an alternative approach to fuzzy control charts: direct fuzzy approach

Information Sciences 177 (2007) 1463–1480

www.elsevier.com/locate/ins

An alternative approach to fuzzy control charts:Direct fuzzy approach

Murat Gulbay *, Cengiz Kahraman

Istanbul Technical University, Department of Industrial Engineering, Macka 34367, Istanbul, Turkey

Received 10 May 2004; received in revised form 27 July 2006; accepted 4 August 2006

Abstract

The major contribution of fuzzy set theory lies in its capability of representing vague data. Fuzzy logic offers a system-atic base to deal with situations, which are ambiguous or not well defined. In the literature, there exist few papers on fuzzycontrol charts, which use defuzziffication methods in the early steps of their algorithms. The use of defuzziffication meth-ods in the early steps of the algorithm makes it too similar to the classical analysis. Linguistic data in those works are trans-formed into numeric values before control limits are calculated. Thus both control limits as well as sample values becomenumeric. In this paper, some contributions to fuzzy control charts based on fuzzy transformation methods are made by theuse of a-cut to provide the ability of determining the tightness of the inspection: the higher the value of a the tighter inspec-tion. A new alternative approach ‘‘Direct Fuzzy Approach (DFA)’’ is also developed in this paper. In contrast to the exist-ing fuzzy control charts, the proposed approach is quite different in the sense it does not require the use of thedefuzziffication. This prevents the loss of information included by the samples. It directly compares the linguistic datain fuzzy space without making any transformation. We use some numeric examples to illustrate the performance of themethod and interpret its results.� 2006 Elsevier Inc. All rights reserved.

Keywords: Fuzzy control charts; Membership approach; Direct fuzzy Approach; a-Cut fuzzy control charts; Linguistic data

1. Introduction

A control chart is a tool that is commonly used to monitor and examine a process. It graphically depicts theaverage value and the upper and lower control limits of a process. The power of control charts lies in theirability to detect process shifts and to indicate abnormal conditions in a production process.

Even though the first control chart was proposed during the 1920s by Shewhart, today they are still subjectto new application areas that deserve further attention. The control charts introduced by Shewhart [27] weredesignated to monitor processes for shifts in the mean or variance of a single quality characteristic. Manyapproaches have been suggested to improve the performance of the control charts proposed by Shewhart:

0020-0255/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2006.08.013

* Corresponding author. Tel.: +90 212 2931300x2073; fax: +90 212 2407260.E-mail address: [email protected] (M. Gulbay).

mailto:[email protected]

1464 M. Gulbay, C. Kahraman / Information Sciences 177 (2007) 1463–1480

CUSUM Charts [22,9,10,34], EWMA Charts [25,5,21], adaptive control charts [24,6,4,7], fuzzy control charts[23,31,19,15,16,11,13,12,14]. A bibliography of control charts for attributes is presented by Woodall [33].

When the quality-related characteristics, such as appearance, softness, color, etc., cannot be represented innumerical values, the control charts for attributes are used. Product units are classified as either conforming ornonconforming, depending upon whether or not they meet specifications. The number of nonconformities(deviations from specifications) can also be counted. The binary classification into conforming and noncon-forming used in the p-chart might not be appropriate in many situations where product quality does notchange abruptly from satisfactory to worthless, and there might be a number of intermediate levels. Withoutfully utilizing such intermediate information, the use of the p-chart usually results in poorer performance thanthat of the �x-chart. This is evidenced by weaker detectability of process shifts and other abnormal conditionssuch as unnatural patterns [32]. To supplement the binary classification, several intermediate levels may beexpressed by using linguistic terms. For example, the quality of a product can be classified into the followingterms: ‘perfect’, ‘good’, ‘medium’, ‘poor’, or ‘bad’ depending on its deviation from specifications. Then, thecontinuous functions selected appropriately can be used to describe the quality characteristic associated witheach linguistic term. In this study, the control charts for number of nonconformities are handled. The type ofavailable data is the imprecise number of nonconformities such as ‘‘between 5 and 8’’ or ‘‘approximately 6’’.The statistical model is based on the classical Shewhart control charts.

A research work incorporating uncertainty into decision analysis is basically done through the probabilitytheory and/or the fuzzy set theory. The former represents the stochastic nature of decision analysis while thelatter captures the subjectivity of human behavior. A rational approach toward decision-making should takehuman subjectivity into account, rather than employing only objective probability measures. The fuzzy settheory is a perfect means for modeling uncertainty (or imprecision) arising from mental phenomena whichis neither random nor stochastic.

In the literature, there are few papers on fuzzy control charts pertained to represent uncertainty in humancognitive processes. When human subjectivity plays an important role in defining the quality characteristics,the classical control charts may not be applicable since they require certain information. The judgment in clas-sical process control results in binary classification as ‘‘in-control’’ or ‘‘out-of-control’’ while fuzzy controlcharts may handle several intermediate decisions. Fuzzy control charts are inevitable to use when the statis-tical data in consideration are uncertain or vague; or available information about the process is incomplete orincludes human subjectivity. A general comparison of traditional Shewhart control charts and fuzzy controlcharts is given in Table 1.

