# Ranking-Theory Methods for Solving Multicriteria Decision-Making Problems

Authors: Joseph Gogodze
Journal: Advances in Operations Research
 (2019)

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## Abstract

The Pareto optimality is a widely used concept for the multicriteria decision-making problems. However, this concept has a significant drawback—the set of Pareto optimal alternatives usually is large. Correspondingly, the problem of choosing a specific Pareto optimal alternative for the decision implementation is arising. This study proposes a new approach to select an “appropriate” alternative from the set of Pareto optimal alternatives. The proposed approach is based on ranking-theory methods used for ranking participants in sports tournaments. In the framework of the proposed approach, we build a special score matrix for a given multicriteria problem, which allows the use of the mentioned ranking methods and to choose the corresponding best-ranked alternative from the Pareto set as a solution of the 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 evaluated previously. The proposed approach is tested on an example of a materials-selection problem for a sailboat mast.


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## Body

## 1. Introduction

This paper considers a novel approach for solving a multicriteria decision-making (MCDM) problem, with a finite number of decision alternatives and criteria. The multicriteria 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 can be used for decision-making purpose.However, unlike single-objective optimizations, a characteristic feature of Pareto optimality is that the set of Pareto optimal alternatives (i.e., set of efficient alternatives) is usually large. In addition, all these Pareto optimal alternatives must be considered as mathematically equal. Correspondingly, the problem of choosing a specific Pareto optimal alternative for implementation arises, because the final decision usually must be unique. Thus, additional factors must be considered to aid a decision-maker the selection of specific or more-favorable alternatives from the set of Pareto optimal solutions.The proposed approach is based on ranking-theory methods that used to rank participants in sports tournaments. In the framework of the proposed approach, we build a special score matrix for a given multicriteria problem, which allows us to use the mentioned ranking methods and choose the corresponding best-ranked alternative from the Pareto set as a solution of the problem. Note that the score matrix is built by the quite natural way—it is composed on the simple calculations of how many times one alternative is better than the other for each of the criteria. Hence, there is hope that the proposed approach yields a “notionally objective” ranking method and provides an “accurate ranking” of the alternatives for MCDM. The proposed approach is particularly useful when no decision-making authority is available, or when the relative importance of various criteria has not been evaluated previously.To demonstrate viability and suitability for applications, the proposed approach illustrated using an example of a materials-selection problem for a sailboat mast. This problem has been addressed by several researchers using various methods and, thus, can be considered as a kind of a benchmark problem. This illustration sheds light on the ranking approach’s applicability to the MCDM problems. Particularly, it is shown that the solutions of the illustrative example obtained by the proposed approach are quite competitive.The rest of this paper is structured as follows. In Section2, preliminaries regarding MCDM and ranking problems are presented, and the proposed methodology is described; Section 3 considers an illustrative example and Section 4 summarizes the article.


## 2. Proposed Methods

In what follows, for a natural numbern, we denote an n-dimensional vector space by Rn and Nn=1,…,n. If not otherwise mentioned, we identify a finite set A with the set Nn=1,…,n, where n=A is the capacity of the set A. By necessity, we also identify the matrix Π∈Rn×m with the map Π:Nn×Nm→R. For a matrix Π∈Rn×m, we denote its transpose by ΠT∈Rm×n.


### 2.1. Preliminaries

#### 2.1.1. Background on Multiobjective Decision-Making Problems

The following notation is drawn from a general treatment of multicriteria optimization theory [5, 6]. Let us consider the MCDM problem A,C, where A=a1,…,am is a set of alternatives and C=c1,…,cn is a set of criteria; i.e., ci:A→R,i=1,…,n, are given function. Without loss of generality, we may assume that the lower value is preferable for each criterion (i.e., each criterion is nonbeneficial), and the goal of the decision-making procedure is to minimize all criteria simultaneously [7].We say furthermore thatA is the set of admissible alternatives and map c→=(c1,…,cn):A→Rn is the criterion map (correspondingly, c→(A)⊂Rn is the set of admissible values of criteria). The following concepts are also associated with the criterion map and the set of alternatives. An alternative a∗∈A is Pareto optimal (i.e., efficient) if there exists no a∈A such that cj(a)≤cj(a∗) for all j∈Nn and ck(a)<ck(a∗) for some k∈Nn. The set of all efficient alternatives is denoted as Ae and is called the Pareto set. Correspondingly, f(Ae) is called the efficient front.Pareto optimality is an appropriate concept for the solutions of MCDM problems. In general, however, the setAe of Pareto optimal alternatives is very large and, moreover, all alternatives from Ae must be considered as “equally good solutions”. On the other hand, the final decision usually must be unique. Hence, additional factors must be considered to aid the selection of specific or more-favorable alternatives from the set Ae. The following subsections describe a novel approach that handles this problem objectively.


