Unacceptable Consistency Research Articles

Attitude- and cost-driven consistency optimization model for decision-making with probabilistic linguistic preference relation

Probabilistic linguistic term set (PLTS) conveys the preference for a linguistic term by assigning a probability distribution, which can more correctly and flexibly reflect the decision-maker (DM)’s evaluation information and make the decision outcome more scientific and rational. Probabilistic linguistic preference relation (PLPR) allows the DM to represent decision information for pairwise comparison of alternatives with PLTS, allowing a more comprehensive representation of the DM’s valid preferences. Consistency is essential in the preference decision-making process and directly influences the dependability of the final results. Therefore, this paper develops a minimum cost consistency adjustment mechanism based on attitudes and emotions for solving the decision-making problem with PLPR. Firstly, a new PLTS normalization model is proposed to handle the unknown information, considering the DM’s preferred attitude and emotional tone. Secondly, the expected parametric multiplicative consistency index for PLPR is defined, and parameter settings improve the flexibility and relevance of the consistency. Then, the concepts of linguistic hesitancy, probabilistic uncertainty, and probabilistic incompleteness are defined for the PLTS that can reflect the psychological characteristics of DMs’ hesitancy; they are then used to measure hesitation and unit consistency adjustment costs. Next, a new distance measure is established between the two PLTSs to measure the difference between the preference information. Subsequently, a direction-guided minimum cost consistency optimization model with unacceptable consistency is constructed for the PLPR. Finally, the usefulness of the suggested decision-making method with PLPR is tested using a numerical example of talent selection, and parametric sensitivity analysis and comparative analysis are utilized to show the benefits of the suggested methodology.

Consensus reaching with minimum adjustment and consistency management in group decision making with intuitionistic multiplicative preference relations

Intuitionistic multiplicative preference relation (IMPR) is an efficient tool for describing the paired comparisons between objectives in group decision making (GDM). Existing GDM methods with IMPRs overlook the hesitancy of IMPRs and lack a feedback mechanism of adjustment in the consensus reaching process (CRP). Thus, to ensure that the adjusted IMPRs can provide reliable information for decision making, the hesitancy measure for IMPRs is first incorporated into the process of enhancing consistency and consensus. A new consistency index for IMPRs is defined based on distance measure. Some goal programming models are constructed for enhancing consistency of IMPR. For a collection of IMPRs with unacceptable consistency and consensus, based on the optimization model and the identification rule and direction rule (IR-DR), an interactive iterative algorithm is designed for enhancing consistency and consensus, which can greatly save the completion time of consensus reaching and maintain primitive information to the maximum. Subsequently, a distance-based aggregation model is constructed to derive the group IMPR by considering different experts’ decision-making principles. Thereby, a new GDM method with IMPRs considering minimum adjustment is proposed. Moreover, the rationality and effectiveness of the proposed method are verified through a numerical example and some related simulation experiments. Meanwhile its merits are illustrated by comparison analyses.

Multi-criteria selection of the mix of generating plants in local energy systems based on a modified analytic hierarchy process

This paper presents a modification of the analytic hierarchy process in order to increase its efficiency for a multi-criteria comparison of mixes of generating plants in local energy areas during their development. The multi-criteria problem of selecting the most effective ratio of rated capacities is considered for power plants representing a single mix of generating plants during the development of a local energy area in the Khabarovsk Krai. The energy sources are represented by thermal, solar, wind and diesel power plants. The following estimation criteria for alternative solutions were accepted: levelized cost of electricity, ecological effeciency estimation; estimation of public opinion about the consequences involved with the establishment of power plants. In order to solve the multi-criteria problem, the analytic hierarchy process (AHP) was used. When using the original AHP for the set problem, a large quantity of alternatives at the stage of pairwise comparisons were found to represent a perceptible load on a decision maker. Thus, already during the estimation of 10 alternatives according to 5 criteria, decision makers should conduct 225 pairwise comparisons, which may eventually result in an unacceptable consistency of the results. In addition, this requires a procedure accounting for the uncertainty of the decision maker's preferences. The proposed solution represents a method of forming matrices of pairwise comparisons upon criteria. This method consists in generating an interval or fuzzy model of the decision maker's preferences for evaluating pairs of estimates according to the criterion. The proposed method was verified using a numerical example of solving the set problem. The obtained optimum mix of power plants consists of thermal, solar and diesel plants with a power of 30, 35 and 39 mW, respectively. The proposed method ensures a high consistency of the results obtained during alternative pairwise comparisons. In addition, the modified analytic hierarchy process takes into account the non-linear nature of the decision maker's preferences for estimating alternatives according to criteria.

