In order to improve the reasonability and effectiveness of the methods on dealing with single valued trapezoidalneutrosophic MCGDM problems, also overcome the limitations of the existing approaches. In this paper, a single valued trapezoidalneutrosophic MCGDM method is proposed form the possibilitydegree of SVTNNs and the single valued trapezoidalneutrosophicpoweraggregationoperators. Firstly, the new operations of SVTNNs are proposed for avoiding information loss and distortion, the possibility degrees of SVTNNs are proposed from the probability viewpoint. Based on the proposed operations and possibility degrees, SVTNPA and SVTNPG operators are proposed. Furthermore, a single valued trapezoidalneutrosophic MCGDM method based on SVTNPA, SVTNPG operator and the possibility degrees of SVTNNs is developed. The prominent advantages of the proposed method are not only its ability to effectively deal with the preference information expressed by SVTNNs but also the consideration of the relationship among the information being aggregated in the process on dealing with the practical MCGDM problems and the advantage of the possibility degrees of SVTNNs, which can avoid information loss and distortion, is combined. Thus, the final results are more scientific and reasonable. Finally, the method is applied to a practical problem on selecting the most appropriate green supplier for SGM Company, meanwhile, the comparison with other method is carried on and demonstrates its feasibility and effectiveness in dealing with MCGDM problems.
Abstract Single-valued trapezoidalneutrosophicnumbers (SVTNNs) have a strong capacity to depict uncertain, inconsistent, and incomplete information about decision- making problems. Preference relations represent a practical tool for presenting decision makers’ preference informa- tion regarding various alternatives. The purpose of this paper is to propose single-valued trapezoidalneutrosophic preference relations (SVTNPRs) as a strategy for tackling multi-criteriadecision-making problems. First, this paper briefly reviews basic concepts about neutrosophic sets and SVTNNs and defines a new comparison method and new operations for SVTNNs. Next, two aggregationoperators, the single-valued trapezoidalneutrosophic weighted arith- metic average operator and the single-valued trapezoidalneutrosophic weighted geometric average operator, are proposed for applications in information fusion. Then, this paper discusses the definitions of completely consistent SVTNPRs and acceptably consistent SVTNPRs. Finally, we outline a decision-makingmethod based on SVTNPRs to address green supplier selection problems, and we con- duct a comparison study and discussion to illustrate the rationality and effectiveness of the decision-makingmethod.
for multi-criteriadecisionmaking problems under simplified neutrosophic environ- ments. Peng et al. [ 26 ] developed some new operations of simplified neutrosophic set and proposed a novel outranking approach for multi-criteriadecision-making (MCDM) prob- lems. Peng et al. [ 27 ] pointed out certain problems regarding the existing operations of simplified neutrosophicnumbers, their aggregationoperators and the comparison method, and defined the new operations of simplified neutrosophic num- bers. They also developed a multi-criteriadecisionmakingmethod and comparison method. Liu et al. [ 19 ] combined Hamacher operations and generalized aggregationoperators to neutrosophic sets, and proposed some new operational rules for neutrosophicnumbers (NNs) based on Hamacher operations. They also proposed some new operators such as generalized neutrosophic number Hamacher weighted aver- aging (GNNHWA) operator, generalized neutrosophic num- ber Hamacher ordered weighted averaging (GNNHOWA) operator and generalized neutrosophic number Hamacher
compared them with the use of those selected by construction industry. Ben-David and Raz  proposed a decision-support model for the risk response phase in software development of electronic devices. Piny  built the response planning chart for threats and opportunities which was divided into six areas to define the overall strategy for each risk. Young  proposed a conceptual framework for the risk response phase on projects. Pan and Chen  presented an economic optimization model based on the model proposed by Ben-David and Raz  for selecting risk reduction actions in CMMI-based software projects with an example taken from a Chinese software industry. Fan et al.  provided a conceptual framework that defined the relationship between risk-handling strategy and relevant project characteristics, and described the quantitative relationships among all variables. Aaltonen and Sivonen  identified and regarded five different types of response strategies through an empirical analysis of four cases in emerging markets in global projects. Kutsch and Mark Hall  proposed the results of a qualitative study of IT project managers, considering their reasons for deeming certain known risks to be irrelevant.
