Fuzzy approach to analyzing a three-factor experiment with binary levels

Rahul Thakur; S.C. Malik; Masum Raj

doi:10.55976/dma.42026151443-59

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Fuzzy approach to analyzing a three-factor experiment with binary levels

Rahul Thakur ¹ ^* , S.C. Malik ² , Masum Raj ³

1. Department of Statistics, Maharshi Dayanand University, Rohtak, Haryana, India
2. Department of Statistics, Maharshi Dayanand University, Rohtak, Haryana, India
3. Department of Mathematics, Institute of Applied Sciences and Humanities, Ganeshi Lal Agrawal University, Mathura, U.P., India

Correspondence: thakurrahul3394@gmail.com

DOI: https://doi.org/10.55976/dma.42026151443-59

Received

01 January 2026
Revised

22 March 2026
Accepted

18 May 2026
Published

04 June 2026

Design of experiments Fuzzy environment Factorial experiment Fuzzy logic Lower-level model Upper-level model

Abstract

A factorial experiment is a widely used statistical analysis technique in experimental studies in agricultural and engineering sciences. The classical factorial experiment design considers several factors at different crisp levels, which are quantitatively measurable. However, factor levels are often imprecise, vague, incomplete, or described linguistically. In such cases, fuzzy statistics are used instead of classical statistics when analyzing data with uncertain observations. In this paper, we propose an extension of the classical factorial design for analyzing experiments with fuzzy observations. The experimental data are represented using triangular fuzzy numbers and transformed using the α-cut interval approach, resulting in lower-level and upper-level factorial models. These models are analyzed separately to evaluate the effects of factors and their interactions under uncertainty. The results are also compared with classical ANOVA applied to defuzzified data to highlight the advantages of incorporating uncertainty through fuzzy representations. Furthermore, the robustness of the proposed approach is examined through sensitivity analysis across different α levels. The findings demonstrate that the proposed fuzzy factorial approach provides a more flexible framework for analyzing experimental data in the presence of imprecision and uncertainty.