Linguistic Z-number preference relation for group decision making and its application in digital transformation assessment of SMEs

In the new phase of technological revolution and industrial transformation, the digital transformation (DT) of small and medium-sized enterprises (SMEs) has become a critical strategic issue. Assessing the DT of SMEs as a crucial link can be viewed as a group decision making (GDM) problem with uncertain linguistic information. Recently, the linguistic Z-number (LZN) has demonstrated benefits in the representation of uncertain linguistic information due to the characteristics of simultaneous representation of linguistic evaluation and corresponding credibility. However, rather of providing direct evaluation, decision makers (DMs) of the DT assessment prefer to pairwise compare alternatives. To this end, we put forward linguistic Z-number preference relations (LZPRs) as an assessment tool for expressing DMs’ preferences and credibilities. Following that, novel operational rules of LZNs are proposed to correspond with LZPRs. After considering the difference between the importance of the two parts A and B in LZNs, we further propose the weighted Euclidean distance and the improved comparison method of LZNs. In addition, the additive consistency of the LZPR is discussed, as well as a consistency index for determining its consistency. The automatic improvement algorithm and the feedback improvement method are used to enhance the consistency of LZPRs that have unacceptable consistency. Furthermore, a GDM model of LZPRs is built using DMs’ combined weights based on the credibility and the novel linguistic Z-number weighted averaging (NLZWA) operator. Finally, a case study and comparative analysis of SMEs’ DT assessment demonstrate the effectiveness, flexibility and superiority of the proposed method.

Towards achieving consistent opinion fusion in group decision making with complete distributed preference relations

Belief distribution (BD) is a scheme of representing qualitative information with subjective uncertainty and imprecision. Distributed preference relation (DPR) extends BDs to the form of pairwise comparison by expressing the preferred, non-preferred, indifferent, and uncertain degrees of one decision alternative over another. However, previous studies on DPR only require comparison of adjacent alternatives, and consensus reaching is not considered fully in the decision making process. To solve this problem, a complete DPR model is presented in this paper to support group decision making (GDM). First, a consistency index is defined to measure the consistency level of the complete DPR representing experts’ judgments. Second, an automatic adjustment algorithm is proposed to improve the consistency of DPRs with unacceptable consistency to an acceptable level. Third, the evidential reasoning (ER) algorithm is utilized to aggregate all the DPRs together, and an optimization model is further constructed to generate experts’ weights, which maximizes the degree of consensus among experts. A GDM example is provided to illustrate the applicability and validity of the proposed DPR model, and comparative analysis demonstrates the potential of the proposed method in supporting real-world GDM problems.

