Applications of Some Rating Methods to Solve Multicriteria Decision-Making Problems

This study proposes a new approach for the solution of multicriteria decision-making problems. The proposed approach is based on using rating/ranking methods. Particularly, in this paper, we investigate the possibility of applying Massey, Colley, Keener, offence-defence, and authority-hub rating methods, which are successfully used in various fields. The proposed approach is useful when no decision-making authority is available or when the relative importance of various criteria has not been previously evaluated. The proposed approach is tested with an example problem to demonstrate its viability and suitability for  application.


Introduction
This paper proposes a novel approach for solving multi criteria decision-making (MCDM) problems based on using rating/ranking methods. The proposed approach is particularly useful when no decision-making authority is available or when the relative importance of various criteria has not been previously evaluated. The essence of the proposed method is briefly explained. The multi criteria formulation is the typical starting point for theoretical and practical analyses of decision-making problems. Thus, the definition of Pareto optimality and a vast arsenal of different Pareto optimization methods widely used for decision-making purposes. Correspondingly, the literature on Pareto optimization methods for decision-making very extensive and we can indicate here only an insignificant number of theoretical and applied works: see e.g. (

[1] [2] [3] [4] [5] [6] [7] ).
However, unlike single-objective optimizations, a characteristic feature of Pareto optimality is that the set of Pareto-optimal alternatives (i.e., the set of efficient alternatives) is usually 'large'. In addition, all these Paretooptimal alternatives must be considered mathematically equal. Hence, the problem of choosing a specific Paretooptimal alternative for implementation arises because the final decision usually must be unique. Thus, additional factors must be considered to aid a decision-maker in the selection of specific or more favorable alternatives from the set of Pareto-optimal solutions.
In this paper, we show that, for any MCDM problem that can be defined naturally, a special kind of score matrix (and some its modifications) can be used to introduce a rating/rank on the set of alternatives. The best option using this rating/rank alternative can be declared a 'solution' of the MCDM problem under consideration. We investigate the possibility of applying different rating/ranking methods. These are a number of well-known rating methods If not otherwise mentioned, we identify the finite set with the set N (n) ,where n = |A| is the capacity of set A.We also identify the matrix Π ∈ R n×m with the map Π : N (n) × N (m) → R. For the matrix Π ∈ R n×m ,we denote its transpose by Π T ∈ R m×n . For a vector ξ = (ξ 1 , . . . , ξ n ) ∈ R n we introduce the matrix Λ(ξ) = [λ ij ] i,j=1,...,n ∈ R n×n such that λ ii = ξ i , i = 1, . . . , n, and λ ij = 0, i, j = 1, . . . , n, i ̸ = j.The Heaviside function is defined as follows: Now, we give a brief overview of some basic notations of ranking theory (see e.g. [8]) which are necessary for further consideration. For a natural number N, Note that we can interpret the elements of N (N ) as athletes, and for each pair of athletes (i, j), 1 ≤ i, j ≤ N we assume that the joint match M (i, j) includes N games. We interpret entry S ij , 1 ≤ i, j ≤ N as the number total wins of athlete i in matches M (i, j) and the result of the match M (i, j) is S ij wins of athlete i, S ji wins of athlete j,and (N − S ij − S ji ) draws. We also introduce the matrix

Pareto optimality
The following concepts is drawn from a general treatment of the multicriteria optimization theory. Alternatives are denoted byA = {a 1 , ..., a m } , criteria are denoted by c j : A → R, j = 1, . . . , n; C = {c 1 , ..., c n } , and the MCDM problem is ordered pair < A, C > . Obviously, we may assume that the criteria are normalized such that the lower value is preferable for each criterion and the goal of the decisionmaking procedure is to minimize all criteria simultaneously (see e.g. [1]). Later in the article, we will adhere to this assumption. Furthermore, A is the set of admissible alternatives, and map − → c = (c 1 , . . . , c n ) : A → R n is the criterion map (correspondingly, − → c (A) ⊂ R n is the set of admissible values of criteria). An alternative a * ∈ A is Pareto optimal (i.e., efficient) if no a ∈ A exists such that c j (a) ≤ c j (a * ) for all j ∈ N (n) and c k (a) < c k (a * ) for some k ∈ N (n) .The set of all efficient alternatives is denoted as A e and is called the Pareto set. Correspondingly, − → c (A e ) is called the efficient front.