<h5>1. Introduction</h5>
In any experiment where more than one factor is used and all levels of each factor are combined with all levels of every other factor, the experiment is called a factorial experiment. The concept of factorial experiments represents a major advancement in the design of experimental systems. In these designs, the effects of individual factors are calculated using the entire set of observations. This method can improve the efficiency of the experiment. Fisher <a href="#reference_list_0">[1]</a> was the first to introduce the concept of factorial experiments, suggesting that factorial design is more efficient than studying one factor at a time. Yates <a href="#reference_list_1">[2]</a> made significant contributions to the analysis of factorial designs through the Yates' method. Williams <a href="#reference_list_2">[3]</a> provided an overview and interpretations of the interaction terms present in factorial experiments. Addelman <a href="#reference_list_3">[4]</a> discussed developments in the design of factorial experiments before the 1960s. Factorial design has been applied and expanded in various fields (e.g., agriculture, medicine, engineering, industry, biology) by many authors. Patterson <a href="#reference_list_4">[5]</a> specified procedures for producing asymmetrical factorial designs based on the relationships between the amounts of treatment and plot components. Liao <a href="#reference_list_5">[6]</a> proposed three classes of optimal designs with equal or unequal block sizes in two-level factorial experiments. Additional literature related to factorial experiments can be found in references <a href="#reference_list_6">[7, 8]</a>.
A crucial limitation is that the factorial experiment is used only when measured data are available in exact form. Observations from any experiment are often uncertain and contain some vagueness caused by human judgment, evaluations and decisions, measurement devices, environmental conditions, or the subjective or linguistic nature of the observed data. Therefore, the existing factorial experiment under classical statistics cannot be used when the observations are uncertain. Methods in fuzzy statistics provide a useful methodology to address these types of uncertainty and imprecision in the data. Numerous approaches have been proposed in the last decade to combine the classical design of experiments with fuzzy statistics. Zadeh <a href="#reference_list_8">[9]</a> was the first to introduce fuzzy set theory. Romer and Kandel <a href="#reference_list_9">[10]</a> investigated the effects of fuzzy data on statistical tests for hypothesis testing of fuzzy hypotheses. Buckley and Eslami <a href="#reference_list_10">[11]</a> presented a method that estimates the variance and mean of fuzzy statistics. This method utilizes a set of confidence intervals to produce an estimator as a triangular number. Dubois and Prade <a href="#reference_list_11">[12]</a> discussed algebraic operations by applying the fuzzification rule to fuzzy numbers. Arnold <a href="#reference_list_12">[13]</a> defined fuzzy hypothesis testing for crisp data and proposed one- and two-sided hypothesis testing under a fuzzy environment for type I and type II errors. The fuzzy test for hypothesis testing under vague circumstances was proposed by Grzegorzewski <a href="#reference_list_13">[14]</a>. The author also developed a fuzzy decision method that shows a scale of acceptability for binary hypotheses. Grzegorzewski and Mrowka <a href="#reference_list_14">[15]</a> formulated operators to approximate fuzzy numbers to trapezoidal expected intervals. Buckley <a href="#reference_list_15">[16]</a> compared the fuzzy means of m populations, where each population was normally distributed with the same but unknown variance. Wu <a href="#reference_list_16">[17]</a> discussed hypothesis testing for fuzzy analysis of variance (ANOVA) using fuzzy data, introducing the concepts of optimistic and pessimistic degrees for h-level sets and solving the associated optimization problem. Lubiano and Trutsching <a href="#reference_list_17">[18]</a> introduced the R package “Statistical Analysis of Fuzzy Data (SAFD)” for fuzzy ANOVA, using the de-fuzzification process for Fuzzy Random Variables. Jorge <a href="#reference_list_18">[19]</a> described fuzzy least squares regression and combined it with two-way ANOVA to present a claim reserving method. Parchami et al. <a href="#reference_list_19">[20]</a> extended the concept of one-way ANOVA for imprecise observations. Rahul et al. <a href="#reference_list_20">[21]</a> proposed a post-hoc test for fuzzy one-way analysis of variance to identify the exact pairs of fuzzy means that are significant. Thakur et al. <a href="#reference_list_21">[22]</a> proposed one-way analysis of covariance under a fuzzy environment and its application in drug yield analysis. Recent developments in fuzzy statistical analysis and uncertainty modeling further highlight the importance of fuzzy-based approaches in handling imprecise data (Garg <a href="#reference_list_22">[23]</a>; Kahraman et al. <a href="#reference_list_23">[24]</a>). Recent studies have also emphasized the role of fuzzy methods in decision-making and experimental analysis under uncertainty (Więckowski et al. <a href="#reference_list_24">[25]</a>; Tang & Ahmad <a href="#reference_list_25">[26]</a>).
