Fuzzy Control Chart A Better Alternative for Shewhart Average Chart

Quality & Quantity (2007) 41:375–385 DOI 10.1007/s11135-006-9007-9

© Springer 2006

Fuzzy Control Chart A Better Alternative for Shewhart Average Chart ALIREZA FARAZ1 and M. BAMENI MOGHADAM2,∗ 1

Department of Quality Management, Aerospace Industrial Organizations, Tehran, Iran; Department of Statistics, Faculty of Economics, Allameh Tabatabaee University, Beheshti Ghasir Ave, Tehran 1513615411, Iran 2

Abstract. This paper through a real illustrative example and a power test shows that designing a fuzzy control chart for process average of a continuous (variable) quality characteristic with a warning line is a better alternative to Shewhart X¯ chart in many respects, like providing better neural view to inspectors, offering different strategic options for company to choose, detecting the desire shifts more quickly, and more sensibility to small shifts without any complexity augmentation to the chart. Key words: fuzzy, control chart, warning line, alarm rate, base rule, ARL

1. Introduction Many quality characteristics (Q.Ch.) are expressed in terms of original or its derived measurement units, like weight, length, pressure, etc. for which they are called continuous or variable. As normality is a usual assumption of control charts of continuous Q.Ch.s and independency of mean and variance is a basic assumption of normal distribution, a separate control chart is prepared for monitoring the process average. In this connection, the most popular control chart for monitoring process mean of a continuous Q.Ch. is Shewhartian X¯ chart, which uses normal probability distribution. In this regards, it seems a more practical method of controlling the process average is a fuzzy control chart, which provides a neural view to inspectors, offers different strategic options for company to choose, detect the desire shifts more quickly, and it is more sensitive to small shifts without any complexity augmentation to the chart. As the literature of a Shewhartian control chart and fuzzy theory is quite famous and available in different related texts and articles, which ∗

Author For correspondence: M. Bameni Moghadam, Department of Statistics, Faculty of Economics, Allameh Tabatabaee University, Beheshti, Ghasir Ave, Tehran 1513615411, Iran. E-mail: [email protected]

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some of them is given in our references we skip the general discussion about them and pay attention to our work through fuzzy description in Section 2, designing fuzzy chart in Section 3, illustrative example and its power test in Section 4, and finally conclusion in Section 5. 2. Fuzzy Description In terms of quality personnel at shop floor, a product either has proper quality or not. This traditional concept of quality may be expressed by a binary quality membership function that takes values 1 and 0 for proper and improper quality, respectively. However, in fuzzy approach, quality is always a value between 0 and 1, and therefore, its quality membership function takes any value in between (Wang and Raz, 1990). Consider a situation where the overall quality of a product can be described on a Q.Ch. X and that this variable can be described by a set of linguistic terms of quality such as negative, zero, and positive depending on its mean deviation from its target and also the overall quality of the product can be described by a set of fuzzy terms such as bad, good, very good, and excellent. Now, we examine the Q.Ch. of the product in the samples and we will assign a membership value to linguistic term zero. As Yager and Filer (1994) mentioned, this fuzziness can be due to the shift in the process mean and/or dispersion, or the measurement error of instruments. We describe the sample mean by linguistic terms zero and good, which imply that how much this sample mean has shifted from the process mean and target value, respectively. After describing subgroup mean and samples by a set of fuzzy quality linguistic terms, an expert team then can describe the overall quality of the product by a set of fuzzy If-then rules with the following format: IF (X¯ i IS Qi ) THEN (subgroup overall quality is Qj ), where X¯ i is the ith sample mean and Qi are the linguistic terms associated with average and Qj are the linguistic terms associated with the overall quality of subgroup. Fuzzy if-then rules are used here to capture the imprecise modes of reasoning that play an essential role in the quality assurance inspector ability to make decision in an environment of uncertainty imprecision encounter with the classification process (Zimmermann, 1996). In this paper we have used, Qi = {zero} and Qj = {good}. In all, we follow the following steps for designing the fuzzy chart as follows: 1. Our aim is to control process by fuzzy chart, so we describe the deviation of the sample means as linguistic variable terms. 2. Gathering data in rational subgroups. 3. Estimating the process mean and dispersion.

