Development of fuzzy process control charts and ... - Semantic Scholar
Computational Statistics & Data Analysis 51 (2006) 434 – 451 www.elsevier.com/locate/csda
Development of fuzzy process control charts and fuzzy unnatural pattern analyses Murat Gülbaya, b,∗ , Cengiz Kahramana a Industrial Engineering Department, Istanbul Technical University, Istanbul, Turkey b Industrial Engineering Department, University of Gaziantep, Gaziantep, Turkey
Available online 15 May 2006
Abstract Many problems in scientific investigation generate nonprecise data incorporating nonstatistical uncertainty. A nonprecise observation of a quantitative variable can be described by a special type of membership function defined on the set of all real numbers called a fuzzy number or a fuzzy interval. A methodology for constructing control charts is proposed when the quality characteristics are vague, uncertain, incomplete or linguistically defined. Fuzzy set theory is an inevitable tool for fuzzy control charts as well as other applications subjected to uncertainty in any form. The vagueness can be handled by transforming incomplete or nonprecise quantities to their representative scalar values such as fuzzy mode, fuzzy midrange, fuzzy median, or fuzzy average. Then crisp methods may be applied to those representative values for control chart decisions as “in control” or “out of control”. Transforming the vague data by using one of the transformation methods may result in biased decisions since the information given by the vague data is lost by the transformation. Such data needs to be investigated as fuzzy sets without transformation, and the decisions based on the vague data should not be concluded with an exact decision. A “direct fuzzy approach (DFA)” to fuzzy control charts for attributes under vague data is proposed without using any transformation method. Then, the unnatural patterns for the proposed fuzzy control charts are defined using the probabilities of fuzzy events. © 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy process control; Run tests; Unnatural patterns; Fuzzy probability
1. Introduction Control charts have been widely used for monitoring process stability and capability. Control charts are based on data representing one or several quality-related characteristics of the product or service. If these characteristics are measurable on numerical scales, then variable control charts are used. If the quality-related characteristics cannot be easily represented in numerical form, then attribute control charts are useful. When a process is in statistical control, a control chart displays the known patterns of variation. When the control chart points deviate from these known patterns, the process is considered to be out of control. The control chart distinguishes between normal and nonnormal variation through the use of statistical tests and control limits. The control limits are calculated using the rules of probability so that when a point is determined to be out of control, it is due to an assignable cause and not due to a normal variation. ∗ Corresponding author. Industrial Engineering Department, Istanbul Technical University, Istanbul, Turkey. Tel.: +90 212 2931300x2073; fax: +90 212 2407260. E-mail address: [email protected] (M. Gülbay).
0167-9473/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2006.04.031
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UCL A CL+2σ B CL+1σ C CL C CL-1σ B CL-2σ A LCL
Fig. 1. The zones of a control chart.
The points outside the control limits are not the only criteria to determine the out of control conditions. However all points are inside the limits, the process may still be out of control if it does not display a normal pattern of variation. The zone tests, which are hypothesis tests in a modified form, are used to determine out of control conditions. They are used to test if the plotted points are following a normal pattern of variation. For a control chart to be effective, some action must be taken as a result of the chart pattern. When the process average is centered where it is supposed to be, and the variability displays a normal pattern, the process is considered to be in control. A normal pattern means that the process is aligned with the probabilities of the normal distribution. A large abnormal variability and unnatural patterns indicate out of control conditions. Out of control conditions usually have assignable causes that must be investigated and resolved. Numerous supplementary rules, like zone tests or run rules have been developed to assist quality practitioners in detection of the unnatural patterns for the crisp control charts. The run rules are based on the premise that a specific run of data has a low probability of occurrence in a completely random stream of data. If a run occurs, then this must mean that something has changed in the process to produce a nonrandom or unnatural pattern. The control charts may indicate an out-of-control condition when either one or more points fall beyond the control limits or plotted points show some nonrandom patterns of behavior. Unnatural (nonrandom) patterns for classical control charts have been extensively studied. Over the years, many rules have been developed to detect nonrandom patterns within the control limits. Under the pattern-recognition approach, numerous researches have defined several types of out-of-control patterns (e.g. trends, cyclic pattern, mixture, etc.) with a specific set of possible causes. When a process exhibits any of these unnatural patterns, it implies that those patterns may provide valuable information for process improvement. The unnatural (nonrandom) patterns for fuzzy control charts have not been studied yet. The zones of a control chart used in the zone tests are bounded by the standard deviations of the data as illustrated in Fig. 1. The probability of each zone based on the normal distribution is depicted in Fig. 2. The main idea behind defining a rule for an unnatural pattern is the probability of the occurrence: these rules are based on the premise that a specific run of data has a low probability of occurrence in a completely random stream of data. In general, probability of occurrence of an unnatural pattern is less than 1%. In the literature, there exist some unnatural patterns defined for the crisp cases. There is no certain rule about which unnatural patterns to use and the selection of a set of rules depends on the user preferences. Unnatural patterns are defined for the short runs, i.e., rules for a 15–20 consecutive points on the chart are investigated. The Western Electric (1956) suggested a set of decision rules for detecting unnatural patterns on control charts. Specifically, it suggested concluding that the process is out of control if any of the following conditions is satisfied. Rule 1: A single point falls outside of the control limits (beyond ±3 limits); Rule 2: Two out of three successive points fall in zone A or beyond (the odd point may be anywhere. Only two points count). Rule 3: Four out of five successive points fall in zone B or beyond (the odd point may be anywhere. Only four points count). Rule 4: Eight successive points fall in zone C or beyond.
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A
B
B
A
2.14%
13.59 %
13.59 %
2.14 %
C
C
34.13 % 34.13 %
-3σ
-2σ
-1σ
µ
+1σ
+2σ
+3σ
Fig. 2. The zones and probabilities of normal distribution.
One-sided probabilities of the rules above are calculated as 0.00135, 0.0015, 0.0027, and 0.0039, respectively. Grant and Leavenworth (1988) recommended that nonrandom variations are likely to be presented if any one of the following sequences of points occurs in the control charts. Rule 1: Seven consecutive points on the same side of the center line. Rule 2: At least 10 of 11 consecutive points on the same side of the center line. Rule 3: At least 12 of 14 consecutive points on the same side of the center line. Rule 4: At least 14 of 17 consecutive points on the same side of the center line. One-sided probabilities of the rules above are calculated as 0.00781, 0.00586, 0.00647, and 0.00636, respectively. Nelson (1985) proposed the following rules for unnatural patterns: Rule 1: One or more points outside of the control limits. Rule 2: Nine consecutive points in the same side of center line. Rule 3: Six points in a row steadily increasing or decreasing. Rule 4: Fourteen points in a row altering up and down. Rule 5: Two out of 3 points in a row in zone A or beyond. Rule 6: Four out of 5 points in zone B or beyond. Rule 7: Fifteen points in a row in zones C, above and below the centerline. Rule 8: Eight points in a row on both sides of the centerline with none in zone C. The unnatural patterns tend to fluctuate too wide or they fail to balance around the centerline. The portrayal of natural and unnatural patterns is what makes the control chart a very useful tool for statistical process and quality control. When a chart is interpreted, we look for special patterns such as cycles, trends, freaks, mixtures, groupings or bunching of measurements, and sudden shifts in levels. A product is generally classified in a binary manner (conforming or nonconforming) in attributed control charts. Binary classification may not be the most suitable if the product quality changes gradually rather than abruptly. Various procedures have been proposed for monitoring procedures in which the data are categorical in nature. In past research, Wang and Raz (1988, 1990), Raz and Wang (1990) and Kanagawa et al. (1993) proposed an assessment of intermediate quality level instead of the traditional binary judgement when the quality characteristics are not numerically measurable. Linguistic variables represent features by linguistic terms instead of numerical measurement. For example, the product quality feature can be classified by one of the terms ‘perfect’, ‘good’, ‘poor’, and ‘bad’, depending on the product’s deviation from specifications. These words or phrases are called linguistic variables. Furthermore, each linguistic variable value can be represented as a membership function. A linguistic variable differs from a numerical variable
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in that its values are not numbers but words or phrases in a language (Wang and Raz, 1990). Linguistic variables are derived from human subjective judgements. Wang and Raz (1988, 1990) and Raz and Wang (1990) provided a potential application of a fuzzy-set theory and constructed a control chart for these linguistic variables under the consideration of a normal distribution. Kanagawa et al. (1993) provided another control chart under an estimated distribution function. Laviolette et al. (1995) pointed out that the control limit proposed by Wang and Raz (1990) had no satisfactory efficiency when it was used to detect a change in the process mean. It is reasonable to treat the linguistic and uncertain data in the light of the fuzzy-set theory. Although the fuzzy methods provide a powerful framework for pattern recognition due to their ability to generate gradual memberships of objects to clusters, a number of rules have been proposed to defuzzify the classification results in order to be able to make a final (crisp) decision about the process. It is necessary to convert the fuzzy sets associated with linguistic values into scalars, which will be referred to as representative values. This conversion may be done in a number of ways, as long as the result is intuitively representative of the range of the base variable included in the fuzzy set, but the concept of the information connected with the concept of uncertainty has been lost upon this conversion. Some of them are fuzzy mode, fuzzy median, and -level fuzzy midrange (Zadeh, 1965). The purpose of data analysis is to gain information from data. By applying a conversion method to the vague data which result in scalar data we have already lost the information at the initial stage. Recently, -level fuzzy control charts for attributes are proposed by Gülbay et al. (2004) in order to reflect the vagueness of the data and tightness of the inspection. The objective of this paper is to develop a monitoring and diagnostic system to indicate natural out-of-control situations without defuzzification and define fuzzy unnatural patterns for the fuzzy control charts. The paper is organized as follows: fuzzy process control charts based on defuzzification using fuzzy transformation methods, and an alternative approach without any defuzzification are presented in Section 2. Probability of fuzzy events is explained in Section 3. Based on the probability of fuzzy events, fuzzy unnatural pattern rules for fuzzy control charts are given in Section 4. A numerical illustrative example is presented in Section 5. Finally, concluding remarks are given in Section 6. 2. Fuzzy process control charts A fuzzy approach is suitable for attributes control charts (p, np, c, and u charts) when the data is linguistic, categorical, uncertain, or human dependent subjective judgement is possible. In classical p charts, products are distinctly classified as “conformed” or “nonconformed” when determining fraction rejected. In fuzzy p control charts (See: Gülbay et al., 2004), when categorizing products, several linguistic terms are used to denote the degree of being nonconformed product such as “standard”, “second choice”, “third choice”, “chipped”, and so on… . A membership degree of being a nonconformed product is assigned to each linguistic term. Sample means for each sample group, Mj , are calculated as: Mj =
t
i=1 kij ri
mj
,
(1)
where t is the total number of linguistic terms, kij the number of products categorized with the linguistic term i in the sample j , ri (0 ri 1) the membership degree of the linguistic term i, and mj the total number of products in sample j . Center line, CL, is the average of the means of the n sample groups and can be determined by n j =1 Mj , (2) CL = Mj = n where n is the number of sample groups initially available. kij and ri in Eq. (1), and so in Eq. (2), are the uncertain values and depend on the human subjective judgment. In another word, a sample can be belonged to the second choice category by a quality controller, while it may be included in the standard or third choice by another quality controller. In the same way, defining a membership degree for a category may depend on the quality controller preferences. Therefore, the value of Mj may lie between 0 and 1, as a result of these human judgments. It is clear that CL in Eq. (2) has a range between 0 and 1 too. To overcome the uncertainty in the determination of the CL, fuzzy set theory can successfully be adopted by defining CL as a triangular fuzzy number (TFN) whose fuzzy mode is CL, as shown in
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j (x)
1
L j (α) Rj (α)
x 0
CL = M = p
1
Fig. 3. TFN representation of M and Mj of the sample j .
