Optimal Tolerance Allocation Based on Fuzzy Comprehensive Evaluation and Genetic Algorithm

Transcription

1 It J Adv Mauf Techol (2000) 16: Spriger-Verlag Lodo Limited Optimal Tolerace Allocatio Based o Fuzzy Comprehesive Evaluatio ad Geetic Algorithm S. Ji, X. Li, Y. Ma ad H. Cai The Advaced Machiig Techology Ceter, Harbi Istitute of Techology, Harbi, Chia It is very importat to kow how to allocate toleraces ecoomically for parts i a CAD/CAM system because this directly affects the machiig costs of the parts. A ew approach based o fuzzy comprehesive evaluatio (FCE) ad a geetic algorithm (GA) is preseted to obtai a ratioal tolerace allocatio for the parts. First, the curret methods for tolerace allocatio are reviewed i detail. The, FCE is used to evaluate the machiability of a part; a ew optimal model, which ca fully exploit DFA (desig for assembly) ad DFM (desig for maufacturig), is established by combiig the fuctioal sesitivity factors ad machiability factors of parts. A geetic algorithm (GA) is developed ad used to optimise the above model. Fially, a actual example is used to verify the feasibility of the above method; the computed result shows that the method ca produce tolerace allocatios ecoomically ad accurately. Keywords: Fuzzy comprehesive evaluatio; Geetic algorithm; Tolerace allocatio 1. Itroductio I mechaical desig, geometric ad dimesioal toleraces are used to specify the rage withi which a part geometry ad size may vary while coformig to the fuctioal requiremets. Assiged toleraces have a direct effect ot oly o the machiig costs but also o the product quality. Uecessarily tight toleraces result i high productio costs, yet the toleraces should esure that the fuctioal performace requiremets of the products stay withi a satisfactory rage. Toleraces which are too loose ca affect the product quality, ad icrease the scrap rate ad productio costs [1,2]. I geeral, desigers allocate toleraces for parts based o their experiece, ad o hadbooks ad stadards, which leads to some errors [3]. I recet years, computer-aided toleracig desig (CATD) has become a importat research directio i CAD systems ad itegrated CAD/CAM systems [4 7]. Tolerace allocatio is oe of the most importat problems for CATD. Give a required tolerace for the assembly, the plaer is first faced with the problem of how to allocate ecoomically suitable tolerace values for the parts by cosiderig the trade-off betwee fuctioal requiremets ad machiig costs. May researchers cosidered tolerace allocatio as a optimisatio problem. The tolerace values of parts were take as the cotrol variables, ad the machiig costs were take as the objective fuctio to be miimised. The tolerace stack-up limits coditios were take as costraits o the variables [8 10]. I this paper, the tolerace allocatio is represeted as a optimisatio problem. A set of methods used is first reviewed i detail. Fuzzy comprehesive evaluatio is used to evaluate the machiablity of parts, the a ew mathematical model is established ad solved usig a geetic algorithm (GA). The tolerace allocatio for a actual idustrial assembly is produced by the above method, ad the results show that the method ca be used ecoomically to desig the tolerace values of parts. 2. Review: Methods for Tolerace Allocatio 2.1 Geeral Allocatio Methods Whe a assembly fuctio requiremet is give, the tolerace values of parts must be solved. Because the give coditios are almost always isufficiet, the toleraces are usually regarded as equal. The methods used ofte iclude the same tolerace method, the costat precisio factor method, the same ifluece method ad the proportioal scalig method Same Tolerace Method Correspodece ad offprit requests to: Dr Xiaoli Li, Departmet of Maufacturig Egieerig, City Uiversity of Hog Kog, 83 Tat Chee Aveue, Kowloo, Hog Kog mel50001 cityu.edu.hk I the same tolerace method, all of the toleraces of parts are equal o the premise of satisfyig the fuctioal requiremet of the assembly, that is,