#### 2.1.2. Ranking Methods

This section gives a brief overview of the basic concepts of ranking theory. References [8, 9] discuss ranking theory in greater detail. For a natural number N, the N×N matrix S=[Sij],1≤i,j≤N, is a score matrix if Sij≥0,Sii=0,1≤i,j≤N. To emphasize that this problem was formulated in the context of competitive sports—note also that we can interpret elements of NN as athletes (or teams) who contest matches among themselves—and for each pair of athletes (i,j),1≤i,j≤N, the joint match Μ(i,j) includes K games. We interpret entry Sij,1≤i,j≤N, as the number of athlete i’s total wins in the match Μ(i,j). We also say that the result of the match Μ(i,j) is Sij wins of athlete i (losses of athlete j), Sji wins of athlete j (losses of athlete i), and K-Sij-Sji draws. Hence Gij(S)=Sij+Sji,(G=[Gij(S)],1≤i,j≤N;G=S+ST), can be interpreted as the number of decisive games that did not end in a draw in the match Μ(i,j),1≤i,j≤N. We also introduce the function gi(S)=∑j=1NGij(S),1≤i≤N, which reflects the number of decisive outcomes in all matches played by athlete i,1≤i≤N.For naturalN and score matrix S=[Sij],1≤i,j≤N, we say that the pair NN,S is the ranking problem. The weak-order (i.e., transitive and complete) relation R(N,S)⊂NN×NN represents the ranking method for the ranking problem NN,S. The vector r∈RN is a rating vector, where each ri,1≤i≤N, is the measure of the performance of player i∈NN in the ranking problem NN,S. For the ranking problem NN,S, a ranking method R(N,S) is induced by the rating vector r∈RN if(1)i,j∈RN,Si.e.,RN,Sranksiweaklyabovejif  and  only  ifri≥rj.In this article, for illustrative purposes, we consider only a few of the many ranking methods discussed in the literature (note also that the ranking methods considered here, based on the ranking problems involved in chess tournaments, go back to the investigations of H. Neustadtl, E. Zermelo, and B. Buckholdz. For detailed explanations see, e.g., [9] and the literature cited therein). All these methods are induced by their corresponding rating vectors. For a given score matrix S=[Sij],1≤i,j≤N, we consider the following ranking methods.Score Method..The rating vector for the score method, rs=(r1s,…,rNs)∈RN, is defined as the average score ris=∑j=1NSij/gi(S),1≤i≤N.Neustadt’s Method..Neustadt’s rating vector, rN∈RN, is defined by the equality rN=S-rs, where S-=[S-ij] and S-ij=Sij/gi(S),1≤i,j≤N.Buchholz’s Method..Buchholz’s rating vector, rB∈RN, is defined by the equality rN=G-(S)+ENrs, where G-(S)=[G-ij(S)],G-ij(S)=Gij(S)/gi(S),1≤i,j≤N.Fair-Bets Method..The rating vector for the fair-bet method, rfb∈Δ∘N, is defined as the unique solution of the following system of linear equations:(2)∑j=1NSijrjfb-∑j=1NSjirifb=0,1≤i≤N.Maximum-Likelihood Method..The rating vector for the maximum-likelihood method, rml=(r1ml,…,rNml)∈RN, is defined by the equality riml=ln(πi),1≤i≤N, where vector π=(π1,…,πN)∈Δ∘N is the unique solution of the following nonlinear system of equations:(3)πi∑j=1j≠iNGijSπi+πj=risgiS,1≤i≤N.


