Hesitant Multiplicative Preference Relation Research Articles

Generalized hesitant multiplicative preference relations and the analytic risk-network process

The hesitant fuzzy preference relation (HFPR) and the hesitant multiplicative preference relation (HMPR) are emerging techniques in decision-making research field. Clearly, an intuitionistic idea to improve the above preference relations is to synthesize the HFPR and the HMPR, which is also a primary contribution of this paper. Based on which, we propose the generalized HMPR (GHMPR) with full desired properties using transformation formulas that combine HFPR with HMPR and convert the HMPR into the GHMPR. Then, the GHMPR not only considers different risk appetites of a decision maker, but also preserves the advantages of both. Moreover, we design the corresponding consistency check algorithms, weight calculation methods and convergence algorithms for the GHMPR. Based on these new methods, we further propose a qualitative decision-making method under the GHMPR environment named as the analytic risk-network process (ARNP). Saaty’s well-known analytic network process case is then utilized as an illustrated example to verify the feasibility of the proposed methods via comparisons. The simulation results show our proposed methods are effective and useful.

Consistency Checking and Improving for Interval-Valued Hesitant Preference Relations

Group decision making (GDM), which aims to obtain a sensible decision result with several decision makers, is a common occurrence in daily life. Since the uncertainty of the objects is a thorny issue in the process of GDM, it is important to eliminate uncertainty in order to achieve an optimal decision result. Considerations of some types of preference relations based on various fuzzy sets have been presented and investigated in previous studies; in this paper, we define the interval-valued hesitant multiplicative preference relation (IVHMPR) and the multiplicative consistency of IVHMPR. Based on these, we provide a detailed discussion on the connections between the interval-valued hesitant fuzzy preference relation (IVHFPR) and the IVHMPR. Then, we give a method to check the for unacceptable consistency of IVHFPR and IVHMPR, and improve them to make the consistency acceptable. Finally, an illustrative example of selecting the optimal treatment for a lung cancer patient is given to demonstrate our work in detail.

Improving the consistency of incomplete hesitant multiplicative preference relation

As it know, most decision processes are depends on preference relations as in the forms by comparing different alternatives over criteria. In the process of decision making, a decision maker may give his/her judgement by means of hesitant multiplicative preference relation (HMPR) using the scale of $$1/9-9$$ for hesitancy and uncertainty. Shortage of of time, experience and lack of the experts’ professional knowledge lead to form an incomplete hesitant multiplicative preference relation. In this paper we have defined a multiplicative transitive property of HMPR that preserves the hesitancy property. Another aim of this paper is to characterise two different approaches to calculate the missing element of incomplete HMPR from the incomplete one. The first approach involves two step, estimating step calculate the initial values of the missing element and the adjusting step minimizing the error. A linear programming model is developed in the second approach to calculate the missing values of incomplete hesitant multiplicative preference relation. The satisfaction degree and the acceptable consistency of complete HMPR is also checked. The whole procedure is explained with a suitable example.

Two Hesitant Multiplicative Decision-Making Algorithms and Their Application to Fog-Haze Factor Assessment Problem

Hesitant multiplicative preference relation (HMPR) is a useful tool to cope with the problems in which the experts utilize Saaty’s 1–9 scale to express their preference information over paired comparisons of alternatives. It is known that the lack of acceptable consistency easily leads to inconsistent conclusions, therefore consistency improvement processes and deriving the reliable priority weight vector for alternatives are two significant and challenging issues for hesitant multiplicative information decision-making problems. In this paper, some new concepts are first introduced, including HMPR, consistent HMPR and the consistency index of HMPR. Then, based on the logarithmic least squares model and linear optimization model, two novel automatic iterative algorithms are proposed to enhance the consistency of HMPR and generate the priority weights of HMPR, which are proved to be convergent. In the end, the proposed algorithms are applied to the factors affecting selection of fog-haze weather. The comparative analysis shows that the decision-making process in our algorithms would be more straight-forward and efficient.

Two regression methods for hesitant multiplicative preference relations with different consistencies