In many real-world experimental situations, observed responses are not always recorded as precise numerical values. Measurement instruments, environmental variability, and subjective evaluations may introduce uncertainty or vagueness into experimental observations. Classical factorial experiment models assume exact observations and therefore may not adequately represent such uncertainty. To address this limitation, fuzzy set theory provides an effective framework for modeling imprecise and uncertain data. By integrating fuzzy numbers with factorial experimental design, it becomes possible to analyze treatment effects while explicitly accounting for uncertainty in observations. The primary reason for conducting this study is to address issues in studies utilizing imprecise and uncertain data that necessitate the application of the fuzzy 23 factorial experiment. For example, to determine the effect of different types of fertilizers on potato crop yield, consider three fertilizers: Nitrogen (N), Potash (K), and Phosphate (P), each having two fuzzy levels, high and low. The quantities at each level are inexact, which creates vagueness in the data. Therefore, a fuzzy 23 factorial experiment is necessary for this type of experiment. Although several researchers have developed statistical methods for fuzzy data, most existing studies focus on fuzzy hypothesis testing, fuzzy regression, and fuzzy one-way analysis of variance. Very little attention has been given to factorial experimental designs under fuzzy environments, particularly for multi-factor experiments involving interactions between factors.
To the best of our knowledge, a systematic framework for analyzing three-factor factorial experiments with fuzzy observations has not been adequately addressed in the literature. This study aims to fill that gap by proposing a fuzzy factorial design methodology. The data are represented as triangular fuzzy numbers. The fuzzy factorial design is defuzzified using the α-cut interval approach, resulting in lower and upper-level models. Both models are discussed separately and fuzzy logic is used to draw conclusions. We implement and explain this method with a numerical example. Compared to existing methods in classical statistics, the proposed fuzzy factorial method is adaptable and efficient in the presence of uncertainty.
1.1 Objectives of the study
The main objectives of this study are:
  1) To extend the classical 23 factorial experiment framework to experimental observations represented as fuzzy numbers.
  2) To develop a methodology based on the α-cut approach for transforming fuzzy observations into lower and upper factorial models.
  3) To analyze main and interaction effects under fuzzy environments using factorial ANOVA.
  4) To evaluate the robustness of the proposed approach through sensitivity analysis across different α levels.
This paper is structured as follows. Section 2 discusses some basic definitions of fuzzy numbers and concepts related to the research. After defuzzification, the separate analysis of the lower and upper fuzzy 23 factorial experiment is presented in Section 3. Section 4 provides and explains the illustration. A comparison with the existing model is made in Section 5 and discussed in Section 6. The final section presents the conclusion of the paper.
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<h5>7. Conclusion</h5>
This paper proposes a fuzzy factorial experiment framework for a design to analyze experimental data in the presence of uncertainty. Unlike classical factorial experiments, which assume precise observations, the proposed methodology uses triangular fuzzy numbers to represent imprecise measurements. The α-cut defuzzification approach converts fuzzy observations into interval-valued data, enabling factorial analysis through lower-level and upper-level models. Results from the numerical illustration demonstrate that the proposed fuzzy factorial design captures the variability in experimental observations and provides a more flexible interpretation of treatment effects under uncertain conditions. By analyzing both the lower and upper bounds of the fuzzy data, the proposed model offers a range of possible outcomes, reducing the risk of misleading conclusions that may result from ignoring measurement uncertainty. The proposed methodology is particularly useful in experimental situations where measurements are affected by instrument limitations, environmental variability, or subjective evaluations. Potential applications include engineering experiments, agricultural field trials, environmental studies, and industrial process optimization, where uncertain observations are common. Future research may extend the proposed fuzzy factorial framework to designs with more factors and levels, as well as explore other fuzzy representations, such as trapezoidal or interval-valued fuzzy numbers.
<h5>Authors' contribution</h5>
Rahul Thakur: Conceptualization, data curation, formal analysis, methodology, writing – original draft. Masum Raj: Validation, investigation, writing – review & editing. S.C. Malik: Supervision, validation, writing – review & editing.
<h5>Conflicts of interest</h5>
The authors declare that there are no known financial or personal conflicts of interest that could have influenced the research findings or conclusions presented in this study.
<h5>Data availability statement</h5>
No data was utilized in the research presented in this article.

