Fuzzy development of Mean and Range control charts using statistical ...
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Journal of Intelligent & Fuzzy Systems 22 (2011) 253–265 DOI:10.3233/IFS-2011-0487 IOS Press
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Fuzzy development of Mean and Range control charts using statistical properties of different representative values H. Moheb Alizadeha and S.M.T. Fatemi Ghomib,∗ a Department
of Industrial Engineering, Tarbiat Moallem University, Tehran, Iran of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran
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b Department
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Abstract. This paper develops Mean and Range control charts in fuzzy environment using different transformation methods. It is assumed that the observations of each sample are fuzzy random variables, which have triangular membership functions. After calculating fuzzy mean and fuzzy range of each sample using fuzzy arithmetic, their representative values are calculated exploiting the transformation methods. Then, using statistical properties of the obtained representative values and basic structure of Shewhart control charts, new Mean and Range control charts are constructed to monitor the process mean and variation. After that, the power of control charts constructed based on different transformation methods are examined applying average run length (ARL) criterion. According to this criterion, it is concluded that, contrary to the previous claim, these different transformation methods lead to mean and rang control charts with various performances. Moreover, it is represented that the value of α – level has significant influence on the performance of obtained control charts. Finally, it is derived that incorporating fuzziness into observations results in less powerful control charts.
1. Introduction
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Keywords: Mean and Range control charts, fuzzy sets, transformation methods, fuzzy random variable, ARL criterion
Statistical process control (SPC) is a collection of methods to achieve continuous improvement in quality. This objective is accomplished by a continuous monitoring of the process under study in order to quickly detect the occurrence of the assignable causes and undertake the necessary corrective actions. The most commonly used SPC tools are control charts. A control chart contains a center line that represents average value of the quality characteristic corresponding to the in ∗ Corresponding author. S.M.T. Fatemi Ghomi, Department of Industrial Engineering, Amirkabir University of Technology, 424 Hafez Avenue,1591634311 Tehran, Iran. Tel.: +98 21 66413034; Fax: +98 21 66413025; E-mail: [email protected].
control state. Two other horizontal lines, called the upper control limit (UCL) and the lower control limit (LCL), are also shown on the chart. These control limits are chosen so that if the process is in control, nearly all samples will fall between them. As long as the points plot within the control limits, the process is assumed to be in control and no action is necessary. However, a point that plots outside of the control limits is interpreted as evidence that the process is out of control. Therefore, investigation and corrective action are required to find and eliminate the assignable cause or causes responsible for this behavior [19]. The traditional control charts are constructed using precise data. However, due to uncertain situations in the real world, it may be difficult to obtain precise data
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H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
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[1] proposed a new p-chart controlling the degree of nonconformity indicated by a fuzzy set that its support is [L, U], where L and U denote lower and upper specification limits for respective quality characteristic. Other approaches to fuzzy control charts can be found in [3, 7, 18, 23, 24]. In non-fuzzy environment, Mean and Range control charts are established following main structure of Shewhart control charts. This structure should be kept when using fuzzy data. However, some previously proposed approaches in the literature such as one given by Sentruk and Erginel [22] have ignored this structure. This ¯ −R paper addresses this issue. In this regard, fuzzy X control charts are constructed using different transformation methods. It is supposed that the observations of each sample are fuzzy random variables, which have triangular membership functions. After calculating fuzzy mean and fuzzy range of each sample using fuzzy arithmetic, the transformation methods are applied for obtained fuzzy mean and fuzzy range to compute their corresponding representative values. Then using statistical properties of the representative values and main ¯ − R control structure of Shewhart control charts, new X charts are constructed to monitor mean and variation of the process. In order to compare the performance of obtained control charts designed by different transformation methods, average run length (ARL) criterion is utilized. Contrary to the proposition of Wang and Raz [25], ARL curves depict that different transformation methods result in control charts with different performances. It is also observed that the value of α – level can significantly influence on the performance of the control charts. Moreover, according to ARL curves, it is concluded that, in spite of all advantages of incorporating fuzziness into observations, it leads to less powerful control charts. The rest of the paper is organized as follows: Section 2 introduces the concept of fuzzy random variables. Section 3 presents representative values of the triangu¯ − R control charts lar fuzzy numbers. In section 4, X are constructed in fuzzy environment using different transformation methods. Section 5 studies the power of these control charts by ARL criterion. Section 6 presents a numerical example and finally, section 7 is devoted to conclusion.
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and it would be better to deal with imprecise or even linguistic data. One way to take into account such an uncertainty in the real world is to apply fuzzy set theory proposed by Zadeh [26]. In the literature, there are some attempts to develop fuzzy control charts. For example, Raz and Wang [21] proposed an approach based on fuzzy set theory by assigning a fuzzy set to each linguistic term. Wang and Raz [25] developed two approaches called fuzzy probabilistic approach and membership approach. In fuzzy probabilistic approach, fuzzy subsets associated with the linguistic terms are transformed into their respective representative values using one of the transformation methods. On the other hand, membership approach is based on fuzzy set theory to combine all observations in only one fuzzy subset using fuzzy arithmetic. Kanagawa et al. [12] introduced an approach for directly controlling the underlying probability distributions of the linguistic data, which were not considered by Wang and Raz [25]. The main difficulty of this approach is that the unknown probability distribution function can not be determined easily. Laviolette et al. [17] and Asai [2] discussed these procedures. Cheng [4] proposed the following approach to deal with expert subjective judgments. Based on the rating scores assigned by individual inspectors to inspected items, fuzzy numbers are constructed to represent the vague outcomes of the process. Then fuzzy control charts are established directly from these fuzzy numbers, thereby retaining the fuzziness of the original vague observations. The out of control conditions are formulated using possibility theory. Engin et al. [6] combined fuzzy sets with genetic algorithms to determine sample size in attribute control charts. Gulbay and kahraman [9] developed a direct fuzzy approach to fuzzy control charts without any deffuzification and then defined unnatural pattern rules based on the fuzzification of crisp rules. Gulbay and Kahraman [10] developed a fuzzy approach to control charts based on the different fuzzy transformation methods and proposed α - cut approach to determine the tightness of inspection. El-Shal and Morris [5] investigated the use of fuzzy logic to modify the statistical process control rules. They aimed at reducing the generation of false alarms and improving detection and detection-speed of real faults. Zarandi et al. [27] presented a new hybrid method based on a combination of fuzzified sensitivity criteria and fuzzy adaptive sampling rules to determine the sample size and sample interval of the control charts. Sentruk and Erginel [22] first transformed traditional ¯ − R and X ¯ − S control charts to fuzzy control charts X ¯ − R ¯ − S ˜ and X and then developed α - cut fuzzy X control charts using α - cut approach. Amirzadeh et al.
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2. Fuzzy random variables Before giving a definition of fuzzy random variables, some preliminary concepts of fuzzy set theory ˜ is a fuzzy subset of the are given. Suppose A
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
3. Representative values for fuzzy sets
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To retain the standard format of control charts and to facilitate plotting of observations on the chart, it is necessary to convert the fuzzy sets associated with fuzzy mean and fuzzy range of each sample into scalars referred to as representative values. There exist four transformation methods, which are similar to the measure of central tendency used in descriptive statistic: α – level fuzzy median, α – level fuzzy midrange, fuzzy average and fuzzy mode. Although it is claimed that there is no theoretical basis supporting any transformation method specifically and that, a transformation method is principally selected based on the ease of computation or preference of the user [25], it is shown by ARL criterion in section 5 that these different transformation methods impact significantly on performance of the control charts. Let x˜ be a continuous fuzzy set with triangular membership function denoted as (xa , xb , xc ). In this case, its representative values are defined as follows: α xmed = (xaα + 2xb + xcα ) 4 (1) α xmid = (xaα + xcα ) 2 (2) xavg = (xa + xb + xc ) 3 (3)
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˜ universal set X. The α -* level cut of A is denoted ˜ by Aα = x : µA˜ (x) ≥ α where µA˜ (x) ∈ [0, 1] is the ˜ A ˜ is called a normal fuzzy membership function of A. ˜ is also set if there exists an x such that µA˜ (x) = 1. A called a convex fuzzy set if and only if µ (λx + (1 − ˜ A λ)y) ≥ min µA˜ (x), µA˜ (y) ; for λ ∈ [0, 1]. In this paper, the universal set X is assumed to be a real number system, i.e. X = R. Let f be a real-valued function on R, then f is said to be upper semi continuous if {x : f (x) ≥ α} is a closed set for each α. Now x˜ is called a fuzzy number if the following conditions are satisfied: i) x˜ is a normal and convex fuzzy set. ii) Its membership function is upper semi continuous. iii) The α – level cut, x˜ α , is bounded for each α ∈ [0, 1]. F (R) ˜ is a convex fuzzy denotes the set of all fuzzy numbers. A set if and only if its α – level cut is a convex set for all α ∈ [0, 1] [26]. Therefore if x˜ is a fuzzy number, the α – level cut x˜ α is a compact and convex set, i.e. x˜ α is a closed interval. Fuzzy random variables represent a well-formulated concept underlying many recent probabilistic and statistical involving data obtained from a random experiment when these data are assumed to be fuzzy set valued. Fuzzy random variables have been considered in the setting of a random experiment to model [8]:
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• Either a fuzzy observation of a mechanism associating a real value with each experiment outcome. • or an essentially fuzzy-valued mechanism, that is, a mechanism associating a fuzzy value with each experiment outcome.
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Kwakernaak [15, 16] introduced a mathematical model for the first situation elaborated later by Kruse and Mayer [14]. On the other hand, Puri and Ralescu [20] gave another approach to the second situation to be modeled. In this paper, the first situation is applied to model fuzziness of observations in each sample, where a fuzzy random variable is viewed as a fuzzy perception/observation of a classical real-valued random variable (following a normal distribution with the mea µ and the variance σ 2 ) stated as follows: Definition 1: Given a probability space (, A, P), a mapping x : → F (R) is said to be fuzzy random variable if the two real-valued mapping xαl → R and xαu → R are real-valued random variables for all α ∈ [0, 1], where x˜ α = [xαl , xαu ].
To get an overview on the development of fuzzy random variable, the interested reader may refer to Gil et al. [8].
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and x mod = xb
(4)
α , xα , x where xmed mid avg
and x mod are representative values calculated by α – level fuzzy median, α – level fuzzy midrange, fuzzy average and fuzzy mode transformation methods respectively, and xaα = xa + α(xb − xa )
(5)
= xc − α(xc − xb )
(6)
xcα
To get a review of deffuzification methods, the interested reader is referred to Hellendoorn and Thomas [11].
