Fuzzy Control Charts based on Ranking of Pentagonal Fuzzy Numbers Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

Fuzzy Control Charts based on Ranking of Pentagonal

Fuzzy Numbers

*Mohammad Ahmad, 2Weihu Cheng, 3Zhao Xu, 4Abdul Kalam, 5Ahteshamul Haq

1,2,3,4 Faculty of Science, Beijing University of Technology, Beijing, China ^Department of Mathematics and Statistics, Integral University Lucknow, Uttar Pradesh India ' [email protected], [email protected], [email protected], [email protected],[email protected] *Corresponding author

Abstract

A Control Chart is a fundamental approach in Statistical Process Control. When uncommon causes of variability are present, sample averages will plot beyond the control boundaries, making the control chart a particularly effective process monitoring approach. Uncertainties are caused by the measuring system, including the gauges operators and ambient circumstances. In this paper, the concept of fuzzy set theory is used for dealing with uncertainty. The control limits are to converted into fuzzy control limits using the membership function. The fuzzy X-R and X-S control chart is developed by using the ranking of the pentagonal fuzzy number system. An illustrative example is done with the discussed technique to make fuzzy X-R and X-R control charts and increase the flexibility of the control limit.

Keywords: Statistical Process Control, Rank Membership Function, Fuzzy Pentagonal Number

I. Introduction

Quality has long been recognized as a significant influencing element in the performance and competitiveness of manufacturing and service organizations in both domestic and global markets. The return on capital is the consequence of well-executed strategies. Appropriate quality techniques provide productive outcomes. Fuzzy number ranking is a component of the quality control planning system. The fuzzy mathematical model for transportation of vegetable diet plan based on the ranking function of fuzzy pentagonal number (FPN) and solved by Vogel's approximation method to minimize the cost is discussed by Venkatesh and Manoj [1]. In constructing an FPN and related arithmetic operations, A. Panda and M. Pal [2] established the logical definition. The construction and fundamental features of pentagonal fuzzy matrices (PFMs) are investigated using FPN. The algebraic natures of several particular types of PFMs (trace of PFM, adjoint of PFM, determinant of PFM, etc.) are addressed. Pathinathan and Ponnivalavan [3] discussed FPN in continuation with the other defined fuzzy numbers and addressed some basic arithmetic operations. A. Chakraborty et al. [4] discuss different measures of interval-valued pentagonal fuzzy numbers (IVFPN) associated with assorted membership functions, and the ranking function is the main feature. The ranking functions of FPN develop real application and comprehend the uncertainty of the parameters more precisely in the evaluation process. A.

Mohammad Ahmad, Weihu Cheng, Zhao Xu, Abdul Kalam, Ahteshamul Haq RT&A, No 4 (76) Fuzzy Control Charts based on Ranking of Pentagonal Fuzzy Numbers_Volume 18, December 2023

Chakraborty et al. [5] dealt with the idea of pentagonal neutrosophic number (PNN) from a

different frame of reference and discussed some properties of PNN with real-life operational

research applications, which is more reliable than the other method. A. Shafqat et al. [6] used the

lower record values for developing the X control chart for the Inverse Rayleigh Distribution (IRD)

is designed under repetitive group sampling. The mean and standard deviation of the Inverse

Rayleigh Distribution based on lower record values are used to determine the width and power of

the X control limits. Lim S. A. H. [7] evolved X-R and X-S chart for the food industry in the UK. The

control charts developed using triangular and trapezoidal fuzzy control for balanced and

unbalanced [8, 9, 10]. Ozdemir [11] developed the fuzzy control chart with a triangular fuzzy

system into three phases for 1-S chart and process capability indices using unbalanced data,

converted the data for each sample into the fuzzy form and then decided the fuzzy limits and

illustrated it for uncertain data. Yeh [12] Shows an example of weighted triangular approximation

of fuzzy numbers, which Zheng and Li propose. Senturk et al. [13] researched the most popular

control chart for univariate data, the exponential weighted moving average control chart under

fuzzy environment and applied this work into real case applications in Turkey. Senturk et al. [14]

consider the control chart for fuzzy nonconformities per unit by using alpha cut and applied this

technique in real case applications for truck engine manufacture. Alipour and Noorossana [15]

created a control chart using a fuzzy multivariate exponentially weighted moving average (F-

