A NEW DIAGNOSIS SYSTEM BASED ON FUZZY REASONING TO DETECT MEAN AND/OR VARIANCE SHIFTS IN A PROCESS. Received August 2010; revised February 2011

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1 International Journal of Innovative Computing, Information and Control ICIC International c 2011 ISSN Volume 7, Number 12, December 2011 pp A NEW DIAGNOSIS SYSTEM BASED ON FUZZY REASONING TO DETECT MEAN AND/OR VARIANCE SHIFTS IN A PROCESS İnci Sariçiçek 1 and Ömer Çimen2 1 Department of Industrial Engineering Eskişehir Osmangazi University 26480, Batı Meşelik, Eskişehir, Turkey incid@ogu.edu.tr 2 ETİ Cookie Plant Eskişehir Organized Industrial District Street 14, Eskişehir, Turkey ocimen@etigida.com.tr Received August 2010; revised February 2011 Abstract. Statistical process control is a very useful method to improve the product quality and reduce reworks and scraps. In a production environment, control charts are the most important tool to determine whether a process is in-control or out-of-control. Control charts are to detect the occurrence of the shifts in a process rapidly so that their causes can be found and the necessary corrective action can be taken before a large quantity of nonconforming products are manufactured. The determination of variability affects the cost and the quality in a process. Considering the cost that is caused by delay in defining the variability, it is important to determine the variation correctly and quickly in a production process. This paper presents a new method based on a fuzzy inference system for determining shifts in the process. The Fuzzy Inference Control System includes four stages to detect and distinguish mean and/or variance shifts in the quality characteristic. Furthermore, the performance of the proposed method is examined and compared with that of Shewhart Control Charts by evaluating Type II error. In addition, the proposed model is evaluated by comparing performances of the joint X-bar and R charts, and X-bar and s charts for different sample sizes. Keywords: Statistical process control, Shewhart control charts, Fuzzy logic, Fuzzy inference system 1. Introduction. It is expected that production processes operate in control every time. However, one or more assignable causes associated with the machines, the operators, or the materials may occur resulting in a shift of a process to an out-of-control state. When that happens, a significant percent of the process output does not conform to required specifications. Therefore, it is critical to detect shifts in a process regarding the quality and cost. If the time between variation occurrence and its determination is considered, the determination of the variation is very important to improve the product quality and reduce rework which is a fundamental industrial problem. The development of intelligent quality control systems is essential. In the near future, control systems will take data from the product and decide whether the process is in control or not. Several studies on this subject have been made by using artificial intelligence techniques. One of the recent research areas is fuzzy logic applications in Statistical Process Control (SPC). Rowlands and Wang [1] explored the integration of fuzzy logic and control charts in order to create and design a fuzzy-spc evaluation and control method. Hsu and Chen [2] 6935