2 462 S. Ji et al. t 1 = t 2 = % = t where t i deotes the tolerace value of part i Costat Precisio Factor Method The costat precisio factor method is based o the rule of thumb that the tolerace of a part icreases as the cubic root of the omial size, thereby t i = P(d i ) 1/3 where t i J, the precisio factor P ca be calculated as P = J d 1/3 i where J deotes the fuctioal requiremet o the assembly Same Ifluece Method The assembly fuctioal requiremet is iflueced by two factors. Oe is the tolerace t i of each part, aother is the fuctioal sesitivity coefficiet i. So the same ifluece method ca be expressed as follows: t 1 1 = t 2 2 =, %, = t Proportioal Scalig Method The part toleraces are first determied usig a database i the proportioal scalig method. If the sum of the part toleraces exceeds the required assembly tolerace J, each part tolerace t i is relatively reduced correspodig to dimesio d i. The geeral expressio is: d 1 = d 2 = % = d i = % = d t 1 t 2 t i t where d i is the dimesio of part i, t i is the tolerace allocated for d i, ad t i J (worst tolerace aalysis), or t 2 i J 2 (statistical aalysis). I short, the above methods are very simple, but ca oly evaluate the tolerace value ad correspodig precisio grade of each part as a whole. These values are ofte used i the iitial stage of tolerace allocatio. 2.2 Miimisig Costs Methods The best rule for evaluatig tolerace allocatio is the machiig costs rule, so the miimisig costs methods have bee importat research topics. I these methods, the relatioship betwee the machiig costs ad the tolerace of a part is expressed usig mathematical formulae, ad miimisig the total machiig costs is take as a optimal objective uder the costraits of the fuctioal requiremets. I the last 50 years, more tha 10 models o the cost tolerace relatioship have bee preseted. They are show i Table 1. These models are all based o the empirical cost tolerace data frequetly used i productio processes [20,21]. The model parameters are determied usig the least-squares method based o these data. Owig to the lack of available productio data about the costs tolerace relatioship, ad the fact that the machiig costs will chage whe the machiig cotext chages, the applicatios of the methods are very limited. 2.3 Comprehesive Factor Method The quatitative evaluatio of the machiability of each part is called the comprehesive factor. Because machiig costs are strogly related to the machiability, the relative machiig costs of each part ca be evaluated usig its comprehesive factor. Usually, after the appropriate weight value of each factor (which has importat effects o machiig costs) is give, the comprehesive factor ca be calculated by: F i = F i1 F i2 % F ij % F im (i = 1, 2, %, ; j = 1, 2, %, m) P i = F i F i where i ( % ) deotes the umber of parts, m is the umber of factors relative to machiability, P i is the percetage scalig which idicates the relative machiability of part i o the whole assembly. Geerally, the factors that ifluece machiig costs iclude machiig methods, part materials, part geometrical structure, part size, etc. Because these factor values ca oly be satisfactorily determied by experts with abudat productio kowledge ad experiece, the method is subjective. 2.4 Artificial Itelliget Method Artificial itelligece techiques are curretly used for CATD. Kopardekar ad Aad [22] preseted a eural etwork-based method for tolerace allocatio, takig ito accout machiability ad mea shifts. The eural etwork ca predict idividual part toleraces, ad is show i Fig. 1. The advatages of the proposed method are: 1. The procedure does ot eed ay assumptio about distributio of the part dimesios, ulike statistical techiques. 2. This method ca be exteded to the assemblies of umbers of parts. However, some disadvatages limit its applicatio, such as: 1. The approach requires some data with kow outputs. 2. The approach is ot very efficiet whe the umber of parts is small. Dupiet [9] preseted a ew method that used fuzzy iferece to evaluate the maufacturig difficulties of a part. All fuzzy rules were give by a expert ad could be chaged to accommodate the kowledge of other compaies or other expert skills, so the method is very simple, but sometimes the experts have great difficulty i defiig the fuzzy rules. 3. Tolerace Allocatio Based o FCE ad GA The major drawback of the geeral allocatio methods is that they do ot take ito accout the machiig costs. I the

