The aim of this work is to present some cases of **aggregation** **operators** with **intuitionistic** **trapezoidal** **fuzzy** **numbers** and study their desirable properties. First, some operational laws of **intuitionistic** **trapezoidal** **fuzzy** **numbers** are introduced. Next, based on these oper- ational laws, we develop some **geometric** **aggregation** **operators** for aggregating intuition- istic **trapezoidal** **fuzzy** **numbers**. In particular, we present the **intuitionistic** **trapezoidal** **fuzzy** weighted **geometric** (ITFWG) operator, the **intuitionistic** **trapezoidal** **fuzzy** ordered weighted **geometric** (ITFOWG) operator, the induced **intuitionistic** **trapezoidal** **fuzzy** ordered weighted **geometric** (I-ITFOWG) operator and the **intuitionistic** **trapezoidal** **fuzzy** hybrid **geometric** (ITFHG) operator. It is worth noting that the aggregated value by using these **operators** is also an **intuitionistic** **trapezoidal** **fuzzy** value. Then, an approach to multi- ple attribute group decision making (MAGDM) problems with **intuitionistic** **trapezoidal** **fuzzy** information is developed based on the ITFWG and the ITFHG **operators**. Finally, an illustrative example is given to verify the developed approach and to demonstrate its prac- ticality and effectiveness.

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In a similar way to TIFNs, Wang [35] dened the **Trapezoidal** IFN (TrIFN) and Interval-Valued Trape- zoidal IFN (IVTrIFN). Both TrIFN and IVTrIFN are extensions of TIFNs. Wang and Zhang [36] investi- gated the weighted arithmetic averaging operator and weighted **geometric** averaging operator on TrIFNs and their applications to MADM problems. Wei [37] in- vestigated some arithmetic **aggregation** **operators** with TrIFNs and their applications to MAGDM problems. Du and Liu [38] extended the **fuzzy** VIKOR method with TrIFNs. Wu and Cao [39] developed some fam- ilies of **geometric** **aggregation** **operators** with TrIFNs and applied them to MAGDM problems. Wan and Dong [40] dened the expectation and expectant score, ordered weighted **aggregation** operator and hybrid **aggregation** operator for TrIFNs and employed them for MAGDM. Ye [41] developed the expected value method for **intuitionistic** **trapezoidal** **fuzzy** multicrite- ria decision-making problems. Ye [42] proposed the MAGDM method using vector similarity measures for TrIFNs. Wan [43] developed four kinds of power av- erage operator of TrIFNs, involving the power average operator, weighted power average operator of, power order weighted average operator of, and power hybrid average operator of TrIFNs. Wan [44] rstly dened some operational laws and the weighted arithmetical average operator of IVTrIFNs. Based on the score function and accurate function, an approach is pre- sented to rank IVTrIFNs. The MAGDM method using IVTrIFNs is then proposed. Wan [45] constructed non- linear fractional programming models to estimate the alternative's relative closeness. After transformation into linear programming models, the interval of the

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Abstract. Soft set theory acts as a fundamental tool for handling the uncertainty in the data by adding a parameterized factor during the process unlike **fuzzy** as well as **intuitionistic** **fuzzy** set theory. In this manuscript, an attempt has been made to compare two **Intuitionistic** **Fuzzy** Soft **Numbers** (IFSNs) and then weighted averaging and **geometric** **aggregation** **operators** for aggregating the dierent input arguments have been presented. Further, their various properties have been established. The eectiveness of these **operators** has been demonstrated through a case study.

