Sage Journals: Discover world-class research

Abstract

Process Capability Analysis (PCA) is a widely used statistical method for assessing whether a process can meet customer-defined specification limits (SLs). However, traditional PCA faces limitations when dealing with rigidly defined SLs or uncertain data types. To address these challenges, fuzzy set theory (FST) has been extensively applied, enhancing PCA sensitivity by capturing imprecision in human judgments. This study proposes a novel PCA method based on Pythagorean fuzzy sets (PFSs) and q-rung orthopair fuzzy sets (q-ROFSs), incorporating two linguistic terms (LTs) for quality evaluation and hesitation. Unlike previous studies, our method replaces traditional α-cuts with a hesitancy-based approach and employs q-ROFSs to represent uncertainty comprehensively. In a real-world case study in the meat industry, the proposed method demonstrated a long-term capability index (Ppk) ranging from 0.42 to 0.73 and a short-term capability index (Cpk) ranging from 1.01 to 1.43. At the same time, traditional approaches showed wider and less informative ranges. These results indicate enhanced sensitivity and robustness in process performance assessment. The novelty of this approach lies in combining dual linguistic terms and the hesitancy estimation via q-ROFSs, offering a more nuanced and accurate depiction of process capability, particularly in contexts involving imprecise human evaluations.

Keywords

fuzzy process capability analysis Pythagorean fuzzy sets q-rung orthopair fuzzy sets hesitancy evaluation meat product evaluation

Get full access to this article

View all access options for this article.

References

Abdolshah

Yusuff

R. M.

Hong

T. S.

Ismail

M. Y. B.

Sadigh

A. N.

(2011). Measuring process capability index Cpmk with fuzzy data and compare it with other fuzzy process capability indices. Expert Systems with Applications, 38(6), 6452–6457. https://doi.org/10.1016/j.eswa.2010.11.101

Ali

(2025). Fairly aggregation operators based on complex p, q-rung orthopair fuzzy sets and their application in decision-making problems. Spectrum of Operational Research, 2(1), 113–131. https://doi.org/10.31181/sor21202514

Alkan

Kahraman

(2024). CODAS Extension using novel decomposed Pythagorean fuzzy sets: Strategy selection for IOT based sustainable supply chain system. Expert Systems with Applications, 237, 121534. https://doi.org/10.1016/j.eswa.2023.121534

Atanassov

Gargov

(1989). Interval valued intuitionistic fuzzy sets. Fuzzy Sets and Systems, 31(3), 343–349. https://doi.org/10.1016/0165-0114(89)90205-4

Atanassov

K. T.

(1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. https://doi.org/10.1016/S0165-0114(86)80034-3

Barukab

Khan

S. A.

(2024). Fermatean fuzzy linguistic term set based on linguistic scale function with dombi aggregation operator and their application to multi criteria group decision –making problem. Heliyon, 10(17), e36563. https://doi.org/10.1016/j.heliyon.2024.e36563

Buckley

J. J.

(2005). Fuzzy statistics: Hypothesis testing. Soft Computing, 9(7), 512–518. https://doi.org/10.1007/s00500-004-0368-5

Camargo

M. E.

Santos

G. M.

Russo

S. L.

(2010). Applied control charts for analysis of quality control. The 40th International Conference on Computers & Indutrial Engineering (pp. 1–3). IEEE.

Chen

C.-C.

Lai

C.-M.

Nien

H.-Y.

(2010). Measuring process capability index C pm with fuzzy data. Quality Quantity, 44(3), 529–535. https://doi.org/10.1007/s11135-008-9211-x

10.

Chen

K.-S.

Chen

S.-C.

Hsu

T.-H.

Lin

M.-Y.

C.-F.

(2021). Fuzzy evaluation model of process improvement capability with costs consideration. Applied Sciences, 11(10), 4344. https://doi.org/10.3390/app11104344

11.

Chen

K.-S.

Wang

C.-H.

Tan

K.-H.

(2019). Developing a fuzzy green supplier selection model using six sigma quality indices. International Journal of Production Economics, 212, 1–7. https://doi.org/10.1016/j.ijpe.2019.02.005

12.

Chen

Hung

(2021). What can fuzziness do for capability analysis based on fuzzy data. Scientia Iranica, 28(2), 1049–1064. https://doi.org/10.24200/sci.2019.52104.2537

13.

Chen

T.-Y.

Chang

C.-H.

J.-f. R.

(2013). The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making. European Journal of Operational Research, 226(3), 615–625. https://doi.org/10.1016/j.ejor.2012.11.038

14.

Choi

Y.-H.

G.-Y.

Yang

(2020). Fuzzy-inference-based decision-making method for the systematization of statistical process capability control. Computers in Industry, 123, 103296. https://doi.org/10.1016/j.compind.2020.103296

15.