MEWMA). The proposed technique is developed using a combination of multivariate statistical

quality control and fuzzy set theory in this study. Erginel and §enturk [16] derived the fuzzy

exponential weighted moving average and cumulative sum control chart (CUSUM) with a suitable

example. Erginel [17] developed a fuzzy P control chart by using the rules that introduces the

fuzzy np chart based on the constant sample size and variable sample size. In addition, the

decision is taken whether it is under control or out of control. In the uncertainty theory for

modelling, fuzzy sets theory plays numerous important roles. An essential consideration is that if

somebody wants to take an FPN, what should its graphical representations (uncertainty

quantification area) look like? How should the membership functions be defined? From this

perspective, we developed the phases of an FPN control chart that may be a good choice for a

decision-maker in a real-world scenario. In this study we proposed a Fuzzy control chart by using

rank membership function of Pentagonal fuzzy number and the example is done with this

proposed technique.

II. Development of Proposed Methodology

There are many articles published related to FPN. Venkatesh and Manoj [1] developed the pentagonal fuzzy model for transportation problems using the ranking membership function. Some definitions of FPN are as follows:

Definition 1: (Zadeh [19]) Let X be a fixed set. A fuzzy set A of X is an object having the form A = {(x, juA (x)): x g X} where ¡u , (x) e [0. l] represents the degree of membership of the element

iel being in A, and //, : X —> [(). l] is called the membership function.

Definition 2: The a -cut of the fuzzy set A is defined as:

Aa= (x £ x/ju^x) > a, where a 6 (0, 1).

Definition 3: A set A is defined on the real numbers ^ is said to be a fuzzy number if its membership function /LI- :X—>[0,l] follows:

(i) A is continuous.

(ii) A is normal such that // , (x) = 1 there exists an x £ R

Mohammad Ahmad, Weihu Cheng, Zhao Xu, Abdul Kalam, Ahteshamul Haq RT&A, No 4 (76)

Fuzzy Control Charts based on Ranking of Pentagonal Fuzzy Numbers_Volume 18, December 2023

Definition 4: A fuzzy number ÄP is an FPN denoted by ÄP = (a1,a2,a3,a4,a5), where a1, a2, a3, a4, a5 are real numbers, and its membership function will be:

=

0

(x-a-i)

(a2-ai)

1 (x-a2)

2 (a3-a.2)

1 , (a4-x) 2(0.4-0.3) ' (a5-x)

(a5-a4) 0

x < a

1

a -, < x < a?

a2 < x < a

3

x = a

3

(1)

a3 < x < a4

a4 < x < a5 , x > a5

Ranking of FPN: Ranking a fuzzy number entails comparing up to two fuzzy numbers, and defuzzification is a technique for converting a fuzzy number to an estimated crisp number. Just as the decision-maker compares two ideas that are the same, we must convert the fuzzy number to a comparable crisp number and compare the numbers based on crisp values in this problem.

Fuzzy numbers become the real line directly by using the ranking method [1]. Let ÁP be a generalized FPN. The ranking of ÁP is symbolised by R(ÁP) and it is calculated as follows:

r(A ) = |"i+2a2+3a3+2a4+n5j

Statistical Process Control: Many quality attributes may be stated numerically. A bearing's diameter, for example, might be measured using a micrometre and represented in mm. A variable is a single quantifiable qualitative attribute, such as a dimension, weight, or volume. Control charts for variables are widely used. When dealing with a variable quality characteristic, it is frequently required to monitor both the mean value and the variability of the quality characteristic. The control chart for means, or the control chart, is typically used to regulate the process average or mean quality level. Process variability may be tracked using either a control chart for the standard deviation known as an S control chart or a control chart for the range, known as an R control chart. The X , R-chart and X , S-chart is the most widely used control chart for the production process [16, 18].

X and R control charts

Table 1: Formula for X and R Control Charts

Chart Lower Control Limit Central Upper Control Limit

(LCL) Line (UCL)

X 1-a2r X % + a2r

R RD3 R rd4

where, X = —, x = Zi-iR(ap(xi))>, & is constant tabulated value for n.

m n

where, R = ^Ran3e(R(Ap(xi)^), and D3,D4 are constant tabulated values for n.

