A fuzzy genetic algorithm for the discovery of process parameter settings using knowledge representation

Expert Systems with Applications 36 (2009) 7964–7974

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

A fuzzy genetic algorithm for the discovery of process parameter settings using knowledge representation H.C.W. Lau, C.X.H. Tang, G.T.S. Ho *, T.M. Chan Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

a r t i c l e

i n f o

Keywords: Evolutionary computing Genetic algorithms Fuzzy set Reactive ion etching Inverted beta loss function Knowledge representation

a b s t r a c t In this paper, we propose a fuzzy genetic algorithm (Fuzzy-GA) approach integrating fuzzy rule sets and their membership function sets, in a chromosome. The proposed approach consists of two processes: knowledge representation and knowledge assimilation. The knowledge of process parameter setting is encoded as a string with a fuzzy rule set and the associated membership functions. The historical process data forming a combined string is used as the initial knowledge population, which is then ready for knowledge assimilation. A genetic algorithm is used to generate an optimal or nearly optimal fuzzy set and membership functions for the process parameters. The originality of this research is that the proposed system is equipped with the ability to take advantage of assessing the loss which is caused by discrepancy with a process target, thereby enabling the identification of the best set of process parameters. The approach is demonstrated by the use of an experimental example drawn from a semiconductor manufacturer and the results show us that the suggested approach is able to achieve an optimal solution for a process parameter setting problem. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction In both industrialized countries and newly industrialized countries, manufacturing firms are facing significant change resulting from mass customization and shortening product life cycles (Prajogo, Laosirihongthong, Sohal, & Boon-itt, 2007). There are various intelligent and information systems to explore the artificial intelligence (AI) techniques in optimizing the processes with better finished product quality or service performance (Huang, Trappey, & Yao, 2006; Tsai, 2006; Tsaih & Lin, 2006). Hybrid architecture for intelligent systems has become a new field of AI research, operating in concert with the development of the next generation of intelligent systems. Current research in this field concentrates mainly on the marriage of genetic algorithms (GA) and fuzzy logic (Caputo, Fratocchi, & Pelagagge, 2006; Feng & Huang, 2005). Exploring the similarities of the essential structures of these two knowledge manipulation methods is where intelligent hybrid systems can possibly play an important role. However, such hybrid systems have not shown great significance in the manufacturing industry. To extend the application of specifically the above hybrid approach, a knowledge framework integrating fuzzy rule sets and their associated membership function sets in a GA chromosome has been proposed, assisted with the mathematical evaluation in terms of the chromosome fitness. This * Corresponding author. E-mail address: [email protected] (G.T.S. Ho). 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.10.088

innovative research approach is intended to accentuate integrating the GA and fuzzy logic so that the related crisp and fuzzy values could coexist in simply one chromosome, while the unavoidable performance trade-off phenomenon occurring in the manufacturing process is to be also taken into consideration. The proposed approach consists of two processes: knowledge representation and knowledge assimilation. In the stage of knowledge representation, the expertise of process parameter setting is encoded as a string with fuzzy rule sets and the associated fuzzy membership function. The historical process data are also included into the strings mentioned above, contributing to the formulation of an initial knowledge population. Then in knowledge assimilation, GA is used to generate an optimal or nearly optimal fuzzy set and membership functions for the entitled process parameters. The approach is demonstrated by the employment of an experimental example, the reactive ion etching (RIE) process, drawn from a semiconductor manufacturer. Two objectives have been set in this paper. Firstly, a comprehensive discussion of scientific control of parameter-based operations with investigation into previously used methods has been conducted. Ideally such an argument can be useful to the improvement of any process with various parameters, since many researchers may not fully realize the importance of the research on process parameters or of performance trade-off issues. Secondly, a set of methodologies by means of numerical analysis for determining the optimal process parameters has been provided. This can be thought of as a guide to achieve the purpose of manufacturing optimization in a

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Nomenclature Pp Dr P D A B yj y0j wj

total number of process parameters total number of defects index set of process parameters, P = {1,2,. . ., Pp} index set of defects, D = {1,2,. . ., Dr} index set of membership functions of process parameters, A = {1,2,. . ., 6Pp} index set of membership functions of defects, B = {1,2,. . ., 6Dr} parametrical value of the generated rules represented in chromosomes parametrical value of the test objects the weight of the jth parameter