But George worked overtime 5 hours, so his membership degree was (40+5)/40 = 45/40 = 1.125 > 1. Thus, we need to make distinction between employees who work overtime, and those who work full-time or part-time. That’s why we need to associate a degree of membership strictly greater than 1 to the overtime workers.
Abstract: This paper introduces a new intuitionistic fuzzy multicriteria deci- sion makingmethod of evaluation based on degree of inclusion of two intuition- istic fuzzy sets. We have called the new technique TOPIIS (Technique to Order Preference by Inclusion of Ideal Solution). The technique is applied to develop an e¤ective employee performance appraisal.
In this paper, we explain the value of both Trapezoidal and Triangular Fuzzy Numbers and develop a new ranking method based on the value of fuzzy number which in turn will be very helpful in decisionmaking situations. Then we propose a MultiCriteriaDecisionMaking (MCDM) Model based on the proposed ranking method. Arithmetic mean operation of fuzzy numbers is used for aggregating experts’ judgments.
The second type is the aggregationoperators of TIFN. Zhang and Liu  dened the concepts of TIFN in which the membership and the non-membership degrees are denoted by triangular fuzzy number. Then, the weighted geometric averaging operator and the weighted arithmetic average operator are presented and used for the decision-making area. Robinson and EC  investigated Triangular Intuitionistic Fuzzy Ordered Weighted Averaging (TIFOWA) operator and the Triangular Intuitionistic Fuzzy Hybrid Aggregation (TIFHA) operator. Chen and Li  developed a new distance measure between two TIFNs to aid in determining attribute weights, and they presented the Weighted Arithmetic Averaging operator on TIFNs (TIFN-WAA), and then proposed a dynamic MADM model with TIFNs. Wang et al.  proposed new arithmetic operations and logic operators for TIFNs and applied them to fault analysis of a printed circuit board assembly system. Yu and Xu  dened the concepts of Intuitionistic Multiplicative Triangular Fuzzy (IMTF) set and intuitionistic multiplicative tri- angular fuzzy number, and then discussed their opera- tional laws and some desirable properties. Based on the operational laws, they developed a series of aggregationoperators for IMTF information. Combining the fuzzy measure and Choquet integral, Wan and Dong  dened the TIF Choquet integral aggregation operator and investigated some desirable properties for this operator.
Therefore, Smarandache ,  firstly proposed Neutrosophic Set (NS), however NS was mainly put forward from a philosophical viewpoint. So Wang et al.  proposed Single-valued Neutrosophic Set (SVNS) with the corresponding properties and operation rules. Similar to IVIFS, Wang et al.  proposed Interval Neutrosophic Set (INS) and gave the set-theoretic operators of INS.
Abstract A generalized trapezoidal-valued intuitionistic fuzzy geometric ag- gregation operator is proposed which is then used to aggregate decision makers’ opinions in groupdecisionmaking process. An extension of TOPSIS, a multi- criteriatrapezoidal-valued intuitionistic fuzzy decisionmaking technique, to a groupdecision environment is also proposed, where inter-dependent or interactive characteristics among criteria and preference of decision makers are under consid- eration. Furthermore, Choquet integral-based distance between trapezoidal-valued intuitionistic fuzzy values is de…ned. Combining the trapezoidal-valued intuitionis- tic fuzzy geometric aggregation operator with Choquet integral-based distance, an extension of TOPSIS method is developed to deal with a multi-criteriatrapezoidal- valued intuitionistic fuzzy groupdecisionmaking problems. Finally, an illustrative example is provided to understand the proposed method.
In section 3 a dissimilarity measure between the opinions of the experts is introduced, coherent with Saaty’s 1 to 9 ratio scale on which the opinions are expressed. This dissimilarity measure induces, in a natural way, a scale transformation, thus allowing a fuzzy formulation of the prob- lem; a soft measure of the consensus is then defined for each criterion, according to the approach of Fedrizzi, Kacprzyk, and Zadro˙zny . More precisely, a fuzzy-logic-based calculus of lin- guistic quantified propositions is used to derive a measure of consensus that expresses the degree to which, for example, “almost all experts agree with the group’s opinions concerning the most important alternatives”.
