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Ranking-Theory Methods for Solving Multicriteria Decision-Making Problems

Ranking-Theory Methods for Solving Multicriteria Decision-Making Problems Hindawi Advances in Operations Research Volume 2019, Article ID 3217949, 7 pages https://doi.org/10.1155/2019/3217949 Research Article Ranking-Theory Methods for Solving Multicriteria Decision-Making Problems Joseph Gogodze Institute of Control Systems, TECHINFORMI, Georgian Technical University, Tbilisi, Georgia Correspondence should be addressed to Joseph Gogodze; jgogodze@gmail.com Received 7 August 2018; Revised 29 January 2019; Accepted 27 February 2019; Published 1 April 2019 Academic Editor: Imed Kacem Copyright © 2019 Joseph Gogodze. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. eTh 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. eTh 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. eTh proposed approach is tested on an example of a materials-selection problem for a sailboat mast. 1. Introduction The proposed approach is based on ranking-theory meth- ods that used to rank participants in sports tournaments. In This paper considers a novel approach for solving a mul- the framework of the proposed approach, we build a special ticriteria decision-making (MCDM) problem, with a finite score matrix for a given multicriteria problem, which allows number of decision alternatives and criteria. The multicriteria us to use the mentioned ranking methods and choose the formulation is the typical starting point for theoretical and corresponding best-ranked alternative from the Pareto set practical analyses of decision-making problems. us, Th the as a solution of the problem. Note that the score matrix is definition of Pareto optimality and a vast arsenal of different built by the quite natural way—it is composed on the simple Pareto optimization methods can be used for decision- calculations of how many times one alternative is better than making purpose. the other for each of the criteria. Hence, there is hope that However, unlike single-objective optimizations, a charac- the proposed approach yields a “notionally objective” ranking teristic feature of Pareto optimality is that the set of Pareto method and provides an “accurate ranking” of the alternatives optimal alternatives (i.e., set of efficient alternatives) is usually for MCDM. The proposed approach is particularly useful large. In addition, all these Pareto optimal alternatives must when no decision-making authority is available, or when the be considered as mathematically equal. Correspondingly, the relative importance of various criteria has not been evaluated problem of choosing a specific Pareto optimal alternative previously. for implementation arises, because the final decision usually To demonstrate viability and suitability for applications, must be unique. us, Th additional factors must be consid- the proposed approach illustrated using an example of a ered to aid a decision-maker the selection of specific or materials-selection problem for a sailboat mast. This problem more-favorable alternatives from the set of Pareto optimal has been addressed by several researchers using various solutions. 2 Advances in Operations Research methods and, thus, can be considered as a kind of a bench- ranking theory in greater detail. For a natural number 𝑁 ,the mark problem. This illustration sheds light on the ranking 𝑁×𝑁 matrix𝑆= [𝑆 ], 1 ≤ 𝑖,𝑗 ≤ 𝑁,is a score matrix if 𝑆 ≥ approach’s applicability to the MCDM problems. Particularly, 0, 𝑆 =0, 1 ≤𝑖,𝑗 ≤𝑁. To emphasize that this problem was 𝑖𝑖 it is shown that the solutions of the illustrative example formulated in the context of competitive sports—note also obtained by the proposed approach are quite competitive. that we can interpret elements of N(𝑁) as athletes (or teams) The rest of this paper is structured as follows. In Section 2, who contest matches among themselves—and for each pair of preliminaries regarding MCDM and ranking problems are athletes (𝑖,),𝑗 1 ≤ 𝑖,𝑗 ≤ 𝑁 ,the joint match M(𝑖, )𝑗 includes 𝐾 presented, and the proposed methodology is described; games. We interpret entry 𝑆 ,1 ≤ 𝑖,𝑗 ≤ 𝑁 ,as the number Section 3 considers an illustrative example and Section 4 of athlete 𝑖 ’s total wins in the match M(𝑖,).𝑗 We also say summarizes the article. that the result of the match M(𝑖,)𝑗 is 𝑆 wins of athlete 𝑖 (losses of athlete 𝑗 ), 𝑆 wins of athlete 𝑗 (losses of athlete 𝑖 ), 𝑗𝑖 and (𝐾 − 𝑆 −𝑆 ) draws. Hence 𝐺 (𝑆) = 𝑆 +𝑆 ,(𝐺 = 𝑗𝑖 𝑗𝑖 2. Proposed Methods [𝐺 (𝑆)], 1 ≤ 𝑖, 𝑗 ≤ 𝑁; 𝐺 = 𝑆 + 𝑆 ), can be interpreted In what follows, for a natural number 𝑛 , wedenotean 𝑛 - as the number of decisive games that did not end in a draw dimensional vector space by R and N(𝑛) = {1,...,𝑛}. If not in the match M(𝑖,),𝑗 1 ≤ 𝑖, 𝑗 ≤ 𝑁. We also introduce the otherwise mentioned, we identify a n fi ite set 𝐴 with the set function 𝑔 (𝑆) = ∑ 𝐺 (𝑆), 1 ≤ 𝑖 ≤ 𝑁, which reflects the 𝑖 𝑗=1 N(𝑛) = {1,...,𝑛} ,where 𝑛=||𝐴 is the capacity of the set 𝐴. number of decisive outcomes in all matches played by athlete 𝑛×𝑚 By necessity, we also identify the matrix Π∈ R with the 𝑖,1 ≤ 𝑖 ≤ 𝑁. 