α-cut fuzzy control charts for linguistic data

a-Cut Fuzzy Control Charts forLinguistic DataMurat Gülbay,1,* Cengiz Kahraman,1,† Da Ruan2,‡

1Istanbul Technical University, Department of Industrial Engineering,Macka 34367, Istanbul, Turkey2Belgian Nuclear Research Centre (SCK•CEN), Boeretang 200,B-2400 Mol, Belgium

The major contribution of fuzzy set theory is its capability of representing vague data. Fuzzylogic offers a systematic base in dealing with situations that are ambiguous or not well defined.In the literature, there exist some fuzzy control charts developed for linguistic data that aremainly based on membership and probabilistic approaches. In this article, a-cut control chartsfor attributes are developed. This approach provides the ability of determining the tightness ofthe inspection by selecting a suitable a-level: The higher a the tighter inspection. The articlealso presents a numerical example and interprets and compares other results with the approachesdeveloped previously. © 2004 Wiley Periodicals, Inc.

1. INTRODUCTION

A control chart is a device for describing in a precise manner what is meantby statistical control and is a widely used tool for monitoring and examining pro-duction processes. The power of control charts lies in their ability to detect pro-cess shifts and to identify abnormal conditions in a production process. It improvesproductivity, prevents unnecessary process adjustments, provides diagnostic infor-mation and information about process capability, and is effective in defect preven-tion. This makes possible the diagnosis and correction of many production problemsand often reduces losses and brings substantial improvements in product quality.A control chart is essentially a time plot (or run chart) of observations with controllimits added. The purpose of the control limits is to indicate when the variabilityof the process is so great that some special cause is likely to be operating. When aprocess observation exceeds the control limits a search for a special cause shouldbe initiated.

*Author to whom all correspondence should be addressed: e-mail: [email protected].†e-mail: [email protected].‡e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 19, 1173–1195 (2004)© 2004 Wiley Periodicals, Inc. Published online in Wiley InterScience(www.interscience.wiley.com). • DOI 10.1002/int.20044

There are four major types of control charts. The first type, the x-chart (andrelated x-bar, r-, and s-charts), is a generic and simple control chart. The x-chart isdesigned to be used primarily with variable data, which are usually measurementssuch as the length of an object, processing time, or the number of objects producedper period. The other types of control charts are the p-charts, which are used withbinomial data, the c-charts, which are used for Poisson processes, and the u-charts,generally designed for counting defects per sample when the sample size variesfor each inspection.

Even though the first control chart was proposed during the 1920s byShewhart,1 who published his work in 1931, today they are still subject to newapplication areas that deserve further attention. Shewhart control charts1 were des-ignated to monitor processes for shifts in the mean or variance of a single qualitycharacteristic, and so-called univariate control charts. Further developments arefocused on the usage of the probability and fuzzy set theory integrated with thecontrol charts.

Marcucci2 proposed two procedures using Shewhart-type control charts. Thefirst type is designed to detect changes in any of the quality proportions and thesecond type uses the multinomial distribution, which can be approximated by amultivariate normal distribution. More recently, Chen and Yang3 proposed a modelof a moving average control chart (MA control chart) with a Weibull failure mech-anism from an economic viewpoint. When the process-failure mechanism followsa Weibull model or other models having increasing hazard rates, it is desirable tohave the sampling interval decreasing with the age of the system. A cost modelutilizing a variable scheme instead of fixed sampling lengths in a continuous flowprocess is studied in this research. The variable sampling scheme is used to main-tain a constant integrated hazard rate over each sampling interval. Optimal valuesfor the design parameter, the moving subgroup size, the sampling interval, and thecontrol limit coefficient are determined by minimizing the loss-cost model. Theperformance of the loss cost with various Weibull parameters is studied. A sensi-tivity analysis shows that the design parameters and loss cost depend on the modelparameters and shift amounts. Wu and Luo4 presented an algorithm for designingthe np control charts on the basis of their statistical performance. The algorithmgives the sample size and control limits for an np chart (the 3-triplet np chart)based on the specified values of (a) the in-control fraction nonconforming valuep0, a downward shifted value p� , and an upward shifted value p� ; (b) the corre-sponding design run lengths ~RL!; and (c) the corresponding probabilities ~W !.Specifically, whenever the process fraction nonconforming equals one of the threespecified values, the 3-triplet np chart will produce a signal in a predetermined runlength with a predetermined probability. This feature makes the operating charac-teristics of the np chart highly specifiable and provides the quality assurance engi-neers with better perception and more control in the design and operation of the npcharts. The 3-triplet np chart is designed by seeking an optimal combination of thesample size and the control limits.