### 2.2. Ranking Methods to Solve MCDM Problems

Assume now thatA,C is a MCDM problem with a set of alternatives A=a1,…,am and a set of nonbeneficial criteria C=c1,…,cn and the decision-making goal is therefore to minimize the criteria simultaneously. Let us consider each element of A as an athlete (e.g., chess player) and assume that, for each pair of athletes a,a′∈A, the match Μ(a,a′) includes m games. The special construction of the score matrix of alternatives, SA, is defined as follows: for any a,a′∈A, we define(4)SAa,a′=∑c∈CscAa,a′,wherescAa,a′=1,ca<ca′;0,ca≥ca′;∀c∈C.Thus, the equality scA(a,a′)=1 means that c(a)<c(a′) for criterion c∈C and the alternative a (“athlete a”) receives one point (i.e., the athlete a wins a game c∈C in the match Μ(a,a′) and, correspondingly, SA(a,a′) indicates the number of total wins of athlete a in the match Μ(a,a′)). Obviously, m≥SA(a,a′)≥0,SA(a,a)=0,∀a,a′∈A. We say that an alternative a has defeated an alternative a′ if SA(a,a′)>SA(a′,a). We also say that the result of the match Μ(a,a′) is SA(a,a′) wins of the alternative a (losses of alternative a′), SA(a′,a) wins of the alternative a′ (losses of alternative a) and number of draws n-SA(a,a′)-SA(a′,a). Obviously matrix SA=SA(a,a′)a,a′∈Ais the score matrix for a set of alternatives in the sense of the definition from the previous subsection.The following procedure is used for solving MCDM problemA,C:(i)
For the MCDM problemA,C, the score matrix SA=SA(a,a′)a,a′∈A is constructed.(ii)
Using the score matrixSA, the alternatives from set A are ranked using a method R.(iii)
The alternative from the Pareto set,Ae, ranked best by method R is declared as the R-solution of the considered MCDM problem.Obviously, it would suffice to rank the Pareto set if Pareto set is known at the beginning of the proposed procedure. Nevertheless, we prefer given above description because it is more convenient in the cases when Pareto set is not known (or partially/approximately known), as it took place usually for the complex MCDM problems.It is clear that, instead of the MCDM problemA,C, we can consider also MCDM problem C,A. Obviously, applying described above procedure to the MCDM problem C,A, we can obtain a ranking of the criteria. However, we omit the corresponding details here.


## 2.1. Preliminaries

### 2.1.1. Background on Multiobjective Decision-Making Problems

The following notation is drawn from a general treatment of multicriteria optimization theory [5, 6]. Let us consider the MCDM problem A,C, where A=a1,…,am is a set of alternatives and C=c1,…,cn is a set of criteria; i.e., ci:A→R,i=1,…,n, are given function. Without loss of generality, we may assume that the lower value is preferable for each criterion (i.e., each criterion is nonbeneficial), and the goal of the decision-making procedure is to minimize all criteria simultaneously [7].We say furthermore thatA is the set of admissible alternatives and map c→=(c1,…,cn):A→Rn is the criterion map (correspondingly, c→(A)⊂Rn is the set of admissible values of criteria). The following concepts are also associated with the criterion map and the set of alternatives. An alternative a∗∈A is Pareto optimal (i.e., efficient) if there exists no a∈A such that cj(a)≤cj(a∗) for all j∈Nn and ck(a)<ck(a∗) for some k∈Nn. The set of all efficient alternatives is denoted as Ae and is called the Pareto set. Correspondingly, f(Ae) is called the efficient front.Pareto optimality is an appropriate concept for the solutions of MCDM problems. In general, however, the setAe of Pareto optimal alternatives is very large and, moreover, all alternatives from Ae must be considered as “equally good solutions”. On the other hand, the final decision usually must be unique. Hence, additional factors must be considered to aid the selection of specific or more-favorable alternatives from the set Ae. The following subsections describe a novel approach that handles this problem objectively.


