**1. Introduction**

In a specific activity, a person consciously and intuitively seeks to find the best solutions to emerging problems or tasks. The action of making the best or most e ffective use of a situation or resource is called optimization. The Simple Additive Weighting (SAW), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), Complex Proportional Assessment Method (COPRAS), Multi-Objective Optimization by Ratio Analysis (MOORA) and Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) methods applied in this paper have been described in di fferent research papers as Multiple Criteria Decision Making (MCDM) [1–6], Multiple Attribute Decision Making (MADM) [7–9], Multiple Criteria Decision Analysis (MCDA) [10] or Multi-Attribute Decision Analysis (MADA) [11], and Multi-Criteria Analysis (MCA) [12,13]. Since this work is focused on decision-making and the number of alternatives are explicitly given and finite, the name MADM will be used to define the above-listed methods.

MADM methods are aimed at identifying the most satisfactory of several comparative alternatives or at ranking options according to their relevance in terms of the evaluated objective [14]. The methods are used for selecting the most satisfactory alternative/solution provided that there is no such alternative for which all criteria values are the best.

To solve an optimization problem with classical optimization methods, the function of its objective is fixed, establishing the set of objects to be optimized or the allowable area to be determined. The minimum or maximum values of the function are sought depending on the purpose of the problem being solved. The theoretical part of this work presents MADM methods as a component of mathematical optimization methods, and evaluation criteria for SAW, TOPSIS, COPRAS, MOORA and PROMETHEE methods appear as objective functions, which is a new form of presenting and interpreting methods. To illustrate the idea of this publication, some of the most widely applied MADM methods have been selected. The presented methodology can be transferred to other methods as well.

The judging matrix and the vector of criterion weights are the components of most of MADM methods. The judging matrix covers statistical data or the values of expert evaluation according to the criteria defining the objective [14]. Since the impact of criteria on the outcome of the problem to be solved is di fferent, the significance (weights) of criteria is determined [15]. Criterion weights can be clarified directly or by employing certain weighting methods. The main idea of most of the used MADM methods is merging criterion values and their weights into a single evaluation characteristic (i.e., the summarized criterion of the method). Data on MADM methods are static, and their values do not vary in the problem-solving process.

Most of the assignments solved by people include problems that do not have su fficient numerical data or problems where the investigated objects are impossible to measure. In such cases, the judgment matrix is supplemented by the data obtained from the expert evaluation. Particular focus is switched to selecting experts in a particular field, considering their characteristics related to professional competence, work experience, scientific degree, research activity and the ability to address specific issues in the field given. MADM methods operate in numerical values, although the criteria themselves can be both quantitative and qualitative. The qualitative meanings of criteria, in some cases, facilitate expert evaluation that can be individual when the expert expresses individual opinions independently of other experts or shared and accepted in a group of professionals.

The research literature does not propose the best or most appropriate MADM method for dealing with a specific problem. This question is relevant, and thus there are many research papers focused on determining the stability of the method on the basis that any mathematical model or method can be applied in practice in the case that they remain stable with respect to the applied parameters [16]. A mathematical model is considered to be stable if a small change in the results is consistent with minor variations in the parameters for the model. Multiple MADM methods are applied in most complex decision-making tasks to ensure the accuracy of the final result. In the cases when several MADM methods are used for evaluation, it becomes unclear what results of which method are reliable. This paper proposes a new approach that helps the expert make the right decision. The core of the suggested approach is to apply several MADM methods and to determine the suitability/impact of the employed MADM methods on the problem solved (i.e., to clarify the stability of the method). The final result consists of the estimates of several methods taking into account the weight of the e ffect of each method.

The paper verifies the stability of multi-criteria methods when slightly changing data in the matrix of the initial solution (i.e., expert evaluations and weights of the vector, fixing recurrence frequency of the best alternative to the initial data). Previous papers of the author considered that the higher the number of imitations, the more accurate the evaluation of the stability of the multi-criteria method (i.e., the range of the varying result decreased). A su fficient number of recurrences was established when the result of evaluating MADM stability remained almost unchanged, because 10<sup>5</sup> times could be treated as an adequate number of estimations [17].

The practical part of the paper combines the results of several MADM methods into a single outcome and shows a few ways to normalize results obtained using MADM methods of di fferent scales.