Some Methods Considering Multiplicative Consistency and Consensus in Group Decision Making with Interval-Valued Intuitionistic Fuzzy Preference Relations

In decision making with interval-valued intuitionistic fuzzy preference relations (IVIFPRs), the consistency and its improvement are key issues. The aim of this paper is to investigate a new multiplicative consistency of IVIFPRs based on interval arithmetic, and propose goal programming approach to improve consistency and consensus. Based on analysis of the drawbacks of the existing consistency definitions of IVIFPR, new definition for multiplicative consistent IVIFPR is introduced. Based on the proposed multiplicative consistency and analysis of the relationship between interval-valued intuitionistic fuzzy weights and multiplicative consistent IVIFPR, goal programming-based model is developed for deriving interval-valued intuitionistic fuzzy weights from IVIFPR. To measure the multiplicative consistency degree of IVIFPRs, a consistency index is introduced and then an acceptable consistency is defined. For an IVIFPR with unacceptable consistency, a mathematical programming approach is proposed to improve its consistency. By considering the experts’ subjective and consensus weights, the experts’ comprehensive weights are determined. Some numerical examples illustrate the proposed models.

Multi-stage consistency optimization algorithm for decision making with incomplete probabilistic linguistic preference relation

Incomplete probabilistic linguistic term sets (InPLTSs) can effectively describe the qualitative pairwise judgment information in uncertain decision-making problems, making them suitable for solving real decision-making problems under time pressure and lack of knowledge. Thus, in this study, an optimization algorithm is developed for preference decision-making with the incomplete probabilistic linguistic preference relation (InPLPR). First, to fully investigate the ability of InPLTSs to express uncertain information, they are divided into two categories. Then, a two-stage mathematical optimization model based on an expected multiplicative consistency for estimating missing information is constructed, which can obtain the complete information more scientifically and effectively than some exiting methods. Subsequently, for the InPLPR with unacceptable consistency, a multi-stage consistency-improving optimization model is proposed for improving the consistency of the InPLPR by minimizing the information distortion and the number of adjusted linguistic terms, which can also minimize the uncertainty of the relationship as small as possible. Afterward, to rank all the alternatives, a mathematical model for deriving the priority weights of the alternatives is constructed and solved, which can obtain the priority weight conveniently and quickly. A decision-making algorithm based on the consistency of the InPLPR is developed, which involves estimating missing information, checking and improving the consistency, and ranking the alternatives. Finally, a numerical case involving the selection of excellent students is presented to demonstrate the application of the proposed algorithm, and a detailed validation test and comparative analysis are presented to highlight the advantages of the proposed algorithm.

Distance-Based Consistency Measure and Priority Weights of Best-Worst Multi-Criteria Decision Making Method

Best-Worst method (BWM) is a new multi-criteria decision-making method based on pairwise comparisons, but only the comparisons concerning the best and the worst alternatives or criteria. This method shows some significant advantages in the simplicity with a less requirement of comparison data and reliability with better consistency. This paper proposes a new consistency measure method based on the distance of the vectors of reference comparisons in BWM because the difference of the preference in two vectors directly affects the reliability of results. Through the establishment of the threshold of consistency ratio, we supplement the definition of satisfactory consistency of the comparisons in BWM. With comparisons satisfying the acceptable consistency, we use linear programming models to find all possible priority weights between the preferences given by decision maker and derive interval weights. For comparisons with unacceptable consistency, another approach is presented to find the interval weights meeting the consistent requirement. At last, several examples are used to illustrate the details of process.

Approach to the consistency and consensus of Pythagorean fuzzy preference relations based on their partial orders in group decision making

<p style='text-indent:20px;'>Although intuitionistic fuzzy preference relations have become powerful techniques to express the decision makers' preference information over alternatives or criteria in group decision making, some limitations of them are pointed out in this paper, then they are overcame by developed the group decision making with Pythagorean fuzzy preference relations (PFPRs). Specially, we provide a partial order on the set of all the PFPRs, based on which, a deviation measure is defined. Then, we check and reach the acceptably multiplicative consistency and consensus of PFPRs associated with the partial order and mathematical programming. Concretely, acceptably multiplicative consistent PRPRs are defined by the deviation between a given PFPR and a multiplicative consistent PFPR constructed by a normal Pythagorean fuzzy priority vector. Then acceptable consensus of a collection of PFPRs is defined by the deviation of each PFPR and the aggregated result from symmetrical Pythagorean fuzzy aggregation operators. Based on which, a method which can simultaneously modify the unacceptable consistency and consensus of PFPRs in a stepwise way is provided. Particularly, we also prove that the collective PFPR obtained by aggregating several individual acceptably consistent PFPRs with various symmetric aggregation operators is still acceptably consistent. Then, a procedure is provided to solve group decision making with PRPRs and a numerical example is given to illustrate the effectiveness of our method.

