A fuzzy logic approach for developing variables control charts and process capability indices under linguistic measurements

(1)

A FUZZY-LOGIC APPROACH FOR DEVELOPING VARIABLES

CONTROL CHARTS AND PROCESS CAPABILITY INDICES

UNDER LINGUISTIC MEASUREMENTS

Abbas Al-Refaie, Areen Obaidat

1

_{, Rami H. Fouad}

1

_{, and Bassel Hanayneh}

2

1

Department of Industrial Engineering, University of Jordan, Amman, Jordan 2Department of Civil Engineering, University of Jordan, Amman, Jordan

E-Mail: [email protected]

ABSTRACT

In the traditional variables control charts, the sample observations are characterized by numerical values. In practice, the uncertainty that comes from the measurement system; including operators, gauges, and environmental conditions, results in linguistic data and thereby fuzzy control charts. In this situation, fuzzy set theory is a useful tool to handle this ambiguity. Therefore, this research develops variables control charts for monitoring process mean and variability under linguistic data using fuzzy logic. In this research, then each observation is represented by a triangular membership function. Then, the comprehensive output measure (COM) is obtained for each sample replicate using fuzzy logic. Finally, the COM values of sample replicates are utilized to establish the appropriate variable control chart. Similarly, each process capability index is represented by a suitable membership function and then estimated using fuzzy logic to assess process capability. This approach was implemented on three case studies; in all of which the developed control charts and estimated process capability were found efficient in monitoring of process condition and assessing its performance. Moreover, the simplicity and ease of interpretation can make this approach be widely used by practitioners. In conclusions, the developed variables control charts and process capability indices may provide a beneficial guide for practitioners in monitoring process parameters and its performance in a wide range of manufacturing applicationsunder linguistic data.

Keywords: fuzzy logic, control charts, process capability, linguistic measurements.

1. INTRODUCTION

In reality, every process performance needs to be measured and evaluated. Statistical process control (SPC) is a powerful collection of problem solving techniques useful in achieving process stability and improving capability through the reduction of variability [1]. Control charts, one of the famous SPC tools, are widely used for process monitoring in manufacturing industry [2]. They provide a graphical depiction of sample data points that are used to control the ongoing process, predict the expected range of quality characteristics from a process, and to determine whether or not the process is in statistical control, by analyzing the patterns of process variation causes by assignable causes. Generally, a control chart, as shown in Figure-1, consists of three parameters, upper control limit (UCL), lower control limit (LCL), and center line (CL).

Figure-1. A schematic of control chart.

When dealing with a quality characteristic that is variable, it is usually essential to monitor both the mean value of the quality characteristic and its variability. Variable control charts are used to monitor and evaluate the performance of a continuous (variable) quality characteristic of a product. The process mean is monitored with the control chart for means; or so called

x

-control chart. Whereas, the process variability is monitored with either a control chart for a standard deviation called (

s

-control chart) or a -control chart for the Range called (

R

-control chart). When the sample size of a quality characteristic is one, the individual and moving range ( I-MR) charts are used.

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different procedures of constructing control charts for linguistic data, based on fuzzy and probability theory. Three sets of membership functions, with different degrees of fuzziness were employed. Then, a comparison between fuzzy and probability approaches based on the average run length and samples under control was conducted on real data. Cheng [5]studied the construction of fuzzy control charts for a defined process by using the fuzzy process control methodology and use of possibility theory. Gulbay and Kahraman [6] proposed fuzzy control charts by using the probability of fuzzy events, and α-cut concept to determine the tightness of the inspection. The direct fuzzy approach was used. Senturk and Erginel [3] introduced the framework of fuzzy

x



R

and

x



s

control charts with



-cuts. The traditional

x



R

and

x



s

control charts by transforming the numeric control limits into fuzzy control limits using triangular membership function. Faraza and Shapiro [7] constructed the fuzzy

x



s

2

control charts. The proposed control chart avoided defuzzification methods; such as, fuzzy mean, fuzzy mode, fuzzy midrange and fuzzy median, by using the fuzzy Random variables. Tannock [8] developed fuzzy control chart for individuals. Khademi and Amirzadeh [9] developed two alternative approaches to

x



R

control chart for monitoring the sample averages and ranges based on fuzzy mode and fuzzy rules methods, when the measures are expressed by non-symmetric triangular fuzzy numbers. Gildeh and Shafiee [10]studied the construction of the

I



MR

control chart for the autocorrelated fuzzy readings. The variance, covariance, and autocorrelation coefficient were calculated by using the distance between fuzzy numbers approach. Then, the control limits were calculated by the use of the autocorrelation coefficient. However, the above-mentioned research developed control charts of fuzzy parameters which make it difficult to assess a process condition and measure its capability. Moreover, the complexity of those approaches hinders it is usage by practitioners.

On the other hand, process capability analysis is usually conducted to assess the performance of manufacturing process [11]. The Cp and Cpk indices are

widely used to assess process capability. Nevertheless, the existence of uncertainty in measurement data results in vague values of process indices and thereby provides confusing conclusions. As a result, developing proper process capability indices to deal with such situations is a real challenge to process engineers. Several studies were, therefore, developed fuzzy capability indices [12-15].

The fuzzy logic principle is widely used to handle vague and uncertain information. Two common types of fuzzy systems are used; Takagi-Sugeno (T-S) and Mamdani fuzzy systems. Mamdani fuzzy systems are special cases of T-S fuzzy systems, which involve mathematical expressions that contain a linear function [11]. The functions of fuzzy logic consist of the fuzzification, fuzzy rule evaluation, membership function of the output and setting fuzzy rules that must contain input variables and rules to be used to compute a comprehensive output, and defuzzification that transforms fuzzy values into a comprehensive output [16-18]. This research, therefore, proposes an effective approach for developing variables control charts and process capability indices utilizing fuzzy logic technique [16-18]. The remaining of this research is outlined in the following sequence. Section two presents the proposed approach for variables control charts. Section three provides illustrative case studies. Section four discusses research results. Conclusions are finally summarized in section five.

2. METHODOLOGY

2.1 Developing variable control charts

The proposed approach for developing variable control charts using linguistic variable is outlined as follows:

Step 1: For a quality characteristic of interest, assume N replicates are taken in each sample. Let

x

_ijk

denotes the kth reading of jth replicate at sample i of a quality characteristic where,

i



1,...

m

,j=1, …, N, and

1,...

k



K

. The replicate observations of each replicate are displayed as shown in Table-1.

