ranking-theory methods for solving multicriteria decision

Research ArticleRanking-Theory Methods for Solving MulticriteriaDecision-Making Problems

Joseph Gogodze

Institute of Control Systems TECHINFORMI Georgian Technical University Tbilisi Georgia

Correspondence should be addressed to Joseph Gogodze jgogodzegmailcom

Received 7 August 2018 Revised 29 January 2019 Accepted 27 February 2019 Published 1 April 2019

Academic Editor Imed Kacem

Copyright copy 2019 Joseph Gogodze This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The Pareto optimality is a widely used concept for the multicriteria decision-making problems However this concept has asignificant drawbackmdashthe set of Pareto optimal alternatives usually is large Correspondingly the problem of choosing a specificPareto optimal alternative for the decision implementation is arisingThis study proposes a new approach to select an ldquoappropriaterdquoalternative from the set of Pareto optimal alternatives The proposed approach is based on ranking-theory methods used forranking participants in sports tournaments In the framework of the proposed approach we build a special score matrix for agiven multicriteria problem which allows the use of the mentioned ranking methods and to choose the corresponding best-rankedalternative from the Pareto set as a solution of the problemThe proposed approach is particularly useful when no decision-makingauthority is available or when the relative importance of various criteria has not been evaluated previouslyThe proposed approachis tested on an example of a materials-selection problem for a sailboat mast

1 Introduction

This paper considers a novel approach for solving a mul-ticriteria decision-making (MCDM) problem with a finitenumber of decision alternatives and criteriaThemulticriteriaformulation is the typical starting point for theoretical andpractical analyses of decision-making problems Thus thedefinition of Pareto optimality and a vast arsenal of differentPareto optimization methods can be used for decision-making purpose

However unlike single-objective optimizations a charac-teristic feature of Pareto optimality is that the set of Paretooptimal alternatives (ie set of efficient alternatives) is usuallylarge In addition all these Pareto optimal alternatives mustbe considered as mathematically equal Correspondingly theproblem of choosing a specific Pareto optimal alternativefor implementation arises because the final decision usuallymust be unique Thus additional factors must be consid-ered to aid a decision-maker the selection of specific ormore-favorable alternatives from the set of Pareto optimalsolutions

The proposed approach is based on ranking-theorymeth-ods that used to rank participants in sports tournaments Inthe framework of the proposed approach we build a specialscore matrix for a given multicriteria problem which allowsus to use the mentioned ranking methods and choose thecorresponding best-ranked alternative from the Pareto setas a solution of the problem Note that the score matrix isbuilt by the quite natural waymdashit is composed on the simplecalculations of how many times one alternative is better thanthe other for each of the criteria Hence there is hope thatthe proposed approach yields a ldquonotionally objectiverdquo rankingmethod and provides an ldquoaccurate rankingrdquo of the alternativesfor MCDM The proposed approach is particularly usefulwhen no decision-making authority is available or when therelative importance of various criteria has not been evaluatedpreviously

To demonstrate viability and suitability for applicationsthe proposed approach illustrated using an example of amaterials-selection problem for a sailboat mastThis problemhas been addressed by several researchers using various

HindawiAdvances in Operations ResearchVolume 2019 Article ID 3217949 7 pageshttpsdoiorg10115520193217949

2 Advances in Operations Research

methods and thus can be considered as a kind of a bench-mark problem This illustration sheds light on the rankingapproachrsquos applicability to the MCDMproblems Particularlyit is shown that the solutions of the illustrative exampleobtained by the proposed approach are quite competitive

The rest of this paper is structured as follows In Section 2preliminaries regarding MCDM and ranking problems arepresented and the proposed methodology is describedSection 3 considers an illustrative example and Section 4summarizes the article

2 Proposed Methods

In what follows for a natural number 119899 we denote an 119899-dimensional vector space by R119899 and N(119899) = 1 119899 If nototherwise mentioned we identify a finite set 119860 with the setN(119899) = 1 119899 where 119899 = |119860| is the capacity of the set 119860By necessity we also identify the matrix Π isin R119899times119898 with themap Π N(119899) times N(119898) 997888rarr R For a matrix Π isin R119899times119898 wedenote its transpose by Π119879 isin R119898times11989921 Preliminaries

211 Background on Multiobjective Decision-Making Prob-lems The following notation is drawn from a general treat-ment of multicriteria optimization theory [5 6] Let usconsider the MCDM problem ⟨119860 119862⟩ where119860 = 1198861 119886119898is a set of alternatives and 119862 = 1198881 119888119899 is a set ofcriteria ie 119888119894 119860 997888rarr R 119894 = 1 119899 are givenfunction Without loss of generality we may assume thatthe lower value is preferable for each criterion (ie eachcriterion is nonbeneficial) and the goal of the decision-making procedure is to minimize all criteria simultaneously[7]

We say furthermore that 119860 is the set of admissiblealternatives and map 997888rarr119888 = (1198881 119888119899) 119860 997888rarr R119899 is thecriterion map (correspondingly 997888rarr119888 (119860) sub R119899 is the set ofadmissible values of criteria) The following concepts are alsoassociated with the criterion map and the set of alternativesAn alternative 119886lowast isin 119860 is Pareto optimal (ie efficient) ifthere exists no 119886 isin 119860 such that 119888119895(119886) le 119888119895(119886lowast) for all119895 isin N(119899) and 119888119896(119886) lt 119888119896(119886lowast) for some 119896 isin N(119899) Theset of all efficient alternatives is denoted as 119860119890 and is calledthe Pareto set Correspondingly 119891(119860119890) is called the efficientfront