¯ − R control charts using 4. Construction X representative values ¯ − R control This section explains how to construct X charts using statistical properties of each representative value. To do this, after drawing a sample of size n including n fuzzy random variables, fuzzy mean and fuzzy range of each sample are calculated using fuzzy arithmetic. Then, one of the above mentioned transformation methods is applied to compute representative values of
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
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¯ − R control charts based on α – level fuzzy 4.1. X median transformation
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After calculating fuzzy mean and fuzzy range of each sample, α – level fuzzy median representative valueof ¯ is calculated as S α xaαj + 2¯xbj + x¯ cαj ) 4, X j ¯ j = (¯ med,X where x¯ aαj = x¯ aj + α(¯xbj − x¯ aj ) and x¯ cαj = x¯ cj − α(¯xcj − x¯ bj ). Equivalently, it is as follows α Smed, xaj + 2(1 + α)¯xbj + (1 − α)¯xcj ) 4 (8) ¯ j = ((1 − α)¯ X
Since it is assumed that the quality characteristic follows a normal distribution with the mean µ and the variance σ 2 , according to the Definition 1 of fuzzy random variables, it is concluded that x¯ aαj and x¯ cαj follow a normal distribution with the mean µ and the variance α σ 2 n. So the mean of Smed, ¯ , is as follows: ¯ , µmed,X X j
µmed,X¯ = µ
+ 4(1 + α)2 var(¯xbj ) + 4(1 − α2 )cov(¯xaj , x¯ bj ) + 2(1 − α)2 cov(¯xaj , x¯ cj )
(9)
α Therefore Smed, ¯ j is an unbiased estimator of the X quality characteristic mean µ. Hence, it can be used α to monitor the changes of µ. The variance of Smed, ¯ , X j
+ 4(1 − α2 )cov(¯xbj , x¯ cj ) 16
(10)
It can be simply proved that, cov(¯xaj , x¯ bj ), cov(¯xaj , x¯ cj ) and cov(¯xbj , x¯ cj ) are equal to 2 σ as . So the2 above equation 2is simplified If the ((1 − n)(6α + 4α) + 10n + 6)σ 16n . values of µ and σ are not known, they should be estimated using samples initially drawn from the process. To do this, suppose m samples of size n have been ¯ ¯ = (¯r , r¯ , r¯ ) are the taken. X = (x¯ a , x¯ b , x¯ c ) and R a b c average of fuzzy mean and fuzzy range of all m samples respectively. The estimates of the process mean and standard deviation are respectively as follows: µ ˆ = (x¯ αa + 2x¯ b + x¯ αc ) 4 (11) ¯ med d2 σˆ = (¯raα + 2¯rb + r¯cα ) 4d2 = R (12)
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and raj = xmax,aj − xmin,cj , rbj = xmax,bj − xmin,bj and rcj = xmax,cj − xmin,aj . Here (xmax,aj , xmax,bj , xmax,cj ) and (xmin,aj , xmin,bj , xmin,cj ) are the maximum and minimum fuzzy numbers in the j-th sample respectively. The ranking method used in this paper involves three ordered criteria proposed by Kaufmann and Gupta [13]. As the first criterion, the greatest associate ordi˜ = (xa + 2xb + xc ) 4 is nary number defined as C1 (X) used. If C1 does not rank the fuzzy numbers, those ˜ = xb which have the best maximal presumption C2 (X) will be chosen as the second criterion. Finally, if C1 and C2 do not rank the fuzzy numbers, the difference ˜ = xc − xa will be used as the third of the spreads C3 (X) criterion.
2 σmed, ¯ , is computed as follows: X 2 2 σmed, xaj ) + var(¯xcj )) ¯ = (1 − α) (var(¯ X
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the obtained fuzzy mean and fuzzy range. Finally, the 3-sigma control charts are constructed using statistical properties of representative values and basic structure of Shewhart control charts. In this paper, each fuzzy random variable is supposed to have triangular membership function. Suppose ˜ ij = (xaij , xbij , xcij ) is the i-th fuzzy observation in the X ¯ = (¯x , x¯ , x¯ ) and R ˜j = j-th sample. In this case, X j aj bj cj (raj , rbj , rcj ) are fuzzy mean and fuzzy range of the j-th sample respectively, where n (7) x¯ kj = xkij n ; k = a, b, c.
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where
x¯ αa = x¯ a + α(x¯ b − x¯ a )
(13)
x¯ αc r¯aα
= x¯ c − α(x¯ c − x¯ b )
(14)
= r¯a + α(¯rb − r¯a )
(15)
r¯cα = r¯c − α(¯rc − r¯b )
(16)
and
The values of d2 can be found based on the size of drown sample in [19]. Therefore the standard deviation α of Smed, ¯ , σmed,X ¯ , is derived as follows: X j
¯ med σmed,X¯ = A2 R
(1 − n)(6α2 + 4α) + 10n + 6 12
(17)
where √ A 2 = 3 d2 n ¯ med R
= (¯raα + 2¯rb + r¯cα ) 4
(18) (19)
Now using basic structure of Shewhart control charts, which is UCL = µw + 3σw , CL = µw and LCL = α µw − 3σw where w is a sample statistic (here Smed, ¯ j) X ¯ measuring some quality characteristic of interest, X
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
control chart based on α – level fuzzy median transvar(rcj ) = (d3 r¯c d2 )2 formation is constructed as follows: ⎧
¯ med (1 − n)(6α2 + 4α) + 10n + 6) 4 ⎪ UCLαmed,X¯ = µmed,X¯ + 3σmed,X¯ = (x¯ αa + 2x¯ b + x¯ αc + A2 R ⎪ ⎪ ⎨ CLαmed,X¯ = µmed,X¯ = (x¯ αa + 2¯xb + x¯ αc ) 4 ⎪
⎪ ⎪ ¯ ⎩ LCLα =µ − 3σ = (x¯ α + 2x¯ + x¯ α − A R (1 − n)(6α2 + 4α) + 10n + 6) 4 ¯ med,X
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The process mean is in control if LCLαmed,X¯ ≤ ≤ UCLαmed,X¯ and otherwise is out of control. As it is observed, the above control limits are natural generalization of crisp control limits into fuzzy environment, because they decrease to the common control limits in crisp environment if α = 1. α Smed, ¯j X
¯ which is plotted on Remark 1: It is worth noting that, X the traditional control charts is an unbiased estimator of the quality characteristic mean. The control limits are calculated using its mean and standard deviation and main structure of Shewhart control chart. This point is α used in construction of control chart for Smed, ¯ . X j
2 med
Hence, after a few computational efforts, the standard α deviation of Smed,R , σmed,R , is derived as follows: j σmed,R = ((1 − α)2 (¯ra2 + r¯c2 ) + 4(1 + α)2 r¯b2 )(d3 d2 )2 0.5 ¯ med d2 )2 4 (27) + 4(5 − 2α − 3α2 )(R where the values of d3 can be found based on the size of sample in [19]. Now R control chart based on α – level fuzzy median transformation is constructed as follows using Eqs (22) and (27), and basic structure of Shewhart control chart: ⎧ ⎪ UCLαmed,R = µmed,R + 3σmed,R ⎪ ⎨ CLαmed,R = µmed,R (28) ⎪ ⎪ ⎩ LCLαmed,R = µmed,R − 3σmed,R
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Now control limits of R chart are computed. After calculating fuzzy range of each sample, α -level fuzzy ˜ j is calculated as median representative valueof R
c
(20)
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¯ med,X
(26)
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¯ med,X
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α Smed,R = (raαj + 2rbj + rcαj ) 4. Equivalently, it is as j follows: α Smed,R = ((1 − α)raj + 2(1 + α)rbj + (1 − α)rcj ) 4 j
(21)
¯ med µmed,R = R
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α The mean of Smed,R , µmed,R , is d2 σ. If σ is not known, j it should be estimated by Eq. (12). Therefore, we have:
(22)
α 2 On the other hand, the variance of Smed,R , σmed,R , j is calculated as: 2 σmed,R = (1 − α)2 (var(raj ) + var(rcj ))
+ 4(1 + α)2 var(rbj ) + 4(1 − α2 )cov(raj , rbj ) + 2(1 − α)2 cov(raj , rcj ) + 4(1 − α2 )cov(rbj , rcj )
16
(23)
It is simply proved that, cov(raj , rbj ), cov(raj , rcj ) and cov(rbj , rcj ) are equal to 2σ 2 . Moreover, it is known [19] that var(raj ) = (d3 r¯a d2 )2 (24) 2 var(rbj ) = (d3 r¯b d2 ) (25)
In this case, the process variation is in control α if LCLαmed,R ≤ Smed,R ≤ UCLαmed,R and otherwise is j out of control. If α = 1, the above control limits also decrease to the common control limits in crisp environment. Therefore, they can be viewed as natural generalization of common control limits of R control chart into fuzzy environment. ¯ − R Control charts based on α – level 4.2. X fuzzy midrange transformation
¯ − R control charts based on Development of X α – level fuzzy midrange transformation is similar to that based on α – level fuzzy median transformation. If α – level fuzzy midrange is used as the transformation ¯ is as S α method, the representative value of X j mid,Xj = (¯xaαj + x¯ cαj ) 2. Equivalently, it is as follows: α Smid, xaj + 2α¯xbj + (1 − α)¯xcj ) 2 (29) ¯ j = ((1 − α)¯ X According to the Definition 1 of fuzzy random variables, x¯ aαj and x¯ cαj follow a normal distribution with the mean µ and the variance σ 2 n. Hence, the mean of
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
α Smid, ¯ is computed as follows: X
˜ j is calculated as sample, the representative value of R
j
µmid,X¯ = µ
(30)
α Therefore, Smid, ¯ j is an unbiased estimator of the qualX ity characteristic mean and can be used to monitor the α 2 changes of µ. The variance of Smid, ¯ j , σmid,X ¯ , is calX culated as: 2 2 σmid, xaj ) + var(¯xcj )) + 4α2 var(¯xbj ) ¯ = (1 − α) (var(¯ X
+ 4α(1 − α)cov(¯xaj , x¯ bj )
+ 4α(1 − α)cov(¯xbj , x¯ cj )
4
(31)
j
¯ mid µmid,R = R
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¯ mid = (¯raα + r¯cα ) 2 R
(38)
2 = (1 − α)2 (var(raj ) + var(rcj )) + 4α2 var(rbj ) σmid,R + 4α(1 − α)cov(raj , rbj ) + 2(1 − α)2 cov(raj , rcj )
(35)
4
(39)
According to the Eqs (24)-(26) and that, cov(raj , rbj ), cov(raj , rcj ) and cov(rbj , rcj ) are equal to 2σ 2 , the stanα , σmid,R , is derived as follows: dard deviation of Smid,R j σmid,R = ((1 − α)2 (¯ra2 + r¯c2 ) + 4α2 r¯b2 )(d3 d2 )2 0.5 ¯ mid d2 )2 2 (40) + 4(1 + 2α − 3α2 )(R
Hence, R control chart based on α – level fuzzy midrange transformation is constructed as follows: ⎧ ⎪ UCLαmid,R = µmid,R + 3σmid,R ⎪ ⎨ CLαmid,R = µmid,R (41) ⎪ ⎪ ⎩ LCLαmid,R = µmid,R − 3σmid,R
¯ control chart based on α – level fuzzy Now, X midrange transformation is constructed using Eqs (30) and (34), and basic structure of Shewhart control charts as follows: ⎧
¯ mid (1 − n)(6α2 − 4α) + 2n + 2) 2 ⎪ UCLαmid,X¯ = µmid,X¯ + 3σmid,X¯ = (x¯ αa + x¯ αc + A2 R ⎪ ⎪ ⎨ CLαmid,X¯ = µmid,X¯ = (x¯ αa + x¯ αc ) 2 ⎪
⎪ ⎪ ¯ mid (1 − n)(6α2 − 4α) + 2n + 2) 2 ⎩ LCLα ¯ = µmid,X¯ − 3σmid,X¯ = (x¯ αa + x¯ αc − A2 R
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+ 4α(1 − α)cov(rbj , rcj )
(1 − n)(6α2 − 4α) + 2n + 2 6) (34)
where
α The mean of Smid,R , µmid,R , is d2 σ. If the value j of σ is not known, it should be estimated by Eq. (33). Therefore, we have:
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α So the standard deviation of Smid, ¯ , σmid,X ¯ , is as X
¯ mid σmid,X¯ = (A2 R
(37)
α 2 The variance of Smid,R , σmid,R , is computed as j
As mentioned previously, it can be simply proved that, cov(¯xaj , x¯ bj ), cov(¯xaj , x¯ cj ) and cov(¯xbj , x¯ cj ) are equal to σ 2 . Therefore, Eq. (31) can bereadily rewritten as [((1 − n)(6α2 − 4α) + 2n + 2)σ 2 4n]. If the values of µ and σ are not known, they should be estimated using m initially drawn samples as follows: µ ˆ = (x¯ αa + x¯ αc ) 2 (32) ¯ mid d2 σˆ = (¯raα + r¯cα ) 2d2 = R (33)
α Smid,R = ((1 − α)raj + 2αrbj + (1 − α)rcj ) 2 j
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+ 2(1 − α)2 cov(¯xaj , x¯ cj )
α Smid,R = (raαj + rcαj ) 2. Equivalently, it is as follows: j
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(36)
mid,X
The process mean is in control if LCLαmid,X¯ ≤ α α Smid, ¯ j ≤ UCLmid,X ¯ and otherwise is out of control. X When α = 1, the above control limits decrease to the common control limits in crisp environment. Thus, they can be regarded as natural generalization of common ¯ control chart into fuzzy environment. control limits of X Now control limits of R control chart are going to be computed. After calculating fuzzy range of each
The process variation is in control if LCLαmid,R ≤ ≤ UCLαmid,R and otherwise is out of control. Since the above control limits decrease to the common control limits in crisp environment when α = 1, they can be viewed as natural generalization of common control limits of R control chart into fuzzy environment.