X and S Control Charts

Table 2: Formula for X and S Control Charts

Charts LCL Central UCL

Line

X X-A3S X X +A3S

5 B3S S B4S

where, X = —, x = ^=lR(Ap(Xi)), s= P=l(Xi *)2, S = and A3,B3,B4 are constant tabulated

m n -\J (n-1) m 334

values for n.

The next step will be complete with three steps for the proposed fuzzy control chart by using the following procedure.

Step 1: Normality Assumption- Check the normality assumption using Anderson Darling Test (Anderson and Darling, 1954).

Step 2: Use of FPN Control Chart for X-R and X-S

A summary of the work on the FPN Control Chart is as follows:

(i) The development of FPN

(ii) The representation of the FPNs in parametric form.

(iii) Apply ranking and defuzzification of FPN for the data.

(iv) Put the calculated crisp value into the control chart formula and set up the FPN control limit.

Step 3: Interpretation of FPN Control Chart for X-R and X-S. The fuzzy CLs of the recommended X-R and X-S control charts are used to assess the fuzzy sample mean and standard deviation. If the fuzzy sample mean and standard deviation are inside the control bounds, the process is under control for the sample. Otherwise, the process will spiral out of control.

III. Illustrative Example In this article, we choose the simulation data shown in Table 3 are the deviations from milling a slot in an aircraft terminal block. A high rate of rejections for many of the components manufactured in an aviation company's machine shop highlighted the necessity for an investigation into the causes of the problems. Because the majority of the rejections were for failing to satisfy dimensional limits, it was decided to utilize X-R and X-S charts to try to pinpoint the source of the problem. These charts, which needed real dimension measurement, were to be utilized just for the dimensions that were creating a high number of rejections. Among many such dimensions, the ones chosen for control charts were those with significant spoilage costs and reworked for those where rejections caused delays in assembly processes. Although the primary objective of all of the X-R and X-S charts was to diagnose problems, it was expected that some of the charts would be kept for routine process control and potentially for acceptance inspection.

Table 3: Data for width of slot in an aircraft terminal block.

Sample X1 X2 X3 X4 X5

1 773 803 780 720 776

2 757 786 734 740 735

3 755 774 720 761 746

4 745 779 755 775 772

5 800 728 747 759 745

6 784 806 787 763 758

7 745 765 754 759 768

8 789 751 782 769 763

9 757 747 741 746 747

10 746 731 762 781 744

11 742 731 754 736 750

12 748 726 764 735 733

13 748 763 779 786 770

14 770 768 784 771 767

15 772 759 770 772 772

16 765 768 773 791 787

17 780 777 742 761 761

18 775 765 746 782 745

19 759 748 781 764 756

20 765 782 739 754 768

21 770 784 730 759 781

22 773 765 777 760 750

23 758 780 761 746 757

24 761 771 756 772 758

25 752 785 764 758 773

26 745 774 786 739 796

27 759 766 790 730 781

28 739 795 750 780 776

29 747 781 763 768 756

30 767 752 774 746 769

The first step is to test the normality assumption. It is shown that Figure 1 holds the normality assumption for the above data.

Figure 1: Normal Probability Plot

Now the data shows normal, and we convert the data into FPN form. Table [4-8] shows the FPN form for the variables X1, X2, X3, X4 and X5.

Table 4: FPN for X1

Xa1 Xb1 Xc1 Xd1 Xe1 R(AP(Xl))

768 770 773 777 785 774

752 754 757 761 771 758.2222

750 752 755 759 768 756.1111

740 742 745 749 757 746

795 797 800 804 812 801

777 781 784 788 796 784.7778

738 741 745 749 757 745.5556

781 786 789 793 801 789.6667

748 754 757 760 769 757.3333

740 743 746 749 752 746

721 727 730 733 741 730.2222

739 745 748 751 758 748.1111

740 744 748 751 759 748.1111

764 767 770 773 780 770.4444

766 769 772 775 778 772

757 761 765 769 786 766.4444

775 777 780 784 792 781

770 772 775 779 788 776.1111

749 756 759 763 771 759.4444

758 763 765 769 778 766.1111

763 767 770 774 783 770.8889

768 770 773 777 785 774

750 755 758 762 770 758.6667

756 758 761 765 774 762.1111

745 749 752 756 764 752.7778

738 743 745 750 759 746.4444

754 756 759 763 772 760.1111

734 736 739 743 751 740

740 745 747 751 760 748.1111

760 764 767 771 779 767.7778

Table 5: FPN for X2

Xa2 Xb2 Xc2 Xd2 Xe2 R(Ap(X2))