### 2.1.2. Ranking Methods

This section gives a brief overview of the basic concepts of ranking theory. References [8, 9] discuss ranking theory in greater detail. For a natural number N, the N×N matrix S=[Sij],1≤i,j≤N, is a score matrix if Sij≥0,Sii=0,1≤i,j≤N. To emphasize that this problem was formulated in the context of competitive sports—note also that we can interpret elements of NN as athletes (or teams) who contest matches among themselves—and for each pair of athletes (i,j),1≤i,j≤N, the joint match Μ(i,j) includes K games. We interpret entry Sij,1≤i,j≤N, as the number of athlete i’s total wins in the match Μ(i,j). We also say that the result of the match Μ(i,j) is Sij wins of athlete i (losses of athlete j), Sji wins of athlete j (losses of athlete i), and K-Sij-Sji draws. Hence Gij(S)=Sij+Sji,(G=[Gij(S)],1≤i,j≤N;G=S+ST), can be interpreted as the number of decisive games that did not end in a draw in the match Μ(i,j),1≤i,j≤N. We also introduce the function gi(S)=∑j=1NGij(S),1≤i≤N, which reflects the number of decisive outcomes in all matches played by athlete i,1≤i≤N.For naturalN and score matrix S=[Sij],1≤i,j≤N, we say that the pair NN,S is the ranking problem. The weak-order (i.e., transitive and complete) relation R(N,S)⊂NN×NN represents the ranking method for the ranking problem NN,S. The vector r∈RN is a rating vector, where each ri,1≤i≤N, is the measure of the performance of player i∈NN in the ranking problem NN,S. For the ranking problem NN,S, a ranking method R(N,S) is induced by the rating vector r∈RN if(1)i,j∈RN,Si.e.,RN,Sranksiweaklyabovejif  and  only  ifri≥rj.In this article, for illustrative purposes, we consider only a few of the many ranking methods discussed in the literature (note also that the ranking methods considered here, based on the ranking problems involved in chess tournaments, go back to the investigations of H. Neustadtl, E. Zermelo, and B. Buckholdz. For detailed explanations see, e.g., [9] and the literature cited therein). All these methods are induced by their corresponding rating vectors. For a given score matrix S=[Sij],1≤i,j≤N, we consider the following ranking methods.Score Method..The rating vector for the score method, rs=(r1s,…,rNs)∈RN, is defined as the average score ris=∑j=1NSij/gi(S),1≤i≤N.Neustadt’s Method..Neustadt’s rating vector, rN∈RN, is defined by the equality rN=S-rs, where S-=[S-ij] and S-ij=Sij/gi(S),1≤i,j≤N.Buchholz’s Method..Buchholz’s rating vector, rB∈RN, is defined by the equality rN=G-(S)+ENrs, where G-(S)=[G-ij(S)],G-ij(S)=Gij(S)/gi(S),1≤i,j≤N.Fair-Bets Method..The rating vector for the fair-bet method, rfb∈Δ∘N, is defined as the unique solution of the following system of linear equations:(2)∑j=1NSijrjfb-∑j=1NSjirifb=0,1≤i≤N.Maximum-Likelihood Method..The rating vector for the maximum-likelihood method, rml=(r1ml,…,rNml)∈RN, is defined by the equality riml=ln(πi),1≤i≤N, where vector π=(π1,…,πN)∈Δ∘N is the unique solution of the following nonlinear system of equations:(3)πi∑j=1j≠iNGijSπi+πj=risgiS,1≤i≤N.


## 2.1.1. Background on Multiobjective Decision-Making Problems

The following notation is drawn from a general treatment of multicriteria optimization theory [5, 6]. Let us consider the MCDM problem A,C, where A=a1,…,am is a set of alternatives and C=c1,…,cn is a set of criteria; i.e., ci:A→R,i=1,…,n, are given function. Without loss of generality, we may assume that the lower value is preferable for each criterion (i.e., each criterion is nonbeneficial), and the goal of the decision-making procedure is to minimize all criteria simultaneously [7].We say furthermore thatA is the set of admissible alternatives and map c→=(c1,…,cn):A→Rn is the criterion map (correspondingly, c→(A)⊂Rn is the set of admissible values of criteria). The following concepts are also associated with the criterion map and the set of alternatives. An alternative a∗∈A is Pareto optimal (i.e., efficient) if there exists no a∈A such that cj(a)≤cj(a∗) for all j∈Nn and ck(a)<ck(a∗) for some k∈Nn. The set of all efficient alternatives is denoted as Ae and is called the Pareto set. Correspondingly, f(Ae) is called the efficient front.Pareto optimality is an appropriate concept for the solutions of MCDM problems. In general, however, the setAe of Pareto optimal alternatives is very large and, moreover, all alternatives from Ae must be considered as “equally good solutions”. On the other hand, the final decision usually must be unique. Hence, additional factors must be considered to aid the selection of specific or more-favorable alternatives from the set Ae. The following subsections describe a novel approach that handles this problem objectively.