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All the above **aggregation** **operators** are based on the algebraic operational rules of IVIFNs, and the keys of the algebraic operations are Algebraic product and Algebraic sum, which are one type of operations that can be chosen to model the intersection and union of IVIFNs. In general, a general T -norm and T - conorm can be used to model the intersection and union of IVIFNs [32,33]. Wang and Liu [34] proposed the **intuitionistic** **fuzzy** Einstein **aggregation** **operators** based on Einstein operations which meet the typical T -norm and T -conorm and have the **same** smooth approximations as the algebraic **operators** such as the **Intuitionistic** **Fuzzy** Einstein Weighted **Geometric** op- erator (IFEWG) and the **Intuitionistic** **Fuzzy** Einstein Ordered Weighted **Geometric** operator (IFEOWG), and established some general properties of these oper- ators such as idempotency, commutativity, and mono- tonicity. Wang and Liu [35] proposed the **Intuitionistic** **Fuzzy** Einstein Weighted Averaging operator (IFEWA) and the **Intuitionistic** **Fuzzy** Einstein Ordered Weighted Averaging operator (IFEOWA), and studied various properties of these **operators** and analyzed the relations between the existing **intuitionistic** **fuzzy** **aggregation** **operators** and them. Maris and Iliadis [36] further explained the advantages of Einstein operations by using some T -norms to unify the risk indices and to produce a unied means of risk measure. The algebraic T -norm estimated the risky areas under average rainfall conditions, and the Einstein T -norm oered a good approach for an overall evaluation. The computer system has proven its ability to work more eectively compared to the older methods.

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Multiple Criteria Group Decision Making Problems (MCGDM) are important parts of modern decision theory due to rapid development of economic and social uncertainties. Today, Decision-Maker (DM) wants to attain more than one goal by satisfying dierent constraints. But, due to the complexity of management environments and decision problems themselves, DMs may provide their rating or judgment in the form of crisp **numbers** without considering the degree of fuzziness or vagueness of the data in the domain of the problem [1]. However, in these days, uncertainties play a dominant role during the decision-making process, and the decision-maker cannot give their preferences to an accurate level without being proper handled. The main objective during an analysis is to handle the proper data so as to minimize the uncertainties level. To handle this, a **fuzzy** set theory [2] has been

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Concerning ranking IF **numbers** some work has been reported in the literature. Grzegorzewski [12] defined two **families** of metrics in the space of IF **numbers** and proposed a method for comparing IF **numbers** based on these metrics. Mitchell [15] extended the natural ordering of real **numbers** to triangular **intuitionistic** **fuzzy** (TrIF) **numbers** by adopting a statistical view point and interpreting each IF number as ensemble of ordinary **fuzzy** **numbers**. Nayagam et al. [19] introduced TrIF **numbers** of special type and described a method to compare them. Although their ranking method appears to be attractive, the definition of TrIF number seems unrealistic. This is because the triangular nonmembership function is defined to geometrically behave in an identical manner as the membership function. Nan and Li [16] proposed a method for comparing TrIF number using lexicographic technique. Nehi [20] proposed a new method for comparing IF **numbers** in which two characteristic values for IF **numbers** are defined by the integral of the inverse **fuzzy** membership and nonmembership functions multiplied by the grade with powered parameter. Almost in parallel, Li [13] introduced a new definition of the TrIF number which has an appealing and logically reasonable interpretation. He defined two concepts of the value and the ambiguity of a TrIF number similar to those for a **fuzzy** number introduced by Delgado et al. [4]. Dubey and Mehra [5] defined a TrIF number which is more general than the one defined in [13, 16]. They extended the definitions of the value and the ambiguity index given by Li [13] to the newly defined TrIF **numbers** and proposed an approach to handle linear programming problems with data as IF **numbers**.

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The objective of this paper is to introduce a distance measure for **intuitionistic** **fuzzy** **numbers**. Firstly the existing distance measures for **intuitionistic** **fuzzy** sets are analyzed and compared with the help of some examples. Then the new distance measure for **intuitionistic** **fuzzy** **numbers** is proposed based on interval diﬀerence. Also in particular the type of distance measure for triangle **intuitionistic** **fuzzy** **numbers** is described. The metric properties of the proposed measure are also studied. Some numerical examples are considered for applying the proposed measure and finally the result is compared with the existing ones.

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This work is mainly focussed on the interpretation of Atanassov’s **intuitionistic** **fuzzy** logic via QC , where not only the **intuitionistic** **fuzzy** sets but also complement, inter- section, union, diﬀerence and codiﬀerence operations are interpreted based on the quantum circuit model, including IFSs obtained by representable (co)implications. Further work aims to consolidate this speciﬁcation including not only other **fuzzy** connectives but also constructors (e.i. automorphisms and reductions) and the corresponding extension of (de)fuzzyﬁcation methodology from formal structures provided by QC .

