Additive Preference Relations Research Articles

And-like-uninorm based consistency analysis and optimized fuzzy weight closed-form solution of triangular fuzzy additive preference relations

This paper focuses on multiplicative consistency of triangular fuzzy additive preference relations (TFAPRs) and deriving a closed-form solution of optimized triangular fuzzy weights (TFWs) from TFAPRs. And-like-uninorm based indices are introduced to measure increasing part vagueness, decreasing part vagueness and overall vagueness for ]0,1[-valued triangular fuzzy preferences and TFAPRs. An index is then defined to measure row vagueness proportionality of TFAPRs. An and-like-uninorm based method is further proposed to generate multiplicatively consistent TFAPRs from ]0,1[-valued TFWs. By discussing equivalency of ]0,1[-valued TFW vectors, the paper presents two frameworks of normalized TFWs called multiplicatively modal-value-normalized TFWs and multiplicatively support-interval-normalized TFWs. Based on crucial properties of consistent TFAPRs, a logarithmic least square (LLS) model is established to seek multiplicatively support-interval-normalized TFWs from TFAPRs. By decomposing its goals and constraints, the LLS model is transformed into two least square models whose closed-form solutions are found by the Lagrange multiplier method. On basis of the obtained closed-form solution of TFWs, an algorithm including acceptable consistency checking is developed for decision making with TFAPRs. The rationality and advantages of the presented models are illustrated by a numerical example with five TFAPRs and a comparative analysis.

A consensus model for group decision making under additive reciprocal matrices with flexibility

A consensus model is proposed for group decision making (GDM) with additive reciprocal preference relations based on the technique for order preference by similarity to an ideal solution (TOPSIS). The initial positions of decision makers are expressed as additive reciprocal matrices. A method of obtaining a collective ideal additive reciprocal matrix is proposed in terms of individual ones. A novel fitness function is constructed through a linear combination of two objective functions involving the consensus degree within the group of experts and the consistency measure of individual preference relations. A new algorithm based on particle swarm optimization (PSO) is proposed to model the dynamic and iterative consensus process of GDM with flexibility. Numerical results are reported to illustrate the proposed model through some comparisons with the existing ones. The weight variance is proposed to characterize the dispersion degree of the weights of alternatives, and analyzed by considering the effects of the flexibility degree and the constraint conditions. The observations reveal that with an increasing of the flexibility degree, the dominant alterative may be chosen, or any solution may not be reached by a group of experts.

Direct Iterative Procedures for Consensus Building with Additive Preference Relations Based on the Discrete Assessment Scale

Individual consistency and group consensus are both important when seeking reliable and satisfying solutions for group decision making (GDM) problems using additive preference relations (APRs). In this paper, two new algorithms are proposed to facilitate the consensus reaching process, the first of which is used to improve the individual consistency level, and the second of which is designed to assist the group to achieve a predefined consensus level. Unlike previous GDM studies for consistency and consensus building, the proposed algorithms are essentially heuristic, modify only some of the elements in APRs to reduce the number of preference modifications in the consistency and consensus process, and have modified preferences that belong to the original evaluation scale to make the generated suggestions easier to understand. In particular, the consensus algorithm ensures that the individual consistency level is still acceptable when the predefined consensus level is achieved. Finally, classical examples and simulations are given to demonstrate the effectiveness of the proposed approaches.

An Optimization-Based Approach to Social Network Group Decision Making with an Application to Earthquake Shelter-Site Selection.

The social network has emerged as an essential component in group decision making (GDM) problems. Thus, this paper investigates the social network GDM (SNGDM) problem and assumes that decision makers offer their preferences utilizing additive preference relations (also called fuzzy preference relations). An optimization-based approach is devised to generate the weights of decision makers by combining two reliable resources: in-degree centrality indexes and consistency indexes. Based on the obtained weights of decision makers, the individual additive preference relations are aggregated into a collective additive preference relation. Further, the alternatives are ranked from best to worst according to the obtained collective additive preference relation. Moreover, earthquakes have occurred frequently around the world in recent years, causing great loss of life and property. Earthquake shelters offer safety, security, climate protection, and resistance to disease and ill health and are thus vital for disaster-affected people. Selection of a suitable site for locating shelters from potential alternatives is of critical importance, which can be seen as a GDM problem. When selecting a suitable earthquake shelter-site, the social trust relationships among disaster management experts should not be ignored. To this end, the proposed SNGDM model is applied to evaluate and select earthquake shelter-sites to show its effectiveness. In summary, this paper constructs a novel GDM framework by taking the social trust relationship into account, which can provide a scientific basis for public emergency management in the major disasters field.

