a new diagnosis system based on fuzzy reasoning …study, some contributions to fuzzy control charts...

International Journal of InnovativeComputing, Information and Control ICIC International c⃝2011 ISSN 1349-4198Volume 7, Number 12, December 2011 pp. 6935–6948

A NEW DIAGNOSIS SYSTEM BASED ON FUZZY REASONINGTO DETECT MEAN AND/OR VARIANCE SHIFTS IN A PROCESS

Inci Saricicek1 and Omer Cimen2

1Department of Industrial EngineeringEskisehir Osmangazi University

26480, Batı Meselik, Eskisehir, [email protected]

2ETI Cookie PlantEskisehir Organized Industrial District

Street 14, Eskisehir, [email protected]

Received August 2010; revised February 2011

Abstract. Statistical process control is a very useful method to improve the productquality and reduce reworks and scraps. In a production environment, control charts arethe 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 thattheir causes can be found and the necessary corrective action can be taken before a largequantity of nonconforming products are manufactured. The determination of variabilityaffects the cost and the quality in a process. Considering the cost that is caused by delay indefining the variability, it is important to determine the variation correctly and quickly ina production process. This paper presents a new method based on a fuzzy inference systemfor determining shifts in the process. The Fuzzy Inference Control System includes fourstages to detect and distinguish mean and/or variance shifts in the quality characteristic.Furthermore, the performance of the proposed method is examined and compared withthat of Shewhart Control Charts by evaluating Type II error. In addition, the proposedmodel is evaluated by comparing performances of the joint X-bar and R charts, and X-barand s charts for different sample sizes.Keywords: Statistical process control, Shewhart control charts, Fuzzy logic, Fuzzyinference 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, orthe materials may occur resulting in a shift of a process to an out-of-control state. Whenthat happens, a significant percent of the process output does not conform to requiredspecifications. Therefore, it is critical to detect shifts in a process regarding the qualityand 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 andreduce 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 incontrol or not. Several studies on this subject have been made by using artificial intelli-gence techniques. One of the recent research areas is fuzzy logic applications in StatisticalProcess Control (SPC).

Rowlands and Wang [1] explored the integration of fuzzy logic and control charts inorder to create and design a fuzzy-SPC evaluation and control method. Hsu and Chen [2]

6935

6936 I. SARICICEK AND O. CIMEN

proposed a new diagnosis system based on fuzzy reasoning for X chart. The performanceof the fuzzy control chart for three typical unnatural patterns (shift, trend and cyclical) isexamined by using a fuzzy control chart for individual observations (X) by Tannock [3].Gulbay and Kahraman [4,5] have developed a direct fuzzy approach to fuzzy control chartsfor attributes of vague data without any defuzziffication. They defined fuzzy unnaturalpattern rules based on the probabilities of fuzzy events. In Gulbay and Kahraman’sstudy, some contributions to fuzzy control charts based on fuzzy transformation methodsare made by the use of α-cut to provide the ability of determining the tightness of theinspection. Faraz and Moghadam [6] compared fuzzy chart and X chart and showed thatfuzzy control chart has better power to detect shifts. Fazel Zarandi et al. [7] presenteda new hybrid method based on a combination of fuzzified sensitivity criteria and fuzzyadaptive sampling rules. A hybrid fuzzy-statistical clustering approach for estimatingthe time of changes in fixed and variable sampling control charts has been studied byAlaeddini et al. [8].Fuzzy logic systems are useful for analysis of unnatural patterns and for the determi-

nation of process shifts. While some of the studies have focused on run-rules/patternanalysis [3,4,7], others have focused on determining the process mean shift [2,6]. In thispaper, authors propose a new approach to the detection of the mean and/or varianceshifts in the process using fuzzy inference systems. The proposed fuzzy inference controlsystem consists of four stages to detect and distinguish mean and/or variance shifts. Thefirst stage determines whether a process is in-control or not. The remaining stages de-termine which shift has occurred in the process. Furthermore, we present a performanceanalysis based on sample sizes used in this study. The system’s performance can be easilymodified by adjusting the thresholds of the defuzzified output variables. Moreover, fuzzyinference 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 ofa process. The detection of the shift in the process mean and/or variance is one of themost important problems in quality engineering. When the shift in the process mean andvariance occurs at the same time, this state needs to be distinguished from the others. Itis important to detect and classify the shift as soon as it occurs and provide correctiveactions 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 possiblecauses. SPC tools provide a graphical display of the characteristic quality and data seriesversus the sample number or time [7]. A typical control chart is shown in Figure 1.

