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Multi-criteria decision-making (MCDM) entails a heterogeneous decision-making problem, which poses challenges for decision-makers (DMs) in generating an optimal solution. To address this, we have proposed a heterogeneous evaluation framework. First, a novel picture fuzzy cross-entropy measure was defined with the simultaneous consideration of uncertainty and hesitancy of picture fuzzy information, overcoming the shortcomings of the existing cross-entropy measure in relation to its validity and properties. Next, an optimisation-model for determining the objective weights of criteria was constructed based on the proposed closeness measurement and the step-wise weight assessment ratio analysis (SWARA) method. This model was constituted from both objective and subjective perspectives under the circumstance of completely unknown criterion information. Additionally, the conventional Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS) method was extended to the picture fuzzy environment, and a normalisation method was developed to transform heterogeneous judgments, including linguistic terms, interval numbers, and picture fuzzy numbers, into a unified representation form. A heterogeneous picture fuzzy SWARA-MARCOS evaluation framework was then established and used to solve a military equipment supplier selection problem. The results demonstrated the validity and feasibility of the proposed evaluation framework, while sensitivity, comparative, and complexity analyses demonstrated the robustness and superiority of it. [ABSTRACT FROM AUTHOR]

With the generalisation of intuitionistic fuzzy sets (IFSs), picture fuzzy sets (PFSs) have been developed based on membership, neutral membership, and non-membership degrees. Compared with IFSs, PFSs can more accurately represent uncertainty in real-world decision-making problems. Recently, the research on decision-making theories and methods under picture fuzzy environments has been rapidly developing. Therefore, this manuscript presents a systematic review of picture fuzzy decision-making theories and methods, including the context in which this field was developed, the current status and advancements of this field, and the main research results obtained with the picture fuzzy information. First, this review introduces the development process of PFSs and the current status of corresponding theories. Next, the basic theories based on PFSs, including operations rules, measures, and aggregation operators are introduced. Afterward, this review summarises the current research on multi-criteria decision-making (MCDM), multi-criteria group decision-making (MCGDM), and large-scale group decision-making (LSGDM) methods with picture fuzzy information. Finally, future research directions of picture fuzzy decision-making theories and methodologies are discussed. [ABSTRACT FROM AUTHOR]

The technology of wind power is being widely developed worldwide. Ensuring the reliable operation of wind turbine systems is of significance. A popular tool to identify the potential risk of an engineering system is the failure mode and effect analysis (FMEA) method. However, when conducting a FMEA in a complex and uncertain environment, experts may find it challenging to select an appropriate linguistic term for the evaluation. The probabilistic linguistic term sets (PLTSs) are a useful fuzzy set to help experts in describing their assessment. However, different experts may assign different semantic values to the same linguistic terms, and this aspect has not been extensively examined in the existing studies. Therefore, considering the psychological behaviour of experts and the semantics of linguistic terms in the risk ranking process, an improved FMEA based on the probabilistic linguistic information and TODIM (an acronym in Portuguese of interactive and multicriteria decision making) method was developed to identify the risks in wind turbine systems. Furthermore, to demonstrate the utility of the proposed model, it was applied in a floating offshore wind turbine system. The effectiveness and the validity of the model were verified by comparing with some other methods. [ABSTRACT FROM AUTHOR]

In the multi‐criteria group decision‐making (MCGDM) problems with great uncertainty, making full use of participants' evaluation information could help improve the accuracy and reliability of decision results. Probabilistic linguistic term set (PLTS) is an effective tool to represent qualitative data and can fully express the hesitation and preference of decision makers. Therefore, this paper aims to propose an MCGDM method based on PLTSs. In the proposed method, the projection of PLTSs is explored to measure the distance and angle differences between two objects, and Bayesian best–worst method (Bayesian BWM) is used to determine the aggregated final weights of criteria. Besides, the elimination and choice translating reality III (ELECTRE III) method combined with distillation algorithm deals with the projection of PLTSs to obtain the alternatives' ranking of each decision maker. Then, the weighted convex median voting rule is developed to integrate the rankings results regarding all decision makers, which can solve the conflict of ranking results among experts and ensure that the comprehensive ranking results are reasonable and practical. Finally, a case study of health‐care waste management is designed and comparative analyses are implemented to show the effectiveness and advantages of the proposed method. [ABSTRACT FROM AUTHOR]

