Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 217406, 8 pages
http://dx.doi.org/10.1155/2015/217406

Research Article

Measuring Loss-Based Process Capability Index and Its Generation with Fuzzy Numbers

Mohammad Abdolshah

Department of Industrial Engineering, Faculty of Engineering, Islamic Azad University, Semnan Branch, Semnan 3519744571, Iran

Received 19 June 2014; Revised 28 October 2014; Accepted 31 October 2014

Academic Editor: Mohamed Abd El Aziz

Copyright © 2015 Mohammad Abdolshah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Loss-based process capability indices are appropriate and realistic tools in order to measure the process capability. Among them, index and its generation are well-known loss-based process capability indices, whose concepts are based on the worth (the opposite concept of loss). Sometimes, in order to calculate and there are some uncertainties in observations, so fuzzy logic can be employed to manage the uncertainties. This paper investigates fuzzification of process capability index and its generation . In order to find the membership function of process capability indices and , the α-cuts of fuzzy observation were employed. Then with an example of fuzzy process capability index, and were calculated and compared. Results showed that fuzzy was more sensitive compared with and was increased while the target departs (asymmetric tolerance). This example also showed that, with departure from the target, variation of fuzzy and consequently its fuzziness were increased.

1. Introduction

Nowadays manufacturers are so interested in understanding the capability of their processes in order to improve them [1]. Process capability can be defined as a measurement of inherent variability in a process compared to the specification requirements of the product [2–4]. The process capability indices (PCIs) , , , , and have been described as five original PCIs by Sullivan [5] who had observed the usage of these indices at Japanese manufacturing facilities. Since PCIs present the capability of process, loss-based PCIs are more appropriate indices in order to measure the real capability of a process. The most well-known loss-based PCIs are , [6], , and [7].

Fuzzy set theory which was introduced by Zadeh [8] has had a wide application in many fields such as industrial and production management [9, 10]. Fuzzy set theory and fuzzy logic also have been applied in measuring of the fuzzy process capability indices and till now there are some papers about some fuzzy PCIs such as , , and . This paper is an extension of Tsai and Chen [11] and Chen et al. [12] for the process capability index and its generation of fuzzy numbers and their differences.

The organization of this paper is as follows. Section 2 reviews the literature of loss-based process capability indices and fuzzy PCIs. Section 3 presents the α-cuts of fuzzy estimation for index with fuzzy data. Similarly Section 4 describes fuzzy and Section 5 illustrates a numerical example. Finally Sections 6 and 7 deal with conclusions and suggestions for further research.

2. Literature Review

Kane [3] developed and indices, which are commonly used in industries to evaluate single quality characteristics in mass production. Since and indices do not take into account the difference between the processes mean and its target value and the loss of process (rejects and reworks), Chan et al. [13] considered this difference to develop and later Pearn et al. [14] developed . The indices and are initial loss-based PCIs which were based on Taguchi loss function [7]. Another well-known loss-based PCI is which was proposed by Johnson [15]. This index was inspired from the process losses similar to and concept of relative expected squared error. This process capability index actually uses the concept of worth (the opposite concept of loss). It was assumed that a quality characteristic can achieve its maximum worth , when . So when moves away from the target, the worth becomes zero and then negative. The worth function can be defined by , for , and it will be zero when . So in order to define the specification limits of , Johson [15] used this concept and defined for specification limits of . Johson [15] defined ratio of worth to maximum worth as follows:

where and are the relative loss. Then is defined as follows [15]:

where . The function is related to ratio of worth. For example, supposing that , then is the index which has the negative relationship with . Because the is a PCI between , then will be a loss function between . The index is defined as the ratio of the expected quadratic loss and the square of the half specification width as follows:

where is a constant parameter derived from standard table. Then can be defined as follows:

Tsui [16] expressed the above equation as . So Tsui [16] interpreted as a summation of potential relative expected loss and the relative off-target squared deviation .