An Alternative Consensus Model of Additive Preference Relations for Group Decision Making Based on the Ordinal Consistency

Pairwise comparison is a useful tool to express decision makers’ (DMs’) preferences in the group decision-making (GDM) problems. However, the preferences provided by pairwise comparisons could be self-contradictory, i.e., ordinal inconsistencies exist. Therefore, before reaching consensus, the first thing is to assure the DM’s judgments that are not contradictory. As the purpose of the GDM is to choose most preferred alternative, the consensus degree for each alternative of all the DMs should be measured. In the present paper, an alternative consensus model for additive preference relations (APRs) based on ordinal consistency (OC) is developed. An algorithm is applied to detect and adjust the ordinally inconsistent elements for APRs. Then the alternative rankings for each ordinally consistent APR and the aggregated APR is obtained, respectively. A model is designed to change the DMs’ importance, which increases the alternative consensus degree. The proposed model does not change the DMs’ preferences, aiming to make full use of the DMs’ judgements. Finally, an illustrative example and comparisons with the current approaches are furnished to demonstrate the effectiveness of the developed method.

Interval-valued fuzzy multi-criteria decision-making based on simple additive weighting and relative preference relation

To encompass imprecise messages, traditional multi-criteria decision-making (MCDM) was generalized into fuzzy multi-criteria decision-making(FMCDM) in solving decision-making problems. In numerous MCDM methods, simple additive weighting(SAW) is the simplest one, but fuzzy multiplication for generalizing SAW under fuzzy environment is still complicated, especially interval-valued fuzzy numbers. In the past, Wang proposed a model based on SAW and relative preference relation to resolve fuzzy multiplication tie. Practically, Wang's model is useful to fuzzy generalization of SAW. However, the model is only used in triangular fuzzy numbers. Recently, more and varied messages are obtained and processed for decision-making problems. Triangular fuzzy numbers are insufficient to express the variance, whereas interval-valued fuzzy numbers are suitable. Therefore, SAW had to be generalized under interval-valued fuzzy environment. In this paper, we combine SAW with a relative preference relation into interval-valued FMCDM to avoid fuzzy multiplication and grasp more messages. Our proposed relative preference relation improved from Lee's fuzzy preference relation is similar to Wang's relative preference relation. The main difference is that the proposed relation is utilized in interval-valued fuzzy numbers, whereas Wang's is used in triangular fuzzy numbers. To sum up, we provide the FMCDM to solve decision-making problems with interval-valued fuzzy numbers easily and quickly.

Optimal consistency and consensus models for interval additive preference relations: A discrete distribution perspective

To assist in the consensus reaching process, this article presents optimization models with interval additive preference relations (APRs) drawn from pairwise comparisons. First, consistency models are proposed to obtain additively consistent interval APRs for continuous and discrete scale cases, after which consensus models are established to arrive at a predefined consensus level for the two cases. These models seek to minimize the amount of preference changes and can be solved using linear or integer linear programming techniques. While the obtained solutions may not be unique, a second stage model is introduced to reduce the uncertainty degrees in the suggested preferences. Compared to existing approaches, the proposed models have two major advantages: the derived solution can be limited to the easy to understand original scales, and refined solutions can be determined using multi-stage optimization. Finally, several numerical examples are given to verify the proposed models, and several simulations are conducted to demonstrate the potential behaviour of the presented models in practical applications.

Managing personalized individual semantics and consensus in linguistic distribution large-scale group decision making

A linguistic distribution assessment is an effective approach to represent uncertain preferences in large-scale group decision making (LSGDM). The same word often signifies different things for different decision makers in linguistic distribution assessments, which is called personalized individual semantics (PIS). Moreover, preference conflicts widely exist in LSGDM. This paper develops a new framework to address PIS and consensus in LSGDM using linguistic distribution preference relations (LDPRs). In the proposed LSGDM framework, a consistency-driven optimization model is put forward to produce the numerical scales with the PIS by maximizing the consistency of the additive preference relation that is transformed from its LDPR. Then, a preference clustering technique is employed to decompose decision makers into different clusters for managing their preferences. Next, the paper devises a two-stage-based consensus reaching model to manage the individual consistency and group consensus, which seeks to minimize the preference information loss. The first stage aims to assist decision makers in achieving a consensus within each obtained cluster, and the second stage is devoted to facilitating the consensus building among the different clusters. Finally, a case study that evaluates water management plans and a comparative analysis with the existing baseline approach are conducted to assess the feasibility and validity of the proposed LSGDM framework.