Different procedures are proposed to monitor multinomial processes when products are classified intomutually exclusive linguistic categories. Bradshaw [2] used the fuzzy set theory as a basis for interpretingthe representation of a graded degree of product conformance. Bradshaw [2] stressed that fuzzy economic con-trol limits would be advantageous over traditional acceptance charts in that fuzzy economic control charts

Table 1Comparison of traditional Shewhart and fuzzy control charts

Comparison issue Traditional Shewhart control charts Fuzzy control charts

Number of qualitycharacteristics

Only one quality characteristic Multiple quality characteristics

Availability and type ofstatistical data

Completely required and certain Vague, uncertain, and incomplete information

Information used inbase period

Historical data Experts’ experience rules

Judgment In control or out of control Further intermediate linguistic decisionsAdvantages 1. Easier for considering one quality

characteristic2. More objective

1. Provide more accurate control standards for the process basedon experts’ experience expressed in degree of membership

2. More flexible for the definitions of the fuzzy inference rulesDisadvantages 1. Inflexible control limits

2. Sample size influences the widthof control limits

3. Historical data are needed toobtain the formal control limits

1. Inference outcomes are based on the subjective experience rules2. Supplemental rules (for systematic changes) of the traditional

control charts cannot be used

M. Gulbay, C. Kahraman / Information Sciences 177 (2007) 1463–1480 1465

provide information on severity as well as the frequency of product nonconformance. Raz and Wang [23] pro-posed an approach based on the fuzzy set theory by assigning a fuzzy set to each linguistic term. Wang andRaz [31] developed two approaches called fuzzy probabilistic approach andmembership approach. The fuzzy

probabilistic approach is based on both the fuzzy set theory and the probability theory and so-called fuzzyprobabilistic approach. In fuzzy probabilistic approach, fuzzy subsets associated with the linguistic termsare transformed into their respective representative values using one of the transformation methods. Centerline (CL) corresponds to the arithmetic mean of representative values of the samples initially available. Mem-

bership approach is based on the fuzzy set theory to combine all observations in only one fuzzy subset usingfuzzy arithmetic. Membership control limits are based on membership functions. In membership approach,center line is located as the value of the representative value of the aggregate fuzzy subset. In the control chartsproposed by Raz and Wang [23], and Wang and Raz [31], the control limits are composed of classical valuessince the samples denoted by linguistic variables are transformed into classical values by the use of fuzzy trans-formation. The control limits are then calculated on the base of these transformed values.

Apart from the fuzzy probabilistic and fuzzy membership approaches, Kanagawa et al. [19] introduced mod-ifications to the construction of control charts given by Wang and Raz [30,31]. Their study aimed at directlycontrolling the underlying probability distributions of the linguistic data, which were not considered by Wangand Raz [31]. Kanagawa et al. [19] proposed control charts for linguistic data from a standpoint different to thatof Wang and Raz in order not only to control the process average but also to control the process variability.They presented new linguistic control charts for process average and process variability based on the estimationof probability distribution existing behind the linguistic data. They defined the center line as the average mean ofthe sample cumulants and then calculated the control limits using Gram–Charlier series. The main difficulty ofthis approach is that the unknown probability distribution function cannot be determined easily. These proce-dures are reviewed by Woodall et al. [35] and discussed by Laviolette et al. [20] and Asai [1]. Wang and Chen [29]presented a fuzzy mathematical programming model and a heuristic solution for the economic design of statis-tical control charts. They argued that under the assumptions of the economic statistical model, the fuzzy pro-cedure presented an improved economic design of control charts by allowing more flexibility in modeling theexisting imprecision. Kahraman et al. [18] used triangular fuzzy numbers in the tests of control charts for unnat-ural patterns. Chang and Aw [3] proposed a neural fuzzy control chart for identifying process mean shifts. Asupervised multilayer backpropagation neural network is trained off-line to detect various mean shifts in a pro-duction process. In identifying mean shifts in real-time usage, the neural network’s outputs are classified intovarious decision regions using a fuzzy set scheme. The approach offers better performance and additional advan-tages over conventional control charts. Woodall et al. [35] gave a review of statistical and fuzzy control chartsbased on categorical data. El-Shal and Morris [8] investigated the use of fuzzy logic to modify the statistical pro-cess control (SPC) rules. They aimed at reducing the generation of false alarms and improving the detection anddetection-speed of real faults. Rowlands and Wang [26] explored the integration of fuzzy logic and controlcharts to create and design a fuzzy-SPC evaluation and control method based on the application of fuzzy logicto the SPC zone rules. Hsu and Chen [17] described a new diagnosis system based on fuzzy reasoning to monitorthe performance of a discrete manufacturing process and to justify the possible causes. The diagnosis systemmainly consists of a knowledge bank and a reasoning mechanism. Taleb and Limam [28] discussed different pro-cedures of constructing control charts for linguistic data, based on the fuzzy set and probability theories. A com-parison between the fuzzy and probabilistic approaches, based on the average run length and the samples undercontrol, is made by using real data. Contrary to the conclusions of Raz and Wang [23] the choice of degree offuzziness affects the sensitivity of control charts. Grzegorzewski and Hryniewicz [13] proposed a new fuzzy con-trol chart based on the necessity index. This work has a significant worth in the literature of the fuzzy controlcharts since this index has natural interpretation and effectiveness in solving real problems.

This paper aims at developing fuzzy approaches to control charts based on fuzzy transformation methods,which are fuzzy mode, fuzzy midrange, and fuzzy median. We use an a-cut approach to provide the ability ofdetermining the tightness of the inspection: the higher the value of a the tighter inspection. We also present anew fuzzy approach to control charts: direct fuzzy approach (DFA). The first three approaches are mainlybased on the fuzzy transformation methods while the proposed DFA is based on a fuzzy comparison method.DFA provides the ability of making linguistic decisions like ‘‘rather in control’’ or ‘‘rather out of control’’. Fur-ther intermediate levels of process control decisions are also possible to introduce.


This paper is organized in the following order. In Section 2, the fuzzy control charts existing in the literatureare integrated with an a-cut approach. In Section 3, a new fuzzy approach to control charts, DFA, is pro-posed. In Section 4, the results of the developed approaches are compared and interpreted with a numericalexample. Finally, in Section 5, the concluding remarks are given.

