On optimal completions of incomplete pairwise comparison matrices

An important variant of a key problem for multi-attribute decision making is considered. We study the extension of the pairwise comparison matrix to the case when only partial information is available: for some pairs no comparison is given. It is natural to define the inconsistency of a partially filled matrix as the inconsistency of its best, completely filled completion. We study here the uniqueness problem of the best completion for two weighting methods, the Eigen-vector Method and the Logarithmic Least Squares Method. In both settings we obtain the same simple graph theoretic characterization of the uniqueness. The optimal completion will be unique if and only if the graph associated with the partially defined matrix is connected. Some numerical experiences are discussed at the end of the paper.

A method for solving LSM problems of small size in the AHP

The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. It provides with ratio-scale measurements of the prioirities of elements on the various leveles of a hierarchy. These priorities are obtained through the pairwise comparisons of elements on one level with reference to each element on the immediate higher level. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM), Logarithmic Least Squares Method (LLSM), Weighted Least Squares Method (WLSM) and Chi Squares Method (X2M) are of the tools for computing the priorities of the alternatives. This paper studies a method for generating all the solutions of the LSM problems for 3 × 3 matrices. We observe non-uniqueness and rank reversals by presenting numerical results.

Solving the Least Squares Method problem in the AHP for 3 X 3 and 4 X 4 matrices

The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM) are of the possible tools for computing the priorities of the alternatives. A method for generating all the solutions of the LSM problem for 3 × 3 and 4 × 4 matrices is discussed in the paper. Our algorithms are based on the theory of resultants.

On pairwise comparison matrices that can be made consistent by the modification of a few elements

Pairwise comparison matrices are often used in Multi-attribute Decision Making forweighting the attributes or for the evaluation of the alternatives with respect to a criteria. Matrices provided by the decision makers are rarely consistent and it is important to index the degree of inconsistency. In the paper, the minimal number of matrix elements by the modification of which the pairwise comparison matrix can be made consistent is examined. From practical point of view, the modification of 1, 2, or, for larger matrices, 3 elements seems to be relevant. These cases are characterized by using the graph representation of the matrices. Empirical examples illustrate that pairwise comparison matrices that can be made consistent by the modification of a few elements are present in the applications.

Több szempontú részvételi döntések a fenntarthatósági értékelésekben. A legnépszerűbb módszerek összehasonlítása = Participatory multi-criteria decision analysis. A comparison of methodologies

A fenntarthatóság értékelése definíciószerűen többdimenziós probléma. A megfelelő alternatíva, forgatókönyv, eljárás stb. kiválasztásakor ugyanis a döntéshozóknak egyszerre kell figyelembe venniük környezetvédelmi, gazdasági és társadalmi szempontokat. Az ilyen döntéseket alátámaszthatják a több szempontú döntéshozatali modellek. A tanulmány a több szempontú döntési eljárások közül a legfontosabb hétnek az alkalmazhatóságát vizsgálja részvételi körülmények között. Az utóbbi évek e témában publikált esettanulmányainak áttekintésével megállapítható, hogy egyik módszer sem uralja a többit, azok különböző feltételek mellett eltérő sikerrel használhatók. Ennek ellenére a különböző módszerek kombinációjával végrehajthatunk olyan eljárásokat, amelyekkel az egyes módszerek előnyeit még jobban kiaknázhatjuk. ________ Measuring and comparing the sustainability of certain actions, scenarios, technologies, etc. is by definition a multidimensional problem. Decision-makers must consider environmental, economic and social aspects when choosing an alternative course of action. Such decisions can be aided by multi-criteria decision analysis (MCDA). This paper investigates seven different MCDA methodologies: MAU, the Analytic Hierarchic Process (AHP), the ELECTRE, PROMETHEE, REGIME, and NAIADE methods, and "Ideal and reference point" approaches). It is based on a series of reports in which over 30 real-world case studies focusing on participatory MCDA were reviewed. It is stressed, however, that there is no "best" choice in the list of MCDA techniques. Some methods fit certain decision problems better than others. Nonetheless, some complementary benefits of the different techniques can be exploited by combining these methodologies.