α Smid,R j
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
¯ − R control charts based on fuzzy average 4.3. X transformation If fuzzy average transformation method is used, the ¯ is calculated as follows: representative value of X j Savg,X¯ j = (¯xaj + x¯ bj + x¯ cj ) 3 (42)
Therefore, we have: ¯ avg µavg,R = R
2 = var(raj ) + var(rbj ) + var(rcj ) σavg,R + 2(cov(raj , rbj ) + cov(raj , rcj ) + cov(rbj , rcj )) 9
the computed as follows: 2 xaj ) + var(¯xbj ) + var(¯xcj ) σavg, ¯ = var(¯ X
(43)
(52)
Hence, R control chart based on fuzzy average transformation is constructed as follows: ⎧ UCLavg,R = µavg,R + 3σavg,R ⎪ ⎨ CLavg,R = µavg,R (53) ⎪ ⎩ LCL avg,R = µavg,R − 3σavg,R
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The above equation can be simplified as σ 2 (2n + 1) 3n. Based on m initial samples, the values of µ and σ are estimated as: µ ˆ = (x¯ a + x¯ b + x¯ c ) 3 (44) ¯ avg d2 σˆ = (¯ra + r¯b + r¯c ) 3d2 = R (45)
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¯ avg d2 ) σmid,R = (¯ra + r¯b + r¯c )(d3 d2 ) + 12(R
+ 2(cov(¯xaj , x¯ bj ) + cov(¯xaj , x¯ cj ) + cov(¯xbj , x¯ cj )) 9
(51)
So the standard deviation of Savg,Rj , σavg,R , is as follows: 2 2 2 0.5 2 2
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α α and Smid, , S ¯ j avg,X ¯ j can be also used to monitor X 2 changes of µ. The variance of Savg,X¯ , σavg, ¯ , is X j
(50)
2 The variance of Savg,Rj , σavg,R , is computed as:
The mean of Savg,X¯ , µavg,X¯ , is µ (note that x¯ aj and j ¯ with α = 0). So similar to x¯ are the end points of X cj α Smed, ¯j X
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¯ avg = (¯ra + r¯b + r¯c ) 3 R
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where
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So the standard deviation of Savg,X¯ , σavg,X¯ , is as j follows √ ¯ avg 6n + 3 9) σavg,X¯ = (A2 R (46)
(47)
The process variation is in control if LCLavg,R ≤ Savg,Rj ≤ UCLavg,R and otherwise is out of control.
¯ − R control charts based on fuzzy mode 4.4. X transformation
If fuzzy mode is used as the transformation method, the constructed control charts are similar to those in ¯ crisp state, established by x¯ b and rb . Therefore, X control chart based on fuzzy mode transformation is calculated as follows:
¯ control chart based on fuzzy average transNow X formation is constructed as follows: ⎧ √ ¯ avg 6n + 3) 3 ⎪ UCLavg,X¯ = µavg,X¯ + 3σavg,X¯ = (x¯ a + x¯ b + x¯ c + A2 R ⎪ ⎨ CLavg,X¯ = µavg,X¯ = (x¯ a + x¯ b + x¯ c ) 3 √ ⎪ ⎪ ¯ avg 6n + 3) 3 ⎩ LCLavg,X¯ = µavg,X¯ − 3σavg,X¯ = (x¯ a + x¯ b + x¯ c − A2 R The process mean is in control if LCLavg,X¯ ≤ Savg,X¯ ≤ UCLavg,X¯ and otherwise is out of control. j After calculating fuzzy range of each sample, the rep˜ j is calculated using the following resentative value of R equation: Savg,Rj = (raj + rbj + rcj ) 3 (49) The mean of Savg,Rj , µavg,R , is d2 σ. If the value of σ is not known, it should be estimated by Eq. (45).
⎧ ¯ ⎪ ⎨ UCL mod ,X¯ = µ mod ,X¯ + 3σ mod ,X¯ = xb + A2 r¯b CL mod ,X¯ = µ mod ,X¯ = x¯ b ⎪ ⎩ LCL ¯ b − A2 r¯b ¯ = µ mod ,X ¯ − 3σ mod ,X ¯ =x mod ,X
(48)
(54)
The process mean is in control if LCL mod ,X¯ ≤ S mod ,X¯ ≤ UCL mod ,X¯ and otherwise is out of j control. Moreover, R control chart based on fuzzy mode
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H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
transformation is as follows: ⎧ ⎪ ⎨ UCL mod,R = µ mod,R + 3σ mod,R = r¯b (1 + 3d3 d2 ) CL mod,R = µ mod,R = r¯b
⎪ ⎩ LCL mod,R = µ mod,R − 3σ mod,R = r¯b (1 − 3d3 d2 )
(55)
¯ control chart designed Application of Eq. (56) to X by α – level fuzzy midrange transformation also leads to: α α βmid = Pr(LCLαmid,X¯ ≤ Smid, ¯ j ≤ UCLmid,X ¯ |µmid X
β = Pr(LCL ≤ w ≤ UCL |µ = µ0 + kσ)
(56)
AU
TH
where, µ0 is the current value of the process mean, µ is the shifted value of mean measured by units of standard deviation of the process, σ, and k determines the amount of this shift. When the process is out of control, ARL is calculated as [19]: (57)
¯ and R ARL criterion can be calculated for both X ¯ control charts. In this paper, it is just computed for X chart. ¯ control chart established by Applying Eq. (56) to X α – level fuzzy median transformation results in: α α βmed = Pr(LCLαmed,X¯ ≤ Smed, ¯ j ≤ UCLmed,X ¯ |µmed X
¯ med d2 )) = ((x¯ αa + 2x¯ b + x¯ αc ) 4) + kR = Pr(−3 − 4k
≤ Z ≤ 3 − 4k
n [(1 − n)(6α2 − 4α) + 2n + 2]
n [(1 − n)(6α2 + 4α) + 10n + 6]
n [(1 − n)(6α2 + 4α) + 10n + 6])
(58) where, Z is a random variable following standard normal distribution.
n [(1 − n)(6α2 − 4α) + 2n + 2])
CO
PY
(59) When fuzzy average and fuzzy mode transformations are used to construct control charts, the probability of type II error is calculated for each transformation method respectively as follows: βavg = Pr(LCLavg,X¯ ≤ S ¯ µavg ¯ ≤ UCLavg,X
avg,Xj
¯ avg d2 ) = ((x¯ a + x¯ b + x¯ c ) 4) + kR
= Pr(−3 − k
3n (2n + 1) ≤ Z ≤ 3 − k
3n (2n + 1))
(60)
and
OR
After developing control charts using different transformation methods, a practitioner may ask this question: which transformation method should be used in practice? To answer this question, four different control charts constructed using four various transformation methods are compared in this section exploiting average run length (ARL) criterion. ARL criterion is defined as the number of samples drawn before identifying an out of control condition. Therefore, the less ARL value, the more powerful control chart in detecting shifts in the quality characteristic we have. For a control chart with the control limits UCL and LCL, the probability of type II error is defined as follows [19]:
= Pr(−3 − 2k
≤ Z ≤ 3 − 2k
5. Performance of control charts based on different transformation methods
ARL = 1/(1 − β)
¯ mid d2 )) = ((x¯ αa + x¯ αc ) 2) + kR
The process variation is in control if LCL mod,R ≤ S mod,Rj ≤ UCL mod,R and otherwise is out of control.
β mod = Pr(LCL mod,X¯ ≤ S mod,X¯ j ≤ UCL mod,X¯ |µ mod
¯ mod d2 ) = x¯ b + kR √ √ = Pr(−3 − k n ≤ Z ≤ 3 − k n)
(61)
After calculating the probability of type II error, the ARL criterion is computed using Eq. (57). It is usually depicted for a specific value of n and different values of k. But this should be done for a predetermined value of α, because for a specific values of n and k, βmed and βmid are functions of α. Fig. 1 shows the ARL curves when n = 4 and α = 0.35, 0.55, 0.75, 0.95 for different transformation methods. In this Figure, the ARL of control chart constructed by fuzzy mode transformation is depicted by number 1, α – level fuzzy median transformation by number 2, α – level fuzzy midrange transformation by number 3 and fuzzy average transformation by number 4. Considering this Figure, some significant conclusions can be drawn. Firstly, it is apparent for all specified values of α that various transformation methods result in different per¯ control chart. This clearly contradicts formances of X the previous proposition [25] that, there is no theoretical difference among different transformation methods and that, a transformation method is chosen based on its ease of computation or user’s preference. The control chart constructed by fuzzy mode and fuzzy average
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
α = 0.35
350
350
300
300
250
250
200
3, 4 1
150
200 150 100
50 0
3 2
50
2
0.5
1
0
1.5
0
K
α = 0.75
350
400 350
4
1.5
1
1.5
α = 0.95
4
250
ARL
250 200 1
3
200
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ARL
1
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300
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1
150
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100 2
0
50
0.5
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50 0
0.5
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0
α = 0.55
400
ARL
ARL
400
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1
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1.5
0 0
2
0.5
K
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Fig. 1. The ARL curves of different control charts for n = 4 and based on: fuzzy Mode (no. 1), α- level fuzzy median (no. 2), α – level fuzzy midrange (no. 3) and fuzzy average (no. 4) transformation.
