Intuitionistic Fuzzy One-Way ANOVA Model and an Application

Öz

Analysis of variance (ANOVA) models are essential techniques used in statistical studies when comparing the means of more than two independence groups. ANOVA models have been widely used in scientific research in almost every field for many years. In the one-way ANOVA model, there is one factor (independent variable) and various levels or experiments of this factor. The aim here is to analyze the effects of the experiments on the dependent variable. Different fuzzy ANOVA models were developed to model observations obtained under uncertainty and represented by fuzzy numbers. When the studies in the literature were examined statistically, it was seen that there were few studies on the intuitionistic fuzzy ANOVA models. In this study, the theoretical structure of the intuitionistic fuzzy one-way ANOVA model was explained and illustrated with an application. The results obtained were supported and interpreted by comparative analysis.

Anahtar Kelimeler

• Intuitionistic Fuzzy Numbers
• One-Way Analysis of Variance (ANOVA)
• Fuzzy Sets

Sezgisel Bulanık Tek-Yönlü ANOVA Modeli ve Bir Uygulama

Öz

Varyans analizi (ANOVA) modelleri ikiden fazla bağımsız grubun ortalamalarının karşılaştırılmasında istatistiksel çalışmalarda kullanılan önemli yöntemlerdir. Uzun yıllardır hemen her alanda, bilimsel araştırmalarda ANOVA modelleri yaygın olarak kullanılmaktadır. Tek-yönlü ANOVA modelinde tek faktör (bağımsız değişken) ve bu faktörün çeşitli düzeyleri ya da denemeleri söz konusu olmaktadır. Burada amaç denemelerin bağımlı değişken üzerindeki etkilerini incelemektir. Belirsizlik altında elde edilen ve bulanık sayılarla temsil edilen gözlemlerin modellenmesi için de farklı bulanık ANOVA modelleri geliştirilmiştir. Literatürde yapılan çalışmalar istatistiksel olarak incelendiğinde, sezgisel bulanık ANOVA modellerine ilişkin çalışmaların az sayıda olduğu görülmüştür. Bu çalışmada sezgisel bulanık tek-yönlü ANOVA modelinin teorik yapısı anlatılmış ve bir örnek uygulama ile açıklanmıştır. Elde edilen sonuçlar karşılaştırma analizi ile desteklenmiş ve yorumlanmıştır.

Anahtar Kelimeler

• Sezgisel Bulanık Sayılar
• Tek-Yönlü Varyans Analizi (ANOVA)
• Bulanık Kümeler

Kaynakça

1. Şenoğlu, B., & Acıtaş, Ş. (2011). İstatistiksel deney tasarımı: sabit etkili modeller. Nobel.
2. Kumar, R., Khepar, J., Yadav, K., Kareri, E., Alotaibi, S. D., Viriyasitavat, W., ... & Dhiman, G. (2022). A systematic review on generalized fuzzy numbers and its applications: past, present and future. Archives of Computational Methods in Engineering, 29(7), 5213-5236.
3. De Garibay, V. G. (1987). Behaviour of fuzzy ANOVA. Kybernetes, 16(2), 107-112.
4. López-Díaz, M., Gil, M. Á., Grzegorzewski, P., Hryniewicz, O., Lawry, J., Montenegro, M., & Casals, M. R. (2004). Introduction to ANOVA with fuzzy random variables. In Soft methodology and random information systems (pp. 487-494). Springer Berlin Heidelberg.
5. Montenegro, M., Colubi, A., Casals, M.R., & Gil, M.A. (2004). Asymptotic and bootstrap techniques for testing the expected value of a fuzzy random variable, Metrika 59, 31–49.
6. Cuevas, A., Febrero, M., & Fraiman, R. (2004). An anova test for functional data. Computational statistics & data analysis, 47(1), 111-122.
7. Konishi, M., Okuda, T., & Asai, K. (2006). Analysis of variance based on fuzzy interval data using moment correction method. International Journal of Innovative Computing, Inf. Control 2 (1), 83–99.
8. Gil, M. Á., Montenegro, M., González-Rodríguez, G., Colubi, A., & Casals, M. R. (2006). Bootstrap approach to the multi-sample test of means with imprecise data. Comput. Stat. Data Anal. 51 (1), 148–162.