Table-1. Arrangement of replicate observations.

Replicate j

Sample i Rep. 1 Rep. 2 … Rep. N

1

x

111

...

x

11K

x

121

...

x

12K …

x

1 1N

...

x

1NK

2

x

₂₁₁

...

x

₂₁_K

x

₂₂₂

...

x

₂₂_K …

x

₂_N₂

...

x

₂_NK

⁞ ⁞ ⁞ ⁞

m

x

_m₁₁

...

x

_{m K}₁

x

_m₂₂

...

x

_{m K}₂ …

x

_mN₁

...

x

_mNK

Step 2: Calculate the average,

x

ij, of the K

(3)

1

_,

ij

K ijk k

x

K

i

j





 

(1)

Step 3: Let

z

ijk denotes the normalized value of

ijk

x

. Normalize the

x

_ijkbetween 0 and 1 using

min

,

max

min

ijk ijk ijk

ijk ijk

x

z

i

j

k

x





  



(2)

where min

x

_ijkand max

x

_ijkare the smallest and largest observations from all replicates, respectively.

Step 4: Use fuzzy logic to convert the fuzzy data into a crisp then construct the variable control charts. Adopt Madani-style fuzzy logic, in which the inputs and output membership functions (MFs) are linear. Its input variables are zijk values, whereas the Comprehensive output (COMi) values are the output. The control charts are then established using fuzzy logic as follows:

Step 4-1: Fuzzification of the inputs

Define the membership function (MF) for each input that is represented from each replicate as

z

_i_jkas shown in Figure-2. The three fuzzy MFs low, middle and high are assigned for each input

z

_i_jk of each of Replicate 1 (Rep. 1) to Replicate N(Rep. N).

Figure-2. The MFs of

z

_i_jkin each replicate.

Step 4-2: Rule evaluation

Set the rules that communicate between the inputs and the output. The fuzzy rule base consists of a group of fuzzy n inputs and one output in the form of (Low (L), Low+ (L+), Middle- (M-) Middle (M), Middle+ (M+), High- (H-) and High (H)). Table-2 displays the fuzzy rules. The rule examples include:

If

z

111is L,

z

112 is L,

z

113 is L, then the COM11 is L.

If

z

111is L,

z

112 is L,

z

113 is M, then the COM11 is L+.

If

z

111is L,

z

112 is M,

z

113 is M, then the COM11 is M-.

If

z

111is M,

z

112 is M,

z

113 is M, then the COM11 is M.

If

z

111_is_M,

z

112_is_M_,

z

113_is_H,_{then the}_COM

11 is M+.

If

z

111_is_M,

z

112_is_H_,

z

11 is H-.

If

z

111_is_H,

z

112_is_H_and

z

11 is H.

Table-2. Generated fuzzy rules for zijk.

1

ij

z z_ij₂ z_ij₃ _COM_ij

LOW LOW LOW LOW

LOW LOW MIDDLE LOW +

LOW MIDDLE LOW LOW +

MIDDLE LOW LOW MIDDLE-

MIDDLE MIDDLE LOW MIDDLE-

MIDDLE LOW MIDDLE MIDDLE-

LOW MIDDLE MIDDLE MIDDLE-

MIDDLE MIDDLE MIDDLE MIDDLE

MIDDLE MIDDLE HIGH MIDDLE+

HIGH HIGH MIDDLE MIDDLE+

MIDDLE HIGH MIDDLE MIDDLE+

HIGH HIGH MIDDLE HIGH-

MIDDLE HIGH HIGH HIGH-

HIGH MIDDLE HIGH HIGH-

HIGH HIGH HIGH HIGH

Applying the rules shown in Table-3 to fuzzy values yields the following results:

low



(input 1)^



_low(input 2) ^



_low(input3)=



_low (output).

low



(input 1)^



_low(input 2) ^



_middle(input3)=



_low_ (output).

low



(input 1)^



_middle(input 2) ^



_middle(input3)=

middle



_(output).

middle



(input 1)^



_middle(input3)=

middle



(output).

middle



(input 1)^



_High(input3)=

middle



_(output).

middle



(input 1)^



_High(input 2) ^



_High(input3)=



_High_ (output).

High



(input 1)^



_High(input 2) ^



_High(input3)=



_High (output).

Step 4-3: Aggregation of the rule outputs.

(4)

the max-min composition operation. The MFs of the

COMij of the fuzzy reasoning can be expressed as shown in Figure-3.

Figure-3. The MFs of COMij.

Step 4-4: Defuzzification of fuzzy value for the output.

Defuzzification is the opposite operation of fuzzification; it is used to convert the fuzzy inference output into a non-fuzzy value. The transformation is carried out using the center of gravity method (COG). The COMij value is calculated using Equation. (3) for each

sample i as displayed in Figure-4.

0

( ).

( )

i j

C F FdF

COM

C F dF







(3)

where



C

₀is the fuzzy inference output, F is the area under the trimmed output.

Figure-4. Defuzzification using COG method.

Step 4: Tabulate the defuzzified COMij values as shown in Table-3.

Table-3. Comprehensive output measurements (COMij).

Sample i COM_i₁ COM_i₂ ……… COM_in

1 COM11 COM12 ……… COM1n

2 COM21 COM22 ……… COM2n

m COM_m₁ COM_m₂ ……… COM_mn

Step 5: Calculate the corresponding actual measurements,

x

ij, for the COMij listed in Table-4 using Equation. (4).

(max

min

) min

ij ij ijk ij k ij k

x

 

COM

x



x



x

(4)

Step6: Construct the appropriate variable control charts of collected data of m samples each of a sample size

of n observations,

x

ij



_{, the variable control charts are} developed as follows:

(i) The

x



R

control charts are used when the sample size, n, is moderate or small (n=3⁓5); where the

x

control chart detects the shift in a process mean and R

chart monitors the variability. Suppose that a quality characteristic is normally distributed with mean, µ, and

standard deviation,



, where both



and



are

unknown. The average (

x

_i



) and the range (

R

_i ) of this

sample are calculated as follows, respectively:

1 N

ij j i

x

N 

  



(5)

max

min

i i i

R



x





x



(6)

Calculate the grand average

( )

x

, which equals

(5)

1 m i i x x m   



(7)

Similarly, calculate the average range

_{( )}

_R

using Equation. (8). 1 m i i R R m  



(8)

Then the estimated process standard deviation

ˆ

( )



is calculated as follows:

2

ˆ

R

d





(9)

where d2 is a constant that depends on the sample

size. Finally, establish the parameters of the

x



R

control charts as follows:

(a) For

x



control chart, the parameters are estimated as:

2

3

x

UCL

x

R

d

n



 

(10)

x

CL

_



x

(11)

2 3 x

LCL x R

d n

  (12)

(b) For the

R

control chart, the parameters are calculated as follows:

3 2 3

R

UCL R d

d

  (13)

R

CL



R

(14)

3 2

3

R

LCL R d

d

  (15)

where

d

3is a constant depends on sample size.