3 Optimal Tolerace Allocatio 463 Table 1. Cost tolerace models preseted. Model Mathematical expressio Referece Expoetial c( ) = a 0 e a i [11, 19] Reciprocal squared model (R-squared) c( ) = a 0 / 2 [12, 19] Reciprocal powers model (R-power) c( ) = a 0 a i [13, 19] Reciprocal powers ad expoetial hybrid model (RP-E hybrid) c( ) = a 0 a ie a 2 [14, 19] Reciprocal model (reciprocal) c( ) = a 0 / [15, 19] Modified expoetial model (M-expoet) c( ) = a 0 e a 1 ( a 2 ) + a 3 mi max [16, 19] Discrete Model (discrete) [17, 19] Combied reciprocal powers ad expoetial model (combied RP-E) c( ) = a 0 + a 1 a 2 + a 3 e a 4 [18, 19] Combied liear ad expoetial model (combied L-E) c( ) = a 0 + a 1 + a 2 e a 3 [18, 19] Cubic ployomial (cubic-p) c( ) = a 0 + a 1 + a a 3 3 [18, 19] Fourth-order polyomial (4th-p) c( ) = a 0 + a 1 + a a a 4 4 [18, 19] Firth-order polyomial (5th-p) c( ) = a 0 + a 1 + a a a a 5 5 [18, 19] Neural etwork model [20] racy. Clearly, these are fuzzy factors. We use two-order fuzzy comprehesive evaluatio to process the fuzzy factors. The method ad theory of fuzzy comprehesive evaluatio is referred to i [23,24]. Fig. 1. Network architecture. miimisig costs method, the machiig costs are take ito accout, but it is difficult to determie the cost tolerace relatioship for every machiig process. I fact, the desiger ca evaluate a o-uiform cost tolerace relatioship oly accordig to previous experiece. The geeral rule is that the machiability of a part determies its machiig costs. The tolerace allocatio of a part depeds o its machiability. Takig ito accout the advatages ad disadvatages of the above methods, we propose a ew method based o fuzzy comprehesive evaluatio ad a geetic algorithm (see Fig. 2). 3.1 Machiability Estimator Based o Fuzzy Comprehesive Evaluatio Accordig to the desig ad machiig criteria, the machiability of parts depeds o the dimesios, the geometrical structure, the material machiability ad the machiig accu- Fig. 2. The whole procedure of tolerace allocatio usig the ew approach Establishig the Fuzzy Factor Set The factors of dimesio size, geometrical structure, material machiability, ad machiig accuracy ca be expressed as: U = {u i, u 2, %, u m } = {DS, GS, MM, PA} (1) where m=4, DS: dimesio size, GS: geometrical structure, MM: material machiability, ad PA: machiig accuracy Grade of Factor For evaluatig accurately the value of each parameter, each factor is divided ito differet fuzzy subsets such that: u i = {u i1, u i2, %, u ii } (2) where u ij (i = 1 m; j = 1 i ) deotes the jth grade of the ith factor, i represets the grade umber of each factor (the detailed value is show i Table 2) Establishig the Evaluatio Set Sice the rage of machiability is betwee 0.0 ad 1.0, it ca be divided ito 10 equal levels, amely, = { 1, 2, %, k, %, 9, 10 } = {0.1, 0.2, %, 0.9, 1.0} (3) where is the fuzzy evaluatio set First-Order Fuzzy Comprehesive Evaluatio Matrix Based o the experiece of experts, the first-level fuzzy comprehesive evaluatio matrix is determied below: R 1 = R 2 =

4 464 S. Ji et al. Table 2. Mai effective factors ad its grade divisio. Factors Grades of each factor Grade 1 Grade 2 Grade 3 Grade 4 u 1 DS 5mm 25 mm 75 mm 120 mm u 2 GS Easy to maufacture Difficult to maufacture u 3 MM Good Medium Poor u 3 PA Normal Medium Accuracy Note: 1 =4, 2 =2, 3 =3, 4 =3. R 3 = R 4 = Establishig the Weight Vector of each Grade The weight vector is composed of membership degrees of the evaluatio object to all grades i each grade of each factor, show as follows: A i = (a i1, a i2, %, a ii ) (4) i where a ij = ij ij (i = 1 % 4; j = 1 % i ) Here, ij is the membership degree of the evaluatio object to the jth grade of the ith factor First-Order Fuzzy Comprehesive Evaluatio Whe the fuzzy comprehesive evaluatio is made for every grade of the ith factor, the first-order fuzzy comprehesive set is obtaied by: B i = A i R i = (a i1, a i2, %, a ii ) ri11, ri12, %, ri1p r i21, r i22, %, r i2p p %%%%%% r ii 1, r ii 2, %, r ii = (bi1, bi2, %, bip) (5) where p=10. Membership degrees are determied directly by the experts, or by membership fuctios. Here the compositio operator M(, +), which ca take ito accout the effects of all factors, but ca also cotai all the iformatio of a idividual factor, is expressed by i b ik = a ij r ijk (i = {1,2,%,4}; k = {1,2,%,10}) (6) j=1 The, the first-order fuzzy comprehesive evaluatio matrix ca be writte as B1 B 2 R = m = B where m=4 ad p=10. b11, b12, %, b1p b 21, b 22, %, b 2p b m1, b m2, %, b mp (7) Determiig the Weight Vector of Factors After obtaiig the first-order fuzzy comprehesive evaluatio matrix, we have to determie the weight vector of factors, which idicates the degrees of importace of factors to the evaluatio object. It ca be writte as A = (a 1, a 2, %, a m ) (8) Whe assumig that DS ad GS are more importat tha MM ad PA, we write Eq. (8) as A = (0.3, 0.3, 0.25, 0.15) Secod-Order Fuzzy Comprehesive Evaluatio Fially, we make the secod-order fuzzy comprehesive evaluatio. Fuzzy set B ca be calculated by B = A R = (a 1, a 2, %, a m ) = (b 1, b 2, %, b p ) where m=4 ad p= Determiig Parameter b11, b12, %, b1p b 21, b 22, %, b 2p b m1, b m2, %, b mp (9) Geerally, the weighted averagig method is used to obtai the accuracy of the evaluatio object. It is show as follows 10 = p=1 b p p b p (10) k=1 Accordig to the above steps, the machiability, which is very importat for tolerace allocatio, ca be determied. 3.2 Modellig of Tolerace Allocatio Machiiability ca be determied based o the FCE method. The assembly respose fuctio is assumed as

5 Optimal Tolerace Allocatio 465 tol asm = g(tol) = tol tol tol 2 + % + tol (11) where i = g/ (tol i ) (12) where, tol asm deotes the assembly fuctio requiremet i tolerace desig, g( ) is the assembly respose fuctio, tol i represets the tolerace value of the ith correspodig part, tol 0 is costat, i, which reflects the degree of importace of each desig tolerace o a assembly, is the assembly sesitivity coefficiet of the ith part. Accordig to the experts experiece, the larger i is, the smaller is the correspodig tolerace allocated. Therefore, a comprehesive factor for tolerace allocatio, i, ca be produced as follows: i = i = 2 i i g 2 (tol i ) (13) where i is the machiability of ith part obtaied by the FCE. Referecig to the reciprocal model, a model of the optimal tolerace allocatio ca be expressed as follows: Mi C = f(tol) = C 0 + Subject to: l i tol i u i, 1 i (14) l tol asm u where C deotes the total machiig costs, f(tol) is the fuctio relatioship of cost tolerace, C 0 is the costs costat, L = (l 1, l 2, %, l ) ad U = (u 1, u 2, %, u ) are costrait vectors for desig tolerace of parts i a assembly, l ad u represet the upper ad lower limit of the assembly requiremet. 3.3 Geetic Algorithm A geetic algorithm, which is a recetly developed heuristic optimisatio strategy, has bee used for global optimisatio i a variety of research fields. A GA is based o the mechaics of atural selectio ad atural geetics ad Darwiia survival of the fittest. Detailed discussio of the mechaisms of GA ca be foud i [25,26]. GA is differet from traditioal search methods ecoutered i egieerig optimisatio problems. GA works with a codig of the desig variables as opposed to the variables themselves cotiuity of parameter space is ot a requiremet. GA searches from a populatio of poits, ot a sigle poit parallel processig of poits reduces the chace of beig trapped ito a local optimum. GA uses probabilistic trasitio rules, ot determiistic trasitio rules, which leads to high-quality solutios, ad GA requires oly the objective fuctio values, these miimal requiremets result i a broad applicability of GA. These importat features of GA, such as the flexibility, global applicatio, parallelism, simplicity, versatility, good problem solvig capability, etc, make geetic algorithms very useful, ad therefore popular. Durig the last decade, geetic algorithms have had icreasig applicatios i a variety of i tol i fields with promisig results. Recetly, some