comprehensive way, and furthermore, as an inspiration showing how appropriate mathematical tools can be adopted to conquer various manufacturing problems. This paper is organized as follows: Section 2 presents a literature review of GA, fuzzy logic, and related applications in the manufacturing domain. The formal description of the problem and model about parameter settings are presented in Section 3. Section 4 provides the details of GA and the performance trade-off function. Section 5 provides a case example for the proposed FuzzyGA approach. In Sections 6, the numerical analysis and conclusions are addressed. 2. Literature reviews Much work has been done in machine learning for classification; the ultimate goal is to attain a more accurate prediction. Artificial intelligence has been widely used in knowledge discovery by considering both cognitive and psychological factors. In some research development, GA, one of the search algorithms based on the mechanics of natural selection and natural genetics (Gen & Cheng, 2000; Holland, 1992), has been regarded as a genetic optimization technique for global optimization, constrained optimization, combinatorial optimizations and multi-objective optimization. Recently, genetic algorithm has been used to enhance industrial engineering for the achievement of high throughput and high quality (Al-Kuzee et al., 2004; Li, Su, & Chiang, 2003; Santos, Spim, Ierardi, & Garcia, 2002). Milfelner, Kopac, Cus, and Zuperl (2005) has deployed a genetic equation in order to optimize the analyzed knowledge of an industrial process and to help to reduce the production costs and manufacturing time as well as to enhance the flexibility of selecting machining parameters. However, the lack of concern about the trade-off of the lifetime of the machine and the quality of the product has shown that GA is not comprehensive enough to describe the whole manufacturing process. Recently it has been an appealing research direction to build up a hybrid artificial intelligence (AI) framework, such as the GA-Fuzzy logic, to better utilize the input-output data, which seems to offer an attractive possibility of solving the above problems (Chen, Taniguchi, & Toyota, 2003; Huang, Yang, & Huang, 1997). Fuzzy logic is characterized as an extension of a binary crisp operation. Each fuzzy rule has an antecedent part which incorporates several preconditions, and a consequent part prescribing the corresponding value (Kandel & Langholz, 1992). The main parameters and the associated terms which represent the linguistic variables can be identified by means of fuzzy sets. For example, a set including linguistic values such as (low, medium, high) may be used. Therefore fuzzy rule sets, together with the associated membership functions, have been proven of great potential in their integration into GA to formulate a compound knowledge processing

n cpiv cdix wpiv wdix lP p uPp lDr uDr

the total number of test objects selected for comparison  center abscissa of the membership function F piv for process parameter  center abscissa of the membership function F dix for defect 

half the spread of the membership function F piv for process parameter  half the spread of the membership function F dix for defect lower bound of process parameter upper bound of process parameter lower bound of defect rate upper bound of defect rate

system (Leung, Lau, & Kwong, 2003; Wang, Hong, & Tseng, 1998). On the other hand, trade-off on parameter settings is also a crucial aspect in the manufacturing process. To achieve this goal, Leung and Spiring (2002) have introduced the concept of the inverted beta loss function (IBLF), which is a further deduction of the Taguchi loss function (Taguchi, 1986) in the industrial domain, helping to balance the possible loss resulting from the use of different process parameter combinations along the entire process. However, another substantial issue, prediction of the parameter settings, has not been strongly emphasized by scholars. In the paper, reactive ion etching (RIE), which is used to demonstrate one of the process parameter setting methods, is a combination of chemical and physical dry etching on which ions are blasted at a wafer’s surface through the acceleration of reactive ions in the direction of the surface to be etched (Fatikow & Rembold, 1997). As the RIE process is highly nonlinear and multivariable, artificial intelligence techniques are being adopted to monitor the quality of the etching operation (Kim & Park, 2002). In summary, a review of contemporary publications signifies that whilst some research studies have been conducted on the use of multiple approaches to optimize RIE systems, the researches on performance trade-off and parametric settings for RIE processes have yet to be considered as adequate. This issue is thus addressed in this paper with the introduction of a performance trade-off function to be fully illustrated in the following sections. 3. Novel model for process parameter setting Engineers need to set the appropriate process parameters for manufacturing processes. The process parameters may affect the quality of the finished products and the life time of the equipment. Apart from satisfying the customer’s requirements, the production engineers need to pay attention to the efficient functioning of the manufacturing process. To ensure that equipment has a long lifetime and to prevent the breakdown of the machines, which may ultimately affect the total production output, process engineers need to set appropriate process parameters. Preventive maintenance including detecting deviation from normal operation can help to alert process engineers to the possible occurrence of breakdowns. In addition, workers can record equipment deterioration so they know when to replace or repair worn parts before they cause system failure. Recent technological advances in tools and systems for inspection and diagnosis have enabled more accurate and effective equipment maintenance (Siddique, Yadava, & Singh, 2003). The manufacturing process consists of a number of parameters. In practice, production engineers will make their decisions based on customer specifications such as shallowness specifications and cavity specifications including etch depth, to produce the goods. From the precedence relationship between these parameters, a set of