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The concept of **fuzzy** set was introduced by Lotfi.A.Zadeh[1]. It is an extension of the classical sets. Many authors extended the idea of **fuzzy** set in different directions. In [2] and [3], Atanassov introduced the concept of **intuitionistic** **fuzzy** set(IFS), using a degree of membership and a degree of non-membership, under the constraint that the sum of the two degrees does not exceed one. IFS is one of the most successful extension of **fuzzy** set used for handling the uncertainties in the data. Modal **operators**, topological **operators**, level **operators**, negation **operators** and **aggregation** **operators** are different groups of **operators** over the IFS due to Atanassov[2]. In [4], T.K.Mondal and S.K.Samanta introduced the concept of generalized **intuitionistic** **fuzzy** set (GIFS)𝑆 = { 𝑥, 𝜇 𝑆 𝑥 , 𝜈 𝑆 (𝑥) : 𝑥 ∈ 𝐸} where 𝜇 𝑆 : 𝐸 → 𝐼 and 𝜈 𝑆 : 𝐸 → 𝐼 satisfy the condition 𝜇 𝑆 𝑥 ∧ 𝜈 𝑆 𝑥 ≤ 0.5, ∀ 𝑥 ∈

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In this paper we propose a new approach for Medical diagnosis with the symptoms of disease using IFS with new **operators**. This **operators** apply to identified the disease of the patient with symptoms in the data. The membership and non-membership values are not always possible upto our satisfaction, but in deterministic(hesitation) part has more important role here, the fact that in decision making, particularly in case of medical diagnosis, there is a fair chance of the existence of a non-zero hesitation part at each moment of evaluation .

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In this paper, we introduce a more general set of operators on M than were given by Taká cˇ, and we study, among other properties, the conditions required to satisfy the axioms of the [r]

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In this paper priority based **fuzzy** goal programming with generalized **trapezoidal** **fuzzy** **numbers** has been proposed. Euclidean distance is used for selecting proper priority structure for obtaining compromise optimal solution. The concept presented, in this paper, is illustrated with multi- objective assignment problems involving generalized **trapezoidal** **fuzzy** **numbers** to check the effectiveness of the proposed method. The proposed method is simple and easy to implement. It may be hoped that proposed method can be applied to solve realistic optimization problems involving generalized **trapezoidal** **fuzzy** **numbers**.

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transforms the given problem in to another FLFPP with fewer **fuzzy** constraints. A relationship between these two problems, which ensure that the solution of the original problem can be recovered from the solution of the transformed problem. A simple numerical example explains the procedure of the proposed method.

E.BalouiJamkhaneh and Nadarajah[8] defined an extension of generalized **intuitionistic** **fuzzy** set. In 2017, BalouiJamkhaneh[9] defined level **operators** 𝑃 𝛼,𝛽 and 𝑄 𝛼,𝛽 over GIFS.BalouiJamkhaneh and NadiGhara[10] defined four new level **operators** over GIFS and established some of their properties.In this paper, we introduce new level **operators** 𝑃 𝛼,𝛽 ∗ and 𝑄

ABSTRACT: In this paper, we investigate **Intuitionistic** **Fuzzy** Transportation Problem with the **Trapezoidal** **Intuitionistic** **fuzzy** **numbers**. Methods are proposed to find the initial solution of **Intuitionistic** **Fuzzy** Transportation Problem and a method to find the optimal solution of **Intuitionistic** **Fuzzy** Transportation Problem is developed. Ranking method based on the magnitude of membership function and non-membership function of a **Intuitionistic** **Fuzzy** Number is utilized to order the **Intuitionistic** **fuzzy** **numbers**. Numerical example is provided to illustrate the new approaches.