Dağıstanlı

H. A.

(2025). Weapon system selection for capability-based defense planning using lanchester models integrated with fuzzy MCDM in computer assisted military experiment. Knowledge Decision Systems with Applications, 1, 11–23. https://doi.org/10.59543/kadsa.v1i.13601

16.

Dağıstanlı

H. A.

Temel Gencer

(2025). Evaluation of medium-lift forest fire helicopter using q-rung orthopair fuzzy set based alternative ranking technique based on adaptive standardized intervals approach. Engineering Applications of Artificial Intelligence, 148, 110468. https://doi.org/10.1016/j.engappai.2025.110468

17.

Deleryd

(1998). On the gap between theory and practice of process capability studies. International Journal of Quality & Reliability Management, 15(2), 178–191. https://doi.org/10.1108/02656719810204892

18.

Dudek-Burlikowska

(2005). Quality estimation of process with usage control charts type XR and quality capability of process Cp, Cpk. Journal of Materials Processing Technology, 162, 736–743. https://doi.org/10.1016/j.jmatprotec.2005.02.210

19.

Garg

(2016). A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making. International Journal of Intelligent Systems, 31(9), 886–920. https://doi.org/10.1002/int.21809

20.

Haktanır

Kahraman

(2021). Design for six sigma and process capability using penthagorean fuzzy sets. Intelligent and Fuzzy Techniques: Smart and Innovative Solutions: Proceedings of the INFUS 2020 Conference, Istanbul, Turkey, July 21–23, 2020 (pp. 1385–1395). Springer.

21.

Hesamian

Akbari

M. G.

(2021). A process capability index for normal random variable with intuitionistic fuzzy information. Operational Research, 21(2), 951–964. https://doi.org/10.1007/s12351-019-00490-4

22.

Kahraman

Kaya

(2010). Fuzzy process capability analysis and applications. In Kahraman

Yavuz

(Eds.), Production engineering and management under fuzziness (pp. 483–513). Springer Berlin Heidelberg.

23.

Kahraman

Parchami

Cevik Onar

Oztaysi

(2017). RETRACTED: Process capability analysis using intuitionistic fuzzy sets. Journal of Intelligent Fuzzy Systems, 32(3), 1659–1671. https://doi.org/10.3233/JIFS-141877

24.

Kane

V. E.

(1986). Process capability indices. Journal of Quality Technology, 18(1), 41–52. https://doi.org/10.1080/00224065.1986.11978984

25.

Kaya

Çolak

(2020). A literature review on fuzzy process capability analysis. Journal of Testing Evaluation, 48(5), 3963–3985. https://doi.org/10.1520/JTE20180038

26.

Kaya

İlbahar

Karaşan

(2023). A design methodology based on two dimensional fuzzy linguistic variables for attribute control charts with real case applications. Engineering Applications of Artificial Intelligence, 126, 106792. https://doi.org/10.1016/j.engappai.2023.106792

27.

Kaya

Kahraman

(2010a). A new perspective on fuzzy process capability indices: Robustness. Expert Systems with Applications, 37(6), 4593–4600. https://doi.org/10.1016/j.eswa.2009.12.049

28.

Kaya

İ.

Kahraman

(2010b). Fuzzy process capability analyses with fuzzy normal distribution. Expert Systems with Applications, 37(7), 5390–5403. https://doi.org/10.1016/j.eswa.2010.01.017

29.

Kaya

İ.

Karaşan

İlbahar

Cebeci

(2020). Design of control charts for number of defects based on pythagorean fuzzy sets. Conference Proceedings of Science and Technology (vol. 3, no. 1, pp. 115–121). Murat Tosun.

30.

Kaya

Kahraman

(2011). Fuzzy process capability indices with asymmetric tolerances. Expert Systems with Applications, 38(12), 14882–14890. https://doi.org/10.1016/j.eswa.2011.05.059

31.

Kotz

Johnson

N. L.

(2002). Process capability indices—a review, 1992–2000. Journal of Quality Technology, 34(1), 2–19. https://doi.org/10.1080/00224065.2002.11980119

32.

Kwong

C.-K.

Bai

(2002). A fuzzy AHP approach to the determination of importance weights of customer requirements in quality function deployment. Journal of Intelligent Manufacturing, 13(5), 367–377. https://doi.org/10.1023/A:1019984626631

33.

Lee

W.-C.

Hong

C.-W.

J.-W.

(2015). Computational procedure of performance assessment of lifetime index of normal products with fuzzy data under the type II right censored sampling plan. Journal of Intelligent Fuzzy Systems, 28(4), 1755–1773. https://doi.org/10.3233/IFS-141463

34.