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rules in form of ‘‘if. . .then” are formulated by experts in order to have a good set of process parameters. An example of a manufacturing process is reactive ion etching which consists of seven main process parameters including: pressure, RF power, CH3, O2, cycle time, air groove and wall angle. Different sets of process parameters will generate products of varying quality described by their defect rates. The proposed model is to encode customer requirements, process parameters, and defect rate (in the form of a fuzzy rule set) into the chromosome of a genetic algorithm. The chromosome consists of five parts which are (1) customer requirement riu, (2) process parameter piv, (3) defect rate dix, (4) fuzzy membership function of process parameter kiy and (5) fuzzy membership function of defect rate qiz shown in Definition 3. The offspring fuzzy rules set, with

piv ¼ randomblPp ; uPp c; dix ¼ randomblDr ; uDr c; ki;s ¼ cpiv ; ki;k ¼ wpiv ; qi;s ¼ cdix ; qi;k ¼ wdix ;

8i 2 C h ; 8v 2 P; 8x 2 D; 8y 2 A; 8z 2 B; s ¼ 1; 3; 5; ::::::; k ¼ 2; 4; 6; ::::::; m ¼ M; b ¼ P p ; s ¼ Dr ; e ¼ 6Pp ; n ¼ 6Dr . An example of Gmw is shown as follows:

   3 3  1 0 2 3  5 5 . . . 12:5  5 0:5 ::: 0:5 6 2 2 . . . 1  1 2 . . . 1  7 6 . . . 12:5  6 0:5 . . . 0:5 7    7 6 ¼6 .  .. .. . . ..  .. ..  .. .. . . .. .. .. 7 7 6 .. .. . . . 4. . . .. . . . . . .. . . . . 5    1 0 . . . 2  1 0 . . . 2  6 6 . . . 12:5  7 0:5 . . . 0:5 2

Gmw

w ¼ b þ s þ e þ n;

2 0

3

their related membership function, evolves until an optimal or nearly optimal set of fuzzy rules is found. The fitness of the chromosome is evaluated according to the accuracy of the fuzzy rule set and the performance trade-off function. 4. Fuzzy knowledge representation in a genetic algorithm Fuzzy-GA is proposed for capturing domain knowledge from an enormous amount of data. The proposed approach is to represent the knowledge with a fuzzy rule set and encode those rules together with the associated membership into a chromosome. A population of chromosomes comes from the past historical data and an individual chromosome represents the fuzzy rule and the related problem. A binary tournament, using roulette wheel selection, is used for picking out the best chromosome between two when a pair of chromosomes is drawn. The fitness value of each individual is calculated using the fitness function by considering the accuracy and the trade-off of the resulting process parameter setting, where the fitter one will remain in the population pool for further mating. After crossover and mutation, the offspring will be evaluated by the fitness function and the optimized solution will then be obtained. 4.1. Notations and definitions The notations and definitions of the Fuzzy-GA are shown as follows: Definition 1. C h ¼ f1; 2; :::; Mg represents the index set of chromosomes where M is the total number of chromosomes in the population. Definition 2. Gmw represents a gene matrix generated for the population where

2

p11

p12

pm1

p22 .. . pm2

6p 6 21 Gmw ¼ 6 6 .. 4 .

 p1b  d11  . . . p2b  d21  .. ..  .. . .  . . . . pmb  dm1 ...

d12 d22 .. . dm2

 d1s  k11 . . . d2s  k21  .. ..  .. . .  . . . . dms  km1 ...

k12 k22 .. . km2

Note that the decoding method of an element in the first submatrix ðpiv Þmb or second sub-matrix ðdix Þms of Gmw to a linguistic variable is given by (i) 0: ignore, (ii) 1: low, (iii) 2: medium, and (iv) 3: high. For any row of the third sub-matrix ðkiy Þme of Gmw , a group of kið6q3Þ; kið6q2Þ; kið6q1Þ; kið6qÞ six consecutive values kið6q5Þ; kið6q4Þ;  in the matrix forms a single set F piv ¼ cpiv  wpiv ; wpiv ; cpiv ; wpiv ; cpiv þ wpiv ; wpiv g for process parameter pv where q ¼ 1; 2; 3; ::::::. Also, for any row of the fourth sub-matrix ðqiz Þmn of Gmw , a group of six consecutive values qið6q5Þ; qið6q4Þ; qið6q3Þ; qið6q2Þ; qið6q1Þ; qið6qÞ in the matrix forms a single set F dix ¼ cdix  wdix ; wdix ; cdix ; wdix ; cdix þ wdix ; wdix g for defect rate dx where q ¼ 1; 2; 3; ::::::. For both two cases, there are totally six genes in the sets of membership functions shown in Fig. 1. The membership functions shown in Fig. 1 have the following properties:  F piv consists of aggregated membership functions which relate to a fuzzy rule set is assumed to be isosceles-triangle functions. cpiv is the center abscissa of F piv .  wpiv represents half the spread of F piv . In ‘‘cpiv ”, ‘‘piv” indicates that the 5th feature test is included, while i specifies the order of all the condition levels of each feature test. For instance, cpi1 stands for the center abscissa of the 1st process test, within the whole membership function matrix. The range of a certain process parameter in different conditions  is denoted by wpiv of F piv i.e., (cpiv ; wpiv ) shown in Fig. 2. For a clearer explanation: The Pressure is within the range of 5–15 m Torr,

cpi1 : ð5 þ 15Þ=2 ¼ 10; wpi1 : 10  5 ¼ 5 or 15  10 ¼ 5: ‘‘Pressure is medium” (cpi1 ; wpi1 ) could be represented as (10, 5) in the format of cpi1 and wpi1 . Therefore a sample illustration is represented in Fig. 3.