The index has some weak points. This index uses expected relative loss and considers the proximity of the target value, while users need to specify the target and the distance from the target at which the product would have zero worth to quantify the process loss. Another weak point of is that it does not take into account the asymmetric tolerances. Pearn et al. [17] with an example showed this weak point and proposed which is derived from . The index is defined as follows:

where , , , and . If the tolerances are symmetric , then , , and , where is the midpoint of tolerance. Pearn et al. [17] also proposed the following estimator for :

where and the mean is estimated by the sample mean and by sample standard deviation. There are some other PCIs such as [18], but since is a loss-based PCI, it can take into account the target and have more reliable and reasonable outputs.

Some research on fuzzy process capability indices has been conducted. An initial study by Yongting [19] defined and explained fuzzy quality and analysis on fuzzy probability. Fuzzy α-cuts method is another methodology that has been used vastly in fuzzy PCIs studies. This methodology was introduced for the first time by Lee [20]. Chen et al. [21] introduced a fuzzy evaluation of for selecting a supplier. They employed Mamdani inference method [22] to interpret the different amount of for fuzzy data. Their methodology was based on the usage of confidence intervals. Tsai and Chen [11] had a survey about making a decision to evaluate a process capability index with fuzzy numbers. Their methodology was using α-cuts method based on Kao and Liu [23] in order to find the membership function of the fuzzy . In another study Chen et al. [12] used this methodology in order to calculate the fuzzy process capability . Buckley’s estimation approach [24–26] is another methodology which was employed by Parchami and Mashinchi [27] to estimate process capability indices , , and .

Recently these fuzzy indices have been used in real projects; for example, Kaya and Kahraman [28] developed fuzzy robust process capability indices for risk assessment of air pollution.

Most of the studies are based on fuzzy data and in this study we also assumed fuzzy observations with the methodology of Tsai and Chen [11] and Chen et al. [12] for fuzzy indices and .

3. Measuring Process Capability Index with Fuzzy Data

Fuzzy logic can be employed to manage the uncertainties. These uncertainties could exist in the concepts of data which are being used to measure the capability of processes. Different relevant uncertainties are as follows:

(1)specification limits cannot always be represented by crisp numbers and are defined by fuzzy numbers [28];(2)interpretation of a capable process is sometimes fuzzy [21];(3)the specification data and observations can be fuzzy [11, 12, 27, 29].

The first uncertainty is not common, because specification limits are usually defined exactly by customers. The second uncertainty also is a kind of estimation and has error, so, in this paper, we considered the last kind of uncertainty. If specification data is fuzzy, the process capability indices will be fuzzy. For simplicity, we supposed that our observations were triangular fuzzy numbers defined as:

where is the number of fuzzy parameters. We can assume fuzzy observations as a fuzzy set. Each observation or number in a fuzzy set has a membership function. So a fuzzy set consists of data and their memberships. For example, set and fuzzy variable are ordered in pairs as follows [30]:

where represents the membership function of variable in . The membership function of a fuzzy variable is the most important concept of a fuzzy variable. So in order to measure , its membership function must be achieved:

where and are defined as

where is the number of fuzzy parameters. According to (10), calculation of and its membership function is so complicated. This complexity has two main reasons:

(1)there are two fuzzy functions ( and ) in (10) which makes this equation more complicated;(2)the relationship between the index and the fuzzy observations is nonlinear.

In order to overcome this problem, according to the methodology of Tsai and Chen [11] and Chen et al. [12], the α-cuts of the fuzzy observation based on Kao and Liu [23] can be used. Consequently, the exact form of the membership function can be derived by taking the membership functions of the α-cut. The α-cuts of the fuzzy observation were presented as follows:

where is the crisp universal set , is alpha cut of lower part of parameter, and is alpha cut of upper part of parameter, in which is defined. For a typical , the yield of will be . On the other hand, stated observations are as triangular fuzzy numbers denoted by the triplet and these observations have the shape of a triangle as shown in the Figure 1.

Figure 1: A triangular fuzzy number .