Solar energy, as a major and least-cost renewable resource, has attracted extensive attention of experts and scholars. However, the establishment of the power station is time-consuming and costly. And once selected, it is difficult to change. So it is crucial to choose the appropriate site of power station. This paper combines data analysis with multi-criteria group decision-making to solve this problem. First of all, K-means clustering method is selected to process the data according to the characteristics of the data. Secondly, the results obtained by K-means method are represented by probabilistic linguistic term sets. Thirdly, Bonferroni Mean operator is used to adjust the weight of the criterion, which considers the consensus among experts. Fourthly, Technique for Order Preference by Similarity to Ideal Solution method is employed to rank the alternatives and select the best one. Finally, sensitivity analysis, comparison analysis and simulation are carried out to further confirm the robustness and advantage of the model. This model can help decision makers to better understand the basic situation of power station sites, make the right decisions, and improve some candidate sites according to the results. [ABSTRACT FROM AUTHOR]

Single-valued neutrosophic hesitant fuzzy sets (SVNHFSs) have recently become a subject of great interest for researchers, and have been applied widely to multi-criteria decision-making (MCDM) problems. In this paper, the single-valued neutrosophic hesitant fuzzy geometric weighted Choquet integral Heronian mean operator, which is based on the Heronian mean and Choquet integral, is proposed, and some special cases and the corresponding properties of the operator are discussed. Moreover, based on the proposed operator, an MCDM approach for handling single-valued neutrosophic hesitant fuzzy information where the weights are unknown is investigated. Furthermore, an illustrative example to demonstrate the applicability of the proposed decision-making approach is provided, together with a sensitivity analysis and comparison analysis, which proves that its results are feasible and credible. [ABSTRACT FROM AUTHOR]

In this paper, an extension Elimination and Choice Translating Reality (ELECTRE) method is introduced to handle multi-valued neutrosophic multi-criteria decision-making (MCDM) problems. First of all, some outranking relations for multi-valued neutrosophic numbers (MVNNs), which are based on traditional ELECTRE methods, are defined, and several properties are analyzed. In the next place, an outranking method to deal with MCDM problems similar to ELECTRE III, where weights and data are in the form of MVNNs, is developed. At last, an example is provided to demonstrate the proposed approach and testify its validity and feasibility. This study is supported by the comparison analysis with other existing methods. [ABSTRACT FROM AUTHOR]

In this paper, multi-criteria decision-making (MCDM) problems based on the qualitative flexible multiple criteria method (QUALIFLEX), in which the criteria values are expressed by multi-valued neutrosophic information, are investigated. First, multi-valued neutrosophic sets (MVNSs), which allow the truth-membership function, indeterminacy-membership function and falsity-membership function to have a set of crisp values between zero and one, are introduced. Then the likelihood of multi-valued neutrosophic number (MVNN) preference relations is defined and the corresponding properties are also discussed. Finally, an extended QUALIFLEX approach based on likelihood is explored to solve MCDM problems where the assessments of alternatives are in the form of MVNNs; furthermore an example is provided to illustrate the application of the proposed method, together with a comparison analysis. [ABSTRACT FROM AUTHOR]