## 2.1.2. Ranking Methods

This section gives a brief overview of the basic concepts of ranking theory. References [8, 9] discuss ranking theory in greater detail. For a natural number N, the N×N matrix S=[Sij],1≤i,j≤N, is a score matrix if Sij≥0,Sii=0,1≤i,j≤N. To emphasize that this problem was formulated in the context of competitive sports—note also that we can interpret elements of NN as athletes (or teams) who contest matches among themselves—and for each pair of athletes (i,j),1≤i,j≤N, the joint match Μ(i,j) includes K games. We interpret entry Sij,1≤i,j≤N, as the number of athlete i’s total wins in the match Μ(i,j). We also say that the result of the match Μ(i,j) is Sij wins of athlete i (losses of athlete j), Sji wins of athlete j (losses of athlete i), and K-Sij-Sji draws. Hence Gij(S)=Sij+Sji,(G=[Gij(S)],1≤i,j≤N;G=S+ST), can be interpreted as the number of decisive games that did not end in a draw in the match Μ(i,j),1≤i,j≤N. We also introduce the function gi(S)=∑j=1NGij(S),1≤i≤N, which reflects the number of decisive outcomes in all matches played by athlete i,1≤i≤N.For naturalN and score matrix S=[Sij],1≤i,j≤N, we say that the pair NN,S is the ranking problem. The weak-order (i.e., transitive and complete) relation R(N,S)⊂NN×NN represents the ranking method for the ranking problem NN,S. The vector r∈RN is a rating vector, where each ri,1≤i≤N, is the measure of the performance of player i∈NN in the ranking problem NN,S. For the ranking problem NN,S, a ranking method R(N,S) is induced by the rating vector r∈RN if(1)i,j∈RN,Si.e.,RN,Sranksiweaklyabovejif  and  only  ifri≥rj.In this article, for illustrative purposes, we consider only a few of the many ranking methods discussed in the literature (note also that the ranking methods considered here, based on the ranking problems involved in chess tournaments, go back to the investigations of H. Neustadtl, E. Zermelo, and B. Buckholdz. For detailed explanations see, e.g., [9] and the literature cited therein). All these methods are induced by their corresponding rating vectors. For a given score matrix S=[Sij],1≤i,j≤N, we consider the following ranking methods.Score Method..The rating vector for the score method, rs=(r1s,…,rNs)∈RN, is defined as the average score ris=∑j=1NSij/gi(S),1≤i≤N.Neustadt’s Method..Neustadt’s rating vector, rN∈RN, is defined by the equality rN=S-rs, where S-=[S-ij] and S-ij=Sij/gi(S),1≤i,j≤N.Buchholz’s Method..Buchholz’s rating vector, rB∈RN, is defined by the equality rN=G-(S)+ENrs, where G-(S)=[G-ij(S)],G-ij(S)=Gij(S)/gi(S),1≤i,j≤N.Fair-Bets Method..The rating vector for the fair-bet method, rfb∈Δ∘N, is defined as the unique solution of the following system of linear equations:(2)∑j=1NSijrjfb-∑j=1NSjirifb=0,1≤i≤N.Maximum-Likelihood Method..The rating vector for the maximum-likelihood method, rml=(r1ml,…,rNml)∈RN, is defined by the equality riml=ln(πi),1≤i≤N, where vector π=(π1,…,πN)∈Δ∘N is the unique solution of the following nonlinear system of equations:(3)πi∑j=1j≠iNGijSπi+πj=risgiS,1≤i≤N.