𝑛×𝑚 map Π: N(𝑛) × N(𝑚) → 󳨀 R.For a matrix Π∈ R ,we For natural 𝑁 and score matrix 𝑆= [𝑆 ],1 ≤ 𝑖, 𝑗 ≤ 𝑁,we 𝑇 𝑚×𝑛 denote its transpose by Π ∈ R . say that the pair (N(𝑁),)𝑆 is the ranking problem. The weak- order (i.e., transitive and complete) relation𝑅(𝑁, 𝑆) ⊂ N(𝑁)× N(𝑁) represents the ranking method for the ranking problem 2.1. Preliminaries (N(𝑁),).𝑆 The vector 𝑟∈ R is a rating vector, where each 2.1.1. Background on Multiobjective Decision-Making Prob- 𝑟 ,1 ≤𝑖 ≤𝑁 , is the measure of the performance of player lems. The following notation is drawn from a general treat- 𝑖∈ N(𝑁) in the ranking problem (N(𝑁),).𝑆 For the ranking ment of multicriteria optimization theory [5, 6]. Let us problem (N(𝑁),)𝑆 ,a ranking method 𝑅(𝑁,𝑆) is induced by consider the MCDM problem ⟨𝐴, ⟩𝐶 ,where𝐴= {𝑎 ,...,𝑎 } 1 𝑚 the rating vector 𝑟∈ R if is a set of alternatives and 𝐶={𝑐 ,...,𝑐 } is a set of 1 𝑛 criteria; i.e., 𝑐 :𝐴 →󳨀 R, 𝑖 = 1,...,𝑛 ,are given (𝑖,)𝑗 ∈ 𝑅 (𝑁,𝑆 ) function. Without loss of generality, we may assume that (1) the lower value is preferable for each criterion (i.e., each (i.e.,𝑅 𝑁, 𝑆 ranks 𝑖 weakly above𝑗) if and only if 𝑟 ≥𝑟 . ( ) 𝑖 𝑗 criterion is nonbeneficial), and the goal of the decision- making procedure is to minimize all criteria simultaneously [7]. In this article, for illustrative purposes, we consider only a We say furthermore that 𝐴 is the set of admissible few of the many ranking methods discussed in the literature →󳨀 𝑛 alternatives and map 𝑐= (𝑐 ,...,𝑐 ): 𝐴󳨀→ R is the 1 𝑛 (note also that the ranking methods considered here, based →󳨀 𝑛 criterion map (correspondingly, 𝑐()𝐴 ⊂ R is the set of on the ranking problems involved in chess tournaments, go admissible values of criteria). The following concepts are also back to the investigations of H. Neustadtl, E. Zermelo, and associated with the criterion map and the set of alternatives. B. Buckholdz. For detailed explanations see, e.g., [9] and the An alternative 𝑎 ∈𝐴 is Pareto optimal (i.e., efficient) if literature cited therein). All these methods are induced by there exists no 𝑎∈𝐴 such that 𝑐 (𝑎) ≤ 𝑐 (𝑎 ) for all their corresponding rating vectors. For a given score matrix 𝑗 𝑗 ∗ 𝑗∈ N(𝑛) and 𝑐 (𝑎) < 𝑐 (𝑎 ) for some 𝑘∈ N(𝑛). The 𝑆= [𝑆 ],1 ≤ 𝑖, 𝑗 ≤ 𝑁, we consider the following ranking 𝑘 𝑘 ∗ set of all efficient alternatives is denoted as 𝐴 and is called methods. the Pareto set. Correspondingly, 𝑓(𝐴 ) is called the efficient front. Score Method. The rating vector for the score method, 𝑟 = Pareto optimality is an appropriate concept for the 𝑠 𝑠 𝑁 𝑠 (𝑟 ,...,𝑟 )∈ R , isdenfi ed asthe averagescore 𝑟 = solutions of MCDM problems. In general, however, the 1 𝑁 𝑖 set 𝐴 of Pareto optimal alternatives is very large and, ∑ 𝑆 /𝑔 (𝑆), 1 ≤ 𝑖 ≤ 𝑁. 𝑒 𝑖 𝑗=1 moreover, all alternatives from 𝐴 must be considered as “equally good solutions”. On the other hand, the n fi al 𝑁 𝑁 decision usually must be unique. Hence, additional factors Neustadt’s Method. Neustadt’s rating vector, 𝑟 ∈ R ,is 𝑁 𝑠 must be considered to aid the selection of specific or more- defined by the equality 𝑟 = 𝑆𝑟 ,where 𝑆= [ 𝑆 ] and 𝑆 = favorable alternatives from the set 𝐴 . The following subsec- 𝑆 /𝑔 (𝑆),1 ≤ 𝑖,𝑗 ≤ 𝑁. tions describe a novel approach that handles this problem objectively. 𝐵 𝑁 Buchholz’s Method. Buchholz’s rating vector, 𝑟 ∈ R ,is 𝑁 𝑠 defined by the equality 𝑟 =[𝐺(𝑆) + 𝐸 ]𝑟 ,where 𝐺(𝑆) = 2.1.2. Ranking Methods. This section gives a brief overview of the basic concepts of ranking theory. References [8, 9] discuss [𝐺 (𝑆)], 𝐺 (𝑆) = 𝐺 (𝑆)/𝑔 (𝑆), 1 ≤ 𝑖,𝑗 ≤ 𝑁. 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 Advances in Operations Research 3 Fair-Bets Method. The rating vector for the fair-bet method, (ii) Using the score matrix 𝑆 ,the alternatives from set 𝐴 are ranked using a method 𝑅 . 𝑟 ∈ Δ , isdenfi ed asthe unique solution of thefollowing system of linear equations: (iii) The alternative from the Pareto set, 𝐴 ,ranked best by method 𝑅 is declared as the 𝑅− solution of the 𝑁 𝑁 considered MCDM problem. ∑𝑆 𝑟 −( ∑𝑆 )𝑟 =0, 1 ≤𝑖 ≤𝑁. (2) 𝑗𝑖 𝑗 𝑖 𝑗=1 𝑗=1 Obviously, it would suffice to rank the Pareto set if Pareto set is known at the beginning of the proposed procedure. Maximum-Likelihood Method. The rating vector for the Nevertheless, we prefer given above description because it is 𝑚𝑙 𝑚𝑙 𝑚𝑙 𝑁 maximum-likelihood method, 𝑟 =(𝑟 ,...,𝑟 )∈ R ,is 1 𝑁 more convenient in the cases when Pareto set is not known 𝑚𝑙 defined by the equality 𝑟 = ln(𝜋 ), 1 ≤ 𝑖 ≤ 𝑁 ,where vector 𝑖 𝑖 (or partially/approximately known), as it took place usually 𝜋=(𝜋 ,...,𝜋 )∈ Δ is theuniquesolution of the following for the complex MCDM problems. 1 𝑁 𝑁 nonlinear system of equations: It is clear that, instead of the MCDM problem ⟨𝐴, ⟩𝐶 , we can consider also MCDM problem ⟨𝐶, 𝐴⟩. Obviously, applying described above procedure to the MCDM problem 𝐺 (𝑆 ) ⟨𝐶, 𝐴⟩ , we can obtain a ranking of the criteria. However, we 𝜋 ∑ =𝑟 𝑔 (𝑆 ),1≤𝑖≤𝑁. 𝑖 𝑖 (3) 𝜋 +𝜋 𝑖 𝑗 𝑗=1 omit the corresponding details here. 