As a variation of fuzzy sets and intuitionistic fuzzy sets, neutrosophic sets have been developed to represent uncertain, imprecise, incomplete and inconsistent information that exists in the real world. Simplified neutrosophic sets (SNSs) have been proposed for the main purpose of addressing issues with a set of specific numbers. However, there are certain problems regarding the existing operations of SNSs, as well as their aggregation operators and the comparison methods. Therefore, this paper defines the novel operations of simplified neutrosophic numbers (SNNs) and develops a comparison method based on the related research of intuitionistic fuzzy numbers. On the basis of these operations and the comparison method, some SNN aggregation operators are proposed. Additionally, an approach for multi-criteria group decision-making (MCGDM) problems is explored by applying these aggregation operators. Finally, an example to illustrate the applicability of the proposed method is provided and a comparison with some other methods is made. [ABSTRACT FROM AUTHOR]

Hesitant fuzzy sets (HFSs), an extension of fuzzy sets, are considered to be useful in solving decision making problems where decision makers are unable to choose between several values when expressing their preferences. The purpose of this paper is to develop two hesitant fuzzy multi-criteria decision making (MCDM) methods based on prospect theory (PT). First, the novel component-wise ordering method for two hesitant fuzzy numbers (HFNs) is defined; however, this method does not consider the length of the two HFNs. Second, by utilizing the directed Hausdorff distance between two imprecise point sets, the generalized hesitant Hausdorff distance is developed, which overcomes the shortcomings of the existing distance measures. Third, based on the proposed comparison method and distance, as well as PT, the extended TODIM and Preference Ranking Organization Method for Enrichment Evaluations (PROMETHEE) approaches are developed in order to solve MCDM problems with hesitant fuzzy information. Finally, a practical example is provided to illustrate the pragmatism and effectiveness of the proposed approaches. Sensitivity and comparison analyses are also conducted using the same example. The findings indicate that the proposed methods do not require complicated computation procedures, yet still yield a reasonable and credible solution. [ABSTRACT FROM AUTHOR]

In this paper, the cross-entropy of intuitionistic hesitant fuzzy sets (IHFSs) is developed by integrating the cross-entropy of intuitionistic fuzzy sets (IFSs) and hesitant fuzzy sets (HFSs). First, several measurement formulae are discussed and their properties are studied. Then, two approaches, which are based on the developed intuitionistic hesitant fuzzy cross-entropy, are proposed for solving multi-criteria decision-making (MCDM) problems within an intuitionistic hesitant fuzzy environment. For both methods, an optimisation model is established in order to determine the weight vector for MCDM problems with incomplete information on criteria weights. Finally, an example is provided in order to illustrate the practicality and effectiveness of the proposed approaches. [ABSTRACT FROM PUBLISHER]

Objective To study the associations of pedestrian injuries with age, income and educational level in Shanghai and to analyze the relative disease burden. Methods Information on pedestrian-related cases and deaths were collected from 494 hospitals and mortality registry systems from 1992 to 2010, and a multistage cluster sampling survey conducted in 2006. Logistic regression model was used in the analyses. Results The age group of 5-9 had the highest mortality and morbidity among children. Mortality increased obviously among those aged 60 or above. Individuals with an educational level under the primary school and with the lower family average income were more likely to suffer pedestrian-related injuries. Multivariate Logistic analysis demonstrated that lower income and lower educational level increased the risk of pedestrian injuries with the odds ratio of 1.40 (95% CI: 1.15-1.71) and 1.70 (95% CI: 1.20-2-40), respectively. About 13.54% of the share of GDP for the healthcare, social security and welfare industries in Shanghai was occupied by the burden of pedestrian-related injuries in 2006. Conclusion Pedestrian-related injury has inverse association with victims' income and educational level. Children of 5-9 years old and adults over 60 with lower educational level and lower monthly income are the target persons to be intervened. [ABSTRACT FROM AUTHOR]

In this paper, a new outranking approach for multi-criteria decision-making (MCDM) problems is developed in the context of a simplified neutrosophic environment, where the truth-membership degree, indeterminacy-membership degree and falsity-membership degree for each element are singleton subsets in [0,1]. Firstly, the novel operations of simplified neutrosophic sets (SNSs) and relational properties are developed. Then some outranking relations for simplified neutrosophic number (SNNs) are defined, based on ELECTRE, and the properties within the outranking relations are further discussed in detail. Additionally, based on the outranking relations of SNNs, a ranking approach is developed in order to solve MCDM problems. Finally, two practical examples are provided to illustrate the practicality and effectiveness of the proposed approach. Moreover, a comparison analysis based on the same example is also conducted. [ABSTRACT FROM AUTHOR]

