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