2. Fuzzy control charts based on a-cuts

In the literature, fuzzy control charts have been developed by converting the fuzzy sets associated with lin-guistic values into scalars referred to as representative values. This conversion, which facilitates the plotting ofobservations on the chart, may be performed in a number of ways as long as the result is intuitively represen-tative of the range of the base variable included in the fuzzy set. Four ways, which are similar to the measuresof central tendency used in descriptive statistics, are fuzzy mode, a-level fuzzy midrange, fuzzy median, andfuzzy average. It should be pointed out that there is no theoretical basis supporting any one specifically.The selection among them should be mainly based on the ease of computation or the user’s preference [31].When a fuzzy sample is asymmetric, these fuzzy transformation methods result in different crisp numbers,and therefore a process control based on these transformations may result in different decisions.

Classical control charts for attributes can be obtained for fraction rejected as nonconforming to specifica-

tions, number of nonconforming items, number of nonconformities, and number of nonconformities per unit. Inthe crisp case, control limits for number of nonconformities are calculated by the following equations:

Fig. 1.

CL ¼ �c ð1ÞLCL ¼ �c� 3

ffiffiffi�cp

ð2ÞUCL ¼ �cþ 3

ffiffiffi�cp

ð3Þ
where �c is the mean of the nonconformities. In the fuzzy case, each sample, or subgroup, is represented by atrapezoidal fuzzy number (a,b,c,d) or a triangular fuzzy number (a,b,d), or (a,c,d) as shown in Fig. 1. Notethat a trapezoidal fuzzy number becomes triangular when b = c. For the ease of representation and calcula-tion, a triangular fuzzy number is also represented as trapezoidal by (a,b,b,d) or (a,c,c,d).
The center line, gCL, given in Eq. (4), is the mean of fuzzy samples, and it is shown as ð�a; �b;�c; �dÞ where �a, �b,�c, and �d are the arithmetic means of a, b, c, and d, respectively.

gCL ¼Pm

j¼1aj

m;

Pmj¼1bj

m;

Pmj¼1cj

m;

Pmj¼1dj

m

� �¼ ð�a; �b;�c; �dÞ ð4Þ

where m is the number of fuzzy samples.Since the gCL is a fuzzy set, it can be represented by a fuzzy number whose fuzzy mode (multimodal) is the

closed interval of ½�b;�c�. gCL, gLCL, and gUCL are calculated as follows:

0

1

μ

α

b c da 0

1

μ

α

b=c daaα dα aαdα

Representation of a sample by trapezoidal and/or triangular fuzzy numbers: (a) Trapezoidal (a,b,c,d) and (b) triangular (a,b,b,d).


gCL ¼ ð�a; �b;�c; �dÞ ¼ ðCL1;CL2;CL3;CL4Þ ð5Þ

gLCL ¼gCL � 3

ffiffiffiffiffiffiffiffigCL

q¼ ð�a; �b;�c; �dÞ � 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�a; �b;�c; �dÞ

q¼ �a� 3

ffiffiffi�d

p; �b� 3

ffiffiffi�cp;�c� 3

ffiffiffi�b

p; �d � 3

ffiffiffi�ap� �

¼ ðLCL1;LCL2;LCL3;LCL4Þ ð6Þ

gUCL ¼gCL þ 3

ffiffiffiffiffiffiffiffigCL

q¼ ð�a; �b;�c; �dÞ þ 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�a; �b;�c; �dÞ

q¼ �aþ 3

ffiffiffi�ap

; �bþ 3ffiffiffi�b

p; cþ 3

ffiffiffi�cp; �d þ 3

ffiffiffi�d

p� �¼ ðUCL1;UCL2;UCL3;UCL4Þ ð7Þ

An a-cut is a nonfuzzy set which comprises all elements whose membership degrees are greater than or equalto a. Applying a-cuts of fuzzy sets (Fig. 1), the values of aa and da are determined as follows:

aa ¼ aþ aðb� aÞ ð8Þda ¼ d � aðd � cÞ ð9Þ

Similarly a-cut fuzzy control limits can be stated as follows:
gCLa ¼ ð�aa; �b;�c; �daÞ ¼ ðCLa1;CL2;CL3;CLa
4Þ ð10Þ

gLCLa ¼ gCLa � 3

ffiffiffiffiffiffiffiffiffigCLa

q¼ ð�aa; �b;�c; �daÞ � 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�aa; �b;�c; �daÞ

q¼ �aa � 3

ffiffiffiffiffi�da

p; �b� 3

ffiffiffi�cp;�c� 3

ffiffiffi�b

p; �da � 3

ffiffiffiffiffi�aap� �

¼ ðLCLa1;LCL2;LCL3;LCLa

4Þ ð11Þ

gUCLa ¼ gCLa þ 3

ffiffiffiffiffiffiffiffiffigCLa

q¼ ð�aa; �b;�c; �daÞ þ 3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�aa; �b;�c; �daÞ

q¼ �aa þ 3

ffiffiffiffiffi�aap

; �bþ 3ffiffiffi�b

p;�cþ 3

ffiffiffi�cp; �da þ 3

ffiffiffiffiffi�da

p� �¼ ðUCLa

1;UCL2;UCL3;UCLa4Þ ð12Þ

The results of these equations can be illustrated as in Fig. 2. The value of a-cut can be interpreted as the tight-ness of the inspection and so can subjectively be defined by the quality manager: the higher the value of a thetighter inspection. When a-cut is set to 1, control chart tends to the classical case.

In the following, some alternative approaches to fuzzy control charts depending on the fuzzy transforma-tion methods are developed and the DFA approach for number of nonconformities is proposed and comparedwith the other approaches.