A consistency and consensus-based method for group decision making with hesitant fuzzy linguistic preference relations

Hesitant fuzzy linguistic preference relations (HFLPRs) can well denote the decision makers (DMs)’ hesitant qualitative comparison judgments. Considering that consistency and consensus are two key issues in group decisionmaking (GDM), this paper discusses GDM with HFLPRs within the context of multiplicative consistency and consensus. First, a multiplicative consistency concept for linguistic preference relations (LPRs) is presented, and a method for obtaining the priority weight vectors of LPRs is offered. Then, a method of deriving the priority weight vector of a HFLPR is proposed. Furthermore, the consistency index of a HFLPR is defined, an acceptable multiplicative consistency definition for HFLPRs is offered. Subsequently, for a HFLPR which has unacceptable consistency, an algorithm to improve its consistency level is developed. With respect to GDM with HFLPRs, a method to calculate the DMs’ weights is provided. Then, individual HFLPRs are integrated into a collective HFLPR. After that, a consensus index is defined, and an algorithm to improve the consensus is proposed that can guarantee the multiplicative consistency of individual HFLPRs. Moreover, an automatic algorithm to GDM with HFLPRs is developed. Lastly, we apply the new methods to two practical problems to verify their operability and applicability.

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.

Group decision making with double hierarchy hesitant fuzzy linguistic preference relations: Consistency based measures, index and repairing algorithms and decision model

Group decision making, refers to inviting a group of decision makers to evaluate, prioritize or select the optimal one among some available alternatives in the actual decision making process. Considering that the double hierarchy hesitant fuzzy linguistic term set can describe natural languages clearly, in this paper, we define the concept of double hierarchy hesitant fuzzy linguistic preference relation (DHHFLPR) and propose some additive consistency measures. To judge whether a DHHFLPR is of acceptable consistency or not, we introduce a consistency index, and develop some novel threshold values for judging whether a DHHFLPR is of acceptable consistency or not. Furthermore, we develop two consistency repairing algorithms based on the automatic improving method and the feedback improving method respectively, to improve the DHHFLPR with unacceptable consistency. Additionally, a method is set up to deal with group decision making problems with double hierarchy hesitant fuzzy linguistic preference information. Finally, the proposed method is validated by a case study that is used to evaluate the water resource situations of some important cities in Sichuan Province, and some comparative analyses are given to show the efficiency of the proposed method.

Deriving the priority weights from probabilistic linguistic preference relation with unknown probabilities.

Generally, the probabilistic linguistic term set (PLTS) provides more accurate descriptive properties than the hesitant fuzzy linguistic term set does. The probabilistic linguistic preference relation (PLPR), which is applied to deal with complex decision-making problems, can be constructed for PLTSs. However, it is difficult for decision makers to provide the probabilities of occurrence for PLPR. To deal with this problem, we propose a definition of expected consistency for PLPR and establish a probability computing model to derive probabilities of occurrence in PLPR with priority weights for alternatives. A consistency-improving iterative algorithm is presented to examine whether or not the PLPR is at an acceptable consistency. Moreover, the consistency-improving iterative algorithm should obtain the satisfaction consistency level for the unacceptable consistency PLPR. Finally, a real-world employment-city selection is used to demonstrate the effectiveness of the proposed method of deriving priority weights from PLPR.

A Consistency and Consensus-Based Goal Programming Method for Group Decision-Making With Interval-Valued Intuitionistic Multiplicative Preference Relations.