 k1e  q11 . . . k2e  q21 .. ..  .. . .  . . . . kme  qm1 ...

q12 q22 .. . qm2

...

q1n

3

. . . q2n 7 7 7 .. .. 7 ¼ ð ðpiv Þmb . . 5 . . . qmn

ðdix Þms

ðkiy Þme

ðqiz Þmn Þ

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Low

Medium

w ¼ b þ s þ e þ n;

High

µ

p0iv ¼ randomblPp ; uPp c; 0

dix ¼ randomblDr ; uDr c;

c piv + wpiv

0

0

ki;s ¼ cpiv ; ki;k ¼ wpiv ;

c piv − wpiv

w piv c piv wpiv

q0i;s ¼ cdix ; q0i;k ¼ wdix ;

wpiv

8i 2 C h c ; 8v 2 P; 8x 2 D; 8y 2 A; 8z 2 B; s ¼ 1; 3; 5; ::::::; k ¼ 2; 4; 6; ::::::;m = S, b = Pp, s = Dr, e = 6Pp, n = 6Dr

Fig. 1. Membership functions of process parameter.

Low

Medium

Definition 6. mask1w indicates a mask matrix generated for uniform crossover where

High

µ (Pressure)

mask1w ¼ ½mp1 :::mpb jmd1 :::mds jmk1 :::mke jmq1 :::mqn  ¼ ð ðmpv Þ1b ðmdx Þ1s ðmky Þ1e ðmqz Þ1n Þ

5

c piv − wpiv

10

wpiv c p

cpi + w p v

15

iv

mpv ¼ randomf0; 1g; mdx ¼ randomf0; 1g; mky ¼ randomf0; 1g; iv

Pressure (mTorr)

wpiv

mqz ¼ randomf0; 1g

8v 2 P; 8x 2 D; 8y 2 A; 8z 2 B; b ¼ Pp ; s ¼ Dr ; e ¼ 6Pp ; n ¼ 6Dr :

wpiv

Fig. 2. Membership functions of pressure.

4.2. Initial population The proposed Fuzzy-GA method is used to optimize the fuzzy rule set (i.e. If piv , then dix ) where the associated fuzzy membership

Definition 3. Bm1 denotes a random number matrix generated for selection and crossover where

2

b1

6b 6 2 Bm1 ¼ 6 6 .. 4 .



7 7 7 ¼ ðbi Þ ; m1 7 5

bm bi ¼ random½0; 1; 8i 2 C h ; m ¼ M: Definition 4. C h c ¼ f1; 2; :::::; Sg denotes the index set of the chosen chromosomes in the crossover where S is the total number of chosen chromosomes.

4.3. Uniform crossover

Definition 5. G0mw indicates the gene matrix in which the Q chromosomes chosen in crossover are stored where

2 G0mw

p011

6 0 6 p21 6 ¼6 . 6 .. 4 p0m1

p012 p022 .. . p0m2

 p01b  d011  0  0 . . . p2b  d21  .. ..  .. . .  . . . . p0mb  d0m1 ...

0

d12 0

d22 .. . 0 dm2

0  0 d1s  k11 0 0 . . . d2s  k21  .. ..  .. . .  . 0  0 . . . dms km1

0

k12

...

0

k22 .. . 0 km2

12.5

110

48.7

13.3

p11

p12

p13

p14

Fig. 4 depicts the operation of uniform crossover. Its implementation steps are shown as follows.

0  k1e  q011 0 . . . k2e  q021  .. ..  .. . .  . 0  . . . kme q0m1

...

…



are F piv and F dix . The population in genetic algorithms was initialized by using two approaches together, (a) 50% of total number of chromosomes was randomly generated, and (b) 50% of total number of chromosomes was acquired from the rules based on past experience of engineers. In our approach, the fuzzy rule sets are mined from corporation databases, sisters companies, machine vendors, customer specifications or information from semiconductor associations. If the initial number of knowledge sources is not sufficient, some trial data will be encoded and some dummy initial rule sets will be formulated for evolution.

3

3

q012

...

q01n

q022 .. .

... .. .

q02n .. .

q0m2

. . . q0mn

7 7 7 7 ¼ ð ðp0iv Þmb 7 5

5

5

10

5

15

5

k11

k12

k13

k14

k15

k16 ~

Pressure: 6 kiy represent 1

F ri

Fig. 3. Representation of a process parameter and the associated membership.