work has bee successfully carried out usig geetic algorithms for optimal egieerig desig problems. The applicatio of GA for tolerace optimisatio allocatio is give i the followig sectios Represetatio Scheme I geetic algorithms, represetatio is a essetial issue because the represetatio scheme liks the real-world problem to the geetic algorithms ad the geetic algorithms directly maipulate the coded represetatio of the problem. There are may kids of represetatios, such as biary digit strig represetatio, floatig poit represetatio, permutatios of a list, etc. The selectio of a appropriate represetatio depeds o the characteristics of the search space. Because machiig precisio is kow i the tolerace allocatio problem, the tolerace variable is cosidered as a discrete oe i the feasible desig domai. We adopt the biary digit strig represetatio. If i is the machiig precisio of the tolerace desig variable tol i, the its strig legth i is determied by the followig iequity: i = log 2 tolu i tol l i i +1 (15) where tol u i ad toli l are upper ad lower bouds of the ith tolerace desig variable. After the strig legth of each tolerace desig variable has bee determied, the legth of the chromosome is computed from: A = i (16) where is the umber of idepedet tolerace desig variables Decodig If the ith biary digit substrig of a chromosome is decoded ito a usiged decimal iteger I i, the the physical value of the ith tolerace desig variable tol i is computed by: tol i = tol l i + I i i (17) Fitess Fuctio Fitess is a quality value, which is a measure of the reproductive efficiecy of livig creatures accordig to the priciple of survival of the fittest. I geetic algorithms, the fitess is used to allocate reproductive trials ad thus is some measure of goodess to be maximised. This meas that strigs with higher fitess values will have a higher probability of beig selected as parets. Tolerace allocatio is a costraied miimisatio problem. If the objective fuctio is also expressed by: Mi f(tol) (18) s.t. b i (tol) 0 (i = 1, 2, %, m) Where b i (tol) is the costraied fuctio, the it ca be trasformed to a ucostraied problem by the exterior pealty fuctio method.

6 466 S. Ji et al. Max F(tol) (19) where the fitess fuctio F is computed by the followig fuctio: F(tol) = G(tol) P(tol) (20) where G(tol) = K (g 0) 1 + (1.1) f K (g 0) 1 + (0.9) f (21) 1 P(tol) = (22) (1.1) (tol) m (tol) = b i (tol) (23) Where P(tol) deotes the pealty fuctio, G(tol) is used to describe the quality of the solutio, K ad are costat. I geeral, K=1, is computed by: 1 (f(tol) 0) = (24) 1 (f(tol) 0) Liear-Fitess Scalig Sice the strigs with higher fitess values have a higher probability of beig selected as the parets, it is importat to cofie the allocatio of selectio to the best strigs, especially i small populatio geetic algorithms. The fitess scalig ca regulate ad tue the umber of times selected to prevet the domiatio of extraordiary idividuals, ad therefore it ca prevet premature covergece. I additio, the fitess scalig ca ecourage a healthy competitio amog ear equals whe the populatio average fitess is close to the populatio best fitess. A liear scalig formula is adopted with a ormalisatio process as follows: F = 1 F max F (25) F max F mi where F is the scaled fitess, F is the raw fitess, F max ad F mi are maximum ad miimum fitess, respectively Selectio The selectio operator determies the set of idividuals, which remai at the ext geeratio. The roulette wheel selectio scheme is used herei. This scheme is implemeted as a liear search through a roulette wheel with each slice weighted i proportio to a scaled fitess value, the selectio operator is obtaied as follows: 1. Sum the fitess values of all the populatio members. The result is the total fitess F total. 