This paper proposes three types of aggregation operators for fuzzy numbers as an extension of the traditional ordered weighted averaging (OWA) operator; namely the fuzzy ordered weighted averaging (F-OWA) operator, the weighted fuzzy ordered weighted averaging (WF-OWA) operator and the ordered weighted fuzzy ordered weighted averaging (OWF-OWA) operator. Moreover, some corresponding properties of the three operators are investigated in detail. Furthermore, two fuzzy multi-criteria decision-making (MCDM) approaches, respectively based on the WF-OWA operator and the OWF-OWA operator, are developed. Finally, an example is given to show their feasibility and validity, together with a comparison analysis. [ABSTRACT FROM AUTHOR]

Abstract: Thioglycolic acid (TGA) capped CdTe quantum dots (QDs) with the diameter of 2–3nm were synthesized. The interaction between CdTe QDs and Nile blue (NB) was investigated by ultraviolet–visible (UV–vis) absorption, resonance Rayleigh scattering (RRS) and fluorescence spectroscopy. UV–vis absorption spectrum of CdTe QDs and NB obviously changed, showing that CdTe QDs could associate with NB to form a new complex. At pH 6.8, NB effectively quenched the fluorescence of CdTe QDs. It was proved that the fluorescence quenching of CdTe QDs by NB was mainly result of the formation of CdTe QDs–NB complex, electrostatic attraction and hydrophobic forces played a major role in stabilizing the complex. The binding molar ratio of CdTe QDs and NB was 5:1 by a mole-ratio method. The interaction between CdTe QDs and NB lead to the remarkable enhancement of RRS and the enchantments were in proportional to the concentration of NB in a certain range. The mechanism of the interaction between CdTe QDs and NB, reasons for the enhancement of RRS intensity were also discussed. The obtained results suggested the more satisfactory mechanism for the interaction between CdTe QDs and NB. [Copyright &y& Elsevier]

This paper aims to construct a sustainability evaluation framework based on the extended VIsekriterijumska optimizacija i KOmpromisno Resenje (VIKOR) method under a picture fuzzy environment. First, a sustainability evaluation index system for water environment treatment public-private-partnership (WET-PPP) projects is constructed from the perspectives of the economy, engineering, environment, management, and society, including five criteria and 20 sub-criteria. The constructed index system considers engineering sustainability and management sustainability, and can thus more comprehensively evaluate the "sustainability" of the project and meet the requirements of the new economic environment. Second, a corresponding method for the sustainability evaluation of WET-PPP projects is proposed based on the picture fuzzy similarity-based VIKOR method. The main advantages of the proposed method include the following: (1) the sustainability evaluation process for WET-PPP projects involves linguistic terms and picture fuzzy numbers (PFNs), which can not only more comprehensively describe the uncertainty of human cognition and decision-making processes in the form of positive, neutral, and negative membership functions, but can also reduce the difficulty for decision-makers (DMs) to provide evaluation information; (2) the proposed evaluation method combines picture fuzzy similarity and the traditional VIKOR method, and can therefore not only obtain compromise solutions but can also ensure the stability and feasibility of the evaluation results. Finally, a case study of WET-PPP projects in China is provided to testify the feasibility and effectiveness of the constructed framework. The results indicate that the proposed framework can improve the accuracy of decision-making and can be applied for handling evaluation problems. Image 1 • This paper constructs a sustainability assessment index system for WET-PPP projects from five perspectives. • This paper uses PFNs to quantify the uncertainty of information provided by DMs. • This paper defines the picture fuzzy biparametric similarity measures and explores the related properties. • This paper constructs an extended picture fuzzy similarity-based VIKOR method. • This paper proposes an integrated evaluation framework to assess the sustainability of WET-PPP projects. [ABSTRACT FROM AUTHOR]

