Priority Weight Vector Research Articles

Two-stage group decision making methodology with hesitant fuzzy preference relations under social network: Multiplicative consistency determination and personalized feedback

Some social network group decision making (SNGDM) researches may overlook two issues: (1) the impact of consistency on decision reliability and decision makers’ (DMs’) status in social network, and (2) DMs’ personalization during consensus reaching process. In response to these two issues, a two-stage SNGDM model is proposed for hesitant fuzzy preference relations (HFPRs). In the first stage, multiplicative consistency determination and improvement of HFPRs are achieved. For complete and incomplete HFPRs, several programming models are constructed to classify consistency types. When it comes to inconsistent HFPRs, an inconsistency fixing method is designed to ensure the original HFPRs as much as possible. With the improvement of DMs’ consistency, their status in social network will be dynamically enhanced. In the second stage, two programming models are devised to elicit the reasonable individual priority weight vector in consistent HFPR. Further, minimum adjustment cost-based personalized feedback mechanism is invented to attain the optimal personalized adjustment coefficient and recommendation, and achieve the harmonious unity of individual personalization and group consensus. Finally, the superiority of our proposal is verified by comparative analysis and illustrative example.

Parameter-Free Interval Priority Weight Estimation Methods Based on Minimum Conceivable Ranges Under a Crisp Pairwise Comparison Matrix

Methods for interval priority weight estimation from a crisp pairwise comparison matrix were proposed in the interval analytic hierarchy process assuming the vagueness of human evaluation. The interval priority weights estimated by the conventional method do not reflect the intrinsic vagueness in the given pairwise comparison matrix (PCM). This paper proposes parameter-free methods based on minimal conceivable ranges for estimating interval priority weights from a crisp pairwise comparison matrix. The estimated interval priority weight vectors are required to satisfy (1) the potential reproducibility, (2) the normality, and (3) the preservation of the perfect consistent data. Estimation methods of interval priority weights are proposed based on the minimum possible range. We show those proposed methods satisfy the required three properties. The estimation problem of interval priority weights potentially has multiple solutions with which the associated interval PCMs are identical to one another. To make the further investigation simpler, we use an interval priority weight vector among multiple solutions such that the sum of the center values of interval priority weights is one. We compare the estimation methods of interval priority weights from the viewpoint of estimation accuracy by numerical experiments. Namely, by generating crisp pairwise comparison matrices randomly under true interval PCMs, we evaluate the accuracies of the estimated interval priority weight vectors by comparing the true interval priority weight vectors.

Stochastic group preference acceptability analysis for interval-valued multiplicative preference relations based on TODIM method

Among many decision problems, interval-valued multiplicative preference relation (IMPR) is widely utilized due to its ability to express uncertain information. A new approach to solve group decision making (GDM) with IMPRs is proposed in this paper, named stochastic group preference acceptability analysis with TODIM (SGPAA-TODIM) method, by combining TODIM method (an acronym in Portuguese of Interactive and multi-criteria Decision Making) with stochastic multi-criteria acceptability analysis (SMAA-2). It effectively circumvents information loss and considers the weight and risk preferences of experts. Firstly, the stochastic multiplicative preference relation is defined through stochastic simulation employing a certain density function, and its priority weight vector is determined using the logarithmic least squares method (LLSM). Then, the priority preference comprehensive matrix is proposed by extracting information of priority vectors. Moreover, the novel SGPAA-TODIM method is developed to analyze the stochastic parameter spaces, with the optimal rank determined based on the analysis of acceptability degrees associated with dominance rank. Finally, to demonstrate the validity and applicability of the proposed method, numerical examples are given.

Multiplicative Consistent q-Rung Orthopair Fuzzy Preference Relations with Application to Critical Factor Analysis in Crowdsourcing Task Recommendation

This paper presents a group decision-making (GDM) method based on q-rung orthopair fuzzy preference relations (q-ROFPRs). Firstly, the multiplicative consistent q-ROFPRs (MCq-ROFPRs) and the normalized q-rung orthopair fuzzy priority weight vectors (q-ROFPWVs) are introduced. Then, to obtain q-ROFPWVs, a goal programming model under q-ROFPRs is established to minimize their deviation from the MCq-ROFPRs and minimize the weight uncertainty. Further, a group goal programming model of ideal MCq-ROFPRs is constructed to obtain the expert weights using the compatibility measure between the ideal MCq-ROFPRs and the individual q-ROFPRs. Finally, a GDM method with unknown expert weights is solved by combining the group goal programming model and the simple q-rung orthopair fuzzy weighted geometric (Sq-ROFWG) operator. The effectiveness and practicality of the proposed GDM method are verified by solving the crucial factors in crowdsourcing task recommendation. The results show that the developed GDM method effectively considers the important measures of experts and identifies the crucial factors that are more reliable than two other methods.