Rating Methods
In this section, we present various ranking methods. The ranking methods presented here were originally proposed in different fields, namely the Massey, Colley, Keener, and offence-defence rating methods originated in the context of team ranking in sports, see ( [9] [10] [11]), and the authority-hub method originated in the context of citation index development and for web ranking purposes, see ( [12] [13] [14]) .

Massey method
To describe the Massey method, we introduce the following notation: m is the number of teams, n G is the number of the games that have already been played, r M i , i = 1, . . . , m is Massey ' s rating of the i team and r M = (r M 1 , . . . , r M m )is Massey ' s rating vector. The result of the k th game is described as an ordered quadruple (t 1 (k), s 1 (k), t 2 (k), s 2 (k)) where t 1 (k), t 2 (k) are teams who played the k th game, s 1 (k), s 2 (k) are the scores of these teams in the game, and the margin of the k th game (we also consider draws) is assumed to be y k = s 1 The main assumption of the Massey method is that the rating vector r M = (r M 1 , . . . , r M m ) can be obtained through least squares approximation, as follows: where X is the correspondingly defined matrix. Hence, we can conclude that ( X T X ) r M = X T y. To avoid ill definiteness of the matrix, one of the possibilities is that a row (for definiteness -the last one) in the matrix X T Xmust be replaced with a row of all ones, and the corresponding component of the vector X T y must be replaced with a zero. The matrix M and the vector v M that are obtained in such a way are called the Massey matrix and Massey vector, respectively. After this, Massey 's rating vector, r M , is defined as solution of the linear equation M r M = v M .

Colley method
The Colley method can be described based on the following notation: m is the number of teams, w i is the number of wins for team i, l i is the number of losses for team i, n i is the total number of games played by team i, and n ij is the number of times teams i, j played, i, j = 1, . . . , m.
From this data, the following objects can be introduced: the Colley matrix, C = [C ij ] i,j=1,...,n , and the Colley vector The Colley rating vector, r C , is obtained as a solution of the equation Cr C = v C .

Keener method
The Keener method can be described based on the following notation: m is the number of teams, and S ij is the score produced by team i against team i, j = 1, . . . , m. Keener's matrix K = [K ij ] i,j=1,...,m is defined as follows: Correspondingly, the rating vector for Keener method, r K , is obtained as a solution of the eigenvalue problem Kr K = λr K . If the matrix K is irreducible, the Perron-Frobenius theorem guarantees the existence and uniqueness of the ratings vector r K , see ( [11]) .

Offence-defence method
We assume that, given a set A of m objects that should be ranked, a (row) stochastic matrix P is defined for reflecting the set of available data on pairwise comparisons of the considered objects. The offense and defense vectors are defined through the following iteration process: It is known that, under certain conditions, the convergence is guaranteed, and ). The aforementioned 'certain condition' is that matrix P has total support. Checking whether a sufficiently large matrix has total support is a difficult task, and it is convenient to realise a small perturbation of matrix P i.e. instead of matrix P we use matrix P + ε1 m 1 T m with sufficiently small ε > 0,. The offense and defence vectors o, d can be considered as rating vectors of elements of set A. Moreover, by equalities, . . , m can be defined new rating vector r od = (r od 1 , . . . , r od m ), called the offencedefence (aggregate) rating vector .

Authority-hub method
As in the previous case, we assume that, given set A of m objects, which should be ranked, a (row) stochastic matrix P is defined for reflecting the set of available data on pairwise comparisons of the considered objects. The authority and hub vectors a, h can be defined through the following iteration process: a (k) = P T P a (k−1) , h (k) = P P T h (k−1) . If matrices P T P, P P T are primitive, this iteration process converges to the eigenvectors of these matrices, respectively: . If matrices P T P, P P T are irreducible but not primitive, the above-described iterative process cannot be used for determining vectors a, h. However, dominant eigenvectors for matrices P T P, P P T exist and are unique up to multiplication by a scalar. The authority and hub vectors a, h can be considered as ranking vectors and will be named further as authority rating vector and hub rating vector, respectively.