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In recent years, with the increasing complexity of the managerial decision making envi- ronment, many managerial decision-making problems contain qualitative properties which are difficult to quantify. Zadeh’s **fuzzy** sets have been greatly successful in dealing with **fuzzy** manage- ment decision making problems [3,4,18,20,22,35]. Zadeh’s **fuzzy** set is characterized by a single scale (membership), which can only characterize the support and opposition of the two aspects of the evidence. But some decision making problems have ambiguous hesitant phenomenon with respect to evaluation of information, and Zadeh’s **fuzzy** set is hard or difficult to depict these sit- uations. Therefore, many scholars developed Zadeh’s **fuzzy** set, and **intuitionistic** **fuzzy** (IF) set is one of the most famous **fuzzy** sets among them. Originally proposed by Atanassov in 1986 [1], IF sets can well describe the hesitation and uncertainty of judgment through the addition of a non-membership parameters, which can describe the vague characters of things comprehensively. Then IF sets have become a powerful and effective tool in dealing with uncertain or vague in- formation in actual applications. In dealing with ambiguity and uncertainty, IF sets are more flexible and practical than **fuzzy** sets, and thus they have been applied widely in decision making. Because of the complexity and uncertainty of objective things and the limitation of decision maker’s knowledge, membership and non-membership functions are sometimes difficult to repre- sent by using the precise **numbers**. But interval number can be very useful to describe this kind

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aggregate all individual decision into a collective decision. Since the individuals in the groups may have different capabilities it is necessary to use weighted **aggregation**. In the present complex socio economic environment and the insufficient knowledge of the problem, under the **fuzzy** environment, individuals in the group may provide their information over alternatives with interval valued **intuitionistic** **fuzzy** number (IVIFN). Chen (1985) introduced a **fuzzy** assignment model that considers all individuals have **same** skills. Huang and Zhang (2006) proposed a mathematical model for the **fuzzy** assignment problem with restriction on qualification. Mukherjee and Basu (2011) proposed **intuitionistic** **fuzzy** assignment problem by using similarity measures and score functions. Xu (2007) defined the concept of interval - valued **intuitionistic** **fuzzy** number. Xu and Chen (2007) define an interval-valued **intuitionistic** **fuzzy** ordered weighted averaging operator and an interval-valued **intuitionistic** hybrid averaging operator. Lin and Wen (2004) concentrate on the assignment problem where costs are not deterministic **numbers** but imprecise ones. Gaurav and Bajaj (2014) proposed interval-valued **intuitionistic** **fuzzy** assignment problem by using similarity measure and score functions.

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Thus, the objective of this manuscript is to present some series of averaging **aggregation** **operators** in an IVIFSs environment. For it, a new operational law on dierent IVIFNs has been proposed by taking the interaction between the pair of membership and non-membership functions. Based on these new op- erational laws, weighted aggregated **operators**, namely Interval-Valued **Intuitionistic** **Fuzzy** (IVIF) Hamacher Interactive Weighted **Aggregation** (IVIFHIWA), IVIF Hamacher Interactive Ordered Weighted **Aggregation** (IVIFHIOWA), and IVIF Hamacher Interactive Hy- brid Weighted **Aggregation** (IVIFHIHWA), have been proposed by properly handling the shortcoming of the existing **operators**. The main signicance of these **operators** is that the inuence of the degree of non-

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49 TOPSIS and DEA to select the units with the most efficiency. First, the alternative evaluation problem is formulated by DEA and separately formulates each pair of units. In the second stage, he used the opinion of experts to be applied to a model of group Decision-Making called the **Intuitionistic** **Fuzzy** TOPSIS method. Gandotra et al. [53] proposed an algorithm to rank DMUs in the presence of **intuitionistic** **fuzzy** weighted entropy. Hajiagha et al. [54] developed a DEA model when input/output data was expressed in the form of IFS. They further extended the model to the case of a weighted aggregated operator for IFS. Puri and Yadav [55] developed optimistic and pessimistic DEA models under **intuitionistic** **fuzzy** input data. They also presented the application of their proposed models through a case from the banking sector in India where some of the inputs were represented as triangular **intuitionistic** **fuzzy** **numbers** in the form of A a a a a a a l , m , u ; l , m , u .

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Ranking **fuzzy** **numbers** is a prerequisite for the decision making problem. In order to rank **fuzzy** quantities many researchers proposed and analyzed different techniques on triangular and **trapezoidal** **fuzzy** **numbers**. However, no one can claim their method is a satisfactory one. In this paper a modified distance based approach called signed distance proposed by Yao and Wu [9] is discussed. This proposed approach is free from computational complexity in the process of decision making, optimization and forecasting problems. Some Numerical examples are used to illustrate the proposed approach.

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