Two-Sided Matching Decision Method of Electricity Sales Package Based on Disappointment Theory

Under the background of the “dual carbon” targets and continuously promoted power system reform, the application of a high proportion of renewable energy is becoming increasingly widespread. All sectors of society have greater demands for more appropriate electricity sales packages to guide the behavior of power users, which will in turn help conserve energy, reduce emissions, and finally achieve low-carbon operation of the power market economy. However, the existing methods of recommending electricity sales packages fail to provide appropriate and accurate recommendations for the users lacking preference information. Therefore, this paper proposes a two-sided matching decision-making method of an electricity sales package based on disappointment theory. First of all, according to the incomplete fuzzy preference relationship provided by the power user and the electricity sales package, the respective priority weight vector is calculated, and then the subjective satisfaction matrix of the power user and the electricity sales package is calculated. Next, the adjusted satisfaction matrix is calculated by adding the influence of the theory of elation and disappointment. Then, on the basis of the adjusted satisfaction matrix, an optimization model aiming at maximizing the total satisfaction of electric power customers and electricity sales packages is established, and the optimal stable matching model of electric power customers and electricity sales packages is obtained. Lastly, taking an industrial park in Zhejiang Province as an example, using the bilateral matching method proposed in this article, the optimal matching schemes for five electric power customers and six electricity sales packages is obtained, which shows the effectiveness and rationality of the two-sided matching decision-making method of electricity sales packages based on the disappointment theory.

Interval-valued intuitionistic fuzzy best-worst method with additive consistency

Best-worst method (BWM) has received extensive attention because of its fewer pairwise comparisons and maintenance of consistency. Extending it to various fuzzy environments has become a hot research spot. This paper initiates a new interval-valued intuitionistic fuzzy (IVIF) BWM with additive consistency. Firstly, an IVIF fuzzy preference relation (IVIFPR) is viewed as a directed graph with IVIF weights. An approach is offered to distinguish the best and worst criteria by the defined score out-degree and accuracy out-degree of criteria. The concept of IVIF reference comparisons is introduced. The additive consistency of IVIF reference comparisons is acquired by that of IVIFPR using its left and right intuitionistic fuzzy preference relations. We derive the relationship between the IVIF reference comparisons and the IVIF priority weight vector. Based on this relationship, a linear goal programming model is constructed to generate the IVIF priority weights. Two consistency indexes are analytically obtained and the consistency ratio is then defined to check the consistency of IVIF reference comparisons. For the unacceptably additive consistency, a new approach is put forward to enhance the consistency. Thereby, the proposed IVIF BWM is summarized step by step. Three examples of applications are furnished to demonstrate its effectiveness and validation.

Incomplete linguistic q-rung orthopair fuzzy preference relations and their application to multi-criteria decision making

With the development of social and economic research, the judgements that people deal with are full of fuzziness and vagueness. It is convenient to use the linguistic q-rung orthopair fuzzy numbers to offer the qualitative uncertain information. Consequently, in this paper, we mainly study linguistic q-rung orthopair fuzzy preference relations (Lq-ROFPRs) and apply them to a multi-criteria decision-making (MCDM) problem. The multiplicative consistency of Lq-ROFPRs is first introduced, then a consistency-based model is established to derive the normalized linguistic q-rung orthopair fuzzy priority weight vector. The concept of acceptably multiplicative consistency is further defined, and an optimization model is formulated to repair the unacceptably multiplicative consistent Lq-ROFPR. To cope with the incomplete Lq-ROFPR, another programming model is built to ascertain the missing values. The weights of criteria are determined by constructing a possibility degree matrix. Subsequently, an MCDM problem with incomplete Lq-ROFPRs is considered and a step-by-step algorithm is put forward. Finally, the proposed method is applied to three examples, and the advantages of this new approach are also demonstrated.

A Fractional Programming Model for Improving Multiplicative Consistency of Intuitionistic Fuzzy Preference Relations

In this paper, we propose a method that improves multiplicative consistency based on a fractional programming model to derive the normalized intuitionistic fuzzy priority weight vector from an intuitionistic fuzzy preference relation. To do so, a new definition is formulated that captures previous definitions for multiplicative consistency of intuitionistic fuzzy preference relations. A transformation formula is proposed to convert the normalized intuitionistic fuzzy priority weight vector into a multiplicative consistent intuitionistic fuzzy preference relation. By using the properties of some function, we construct a deviation matrix and prove that the elements in the intuitionistic fuzzy preference relation corresponding to the largest value in the deviation matrix is the most inconsistent. This method not only preserves a lot of the original preference judgments, but also reduce many operations in comparison with previous methods to improve multiplicative inconsistency. Several numerical examples are given to convince the proposed model and method.