## 2.2. Ranking Methods to Solve MCDM Problems

Assume now thatA,C is a MCDM problem with a set of alternatives A=a1,…,am and a set of nonbeneficial criteria C=c1,…,cn and the decision-making goal is therefore to minimize the criteria simultaneously. Let us consider each element of A as an athlete (e.g., chess player) and assume that, for each pair of athletes a,a′∈A, the match Μ(a,a′) includes m games. The special construction of the score matrix of alternatives, SA, is defined as follows: for any a,a′∈A, we define(4)SAa,a′=∑c∈CscAa,a′,wherescAa,a′=1,ca<ca′;0,ca≥ca′;∀c∈C.Thus, the equality scA(a,a′)=1 means that c(a)<c(a′) for criterion c∈C and the alternative a (“athlete a”) receives one point (i.e., the athlete a wins a game c∈C in the match Μ(a,a′) and, correspondingly, SA(a,a′) indicates the number of total wins of athlete a in the match Μ(a,a′)). Obviously, m≥SA(a,a′)≥0,SA(a,a)=0,∀a,a′∈A. We say that an alternative a has defeated an alternative a′ if SA(a,a′)>SA(a′,a). We also say that the result of the match Μ(a,a′) is SA(a,a′) wins of the alternative a (losses of alternative a′), SA(a′,a) wins of the alternative a′ (losses of alternative a) and number of draws n-SA(a,a′)-SA(a′,a). Obviously matrix SA=SA(a,a′)a,a′∈Ais the score matrix for a set of alternatives in the sense of the definition from the previous subsection.The following procedure is used for solving MCDM problemA,C:(i)
For the MCDM problemA,C, the score matrix SA=SA(a,a′)a,a′∈A is constructed.(ii)
Using the score matrixSA, the alternatives from set A are ranked using a method R.(iii)
The alternative from the Pareto set,Ae, ranked best by method R is declared as the R-solution of the considered MCDM problem.Obviously, it would suffice to rank the Pareto set if Pareto set is known at the beginning of the proposed procedure. Nevertheless, we prefer given above description because it is more convenient in the cases when Pareto set is not known (or partially/approximately known), as it took place usually for the complex MCDM problems.It is clear that, instead of the MCDM problemA,C, we can consider also MCDM problem C,A. Obviously, applying described above procedure to the MCDM problem C,A, we can obtain a ranking of the criteria. However, we omit the corresponding details here.