𝑗 =𝑖̸ 3. Example 2.2. Ranking Methods to Solve MCDM Problems. Assume now that ⟨𝐴, ⟩𝐶 is a MCDM problem with a set of alternatives This section discusses the example problem that was solved 𝐴= {𝑎 ,...,𝑎 } and a set of nonbeneficial criteria 𝐶= 1 𝑚 to demonstrate the practicality of the proposed in Section 2.2 {𝑐 ,...,𝑐 } and the decision-making goal is therefore to 1 𝑛 procedure. All the necessary calculations were performed minimize the criteria simultaneously. Let us consider each in the MATLAB computing environment. The example element of 𝐴 as an athlete (e.g., chess player) and assume that, considered here is the problem of selecting the material 󸀠 󸀠 for each pair of athletes 𝑎, 𝑎 ∈𝐴 ,the matchM(𝑎,𝑎 ) includes for the mast of a sailing boat. This problem has been 𝑚 games. The special construction of the score matrix of addressed by several researchers, using various methods 𝐴 󸀠 alternatives, 𝑆 , isdenfi ed asfollows: for any 𝑎, 𝑎 ∈𝐴 ,we and, thus, can be considered as a kind of benchmark define problem. The component to be optimized, the mast, is modeled as 𝐴 󸀠 𝐴 󸀠 a hollow cylinder that is subjected to axial compression. It 𝑆 (𝑎, 𝑎 )= ∑ 𝑠 (𝑎, 𝑎 ), 𝑐∈𝐶 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 (4) 1, 𝑐 (𝑎 ) <𝑐(𝑎 ); compressive force of 153 kN [2]. The following criteria are 𝐴 󸀠 where 𝑠 (𝑎, 𝑎 )= ∀𝑐 ∈ .𝐶 󸀠 chosen for the ranking problem at hand: specific strength 0, 𝑐 𝑎 ≥𝑐(𝑎 ); ( ) (SS), specicfi modulus (SM), corrosion resistance (CR), and cost category (CC) [2]. The choice must be made from 15 𝐴 󸀠 󸀠 alternative materials. The corresponding decision-making Thus, the equality 𝑠 (𝑎, 𝑎 )= 1 means that 𝑐(𝑎) < 𝑐(𝑎 ) for data are given in Table 3 of the Appendix, and the normalized criterion 𝑐∈ 𝐶 and the alternative 𝑎 (“athlete 𝑎”) receives decision matrix is given in Table 4 of the Appendix. Note also one point (i.e., the athlete 𝑎 wins a game 𝑐∈𝐶 in the match 󸀠 𝐴 󸀠 that, for the problem under consideration, the upper-lower- M(𝑎, 𝑎 ) and, correspondingly, 𝑆 (𝑎,𝑎 ) indicates the number boundapproach wasusedfor normalization ofthe decision of total wins of athlete 𝑎 in the match M(𝑎, 𝑎 )). Obviously, 𝐴 󸀠 𝐴 󸀠 matrix [7]. The Pareto set for the considered problem is 𝐴 = 𝑚≥𝑆 (𝑎, 𝑎 )≥ 0,𝑆 (,𝑎𝑎 ) = 0,∀,𝑎𝑎 ∈𝐴. We say that {2,3, 4, 7, 9, 11,12, 13,14, 15}. 󸀠 𝐴 󸀠 an alternative 𝑎 has defeated an alternative 𝑎 if 𝑆 (𝑎,𝑎 )> 𝐴 󸀠 󸀠 The following methods were used to solve the prob- 𝑆 (𝑎 ,𝑎). We also say that the result of the match M(𝑎, 𝑎 ) is lem by previous investigators: WPM (weighted-properties 𝐴 󸀠 󸀠 𝑆 (𝑎, 𝑎 ) wins of the alternative 𝑎 (losses of alternative 𝑎 ), method), VIKOR (multicriteria optimization through the 𝐴 󸀠 󸀠 𝑆 (𝑎 ,𝑎) wins of the alternative 𝑎 (losses of alternative 𝑎) concept of a compromise solution), CVIKOR (comprehen- 𝐴 󸀠 𝐴 󸀠 and number of draws (𝑛 − 𝑆 (𝑎,𝑎 )− 𝑆 (𝑎 ,)). 𝑎 Obviously sive VIKOR), FLA (fuzzy-logic approach), MOORA (mul- 𝐴 𝐴 󸀠 matrix 𝑆 =[𝑆 (𝑎, 𝑎 )] 󸀠 is the score matrix for a set of 𝑎,𝑎 ∈𝐴 tiobjective optimization based on ratio analysis), MULTI- alternatives in the sense of the definition from the previous MOORA (a multiplicative form of MOORA), RPA (the subsection. reference-point approach), and a recently proposed game- The following procedure is used for solving MCDM theoretic method GTM [1–4, 10, 11]. Note also that the problem ⟨𝐴, ⟩𝐶 : material-selection problem is an important application of MCDM [12, 13]. Table 5 of the Appendix presents the mate- (i) For the MCDM problem⟨𝐴, ⟩𝐶 ,thescorematrix𝑆 = rials ranked by methods other than the one proposed in this 𝐴 󸀠 [𝑆 (𝑎,𝑎 )] 󸀠 is constructed. paper. 𝑎,𝑎 ∈𝐴 𝑖𝑗 𝑖𝑗 𝑓𝑏 𝑓𝑏 𝑓𝑏 4 Advances in Operations Research Table 1: Materials ranked by proposed methods. 𝑆 𝑁 𝐵 𝑚𝑙 𝑟 𝑟 𝑟 𝑟 𝑟 Material 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. Direct calculations show that the score matrix 𝑆 in the unefficient alternative 6). However, we should not consider considered case is this as contradiction because the Pareto set and the ranking methods are independent objects and only the restriction of 00 013 112 222 111 1 the ranking method on the Pareto set is essential. [ ] For comparison, Table 2 presents the correlation coef- 10 113 112 222 111 1 [ ] [ ] ficients of the alternative ranks as calculated by different [ ] 21 013 112 222 111 1 [ ] methods. As we can see, the results of the proposed ranking [ ] [ ] methods correlate well with the rankings obtained by FLA, 33 202 113 333 111 1 [ ] [ ] CVIKOR, and VIKOR; they are somewhat correlated with the [ ] 11 100 001 111 111 1 [ ] rankings returned by MOORA, MULTIMOORA, RPA, and [ ] 33 312 002 222 221 2 WPM and are poorly correlated with the ranking obtained [ ] 𝑆 𝑁 𝐵 𝑓𝑏 𝑚𝑙 [ ] by GTM. Meanwhile, the methods 𝑟 , 𝑟 , 𝑟 , 𝑟 ,and 𝑟 are [ 33 311 102 222 221 2] [ ] very strongly correlated between themselves. [ ] 𝑆 = 22 213 220 000 111 1 (5) [ ] [ ] [ ] 22 213 222 011 111 1 [ ] 4. Conclusions [ ] [ ] 22 213 221 100 111 1 [ ] In this study, we have proposed a new approach for solv- [ ] [ ] 22 213 222 120 111 1 ing MCDM problems. The proposed approach is based on [ ] [ ] ranking-theory methods which are used in the competitive [ 33 333 223 333 011 2 [ ] sports tournaments. In the framework of the proposed [ ] [ 33 322 113 333 302 2] approach, we build a special score matrix for a given mul- [ ] [ ] ticriteria problem, which allows us to use an appropriate 33 322 223 333 210 1 [ ] ranking method and choose the corresponding best-ranked 33 322 113 333 111 0 [ ] alternative from the Pareto set as a solution of the MCDM problem. The proposed approach is particularly useful when Using the score matrix 𝑆 , we rank the materials with each no decision-making authority is available, or when the of the vfi e methods described in Section 2.1.2. The ranking relative importance of various criteria has not been