3.ECE (1)

International Journal of Electronics, Communication

& Instrumentation Engineering Research and

Development (IJECIERD)

ISSN 2249-684X

Vol. 2 Issue 4 Dec 2012 17-30

© TJPRC Pvt. Ltd.,

ISSN 2249–6939

Vol.2, Issue 2 (2012) 1-16

© TJPRC Pvt. Ltd.,

α - CUT FUZZY CONTROL CHARTS FOR BOTTLE BURSTING STRENGTH DATA

1A. SARAVANAN &

2P. NAGARAJAN

1Assistant Professor, Department of Instrumentation Technology, MSRIT, Bangalore, India

2Associate Professor, Department of Chemical Engineering, Annamalai University, India

ABSTRACT

Quality has become one of the most important consumer decision factors in the selection among competing

products and services. Statistical Process Control (SPC) is a technique applied towards improving the quality of

characteristics by monitoring the process under study continuously, in order to detect assignable causes and take required

actions as quickly as possible. A traditional variable control chart consists of three lines namely Center Line (average

value) Upper Control Limit and Lower Control Limit (other two horizontal lines). These limits are represented by the

numerical values. The process is either “in-control” or “out-of-control” depending on numerical observations. For many

problems, control limits could not be so precise. Uncertainty comes from the measurement system including operators and

gauges and environmental conditions. In this situation, fuzzy set theory is a useful tool to handle this uncertainty. Fuzzy

control limits provide a more accurate and flexible evaluation. In this paper, the fuzzy α cut control charts are constructed

and applied in bottle bursting strength data.

KEYWORDS: Statistical Process Control, Fuzzy Control Charts, -cutand- Level Fuzzy Midrange

INTRODUCTION

Statistical Process Control (SPC) is used to monitor the process stability which ensures the predictability of the

process. Control charts are viewed as the most commonly applied SPC tools. A control chart consists of three horizontal

lines called; Upper Control Limit (UCL), Center Line (CL) and Lower Control Limit (LCL). The center line in a control

chart denotes the average value of the quality characteristic under study. If a point lies within UCL and LCL, then the

process is deemed to be under control. Otherwise, a point plotted outside the control limits can be regarded as evidence

representing that the process is out of control and, hence preventive or corrective actions are necessary in order to find and

eliminate the assignable cause or causes, which subsequently result in improving quality characteristics [7]. The control

chart may be classified into two types namely variable and attribute control charts.

The fuzzy set theory was first introduced by Zadeh and studied by many authors [2], [3], [4], [5] . It is mostly

used when the data is attribute in nature and these types of data may be expressed in linguistic terms such as “very good”,

“good”, “medium”, “bad” and “very bad”. The measures of central tendency in descriptive Statistics are used in variable

control charts. These measures can be used to convert fuzzy sets into scalars which are fuzzy mode, -level fuzzy

midrange, and fuzzy median and fuzzy average. There is no theoretical basis to select the appropriate fuzzy measures

among these four.

The objective of this study is first to construct the fuzzy and control charts with α cuts by using

α -level fuzzy midrange. The following procedures are used to construct the fuzzy and control charts.

18 A. Saravanan & P. Nagarajan

1. First transform the traditional and control charts to fuzzy control charts. To obtain fuzzy

and control charts, the trapezoidal fuzzy number (a, b, c, d) are used.

2. The cut fuzzy control charts and cut fuzzy control charts are developed by using cut

approach.

3. The -level fuzzy and midrange for fuzzy control

charts are calculated by using - level fuzzy midrange transformation techniques

4. Finally, the application of control charts is highlighted by using bottle bursting strength data.

FUZZY TRANSFORMATION TECHNIQUES

Mainly four fuzzy transformation techniques, which are similar to the measures of central tendency, used in

descriptive statistics: - level fuzzy midrange, fuzzy median, fuzzy average, and fuzzy mode are used. In this paper,

among the above four transformation techniques, the - level fuzzy midrange transformation technique is used for the

construction of fuzzy and control charts based on fuzzy trapezoidal number.

- LEVEL FUZZY MIDRANGE

This is defined as the midpoint of the ends of the - level cuts, denoted by , is a non fuzzy set that comprises

all elements whose membership is greater than or equal to. If and are the end points of , then

( )

In fact, the fuzzy mode is a special case of - level fuzzy midrange when =1.- level fuzzy midrange of

sample j, is used to transform the fuzzy control limits into scalar and is determined as follows.

FUZZY CONTROL CHART BASED ON RANGES

In monitoring the production process, the control of process averages or quality level is usually done by charts.