transformation methods are the most and the least powerful control chart respectively. In addition, it is observed that the value of α can have significant influence on the control chart performance when either fuzzy median or fuzzy midrange transformation method is used. In the other words, although a large value of α results in more tightness of inspection, it leads to a more powerful control chart when one of two abovementioned transformation methods is applied. Moreover, since the α – level fuzzy median and α – level fuzzy midrange approach to fuzzy mode when increasing the value of α, the ARL curves of their respective control charts approach to ARL curve of control chart constructed by fuzzy mode transformation method when the value of α increases. Finally, it can be inferred that in spite of all beneficial points of incorporating
fuzziness into control charts and using fuzzy information such as taking into account variability caused by human subjectivity or measurement devices, or environmental conditions, it results in a less powerful control chart. The later conclusion is drown by comparing ARL curves of control charts established by α – level fuzzy median, α – level fuzzy midrange and fuzzy average transformations with that of control chart made by fuzzy mode transformation, which is equivalent to crisp control chart. Nonetheless, we know that applying fuzzy mode transformation deprives us of advantages of using fuzzy sets. It may also lead to a biased result when the membership function is extremely asymmetrical. Our proposition in such a dilemma is to exploit fuzzy median transformation with a large value of α. Because, not
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H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts Table 1 The fuzzy measurement values and their fuzzy means and ranges Fuzzy mean 3
(17.40, 17.44, 17.46) (18.06, 18.08, 18.11) (18.45, 18.49, 18.51) (17.66, 17.67, 17.68) (18.25, 18.30, 18.32) (18.30, 18.34, 18.37) (18.00, 18.02, 18.03) (17.59, 17.61, 17.63) (17.80, 17.81, 17.83) (17.98, 18.00, 18.02) (18.50, 18.53, 18.55) (18.04, 18.05, 18.08) (17.92, 17.95, 17.98) (18.00, 18.02, 18.06) (17.40, 17.42, 17.45) (18.42, 18.44, 18.46) (17.65, 17.68, 17.71) (17.73, 17.74, 17.76) (18.06, 18.09, 18.11) (17.96, 18.00, 18.02) (17.55, 17.57, 17.60) (18.52, 18.54, 18.55) (17.96, 18.00, 18.01) (18.40, 18.44, 18.45) (17.45, 17.47, 17.50)
(18.07, 18.10, 18.13) (17.46, 17.50, 17.51) (18.63, 18.67, 18.68) (18.10, 18.13, 18.16) (18.35, 18.37, 18.38) (17.35, 17.37, 17.41) (18.38, 18.39, 18.41) (18.26, 18.28, 18.30) (17.28, 17.31, 17.34) (17.35, 17.40, 17.41) (17.63, 17.64, 17.66) (18.10, 18.13, 18.16) (18.33, 18.37, 18.41) (17.41, 17.43, 17.46) (18.52, 18.55, 18.57) (18.47, 18.49, 18.50) (18.34, 18.38, 18.39) (17.36, 17.37, 17.39) (17.85, 17.87, 17.91) (18.28, 18.31, 18.33) (17.32, 17.35, 17.36) (18.05, 18.08, 18.11) (17.81, 17.84, 17.87) (18.30, 18.32, 18.33) (18.26, 18.30, 18.32)
(17.26, 17.31, 17.32) (17.45, 17.47, 17.50) (18.08, 18.10, 18.12) (17.66, 17.68, 17.70) (18.39, 18.42, 18.43) (18.14, 18.16, 18.18) (18.25, 18.27, 18.28) (17.64, 17.66, 17.67) (17.43, 17.44, 17.46) (18.25, 18.28, 18.29) (17.59, 17.61, 17.64) (17.25, 17.27, 17.31) (18.63, 18.64, 18.65) (18.26, 18.29, 18.33) (18.41, 18.44, 18.46) (17.73, 17.76, 17.77) (17.67, 17.70, 17.74) (17.78, 17.81, 17.84) (18.50, 18.54, 18.56) (17.94, 17.98, 17.99) (18.16, 18.18, 18.20) (17.80, 17.84, 17.87) (17.96, 17.97, 17.98) (18.22, 18.24, 18.26) (18.05, 18.07, 18.08)
Fuzzy range
4 (18.18, 18.20, 18.22) (17.73, 17.76, 17.78) (18.13, 18.14, 18.15) (17.78, 17.80, 17.82) (18.25, 18.27, 18.30) (18.35, 18.38, 18.40) (18.02, 18.04, 18.06) (17.86, 17.88, 17.90) (18.21, 18.23, 18.24) (18.30, 18.33, 18.34) (18.55, 18.58, 18.60) (18.09, 18.11, 18.14) (17.77, 17.78, 17.80) (18.10, 18.12, 18.13) (18.70, 18.73, 18.74) (18.05, 18.07, 18.09) (17.34, 17.37, 17.40) (17.46, 17.49, 17.51) (18.10, 18.13, 18.15) (17.90, 17.95, 17.97) (18.19, 18.21, 18.25) (17.98, 18.00, 18.03) (17.50, 17.53, 17.56) (17.72, 17.75, 17.78) (17.35, 17.36, 17.37) (18.06, 18.08, 18.10) (18.54, 18.56, 18.59) (18.05, 18.07, 18.11) (18.56, 18.59, 18.60) (18.22, 18.25, 18.27) (18.70, 18.73, 18.74) (18.33, 18.36, 18.37) (17.85, 17.89, 17.90) (17.88, 17.91, 17.94) (17.39, 17.42, 17.45) (17, 57, 17.58, 17.61) (17.70, 17.74, 17.75) (18.03, 18.06, 18.08) (18.52, 18.56, 18.58) (18.18, 18.21, 18.23) (18.33, 18.36, 18.37) (17.84, 17.87, 17.88) (17.85, 17.90, 17.92) (18.06, 18.09, 18.11) (18.00, 18.02, 18.05) (17.93, 17.96, 17.98) (17.65, 17.68, 17.70) (18.14, 18.17, 18.19) (17.11, 17.14, 17.17) (17.72, 17.75, 17.77)
OR
CO
2
(0.86, 0.89, 0.96) (0.63, 0.67, 0.70) (0.51, 0.57, 0.60) (0.42, 0.46, 0.50) (0.15, 0.19, 0.22) (1.14, 1.21, 1.25) (0.58, 0.61, 0.64) (1.07, 1.12, 1.15) (0.46, 0.50, 0.55) (0.84, 0.88, 0.94) (0.86, 0.92, 0.96) (0.79, 0.86, 0.91) (1.26, 1.28, 1.30) (1.08, 1.13, 1.18) (1.11, 1.17, 1.20) (0.93, 0.97, 1.01) (0.63, 0.70, 0.74) (0.39, 0.44, 0.48) (0.75, 0.80, 0.86) (0.53, 0.58, 0.64) (0.97, 1.01, 1.05) (0.65, 0.70, 0.75) (0.13, 0.18, 0.24) (0.70, 0.76, 0.80) (1.09, 1.16, 1.21)
Table 2 Representative values of fuzzy mean and fuzzy ranges 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
Fuzzy mean (17.73, 17.76, 17.78) (17.78, 17.80, 17.82) (18.35, 18.38, 18.40) (17.86, 17.88, 17.90) (18.30, 18.33, 18.34) (18.09, 18.11, 18.14) (18.10, 18.12, 18.13) (18.05, 18.07, 18.09) (17.46, 17.49, 17.51) (17.92, 17.95, 17.97) (17.98, 18.00, 18.03) (17.72, 17.75, 17.78) (18.06, 18.08, 18.10) (18.05, 18.07, 18.11) (18.22, 18.25, 18.27) (18.33, 18.36, 18.37) (17.88, 17.91, 17.94) (17, 57, 17.58, 17.61) (18.03, 18.06, 18.08) (18.18, 18.21, 18.23) (17.84, 17.87, 17.88) (18.06, 18.09, 18.11) (17.93, 17.96, 17.98) (18.14, 18.17, 18.19) (17.72, 17.75, 17.77)
R.V. of fuzzy mean
Fuzzy range
R.V. of fuzzy range
17.762 17.798 18.384 17.879 18.328 18.112 18.117 18.071 17.485 17.952 17.998 17.746 18.081 18.074 18.249 18.355 17.913 17.585 18.057 18.214 17.865 18.091 17.957 18.170 17.747
(0.86, 0.89, 0.96) (0.63, 0.67, 0.70) (0.51, 0.57, 0.60) (0.42, 0.46, 0.50) (0.15, 0.19, 0.22) (1.14, 1.21, 1.25) (0.58, 0.61, 0.64) (1.07, 1.12, 1.15) (0.46, 0.50, 0.55) (0.84, 0.88, 0.94) (0.86, 0.92, 0.96) (0.79, 0.86, 0.91) (1.26, 1.28, 1.30) (1.08, 1.13, 1.18) (1.11, 1.17, 1.20) (0.93, 0.97, 1.01) (0.63, 0.70, 0.74) (0.39, 0.44, 0.48) (0.75, 0.80, 0.86) (0.53, 0.58, 0.64) (0.97, 1.01, 1.05) (0.65, 0.70, 0.75) (0.13, 0.18, 0.24) (0.70, 0.76, 0.80) (1.09, 1.16, 1.21)
0.8905 0.6699 0.5696 0.4600 0.1899 1.2096 0.6100 1.1198 0.5001 0.8803 0.9198 0.8598 1.2800 1.1300 1.1696 0.9700 0.6996 0.4399 0.8001 0.5801 1.0100 0.7000 0.1801 0.7598 1.1598
TH
Sample no.
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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
Part no. 1
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Sample no.
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
X bar control chart based on fuzzy median transformation with =0.95
18.8
UCL=18.62
18.4 18.2 18
CL=18
17.8 17.6 17.4 17.2 17 0
5
10
15
25
R control chart based on fuzzy median transformation with =0.95
2 1.8 1.6
UCL=1.84
1.2 1 0.8 0.6 0.4 0.2 0
5
TH
OR
1.4
0
20
LCL=17.38
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Subgroup Number
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Representative value of the Mean
18.6
Representative value og the Rang
263
10
CL=0.79
LCL=0 15
20
25
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Subgroup Number
¯ and R control charts based on fuzzy median transformation with α = 0.95. Fig. 2. X
only do we benefit the advantages of incorporating fuzziness, but also the ARL curve of control chart constructed using α – level fuzzy median transformation with a large value of α is very close to that established by fuzzy mode transformation. In this case, the control chart performance is very close to the best performance obtained from fuzzy mode transformation.
6. Numerical example The pervious section described how to determine the most powerful control chart. In this section, a numerical
example is given in which fuzzy median transformation at the level of α = 0.95 is utilized. As mentioned before, we retain fuzziness in data in such level and that, performance of the constructed control chart is very close to the best performance obtained by fuzzy mode transformation method. The example presents a process producing a shaft of a gearbox. The quality characteristic that experts are going to examine is outer diameter of this shaft. 25 samples with a sample size of 4 were taken from the production process. Each observation was measured and recorded as a triangular fuzzy number. The fuzzy measures and their fuzzy means and fuzzy ranges are given in Table 1. Table 2 presents the representative values of fuzzy mean and fuzzy range for each
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
0.95 18.62, CL0.95 ¯ = 18 and LCLmed,X ¯ = 17.38. The med,X
limits of R control chart are obtained as UCL0.95 med,R = 0.95 0.95 1.84, CLmed,R = 0.79 and LCLmed,R = 0. In Table 2, R.V. stands for representative value. Now the representative values obtained from Table 2 can be plotted on a control chart with the given control limits. ¯ and R control charts for this example Figure 2 shows X respectively. Since the above control charts are constructed using basic structure of Shewhart control charts, they present the process status more accurately. Moreover, they are sufficiently powerful for detecting changes in the process mean, because the 0.95-level fuzzy median transformation method is applied to construct them.
powerful control charts. Finally, it was concluded that in spite of all advantages of using fuzzy data in the real world, it resulted in less powerful control charts. Therefore, it is better for a practitioner to apply fuzzy mode transformation method. However, in order to deal with the disadvantages of using fuzzy mode such as lacking flexibility in measuring and recording data and ignoring variability caused by human subjectivity or measurement devices, we suggested exploiting α – level median transformation with a large value of α. In this case, the fuzziness of data is retained and the power of obtained control charts is very close to the best power obtained by fuzzy mode transformation method.
PY
sample, which are computed by fuzzy median transformation method with α = 0.95. The control limits of ¯ − R control charts can be calculated using Eqs (20) X and (28). It should be noted for n = 4 that, A2 = 0.729, d3 = 0.88 and d2 = 2.059. Using these factors, the lim¯ control chart are obtained as UCL0.95 ¯ = its of X med,X
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7. Conclusion
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[3]
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[4]
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¯ − R control charts were developed In this paper, X in fuzzy environment using different transformation methods. Observations of each sample were assumed to be fuzzy random variables with triangular membership functions. After calculating fuzzy mean and fuzzy range of each sample using fuzzy arithmetic, their representative values were obtained applying the transformation methods. Then, using statistical properties ¯ − R control charts of the representative values, new X were constructed to monitor the process mean and variation. In this case, since the obtained control charts are established based on main structure of Shewhart control charts, they reveal the process status more correctly. After that, in order to distinguish and compare different control charts constructed by various transformation methods, their powers were examined applying average run length (ARL) criterion. Some significant conclusions were drawn considering the obtained ARL curves. Firstly, contrary to a previous proposition, it was derived that different transformation methods affect significantly on performances of the constructed control charts. Fuzzy mode and fuzzy average transformation methods resulted in the most and the least powerful control charts respectively. Moreover, when either α – level fuzzy median or α – level fuzzy midrange transformation method was applied, a larger α value led to more
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Journal of Intelligent & Fuzzy Systems 22 (2011) 253–265 DOI:10.3233/IFS-2011-0487 IOS Press
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Fuzzy development of Mean and Range control charts using statistical properties of different representative values H. Moheb Alizadeha and S.M.T. Fatemi Ghomib,∗ a Department
of Industrial Engineering, Tarbiat Moallem University, Tehran, Iran of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran
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b Department
TH
OR
Abstract. This paper develops Mean and Range control charts in fuzzy environment using different transformation methods. It is assumed that the observations of each sample are fuzzy random variables, which have triangular membership functions. After calculating fuzzy mean and fuzzy range of each sample using fuzzy arithmetic, their representative values are calculated exploiting the transformation methods. Then, using statistical properties of the obtained representative values and basic structure of Shewhart control charts, new Mean and Range control charts are constructed to monitor the process mean and variation. After that, the power of control charts constructed based on different transformation methods are examined applying average run length (ARL) criterion. According to this criterion, it is concluded that, contrary to the previous claim, these different transformation methods lead to mean and rang control charts with various performances. Moreover, it is represented that the value of α – level has significant influence on the performance of obtained control charts. Finally, it is derived that incorporating fuzziness into observations results in less powerful control charts.