## 3. Example

This section discusses the example problem that was solved to demonstrate the practicality of the proposed in Section2.2 procedure. All the necessary calculations were performed in the MATLAB computing environment. The example considered here is the problem of selecting the material for the mast of a sailing boat. This problem has been addressed by several researchers, using various methods and, thus, can be considered as a kind of benchmark problem.The component to be optimized, the mast, is modeled as a hollow cylinder that is subjected to axial compression. It has a length of 1,000 mm, an outer diameter ≤ 100 mm, an inner diameter ≥ 84 mm, a mass ≤ 3 kg, and a total axial compressive force of 153 kN [2]. The following criteria are chosen for the ranking problem at hand: specific strength (SS), specific modulus (SM), corrosion resistance (CR), and cost category (CC) [2]. The choice must be made from 15 alternative materials. The corresponding decision-making data are given in Table 3 of the Appendix, and the normalized decision matrix is given in Table 4 of the Appendix. Note also that, for the problem under consideration, the upper-lower-bound approach was used for normalization of the decision matrix [7]. The Pareto set for the considered problem is Ae=2,3,4,7,9,11,12,13,14,15.The following methods were used to solve the problem by previous investigators: WPM (weighted-properties method), VIKOR (multicriteria optimization through the concept of a compromise solution), CVIKOR (comprehensive VIKOR), FLA (fuzzy-logic approach), MOORA (multiobjective optimization based on ratio analysis), MULTIMOORA (a multiplicative form of MOORA), RPA (the reference-point approach), and a recently proposed game-theoretic method GTM [1–4, 10, 11]. Note also that the material-selection problem is an important application of MCDM [12, 13]. Table 5 of the Appendix presents the materials ranked by methods other than the one proposed in this paper.Direct calculations show that the score matrixSA in the considered case is(5)SA=000131122221111101131122221111210131122221111332021133331111111000011111111333120022222212333111022222212222132200001111222132220111111222132211001111222132221201111333332233330112333221133333022333222233332101333221133331110Using the score matrix SA, we rank the materials with each of the five methods described in Section 2.1.2. The ranking results are presented in Table 1. These results show that material 14 (Epoxy–63% carbon fabric) is ranked best by ranking methods rS, rB, and rml and material 13(Epoxy–70% glass fabric) is ranked best by ranking methods rN and rfb.Table 1
Materials ranked by proposed methods.
Material r S r N r B r f b rml Rating Rank Rating Rank Rating Rank Rating Rank Rating Rank 1 0,3529 14 0,1666 14 0,8752 14 0,0335 14 -3,408 14 2 0,3922 12 0,1816 12 0,9136 12 0,0380 12 -3,231 12 3 0,4118 10 0,1882 11 0,9274 11 0,0403 11 -3,167 10 4 0,6087 5 0,2693 7 1,0945 5 0,0819 7 -2,424 5 5 0,2340 15 0,1164 15 0,7342 15 0,0201 15 -4,072 15 6 0,5870 7 0,2716 6 1,0694 7 0,0822 6 -2,532 7 7 0,6087 6 0,2843 4 1,0907 6 0,0913 4 -2,438 6 8 0,3673 13 0,1767 13 0,8888 13 0,0371 13 -3,349 13 9 0,4400 9 0,2052 9 0,9563 9 0,0470 9 -3,043 9 10 0,4082 11 0,1931 10 0,9288 10 0,0425 10 -3,167 11 11 0,4600 8 0,2130 8 0,9755 8 0,0502 8 -2,959 8 12 0,6481 3 0,2969 3 1,1322 3 0,1022 3 -2,256 3 13 0,6800 2 0,3190 1 1,1595 2 0,1232 1 -2,125 2 14 0,6875 1 0,3161 2 1,1600 1 0,1222 2 -2,115 1 15 0,6250 4 0,2790 5 1,1001 4 0,0884 5 -2,395 4 Note: italic corresponds to the Pareto optimal (efficient) alternatives.Table1 also shows that sometimes the case when the alternative which does not belong to the Pareto set is ranked better than some set of the efficient alternatives can be observed (e.g., the efficient alternatives 11,3 and the unefficient alternative 6). However, we should not consider this as contradiction because the Pareto set and the ranking methods are independent objects and only the restriction of the ranking method on the Pareto set is essential.For comparison, Table2 presents the correlation coefficients of the alternative ranks as calculated by different methods. As we can see, the results of the proposed ranking methods correlate well with the rankings obtained by FLA, CVIKOR, and VIKOR; they are somewhat correlated with the rankings returned by MOORA, MULTIMOORA, RPA, and WPM and are poorly correlated with the ranking obtained by GTM. Meanwhile, the methods rS, rN, rB, rfb, and rml are very strongly correlated between themselves.Table 2
Correlation between methods.
r S r N r B r fb r ml MOORA∗ 0,564286 0,603571 0,578571 0,603571 0,564286 MULTIMOORA∗ 0,496429 0,503571 0,521429 0,503571 0,496429 RPA∗ 0,467857 0,492857 0,485714 0,492857 0,467857 FLA∗ 0,764286 0,717857 0,792857 0,717857 0,764286 Wpm∗∗ 0,403571 0,410714 0,442857 0,410714 0,403571 CVIKOR∗∗∗ 0,742857 0,646429 0,739286 0,646429 0,742857 VIKOR∗∗∗ 0,892857 0,871429 0,907143 0,871429 0,892857 GTM∗∗∗∗ -0,12143 -0,07857 -0,09286 -0,07857 -0,12143 Sources:∗[1]; ∗∗[2]; ∗∗∗[3]; ∗∗∗∗[4].Table 3
Decision matrix for selecting material for a sailing boat mast.
# Material Criteria Specific strength (MPa) Specific modulus (GPa) Corrosion resistance Cost Category SS SM CR CC 1 2 3 4 1 AISI 1020 35.9 26.9 1 5 2 AISI 1040 51.3 26.9 1 5 3 ASTM A242 type 1 42.3 27.2 1 5 4 AISI 4130 194.9 27.2 4 3 5 AISI 316 25.6 25.1 4 3 6 AISI 416 heat treated 57.1 28.1 4 3 7 AISI 431 heat treated 71.4 28.1 4 3 8 AA 6061 T6 101.9 25.8 3 4 9 AA 2024 T6 141.9 26.1 3 4 10 AA 2014 T6 148.2 25.8 3 4 11 AA 7075 T6 180.4 25.9 3 4 12 Ti–6Al–4V 208.7 27.6 5 1 13 Epoxy–70% glass fabric 604.8 28.0 4 2 14 Epoxy–63% carbon fabric 416.2 66.5 4 1 15 Epoxy–62% aramid fabric 637.7 27.5 4 1 Source: [1]. Notes: CR scale: 1 = poor; 2 = fair; 3 = good; 4 = very good; 5 = excellent. CC scale: 1 = very high; 2 = high; 3 = moderate; 4 = low; 5 = very low.Table 4
Normalized decision matrix for the material selection problem.
Criteria 1 2 3 4 Materials 1 0.9832 0.9565 1.0000 0.0000 2 0.9580 0.9565 1.0000 0.0000 3 0.9727 0.9493 1.0000 0.0000 4 0.7234 0.9493 0.2500 0.5000 5 1.0000 1.0000 0.2500 0.5000 6 0.9485 0.9275 0.2500 0.5000 7 0.9252 0.9275 0.2500 0.5000 8 0.8753 0.9831 0.5000 0.2500 9 0.8100 0.9758 0.5000 0.2500 10 0.7997 0.9831 0.5000 0.2500 11 0.7471 0.9807 0.5000 0.2500 12 0.7009 0.9396 0.0000 1.0000 13 0.0537 0.9300 0.2500 0.7500 14 0.3619 0.0000 0.2500 1.0000 15 0.0000 0.9420 0.2500 1.0000 Note: italic denotes Pareto optimal (efficient) alternatives.Table 5
Materials ranked by comparable methods.
Material MOORA∗ MULTIMOORA∗ RPA∗ FLA∗ Wpm∗∗ CVIKOR∗∗∗ VIKOR∗∗∗ GTM∗∗∗∗ 1 14 14 14 14 14 12 14 14 2 15 15 13 13 13 6 11 10 3 13 13 12 15 15 9 13 11 4 12 12 15 4 11 4 4 2 5 4 4 4 11 10 15 15 9 6 7 11 11 9 9 14 10 8 7 6 10 10 10 8 11 5 7 8 11 9 9 8 7 13 12 5 9 10 7 8 12 2 8 7 4 10 9 6 7 7 4 10 9 3 11 5 8 6 6 6 5 6 1 12 8 5 2 5 3 7 8 12 13 2 2 3 3 12 2 2 6 14 3 3 1 2 1 1 1 15 15 1 1 5 1 5 3 3 13 Sources:∗[1]; ∗∗[2]; ∗∗∗[3]; ∗∗∗∗ [4].