Figure 1. A typical control chart

A NEW DIAGNOSIS SYSTEM BASED ON FUZZY REASONING 6937

Shewhart control chart monitors whether the process is in-control or not. The controlchart is a graphical display of a quality characteristic that has been measured or computedfrom 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 in-control 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 samplepoints will plot between them. As long as a point plots within the control limits, theprocess is assumed to be in-control, and no action is necessary. However, a point thatplots outside of the control limits is interpreted as evidence that the process is out-of-control. Therefore, investigation and corrective action are required to find and eliminatethe assignable cause(s) responsible for this behaviour [9].

Traditionally, X and R (or s) control charts are used to monitor the stability of aproduction process and are among the most useful SPC tools. The process mean iscontrolled 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 andprocess variance to vary simultaneously during a production cycle. Joint X and R chartsare 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 sizeis large (n ≥ 10), some organizations prefer to estimate and control the sample standarddeviation (s chart) directly as the measure of the subgroup dispersion. When subgroupsizes 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 theR 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. Thismeans that an assignable cause of variation is present. Another way of viewing the out-of-control point is to think of the subgroup value as coming from a different populationthan the one from which the control limits were obtained. The relationship betweennormal 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 nominalvalues (µ0, σ0). This means that most of the process output plot is contained within thecontrol limits. In Figure 2(b), the mean has shifted to a value µ1 (µ1 > µ0), leadingto a higher fraction of non-conforming product. Similarly, in Figures 2(c) and 2(d),the standard deviation has shifted to a value σ1 (σ1 > σ0), also resulting in more non-conforming products.

Over the years, fuzzy logic has been successful mathematical tool for various types ofscientific applications. Fuzzy sets are a generalized form of conventional set theory torepresent vagueness existing in various phenomena which involve a decision somewherein between perfectly true and completely false [11]. They hold the potential for theapplication in statistical process control.

3. The Proposed Fuzzy Inference Control System. Recently, many researchershave studied fuzzy theory and its applications, because vague concepts and linguistic in-formation 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 areassince then, although there are only a few studies in the field of intelligent quality con-trol 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 0and 1 conform to no membership and full membership, respectively, just as the indicatorfunction does for crisp sets, but where the infinite number of values in between the end-points can represent various degrees of membership for an element X in some set on the


(a) (b)

(c) (d)

Figure 2. Illustration of the relationship between normal distribution andshifts on control chart. (a) Nominal mean and standard deviation; (b) meanshift to µ1 > µ0; (c) standard deviation shift to σ1 > σ0 [9]; (d) mean shiftto µ1 > µ0 and standard deviation shift to σ1 > σ0.

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 theknowledge base to make inferences by means of a reasoning method; and a defuzzifierthat translates the fuzzy control action to a real control action. After the rules have beenestablished, the FIS can be viewed as a mapping from inputs to outputs, and this mappingcan 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 inferencesystem does not yield the desired results in classifying all of the possible states, the fuzzysystem was designed in a way in which it could make decisions gradually. The frameworkof the FICS is given in Figure 4. A four-stage fuzzy classification system is developed byusing 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 whichprocess shift has occurred in the out-of-control state in the process. The detailed stagesof 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.


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 processmean and the state in which there is a positive shift in it.


FIS-4 distinguishes between the state in which there is shift in variance and the statein which there is no shift in variance for the state in which there is a negative shift in theprocess mean and between the state in which there is a shift in variance and the statein which there is no shift in variance for the state in which there is positive shift in theprocess 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 inferencerules 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

6”.