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4. Defining proper linguistic variables and associated terms for Q.Ch. subgroup mean, overall quality of the product and overall quality of subgroup. 5. Defining proper membership functions for each linguistic variable term defined above. 6. Defining base rule that must be complete, consistent, and adaptive. 7. Defining inference engine. 8. Defining the defuzzification method and setting crisp value to the overall quality fuzzy set. 9. Classifying the overall quality with respect to the crisp value and associated cut off points. 10. Running and improving rule base by reducing fuzzy errors. In this paper, and for our example, we have chosen followings: Quality linguistic variable as “a shift in the process mean” with associated terms “zero ” and subgroup overall quality with associated term “good” and overall quality product with relative terms “excellent, very good, good, bad” with associated membership functions that define later in this paper. We have used only one linguistic term for the subgroup sample and overall quality of subgroup, because we want to control sample mean rather than to classify it. We will use following rules, for controlling the process mean, because we don’t want to classify the overall quality of the product and we just want to control the process mean. If x¯i is zero then x¯i is good so the process is in control (Table I). This forms our base rule. For defining the initial membership functions of the fuzzy sets utilized in the premise part of the fuzzy rules, we classify the ith sample observation into quality categories base on following rules (Table II): Table I. Base rule for controlling process mean Rule

X¯ is

1

Zero

Then µzero (x¯i )

Overall quality is µGood (x¯i )

Good

Table II. Base rule for controlling variability

Rules

If

1 2 3

µquality (xij ) ≥ µquality (µˆ ± σˆ ) µquality (µˆ ± σˆ ) > µquality (xij ) ≥ µquality (µˆ ± 2σˆ ) µquality (µˆ ± 2σˆ ) > µquality (xij ) ≥ µquality (µˆ ± 3σˆ )

then

Observation quality category Good Normal Not bad

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If µquality (xij ) ≥ µquality (µˆ ± σˆ ) then xij is good. If µquality (µˆ ± σˆ ) > µquality (xij ) ≥ µquality (µˆ ± 2σˆ ) then xij is normal. If µquality (µˆ ± 2σˆ ) > µquality (xij ) ≥ µquality (µˆ ± 3σˆ ) then xij is not bad. Else xij is bad and a shift in the process has occurred. where, xij is the jth sample observation ith subgroups. Consider that each quality team can have its own base rule for fuzzy charts that covers the reality and process. Then we will classify our observations into one of the final overall quality categories based on their membership values. After creating base rule, we should define the membership functions for each quality linguistic term. This membership function can be any function that corresponds to the product and reality (Driankov et al., 1993). Also we can get help from the sample histogram. In this paper, we have used the Gaussian membership function for our example, Since the data follows normal distribution. An expert team who is familiar with the product to be classified determines the number of rules and the number and type of membership functions associated with each linguistic Q.Ch. In this paper, we have used Gaussian type of membership functions as shown in Figure 1. Since the normal distributions are utilized to determine the specifications of Q.Ch. and our example data follows the normal distribution, the following Gaussian membership function is the appropriate membership function to describe linguistic terms:  2   x¯i − µˆ  µZero (x¯i ) = exp − ,   δˆµˆ

x¯i − τ 2 µGood (x¯i ) = exp − , δµ

xj i − µˆ 2 . µquality (xij ) = exp − δˆ Determining the initial memberships, the overall quality of the product can be described by fuzzy if-then rules. Consider following observation form below of a sample of size n:

µzero (x¯i ) µGood (x¯i ) ¯ (Xi = x¯i ) => , Zero Good for which each membership is selected by membership functions. Using base rule, we will obtain following fuzzy set that describes the overall quality of subgroup mean

379

FUZZY CONTROL CHART

Figure 1. Trangular and gaussian membership functions for linguistic term zero.

µ∗Good (x¯i ) , Q(x¯i ) = good

where µ∗Good (x¯i ) = min{µzero (x¯i ), µgood (x¯i )}. For each observation in the subgroup, we obtain quality membership by which we will classify the sample through

xj i − µˆ 2 µquality (xij ) = exp − δˆ so that to get the following table for each subgroup: Sample

1

2

...