(2013). A novel Likert scale based on fuzzy sets theory. Expert Systems with Applications, 40(5), 1609–1618. https://doi.org/10.1016/j.eswa.2012.09.015

35.

Liao

M.-Y.

C.-W.

J.-W.

(2013). Fuzzy inference to supplier evaluation and selection based on quality index: A flexible approach. Neural Computing Applications, 23(Suppl 1), 117–127. https://doi.org/10.1007/s00521-012-1266-x

36.

Likert

(1932). A technique for the measurement of attitudes. Archives of psychology, 140, 5–53.

37.

Memmedova

Ertuna

(2024). Development of a fuzzy Likert scales to measure variables in social sciences. Information Sciences, 654, 119792. https://doi.org/10.1016/j.ins.2023.119792

38.

Montgomery

D. C.

(2020). Introduction to statistical quality control. John Wiley & Sons.

39.

Parchami

Onar

SÇ

Öztayşi

Kahraman

(2017). Process capability analysis using interval type-2 fuzzy sets. International Journal of Computational Intelligence Systems, 10(1), 721–733. https://doi.org/10.2991/ijcis.2017.10.1.48

40.

Parimala

Ozel

Shumrani

M. A. A.

Kaymakci

A. K.

(2022). q-Rung orthopair fuzzy supra topological applications in data mining process. In Garg

(Ed.), q-Rung orthopair fuzzy sets: Theory and applications (pp. 1–15). Springer Nature Singapore.

41.

Pearn

W. L.

C.-W.

(2007). An effective decision making method for product acceptance. Omega, 35(1), 12–21. https://doi.org/10.1016/j.omega.2005.01.018

42.

Senvar

Kahraman

(2014a). Fuzzy Process Capability Indices Using Clements’ Method for Non-Normal Processes. Journal of Multiple-Valued Logic Soft Computing, 22(1-2), p.95.

43.

Senvar

Kahraman

(2014b). Type-2 fuzzy process capability indices for non-normal processes. Journal of Intelligent Fuzzy Systems, 27(2), 769–781. https://doi.org/10.3233/IFS-131035

44.

Shamsuzzaman

Alsyouf

Ali

(2015). Optimization design of X &EWMA control chart for minimizing mean number of defective units per out-of-control case. 2015 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM) (pp. 391–395). IEEE.

45.

Sheng

Cheng

(2016). A method to construct the confidence intervals for process capability indices based on fuzzy set theory. 2016 3rd International Conference on Information Science and Control Engineering (ICISCE) (pp. 758–762). IEEE.

46.

Shu

M.-H.

H.-C.

(2011). Fuzzy X and R control charts: Fuzzy dominance approach. Computers Industrial Engineering, 61(3), 676–685. https://doi.org/10.1016/j.cie.2011.05.001

47.

Shu

M.-H.

H.-C.

(2012). Manufacturing process performance evaluation for fuzzy data based on loss-based capability index. Soft Computing, 16(1), 89–99. https://doi.org/10.1007/s00500-011-0736-x

48.

Tran

Duckstein

(2002). Comparison of fuzzy numbers using a fuzzy distance measure. Fuzzy Sets Systems, 130(3), 331–341. https://doi.org/10.1016/S0165-0114(01)00195-6

49.

Uehara

Hirota

(1998). Parallel and multistage fuzzy inference based on families of α-level sets. Information Sciences, 106(1–2), 159–195. https://doi.org/10.1016/S0020-0255(97)10002-0

50.

Wang

C.-H.

Chen

K.-S.

(2020). New process yield index of asymmetric tolerances for bootstrap method and six sigma approach. International Journal of Production Economics, 219, 216–223. https://doi.org/10.1016/j.ijpe.2019.05.004

51.

Wang

Wei, G.

Lu, J.

Alsaadi, F.E.

Hayat, T.

Wei

, Zhang, Y. (2019a). Some q-rung orthopair fuzzy Hamy mean operators in multiple attribute decision-making and their application to enterprise resource planning systems selection. International Journal of Intelligent Systems, 34(10), 2429–2458. https://doi.org/10.1002/int.22155

52.

Wang

Zhang

Zhu

Shang

(2019b). RETRACTED: A novel approach to multi-attribute group decision making based on q-rung orthopair uncertain linguistic information. Journal of Intelligent Fuzzy Systems, 36(6), 5565–5581. https://doi.org/10.3233/JIFS-181425

53.

Wang

W.-J.

Luoh

(2000). Simple computation for the defuzzifications of center of sum and center of gravity. Journal of Intelligent Fuzzy Systems, 9(1–2), 53–59. https://doi.org/10.3233/IFS-2000-118

54.

Wang

Y.-M.