1. Introduction
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Keywords: Mean and Range control charts, fuzzy sets, transformation methods, fuzzy random variable, ARL criterion
Statistical process control (SPC) is a collection of methods to achieve continuous improvement in quality. This objective is accomplished by a continuous monitoring of the process under study in order to quickly detect the occurrence of the assignable causes and undertake the necessary corrective actions. The most commonly used SPC tools are control charts. A control chart contains a center line that represents average value of the quality characteristic corresponding to the in ∗ Corresponding author. S.M.T. Fatemi Ghomi, Department of Industrial Engineering, Amirkabir University of Technology, 424 Hafez Avenue,1591634311 Tehran, Iran. Tel.: +98 21 66413034; Fax: +98 21 66413025; E-mail: [email protected].
control state. Two other horizontal lines, called the upper control limit (UCL) and the lower control limit (LCL), are also shown on the chart. These control limits are chosen so that if the process is in control, nearly all samples will fall between them. As long as the points plot within the control limits, the process is assumed to be in control and no action is necessary. However, a point that plots outside of the control limits is interpreted as evidence that the process is out of control. Therefore, investigation and corrective action are required to find and eliminate the assignable cause or causes responsible for this behavior [19]. The traditional control charts are constructed using precise data. However, due to uncertain situations in the real world, it may be difficult to obtain precise data
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H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
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[1] proposed a new p-chart controlling the degree of nonconformity indicated by a fuzzy set that its support is [L, U], where L and U denote lower and upper specification limits for respective quality characteristic. Other approaches to fuzzy control charts can be found in [3, 7, 18, 23, 24]. In non-fuzzy environment, Mean and Range control charts are established following main structure of Shewhart control charts. This structure should be kept when using fuzzy data. However, some previously proposed approaches in the literature such as one given by Sentruk and Erginel [22] have ignored this structure. This ¯ −R paper addresses this issue. In this regard, fuzzy X control charts are constructed using different transformation methods. It is supposed that the observations of each sample are fuzzy random variables, which have triangular membership functions. After calculating fuzzy mean and fuzzy range of each sample using fuzzy arithmetic, the transformation methods are applied for obtained fuzzy mean and fuzzy range to compute their corresponding representative values. Then using statistical properties of the representative values and main ¯ − R control structure of Shewhart control charts, new X charts are constructed to monitor mean and variation of the process. In order to compare the performance of obtained control charts designed by different transformation methods, average run length (ARL) criterion is utilized. Contrary to the proposition of Wang and Raz [25], ARL curves depict that different transformation methods result in control charts with different performances. It is also observed that the value of α – level can significantly influence on the performance of the control charts. Moreover, according to ARL curves, it is concluded that, in spite of all advantages of incorporating fuzziness into observations, it leads to less powerful control charts. The rest of the paper is organized as follows: Section 2 introduces the concept of fuzzy random variables. Section 3 presents representative values of the triangu¯ − R control charts lar fuzzy numbers. In section 4, X are constructed in fuzzy environment using different transformation methods. Section 5 studies the power of these control charts by ARL criterion. Section 6 presents a numerical example and finally, section 7 is devoted to conclusion.
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and it would be better to deal with imprecise or even linguistic data. One way to take into account such an uncertainty in the real world is to apply fuzzy set theory proposed by Zadeh [26]. In the literature, there are some attempts to develop fuzzy control charts. For example, Raz and Wang [21] proposed an approach based on fuzzy set theory by assigning a fuzzy set to each linguistic term. Wang and Raz [25] developed two approaches called fuzzy probabilistic approach and membership approach. In fuzzy probabilistic approach, fuzzy subsets associated with the linguistic terms are transformed into their respective representative values using one of the transformation methods. On the other hand, membership approach is based on fuzzy set theory to combine all observations in only one fuzzy subset using fuzzy arithmetic. Kanagawa et al. [12] introduced an approach for directly controlling the underlying probability distributions of the linguistic data, which were not considered by Wang and Raz [25]. The main difficulty of this approach is that the unknown probability distribution function can not be determined easily. Laviolette et al. [17] and Asai [2] discussed these procedures. Cheng [4] proposed the following approach to deal with expert subjective judgments. Based on the rating scores assigned by individual inspectors to inspected items, fuzzy numbers are constructed to represent the vague outcomes of the process. Then fuzzy control charts are established directly from these fuzzy numbers, thereby retaining the fuzziness of the original vague observations. The out of control conditions are formulated using possibility theory. Engin et al. [6] combined fuzzy sets with genetic algorithms to determine sample size in attribute control charts. Gulbay and kahraman [9] developed a direct fuzzy approach to fuzzy control charts without any deffuzification and then defined unnatural pattern rules based on the fuzzification of crisp rules. Gulbay and Kahraman [10] developed a fuzzy approach to control charts based on the different fuzzy transformation methods and proposed α - cut approach to determine the tightness of inspection. El-Shal and Morris [5] investigated the use of fuzzy logic to modify the statistical process control rules. They aimed at reducing the generation of false alarms and improving detection and detection-speed of real faults. Zarandi et al. [27] presented a new hybrid method based on a combination of fuzzified sensitivity criteria and fuzzy adaptive sampling rules to determine the sample size and sample interval of the control charts. Sentruk and Erginel [22] first transformed traditional ¯ − R and X ¯ − S control charts to fuzzy control charts X ¯ − R ¯ − S ˜ and X and then developed α - cut fuzzy X control charts using α - cut approach. Amirzadeh et al.
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2. Fuzzy random variables Before giving a definition of fuzzy random variables, some preliminary concepts of fuzzy set theory ˜ is a fuzzy subset of the are given. Suppose A
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
3. Representative values for fuzzy sets
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To retain the standard format of control charts and to facilitate plotting of observations on the chart, it is necessary to convert the fuzzy sets associated with fuzzy mean and fuzzy range of each sample into scalars referred to as representative values. There exist four transformation methods, which are similar to the measure of central tendency used in descriptive statistic: α – level fuzzy median, α – level fuzzy midrange, fuzzy average and fuzzy mode. Although it is claimed that there is no theoretical basis supporting any transformation method specifically and that, a transformation method is principally selected based on the ease of computation or preference of the user [25], it is shown by ARL criterion in section 5 that these different transformation methods impact significantly on performance of the control charts. Let x˜ be a continuous fuzzy set with triangular membership function denoted as (xa , xb , xc ). In this case, its representative values are defined as follows: α xmed = (xaα + 2xb + xcα ) 4 (1) α xmid = (xaα + xcα ) 2 (2) xavg = (xa + xb + xc ) 3 (3)
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˜ universal set X. The α -* level cut of A is denoted ˜ by Aα = x : µA˜ (x) ≥ α where µA˜ (x) ∈ [0, 1] is the ˜ A ˜ is called a normal fuzzy membership function of A. ˜ is also set if there exists an x such that µA˜ (x) = 1. A called a convex fuzzy set if and only if µ (λx + (1 − ˜ A λ)y) ≥ min µA˜ (x), µA˜ (y) ; for λ ∈ [0, 1]. In this paper, the universal set X is assumed to be a real number system, i.e. X = R. Let f be a real-valued function on R, then f is said to be upper semi continuous if {x : f (x) ≥ α} is a closed set for each α. Now x˜ is called a fuzzy number if the following conditions are satisfied: i) x˜ is a normal and convex fuzzy set. ii) Its membership function is upper semi continuous. iii) The α – level cut, x˜ α , is bounded for each α ∈ [0, 1]. F (R) ˜ is a convex fuzzy denotes the set of all fuzzy numbers. A set if and only if its α – level cut is a convex set for all α ∈ [0, 1] [26]. Therefore if x˜ is a fuzzy number, the α – level cut x˜ α is a compact and convex set, i.e. x˜ α is a closed interval. Fuzzy random variables represent a well-formulated concept underlying many recent probabilistic and statistical involving data obtained from a random experiment when these data are assumed to be fuzzy set valued. Fuzzy random variables have been considered in the setting of a random experiment to model [8]:
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• Either a fuzzy observation of a mechanism associating a real value with each experiment outcome. • or an essentially fuzzy-valued mechanism, that is, a mechanism associating a fuzzy value with each experiment outcome.
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Kwakernaak [15, 16] introduced a mathematical model for the first situation elaborated later by Kruse and Mayer [14]. On the other hand, Puri and Ralescu [20] gave another approach to the second situation to be modeled. In this paper, the first situation is applied to model fuzziness of observations in each sample, where a fuzzy random variable is viewed as a fuzzy perception/observation of a classical real-valued random variable (following a normal distribution with the mea µ and the variance σ 2 ) stated as follows: Definition 1: Given a probability space (, A, P), a mapping x : → F (R) is said to be fuzzy random variable if the two real-valued mapping xαl → R and xαu → R are real-valued random variables for all α ∈ [0, 1], where x˜ α = [xαl , xαu ].
To get an overview on the development of fuzzy random variable, the interested reader may refer to Gil et al. [8].
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and x mod = xb
(4)
α , xα , x where xmed mid avg
and x mod are representative values calculated by α – level fuzzy median, α – level fuzzy midrange, fuzzy average and fuzzy mode transformation methods respectively, and xaα = xa + α(xb − xa )
(5)
= xc − α(xc − xb )
(6)
xcα
To get a review of deffuzification methods, the interested reader is referred to Hellendoorn and Thomas [11].