…

0

ðdix Þms

0

ðkiy Þme

ðq0iz Þmn Þ

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Fig. 4. Uniform crossover.

Step 1 : Generate mask1w Step 2a: If mpv ¼ 1, exchange p0ð2i1Þv and p0ð2iÞv , i ¼ 1; 2; :::; 2S ; v 2 P where i is the index of a chromosome pair. 0 0 Step 2b: If mdx ¼ 1, exchange dð2i1Þx and dð2iÞx , i ¼ 1; 2; :::; 2S ; x 2 D where i is the index of a chromosome pair. 0 0 Step 2c: If mky ¼ 1; exchange kð2i1Þy and kð2iÞy , i ¼ 1; 2; :::; 2S ; y 2 A where i is the index of a chromosome pair. Step 2d: If mqz ¼ 1; exchange q0ð2i1Þz and q0ð2iÞz ; i ¼ 1; 2; :::; 2S ; z 2 B where i is the index of a chromosome pair. For example, in Fig. 4, as the values of mp1 ; mp3 ; md1 ; mk1 ; mk2 ; mq2 ; mq4 ; mq6 are 1, genes of p1 ; p3 ; d1 ; k1 ; k2 ; q2 ; q4 ; q6 in the parent 1 and 2 are exchanged.It is found that the membership function is not in ascending order. The new offspring should be modified by exchanging k1 and k3 since    F piv ¼ cpiv  wpiv ; wpiv ; cpiv ; wpiv ; cpiv þ wpiv ; wpiv :

ð1Þ

Therefore, k2 should be modified by subtracting k1 from k3 and k4 should be modified by subtracting k3 from k5 . The modification procedure of chromosomes is shown in Fig. 5.

Accurancy ¼

4.4. Fitness and selection To have a good set of process parameters, the genetic algorithm selects the best chromosome for mating according to the fitness function suggested below. The fitness function is to optimize the accuracy of the fuzzy rules and the performance trade-off function. An evaluation function is used to evaluate the derived fuzzy rule set during the evolution process. Two important factors are used in evaluating derived fuzzy rule sets with their associated membership functions: the accuracy and the complexity of the resulting knowledge structure.

Fitness funtion ¼

accurancy with error rate performance trade-off

ð2Þ

4.4.1. Accuracy with error rate Accuracy of a fuzzy rule is evaluated using test objects as follows with acceptable error rate (Wang et al., 1998).

total number of objects correctly matched by rule set within error range total number of objects

Fig. 5. Modification after crossover.

ð3Þ

H.C.W. Lau et al. / Expert Systems with Applications 36 (2009) 7964–7974

where error rate ðeÞ ¼

m X j¼1

wj

ðyj  y0j Þ2 2n

:

ð4Þ

Based on the finding of Wang et al. (1998), the more data imported the more accurate the evaluation is. In the newly derived accuracy function, the accuracy greatly depends on the defined error rate. If the tolerance of error range is slack, more objects will be matched by the rule set while the accuracy may be reduced. 4.4.2. Performance trade-off In statistics, decision theory and economics, a loss function maps an event onto a real number so as to represent the economic cost associated with the event (Taguchi, 1986). The more data used, the more objective and accurate the evaluation is while the loss index of the resulting rule set is based on Taguchi’s method.

k¼

L ; D2

ð5Þ

L ¼ kðy  mÞ2 :

ð6Þ

Referring to Fig. 6, L1 is the quality loss in dollars when the parameter is equal to Y1, Y is the value of parameter setting (i.e. temperature, pressure etc). Meanwhile L is the maximum loss possibly occurred in the production process, m is the target value and k the coefficient value. In actual industrial situations, a symmetric loss off the target may not always be true. Thus the Taguchi loss function is further developed by the inverted beta loss function (Leung & Spiring, 2002) to better describe the following circumstances. Therefore a family of symmetric, asymmetric and half-bell loss functions could be plotted based on an inverted beta pdf. The shape of the IBLF which can be modified to fit the practitioner’s needs is a strong mapping method for fitting/selecting an appropriate IBLF to the required industrial process curve for prediction. In brief, the loss function associated with inverting the beta pdf is referred to as the IBLF is represented as the following:

  pðx; TÞ Lðx; TÞ ¼ K 1  m

ð7Þ

where K is the maximum loss, T is the ideal target of the process, p(x, T) denotes a function of a beta probability density function, m denotes the supremum of p(x, T). When x = T, the supremum value could be achieved.Substitute pðx; TÞ ¼ Bða1;bÞ xa1 ð1  xÞb1 and m ¼ sup pðx; TÞ ¼ Bða1;bÞ T a1 ð1  TÞb1 into (7)