Fig. 3. Then, for each sample mean, Lj () and Rj () can be calculated using Eqs. (3) and (4), respectively Lj () = Mj , Rj () = 1 − 1 − Mj . The membership function of the M, or CL, can be written as ⎧ 0 if x 0, ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ if 0 x M, ⎪ ⎨M Mj (x) = 1−x ⎪ ⎪ ⎪ if M x 1, ⎪ ⎪ 1−M ⎪ ⎪ ⎩ 0 if x 1.
(3) (4)
(5)
The control limits for -cut is also a fuzzy set and can be represented by TFNs. Since the membership function of CL is divided into two components, then, each component will have its own CL, LCL, and U CL. The membership function of the control limits depending upon the value of is given in Eq. (6) and illustrated in Fig. 4 ⎫ ⎧ ⎧ CLL = M ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫
⎪ ⎪ ⎪ ⎪ ⎪ L ⎬⎪ ⎪ ⎪ ⎪ ⎨ L ⎪ ⎪ ⎪ ⎪ CL 1 − CL ⎪ ⎪ ⎪ ⎪ ⎪ L L ⎪ ⎪ ⎪ = max CL − 3 , 0 LCL ⎪ ⎨ ⎬ ⎪ ⎩ ⎭ n ⎪ ⎪ if 0 Mj M, ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫⎪ ⎪ ⎪
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬⎪ ⎪ ⎪ ⎪ ⎪ CLL 1 − CLL ⎪ ⎪ ⎪ L = min CLL + 3 ⎪ ⎪ ⎪ , 1 U CL ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ ⎭⎭ n ⎨ Control limits () = ⎧ (6) ⎫ CLR = 1 − 1 − M ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧
⎪ ⎪ ⎪ ⎪ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ R R ⎪ ⎪ ⎪ CL 1 − CL ⎪ ⎪ ⎪ ⎪ R = max CLR − 3 ⎪ ⎪ ⎪ , 0 LCL ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ n ⎪ ⎪ if M Mj 1, ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎫ ⎧ ⎪
⎪ ⎪ ⎪ ⎪ R ⎬⎪ ⎪ ⎪ ⎨ ⎪⎪ ⎪ ⎪ ⎪ CL 1 − CLR ⎪ ⎪ ⎪ ⎪ ⎪ R R ⎪ ⎪ , 1 = min CL + 3 U CL ⎩⎪ ⎩ ⎭⎭ ⎩ n where n is the average sample size (ASS). When the ASS is used, the control limits do not change with the sample size. Hence, the control limits for all samples are the same.
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439
α-level R LCL ()
1
R CL ()
UCLR () LCLL ()
UCLL () L CL ()
M Fig. 4. Illustration of the -cut control limits.
1
1
a a b (a)
c d
d
a a b=c d (b)
d
number of nonconformity
Fig. 5. Representation of number of nonconformities by fuzzy numbers: (a) trapezoidal (a, b, c, d); (b) triangular (a, b, b, d).
For the variable sample size (VSS), n should be replaced by the size of the j th sample, nj . Hence, control limits change for each sample depending upon the size of the sample. Therefore, each sample has its own control limits. The decision that whether process is in control (1) or out of control (0) for both ASS and VSS is as follows: 1 if LCLL ()Lj ()U CLL () ∧ LCLR () Rj () U CLR (), (7) Process control = 0 otherwise. The value of -cut is decided with respect to the tightness of inspection such that for a tight inspection, values close to 1 may be used. As can be seen from Fig. 4, while reduces to 0 (decreasing the tightness of inspection), the range where the process is in control (difference between U CL and LCL) increases. In the crisp case, the control limits for number of nonconformities are calculated by the Eqs. (8)–(10) CL = c,
√ LCL = c − 3 c, √ U CL = c + 3 c,
(8) (9) (10)
where c is the mean of the nonconformities. In the fuzzy case, where number of nonconformity includes human subjectivity or uncertainty, uncertain values such as “between 10 and 14” or “approximately 12” can be used to define number of nonconformities in a sample. Then number of nonconformity in each sample, or subgroup, can be represented by a trapezoidal fuzzy number (a, b, c, d) or a triangular fuzzy number (a, b, d) as shown in Fig. 5. Note that a trapezoidal fuzzy number becomes triangular when b = c. For the ease of representation and calculation, a triangular fuzzy number is also represented as trapezoidal by (a, b, b, d) or (a, c, c, d). Here, we propose a direct fuzzy approach (DFA) to deal with the vague data for the control charts. Transforming the vague data by representing them with their representative values may result in biased decisions for particular data especially when they are represented by asymmetrical fuzzy numbers. The center line, CL, given in Eq. (8), is the mean
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of the samples. For fuzzy case, where the numbers of nonconformities are represented by trapezoidal fuzzy numbers, ˜ can be determined using the arithmetic mean of the fuzzy numbers and written as in Eq. (11) fuzzy center line, CL, (see Chen and Hwang, 1992 for the fuzzy arithmetics performed in this paper) n n n n aj bj cj dj j =1 j =1 j =1 j =1 ˜ = CL , , , = a, b, c, d , (11) n n n n where n is the number of fuzzy samples and a, b, c and d are the arithmetic means of the a, b, c, and d, respectively. ˜ can be rewritten as in Eq. (12). Then LCL ˜ and U CL ˜ are calculated using fuzzy arithmetics as given in Eqs. (13) CL and (14), respectively ˜ = a, b, c, d = (CL1 , CL2 , CL3 , CL4 ) , CL (12) ˜ = CL ˜ ˜ LCL − 3 CL = CL1 − 3 CL4 , CL2 − 3 CL3 , CL3 − 3 CL2 , CL4 − 3 CL1 = (LCL1 , LCL2 , LCL3 , LCL4 ) , ˜ = CL ˜ + 3 CL ˜ U CL = CL1 + 3 CL1 , CL2 + 3 CL2 , CL3 + 3 CL3 , CL4 + 3 CL4 = (U CL1 , U CL2 , U CL3 , U CL4 ) .
(13)
(14)
An -cut is a nonfuzzy set which comprises all elements whose membership is greater than or equal to . Applying -cuts of fuzzy sets (Fig. 5), values of a and d for samples and CL1 and CL4 (start and end points of the -cut of CL) for center line are determined by Eqs. (15) and (16), respectively a = a + (b − a),
CL1 = CL1 + (CL2 − CL1 ) ,
(15)
d = d − (d − c),
CL4 = CL4 − (CL4 − CL3 ) .
(16)
Using -cut representations, fuzzy control limits can be rewritten as given in Eqs. (17)–(19) ˜ = CL , CL2 , CL3 , CL , CL 1 4 ˜ = LCL , LCL2 , LCL3 , LCL , LCL 1 4 ˜ = U CL , U CL2 , U CL3 , U CL . U CL 1 4
(17) (18) (19)
The results of these equations can be illustrated as in Fig. 6. To retain the standard format of control charts and to facilitate the plotting of observations on the chart, it is necessary to convert the fuzzy sets associated with linguistic values into scalars referred to as representative values. This conversion may be performed in a number of ways as long as the result is intuitively representative of the range of the base variable included in the fuzzy set. The four ways, which are similar in principle to the measures of central tendency used in descriptive statistics, are fuzzy mode, -level fuzzy midrange, fuzzy median, and fuzzy average. It should be pointed out that there is no theoretical basis supporting any one specifically and the selection between them should be mainly based on the ease of computation or preference of the user (Wang and Raz, 1990). The conversion of fuzzy sets into crisp values results in loss of information in linguistic data. To retain the information of the linguistic data, we prefer to keep fuzzy sets as themselves and to compare fuzzy samples with the fuzzy control limits. For this reason, a direct fuzzy approach (DFA) based on the area measurement is proposed for the fuzzy control charts. The decision about whether the process is in control can be made according to the percentage area of the sample ˜ and/or LCL ˜ defined as fuzzy sets. When the fuzzy sample is completely involved by which remains inside the U CL the fuzzy control limits, the process is said to be “in-control”. If a fuzzy sample is totally excluded by the fuzzy control limits, the process is said to be “out of control”. Otherwise, a sample is partially included by the fuzzy control limits. In this case, if the percentage area (j ) which remains inside the fuzzy control limits is equal or greater than a predefined acceptable percentage (), 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 illustrated in Fig. 7. The parameters for determination
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UCL4
UCL4 UCL3 UCL2
~ UCL
UCL1 CL4 UCL1
CL4 CL3 ~ CL
CL2
CL1 LCL4 CL1
LCL4
LCL3 ~ LCL LCL 2
LCL1
LCL 1 µ
α
1
Fig. 6. Representation of fuzzy control limits.
d c b a ~ UCL
α
1 Type U1
t1 Type U2
Type U3
t1 t2 Type U4
Type U5
Type U6
Type U7
Type L4
Type L5
Type L6
Type L7
~ LCL
Type L1
Type L2
Type L3
Fig. 7. Illustration of all possible sample areas outside the fuzzy control limits at -level cut.
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of the sample area outside the control limits for -level fuzzy cut are LCL1 , LCL2 , U CL3 , U CL4 , a, b, c, d, and . The shape of the control limits and fuzzy sample are formed by the lines of LCL1 LCL2 , U CL3 U CL4 , ab, and cd. A flowchart to calculate area of the fuzzy sample outside the control limits is given in Fig. 8. The sample area above the L upper control limits, AU out , and the sample area falling below the lower control limits, Aout , are calculated. Equations U L to compute Aout and Aout are given in Appendix A. Then, the total sample 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. (20) j =
Sj − Aout,j Sj
,
(20)
where Sj is the sample area at -level cut. The DFA provides the possibility of obtaining linguistic decisions like “rather in control” or “rather out of control”. Further intermediate levels of process control decisions are also possible by defining in stages. For instance, it may be defined as given below which is more distinguished ⎧ in control, 0.85 j 1, ⎪ ⎪ ⎪ ⎨ rather in control, 0.60 j < 0.85, Process control = (21) ⎪ rather out of control, 0.10 j < 0.60, ⎪ ⎪ ⎩ out of control, 0 j < 0.10. The intermediate levels of process control decisions are subjectively defined by the quality expert. In binary classification (crisp case), the quality expert may only know if the process is in control or out of control. These predefined levels may refer to the truthful of the out of control. It can be used as a tracking and may give valuable information before the process is out of control. However intermediate levels are subjectively defined, it should refer to the depth of information the quality expert needs to take some preventive actions. 3. Probability of fuzzy events Analysis of fuzzy unnatural patterns for fuzzy control charts is necessary to develop. The formula for calculating the probability of a fuzzy event A is a generalization of the probability theory: in the case which a sample space X is a continuum or discrete, the probability of a fuzzy event P (A) is given by (Yen and Langari, 1999): (x)PX (x) dx if X is continuous, (22) P (A) = A i A (xi )PX (xi ) if X is discrete, where PX denotes a classical probability distribution function of X for continuous sample space and probability function for discrete sample space, and A is a membership function of the event A. The membership degree of a fuzzy sample that belongs to a region is directly related to its percentage area falling in that region, and therefore, it is continuous. For example, a fuzzy sample may be in zone B with a membership degree of 0.4 and in zone C with a membership degree of 0.6. While counting fuzzy samples in zone B, that sample is counted as 0.4. 4. Generation of fuzzy rules for unnatural patterns Numerous supplementary rules, like zone tests or run rules (Western Electric, 1956; Nelson, 1984, 1985; Duncan, 1986; Grant and Leavenworth, 1988) have been developed to assist quality practitioners in detection of unnatural patterns for the crisp control charts. The run rules are based on the premise that a specific run of data has a low probability of occurrence in a completely random stream of data. If a run occurs, then this must mean that something has changed in the process to produce a nonrandom or unnatural pattern. Based on the expected percentages in each zone, sensitive run tests can be developed for analyzing the patterns of variation in the various zones. For the fuzzy control charts, based on the Western Electric rules (1956), the following fuzzy unnatural pattern rules can be defined. The probabilities of these fuzzy events are calculated using normal approach to binomial distribution.