Multiplicative preference relation (MPR) is an efficient and widely used tool in describing the preferences of decision makers. The hesitant multiplicative preference relation (HMPR), as an extension of the MPR, is commonly used in collecting and representing all hesitant preferences and judgements of the decision makers. Considering that there usually appear some random or illogical preference degrees in the process of constructing the HMPRs, so it is necessary to derive a pithy preference relation (such as MPR) from the original HMPR through selecting the most proper and abandoning improper preference degrees, at the same time, the derived MPR is with the advantage of both terseness and the best consistency. Based on the multiplicative consistency measure of HMPRs, this paper proposes two regression methods to transform the HMPR into MPRs (called reduced MPRs). With the error analysis, the first proposed regression method provides some steps to not only extract the reduced MPRs from original HMPR but also calculate the consistencies of both the HMPR and the reduced MPRs. A case study reflects that the reduced MPR is with the highest consistency degree among those possible separated MPRs from HMPR. To apply this method to solve practical problems conveniently, a group decision-making procedure with hesitant multiplicative information is further given based on the first regression method. Time complexity comparison between our proposed method and an existing method indicates the effectiveness of our method. In addition, according to the weak consistency, we develop the second regression method and design an algorithm to obtain the reduced MPRs from the HMPR. Furthermore, we provide a method to check the weak consistency of the HMPR and repair the inconsistent one. Numerical examples verify that the second regression method proposed in this paper is an effective technique for checking the weak consistency and modifying the inconsistency HMPR to the one with weak consistency.

Group Decision-Making Model With Hesitant Multiplicative Preference Relations Based on Regression Method and Feedback Mechanism

The hesitant multiplicative preference relation (HMPR) was initially put forward in 2013. Utilizing the HMPR, the decision makers can give some possible preference values from the Saaty’s 1–9 scale for pairwise comparisons over alternatives. However, until now, there is little research on the consistency and consensus of HMPRs. In this paper, we focus on exploiting the regression method and the feedback mechanism to improve the consistency and consensus for HMPRs and developing an efficient group decision-making model with HMPRs. First, a regression method based on complete consistency is developed to reduce HMPRs to multiplicative preference relations (MPRs). After that, a novel consistency checking and revising method based on the threshold estimation method and the feedback mechanism is given to improve the consistency level of the reduced MPRs. Then, a group consensus index is put forward to calculate the deviation degree between the reduced MPR and the group MPR and it is utilized to develop a consensus reaching process based on the feedback mechanism to improve the group consensus level of the reduced MPRs. Next, a complete group decision-making model with HMPRs is developed to rank all the alternatives and select the best one. Finally, a numerical example with respect to the investment of shared bikes is presented to demonstrate the proposed group decision-making model and then we also compare our proposed model with the existing one.

An approach to group decision making with hesitant information and its application in credit risk evaluation of enterprises

Hesitant multiplicative preference relation (HMPR) contains much more comprehensive information than the traditional multiplicative preference relations. The HMPR is a useful tool to help the decision makers express their preferences in group decision making under uncertainty. The key of group decision making with the HMPR is to derive the priority weights from the HMPR. Thus, an efficient and practical priority method should be put forward so as to ensure the reasonability of the final decision result. In order to do that, in this paper, we first introduce the expected value and the geometric average value of hesitant multiplicative element (HME) which is the component of the HMPR. Then from different perspectives, we utilize the error-analysis technique to put forward three novel methods for the priorities of the HMPR, i.e., the expectation value method, the geometric average value method, and the multiplicative deviation method. We also investigate the relationships among these methods, and develop an approach to group decision making with the HMPR by using the methods and the possibility degree formula. Finally, by constructing the indicator system for credit risk evaluation of supply chain enterprises, we make a detailed case study concerning Lu-Zhou-Lao-Jiao (the well-known liquor enterprise in China) to demonstrate our approach.

Some Hesitant Multiplicative Aggregation Operators and Their Application in Group Decision Making with Hesitant Multiplicative Preference Relations

The hesitant multiplicative preference relation (HMPR) is a newly developed preference structure that uses an unsymmetrical scale (Saaty’s 1–9 scale) to express the decision makers’ (DMs’) preferences instead of the symmetrical scale (0.1–0.9 scale) in a hesitant fuzzy preference relation. This new preference relation is suitable for describing the situation in which the DMs are hesitant about several possible multiplicative values for the preference degrees over paired comparisons of alternatives. In this paper, we first define some new operational laws for hesitant multiplicative elements. Then, based on these operational laws, we develop two types of hesitant multiplicative aggregation operators for aggregating the hesitant multiplicative preference information, which we call the independent hesitant multiplicative aggregation operators and the correlative hesitant multiplicative aggregation operators. The independent hesitant multiplicative aggregation operators are based on the assumption that the aggregated arguments are independent, and the operators can be used to address situations where the preferences of the DMs are independent. The correlative hesitant multiplicative aggregation operators are developed to reflect the correlations of the aggregated arguments based on the Choquet integral and the power average, and the operators can be used to address the situation where the preferences of DMs are dependent on each other. We further give some desirable properties and special cases of the developed operators and apply these operators to develop several approaches to group decision making based on HMPRs. Finally, several practical examples are provided to illustrate the developed operators and approaches.