Multi-criteria (MC) problems involve makingdecision over alternatives that are characterized by several criteria. These criteria represent basis of evaluation in MC evaluation models or goal aspiration in MC optimization models. In most of MC models, criteria weights must be predetermined before the problem can be solved. These weights are interpreted differently but mostly as relative importance of criteria. There are many weighting methods available, but are generally categorized as subjective or objective methods. The subjective methods involve evaluator(s) to evaluate the relative importance of the criteria. Even though multi-person may involve in evaluating the criteria, the final weights must be represented as only one set of weights. Many aggregation methods have been proposed to compose the evaluations. However, these evaluators may have different degree of credibility since they may come from different background or may have different degree of superiority. The aim of this paper is to propose a different concept of weights that would represent the degree of credibility of the evaluators. Furthermore, several aggregation approaches are suggested on how to include these ‘new’ weights in order to produce new criteria weights that also take the credibility of the evaluators into considerations. A numerical example is used to show how these weights of credibility can be used to solve a MC problem in particular to determine the criteria relative importance. This new concept of weight signifies a different insight to the domain of MC decisionmaking (MCDM).
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In real life, a large number of natural and social phenomenon obey normal distribution [8,49], such as “product life span”, “stock price”, "commodity customer experience evaluation" and so on. In view of these phenomenon, Yang and Ko  proposed normal fuzzy number (NFN) to characterize them. Li and Liu  pointed out that compared with triangular and trapezoidal fuzzy numbers, NFN has higher derivative continuity, which can characterize natural and social phenomenon more extensively, and their membership functions are closer to human thinking. Li and Liu  used examples to prove that the extension of intuitionistic fuzzy number (IFN) based on NFN is better than IFN based on triangular, trapezoidal fuzzy numbers, etc. Therefore, Wang et al.  and Wang et al.  defined intuitionistic normal fuzzy (INF) number and its operation rules and some information aggregators. Based on these, the concept of interval-valued INF was defined , and the family of INF-induced ordered operators , and a series of aggregationoperators based on classic operators under INF information environment were proposed [54–58].
According to the result of comparing these exist- ing approaches with general intuitionistic sets (IVIFSs or IFSs), the proposed DM method under the CIFS environment contains much more evaluation infor- mation on the alternatives by considering both the IVIFSs and IFSs simultaneously, while the existing approaches contain either IFS or IVIFS information. Therefore, the approaches under the IVIFSs or IFSs may lose some useful information, either IVIFNs or IFNs, of alternatives which may aect the decision results. Furthermore, it is noted from the study that the computational procedure of the proposed approach is dierent from the existing approaches under the dierent environment; however, the proposed result in this paper is more rational to reality in the decision pro- cess due to the consideration of the consistent priority degree between the pairs of the arguments and between dierent experts. Moreover, the corresponding studies under the IVIFS or IFS environment can be considered as the special case of the proposed operators. Finally, the existing DM methods under IVIFSs or IFSs cannot deal with the DM problem by CIFS. Therefore, the proposed approach is more generalized and suitable to capture the real-life fuzziness more accurately than the existing ones.
Type-1 fuzzy sets are effi cient in modeling the multi-criteriadecision-making problems and have many applications for extending MCDM methods in an uncertain environment. However, we may confront with situations that more degrees of fl exibility are needed to deal with the decision-making process. Type-2 fuzzy sets (T2FSs), which was introduced by Zadeh (1975), are more fl exible than type-1 fuzzy sets in the modeling of uncertainty. Interval type-2 fuzzy sets (IT2FSs) are a special type of T2FSs. Some basic defi nitions of IT2FSs were proposed by Mendel et al. (2006). IT2FSs have increasingly been considered by researchers in applications and extensions of multi-criteriadecision-making methods. Chen and Lee (2010) developed a new ranking method for interval type-2 fuzzy sets and used it in a new fuzzy MCDM method. Chen et al. (2012) proposed a new ranking method and a new multi-criteriadecision-makingmethod with interval type-2 fuzzy sets. Wang et al. (2012) introduced a multi-criteriagroupdecision-making (MCGDM) method in type- 2 fuzzy environment, which can be used with incomplete information about criteria weights. Hu et al. (2013) developed a new ranking method based on the possibilitydegree for IT2FSs and applied it in multi-criteriadecision- making process. Keshavarz Ghorabaee et al. (2014) presented a new fuzzy ranking method and extended COPRAS (Complex Proportional Assessment) method in the context of IT2FSs to evaluate suppliers in a supply chain. Celik et al. (2014) proposed an interval type-2 fuzzy MCDM method to identify and evaluate critical success factors for humanitarian relief logistics management. Balin and Baraçli (2015) developed a fuzzy MCDM methodology based on the IT2FSs for evaluating renewable energy alternatives in Turkey. Chen (2015) proposed a new likelihood-based interval type-2 fuzzy
Abstract: To address the problem of multi-attribute groupdecisionmaking with interval grey numbers, decision matrices are adjusted using kernels of interval grey numbers to reduce the psychological effects of decision makers. The comprehensive weights of attributes are obtained by aggregating the subjective weights with objective weights, which are calculated based on the accuracy and difference of attributes. Considering the consistent, best and worst decision-making abilities of decision makers, grey incidence models are established to obtain the consistency weights and individual bipolar weights of decision makers; then, the comprehensive weights of decision makers are determined. A clustering approach of interval grey numbers is presented, and overall evaluations are obtained. Finally, an example is provided and its validity is tested to verify the feasibility of the proposed method.