Pareto optimality is an appropriate concept for thesolutions of MCDM problems In general however theset 119860119890 of Pareto optimal alternatives is very large andmoreover all alternatives from 119860119890 must be considered asldquoequally good solutionsrdquo On the other hand the finaldecision usually must be unique Hence additional factorsmust be considered to aid the selection of specific or more-favorable alternatives from the set 119860119890The following subsec-tions describe a novel approach that handles this problemobjectively

212 RankingMethods This section gives a brief overview ofthe basic concepts of ranking theory References [8 9] discuss

ranking theory in greater detail For a natural number119873 the119873times119873matrix 119878 = [119878119894119895] 1 le 119894 119895 le 119873 is a score matrix if 119878119894119895 ge0 119878119894119894 = 0 1 le 119894 119895 le 119873 To emphasize that this problem wasformulated in the context of competitive sportsmdashnote alsothat we can interpret elements of N(119873) as athletes (or teams)who contest matches among themselvesmdashand for each pair ofathletes (119894 119895) 1 le 119894 119895 le 119873 the joint match M(119894 119895) includes 119870games We interpret entry 119878119894119895 1 le 119894 119895 le 119873 as the numberof athlete 119894rsquos total wins in the match M(119894 119895) We also saythat the result of the match M(119894 119895) is 119878119894119895 wins of athlete 119894(losses of athlete 119895) 119878119895119894 wins of athlete 119895 (losses of athlete 119894)and (119870 minus 119878119894119895 minus 119878119895119894) draws Hence 119866119894119895(119878) = 119878119894119895 + 119878119895119894 (119866 =[119866119894119895(119878)] 1 le 119894 119895 le 119873 119866 = 119878 + 119878119879) can be interpretedas the number of decisive games that did not end in a drawin the match M(119894 119895) 1 le 119894 119895 le 119873 We also introduce thefunction 119892119894(119878) = sum119873119895=1 119866119894119895(119878) 1 le 119894 le 119873 which reflects thenumber of decisive outcomes in all matches played by athlete119894 1 le 119894 le 119873

For natural 119873 and score matrix 119878 = [119878119894119895] 1 le 119894 119895 le 119873 wesay that the pair (N(119873) 119878) is the ranking problemThe weak-order (ie transitive and complete) relation119877(119873 119878) sub N(119873)timesN(119873) represents the ranking method for the ranking problem(N(119873) 119878) The vector 119903 isin R119873 is a rating vector where each119903119894 1 le 119894 le 119873 is the measure of the performance of player119894 isin N(119873) in the ranking problem (N(119873) 119878) For the rankingproblem (N(119873) 119878) a ranking method 119877(119873 119878) is induced bythe rating vector 119903 isin R119873 if

(119894 119895) isin 119877 (119873 119878)(ie 119877 (119873 119878) ranks 119894 weakly above119895) if and only if 119903119894 ge 119903119895 (1)

In this article for illustrative purposes we consider only afew of the many ranking methods discussed in the literature(note also that the ranking methods considered here basedon the ranking problems involved in chess tournaments goback to the investigations of H Neustadtl E Zermelo andB Buckholdz For detailed explanations see eg [9] and theliterature cited therein) All these methods are induced bytheir corresponding rating vectors For a given score matrix119878 = [119878119894119895] 1 le 119894 119895 le 119873 we consider the following rankingmethods

Score Method The rating vector for the score method 119903119904 =(1199031199041 119903119904119873) isin R119873 is defined as the average score 119903119904119894 =sum119873119895=1 119878119894119895119892119894(119878) 1 le 119894 le 119873Neustadtrsquos Method Neustadtrsquos rating vector 119903119873 isin R119873 isdefined by the equality 119903119873 = 119878119903119904 where 119878 = [119878119894119895] and 119878119894119895 =119878119894119895119892119894(119878) 1 le 119894 119895 le 119873Buchholzrsquos Method Buchholzrsquos rating vector 119903119861 isin R119873 isdefined by the equality 119903119873 = [119866(119878) + 119864119873]119903119904 where 119866(119878) =[119866119894119895(119878)] 119866119894119895(119878) = 119866119894119895(119878)119892119894(119878) 1 le 119894 119895 le 119873

Advances in Operations Research 3

Fair-Bets Method The rating vector for the fair-bet method119903119891119887 isin ∘Δ119873 is defined as the unique solution of the followingsystem of linear equations

119873sum119895=1

119878119894119895119903119891119887119895 minus ( 119873sum119895=1

119878119895119894)119903119891119887119894 = 0 1 le 119894 le 119873 (2)

Maximum-Likelihood Method The rating vector for themaximum-likelihood method 119903119898119897 = (1199031198981198971 119903119898119897119873 ) isin R119873 isdefined by the equality 119903119898119897119894 = ln(120587119894) 1 le 119894 le 119873 where vector120587 = (1205871 120587119873) isin ∘Δ119873 is the unique solution of the followingnonlinear system of equations