4. Experimental Results. Normal distribution can be used in order to assign the sam-ples from each case [9].

4.1. Dataset. This study includes the following states which were used by Dedeakayogullarıand Burnak [16]. Samples were generated for each case by using normal distribution. Nor-mally distributed data with the average 0 and variance 1 refers to the state in which theprocess is in-control. It is convenient for input data to have significant properties for eachcase which we want to classify. Six cases are defined as:

Case 1: Process is in-control: N(µx = 0, σ2x = 1),

Case 2: Shift in the variance: N(µx = 0, σ2x = 32),

Case 3: Positive shift in the mean: N(µx = 3, σ2x = 1),

Case 4: Positive shift in the mean and shift in the variance together: N(µx = 3, σ2x = 32),

Case 5: Negative shift in the mean: N(µx = −3, σ2x = 1),

Case 6: Negative shift in the mean and shift in the variance together: N(µx = −3, σ2x =

32).

A total of 2250 data points are derived randomly as 375 data points (observations) foreach state. Three different randomly derived data sets are used to minimize the effect ofthe data on the results. Sample size is repeated for n = 2, n = 5 and n = 10 so that theperformance 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. Whendestructive testing is used and the item is expensive, a small subgroup size of two or threeis necessary, since it will minimize the destruction of the expensive product. When thesubgroup size exceeds 10, the s chart should be used instead of the R chart for the controlof the dispersion. For that reason, our system is evaluated according to sample sizes ofn = 2, n = 5 and n = 10. In this study, X, R and s statistics are used in addition to


the samples. For sample sizes of two, five and ten, the system has five, eight and thirteeninputs, respectively. X, R and s statistics of six cases for sample size of five are given inFigure 5.

Figure 5. X, R and s statistics for samples (n = 5)

If x1, x2, ..., xn is a sample of size n, then the average of this sample is

X =x1 + x2 + ...+ xn

n

The range of the sample, R, is defined as the difference between the largest and thesmallest observations;

R = xmax − xmin

The sample standard deviation, s, is defined as

s =

√√√√√ n∑i=1

(xi − x)2

n− 1

4.2. Membership functions (MFs). In FIS-1, the fuzzy sets and membership func-tions (MFs) of observations and X are the same. These fuzzy sets have three basic MFsthat are shown for sample size of five in Figure 6.


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 inmean” and “positive shift in mean” conditions, respectively. Because of the clear charac-teristic of the points around µ = 0, gauss2 curve is used to membership function. Gauss2curve is composed of two Gaussian curves. The membership degree of the NoMS is equalto 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” condi-tion 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 thelast point that the membership degree is equal to 1. The second parameter representsthe first point that the membership degree is equal to 0. The membership degree ofthe 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 curveand reach 0 at the second parameter. The SMF is the opposite of the ZMF, and themembership of the SMF is equal to 0 for values that are smaller than or equal to the firstparameter. The membership degrees increase as a gaussian curve between the first andsecond parameter, and they reach 1 at the second parameter. For the ZMF and SMFmembership functions, the points that the membership degree is equal to 1 are X = −3and X = 3 points. These points are the mean of the distributions that represent positiveand negative mean shift. The membership degree of the ZMF and SMF is equal to 0 forvalues 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 usingdifferent 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 Figures8(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 Figure9.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.



(a) FIS-4 (neg)

(b) FIS-4 (pos)


In the “RLow” membership function, first parameter is the center line of the R controlchart (σ = 1) and second parameter is the maximum R value of the samples which havegenerated with σ = 1. Similarly, in the “NoVS” membership function, first parameter isthe center line of the s control chart (σ = 1) and second parameter is the maximum svalue 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


(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 firstparameter of the “NoVS” membership function and second parameter is the center lineof the s control chart (σ = 3).

4.3. Comparison of results. The X chart is the most widely used chart for controllingtendency, µ, while charts based on either the sample range, R, or the sample standarddeviation, 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 characteristicis normally distributed with mean µ and standart deviation σ, where both µ and σ areknown. The control limits of the X, R and s charts are obtained by using these standardsand 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 aboutvariance 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 classifi-

cation it belongs to. For example, in Case 4, every decision is considered as wrong exceptfor “there is a positive shift in the average and there is a shift in variance” condition fora sample from N(3, 32). The wrong decision percentages of X and R control charts were


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, 32)condition, the “out-of-control” decision from the X control chart indicates that there isa shift in the mean and the “out of control” decision from the R control chart indicatesthat there is a shift in the variance. Otherwise, the decision is considered as wrong sinceit would not include the correct classification.

