**Irina Vinogradova**

Department of Information Technologies, Vilnius Gediminas Technical University, 10223 Vilnius, Lithuania; irina.vinogradova@vgtu.lt; Tel.: +37-052-745-035

Received: 21 August 2019; Accepted: 26 September 2019; Published: 2 October 2019

**Abstract:** Optimization problems are relevant to various areas of human activity. In di fferent cases, the problems are solved by applying appropriate optimization methods. A range of optimization problems has resulted in a number of di fferent methods and algorithms for reaching solutions. One of the problems deals with the decision-making area, which is an optimal option selected from several options of comparison. Multi-Attribute Decision-Making (MADM) methods are widely applied for making the optimal solution, selecting a single option or ranking choices from the most to the least appropriate. This paper is aimed at providing MADM methods as a component of mathematics-based optimization. The theoretical part of the paper presents evaluation criteria of methods as the objective functions. To illustrate the idea, some of the most frequently used methods in practice—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)—were chosen. These methods use a finite number of explicitly given alternatives. The research literature does not propose the best or most appropriate MADM method for dealing with a specific task. Thus, several techniques are frequently applied in parallel to make the right decision. Each method di ffers in the data processing, and therefore the results of MADM methods are obtained on di fferent scales. The practical part of this paper demonstrates how to combine the results of several applied methods into a single value. This paper proposes a new approach for evaluating that involves merging the results of all applied MADM methods into a single value, taking into account the suitability of the methods for the task to be solved. Taken as a basis is the fact that if a method is more stable to a minor data change, the greater importance (weight) it has for the merged result. This paper proposes an algorithm for determining the stability of MADM methods by applying the statistical simulation method using a sequence of random numbers from the given distribution. This paper shows the di fferent approaches to normalizing the results of MADM methods. For arranging negative values and making the scales of the results of the methods equal, Weitendorf's linear normalization and classical and author-proposed transformation techniques have been illustrated in this paper.

**Keywords:** optimization; decision-making; MADM; SAW; COPRAS; TOPSIS; PROMETHEE; MOORA; normalization; stability