(ii)The

x



s

control charts are used when the sample size (n) is moderately large (n>10) or when (n) is variable. The



ˆ

and



ˆ

values are calculated for the

x s



control charts as follows:

ˆ

x





₍₁₆₎

1 4 4 ˆ m i i s m c c

s

   



(17)

where si and

s

are the sample standard deviation and the

average standard deviation, respectively. Mathematically,

2 ( ) 1 1 i i N N x x i s     

 ₍₁₈₎

1 i

m i s

m s  

(19)

Then, the parameters of

x



s

control charts are estimated as follows:

(a) For the

x

chart, the parameters are calculated as follows: 4 3 x s UCL x c n

   (20)

x

CL

_



x

(21)

4 3 x s LCL x c n

  (22)

(b) For the

s

chart, the parameters are calculated as follows: 2 4 4 3 1 s s

UCL s c

c

   (23)

s

CL



s

(24) 2 4 4 3 1 s s

LCL s c

c

   (25)

where

c

₄ is a constant that depends on sample size (n).

(iii) The Individual-Moving Range (I-MR) control charts are employed when the sample size equals one (n=1). The moving range, MRi, for sample i is defined as:

1

i i i

MR



x





x



_

(6)

1 2 2 ˆ m i i MR MR _m d d



  



(28)

Then, the parameters of the

I



MR

control charts are estimated as follows:

(a) The parameters of the I-control chart

2 3 I MR UCL x d    (29) I

CL



x



(30) 2

3

I

MR

LCL

x

d



 

(31)

(b) For

MR

chart, the parameters are calculated as follows:

4

MR

UCL



MR



D MR

(32)

MR

CL



MR

(33)

3

MR

LCL



MR



D MR

(34)

where d2, D3, and D4 are constants that

correspond to n equals two.

2.2 Estimating the process capability indices

Suppose that a fuzzy process with fixed



and , for which the product’s upper and lower specification limits are defined by TFNs; that is,

1 2 3

(

,

)

USL



USL USL USL

and

1 2 3

(

,

)

LSL



LSL LSL LSL

. This results in fuzzy

process capability indices,

C

_pand

C

_pk, which measure potential and actual capability, respectively. Then, the proposed approach that will be used to assess process performance is outlined in the following steps:

Step 1: Calculate the estimated process capability indices, using the fuzzy specification limits

(

USL LSL

,

)

and crisp mean



ˆ

, standard deviation



ˆ

as follows:

1 3 2 2 3 1

1 2 3

ˆ ˆ ˆ

( , , ) ( , , )

ˆ ˆ ˆ ˆ

6 6 6 6

p p p p

USL LSL USL LSL USL LSL

USL LSL

C C C C

   



  

(35)

3 2 1

1 2 3

ˆ ˆ ˆ

ˆ _ˆ _ˆ _ˆ

( , , ) ( , , )

ˆ ˆ ˆ ˆ

3 3 3 3

pl pl pl pl

LSL LSL LSL

LSL

C     C C C

   

  



   (36)

3

1 2

1 2 3

ˆ

ˆ ˆ

ˆ _ˆ _ˆ _ˆ

( , , ) ( , , )

ˆ ˆ ˆ ˆ

3 3 3 3

pu pu pu pu

USL

USL USL

USL

C     C C C

   



 



   (37)

1 2 3

ˆ

min(

,

)

(

,

)

pk pu pl pk pk pk

C



C



C

(38)

Step 2: Use Mamdani-style fuzzy logic to convert fuzzy indices into crisp indices as follows:

A. Fuzzification of the inputs: Define the MFs for each input of

C

_p

and

C

_pkvaluesas illustrated in Figure-5.

Figure-5. The MFs for

C

_pand

C

_pk.

B. Rule evaluation: Set the rules that communicate between the inputs and the output. The fuzzy

rule base consists of a group of fuzzy three inputs and one output in the form of (Poor, Inadequate, Capable, Satisfactory, Excellent, and Super- Excellent). The fuzzy rules are set to be fifteen rules as shown in Table-4.

Table-4. Generated fuzzy rules for

C

ˆ

_pand

C

ˆ

_pk.

1

,

1

p pk

C

_p₂

,

C

_pk₂

C

_p₃

,

C

_pk₃ COM

Low Low Low Poor

Middle Inadequate

Middle Low

Middle Capable

Low Middle

Middle Middle

High Satisfactory

High Middle

(7)

High Excellent

Middle High High

High Middle

High Super

Excellent

C. Aggregation of the rule outputs: The MFs of the COM value of the fuzzy reasoning is expressed as

shown in Figure-6. Figure-6. The MFs for the output

C C

ˆ ˆ

p

,

pk_.

D. Defuzzification of fuzzy value for the output. The transformation is carried out using the COG. The COM values are calculated as shown in Figure-7.

Figure-7. Defuzzification using COG method for

C C

_p

,

_pk.

3. ILLUSTRATIVE CASE STUDIES

3.1 Case Study: Monitoring the piston diameter ring

Kaya and Kahraman [15] developed the fuzzy set theory to calculate fuzzy process capability indices and construct

x



R

control charts. The quality characteristic of interest was the inside piston diameter ring. Twenty-five samples were collected each with Twenty-five replicates. Linguistic variable observations were measured as shown in Table 6. The proposed approach was implemented and is outlined as follows. The replicate observations are ranked from smallest to largest. The averages of observations in each replicate are calculated for all

samples and then tabulated as shown in Table-5. The fuzzy data is then normalized between 0 and 1 by using the formula as shown in Table-6. The COMij values are calculated by using the Mamdani-style fuzzy for all samples. The COG defuzzification method is used to convert the fuzzy value of the COMij to a crisp value. The fuzzy rules used for computing the COMij value are shown in Figure-7. Table-8 displays the fuzzy logic results for

1

i

(8)

Table-5. The inside diameter ring measurements for case study (1).