It is not surprising that uncertainty always exists in the human world. Researchthat attempts to model uncertainty into decision analysis is done basically throughprobability theory and/or fuzzy set theory. The former represents the stochastic

1174 GÜLBAY, KAHRAMAN, AND RUAN

nature of decision analysis whereas the latter captures the subjectivity of humanbehavior. Dubois and Prade5 suggest that a stochastic decision method such asstatistical decision analysis does not measure the imprecision in human behavior;rather, this method is a way to model incomplete knowledge about the externalenvironment surrounding human beings. Fuzzy set theory, on the other hand, is aperfect means for modeling uncertainty (or imprecision) arising from mental phe-nomena that are neither random nor stochastic. Human beings are heavily involvedin the process of decision analysis. A rational approach toward decision makingshould take into account human subjectivity, rather than employing only objectiveprobability measures. This attitude toward the uncertainty of human behavior ledto the study of a relatively new decision analysis field: fuzzy decision making.

If the quality-related characteristics cannot be represented in numerical form,such as characteristics for appearance, softness, color, and so forth, then controlcharts for attributes are used. Product units are either classified as conforming ornonconforming, depending upon whether or not they meet specifications, or thenumber of nonconformities (deviations from specifications) per unit is counted.The binary classification into conforming and nonconforming used in the p-chartmight not be appropriate in many situations where product quality does not changeabruptly from satisfactory to worthless, and there might be a number of intermedi-ate levels. Without fully utilizing such intermediate information, the use of thep-chart usually results in poorer performance than that of the x-chart. This is evi-denced by weaker detectability of process shifts and other abnormal conditions.To supplement the binary classification, several intermediate levels may beexpressed in the form of linguistic terms. For example, the quality of a product canbe classified by one of the following terms: “perfect,” “good,” “medium,” “poor,”and “bad”; depending on its deviation from specifications, appropriately selectedcontinuous functions can then be used to describe the quality characteristic asso-ciated with each linguistic term. A comparison of traditional Shewhart control chartsand fuzzy inference control charts is given in Table I.

Bradshaw6 used fuzzy set theory as a basis for interpreting the representationof a graded degree of product conformance with quality standard. When the costresulting from substandard quality is related to the extent of nonconformance, acompatibility function describing the grade of nonconformance associated withany given value of the quality characteristics exists. This compatibility functioncan be used to construct fuzzy economic control charts on an acceptance chart.Bradshaw6 stressed that fuzzy economic control chart limits would be advanta-geous over traditional acceptance charts in that fuzzy economic control charts pro-vide information on severity as well as the frequency of product nonconformance.Raz and Wang7 and Wang and Raz8 proposed an approach based on fuzzy settheory by assigning fuzzy sets to each linguistic term, and then combining foreach sample using rules of fuzzy arithmetic. The result is a single fuzzy set. Ameasure of centrality of this aggregate fuzzy set is then plotted on a Shewhart-type control chart. Kanagawa et al.9 introduced modifications to the constructionof control charts given by Wang and Raz8 and presented a control chart based onthe probability density function existing behind the linguistic data. Wang and Chen10

presented a fuzzy mathematical programming model and solution heuristic for the

a-CUT FUZZY CONTROL CHARTS 1175

economic design of statistical control charts. They argued that under the assump-tions of the economic statistical model, the fuzzy set theory procedure presentedimproved the economic design of control charts by allowing more flexibility inthe modeling of the imprecision that exists when satisfying type I and type II errorconstraints. Kahraman et al.11 used triangular fuzzy numbers in the tests of controlcharts for unnatural patterns. Chang and Aw12 proposed a neural fuzzy (NF) con-trol chart for identifying process mean shifts. A supervised multilayer backpropa-gation neural network is trained off-line to detect various mean shifts in a productionprocess. In identifying mean shifts in real-time usage, the neural network’s out-puts are classified into various decision regions using a fuzzy set scheme. Theapproach offers better performance and additional advantages over conventionalcontrol charts. Simulation results show that the proposed NF control charts aresuperior to conventional x-bar charts and cumulative sum (CuSum) charts in termsof the average run lengths (ARL). The proposed system also has the ability toidentify the magnitude of a mean shift, in addition to the Shewhart-type controlchart heuristic rules. Correct classification percentages are studied. Furthermore,general guidelines are given for the proper use of the proposed NF charts.