An Approach for Achieving Consistency for Symmetric Trapezoidal Interval Type-2 Fuzzy Sets

Interval type-2 fuzzy sets (IT2 FSs) are powerful tools for dealing with linguistic information in decision making. However, there is a dearth of research regarding the consistency of preference relations based on IT2 FSs. In this paper, symmetric IT2 FSs and IT2 additive preference relations are defined, whilst at the same time a mapping method is proposed to convert IT2 numbers into the corresponding linguistic terms based on the ranking values for IT2 FSs, and some properties for symmetric IT2 FSs are proved. Then, we discuss the process for achieving consistency for IT2 additive preference relations. An algorithm is developed for the IT2 additive preference relation process for achieving consistency, and some desired algorithmic properties are proved. Finally, an actual case study is used in order to demonstrate the effectiveness of the proposed approach.

Goal programming models for incomplete interval additive reciprocal preference relations with permutations

Within the framework of the fuzzy analytic hierarchy process, the importance of alternatives could be expressed as interval-valued comparison matrices to model some uncertainty experienced by decision makers (DMs). Owing to the complexity of the considered problem and the lack of DMs’ knowledge, the given interval-valued comparison matrix may be incomplete. It is of importance to develop mathematical models for incomplete interval-valued comparison matrices, so that the missing entries can be estimated. In this study, it is considered that a missing value may be one of the bounds of interval-valued entries. Acceptable incomplete interval additive reciprocal preference relations (IARPRs) are redefined and the existing drawback is overcome. By considering the random behavior of decision makers in comparing alternatives, the novel definitions of multiplicative and additive approximate consistency for incomplete IARPRs are introduced. Then, we propose two goal programming models to estimate the missing values of incomplete IARPRs. The permutations of alternatives are incorporated into the proposed mathematical models. It is found that when the permutations of alternatives are different, the estimated preference values could be different. Some numerical results are reported to illustrate the given algorithms for solving decision-making problems with incomplete IARPRs. The observations reveal that the proposed mathematical models can be used to capture the randomness experienced by decision makers in comparing alternatives.

Group Decision Making with Heterogeneous Preference Structures: An Automatic Mechanism to Support Consensus Reaching

In real-world decision problems, decision makers usually express their opinions with different preference structures. In order to deal with the heterogeneous preference information in group decision making, this paper presents an optimization-based consensus model for group decision making with heterogeneous preference structures (utility values, preference orderings, multiplicative preference relations and additive preference relations). This proposal seeks to minimize the information loss between decision makers’ heterogeneous preference information and individual preference vectors and also seeks the collective solution with a consensus. Meanwhile, in order to justify the consensus model, we discuss its internal aggregation operator between the obtained individual and group preference vectors, demonstrate that the proposed model satisfies the Pareto principle of social choice theory, and prove the uniqueness of the solution to the optimization model. Furthermore, based on the proposed optimization-based consensus model, we present an automatic mechanism to support consensus reaching in the group decision making with heterogeneous preference structures. In the consensus reaching process, the obtained individual and group preference vectors are considered as a decision aid which decision makers can use as a reference to adjust their preference opinions. Finally, detailed simulation experiments and comparison analysis are conducted to demonstrate the feasibility and effectiveness of our proposed model.

Nature Disaster Risk Evaluation with a Group Decision Making Method Based on Incomplete Hesitant Fuzzy Linguistic Preference Relations.

Because the natural disaster system is a very comprehensive and large system, the disaster reduction scheme must rely on risk analysis. Experts’ knowledge and experiences play a critical role in disaster risk assessment. The hesitant fuzzy linguistic preference relation is an effective tool to express experts’ preference information when comparing pairwise alternatives. Owing to the lack of knowledge or a heavy workload, information may be missed in the hesitant fuzzy linguistic preference relation. Thus, an incomplete hesitant fuzzy linguistic preference relation is constructed. In this paper, we firstly discuss some properties of the additive consistent hesitant fuzzy linguistic preference relation. Next, the incomplete hesitant fuzzy linguistic preference relation, the normalized hesitant fuzzy linguistic preference relation, and the acceptable hesitant fuzzy linguistic preference relation are defined. Afterwards, three procedures to estimate the missing information are proposed. The first one deals with the situation in which there are only known judgments involving all the alternatives; the second one is used to estimate the missing information of the hesitant fuzzy linguistic preference relation with more known judgments; while the third procedure is used to deal with ignorance situations in which there is at least one alternative with totally missing information. Furthermore, an algorithm for group decision making with incomplete hesitant fuzzy linguistic preference relations is given. Finally, we illustrate our model with a case study about flood disaster risk evaluation. A comparative analysis is presented to testify the advantage of our method.