The process variability or dispersion can controlled by either a control chart for the range, called R chart, or a control chart

for the standard deviation, called S chart. In this section, fuzzy control charts are introduced based on fuzzy

trapezoidal number. The fuzzy control charts are presented in the next section. Montgomery [7] has proposed the

control limits for control chart based on sample range is given below

Where is a control chart co-efficient and is the average of Ri that are the ranges of samples. In the case of

fuzzy control chart, each sample or subgroup is represented by a trapezoidal fuzzy number (a, b, c, d) as shown in Fig. 1.

α - Cut Fuzzy Control Charts for Bottle Bursting Strength Data 19

In this study, trapezoidal fuzzy numbers are represented as ( , ) for each observation. Note that a

trapezoidal fuzzy number becomes triangular when b=c. For the case of representation and calculation, a triangular fuzzy

number is also represented as a trapezoidal fuzzy number by (a, b, b, d ) or (a, c, c, d).The center line C is the arithmetic

mean of the fuzzy sample means, which are represented by ( )

.Here are called the overall means and is calculated as follows.

; r =a,b,c,d; i=1,2,3,…….n ; j =1,2,3,……m.

; r=a,b,c,d; j=1,2,3 ………m.

=( )= { , , , }

Where „n‟ is the fuzzy sample size, „m‟ is the number of fuzzy samples and

is the center line for fuzzy control chart.

Control Limits for Fuzzy Control Chart

By using the traditional control chart procedure, the control limits of fuzzy control charts with ranges based on

fuzzy trapezoidal number are calculated as follows

= + = ( ) + A2 (

= ( )

= ( ) = (

C - = ( ) – A2 (

= (

Where ; r=a,b,c,d; j=1,2,3 ………m the proceduce for calculating is as follows

j= 1, 2, 3,….m.

Where ( is the maximum fuzzy number in the sample and


( is the minimum fuzzy number in the sample .

Fig.1: Representation of a Sample by Trapezoidal Fuzzy Numbers

Control Limits for α- Cut Fuzzy Control Chart

Introducing the α - cut procedure to the above fuzzy control limits, it can be rewritten as follows (the value of α

can be selected according to the nature of the given problem and the selected α value must should lies between0 and 1)

= ( ) + A2 (

= ( )

= ( ) =

( ) - A2 (

= (

Where

aα = a+ α(b – a) ; d

α = d+ α(d – c)

The α - cut fuzzy control limits based on ranges are shown in fig.2

Fig.2: α - Cut Fuzzy Control Chart Based on Ranges using Fuzzy Trapezoidal Number


α - Level Fuzzy Midrange for α- Cut Fuzzy Control Chart Based on Ranges

The α - level fuzzy midrange is one of the transformation techniques (among the four) used to transform the

fuzzy set into scalar. It is used to check the production process, whether the process is “in-control” or “out-of-control”.

The control limits for α - level fuzzy midrange for α -Cut Fuzzy control chart based on ranges can be obtained as

follows.

The definition of α - level fuzzy midrange of sample j for fuzzy control chart is

Then, the condition of process control for each sample can be defined as:

Process control = {in control; for

Out –of –control; otherwise}

FUZZY CONTROL CHART

The control limits for Shewhart R control chart is given by

UCLR = D4 ; CLR = ; UCLR = D3

Where and are control chart co-efficient [6].

By using the traditional R control chart procedure, the control limits for fuzzy control chart with trapezoidal fuzzy

number is obtained as follows.

Control Limits for α – Cut Fuzzy Control Chart

The control limits of α - cut fuzzy control chart based on trapezoidal fuzzy numbers are obtained as follows


)

)

)

α - Level Fuzzy Midrange for α - Cut Fuzzy Control Chart

The control limits of α - Level fuzzy midrange for α - Cut Fuzzy Control chart based on fuzzy Trapezoidal number can

be calculated as follows

The definition of α - level fuzzy midrange of sample j for fuzzy control chart can be calculated as follows


Process control ={ in control; for

Out –of –control; otherwise}

FUZZY CONTROL CHART BASED ON STANDARD DEVIATION

The R chart is used to monitor the dispersion associated with a quality characteristic. Its simplicity of construction

and maintenance make the R chart very commonly used and the range is a good measure of variation for small subgroup

sizes. When the sample size increases (n>10), the utility of the range as a measure of dispersion falls off and the standard

deviation measure is preferred (Montgomery 2002)

The Shewhart chart based on standard deviation is given below

Where is a control chart co-efficient (Kolarik 1995)

The value of is

=

Where is the standard deviation of sample j and is the average of s.