A multi-criteria decision-making (MCDM) method with single-valued neutrosophic information is developed based on the Partitioned Heronian Mean (PHM) operator and the Shapley fuzzy measure, which recognizes correlation among the selection criteria. Motivated by the PHM operator and Shapley fuzzy measure, two new aggregation operators, namely the single-valued neutrosophic PHM operator and the weighted single-valued neutrosophic Shapley PHM operator, are defined, and their corresponding properties and some special cases are investigated. An MCDM model is applied to solve the single-valued neutrosophic problem where weight information is not completely known. An example is provided to validate the proposed method. [ABSTRACT FROM AUTHOR]

The aggregation operator is one of the most common techniques to solve multi-criteria decision-making (MCDM) problems. The aim of this paper is to propose an MCDM method based on the improved single-valued neutrosophic weighted geometric (ISVNWG) operator. First, the defects of several existing single-valued neutrosophic weighted geometric aggregation operators in terms of producing uncertain results in some special cases are analyzed. Second, an ISVNWG operator is proposed to avoid the defects of existing operators. Further, the properties of the proposed ISVNWG operator, including idempotency, boundedness, monotonicity, and commutativity, are discussed. Finally, a single-valued neutrosophic MCDM method based on the developed ISVNWG operator is proposed to overcome the defects of existing MCDM methods based on existing operators. Application examples demonstrate that our proposed operator and corresponding MCDM method are effective and rational for avoiding uncertain results in some special cases. [ABSTRACT FROM AUTHOR]

Single-valued neutrosophic sets (SVNSs), which involve in truth-membership, indeterminacy-membership and falsity-membership, play a significant role in describing the decision-makers' preference information. In this study, a single-valued neutrosophic multi-criteria decision-making (MCDM) approach is developed based on Shapley fuzzy measures and power aggregation operator that takes a correlative relationship among criteria into account and also simultaneously reduces the effects of abnormal preference information. Firstly, two aggregation operators, namely, generalized weighted single-valued neutrosophic power Shapley Choquet average (GWSVNPSCA) operator and generalized weighted single-valued neutrosophic power Shapley Choquet geometric (GWSVNPSCG) operator, are accordingly defined, and the corresponding properties are discussed as well. Secondly, based on the proposed aggregation operators, an integrated MCDM approach is proposed to effectively solve single-valued neutrosophic problems where the weight information is incompletely known. A programming model is constructed to obtain the optimal Shapley fuzzy measure. Next, the proposed operators are utilized to aggregate the decision-makers' preference information. Finally, a theoretical example with tourism attraction selection is provided to examine the efficacy of the developed approach, in which the results is found reasonable and credible. [ABSTRACT FROM AUTHOR]

With respect to multi-criteria decision-making (MCDM) problems in which the criteria denote the form of single-valued neutrosophic sets (SVNSs), and the weight information is also fully unknown, a novel MCDM method based on qualitative flexible multiple criteria (QUALIFLEX) is developed. Firstly, the improved cosine measure of the included angle between two SVNSs is defined. Then, the improved single-valued neutrosophic projection is developed, and the corresponding improved single-valued neutrosophic bidirectional projection and single-valued neutrosophic bidirectional projection difference are investigated. Moreover, the partial ordering relation of SVNSs is developed. Secondly, an extended QUALIFLEX method based on an improved single-valued neutrosophic projection measure is proposed to handle MCDM problems in which the weights of criteria are completely unknown. Finally, an example for selection of a green supplier, as well as a performance comparison analysis, are provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Multi-valued neutrosophic sets (MVNSs) consider the truth-membership, indeterminacy-membership, and falsity-membership simultaneously, which can more accurately express the preference information of decision-makers. In this paper, the normalized multi-valued neutrosophic distance measure is developed firstly and the corresponding properties are investigated as well. Secondly, the normalized multi-valued neutrosophic distance difference is defined and the corresponding partial ordering relation is discussed. Thirdly, based on the developed distances and comparison method, an extended multi-valued neutrosophic QUALItative FLEXible multiple criteria (QUALIFLEX) method is proposed to handle MCDM problems where the weights of criteria are completely unknown. Finally, an example for selection of medical diagnostic plan is provided to demonstrate the proposed method, together with sensitivity analysis and comparison analysis. [ABSTRACT FROM AUTHOR]