N

number I

Sample

Good

Normal

Not bad

Bad

F1

F2

F3

F4

mean Xi1

Xi2

...

Xin

X¯ i

Thus, if the process mean has shifted, this cause decrease the membership value linguistic term zero and influence µ∗Good (x¯i ) of the sample. 3. Designing the Fuzzy Chart Designing a fuzzy chart to control the mean process, the following five steps should be taken: 1. Selecting the amount of minimum shift in the process mean that is important and must be detected. This is called θ. 2. Selecting the warning line (WL). If a shift at least equal to the θ is occurred the chart will alarm quickly using WL. 3. Creating a base rule for warning line alarms. 4. Calculating the false alarm rate and average run lengths.

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5. Choosing the best rule that produce minimum false alarm rate and adequate average run length. Then, we plot di = 1 − µ∗Good (x¯i ) in a chart, which has the upper control limit equal to UCL= 1 − µ∗Good (µˆ + 3σˆ µˆ ) where σˆ µˆ is the estimation of mean process variability and a WL. In the next section, we describe how to set WL. As it is proved that a fuzzy control chart of process mean is insensitive to changes in dispersion, the derived warnings will be purely due to shift in mean. For controlling the variability, we can use the Pearson goodness of fit statistic with the upper control limit at appropriate quantile χα2 (2). Now if the fuzzy chart for the process mean doesn’t show an out of control signal but the fuzzy chart for the variability shows an out of control signal, we may conclude that the variability is out of control. Otherwise, the signal in the second chart may be just for the shift in the mean which has detected by the first chart. 3.1. setting the warning line For calculating the WL we used the training data generated from normal distribution. For this purpose, consider that we want to detect a shift in the process at least equal to θ by a sample of size n. So, when the process is in control the sample has a normal distribution with mean µ, but when there is a shift in the mean (process is out of control) the sample has a normal distribution with mean µ + θ . Now, we follow the algorithm as below as follows: 1. We generate m samples of size n from out of control process and then calculate the sample mean. Our experiences suggest that m = 100,000 is a sufficient value for calculating the WL. 2. We calculate the following statistic for ith sample mean. di = 1 − µ∗Good (x¯i ),

i = 1, 2, . . ., 100,000.

3. We calculate the mean of above memberships. So the warning line is: WL= di/m. As a matter of fact WL is our expectation of chart statistic, when there is a shift in the process at specified level. 3.2. setting the base rule for warning rule Above WL is used for detecting a shift in the process mean at least equal to θ . The UCL is set like the Shewhart chart. If a point draws out of UCL, the process is out of control. If some points draw out of WL successively,

FUZZY CONTROL CHART

381

a shift at least equal to θ is detected. So, we need some rules. We have recommended here two rules as below: 1. If two successive points draw out of WL (and still are below UCL), a shift at least equal to θ is occurred. 2. If three successive points draw out of WL (and still are below UCL), a shift at least equal to θ is occurred. So (if we used rule 1), we plot the sample mean membership and if it draws out of WL, but still below UCL, we take into account the next sample. If it draws out of WL, so we may conclude that process is out of control and a shift at least equal to the specified level has occurred. 3.3. setting the false alarm rate (α) According to the above rules, the false alarm rates are called α(2), α(3) and have the same concepts in Shewhart chart. α(2): The proportion of plotting two successive points out of WL or one point out of UCL, when the process is in control. α(3): The proportion of plotting three successive points out of WL, or one point out of UCL, when the process is in control. So we can use one of the above rules that has minimum false alarm rate. Consider that: α(2) = P (di+1 >U CL |θ=◦ )+P (W L < di+1 < U CL|W L < di U CL |θ = ◦ ) + P (W L < di+2 < U CL|W L < di+1 < U CL|W L < di < U CL|θ = ◦). 3.4. setting the average run lengths The ARL base on the two above rules are called ARL (2), ARL (3) and have the same concept in Shewhart chart. We wrote a computer program for calculating the above terms with the SAS software and it is available upon request. The user can choose the proper base rule that has minimum false alarm rate and average run length. We will illustrate above concepts through an illustrative example. 4. Example and Power Test A forging process produces piston rings for an automotive engine. We wish to establish statistical control of the inside diameter of the rings manufactured by this process using fuzzy quality control. Twenty-five samples, each of size five, have been taken when we think the process is in control. The quality proportions for products estimated as: πexcellent = 0.68,