Woodall et al.13 gave a review of statistical and fuzzy control charts basedon categorical data. An investigation into the use of fuzzy logic to modify sta-tistical process control (SPC) rules, with the aim of reducing the generation offalse alarms and also improving the detection and detection speed of real faultsis studied by El-Shal and Morris.14 Rowlands and Wang15 explored the integra-tion of fuzzy logic and control charts to create and design a fuzzy-SPC evalua-tion and control (FSEC) method based on the application of fuzzy logic to the

Table I. Comparison of traditional Shewhart and fuzzy inherence control charts.

Traditional Shewhartcontrol charts Fuzzy inference control charts

Number of quality characteristics Only one quality characteristic Multiple quality characteristics

Information used in base period Historical data Experts’ experience rules

Judgment Process is in statistical control ornot.

Current process information andprocess are in statistical control ornot.

Advantages 1. Easier for considering onequality characteristic

2. More objective

1. Provide more accurate controlstandards for the process basedon experts’ experience

2. More flexible for the defini-tions of the fuzzy inferencerules in control charts

Disadvantages 1. Control limits are not flexible.2. Sample size influences the

width of control limits.3. Historical data need to be ver-

ified to obtain the formal con-trol limits.

1. Inference outcomes are basedon the subjective experiencerules.

2. Supplemental rules (for sys-tematic changes) of the tradi-tional control chart cannot beused.


SPC zone rules. A simulation program implementing FSEC was written in Bor-land C�� 5.0 and simulation results were obtained and analyzed. The abnormalprocesses simulated were automatically adjusted for each of the zone rules testedand showed an improved performance after the control action, thus confirmingthe merit of the technique as a special method with the specific numerical con-trol action based on a quality evaluation criterion. Hsu and Chen16 describeda new diagnosis system based on fuzzy reasoning to monitor the performanceof a discrete manufacturing process and to justify the possible causes. The diag-nosis system consists chiefly of a knowledge bank and a reasoning mechanism.The knowledge bank provides knowledge of the membership functions of unnat-ural symptoms that are described by Nelson’s rules on (X) over bar controlcharts and knowledge of cause–symptom relations. They developed an approachcalled the maximal similarity method (MSM) for knowledge acquisition toconstruct the fuzzy cause–symptom relation matrix. Through the knowledgebank, the diagnosis system can first determine the degrees of an observationfitting each unnatural symptom. Then, using the fuzzy cause–symptom relationmatrix, we can diagnose the causes of process instability. Taleb and Limam17

discussed different procedures of constructing control charts for linguistic data,based on fuzzy and probability theory. Three sets of membership functions, withdifferent degrees of fuzziness, are proposed for fuzzy approaches. A comparisonbetween fuzzy and probability approaches, based on the average run length andsamples under control, is conducted for real data. Contrary to the conclusionsof Raz and Wang,7 the choice of degree of fuzziness affected the sensitivity ofcontrol charts.

This article aims at developing a-cut control charts to regulate the tightnessof the inspection: The higher a-cut, the tighter inspection. The results obtainedusing a-cut control charts are compared with the fuzzy control charts developedpreviously. This article is organized in the following order. In Section 2, represen-tative values for fuzzy sets are introduced. Fuzzy control charts in the literatureare summarized in Section 3. In Section 4, a-cut fuzzy control charts are devel-oped. Then a numerical example is given in Section 5. Section 6 provides theconcluding remarks.

2. REPRESENTATIVE VALUES FOR FUZZY SETS

To retain the standard format of control charts and to facilitate the plotting ofobservations on the chart, it is necessary to convert the fuzzy sets associated withlinguistic values into scalars referred to as representative values. This conversionmay be performed in a number of ways as long as the result is intuitively repre-sentative of the range of the base variable included in the fuzzy set. Four ways,which are similar in principle to the measures of central tendency used in descrip-tive statistics, are fuzzy mode, a-level fuzzy midrange, fuzzy median, and fuzzyaverage. It should be pointed out that there is no theoretical basis supporting anyone specifically and the selection between then should be mainly based on theease of computation or preference of the user.8


The fuzzy mode of a fuzzy set F is the value of the base variable where themembership function equals to 1. This is stated as

fmode � $x 6m f ~x!� 1%, ∀x � F (1)