Each α-cut of the triangular fuzzy number is which is a closed interval as follows:

Similarly are the α-cuts of , are the α-cuts of , and are the α-cuts of . According to (10) and (12) and Kao and Liu [23], lower and upper bounds of the α-cuts of can be obtained as follows:

Since (14a) and (14b) denote nonlinear relationship between and α-cuts of , nonlinear programming package, LINGO [31], can be employed. It was shown in part 5 that each α-cut of for different α from 0 to 1 must be calculated. Finally, the membership function is constructed as

4. Measuring the Process Capability Index with Fuzzy Data

As stated in (5) Pearn et al. [17] proposed   which is derived from . The index , which is defined as follows, is a combination of two separate indices and :

In order to estimate the fuzzy membership of , the fuzzy membership of and was calculated and then was combined to each other. Using (16), the fuzzy can be calculated as follows:

where the statistics and are defined as stated in (11). Since, in the above equation, there are two fuzzy functions ( and ) and also nonlinear relationship between the index and the fuzzy observations , the calculation of is so complicated, and so, according to the methodology of Tsai and Chen [11] and Chen et al. [12], the α-cuts of the fuzzy observation can be used. Consequently, the exact form of the membership function can be derived by taking the membership functions of the α-cut. Similarly are the α-cuts of and according to Kao and Liu [23] lower and upper bounds of the α-cuts of can be obtained as follows:

Similarly LINGO [31] can be employed in order to calculate and for various α and then the membership function, , is constructed as

5. A Numerical Example-Comparison with Other Fuzzy PCIs

In order to clarify the fuzzy process capability indices of and , we used the numerical example of Chen et al. [12] which was for the fuzzy PCI . In this example, there are 30 fuzzy observations which are shown in Table 1. The production specifications for a particular product are as follows: , , , and . In order to measure the fuzzy process capability indices and , α-cuts of the fuzzy observations must be calculated. So, based on (9), α-cuts of these fuzzy observations for 11 various α were calculated and shown in Table 2. Note that in Table 2 just α-cuts of first 5 observations were mentioned and because of their similarity other α-cuts were omitted.

Table 1: An example consisted of 30 fuzzy triangular observations.

= [5.85 6.15 6.35]
= [5.86 6.04 6.25]
= [5.50 5.81 5.99]

= [5.79 5.9 5.98]
= [6.13 6.23 6.33]
= [5.60 5.92 6.05]

= [5.71 5.83 5.99]
= [5.95 6.05 6.19]
= [5.50 5.75 5.95]

= [6.05 6.18 6.32]
= [5.6 5.65 5.70]
= [5.84 6.03 6.15]

= [5.89 6.06 6.23]
= [5.65 5.74 5.84]
= [6.05 6.30 6.50]

= [6.01 6.10 6.25]
= [5.70 5.77 5.83]
= [6.25 6.35 6.45]

= [6.15 6.20 6.30]
= [6.23 6.32 6.40]
= [5.65 5.86 6.05]

= [5.64 5.81 6.05]
= [5.60 5.70 5.80]
= [5.70 5.87 5.95]

= [5.8 5.90 5.98]
= [5.85 5.95 6.05]
= [5.75 5.95 6.15]

= [6.01 6.12 6.24]
= [5.90 6.00 6.10]
= [6.10 6.23 6.46]

Table 2: Different α-cuts of the first five observations.

Mid

α
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

5.85
6.00
6.15

5.85
5.88
5.91
5.94
5.97
6
6.03
6.06
6.09
6.12
6.15

6.35
6.33
6.31
6.29
6.27
6.25
6.23
6.21
6.19
6.17
6.15

5.79
5.845
5.9

5.79
5.8
5.81
5.82
5.83
5.85
5.86
5.87
5.88
5.89
5.9

5.98
5.972
5.964
5.956
5.948
5.94
5.932
5.924
5.916
5.908
5.9

5.71
5.77
5.83

5.71
5.72
5.73
5.75
5.76
5.77
5.78
5.79
5.81
5.82
5.83

5.99
5.974
5.958
5.942
5.926
5.91
5.894
5.878
5.862
5.846
5.83

6.05
6.115
6.18

6.05
6.06
6.08
6.09
6.1
6.12
6.13
6.14
6.15
6.17
6.18

6.32
6.306
6.292
6.278
6.264
6.25
6.236
6.222
6.208
6.194
6.18

5.89
597.5
6.06

5.89
5.91
5.92
5.94
5.96
5.98
5.99
6.01
6.03
6.04
6.06

6.23
6.213
6.196
6.179
6.162
6.145
6.128
6.111
6.094
6.077
6.06

According to (11) the functions and can be calculated. Equation (7) stated the α-cut of fuzzy number or variable. So some functions such as , , and also can have fuzzy α-cuts. Calculation of these functions is not useful directly, but it can help in order to define the appropriate operations research model in LINGO [31]. The results of α-cuts of , , and are shown in Table 3.