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Fig. 8. Flowchart to compute the area outside the fuzzy control limits.
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The probability of each fuzzy rule (event) below depends on the definition of the membership function which is subjectively defined so that the probability of each of the fuzzy rules is as close as possible to the corresponding classical rule for unnatural patterns. The idea behind this approach may justify the following rules. Rule 1: Any fuzzy data falling outside the three-sigma control limits with a ratio of more than predefined percentage () of sample area at desired -level. The membership function for this rule can subjectively be defined as below: ⎧ 0, 0.85x 1, ⎪ ⎨ (x − 0.60)/0.25, 0.60 x 0.85, 1 (x) = (23) ⎪ ⎩ (x − 0.10)/0.50, 0.10 x 0.60, 1, 0 x 0.10. Rule 2: A total membership degree around 2 from three consecutive points in zone A or beyond. Probability of a sample being in zone A (0.0214) or beyond (0.00135) is 0.02275. Let the membership function for this rule be defined as follows: 0, 0 x 0.59, 2 (x) = (x − 0.59)/1.41, 0.59 x 2, (24) 1, 2 x 3. Using the membership function above, the fuzzy probability given in Eq. (22) can be determined by Eq. (25) 3 x1 x2 3 2 (x)P2 (x) dx = 2 (x)P2 (x) dx + 2 (x)P2 (x) dx + 2 (x)P2 (x) dx 0
0
=
x2
x1
where
x1 3
x2
2 (x)P2 (x) dx +
x2
2 (x)P2 (x) dx,
x − np PX (x) = PX z √ . npq
(25)
(26)
To integrate the equation above, the membership function is divided into sections each with a 0.05 width and 2 (x)Px (x) values for each section are added. For x1 = 0.59 and x2 = 2, the probability of the fuzzy event, rule 2, is determined as 0.0015, which corresponds to the crisp case of this rule. In the following rules, the membership functions are set in the same way. Rule 3: A total membership degree around 4 from five consecutive points in zone C or beyond: 0, 0 x 2.42, 3 (x) = (x − 2.42)/1.58, 2.42 x 4, (27) 1, 4 x 5. The fuzzy probability for this rule is calculated as 0.0027. Rule 4: A total membership degree around 8 from eight consecutive points on the same side of the centerline with the membership function below: 0, 0 x 2.54, 4 (x) = (28) (x − 2.54)/5.46, 2.54 x 8. The fuzzy probability for the rule above is then determined as 0.0039. Based on Grant and Leavenworth’s rules (1988), the following fuzzy unnatural pattern rules can be defined. Rule 1: A total membership degree around 7 from seven consecutive points on the same side of the center line. The fuzzy probability of this rule is 0.0079 when membership function is defined as below: 0, 0 x 2.48, 1 (x) = (29) (x − 2.48)/4.52, 2.48 x 7.
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Rule 2: At least a total membership degree around 10 from 11 consecutive points on the same side of the center line. The fuzzy probability of this rule is 0.0058 when the membership function is defined as below: 0, 0 x 9.33, 2 (x) = (x − 9.33)/0.77, 9.33x 10, (30) 1, 10 x 11. Rule 3: At least a total membership degree around 12 from 14 consecutive points on the same side of the center line. If the membership function is set as given below, then the fuzzy probability of the rule is equal to 0.0065 0, 0 x 11.33, 3 (x) = (x − 11.33)/0.67, 11.33x 12, (31) 1, 12 x 14. Rule 4: At least a total membership degree around 14 from 17 consecutive points on the same side of the center line. The probability of this fuzzy event with the membership function below is 0.0062 0, 0 x 13.34, 4 (x) = (x − 13.34)/0.66, 13.34 x 14, (32) 1, 14 x 17. The fuzzy unnatural pattern rules based on Nelson’s Rules (1985) can be defined in the same way. Some of Nelson’s rules (Rules 3 and 4) are different from the Western Electric Rules and Grant and Leavenworth’s rules. In order to apply these rules to fuzzy control charts, fuzzy samples can be defuzzified using -level fuzzy midranges of the samples. The , is defined as the midpoint of the ends of the -cut. If a and d are the end points of -cut, -level fuzzy midrange, fmr then, fmr = 21 a + d . (33) Then Nelson’s 3rd and 4th rules are fuzzified as follows: Rule 3: Six points in a row steadily increasing or decreasing with respect to the desired -level fuzzy midranges. Rule 4: Fourteen points in a row altering up and down with respect to the desired -level fuzzy midranges. 5. An illustrative example Samples of 200 units are taken every 4 h to control number of nonconformities. Data collected from 30 subgroups shown in Table 1 are linguistic such as “approximately 30” or “between 25 and 30”. The linguistic expressions in Table 1 are represented by fuzzy numbers as shown in Table 3. These numbers are subjectively identified by the quality control expert who also sets = 0.60 and minimum acceptable ratio as = 0.70. As decreases, membership degree of the linguistic terms decreases as well as the truthful of the decisions. Therefore, process control decisions may change by choosing different levels of and . The quality control expert also set the acceptable membership degree of unnaturalness as 0.95, that is, when a sample refers to an unnatural sample with respect to any rule, it should refer a membership degree of unnaturalness more than 0.95 with respect to the membership functions defined for that rule. ˜ LCL, ˜ and U CL ˜ are determined as follows: Using Eqs. (11)–(14), CL, ˜ = (18.13, 22.67, 26.93, 32.07), CL ˜ = (1.15, 7.10, 12.65, 19.29), LCL ˜ = (30.91, 36.95, 42.50, 49.05). U CL Applying -cut of 0.60, values of CL˜=0.60 , LCL˜=0.60 , and U CL˜=0.60 are calculated CL˜=0.60 = (20.85, 22.67, 26.93, 28.99), LCL˜=0.60 = (4.72, 7.10, 12.65, 15.31), U CL˜=0.60 = (34.53, 36.95, 42.50, 45.12).
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Table 1 Number of nonconformities for 30 subgroups No.
Approximately
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
30
Between
20–30 5–12 6 38 20–24 4–8 36–44 11–15 10–13 6 32 13 50–52 38–41
No.
Approximately
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
40
Between
32–50 39 15–21 28 32–35 10–25 30 25 31–41 10–25 5–14 28–35 20–25 8
Table 2 Fuzzy number (a, b, c, d) representation of 30 subgroups No.
a
b
c
d
=0.60 fmr
No.
a
b
c
d
=0.60 fmr
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
25 15 4 3 32 16 3 27 9 7 3 27 11 39 28
30 20 5 6 38 20 4 36 11 10 6 32 13 50 38
30 30 12 6 38 24 8 44 15 13 6 32 13 52 41
35 35 15 8 45 28 12 50 20 15 10 37 15 55 45
28 18 4.6 4.8 35.6 18.4 3.6 32.4 10.2 8.8 4.8 30 12.2 45.6 34
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
33 28 33 12 23 28 14 24 20 25 7 3 23 17 5
40 32 39 15 28 32 18 30 25 31 10 5 28 20 8
40 50 39 21 28 35 28 30 25 41 25 14 35 25 8
44 60 43 38 36 42 33 34 31 46 28 20 38 29 15
37.2 30.4 36.6 13.8 26 30.4 16.4 27.6 23 28.6 8.8 4.2 26 18.8 6.8
18.13
22.67
26.93
32.07
Average
Table 3 Fuzzy zones calculated for the example Zone
a
b
c
d
U CL +2 +1 CL −1 −2 LCL
34.53 29.97 25.41 20.85 15.47 10.10 4.72
36.95 32.19 27.43 22.67 17.48 12.29 7.10
42.50 37.31 32.12 26.93 22.17 17.41 12.65
45.12 39.74 34.37 28.99 24.43 19.87 15.31
Based on the Western Electric Rules 1–4, membership functions in Eqs. (23), (24), (27), and (28) are used. These membership functions set the degree of unnaturalness for each rule. As an example, when a total membership degree of 1.90 is calculated for the rule 2, its degree of unnaturalness is determined from 2 (x) as 0.9291.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0 0 0 0 0 0 0 0.13 0 0 0 0 0 1 0 0 0.39 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0.73 0 0 0 0 0 0 0.98 1 0.39 1 0 0 0.2 0 0 0 0.51 0 0 0.04 0 0
1 1 1 1 0 1 1 0.14 1 1 1 1 1 0 0.02 0 0.22 0 1 1 0.8 1 1 1 0.49 1 1 0.96 1 1
0 0 0 0 0.32 0 0 0.61 0 0 0 0 0 1 0.56 0.72 0.65 0.49 0 0 0 0 0 0 0.28 0 0 0 0 0
0.24 0.04 0 0 0.68 0 0 0.39 0 0 0 0.96 0 0 0.44 0.28 0.35 0.51 0 0.05 0.99 0 0.17 0 0.61 0 0 0.53 0 0
0.76 0.96 1 1 0 1 1 0 1 1 1 0.04 1 0 0 0 0 0 1 0.95 0.01 1 0.83 1 0.1 1 1 0.47 1 1
0 0 0 0 1 0 0 0.97 0 0 0 0 0 1 1 1 0.9 1 0 0 0.61 0 0 0 0.72 0 0 0.27 0 0
1 0.38 0 0 0 0 0 0.03 0 0 0 1 0 0 0 0 0.1 0 0.03 1 0.39 0.22 1 0.2 0.28 0.01 0 0.73 0.03 0
0 0.62 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0.97 0 0 0.78 0 0.8 0 0.99 1 0 0.97 1
0.94 0.25 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 0.58 1 0.09 0.89 0 1 0 0 0.87 0 0
0.06 0.52 0 0 0 0.54 0 0 0 0 0 0 0 0 0 0 0 0 0.28 0.42 0 0.53 0.11 1 0 0.24 0 0.13 0.63 0
0 0.23 1 1 0 0.46 1 0 1 1 1 0 1 0 0 0 0 0 0.72 0 0 0.39 0 0 0 0.76 1 0 0.37 1
1 0.64 0 0 1 0.27 0 1 0 0 0 1 0 1 1 1 1 1 0.13 1 1 0.48 1 0.89 1 0.14 0 1 0.39 0
0 0.36 0 0 0 0.73 0 0 0.05 0 0 0 0 0 0 0 0 0 0.67 0 0 0.52 0 0.11 0 0.42 0.01 0 0.61 0
0 0 1 1 0 0 1 0 0.95 1 1 0 1 0 0 0 0 0 0.21 0 0 0 0 0 0 0.44 0.99 0 0 1
1 0.97 0 0 1 0.95 0 1 0 0 0 1 0 1 1 1 1 1 0.58 1 1 0.87 1 1 1 0.43 0 1 0.98 0
0 0.03 0.18 0 0 0.05 0 0 0.89 0.55 0 0 1 0 0 0 0 0 0.42 0 0 0.13 0 0 0 0.46 0.38 0 0.02 0.02
0 0 0.82 1 0 0 1 0 0.11 0.45 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.11 0.62 0 0 0.98
1 1 0 0 1 1 0 1 0.37 0.01 0 1 0.02 1 1 1 1 1 0.97 1 1 1 1 1 1 0.72 0.12 1 1 0
0 0 0.86 0.68 0 0 0.58 0 0.63 0.99 0.74 0 0.98 0 0 0 0 0 0.03 0 0 0 0 0 0 0.28 0.76 0 0 1
0 0 0.14 0.32 0 0 0.42 0 0 0 0.26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.12 0 0 0
No. A +3 in +3 B +3 A +2 in +2 B +2 A +1 in +1 B +1 A CL in CL B CL A −1 in −1 B −1 A −2 in −2 B −2 A −3 in −3 B −3
Table 4 Membership degrees of fuzzy samples for different zones (A: Above, B: Below)
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Table 5 Total membership degrees of the fuzzy samples in zones for the fuzzified Western Electric Rules Sample no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 a Unnatural
Beyond ±3
0.00 0.00 0.14 0.32 0.00 0.00 0.42 0.13 0.00 0.00 0.26 0.00 0.00 1.00 0.00 0.00 0.39 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.12 0.00 0.00 0.00
In or above fuzzy CL
In or below fuzzy CL
Rule 2
Rule 3
Rule 4
Rule 2
Rule 3
Rule 4
0.24 0.04 0 0 1 0 0 0.87 0 0 0 0.96 0 0 1 1 (µ = 1)a 0.61 1 0 0.05 0.99 0 0.17 0 0.9 0 0 0.53 0 0
1 0.38 0 0 1 0 0 0.87 0 0 0 1 0 0 1 1 0.61 1 0.03 1 1 0.22 1 0.2 1 0.01 0 1 0.03 0
1 0.77 0 0 1 0.54 0 0.87 0 0 0 1 0 0 1 1 0.61 1 0.28 1 1 0.61 1 1 1 0.24 0 1 0.63 0
0 0.03 0.86 0.68 0 0.05 0.58 0 1 1 (µ = 1) 0.74 0 1 0 0 0 0 0 0.42 0 0 0.13 0 0 0 0.57 0.88 0 0.02 1
0 0.36 0.86 0.68 0 0.73 0.58 0 1 1 0.74 0 1 0 0 0 0 0 0.87 0 0 0.52 0 0.11 0 0.86 0.88 0 0.61 1
0.06 0.75 0.86 0.68 0 1 0.58 0 1 1 0.74 0 1 0 0 0 0 0 1 0.42 0 0.91 0.11 1 0 1 0.88 0.13 1 1
sample with the corresponding degree of unnaturalness defined by the membership functions for each rule.