If the membership function is unimodal, the fuzzy mode is unique.The a-level fuzzy midrange, fmr ~a!, is defined as the midpoint of the ends of

the a-level cut. An a-level cut, denoted by Aa , is a nonfuzzy set that comprises allelements whose membership is greater than or equal to a. If aa and ba are the endpoints of Aa , then

fmr ~a! �1

2~aa� ba! (2)

In fact, the fuzzy mode is a special case of a-level fuzzy midrange when a� 1.The fuzzy median, fmed , is the point that partitions the curve under the mem-

bership function of a fuzzy set into two equal regions satisfying the followingequation:

�a

fmed

m f ~x! dx � �fmed

b

m f ~x! dx �1

2�

a

b

m f ~x! dx (3)

where a and b are the end points in the base variable of the fuzzy set F such thata � b.

The fuzzy average, favg , is

favg � Av~x : F!�

�x�0

1

xm f ~x! dx

�x�0

1

m f ~x! dx

(4)

In general, the first two methods are easier to calculate than the last two,especially when the membership function is nonlinear. However, the fuzzy modemay lead to a biased result when the membership function is extremely asymmet-rical. The fuzzy midrange is more flexible because one can choose different levelsof membership (a) of interest. If the area under the membership function is con-sidered to be an appropriate measure of fuzziness, the fuzzy median may be thoughtto be suitable. When anyone wants to account for the shape of the membershipfunction as well as its location, the fuzzy average will then be a better choicebecause it is derived from the extension principle and is basically a weighted aver-age of the base variable.8

A sample consists of several observations selected for inspection. Each obser-vation is classified by a linguistic term (or value) and is associated with a knownmembership function. These linguistic values need to be combined to yield a sin-gle value for the sample. This combination may be done either before or after theconversion of fuzzy subsets associated with linguistic terms into their representa-tive values.


3. FUZZY CONTROL CHARTS

In the literature, different procedures are proposed to monitor multinomialprocesses when products are classified into mutually exclusive linguistic catego-ries. Marcucci2 proposed two procedures using Shewhart-type control charts. Thefirst type is used when quality proportions are designed to be specific values, whereany change in these proportions must be detected by the monitoring procedure. Inthis case, a standard statistical procedure used to monitor such a multinomial pro-cess is Pearson goodness-of-fit statistic:

Yi2 � (

j�1

t ~Xij � nipj !2

nipj

(5)

where pjs are proportions, ni is the ith sample size, and Xijs are the numbers ofobservations in categories 1,2, . . , t. When the process is in control, the asymptoticdistribution of Yi

2 is x~t�1!2 , a chi-squared distribution with ~t � 1) degrees of

freedom.The second type allows for one-sided monitoring of quality proportions and

is designed to detect only an increase in all but one quality proportion. When spe-cific values of process proportions are not known, the Pearson goodness-of-fitstatistic is not applicable. An appropriate statistical procedure that is a test ofhomogeneity of proportions between the base period (0) and each monitoring periodi is defined as follows:

Zi2 � (

k�i,0(j�1

tnk� Xkj

nk

�Xij � X0j

ni � n0�2

Xij � X0j

ni � n0

� ni n0(j�1

t ~ pij � p0j !2

Xij � X0j

(6)

where k � $0, i %, pkj � Xkj /nk , j � 1,2, . . . , t, are the sample proportions, and ni

is the sample size.Wang and Raz8 developed two approaches called the membership approach

and the fuzzy probabilistic approach. In the membership approach, membershipcontrol limits are based on membership functions. For m samples of size n, Wangand Raz8 described the center line, CL, as the grand average of the means of thesamples initially available (Equation 7), and, as illustrated in Figure 1, calculatedthe mean deviation for a given fuzzy set A ~d~A!! (Equation 9), by using the sumof the left mean deviation dl , and the right mean deviation dr :

CL � Mj �

(j�1

m

Mj

m(7)

where Mj is the sample mean of the j th sample, RMj is the average sample mean,and m is the number of sample initially available:


Mj �

(j�1

m

kij ri

nj

i � 1,2, . . . , t (8)

where

kij � number of products categorized with the linguistic term i in the samplej,

ri � fuzzy representative value of the linguistic term i ,

nj � size of the sample j;

d~A! � dl ~A!� dr ~A!

��a�0

1

@xm � xl ~a!# da��a�0

1

@xr ~a!� xm # da

��a�0

1

@xr ~a!� xl ~a!# da (9)

where a is the value of membership.The control limits are located below and above the central line at distances

expressed as multiples of the mean deviation. Because the representative value ofeach sample will be in the range [0, 1],

Membership LCL � Max$0, ~CL � kd!%

MembershipUCL � Min$1, ~CL � kd!%(10)

where CL is the center line and k is the number of mean deviations that the controllimits will be located away from the center line.