Table 3: Results of α-cuts for 11 various α.

PCI
α

0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

0.0124
0.0138
0.0153
0.0171
0.0191
0.0214
0.024
0.0269
0.0303
0.0341
0.0383

0.1065
0.097
0.0881
0.0798
0.072
0.0678
0.0607
0.0547
0.0489
0.0434
0.0383

5.8436
5.8585
5.8734
5.8882
5.9031
5.9180
5.9328
5.9477
5.9626
5.9774
5.9923

6.1293
6.1156
6.1019
6.0882
6.0745
6.0608
6.0471
6.0334
6.0197
6.006
5.9923

0.1455
0.1301
0.1146
0.0992
0.0838
0.0684
0.0530
0.0376
0.0222
0.0068
0.0087

0.1760
0.1592
0.1424
0.1258
0.1090
0.0922
0.0756
0.0588
0.0421
0.0254
0.0087

0.1164
0.1040
0.0917
0.0794
0.0670
0.0547
0.0424
0.0301
0.0177
0.0054
0.0069

0.1408
0.1274
0.1139
0.1006
0.0872
0.0738
0.0605
0.0471
0.0337
0.0203
0.0069

0.1455
0.1301
0.1146
0.0992
0.0838
0.0684
0.0530
0.0376
0.0222
0.0068
0.0087

0.1760
0.1592
0.1424
0.1258
0.1090
0.0922
0.0756
0.0588
0.0421
0.0254
0.0087

Using (11), α-cuts of could be calculated. The results were shown in Table 4. In order to calculate α-cuts of , at first, α-cuts of and were calculated and then they were added together. Figure 2 shows the relation of and more clearly.

Table 4: Results of α-cuts of for 11 various α.

PCI
α

0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

0.0771
0.0854
0.0948
0.1056
0.1178
0.1315
0.1473
0.1651
0.1851
0.2074
0.2323

0.6100
0.5600
0.5124
0.4673
0.4249
0.4010
0.3615
0.3275
0.2944
0.2622
0.2323

0.0111
0.0122
0.0138
0.0154
0.0173
0.0197
0.0222
0.0249
0.0287
0.0330
0.0377

0.1398
0.1228
0.1078
0.0944
0.0824
0.0763
0.0663
0.0581
0.0506
0.0438
0.0377

0.0882
0.0976
0.1086
0.1210
0.1351
0.1512
0.1695
0.190
0.2138
0.2404
0.27

0.7498
0.6828
0.6202
0.5617
0.5073
0.4773
0.4278
0.3856
0.3450
0.306
0.27

Figure 2: Membership function of , , and .

Now in order to calculate , we calculate α-cuts 9 of , , , and . The results are shown in Table 5 and Figure 3 for various .

Table 5: Results of α-cuts of for 11 various α.

PCI
α

0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

0.0769
0.0851
0.0945
0.1052
0.1173
0.1309
0.1465
0.1642
0.1839
0.2058
0.2303

0.5951
0.5477
0.5022
0.4589
0.4181
0.3950
0.3567
0.3236
0.2913
0.2597
0.2303

0.0010
0.0012
0.0015
0.0018
0.0023
0.0029
0.0036
0.0045
0.0057
0.0073
0.0092

0.0709
0.0588
0.0485
0.0398
0.0324
0.0287
0.0230
0.0187
0.0149
0.0118
0.0092

0.0779
0.0863
0.0960
0.1070
0.1196
0.1338
0.1501
0.1687
0.1896
0.2131
0.2395

0.6660
0.6065
0.5507
0.4987
0.4505
0.4237
0.3797
0.3423
0.3062
0.2715
0.2395

Figure 3: Membership function of , , and .