A group decision making model based on triangular fuzzy additive reciprocal matrices with additive approximation-consistency

A group decision making (GDM) model is proposed when the experts evaluate their opinions through triangular fuzzy numbers. First, it is pointed out that the preference relations with triangular fuzzy numbers are inconsistent in nature. In order to distinguish the typical consistency, the concept of additive approximation-consistency is proposed for triangular fuzzy additive reciprocal matrices. The properties of triangular fuzzy additive reciprocal matrices with additive approximation-consistency are studied in detail. Second, using (n − 1) restricted preference values, a triangular fuzzy additive reciprocal preference relation with additive approximation-consistency is constructed. Third, a novel compatibility degree among triangular fuzzy additive reciprocal preference relations is defined. It is further applied to introduce the compatibility-degree induced ordered weighted averaging (CD-IOWA) operator for generating a collective triangular fuzzy additive reciprocal matrix with additive approximation-consistency. Finally, a new algorithm for the group decision-making problem with triangular fuzzy additive reciprocal preference relations is presented. A numerical example is carried out to illustrate the proposed definitions and algorithm.

A novel group decision-making model based on triangular neutrosophic numbers

In group decision-making (GDM) model, experts often evaluate their opinion by using triangular fuzzy numbers. The preference relations with triangular fuzzy numbers are in consistent in nature, so we turned to neutrosophic in this paper. It is very important to take into account consistency of expert opinion and consensus degree in GDM. In order to distinguish the typical consistency, the concept of additive approximation consistency is proposed for triangular neutrosophic additive reciprocal matrices. The properties of triangular neutrosophic additive reciprocal matrices with additive approximation consistency are studied in detail. Second, by using $$(n-1)$$ restricted preference values, a triangular neutrosophic additive reciprocal preference relation with additive approximation consistency is constructed. The differences among expert’s opinions are measured using consensus degree. For generating a collective triangular neutrosophic additive reciprocal matrix with additive approximation consistency, the neutrosophic triangular weighted aggregation operator is used. Finally, a novel algorithm for the group decision-making problem with triangular neutrosophic additive reciprocal preference relations is presented. A numerical example is carried out to illustrate the proposed definitions and algorithm.

Additive consistent interval-valued Atanassov intuitionistic fuzzy preference relation and likelihood comparison algorithm based group decision making

This paper investigates a group decision making (GDM) method based on additive consistent interval-valued Atanassov intuitionistic fuzzy (IVAIF) preference relations (IVAIFPRs) and likelihood comparison algorithm. Firstly, the likelihood of IVAIF values (IVAIFVs) is defined by the likelihood of intervals. Then a likelihood comparison algorithm is designed to rank IVAIFVs. According to the additive consistent interval fuzzy preference relation, we define the additive consistency of an IVAIFPR. Two special interval fuzzy preference relations are extracted from an IVAIFPR. They can be regarded as the lowest and highest preferred matrices of the IVAIFPR, respectively. Using a parametric linear program, the IVAIF priority weights of an IVAIFPR are generated from these two extracted special interval fuzzy preference relations. For the GDM with IVAIFPRs, the group consensus is defined by the distances between the individual IVAIFPRs and the collective one. To derive decision makers' weights, an optimization model is constructed by maximizing the group consensus and transformed into a linear program to resolve. Subsequently, utilizing the IVAIF weighted averaging operator, the collective IVAIFPR is obtained and applied to obtain the IVAIF priority weights. The order of alternatives is generated by ranking the IVAIF priority weights. At length, an enterprise resource planning system selection example is analyzed to verify the effectiveness of the proposed method.

On possibility-degree formulae for ranking interval numbers

Since interval numbers are used to evaluate the opinions of decision makers and express the weights of alternatives in various decision making problems, it is requisite to give a feasible method to rank them. In the present paper, the two existing possibility degree formulae for ranking interval numbers are proved to be equivalent. A generalized possibility degree formula is proposed by considering the attitude of decision makers with a prescribed function. Some known possibility degree formulae for ranking interval numbers can be recovered by choosing a special function. The proposed method is applied to uniformly define the weak transitivity of interval multiplicative and additive reciprocal preference relations. Numerical examples are carried out to illustrate the new formula.

