a fuzzy-logic approach for developing variables control ... · in conclusions, the developed...

VOL. 14, NO. 6, MARCH 2019 ISSN 1819-6608

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A FUZZY-LOGIC APPROACH FOR DEVELOPING VARIABLES

CONTROL CHARTS AND PROCESS CAPABILITY INDICES

UNDER LINGUISTIC MEASUREMENTS

Abbas Al-Refaie, Areen Obaidat1, Rami H. Fouad1, and Bassel Hanayneh

2

1Department 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.

In the traditional variable control charts, it is

assumed that precise measurements are obtained. Then,

the CL ,UCL and LCL are exemplified by numeric

values, then a process is judged either ‘‘in control” or ‘‘out

of control” depending on numeric observation values. But

for many real problems, observations of continuous

quantities are not precise numbers; such observations are

called fuzzy. As a result, the control limits could not be so

precise and certain [3]. Uncertainty is usually resulted

from measurement system, including operators and

gauges, environmental conditions, or from the process

itself. In this case, fuzzy set theory is a valuable tool to

handle this uncertainty, by transforming the numeric

control limits into a fuzzy control one by using

membership functions. Fuzzy control charts are preferable

over the traditional ones, since they are more sensitive in

detecting small shifts without any complexity

augmentation to the chart. Taleb and Limamy [4] proposed

mailto:[email protected]

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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 2

x scontrol 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. Letijk

x

denotes the kth reading of jth replicate at sample i of a

quality characteristic where, 1,...i 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

1 111 11... Kx x 121 12... Kx x … 1 1 1...N NKx x

2 211 21... Kx x 222 22... Kx x … 2 2 2...N NKx x

⁞ ⁞ ⁞ ⁞

m 11 1...m m Kx x 22 2...m m Kx x … 1...mN mNKx x

Step 2: Calculate the average, ijx , of the K

observations,ijk

x , of the jth replicate for the ith sample

using Equation. (1).

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1 , ij

K

ijk

k

x

xK

i j

(1)

Step 3: Let ijkz denotes the normalized value of

ijkx . Normalize the

ijkx between 0 and 1 using

min , ,

max min

ijk ijk

ijk

ijk ijk

x xz i j k

x x

(2)

where min ijkx and max ijk

x are 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 jkiz as

shown in Figure-2. The three fuzzy MFs low, middle and

high are assigned for each inputjkiz of each of Replicate 1

(Rep. 1) to Replicate N(Rep. N).

Figure-2. The MFs of jkiz in 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 111z is L, 112z is L, 113z is L, then the COM11 is L.

If 111z is L, 112z is L, 113z is M, then the COM11 is L+.

If 111z is L, 112z is M, 113z is M, then the COM11 is M-.

If 111z is M, 112z is M, 113z is M, then the COM11 is M.

If 111zis M, 112z

is M, 113z is H, then the COM11 is M+.

If 111zis M, 112z

is H, 113z is H, then the COM11 is H-.

If 111zis H, 112z

is H and 113z is H, then the COM11 is H.

Table-2. Generated fuzzy rules for zijk.

1ijz 2ijz

3ijz COMij

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 (input 2) ^ middle (input3)=

middle (output).

middle (input 1)^ middle (input 2) ^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.

From the fuzzy rules, the MFs of the COMij are

identified using the zijk values input variables of the rules.

The fuzzy reasoning of the rules yields the COMij by using

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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

0

( ).

( )i j

C F FdFCOM

C F dF

(3)

where 0C is 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 1iCOM 2iCOM ……… inCOM

1 11COM 12COM ………

1nCOM

2 21COM 22COM ……… 2nCOM

m 1mCOM 2mCOM ………

mnCOM

Step 5: Calculate the corresponding actual

measurements, ijx , for the COMij listed in Table-4 using

Equation. (4).

(max min ) minij ij k ijk ijkijx 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, ijx

, 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 (ix ) and the range ( iR ) of this

sample are calculated as follows, respectively:

1

N

ij

j

i

x

xN

(5)

max mini i iR x x (6)

Calculate the grand average ( )x , which equals

the estimated mean ˆ( )

, as follows:

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1

m

i

i

x

xm

(7)

Similarly, calculate the average range ( )R using

Equation. (8).