Sample i x_i₁₁ x_i₁₂

21

i

x x_i₂₂

31

i

x x_i₃₂

41

i

x x_i₄₂ x_i₅₁

52

i

x

1 74.002 74.003 74.001 74.002 74.003 74.004 73.985 73.986 73.996 73.997 2 74.006 74.007 73.993 73.994 74.016 74.017 73.999 74.000 74.017 74.018 3 74.008 74.009 74.007 74.008 73.996 73.997 74.017 74.018 74.016 74.017 4 73.990 73.991 74.013 74.014 73.991 73.992 74.018 74.019 73.995 73.996 5 74.014 74.015 73.987 73.988 74.011 74.012 74.001 74.002 73.992 73.993 6 73.986 73.987 73.996 73.997 73.985 73.986 73.998 73.999 73.996 73.997 7 73.998 73.999 74.003 74.004 73.987 73.988 74.000 74.001 74.012 74.013 8 74.004 74.005 74.021 74.022 73.996 73.997 74.018 74.019 74.009 74.010 9 73.981 73.982 74.002 74.003 73.996 73.997 74.012 74.013 74.007 74.008 10 73.991 73.992 73.989 73.990 74.009 74.010 74.000 74.001 74.002 74.003

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

20 74.009 74.010 73.991 73.992 73.999 74.000 74.005 74.006 73.985 73.986 21 74.007 74.008 74.001 74.002 73.990 73.991 73.998 73.999 73.988 73.989 22 73.989 73.990 74.000 74.001 74.013 74.014 74.013 74.014 74.009 74.010 23 74.013 74.014 73.993 73.994 73.994 73.995 73.987 73.988 73.997 73.998 24 74.003 74.004 74.008 74.009 73.992 73.993 74.006 74.007 73.999 74.000 25 73.988 73.989 74.012 74.013 74.002 74.003 73.994 73.995 74.010 74.011

Table-6. The calculated observation averages for case study (1).

Sample

i

Rep.1

1min i

x x_i₁ x_i_1max

Rep. 2

2 min

i

x x_i₂ x_i_{2 max}

Rep.3

3 min

i

x x_i₃ x_i_{3 max}

Rep.4

4 max

i

x x_i₄ x_{4 min} i

Rep.5

5 min

i

x x_i₅ x_i_{5 max}

1 74.002 74.0025 74.003 74.001 74.0015 74.002 74.003 74.0035 74.004 73.985 73.9855 73.986 73.996 73.9965 73.997

2 74.006 74.0065 74.007 73.993 73.9935 73.994 74.016 74.0165 74.017 73.999 73.9995 74.000 74.017 74.0175 74.018

3 74.008 74.0085 74.009 74.007 74.0075 74.008 73.996 73.9965 73.997 74.017 74.0175 74.018 74.016 74.0165 74.017

4 73.990 73.9905 73.991 74.013 74.0135 74.014 73.991 73.9915 73.992 74.018 74.0185 74.019 73.995 73.9955 73.996

5 74.014 74.0145 74.015 73.987 73.9875 73.988 74.011 74.0115 74.012 74.001 74.0015 74.002 73.992 73.9925 73.993

6 73.986 73.9865 73.987 73.996 73.9965 73.997 73.985 73.9855 73.986 73.998 73.9985 73.999 73.996 73.9965 73.997

7 73.998 73.9985 73.999 74.003 74.0035 74.004 73.987 73.9875 73.988 74.000 74.0005 74.001 74.012 74.0125 74.013

8 74.004 74.0045 74.005 74.021 74.0215 74.022 73.996 73.9965 73.997 74.018 74.0185 74.019 74.009 74.0095 74.010

9 73.981 73.9815 73.982 74.002 74.0025 74.003 73.996 73.9965 73.997 74.012 74.0125 74.013 74.007 74.0075 74.008

10 73.991 73.9915 73.992 73.989 73.9895 73.990 74.009 74.0095 74.010 74.000 74.0005 74.001 74.002 74.0025 74.003

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

20 74.009 74.0095 74.010 73.991 73.9915 73.992 73.999 73.9995 74.000 74.005 74.0055 74.006 73.985 73.9855 73.986

21 74.007 74.0075 74.008 74.001 74.0015 74.002 73.990 73.9905 73.991 73.998 73.9985 73.999 73.988 73.9885 73.989

22 73.989 73.9895 73.990 74.000 74.0005 74.001 74.013 74.0135 74.014 74.013 74.0135 74.014 74.009 74.0095 74.010

23 74.013 74.0135 74.014 73.993 73.9935 73.994 73.994 73.9945 73.995 73.987 73.9875 73.988 73.997 73.9975 73.998

24 74.003 74.0035 74.004 74.008 74.0085 74.009 73.992 73.9925 73.993 74.006 74.0065 74.007 73.999 73.9995 74.000

(9)

Table-7. The inside diameter ring measurements (in normalized form) for case study (1).

Sample

No. i Rep. 1 xi11xi1xi12 Rep. 2xi21xi2 xi22 Rep. 3 xi31 xi3 xi32 Rep. 4xi41xi4 xi42 Rep.5 xi51 xi5 xi52 1 0.523 0.534 0.545 0.500 0.511 0.523 0.545 0.557 0.568 0.136 0.148 0.159 0.386 0.398 0.409

2 0.614 0.625 0.636 0.318 0.330 0.341 0.841 0.852 0.864 0.455 0.466 0.477 0.864 0.875 0.886

3 0.659 0.670 0.682 0.636 0.648 0.659 0.386 0.398 0.409 0.864 0.875 0.886 0.841 0.852 0.864

4 0.250 0.261 0.273 0.773 0.784 0.795 0.273 0.284 0.295 0.886 0.898 0.909 0.364 0.375 0.386

5 0.795 0.807 0.818 0.182 0.193 0.205 0.727 0.739 0.750 0.500 0.511 0.523 0.295 0.307 0.318

6 0.159 0.170 0.182 0.386 0.398 0.409 0.136 0.148 0.159 0.432 0.443 0.455 0.386 0.398 0.409