2.1. Fuzzy control charts based on fuzzy mode transformation

The fuzzy mode of a fuzzy set f is the value of the base variable, X, where the membership value equals to 1.This is stated as

fmod ¼ fx 2 X jlf ðxÞ ¼ 1g ð13Þ

If the membership function is unimodal, the fuzzy mode is unique. The fuzzy numbers of nonconformities ofinitially available and incoming samples are transformed to crisp numbers via the fuzzy mode transformation.Since a trapezoidal membership function is multimodal, the fuzzy mode is the set of points between b and c,

[b,c]. gUCL,gCL, and gLCL given in Eqs. (4)–(6) are transformed to their representative values using fuzzy modeto determine control limits for these charts. Since gUCL, gCL, and gLCL are multimodal, their fuzzy modes areclosed intervals whose membership degrees are 1. Referring to the representation in Fig. 2, the fuzzy mode ofsample j, Smod,j, and the corresponding closed intervals of CL, LCL, and UCL are determined by the followingequations:

Smod;j ¼ ½bj; cj� ð14ÞCLmod ¼ fmodðgCLÞ ¼ ½CL2;CL3� ð15Þ

LCLmod ¼ CLmod � 3ffiffiffiffiffiffiffiffiffiffiffiffiffiCLmod

p¼ ðCL2 � 3

ffiffiffiffiffiffiffiffiffiCL2

pÞ; ðCL3 � 3


pÞ

h i¼ ½LCL2;LCL3� ð16Þ

UCLmod ¼ CLmod þ 3ffiffiffiffiffiffiffiffiffiffiffiffiffiCLmod

p¼ ðCL2 þ 3


pÞ; ðCL3 þ 3


pÞ

h i¼ ½UCL2;UCL3� ð17Þ

10

3UCL

2UCL

1UCL

4CL

4LCL

1CL

3CL

3LCL

2LCL

4UCLα

1UCLα

4CLα

1CLα

α4LCL

α1LCL

4UCL

μ

~UCL

~CL

~LCL

2CL

1LCL

α

Fig. 2. Representations of gUCL, gCL, and gLCL.


where fmodðgCLÞ is the fuzzy mode of the fuzzy center line. After the calculation of control limits, the fuzzymode of each sample is compared to these intervals to determine whether the sample indicates in-controlor out-of-control situation for the process. If the set of Smod,j is totally covered by the control limits, then sam-ple j is strictly ‘‘in-control’’. Reversely, a sample whose fuzzy mode set is completely outside the control limitsis strictly ‘‘out-of-control’’. Alternatively, for a sample whose fuzzy mode set is partially included in the set ofcontrol limits, the percentage (bj) of the set falling into the interval of fuzzy control limits can be compared toa predefined acceptable percentage (b), and then can be decided as ‘‘rather in control’’ if bj P b or ‘‘rather out

of control’’ if bj < b where

bj ¼

0; for bj P UCL3

UCL3�bj

cj�bj; for ðLCL2 6 bj 6 UCL3Þ ^ ðcj P UCL3Þ

1; for ðbj P LCL2Þ ^ ðcj 6 UCL3ÞLCL2�bj

cj�bj; for ðbj 6 LCL2Þ ^ ðLCL2 6 cj 6 UCL3Þ

0; for cj 6 LCL2

8>>>>>>><>>>>>>>:ð18Þ

As b approaches to 1, the tightness of the inspection increases. When b = 1, the process control decisionresults in ‘‘in-control’’ or ‘‘out-of-control’’. The value of the b can subjectively be defined by the experts’experiences. Process control decisions for some samples are illustrated in Fig. 3.

LCL2

LCL1

LCL3

CL1

LCL4

CL2

CL3

CL4

UCL2

UCL3

UCL4

UCL1

a1

d1

c1

b1

S1

a2

d2

c2

b2

S2

S3S4

S5

S6S7

a3

d3

c3

b3

a4

d4

c4

b4

a5

d5

c5

b5

a6

d6

c6

b6

a7

d7

c7

b7

Fig. 3. Samples (Sj, j = 1,2, . . . , 7) resulting in four types of different decisions for b = 0.50: S1, S7: ‘‘out-of-ontrol’’ (the mode set of S1 andS7 are completely outside the fuzzy control limits, b1 = b7 = 0), S2, S6: ‘‘rather out of control’’, (b = 0.50,b2,b6 6 0.50), S3, S5: ‘‘rather incontrol’’ (b = 0.50,b3,b5 P 0.50), S4: ‘‘in control’’ (mode set of S4 is completely inside the fuzzy control limits, b4 = 1).


The conditions of the process control can be stated as below.

Process control ¼

in-control; for b ¼ 1 ðbj P LCL2 ^ cj 6 UCL3Þout-of-control; for b ¼ 0 ðbj P UCL3 _ cj 6 LCL2Þrather in-control; for bj P b

rather out-of-control; for bj < b

)otherwise

8>>>><>>>>: ð19Þ

2.2. Fuzzy control charts based on a-level fuzzy midrange transformation

The a-level fuzzy midrange, f amr, is defined as the midpoint of the ends of the a-cut. If aa and d a are the end

points of a-cut, then

f amr ¼

1

2ðaa þ daÞ ð20Þ

In fact, the fuzzy mode is a special case of a-level fuzzy midrange when a = 1. Using the representation givenin Fig. 2, a-level fuzzy midrange of sample j, Sa

mr;j, is determined by Eq. (21).

Samr;j ¼

aaj þ da

j

2¼ ðaj þ djÞ þ a½ðbj � ajÞ � ðdj � cjÞ�

2ð21Þ

In this approach, a-level fuzzy midrange is used as the fuzzy transformation method when calculating controllimits (Eqs. (22)–(24)).

