Incomplete Preference Relations Research Articles

Incomplete interval-valued probabilistic uncertain linguistic preference relation in group decision making

In group decision making (GDM), due to complexity of various factors, decision makers (DMs) often provide incomplete preference relations (PRs) in preference matrix. Interval-valued probabilistic uncertain linguistic term set (IVPULTS) is a flexible and accurate tool to depict evaluation information of experts. In this paper, we mainly propose a new concept pertaining to interval-valued probabilistic uncertain linguistic preference relation (IVPULPR) that applies the IVPULTS to preference relations. Firstly, some new basic theoretical concepts of IVPULTS are developed including ordered IVPULTS, normalization method and new expectation function. Secondly, we establish several goal programming models to estimate the unknown elements in incomplete IVPULPR and propose the expected additive consistency of IVPULPR. To improve the consistency level, two optimization models are constructed based on the idea of minimum adjustment. Thirdly, we derive the experts’ weights in terms of information uncertainty, where a new method to measure information uncertainty of IVPULTS is proposed. For the sake of improving group consensus, we construct a group consensus index (GCI) and two optimization models depending on the adjustment mechanism of expert weight. Finally, a complete GDM framework with incomplete IVPULPR is devised based on the analysis of IVPULPR consistency and group consensus. Through an experiment analysis by using an UCI dataset, we find that the proposed GDM model can not only precisely express fuzzy preference information of DMs, but also ensure achievement of acceptable consistency and group consensus under the condition of not changing the initial preference as much as possible.

Consistency-Driven Methodology to Manage Incomplete Linguistic Preference Relation: A Perspective Based on Personalized Individual Semantics.

In linguistic decision-making problems, there may be cases when decision makers will not be able to provide complete linguistic preference relations. However, when estimating unknown linguistic preference values in incomplete preference relations, the existing research approaches ignore the fact that words mean different things for different people, that is, decision makers have personalized individual semantics (PISs) regarding words. To manage incomplete linguistic preference relations with PISs, in this article, we propose a consistency-driven methodology both to estimate the incomplete linguistic preference values and to obtain the personalized numerical meanings of linguistic values of the different decision makers. The proposed incomplete linguistic preference estimation method combines the characteristic of the personalized representation of decision makers and guarantees the optimum consistency of incomplete linguistic preference relations in the implementation process. Numerical examples and a comparative analysis are included to justify the feasibility of the PISs-based incomplete linguistic preference estimation method.

A new group decision-making method with incomplete intuitionistic multiplicative preference relations

PurposeConsistency and consensus are two important research issues in group decision-making (GDM). Considering some drawbacks associated with these two issues in existing GDM methods with intuitionistic multiplicative preference relations (IMPRs), a new GDM method with complete IMPRs (CIMPRs) and incomplete IMPRs (ICIMPRs) is proposed in this paper.Design/methodology/approachA mathematically programming model is constructed to judge the consistency of CIMPRs. For the unacceptably consistent CIMPRs, a consistency-driven optimization model is constructed to improve the consistency level. Meanwhile, a consistency-driven optimization model is constructed to supplement the missing values and improve the consistency level of the ICIMPRs. As to GDM with CIMPRs, first, a mathematically programming model is built to obtain the experts' weights, after that a consensus-driven optimization model is constructed to improve the consensus level of CIMPRs, and finally, the group priority weights of alternatives are obtained by an intuitionistic fuzzy programming model.FindingsThe case analysis of the international exchange doctoral student selection problem shows the effectiveness and applicability of this GDM method with CIMPRs and ICIMPRs.Originality/valueFirst, a novel consistency definition of CIMPRs is presented. Then, a consistency-driven optimization model is constructed, which supplements the missing values and improves the consistency level of ICIMPRs simultaneously. Therefore, this model greatly improves the efficiency of consistency improving. Experts' weights determination method considering the subjective and objective information is proposed. The priority weights of alternatives are determined by an intuitionistic fuzzy (IF) programming model considering the risk preference of experts, so the method determining priority weights is more flexible and agile. Based on the above theoretical basis, a new GDM method with CIMPRs and ICIMPRs is proposed in this paper.

Transitive full covers of incomplete preference relations

When a decision maker is asked to compare a set of alternatives, it may happen that the information provided is incomplete because she has no time to compare all the options or is unable to compare some alternatives against others. This contribution departs from an incomplete fuzzy weak preference relation by completing it on a consistent way with the known information. Herein the original notion of fuzzy transitivity, min-transitivity, is considered. If the decision maker is assumed to be coherent, i.e. under the assumption that the known preferences satisfy transitivity, a complete transitive preference relation that preserves the information given by the coherent decision maker is derived. In the case of the given preference values violate transitivity, a degree of transitivity is defined, and an algorithm is presented to provide the most coherent preference relation that preserves the original information according to that degree of transitivity.