2 6936 İ. SARIÇIÇEK AND Ö. ÇIMEN proposed a new diagnosis system based on fuzzy reasoning for X chart. The performance of the fuzzy control chart for three typical unnatural patterns (shift, trend and cyclical) is examined by using a fuzzy control chart for individual observations (X) by Tannock [3]. Gülbay and Kahraman [4,5] have developed a direct fuzzy approach to fuzzy control charts for attributes of vague data without any defuzziffication. They defined fuzzy unnatural pattern rules based on the probabilities of fuzzy events. In Gülbay and Kahraman s study, some contributions to fuzzy control charts based on fuzzy transformation methods are made by the use of α-cut to provide the ability of determining the tightness of the inspection. Faraz and Moghadam [6] compared fuzzy chart and X chart and showed that fuzzy control chart has better power to detect shifts. Fazel Zarandi et al. [7] presented a new hybrid method based on a combination of fuzzified sensitivity criteria and fuzzy adaptive sampling rules. A hybrid fuzzy-statistical clustering approach for estimating the time of changes in fixed and variable sampling control charts has been studied by Alaeddini et al. [8]. Fuzzy logic systems are useful for analysis of unnatural patterns and for the determination of process shifts. While some of the studies have focused on run-rules/pattern analysis [3,4,7], others have focused on determining the process mean shift [2,6]. In this paper, authors propose a new approach to the detection of the mean and/or variance shifts in the process using fuzzy inference systems. The proposed fuzzy inference control system consists of four stages to detect and distinguish mean and/or variance shifts. The first stage determines whether a process is in-control or not. The remaining stages determine which shift has occurred in the process. Furthermore, we present a performance analysis based on sample sizes used in this study. The system s performance can be easily modified by adjusting the thresholds of the defuzzified output variables. Moreover, fuzzy inference rules of the system can be used to solve some potential problems in practice. 2. Shifts in Process and Control Charts. Shifts may occur mean and/or variance of a process. The detection of the shift in the process mean and/or variance is one of the most important problems in quality engineering. When the shift in the process mean and variance occurs at the same time, this state needs to be distinguished from the others. It is important to detect and classify the shift as soon as it occurs and provide corrective actions to eliminate or minimize future occurrences of similar shifts. SPC is a widely used method to measure, classify, analyze and interpret the process data to improve the quality of the product and service by detecting instabilities and possible causes. SPC tools provide a graphical display of the characteristic quality and data series versus the sample number or time [7]. A typical control chart is shown in Figure 1. Figure 1. A typical control chart

3 A NEW DIAGNOSIS SYSTEM BASED ON FUZZY REASONING 6937 Shewhart control chart monitors whether the process is in-control or not. The control chart is a graphical display of a quality characteristic that has been measured or computed from a sample versus the sample number or time. The chart consists of: (1) a centerline, which represents the average value of the quality characteristic corresponding to an incontrol state, (2) an upper control limit (UCL) and (3) a lower control limit (LCL). The control limits are chosen such that if the process is in-control, nearly all the sample points will plot between them. As long as a point plots within the control limits, the process is assumed to be in-control, and no action is necessary. However, a point that plots outside of the control limits is interpreted as evidence that the process is out-ofcontrol. Therefore, investigation and corrective action are required to find and eliminate the assignable cause(s) responsible for this behaviour [9]. Traditionally, X and R (or s) control charts are used to monitor the stability of a production process and are among the most useful SPC tools. The process mean is controlled using the X chart and variance can be controlled using either the R (range) chart or the s (standard deviation) chart. It is possible for both the process mean and process variance to vary simultaneously during a production cycle. Joint X and R charts are used to control both the process mean and the process variance simultaneously. While X and R charts are the most common charts for variables, when the sample size is large (n 10), some organizations prefer to estimate and control the sample standard deviation (s chart) directly as the measure of the subgroup dispersion. When subgroup sizes are less than 10, both charts will graphically portray the same variation; however, as subgroup sizes increase to 10 or more, extreme values have an undue influence on the R chart [10]. Therefore, the s chart is desirable at larger subgroup sizes. When a point falls outside of the control limits, the process is out-of-control. This means that an assignable cause of variation is present. Another way of viewing the outof-control point is to think of the subgroup value as coming from a different population than the one from which the control limits were obtained. The relationship between normal distribution and shifts on control chart is demonstrated in Figure 2. In Figure 2(a), both the mean and the standard deviation are in-control at their nominal values (µ 0, σ 0 ). This means that most of the process output plot is contained within the control limits. In Figure 2(b), the mean has shifted to a value µ 1 (µ 1 > µ 0 ), leading to a higher fraction of non-conforming product. Similarly, in Figures 2(c) and 2(d), the standard deviation has shifted to a value σ 1 (σ 1 > σ 0 ), also resulting in more nonconforming products. Over the years, fuzzy logic has been successful mathematical tool for various types of scientific applications. Fuzzy sets are a generalized form of conventional set theory to represent vagueness existing in various phenomena which involve a decision somewhere in between perfectly true and completely false [11]. They hold the potential for the application in statistical process control. 3. The Proposed Fuzzy Inference Control System. Recently, many researchers have studied fuzzy theory and its applications, because vague concepts and linguistic information can be dealt quantitatively in this theory [12]. Fuzzy inference system (FIS) was first introduced by Zadeh in 1965, and has been successfully applied in many areas since then, although there are only a few studies in the field of intelligent quality control systems. Zadeh extended the notion of binary membership to accommodate various degrees of membership on the real continuous interval [0, 1], where the endpoints of 0 and 1 conform to no membership and full membership, respectively, just as the indicator function does for crisp sets, but where the infinite number of values in between the endpoints can represent various degrees of membership for an element X in some set on the