CLamr ¼ f a

mrðgCLÞ ¼ CLa1 þ CLa

4

2¼ CL1 þ CL4 þ a½ðCL2 � CL1Þ � ðCL3 � CL4Þ�

2ð22Þ

LCLamr ¼ CLa

mr � 3ffiffiffiffiffiffiffiffiffiffiffiCLa

mr

qð23Þ

UCLamr ¼ CLa

mr þ 3ffiffiffiffiffiffiffiffiffiffiffiCLa

mr

qð24Þ

S7

S6

S5

S4

S3

S2

S1

LCL2

LCL1

LCL3

CL1

LCL4

CL2

CL3

CL4

UCL2

UCL3

UCL4

UCL1

a1

d1

c1

b1

a2

d2

c2

b2

a3

d3

c3

b3

a4

d4

c4

b4

a5

d5

c5

b5

a6

d6

c6

b6

a7

d7

c7

b7

0 1 0 1 0 1 0 1 0 1 0 1 0 10 1α α α α α α α

Fig. 4. Samples (Sj, j = 1,2, . . . , 7) resulting in two types of different decisions for a = 0.50: S1, S2, S5, S6, S7: ‘‘out-of-control’’ (a-levelfuzzy midrange (d) of S1, S2, S5, S6, and S7 outside the control limits), S3, S4: ‘‘in control’’ (a-level midrange of S3 and S4 are inside thecontrol limits).


Possible process control decisions based on the a-level (a = 0.70) fuzzy midrange for some samples are illus-trated in Fig. 4.

The conditions of process control for each sample can be defined as,

Process control ¼in-control; for LCLa

mr 6 Samr;j 6 UCLa

mr

out-of-control; otherwise

�ð25Þ

2.3. Fuzzy control charts based on a-level fuzzy median transformation

The a-level fuzzy median, f amed, is the point which partitions the membership function of a fuzzy set into two

equal regions at a-level. For a sample j, a-level fuzzy median, Samed;j, as illustrated in Fig. 5 can be calculated

0

1

μ

α

jb jcjdja

jaαjdα

,med jSα

A1 A2

Fig. 5. Illustration of fuzzy median (A1 = A2).

S7

S6

S5

S4

S3

S2

S1

LCL2

LCL1

LCL3

CL1

LCL4

CL2

CL3

CL4

UCL2

UCL3

UCL4

UCL1

a1

d1

c1

b1

a2

d2

c2

b2

a3

d3

c3

b3

a4

d4

c4

b4

a5

d5

c5

b5

a6

d6

c6

b6

a7

d7

c7

b7

0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 α α α α α α α α

Fig. 6. Samples (Sj, j = 1,2, . . . ,7) resulting in two types of different decisions for a = 0.50: S1, S2, S5, S6, S7: ‘‘out-of-control’’ (a-levelfuzzy median (d) of S1, S2, S5, S6, and S7 outside the control limits), S3, S4: ‘‘in control’’ (the a-level fuzzy medians of S3 and S4 are insidethe control limits).


using Eq. (26). It determines the place of a vertical line on the x-axis dividing the area of (aa,b,c,da) into twoequal regions.

Samed;j ¼

1

4ðaa

j þ bj þ cj þ daj Þ ð26Þ

where aa and da are the a-level end points of the fuzzy set.Using the a-level fuzzy median as the representative value, control limits are determined as given in Eqs.

(27)–(29).

CLamed ¼ f a

medðgCLÞ ¼ 1

4ðCLa

1 þ CL2 þ CL3 þ CLa4Þ ð27Þ

LCLamed ¼ CLa

med � 3ffiffiffiffiffiffiffiffiffiffiffiffiffiCLa

med

qð28Þ

UCLamed ¼ CLa

med þ 3ffiffiffiffiffiffiffiffiffiffiffiffiffiCLa

med

qð29Þ

Possible process control decisions based on the a-level fuzzy median for some samples are illustrated in Fig. 6.The conditions of process control for each sample can be defined as,

Process control ¼in-control; for LCLa

med 6 Samed;j 6 UCLa

med

out-of-control; otherwise

�ð30Þ

3. A new approach to fuzzy control charts: direct fuzzy approach

In this approach, the linguistic data are not transformed into representative values using fuzzy transforma-tion in order not to lose any information included in the fuzzy samples. So, both samples and control limits arerepresented in fuzzy numbers along the whole process. The a-level fuzzy control limits, gUCLa , gCLa , and,gLCLa , can be determined by fuzzy arithmetic as shown in Eqs. (4)–(12).

11t1t2

1

LCL1

LCL2

t

d

ab

c

UCL4

UCL3

1 1t1t 1t1t1t2 t1

1

LCL1

LCL2

t

d

abc

UCL4

UCL3

1t 1 1t 1

Type U1 Type U2 Type U3 Type U4 Type U5 Type U6 Type U7

Type L1 Type L2 Type L3 Type L4 Type L5 Type L6 Type L7

α α α α α α α

ααααααα

Fig. 7. Illustration of the possible areas outside the fuzzy control limits at a-level cut.


A decision about whether the process is in control can be made according to the percentage area of thesample which remains inside the gUCL and/or gLCL. When the fuzzy sample is completely involved by thefuzzy control limits, the process is said to be ‘‘in-control’’. If a fuzzy sample is totally excluded by the fuzzycontrol limits, the process is said to be ‘‘out-of-control’’. Otherwise, a sample is partially included by the fuzzycontrol limits. In this case, if the percentage area which remains inside the fuzzy control limits (bj) is equal orgreater than a predefined acceptable percentage (b), then the process can be accepted as ‘‘rather in-control’’.Otherwise, it can be stated as ‘‘rather out of control’’. The possible decisions resulting from DFA are illustratedin Fig. 7. The parameters to determine the sample’s area outside the control limits for any a-level cut areLCL1, LCL2, UCL3, UCL4, a, b, c, d, and a. The shapes of the control limits and fuzzy samples are formedby the lines of LCL1LCL2, UCL3LCL4, ab, and cd. A flowchart to calculate area of the fuzzy sample outsidethe control limits is given in Fig. 8. The sample’s area above the upper control limits, AU

out, and sample areafalling below the lower control limits, AL

out, can be calculated according to the flowchart given in Fig. 8. Theequations to compute AU

out and ALout are given in Appendix A. Then, the total area outside the fuzzy control

limits, Aout, is the sum of the areas below the fuzzy lower control limit and above the fuzzy upper control limit.The percentage sample area within the control limits is calculated as given in Eq. (31).

baj ¼

Saj � Aa

out;j

Saj

ð31Þ

where Saj is the sample’s area at a-level cut.