A social ties-based approach for group decision-making problems with incomplete additive preference relations

With the rapid growth of Web 2.0 technology, a new paradigm has been developed that allows many users to participate in decision-making processes within online social networks. The social information (i.e., social ties and social influence) of the members that is stored in online social networks provides a new perspective for investigating group decision-making (GDM) problems. In this paper, a new interactive GDM approach, based on online social networks, is proposed to address a ranking problem with incomplete additive preference relations (IAPRs). This approach incorporates the strength of social ties and social influence calculated by social network analysis methods regarding the decision-making process. After decision makers (DMs) provide IAPRs, a searching algorithm is developed to identify the optimal preference information transfer path from DMs to the decision supporters who can provide the corresponding preference information. Next, a linear programming model is constructed to complete the missing preference values of the IAPRs. The main features of the linear programming model include its ability to account for other DMs’ preference information and to maintain consistency. To help the group reach an agreement on the ranking of alternatives, a consensus reaching process is proposed. The strength of social ties and social influence are used to calculate the acceptable adjustment coefficients for DMs in the feedback mechanism. Finally, an illustrative example and further discussion demonstrate the validity of the proposed approach.

Consistency analysis and group decision making based on triangular fuzzy additive reciprocal preference relations

Triangular fuzzy numbers are effective in modeling imprecise and uncertain information, and have been widely applied in decision making. This paper uses a cross-ratio-expressed triplet to characterize a positive triangular fuzzy number, and introduces notions of cross-ratio-expressed triangular fuzzy numbers (CRETFNs) and triangular fuzzy additive reciprocal preference relations (TFARPRs). We present transformation methods between TFARPRs and triangular fuzzy multiplicative reciprocal preference relations, and develop operational laws of CRETFNs, such as complement, addition, multiplication and power. A cross-ratio-expressed triangular fuzzy multiplication based transitivity equation is established to define multiplicative consistency of TFARPRs. The new consistency captures Tanino's multiplicative consistency among the cross-ratio-expressed modal values, and geometric consistency of the interval fuzzy preference relation constructed from lower and upper support values of cross-ratio-expressed triangular fuzzy judgments. Some desirable properties are furnished for multiplicatively consistent TFARPRs. We propose a cross-ratio-expressed triangular fuzzy weighted geometric operator to aggregate CRETFNs, and extend it to fuse TFARPRs. Score and uncertainty index functions are defined and employed to devise a novel comparison method for CRETFNs. A detailed procedure is put forward to solve group decision making problems with TFARPRs. Six numerical examples are provided to illustrate the validity and applicability of the proposed models.

Multi-criteria group decision making with incomplete hesitant fuzzy preference relations

In order to simulate the hesitancy and uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of hesitant fuzzy preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. This paper defines the concept of additive consistent hesitant fuzzy preference relations. The characterizations of additive consistent hesitant fuzzy preference relations are studied in detail. Owing to the limitations of the experts’ professional knowledge and experience, the provided preferences in a hesitant fuzzy preference relation are usually incomplete. Consequently, this paper introduces the concepts of incomplete hesitant fuzzy preference relation, acceptable incomplete hesitant fuzzy preference relation, and additive consistent incomplete hesitant fuzzy preference relation. Then, two estimation procedures are developed to estimate the missing information in an expert's incomplete hesitant fuzzy preference relation. The first procedure is used to construct an additive consistent hesitant fuzzy preference relation from the lowest possible number, (n−1), of pairwise comparisons. The second one is designed for the estimation of missing elements of the acceptable incomplete hesitant fuzzy preference relations with more known judgments. Moreover, an algorithm is given to solve the multi-criteria group decision making problem with incomplete hesitant fuzzy preference relations. Finally, a numerical example is provided to illustrate the solution processes of the developed algorithm and to verify its effectiveness and practicality.

A fuzzy multi-criteria decision-making model based on simple additive weighting method and relative preference relation

In the past, approaches often generalized classical multi-criteria decision-making (MCDM) methods under fuzzy environment to solve fuzzy multi-criteria decision-making (FMCDM) problems and encompass decision-making messages uncertainty and vagueness. These MCDM methods included analytic hierarchy process (AHP), simple additive weighting method (SAW), technique for order preference by similarity to ideal solution (TOPSIS), etc. In the MCDM methods, SAW is a famous method that is applied in fuzzy environment, but fuzzy multiplication is a drawback to generalize SAW under fuzzy environment. To resolve the multiplication drawback, we utilize a relative preference relation that is from fuzzy preference relation in fuzzy generalized SAW. Generally, fuzzy preference relation is an option to reserve lots of messages. However, pair-wise comparison for fuzzy preference relation is complex on operation. Through the description above, the relative preference relation is improved form fuzzy preference relation to avoid comparing fuzzy numbers on pair-wise and reserving fuzzy messages. Through the relative preference relation, we can generalize SAW under fuzzy environment. It is said that we propose a FMCDM model based on SAW and the relative preference relation to easily and quickly solve FMCDM problems.