evaluated results are presented in Table 1. These results show that previously. material 14 (Epoxy–63% carbon fabric)isranked best by To demonstrate the viability and suitability for applica- 𝑆 𝐵 𝑚𝑙 ranking methods 𝑟 , 𝑟 ,and 𝑟 and material 13 (Epoxy–70% tions, the proposed approach illustrated using an example 𝑁 𝑓𝑏 of a materials-selection problem. It is shown that the solu- glass fabric) is ranked best by ranking methods 𝑟 and 𝑟 . Table 1 also shows that sometimes the case when the tions of the illustrative example obtained by the proposed alternative which does not belong to the Pareto set is approach are quite competitive. Note also that the proposed ranked better than some set of the efficient alternatives approach seems numerically efficient. Namely, our prelim- can be observed (e.g., the efficient alternatives 11,3 and the inary numerical experiments (unpublished) show that that 𝑓𝑏 Advances in Operations Research 5 Table 2: Correlation between methods. S N B fb ml r r r r r 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. Criteria #Material Specific strength (MPa) Specific modulus (GPa) Corrosion resistance Cost Category SS SM CR CC 12 3 4 1AISI1020 35.9 26.9 1 5 2AISI1040 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= veryhigh; 2 =high; 3 = moderate; 4 =low; 5 =verylow. MCDM problems with the number of alternatives of the evolutionary) Pareto optimization algorithms. However, we order of 1.5 hundred and with the number of criteria of the will limit ourselves here only to mention these directions for order of ten can be solved by the proposed method in a few further investigations. minutes (∼5 min, the calculations were conducted on a laptop with 2.59GHz, 8GB RAM, 64-bit operation system, MATLAB Appendix environment, and not making any effort to optimize the code). See Tables 3,4,and 5. Due to the simplicity and flexibility of the implementa- tion, the proposed approach can be also used in a few interest- Data Availability ing directions. For example, if we consider the “transposed” MCDM problem (i.e., the problem, for which the criteria Previously reported data were used to support this study. of the original problem are alternatives and the alternatives These prior studies are cited at relevant places within the text of the original problem are criteria), the proposed approach as references. 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 Conflicts of Interest for the Pareto optimization. It also seems possible that the proposed approach will find applications in the (e.g., The author declares that he has no conflicts of interest. 6 Advances in Operations Research Table 4: Normalized decision matrix for the material selection problem. Criteria 12 3 4 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 Materials 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 114 14 14 14 14 12 14 14 2 15 15 13 13 13 6 11 10 3 13 13 12 15 15 9 13 11 412 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 7 811 9 9 8 7 13 12 5 910 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]. References [6] K.M.Miettinen, Nonlinear Multiobjective Optimization,Kluwer Academic Publishers, 1999. [1] P. Karande and S. Chakraborty, “Application of multi-objective [7] R. T. Marler and J. S. Arora, “Function-transformation methods optimization on the basis of ratio analysis (MOORA) method for multi-objective optimization,” Engineering Optimization, for materials selection,” Materials & Design, vol.37,pp.317–324, vol. 37, no. 6, pp. 551–570, 2005. [2] M. M. Farag, “Quantitative methods of materials selection,” in [8] A. Y. Govan, Ranking eTh ory with Application to Popular Sports Handbook of Materials Selection,M. Kutz,Ed., 2002. [PhD thesis], North Carolina State, University, 2008. [3] A. Jahan, F. Mustapha, M. Y. Ismail, S. M. Sapuan, and M. [9] J. Gonzalez ´ -D´ıaz, R. Hendrickx, and E. Lohmann, “Paired com- Bahraminasab, “A comprehensive VIKOR method for material parisons analysis: an axiomatic approach to ranking methods,” selection,” Materials & Design, vol. 32, no. 3, pp. 1215–1221, 2011. Social Choice and Welfare,vol. 42, no.1, pp. 139–169, 2014. [4] J. Gogodze, “Using a two-person zero-sum game to solve a decision-making problem,” Pure and Applied Mathematics [10] P. Chatterjee, V. M. Athawale, and S. Chakraborty, “Selection of Journal,vol.7,no.2, pp. 11–19, 2018. materials using compromise ranking and outranking methods,” [5] M. Ehrgott, Multicriteria Optimization,Springer,2005. Materials and Corrosion, vol. 30, no. 10, pp. 4043–4053, 2009. Advances in Operations Research 7 [11] R. Sarfaraz Khabbaz, B. Dehghan Manshadi, A. Abedian, and R. Mahmudi, “A simplified fuzzy logic approach for materi- als selection in mechanical engineering design,” Materials & Design, vol.30, no.3,pp.687–697,2009. [12] M. Yazdani, “New approach to select materials using MADM tools,” International Journal of Business and Systems Research, vol. 12, no. 1, pp. 25–42, 2018. [13] K. Anyfantis, P. Foteinopoulos, and P. Stavropoulos, “Design for manufacturing of multi-material mechanical parts: a computa- tional based approach,” Procedia CIRP,vol.66,pp.22–26,2017. 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Ranking-Theory Methods for Solving Multicriteria Decision-Making Problems

Gogodze, Joseph
