen, here are eigh workers being assigned o eigh differen worker locaions, and each worker mus be a he same worker locaion for 8 hours. Locaion coordinaes of he eigh worker locaions are also shown in Table 7. Ambien noise level in his faciliy is assumed o be 70 dba. When no engineering noise conrol is implemened, 8-hour TWAs a he eigh worker locaions are as shown in Table 8. From Eq. (), he maximum daily noise load l max is found o be.038 (a worker locaion WL5). Noise conrol daa for his example is as shown below. - Noise conrol budge EB = 0,000 bah. - There are wo mehods for blocking he noise ransmission pah. When applied, noise reducion occurs a worker locaions WL5 and WL6. The amoun of noise reducion is 7 dba a each locaion. The barrier cos is 3,800 bah, and is he same for boh mehods. - There are wo mehods for conrolling noise a he machines. Noise reducion daa (in dba) a he eigh machines when each mehod is uilized (wherever applicable) and noise conrol coss (in bah) are as follows: GA is applied o find feasible engineering conrols ha will minimize he maximum daily noise load such ha he oal noise conrol cos does no exceed 0,000 bah. GA parameers are Pc = 0.5, Pm = 0.05, Popsize = 50, and Max_gen =,000 generaions. Enlarged sampling space, roulee wheel selecion wih eliis selecion, single-poin crossover, single-poin muaion, and ordered repair procedure are seleced as GA operaions. A noise conrol soluion recommended by GA requires he following engineering conrols: - Reducing noise a machine M5 using engineering conrol mehod - Reducing noise a machine M6 using engineering conrol mehod - Reducing noise a machine M7 using engineering conrol mehod - Reducing noise a machine M8 using engineering conrol mehod The oal noise conrol cos is 0,000 bah. As a resul, he reduced daily noise loads a he eigh worker locaions are.73,.0570,.0570,.3, 0.835, 0.80, , and , respecively. Noe ha he maximum daily noise load l max =.3 (a worker locaion WL4) is he minimum among hose feasible soluions found by GA. Since here are several daily noise loads ha exceed, noise hazard has no ye been eliminaed. For ease of comparison, updaed 8-hour TWAs a he eigh worker locaions afer implemening he recommended engineering conrols are also shown in Table 8. Table 7. Locaion coordinaes and noise levels of he eigh machines and locaion coordinaes of he eigh worker locaions Machine Locaion Coordinae (m) Locaion Coordinae (m) x-coor -coordinae y-coor -coordinae Noise Level (dba) ) Worker Locaion x-coordinae y-coordinae M 3 90 WL 3 3 M 6 89 WL 6 3 M WL3 9 3 M4 9 WL4 3 M WL5 3 5 M WL6 6 5 M WL7 9 5 M WL8 5

11 ScienceAsia 33 (007) 99 To eliminae noise hazard, he noise conrol budge EB has o be increased. Using a rial-and-error approach, i is found when EB is se a 8,000 bah, he 8-hour TWAs a all worker locaions do no exceed 90 dba (see Table 8). The new recommended engineering conrols are as follows: - Reducing noise a machine M using engineering conrol mehod - Reducing noise a machine M4 using engineering conrol mehod - Reducing noise a machine M5 using engineering conrol mehod - Reducing noise a machine M6 using engineering conrol mehod - Reducing noise a machine M7 using engineering conrol mehod - Reducing noise a machine M8 using engineering conrol mehod CONCLUSION AND DISCUSSION The selecion of engineering conrols for noise hazard prevenion is examined. The ENCP is mahemaically formulaed as a zero-one nonlinear programming problem. The problem objecive is o find a se of engineering noise conrols wihou exceeding he given budge such ha he maximum daily noise load is minimized. The ENCP is a varian of he knapsack problem. Geneic algorihm (GA) is developed o provide he opimal or near-opimal soluion for he ENCP. To selec he appropriae GA parameers and operaions, wo compuaional experimens are carried ou. The firs experimen invesigaes he effecs of GA parameers, namely, crossover probabiliy Pc, muaion probabiliy Pm, populaion size Popsize, and maximum generaion Max_gen on he maximum daily noise load l max. The resuls show ha all of he above GA parameers have significan effecs on l max. The second experimen is inended o find he proper GA operaions for solving he ENCP. Two repair procedures are developed o assis GA in enhancing he qualiy of he soluion. Four hundred and fify problems (9 problem sizes 5 subproblems 0 replicaes) are analyzed by GA. The problems are grouped ino six combinaions of seleced GA operaions (called reamens). I is found ha he combinaion which employs enlarged sampling space, roulee wheel selecion wih eliis selecion, singlepoin crossover, single-poin muaion, and ordered repair procedure demonsraes he bes performance (i.e., achieving he lowes average l max ). When comparing beween he average l max s obained from GA and LINGO (only in small-sized problems for which LINGO can find he opimal soluions), i is seen ha GA is excepionally effecive since i is able o yield he average l max s ha are idenical o hose obained from LINGO. When he problem size is large (e.g., 0 0 and 5 5), he average l max obained from GA is slighly greaer han ha from LINGO (he% deviaion is found o be small). When he problem size is very large, LINGO will have difficuly finding he opimal soluion wihin he given ime limi of 50,000 seconds. Depending on he problem size, LINGO may or may no be able o find he bes feasible soluion wihin he ime limi. GA, on he oher hand, is able o yield he feasible soluion in relaively shor ime irrespecive of he problem size. These findings confirm he effeciveness of GA in solving he ENCP. From he given numerical example, when he noise conrol budge is se a 0,000 bah, he recommended engineering conrols canno compleely eliminae noise hazard since daily noise loads a some worker locaions are sill greaer han he permissible level. By increasing he budge o 8,000 bah, he new noise conrol soluion ha is effecive can now be obained. In mos real siuaions, he noise conrol budge is limied and fixed. As such, oher noise conrol approaches should be considered. For insance, job roaion can be Table 8. 8-hour TWAs a he eigh worker locaions Worker Locaion 8-hour TWA (dba) Before Implemening Afer Implemening Afer Implemening Engineering Conrols Engineering Conrols Engineering Conrols (EB = 0,000 bah) (EB = 8,000 bah) WL 9.3 a 90.8 a 84.8 WL 9. a 90.4 a 90.0 WL3 9.0 a 90.4 a 89.9 WL4 9.6 a 9.5 a 84.8 WL a WL6 95. a WL a WL a a Exceeding he daily permissible level.

12 00 ScienceAsia 33 (007) implemened o roae workers among worker locaions so as o reduce heir noise hazard exposures. The use of hearing proecion devices (HPDs) can be addiionally enforced o reduce he amouns of perceived noise a seleced worker locaions. I should be noed ha job roaion and he use of HPDs are no as effecive as engineering noise conrols, bu hey usually are less expensive. In pracice, a combinaion of noise conrol approaches should be implemened o keep he oal noise conrol cos from exceeding he budge and o achieve safey daily noise exposures in all workers. ACKNOWLEDGMENTS The auhors are indebed o he Thailand Research Fund (TRF) for sponsoring his research sudy hrough he RGJ-PHD gran No.PHD/084/545. REFERENCES. Naional Insiue for Occupaional Safey and Healh (988) Crieria for a recommended sandard occupaional noise exposure. DHHS (NIOSH) Publicaion NO. 98-6, Cincinnai, Ohio.. 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