¯ − R control charts using 4. Construction X representative values ¯ − R control This section explains how to construct X charts using statistical properties of each representative value. To do this, after drawing a sample of size n including n fuzzy random variables, fuzzy mean and fuzzy range of each sample are calculated using fuzzy arithmetic. Then, one of the above mentioned transformation methods is applied to compute representative values of
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
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¯ − R control charts based on α – level fuzzy 4.1. X median transformation
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After calculating fuzzy mean and fuzzy range of each sample, α – level fuzzy median representative valueof ¯ is calculated as S α xaαj + 2¯xbj + x¯ cαj ) 4, X j ¯ j = (¯ med,X where x¯ aαj = x¯ aj + α(¯xbj − x¯ aj ) and x¯ cαj = x¯ cj − α(¯xcj − x¯ bj ). Equivalently, it is as follows α Smed, xaj + 2(1 + α)¯xbj + (1 − α)¯xcj ) 4 (8) ¯ j = ((1 − α)¯ X
Since it is assumed that the quality characteristic follows a normal distribution with the mean µ and the variance σ 2 , according to the Definition 1 of fuzzy random variables, it is concluded that x¯ aαj and x¯ cαj follow a normal distribution with the mean µ and the variance α σ 2 n. So the mean of Smed, ¯ , is as follows: ¯ , µmed,X X j
µmed,X¯ = µ
+ 4(1 + α)2 var(¯xbj ) + 4(1 − α2 )cov(¯xaj , x¯ bj ) + 2(1 − α)2 cov(¯xaj , x¯ cj )
(9)
α Therefore Smed, ¯ j is an unbiased estimator of the X quality characteristic mean µ. Hence, it can be used α to monitor the changes of µ. The variance of Smed, ¯ , X j
+ 4(1 − α2 )cov(¯xbj , x¯ cj ) 16
(10)
It can be simply proved that, cov(¯xaj , x¯ bj ), cov(¯xaj , x¯ cj ) and cov(¯xbj , x¯ cj ) are equal to 2 σ as . So the2 above equation 2is simplified If the ((1 − n)(6α + 4α) + 10n + 6)σ 16n . values of µ and σ are not known, they should be estimated using samples initially drawn from the process. To do this, suppose m samples of size n have been ¯ ¯ = (¯r , r¯ , r¯ ) are the taken. X = (x¯ a , x¯ b , x¯ c ) and R a b c average of fuzzy mean and fuzzy range of all m samples respectively. The estimates of the process mean and standard deviation are respectively as follows: µ ˆ = (x¯ αa + 2x¯ b + x¯ αc ) 4 (11) ¯ med d2 σˆ = (¯raα + 2¯rb + r¯cα ) 4d2 = R (12)
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and raj = xmax,aj − xmin,cj , rbj = xmax,bj − xmin,bj and rcj = xmax,cj − xmin,aj . Here (xmax,aj , xmax,bj , xmax,cj ) and (xmin,aj , xmin,bj , xmin,cj ) are the maximum and minimum fuzzy numbers in the j-th sample respectively. The ranking method used in this paper involves three ordered criteria proposed by Kaufmann and Gupta [13]. As the first criterion, the greatest associate ordi˜ = (xa + 2xb + xc ) 4 is nary number defined as C1 (X) used. If C1 does not rank the fuzzy numbers, those ˜ = xb which have the best maximal presumption C2 (X) will be chosen as the second criterion. Finally, if C1 and C2 do not rank the fuzzy numbers, the difference ˜ = xc − xa will be used as the third of the spreads C3 (X) criterion.
2 σmed, ¯ , is computed as follows: X 2 2 σmed, xaj ) + var(¯xcj )) ¯ = (1 − α) (var(¯ X
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the obtained fuzzy mean and fuzzy range. Finally, the 3-sigma control charts are constructed using statistical properties of representative values and basic structure of Shewhart control charts. In this paper, each fuzzy random variable is supposed to have triangular membership function. Suppose ˜ ij = (xaij , xbij , xcij ) is the i-th fuzzy observation in the X ¯ = (¯x , x¯ , x¯ ) and R ˜j = j-th sample. In this case, X j aj bj cj (raj , rbj , rcj ) are fuzzy mean and fuzzy range of the j-th sample respectively, where n (7) x¯ kj = xkij n ; k = a, b, c.
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where
x¯ αa = x¯ a + α(x¯ b − x¯ a )
(13)
x¯ αc r¯aα
= x¯ c − α(x¯ c − x¯ b )
(14)
= r¯a + α(¯rb − r¯a )
(15)
r¯cα = r¯c − α(¯rc − r¯b )
(16)
and
The values of d2 can be found based on the size of drown sample in [19]. Therefore the standard deviation α of Smed, ¯ , σmed,X ¯ , is derived as follows: X j
¯ med σmed,X¯ = A2 R
(1 − n)(6α2 + 4α) + 10n + 6 12
(17)
where √ A 2 = 3 d2 n ¯ med R
= (¯raα + 2¯rb + r¯cα ) 4
(18) (19)
Now using basic structure of Shewhart control charts, which is UCL = µw + 3σw , CL = µw and LCL = α µw − 3σw where w is a sample statistic (here Smed, ¯ j) X ¯ measuring some quality characteristic of interest, X
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
control chart based on α – level fuzzy median transvar(rcj ) = (d3 r¯c d2 )2 formation is constructed as follows: ⎧
¯ med (1 − n)(6α2 + 4α) + 10n + 6) 4 ⎪ UCLαmed,X¯ = µmed,X¯ + 3σmed,X¯ = (x¯ αa + 2x¯ b + x¯ αc + A2 R ⎪ ⎪ ⎨ CLαmed,X¯ = µmed,X¯ = (x¯ αa + 2¯xb + x¯ αc ) 4 ⎪
⎪ ⎪ ¯ ⎩ LCLα =µ − 3σ = (x¯ α + 2x¯ + x¯ α − A R (1 − n)(6α2 + 4α) + 10n + 6) 4 ¯ med,X
b
a
The process mean is in control if LCLαmed,X¯ ≤ ≤ UCLαmed,X¯ and otherwise is out of control. As it is observed, the above control limits are natural generalization of crisp control limits into fuzzy environment, because they decrease to the common control limits in crisp environment if α = 1. α Smed, ¯j X
¯ which is plotted on Remark 1: It is worth noting that, X the traditional control charts is an unbiased estimator of the quality characteristic mean. The control limits are calculated using its mean and standard deviation and main structure of Shewhart control chart. This point is α used in construction of control chart for Smed, ¯ . X j
2 med
Hence, after a few computational efforts, the standard α deviation of Smed,R , σmed,R , is derived as follows: j σmed,R = ((1 − α)2 (¯ra2 + r¯c2 ) + 4(1 + α)2 r¯b2 )(d3 d2 )2 0.5 ¯ med d2 )2 4 (27) + 4(5 − 2α − 3α2 )(R where the values of d3 can be found based on the size of sample in [19]. Now R control chart based on α – level fuzzy median transformation is constructed as follows using Eqs (22) and (27), and basic structure of Shewhart control chart: ⎧ ⎪ UCLαmed,R = µmed,R + 3σmed,R ⎪ ⎨ CLαmed,R = µmed,R (28) ⎪ ⎪ ⎩ LCLαmed,R = µmed,R − 3σmed,R
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Now control limits of R chart are computed. After calculating fuzzy range of each sample, α -level fuzzy ˜ j is calculated as median representative valueof R
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(20)
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¯ med,X
(26)
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¯ med,X
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α Smed,R = (raαj + 2rbj + rcαj ) 4. Equivalently, it is as j follows: α Smed,R = ((1 − α)raj + 2(1 + α)rbj + (1 − α)rcj ) 4 j
(21)
¯ med µmed,R = R
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α The mean of Smed,R , µmed,R , is d2 σ. If σ is not known, j it should be estimated by Eq. (12). Therefore, we have:
(22)
α 2 On the other hand, the variance of Smed,R , σmed,R , j is calculated as: 2 σmed,R = (1 − α)2 (var(raj ) + var(rcj ))
+ 4(1 + α)2 var(rbj ) + 4(1 − α2 )cov(raj , rbj ) + 2(1 − α)2 cov(raj , rcj ) + 4(1 − α2 )cov(rbj , rcj )
16
(23)
It is simply proved that, cov(raj , rbj ), cov(raj , rcj ) and cov(rbj , rcj ) are equal to 2σ 2 . Moreover, it is known [19] that var(raj ) = (d3 r¯a d2 )2 (24) 2 var(rbj ) = (d3 r¯b d2 ) (25)
In this case, the process variation is in control α if LCLαmed,R ≤ Smed,R ≤ UCLαmed,R and otherwise is j out of control. If α = 1, the above control limits also decrease to the common control limits in crisp environment. Therefore, they can be viewed as natural generalization of common control limits of R control chart into fuzzy environment. ¯ − R Control charts based on α – level 4.2. X fuzzy midrange transformation
¯ − R control charts based on Development of X α – level fuzzy midrange transformation is similar to that based on α – level fuzzy median transformation. If α – level fuzzy midrange is used as the transformation ¯ is as S α method, the representative value of X j mid,Xj = (¯xaαj + x¯ cαj ) 2. Equivalently, it is as follows: α Smid, xaj + 2α¯xbj + (1 − α)¯xcj ) 2 (29) ¯ j = ((1 − α)¯ X According to the Definition 1 of fuzzy random variables, x¯ aαj and x¯ cαj follow a normal distribution with the mean µ and the variance σ 2 n. Hence, the mean of
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
α Smid, ¯ is computed as follows: X
˜ j is calculated as sample, the representative value of R
j
µmid,X¯ = µ
(30)
α Therefore, Smid, ¯ j is an unbiased estimator of the qualX ity characteristic mean and can be used to monitor the α 2 changes of µ. The variance of Smid, ¯ j , σmid,X ¯ , is calX culated as: 2 2 σmid, xaj ) + var(¯xcj )) + 4α2 var(¯xbj ) ¯ = (1 − α) (var(¯ X
+ 4α(1 − α)cov(¯xaj , x¯ bj )
+ 4α(1 − α)cov(¯xbj , x¯ cj )
4
(31)
j
¯ mid µmid,R = R
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¯ mid = (¯raα + r¯cα ) 2 R
(38)
2 = (1 − α)2 (var(raj ) + var(rcj )) + 4α2 var(rbj ) σmid,R + 4α(1 − α)cov(raj , rbj ) + 2(1 − α)2 cov(raj , rcj )
(35)
4
(39)
According to the Eqs (24)-(26) and that, cov(raj , rbj ), cov(raj , rcj ) and cov(rbj , rcj ) are equal to 2σ 2 , the stanα , σmid,R , is derived as follows: dard deviation of Smid,R j σmid,R = ((1 − α)2 (¯ra2 + r¯c2 ) + 4α2 r¯b2 )(d3 d2 )2 0.5 ¯ mid d2 )2 2 (40) + 4(1 + 2α − 3α2 )(R
Hence, R control chart based on α – level fuzzy midrange transformation is constructed as follows: ⎧ ⎪ UCLαmid,R = µmid,R + 3σmid,R ⎪ ⎨ CLαmid,R = µmid,R (41) ⎪ ⎪ ⎩ LCLαmid,R = µmid,R − 3σmid,R
¯ control chart based on α – level fuzzy Now, X midrange transformation is constructed using Eqs (30) and (34), and basic structure of Shewhart control charts as follows: ⎧
¯ mid (1 − n)(6α2 − 4α) + 2n + 2) 2 ⎪ UCLαmid,X¯ = µmid,X¯ + 3σmid,X¯ = (x¯ αa + x¯ αc + A2 R ⎪ ⎪ ⎨ CLαmid,X¯ = µmid,X¯ = (x¯ αa + x¯ αc ) 2 ⎪
⎪ ⎪ ¯ mid (1 − n)(6α2 − 4α) + 2n + 2) 2 ⎩ LCLα ¯ = µmid,X¯ − 3σmid,X¯ = (x¯ αa + x¯ αc − A2 R
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+ 4α(1 − α)cov(rbj , rcj )
(1 − n)(6α2 − 4α) + 2n + 2 6) (34)
where
α The mean of Smid,R , µmid,R , is d2 σ. If the value j of σ is not known, it should be estimated by Eq. (33). Therefore, we have:
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α So the standard deviation of Smid, ¯ , σmid,X ¯ , is as X
¯ mid σmid,X¯ = (A2 R
(37)
α 2 The variance of Smid,R , σmid,R , is computed as j
As mentioned previously, it can be simply proved that, cov(¯xaj , x¯ bj ), cov(¯xaj , x¯ cj ) and cov(¯xbj , x¯ cj ) are equal to σ 2 . Therefore, Eq. (31) can bereadily rewritten as [((1 − n)(6α2 − 4α) + 2n + 2)σ 2 4n]. If the values of µ and σ are not known, they should be estimated using m initially drawn samples as follows: µ ˆ = (x¯ αa + x¯ αc ) 2 (32) ¯ mid d2 σˆ = (¯raα + r¯cα ) 2d2 = R (33)
α Smid,R = ((1 − α)raj + 2αrbj + (1 − α)rcj ) 2 j
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+ 2(1 − α)2 cov(¯xaj , x¯ cj )
α Smid,R = (raαj + rcαj ) 2. Equivalently, it is as follows: j
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(36)
mid,X
The process mean is in control if LCLαmid,X¯ ≤ α α Smid, ¯ j ≤ UCLmid,X ¯ and otherwise is out of control. X When α = 1, the above control limits decrease to the common control limits in crisp environment. Thus, they can be regarded as natural generalization of common ¯ control chart into fuzzy environment. control limits of X Now control limits of R control chart are going to be computed. After calculating fuzzy range of each
The process variation is in control if LCLαmid,R ≤ ≤ UCLαmid,R and otherwise is out of control. Since the above control limits decrease to the common control limits in crisp environment when α = 1, they can be viewed as natural generalization of common control limits of R control chart into fuzzy environment.