796 799 803 808 816 803.8889

778 781 786 790 798 786.2222

768 771 774 779 785 775

771 776 779 784 791 779.8889

719 724 728 735 742 729.2222

799 802 806 811 819 806.8889

758 762 765 770 776 765.8889

746 749 751 758 765 753.1111

741 745 747 752 759 748.3333

725 729 731 736 744 732.4444

724 728 731 735 745 732

719 723 726 730 737 726.6667

757 760 763 769 775 764.3333

760 765 768 773 778 768.6667

752 755 759 764 772 759.8889

760 765 768 774 780 769.1111

770 774 777 781 789 777.7778

760 762 765 769 778 766.1111

743 745 748 752 760 749

774 779 782 786 794 782.6667

777 782 784 788 797 785.1111

758 762 765 769 777 765.7778

772 777 780 784 792 780.6667

764 769 771 775 780 771.6667

780 782 785 789 796 785.8889

768 771 774 779 789 775.4444

759 761 766 771 782 767

790 793 795 799 808 796.3333

776 779 781 786 794 782.5556

747 750 752 757 765 753.5556

Table 6: FPN for X3

Xa3 Xb3 Xc3 Xd3 Xe3 R(AP(x3))

773 777 780 783 788 780.1111

727 731 734 740 744 735

715 717 720 725 728 720.7778

750 753 755 760 764 756.1111

741 745 747 753 755 748.1111

782 785 787 793 796 788.3333

749 752 754 760 763 755.3333

777 779 782 788 791 783.1111

737 740 741 747 750 742.6667

756 759 762 765 769 762.1111

748 751 754 760 763 755

758 761 764 770 773 765

772 775 779 785 787 779.5556

778 781 784 791 793 785.2222

764 767 770 776 779 771

768 771 773 780 782 774.5556

737 740 742 748 756 743.8889

740 744 746 750 758 747.1111

775 779 781 786 794 782.4444

734 737 739 746 754 741.2222

725 728 730 737 746 732.3333

771 774 777 783 791 778.5556

757 759 761 768 776 763.3333

751 754 756 760 768 757.2222

759 762 764 769 777 765.5556

781 784 786 790 798 787.2222

785 788 790 795 805 791.7778

745 747 750 758 768 752.5556

758 760 763 768 775 764.2222

768 771 774 779 787 775.2222

Table 7: FPN for X4

Xa4 Xb4 Xc4 Xd4 Xe4 R(Ap(X4))

715 717 720 727 729 721.3333

736 738 740 749 751 742.3333

756 759 761 768 771 762.6667

770 773 775 781 784 776.3333

753 756 759 766 768 760.2222

758 761 763 769 771 764.2222

753 756 759 766 768 760.2222

764 767 769 775 778 770.3333

740 743 746 753 756 747.3333

776 779 781 787 789 782.2222

728 733 736 742 743 736.5556

729 732 735 741 743 735.8889

781 783 786 793 795 787.3333

766 769 771 778 781 772.6667

768 770 772 778 782 773.5556

787 789 791 797 799 792.3333

755 758 761 768 775 762.7778

777 780 782 787 794 783.4444

757 760 764 769 778 765

748 752 754 759 767 755.4444

754 757 759 765 773 760.8889

755 758 760 765 774 761.6667

741 744 746 750 758 747.2222

767 770 772 777 784 773.4444

753 755 758 764 773 759.7778

734 736 739 745 755 740.8889

725 728 730 735 743 731.5556

775 778 780 785 794 781.6667

763 765 768 772 780 769

741 744 746 751 760 747.6667

Table 8: FPN for X5

Xa5 Xb5 Xc5 Xd5 Xe5 R(AP(xs))