Comments on “A prioritization approach of non-reciprocal fuzzy preference relations and its extension”

A recently published paper by Liu, Yang and Hu (2022) introduced a concept of non-reciprocal fuzzy preference relations (NrFPRs) and brought forward double-variance-based measuring formulas to compute additive consistency indices of NrFPRs and interval reciprocal fuzzy preference relations as well as built a double-variance-based minimization model with three parameters to acquire priority weight vectors from NrFPRs. This study employs five numerical illustrations to expose that some properties proposed by Liu, Yang and Hu (2022) are not true and the measuring formulas fail to obtain correct outcomes as well as the double-variance-based minimization model is based on an incorrect assumption. NrFPRs are redefined as particular cases of fuzzy preference relations. An additive transitivity equation is established to define additive consistency of NrFPRs and a square deviation based formula is developed to calculate additive consistency indices of NrFPRs. A quadratic programming model is set up to acquire priority weight vectors with interval fuzzy elements from NrFPRs. A closed-form solution based approach is put forward to acquire optimal priority weight vectors from additively consistent NrFPRs.

A method for improving the multiplicative inconsistency based on indeterminacy of an intuitionistic fuzzy preference relation

In this paper, we propose a method for improving the multiplicative inconsistency of an intuitionistic fuzzy preference relation (IFPR) without computing any model to derive an underlying priority weight's vector with respect to alternatives. For this, a necessary and sufficient condition for the IFPR to be multiplicative consistent is proposed and proved. Based on it, a ratio-based deviation identifying matrix that takes an accurate measurement of deviation of every element in the IFPR is constructed. We prove that the greater the value of the deviation matrix is, the more inconsistent is its corresponding element in the IFPR and based on it, a convergent iterative algorithm of improving the multiplicative consistency is presented. The algorithm uses the fact that all the indeterminacy degrees of the IFPR are never changed in the revising process of multiplicative inconsistency and as a result, the most inconsistent elements are uniquely determined by suitable elements in the IFPR, respectively. The proposed method makes a great difference from the previous methods that derived the underlying priority weight's vector with respect to alternatives based on the given IFPR for improving consistency. A numerical example is provided to show the feasibility and efficiency of the proposed method.

Group decision making based on multiplicative consistency and consensus of Pythagorean fuzzy preference relations

Pythagorean fuzzy sets (PFSs) become a useful tool to describe the complex cognition of decision makers (DMs). In this paper, Pythagorean fuzzy preference relations (PFPRs) whose elements are PFSs are used for group decision making (GDM). First, a novel multiplicative consistency of PFPRs is proposed. Then, a programming model is constructed to derive the priority weight vector of PFPRs. Then, an inconsistency-repairing method of PFPRs is designed. Moreover, a group consensus index to calculate the degrees of similarity among PFPRs is proposed and an iterative consensus reaching procedure with PFPRs is developed. By maximizing the group consensus level of PFPRs, a model is built to determine DMs’ weights. Furthermore, a new GDM method based on PFPRs is proposed. Finally, we offer an example to illustrate the proposed GDM method and complete a comparative analysis. The proposed GDM method outperforms the existing GDM methods for GDM in Pythagorean fuzzy environments.

Organizational and technological platform of construction

Introduction. At present, due to the active introduction of digital technologies, technological platforms are actively developing, the purpose of which is the implementation of various products. The presented study considers the organizational and technological platform of large-scale construction projects — a set of material and labor resources, organizational and technological solutions that allow, through their interaction, using modern information modeling technologies, to create construction objects with specified technical and economic characteristics.
 
 Materials and methods. For the formation of the organizational and technological platform, the methods of the goal tree and the analysis of hierarchies are used. A decomposition of a complex system is presented, presented at three levels of the goal tree and described using the analysis of hierarchies. At the top level, the goal of the project, followed by the problem (task), and at the last, lower level of the tree of goals are directions for solutions.
 
 Results. For the most adequate analytical description, it is proposed to use the hierarchy analysis method, which makes it possible to decompose the above factors that make up the goal tree and obtain specific criteria and alternatives inherent in large-scale construction projects, based on the principle of smart criteria formation. The creation of a mathematical model of an organizational and technological platform should be carried out using the possibility of constructing priority weight vectors that take into account the significance of all criteria, and each criterion is weighted based on these data. The integral assessment of indicators weighted for each criterion can be the final weight in the criterion structure. Detailed development of the mathematical apparatus for obtaining an integral estimate will be the subject of further research.
 