In order to make calculations easy and mine our sample database for unnaturalness a computer program is coded using Fortran 90 programming language. Table 4 gives the total membership degrees of the fuzzy samples in various zones. The total membership degrees of the fuzzy samples (and degree of unnaturalness) for the fuzzified Western Electric Rules are given in Table 5. Sample 14 shows an out of control situation with respect to Rule 1, while samples 10 and 16 indicate an unnatural pattern with respect to the Rule 2. Their degrees of unnaturalness are determined from the membership function of the Rule 2. As an example, considering Rule 2 of the fuzzified Western Electric Rules, the membership degrees of samples 15 and 16 become 2 and it corresponds to a degree of unnaturalness of 1 (See Eq. (24)). The total membership degrees of the fuzzy samples in zones for fuzzified Grant and Leavenworth’s rules are represented in Table 6. As can be seen from Table 6, no samples indicate an unnatural pattern with respect to the fuzzified Grant and Leavenworth’s rules. The rules 1, 2, 5, 6, 7, and 8 among Nelson’s rules can be examined in the same way. For Nelson’s Rules 3 and 4, the fuzzy samples are defuzzified by using -level fuzzy midranges (given in Table 2) in order to check whether next sample shows an increment or decrement or alternating. -level fuzzy midranges for = 0.60 are illustrated in Fig. 9, which refers no unnaturalness with respect to the Nelson’s Rules 3 and 4.
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Table 6 Total membership degrees of the fuzzy samples in zones for fuzzified Grant and Leavenworth’s rules Sample no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
In or above CL
1.00 0.77 0.00 0.00 1.00 0.54 0.00 0.87 0.00 0.00 0.00 1.00 0.00 0.00 1.00 1.00 0.61 1.00 0.28 1.00 1.00 0.61 1.00 1.00 1.00 0.24 0.00 1.00 0.63 0.00
In or below CL
0.06 0.75 0.86 0.68 0.00 1.00 0.58 0.00 1.00 1.00 0.74 0.00 1.00 0.00 0.00 0.00 0.00 0.00 1.00 0.42 0.00 0.91 0.11 1.00 0.00 1.00 0.88 0.13 1.00 1.00
In or above CL
In or below CL
Rule 1
Rule 2
Rule 3
Rule 4
Rule 1
Rule 2
Rule 3
Rule 4
– – – – – – 3.31 3.18 2.41 2.41 2.41 2.41 1.87 1.87 2.00 3.00 3.61 4.61 3.89 4.89 5.89 5.51 5.51 5.90 5.90 5.86 4.86 4.86 4.87 3.87
– – – – – – – – – – 4.18 4.18 3.41 3.41 4.41 4.41 4.48 5.48 4.89 5.89 6.89 7.51 7.51 8.51 9.51 8.75 7.75 8.14 7.77 7.49
– – – – – – – – – – – – – 5.18 5.18 5.41 6.02 7.02 6.30 6.76 7.76 7.51 8.51 9.51 10.51 9.75 9.75 10.75 10.38 9.38
– – – – – – – – – – – – – – – – 7.79 7.79 7.30 8.30 9.30 8.92 9.38 10.38 10.51 10.75 10.75 11.75 11.38 11.38
– – – – – – 3.93 3.87 4.12 4.26 4.32 4.32 4.32 3.74 3.74 2.74 1.74 1.00 2.00 1.42 1.42 2.34 2.45 3.45 3.45 3.45 3.91 4.04 4.12 5.01
– – – – – – – – – – 6.67 6.61 6.86 6.00 5.32 5.32 4.32 3.74 4.74 4.16 3.16 3.34 3.45 3.45 3.45 4.45 5.33 5.46 6.46 6.46
– – – – – – – – – – – – – 7.67 7.61 6.86 6.00 5.32 6.32 5.74 5.16 6.08 5.19 5.19 4.45 5.45 5.33 5.46 6.46 7.46
– – – – – – – – – – – – – – – – 7.67 7.61 7.86 7.42 6.74 7.66 6.77 7.19 7.19 7.19 7.07 6.46 7.46 7.46
Fig. 9. -level ( = 0.60) fuzzy midranges of the fuzzy samples.
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6. Conclusion In the literature, numerous zone tests or run rules have been developed to assist quality practitioners in the detection of unnatural patterns for the crisp control charts. The fuzzy control charts in the literature are commonly based on the fuzzy transformations to crisp cases. The unnatural patterns analyses for fuzzy control charts have not been studied yet. In this paper, we have developed a direct fuzzy approach to fuzzy control charts without any defuzziffication, and then defined fuzzy unnatural pattern rules based on the probabilities of fuzzy events. The proposed fuzzy control chart is illustrated with a numerical example. Then fuzzy unnatural pattern rules are developed and applied to the test problem. We have defined fuzzy unnatural pattern rules based on the fuzzification of the crisp rules. The probability of each fuzzy unnatural pattern rule is calculated using the probability of fuzzy events. For further research, new fuzzy unnatural pattern rules can be developed and tested using fuzzy random variables. Useful and detailed information about the fuzzy random variables can be seen in Puri and Ralescu (1986). Appendix A Equations to compute sample area outside the control the limits t t 1 AU out = 2 d − U CL4 + d − U CL4 (max(t − , 0)) + 21 d z − a z + (c − b) (min(1 − t, 1 − )),
(A-U1)
where U CL4 − a and z = max(t, ), (b − a) + (c − b) 1 AU out = 2 d − U CL4 + (c − U CL3 ) (1 − ), 1 AU out = 2 d − U CL4 (max(t − , 0)),
(A-U3)
U CL4 − d , (U CL4 − U CL3 ) − (d − c) z z 1 AU out = 2 (c − U CL3 ) + d − U CL4 (min(1 − t, 1 − )),
(A-U4)
U CL4 − d and z = max(t, ), (U CL4 − U CL3 ) − (d − c) z t z2 t1 1 2 1 AU (min (max (t1 − , 0) , t1 − t2 )) out = 2 d − U CL4 + d − U CL4 z z 1 + 2 d 1 − a 1 + (c − b) (min (1 − t1 , 1 − )) ,
(A-U5)
t=
(A-U2)
where t=
where t=
where t1 =
U CL4 − a , (b − a) + (U CL4 − U CL3 )
z1 = max (, t1 ) AU out = 0,
and
t2 =
U CL4 − d , (U CL4 − U CL3 ) − (d − c)
z2 = max (, t2 )
1 AU out = 2 d − a + (c − b) (1 − ), t t 1 AL (max(t − , 0)) out = 2 LCL1 − a + LCL1 − a z z 1 + 2 d − a + (c − b) (min(1 − t, 1 − )),
(A-U6) (A-U7)
(A-L1)
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451
where d − LCL1 and z = max(, t), (LCL2 − LCL1 ) + (d − c) 1 AL out = 2 d − a + (c − b) (1 − ), 1 AL out = 2 LCL1 − a + (LCL2 − b) (1 − ), z2 z2 1 + LCLt11 − a t1 (min (max (t1 − , 0) , t1 − t2 )) AL out = 2 LCL1 − a + 21 d z1 − a z1 + (c − b) (min(1 − t, 1 − )), t=
(A-L2) (A-L3)
(A-L4)
where t1 =
d − LCL1 , (LCL2 − LCL1 ) + (d − c)
t2 =
a − LCL1 , (LCL2 − LCL1 ) − (b − a)
z1 = max (, t1 ) and z2 = max (, t2 ) , z z 1 AL out = 2 LCL1 − a + (LCL2 − b) (min(1 − t, 1 − )),
(A-L5)
where t=
a − LCL1 (LCL2 − LCL1 ) − (b − a)
AL out = 0,
and
1 AL out = 2 d − a + (c − b) (1 − ).
z = max(, t), (A-L6) (A-L7)
References Chen, S.-J., Hwang, C.-L., 1992. Fuzzy Multi Attribute Decision Making: Methods and Applications. Springer, Germany. Duncan, A.J., 1986. Quality Control and Industrial Statistics. fifth ed. Irwin Book Company, IL. Grant, E.L., Leavenworth, R.S., 1988. Statistical Quality Control. sixth ed. McGraw-Hill, New York. Gülbay, M., Kahraman, C., Ruan, D., 2004. -cut fuzzy control charts for linguistic data. Int. J. Intell. Systems 19, 1173–1196. Kanagawa, A., Tamaki, F., Ohta, H., 1993. Control charts for process average and variability based on linguistic data. Int. J. Prod. Res. 31, 913–922. Laviolette, M., Seaman, J.W., Barrett, J.D., Woodall, W.H., 1995. A probabilistic and statistical view of fuzzy methods. Technometrics 37, 249–262. Nelson, L.S., 1984. The Shewhart control chart-tests for special causes. J. Qual. Technol. 16, 237–239. Nelson, L.S., 1985. Interpreting Shewhart x-bar control charts. J. Qual. Technol. 17, 114–116. Puri, M.L., Ralescu, D.A., 1986. Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422. Raz, T., Wang, J.H., 1990. Probabilistic and membership approaches in the construction of control charts for linguistic data. Prod. Plann. Control 1, 147–157. Wang, J.H., Raz, T., 1988. Applying fuzzy set theory in the development of quality control chart. Proceedings of the International Industrial Engineering Conference, Orlando, FL, pp. 30–35. Wang, J.H., Raz, T., 1990. On the construction of control charts using linguistic variables. Int. J. Prod. Res. 28, 477–487. Western Electric, 1956. Statistical Quality Control Handbook. Western Electric, New York. Yen, J., Langari, R., 1999. Fuzzy Logic: Intelligence, Control, and Information. Prentice-Hall, NJ. Zadeh, L.A., 1965. Fuzzy set. Inform. Control 8, 338–353.