7 0.432 0.443 0.455 0.545 0.557 0.568 0.182 0.193 0.205 0.477 0.489 0.500 0.750 0.761 0.773

8 0.568 0.580 0.591 0.955 0.966 0.977 0.386 0.398 0.409 0.886 0.898 0.909 0.682 0.693 0.705

9 0.045 0.057 0.068 0.523 0.534 0.545 0.386 0.398 0.409 0.750 0.761 0.773 0.636 0.648 0.659

10 0.273 0.284 0.295 0.227 0.239 0.250 0.682 0.693 0.705 0.477 0.489 0.500 0.523 0.534 0.545

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

20 0.682 0.693 0.705 0.273 0.284 0.295 0.455 0.466 0.477 0.591 0.602 0.614 0.136 0.148 0.159

21 0.636 0.648 0.659 0.500 0.511 0.523 0.250 0.261 0.273 0.432 0.443 0.455 0.205 0.216 0.227

22 0.227 0.239 0.250 0.477 0.489 0.500 0.773 0.784 0.795 0.773 0.784 0.795 0.682 0.693 0.705

23 0.773 0.784 0.795 0.318 0.330 0.341 0.341 0.352 0.364 0.182 0.193 0.205 0.409 0.420 0.432

24 0.545 0.557 0.568 0.659 0.670 0.682 0.295 0.307 0.318 0.614 0.625 0.636 0.455 0.466 0.477

25 0.205 0.216 0.227 0.750 0.761 0.773 0.523 0.534 0.545 0.341 0.352 0.364 0.705 0.716 0.727

(10)

Table-8. The calculated COMij values for case study (1).

Sample No. i COMi1 COMi2 COMi3 COMi4 COMi5

1 0.525 0.525 0.525 0.100 0.317

2 0.525 0.100 0.950 0.525 0.950

3 0.623 0.540 0.317 0.317 0.950

4 0.100 0.950 0.100 0.950 0.243

5 0.950 0.100 0.872 0.525 0.100

6 0.100 0.317 0.100 0.482 0.317

7 0.482 0.525 0.100 0.525 0.950

8 0.525 0.950 0.317 0.950 0.695

9 0.100 0.525 0.317 0.950 0.540

10 0.100 0.100 0.695 0.525 0.525

⸽ ⸽ ⸽ ⸽ ⸽ ⸽

20 0.695 0.100 0.525 0.525 0.100

21 0.540 0.525 0.100 0.482 0.100

22 0.100 0.525 0.950 0.950 0.695

23 0.950 0.100 0.124 0.100 0.385

24 0.525 0.623 0.100 0.525 0.525

25 0.100 0.950 0.525 0.124 0.752

Table-9. The corresponding actual measurements

x

ij



for COMij values for case study (1).

Sample No.

I

x

i



1 xi2 xi3 xi4 xi5

x

i



R

i

1 74.0021 74.0021 74.0021 73.9834 73.9930 73.9965 0.0187 2 74.0021 73.9834 74.0208 74.0021 74.0208 74.0058 0.0374 3 74.0064 74.0028 73.9930 73.9930 74.0208 74.0032 0.0279 4 73.9834 74.0208 73.9834 74.0208 73.9897 73.9996 0.0374 5 74.0208 73.9834 74.0174 74.0021 73.9834 74.0014 0.0374 6 73.9834 73.9930 73.9834 74.0002 73.9930 73.9906 0.0168 7 74.0002 74.0021 73.9834 74.0021 74.0208 74.0017 0.0374 8 74.0021 74.0208 73.9930 74.0208 74.0096 74.0093 0.0279 9 73.9834 74.0021 73.9930 74.0208 74.0028 74.0004 0.0374 10 73.9834 73.9834 74.0096 74.0021 74.0021 73.9961 0.0262

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

20 74.0096 73.9834 74.0021 74.0021 73.9834 73.9961 0.0262 21 74.0028 74.0021 73.9834 74.0002 73.9834 73.9944 0.0194 22 73.9834 74.0021 74.0208 74.0208 74.0096 74.0073 0.0374 23 74.0208 73.9834 73.9845 73.9834 73.9959 73.9936 0.0374 24 74.0021 74.0064 73.9834 74.0021 74.0021 73.9992 0.0230 25 73.9834 74.0208 74.0021 73.9845 74.0121 74.0006 0.0374

x



73.99915

R



(11)

The

x





R



control chart is constructed as illustrated in Figure-9.

25 23 21 19 17 15 13 11 9 7 5 3 1 74.02 74.01 74.00 73.99 73.98

Sample

S

a

m

p

le

M

e

a

n

_ _ X=73.99915 U C L=74.01633

LC L=73.98196

25 23 21 19 17 15 13 11 9 7 5 3 1 0.060 0.045 0.030 0.015 0.000

Sample

S

a

m

p

le

R

a

n

g

e

_ R=0.02979 U C L=0.06300

LC L=0

Figure-9. The

x





R



control charts for case study (1).

From Figure-9, it can be concluded that the process is in control because there are no points fall beyond outside the control limits and the plotted points exhibit a random pattern of behaviour.

3.2 Case Study: Monitoring crown cap production line

This case study was conducted by authors in a cans manufacturing industry to evaluate the performance of the crown cap production line using appropriate control charts. The crown cap production line machine produces at

each round 13 caps one shot, the study measures the crown cap angle. According to the specifications the angle is

(12)

Table-10. The caps angle measurements.