 Conclusions. The organizational and technological platform of large-scale projects in construction is a tool for the interaction of all participants in the implementation of such projects, aimed at the successful implementation of the set goals — achieving design indicators in terms of duration.

Decision-making models based on satisfaction degree with incomplete hesitant fuzzy preference relation

To address the situation where incomplete hesitant fuzzy preference relation (IHFPR) is necessary, this paper develops decision-making models taking into account decision makers’ satisfaction degree. First, the consistency measures, respectively, from the perspectives of additive and multiplicative consistent IHFPR are defined, which is based on the relationships of the IHPFRs and their corresponding priority weight vector. Second, two decision-making models are developed, respectively, in view of the proposed additive and multiplicative consistency measures. The main characteristic of the constructed models is they taking into account the decision makers’ satisfaction degree. The objective functions of the models are developed by maximizing the parameter of the satisfaction degree. Third, a square programming model is developed to obtain the decision makers’ weights by utilizing the optimal priority weight vectors information, the solution of the model is obtained by solving the partial derivatives of Lagrange function. Finally, a procedure for multi-criteria decision-making (MCDM) problems with IHFPRs is given, and an illustrative example in conjunction with comparative analysis is used to demonstrate the proposed models are feasible and efficiency for practical MCDM problems.

A signed distance based ranking approach with unknown fuzzy priority vectors for medical diagnosis involving interval type-2 trapezoidal pythagorean fuzzy preference relations

In many of our real life problems, we often come across situations where there is no information about the priority weights which make it difficult to analyze the objects under consideration. Instead of employing simple fuzzy sets, “interval type-2 trapezoidal pythagorean fuzzy preference relations (IT2TrPFPRs)” can be used which have better representational power and ability to cope with uncertain situations. The approach discussed in this article is an effective tool for managing multiple criteria group decision-making situations with completely unknown priority weights modeled as IT2TrPFPRs. To aggregate the opinion of multiple decision-makers, a hybrid averaging operation based on weighted averaging and ordered weighted averaging (OWA) operations is employed for a collective decision environment. To calculate the fuzzy priority weight vectors in case of completely unknown environment, we construct a non-linear optimization model. An integrated optimization model based on a new signed distance-based closeness coefficients approach is employed to determine the priority ranking of alternatives. Feasibility of the proposed technique is discussed with an implementation of patient centered medicine system for choosing the appropriate treatment method. Moreover, a comparative investigation with previous approaches is conducted to demonstrate the effectiveness of the given approach.

A new AHP‐prioritization method based on linear programming for crisp and interval preference relations

AbstractObtaining a priority (weight) vector is a crucial topic in the analytic hierarchy process (AHP). This paper proposes a novel prioritization approach based on linear programming (LP) for obtaining priorities in the AHP. We apply this approach to group decision making and modify it to include uncertainty in the decision maker's subjective judgments when the pairwise comparison matrix (PCM) elements are intervals. The proposed LP approach has a high potential in ranking the elements (decision criteria or alternatives). It is always feasible and produces interval priority vectors with positive lower and upper bounds for interval PCMs. These are important advantages of the proposed model in comparison to some existing models. To illustrate the potential applications and advantages of this approach, some numerical examples, including a multiple criteria decision making problem with a hierarchical structure, are examined.

Approaches for multicriteria decision-making based on the hesitant fuzzy best\u2013worst method

Hesitant fuzzy preference relations (HFPRs) have been widely applied in multicriteria decision-making (MCDM) for their ability to efficiently express hesitant information. To address the situation where HFPRs are necessary, this paper develops several decision-making models integrating HFPRs with the best worst method (BWM). First, consistency measures from the perspectives of additive/multiplicative consistent hesitant fuzzy best worst preference relations (HFBWPRs) are introduced. Second, several decision-making models are developed in view of the proposed additive/multiplicatively consistent HFBWPRs. The main characteristic of the constructed models is that they consider all the values included in the HFBWPRs and consider the same and different compromise limit constraints. Third, an absolute programming model is developed to obtain the decision-makers’ objective weights utilizing the information of optimal priority weight vectors and provides the calculation of decision-makers’ comprehensive weights. Finally, a framework of the MCDM procedure based on hesitant fuzzy BWM is introduced, and an illustrative example in conjunction with comparative analysis is provided to demonstrate the feasibility and efficiency of the proposed models.