In contrast to the methods, using fuzzy transformations, this approach is very flexible and more accuratesince both the linguistic data and control limits are not transformed into representative values to prevent theloss of information included in the samples.

4a UCL≥?

T

F

4d UCL≥?

T

F

3b UCL≥?

T

F

3c UCL≤?

T

F

3c UCL≥

?

T

F

3b UCL≥?

F

T

Use Eq. A-U7

UoutA

UseEq. A-U6

UoutA

UseEq. A-U5

UoutA

UseEq. A-U4

UoutA

UseEq. A-U1

UoutA

UseEq. A-U2

UoutA

UseEq. A-U3

UoutA

L Uout out outA A A= +

1d LCL≤?

T

F

1a LCL≤?

T

F

2c LCL≤?

T

F

2b LCL≤?

T

F

2b LCL≤?

T

F

2c LCL≤?

F

T

Use Eq. A-L7

LoutA

UseEq. A-L6

LoutA

Use Eq. A-L4

LoutA

UseEq. A-L5

LoutA

UseEq. A-L1

LoutA

UseEq. A-L3

LoutA

UseEq. A-L2

LoutA

Fig. 8. Flowchart to compute the area of a fuzzy sample (a,b,c,d) falling outside the fuzzy control limits. (See appendix for the equations).


4. A numerical example for number of nonconformities

The samples from a toy company producing large-sized toys are taken every 4 h to control number of non-conformities. Because of the large dimensions of the toys, the number of nonconformities may also be large.

The data collected from 30 subgroups are linguistic as shown in Table 2.The linguistic expressions in Table 2 are represented by fuzzy numbers as shown in Table 3. These numbers

are subjectively identified by the quality control expert who also sets a = 0.60 and minimum acceptable ratioas b = 0.70.

Table 2Number of nonconformities for 30 subgroups

Sample no. Approximately Between

1 302 20–303 5–124 65 386 20–247 4–88 36–449 11–15

10 10–1311 612 3213 1314 50–5215 38–4116 4017 32–5018 3919 15–2120 2821 32–3522 10–2523 3024 2525 31–4126 10–2527 5–1428 28–3529 20–2530 8


Using Eqs. (4)–(7), gCL, gLCL, and gUCL are determined as follows:
gCL ¼ ð18:13; 22:67; 26:93; 32:07ÞgLCL ¼ ð1:15; 7:10; 12:65; 19:29ÞgUCL ¼ ð30:91; 36:95; 42:50; 49:05Þ
Applying an a-cut of 0.60, values of gCLa¼0:60, gLCLa¼0:60, and gUCLa¼0:60 are calculated as follows (see Eqs.(10)–(12)).
gCLa¼0:60 ¼ ð20:85; 22:67; 26:93; 28:99ÞgLCLa¼0:60 ¼ ð4:72; 7:10; 12:65; 15:31ÞgUCLa¼0:60 ¼ ð36:95; 36:95; 42:50; 45:12Þ
The fuzzy modes, a-level fuzzy midranges, and a-level fuzzy medians of the fuzzy control limits above are sum-marized in Table 4.

The decisions about the process control resulted from each sample based on the fuzzy mode, a-level fuzzymidrange, and a-level fuzzy median are given in Table 5.

4.1. Application of direct fuzzy approach (DFA)

The a-cut fuzzy control limits are summarized in Table 6. The numerical results and process control deci-sions based on the DFA are given in Table 7.

Table 3Fuzzy number (a,b,c,d) representation of 30 subgroups

No. a b c d

1 25 30 30 352 15 20 30 353 4 5 12 154 3 6 6 85 32 38 38 456 16 20 24 287 3 4 8 128 27 36 44 509 9 11 15 20

10 7 10 13 1511 3 6 6 1012 27 32 32 3713 11 13 13 1514 39 50 52 5515 28 38 41 4516 33 40 40 4417 28 32 50 6018 33 39 39 4319 12 15 21 3820 23 28 28 3621 28 32 35 4222 14 18 28 3323 24 30 30 3424 20 25 25 3125 25 31 41 4626 7 10 25 2827 3 5 14 2028 23 28 35 3829 17 20 25 2930 5 8 8 15

Table 4Control limits and their representative values based on fuzzy mode, fuzzy midrange, and fuzzy median

Fuzzy number Fuzzy transformation method

a b c d Mode Midrange (a = 0.60) Median (a = 0.60)

CL 18, 13 22, 67 26, 93 32, 07 [22.67,26.93] 24.95 24.88LCL 1, 15 7, 10 12, 65 19, 29 [7.10,12.65] 10.05 9.96UCL 30, 91 36, 95 42, 5 49, 05 [36.95,42.50] 38.95 39.79


The overall results of these approaches are summarized in Table 8. As it is clearly seen, some different deci-sions are obtained. For example, sample 3 indicates ‘‘Rather in control’’ when fuzzy mode transformation orDFA (85.81% of the sample is inside the control limits) is used, but it also indicates ‘‘out-of-control’’ whenfuzzy midrange or fuzzy median is used. On the other hand, while sample 11 indicates an ‘‘out-of-control’’situation when fuzzy mode, fuzzy midrange, or fuzzy median is used, DFA results in ‘‘Rather in control’’ since74.38% of the fuzzy sample is inside the fuzzy control limits. Another typical result is sample 27’s, whichreveals three different process control decisions. According to the fuzzy mode transformation and DFA, thissample indicates ‘‘rather in control’’, while fuzzy midrange transformation results in ‘‘In control’’ and fuzzymedian results in ‘‘out-of-control’’. DFA shows that 87.67% of this sample is within the fuzzy control limitsand it is strongly ‘‘rather in control’’ for b = 0.70. Sample 30 is another example that reveals differentdecisions.