Advances in Operations Research , Volume 2019 – Apr 1, 2019
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Copyright © 2019 Joseph Gogodze. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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10.1155/2019/3217949
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Hindawi Advances in Operations Research Volume 2019, Article ID 3217949, 7 pages https://doi.org/10.1155/2019/3217949 Research Article Ranking-Theory Methods for Solving Multicriteria Decision-Making Problems Joseph Gogodze Institute of Control Systems, TECHINFORMI, Georgian Technical University, Tbilisi, Georgia Correspondence should be addressed to Joseph Gogodze; jgogodze@gmail.com Received 7 August 2018; Revised 29 January 2019; Accepted 27 February 2019; Published 1 April 2019 Academic Editor: Imed Kacem Copyright © 2019 Joseph Gogodze. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. eTh 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. eTh 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. eTh proposed approach is tested on an example of a materials-selection problem for a sailboat mast. 1. Introduction The proposed approach is based on ranking-theory meth- ods that used to rank participants in sports tournaments. In This paper considers a novel approach for solving a mul- the framework of the proposed approach, we build a special ticriteria decision-making (MCDM) problem, with a finite score matrix for a given multicriteria problem, which allows number of decision alternatives and criteria. The multicriteria us to use the mentioned ranking methods and choose the formulation is the typical starting point for theoretical and corresponding best-ranked alternative from the Pareto set practical analyses of decision-making problems. us, Th the as a solution of the problem. Note that the score matrix is definition of Pareto optimality and a vast arsenal of different built by the quite natural way—it is composed on the simple Pareto optimization methods can be used for decision- calculations of how many times one alternative is better than making purpose. the other for each of the criteria. Hence, there is hope that However, unlike single-objective optimizations, a charac- the proposed approach yields a “notionally objective” ranking teristic feature of Pareto optimality is that the set of Pareto method and provides an “accurate ranking” of the alternatives optimal alternatives (i.e., set of efficient alternatives) is usually for MCDM. The proposed approach is particularly useful large. In addition, all these Pareto optimal alternatives must when no decision-making authority is available, or when the be considered as mathematically equal. Correspondingly, the relative importance of various criteria has not been evaluated problem of choosing a specific Pareto optimal alternative previously. for implementation arises, because the final decision usually To demonstrate viability and suitability for applications, must be unique. us, Th additional factors must be consid- the proposed approach illustrated using an example of a ered to aid a decision-maker the selection of specific or materials-selection problem for a sailboat mast. This problem more-favorable alternatives from the set of Pareto optimal has been addressed by several researchers using various solutions. 2 Advances in Operations Research methods and, thus, can be considered as a kind of a bench- ranking theory in greater detail. For a natural number 𝑁 ,the mark problem. This illustration sheds light on the ranking 𝑁×𝑁 matrix𝑆= [𝑆 ], 1 ≤ 𝑖,𝑗 ≤ 𝑁,is a score matrix if 𝑆 ≥ approach’s applicability to the MCDM problems. Particularly, 0, 𝑆 =0, 1 ≤𝑖,𝑗 ≤𝑁. To emphasize that this problem was 𝑖𝑖 it is shown that the solutions of the illustrative example formulated in the context of competitive sports—note also obtained by the proposed approach are quite competitive. that we can interpret elements of N(𝑁) as athletes (or teams) The rest of this paper is structured as follows. In Section 2, who contest matches among themselves—and for each pair of preliminaries regarding MCDM and ranking problems are athletes (𝑖,),𝑗 1 ≤ 𝑖,𝑗 ≤ 𝑁 ,the joint match M(𝑖, )𝑗 includes 𝐾 presented, and the proposed methodology is described; games. We interpret entry 𝑆 ,1 ≤ 𝑖,𝑗 ≤ 𝑁 ,as the number Section 3 considers an illustrative example and Section 4 of athlete 𝑖 ’s total wins in the match M(𝑖,).𝑗 We also say summarizes the article. that the result of the match M(𝑖,)𝑗 is 𝑆 wins of athlete 𝑖 (losses of athlete 𝑗 ), 𝑆 wins of athlete 𝑗 (losses of athlete 𝑖 ), 𝑗𝑖 and (𝐾 − 𝑆 −𝑆 ) draws. Hence 𝐺 (𝑆) = 𝑆 +𝑆 ,(𝐺 = 𝑗𝑖 𝑗𝑖 2. Proposed Methods [𝐺 (𝑆)], 1 ≤ 𝑖, 𝑗 ≤ 𝑁; 𝐺 = 𝑆 + 𝑆 ), can be interpreted In what follows, for a natural number 𝑛 , wedenotean 𝑛 - as the number of decisive games that did not end in a draw dimensional vector space by R and N(𝑛) = {1,...,𝑛}. If not in the match M(𝑖,),𝑗 1 ≤ 𝑖, 𝑗 ≤ 𝑁. We also introduce the otherwise mentioned, we identify a n fi ite set 𝐴 with the set function 𝑔 (𝑆) = ∑ 𝐺 (𝑆), 1 ≤ 𝑖 ≤ 𝑁, which reflects the 𝑖 𝑗=1 N(𝑛) = {1,...,𝑛} ,where 𝑛=||𝐴 is the capacity of the set 𝐴. number of decisive outcomes in all matches played by athlete 𝑛×𝑚 By necessity, we also identify the matrix Π∈ R with the 𝑖,1 ≤ 𝑖 ≤ 𝑁. 