A group decision making method considering both the consistency and consensus of intuitionistic multiplicative preference relations

This paper aims to develop a new group decision making (GDM) approach with intuitionistic multiplicative preference relations (IMPRs) by considering the consistency and consensus. Using the distance between a given IMPR and its corresponding underlying consistent IMPR, the concept of acceptably consistent IMPR is introduced, then an automatic algorithm is designed to repair the inconsistent IMPR to be of acceptable consistency. Meanwhile, each decision maker's consensus level is evaluated by the deviation between his/her individual IMPR and the group IMPR, and another algorithm for reaching acceptable level of consensus is provided. Moreover, the consensus improving process can guarantee that the modified IMPRs still be acceptably consistent, then the normalized intuitionistic multiplicative priority weight vector can be obtained from a mathematical programming model. A step-by-step algorithm based on the consistency and consensus of IMPRs is offered. Finally, two examples and the corresponding comparative analyses are presented to demonstrate the effectiveness of the proposed method.

A novel probabilistic linguistic decision-making method with consistency improvement algorithm and DEA cross-efficiency

Probabilistic linguistic term set (PLTS) is highly useful for decision-makers (DMs) to describe qualitative and uncertain information in the decision-making process. This paper proposes a novel probabilistic linguistic decision-making method with consistency improvement algorithm and data envelopment analysis (DEA) cross-efficiency. Firstly, we put forward the concept of order consistency of probabilistic linguistic preference relation (PLPR). The order consistency is helpful for DMs to make quick and efficient decision in certain situations. Then, based on the defined multiplicative consistency of PLPR, we develop a consistency improvement algorithm to transform the unacceptable multiplicative consistent PLPRs into the acceptable ones. Furthermore, a DEA model is established to derive the priority weight vector of alternatives from the acceptable multiplicative consistent PLPR. Meanwhile, for the alternatives that have equal priority weights, we use a DEA cross-efficiency model to further differentiate and obtain the final ranking of alternatives. Finally, a numerical example of emergency logistics distribution selection is given to illustrate the effectiveness and applicability of the proposed method.

Sustainable e-bike sharing recycling supplier selection: An interval-valued Pythagorean fuzzy MAGDM method based on preference information technology

Amid the trend of surging demand, e-bike sharing provides substantial convenience for travel but also causes a large number of abandoned bicycles, parking issues, and other problems, which create significant challenges to recycling management. In this regard, a Pythagorean fuzzy decision-making method is developed to determine the selection of recycling suppliers. First, the concept of interval-valued Pythagorean fuzzy preference relation (IVPFPR) is defined, and the most optimistic and pessimistic acceptably consistent PFPRs are extracted for a given IVPFPR. Subsequently, an acceptably consistent PFPR that integrates the attitude characteristics is derived, and the priority weight vector is computed by constructing an optimization model. Second, a subjective and objective weight method of experts considering trust degree is constructed, and the weighted quadratic operator is defined to aggregate IVPFPRs. The linguistic weights of attributes are quantified and standardized by operator and score function. Third, a decision-making method is proposed and applied to a sustainable e-bike sharing recycling supplier selection example to show the feasibility of the method. Finally, sensitivity analysis shows that a relatively strict threshold can be set to improve the consensus of experts in order to obtain more reliable results when managers lack the knowledge of experts. In addition, in the process of recycling, managers need to urge and cooperate with suppliers to improve their performance in pollution and emission, green image, and recovery capacity.

Geometric consistency of triangular fuzzy multiplicative preference relation and its application to group decision making

The triangular fuzzy multiplicative preference relation (TFMPR) has attracted the attention of many scholars. This paper investigates the geometric consistency of TFMPR and applies it to group decision making (GDM). Firstly, by introducing two parameters, a triangular fuzzy number is transformed into an interval. According to the geometric consistency of interval multiplicative preference relation (IMPR), the geometric consistency of TFMPR is defined. Then, two corresponding IMPRs are extracted from the TFMPR by programming models in the majority case and minority case, respectively. Using the constructed linear programming models, two interval priority weight vectors are obtained from the two extracted IMPRs, respectively. Combining two interval priority weight vectors, a linear programming model is established to derive the triangular fuzzy priority weights. Subsequently, the closeness degrees of alternatives by experts are defined to obtain the group utility indices and individual regret indices of alternatives. Then, the compromise indices of alternatives are calculated considering experts’ compromise attitude. By minimizing the compromise indices of alternatives, a multi-objective programming model is constructed to obtain experts’ weights. By aggregating the individual TFMPRs, the collective TFMPR is obtained to derive the triangular fuzzy priority weights. Using the arithmetic mean values, the ranking order of alternatives is generated. Therefore, a method is proposed to solve GDM with TFMPRs. Finally, a performance evaluation example of precise poverty alleviation is provided to illustrate the advantage of the proposed method.