Interval-valued intuitionistic multiplicative preference relations (IVIMPRs) form a suitable conceptual framework to represent and process simultaneously uncertain preferred and nonpreferred judgments of decision makers (DMs). The focus of this paper is on group decision-making (GDM) problems realized with IVIMPRs. First, a consistency index is introduced to evaluate the consistency degree for intuitionistic multiplicative preference relations (IMPRs), and a consistency optimization approach is presented to jointly improve the consistency degrees of several IMPRs that do not satisfy the predefined consistency threshold. Then, a consistency definition and an acceptable consistency definition for IVIMPRs are established by splitting an IVIMPR into two IMPRs. For several IVIMPRs with unacceptable consistency, a goal program-based approach is proposed to simultaneously improve their consistency. Subsequently, by minimizing the degree to which the opinions of individual DMs deviate from those of the group, a maximum consensus-based goal program is established to determine the DMs' weights. Furthermore, an aggregation approach is applied to integrate individual IVIMPRs into a collective one. A linear program is then built to determine the interval-valued intuitionistic multiplicative priority weights of alternatives coming from the collective IVIMPR. A consistency-based GDM algorithm is proposed. Finally, a practical example is offered to show the application of the new algorithm, and a comparative analysis is presented to highlight the advantages of the new method.

A group consensus decision support model for hesitant 2-tuple fuzzy linguistic preference relations with additive consistency

Hesitant fuzzy linguistic term sets suit the modelling of qualitative setting so that experts provide several possible linguistic values than a single term for an indicator, alternative, variable, etc. This paper proposes a consistency and consensus based decision support model for group decision m aking with hesitant 2-tuple fuzzy linguistic preference relations (H2TFLPRs). A revised definition of hesitant fuzzy linguistic preference relation (HFLPR) is presented. In order to measure the consensus degrees, two indexes, an individual to group consensus index (ICI) and a group consensus index (GCI) are introduced, respectively. A consistency control process is developed to transform an unacceptable consistency H2TFLPR to an acceptable one. Then, an iterative group consensus reaching process is designed to conduct the decision makers (DMs) to achieve the predefined consensus level. In the consistency and consensus improving process, both the feedback mechanism and no-feedback mechanism are devised to conduct the DMs how to modify their original judgments, and both mechanisms can retain the DMs’ original information as much as possible. Finally, a numerical example is illustrated to demonstrate the effectiveness of the proposed method.

The additive consistency measure of fuzzy reciprocal preference relations

Fuzzy reciprocal preference relations (FPR) are one of the most common preference relations which decision makers (DMs) express their comparison information in decision making, and the consistency of preference relations is an important step for reasonable and reliable decision making. Based on the concept of deviation between two matrices, we develop some consistency measures for FPRs to ensure that the DMs are being neither random nor illogical. A consistency index $$(CI)$$ and the threshold $$(\overline{CI})$$ of FPRs are defined to measure whether a FPR is of acceptable consistency. For FPRs with unacceptable consistency, an optimization method and two iterative algorithms are presented to improve its consistency and the process terminates until the $$CI$$ is controlled within the threshold $$\overline{CI}.$$ Furthermore, one of algorithms is extended to handle group decision making (GDM) of FPRs. Finally, two examples and comparative analysis are furnished to demonstrate the effectiveness of the developed methods.

Usefulness of MRI-CBCT image registration in the evaluation of temporomandibular joint internal derangement by novice examiners