𝑛×𝑚 map Π: N(𝑛) × N(𝑚) → 󳨀 R.For a matrix Π∈ R ,we For natural 𝑁 and score matrix 𝑆= [𝑆 ],1 ≤ 𝑖, 𝑗 ≤ 𝑁,we 𝑇 𝑚×𝑛 denote its transpose by Π ∈ R . say that the pair (N(𝑁),)𝑆 is the ranking problem. The weak- order (i.e., transitive and complete) relation𝑅(𝑁, 𝑆) ⊂ N(𝑁)× N(𝑁) represents the ranking method for the ranking problem 2.1. Preliminaries (N(𝑁),).𝑆 The vector 𝑟∈ R is a rating vector, where each 2.1.1. Background on Multiobjective Decision-Making Prob- 𝑟 ,1 ≤𝑖 ≤𝑁 , is the measure of the performance of player lems. The following notation is drawn from a general treat- 𝑖∈ N(𝑁) in the ranking problem (N(𝑁),).𝑆 For the ranking ment of multicriteria optimization theory [5, 6]. Let us problem (N(𝑁),)𝑆 ,a ranking method 𝑅(𝑁,𝑆) is induced by consider the MCDM problem ⟨𝐴, ⟩𝐶 ,where𝐴= {𝑎 ,...,𝑎 } 1 𝑚 the rating vector 𝑟∈ R if is a set of alternatives and 𝐶={𝑐 ,...,𝑐 } is a set of 1 𝑛 criteria; i.e., 𝑐 :𝐴 →󳨀 R, 𝑖 = 1,...,𝑛 ,are given (𝑖,)𝑗 ∈ 𝑅 (𝑁,𝑆 ) function. Without loss of generality, we may assume that (1) the lower value is preferable for each criterion (i.e., each (i.e.,𝑅 𝑁, 𝑆 ranks 𝑖 weakly above𝑗) if and only if 𝑟 ≥𝑟 . ( ) 𝑖 𝑗 criterion is nonbeneficial), and the goal of the decision- making procedure is to minimize all criteria simultaneously [7]. In this article, for illustrative purposes, we consider only a We say furthermore that 𝐴 is the set of admissible few of the many ranking methods discussed in the literature →󳨀 𝑛 alternatives and map 𝑐= (𝑐 ,...,𝑐 ): 𝐴󳨀→ R is the 1 𝑛 (note also that the ranking methods considered here, based →󳨀 𝑛 criterion map (correspondingly, 𝑐()𝐴 ⊂ R is the set of on the ranking problems involved in chess tournaments, go admissible values of criteria). The following concepts are also back to the investigations of H. Neustadtl, E. Zermelo, and associated with the criterion map and the set of alternatives. B. Buckholdz. For detailed explanations see, e.g., [9] and the An alternative 𝑎 ∈𝐴 is Pareto optimal (i.e., efficient) if literature cited therein). All these methods are induced by there exists no 𝑎∈𝐴 such that 𝑐 (𝑎) ≤ 𝑐 (𝑎 ) for all their corresponding rating vectors. For a given score matrix 𝑗 𝑗 ∗ 𝑗∈ N(𝑛) and 𝑐 (𝑎) < 𝑐 (𝑎 ) for some 𝑘∈ N(𝑛). The 𝑆= [𝑆 ],1 ≤ 𝑖, 𝑗 ≤ 𝑁, we consider the following ranking 𝑘 𝑘 ∗ set of all efficient alternatives is denoted as 𝐴 and is called methods. the Pareto set. Correspondingly, 𝑓(𝐴 ) is called the efficient front. Score Method. The rating vector for the score method, 𝑟 = Pareto optimality is an appropriate concept for the 𝑠 𝑠 𝑁 𝑠 (𝑟 ,...,𝑟 )∈ R , isdenfi ed asthe averagescore 𝑟 = solutions of MCDM problems. In general, however, the 1 𝑁 𝑖 set 𝐴 of Pareto optimal alternatives is very large and, ∑ 𝑆 /𝑔 (𝑆), 1 ≤ 𝑖 ≤ 𝑁. 𝑒 𝑖 𝑗=1 moreover, all alternatives from 𝐴 must be considered as “equally good solutions”. On the other hand, the n fi al 𝑁 𝑁 decision usually must be unique. Hence, additional factors Neustadt’s Method. Neustadt’s rating vector, 𝑟 ∈ R ,is 𝑁 𝑠 must be considered to aid the selection of specific or more- defined by the equality 𝑟 = 𝑆𝑟 ,where 𝑆= [ 𝑆 ] and 𝑆 = favorable alternatives from the set 𝐴 . The following subsec- 𝑆 /𝑔 (𝑆),1 ≤ 𝑖,𝑗 ≤ 𝑁. tions describe a novel approach that handles this problem objectively. 𝐵 𝑁 Buchholz’s Method. Buchholz’s rating vector, 𝑟 ∈ R ,is 𝑁 𝑠 defined by the equality 𝑟 =[𝐺(𝑆) + 𝐸 ]𝑟 ,where 𝐺(𝑆) = 2.1.2. Ranking Methods. This section gives a brief overview of the basic concepts of ranking theory. References [8, 9] discuss [𝐺 (𝑆)], 𝐺 (𝑆) = 𝐺 (𝑆)/𝑔 (𝑆), 1 ≤ 𝑖,𝑗 ≤ 𝑁. 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 𝑖𝑗 Advances in Operations Research 3 Fair-Bets Method. The rating vector for the fair-bet method, (ii) Using the score matrix 𝑆 ,the alternatives from set 𝐴 are ranked using a method 𝑅 . 𝑟 ∈ Δ , isdenfi ed asthe unique solution of thefollowing system of linear equations: (iii) The alternative from the Pareto set, 𝐴 ,ranked best by method 𝑅 is declared as the 𝑅− solution of the 𝑁 𝑁 considered MCDM problem. ∑𝑆 𝑟 −( ∑𝑆 )𝑟 =0, 1 ≤𝑖 ≤𝑁. (2) 𝑗𝑖 𝑗 𝑖 𝑗=1 𝑗=1 Obviously, it would suffice to rank the Pareto set if Pareto set is known at the beginning of the proposed procedure. Maximum-Likelihood Method. The rating vector for the Nevertheless, we prefer given above description because it is 𝑚𝑙 𝑚𝑙 𝑚𝑙 𝑁 maximum-likelihood method, 𝑟 =(𝑟 ,...,𝑟 )∈ R ,is 1 𝑁 more convenient in the cases when Pareto set is not known 𝑚𝑙 defined by the equality 𝑟 = ln(𝜋 ), 1 ≤ 𝑖 ≤ 𝑁 ,where vector 𝑖 𝑖 (or partially/approximately known), as it took place usually 𝜋=(𝜋 ,...,𝜋 )∈ Δ is theuniquesolution of the following for the complex MCDM problems. 1 𝑁 𝑁 nonlinear system of equations: It is clear that, instead of the MCDM problem ⟨𝐴, ⟩𝐶 , we can consider also MCDM problem ⟨𝐶, 𝐴⟩. Obviously, applying described above procedure to the MCDM problem 𝐺 (𝑆 ) ⟨𝐶, 𝐴⟩ , we can obtain a ranking of the criteria. However, we 𝜋 ∑ =𝑟 𝑔 (𝑆 ),1≤𝑖≤𝑁. 𝑖 𝑖 (3) 𝜋 +𝜋 𝑖 𝑗 𝑗=1 omit the corresponding details here. 