120587119894 119873sum119895=1119895 =119894

119866119894119895 (119878)120587119894 + 120587119895 = 119903119904119894119892119894 (119878) 1 le 119894 le 119873 (3)

22 Ranking Methods to Solve MCDM Problems Assumenow that ⟨119860 119862⟩ is aMCDMproblemwith a set of alternatives119860 = 1198861 119886119898 and a set of nonbeneficial criteria 119862 =1198881 119888119899 and the decision-making goal is therefore tominimize the criteria simultaneously Let us consider eachelement of119860 as an athlete (eg chess player) and assume thatfor each pair of athletes 119886 1198861015840 isin 119860 thematchM(119886 1198861015840) includes119898 games The special construction of the score matrix ofalternatives 119878119860 is defined as follows for any 119886 1198861015840 isin 119860 wedefine

119878119860 (119886 1198861015840) = sum119888isin119862

119904119860119888 (119886 1198861015840)

where 119904119860119888 (119886 1198861015840) = 1 119888 (119886) lt 119888 (1198861015840) 0 119888 (119886) ge 119888 (1198861015840) forall119888 isin 119862

(4)

Thus the equality 119904119860119888 (119886 1198861015840) = 1 means that 119888(119886) lt 119888(1198861015840) forcriterion 119888 isin 119862 and the alternative 119886 (ldquoathlete 119886rdquo) receivesone point (ie the athlete 119886 wins a game 119888 isin 119862 in the matchM(119886 1198861015840) and correspondingly 119878119860(119886 1198861015840) indicates the numberof total wins of athlete 119886 in the match M(119886 1198861015840)) Obviously119898 ge 119878119860(119886 1198861015840) ge 0 119878119860(119886 119886) = 0 forall119886 1198861015840 isin 119860 We say thatan alternative 119886 has defeated an alternative 1198861015840 if 119878119860(119886 1198861015840) gt119878119860(1198861015840 119886)We also say that the result of the match M(119886 1198861015840) is119878119860(119886 1198861015840) wins of the alternative 119886 (losses of alternative 1198861015840)119878119860(1198861015840 119886) wins of the alternative 1198861015840 (losses of alternative 119886)and number of draws (119899 minus 119878119860(119886 1198861015840) minus 119878119860(1198861015840 119886)) Obviouslymatrix 119878119860 = [119878119860(119886 1198861015840)]1198861198861015840isin119860is the score matrix for a set ofalternatives in the sense of the definition from the previoussubsection

The following procedure is used for solving MCDMproblem ⟨119860 119862⟩

(i) For theMCDMproblem ⟨119860 119862⟩ the scorematrix 119878119860 =[119878119860(119886 1198861015840)]1198861198861015840isin119860 is constructed

(ii) Using the score matrix 119878119860 the alternatives from set 119860are ranked using a method 119877

(iii) The alternative from the Pareto set 119860119890 ranked bestby method 119877 is declared as the 119877minussolution of theconsidered MCDM problem

Obviously it would suffice to rank the Pareto set if Paretoset is known at the beginning of the proposed procedureNevertheless we prefer given above description because it ismore convenient in the cases when Pareto set is not known(or partiallyapproximately known) as it took place usuallyfor the complex MCDM problems

It is clear that instead of the MCDM problem ⟨119860 119862⟩we can consider also MCDM problem ⟨119862 119860⟩ Obviouslyapplying described above procedure to the MCDM problem⟨119862 119860⟩ we can obtain a ranking of the criteria However weomit the corresponding details here

3 Example

This section discusses the example problem that was solvedto demonstrate the practicality of the proposed in Section 22procedure All the necessary calculations were performedin the MATLAB computing environment The exampleconsidered here is the problem of selecting the materialfor the mast of a sailing boat This problem has beenaddressed by several researchers using various methodsand thus can be considered as a kind of benchmarkproblem

The component to be optimized the mast is modeled asa hollow cylinder that is subjected to axial compression Ithas a length of 1000 mm an outer diameter le 100 mm aninner diameter ge 84 mm a mass le 3 kg and a total axialcompressive force of 153 kN [2] The following criteria arechosen for the ranking problem at hand specific strength(SS) specific modulus (SM) corrosion resistance (CR) andcost category (CC) [2] The choice must be made from 15alternative materials The corresponding decision-makingdata are given in Table 3 of the Appendix and the normalizeddecision matrix is given in Table 4 of the Appendix Note alsothat for the problem under consideration the upper-lower-bound approach was used for normalization of the decisionmatrix [7]The Pareto set for the considered problem is119860119890 =2 3 4 7 9 11 12 13 14 15

The following methods were used to solve the prob-lem by previous investigators WPM (weighted-propertiesmethod) VIKOR (multicriteria optimization through theconcept of a compromise solution) CVIKOR (comprehen-sive VIKOR) FLA (fuzzy-logic approach) MOORA (mul-tiobjective optimization based on ratio analysis) MULTI-MOORA (a multiplicative form of MOORA) RPA (thereference-point approach) and a recently proposed game-theoretic method GTM [1ndash4 10 11] Note also that thematerial-selection problem is an important application ofMCDM [12 13] Table 5 of the Appendix presents the mate-rials ranked by methods other than the one proposed in thispaper


