**1. Introduction**

A mathematical model makes practical sense if its results are stable concerning the model parameters. If an insignificant variation in the values of the resulting characteristics of the model corresponds to slight variations in the model parameters. Components of Multicriteria Decision-Making (MCDM) models are represented by the criteria characterizing the process under evaluation, and these criteria' weights.

The criteria weights provide a quantitative estimation of the importance of the criteria. Given that the use of criteria weights in MCDM methods has an essential influence on the result of the evaluations and on the making of the proper decision, an investigation of the accuracy of such evaluations is interesting and important from both the theoretical and the practical point of view. This paper contains an investigation of the stability of the evaluations of the subjective weights of the criteria and the influence of data uncertainty upon the results.

**Citation:** Vinogradova-Zinkeviˇc, I.; Podvezko, V.; Zavadskas, E.K. Comparative Assessment of the Stability of AHP and FAHP Methods. *Symmetry* **2021**, *13*, 479. https:// doi.org/10.3390/sym13030479

Academic Editor: José Carlos R. Alcantud

Received: 31 January 2021 Accepted: 9 March 2021 Published: 15 March 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Uncertainty of data may result from the evaluation of a subjective expert, the extent of the expert's interest, ambiguity, inaccuracy of measurements, or improperly applied methods. Various approaches, such as, in particular, fuzzy set theory and the methods of mathematical statistics, Boolean logic, logistic regression, Monte Carlo simulation, Bayesian networks, and neural networks, are used to evaluate the influence of the degree of uncertainty [1].

So-called subjective weights, obtained based on peer reviews, are most frequently applied in practice [2–6]. For that reason, the subjective evaluation carries an uncertainty in itself. Despite the experience and competence of the expert, the evaluations provided by the same expert may vary when solving complex problems with a large number of criteria. For example, if an expert fills in the same questionnaire several times at different times, these evaluations are usually different from each other.

There are various techniques applied in the evaluation of the weight criteria. The simplest methods are based on a ranking of the criteria depending on their significance and on the direct evaluation of the weights when the sum of the evaluations is equal to one or to 100%. The use of other scales with normalization of the results is also possible. More complex subjective methods for weight evaluation, like the AHP and FAHP, use mathematical theories and verify the consistency of the expert evaluations.

There is another approach to the quantitative evaluation of the importance of the criteria. This approach evaluates the structure of the data array—the criteria values for all the alternatives [3,7,8]. Methods like this are called objective. Objective weights are applied rarely in practice, and we disregard them in this paper. A combined evaluation of the weights, which is based on the integration of subjective and objective evaluations, is also possible [9–12].

Evans [13] related the concept of sensitivity analysis in decision-making theory to the stability of an optimal solution under variation of the model parameters and the accurate evaluation of the values of such parameters. The first significant papers in the sector of sensitivity analysis were written on the basis of using the concepts of sensitivity analysis in linear programming for the development of an optimal approach that could be applied to the classical problems of decision theory [13], and on using entropy and the least squares method [14].

Zhou et al. [15] suggested a method for the calculation of the entropy weights in the situation when the evaluation of the criteria might contain uncertainties such as, for example, interval values, and when it contains both uncertainties and incompleteness, for example, with the distribution of judgements. Wolters and Mareschal suggested three types of sensitivity analysis: (1) a fixed relation between the variation of the ranking and the variations of the alternatives based on certain criteria, (2) the influence exerted by specific variations of the points/criteria of the alternative, and (3) the minimum modification of the weights necessary to provide for the alternative to take first place [16]. The analysis was focused on and developed for the preference ranking organization method for enrichment evaluation (PROMETHEE) methods. Triantaphyllou and Sánchez [17] presented the methodology for the performance of sensitivity analysis of the weights of the decision-making criteria and the alternative efficiency values for the weighted sum model (WSM), the weighted product model (WPM), and the analytic hierarchy process (AHP) methods. Evaluation of the influence exerted by uncertainty in the SAW method was performed by Podvezko [18], who evaluated the ranges of weight intervals for the process criteria, the levels of matching and stability of the expert evaluations, and the influence of uncertainty on the ranking of the matched objects. The influence of weight variation upon the final result in the SAW method was studied by Zavadskas et al. [19], and sensitivity analysis of the SAW, TOP-SIS, MOORA, and PROMETHEE methods was studied by Vinogradova [20]. Memariani et al. [21] and Alinezhad [22] studied the influence of the values of the decision matrix elements upon the results of the ranking. Moghassem [23] increased and decreased the weights of all the criteria by 5, 10, 15, and 20 percent in sensitivity analysis of the TOPSIS and VIKOR methods. Hsu et al. performed sensitivity analysis of the TOPSIS method by