Overall wrong decision rate of control charts and the proposed FICS are shown in Table2 for each input data set.

For all sample sizes, the wrong decision percentages of the proposed FICS are the lowestcompared to that of the control charts. The proposed FICS outperformed the X −R andX − s control charts and proved to be quite successful in determining the states in whichthere is a shift in the variation.

When a point is outside of the control limits, it is assumed to be due to an assignablecause. In statistical process control, Type II error is defined as incorrectly inferring theprocess is in control when the process is actually out-of-control. Table 3 illustrates theType II errors of the control charts and the FICS.

Note that the Type II errors of the designed fuzzy system are quite low comparedwith that of the control charts. For sample size n = 2, the Type II errors occurredapproximately 21% for control charts and 6.3% for the proposed FICS. For sample sizen = 5, Type II errors occurred 3-4% for control charts and 0.9% for the FICS. And forsample size n = 10, Type II errors occurred 0.7% for the X −R control chart while thereis no Type II error for both the X − s control chart and the FICS.

The proposed FICS performs significantly more efficient than traditional control chartsfor wrong decision percentages and Type II error. It is also observed that the wrongdecision percentages and Type II errors get smaller as the sample size increased for thecontrol charts and the FICS. See Figure 10.


Table 2. Wrong decision percentages of the control charts and the pro-posed FICS

Sample size Data setControl Charts

FICSX −R X − s

n = 2

data set-1 0.417 0.417 0.371

data set-2 0.411 0.411 0.365

data set-3 0.400 0.400 0.347

Average 0.409 0.409 0.361

n = 5

data set-1 0.182 0.169 0.153

data set-2 0.180 0.169 0.129

data set-3 0.169 0.153 0.142

Average 0.177 0.164 0.141

n = 10

data set-1 0.066 0.040 0.044

data set-2 0.075 0.075 0.040

data set-3 0.083 0.061 0.031

Average 0.075 0.058 0.038

Table 3. Type II errors of the control charts and the proposed FICS

Sample size Data setControl Charts

FICSX −R X − s

n = 2

data set-1 0.225 0.225 0.062

data set-2 0.199 0.199 0.060

data set-3 0.202 0.202 0.067

Average 0.209 0.209 0.063

n = 5

data set-1 0.048 0.035 0.008

data set-2 0.035 0.032 0.011

data set-3 0.045 0.035 0.008

Average 0.043 0.034 0.009

n = 10

data set-1 0.011 0.000 0.000

data set-2 0.000 0.000 0.000

data set-3 0.011 0.000 0.000

Average 0.007 0.000 0.000

The proposed FICS especially outperformed the X − R and X − s charts for Type IIerror, while the X − s chart outperformed the X − R chart in detecting the shift in theprocess mean and/or variance.

5. Conclusions. In this paper, a fuzzy inference control system has been proposed fordetecting 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


(a) (b)

Figure 10. Wrong decision percentages (a) and Type II error (b) for dif-ferent sample sizes

through two measures, the wrong decision percentages and Type II error. It is foundthat for both measures, the proposed method outperforms the traditional control charts.The proposed method is intelligent, does not need a training process and captures pastinformation. We showed that the fuzzy inference system is applicable to detecting themean and/or variance shifts in a process. Detecting variability occurring in mean and/orvariance in a process and investigating the causes of this variability can help to improvethe product quality and to reduce costs.

In a traditional system, when the observations are close to the control limits, this maycause false alarms. A Fuzzy Inference Control System can provide a more robust controlprocess. FICS’s performance can be easily modified by adjusting the thresholds of thedefuzzified output variables. In addition, the sensitivity of this system can be adjustedby using various kinds of membership functions and changing membership function pa-rameters.

For future studies, different pattern recognition methods can be used by involving thesamples gathered cumulatively in addition to making decisions based on a single sampletaken from the process.

Acknowledgment. The authors would like to thank the associate editor and reviewersfor their invaluable comments and suggestions.

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a new diagnosis system based on fuzzy reasoning …study, some contributions to fuzzy control charts...

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