Multi-Attribute Decision-Making Methods as a Part of Mathematical Optimization

Abstract

Optimization problems are relevant to various areas of human activity. In different cases, the problems are solved by applying appropriate optimization methods. A range of optimization problems has resulted in a number of different 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 differs in the data processing, and therefore the results of MADM methods are obtained on different 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 different 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.

Comparative Assessment of the Stability of AHP and FAHP Methods

“…Thus, for the evaluation of the stabilities of the method for the MCDM model, we can apply the percentage of loss as the best alternative to the leading position, the maximum inconsistencies of the evaluations of the method, the percentage of variation of the order for the ranking of the alternatives, etc. [20].…”

“…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, TOPSIS, 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.…”

Optimal location of electronic charity boxes in charity NGOs by proposing a combined mathematical model

“…3. According to Equation ( 10), the final value of each alternative is calculated and the best alternative (A * ) is chosen [32].…”

TOPSIS Decision on Approximate Pareto Fronts by Using Evolutionary Algorithms: Application to an Engineering Design Problem

“…When real Multi-objective Optimization Problems (MOPs) are tackled, two different working approaches can be identified in the literature. The first, known as Multiple Criteria Decision-Making (MCDM) [1][2][3][4][5][6][7][8][9][10][11][12][13], is essentially interested in decision-making, for example in helping a human Decision-Maker (DM) to choose between various alternatives or solutions in accordance with several conflicting criteria or objectives. The main representatives of this approach can be found in schools of economics, management and finance and the role and participation of the DM before and during the decision-making process are decisive.…”