## 4. Conclusions

In this study, we have proposed a new approach for solving MCDM problems. The proposed approach is based on ranking-theory methods which are used in the competitive sports tournaments. In the framework of the proposed approach, we build 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 evaluated previously.To demonstrate the viability and suitability for applications, the proposed approach illustrated using an example of a materials-selection problem. It is shown that the solutions of the illustrative example obtained by the proposed approach are quite competitive. Note also that the proposed approach seems numerically efficient. Namely, our preliminary numerical experiments (unpublished) show that that MCDM problems with the number of alternatives of the order of 1.5 hundred and with the number of criteria of the order of ten can be solved by the proposed method in a few minutes (~5 min, the calculations were conducted on a laptop with 2.59GHz, 8GB RAM, 64-bit operation system, MATLAB environment, and not making any effort to optimize the code).Due to the simplicity and flexibility of the implementation, the proposed approach can be also used in a few interesting directions. For example, if we consider the “transposed” MCDM problem (i.e., the problem, for which the criteria of the original problem are alternatives and the alternatives of the original problem are criteria), the proposed approach also allows ranking the criteria and identified a “leading criterion”. On the other hand, an “objective” ranking of the criteria may stimulate the development of other instruments for the Pareto optimization. It also seems possible that the proposed approach will find applications in the (e.g., evolutionary) Pareto optimization algorithms. However, we will limit ourselves here only to mention these directions for further investigations.



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