770 773 776 782 784 776.8889

729 733 735 742 745 736.5556

741 744 746 753 755 747.5556

767 769 772 778 780 773

739 743 745 751 753 746.1111

752 754 758 764 767 758.7778

762 766 768 774 776 769.1111

758 760 763 769 772 764.1111

741 745 747 754 758 748.6667

738 741 744 750 752 744.8889

744 747 750 756 758 750.8889

727 730 733 739 743 734.1111

764 768 770 776 779 771.2222

761 765 767 774 778 768.6667

767 770 772 778 781 773.3333

782 785 787 793 795 788.2222

756 758 761 766 772 762.1111

740 743 745 750 759 746.6667

751 754 756 760 768 757.2222

763 766 768 771 779 768.8889

776 779 781 785 793 782.2222

745 748 750 755 762 751.4444

752 754 757 761 769 758

753 755 758 763 770 759.2222

768 770 773 778 784 774.1111

790 793 796 800 808 796.8889

776 779 781 786 794 782.5556

771 774 776 780 788 777.2222

751 753 756 760 767 756.8889

765 767 769 773 780 770.2222

After the FPN form, we use the rank membership function (equation (1)) for the crisp value.

Table 9: The crisp value for X1, X2, X3, X4, X5

R(AP(Xl)) R(Ap(X2)) R(AP(x3)) R(Ap(X4)) R(Ap(X5))

774 803.8889 780.1111 721.3333 776.8889

758.2222 786.2222 735 742.3333 736.5556

756.1111 775 720.7778 762.6667 747.5556

746 779.8889 756.1111 776.3333 773

801 729.2222 748.1111 760.2222 746.1111

784.7778 806.8889 788.3333 764.2222 758.7778

745.5556 765.8889 755.3333 760.2222 769.1111

789.6667 753.1111 783.1111 770.3333 764.1111

757.3333 748.3333 742.6667 747.3333 748.6667

746 732.4444 762.1111 782.2222 744.8889

730.2222 732 755 736.5556 750.8889

748.1111 726.6667 765 735.8889 734.1111

748.1111 764.3333 779.5556 787.3333 771.2222

770.4444 768.6667 785.2222 772.6667 768.6667

772 759.8889 771 773.5556 773.3333

766.4444 769.1111 774.5556 792.3333 788.2222

781 777.7778 743.8889 762.7778 762.1111

776.1111 766.1111 747.1111 783.4444 746.6667

759.4444 749 782.4444 765 757.2222

766.1111 782.6667 741.2222 755.4444 768.8889

770.8889 785.1111 732.3333 760.8889 782.2222

774 765.7778 778.5556 761.6667 751.4444

758.6667 780.6667 763.3333 747.2222 758

762.1111 771.6667 757.2222 773.4444 759.2222

752.7778 785.8889 765.5556 759.7778 774.1111

746.4444 775.4444 787.2222 740.8889 796.8889

760.1111 767 791.7778 731.5556 782.5556

740 796.3333 752.5556 781.6667 777.2222

748.1111 782.5556 764.2222 769 756.8889

767.7778 753.5556 775.2222 747.6667 770.2222

All the crisp calculated value of the simulated data is in Table 9. Now we calculate the mean of

crisps value and then put it into the formula of X- R and X-S, which is given in Table [1&2]. Now

we found that the Control limits for the data are as follows:

UCL=785.65, LCL=741.39 and CL=763.52 for X chart,

UCL=81.11, LCL=0 and CL=38.36 for R chart and

UCL=32.38, LCL=0 and CL=15.50 for S chart.

Figure 2: Pentagonal Fuzzy X — R Chart

Peutngoiml Fuzzy Xbnr-S Chut

UCL= 785,65

3t.rS-S.S2

3 6 9 12 IS IE Z1 24 27 30

llllll

Figure 3: Pentagonal Fuzzy X-S Chart

It is shown that there is no point out of control after plotting the X-R and X-S control charts.

IV. Conclusion

In this paper, it is shown that FPN is suitable for traditional variable control charts. If uncertainty is presented in the data, the FPN control chart theory should control the process. In this study, the population parameter (p and a) is unknown, and we develop the theory for fuzzy X-R and X-S chart by using the rank membership function of FPN. More fuzzy control charts for variable and attribute data were done by a — cut, triangular and trapezoidal fuzzy numbers in several published articles. The result of the illustrative example is done with this proposed technique, and it is shown that the fuzzy process is under control and capable. The proposed FPN control chart effectively increases the process flexibility. For further studies, the process capability indices, p-np chart, fuzzy CUSUM and fuzzy EWMA chart can be used to detect the slight shifting in the FPN process control with fuzzy data.