Proposed Methods
In this section, we show that, with each MCDM problem, special matrices are associated, and these matrices can be successfully utilized for ranking alternatives in the MCDM problem. Namely, we will show how these matrices can be used in the ranking methods described above. In the final part of this section, we provide a general formulation of the proposed approach for solving MCDM problems.
We assume further that an MCDM problem < A, C >, A = {a 1 , . . . , a m } , C = {c 1 , . . . , c n }is given, and the decision-making goal is to minimize the criteria simultaneously.

Special Matrices Associated with an MCDM Problem
We propose a special construction of the score matrices of alternatives, S A , for the MCDM problem < A, C > . For clarity, we imagine that the elements of A are athletes who conduct matches with each other and that, for each pair of athletes a, a ′ ∈ A their match M (a, a ′ )includes m games. Now, for any a, a ′ ∈ A,we define the following: Thus, the equality s A c (a, a ′ ) = 1 means that c(a) < c(a ′ ) for criterion c ∈ C and the alternative (athlete) a receives one point (i.e., athlete awins a game c ∈ C in match M (a, a ′ )). Correspondingly, S A (a, a ′ )indicates the number of total wins of alternative (athlete) a in match M (a, a ′ ). An alternative a has defeated an alternative a a) wins of the alternative a ′ (losses of alternative a), and number of draws ) .
is the score matrix.
can be transformed into a (row) stochastic matrix : )is a vector defined as follows: To justify its name, matrix Π can be interpreted as an adjacency matrix for a directed graph Γ(A, C) associated with the MCDM problem < A, C > and, correspondingly, matrix P (Π) can be interpreted as a transition probability matrix for the Markov chain determined by the graph Γ(A, C).

Adaptation of the Ranking Methods to the MCDM Problems
Below, we assume that < A, C >, i,j=1,...,m is the score matrix, Π = (π ij ] i,j=1,...,m is the adjacency matrix, and P (Π) is the transition probability matrix for it. By definition, matrix Π is a matrix of points scored (considering a draw, estimated at 0.5 points) by alternatives (teams and athletes) A in the 'tournament' determined by the set of criteria C. Now we propose the following adaptations of the described above ranking methods for the MCDM problem < A, C > .

Massey ranking
In the Massey method for the MCDM problem < A, C >, we have m alternatives/teams, and we can define the number of games played (in the sense of matrix Π i.e., we also consider draws ) as n G = m(m − 1)/2. We are considered the set N(n G ) to be the set of games played and construct the mappings t 1 , t 2 : N(m) → N(n G ), such that the inequality y k = s 1 (k) − s 2 (k) ≥ 0 holds for the scores s 1 (k) = π t 1 (k)t 2 (k) , s 2 (k) = π t 2 (k)t 1 (k) , k = 1, . . . , n G .We also construct the matrix X corresponding to the mappings t 1 , t 2

Colley ranking
In the Colley method, for the MCDM problem < A, C > we have m alternatives/teams, and the number of wins and number of losses for the alternative a i ∈ A can be defined as .., m, respectively. Correspondingly, the total numbers of no draw games played by alternative is the number of times team i, j played no draw games, i, j = 1, . . . , m. Hence, for the MCDM problem < A, C >, the Colley matrix C and Colley vector v C can be obtained as described in Section 2.3.2. Correspondingly, the Colley rating vector, r C , for the MCDM problem < A, C > can be obtained as a solution of the equation

Keener ranking
In the Keener method, for the MCDM problem < A, C > we have m lternatives/teams and the score produced by team igainst team j an be defined as S ij = s A ij , i, j = 1, . . . , m. Hence, for the MCDM problem < A, C >, the Keener matrix K can be obtained as described in Section 2.3.3. Correspondingly, the rating vector for the Keener method, r K , is obtained as a solution of the eigenvalue problem Kr K = λr K .