Table 1 Materials ranked by proposed methods

Material 119903119878

119903119873

119903119861

119903119891119887

119903119898119897

Rating Rank Rating Rank Rating Rank Rating Rank Rating Rank1 03529 14 01666 14 08752 14 00335 14 -3408 142 03922 12 01816 12 09136 12 00380 12 -3231 123 04118 10 01882 11 09274 11 00403 11 -3167 104 06087 5 02693 7 10945 5 00819 7 -2424 55 02340 15 01164 15 07342 15 00201 15 -4072 156 05870 7 02716 6 10694 7 00822 6 -2532 77 06087 6 02843 4 10907 6 00913 4 -2438 68 03673 13 01767 13 08888 13 00371 13 -3349 139 04400 9 02052 9 09563 9 00470 9 -3043 910 04082 11 01931 10 09288 10 00425 10 -3167 1111 04600 8 02130 8 09755 8 00502 8 -2959 812 06481 3 02969 3 11322 3 01022 3 -2256 313 06800 2 03190 1 11595 2 01232 1 -2125 214 06875 1 03161 2 11600 1 01222 2 -2115 115 06250 4 02790 5 11001 4 00884 5 -2395 4Note italic corresponds to the Pareto optimal (efficient) alternatives

Direct calculations show that the score matrix 119878119860 in theconsidered case is

119878119860 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

0 0 0 1 3 1 1 2 2 2 2 1 1 1 11 0 1 1 3 1 1 2 2 2 2 1 1 1 12 1 0 1 3 1 1 2 2 2 2 1 1 1 13 3 2 0 2 1 1 3 3 3 3 1 1 1 11 1 1 0 0 0 0 1 1 1 1 1 1 1 13 3 3 1 2 0 0 2 2 2 2 2 2 1 23 3 3 1 1 1 0 2 2 2 2 2 2 1 22 2 2 1 3 2 2 0 0 0 0 1 1 1 12 2 2 1 3 2 2 2 0 1 1 1 1 1 12 2 2 1 3 2 2 1 1 0 0 1 1 1 12 2 2 1 3 2 2 2 1 2 0 1 1 1 13 3 3 3 3 2 2 3 3 3 3 0 1 1 23 3 3 2 2 1 1 3 3 3 3 3 0 2 23 3 3 2 2 2 2 3 3 3 3 2 1 0 13 3 3 2 2 1 1 3 3 3 3 1 1 1 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(5)

Using the score matrix 119878119860 we rank the materials with eachof the five methods described in Section 212 The rankingresults are presented in Table 1 These results show thatmaterial 14 (Epoxyndash63 carbon fabric) is ranked best byranking methods 119903119878 119903119861 and 119903119898119897 and material 13 (Epoxyndash70glass fabric) is ranked best by ranking methods 119903119873 and 119903119891119887

Table 1 also shows that sometimes the case when thealternative which does not belong to the Pareto set isranked better than some set of the efficient alternativescan be observed (eg the efficient alternatives 113 and the

unefficient alternative 6) However we should not considerthis as contradiction because the Pareto set and the rankingmethods are independent objects and only the restriction ofthe ranking method on the Pareto set is essential

For comparison Table 2 presents the correlation coef-ficients of the alternative ranks as calculated by differentmethods As we can see the results of the proposed rankingmethods correlate well with the rankings obtained by FLACVIKOR andVIKOR they are somewhat correlatedwith therankings returned by MOORA MULTIMOORA RPA andWPM and are poorly correlated with the ranking obtainedby GTMMeanwhile the methods 119903119878 119903119873 119903119861 119903119891119887 and 119903119898119897 arevery strongly correlated between themselves

4 Conclusions

In this study we have proposed a new approach for solv-ing MCDM problems The proposed approach is based onranking-theory methods which are used in the competitivesports tournaments In the framework of the proposedapproach we build a special score matrix for a given mul-ticriteria problem which allows us to use an appropriateranking method and choose the corresponding best-rankedalternative from the Pareto set as a solution of the MCDMproblem The proposed approach is particularly useful whenno decision-making authority is available or when therelative importance of various criteria has not been evaluatedpreviously

To demonstrate the viability and suitability for applica-tions the proposed approach illustrated using an exampleof a materials-selection problem It is shown that the solu-tions of the illustrative example obtained by the proposedapproach are quite competitive Note also that the proposedapproach seems numerically efficient Namely our prelim-inary numerical experiments (unpublished) show that that


Table 2 Correlation between methods

rS rN rB rfb rml

MOORAlowast 0564286 0603571 0578571 0603571 0564286MULTIMOORAlowast 0496429 0503571 0521429 0503571 0496429RPA lowast 0467857 0492857 0485714 0492857 0467857FLAlowast 0764286 0717857 0792857 0717857 0764286Wpm lowastlowast 0403571 0410714 0442857 0410714 0403571CVIKOR lowast lowast lowast 0742857 0646429 0739286 0646429 0742857VIKOR lowast lowast lowast 0892857 0871429 0907143 0871429 0892857GTM lowast lowast lowastlowast -012143 -007857 -009286 -007857 -012143Sourceslowast[1] lowastlowast[2] lowast lowast lowast[3] lowastlowastlowastlowast[4]