𝑗 =𝑖̸ 3. Example 2.2. Ranking Methods to Solve MCDM Problems. Assume now that ⟨𝐴, ⟩𝐶 is a MCDM problem with a set of alternatives This section discusses the example problem that was solved 𝐴= {𝑎 ,...,𝑎 } and a set of nonbeneficial criteria 𝐶= 1 𝑚 to demonstrate the practicality of the proposed in Section 2.2 {𝑐 ,...,𝑐 } and the decision-making goal is therefore to 1 𝑛 procedure. All the necessary calculations were performed minimize the criteria simultaneously. Let us consider each in the MATLAB computing environment. The example element of 𝐴 as an athlete (e.g., chess player) and assume that, considered here is the problem of selecting the material 󸀠 󸀠 for each pair of athletes 𝑎, 𝑎 ∈𝐴 ,the matchM(𝑎,𝑎 ) includes for the mast of a sailing boat. This problem has been 𝑚 games. The special construction of the score matrix of addressed by several researchers, using various methods 𝐴 󸀠 alternatives, 𝑆 , isdenfi ed asfollows: for any 𝑎, 𝑎 ∈𝐴 ,we and, thus, can be considered as a kind of benchmark define problem. The component to be optimized, the mast, is modeled as 𝐴 󸀠 𝐴 󸀠 a hollow cylinder that is subjected to axial compression. It 𝑆 (𝑎, 𝑎 )= ∑ 𝑠 (𝑎, 𝑎 ), 𝑐∈𝐶 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 (4) 1, 𝑐 (𝑎 ) <𝑐(𝑎 ); compressive force of 153 kN [2]. The following criteria are 𝐴 󸀠 where 𝑠 (𝑎, 𝑎 )= ∀𝑐 ∈ .𝐶 󸀠 chosen for the ranking problem at hand: specific strength 0, 𝑐 𝑎 ≥𝑐(𝑎 ); ( ) (SS), specicfi modulus (SM), corrosion resistance (CR), and cost category (CC) [2]. The choice must be made from 15 𝐴 󸀠 󸀠 alternative materials. The corresponding decision-making Thus, the equality 𝑠 (𝑎, 𝑎 )= 1 means that 𝑐(𝑎) < 𝑐(𝑎 ) for data are given in Table 3 of the Appendix, and the normalized criterion 𝑐∈ 𝐶 and the alternative 𝑎 (“athlete 𝑎”) receives decision matrix is given in Table 4 of the Appendix. Note also one point (i.e., the athlete 𝑎 wins a game 𝑐∈𝐶 in the match 󸀠 𝐴 󸀠 that, for the problem under consideration, the upper-lower- M(𝑎, 𝑎 ) and, correspondingly, 𝑆 (𝑎,𝑎 ) indicates the number boundapproach wasusedfor normalization ofthe decision of total wins of athlete 𝑎 in the match M(𝑎, 𝑎 )). Obviously, 𝐴 󸀠 𝐴 󸀠 matrix [7]. The Pareto set for the considered problem is 𝐴 = 𝑚≥𝑆 (𝑎, 𝑎 )≥ 0,𝑆 (,𝑎𝑎 ) = 0,∀,𝑎𝑎 ∈𝐴. We say that {2,3, 4, 7, 9, 11,12, 13,14, 15}. 󸀠 𝐴 󸀠 an alternative 𝑎 has defeated an alternative 𝑎 if 𝑆 (𝑎,𝑎 )> 𝐴 󸀠 󸀠 The following methods were used to solve the prob- 𝑆 (𝑎 ,𝑎). We also say that the result of the match M(𝑎, 𝑎 ) is lem by previous investigators: WPM (weighted-properties 𝐴 󸀠 󸀠 𝑆 (𝑎, 𝑎 ) wins of the alternative 𝑎 (losses of alternative 𝑎 ), method), VIKOR (multicriteria optimization through the 𝐴 󸀠 󸀠 𝑆 (𝑎 ,𝑎) wins of the alternative 𝑎 (losses of alternative 𝑎) concept of a compromise solution), CVIKOR (comprehen- 𝐴 󸀠 𝐴 󸀠 and number of draws (𝑛 − 𝑆 (𝑎,𝑎 )− 𝑆 (𝑎 ,)). 𝑎 Obviously sive VIKOR), FLA (fuzzy-logic approach), MOORA (mul- 𝐴 𝐴 󸀠 matrix 𝑆 =[𝑆 (𝑎, 𝑎 )] 󸀠 is the score matrix for a set of 𝑎,𝑎 ∈𝐴 tiobjective optimization based on ratio analysis), MULTI- alternatives in the sense of the definition from the previous MOORA (a multiplicative form of MOORA), RPA (the subsection. reference-point approach), and a recently proposed game- The following procedure is used for solving MCDM theoretic method GTM [1–4, 10, 11]. Note also that the problem ⟨𝐴, ⟩𝐶 : material-selection problem is an important application of MCDM [12, 13]. Table 5 of the Appendix presents the mate- (i) For the MCDM problem⟨𝐴, ⟩𝐶 ,thescorematrix𝑆 = rials ranked by methods other than the one proposed in this 𝐴 󸀠 [𝑆 (𝑎,𝑎 )] 󸀠 is constructed. paper. 𝑎,𝑎 ∈𝐴 𝑖𝑗 𝑖𝑗 𝑓𝑏 𝑓𝑏 𝑓𝑏 4 Advances in Operations Research Table 1: Materials ranked by proposed methods. 𝑆 𝑁 𝐵 𝑚𝑙 𝑟 𝑟 𝑟 𝑟 𝑟 Material 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. Direct calculations show that the score matrix 𝑆 in the unefficient alternative 6). However, we should not consider considered case is this as contradiction because the Pareto set and the ranking methods are independent objects and only the restriction of 00 013 112 222 111 1 the ranking method on the Pareto set is essential. [ ] For comparison, Table 2 presents the correlation coef- 10 113 112 222 111 1 [ ] [ ] ficients of the alternative ranks as calculated by different [ ] 21 013 112 222 111 1 [ ] methods. As we can see, the results of the proposed ranking [ ] [ ] methods correlate well with the rankings obtained by FLA, 33 202 113 333 111 1 [ ] [ ] CVIKOR, and VIKOR; they are somewhat correlated with the [ ] 11 100 001 111 111 1 [ ] rankings returned by MOORA, MULTIMOORA, RPA, and [ ] 33 312 002 222 221 2 WPM and are poorly correlated with the ranking obtained [ ] 𝑆 𝑁 𝐵 𝑓𝑏 𝑚𝑙 [ ] by GTM. Meanwhile, the methods 𝑟 , 𝑟 , 𝑟 , 𝑟 ,and 𝑟 are [ 33 311 102 222 221 2] [ ] very strongly correlated between themselves. [ ] 𝑆 = 22 213 220 000 111 1 (5) [ ] [ ] [ ] 22 213 222 011 111 1 [ ] 4. Conclusions [ ] [ ] 22 213 221 100 111 1 [ ] In this study, we have proposed a new approach for solv- [ ] [ ] 22 213 222 120 111 1 ing MCDM problems. The proposed approach is based on [ ] [ ] ranking-theory methods which are used in the competitive [ 33 333 223 333 011 2 [ ] sports tournaments. In the framework of the proposed [ ] [ 33 322 113 333 302 2] approach, we build a special score matrix for a given mul- [ ] [ ] ticriteria problem, which allows us to use an appropriate 33 322 223 333 210 1 [ ] ranking method and choose the corresponding best-ranked 33 322 113 333 111 0 [ ] alternative from the Pareto set as a solution of the MCDM problem. The proposed approach is particularly useful when Using the score matrix 𝑆 , we rank the materials with each no decision-making authority is available, or when the of