As we know, if matrix P (Π) is sufficiently good then for
Hence, we can define the offense, defense, and aggregate offencedefence rating vectors as follows: , andr od = (r od 1 , . . . , r od m ), respectively, 3.2.6. Proposed MCDM procedure Now, we can establish the following procedure: (i) the matrix S ( Π, P (Π) ) is calculated and the rating method, ρ, is the choice for the MCDM problem < A, C >;

Authority -hub ranking
ii) the alternative from the Pareto set, A e , that is ρ ranked best is declared the solution of the considered MCDM problem.
The following remarks are important in connection with this procedure. Obviously, it would suffice to rank the Pareto set if it is known at the beginning of the proposed procedure. Nevertheless, we prefer the above description because it is more convenient in cases in which the Pareto set is unknown (or partially or approximately known), as is usually the case for complex MCDM problems.
Note also that, instead of the MCDM problem < A, C >, the MCDM problem < C, A > can be considered. Applying the described procedure to the MCDM problem < C, A >, we can obtain a ranking of the criteria and identify a 'leading criterion'. This possibility may be useful for the development of some hierarchical procedures for solution s to MCDM problems.
In Section 4, we will discuss a particular problem that can be solved using the procedure described above and show that appropriate and competitive results are obtained for it.

The problem statement
This section discusses the example that was considered to demonstrate the practicality of the proposed methods. The example considered here is the problem of selecting material for the mast of a sailboat, see [15] for a detailed description (the decision-making matrix is given in Table 1 ) . This problem was investigated by several researchers using various methods , see e.g. (  The ranks of the materials obtained in the mentioned articles using the following methods: the multi-objective optimization based on ratio analysis (MOORA), the multiplicative form of MOORA (MULTIMOORA), the reference-point approach (RPA), the fuzzy-logic approach (FLA), the weighted-properties method (WPM), the multicriteria optimization through the concept of a compromise solution (VIKOR), the comprehensive VIKOR (CVIKOR), and the game-theoretic method ( GTM).
The auxiliary results are presented in the Appendix: the normalized decision matrix ( the upper-lower bound approach was used for normalization, see [1]),  [10] ) are presented in Table 6; the Massey matrix, Massey vector, and Massey rating vector, Table 7; the Colley matrix, Colley vector, and Colley rating vector, Table 8 ; the Keener matrix and Keener rating vector, Table 9 ; the offense, defense, aggregate offense-defense, authority and hub rating vectors, Table 10 . Note also that the Pareto set for the problem under consideration is A e = {2, 3, 4, 7, 9, 11, 12, 13, 14, 15} .

The obtained results
The obtained results are summarized inTable 2 , where the corresponding rankings for the obtained rating vectors are presented. Table 3 presents the results of the correlation analysis between the considered rankings. As we can see, the considered methods are significantly correlated among themselves but sometimes may have an 'opposite orientation' (thus, we will further consider absolute values of correlation coefficients): the ranks corresponding to r M , r C , r K , r o , r d , r aod , r a , and r h are strongly correlated (absolute value of correlation coefficients no less than 42 APPLICATIONS OF SOME RATING METHODS TO SOLVE MULTICRITERIA DECISION-MAKING PROBLEMS For material codes see Table 1 . Gray denote Pareto-optimal (efficient) materials.
0.75) among themselves; at the same time, the ranks corresponding to r M , r C , r K , and r h have one direction, and the ranks corresponding to r o , r d , r aod , and r a have opposite directions. Finally, Table 4 presents correlations between the ranks obtained previously in other studies and the ranks proposed in this article. As Table 4 shows, the ranks proposed in this study are quite competitive. Particularly, all ranks introduced in this study (except cases related to weak correlations, i.e., the absolute value of correlation coefficients less than 0.35, with GTM and few cases with MULTIMOORA, RPA, and WPM) are moderately (absolute value of correlation coefficients between 0.35 and 0.75) or strongly correlated (absolute value of correlation coefficients no less than 0.75) with the previously obtained ranks.

Conclusions
In this study, we proposed a new approach for solving MCDM problems. In the framework of the proposed approach, we built a special score matrix for a given multicriteria problem, which allows us to use an appropriate ranking method and choose the corresponding best-ranked alternative from the Pareto set as a solution of the MCDM problem. The proposed approach is particularly useful when no decision-making authority is available or when the relative importance of various criteria has not been previously evaluated. To demonstrate the viability and suitability for applications, the proposed approach is illustrated using an example of a material-selection problem. It is shown that the solutions obtained using the proposed approach are quite competitive. All our numerical experiments were conducted on standard equipment (laptop with 2.59 GHz, 8 GB RAM, and a 64-bit operation system) and required~1 min for all eight considered rankings in the MATLAB environment without making any effort to optimist the code.