Development of fuzzy process control charts and fuzzy unnatural pattern analyses Murat Gülbaya, b,∗ , Cengiz Kahramana a Industrial Engineering Department, Istanbul Technical University, Istanbul, Turkey b Industrial Engineering Department, University of Gaziantep, Gaziantep, Turkey
Available online 15 May 2006
Abstract Many problems in scientific investigation generate nonprecise data incorporating nonstatistical uncertainty. A nonprecise observation of a quantitative variable can be described by a special type of membership function defined on the set of all real numbers called a fuzzy number or a fuzzy interval. A methodology for constructing control charts is proposed when the quality characteristics are vague, uncertain, incomplete or linguistically defined. Fuzzy set theory is an inevitable tool for fuzzy control charts as well as other applications subjected to uncertainty in any form. The vagueness can be handled by transforming incomplete or nonprecise quantities to their representative scalar values such as fuzzy mode, fuzzy midrange, fuzzy median, or fuzzy average. Then crisp methods may be applied to those representative values for control chart decisions as “in control” or “out of control”. Transforming the vague data by using one of the transformation methods may result in biased decisions since the information given by the vague data is lost by the transformation. Such data needs to be investigated as fuzzy sets without transformation, and the decisions based on the vague data should not be concluded with an exact decision. A “direct fuzzy approach (DFA)” to fuzzy control charts for attributes under vague data is proposed without using any transformation method. Then, the unnatural patterns for the proposed fuzzy control charts are defined using the probabilities of fuzzy events. © 2006 Elsevier B.V. All rights reserved. Keywords: Fuzzy process control; Run tests; Unnatural patterns; Fuzzy probability
1. Introduction Control charts have been widely used for monitoring process stability and capability. Control charts are based on data representing one or several quality-related characteristics of the product or service. If these characteristics are measurable on numerical scales, then variable control charts are used. If the quality-related characteristics cannot be easily represented in numerical form, then attribute control charts are useful. When a process is in statistical control, a control chart displays the known patterns of variation. When the control chart points deviate from these known patterns, the process is considered to be out of control. The control chart distinguishes between normal and nonnormal variation through the use of statistical tests and control limits. The control limits are calculated using the rules of probability so that when a point is determined to be out of control, it is due to an assignable cause and not due to a normal variation. ∗ Corresponding author. Industrial Engineering Department, Istanbul Technical University, Istanbul, Turkey. Tel.: +90 212 2931300x2073; fax: +90 212 2407260. E-mail address: [email protected] (M. Gülbay).
0167-9473/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.csda.2006.04.031
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435
UCL A CL+2σ B CL+1σ C CL C CL-1σ B CL-2σ A LCL
Fig. 1. The zones of a control chart.
The points outside the control limits are not the only criteria to determine the out of control conditions. However all points are inside the limits, the process may still be out of control if it does not display a normal pattern of variation. The zone tests, which are hypothesis tests in a modified form, are used to determine out of control conditions. They are used to test if the plotted points are following a normal pattern of variation. For a control chart to be effective, some action must be taken as a result of the chart pattern. When the process average is centered where it is supposed to be, and the variability displays a normal pattern, the process is considered to be in control. A normal pattern means that the process is aligned with the probabilities of the normal distribution. A large abnormal variability and unnatural patterns indicate out of control conditions. Out of control conditions usually have assignable causes that must be investigated and resolved. Numerous supplementary rules, like zone tests or run rules have been developed to assist quality practitioners in detection of the unnatural patterns for the crisp control charts. The run rules are based on the premise that a specific run of data has a low probability of occurrence in a completely random stream of data. If a run occurs, then this must mean that something has changed in the process to produce a nonrandom or unnatural pattern. The control charts may indicate an out-of-control condition when either one or more points fall beyond the control limits or plotted points show some nonrandom patterns of behavior. Unnatural (nonrandom) patterns for classical control charts have been extensively studied. Over the years, many rules have been developed to detect nonrandom patterns within the control limits. Under the pattern-recognition approach, numerous researches have defined several types of out-of-control patterns (e.g. trends, cyclic pattern, mixture, etc.) with a specific set of possible causes. When a process exhibits any of these unnatural patterns, it implies that those patterns may provide valuable information for process improvement. The unnatural (nonrandom) patterns for fuzzy control charts have not been studied yet. The zones of a control chart used in the zone tests are bounded by the standard deviations of the data as illustrated in Fig. 1. The probability of each zone based on the normal distribution is depicted in Fig. 2. The main idea behind defining a rule for an unnatural pattern is the probability of the occurrence: these rules are based on the premise that a specific run of data has a low probability of occurrence in a completely random stream of data. In general, probability of occurrence of an unnatural pattern is less than 1%. In the literature, there exist some unnatural patterns defined for the crisp cases. There is no certain rule about which unnatural patterns to use and the selection of a set of rules depends on the user preferences. Unnatural patterns are defined for the short runs, i.e., rules for a 15–20 consecutive points on the chart are investigated. The Western Electric (1956) suggested a set of decision rules for detecting unnatural patterns on control charts. Specifically, it suggested concluding that the process is out of control if any of the following conditions is satisfied. Rule 1: A single point falls outside of the control limits (beyond ±3 limits); Rule 2: Two out of three successive points fall in zone A or beyond (the odd point may be anywhere. Only two points count). Rule 3: Four out of five successive points fall in zone B or beyond (the odd point may be anywhere. Only four points count). Rule 4: Eight successive points fall in zone C or beyond.
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A
B
B
A
2.14%
13.59 %
13.59 %
2.14 %
C
C
34.13 % 34.13 %
-3σ
-2σ
-1σ
µ
+1σ
+2σ
+3σ
Fig. 2. The zones and probabilities of normal distribution.
One-sided probabilities of the rules above are calculated as 0.00135, 0.0015, 0.0027, and 0.0039, respectively. Grant and Leavenworth (1988) recommended that nonrandom variations are likely to be presented if any one of the following sequences of points occurs in the control charts. Rule 1: Seven consecutive points on the same side of the center line. Rule 2: At least 10 of 11 consecutive points on the same side of the center line. Rule 3: At least 12 of 14 consecutive points on the same side of the center line. Rule 4: At least 14 of 17 consecutive points on the same side of the center line. One-sided probabilities of the rules above are calculated as 0.00781, 0.00586, 0.00647, and 0.00636, respectively. Nelson (1985) proposed the following rules for unnatural patterns: Rule 1: One or more points outside of the control limits. Rule 2: Nine consecutive points in the same side of center line. Rule 3: Six points in a row steadily increasing or decreasing. Rule 4: Fourteen points in a row altering up and down. Rule 5: Two out of 3 points in a row in zone A or beyond. Rule 6: Four out of 5 points in zone B or beyond. Rule 7: Fifteen points in a row in zones C, above and below the centerline. Rule 8: Eight points in a row on both sides of the centerline with none in zone C. The unnatural patterns tend to fluctuate too wide or they fail to balance around the centerline. The portrayal of natural and unnatural patterns is what makes the control chart a very useful tool for statistical process and quality control. When a chart is interpreted, we look for special patterns such as cycles, trends, freaks, mixtures, groupings or bunching of measurements, and sudden shifts in levels. A product is generally classified in a binary manner (conforming or nonconforming) in attributed control charts. Binary classification may not be the most suitable if the product quality changes gradually rather than abruptly. Various procedures have been proposed for monitoring procedures in which the data are categorical in nature. In past research, Wang and Raz (1988, 1990), Raz and Wang (1990) and Kanagawa et al. (1993) proposed an assessment of intermediate quality level instead of the traditional binary judgement when the quality characteristics are not numerically measurable. Linguistic variables represent features by linguistic terms instead of numerical measurement. For example, the product quality feature can be classified by one of the terms ‘perfect’, ‘good’, ‘poor’, and ‘bad’, depending on the product’s deviation from specifications. These words or phrases are called linguistic variables. Furthermore, each linguistic variable value can be represented as a membership function. A linguistic variable differs from a numerical variable
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437
in that its values are not numbers but words or phrases in a language (Wang and Raz, 1990). Linguistic variables are derived from human subjective judgements. Wang and Raz (1988, 1990) and Raz and Wang (1990) provided a potential application of a fuzzy-set theory and constructed a control chart for these linguistic variables under the consideration of a normal distribution. Kanagawa et al. (1993) provided another control chart under an estimated distribution function. Laviolette et al. (1995) pointed out that the control limit proposed by Wang and Raz (1990) had no satisfactory efficiency when it was used to detect a change in the process mean. It is reasonable to treat the linguistic and uncertain data in the light of the fuzzy-set theory. Although the fuzzy methods provide a powerful framework for pattern recognition due to their ability to generate gradual memberships of objects to clusters, a number of rules have been proposed to defuzzify the classification results in order to be able to make a final (crisp) decision about the process. It is necessary to convert the fuzzy sets associated with linguistic values into scalars, which will be referred to as representative values. This conversion may be done in a number of ways, as long as the result is intuitively representative of the range of the base variable included in the fuzzy set, but the concept of the information connected with the concept of uncertainty has been lost upon this conversion. Some of them are fuzzy mode, fuzzy median, and -level fuzzy midrange (Zadeh, 1965). The purpose of data analysis is to gain information from data. By applying a conversion method to the vague data which result in scalar data we have already lost the information at the initial stage. Recently, -level fuzzy control charts for attributes are proposed by Gülbay et al. (2004) in order to reflect the vagueness of the data and tightness of the inspection. The objective of this paper is to develop a monitoring and diagnostic system to indicate natural out-of-control situations without defuzzification and define fuzzy unnatural patterns for the fuzzy control charts. The paper is organized as follows: fuzzy process control charts based on defuzzification using fuzzy transformation methods, and an alternative approach without any defuzzification are presented in Section 2. Probability of fuzzy events is explained in Section 3. Based on the probability of fuzzy events, fuzzy unnatural pattern rules for fuzzy control charts are given in Section 4. A numerical illustrative example is presented in Section 5. Finally, concluding remarks are given in Section 6. 2. Fuzzy process control charts A fuzzy approach is suitable for attributes control charts (p, np, c, and u charts) when the data is linguistic, categorical, uncertain, or human dependent subjective judgement is possible. In classical p charts, products are distinctly classified as “conformed” or “nonconformed” when determining fraction rejected. In fuzzy p control charts (See: Gülbay et al., 2004), when categorizing products, several linguistic terms are used to denote the degree of being nonconformed product such as “standard”, “second choice”, “third choice”, “chipped”, and so on… . A membership degree of being a nonconformed product is assigned to each linguistic term. Sample means for each sample group, Mj , are calculated as: Mj =
t
i=1 kij ri
mj
,
(1)
where t is the total number of linguistic terms, kij the number of products categorized with the linguistic term i in the sample j , ri (0 ri 1) the membership degree of the linguistic term i, and mj the total number of products in sample j . Center line, CL, is the average of the means of the n sample groups and can be determined by n j =1 Mj , (2) CL = Mj = n where n is the number of sample groups initially available. kij and ri in Eq. (1), and so in Eq. (2), are the uncertain values and depend on the human subjective judgment. In another word, a sample can be belonged to the second choice category by a quality controller, while it may be included in the standard or third choice by another quality controller. In the same way, defining a membership degree for a category may depend on the quality controller preferences. Therefore, the value of Mj may lie between 0 and 1, as a result of these human judgments. It is clear that CL in Eq. (2) has a range between 0 and 1 too. To overcome the uncertainty in the determination of the CL, fuzzy set theory can successfully be adopted by defining CL as a triangular fuzzy number (TFN) whose fuzzy mode is CL, as shown in
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j (x)
1
L j (α) Rj (α)
x 0
CL = M = p
1
Fig. 3. TFN representation of M and Mj of the sample j .