4 6938 İ. SARIÇIÇEK AND Ö. ÇIMEN (a) (b) (c) (d) Figure 2. Illustration of the relationship between normal distribution and shifts on control chart. (a) Nominal mean and standard deviation; (b) mean shift to µ 1 > µ 0 ; (c) standard deviation shift to σ 1 > σ 0 [9]; (d) mean shift to µ 1 > µ 0 and standard deviation shift to σ 1 > σ 0. universe. The sets on the universe X that can accommodate degrees of membership were termed by Zadeh as fuzzy sets [13,14]. FIS contains linguistic control rules; a fuzzifier that has the effect of transforming crisp data into fuzzy sets; an inference engine that uses the fuzzy sets in conjunction with the knowledge base to make inferences by means of a reasoning method; and a defuzzifier that translates the fuzzy control action to a real control action. After the rules have been established, the FIS can be viewed as a mapping from inputs to outputs, and this mapping can be expressed quantitatively as y = f(x) [15]. A FIS is demonstrated in Figure 3. In the present work, the authors propose a new a fuzzy inference control system (FICS) in order to detect and determine the shifts in a process. As the use of a single inference system does not yield the desired results in classifying all of the possible states, the fuzzy system was designed in a way in which it could make decisions gradually. The framework of the FICS is given in Figure 4. A four-stage fuzzy classification system is developed by using the elimination of conditions by means of binary comparisons. At the first stage, the in-control state is separated from the other states. The other stages determine which process shift has occurred in the out-of-control state in the process. The detailed stages of FICS are as follows: FIS-1 distinguishes the state in which the process is in-control from those in which the process is out-of-control. FIS-2 distinguishes the state in which there is no shift in the process mean but there is a shift in variance from the state in which there is a shift in the process mean.

5 A NEW DIAGNOSIS SYSTEM BASED ON FUZZY REASONING 6939 Figure 3. A fuzzy inference system [15] Figure 4. The framework of the proposed fuzzy inference control system FIS-3 distinguishes between the state in which there is a negative shift in the process mean and the state in which there is a positive shift in it.