Sample i

Rep. 1

1max

x

i xi1min

Rep. 2

2 max

i

x

2 min

i

x

Rep.3

3 max

i

x

3min

i

x …

Rep. 11

11max

i

x

11min

i

x

Rep.12

12 max

i

x

12 min

i

x

Rep. 13

13 max

i

x

13 min

i

x

1 17 16 17 14 14 13 … … 17 15 16 15 15 13

2 15 14 16 15 16 15 … … 16 14 16 15 16 15

3 16 15 16 15 14 12 … … 16 15 17 15 16 15

4 15 13 16 15 16 15 … … 15 14 16 15 16 15

5 15 14 16 15 16 13 … … 16 13 14 12 16 14

6 15 14 15 12 16 12 … … 15 12 15 13 16 14

7 16 15 16 13 18 16 … … 17 15 18 16 15 14

8 16 15 16 14 16 15 … … 16 15 18 16 17 15

9 16 14 17 15 15 14 … … 16 13 16 15 16 15

10 15 12 17 14 16 14 … … 16 15 16 15 16 15

11 14 12 17 16 15 13 … … 15 13 16 14 17 15

12 15 14 17 15 16 15 … … 16 15 15 14 16 15

13 16 15 17 15 17 16 … … 16 14 16 14 15 14

14 17 16 16 14 18 15 … … 17 15 17 15 16 14

15 15 14 16 15 16 14 … … 16 14 16 15 16 13

16 15 14 15 14 15 14 … … 15 14 17 15 15 14

17 15 13 16 13 17 16 … … 15 14 15 14 16 13

18 15 14 16 15 16 13 … … 18 16 14 12 16 15

19 16 13 16 15 15 14 … … 16 14 16 14 16 15

20 16 15 16 14 16 14 … … 17 14 15 13 15 12

The averages of measurements in each replicate are calculated for all samples as shown in Table-11. The fuzzy replicate observations are normalized for all sample replicates. Table-12 displays the normalized replicate averages for all samples. The Mamdani-style fuzzy logic is then implemented to calculate the COMij values for all sample replicates. The obtained COMij results are shown in Table-13. The corresponding actual values of the COMij

(13)

Table-11. The calculated averages of every sample measurement for case study (2).

Sample

i

x

i1

x

i2

x

i3

x

i4

x

i5

x

i6

x

i7

x

i8

x

i9

x

i10

x

i11

x

i12

x

i13

1 16.5 15.5 13.5 14.0 15.0 15.5 14.0 15.5 15.5 16.5 16.0 15.5 14.0 2 14.5 15.5 15.5 15.0 14.5 15.5 14.0 15.0 17.0 15.5 15.0 16.0 15.5 3 15.5 15.5 13.0 15.5 16.0 15.5 14.5 16.0 16.5 17.0 15.5 16.5 15.5 4 14.0 15.5 15.5 14.5 15.0 14.5 14.5 17.0 16.5 15.5 14.5 16.0 15.5 5 14.5 15.5 14.5 15.0 15.5 14.5 14.0 15.5 15.5 15.5 14.5 15.0 15.0

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

15 14.5 15.5 15.0 15.5 15.5 15.5 14.5 16.5 14.0 15.0 15.0 16.0 14.5 16 14.5 14.5 14.5 14.5 15.0 14.5 14.5 15.0 15.5 14.5 14.5 16.0 14.5 17 14.0 14.5 16.5 14.0 15.5 14.5 15.5 15.0 14.5 14.0 14.5 15.5 14.5 18 14.5 15.5 14.5 15.5 16.0 14.5 14.5 14.5 15.5 14.0 17.0 15.0 15.5 19 14.5 15.5 14.5 16.0 15.5 15.0 13.0 14.5 16.0 14.5 15.0 16.0 15.5 20 15.5 15.0 15.0 16.5 15.5 15.0 16.0 14.0 14.5 13.0 15.5 15.0 13.5

Table-12. The normalized values for the measurement averages for case study (2).

Sample i zi1 zi2 zi3 zi4 zi5 zi6 zi7 zi8 zi9 zi10 zi11 zi12 zi13

1 0.750 0.583 0.250 0.333 0.500 0.583 0.333 0.583 0.583 0.750 0.667 0.583 0.333 2 0.417 0.583 0.583 0.500 0.417 0.583 0.333 0.500 0.833 0.583 0.500 0.667 0.583 3 0.583 0.583 0.167 0.583 0.667 0.583 0.417 0.667 0.750 0.833 0.583 0.750 0.583 4 0.333 0.583 0.583 0.417 0.500 0.417 0.417 0.833 0.750 0.583 0.417 0.667 0.583 5 0.417 0.583 0.417 0.500 0.583 0.417 0.333 0.583 0.583 0.583 0.417 0.500 0.500

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

(14)

Table-13. The COMij values for case study (2).

Sample i COMi1 COMi2 COMi3 … COMi10 COMi11 COMi12 COMi13

1 0.858 0.500 0.100 … 0.858 0.729 0.552 0.200

2 0.351 0.552 0.552 … 0.552 0.400 0.552 0.200

3 0.552 0.552 0.500 … 0.500 0.552 0.500 0.200

4 0.200 0.552 0.552 … 0.552 0.351 0.552 0.552

5 0.351 0.552 0.351 … 0.552 0.351 0.500 0.400

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

15 0.351 0.552 0.400 … 0.400 0.400 0.552 0.351

16 0.351 0.351 0.351 … 0.351 0.351 0.700 0.351

17 0.200 0.351 0.858 … 0.100 0.351 0.400 0.351

18 0.351 0.552 0.351 … 0.200 0.500 0.500 0.552

19 0.351 0.552 0.351 … 0.351 0.400 0.400 0.552

20 0.552 0.400 0.400 … 0.500 0.500 0.400 0.500

Table-14. The

I





MR



control chart parameters for the actual values for case study (2).

Sample i xi1 xi2 xi3 xi4 xi5 … xi11 xi12 xi13

x

_i

 



x

_i

MR

_i

1 17.148 15.000 12.600 13.200 15.000 … 16.374 15.312 13.200 14.908

2 14.106 15.312 15.312 14.400 14.106 … 14.400 15.312 13.200 14.567 0.341

3 15.312 15.312 15.000 15.312 16.200 … 15.312 15.000 13.200 15.318 0.751

4 13.200 15.312 15.312 14.106 14.400 … 14.106 15.312 15.312 14.868 0.450

5 14.106 15.312 14.106 14.400 15.312 … 14.106 15.000 14.400 14.614 0.254

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

15 14.106 15.312 14.400 15.312 15.312 … 14.400 15.312 14.106 14.637 0.276

16 14.106 14.106 14.106 14.106 14.400 … 14.106 16.200 14.106 14.405 0.232

17 13.200 14.106 17.148 13.200 15.312 … 14.106 14.400 14.106 14.316 0.090

18 14.106 15.312 14.106 15.312 16.200 … 15.000 15.000 15.312 14.706 0.390

19 14.106 15.312 14.106 16.374 15.312 … 14.400 14.400 15.312 14.870 0.164

20 15.312 14.400 14.400 15.000 15.312 … 15.000 14.400 15.000 14.762 0.108

x

 

14.817

(15)

19 17

15 13

11 9

7 5

3 1

15.6

15.2

14.8

14.4

14.0

O bser vation

I

n

d

iv

id

u

a

l

V

a

lu

e

_ X=14.817 U C L=15.586

LC L=14.048

19 17

15 13

11 9

7 5

3 1

1.00

0.75

0.50

0.25

0.00

O bser vation

M

o

v

in

g

R

a

n

g

e

__ M R=0.289 U C L=0.945

LC L=0

Figure-10. The

I





MR



control chart for case study (2).

