INCOMPLETE PREFERENCERELATIONS IN MCDM

Abstract

Multi-citeriadecisionmakingisconcernedwithstructuringandsolvingdecisionandplanningproblemsinvolvingmultiplecriteria. Inadecisionscenario,adecisionmakerisgenerally required to provide his/her assessments of choices. To communicate the decision maker’spreferencedata,preferencerelationsareexceptionallyhelpfulinvariousfieldsof decision-making problem, for example, legislative issues, social brain science, designing, administration, business, and financial aspects, and so on. Sometimes, it has witnessed that in a situation a decision maker might not have a decent comprehension of a specific query, thus he/she can not make an instaneous contrast between each two objects. Consequently it necessities to allow the decision maker to avoid some questionable comparison adaptably. Therefore sometimes, due to lack of time and busy schedule of the decision maker, incomplete preference relations are obtained. In this case, incomplete preference relationsareobtained,andthewholeprocessmayslowdown. Inthisworkwehavedeveloped methods for complementing types of incomplete preference relations. Applications of different preference relations in MCDM, are also discussed. IntroductorychapterpresentsabriefreviewofuncertaintytheorythatisFuzzysettheory and some extension of the fuzzy set theory, to fuzzy relation. Also, this chapter discussestypeofpreferencerelationsforaddressingmulti-criteriadecisionmaking(MCDM). Thus, the present chapter creates a background, gives the motive of thesis work. Chapter2 define (e α ,e β )-cuts and the resolution form of the interval-valued intuitionisticfuzzy(IVIF)relationstodevelopaprocedureforconstructingahierarchicalclustering for IVIF max-min similarity relations. The advantage of the proposed scheme is illustrated in determining the criteria weights in MCDM problems involving IVIF numbers. A complete procedure is drawn to generate criteria weights starting from the criteriaalternative matrix of the MCDM problem with entries provided by a decision maker as interval-valued intuitionistic fuzzy numbers. The chapter is based on a research paper “Hierarchical clustering of interval-valued intuitionistic fuzzy relations and its applica xiii tion to elicit criteria weights in MCDM problems”, published in Opsearch, springer, 54, 388–416, (2017). Chapter 3 propose a characterization of the consistency property using newly defined transitivity property for intuitionistic multiplicative preference relations (IMPR) together withcomplementingmissingelementsforincompleteIMPR.Usingnewtransitivityproperty of IMPR, we have developed two different methods to find the missing element of IMPRs. Acceptably consistent with complete IMPRs is also checked. The another goal of this chapter is to achieve the consistent intuitionistic multiplicative preference relation using graphical approach. We have proposed two different characterization of the consistency for intuitionistic multiplicative preference relation(IMPR). In the first approach, we design an algorithm to achieve the consistency of IMPR by using the cycles of various length in a directed graph. The second approach proves isomorphism between the set of IMPRs and the set of asymmetric multiplicative preference relations. That result is explored to use the methodologies developed for asymmetric multiplicative preference relations to the case of IMPRs and achieve the consistency of asymmetric multiplicative preference relation using directed graph. Also, the above said method is applied for incompleteIMPR,hereconsistencyplayanimportantrole. Theillustrationsareprovidedto exemplify the designed methods. The chapter is based on a research paper “New transitivity property of intuitionistic fuzzy multiplicative preference relation and its application in missing value estimation”, published in Annals of Fuzzy Mathematics and Informatics, 16 (1), 71–86 (2018) and “Two different approaches for consistency of intuitionistic multiplicative preference relation using directed graph”, is communicated in Asia-pacific journal of operational research. Chapter 4 study the consistency property, and especially the acceptably consistent property, for incomplete interval-valued intuitionistic multiplicative preference relations. We propose a technique to evaluate missing elements which first estimates the initial values for all missing entries in an incomplete interval-valuedintuitionistic multiplicative preference relation and then improves them by a local optimization method. A method is developed to estimate the importance of the experts to achieve resultant consistent decision matrix in group decision situations. The proposed method is illustrated using two examples involving group decision scenario. The chapter is based on a research paper “Acceptably consistent incomplete interval-valued intuitionistic multiplicative preference relations”, is published in Soft Computing, Springer,22, 7463–7477 (2018). In Chapter 5, a new definition of additive consistency property of hesitant fuzzy pref xiv erence relation (HFPR) is given that preserves the property of hesitancy and is used to constructthecompleteHFPRfromincompleteone. ThesignificanceofconsistencymeasureforHFPRmakesurethattheDMsareneitherarbitrarynorunreasonable. Wedevelop a method to check consistency level of incomplete HFPR. A numerical example is illustrated to show the applicability of the designed methodology. Group decision-making problemwithincompleteHFPRisalsoconsidered. Thechapterisbasedonaresearchpaper“IncompleteHesitantFuzzyPreferenceRelation”,ispublishedinJournalofStatistics & Management Systems, Taylor & Francis, 21, (8) 1459–1479 (2018). Chapter 6 developed a method to complete incomplete hesitant multiplicative preference relations (HMPRs). A new definition of multiplicative transitive property of HMPR has given that preserve the hesitancy property and is used to construct the complete HMPR from incomplete one. An optimization model is developed to minimize the error. Also a linear programming model is developed to directly calculate the missing elements of incomplete HMPR. The satisfaction degree and the acceptably consistent of complete HMPR is also checked. The whole procedure is explained with a suitable example. The chapterisbasedonresearchpaper“AMethodtoComplementIncompleteHesitantMultiplicativePreferenceRelation”,publishedinInternationalJournalofResearchandAnalyticalReviews,5,(2)1421–1429(2018)and“IncompleteHesitantMultiplicativePreference Relation”,revisedversionsubmittedin Opsearch, Springer. After chapter 6 we present the summary of the research work carried out in this thesis. Further, the future research plan has been discussed in brief. Finally,thebibliographyandlistofpublicationshavebeengivenattheendofthethesis.