In the fuzzy probabilistic approach, fuzzy subsets Fi associated with thelinguistic terms Li are transformed into their respective representative values ri

with one of the transformation methods. The sample mean Mj is calculated as theaverage of the sample linguistic representative values, ri . For each sample j, the

Figure 1. Mean deviation of a fuzzy subset.


standard deviation SDj is calculated as the standard deviation of the representativevalues of the observations in the sample:

SDj � � 1

n � 1 (i�1

t

kij ~ri � Mj !2 (11)

where t is the number of linguistic terms in the term set, ri is the representativevalue of the fuzzy set associated with linguistic term Li , and Mj is the mean of therepresentative values in the sample j. The mean of the standard deviations of the msamples, MSD, is then

MSD �1

m (j�1

m

SDj (12)

The center line is calculated as the grand mean of the sample means Mj as follows:

CL �

(j�1

m

Mj

m�

(j�1

m

(i�1

t

ri kij

mn(13)

Because the points plotted on the charts are sample means of representative val-ues, they should lie within the range [0,1]. Consequently, assuming the samplingdistribution is approximately normal, or the sample size n is relatively large (�25),by applying the standard formulae from variables control charts control limits arederived as

Probabilistic LCL � Max$0, ~CL � A3 MSD!%

Probabilistic UCL � Min$1, ~CL � A3 MSD!%(14)

where

A3 �3

c4Mnand c4 � � 2

n � 1

� n � 2

2 �!

� n � 3

2 �!

(15)

Kanagawa et al.9 introduced new linguistic control charts for process averageand process variability based on the estimation of probability distribution exist-ing behind the linguistic data. They adopted the probabilistic approach devel-oped by Wang and Raz8,18 and designed control charts by using upper andlower confidence limit of level 1 � a. Their study aimed at directly control-ling the underlying probability distributions of the linguistic data, which werenot considered by Wang and Raz.8 They recommended that skewness and kurto-sis be used instead of cumulants kr , because they play a role of the indices ofnormality.


Control limits for a sample of n items is calculated as

UCL � CL* �SD *

Mn �ua/2 �SKW *

6Mn~ua/22 � 1!�

KUT *

24n~ua/23 � 3ua/2 !

�~SKW * !2

36n~2ua/23 � 5ua/2 !

�LCL � CL* �

SD *

Mn �u~1�a/2!�

SKW *

6Mn~u~1�a/2!

2 � 1!

�KUT *

24n~u~1�a/2!

3 � 3u~1�a/2! !

�~SKW * !2

36n~2u~1�a/2!

3 � 5ua/2 !� (16)

where CL, SD*, SKW * , and KUT * are average mean (center line), average stan-dard deviation, average skewness, and average kurtosis, respectively, and deter-mined as

CL* �1

m (j�1

m

kj1 (17)

~SD * !2 �1

m (j�1

m

kj2 (18)

SKW * �1

m

1

~SD * !3 (j�1

m

kj3 (19)

KUT * �1

m

1

~SD * !4 (j�1

m

kj4 (20)

As for plotting points on this control chart, representative values of each linguisticvariable are calculated by the use of the barycenter concerned with Zadeh’s prob-ability measure of fuzzy events as follows:

xi � Rep~Fi !�

��`

`

xm i ~x! f ~x! dx

��`

`

m i ~x! f ~x! dx

(21)

To control the process more precisely, they developed control charts for control-ling process variability. Standard deviation of the j th sample is defined as follows:

SDj � � 1

n � 1 (i�1

t

kij ~xi � Mj !2� 1/2

(22)


where

Mj �1

n (i�1

t

kij xi (23)

Kanagawa et al.9 proposed control charts for linguistic data with the above valuesof UCL, LCL, and CL, from a standpoint different to that of Wang and Raz notonly to control the process average but also to control the process variability.

4. a-CUT FUZZY CONTROL CHARTS FOR ATTRIBUTES

Control charts for attributes are obtained for the fraction rejected as noncon-forming to specifications, number of nonconforming items, number of nonconfor-mities, and number of nonconformities per unit. In the following, first, fuzzy controlcharts for the fraction rejected are obtained. Later, the membership function anda-cut control limits for number of nonconformities are given. The other controllimits for attributes can be obtained in the same way.