Fusion of heterogeneous incomplete hesitant preference relations in group decision making

AbstractIn this paper, we focus on the fusion of heterogeneous incomplete hesitant preference relations (including hesitant fuzzy preference relations and hesitant multiplicative preference relations) under group decision making settings. First, some simple formulae are developed to derive a priority weight vector from an incomplete hesitant fuzzy preference relation or an incomplete hesitant multiplicative preference relation based on the logarithmic least squares method. Based on the priority weight vector, an induced fuzzy or multiplicative preference relation can be derived for an incomplete hesitant preference relation. Moreover, the consistency indices of hesitant fuzzy preference relations and hesitant multiplicative preference relations are defined. Afterwards, an approach to group decision making based on incomplete hesitant fuzzy preference relations and incomplete hesitant multiplicative preference relations is developed to deal with group decision making problems with multiple decision organiz...

A decision support model for group decision making with hesitant multiplicative preference relations

The hesitant fuzzy preference relation (HFPR) was recently introduced by Zhu and Xu to allow the decision makers (DMs) to offer several possible preference values over two alternatives. In this paper, we use an asymmetrical scale (Saaty’s 1–9 scale) to express the decision makers’ preference information instead of the symmetrical scale that is found in a HFPR, and we introduce a new preference structure that is known as the hesitant multiplicative preference relation (HMPR). Each element of the HMPR is characterized by several possible preference values from the closed interval [1/9, 9]; thus, it can model decision makers’ hesitation more accurately and reflect people’s intuitions more objectively. Furthermore, we develop a consistency- and consensus-based decision support model for group decision making (GDM) with hesitant multiplicative preference relations (HMPRs). In this model, an individual consistency index is defined to measure the degree of deviation between an HMPR and its consistent HMPR, and a consistency improving process is designed to convert an unacceptably consistent HMPR to an acceptably consistent HMPR. Additionally, a group consensus index is introduced to measure the degree of deviation between the individual HMPRs and the group HMPR, and a consensus-reaching process is provided to help the individual HMPRs achieve a predefined consensus level. Finally, a numerical example is provided to demonstrate the practicality and effectiveness of the developed model.

Interval-valued Hesitant Multiplicative Preference Relations and Their Application to Multi-criteria Decision Making

Aims: The aim of this paper is to investigate interval -valued hesitant multiplicative preference relations and their application to multi-criteria decision making. Study Design: Based on pseudo-multiplication, we define some basic operations for the interval-valued hesitant multiplicative sets (IVHMSs) an d develop several aggregation operators for aggregating the interval-valued hesitant multiplicati ve information. Some desired properties and special cases of the developed operators are also inv estigated. Furthermore, we present a new preference structure named as the interval-valued hesitant multiplicative preference relation (IVHMPR), each element of which is an IVHMS, denoting all the possible interval multiplicative preference values offered by the decision mak ers for a paired comparison of alternatives. Place and Duration of Study: Interval-valued hesitant fuzzy set (IVHFS), recently introduced by Chen et al., permits the membership degree of an eleme nt to a set to be represented as several possible interval values. However, it is noted that IVHFS us es 0.1‐0.9 scale, which is inconsistent with some practical problems (e.g. the la w of diminishing marginal utility in economics). Methodology: We use the unsymmetrical 1‐9 scale instead of the symmetrical 0.1‐0.9 scale to express the membership degree information in the IVHFS and intr oduce the concept of intervalvalued hesitant multiplicative set (IVHMS). Results: An approach for multi-criteria decision making based on the interval-valued hesitant multiplicative preference relations (IVHMPRs) is developed a nd some numerical examples are provided to illustrate the developed approach. Conclusion: We compare the IVHMPR with the interval-valued hesitant preference relation (IVHPR) and the interval multiplicative preference relat ion (IMPR), and show the effectiveness and practicality of the IVHMPR. Original Research Article

MANAGING HESITANT INFORMATION IN GDM PROBLEMS UNDER FUZZY AND MULTIPLICATIVE PREFERENCE RELATIONS

In the process of group decision making (GDM), preference relations (e.g. fuzzy preference relation and multiplicative preference relation) are very popular tools to express decision makers' preferences, especially when a set of alternatives (or criteria) are compared. However, most of the existing preference relations don't consider the hesitancy information, which allows the decision makers to provide all the possible values when comparing two alternatives (or criteria), and is a common situation in daily life. In this paper, we first define the concept of hesitant fuzzy preference relation and study its properties, based on which we give an approach to GDM. Motivated by the multiplicative preference relation and the hesitant fuzzy set, we introduce the hesitant multiplicative set and develop a series of hesitant multiplicative aggregation operators. Hesitant multiplicative preference relation is also defined to provide decision makers a very useful tool to express their hesitant preferences over alternatives, and then is applied to GDM. Additionally, an example is given to illustrate our results.