the vfi e methods described in Section 2.1.2. The ranking relative importance of various criteria has not been evaluated results are presented in Table 1. These results show that previously. material 14 (Epoxy–63% carbon fabric)isranked best by To demonstrate the viability and suitability for applica- 𝑆 𝐵 𝑚𝑙 ranking methods 𝑟 , 𝑟 ,and 𝑟 and material 13 (Epoxy–70% tions, the proposed approach illustrated using an example 𝑁 𝑓𝑏 of a materials-selection problem. It is shown that the solu- glass fabric) is ranked best by ranking methods 𝑟 and 𝑟 . Table 1 also shows that sometimes the case when the tions of the illustrative example obtained by the proposed alternative which does not belong to the Pareto set is approach are quite competitive. Note also that the proposed ranked better than some set of the efficient alternatives approach seems numerically efficient. Namely, our prelim- can be observed (e.g., the efficient alternatives 11,3 and the inary numerical experiments (unpublished) show that that 𝑓𝑏 Advances in Operations Research 5 Table 2: Correlation between methods. S N B fb ml r r r r r 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. Criteria #Material Specific strength (MPa) Specific modulus (GPa) Corrosion resistance Cost Category SS SM CR CC 12 3 4 1AISI1020 35.9 26.9 1 5 2AISI1040 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= veryhigh; 2 =high; 3 = moderate; 4 =low; 5 =verylow. MCDM problems with the number of alternatives of the evolutionary) Pareto optimization algorithms. However, we order of 1.5 hundred and with the number of criteria of the will limit ourselves here only to mention these directions for order of ten can be solved by the proposed method in a few further investigations. minutes (∼5 min, the calculations were conducted on a laptop with 2.59GHz, 8GB RAM, 64-bit operation system, MATLAB Appendix environment, and not making any effort to optimize the code). See Tables 3,4,and 5. Due to the simplicity and flexibility of the implementa- tion, the proposed approach can be also used in a few interest- Data Availability ing directions. For example, if we consider the “transposed” MCDM problem (i.e., the problem, for which the criteria Previously reported data were used to support this study. of the original problem are alternatives and the alternatives These prior studies are cited at relevant places within the text of the original problem are criteria), the proposed approach as references. 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 Conflicts of Interest for the Pareto optimization. It also seems possible that the proposed approach will find applications in the (e.g., The author declares that he has no conflicts of interest. 6 Advances in Operations Research Table 4: Normalized decision matrix for the material selection problem. Criteria 12 3 4 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 Materials 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 114 14 14 14 14 12 14 14 2 15 15 13 13 13 6 11 10 3 13 13 12 15 15 9 13 11 412 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 7 811 9 9 8 7 13 12 5 910 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]. References [6] K.M.Miettinen, Nonlinear Multiobjective Optimization,Kluwer Academic Publishers, 1999. [1] P. Karande and S. Chakraborty, “Application of multi-objective [7] R. T. Marler and J. S. Arora, “Function-transformation methods optimization on the basis of ratio analysis (MOORA) method for multi-objective optimization,” Engineering Optimization, for materials selection,” Materials & Design, vol.37,pp.317–324, vol. 37, no. 6, pp. 551–570, 2005. [2] M. M. Farag, “Quantitative methods of materials selection,” in [8] A. Y. Govan, Ranking eTh ory with Application to Popular Sports Handbook of Materials Selection,M. Kutz,Ed., 2002. [PhD thesis], North Carolina State, University, 2008. [3] A. Jahan, F. Mustapha, M. Y. Ismail, S. M. Sapuan, and M. [9] J. Gonzalez ´ -D´ıaz, R. Hendrickx, and E. Lohmann, “Paired com- Bahraminasab, “A comprehensive VIKOR method for material parisons analysis: an axiomatic approach to ranking methods,” selection,” Materials & Design, vol. 32, no. 3, pp. 1215–1221, 2011. Social Choice and Welfare,vol. 42, no.1, pp. 139–169, 2014. [4] J. Gogodze, “Using a two-person zero-sum game to solve a decision-making problem,” Pure and Applied Mathematics [10] P. Chatterjee, V. M. Athawale, and S. Chakraborty, “Selection of Journal,vol.7,no.2, pp. 11–19, 2018. materials using compromise ranking and outranking methods,” [5] M. Ehrgott, Multicriteria Optimization,Springer,2005. Materials and Corrosion, vol. 30, no. 10, pp. 4043–4053, 2009. Advances in Operations Research 7 [11] R. Sarfaraz Khabbaz, B. Dehghan Manshadi, A. Abedian, and R. Mahmudi, “A simplified fuzzy logic approach for materi- als selection in mechanical engineering design,” Materials & Design, vol.30, no.3,pp.687–697,2009. [12] M. Yazdani, “New approach to select materials using MADM tools,” International Journal of Business and Systems Research, vol. 12, no. 1, pp. 25–42, 2018. [13] K. Anyfantis, P. Foteinopoulos, and P. Stavropoulos, “Design for manufacturing of multi-material mechanical parts: a computa- tional based approach,” Procedia CIRP,vol.66,pp.22–26,2017. 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