Fuzzy Control Chart Based on Standard Deviation

The theoretical structure of fuzzy control chart and fuzzy control chart has been developed by Senturk and Erginel

(2009). The fuzzy is the standard deviation of sample j and it is calculated as follows

and the fuzzy average is calculated by using standard deviation represented by the following Trapezoidal fuzzy number

= { , }=(

And the control limits of fuzzy control chart based on standard deviation are defined as follows

= + = ( ) + )

, ) = ( )

= ( ) =(

C - = ( ) - )

, ) = (

Control Limits for α – Cut Fuzzy

The control limits for α - Cut Fuzzy control chart based on standard deviation are obtained as follows

= ( ) + )

, )

= ( )

= ( ) =(

( ) - )

, )

= (


Where

α - Level Fuzzy Midrange for α - Cut Fuzzy Control Chart Based on Standard Deviation

The control limits and centre line for α - Cut Fuzzy control chart based on standard deviation using α – Level fuzzy

midrange are

The definition of α - level fuzzy midrange of sample j for fuzzy control chart is


Process control = {in control; for

Out –of –control;otherwise }

FUZZY CONTROL CHART

The control limits for Shewhart control chart is given by

Where and are control chart co-efficient . Then the Fuzzy control chart limits can be obtained as follows

)

)

)

α - Cut Fuzzy Control Chart

The control limits of α - Cut Fuzzy control chart can be obtained as follows:

)


)

)

α - Level Fuzzy Midrange for α - Cut Fuzzy Control Chart

The control limits of α - Level fuzzy midrange for α - Cut Fuzzy control chart can be obtained in a similar way to α -

Cut Fuzzy control chart.

The definition of α - level fuzzy midrange of sample j for fuzzy control chart can be calculated as follows


Decision ={ in control; for

Out –of –control; otherwise }

Application: Different Observation data for Bottle bursting strength have been considered with 10 samples. Fuzzy

control limits are calculated according to the procedures given in the previous section. For n=5, A2= 0.577 Where A2 is

obtained from the coefficients table for variable control charts

Table: 1 Sa

mp

le

no

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1 176 221 242 253 260 265 271 278 286 301 265 205 263 307 220 200 235 246 328 296

2 187 223 243 254 261 265 272 278 287 307 268 260 234 299 215 276 264 269 235 290

3 197 228 245 254 262 265 274 280 290 308 197 286 274 243 231 221 176 248 263 231

4 200 231 246 257 262 267 274 280 293 317 267 281 265 214 318 334 280 260 272 283

5 205 231 248 258 263 267 274 280 294 318 346 317 242 258 276 221 262 271 245 301

6 208 234 248 258 263 268 274 280 296 321 300 208 187 264 271 334 274 253 287 258

7 210 235 250 258 264 269 275 281 298 328 280 242 260 321 228 265 248 260 274 337

8 214 235 250 260 264 269 276 281 299 334 250 299 258 267 293 280 250 278 254 274

9 215 235 250 260 265 270 276 283 299 337 265 254 281 294 223 261 278 250 265 270

10 220 242 251 260 265 271 277 283 300 346 260 308 235 283 277 257 210 280 269 251

The values for „r‟ and is given below, where r = a, b, c, d


(Note: Refer To Appendices)

Fuzzy Control Chart Based on Range

By using the above and , the control limits of fuzzy control charts with ranges based on fuzzy trapezoidal

number are calculated as follows

=C + = ( ) + A2

=(240.42,287.64,263.1,264.4) + 0.577(31.9, 54.1, 86.8, 96.7)

= ( )

= (258.82, 318.88, 313.19, 320.19)

= ( ) = ( ) )

= (240.42, 287.64, 263.1, 264.4)

C - = ( ) - A2

= (240.42, 287.64, 263.1, 264.4)- 0.577(31.9,54.1,86.8,96.7)

= ( )

= (220.02, 256.42, 213.02, 208.61)

α - Cut Fuzzy Control Chart Based on Ranges

α - Cuts in the control limits provide the ability of determining the tightness of the sampling process. α - Level can be

selected according to the nature of the production process. α - level was defined as 0.6 this production process

= 263.62

= d+ α (d – c) =

= ( ) +

= (268.75,287.64 ,263.1,263.62) + 0.577(45.22,54.1,86.8,90.76)

= ( )


= (294.84,318.85 ,313.19,315.9)

= ( ) = ( )

= (268.75,287.64,263.1,263.62)

( ) -

= (268.75,287.64,263.1) – 0.577(45.22,54.1,86.8,90.76)

= (

= (242.66, 256.43, 213.02, 211.26)

α - LEVEL FUZZY MIDRANGE FOR α CUT FUZZY CONTROL CHART BASED ON RANGES

The control limits for α - level fuzzy midrange for - α Cut Fuzzy control chart based on ranges can be obtained as

follows

= 266.18 + 0.577[ ] = 305.41

= = 266.18

= 266.18 - 0.577[ ] = 226.95

FUZZY CONTROL CHART

= (67.46, 114.42, 183.58, 204.52)

= (31.9, 54.1, 86.8, 96.7)

= (0, 0, 0, 0)

Where , n =5, and are obtained from the coefficients table for variable control charts.