3.3 Monitoring tableting process

This case study mainly aimed at monitoring tableting process, for which the main quality characteristics were hardness, weight, thickness, diameter, and shape. This research established the proper control charts for tablet weight. Thirty samples each of nine replicates were chosen randomly every ten minutes. Ten weight observations were collected in each replicates in linguistic form as shown in Table-15. The

x



s

control

(16)

Table-15. Collected replicate observations of tablet’s weight for case study (3).

Sample

i

Rep. 1

1max

i

x

_i_1min

Rep. 2

2max

i

x

_i_2min

Rep. 3

3max

i

x

_i_3min

Rep. 4

4max

i

x

_i_4min …

Rep. 8

8max

i

x

_i_8min

Rep. 9

9max

i

x

_i_9min

1 187.0 186.1 184.9 184.0 183.8 182.9 182.7 181.8 … 184.3 183.4 185.2 184.3

2 183.3 182.4 183.0 182.1 186.5 185.6 183.3 182.4 … 187.3 186.4 185.2 184.3

3 185.8 184.9 184.0 183.1 183.4 182.5 185.2 184.3 … 183.7 182.8 184.6 183.7

4 184.2 183.3 185.6 184.7 182.9 182.0 183.9 183.0 … 189.4 188.5 184.8 183.9

5 184.5 183.6 188.1 187.2 184.6 183.7 183.9 183.0 … 183.9 183.0 182.6 181.7

6 186.8 185.9 186.0 185.1 184.8 183.9 181.8 180.9 … 186.6 185.7 183.6 182.7

7 186.4 185.5 184.2 183.3 184.1 183.2 185.7 184.8 … 183.3 182.4 184.8 183.9

8 189.7 188.8 185.8 184.9 182.3 181.4 183.0 182.1 … 187.6 186.7 185.1 184.2

9 186.4 185.5 183.3 182.4 185.8 184.9 182.0 181.1 … 186.6 185.7 184.2 183.3

10 181.8 180.9 187.5 186.6 185.9 185.0 184.6 183.7 … 185.3 184.4 184.8 183.9

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

27 183.8 182.9 185.2 184.3 185.0 184.1 182.1 181.2 … 184.2 183.3 185.3 184.4

28 182.3 181.4 183.6 182.7 184.9 184.0 184.2 183.3 … 183.4 182.5 183.3 182.4

29 184.6 183.7 184.7 183.8 187.1 186.2 181.9 181.0 … 185.9 185.0 183.5 182.6

30 181.0 180.1 186.6 185.7 185.8 184.9 183.3 182.4 … 182.9 182.0 187.4 186.5

Table-16. The normalized averages for all samples for case study (3).

Sample

i

x

i1

x

i2

x

i3

x

i4

x

i5

x

i6

x

i7

x

i8

x

i9

1 0.697 0.495 0.389 0.284 0.293 0.341 0.688 0.438 0.524

2 0.341 0.313 0.649 0.341 0.159 0.389 0.466 0.726 0.524

3 0.582 0.409 0.351 0.524 0.178 0.361 0.322 0.380 0.466

4 0.428 0.562 0.303 0.399 0.370 0.380 0.813 0.928 0.486

5 0.457 0.803 0.466 0.399 0.351 0.611 0.063 0.399 0.274

6 0.678 0.601 0.486 0.197 0.591 0.139 0.380 0.659 0.370

7 0.639 0.428 0.418 0.572 0.178 0.476 0.620 0.341 0.486

8 0.957 0.582 0.245 0.313 0.476 0.284 0.399 0.755 0.514

9 0.639 0.341 0.582 0.216 0.543 0.524 0.313 0.659 0.428

10 0.197 0.745 0.591 0.466 0.889 0.226 0.332 0.534 0.486

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

25 0.543 0.572 0.611 0.264 0.245 0.409 0.313 0.543 0.351

26 0.341 0.264 0.601 0.226 0.284 0.764 0.457 0.274 0.351

27 0.389 0.524 0.505 0.226 0.063 0.562 0.091 0.428 0.534

28 0.245 0.370 0.495 0.428 0.486 0.553 0.688 0.351 0.341

29 0.466 0.476 0.707 0.207 0.418 0.476 0.447 0.591 0.361

(17)

Table-17. The COMij values for tableting process (case study 3).

Sample

No. i COMi1 COMi2 COMi3 COMi4 COMi5 COMi6 COMi7 COMi8 COMi9

1 0.744 0.525 0.283 0.100 0.100 0.134 0.546 0.421 0.525

2 0.134 0.106 0.590 0.134 0.100 0.283 0.319 0.826 0.525

3 0.525 0.354 0.148 0.525 0.100 0.198 0.114 0.256 0.485

4 0.379 0.525 0.100 0.319 0.229 0.256 0.950 0.950 0.513

5 0.471 0.950 0.485 0.319 0.148 0.531 0.100 0.319 0.100

6 0.681 0.525 0.513 0.100 0.525 0.100 0.256 0.623 0.229

7 0.576 0.397 0.377 0.525 0.100 0.500 0.546 0.134 0.513

8 0.500 0.525 0.100 0.106 0.500 0.100 0.319 0.889 0.525

9 0.576 0.134 0.525 0.100 0.525 0.525 0.106 0. 623 0.397

10 0.100 0.867 0.525 0.485 0.950 0.100 0.124 0.525 0.513

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

25 0.525 0.525 0.531 0.100 0.100 0.354 0.106 0.525 0.148

26 0.134 0.100 0.525 0.100 0.100 0.950 0.471 0.100 0.148

27 0.283 0.525 0.525 0.100 0.100 0.525 0.100 0.397 0.525

28 0.100 0.229 0.525 0.397 0.513 0.525 0.710 0.148 0.134

29 0.485 0.500 0.792 0.100 0.377 0.500 0.451 0.525 0.198

30 0.100 0.623 0.525 0.134 0.100 0.256 0.525 0.100 0.844

Table-18. The

x

 



s

control chart calculations for the COM values for case study (3).