α Smid,R j
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
¯ − R control charts based on fuzzy average 4.3. X transformation If fuzzy average transformation method is used, the ¯ is calculated as follows: representative value of X j Savg,X¯ j = (¯xaj + x¯ bj + x¯ cj ) 3 (42)
Therefore, we have: ¯ avg µavg,R = R
2 = var(raj ) + var(rbj ) + var(rcj ) σavg,R + 2(cov(raj , rbj ) + cov(raj , rcj ) + cov(rbj , rcj )) 9
the computed as follows: 2 xaj ) + var(¯xbj ) + var(¯xcj ) σavg, ¯ = var(¯ X
(43)
(52)
Hence, R control chart based on fuzzy average transformation is constructed as follows: ⎧ UCLavg,R = µavg,R + 3σavg,R ⎪ ⎨ CLavg,R = µavg,R (53) ⎪ ⎩ LCL avg,R = µavg,R − 3σavg,R
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The above equation can be simplified as σ 2 (2n + 1) 3n. Based on m initial samples, the values of µ and σ are estimated as: µ ˆ = (x¯ a + x¯ b + x¯ c ) 3 (44) ¯ avg d2 σˆ = (¯ra + r¯b + r¯c ) 3d2 = R (45)
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¯ avg d2 ) σmid,R = (¯ra + r¯b + r¯c )(d3 d2 ) + 12(R
+ 2(cov(¯xaj , x¯ bj ) + cov(¯xaj , x¯ cj ) + cov(¯xbj , x¯ cj )) 9
(51)
So the standard deviation of Savg,Rj , σavg,R , is as follows: 2 2 2 0.5 2 2
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α α and Smid, , S ¯ j avg,X ¯ j can be also used to monitor X 2 changes of µ. The variance of Savg,X¯ , σavg, ¯ , is X j
(50)
2 The variance of Savg,Rj , σavg,R , is computed as:
The mean of Savg,X¯ , µavg,X¯ , is µ (note that x¯ aj and j ¯ with α = 0). So similar to x¯ are the end points of X cj α Smed, ¯j X
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¯ avg = (¯ra + r¯b + r¯c ) 3 R
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where
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So the standard deviation of Savg,X¯ , σavg,X¯ , is as j follows √ ¯ avg 6n + 3 9) σavg,X¯ = (A2 R (46)
(47)
The process variation is in control if LCLavg,R ≤ Savg,Rj ≤ UCLavg,R and otherwise is out of control.
¯ − R control charts based on fuzzy mode 4.4. X transformation
If fuzzy mode is used as the transformation method, the constructed control charts are similar to those in ¯ crisp state, established by x¯ b and rb . Therefore, X control chart based on fuzzy mode transformation is calculated as follows:
¯ control chart based on fuzzy average transNow X formation is constructed as follows: ⎧ √ ¯ avg 6n + 3) 3 ⎪ UCLavg,X¯ = µavg,X¯ + 3σavg,X¯ = (x¯ a + x¯ b + x¯ c + A2 R ⎪ ⎨ CLavg,X¯ = µavg,X¯ = (x¯ a + x¯ b + x¯ c ) 3 √ ⎪ ⎪ ¯ avg 6n + 3) 3 ⎩ LCLavg,X¯ = µavg,X¯ − 3σavg,X¯ = (x¯ a + x¯ b + x¯ c − A2 R The process mean is in control if LCLavg,X¯ ≤ Savg,X¯ ≤ UCLavg,X¯ and otherwise is out of control. j After calculating fuzzy range of each sample, the rep˜ j is calculated using the following resentative value of R equation: Savg,Rj = (raj + rbj + rcj ) 3 (49) The mean of Savg,Rj , µavg,R , is d2 σ. If the value of σ is not known, it should be estimated by Eq. (45).
⎧ ¯ ⎪ ⎨ UCL mod ,X¯ = µ mod ,X¯ + 3σ mod ,X¯ = xb + A2 r¯b CL mod ,X¯ = µ mod ,X¯ = x¯ b ⎪ ⎩ LCL ¯ b − A2 r¯b ¯ = µ mod ,X ¯ − 3σ mod ,X ¯ =x mod ,X
(48)
(54)
The process mean is in control if LCL mod ,X¯ ≤ S mod ,X¯ ≤ UCL mod ,X¯ and otherwise is out of j control. Moreover, R control chart based on fuzzy mode
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transformation is as follows: ⎧ ⎪ ⎨ UCL mod,R = µ mod,R + 3σ mod,R = r¯b (1 + 3d3 d2 ) CL mod,R = µ mod,R = r¯b
⎪ ⎩ LCL mod,R = µ mod,R − 3σ mod,R = r¯b (1 − 3d3 d2 )
(55)
¯ control chart designed Application of Eq. (56) to X by α – level fuzzy midrange transformation also leads to: α α βmid = Pr(LCLαmid,X¯ ≤ Smid, ¯ j ≤ UCLmid,X ¯ |µmid X
β = Pr(LCL ≤ w ≤ UCL |µ = µ0 + kσ)
(56)
AU
TH
where, µ0 is the current value of the process mean, µ is the shifted value of mean measured by units of standard deviation of the process, σ, and k determines the amount of this shift. When the process is out of control, ARL is calculated as [19]: (57)
¯ and R ARL criterion can be calculated for both X ¯ control charts. In this paper, it is just computed for X chart. ¯ control chart established by Applying Eq. (56) to X α – level fuzzy median transformation results in: α α βmed = Pr(LCLαmed,X¯ ≤ Smed, ¯ j ≤ UCLmed,X ¯ |µmed X
¯ med d2 )) = ((x¯ αa + 2x¯ b + x¯ αc ) 4) + kR = Pr(−3 − 4k
≤ Z ≤ 3 − 4k
n [(1 − n)(6α2 − 4α) + 2n + 2]
n [(1 − n)(6α2 + 4α) + 10n + 6]
n [(1 − n)(6α2 + 4α) + 10n + 6])
(58) where, Z is a random variable following standard normal distribution.
n [(1 − n)(6α2 − 4α) + 2n + 2])
CO
PY
(59) When fuzzy average and fuzzy mode transformations are used to construct control charts, the probability of type II error is calculated for each transformation method respectively as follows: βavg = Pr(LCLavg,X¯ ≤ S ¯ µavg ¯ ≤ UCLavg,X
avg,Xj
¯ avg d2 ) = ((x¯ a + x¯ b + x¯ c ) 4) + kR
= Pr(−3 − k
3n (2n + 1) ≤ Z ≤ 3 − k
3n (2n + 1))
(60)
and
OR
After developing control charts using different transformation methods, a practitioner may ask this question: which transformation method should be used in practice? To answer this question, four different control charts constructed using four various transformation methods are compared in this section exploiting average run length (ARL) criterion. ARL criterion is defined as the number of samples drawn before identifying an out of control condition. Therefore, the less ARL value, the more powerful control chart in detecting shifts in the quality characteristic we have. For a control chart with the control limits UCL and LCL, the probability of type II error is defined as follows [19]:
= Pr(−3 − 2k
≤ Z ≤ 3 − 2k
5. Performance of control charts based on different transformation methods
ARL = 1/(1 − β)
¯ mid d2 )) = ((x¯ αa + x¯ αc ) 2) + kR
The process variation is in control if LCL mod,R ≤ S mod,Rj ≤ UCL mod,R and otherwise is out of control.
β mod = Pr(LCL mod,X¯ ≤ S mod,X¯ j ≤ UCL mod,X¯ |µ mod
¯ mod d2 ) = x¯ b + kR √ √ = Pr(−3 − k n ≤ Z ≤ 3 − k n)
(61)
After calculating the probability of type II error, the ARL criterion is computed using Eq. (57). It is usually depicted for a specific value of n and different values of k. But this should be done for a predetermined value of α, because for a specific values of n and k, βmed and βmid are functions of α. Fig. 1 shows the ARL curves when n = 4 and α = 0.35, 0.55, 0.75, 0.95 for different transformation methods. In this Figure, the ARL of control chart constructed by fuzzy mode transformation is depicted by number 1, α – level fuzzy median transformation by number 2, α – level fuzzy midrange transformation by number 3 and fuzzy average transformation by number 4. Considering this Figure, some significant conclusions can be drawn. Firstly, it is apparent for all specified values of α that various transformation methods result in different per¯ control chart. This clearly contradicts formances of X the previous proposition [25] that, there is no theoretical difference among different transformation methods and that, a transformation method is chosen based on its ease of computation or user’s preference. The control chart constructed by fuzzy mode and fuzzy average
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
α = 0.35
350
350
300
300
250
250
200
3, 4 1
150
200 150 100
50 0
3 2
50
2
0.5
1
0
1.5
0
K
α = 0.75
350
400 350
4
1.5
1
1.5
α = 0.95
4
250
ARL
250 200 1
3
200
OR
ARL
1
K
300
300
3
1
150
150
100
100 2
0
50
0.5
TH
50 0
0.5
CO
400
4
1
PY
100
0
α = 0.55
400
ARL
ARL
400
261
1
K
1.5
0 0
2
0.5
K
AU
Fig. 1. The ARL curves of different control charts for n = 4 and based on: fuzzy Mode (no. 1), α- level fuzzy median (no. 2), α – level fuzzy midrange (no. 3) and fuzzy average (no. 4) transformation.