References

[1]Fisher RA. Statistical methods for research workers. Edinburgh: Oliver and Boyd; 1925.

[2]Yates F. The design and analysis of factorial experiments. London: Imperial Bureau of Soil Science; 1937.

[3]Williams EJ. The interpretation of interactions in factorial experiments. Biometrika. 1952;39(1-2): 65-81. doi: 10.1093/biomet/39.1-2.65.

[4]Addelman S. Recent developments in the design of factorial experiments. Journal of the American Statistical Association. 1972;67(337): 103-111. doi: 10.1080/01621459.1972.10481211.

[5]Patterson HD. Generation of factorial designs. Journal of the Royal Statistical Society Series B: Statistical Methodology. 1976;38(2): 175-179. doi: 10.1111/j.2517-6161.1976.tb01583.x.

[6]Liao CT. Partially replicated block designs for two-level factorial experiments. Statistics & Probability Letters. 2020;158: 108653. doi: 10.1016/j.spl.2019.108653.

[7]Montgomery DC. Design and analysis of experiments. 9th ed. Hoboken: Wiley; 2017.

[8]Hoshmand R. Design of experiments for agriculture and the natural sciences. 2nd ed. Boca Raton: CRC Press; 2018.

[9]Zadeh LA. Fuzzy sets. Information and control. 1965;8(3): 338-353. doi: 10.1016/S0019-9958(65)90241-X.

[10]Römer C, Kandel A. Statistical tests for fuzzy data. Fuzzy Sets and Systems. 1995;72(1): 1-26. doi: 10.1016/0165-0114(94)00270-H.

[11]Buckley JJ, Eslami E. Uncertain probabilities II: the continuous case. Soft computing. 2004;8(3): 193-199. doi: 10.1007/s00500-002-0262-y.

[12]Dubois D, Prade H. Operations on fuzzy numbers. International Journal of systems science. 1978;9(6): 613-626. doi: 10.1080/00207727808941724.

[13]Arnold BF. Testing fuzzy hypotheses with crisp data. Fuzzy sets and systems. 1998;94(3): 323-333. doi: 10.1016/S0165-0114(96)00258-8.

[14]Grzegorzewski P. Testing statistical hypotheses with vague data. Fuzzy sets and Systems. 2000;112(3): 501-510. doi: 10.1016/S0165-0114(98)00061-X.

[15]Grzegorzewski P, Mrówka E. Trapezoidal approximations of fuzzy numbers. Fuzzy sets and systems. 2005;153(1): 115-135. doi: 10.1016/j.fss.2004.02.015.

[16]Buckley JJ. Fuzzy one-way ANOVA. In: Buckley JJ, editor. Fuzzy probability and statistics. Berlin: Springer; 2006. p. 203-207.

[17]Wu HC. Statistical confidence intervals for fuzzy data. Expert Systems with Applications. 2009;36(2): 2670-2676. doi: 10.1016/j.eswa.2008.01.022.

[18]Lubiano MA, Trutschnig W. ANOVA for fuzzy random variables using the R-package SAFD. In: Advances in intelligent and soft computing. Vol 77. Berlin: Springer; 2010. p. 449-456.

[19]De Andrés-Sánchez J. Claim reserving with fuzzy regression and the two ways of ANOVA. Applied Soft Computing. 2012;12(8): 2435-2441. doi: 10.1016/j.asoc.2012.03.033.

[20]Parchami A, Nourbakhsh M, Mashinchi M. Analysis of variance in uncertain environments. Complex & Intelligent Systems. 2017;3(3): 189-196. doi: 10.1007/s40747-017-0046-8.

[21]Rahul, Tiwari P, Gupta P. POST--HOC TEST FOR ONE-FACTOR FUZZY ANALYSIS OF VARIANCE. International Journal of Agricultural & Statistical Sciences. 2022;18: 983.

[22]Thakur R, Malik SC, Raj M. Fuzzy ANCOVA and its application in drug yield analysis. Songklanakarin Journal of Science & Technology. 2026;48(1): 1.

[23]Garg H. New exponential operation laws and operators for interval-valued q-rung orthopair fuzzy sets in group decision making process. Neural Computing and Applications. 2021;33(20): 13937-13963.

[24]Kahraman C, Onar S C, Oztaysi B. Fuzzy multicriteria decision-making: a literature review. International journal of computational intelligence systems. 2015;8(4): 637-666. doi: 10.1080/18756891.2015.1046325.

[25]Więckowski J, Kizielewicz B, Sałabun W. Fuzzy RANCOM: A novel approach for modeling uncertainty in decision-making processes. Information Sciences. 2025;694: 121716. doi: 10.1016/j.ins.2024.121716.

[26]Tang HH, Ahmad NS. Fuzzy logic approach for controlling uncertain and nonlinear systems: a comprehensive review of applications and advances. Systems Science & Control Engineering. 2024;12(1):2394429. doi: 10.1080/21642583.2024.2394429.

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Thakur, R., Malik, S., & Raj, M. (2026). Fuzzy approach to analyzing a three-factor experiment with binary levels. Decision Making and Analysis, 4(1), 43–59. https://doi.org/10.55976/dma.42026151443-59

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Fuzzy approach to analyzing a three-factor experiment with binary levels

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