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πveryGood = 0.288, and πGood = 0.032 and the specified limits are: usl= 74.05, and τ = 74.00, and lsl= 73.95. After data analysis the quality teams found the proper membership functions and have used the Gaussian membership functions as follows:

x¯i − 74.001 2 µZero (x¯i ) = exp − . 0.0044 And the quality team defined following membership functions for the overall quality Linguistic variable term good for the process mean as µGood (x¯i ) = exp −(x¯i − 74/0.0075)2 So,

x¯i − 74 2 x¯i − 74.001 2 ∗ , exp − . µ (x¯i ) = MIN exp − 0.0044 0.0075 Then we set the warning lines and false alarm rates and average run lengths as below: θ

WL

α(2)

α(3)

ARL(2)

ARL(3)

0.010 0.011 0.012 0.013 0.014 0.015

0.8594 0.8950 0.9264 0.9502 0.9674 0.9794

0.030 0.020 0.014 0.0098 0.0071 0.0055

0.0076 0.0057 0.0045 0.0040 0.0037 0.0035

2.60 2.31 2.06 1.85 1.70 1.52

3.75 3.18 2.73 2.37 2.06 1.81

Please consider that the fuzzy chart present a way to detect shifts so quickly, since the ARL is really small and also false alarm rates. So base on above table the quality manager can choose the best strategy. We have used the WL= 0.9264 for detecting a shift at least equal to 0.012. Consider following new samples shown in Table III. The upper control limit for the fuzzy chart d is 0.999 and the quality teams have used WL for detecting a shift in the mean at least equal to the 0.012 as soon as possible. The upper control limit for the fuzzy dispersion chart is 11.83, which is an appropriate 2 quantile χα=0.0027 (2). As you see, the fuzzy chart d shows an out of control signals in subgroups that represent probably a shift in the mean around the time that sample 34 or 35 was taken. The average of the sample means of 34 through 40 is 74.015 mm. So a shift at least equal to 0.014 has occurred. The out of control signal in the χ 2 chart is because of the shift in the mean. Now we want to compare this new method with Shewhart type. The Shewhart chart has the fixed type one error but the new method has different type one error rates associated with various WL.

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FUZZY CONTROL CHART

Table III. New samples Sample 1 number

2

3

4

5

Sample di mean

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

74.002 73.992 73.999 74.010 74.000 74.003 74.002 74.004 74.000 74.005 73.990 74.020 74.010 74.013 74.005

74.019 74.001 73.985 74.003 74.001 74.015 74.018 73.990 74.016 74.000 73.995 74.024 74.012 74.036 74.029

73.992 74.011 74.000 73.991 73.986 74.020 73.995 73.996 74.025 74.016 74.010 74.005 74.015 74.025 74.000

74.008 74.004 73.990 74.006 73.997 74.004 74.005 73.998 74.000 74.012 74.024 74.019 74.026 74.026 74.020

74.010 74.001 73.992 74.004 73.997 74.007 74.006 73.998 74.011 74.013 74.004 74.017 74.020 74.023 74.015

∗ ∗∗

74.03 73.995 73.987 74.008 74.003 73.994 74.008 74.001 74.015 74.030 74.001 74.015 74.035 74.017 74.001

0.983953∗ 0.0176207 0.983953∗ 0.3681712 0.5579106 0.8406333 0.7206752 0.3681712 0.9939125∗ 0.999355∗∗ 0.3681712 0.9999979∗∗ 1∗∗ 1∗∗ 0.9999546∗∗

Excel- Very Good Bad χ 2 lent good (3.4) (1.44) (0.16) (0) (11.83) 3 4 2 5 4 3 4 4 2 2 3 1 1 0 3

1 1 3 0 1 2 1 1 3 2 1 3 2 2 1

1 0 0 0 0 0 0 0 0 1 1 1 1 2 1

0 0 0 0 0 0 0 0 0 0 0 0 1 1 0

4.59 0.40 2.43 2.35 0.40 0.42 0.40 0.40 2.43 5.20 4.59 7.79 7.32 25.8∗ 4.59

A warning has received since the point draws out of WL. Process is out of control.

