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.,

α - CUT FUZZY CONTROL CHARTS FOR BOTTLE BURSTING STRENGTH DATA 1

A. SARAVANAN & 2P. NAGARAJAN

ISSN 2249–6939

1

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

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

Professor, Department of Chemical Engineering, Annamalai University, India

© TJPRC Pvt. Ltd.,

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

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

control charts with α cuts by using and

control charts.

18

A. Saravanan & P. Nagarajan

1. First transform the traditional and 2.

and

control charts to fuzzy control charts. To obtain fuzzy

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

The cut fuzzy

control charts and cut fuzzy

control charts are developed by using cut

approach. 3.

-level

The

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 all elements whose membership is greater than or equal to. If (

and

, is a non fuzzy set that comprises

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

is the arithmetic

)

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 Limits for Fuzzy

control chart.

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

20

A. Saravanan & P. Nagarajan

(

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) ; dα = 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 - Cut Fuzzy Control Charts for Bottle Bursting Strength Data

α - Level Fuzzy Midrange for α- Cut Fuzzy

21

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 Where

and

; CLR =

; UCLR = D3

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 The control limits of α - cut fuzzy

Control Chart control chart based on trapezoidal fuzzy numbers are obtained as follows

22

A. Saravanan & P. Nagarajan

) ) ) 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

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 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

Where

chart based on standard deviation is given below

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.

α - Cut Fuzzy Control Charts for Bottle Bursting Strength Data

Fuzzy

23

Control Chart Based on Standard Deviation

The theoretical structure of fuzzy (2009). The fuzzy

control chart and fuzzy

control chart has been developed by Senturk and Erginel

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

)+

)

, =(

) = (

) =(

(

)-

)

, =(

)

)

24

A. Saravanan & P. Nagarajan

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

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

Where

}

and

control chart is given by

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 - Cut Fuzzy Control Charts for Bottle Bursting Strength Data

25

) ) α - Level Fuzzy Midrange for α - Cut Fuzzy

Control Chart

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

control chart can be obtained in a similar way to α -

control chart.

The definition of α - level fuzzy midrange of sample j for fuzzy

control chart can be calculated as follows

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

26

A. Saravanan & P. Nagarajan

(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)

=

(

)

α - Cut Fuzzy Control Charts for Bottle Bursting Strength Data

27

= (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[

FUZZY

] = 226.95

CONTROL CHART = (67.46, 114.42, 183.58, 204.52) = (31.9, 54.1, 86.8, 96.7) = (0, 0, 0, 0)

Where α – Cut Fuzzy

, n =5, and

are obtained from the coefficients table for variable control charts.

Control Chart

The control limits of α - cut fuzzy

control chart based on trapezoidal fuzzy numbers are obtained as follows

28

A. Saravanan & P. Nagarajan

= (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 the values are given in Table 2.

control chart respectively and

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

2 3

261.18 253.38

In Control In Control

63.6 70

In Control In Control

4

271.64

In Control

73.4

In Control

5

272.52

In Control

74.1

In Control

6

264.82

In Control

71.8

In Control

7

270.96

In Control

70

In Control

8

271.92

In Control

50.2

In Control

9

268.88

In Control

57

In Control

10

270.6

In Control

67.4

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 - Cut Fuzzy Control Charts for Bottle Bursting Strength Data

29

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 1.

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

30

A. Saravanan & P. Nagarajan

= 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