6 6940 İ. SARIÇIÇEK AND Ö. ÇIMEN FIS-4 distinguishes between the state in which there is shift in variance and the state in which there is no shift in variance for the state in which there is a negative shift in the process mean and between the state in which there is a shift in variance and the state in which there is no shift in variance for the state in which there is positive shift in the process mean. Let samples, sample average ( X), range of samples (R) and standard deviation of samples (s) be input variables. Let case be an output. Then the general fuzzy inference rules can be defined as follows: IF sample data and X is centered and R is low and s is low THEN the output is Case 1. IF X is centered and R is high and s is high THEN the output is Case 2. IF X has a positive shift and R is high or low and s is high or low THEN the output is Case 3 and Case 4 (positive shift in mean). IF X has a negative shift and R is high or low and s is high or low THEN the output is Case 5 and Case 6 (negative shift in mean). IF X has a positive shift and R is low and s is low THEN the output is Case 3. IF X has a positive shift and R is high and s is high THEN the output is Case 4. IF X has a negative shift and R is low and s is low THEN the output is Case 5. IF X has a negative shift and R is high and s is high THEN the output is Case Experimental Results. Normal distribution can be used in order to assign the samples from each case [9] Dataset. This study includes the following states which were used by Dedeakayoğulları and Burnak [16]. Samples were generated for each case by using normal distribution. Normally distributed data with the average 0 and variance 1 refers to the state in which the process is in-control. It is convenient for input data to have significant properties for each case which we want to classify. Six cases are defined as: Case 1: Process is in-control: N(µ x = 0, σ 2 x = 1), Case 2: Shift in the variance: N(µ x = 0, σ 2 x = 3 2 ), Case 3: Positive shift in the mean: N(µ x = 3, σ 2 x = 1), Case 4: Positive shift in the mean and shift in the variance together: N(µ x = 3, σ 2 x = 3 2 ), Case 5: Negative shift in the mean: N(µ x = 3, σ 2 x = 1), Case 6: Negative shift in the mean and shift in the variance together: N(µ x = 3, σ 2 x = 3 2 ). A total of 2250 data points are derived randomly as 375 data points (observations) for each state. Three different randomly derived data sets are used to minimize the effect of the data on the results. Sample size is repeated for n = 2, n = 5 and n = 10 so that the performance of the system could be tested based on sample size as well. Decisions on the size of the sample are given according to some helpful guidelines [10]: for ease of computation, a sample size of five is quite common in industry. When destructive testing is used and the item is expensive, a small subgroup size of two or three is necessary, since it will minimize the destruction of the expensive product. When the subgroup size exceeds 10, the s chart should be used instead of the R chart for the control of the dispersion. For that reason, our system is evaluated according to sample sizes of n = 2, n = 5 and n = 10. In this study, X, R and s statistics are used in addition to

7 A NEW DIAGNOSIS SYSTEM BASED ON FUZZY REASONING 6941 the samples. For sample sizes of two, five and ten, the system has five, eight and thirteen inputs, respectively. X, R and s statistics of six cases for sample size of five are given in Figure 5. Figure 5. X, R and s statistics for samples (n = 5) If x 1, x 2,..., x n is a sample of size n, then the average of this sample is X = x 1 + x x n n The range of the sample, R, is defined as the difference between the largest and the smallest observations; R = x max x min The sample standard deviation, s, is defined as n (x i x) 2 i=1 s = n Membership functions (MFs). In FIS-1, the fuzzy sets and membership functions (MFs) of observations and X are the same. These fuzzy sets have three basic MFs that are shown for sample size of five in Figure 6.

8 6942 İ. SARIÇIÇEK AND Ö. ÇIMEN In classification, the µ = 0, σ = 1 gaussian curve represents the no mean shift (NoMS) condition as the target function. NegMS and PosMS represent the negative shift in mean and positive shift in mean conditions, respectively. Because of the clear characteristic of the points around µ = 0, gauss2 curve is used to membership function. Gauss2 curve is composed of two Gaussian curves. The membership degree of the NoMS is equal to 1 for values between the means of two Gaussian curves. Figure 6. Membership functions for observations and X in FIS-1 The Z-shaped built-in membership function (ZMF) for a negative mean shift condition and the S-shaped built-in membership function (SMF) for a positive mean shift condition were used. The ZMF has two parameters. The first parameter represents the last point that the membership degree is equal to 1. The second parameter represents the first point that the membership degree is equal to 0. The membership degree of the ZMF is equal to 1 for values that are smaller than and equal to the first parameter. Membership degrees between the first and second parameter decrease as a gaussian curve and reach 0 at the second parameter. The SMF is the opposite of the ZMF, and the membership of the SMF is equal to 0 for values that are smaller than or equal to the first parameter. The membership degrees increase as a gaussian curve between the first and second parameter, and they reach 1 at the second parameter. For the ZMF and SMF membership functions, the points that the membership degree is equal to 1 are X = 3 and X = 3 points. These points are the mean of the distributions that represent positive and negative mean shift. The membership degree of the ZMF and SMF is equal to 0 for values that the membership degree of the NoMS is equal to 1. In FIS-2, the case no mean shift but shift in variance is separated from the other cases. Fuzzy sets and MFs of observations and X are the same as in FIS-1. In FIS-3, the case negative shift in mean is distinguished from the case positive shift in mean. Membership functions for sample size of five are given in Figure 7. In FIS-4, the positive shift in mean and negative shift in mean cases are identified by checking the shift in variance. MFs are created with similar logic in FIS-1 but using different fuzzy set target function parameters (µ = 3, σ = 1 and µ = 3, σ = 1). Membership functions for observations and X for FIS-4 (neg) and FIS-4 (pos) in Figures 8(a) and 8(b). Membership functions for range and variance values were used in every FIS of the system without any change or revision. They are shown for sample size of five in Figure 9. Two MFs were created for range and variance values as representing the conditions range is low (RLow), range is high (RHigh) and no variance shift (NoVS), and shift in variance (YesVS), respectively.