9. González-Rodríguez, G., Colubi, A., & Gil, M. Á. (2012). Fuzzy data treated as functional data: A one-way ANOVA test approach. Computational Statistics & Data Analysis, 56(4), 943-955.
10. Wu, H. C. (2007). Analysis of variance for fuzzy data. International Journal of Systems Science, 38(3), 235-246.
11. Nourbakhsh, M., Mashinchi, M., & Parchami, A. (2013). Analysis of variance based on fuzzy observations. International Journal of Systems Science, 44(4), 714-726.
12. Parthiban, S., & Gajivaradhan, P. A. (2016). Comparative Study of LSD Under Fuzzy Environments Using Trapezoidal Fuzzy Numbers. IOSR Journal of Mathematics (IOSR-JM), 12(1), 57-75.
13. Anuradha, D., & Kalpanapriya, D. (2018). Intuitionistic fuzzy ANOVA and its application in medical diagnosis. Research Journal of Pharmacy and Technology, 11(2), 653-656.
14. Aslam, M. (2019). Neutrosophic analysis of variance: application to university students. Complex & intelligent systems, 5(4), 403-407.
15. Nortey, E. N., Eric, W. N., & Eunice, O. A. (2022). Neutrosophic-principal component analysis of causes of performance gap among private and public school students in the basic education certificate examination. Asian Journal of Probability and Statistics, 20(3), 132-149.
16. Miari, M., Anan, M. T., & Zeina, M. B. (2022). Neutrosophic two way ANOVA. International Journal of Neutrosophic Science, 18(3), 73-83.
17. Parchami, A., Mashinchi, M., & Kahraman, C. (2021). A case study on vehicle battery manufacturing using fuzzy analysis of variance. In Intelligent and Fuzzy Techniques: Smart and Innovative Solutions: Proceedings of the INFUS 2020 Conference, Istanbul, Turkey, July 21-23, 2020 (pp. 916-923). Springer International Publishing.
18. Parchami, A., Mashinchi, M., & Kahraman, C. (2021). An Implication of Fuzzy ANOVA in Vehicle Battery Manufacturing. Journal of Mahani Mathematical Research, 10(2), 33-47.
19. Gökkuş, Z., Şentürk, S., Yildiz, T., & Cevher, E. Y. (2023, August). An Application of Fuzzy ANOVA on Field of Agricultural Machinery. In International Conference on Intelligent and Fuzzy Systems (pp. 560-571). Cham: Springer Nature Switzerland.
20. Gökkuş, Z., Şentürk, S., Yildiz, T., & Yeşiloğlu Cevher, E. (2023, August). Two-Way ANOVA for Fuzzy Observations and an Implication. In International Conference on Intelligent and Fuzzy Systems (pp. 548-559). Cham: Springer Nature Switzerland.
21. URL. (05.09.2024) https://www.scopus.com/term/analyzer.uri?sort=plf-f&src=s&sid=f733d3dcf9b798351df901e1d2d5c00a&sot=a&sdt=a&sl=55&s=%28TITLE-ABS-KEY%28fuzzy%29+AND+TITLE-ABS-KEY%28one+way+ANOVA%29%29&origin=resultslist&count=10&analyzeResults=Analyze+results
22. Atanassov, K.T. (1983). Intuitionistic fuzzy sets. VII, ITKR’s Session, Sofia.
23. Atanassov, K.T. (1986). Intuitionistic fuzzy sets. Fuzzy sets and Systems.
24. Atanassov, K.T. (1994). New operations defined over intuitionistic Fuzzy sets. Fuzzy sets and Systems, 6, 137-142.
25. Atanassov, K.T. (1999). Intuitionistic fuzzy sets: theory and application, Springer.
26. Malik, M., & Gupta, S. K. (2023). On optimistic, pessimistic and mixed fuzzy-programming based approaches to solve multi-objective fully intuitionistic fuzzy linear fractional programming problems. Annals of Operations Research, 1-45.