Table 3 Decision matrix for selecting material for a sailing boat mast

MaterialCriteria

Specific strength (MPa) Specific modulus (GPa) Corrosion resistance Cost CategorySS SM CR CC1 2 3 4

1 AISI 1020 359 269 1 52 AISI 1040 513 269 1 53 ASTM A242 type 1 423 272 1 54 AISI 4130 1949 272 4 35 AISI 316 256 251 4 36 AISI 416 heat treated 571 281 4 37 AISI 431 heat treated 714 281 4 38 AA 6061 T6 1019 258 3 49 AA 2024 T6 1419 261 3 410 AA 2014 T6 1482 258 3 411 AA 7075 T6 1804 259 3 412 Tindash6Alndash4V 2087 276 5 113 Epoxyndash70 glass fabric 6048 280 4 214 Epoxyndash63 carbon fabric 4162 665 4 115 Epoxyndash62 aramid fabric 6377 275 4 1Source [1] Notes CR scale 1 = poor 2 = fair 3 = good 4 = very good 5 = excellentCC scale 1 = very high 2 = high 3 = moderate 4 = low 5 = very low

MCDM problems with the number of alternatives of theorder of 15 hundred and with the number of criteria of theorder of ten can be solved by the proposed method in a fewminutes (sim5min the calculations were conducted on a laptopwith 259GHz 8GBRAM 64-bit operation systemMATLABenvironment and not making any effort to optimize thecode)

Due to the simplicity and flexibility of the implementa-tion the proposed approach can be also used in a few interest-ing directions For example if we consider the ldquotransposedrdquoMCDM problem (ie the problem for which the criteriaof the original problem are alternatives and the alternativesof the original problem are criteria) the proposed approachalso allows ranking the criteria and identified a ldquoleadingcriterionrdquo On the other hand an ldquoobjectiverdquo ranking of thecriteria may stimulate the development of other instrumentsfor the Pareto optimization It also seems possible thatthe proposed approach will find applications in the (eg

evolutionary) Pareto optimization algorithms However wewill limit ourselves here only to mention these directions forfurther investigations

Appendix

See Tables 3 4 and 5

Data Availability

Previously reported data were used to support this studyThese prior studies are cited at relevant places within the textas references

Conflicts of Interest

The author declares that he has no conflicts of interest


Table 4 Normalized decision matrix for the material selection problem

Criteria1 2 3 4

Materials

1 09832 09565 10000 000002 09580 09565 10000 000003 09727 09493 10000 000004 07234 09493 02500 050005 10000 10000 02500 050006 09485 09275 02500 050007 09252 09275 02500 050008 08753 09831 05000 025009 08100 09758 05000 0250010 07997 09831 05000 0250011 07471 09807 05000 0250012 07009 09396 00000 1000013 00537 09300 02500 0750014 03619 00000 02500 1000015 00000 09420 02500 10000

Note italic denotes Pareto optimal (efficient) alternatives

Table 5 Materials ranked by comparable methods

Material MOORAlowast MULTIMOORAlowast RPA lowast FLAlowast Wpm lowastlowast CVIKOR lowastlowastlowast VIKOR lowastlowastlowast GTMlowastlowastlowastlowast

1 14 14 14 14 14 12 14 142 15 15 13 13 13 6 11 103 13 13 12 15 15 9 13 114 12 12 15 4 11 4 4 25 4 4 4 11 10 15 15 96 7 11 11 9 9 14 10 87 6 10 10 10 8 11 5 78 11 9 9 8 7 13 12 59 10 7 8 12 2 8 7 410 9 6 7 7 4 10 9 311 5 8 6 6 6 5 6 112 8 5 2 5 3 7 8 1213 2 2 3 3 12 2 2 614 3 3 1 2 1 1 1 1515 1 1 5 1 5 3 3 13Sources lowast[1] lowastlowast[2] lowast lowast lowast[3] lowastlowastlowastlowast [4]

References

[1] P Karande and S Chakraborty ldquoApplication of multi-objectiveoptimization on the basis of ratio analysis (MOORA) methodfor materials selectionrdquoMaterials amp Design vol 37 pp 317ndash3242012

[2] M M Farag ldquoQuantitative methods of materials selectionrdquo inHandbook of Materials Selection M Kutz Ed 2002

[3] A Jahan F Mustapha M Y Ismail S M Sapuan and MBahraminasab ldquoA comprehensive VIKOR method for materialselectionrdquoMaterials amp Design vol 32 no 3 pp 1215ndash1221 2011

[4] J Gogodze ldquoUsing a two-person zero-sum game to solvea decision-making problemrdquo Pure and Applied MathematicsJournal vol 7 no 2 pp 11ndash19 2018

[5] M EhrgottMulticriteria Optimization Springer 2005

[6] KMMiettinenNonlinearMultiobjectiveOptimization KluwerAcademic Publishers 1999

[7] R TMarler and J S Arora ldquoFunction-transformationmethodsfor multi-objective optimizationrdquo Engineering Optimizationvol 37 no 6 pp 551ndash570 2005