To assess whether novice examiners can more reliably determine temporomandibular joint (TMJ) disk derangements with the use of fused images of magnetic resonance imaging and cone beam computed tomography (MRI-CBCT) compared with MRI images alone. Thirty dental students with minimal exposure to TMJ imaging received a 30-minute calibration session in which TMJ diagnostic imaging and the normal anatomy of the TMJ internal derangement were explained. The students evaluated the disk positions of 16 TMJs in two sets of images (MRI images alone and MRI-CBCT images) randomly and independently. The disk positions evaluated by two experienced radiologists were used for comparison. The internal consistency among all students improved from an unacceptable consistency (α=0.40) with MRI images alone to a good consistency (α=0.84) with MRI-CBCT images. The agreement in evaluation between the students and the radiologists improved from a poor agreement with MRI images alone (k mean=0.07±0.12) to a moderate agreement with MRI-CBCT images (k mean=0.55±0.25). This improvement in the agreement was significant (P<.001). The use of fused MRI and CBCT images to visualize the TMJ in a single display significantly improved the examiners' reliability and accuracy of assessment of disk positions. The improvement of novice readers in assessing the disk positions highlights the potential use of MRI-CBCT image fusion as an educational tool.

A novel method for group decision making with interval-valued Atanassov intuitionistic fuzzy preference relations

This paper investigates the group decision making (GDM) problems with interval-valued Atanassov intuitionistic fuzzy preference relations (IV-AIFPRs) and develops a novel method for solving such problems. A new consistency index of an Atanassov intuitionistic fuzzy preference relation (AIFPR) is introduced to judge the consistency of an AIFPR and then a convergent iterative Algorithm I is designed to repair the consistency of an AIFPR with unacceptable consistency. Subsequently, the consistency and acceptable consistency of an IV-AIFPR are defined through separating an IV-AIFPR into two AIFPRs. Based on Algorithm I, a new iterative Algorithm II is devised to repair the consistency of an IV-AIFPR with unacceptable consistency. Afterwards, to determine decision makers’ (DMs’) weights objectively, an optimization model is established by minimizing the deviations between each individual IV-AIFPR and the collective one. This model is skillfully transformed into a linear goal program to resolve sufficiently considering different principles of decision making. A linear program is built to derive interval-valued Atanassov intuitionistic fuzzy (IVAIF) priority weights of alternatives. Then, a TOPSIS (technique for order performance by similarity to an ideal solution) based approach is proposed to rank such IVAIF priority weights. Thereby, a method for GDM with IV-AIFPRs is put forward. At length, a practical example of a virtual enterprise partner selection is provided to illustrate the feasibility and validity of the proposed method.

Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations

In group decision making (GDM) with intuitionistic fuzzy preference relations (IFPRs), the consistency and consensus are two key issues. This paper develops a novel method for checking and improving the consistency of individual IFPRs and the consensus among experts. To measure the consistency degree of IFPRs, a consistency index is introduced and then an acceptable consistency is defined. For an IFPR with unacceptable consistency, a mathematical programming approach is developed to improve its consistency. To evaluate the consensus degree among experts, a consensus measure is presented by the proximity degree between one expert and other experts. When several individual IFPRs are unacceptable consistent or consensus is unacceptable, a goal program is built to improve the consistency and consensus simultaneously. By the consistency and proximity degrees of individual IFPRs, experts’ objective weights are determined. Combining the experts’ subjective weights, the experts’ comprehensive weights are derived. Then, an intuitionistic fuzzy geometric weighted mean (IFGWM) operator is proposed to integrate individual IFPRs into a collective one. Moreover, an attractive property is proved that the collective IFPR is acceptable consistent if all individual IFPRs are acceptable consistent. Two examples are provided to illustrate the validity of the proposed method.

A consistency model for group decision making problems with interval multiplicative preference relations

The main aim of this paper is to present a consistency model for interval multiplicative preference relation (IMPR). To measure the consistency level for IMPR, a referenced consistent IMPR of a given IMPR is defined, which has the minimum logarithmic distance from the given IMPR. Based on the referenced consistent IMPR, the consistency level of an IMPR can be measured and an IMPR with unacceptable consistency can be adjusted by a proposed algorithm such that the revised IMPR is of acceptable consistency. A consistency model for group decision making (GDM) problems with IMPRs is proposed to obtain the collective IMPR with highest consistency level. Numerical examples are provided to illustrate the validity of the proposed approaches in decision making.