Table 5Decisions based on fuzzy mode, fuzzy midrange, and fuzzy median (a = 0.60, b = 0.70)

Sj fmod,j bj fmod,j decision f a¼0:60mr;j f a¼0:60

mr;j decision f a¼0:60med;j f a¼0:60

med;j decision

1 30 30 100.00 In control 30.00 In control 30.00 In control2 20 30 100.00 In control 25.00 In control 25.00 In control3 5 12 70.04 Rather in control 8.90 Out of control 8.70 Out of control4 6 6 0.00 Out of control 5.80 Out of control 5.90 Out of control5 38 38 100.00 In control 38.20 In control 38.10 In control6 20 24 100.00 In control 22.00 In control 22.00 In control7 4 8 22.56 Rather out of control 6.60 Out of control 6.30 Out of control8 36 44 81.28 Rather in control 39.40 In control 39.70 In control9 11 15 100.00 In control 13.60 In control 13.30 In control10 10 13 100.00 In control 11.30 In control 11.40 In control11 6 6 0.00 Out of control 6.20 Out of control 6.10 Out of control12 32 32 100.00 In control 32.00 In control 32.00 In control13 13 13 100.00 In control 13.00 In control 13.00 In control14 50 52 0.00 Out of control 49.40 Out of control 50.20 Out of control15 38 41 100.00 In control 38.30 In control 38.90 In control16 40 40 100.00 In control 39.40 In control 39.70 In control17 32 50 58.35 Rather out of control 42.20 Out of control 41.60 Out of control18 39 39 100.00 In control 38.60 In control 38.80 In control19 15 21 100.00 In control 20.80 In control 19.40 In control20 28 28 100.00 In control 28.60 In control 28.30 In control21 32 35 100.00 In control 34.10 In control 33.80 In control22 18 28 100.00 In control 23.20 In control 23.10 In control23 30 30 100.00 In control 29.60 In control 29.80 In control24 25 25 100.00 In control 25.20 In control 25.10 In control25 31 41 100.00 In control 35.80 In control 35.90 In control26 10 25 100.00 In control 17.50 In control 17.50 In control27 5 14 76.69 Rather in control 10.30 In control 9.90 Out of control28 28 35 100.00 In control 31.10 In control 31.30 In control29 20 25 100.00 In control 22.70 In control 22.60 In control30 8 8 100.00 In control 8.80 Out of control 8.40 Out of control

Table 6Control limits for DFA (a = 0.60)

a b c d

CL 20.85 22.67 26.93 28.99LCL 4.72 7.10 12.65 15.31UCL 34.53 36.95 42.50 45.12


DFA provides the possibility of making linguistic decisions like ‘‘rather in control’’ or ‘‘rather out-of-control’’. Further intermediate levels of process control decisions are also possible by defining different inter-vals for b. For instance, it may be defined as in Eq. (32).

Process control ¼

in-control; 0:85 6 bj 6 1

rather in control; 0:60 6 bj < 0:85

rather out of control; 0:10 6 bj < 0:60

out-of-control; 0 6 bj < 0:10

8>>><>>>: ð32Þ

More intervals for the process control decisions can be subjectively defined by the decision-maker.

5. Conclusion

As can be seen from Table 8, the decisions for the process control results in slight differences according tothe approaches. This is due to the loss of information of the fuzzy samples when they are transformed into

Table 7Decisions based on direct fuzzy approach (a = 0.60, b = 0.70)

Sj aa b c da Area out Sample’s area bj DFA decision

1 28 30 30 32 0.00 0.80 100.00 In control2 18 20 30 32 0.00 4.80 100.00 In control3 4.6 5 12 13.2 0.44 3.12 85.81 Rather in control4 4.8 6 6 6.8 0.13 0.40 67.94 Rather out of control5 35.6 38 38 40.8 0.00 1.04 100.00 In control6 18.4 20 24 25.6 0.00 2.24 100.00 In control7 3.6 4 8 9.6 0.84 2.00 57.86 Rather out of control8 32.4 36 44 46.4 0.55 4.40 87.39 Rather in control9 10.2 11 15 17 0.00 2.16 100.00 In control10 8.8 10 13 13.8 0.00 1.60 100.00 In control11 4.8 6 6 7.6 0.14 0.56 74.38 Rather in control12 30 32 32 34 0.00 0.80 100.00 In control13 12.2 13 13 13.8 0.00 0.32 100.00 In control14 45.6 50 52 53.2 1.92 1.92 0.00 Out of control15 34 38 41 42.6 0.00 2.32 100.00 In control16 37.2 40 40 41.6 0.00 0.88 100.00 In control17 30.4 32 50 54 3.27 8.32 60.64 Rather out of control18 36.6 39 39 40.6 0.00 0.80 100.00 In control19 13.8 15 21 27.8 0.00 4.00 100.00 In control20 26 28 28 31.2 0.00 1.04 100.00 In control21 30.4 32 35 37.8 0.00 2.08 100.00 In control22 16.4 18 28 30 0.00 4.72 100.00 In control23 27.6 30 30 31.6 0.00 0.80 100.00 In control24 23 25 25 27.4 0.00 0.88 100.00 In control25 28.6 31 41 43 0.00 4.88 100.00 In control26 8.8 10 25 26.2 0.00 6.48 100.00 In control27 4.2 5 14 16.4 0.52 4.24 87.67 Rather in control28 26 28 35 36.2 0.00 3.44 100.00 In control29 18.8 20 25 26.6 0.00 2.56 100.00 In control30 6.8 8 8 10.8 0.00 0.80 100.00 In control


crisp numbers. If the fuzzy number representations of linguistic variables are symmetric, the fuzzy mode, fuzzymidrange, and fuzzy median of a fuzzy sample become equal to each other and therefore, give the same con-trol decisions. Fuzzy average is another method that can be used to transform linguistic data to a represen-tative value. In general, fuzzy mode and fuzzy midrange methods are easier to calculate than fuzzy medianand fuzzy average, especially when the membership function is nonlinear. However, the fuzzy mode may leadto a biased result when the membership function is extremely asymmetric. If the area under the membershipfunction is considered to be an appropriate measure of fuzziness, the fuzzy median may be more suitable.