9 A NEW DIAGNOSIS SYSTEM BASED ON FUZZY REASONING 6943 Figure 7. Membership functions for observations and X in FIS-3 (a) FIS-4 (neg) (b) FIS-4 (pos) Figure 8. Membership functions for observations and X in FIS-4 In the RLow membership function, first parameter is the center line of the R control chart (σ = 1) and second parameter is the maximum R value of the samples which have generated with σ = 1. Similarly, in the NoVS membership function, first parameter is the center line of the s control chart (σ = 1) and second parameter is the maximum s value of the samples which have generated with σ = 1. In the RHigh membership function, first parameter is the first parameter of the RLow membership function and second parameter is the center line of the R control

10 6944 İ. SARIÇIÇEK AND Ö. ÇIMEN (a) (b) Figure 9. Membership functions for range (a) and variance (b) chart (σ = 3). Similarly, in the YesVS membership function, first parameter is the first parameter of the NoVS membership function and second parameter is the center line of the s control chart (σ = 3) Comparison of results. The X chart is the most widely used chart for controlling tendency, µ, while charts based on either the sample range, R, or the sample standard deviation, s, are used to control process variability, σ [9]. To determine the number of wrong decisions and Type II error, control limits need to be calculated for X, R and s control charts. Suppose that a quality characteristic is normally distributed with mean µ and standart deviation σ, where both µ and σ are known. The control limits of the X, R and s charts are obtained by using these standards and they have been tabulated in Table 1. For X control chart, the decisions about mean shift are made based on whether the X value is outside the X control chart limits. For R control chart, the decisions about variance shift are made based on whether the R value is outside the R control chart limits. Formation of a wrong decision depends on whether the sample could yield the classification it belongs to. For example, in Case 4, every decision is considered as wrong except for there is a positive shift in the average and there is a shift in variance condition for a sample from N(3, 3 2 ). The wrong decision percentages of X and R control charts were

11 A NEW DIAGNOSIS SYSTEM BASED ON FUZZY REASONING 6945 Table 1. Control limits of the X, R and s charts for µ = 0 and σ = 1 determined assuming that they were used together. Again for a sample from the N(3, 3 2 ) condition, the out-of-control decision from the X control chart indicates that there is a shift in the mean and the out of control decision from the R control chart indicates that there is a shift in the variance. Otherwise, the decision is considered as wrong since it would not include the correct classification. Overall wrong decision rate of control charts and the proposed FICS are shown in Table 2 for each input data set. For all sample sizes, the wrong decision percentages of the proposed FICS are the lowest compared to that of the control charts. The proposed FICS outperformed the X R and X s control charts and proved to be quite successful in determining the states in which there is a shift in the variation. When a point is outside of the control limits, it is assumed to be due to an assignable cause. In statistical process control, Type II error is defined as incorrectly inferring the process is in control when the process is actually out-of-control. Table 3 illustrates the Type II errors of the control charts and the FICS. Note that the Type II errors of the designed fuzzy system are quite low compared with that of the control charts. For sample size n = 2, the Type II errors occurred approximately 21% for control charts and 6.3% for the proposed FICS. For sample size n = 5, Type II errors occurred 3-4% for control charts and 0.9% for the FICS. And for sample size n = 10, Type II errors occurred 0.7% for the X R control chart while there is no Type II error for both the X s control chart and the FICS. The proposed FICS performs significantly more efficient than traditional control charts for wrong decision percentages and Type II error. It is also observed that the wrong decision percentages and Type II errors get smaller as the sample size increased for the control charts and the FICS. See Figure 10.