In the crisp case, control limits for the fraction rejected are calculated by thefollowing equations:

CL � Tp

LCL � Tp � 3� pq

n(24)

UCL � Tp � 3� pq

n

In the fuzzy case, the mean of each sample, Mj , and center line, CL, are calculatedby Equations 7 and 8, respectively. Because the CL is a fuzzy set, it can be repre-sented by triangular fuzzy numbers (TFNs) whose fuzzy mode is CL, as shown inFigure 2.

Then for each sample mean, Mj , Lj~a! and Rj~a! can be calculated as below:

Lj ~a! � Mja

Rj ~a! � 1 � @~1 � Mj !a#(25)

Membership function of the RM, or CL, can be written as

mMj~x! � �

0, if x � 0

x

RM, if 0 � x � RM

1 � x

1 � RM, if RM � x � 1

0, if x � 1

(26)


The control limits for a-cut are also a fuzzy set and can be represented byTFNs. Because the membership function of CL is divided into two components,then each component will have its own CL, LCL, and UCL. The membershipfunction of the control limits depending upon the value of a is given below:

Control Limits~a!

�

�CLL � RMa

LCLL � max CLL � 3� ~CLL !~1 � CLL !

Sn,0

UCLL � min CLL � 3� ~CLL !~1 � CLL !

Sn,1� , if 0 � Mj � RM

�CLR � 1 � @~1 � RMa!a#

LCLR � max CLR � 3� ~CLR !~1 � CLR !

Sn,0

UCLR � min CLR � 3� ~CLR !~1 � CLR !

Sn,1� , if RM � Mj � 1

(27)

where Sn is the average sample size ~ASS!. When the ASS is used, the control limitsdo not change with the sample size. Hence, the control limits for all samples arethe same. A general illustration of these control limits is shown in Figure 3.

For the variable sample size ~VSS!, Sn should be replaced by the size of thej th sample, nj . Hence, control limits change for each sample depending upon thesize of the sample. Therefore, each sample has its own control limits.

The decision whether the process is in control (1) or out of control (0) forboth ASS and VSS is as follows:

Figure 2. TFN representation of RM and Mj of the sample j.


Process Control � 1, if LCLL~a!� Lj ~a!� UCLL~a! ∧ LCLR~a!

� Rj ~a!� UCLR~a!

0, otherwise

(28)

The value of a-cut is decided with respect to the tightness of inspectionsuch that for a tight inspection, a values close to 1 may be used. As can be seenfrom Figure 3, whereas a reduces to 0 (decreasing the tightness of inspection),the range where the process is in control (difference between UCL and LCL!increases.

Similarly, the crisp control limits for c charts are as follows:

CL � Sc

LCL � Sc � 3M Sc (29)

UCL � Sc � 3M Sc

where Sc is the number of average nonconformities for the samples initially available.In the fuzzy case, TFN representation of Sc is as shown in Figure 4.Membership function of the Sc, or CL, can be written as

mcj~x! � �

x � cmin

Sc � cmin, if cmin � x � Sc

x � cmax

Sc � cmax, if Sc � x � cmax

(30)

Figure 3. Illustration of the a-cut control limits ~ASS!.


Corresponding control limits for Sc charts are

Control Limits ~a!

� � �CLL � cmin � ~ Sc � cmin!a

LCLL � max$CLL � 3Mcmin � ~ Sc � cmin!a,0%

UCLL � CLL � 3Mcmin � ~ Sc � cmin!a� , if cmin � cj � Sc

�CLR � cmax � ~ Sc � cmax!a

LCLR � max$CLR � 3Mcmax � ~ Sc � cmax!a,0%

UCLR � CLR � 3Mcmax � ~ Sc � cmax!a� , if Sc � cj � cmax

(31)

where cj is the number of nonconformities in the sample j.The decision whether the process is in control (1) or out of control (0) is as

follows:

Process Control � 1, if LCLL~a!� Lj ~a!� UCLL~a! ∧ LCLR~a!

� Rj ~a!� UCLR~a!

0, otherwise

(32)

where j is the sample number and

Lj ~a! � cjmin� ~cj � cjmin

!a

Rj ~a! � cjmax� ~cj � cjmax

!a(33)

~cjmin, cj , cjmax

) is the TFN of the number of nonconformities for the sample j.

Figure 4. TFN representation of Sc.