For the linguistic data represented by asymmetric fuzzy numbers, different possible decisions can be faced.This kind of linguistic data should be processed by DFA to prevent the loss of information in linguistic data. Itis very useful to define an a-cut and an acceptable percentage (b) which enable the quality expert to set thetightness of inspection. We prefer a fuzzy comparison method based on area measurement in DFA approachsince this method examines the degree of being outside the control limits. For further research, we propose theother fuzzy comparison methods like the center of gravity (centroid) to be used.

Appendix A

The equations to compute sample area outside the control the limits.

AUout ¼

1

2½ðda �UCLa

4Þ þ ðdt �UCLt4Þ�ðmaxðt � a; 0ÞÞ þ 1

2½ðdz � azÞ þ ðc� bÞ�ðminð1� t; 1� aÞÞ

ðA:U1Þ

Table 8Comparison of alternative approaches: Fuzzy mode, fuzzy midrange, fuzzy median, and DFA (a = 0.60 and b = 0.70)

j fmod,j decision f a¼0:60mr;j decision f a¼0:60

med;j decision DFA (a = 0.60) decision

1 In control In control In control In control2 In control In control In control In control3 Rather in control Out of control Out of control Rather in control4 Out of control Out of control Out of control Rather out of control5 In control In control In control In control6 In control In control In control In control7 Rather out of control Out of control Out of control Rather out of control8 Rather in control In control In control Rather in control9 In control In control In control In control10 In control In control In control In control11 Out of control Out of control Out of control Rather in control12 In control In control In control In control13 In control In control In control In control14 Out of control Out of control Out of control Out of control15 In control In control In control In control16 In control In control In control In control17 Rather out of control Out of control Out of control Rather out of control18 In control In control In control In control19 In control In control In control In control20 In control In control In control In control21 In control In control In control In control22 In control In control In control In control23 In control In control In control In control24 In control In control In control In control25 In control In control In control In control26 In control In control In control In control27 Rather in control In control Out of control Rather in control28 In control In control In control In control29 In control In control In control In control30 In control Out of control Out of control In control


where

t ¼ UCL4 � aðb� aÞ þ ðc� bÞ and z ¼ maxðt; aÞ

AUout ¼

1

2½ðda �UCLa

4Þ þ ðc�UCL3Þ�ð1� aÞ ðA:U2Þ

AUout ¼

1

2ðda �UCLa

4Þðmaxðt � a; 0ÞÞ ðA:U3Þ

where

t ¼ UCL4 � dðUCL4 �UCL3Þ � ðd � cÞ

AUout ¼

1

2½ðc�UCL3Þ þ ðdz �UCLz

4Þ�ðminð1� t; 1� aÞÞ ðA:U4Þ

where

t ¼ UCL4 � dðUCL4 �UCL3Þ � ðd � cÞ and z ¼ maxðt; aÞ

AUout ¼

1

2½ðdz2 �UCLz2

4 Þ þ ðdt1 �UCLt14 Þ�ðminðmaxðt1 � a; 0Þ; t1 � t2ÞÞ þ

1

2½ðdz1 � az1Þ þ ðc� bÞ�

� ðminð1� t1; 1� aÞÞ ðA:U5Þ


where

t1 ¼UCL4 � a

ðb� aÞ þ ðUCL4 �UCL3Þ; t2 ¼

UCL4 � dðUCL4 �UCL3Þ � ðd � cÞ ;

z1 ¼ maxða; t1Þ; and z2 ¼ maxða; t2Þ

AUout ¼ 0 ðA:U6Þ

AUout ¼

1

2½ðda � aaÞ þ ðc� bÞ�ð1� aÞ ðA:U7Þ

ALout ¼

1

2½ðLCLa

1 � aaÞ þ ðLCLt1 � atÞ�ðmaxðt � a; 0ÞÞ þ 1

2½ðdz � azÞ þ ðc� bÞ�ðminð1� t; 1� aÞÞ ðA:L1Þ

where

t ¼ d � LCL1

ðLCL2 � LCL1Þ þ ðd � cÞ and z ¼ maxða; tÞ

ALout ¼

1

2½ðda � aaÞ þ ðc� bÞ�ð1� aÞ ðA:L2Þ

ALout ¼

1

2½ðLCLa

1 � aaÞ þ ðLCL2 � bÞ�ð1� aÞ ðA:L3Þ

ALout ¼

1

2½ðLCLz2

1 � az2Þ þ ðLCLt11 � at1Þ�ðminðmaxðt1 � a; 0Þ; t1 � t2ÞÞ

þ 1

2½ðdz1 � az1Þ þ ðc� bÞ�ðminð1� t; 1� aÞÞ ðA:L4Þ

where

t1 ¼d � LCL1

ðLCL2 � LCL1Þ þ ðd � cÞ ; t2 ¼a� LCL1

ðLCL2 � LCL1Þ � ðb� aÞz1 ¼ maxða; t1Þ; and z2 ¼ maxða; t2Þ

ALout ¼

1

2½ðLCLz

1 � azÞ þ ðLCL2 � bÞ�ðminð1� t; 1� aÞÞ ðA:L5Þ

where

t ¼ a� LCL1

ðLCL2 � LCL1Þ � ðb� aÞ ; and z ¼ maxða; tÞ

ALout ¼ 0 ðA:L6Þ

ALout ¼

1

2½ðda � aaÞ þ ðc� bÞ�ð1� aÞ ðA:L7Þ

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an alternative approach to fuzzy control charts: direct fuzzy approach

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