transformation methods are the most and the least powerful control chart respectively. In addition, it is observed that the value of α can have significant influence on the control chart performance when either fuzzy median or fuzzy midrange transformation method is used. In the other words, although a large value of α results in more tightness of inspection, it leads to a more powerful control chart when one of two abovementioned transformation methods is applied. Moreover, since the α – level fuzzy median and α – level fuzzy midrange approach to fuzzy mode when increasing the value of α, the ARL curves of their respective control charts approach to ARL curve of control chart constructed by fuzzy mode transformation method when the value of α increases. Finally, it can be inferred that in spite of all beneficial points of incorporating
fuzziness into control charts and using fuzzy information such as taking into account variability caused by human subjectivity or measurement devices, or environmental conditions, it results in a less powerful control chart. The later conclusion is drown by comparing ARL curves of control charts established by α – level fuzzy median, α – level fuzzy midrange and fuzzy average transformations with that of control chart made by fuzzy mode transformation, which is equivalent to crisp control chart. Nonetheless, we know that applying fuzzy mode transformation deprives us of advantages of using fuzzy sets. It may also lead to a biased result when the membership function is extremely asymmetrical. Our proposition in such a dilemma is to exploit fuzzy median transformation with a large value of α. Because, not
262
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts Table 1 The fuzzy measurement values and their fuzzy means and ranges Fuzzy mean 3
(17.40, 17.44, 17.46) (18.06, 18.08, 18.11) (18.45, 18.49, 18.51) (17.66, 17.67, 17.68) (18.25, 18.30, 18.32) (18.30, 18.34, 18.37) (18.00, 18.02, 18.03) (17.59, 17.61, 17.63) (17.80, 17.81, 17.83) (17.98, 18.00, 18.02) (18.50, 18.53, 18.55) (18.04, 18.05, 18.08) (17.92, 17.95, 17.98) (18.00, 18.02, 18.06) (17.40, 17.42, 17.45) (18.42, 18.44, 18.46) (17.65, 17.68, 17.71) (17.73, 17.74, 17.76) (18.06, 18.09, 18.11) (17.96, 18.00, 18.02) (17.55, 17.57, 17.60) (18.52, 18.54, 18.55) (17.96, 18.00, 18.01) (18.40, 18.44, 18.45) (17.45, 17.47, 17.50)
(18.07, 18.10, 18.13) (17.46, 17.50, 17.51) (18.63, 18.67, 18.68) (18.10, 18.13, 18.16) (18.35, 18.37, 18.38) (17.35, 17.37, 17.41) (18.38, 18.39, 18.41) (18.26, 18.28, 18.30) (17.28, 17.31, 17.34) (17.35, 17.40, 17.41) (17.63, 17.64, 17.66) (18.10, 18.13, 18.16) (18.33, 18.37, 18.41) (17.41, 17.43, 17.46) (18.52, 18.55, 18.57) (18.47, 18.49, 18.50) (18.34, 18.38, 18.39) (17.36, 17.37, 17.39) (17.85, 17.87, 17.91) (18.28, 18.31, 18.33) (17.32, 17.35, 17.36) (18.05, 18.08, 18.11) (17.81, 17.84, 17.87) (18.30, 18.32, 18.33) (18.26, 18.30, 18.32)
(17.26, 17.31, 17.32) (17.45, 17.47, 17.50) (18.08, 18.10, 18.12) (17.66, 17.68, 17.70) (18.39, 18.42, 18.43) (18.14, 18.16, 18.18) (18.25, 18.27, 18.28) (17.64, 17.66, 17.67) (17.43, 17.44, 17.46) (18.25, 18.28, 18.29) (17.59, 17.61, 17.64) (17.25, 17.27, 17.31) (18.63, 18.64, 18.65) (18.26, 18.29, 18.33) (18.41, 18.44, 18.46) (17.73, 17.76, 17.77) (17.67, 17.70, 17.74) (17.78, 17.81, 17.84) (18.50, 18.54, 18.56) (17.94, 17.98, 17.99) (18.16, 18.18, 18.20) (17.80, 17.84, 17.87) (17.96, 17.97, 17.98) (18.22, 18.24, 18.26) (18.05, 18.07, 18.08)
Fuzzy range
4 (18.18, 18.20, 18.22) (17.73, 17.76, 17.78) (18.13, 18.14, 18.15) (17.78, 17.80, 17.82) (18.25, 18.27, 18.30) (18.35, 18.38, 18.40) (18.02, 18.04, 18.06) (17.86, 17.88, 17.90) (18.21, 18.23, 18.24) (18.30, 18.33, 18.34) (18.55, 18.58, 18.60) (18.09, 18.11, 18.14) (17.77, 17.78, 17.80) (18.10, 18.12, 18.13) (18.70, 18.73, 18.74) (18.05, 18.07, 18.09) (17.34, 17.37, 17.40) (17.46, 17.49, 17.51) (18.10, 18.13, 18.15) (17.90, 17.95, 17.97) (18.19, 18.21, 18.25) (17.98, 18.00, 18.03) (17.50, 17.53, 17.56) (17.72, 17.75, 17.78) (17.35, 17.36, 17.37) (18.06, 18.08, 18.10) (18.54, 18.56, 18.59) (18.05, 18.07, 18.11) (18.56, 18.59, 18.60) (18.22, 18.25, 18.27) (18.70, 18.73, 18.74) (18.33, 18.36, 18.37) (17.85, 17.89, 17.90) (17.88, 17.91, 17.94) (17.39, 17.42, 17.45) (17, 57, 17.58, 17.61) (17.70, 17.74, 17.75) (18.03, 18.06, 18.08) (18.52, 18.56, 18.58) (18.18, 18.21, 18.23) (18.33, 18.36, 18.37) (17.84, 17.87, 17.88) (17.85, 17.90, 17.92) (18.06, 18.09, 18.11) (18.00, 18.02, 18.05) (17.93, 17.96, 17.98) (17.65, 17.68, 17.70) (18.14, 18.17, 18.19) (17.11, 17.14, 17.17) (17.72, 17.75, 17.77)
OR
CO
2
(0.86, 0.89, 0.96) (0.63, 0.67, 0.70) (0.51, 0.57, 0.60) (0.42, 0.46, 0.50) (0.15, 0.19, 0.22) (1.14, 1.21, 1.25) (0.58, 0.61, 0.64) (1.07, 1.12, 1.15) (0.46, 0.50, 0.55) (0.84, 0.88, 0.94) (0.86, 0.92, 0.96) (0.79, 0.86, 0.91) (1.26, 1.28, 1.30) (1.08, 1.13, 1.18) (1.11, 1.17, 1.20) (0.93, 0.97, 1.01) (0.63, 0.70, 0.74) (0.39, 0.44, 0.48) (0.75, 0.80, 0.86) (0.53, 0.58, 0.64) (0.97, 1.01, 1.05) (0.65, 0.70, 0.75) (0.13, 0.18, 0.24) (0.70, 0.76, 0.80) (1.09, 1.16, 1.21)
Table 2 Representative values of fuzzy mean and fuzzy ranges 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
Fuzzy mean (17.73, 17.76, 17.78) (17.78, 17.80, 17.82) (18.35, 18.38, 18.40) (17.86, 17.88, 17.90) (18.30, 18.33, 18.34) (18.09, 18.11, 18.14) (18.10, 18.12, 18.13) (18.05, 18.07, 18.09) (17.46, 17.49, 17.51) (17.92, 17.95, 17.97) (17.98, 18.00, 18.03) (17.72, 17.75, 17.78) (18.06, 18.08, 18.10) (18.05, 18.07, 18.11) (18.22, 18.25, 18.27) (18.33, 18.36, 18.37) (17.88, 17.91, 17.94) (17, 57, 17.58, 17.61) (18.03, 18.06, 18.08) (18.18, 18.21, 18.23) (17.84, 17.87, 17.88) (18.06, 18.09, 18.11) (17.93, 17.96, 17.98) (18.14, 18.17, 18.19) (17.72, 17.75, 17.77)
R.V. of fuzzy mean
Fuzzy range
R.V. of fuzzy range
17.762 17.798 18.384 17.879 18.328 18.112 18.117 18.071 17.485 17.952 17.998 17.746 18.081 18.074 18.249 18.355 17.913 17.585 18.057 18.214 17.865 18.091 17.957 18.170 17.747
(0.86, 0.89, 0.96) (0.63, 0.67, 0.70) (0.51, 0.57, 0.60) (0.42, 0.46, 0.50) (0.15, 0.19, 0.22) (1.14, 1.21, 1.25) (0.58, 0.61, 0.64) (1.07, 1.12, 1.15) (0.46, 0.50, 0.55) (0.84, 0.88, 0.94) (0.86, 0.92, 0.96) (0.79, 0.86, 0.91) (1.26, 1.28, 1.30) (1.08, 1.13, 1.18) (1.11, 1.17, 1.20) (0.93, 0.97, 1.01) (0.63, 0.70, 0.74) (0.39, 0.44, 0.48) (0.75, 0.80, 0.86) (0.53, 0.58, 0.64) (0.97, 1.01, 1.05) (0.65, 0.70, 0.75) (0.13, 0.18, 0.24) (0.70, 0.76, 0.80) (1.09, 1.16, 1.21)
0.8905 0.6699 0.5696 0.4600 0.1899 1.2096 0.6100 1.1198 0.5001 0.8803 0.9198 0.8598 1.2800 1.1300 1.1696 0.9700 0.6996 0.4399 0.8001 0.5801 1.0100 0.7000 0.1801 0.7598 1.1598
TH
Sample no.
AU
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
Part no. 1
PY
Sample no.
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
X bar control chart based on fuzzy median transformation with =0.95
18.8
UCL=18.62
18.4 18.2 18
CL=18
17.8 17.6 17.4 17.2 17 0
5
10
15
25
R control chart based on fuzzy median transformation with =0.95
2 1.8 1.6
UCL=1.84
1.2 1 0.8 0.6 0.4 0.2 0
5
TH
OR
1.4
0
20
LCL=17.38
CO
Subgroup Number
PY
Representative value of the Mean
18.6
Representative value og the Rang
263
10
CL=0.79
LCL=0 15
20
25
AU
Subgroup Number
¯ and R control charts based on fuzzy median transformation with α = 0.95. Fig. 2. X
only do we benefit the advantages of incorporating fuzziness, but also the ARL curve of control chart constructed using α – level fuzzy median transformation with a large value of α is very close to that established by fuzzy mode transformation. In this case, the control chart performance is very close to the best performance obtained from fuzzy mode transformation.
6. Numerical example The pervious section described how to determine the most powerful control chart. In this section, a numerical
example is given in which fuzzy median transformation at the level of α = 0.95 is utilized. As mentioned before, we retain fuzziness in data in such level and that, performance of the constructed control chart is very close to the best performance obtained by fuzzy mode transformation method. The example presents a process producing a shaft of a gearbox. The quality characteristic that experts are going to examine is outer diameter of this shaft. 25 samples with a sample size of 4 were taken from the production process. Each observation was measured and recorded as a triangular fuzzy number. The fuzzy measures and their fuzzy means and fuzzy ranges are given in Table 1. Table 2 presents the representative values of fuzzy mean and fuzzy range for each
H. Moheb Alizadeh and S.M.T. Fatemi Ghomi / Fuzzy development of Mean and Range control charts
0.95 18.62, CL0.95 ¯ = 18 and LCLmed,X ¯ = 17.38. The med,X
limits of R control chart are obtained as UCL0.95 med,R = 0.95 0.95 1.84, CLmed,R = 0.79 and LCLmed,R = 0. In Table 2, R.V. stands for representative value. Now the representative values obtained from Table 2 can be plotted on a control chart with the given control limits. ¯ and R control charts for this example Figure 2 shows X respectively. Since the above control charts are constructed using basic structure of Shewhart control charts, they present the process status more accurately. Moreover, they are sufficiently powerful for detecting changes in the process mean, because the 0.95-level fuzzy median transformation method is applied to construct them.
powerful control charts. Finally, it was concluded that in spite of all advantages of using fuzzy data in the real world, it resulted in less powerful control charts. Therefore, it is better for a practitioner to apply fuzzy mode transformation method. However, in order to deal with the disadvantages of using fuzzy mode such as lacking flexibility in measuring and recording data and ignoring variability caused by human subjectivity or measurement devices, we suggested exploiting α – level median transformation with a large value of α. In this case, the fuzziness of data is retained and the power of obtained control charts is very close to the best power obtained by fuzzy mode transformation method.
PY
sample, which are computed by fuzzy median transformation method with α = 0.95. The control limits of ¯ − R control charts can be calculated using Eqs (20) X and (28). It should be noted for n = 4 that, A2 = 0.729, d3 = 0.88 and d2 = 2.059. Using these factors, the lim¯ control chart are obtained as UCL0.95 ¯ = its of X med,X
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¯ − R control charts were developed In this paper, X in fuzzy environment using different transformation methods. Observations of each sample were assumed to be fuzzy random variables with triangular membership functions. After calculating fuzzy mean and fuzzy range of each sample using fuzzy arithmetic, their representative values were obtained applying the transformation methods. Then, using statistical properties ¯ − R control charts of the representative values, new X were constructed to monitor the process mean and variation. In this case, since the obtained control charts are established based on main structure of Shewhart control charts, they reveal the process status more correctly. After that, in order to distinguish and compare different control charts constructed by various transformation methods, their powers were examined applying average run length (ARL) criterion. Some significant conclusions were drawn considering the obtained ARL curves. Firstly, contrary to a previous proposition, it was derived that different transformation methods affect significantly on performances of the constructed control charts. Fuzzy mode and fuzzy average transformation methods resulted in the most and the least powerful control charts respectively. Moreover, when either α – level fuzzy median or α – level fuzzy midrange transformation method was applied, a larger α value led to more
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