1

m

i

i

R

Rm

(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

3x

UCL x Rd n

(10)

xCL x (11)

2

3x

LCL x Rd n

(12)

(b) For the R control chart, the parameters are

calculated as follows:

3

2

3R

RUCL R d

d (13)

RCL R (14)

3

2

3R

RLCL R d

d (15)

where 3d is 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 ̂ and̂ 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 xis

(18)

1i

m

is

ms

(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

3x

sUCL x

c n (20)

xCL x (21)

4

3x

sLCL x

c n (22)

(b) For the s chart, the parameters are calculated

as follows:

2

4

4

31

s

sUCL s c

c (23)

sCL s (24) 2

4

4

31

s

sLCL s c

c

(25)

where 4c 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:

1i i iMR x x

(26)

The values of ̂ and ̂ are calculated as

follows:

1ˆ

m

i

i

x

xm

(27)

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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

3I

MRUCL x

d

(29)

ICL x (30)

2

3I

MRLCL x

d

(31)

(b) For MR chart, the parameters are calculated

as follows:

4MRUCL MR D MR (32)

MRCL MR (33)

3MRLCL 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,pC and

pkC , 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 ̂ , standard deviation̂ as follows:

1 3 3 12 21 2 3

ˆ ˆ ˆ( , , ) ( , , )ˆ ˆ ˆ ˆ6 6 6 6

p p p p

USL LSL USL LSLUSL LSLUSL LSLC C C C

(35)

3 2 11 2 3

ˆ ˆ ˆˆ ˆ ˆ ˆ( , , ) ( , , )ˆ ˆ ˆ ˆ3 3 3 3

pl pl pl pl

LSL LSL LSLLSLC C C C

(36)

31 21 2 3

ˆˆ ˆˆ ˆ ˆ ˆ( , , ) ( , , )ˆ ˆ ˆ ˆ3 3 3 3

pu pu pu pu

USLUSL USLUSLC C C C

(37)

1 2 3ˆ ˆ ˆmin( , ) ( , , )pk pu pl pk pk pkC C C 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 and p pkC C valuesas illustrated in

Figure-5.

Figure-5. The MFs for pC and pkC .

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 ˆpC and ˆ

pkC .

1 1,p pkC C 2 2,p pkC C 3 3,p pkC C COM

Low Low Low Poor

Middle Inadequate

Middle Low

Middle Low

Middle Capable

Low Middle

Low Middle

Middle Middle

High Satisfactory

High Middle

Middle High Middle

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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 ˆ ˆ,p pkC C.

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 ,p pkC C .

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 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

1iCOM , 2iCOM 3iCOM 4iCOM and 5iCOM for

Rep. 1 to Rep. 5, respectively. The corresponding actual

measurements, ijx , are calculated as shown in Table-9.

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

Sample i 11i

x 12i

x 21i

x 22i

x 31i

x 32i

x 41i

x 42i

x 51i

x 52i

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

1minix 1ix 1maxix

Rep. 2

2 minix

2ix

2maxix

Rep.3

3minix

3ix

3maxix

Rep.4

4maxix

4ix

4 minxi

Rep.5

5minix

5ix

5maxix

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

25 73.988 73.9885 73.989 74.012 74.0125 74.013 74.002 74.0025 74.003 73.994 73.9945 73.995 74.010 74.0105 74.011

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

Sample

No. i Rep. 1 11xi 1xi 12xi Rep. 2 21xi 2xi 22xi

Rep. 3 31xi 3xi 32xi Rep. 4 41xi 4xi 42xi

Rep.5 51xi 5xi 52xi

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

Figure-8. Fuzzy rules for the first replicate in the first sample for case study (1).

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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 ijx for COMij values for case study (1).

Sample No.

I 1ix 2ix 3ix 4ix 5ix ix

iR

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

0.028382

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The x R control chart is constructed as illustrated in Figure-9.

252321191715131197531

74.02

74.01

74.00

73.99

73.98

Sample

Sa

mp

le M

ea

n

__X=73.99915

UC L=74.01633

LC L=73.98196

252321191715131197531

0.060

0.045

0.030

0.015

0.000

Sample

Sa

mp

le R

an

ge

_R=0.02979

UC 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

limited between 012 and

018 as shown in Table-10. The

application case adopted with twenty samples in thirteen

replicates to be measured. The individual and moving

range (I-MR) control chart was used, because of the

repeated measurements on the samples differ due to

laboratory or analysis error, and multiple measurements

are taken on the same unit of product.

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Table-10. The caps angle measurements.