So, the first step for having a good comparison is that two schemes have the same false alarm rates. As the Shewhart charts have the amount 0.0027, so we looking for fuzzy chart that has nearly above error rates. In Table IV we √ designed some fuzzy charts for detecting specified shift equal to θ = Kσ/ n. So, we choose the fuzzy chart based on K = 1.3. And now, we make a comparison between following two charts as below: (a) A fuzzy chart with parameters WL= 0.9654 and UCL= 0.9998, n = 5 and we used the α(3) = 0.0036 (b) Shewhart chart. Remember that this fuzzy chart will not be appropriate for detecting shifts less that K = 1.3. Table V shows the ARL values for two charts when there is a shift with K, in the process mean. We realize that the fuzzy chart shows better power with respect to the Shewhart type, however, when the k increases the difference between two schemes are not much. Remember that we assumed that√we want to detect a shift in the process mean at least equal to the 1.3σ/ 5, as we see the fuzzy chart is more powerful than Shewhart chart. The advantage of this new method is that we can control the process for detecting a specified level of shifts in the process.

384


Table IV. Calculated WL K

WL

α(2)

α(3)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0.4321 0.4600 0.5031 0.5577 0.6193 0.6832 0.7450 0.8015 0.8503 0.8909 0.9232 0.9476 0.9654 0.9780 0.9864 0.9919 0.9953 0.9973 0.9986 0.9992

0.2031 0.1867 0.1624 0.1352 0.1076 0.0819 0.0606 0.0432 0.0306 0.0213 0.0146 0.0103 0.0075 0.0057 0.0046 0.0041 0.0037 0.0037 0.0037 0.0037

0.0942 0.0828 0.0672 0.0512 0.0369 0.0257 0.0174 0.0117 0.0079 0.0058 0.0047 0.0040 0.0036 0.0036 0.0036 0.0036 0.0036 0.0036 0.0036 0.0036

Table V. ARL values K

ARL (3)

ARL(Shewhart)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

206.16 128.61 69.30 40.58 23.21 13.91 9.34 6.22 4.60 3.60 2.80 2.20 1.70 1.50 1.40

297.62 176.37 99.80 55.83 33.28 20.64 13.09 8.84 6.11 4.47 3.39 2.67 2.15 1.81 1.57

FUZZY CONTROL CHART

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5. Conclusion In this paper we introduced a fuzzy chart for controlling the process mean. Also we classified the observations in the rational groups and that helps to the inspector that has a neural view of the shifts in the process mean and also can control the dispersion by this classification. Also as many inspectors want to control the process in a specific level of shifts in the process mean, so we designed a fuzzy chart that has a warning line besides upper control limit. The control limit, UCL, controls the process in all. The warning line is designed for detecting desired shift in the process that is important to the company. As Aparisi (1997) mentioned, we should not forget that the importance of a process shift depends on the process capability. If a process is very capable, small process shifts hardly influence the amount of nonconforming items. If we have to choose a plan for these very capable processes we should select the one able to quickly detect large process shifts. On the other hand, for a process of small capability even a small shift can produce a large amount of nonconforming items. Therefore, we should be quick to detect these small shifts. Thus, we need to clarify which shift sizes are “important” for control purposes. Also we calculate the false alarm rates and warning lines besides average run length values. This fuzzy chart shows the best value of ARL for detecting the specified shift. Also we made a comparison between fuzzy chart and Shewhart chart. And we showed that this new method has better power to detect shifts. References Aparisi, F. (1997). Sampling plans for the multivariate T 2 control chart. Quality Engineering 10(1). Driankov, D., Hellendoorn, H. & Reinfrank, M. (1993). An Introduction to Fuzzy Control. Berlin, Heidelberg, New York: Springer. Wang, J. H. & Raz, T. (1990). On the construction of control charts using linguistic variables. International Journal of Production Research. 28(3): 477–487. Yager, R. R. & Filer, D. P. (1994). Essentials of Fuzzy Modeling and Control. New York: A Willey Interscience Publication. Zimmermann, H. J. (1996). Fuzzy Set Theory and its Applications, 3rd edn. New York: Kluwer Academic Publishers.