Identifying the priority of risks as regards energy performance contracting (EPC) projects is one of the main issues faced by energy service companies. Considering the complexity and uncertainty of EPC mechanism, this study attempts to formulate a hybrid fuzzy multi-criteria decision-making framework with picture fuzzy information to rank the risk factors of EPC projects. The proposed framework not only considers the interrelationship among criteria but also considers the decision-maker's bounded rationality and behavioural psychology. First, to effectively express the uncertainty and fuzziness of risk and environment, risk evaluation is expressed as picture fuzzy numbers. Subsequently, the new distances of picture fuzzy sets are proposed to fully use picture fuzzy information. Meanwhile, considering the interrelationship among criteria, an optimisation model based on the maximizing deviation method and Bonferroni mean distance of picture fuzzy sets is established to determine the weight vector of criteria with incomplete weight information. Furthermore, a prospect theory-based multi-attributive border approximation area comparison (MABAC) method is proposed to rank the risks and identify the priority of risks by reflecting the decision-maker's bounded rationality and behaviour psychology. In addition, the proposed framework is successfully implemented in a case study of a hotel's energy efficiency retrofit. Results show that the proposed framework is an effective and practical decision tool for risk ranking problems under picture fuzzy environment. Finally, the detailed risk mitigation, energy-saving performance and implications are presented to provide a reference and suggestion for other projects. [ABSTRACT FROM AUTHOR]

The definition of hesitant interval-valued intuitionistic fuzzy sets (HIVIFSs) is developed based on interval-valued intuitionistic fuzzy sets (IVIFSs) and hesitant fuzzy sets (HFSs). Then, some operations on HIVIFSs are introduced in detail, and their properties are further discussed. In addition, some hesitant interval-valued intuitionistic fuzzy number aggregation operators based on t-conorms and t-norms are proposed, which can be used to aggregate decision-makers' information in multicriteria decision-making (MCDM) problems. Some valuable proposals of these operators are studied. In particular, based on algebraic and Einstein t-conorms and t-norms, some hesitant interval-valued intuitionistic fuzzy algebraic aggregation operators and Einstein aggregation operators can be obtained, respectively. Furthermore, an approach of MCDM problems based on the proposed aggregation operators is given using hesitant interval-valued intuitionistic fuzzy information. Finally, an illustrative example is provided to demonstrate the applicability and effectiveness of the developed approach, and the study is supported by a sensitivity analysis and a comparison analysis. [ABSTRACT FROM AUTHOR]

The definition of hesitant interval-valued intuitionistic fuzzy sets (HIVIFSs) is developed based on interval-valued intuitionistic fuzzy sets (IVIFSs) and hesitant fuzzy sets (HFSs). Then, some operations on HIVIFSs are introduced in detail, and their properties are further discussed. In addition, some hesitant interval-valued intuitionistic fuzzy number aggregation operators based on t-conorms and t-norms are proposed, which can be used to aggregate decision-makers' information in multicriteria decision-making (MCDM) problems. Some valuable proposals of these operators are studied. In particular, based on algebraic and Einstein t-conorms and t-norms, some hesitant interval-valued intuitionistic fuzzy algebraic aggregation operators and Einstein aggregation operators can be obtained, respectively. Furthermore, an approach of MCDM problems based on the proposed aggregation operators is given using hesitant interval-valued intuitionistic fuzzy information. Finally, an illustrative example is provided to demonstrate the applicability and effectiveness of the developed approach, and the study is supported by a sensitivity analysis and a comparison analysis.