Sample i xi1 xi2 xi3 xi4 … xi7 xi8 xi9

x

_i



s

_i 1 187.038 184.760 182.243 180.340 … 184.978 183.678 184.760 183.203 2.287 2 180.694 180.402 185.436 180.694 … 182.618 187.890 184.760 182.786 2.654 3 184.760 182.982 180.839 184.760 … 180.486 181.962 184.344 182.426 1.833 4 183.242 184.760 180.340 182.618 … 189.180 189.180 184.635 184.178 3.158 5 184.198 189.180 184.344 182.618 … 180.340 182.618 180.340 183.255 2.813 6 186.382 184.760 184.635 180.340 … 181.962 185.779 181.682 183.405 2.238 7 185.290 183.429 183.221 184.760 … 184.978 180.694 184.635 183.539 1.844 8 184.500 184.760 180.340 180.402 … 182.618 188.546 184.760 183.418 2.911 9 185.290 180.694 184.760 180.340 … 180.402 185.779 183.429 183.357 2.224 10 180.340 188.317 184.760 184.344 … 180.590 184.760 184.635 184.141 3.323

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽

25 184.760 184.760 184.822 180.340 … 180.402 184.760 180.839 182.667 2.156 26 180.694 180.340 184.760 180.340 … 184.198 180.340 180.839 182.337 3.104 27 182.243 184.760 184.760 180.340 … 180.340 183.429 184.760 182.859 2.067 28 180.340 181.682 184.760 183.429 … 186.684 180.839 180.694 183.091 2.153 29 184.344 184.500 187.537 180.340 … 183.990 184.760 181.359 183.839 1.981 30 180.340 185.779 184.760 180.694 … 184.760 180.340 188.078 183.006 2.884

x



(18)

The

x

 



s

control chart is constructed as shown in Figure-11.

28 25 22

19 16

13 10 7

4 1

186

184

182

Sample

S

a

m

p

le

M

e

a

n

_ _ X=183.455 U C L=185.988

LC L=180.922

28 25 22

19 16

13 10 7

4 1

4

3

2

1

Sample

S

a

m

p

le

S

tD

e

v

_ S =2.455 U C L=4.324

LC L=0.587

Figure-11. The

x

 



s

control chart for monitoring tablet weight.

From Figure-11, it is concluded that the process is in statistical control because the plotted points exhibit a random pattern of behaviour.

4. RESULTS

4.1 The results of

x





R



control charts

The parameters of the

x





R



control chart are as given in Table-19.

Table-19. The estimated parameters of the

x





R



control charts.

Control limits

x



chart

R



chart

UCL

74.01633 0.06300

CL

73.99915 0.02979

LCL

73.98196 0

Because the

x





R



control charts are found in statistical control, the estimated process mean and standard deviation are calculated and found to be 73.99915 and 0.012202, respectively. The fuzzy process capability

indices

(

C C

ˆ ˆ

p

,

pk

)

are estimated using the fuzzy logic as follows. The fuzzy upper and lower specification limits are given as:

(74.0340, 74.0346, 74.0360)

USL .

(73.9640, 73.9651, 73.9660)

LSL .

Then, the estimated fuzzy process capability indices are obtained as follows:

0.983449,0.949301,0.92 81 )

ˆ

₍

₈

₂

p

C



0.960196, 0.930146, 0.90556

ˆ

₍

₎

pl

C



ˆ

_{(0.90559, 0.93018, 0.96023)}

pu

C



ˆ

_{(0.90559, 0.93018, 0.96023)}

pk

C



Utilizing the fuzzy logic, the estimated process

capability indices are and found to be

C

ˆ

_p



0.999

and

ˆ

_0.999

pk

C



. These values indicate that the process is inadequate.

4.2 The results of

I





MR



control charts

(19)

I





MR



control limits.

Control limits

_I



chart

MR



chart

UCL

15.586 0.945

CL

14.817 0.289

LCL

14.048 0

The



ˆ

and



ˆ

are equal to 14.817 and 0.256, respectively. For this case study, the fuzzy upper and lower specification limits are decided as:

(18,17.5,17.25)

USL



(12.75,12.25,12)

LSL



The estimated p

C

and

pk

C

are then calculated

using fuzzy specification limits and found to be



3.90625,3.41797, 2.9 7



ˆ

₂₉

p

C



, 3.3

ˆ

_(3.167

₉₇

_{4245, 2.69}

_{14 )}

₁

pk

C



Utilizing fuzzy logic, the

C

ˆ

_pand

C

ˆ

_pkvalues are both found equal to be 2.83, which that the process is super excellent.

4.3 The results of constructing

_{x s}

 



control charts

The parameters of the

x s

 



control chart are obtained as listed in Table-21.

x

 



s

control limits.

Control limits

_x



chart

s



chart

UCL

185.988 4.324

CL

183.455 2.455

LCL

180.922 0.587

From the

x

 



s

control, the



ˆ and



ˆ

values are calculated 183.445 and 2.533, respectively. The fuzzy upper and lower specification limits are given as:

(189.7,189.5,189.2)

USL



(179.8,179.5,179.3)

LSL



The estimated p

C

and

pk

C

are then calculated

using fuzzy specification limits and found to be

0.980104, 0.875808, 0.831

(

555)

p

C





0.894578, 0.798036, 0.717361



pk

C



Finally, the

C

ˆ

_pand

C

ˆ

_pkvalues are calculated 0.732 and 0.336, respectively. As a result, the process capability is judged as poor.

5. CONCLUSIONS

This study utilizes the fuzzy logic to deal with the uncertainty; i.e., under linguistic data, in the measurement system during the development of variables control charts and process capability analysis. The observation is represented by a triangular membership function. Then, the COM value is obtained for each sample replicate using fuzzy logic approach. The appropriate variable control chart and process capability indices are then established. Three case studies were utilized to illustrate the proposed procedures. Results showed that the constructed variables control charts and the corresponding estimates of process capability indices are found efficient in monitoring process mean and variability, and process’s capability assessment, respectively. In conclusions, the developed control charts and capability indices can be easily interpreted and understood by practitioners, which shall make it widely used in monitoring process performance in business applications. Future research will consider developing attributes control charts under uncertainty.

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