5. A NUMERICAL EXAMPLE FOR THE FRACTION REJECTED

To compare our approach, a numerical example of the Tunisie Porcelaine prob-lem stated by Wang and Raz8 and Taleb and Limam17 will be handled. In theexample presented, Taleb and Limam17 classified porcelain products into four cat-egories with respect to the quality. When a product represents no default or aninvisible minor default, it is classified as a standard product ~S!. If it presents avisible minor default that does not affect the use of the product, then it is classifiedas second choice ~SC!. When there is a visible major default that does not affectthe product use, it is denoted third choice ~TC!. Finally, when the use is affected,the item is considered as chipped ~C!. Data for 30 samples of different sizes takenevery half hour are shown in Table II.

Table II. Data of the porcelain process.

Sample Standard Second choice Third choice Chipped Size

1 144 46 12 5 2072 142 50 9 5 2063 142 35 16 6 1994 130 70 19 10 2295 126 60 15 10 2116 112 47 9 8 1767 151 28 22 9 2108 127 43 45 30 2459 102 79 20 3 204

10 137 64 24 5 23011 147 59 16 6 22812 146 30 6 6 18813 135 51 16 8 21014 186 82 23 7 29815 183 53 11 9 25616 137 65 26 4 23217 140 70 10 3 22318 135 48 15 9 20719 122 52 23 10 20720 109 42 28 9 18821 140 31 9 4 18422 130 22 3 8 16323 126 29 11 8 17424 90 23 16 2 13125 80 29 19 8 13626 138 55 12 12 21727 121 35 18 10 18428 140 35 15 6 19629 110 15 9 1 13530 112 37 28 11 188


5.1. Wang and Raz’s Approaches

5.1.1. Probablistic Approach

We will use the fuzzy mode as the representative value of the fuzzy subset.For each sample j, sample mean Mj and the standard deviation SDj are determined.The results of these values, their means, and the corresponding control limits, areshown in Table III. The control limits change when the sample size changes. Onlyon two occasions the process is deemed to be out of control: samples 8 and 29 asshown in Figure 5.

LCL and UCL for each sample is calculated using Equation 14.

5.1.2. Membership Approach

For each sample, the membership function of the fuzzy subset correspondingto the sample observations is determined. Membership function for the porcelainprocess is as follows:

mS ~x! � �0, if x � 0

�x � 1, if 0 � x � 1

0, if x � 1

mSC ~x!� �0, if x � 0

4x, if 0 � x �1

4

�4

3x �

4

3, if

1

4� x � 1

0, if x � 1

(34)

Table III. Determined values of Mj , SDj , UCLj , LCLj for 30-item subgroup.

Sample j Mj SDj UCLj LCLj Sample j Mj SDj UCLj LCLj

1 0.109 0.20 0.184 0.088 16 0.143 0.21 0.170 0.0912 0.107 0.20 0.164 0.088 17 0.114 0.18 0.174 0.0903 0.114 0.22 0.183 0.085 18 0.138 0.24 0.177 0.0884 0.162 0.24 0.178 0.091 19 0.167 0.25 0.178 0.0885 0.154 0.24 0.182 0.089 20 0.178 0.26 0.182 0.0866 0.138 0.24 0.181 0.084 21 0.088 0.19 0.182 0.0857 0.129 0.25 0.179 0.089 22 0.092 0.23 0.188 0.0828 0.258 0.34 0.179 0.092 23 0.119 0.24 0.186 0.0849 0.161 0.19 0.180 0.088 24 0.120 0.21 0.198 0.076

10 0.143 0.21 0.174 0.091 25 0.182 0.27 0.195 0.07711 0.126 0.21 0.178 0.090 26 0.146 0.25 0.190 0.08912 0.088 0.21 0.179 0.086 27 0.151 0.26 0.193 0.08513 0.137 0.23 0.177 0.089 28 0.114 0.22 0.180 0.08714 0.131 0.21 0.174 0.096 29 0.069 0.16 0.193 0.07715 0.108 0.22 0.175 0.093 30 0.182 0.27 0.183 0.086

Average 0.136 0.229


mTC ~x! � �0, if x � 0

2x, if 0 � x �1

2

2 � 2x, if1

2� x � 1

0, if x � 1

mC ~x!� �0, if x � 0

x, if 0 � x � 1

0, if x � 1

By the use of the fuzzy mode transformation, the representative values for fuzzysubsets shown in Table IV are determined.