[8] A Y Govan RankingTheory with Application to Popular Sports[PhD thesis] North Carolina State University 2008

[9] J Gonzalez-Dıaz R Hendrickx and E Lohmann ldquoPaired com-parisons analysis an axiomatic approach to ranking methodsrdquoSocial Choice andWelfare vol 42 no 1 pp 139ndash169 2014

[10] P Chatterjee V M Athawale and S Chakraborty ldquoSelection ofmaterials using compromise ranking and outranking methodsrdquoMaterials and Corrosion vol 30 no 10 pp 4043ndash4053 2009


[11] R Sarfaraz Khabbaz B Dehghan Manshadi A Abedian andR Mahmudi ldquoA simplified fuzzy logic approach for materi-als selection in mechanical engineering designrdquo Materials ampDesign vol 30 no 3 pp 687ndash697 2009

[12] M Yazdani ldquoNew approach to select materials using MADMtoolsrdquo International Journal of Business and Systems Researchvol 12 no 1 pp 25ndash42 2018

[13] K Anyfantis P Foteinopoulos and P Stavropoulos ldquoDesign formanufacturing of multi-material mechanical parts a computa-tional based approachrdquo Procedia CIRP vol 66 pp 22ndash26 2017

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methods and thus can be considered as a kind of a bench-mark problem This illustration sheds light on the rankingapproachrsquos applicability to the MCDMproblems Particularlyit is shown that the solutions of the illustrative exampleobtained by the proposed approach are quite competitive

The rest of this paper is structured as follows In Section 2preliminaries regarding MCDM and ranking problems arepresented and the proposed methodology is describedSection 3 considers an illustrative example and Section 4summarizes the article

2 Proposed Methods

In what follows for a natural number 119899 we denote an 119899-dimensional vector space by R119899 and N(119899) = 1 119899 If nototherwise mentioned we identify a finite set 119860 with the setN(119899) = 1 119899 where 119899 = |119860| is the capacity of the set 119860By necessity we also identify the matrix Π isin R119899times119898 with themap Π N(119899) times N(119898) 997888rarr R For a matrix Π isin R119899times119898 wedenote its transpose by Π119879 isin R119898times11989921 Preliminaries

211 Background on Multiobjective Decision-Making Prob-lems The following notation is drawn from a general treat-ment of multicriteria optimization theory [5 6] Let usconsider the MCDM problem ⟨119860 119862⟩ where119860 = 1198861 119886119898is a set of alternatives and 119862 = 1198881 119888119899 is a set ofcriteria ie 119888119894 119860 997888rarr R 119894 = 1 119899 are givenfunction Without loss of generality we may assume thatthe lower value is preferable for each criterion (ie eachcriterion is nonbeneficial) and the goal of the decision-making procedure is to minimize all criteria simultaneously[7]

We say furthermore that 119860 is the set of admissiblealternatives and map 997888rarr119888 = (1198881 119888119899) 119860 997888rarr R119899 is thecriterion map (correspondingly 997888rarr119888 (119860) sub R119899 is the set ofadmissible values of criteria) The following concepts are alsoassociated with the criterion map and the set of alternativesAn alternative 119886lowast isin 119860 is Pareto optimal (ie efficient) ifthere exists no 119886 isin 119860 such that 119888119895(119886) le 119888119895(119886lowast) for all119895 isin N(119899) and 119888119896(119886) lt 119888119896(119886lowast) for some 119896 isin N(119899) Theset of all efficient alternatives is denoted as 119860119890 and is calledthe Pareto set Correspondingly 119891(119860119890) is called the efficientfront

Pareto optimality is an appropriate concept for thesolutions of MCDM problems In general however theset 119860119890 of Pareto optimal alternatives is very large andmoreover all alternatives from 119860119890 must be considered asldquoequally good solutionsrdquo On the other hand the finaldecision usually must be unique Hence additional factorsmust be considered to aid the selection of specific or more-favorable alternatives from the set 119860119890The following subsec-tions describe a novel approach that handles this problemobjectively

212 RankingMethods This section gives a brief overview ofthe basic concepts of ranking theory References [8 9] discuss

ranking theory in greater detail For a natural number119873 the119873times119873matrix 119878 = [119878119894119895] 1 le 119894 119895 le 119873 is a score matrix if 119878119894119895 ge0 119878119894119894 = 0 1 le 119894 119895 le 119873 To emphasize that this problem wasformulated in the context of competitive sportsmdashnote alsothat we can interpret elements of N(119873) as athletes (or teams)who contest matches among themselvesmdashand for each pair ofathletes (119894 119895) 1 le 119894 119895 le 119873 the joint match M(119894 119895) includes 119870games We interpret entry 119878119894119895 1 le 119894 119895 le 119873 as the numberof athlete 119894rsquos total wins in the match M(119894 119895) We also saythat the result of the match M(119894 119895) is 119878119894119895 wins of athlete 119894(losses of athlete 119895) 119878119895119894 wins of athlete 119895 (losses of athlete 119894)and (119870 minus 119878119894119895 minus 119878119895119894) draws Hence 119866119894119895(119878) = 119878119894119895 + 119878119895119894 (119866 =[119866119894119895(119878)] 1 le 119894 119895 le 119873 119866 = 119878 + 119878119879) can be interpretedas the number of decisive games that did not end in a drawin the match M(119894 119895) 1 le 119894 119895 le 119873 We also introduce thefunction 119892119894(119878) = sum119873119895=1 119866119894119895(119878) 1 le 119894 le 119873 which reflects thenumber of decisive outcomes in all matches played by athlete119894 1 le 119894 le 119873