Sample i Rep. 1

1maxxi 1min

xi

Rep. 2

2 maxix

2 minix

Rep.3

3maxix

3minix

… Rep. 11

11maxix

11minix

Rep.12

12 maxix

12 minix

Rep. 13

13maxix

13minix

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

values are then calculated and then displayed in Table-14.

Finally, the I MR control chart is constructed as

shown in Figure-10, where it is concluded that the process

is in control because there is no points fall beyond outside

the control limits and the plotted points exhibit a random

pattern of behaviour.

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

Sample

i 1ix 2ix 3ix 4ix 5ix 6ix 7ix 8ix 9ix 10ix 11ix 12ix 13ix

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

⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ ⸽ 15 0.417 0.583 0.500 0.583 0.583 0.583 0.417 0.750 0.333 0.500 0.500 0.667 0.417

16 0.417 0.417 0.417 0.417 0.500 0.417 0.417 0.500 0.583 0.417 0.417 0.667 0.417

17 0.333 0.417 0.750 0.333 0.583 0.417 0.583 0.500 0.417 0.333 0.417 0.583 0.417

18 0.417 0.583 0.417 0.583 0.667 0.417 0.417 0.417 0.583 0.333 0.833 0.500 0.583

19 0.417 0.583 0.417 0.667 0.583 0.500 0.167 0.417 0.667 0.417 0.500 0.667 0.583

20 0.583 0.500 0.500 0.750 0.583 0.500 0.667 0.333 0.417 0.167 0.583 0.500 0.250

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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 1ix

2ix

3ix

4ix

5ix

… 11ix

12ix

13ix

i ix x iMR

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 MR 0.289

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191715131197531

15.6

15.2

14.8

14.4

14.0

Observation

In

div

idu

al

Va

lue

_X=14.817

UC L=15.586

LC L=14.048

191715131197531

1.00

0.75

0.50

0.25

0.00

Observation

Mo

vin

g R

an

ge

__MR=0.289

UC 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

chart was implemented as follows. The collected data is

normalized for all replicates. The normalized values for

the replicate averages are listed in Table-16. The

corresponding replicate’s COMij values are estimated

using fuzzy logic for all samples as shown in Table-17.

The x s control chart parameters are calculated as

shown in Table-18.

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

Sample

i

Rep. 1

1maxix 1minix

Rep. 2

2maxix 2minix

Rep. 3

3maxix 3minix

Rep. 4

4maxix 4minix …

Rep. 8

8maxix 8minix

Rep. 9

9maxix 9minix

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 1ix 2ix 3ix 4ix 5ix 6ix 7ix 8ix 9ix

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

30 0.120 0.659 0.582 0.341 0.274 0.380 0.543 0.303 0.736

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

Sample

No. i 1iCOM

2iCOM

3iCOM

4iCOM

5iCOM

6iCOM

7iCOM

8iCOM

9iCOM

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 1ix

2ix

3ix

4ix

… 7ix

8ix

9ix

ix is

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

183.445

s

2.455

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The x s control chart is constructed as shown in Figure-11.

28252219161310741

186

184

182

Sample

Sa

mp

le M

ea

n

__X=183.455

UC L=185.988

LC L=180.922

28252219161310741

4

3

2

1

Sample

Sa

mp

le S

tDe

v

_S=2.455

UC 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 Rchart

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 ˆ ˆ( , )p pk

C C 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 2( 8p

C

0.960196,0.930146,0.90556ˆ ( )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 ˆ 0.999pC and

ˆ 0.999pkC . These values indicate that the process is

inadequate.

4.2 The results of I MR control charts The estimated crisp I-MR control chart

parameters are given as shown in Table-20.

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1138

Table-20. The estimated parameters of the 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 ̂ 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 pC and

pkC are then calculated

using fuzzy specification limits and found to be

3.90625,3.41797,2.9 7ˆ 29p

C

,3.3ˆ 4245,2.69(3.167 197 14 )pk

C

Utilizing fuzzy logic, the ˆpC and ˆ

pkC values 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.

Table-21. The estimated parameters of the 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 ̂ and̂ 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 pC and

pkC are then calculated

using fuzzy specification limits and found to be

0.980104,0.875808,0.831( 555)pC

0.894578,0.798036,0.717361pkC

Finally, the ˆpC and ˆ

pkC values 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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a fuzzy-logic approach for developing variables control ... · in conclusions, the developed...

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