increasing the three maximum weights by 10% and decreasing the three minimum ones by 10% [24].

Erkut and Tarimcilar [25] suggested dividing the problems of the stability of the AHP method that are to be solved into two groups. The approach of group one assumes operations over the criteria proper, by means of calculating the alternative evaluation as the sum of the alternative evaluations multiplied by the correspondent weights, based on the criteria under evaluation. The problems in group two are solved as decision-making problems under conditions of uncertainty, with the uncertainty meaning that there are a number of possible states of the nature and only one of them can be transformed into a true state. The very meaning of state probability is directly related to the meaning of the uncertainty of the problem to be solved within the framework of risk evaluation. The authors of the paper follow the first approach to the problems, solving them graphically by creating the weight space, which is represented as all the possible combinations of the weights for the purposes of tier one of the hierarchy. Consequently, separating the weight space into sets, the spatial data can be generated from the space. Any of the alternatives possesses the highest evaluation ranking in every one of the subsets [25].

In his paper, Masuda [26] studied how variations of the entire columns of the decisionmaking matrix might influence the values of the alternative priorities. He suggested using the sensitivity coefficient of the finite vector of the alternative priorities for each of the column vectors in the decision matrix to show how significantly the values of the finite alternative priorities vary. Warren [27] studied in more detail the theoretical aspects of the AHP method related to the evaluation scale, the determination of the vector of eigenvalues, the issue of normalization of the weights, and so on. Mimovi´c et al. [28] suggested an integrated application of the analytic hierarchy process (AHP) and Bayesian analysis. The Bayesian formula managed to increase the input data accuracy for the analytic hierarchy process. The AHP method was used in this paper for the representation of the objectivized input data for the Bayesian formula in situations in which statistical evaluations of the probability are not possible. In the same way, Wu et al. [29], who generated pairwise comparison matrices and verified their stability, also suggested one of the methods for verification of the stability of the AHP method. The paper by Aguarón et al. shows the development of the theoretical basis for improvement of the AHP matrix inconsistency, when the Row Geometric Mean (RGM) is used as the prioritization procedure and the Geometric Consistency Index (GCI) as the inconsistency measure [30].

A number of papers with a genuine use of the MCDM model have appeared recently in which the stability of results of the methods is studied. The paper by Chen et al. [31] evaluated the stability of the multicriteria weights by studying the GIS-based MCDM model, showing the influence of the variation of the criteria weights upon the model results in the spatial dimension and graphically. The weights were determined with the help of the AHP method and were varied from their initial values within limits of 20%. This range of variation for the initial weights was applied either to all the criteria or to each criterion, as required [31]. The paper by Deepa and Swamynathan [32] facilitated an improvement in the efficiency of internet networks through increasing their throughput capacity, by suggesting a mathematical model of a clustering protocol known as AETCP (a clustering protocol based on AHP-Entropy-TOPSIS). The mobile nodes were hierarchically organized into different clusters based on certain criteria. The integrated method for evaluation of the subjective and objective weights was applied to the evaluation of the mobile nodes. Later on, ranking of the sets of nodes was performed with the purpose selecting the nodes with the largest weight as the correspondent nodes of the cluster-head [32]. The paper by Zyoud and Fuchs-Hanusch [33] address a severe problem of a water deficit in the water supply networks. The FAHP method was used to evaluate the factors influencing the loss of water. The decision was made by diagnosing the loss-of-water risk index at the level of the pipes and the areas. The Fuzzy Synthetic Evaluation Technique (FSET) was used to evaluate the water loss index at the level of the water supply system, and Ordered Weighted Averaging (OWA) was used to aggregate individual values of the index