Certainty equivalent and utility indifference pricing for incomplete preferences via convex vector optimization

For incomplete preference relations that are represented by multiple priors and/or multiple -- possibly multivariate -- utility functions, we define a certainty equivalent as well as the utility buy and sell prices and indifference price bounds as set-valued functions of the claim. Furthermore, we motivate and introduce the notion of a weak and a strong certainty equivalent. We will show that our definitions contain as special cases some definitions found in the literature so far on complete or special incomplete preferences. We prove monotonicity and convexity properties of utility buy and sell prices that hold in total analogy to the properties of the scalar indifference prices for complete preferences. We show how the (weak and strong) set-valued certainty equivalent as well as the indifference price bounds can be computed or approximated by solving convex vector optimization problems. Numerical examples and their economic interpretations are given for the univariate as well as for the multivariate case.

Finite expected multi-utility representation

I study the problem of obtaining a finite expected multi-utility representation for an incomplete preference relation over lotteries. It is shown that, when the prize space is a compact metric space, a preference relation admits such a multi-utility representation provided that it satisfies the standard axioms of expected utility theory and an additional necessary and sufficient axiom.

Additive and Multiplicative Consistency Modeling for Incomplete Linear Uncertain Preference Relations and Its Weight Acquisition

The use of consistency methods to supplement or generate missing information so as to obtain a rational and logical ranking list has been a key point in studies of the incomplete preference relation. Existing works on incomplete interval fuzzy preference relations (IFPRs) utilize only endpoints of the original intervals and generate missing values using interval operations, resulting in the loss and distortion of information. This article considers IFPRs as linear uncertain preference relations (LUPRs), and introduces belief degree and inverse uncertainty distributions to explore the consistency and ranking problems with LUPRs. Definitions related to uncertain preference relations (UPRs) as well as their additive and multiplicative consistency are first given. Then, models to solve weight vectors are proposed under both consistency scenarios. Finally, algorithms that generate missing information in incomplete LUPRs are provided. By building mathematical relationships between the minimum deviation and the belief degree, analytic formulas for missing values are obtained when the minimum deviation is achieved. It is verified that consistent UPRs and the proposed weight-vector solving models under incomplete preference relations are extensions of the traditional IFPRs.

Two new methods for decision-making with incomplete reciprocal fuzzy preference relations based on additive consistency

ABSTRACT In this paper, we propose two new methods to solve the problem of incomplete reciprocal fuzzy preference relations. Sometimes decision-maker(s) might not be able to provide complete information about their preferences on the alternatives. In general, an incomplete preference relation can be completed if at least a set of nonleading diagonal preference values are known and each one of the alternatives is compared directly or indirectly at least once. However, this is not always the case. Sometimes, the decision-maker(s) might not be able to provide any information with regards to at least one of the alternatives, which is called the ignorance situation. We propose two new methods based on additive consistency for solving these two problems in a multi-attributes/group decision-making environment. The first method is based on a system of equations, which is suitable for estimating missing information in the general case. The second method, which is an extension of the system of equations, utilizes a goal programming model to address the ignorance situations. Our validation of these two methods shows that the proposed methods generate a high consistency level irrespective of the nature of the problem under study. The proposed methods outperform other comparable methods without a need to modify or change the original decision-maker(s) preferences.

Are incomplete and self-confident preference relations better in multicriteria decision making? A simulation-based investigation

Incomplete preference relations and self-confident preference relations have been widely used in multicriteria decision-making problems. However, there is no strong evidence to validate their use in decision-making in the current literature. This paper reports on the design of two bounded rationality principle based simulation methods, and detailed experimental results that aim at providing evidence to answer the following two questions: (1) what are the conditions under which incomplete preference relations are better than complete preference relations?; and (2) can self-confident preference relations improve the quality of decisions? The experimental results show that when the decision-maker is of medium rational degree, incomplete preference relations with a degree of incompleteness between 20% and 40% outperform complete preference relations; otherwise, the opposite happens. Furthermore, in most cases the quality of the decision making improves when using self-confident preference relations instead of incomplete preference relations. The paper ends with the presentation of a sensitivity analysis that contributes to the robustness of the experimental conclusions.