" #) xa1 1  x b1 : Lðx; TÞ ¼ K 1  T 1T

Although the IBLF could, to a certain extent, describe the loss occurring in the process, it cannot predict the loss associated with the deterioration of the machine. On the other hand, the monetary loss calculated by either Taguchi loss function or IBLFs is not a generic equation for describing loss of the daily deterioration of the machine. Besides, it may be too troublesome to quantify the monetary loss associated with deviation from desired target values because there may be various kinds of losses to be calculated, for example, machine depreciation, cost of raw materials, labor cost, risk cost. After conducting a wide variety of experiments and relevant studies, it is found that with modification of IBLF, a revised function could fit the real industrial data in a better way. Performance trade-off function (PToF) and average performance loss index (PLI) derived from the application of IBLF are proposed for industrial processes associated with the parameter-based operations. The new concept of PToF shown in (9) is introduced to describe the challenges occurred in the manufacturing process shown in Fig. 7.

  x c 1  x c  li ¼ 1 þ K 1  T 1T Average PLI ¼

n X

li  xi

ð9Þ

ð10Þ

i¼1 n X

xi ¼ 1

ð11Þ

i¼1

where T is ideal target of the process, K is the constant representing industrial process, which is equal to 0.5, x is the historical data of a process parameter (e.g. pressure), c is the machine specific loss constant, which differentiates the loss generated by various machines produced by different manufacturers, xi is weighting assigned for ith process parameter. It is noticed that contradictions occur in various kinds of manufacturing processes in terms of actions, methods, approaches, machine and materials. For instance, in the RIE process, even though the pressure 20 m Torr could reach the etching goal and make the product with acceptable quality, the high pressure would be a danger signal to some machines involved, which may finally cause serious machine depreciation or even breakdown over time. As a result, this strategy of ‘‘peak” parameter setting is not recommended for obvious reasons. However, if a comparatively low pressure, within 3–7 m Torr, is pre-set, say 5 m Torr for example, the

(

Fig. 6. Taguchi loss function.

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ð8Þ

Fig. 7. PToF demonstration.

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product will still comply with the quality standards, and the possibility of machine depreciation is now greatly reduced. As a result, to better utilize the loss function and overcome the above shortage, the concept of PToF is initiated by converting the monetary loss into some simple data representing the degree of loss. To achieve the particular objective at the least cost, the Taguchi loss function is introduced, in terms of the trade-off between production loss and product satisfaction that occurs during the manufacturing process. In addition, the manufacturers are interested in determining their exposure to risk during normal operations. In addition, average PLI can act as an indicator to show how the settings of the parameters can affect the depreciation rate of the machine and affect the likelihood to cause machine breakdown. Average PLI can enable engineers to detect the loss associated with the machine breakdown in advance and further analyze the economic possibilities of utilizing the machine for production.

The evaluation of every chromosome, to find out the fitness value used by the optimum fuzzy rule set and associate membership, can be calculated with (1). This consists of two major parts which are accuracy and the performance trade-off function. The objective function is to maximize the accuracy and minimize the performance trade-off. 5.1. Testing instance based on accuracy Suppose there is only one sample rule in the chromosome1 and the output of E11 and E3 are both 7.5 based on expertise advice. Fig. 8 shows the membership function of RIE with MathLab. The ideal case (#) for E11 (y0 ) = 7.5 and E3 is 7.5, n = 4, weight of E11 = 0.6 and E13 = 0.4Error rate is calculated based on the data of Table 2.

Error rateðeÞ ¼

j¼1

5. Case examples of knowledge assimilation

wj

ðyj y0j Þ2 2n

¼ 0:6ð0

2

þ02 þ1:282 þ1:322 Þþ0:4ð02 þ02 þ1:282 þ1:322 Þ 24

¼ 0:4226

For the determination of the relationship between defect rate and process parameter in reactive ion etching, the knowledge is extracted and represented in the form of a rule. Customer requirements for the production of silicon chips are shown in Table 1. Based on the knowledge extracted from a corporate database, a sample fuzzy rule is formulated as follows:If the pressure is high And RF power is medium And the usage of CHF3 is medium And the usage of O2 is high, THEN the vacuum defect E11 will be low and the vacuum defect E3 will be high.The above fuzzy rule in verbal format is encoded as the following real number chromosome. if-part of rule in Chromosome1

m P

Membership function in Chromosome1

To test the accuracy of the offspring, rule viewer shown in Fig. 9 is used to check the offspring suitable for evolve or not. The 1st offspring shown in Table 3 is perfectly matched since the chromosome has 7.5 and 7.5 for defect rate of E11 and E3 which is matched with the output generated by rule viewer of MathLab. The 2nd offspring shown in Table 3 is not matched since the output generated by rule viewer of MathLab of E11 is 6.03 and E3 is 8.97, respectively, but the offspring value is 10 and 13 which is out of 0.4226 error rate. Similarly, the 3rd and 4th shown in Table 3 offspring are inserted within 0.4226 error rate for all consequence parts of the fuzzy rule. As 3 out of 4 have been matched within the error rate, the accuracy with error rate is 0.75. 5.2. Performance trade-off for machine lifetime In the RIE process, RF power and pressure both play an important role in controlling the quality of the final product. Referring to Table 4, 100 Watts is the ideal target for RF power, while for pressure the optimal value is 10 m Torr. When RFP is 110 Watts, the loss index of RFP is 1.08, extracted from the historical data. Since xP1 RF power (50, 150) is far away from (0, 1), the equation X 0 ¼ P2P1 to transform the X0 within the range between 0 and 1 is employed.