Reference

[1] Venkatesh, A., & Manoj, A. B. (2020). A fuzzy mathematical model for vegetable diet plan using ranking of pentagonal fuzzy number. Malaya Journal of Matematik (MJM), 8(4, 2020), 19951999.

[2] Panda, A., & Pal, M. (2015). A study on pentagonal fuzzy number and its corresponding matrices. Pacific Science Review B: Humanities and Social Sciences, 1(3), 131-139.

[3] Pathinathan, T., & Ponnivalavan, K. (2014). Pentagonal fuzzy number. International journal of computing algorithm, 3, 1003-1005.

[4] Chakraborty, A., Mondal, S. P., Alam, S., Ahmadian, A., Senu, N., De, D., & Salahshour, S. (2019). The pentagonal fuzzy number: its different representations, properties, ranking, defuzzification and application in game problems. Symmetry, 11(2), 248.

[5] Chakraborty, A., Broumi, S., & Singh, P. K. (2019). Some properties of pentagonal neutrosophic numbers and its applications in transportation problem environment. Infinite Study.

[6] Shafqat, A., Huang, Z., & Aslam, M. (2021). Design of X-bar control chart based on Inverse Rayleigh Distribution under repetitive group sampling. Ain Shams Engineering Journal, 12(1), 943-953.

[7] Lim, S. A. H., Antony, J., He, Z., & Arshed, N. (2017). Critical observations on the statistical process control implementation in the UK food industry: A survey. International Journal of

Mohammad Ahmad, Weihu Cheng, Zhao Xu, Abdul Kalam, Ahteshamul Haq RT&A, No 4 (76) Fuzzy Control Charts based on Ranking of Pentagonal Fuzzy Numbers_Volume 18, December 2023

Quality & Reliability Management.

[8] Ahmad, M., & Cheng, W. (2022). A novel approach of fuzzy control chart with fuzzy process capability indices using alpha cut triangular fuzzy number. Mathematics, 10(19), 3572.

[9] Ahmad, M., Cheng, W., Haq, A., & Shah, S. K. (2023). Construction of fuzzy X"- S control chart using trapezoidal fuzzy number with unbalanced data. Journal of Statistical Computation and Simulation, 93(4), 634-645.

[10] Jabbar, R. R., & Alkhafaji, A. A. A. (2023). Analysis of Traditional and Fuzzy Quality Control Charts to Improve Short-Run Production in the Manufacturing Industry. Journal of Engineering, 29(6), 159-176.

[11] Özdemir, A. (2021). Development of fuzzy $$ \bar {X}-S $$ X"-S control charts with unbalanced fuzzy data. Soft Computing, 25(5), 4015-4025.

[12] Yeh, C. T. (2008). On improving trapezoidal and triangular approximations of fuzzy numbers. International Journal of Approximate Reasoning, 48(1), 297-313.

[13] gentürk, S., Erginel, N., Kaya, I., & Kahraman, C. (2014). Fuzzy exponentially weighted moving average control chart for univariate data with a real case application. Applied Soft Computing, 22, 1-10.

[14] gentürk, S., Erginel, N., Kaya, I., & Kahraman, C. (2011). Design of fuzzy ü control charts. Journal of Multiple-Valued Logic & Soft Computing, 17(5-6), 459-473.

[15] Alipour, H., & Noorossana, R. (2010). Fuzzy multivariate exponentially weighted moving average control chart. The International Journal of Advanced Manufacturing Technology, 48(9), 1001-1007.

[16] Erginel, N., & gentürk, S. (2016). Fuzzy EWMA and fuzzy CUSUM control charts. In Fuzzy Statistical Decision-Making (pp. 281-295). Springer, Cham.

[17] Erginel, N. (2014). Fuzzy rule-based $\tilde p $ and $ n\tilde p $ control charts. Journal of intelligent & fuzzy systems, 27(1), 159-171.

[18] Montgomery, D. C. (2009). Introduction to statistical quality control. John Wiley & Sons.

[19] Zadeh, L. A., "Fuzzy sets," Information and control, vol. 8, no. 3, pp. 338-353, 1965.