Heterogeneous group decision making in the setting of incomplete preference relations

This paper focuses on heterogeneous group decision making (GDM) in the setting of incomplete preference relations, including incomplete fuzzy preference relations, incomplete multiplicative preference relations, incomplete additive linguistic preference relations, and incomplete multiplicative linguistic preference relations. Following the consistency-consensus optimization analysis, a new GDM method is proposed which can address inconsistent and incomplete preference relations. Firstly, we consider how to determine unknown information and how to obtain consistent preference relations. In contrast to the existing methods which rank the alternatives from fuzzy preference relations using the additive or multiplicative consistency analysis, this paper follows the consistency level to decide to use the additive or multiplicative consistency. Following the consensus analysis for each type of consistent preference relations, we calculate their comprehensively consistent preference relations and the priority vectors. After that, we build a consistency-based model for calculating the comprehensive priority vector. Furthermore, we offer an ordered position consensus-based method using the built optimization models. The proposed heterogeneous GDM method with incomplete preference relations provides us with a very useful way for dealing with GDM problems in heterogeneous environments.

Expert Selection for Humanitarian Projects Development: A Group Decision Making approach with Incomplete Information Relations.

This paper proposes a methodology to select an expert for the development of humanitarian and social projects based on multi-criteria decision-making with incomplete information relations to add more objectivity to the recruitment process. The proposed approach starts with assessing the decision makers (DMs) by means of Fuzzy Analytic Hierarchy Process (AHP). Then, Analytic Network Process (ANP) is used to weigh the criteria in order to take into account the interdependencies between criteria. In the recruiting process, an expert could be unable to assess all criteria. Therefore, the incomplete preference relations is used when a DM is unable to express his judgment. At a later stage, Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is addressed to rank the different candidates. The proposed methodology is applied to a situation where an extra DM evaluates four DMs, and six criteria are used to select one candidate among five. The obtained criteria weights and the final ranking of the candidates are analysed and compared to an approach where there is no lack of information in the decision maker’s preferences.

Models for pessimistic or optimistic decisions under different uncertain scenarios

In many decision problems under uncertainty, agents are only able to provide a possibly incomplete preference relation on gambles, that can be “irrational” according to the classical expected utility paradigm. Furthermore, agents can find it easier to express their preference relation taking one or more specific scenarios as hypothesis. In order to handle these situations two betting scheme rationality conditions are introduced, which characterize those preference relations representable by a conditional Choquet expected value with respect to a conditional belief or plausibility function. Such conditions encode a pessimistic or an optimistic attitude towards uncertainty, respectively, and allow to take into account (possible) “null” scenarios by means of a weak ordering of “unexpectation”. Moreover, they are the basis of a completion technique for enlarging the set of preferences by preserving rationality.

On the indifference relation in Bewley preferences

Bewley’s (2002) influential model of preferences over uncertain prospects features an incomplete preference relation and a set of priors over the states of the world such that one act is preferred to another if and only if its expected utility is higher under every prior in that set. This note shows that, under general conditions on preferences, the decision maker in the Bewley model cannot be indifferent between distinct monetary acts whenever the set of priors is fully-dimensional. In the special case of two states, in particular, such “objectively rational” preferences are incomplete if and only if the indifference relation is trivial in the above sense.

SET-VALUED SHORTFALL AND DIVERGENCE RISK MEASURES

Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate risk measures is constructed via a recent Lagrange duality for set optimization. In particular, it is shown that a shortfall risk measure can be written as an intersection over a family of divergence risk measures indexed by a scalarization parameter. Examples include set-valued versions of the entropic risk measure and the average value at risk. As a second step, the minimization of these risk measures subject to trading opportunities is studied in a general convex market in discrete time. The optimal value of the minimization problem, called the market risk measure, is also a set-valued risk measure. A dual representation for the market risk measure that decomposes the effects of the original risk measure and the frictions of the market is proved.

Group prioritisation with unknown expert weights in incomplete linguistic context

ABSTRACTIn this paper, we study a group prioritisation problem in situations when the expert weights are completely unknown and their judgement preferences are linguistic and incomplete. Starting from the theory of relative entropy (RE) and multiplicative consistency, an optimisation model is provided for deriving an individual priority vector without estimating the missing value(s) of an incomplete linguistic preference relation. In order to address the unknown expert weights in the group aggregating process, we define two new kinds of expert weight indicators based on RE: proximity entropy weight and similarity entropy weight. Furthermore, a dynamic-adjusting algorithm (DAA) is proposed to obtain an objective expert weight vector and capture the dynamic properties involved in it. Unlike the extant literature of group prioritisation, the proposed RE approach does not require pre-allocation of expert weights and can solve incomplete preference relations. An interesting finding is that once all the experts express their preference relations, the final expert weight vector derived from the DAA is fixed irrespective of the initial settings of expert weights. Finally, an application example is conducted to validate the effectiveness and robustness of the RE approach.