As shown in Figure 2 the indices , , and at have the intersection points with -axis. For example, varies from 0.011 to 0.139 and its highest membership is in 0.077. So the membership functions of them are as follows:

Similarly as shown in Figure 3 the indices , , and at have the intersection points with -axis. So their membership functions are as follows:

As shown in Figures 2 and 3, despite the triangular fuzzy observations, the membership functions of and are not triangular functions, but as concluded from Figures 2 and 3 the indices and such as other fuzzy PCIs have fuzzy behavior. In this example, fuzzy index varies from 0.088 to 0.794. Similarly, fuzzy index varies from 0.07 to 0.666. The most probable value of is 0.232, because this point has the maximum membership at 1 and the most probable quantity of is 0.239.

In order to compare the behavior of and , we shifted the target point (asymmetric tolerance). Initial target was 6 and in this example we supposed that target departs from 6 to 5.70. Then will vary from 0.05 to 0.3. Similar methodology was employed in order to calculate fuzzy α-cuts of and . Results showed that was constant and was not sensitive to the departure from target, while fuzzy index was sensitive to the changes of target. The results of fuzzy α-cuts of are shown in Table 6 and Figure 4.

Table 6: Results of α-cuts of for various α for different .

α

0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

0.35
0.068
0.076
0.084
0.094
0.105
0.117
0.131
0.148
0.166
0.186
0.210

0.35
0.583
0.531
0.482
0.436
0.394
0.371
0.332
0.300
0.268
0.238
0.210

0.3
0.058
0.065
0.072
0.080
0.090
0.100
0.113
0.127
0.142
0.160
0.180

0.3
0.500
0.455
0.413
0.374
0.338
0.318
0.285
0.257
0.230
0.204
0.180

0.25
0.049
0.054
0.060
0.067
0.075
0.084
0.094
0.105
0.119
0.133
0.150

0.25
0.416
0.379
0.344
0.312
0.282
0.265
0.237
0.214
0.191
0.170
0.150

0.2
0.039
0.043
0.048
0.054
0.060
0.067
0.075
0.084
0.095
0.107
0.120

0.2
0.333
0.303
0.275
0.249
0.225
0.212
0.190
0.171
0.153
0.136
0.120

0.15
0.029
0.032
0.036
0.040
0.045
0.050
0.056
0.063
0.071
0.080
0.090

0.15
0.250
0.227
0.207
0.187
0.169
0.159
0.142
0.128
0.115
0.102
0.090

0.1
0.019
0.022
0.024
0.027
0.030
0.033
0.038
0.042
0.047
0.053
0.060

0.1
0.167
0.152
0.138
0.125
0.113
0.106
0.095
0.086
0.077
0.068
0.060

0.05
0.010
0.011
0.012
0.013
0.015
0.017
0.019
0.021
0.024
0.027
0.030

0.05
0.083
0.076
0.069
0.062
0.056
0.053
0.047
0.043
0.038
0.034
0.030

Figure 4: Results for fuzzy α-cuts of , whereas the target departs.

As stated the index was more sensitive in process yield, so in this example, because of shift in process mean, is higher than . Figure 4 showed that with increasing the deviation of these fuzzy data the fuzzy was increased. Fuzzy was grown while the variation of this index also increased. For instance for the fuzzy varies from 0.01 to 0.083 with yield of 0.073. When , then the fuzzy varies from 0.078 to 0.666 with yield of 0.588. This variation for a fuzzy index such as is so much and shows more fuzziness in results.

6. Conclusions

Loss-based process capability indices such as and are more realistic and suitable tools to measure the capability of a process. This study investigated fuzzy process capability indices and . In order to find the membership functions of fuzzy process capability PCIs and , the methodology of Kao and Liu [23] for α-cuts of the fuzzy observations and converting them to a nonlinear model was employed. Moreover, with an example, the behaviors of fuzzy indices and were investigated. In this example the target departed (asymmetric tolerance) gradually and results showed that fuzzy was increased and it was more sensitive compared with (in case of asymmetric tolerance), but in this case when the target departed, was not a fully reliable fuzzy PCI because with departure from target, variation of fuzzy and consequently its fuzziness were increased.

7. Further Research

Fuzzy indices and in the case of symmetric tolerance are the same. When tolerance is asymmetric, the fuzzy index is more sensitive because it takes into account the target, but with departure from the target (more asymmetric tolerance) variation of fuzzy , and consequently its fuzziness was increased. So this fuzzy loss-based PCI also is not a reliable PCI and there is a necessity to develop a more accurate fuzzy loss-based PCI.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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