then-part of rule in Chromosome1

Table 1 Fuzzy term of RIE process parameter. RIEC customer requirement

Etch depth (D) Shallow specification (W)

12550 nm 1450 nm

RIE process parameter

Pressure (PR)

5–15 m Torr Ignore (0), low high (3) 50–150 Watts Ignore (0), low high (3) 1–60 sccm Ignore (0), low high (3) 1–60 sccm Ignore (0), low high (3) 19–26 Ignore (0), low high (3) 1–1.5 h Ignore (0), low high (3) 35–60 nm Ignore (0), low high (3)

RF power (RFP)

CHF3

O2

Wall angle(WA)

Cycle time(CT)

Airgroove roughness (AR) Defect

Vacuum defect E11 Vacuum defect E3

(1), medium (2),

(1), medium (2),

(1), medium (2),

(1), medium (2),

(1), medium (2),

(1), medium (2),

(1), medium (2),

5–10% (0) low (1) medium (2) high 5–10% (0) low (1) medium (2) high

5.2.1. RF power According to Fig. 9, with T = 100 Watts, K = 0.5, x1 = 110 Watts, = 0.5, K = 0.5, x01 = 0.6, l1(x1, T) = 1.075. l1(x1, T) = 1.075, thus T0 = 10050 100 To check the fitness of the mapping function, a set of seven samples is randomly picked out from a pool of 100 items of historical data and is examined by the well-known chi-square. It is proved that the data appears to follow the proposed PToF. (Verified by the chi-square goodness-of-fit test (p-value = 0.2235)). By adjusting c, the RFP–PToF curve could be made to represent the idea case so that the formula finds out a more precise result. Fig. 10 shows RIE sample data with c1 = 4. In addition, it should be noted that further prediction based on the generic formula could also be used to fully describe the trade-off between loss and quality during the manufacturing processes. 5.2.2. Pressure Supposed 10 m Torr is the target for pressure. When pressure is 12.5 m Torr, the loss index of pressure is 1.08, extracted from the historical data.

X0 ¼

x  P1 x5 x5 ¼ ¼ ; P2  P1 15  5 10

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Fig. 8. Membership function of RIE.

Table 2 Extracted past record to find out the error range. Input

#

Output Pressure (PR)

RF power (RFP)

CHF3

O2

CT

AR

WA

E11

E3

10 10 10.1 10.4

78 70.1 130 116

20 38.4 41.9 30.5

35 53.4 48.1 33.1

1.25 1.25 1.25 1.13

47.5 43.4 43.4 46

22.5 21.5 21.5 22.5

7.5 7.5 6.22 6.18

7.5 7.5 8.78 8.82

Fig. 9. Testing the instance with the fuzzy rule.

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Table 3 New generated offspring extracted from the latest population. Outputa

Input

1 2 3 4 a

Pressure (PR)

RF power (RFP)

CHF3

O2

CT

AR

WA

E11

E3

9.7 12.2 12.2 11

101 92.5 136 106

24.3 20.8 14.6 41.9

44.6 41.1 32.3 48.1

1.13 1.35 1.35 1.24

46 49.7 49.7 39.7

22.5 25 25 23.3

7.5 (7.5) 10 (6.03) 6.18 (6.2) 6 (6.11)

7.5 (7.5) 13 (8.97) 8.7 (8.8) 8.9 (8.89)

() is test result come from fuzzy rule.

Table 4 List of process parameters. No

Symbol

Process parameter

Range

1 2 3 4 5 6 7

PR RFP CHF3 O2 WA CT AR

Pressure RF power CHF3 O2 Wall angle Cycle time Airgroove roughness

5–15 m Torr 50–150 Watts 1–60 sccm 1–60 sccm 19–26° 1–1.5 h 35–60 nm

Fig. 11. Data transformation with c2 = 6.7.

6. Numerical analysis

Fig. 10. Data transformation with RIE sample data with c1 = 4.