applied to each area. A Monte Carlo simulation model was used to generate the final ranking of the areas. The results of this modeling provided sufficient stability from the point of view of the ranking of the investigated areas. Xue et al. [34] suggested a method for the evaluation of stability and safety in the construction of engineering facilities for a protective tunnel under a river using the AHP-entropy weight method and the ideal point evaluation model. The paper by Kumar et al. [35] contains an evaluation of the stability of the factor model for the environmental impact risk for materials (products/services) related to pharmaceutical drugs. The model sensitivity was checked in terms of the pro rata variation of the considered risk factor with respect to other factors, varying the weight value from 0.9 to 0.1; variations of inconsistencies were also observed for other risks. Evaluation of the supplier selection stability model was suggested by Stevi´c et. al. [36] and executed by means of varying the weight and recording the variations in the ranking of the alternatives. The weights varied in a manner that on the increase of one criterion by a conditional unit (for example, 12%), the other criterion was, naturally, decreased correspondingly in order to satisfy the condition under which the sum of the values of all the criteria remained unchanged. Continuing with the topic of the estimation of supplier model quality, Stoji´c et al. [37] used the WASPAS method and suggested the calculation of the coefficient α to generate the number of relative values of the alternative; the coefficient depended on the weight parameter (α lay within the limits of 0 to 1, with increments of 0.1).

The generation of individual values of the criteria weights in the SWARA method was suggested by Zavadskas et al. [38]. The paper by Pamucar et al. [39] also evaluated the influence of the criteria (and the sub-criteria) upon the order of ranking of the alternatives, which were represented by suppliers, within the framework of the problem of increasing the service quality of third-party logistics providers. To process uncertain data under the procedure of group decision-making, the paper considered interval rough numbers (IRN) and the IRN-BWM (best worst method). The stability of the ranking of alternatives that was obtained was checked by varying the values of the coefficients of the linear combination and by the application of the operational competitiveness rankings analysis (OCRA) method [40]. Sensitivity analysis has also been used to confirm the stability of the final rankings of the results [41,42], or to verify and evaluate the feasibility of the optimal alternative [43], as well as to study the influence of the variation of the parameters and criteria weights upon the final results of ranking of the alternatives [44].

The stochastic approach to determining the uncertainty of the AHP weights has been used in different ways. Janssen [45] studied the sensitivity of the ranking of the alternatives using the effects table and compared this with the maximum evaluation of the decision-making person. The sensitivities of the rankings of alternatives to overall uncertainty in scores and priorities were analyzed using a Monte Carlo approach. Eskandari and Rabelo [46] followed another stochastic approach, and this gave these authors the opportunity to calculate the AHP weight dispersions, and to process their uncertain behavior with the help of a Monte Carlo simulation. An approach that applied fuzzy logic, an analytic hierarchy process, and a Monte Carlo simulation was used to solve the problem of the over-expenditure of funds within the framework of urban transit projects, to facilitate the effective planning of the future budget by the decision-making persons [47].

The stability of models related to calculations of subjective criteria weights is investigated in this paper. The most frequently used methods in the MCDM model—the analytic hierarchy process (AHP) method and the FAHP method, when fuzzy numbers are used under the conditions of data uncertainty—are taken for the analysis. The method of statistical simulation (Monte Carlo) is used in order to investigate the stability issues. The practical realization of the algorithm is written in the Python programming language. This paper's results can be used as an integral part of the study of the stability of MCDM methods in the ranking of alternatives.