2. Geerate a radom umber R betwee 0 ad 1, the multiply F total by R to geerate a idex umber N: N = R F total (26) 3. Retur the first populatio member whose fitess, added to the fitess of the processig populatio members, is greater tha or equal to N Crossover Crossover is a primary operator i geetic algorithms ad is the key to the success of geetic algorithms. We geerate ew idividuals I a ad I b from paret idividuals I a ad I b which are selected radomly from the populatio. They are divided ito subparts at multiple poits of crossover, ad the ew idividuals are obtaied by swappig them betwee these idividuals Mutatio The mutatio operator arbitrarily alters the gee value accordig to a predetermied probability. The mutatio probability should be carefully prescribed. If the mutatio probability is low, the the algorithm is ofte trapped at a local optimum. However, if the mutatio probability is high, the the propagatio of good schemata will be uduly hidered ad the algorithms will degeerate to a radom search method. We use a ew strategy to decide if the mutatio operator is ivoked. First, the umber N of the same idividual i a populatio is computed, the if N is greater tha or equal to N same, the the mutatio operator is ivoked, otherwise, the crossover operator is used. Here, N same is a ew cotrol parameter for the algorithm ad represets the permissible umber of the same idividuals i the populatio Memory Tool We use memory tool (MT) to remember the best idividual of a geeratio. Whe the best idividual of the ext geeratio is better tha the oe i MT, MT is updated usig the ew idividual. Fially, the solutio of GA is obtaied from MT Parameters For the tolerace allocatio problem, the parameters are selected as follows: Populatio size = 100 Geeratio = 500 Crossover probability = 0.95 N same = 30 The procedure of the modified geetic algorithm is described as follows: Start Program: a. Iitialise (some essetial parameters (GS, P c, N same, )); b. Iitialise (MT); c. Geerate (the first geeratio populatio radomly); d. Compute (their fitess); e. For i = 1toGSdo Begi If (the radom umber r is greater tha P c ) cotiuous; Else

7 Optimal Tolerace Allocatio 467 Table 4. Membership degrees of thickess betwee ed faces of left or right bearig. Factor Grades of each factor I II III IV u 1 Dimesio size u 2 Geometrical structure u 3 Material machiability u 3 Process accuracy Table 5. Membership degrees of thickess betwee ed faces of carriers. Fig. 3. The drivig device ad its dimesio loop chai. Factor Grades of each factor Calculate (the umber of the same idividual i populatio); If ( greater tha or equal to N same ) Do (mutatio operatio); Else Do (crossover operatio); Update (MT); Do (reproductio operatio to product ext geeratio); Ed f. Output (the idividual of memory tool). Ed (Program). Where GS deotes the termial geeratio umber, is the variable of coutig, ad P c represets the crossover probability. 4. Example I order to verify the proposed method, a drivig device (see Fig. 3) is take as a example. I Fig. 3, L 1 =160 mm, L 2 =L 4 =5 mm, L 3 =150 mm. Clearace L 0, which is the assembly fuctio requiremet, is eeded to assemble the product without iterferece. To solve the tolerace allocatio problem, Tables 3 5 list the membership degrees of the four correspodig dimesios o the assembly, accordig to the kow coditios. Usig the above method, the machiability vector of four correspodig part dimesios is obtaied as follows: = {0.66, 0.41, 0.76, 0.41} Table 3. Membership degrees of legth of the shaft shoulder. Factor Grades of each factor I II III IV u 1 Dimesio size u 2 Geometrical structure u 3 Material machiability u 3 Process accuracy I II III IV u 1 Dimesio size u 2 Geometrical structure u 3 Material machiability u 3 Process accuracy where, part 1 is the shaft shoulder; parts 2 ad 4 are the left or right bearig, respectively; part 3 is the carrier. The result shows that the carrier is the most difficult to machie, ad the left ad right bearigs are the easiest. This coicides with the practical machiig case, so the approach is proved to be correct. It is assumed that the two clearaces are combied, the whole clearace is from 0.10 mm to 0.40 mm ad its tolerace is 0.03 mm. The fuctio equatio is writte as: tol asm = g(tol) = tol 1 tol 2 tol 3 tol 4 Usig Eq. (12), the vector of the fuctio sesitivity factors is solved as follows: = {1.0, 1.0, 1.0, 1.0} A comprehesive factor vector of tolerace allocatio is obtaied usig Eq. (13): = {0.66, 0.41, 0.74, 0.41} Therefore, a mathematical model for the tolerace allocatio is established usig Eq. (14): Mi C = C tol 1 tol 2 tol 3 tol 4 Subject to: 0.0 tol i 0.30, 1 i tol asm 0.30 The tolerace vector computed by the improved GA is: tol = {0.8, 0.06, 0.10, 0.06} Accordig to the priciple that determies the bias of parts, the tolerace desig result is: L 1 = , L 2 = , L 3 = , L 4 =