Cities therefore play a significant role in resolving acute challenges related to climate change and energy transition (IEA, 2016). With the increasing importance of urban areas, among the 17 Sustainable Development Goals (SDGs) identified by UN Agenda 2030, goal 11 is completely dedicated to sustainable cities and communities. Particularly, Goal 11 emphasises the better urban planning and management of cities and human settlements with the aim at making them inclusive, safe, resilient and sustainable (UN General, 2015). In fact, cities are the decisive framework for the development of new strategies and approach in facing climate change and energy transition giving concrete and rapid solutions for more sustainable and eco-friendly human development (UN General, 2015). However, most urban planning systems do not have evaluation and monitoring as an integral part of their operations (UN-HABITAT, 2009). Clear indicators are needed to be integrated within each urban planning systems to monitor and evaluate tactics, strategies and processes. Although a larger scale approach is preferable to a building scale, this concept requires considering an all-new set of sustainability variables, and involving numerous new stakeholders, thus extremely complexifying the decision-making activities. In fact, due to many influences and factors in large-scale plans, their impacts are very difficult to be assessed.
As per our knowledge, there are no articles published in neutro- sophic sets and its hybridizations, where experts’ relative weights have been considered. Experts opinions are vital for any decisionmaking process. When experts prescribe their preferences using soft sets, they provide their opinions only about their known set of attributes/parameters. As the domains of expertise of different experts are different, they may be interested in different subset of attributes and remain silent about the rest of attributes. More specifically, one expert may provide opinion about more number of attributes; whereas other expert may be interested in less num- ber of attributes. In another way, one expert may be more confi- dent on his/her opinion than the other on the same set of attributes. For this type of environment, equal weights assignment to different experts may lead to improper and biased solution. Pre- ceding situation motivated us to develop the relative weight assignment procedure, where more confident opinions are given more importance. Suppose Mr. X and Mr. Y provide their opinions about an attribute using neutrosophic set. According to Mr. X, the opinion is (0.3, 0.4, 0.6) and opinion of Mr. Y is (0.9, 0.1, 0.2). Here more importance will be given to Mr. Y as he/she is more confident, since the truth membership value for his/her is close to 1 which is more than that of Mr. X. In this article, we propose an algorithmic approach for GDM using NSMs and relative weights of experts. Ini- tially experts provide their opinions using NSMs, which are nor- malized by the relative weights of the corresponding experts. The relative weights are assigned to individual experts based on their information, which reduces the chances of biasness. The proposed approach focuses on the choice parameters/attributes of various experts to find out the neutrosophic choice matrix (NCM) and com- bined NCM for individual decision maker/expert. In the process, the combined NCMs are multiplied with the normalized NSMs to obtain the product NSMs, which are aggregated to produce the resultant NSM. Then cross-entropy measure is applied on the alter- natives of the resultant NSM to rank the alternatives. The case study is related to investment in business sectors, where two cases have been considered. Case I shows the final outcome without assigning any weight and Case II shows the result by assigning weights.
Traditional project management is, by and large, a process whereby each project is approved and managed independently. In this arena, the focus is on a single project and the triple constraint-Scope, time and cost-of that project separate from other projects. Typically, the project manager is responsible for evaluating the performance of the project. At times, given the importance of the project, the project might be evaluated or reviewed at the executive level, but this review is still conducted in isolation of other projects.