For natural 119873 and score matrix 119878 = [119878119894119895] 1 le 119894 119895 le 119873 wesay that the pair (N(119873) 119878) is the ranking problemThe weak-order (ie transitive and complete) relation119877(119873 119878) sub N(119873)timesN(119873) represents the ranking method for the ranking problem(N(119873) 119878) The vector 119903 isin R119873 is a rating vector where each119903119894 1 le 119894 le 119873 is the measure of the performance of player119894 isin N(119873) in the ranking problem (N(119873) 119878) For the rankingproblem (N(119873) 119878) a ranking method 119877(119873 119878) is induced bythe rating vector 119903 isin R119873 if

(119894 119895) isin 119877 (119873 119878)(ie 119877 (119873 119878) ranks 119894 weakly above119895) if and only if 119903119894 ge 119903119895 (1)

In this article for illustrative purposes we consider only afew of the many ranking methods discussed in the literature(note also that the ranking methods considered here basedon the ranking problems involved in chess tournaments goback to the investigations of H Neustadtl E Zermelo andB Buckholdz For detailed explanations see eg [9] and theliterature cited therein) All these methods are induced bytheir corresponding rating vectors For a given score matrix119878 = [119878119894119895] 1 le 119894 119895 le 119873 we consider the following rankingmethods

Score Method The rating vector for the score method 119903119904 =(1199031199041 119903119904119873) isin R119873 is defined as the average score 119903119904119894 =sum119873119895=1 119878119894119895119892119894(119878) 1 le 119894 le 119873Neustadtrsquos Method Neustadtrsquos rating vector 119903119873 isin R119873 isdefined by the equality 119903119873 = 119878119903119904 where 119878 = [119878119894119895] and 119878119894119895 =119878119894119895119892119894(119878) 1 le 119894 119895 le 119873Buchholzrsquos Method Buchholzrsquos rating vector 119903119861 isin R119873 isdefined by the equality 119903119873 = [119866(119878) + 119864119873]119903119904 where 119866(119878) =[119866119894119895(119878)] 119866119894119895(119878) = 119866119894119895(119878)119892119894(119878) 1 le 119894 119895 le 119873



119873sum119895=1

119878119894119895119903119891119887119895 minus ( 119873sum119895=1

119878119895119894)119903119891119887119894 = 0 1 le 119894 le 119873 (2)


120587119894 119873sum119895=1119895 =119894

119866119894119895 (119878)120587119894 + 120587119895 = 119903119904119894119892119894 (119878) 1 le 119894 le 119873 (3)


119878119860 (119886 1198861015840) = sum119888isin119862

119904119860119888 (119886 1198861015840)


(4)








3 Example






Material 119903119878

119903119873

119903119861

119903119891119887

119903119898119897



119878119860 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

0 0 0 1 3 1 1 2 2 2 2 1 1 1 11 0 1 1 3 1 1 2 2 2 2 1 1 1 12 1 0 1 3 1 1 2 2 2 2 1 1 1 13 3 2 0 2 1 1 3 3 3 3 1 1 1 11 1 1 0 0 0 0 1 1 1 1 1 1 1 13 3 3 1 2 0 0 2 2 2 2 2 2 1 23 3 3 1 1 1 0 2 2 2 2 2 2 1 22 2 2 1 3 2 2 0 0 0 0 1 1 1 12 2 2 1 3 2 2 2 0 1 1 1 1 1 12 2 2 1 3 2 2 1 1 0 0 1 1 1 12 2 2 1 3 2 2 2 1 2 0 1 1 1 13 3 3 3 3 2 2 3 3 3 3 0 1 1 23 3 3 2 2 1 1 3 3 3 3 3 0 2 23 3 3 2 2 2 2 3 3 3 3 2 1 0 13 3 3 2 2 1 1 3 3 3 3 1 1 1 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(5)





4 Conclusions





rS rN rB rfb rml



MaterialCriteria






Appendix


Data Availability






Criteria1 2 3 4

Materials

1 09832 09565 10000 000002 09580 09565 10000 000003 09727 09493 10000 000004 07234 09493 02500 050005 10000 10000 02500 050006 09485 09275 02500 050007 09252 09275 02500 050008 08753 09831 05000 025009 08100 09758 05000 0250010 07997 09831 05000 0250011 07471 09807 05000 0250012 07009 09396 00000 1000013 00537 09300 02500 0750014 03619 00000 02500 1000015 00000 09420 02500 10000





References






















Journal of












Journal of







Volume 2018






Volume 2018










119873sum119895=1

119878119894119895119903119891119887119895 minus ( 119873sum119895=1

119878119895119894)119903119891119887119894 = 0 1 le 119894 le 119873 (2)


120587119894 119873sum119895=1119895 =119894

119866119894119895 (119878)120587119894 + 120587119895 = 119903119904119894119892119894 (119878) 1 le 119894 le 119873 (3)