Fig. 3. Then, for each sample mean, Lj () and Rj () can be calculated using Eqs. (3) and (4), respectively Lj () = Mj , Rj () = 1 − 1 − Mj . The membership function of the M, or CL, can be written as ⎧ 0 if x 0, ⎪ ⎪ ⎪ ⎪ x ⎪ ⎪ if 0 x M, ⎪ ⎨M Mj (x) = 1−x ⎪ ⎪ ⎪ if M x 1, ⎪ ⎪ 1−M ⎪ ⎪ ⎩ 0 if x 1.
(3) (4)
(5)
The control limits for -cut is also a fuzzy set and can be represented by TFNs. Since the membership function of CL is divided into two components, then, each component will have its own CL, LCL, and U CL. The membership function of the control limits depending upon the value of is given in Eq. (6) and illustrated in Fig. 4 ⎫ ⎧ ⎧ CLL = M ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫
⎪ ⎪ ⎪ ⎪ ⎪ L ⎬⎪ ⎪ ⎪ ⎪ ⎨ L ⎪ ⎪ ⎪ ⎪ CL 1 − CL ⎪ ⎪ ⎪ ⎪ ⎪ L L ⎪ ⎪ ⎪ = max CL − 3 , 0 LCL ⎪ ⎨ ⎬ ⎪ ⎩ ⎭ n ⎪ ⎪ if 0 Mj M, ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫⎪ ⎪ ⎪
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬⎪ ⎪ ⎪ ⎪ ⎪ CLL 1 − CLL ⎪ ⎪ ⎪ L = min CLL + 3 ⎪ ⎪ ⎪ , 1 U CL ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ ⎭⎭ n ⎨ Control limits () = ⎧ (6) ⎫ CLR = 1 − 1 − M ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎧
⎪ ⎪ ⎪ ⎪ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ R R ⎪ ⎪ ⎪ CL 1 − CL ⎪ ⎪ ⎪ ⎪ R = max CLR − 3 ⎪ ⎪ ⎪ , 0 LCL ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ n ⎪ ⎪ if M Mj 1, ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎫ ⎧ ⎪
⎪ ⎪ ⎪ ⎪ R ⎬⎪ ⎪ ⎪ ⎨ ⎪⎪ ⎪ ⎪ ⎪ CL 1 − CLR ⎪ ⎪ ⎪ ⎪ ⎪ R R ⎪ ⎪ , 1 = min CL + 3 U CL ⎩⎪ ⎩ ⎭⎭ ⎩ n where n is the average sample size (ASS). When the ASS is used, the control limits do not change with the sample size. Hence, the control limits for all samples are the same.
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α-level R LCL ()
1
R CL ()
UCLR () LCLL ()
UCLL () L CL ()
M Fig. 4. Illustration of the -cut control limits.
1
1
a a b (a)
c d
d
a a b=c d (b)
d
number of nonconformity
Fig. 5. Representation of number of nonconformities by fuzzy numbers: (a) trapezoidal (a, b, c, d); (b) triangular (a, b, b, d).
For the variable sample size (VSS), n should be replaced by the size of the j th sample, nj . Hence, control limits change for each sample depending upon the size of the sample. Therefore, each sample has its own control limits. The decision that whether process is in control (1) or out of control (0) for both ASS and VSS is as follows: 1 if LCLL ()Lj ()U CLL () ∧ LCLR () Rj () U CLR (), (7) Process control = 0 otherwise. The value of -cut is decided with respect to the tightness of inspection such that for a tight inspection, values close to 1 may be used. As can be seen from Fig. 4, while reduces to 0 (decreasing the tightness of inspection), the range where the process is in control (difference between U CL and LCL) increases. In the crisp case, the control limits for number of nonconformities are calculated by the Eqs. (8)–(10) CL = c,
√ LCL = c − 3 c, √ U CL = c + 3 c,
(8) (9) (10)
where c is the mean of the nonconformities. In the fuzzy case, where number of nonconformity includes human subjectivity or uncertainty, uncertain values such as “between 10 and 14” or “approximately 12” can be used to define number of nonconformities in a sample. Then number of nonconformity in each sample, or subgroup, can be represented by a trapezoidal fuzzy number (a, b, c, d) or a triangular fuzzy number (a, b, d) as shown in Fig. 5. Note that a trapezoidal fuzzy number becomes triangular when b = c. For the ease of representation and calculation, a triangular fuzzy number is also represented as trapezoidal by (a, b, b, d) or (a, c, c, d). Here, we propose a direct fuzzy approach (DFA) to deal with the vague data for the control charts. Transforming the vague data by representing them with their representative values may result in biased decisions for particular data especially when they are represented by asymmetrical fuzzy numbers. The center line, CL, given in Eq. (8), is the mean
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of the samples. For fuzzy case, where the numbers of nonconformities are represented by trapezoidal fuzzy numbers, ˜ can be determined using the arithmetic mean of the fuzzy numbers and written as in Eq. (11) fuzzy center line, CL, (see Chen and Hwang, 1992 for the fuzzy arithmetics performed in this paper) n n n n aj bj cj dj j =1 j =1 j =1 j =1 ˜ = CL , , , = a, b, c, d , (11) n n n n where n is the number of fuzzy samples and a, b, c and d are the arithmetic means of the a, b, c, and d, respectively. ˜ can be rewritten as in Eq. (12). Then LCL ˜ and U CL ˜ are calculated using fuzzy arithmetics as given in Eqs. (13) CL and (14), respectively ˜ = a, b, c, d = (CL1 , CL2 , CL3 , CL4 ) , CL (12) ˜ = CL ˜ ˜ LCL − 3 CL = CL1 − 3 CL4 , CL2 − 3 CL3 , CL3 − 3 CL2 , CL4 − 3 CL1 = (LCL1 , LCL2 , LCL3 , LCL4 ) , ˜ = CL ˜ + 3 CL ˜ U CL = CL1 + 3 CL1 , CL2 + 3 CL2 , CL3 + 3 CL3 , CL4 + 3 CL4 = (U CL1 , U CL2 , U CL3 , U CL4 ) .
(13)
(14)
An -cut is a nonfuzzy set which comprises all elements whose membership is greater than or equal to . Applying -cuts of fuzzy sets (Fig. 5), values of a and d for samples and CL1 and CL4 (start and end points of the -cut of CL) for center line are determined by Eqs. (15) and (16), respectively a = a + (b − a),
CL1 = CL1 + (CL2 − CL1 ) ,
(15)
d = d − (d − c),
CL4 = CL4 − (CL4 − CL3 ) .
(16)
Using -cut representations, fuzzy control limits can be rewritten as given in Eqs. (17)–(19) ˜ = CL , CL2 , CL3 , CL , CL 1 4 ˜ = LCL , LCL2 , LCL3 , LCL , LCL 1 4 ˜ = U CL , U CL2 , U CL3 , U CL . U CL 1 4
(17) (18) (19)
The results of these equations can be illustrated as in Fig. 6. To retain the standard format of control charts and to facilitate the plotting of observations on the chart, it is necessary to convert the fuzzy sets associated with linguistic values into scalars referred to as representative values. This conversion may be performed in a number of ways as long as the result is intuitively representative of the range of the base variable included in the fuzzy set. The four ways, which are similar in principle to the measures of central tendency used in descriptive statistics, are fuzzy mode, -level fuzzy midrange, fuzzy median, and fuzzy average. It should be pointed out that there is no theoretical basis supporting any one specifically and the selection between them should be mainly based on the ease of computation or preference of the user (Wang and Raz, 1990). The conversion of fuzzy sets into crisp values results in loss of information in linguistic data. To retain the information of the linguistic data, we prefer to keep fuzzy sets as themselves and to compare fuzzy samples with the fuzzy control limits. For this reason, a direct fuzzy approach (DFA) based on the area measurement is proposed for the fuzzy control charts. The decision about whether the process is in control can be made according to the percentage area of the sample ˜ and/or LCL ˜ defined as fuzzy sets. When the fuzzy sample is completely involved by which remains inside the U CL the fuzzy control limits, the process is said to be “in-control”. If a fuzzy sample is totally excluded by the fuzzy control limits, the process is said to be “out of control”. Otherwise, a sample is partially included by the fuzzy control limits. In this case, if the percentage area (j ) which remains inside the fuzzy control limits is equal or greater than a predefined acceptable percentage (), 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 illustrated in Fig. 7. The parameters for determination
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UCL4
UCL4 UCL3 UCL2
~ UCL
UCL1 CL4 UCL1
CL4 CL3 ~ CL
CL2
CL1 LCL4 CL1
LCL4
LCL3 ~ LCL LCL 2
LCL1
LCL 1 µ
α
1
Fig. 6. Representation of fuzzy control limits.
d c b a ~ UCL
α
1 Type U1
t1 Type U2
Type U3
t1 t2 Type U4
Type U5
Type U6
Type U7
Type L4
Type L5
Type L6
Type L7
~ LCL
Type L1
Type L2
Type L3
Fig. 7. Illustration of all possible sample areas outside the fuzzy control limits at -level cut.