With T = 10 m Torr, K = 0.5, x2 = 12.5 m Torr, l2(x2, T) = 1.427, k. Thus = 0.5, K 8 0.5, x02 = 0.75, l2(x2, T) = 1.427T = 0.5, K = 0.5, x02 = T0 = 105 10 0.75, l2(x2, T) = 1.427. Using PToF Function, c2 = 6.7 (verified by the chi-square goodness-of-fit test (p-value = 0.2157)). The curve is shown in Fig. 11. Using the similar method, the loss of CHF3 and O2 can be found.For example, set the weights of pressure, RF power, CHF3 and O2 as 03, 0.3, 0.2 and 0.2, respectively, and apply the following data,

lRFP ¼ 1:4; lPR ¼ 1:1; lCHF3 ¼ 1:5; lO2 ¼ 1:2; Average PLI ¼

n X

l i  xi

i¼1

¼ 1:4  0:3 þ 1:1  0:3 þ 1:5  0:2 þ 1:2  0:2 ¼ 1:29; Fitness function ¼

accurancy with error rate 0:75 ¼ ¼ 0:581: performance trade-off 1:29

In order to illustrate the effectiveness of the proposed Fuzzy-GA algorithm for knowledge discovery, the algorithm has been applied for setting the parameters for the reactive ion etching process. The process parameter domain contains 37 cases from a manufacturer of magnetic hard disks. The proposed approach was implemented in MathLab, and the code is executed by a regular PC. The results of the proposed method were compared with the physical experimental result. The Fuzzy-GA approach was used to find out the optimal process parameter settings with minimum defects. The goal of the experiment was to simulate the process parameter settings to find out the defect rate. The process parameters are shown in Table 1 together with the associated membership functions of process parameters and defects. Each rule consists of seven RIE process parameters and two defect rates. The accuracy of the ten initial rules was measured using the test instances. The results are shown in Table 2. It is realized that different rule have different degrees of accuracy. The accuracy of the rule will be evaluated by comparing the predefined expertise rule set with the test instance. Apart from accuracy of the rule, it is also necessary to consider the performance trade-off among process parameters. GA will be deployed to find out the optimal process parameter settings. In the experiments, the operation frequency for crossover and uniform mutation was set at 0.8 and 0.01, respectively. The stopping criterion is set as 100 generations, 50 stall generations and 20 s for the stall time limit. The fitness value is 0.535, as shown in Fig. 12. Fig. 13 shows the result for different numbers of generations with respect to (i) best individual, (ii) the average distance between individuals, (iii) minimum, maximum, and means fitness function values in each generation, and (iv) genealogy.

H.C.W. Lau et al. / Expert Systems with Applications 36 (2009) 7964–7974

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Fig. 12. Relationship between the fitness values and the number of generations in the RIE process.

Fig. 13. Numbers of generations with respect to (i) best individual, (ii) the average distance between individuals, (iii) minimum, maximum, and means fitness function values in each generation and (iv) genealogy.

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 The top left graph of Fig. 13 plots the expected number of children versus the raw scores at each generation and the genealogy of individuals. The best individual is obtained by plotting the vector entries of the individual with the best fitness function value in each generation. It is found that RF Power (variable 1) and O2 (variable 2) are the best individuals.  The top right graph of Fig. 13 shows the average distance between individuals at each generation. The average distance decreases and converges to 0 when the number of generations approaches 80.  Bottom left graph of Fig. 13 plots the genealogy of individuals.  Bottom right graph of Fig. 13 plots the minimum, maximum, and mean fitness function values in each generation. The maximum etch rate of 38 Angstroms/minute was obtained with the following process parameters and is shown below. RF power (W) Pressure (mtorr)

99 10

CHF3 flow (sccm) O2 (sccm)

30 30

The process parameter generated by Fuzzy-GA is above 95% matched with the experiment done by Winnall and Winderbaum (2000) and it shows that the result is promising. 7. Conclusion In this paper, a Fuzzy-GA has demonstrated how knowledge is encoded and represented with fuzzy logic in order to find out the optimal process parameters in industrial processes. Experimental results have also shown that our Fuzzy-GA approach helps to encode the fuzzy rule and the associated membership functions such that the simulated result is quite near to the actual experimental results. The significance of this paper is related to the introduction of a knowledge discovery approach to support the optimization process based on expert advice derived from past experience, capitalizing on the essential features and capabilities of the essential features of a knowledge representation technique and optimization technology. It is expected that this proposed Fuzzy-GA can contribute to mastering the subsequent management of relevant experience as well as useful knowledge so as to enable substantial growth of company business to take place. The significance of this research is to introduce PtoF for realizing the loss acquired by improper setting of process parameters. In order to have an optimal parameter setting, this paper provides an innovative and new research methodology to incorporate both fuzzy logic and genetic algorithm to find out the relationship between processes parameters and production yield. Future work will entail applying the proposed approach to different manufacturing processes to have a more generic fitness function, by considering the performance trade-off and the accuracy of the fuzzy rules. Acknowledgement The authors wish to thank the Research Committee of the Hong Kong Polytechnic University for the support of this project.

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