8 468 S. Ji et al. 5. Summary This paper presets a ew approach based o fuzzy comprehesive evaluatio ad GA for tolerace allocatio. I the tolerace allocatio, the machiability, which depeds o the fuzzy comprehesive evaluatio ad the fuctio sesitivity factor, is cosidered, so the ideas of DFA ad DFM are ivolved. The approach ot oly esures the correctess of tolerace desig, but also saves the machiig costs. I additio, the improved GA is used to optimise the model based o DFA ad DFM. The result of a detailed example shows that the method preseted ca obtai tolerace allocatios with ecoomically attaiable accuracy. Refereces 1. G. Yaxi, Basic kowledge of iterchageability ad measure techiques, Harbi Istitute of Techology, Sog Shaomig, The tolerace aalysis ad sythesis i CAD, PhD dissertatio of HIT, September A. Ballu ad L. Matheu, Aalysis of dimesioal ad geometrical specificatios: stadards ad models, Proceedigs of 3rd CIRP Semiars o Computer Aided Toleracig, pp , Z. Gebao, The survey of computer aided toleracig, Chia Mechaics Egieerig, 7(5), pp , F. Hogfag ad W. Zhaotog, A ovel mathematical model for tolerace desig, Mechaical Egieerig ad Automatio, 19(4), pp , L. Yusheg, W. Zhaotog ad Y. Jiagxi, A research of computer aided tolerace optimal desig based o costs model, Zhejiag Uiversity Joural, 29(6), pp , J. Lee ad G. E. Johso, Optimal tolerace allotmet usig a geetic algorithm ad trucated Mote Carlo simulatio, Computer-Aided Desig, 25(9), pp , L. Chupu, Statistical tolerace ad mechaical precisio, Mechaical Egieerig, E. Dupiet, M. Balaziski ad E. Caogala, Tolerace allocatio based o fuzzy logic ad simulated aealig, Joural of Itelliget Maufacturig, 7, pp , S. Qi ad Z. Quer, Aalysis of tolerace distributio method ad machiig costs, Aeroautical Machiig Techology, 2, pp , F. H. Speckhart, Calculatio of tolerace based o a miimum costs approach, Joural of Egieerig for Idustry, 5, pp , M. F. Spotts, Allocatio of tolerace to miimize costs of assembly, Joural of Egieerig for Idustry, pp , August G. H. Sutherlad ad B. Roth, Mechaism desig: accoutig for machiig toleraces ad costs i fuctio geeratig problems, Joural of Egieerig for Idustry, 2, pp , W. Michael ad J. N. Siddall, Optimizatio problem with tolerace assigmet ad full acceptace, ASME Joural of Mechaical Desig, 103, pp , K. W. Chase ad W. H. Greedwood, Desig issues i mechaical tolerace aalysis, Machiig Review, 1(1), pp , Z. Dog ad A. Soom, Automatic optimal tolerace desig for related dimesio chais, ASME Machiig Review, 3(4), pp , December W. J. Lee ad T. C. Woo, Optimum selectio of discrete toleraces, Joural of Mechaisms, Trasmissios, ad Automatio i Desig, 111, pp , Z. Dog ad W. Hu, Optimal process sequece idetificatio ad optimal process tolerace assigmet i computer-aided process plaig, Computers i Idustry, 17, pp , Z. Dog, W. Hu ad D. Xue, New productio costs tolerace models for tolerace sythesis, Joural of Egieerig for Idustry, 116, pp , Y. Jiagxi, G. Daqiag ad W. Zhaotog, Tolerace costs model base o eural etwork, Chia Mechaical Egieerig, 7(6), pp , H. E. Trucks ad H. B. Smith, Desigig for ecoomical productio, Society of Machiig Egieers, Dearbor, Michiga, P. Kopardekar ad S. Aad, Tolerace allocatio usig eural etworks, Iteratioal Joural of Advaced Machiig Techology, 10, pp , W. Caihua ad S. Liatia, The methodology of fuzziess, Chia Architecture Egieerig, C. Qighog, Fuzzy comprehesive evaluatio of desig variables i mechaical reliability, Fuzzy Techiques ad Applicatio(I), pp , Beijig Aerospace Uiversity, M. Pirlot, Geeral local search methods, Europea Joural of Operatioal Research, 92, pp , Z. Michalewica, Geetic Algorithms + Data Structures = Evolutio Programs, Spriger-Verlag, 1998.