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of the sample area outside the control limits for -level fuzzy cut are LCL1 , LCL2 , U CL3 , U CL4 , a, b, c, d, and . The shape of the control limits and fuzzy sample are formed by the lines of LCL1 LCL2 , U CL3 U CL4 , ab, and cd. A flowchart to calculate area of the fuzzy sample outside the control limits is given in Fig. 8. The sample area above the L upper control limits, AU out , and the sample area falling below the lower control limits, Aout , are calculated. Equations U L to compute Aout and Aout are given in Appendix A. Then, the total sample 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. (20) j =
Sj − Aout,j Sj
,
(20)
where Sj is the sample area at -level cut. The DFA provides the possibility of obtaining linguistic decisions like “rather in control” or “rather out of control”. Further intermediate levels of process control decisions are also possible by defining in stages. For instance, it may be defined as given below which is more distinguished ⎧ in control, 0.85 j 1, ⎪ ⎪ ⎪ ⎨ rather in control, 0.60 j < 0.85, Process control = (21) ⎪ rather out of control, 0.10 j < 0.60, ⎪ ⎪ ⎩ out of control, 0 j < 0.10. The intermediate levels of process control decisions are subjectively defined by the quality expert. In binary classification (crisp case), the quality expert may only know if the process is in control or out of control. These predefined levels may refer to the truthful of the out of control. It can be used as a tracking and may give valuable information before the process is out of control. However intermediate levels are subjectively defined, it should refer to the depth of information the quality expert needs to take some preventive actions. 3. Probability of fuzzy events Analysis of fuzzy unnatural patterns for fuzzy control charts is necessary to develop. The formula for calculating the probability of a fuzzy event A is a generalization of the probability theory: in the case which a sample space X is a continuum or discrete, the probability of a fuzzy event P (A) is given by (Yen and Langari, 1999): (x)PX (x) dx if X is continuous, (22) P (A) = A i A (xi )PX (xi ) if X is discrete, where PX denotes a classical probability distribution function of X for continuous sample space and probability function for discrete sample space, and A is a membership function of the event A. The membership degree of a fuzzy sample that belongs to a region is directly related to its percentage area falling in that region, and therefore, it is continuous. For example, a fuzzy sample may be in zone B with a membership degree of 0.4 and in zone C with a membership degree of 0.6. While counting fuzzy samples in zone B, that sample is counted as 0.4. 4. Generation of fuzzy rules for unnatural patterns Numerous supplementary rules, like zone tests or run rules (Western Electric, 1956; Nelson, 1984, 1985; Duncan, 1986; Grant and Leavenworth, 1988) have been developed to assist quality practitioners in detection of unnatural patterns for the crisp control charts. The run rules are based on the premise that a specific run of data has a low probability of occurrence in a completely random stream of data. If a run occurs, then this must mean that something has changed in the process to produce a nonrandom or unnatural pattern. Based on the expected percentages in each zone, sensitive run tests can be developed for analyzing the patterns of variation in the various zones. For the fuzzy control charts, based on the Western Electric rules (1956), the following fuzzy unnatural pattern rules can be defined. The probabilities of these fuzzy events are calculated using normal approach to binomial distribution.
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Fig. 8. Flowchart to compute the area outside the fuzzy control limits.
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The probability of each fuzzy rule (event) below depends on the definition of the membership function which is subjectively defined so that the probability of each of the fuzzy rules is as close as possible to the corresponding classical rule for unnatural patterns. The idea behind this approach may justify the following rules. Rule 1: Any fuzzy data falling outside the three-sigma control limits with a ratio of more than predefined percentage () of sample area at desired -level. The membership function for this rule can subjectively be defined as below: ⎧ 0, 0.85x 1, ⎪ ⎨ (x − 0.60)/0.25, 0.60 x 0.85, 1 (x) = (23) ⎪ ⎩ (x − 0.10)/0.50, 0.10 x 0.60, 1, 0 x 0.10. Rule 2: A total membership degree around 2 from three consecutive points in zone A or beyond. Probability of a sample being in zone A (0.0214) or beyond (0.00135) is 0.02275. Let the membership function for this rule be defined as follows: 0, 0 x 0.59, 2 (x) = (x − 0.59)/1.41, 0.59 x 2, (24) 1, 2 x 3. Using the membership function above, the fuzzy probability given in Eq. (22) can be determined by Eq. (25) 3 x1 x2 3 2 (x)P2 (x) dx = 2 (x)P2 (x) dx + 2 (x)P2 (x) dx + 2 (x)P2 (x) dx 0
0
=
x2
x1
where
x1 3
x2
2 (x)P2 (x) dx +
x2
2 (x)P2 (x) dx,
x − np PX (x) = PX z √ . npq
(25)
(26)
To integrate the equation above, the membership function is divided into sections each with a 0.05 width and 2 (x)Px (x) values for each section are added. For x1 = 0.59 and x2 = 2, the probability of the fuzzy event, rule 2, is determined as 0.0015, which corresponds to the crisp case of this rule. In the following rules, the membership functions are set in the same way. Rule 3: A total membership degree around 4 from five consecutive points in zone C or beyond: 0, 0 x 2.42, 3 (x) = (x − 2.42)/1.58, 2.42 x 4, (27) 1, 4 x 5. The fuzzy probability for this rule is calculated as 0.0027. Rule 4: A total membership degree around 8 from eight consecutive points on the same side of the centerline with the membership function below: 0, 0 x 2.54, 4 (x) = (28) (x − 2.54)/5.46, 2.54 x 8. The fuzzy probability for the rule above is then determined as 0.0039. Based on Grant and Leavenworth’s rules (1988), the following fuzzy unnatural pattern rules can be defined. Rule 1: A total membership degree around 7 from seven consecutive points on the same side of the center line. The fuzzy probability of this rule is 0.0079 when membership function is defined as below: 0, 0 x 2.48, 1 (x) = (29) (x − 2.48)/4.52, 2.48 x 7.
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Rule 2: At least a total membership degree around 10 from 11 consecutive points on the same side of the center line. The fuzzy probability of this rule is 0.0058 when the membership function is defined as below: 0, 0 x 9.33, 2 (x) = (x − 9.33)/0.77, 9.33x 10, (30) 1, 10 x 11. Rule 3: At least a total membership degree around 12 from 14 consecutive points on the same side of the center line. If the membership function is set as given below, then the fuzzy probability of the rule is equal to 0.0065 0, 0 x 11.33, 3 (x) = (x − 11.33)/0.67, 11.33x 12, (31) 1, 12 x 14. Rule 4: At least a total membership degree around 14 from 17 consecutive points on the same side of the center line. The probability of this fuzzy event with the membership function below is 0.0062 0, 0 x 13.34, 4 (x) = (x − 13.34)/0.66, 13.34 x 14, (32) 1, 14 x 17. The fuzzy unnatural pattern rules based on Nelson’s Rules (1985) can be defined in the same way. Some of Nelson’s rules (Rules 3 and 4) are different from the Western Electric Rules and Grant and Leavenworth’s rules. In order to apply these rules to fuzzy control charts, fuzzy samples can be defuzzified using -level fuzzy midranges of the samples. The , is defined as the midpoint of the ends of the -cut. If a and d are the end points of -cut, -level fuzzy midrange, fmr then, fmr = 21 a + d . (33) Then Nelson’s 3rd and 4th rules are fuzzified as follows: Rule 3: Six points in a row steadily increasing or decreasing with respect to the desired -level fuzzy midranges. Rule 4: Fourteen points in a row altering up and down with respect to the desired -level fuzzy midranges. 5. An illustrative example Samples of 200 units are taken every 4 h to control number of nonconformities. Data collected from 30 subgroups shown in Table 1 are linguistic such as “approximately 30” or “between 25 and 30”. The linguistic expressions in Table 1 are represented by fuzzy numbers as shown in Table 3. These numbers are subjectively identified by the quality control expert who also sets = 0.60 and minimum acceptable ratio as = 0.70. As decreases, membership degree of the linguistic terms decreases as well as the truthful of the decisions. Therefore, process control decisions may change by choosing different levels of and . The quality control expert also set the acceptable membership degree of unnaturalness as 0.95, that is, when a sample refers to an unnatural sample with respect to any rule, it should refer a membership degree of unnaturalness more than 0.95 with respect to the membership functions defined for that rule. ˜ LCL, ˜ and U CL ˜ are determined as follows: Using Eqs. (11)–(14), CL, ˜ = (18.13, 22.67, 26.93, 32.07), CL ˜ = (1.15, 7.10, 12.65, 19.29), LCL ˜ = (30.91, 36.95, 42.50, 49.05). U CL Applying -cut of 0.60, values of CL˜=0.60 , LCL˜=0.60 , and U CL˜=0.60 are calculated CL˜=0.60 = (20.85, 22.67, 26.93, 28.99), LCL˜=0.60 = (4.72, 7.10, 12.65, 15.31), U CL˜=0.60 = (34.53, 36.95, 42.50, 45.12).
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Table 1 Number of nonconformities for 30 subgroups No.
Approximately
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
30
Between
20–30 5–12 6 38 20–24 4–8 36–44 11–15 10–13 6 32 13 50–52 38–41
No.
Approximately
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
40
Between
32–50 39 15–21 28 32–35 10–25 30 25 31–41 10–25 5–14 28–35 20–25 8
Table 2 Fuzzy number (a, b, c, d) representation of 30 subgroups No.
a
b
c
d
=0.60 fmr
No.