119878119860 (119886 1198861015840) = sum119888isin119862

119904119860119888 (119886 1198861015840)


(4)








3 Example






Material 119903119878

119903119873

119903119861

119903119891119887

119903119898119897



119878119860 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

0 0 0 1 3 1 1 2 2 2 2 1 1 1 11 0 1 1 3 1 1 2 2 2 2 1 1 1 12 1 0 1 3 1 1 2 2 2 2 1 1 1 13 3 2 0 2 1 1 3 3 3 3 1 1 1 11 1 1 0 0 0 0 1 1 1 1 1 1 1 13 3 3 1 2 0 0 2 2 2 2 2 2 1 23 3 3 1 1 1 0 2 2 2 2 2 2 1 22 2 2 1 3 2 2 0 0 0 0 1 1 1 12 2 2 1 3 2 2 2 0 1 1 1 1 1 12 2 2 1 3 2 2 1 1 0 0 1 1 1 12 2 2 1 3 2 2 2 1 2 0 1 1 1 13 3 3 3 3 2 2 3 3 3 3 0 1 1 23 3 3 2 2 1 1 3 3 3 3 3 0 2 23 3 3 2 2 2 2 3 3 3 3 2 1 0 13 3 3 2 2 1 1 3 3 3 3 1 1 1 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(5)





4 Conclusions





rS rN rB rfb rml



MaterialCriteria






Appendix


Data Availability






Criteria1 2 3 4

Materials

1 09832 09565 10000 000002 09580 09565 10000 000003 09727 09493 10000 000004 07234 09493 02500 050005 10000 10000 02500 050006 09485 09275 02500 050007 09252 09275 02500 050008 08753 09831 05000 025009 08100 09758 05000 0250010 07997 09831 05000 0250011 07471 09807 05000 0250012 07009 09396 00000 1000013 00537 09300 02500 0750014 03619 00000 02500 1000015 00000 09420 02500 10000





References






















Journal of












Journal of







Volume 2018






Volume 2018










Material 119903119878

119903119873

119903119861

119903119891119887

119903119898119897



119878119860 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

0 0 0 1 3 1 1 2 2 2 2 1 1 1 11 0 1 1 3 1 1 2 2 2 2 1 1 1 12 1 0 1 3 1 1 2 2 2 2 1 1 1 13 3 2 0 2 1 1 3 3 3 3 1 1 1 11 1 1 0 0 0 0 1 1 1 1 1 1 1 13 3 3 1 2 0 0 2 2 2 2 2 2 1 23 3 3 1 1 1 0 2 2 2 2 2 2 1 22 2 2 1 3 2 2 0 0 0 0 1 1 1 12 2 2 1 3 2 2 2 0 1 1 1 1 1 12 2 2 1 3 2 2 1 1 0 0 1 1 1 12 2 2 1 3 2 2 2 1 2 0 1 1 1 13 3 3 3 3 2 2 3 3 3 3 0 1 1 23 3 3 2 2 1 1 3 3 3 3 3 0 2 23 3 3 2 2 2 2 3 3 3 3 2 1 0 13 3 3 2 2 1 1 3 3 3 3 1 1 1 0

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

(5)





4 Conclusions





rS rN rB rfb rml



MaterialCriteria






Appendix


Data Availability






Criteria1 2 3 4

Materials

1 09832 09565 10000 000002 09580 09565 10000 000003 09727 09493 10000 000004 07234 09493 02500 050005 10000 10000 02500 050006 09485 09275 02500 050007 09252 09275 02500 050008 08753 09831 05000 025009 08100 09758 05000 0250010 07997 09831 05000 0250011 07471 09807 05000 0250012 07009 09396 00000 1000013 00537 09300 02500 0750014 03619 00000 02500 1000015 00000 09420 02500 10000





References






















Journal of












Journal of







Volume 2018






Volume 2018










rS rN rB rfb rml



MaterialCriteria






Appendix


Data Availability






Criteria1 2 3 4

Materials

1 09832 09565 10000 000002 09580 09565 10000 000003 09727 09493 10000 000004 07234 09493 02500 050005 10000 10000 02500 050006 09485 09275 02500 050007 09252 09275 02500 050008 08753 09831 05000 025009 08100 09758 05000 0250010 07997 09831 05000 0250011 07471 09807 05000 0250012 07009 09396 00000 1000013 00537 09300 02500 0750014 03619 00000 02500 1000015 00000 09420 02500 10000





References






















Journal of












Journal of







Volume 2018






Volume 2018










Criteria1 2 3 4

Materials

1 09832 09565 10000 000002 09580 09565 10000 000003 09727 09493 10000 000004 07234 09493 02500 050005 10000 10000 02500 050006 09485 09275 02500 050007 09252 09275 02500 050008 08753 09831 05000 025009 08100 09758 05000 0250010 07997 09831 05000 0250011 07471 09807 05000 0250012 07009 09396 00000 1000013 00537 09300 02500 0750014 03619 00000 02500 1000015 00000 09420 02500 10000





References






















Journal of












Journal of







Volume 2018






Volume 2018



















Journal of












Journal of







Volume 2018






Volume 2018








ranking-theory methods for solving multicriteria decision

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