Choice deferral, indecisiveness and preference for flexibility

In a standard model of menu choice, we examine the behavior of an agent who applies the following Cautious Deferral rule: “Whenever in doubt, don't commit; just leave options open.” Our primitive is a complete preference relation ≽ that represents the agent's choice behavior. The agent's indecisiveness is captured by means of a possibly incomplete (but otherwise rational) preference relation ≽ˆ. We ask when ≽ can be viewed as a Cautious Deferral completion of some incomplete ≽ˆ. Under the independence and continuity assumptions commonly used in the menu choice literature, we find that even the smallest amount of indecisiveness is enough to force ≽, through the above deferral rule, to exhibit preference for flexibility on its entire domain. Thus we highlight a fundamental tension between non-monotonic preferences, such as preferences for self-control, and tendency to defer choice due to indecisiveness.

A method considering and adjusting individual consistency and group consensus for group decision making with incomplete linguistic preference relations

For group decision making with incomplete linguistic preference relations, this paper first make an analysis about individual consistency, based on which a goal programming model is established to estimate the missing values for incomplete linguistic preference relations. Different to the existing methods, the proposed model can cope with the problem in ignorance situations. Then, a multi-stage algorithm which takes both the individual consistency and group consensus into consideration is presented, including the following main stages: initial check for the incomplete preference relations, completion, check and adjustment of individual consistency, aggregation, check and adjustment of group consensus and final selection. In the aggregation stage of the algorithm, an experience and quality hybrid weight vector (EQWV) is defined to obtain the collective linguistic preference relation in group decision making. To prove the feasibility and efficiency of the proposed method, some examples and the corresponding comparisons with other methods are provided. Besides, the analysis on the threshold values proposed by the multi- stage algorithm is made, which can help the decision maker determine the threshold values according to his/her expectations. Finally, a further discussion about comparison analysis, extensible application and some details of the proposed method is presented.

Linguistic fuzzy consensus model for collaborative development of fuzzy cognitive maps: a case study in software development risks

Fuzzy cognitive maps (FCMs) have been widely used in several domains for information processing, such as, data fusion, decision making. Although several methods to automatically learn FCMs are recognized from the scientific literature, the most used approach to build an FCM relies on a collaborative task involving single person or, more suitably, group of experts. Collaborative development increases reliability and robustness of the resulting FCM, but rises some problems in terms of group decision making to aggregate different perspectives of the problem representation. This paper proposes to support collaborative development of FCMs introducing knowledge engineering process that relies on Linguistic Fuzzy Consensus Model. In the proposed approach, each expert builds the own version of the FCM. When all different versions are available, a Group Decision Making process is activated in order to reach the consensus on conflictual modeling opinions. The result is a unique final version of the FCM that is not a simple aggregation of the versions provided by the experts but is the result of a well-suited mathematical model. In addition, this work adopts consensus model with incomplete preference relations scheme to address knowledge harmonization issues. Finally, advantages and the limitations of the proposed framework are argued.

Confidence models of incomplete preferences

This paper introduces a new class of representations for incomplete preferences called confidence models. Confidence models describe decision makers who behave as if they have probabilistic uncertainty over their true preferences, and are only willing to express a binary preference if it is sufficiently likely to hold. Confidence models provide a natural way to connect incomplete preferences with stochastic choice. This connection is characterized by a simple joint condition on an incomplete preference relation and a random choice rule.

Managing incomplete preference relations in decision making: A review and future trends

In decision making, situations where all experts are able to efficiently express their preferences over all the available options are the exception rather than the rule. Indeed, the above scenario requires all experts to possess a precise or sufficient level of knowledge of the whole problem to tackle, including the ability to discriminate the degree up to which some options are better than others. These assumptions can be seen unrealistic in many decision making situations, especially those involving a large number of alternatives to choose from and/or conflicting and dynamic sources of information. Some methodologies widely adopted in these situations are to discard or to rate more negatively those experts that provide preferences with missing values. However, incomplete information is not equivalent to low quality information, and consequently these methodologies could lead to biased or even bad solutions since useful information might not being taken properly into account in the decision process. Therefore, alternative approaches to manage incomplete preference relations that estimates the missing information in decision making are desirable and possible. This paper presents and analyses methods and processes developed on this area towards the estimation of missing preferences in decision making, and highlights some areas for future research.