a
b
c
d
=0.60 fmr
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
25 15 4 3 32 16 3 27 9 7 3 27 11 39 28
30 20 5 6 38 20 4 36 11 10 6 32 13 50 38
30 30 12 6 38 24 8 44 15 13 6 32 13 52 41
35 35 15 8 45 28 12 50 20 15 10 37 15 55 45
28 18 4.6 4.8 35.6 18.4 3.6 32.4 10.2 8.8 4.8 30 12.2 45.6 34
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
33 28 33 12 23 28 14 24 20 25 7 3 23 17 5
40 32 39 15 28 32 18 30 25 31 10 5 28 20 8
40 50 39 21 28 35 28 30 25 41 25 14 35 25 8
44 60 43 38 36 42 33 34 31 46 28 20 38 29 15
37.2 30.4 36.6 13.8 26 30.4 16.4 27.6 23 28.6 8.8 4.2 26 18.8 6.8
18.13
22.67
26.93
32.07
Average
Table 3 Fuzzy zones calculated for the example Zone
a
b
c
d
U CL +2 +1 CL −1 −2 LCL
34.53 29.97 25.41 20.85 15.47 10.10 4.72
36.95 32.19 27.43 22.67 17.48 12.29 7.10
42.50 37.31 32.12 26.93 22.17 17.41 12.65
45.12 39.74 34.37 28.99 24.43 19.87 15.31
Based on the Western Electric Rules 1–4, membership functions in Eqs. (23), (24), (27), and (28) are used. These membership functions set the degree of unnaturalness for each rule. As an example, when a total membership degree of 1.90 is calculated for the rule 2, its degree of unnaturalness is determined from 2 (x) as 0.9291.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0 0 0 0 0 0 0 0.13 0 0 0 0 0 1 0 0 0.39 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0.73 0 0 0 0 0 0 0.98 1 0.39 1 0 0 0.2 0 0 0 0.51 0 0 0.04 0 0
1 1 1 1 0 1 1 0.14 1 1 1 1 1 0 0.02 0 0.22 0 1 1 0.8 1 1 1 0.49 1 1 0.96 1 1
0 0 0 0 0.32 0 0 0.61 0 0 0 0 0 1 0.56 0.72 0.65 0.49 0 0 0 0 0 0 0.28 0 0 0 0 0
0.24 0.04 0 0 0.68 0 0 0.39 0 0 0 0.96 0 0 0.44 0.28 0.35 0.51 0 0.05 0.99 0 0.17 0 0.61 0 0 0.53 0 0
0.76 0.96 1 1 0 1 1 0 1 1 1 0.04 1 0 0 0 0 0 1 0.95 0.01 1 0.83 1 0.1 1 1 0.47 1 1
0 0 0 0 1 0 0 0.97 0 0 0 0 0 1 1 1 0.9 1 0 0 0.61 0 0 0 0.72 0 0 0.27 0 0
1 0.38 0 0 0 0 0 0.03 0 0 0 1 0 0 0 0 0.1 0 0.03 1 0.39 0.22 1 0.2 0.28 0.01 0 0.73 0.03 0
0 0.62 1 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0.97 0 0 0.78 0 0.8 0 0.99 1 0 0.97 1
0.94 0.25 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 0.58 1 0.09 0.89 0 1 0 0 0.87 0 0
0.06 0.52 0 0 0 0.54 0 0 0 0 0 0 0 0 0 0 0 0 0.28 0.42 0 0.53 0.11 1 0 0.24 0 0.13 0.63 0
0 0.23 1 1 0 0.46 1 0 1 1 1 0 1 0 0 0 0 0 0.72 0 0 0.39 0 0 0 0.76 1 0 0.37 1
1 0.64 0 0 1 0.27 0 1 0 0 0 1 0 1 1 1 1 1 0.13 1 1 0.48 1 0.89 1 0.14 0 1 0.39 0
0 0.36 0 0 0 0.73 0 0 0.05 0 0 0 0 0 0 0 0 0 0.67 0 0 0.52 0 0.11 0 0.42 0.01 0 0.61 0
0 0 1 1 0 0 1 0 0.95 1 1 0 1 0 0 0 0 0 0.21 0 0 0 0 0 0 0.44 0.99 0 0 1
1 0.97 0 0 1 0.95 0 1 0 0 0 1 0 1 1 1 1 1 0.58 1 1 0.87 1 1 1 0.43 0 1 0.98 0
0 0.03 0.18 0 0 0.05 0 0 0.89 0.55 0 0 1 0 0 0 0 0 0.42 0 0 0.13 0 0 0 0.46 0.38 0 0.02 0.02
0 0 0.82 1 0 0 1 0 0.11 0.45 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.11 0.62 0 0 0.98
1 1 0 0 1 1 0 1 0.37 0.01 0 1 0.02 1 1 1 1 1 0.97 1 1 1 1 1 1 0.72 0.12 1 1 0
0 0 0.86 0.68 0 0 0.58 0 0.63 0.99 0.74 0 0.98 0 0 0 0 0 0.03 0 0 0 0 0 0 0.28 0.76 0 0 1
0 0 0.14 0.32 0 0 0.42 0 0 0 0.26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.12 0 0 0
No. A +3 in +3 B +3 A +2 in +2 B +2 A +1 in +1 B +1 A CL in CL B CL A −1 in −1 B −1 A −2 in −2 B −2 A −3 in −3 B −3
Table 4 Membership degrees of fuzzy samples for different zones (A: Above, B: Below)
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Table 5 Total membership degrees of the fuzzy samples in zones for the fuzzified Western Electric Rules Sample no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 a Unnatural
Beyond ±3
0.00 0.00 0.14 0.32 0.00 0.00 0.42 0.13 0.00 0.00 0.26 0.00 0.00 1.00 0.00 0.00 0.39 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.12 0.00 0.00 0.00
In or above fuzzy CL
In or below fuzzy CL
Rule 2
Rule 3
Rule 4
Rule 2
Rule 3
Rule 4
0.24 0.04 0 0 1 0 0 0.87 0 0 0 0.96 0 0 1 1 (µ = 1)a 0.61 1 0 0.05 0.99 0 0.17 0 0.9 0 0 0.53 0 0
1 0.38 0 0 1 0 0 0.87 0 0 0 1 0 0 1 1 0.61 1 0.03 1 1 0.22 1 0.2 1 0.01 0 1 0.03 0
1 0.77 0 0 1 0.54 0 0.87 0 0 0 1 0 0 1 1 0.61 1 0.28 1 1 0.61 1 1 1 0.24 0 1 0.63 0
0 0.03 0.86 0.68 0 0.05 0.58 0 1 1 (µ = 1) 0.74 0 1 0 0 0 0 0 0.42 0 0 0.13 0 0 0 0.57 0.88 0 0.02 1
0 0.36 0.86 0.68 0 0.73 0.58 0 1 1 0.74 0 1 0 0 0 0 0 0.87 0 0 0.52 0 0.11 0 0.86 0.88 0 0.61 1
0.06 0.75 0.86 0.68 0 1 0.58 0 1 1 0.74 0 1 0 0 0 0 0 1 0.42 0 0.91 0.11 1 0 1 0.88 0.13 1 1
sample with the corresponding degree of unnaturalness defined by the membership functions for each rule.
In order to make calculations easy and mine our sample database for unnaturalness a computer program is coded using Fortran 90 programming language. Table 4 gives the total membership degrees of the fuzzy samples in various zones. The total membership degrees of the fuzzy samples (and degree of unnaturalness) for the fuzzified Western Electric Rules are given in Table 5. Sample 14 shows an out of control situation with respect to Rule 1, while samples 10 and 16 indicate an unnatural pattern with respect to the Rule 2. Their degrees of unnaturalness are determined from the membership function of the Rule 2. As an example, considering Rule 2 of the fuzzified Western Electric Rules, the membership degrees of samples 15 and 16 become 2 and it corresponds to a degree of unnaturalness of 1 (See Eq. (24)). The total membership degrees of the fuzzy samples in zones for fuzzified Grant and Leavenworth’s rules are represented in Table 6. As can be seen from Table 6, no samples indicate an unnatural pattern with respect to the fuzzified Grant and Leavenworth’s rules. The rules 1, 2, 5, 6, 7, and 8 among Nelson’s rules can be examined in the same way. For Nelson’s Rules 3 and 4, the fuzzy samples are defuzzified by using -level fuzzy midranges (given in Table 2) in order to check whether next sample shows an increment or decrement or alternating. -level fuzzy midranges for = 0.60 are illustrated in Fig. 9, which refers no unnaturalness with respect to the Nelson’s Rules 3 and 4.
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Table 6 Total membership degrees of the fuzzy samples in zones for fuzzified Grant and Leavenworth’s rules Sample no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
In or above CL
1.00 0.77 0.00 0.00 1.00 0.54 0.00 0.87 0.00 0.00 0.00 1.00 0.00 0.00 1.00 1.00 0.61 1.00 0.28 1.00 1.00 0.61 1.00 1.00 1.00 0.24 0.00 1.00 0.63 0.00
In or below CL
0.06 0.75 0.86 0.68 0.00 1.00 0.58 0.00 1.00 1.00 0.74 0.00 1.00 0.00 0.00 0.00 0.00 0.00 1.00 0.42 0.00 0.91 0.11 1.00 0.00 1.00 0.88 0.13 1.00 1.00
In or above CL
In or below CL
Rule 1
Rule 2
Rule 3
Rule 4
Rule 1
Rule 2
Rule 3
Rule 4
– – – – – – 3.31 3.18 2.41 2.41 2.41 2.41 1.87 1.87 2.00 3.00 3.61 4.61 3.89 4.89 5.89 5.51 5.51 5.90 5.90 5.86 4.86 4.86 4.87 3.87
– – – – – – – – – – 4.18 4.18 3.41 3.41 4.41 4.41 4.48 5.48 4.89 5.89 6.89 7.51 7.51 8.51 9.51 8.75 7.75 8.14 7.77 7.49
– – – – – – – – – – – – – 5.18 5.18 5.41 6.02 7.02 6.30 6.76 7.76 7.51 8.51 9.51 10.51 9.75 9.75 10.75 10.38 9.38
– – – – – – – – – – – – – – – – 7.79 7.79 7.30 8.30 9.30 8.92 9.38 10.38 10.51 10.75 10.75 11.75 11.38 11.38
– – – – – – 3.93 3.87 4.12 4.26 4.32 4.32 4.32 3.74 3.74 2.74 1.74 1.00 2.00 1.42 1.42 2.34 2.45 3.45 3.45 3.45 3.91 4.04 4.12 5.01
– – – – – – – – – – 6.67 6.61 6.86 6.00 5.32 5.32 4.32 3.74 4.74 4.16 3.16 3.34 3.45 3.45 3.45 4.45 5.33 5.46 6.46 6.46
– – – – – – – – – – – – – 7.67 7.61 6.86 6.00 5.32 6.32 5.74 5.16 6.08 5.19 5.19 4.45 5.45 5.33 5.46 6.46 7.46
– – – – – – – – – – – – – – – – 7.67 7.61 7.86 7.42 6.74 7.66 6.77 7.19 7.19 7.19 7.07 6.46 7.46 7.46
Fig. 9. -level ( = 0.60) fuzzy midranges of the fuzzy samples.
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6. Conclusion In the literature, numerous zone tests or run rules have been developed to assist quality practitioners in the detection of unnatural patterns for the crisp control charts. The fuzzy control charts in the literature are commonly based on the fuzzy transformations to crisp cases. The unnatural patterns analyses for fuzzy control charts have not been studied yet. In this paper, we have developed a direct fuzzy approach to fuzzy control charts without any defuzziffication, and then defined fuzzy unnatural pattern rules based on the probabilities of fuzzy events. The proposed fuzzy control chart is illustrated with a numerical example. Then fuzzy unnatural pattern rules are developed and applied to the test problem. We have defined fuzzy unnatural pattern rules based on the fuzzification of the crisp rules. The probability of each fuzzy unnatural pattern rule is calculated using the probability of fuzzy events. For further research, new fuzzy unnatural pattern rules can be developed and tested using fuzzy random variables. Useful and detailed information about the fuzzy random variables can be seen in Puri and Ralescu (1986). Appendix A Equations to compute sample area outside the control the limits t t 1 AU out = 2 d − U CL4 + d − U CL4 (max(t − , 0)) + 21 d z − a z + (c − b) (min(1 − t, 1 − )),
(A-U1)
where U CL4 − a and z = max(t, ), (b − a) + (c − b) 1 AU out = 2 d − U CL4 + (c − U CL3 ) (1 − ), 1 AU out = 2 d − U CL4 (max(t − , 0)),
(A-U3)
U CL4 − d , (U CL4 − U CL3 ) − (d − c) z z 1 AU out = 2 (c − U CL3 ) + d − U CL4 (min(1 − t, 1 − )),
(A-U4)
U CL4 − d and z = max(t, ), (U CL4 − U CL3 ) − (d − c) z t z2 t1 1 2 1 AU (min (max (t1 − , 0) , t1 − t2 )) out = 2 d − U CL4 + d − U CL4 z z 1 + 2 d 1 − a 1 + (c − b) (min (1 − t1 , 1 − )) ,
(A-U5)
t=
(A-U2)
where t=
where t=
where t1 =
U CL4 − a , (b − a) + (U CL4 − U CL3 )
z1 = max (, t1 ) AU out = 0,
and
t2 =
U CL4 − d , (U CL4 − U CL3 ) − (d − c)
z2 = max (, t2 )
1 AU out = 2 d − a + (c − b) (1 − ), t t 1 AL (max(t − , 0)) out = 2 LCL1 − a + LCL1 − a z z 1 + 2 d − a + (c − b) (min(1 − t, 1 − )),
(A-U6) (A-U7)
(A-L1)
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where d − LCL1 and z = max(, t), (LCL2 − LCL1 ) + (d − c) 1 AL out = 2 d − a + (c − b) (1 − ), 1 AL out = 2 LCL1 − a + (LCL2 − b) (1 − ), z2 z2 1 + LCLt11 − a t1 (min (max (t1 − , 0) , t1 − t2 )) AL out = 2 LCL1 − a + 21 d z1 − a z1 + (c − b) (min(1 − t, 1 − )), t=
(A-L2) (A-L3)
(A-L4)
where t1 =
d − LCL1 , (LCL2 − LCL1 ) + (d − c)
t2 =
a − LCL1 , (LCL2 − LCL1 ) − (b − a)
z1 = max (, t1 ) and z2 = max (, t2 ) , z z 1 AL out = 2 LCL1 − a + (LCL2 − b) (min(1 − t, 1 − )),
(A-L5)
where t=
a − LCL1 (LCL2 − LCL1 ) − (b − a)
AL out = 0,
and
1 AL out = 2 d − a + (c − b) (1 − ).
z = max(, t), (A-L6) (A-L7)
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