(2009). Centroid defuzzification and the maximizing set and minimizing set ranking based on alpha level sets. Computers & Industrial Engineering, 57(1), 228–236. https://doi.org/10.1016/j.cie.2008.11.014

55.

Wei

Gao

Wei

(2018). Some q-rung orthopair fuzzy Heronian mean operators in multiple attribute decision making. International Journal of Intelligent Systems, 33(7), 1426–1458. https://doi.org/10.1002/int.21985

56.

C.-W.

Pearn

Kotz

(2009). An overview of theory and practice on process capability indices for quality assurance. International Journal of Production Economics, 117(2), 338–359. https://doi.org/10.1016/j.ijpe.2008.11.008

57.

C.-W.

Shu

M.-H.

Wang

T.-C.

Chen

Y.-L.

(2024a). Integrating capability Index and generalized rule-switching mechanism for enhanced quick-switch sampling systems. International Journal of Production Economics, 276, 109366. https://doi.org/10.1016/j.ijpe.2024.109366

58.

Zhong

Chen

Wang

Min

(2024b). A multi-strategy three-way decision approach for tri-state risk loss under q-rung orthopair fuzzy environment. Applied Soft Computing, 167, 112197. https://doi.org/10.1016/j.asoc.2024.112197

59.

Xiao

(2019). EFMCDM: Evidential fuzzy multicriteria decision making based on belief entropy. IEEE Transactions on Fuzzy Systems, 28(7), 1477–1491. https://doi.org/10.1109/TFUZZ.2019.2936368

60.

Yager

R. R.

(2013). Pythagorean fuzzy subsets. 2013 joint IFSA world congress and NAFIPS annual meeting (IFSA/NAFIPS) (pp. 57–61). IEEE.

61.

Yager

R. R.

(2016). Generalized orthopair fuzzy sets. IEEE transactions on Fuzzy Systems, 25(5), 1222–1230. https://doi.org/10.1109/TFUZZ.2016.2604005

62.

Yager

R. R.

Alajlan

(2017). Approximate reasoning with generalized orthopair fuzzy sets. Information Fusion, 38, 65–73. https://doi.org/10.1016/j.inffus.2017.02.005

63.

Yalcin

Kaya

(2022b). Design and analysis of process capability indices Cpm and Cpmk by neutrosophic sets. Journal of Fuzzy Systems, 19(1), 13–30. https://doi.org/10.22111/ijfs.2022.6548

64.

Yalçın

Kaya

İ.

(2022c). Two-dimensional uncertainty analysis for Cp and Cpk process capability indices. 2022 International Conference on Innovation and Intelligence for Informatics, Computing, and Technologies (3ICT) (pp. 419–423). IEEE.

65.

Yalçın

Kaya

İ.

(2024). Design and analysis of C pm and C pmk indices for uncertainty environment by using two dimensional fuzzy sets. Journal of Intelligent Fuzzy Systems, 46(1), 2331–2355. https://doi.org/10.3233/JIFS-234683

66.

Yalçın

Kaya

(2022a). Analyzing of process capability indices based on neutrosophic sets. Computational Applied Mathematics, 41(6), 287. https://doi.org/10.1007/s40314-022-01973-5

67.

Yang

Meng

Huang

Cui

(2020). Process capability analysis for manufacturing processes based on the truncated data from supplier products. International Journal of Production Research, 58(20), 6235–6251. https://doi.org/10.1080/00207543.2019.1675916

68.

Yang

Pang

(2019). New q-rung orthopair fuzzy partitioned Bonferroni mean operators and their application in multiple attribute decision making. International Journal of Intelligent Systems, 34(3), 439–476. https://doi.org/10.1002/int.22060

69.

Yongting

(1996). Fuzzy quality and analysis on fuzzy probability. Fuzzy Sets Systems, 83(2), 283–290. https://doi.org/10.1016/0165-0114(95)00383-5

70.

Zadeh

L. A.

(1965). Fuzzy sets. Information and Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X

71.

Zaman

Lee

M. H.

Riaz

(2020). An improved process monitoring by mixed multivariate memory control charts: An application in wind turbine field. Computers & Industrial Engineering, 142, 106343. https://doi.org/10.1016/j.cie.2020.106343

72.

Zhang

Chen

S.-M.

(2021). Group decision making with incomplete q-rung orthopair fuzzy preference relations. Information Sciences, 553, 376–396. https://doi.org/10.1016/j.ins.2020.10.015

73.

Zhiyuan

Jinsheng

(2015). Optimal design of AEWMA control chart with new sampling strategy. The 27th Chinese Control and Decision Conference (2015 CCDC) (pp. 13–18). IEEE.

Linguistic Fuzzy Process Capability Analysis Based on Pythagorean and q-Rung Orthopair Fuzzy Sets

Abstract

Keywords

Get full access to this article

References