α – Cut Fuzzy Control Chart

The control limits of α - cut fuzzy control chart based on trapezoidal fuzzy numbers are obtained as follows


= (95.6, 114.42, 183.5, 191.9)

= (45.22, 54.1, 86.8, 90.76)

= (0, 0, 0, 0)

α - LEVEL FUZZY MIDRANGE FOR α - CUT FUZZY

CONTROL CHART

The control limits of α - Level fuzzy midrange for α - Cut Fuzzy

Control chart based on fuzzy Trapezoidal number can be calculated as follows

= 2.115[ ] = 146.15

= 67.99

= 0

The values of and have been calculated by using the formula of α - Level fuzzy midrange for α - Cut

Fuzzy control chart based on ranges and α - Level fuzzy midrange for α - Cut Fuzzy control chart respectively and

the values are given in Table 2.

Control Limits using α- Level Fuzzy Mid Range for α -cut Fuzzy Control Chart Based on Ranges and α- Level

Fuzzy Mid Range for α -Cut Fuzzy Control Chart

Table: 2

Sample

No

1 257.44 In Control 82.4 In Control


3 253.38 In Control 70 In Control








CONCLUSIONS

This paper shows that this process was in control with respect to and for each sample as

shown in table 2. So, these control limits can be used to control the production process. Since the Plotted values are close

to the control limits .Fuzzy observations & Fuzzy control limits can provide more flexibility for controlling a process. The


α - Level fuzzy midrange transformation techniques are used to illustrate applications in a production process. The

methodology can be extended to variable samples for production processes.

REFERENCES

1. A.Pandurangan,R.Varadharajan. (2011)

2. Cheng, C.B. (2005). Fuzzy Process Control: Construction of control charts with fuzzy number. Fuzzy Sets and

Systems, 154, 287-303.

3. El – Shal, S. M., Morris A. S. (2000). A fuzzy rule -based algorithm to improve the performance of statistical

process control in quality Systems, Journal of Intelligent Fuzzy Systems, 9, 20 7 – 223.

4. Gulbay, M., Kahraman, C and Ruan D. (2004). α - Cut fuzzy control charts for linguistic data.International

Journal of Intelligent Systems, 19, 1173-1196.

5. Gulbay, M and Kahraman, C. (2006) . Development of fuzzy process control charts and fuzzy unnatural pattern

analysis”. Computational Statistics and Data Analysis, 51, 434-451.

6. Gulbay, M and Kahraman, C. (2006). An alternative approach to fuzzy control charts: direct fuzzy

approach.Information Sciences, 77(6), 1463-1480.

7. Kolarik, W.J, (1995). Creating Quality- Concepts, Systems Strategies and Tools, McGraw – Hill.

8. Montgomery, D.C., (2002). Introduction to Statistical Quality Control, John Wiley and Sons, New York

9. Rowlands, H and Wang, L.R (2000). An approach of fuzzy logic evaluation and control in SPC. Quality

Reliability Engineering Intelligent, 16, 91-98.

10. Sentruk, S and Erginel, N. (2009). Development of Fuzzy and charts using α- cuts. Information

Sciences, 179(10),1542-1551.

APPENDIX

The fuzzy ranges for the ; r = a, b, c, d values for the 10 samples are calculated as follows

1. = 253 – 200 = 53

= 301 - 265 = 36

= 307 -205 = 102

= 328 – 176 = 152

2. = 261 – 235 = 26

= 307 – 265 = 42

= 299 – 215 = 84

= 290 – 187 = 103

3. = 262 – 176 =86

= 308 – 265 = 43

= 286 - 197 =89

= 263 – 197 = 66

4. = 262 – 260 = 2

= 317 - 267 = 50

= 318 – 214 = 104


= 334 – 200 = 134

5. = 263 – 221 = 42

= 318 – 267 = 51

= 346 – 242 = 104

= 301 – 205 = 96

6. = 263 – 253 = 10

= 321 – 268 = 53

= 300 – 187 = 113

= 334 – 208 = 126

7. = 264 – 248 = 16

= 328 – 269 = 59

= 321 – 242 = 79

= 337 – 210 = 127

8. = 264 – 250 = 14

= 334 – 269 = 65

= 299 – 250 = 49

= 280 – 214 = 66

9. = 265 – 250 = 15

= 337 – 270 = 67

= 294 – 223 = 71

= 278 – 215 = 63

10. = 265 – 210 =55

= 346 – 271 = 75

= 308 – 235 = 73

= 280 – 220 = 60

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