12 6946 İ. SARIÇIÇEK AND Ö. ÇIMEN Table 2. Wrong decision percentages of the control charts and the proposed FICS Sample size n = 2 n = 5 n = 10 Data set Control Charts X R X s FICS data set data set data set Average data set data set data set Average data set data set data set Average Table 3. Type II errors of the control charts and the proposed FICS Sample size n = 2 n = 5 n = 10 Data set Control Charts X R X s FICS data set data set data set Average data set data set data set Average data set data set data set Average The proposed FICS especially outperformed the X R and X s charts for Type II error, while the X s chart outperformed the X R chart in detecting the shift in the process mean and/or variance. 5. Conclusions. In this paper, a fuzzy inference control system has been proposed for detecting the mean and/or variance shifts in a process. Through statistical measures, the performance of the proposed method has been compared to traditional control charts

13 A NEW DIAGNOSIS SYSTEM BASED ON FUZZY REASONING 6947 (a) (b) Figure 10. Wrong decision percentages (a) and Type II error (b) for different sample sizes through two measures, the wrong decision percentages and Type II error. It is found that for both measures, the proposed method outperforms the traditional control charts. The proposed method is intelligent, does not need a training process and captures past information. We showed that the fuzzy inference system is applicable to detecting the mean and/or variance shifts in a process. Detecting variability occurring in mean and/or variance in a process and investigating the causes of this variability can help to improve the product quality and to reduce costs. In a traditional system, when the observations are close to the control limits, this may cause false alarms. A Fuzzy Inference Control System can provide a more robust control process. FICS s performance can be easily modified by adjusting the thresholds of the defuzzified output variables. In addition, the sensitivity of this system can be adjusted by using various kinds of membership functions and changing membership function parameters. For future studies, different pattern recognition methods can be used by involving the samples gathered cumulatively in addition to making decisions based on a single sample taken from the process. Acknowledgment. The authors would like to thank the associate editor and reviewers for their invaluable comments and suggestions. REFERENCES [1] H. Rowlands and L. R. Wang, An approach of fuzzy logic evaluation and control in SPC, Quality and Reliability Engineering International, vol.16, no.2, pp.91-98, [2] H. M. Hsu and Y. K. Chen, A fuzzy reasoning based diagnosis system for X control chats, Journal of Intelligent Manufacturing, vol.12, no.1, pp.57-64, [3] J. T. D. Tannock, A fuzzy control charting method for individuals, International Journal of Production Research, vol.41, no.5, pp , [4] M. Gülbay and C. Kahraman, Development of fuzzy process control charts and fuzzy unnatural pattern analyses, Computational Statistics & Data Analyses, vol.51, no.1, pp , [5] M. Gülbay and C. Kahraman, An alternative approach to fuzzy control charts: Direct fuzzy approach, Information Sciences, vol.177, no.6, pp , [6] A. Faraz and B. Moghadam, Fuzzy control chart a better alternative for Shewhart average chart, Quality and Quantity, vol.41, no.3, pp , [7] M. H. F. Zarandi, A. Alaeddini and I. B. Turksen, A hybrid fuzzy adaptive sampling Run rules for Shewhart control charts, Information Sciences, vol.178, no.4, pp , 2008.

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