The membership functions for the porcelain data are also illustrated in Figure 6.By applying the membership approach to the porcelain data, a fuzzy member-

ship control chart is obtained as in Figure 7. As can be seen from Figure 7, only

Figure 5. Fuzzy probabilistic control chart with fuzzy mode.

Table IV. Representative values oflinguistic terms.

Linguistic term Representative value

S 0SC 0.25TC 0.5C 1


samples 8 and 29 are in an out of control state. Note that fuzzy control limits, here,are calculated as follows:

LCL � max$0, @CL � ks#%

UCL � min$1, @CL � ks#%(35)

Figure 6. Membership functions for the porcelain data.

Figure 7. Fuzzy membership control chart with fuzzy mode transformation.


where

s �1

m (j�1

m

SDj (36)

and

SDj � � 1

n � 1 (i�1

t

kij ~rj � Mj !2 (37)

The value of k is calculated by the use of Monte Carlo simulation so that aprespecified type I error probability yields. In this example, the value of k, usedhere, is approximately 0.2795, CL is 0.136, and s is 0.229. Then UCL and LCL arecalculated as 0.200 and 0.072, respectively.

Assume that the quality control expert decides for reduced inspection, saya � 0.30. If we apply the ASS approach, then center lines and control limits fora� 0.30 can be determined using Equation 31 as

CLL~a � 0.30!� 0.040800 CLR~a� 0.30!� 0.740800

LCLL~a � 0.30!� 0.0 LCLR~a� 0.30!� 0.648321

UCLL~a � 0.30!� 0.082550 UCLR~a� 0.30!� 0.833279

As can be seen from Figure 8, the corresponding control chart for a� 0.30, all thesamples are in control.

For a tighter inspection with a� 0.50, the control chart is obtained as shownin Figure 9. Note that sample 8 begins to be out of control when a is chosen as0.39 or greater.

Figure 8. a-Cut fuzzy control chart for a� 0.30 ~ASS approach).


When a� 1.0, the crisp control limits are obtained as in Figure 10.If we use the VSS approach for the same example, control charts for a �

0.30, a � 0.50, and a � 1.0 are obtained as in Figures 11–13. When increasinga-cut, namely tightening the inspection, sample 8 starts to be out of control whena � 0.33.

Wang and Raz8 attempted to extend the use of control charts to linguisticvariables by presenting several ways for determining the center line and the

Figure 9. a-Cut fuzzy control chart for a� 0.50 ~ASS approach).

Figure 10. a-Cut fuzzy control chart for a� 1.0 (Crisp Case, ASS approach).


control limits. Kanagawa et al.9 proposed control charts for linguistic data withthe above values of UCL, LCL, and CL, from a standpoint different to that ofWang and Raz not only to control the process average but also to control theprocess variability.

Figure 11. a-Cut fuzzy control chart for a� 0.30 ~VSS approach).

Figure 12. a-Cut fuzzy control chart for a� 0.50 ~VSS approach).


Our approach differs from previous studies from the point of view of inspec-tion tightness. The quality controller is able to define the tightness of the inspec-tion depending on the nature of the products and manufacturing processes. This ispossible by selecting the value of a-cut freely. The quality controller may decideto use higher values of a-cut for products that require a tighter inspection. Ourapproach is very easy in computation and similar to the crisp control charts.

6. CONCLUSION

In crisp control charts, all parameters should be well defined. Very often,such an assumption appears too rigid for the real-life problems, especially whendealing with linguistic or imprecise data. To relax this rigidity, fuzzy methods areincorporated into control charts. Especially, in the control charts for attributes,fuzzy methods should be used because the data are usually linguistic. The repre-sentation of linguistic variables as fuzzy sets, which can be manipulated with therules of fuzzy arithmetics, retains the ambiguity and vagueness inherent in naturallanguages and improves the expressive ability of quality assurance inspectors. Inthis article, we have defined the tightness of inspection by the use of a-cut controlcharts. a-Level is determined according to the nature of the products and manu-facturing processes. For a toy factory, a-level might be accepted between 0.6 and0.8 whereas it is between 0.95 and 1.0 for a motor factory. In the case of fuzzydata, our approach is more usable. It is flexible, not complex, easy in computation,and similar to the crisp control charts for attributes. It has the ability of detectingout of control points at least as effectively as the other approaches do. Future

Figure 13. a-Cut fuzzy control chart for a� 1.0 (Crisp Case, VSS approach).


studies may include the development of a-cut fuzzy control charts for variables inthe case of fuzzy data.

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α-cut fuzzy control charts for linguistic data

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