Introduction

Decision-making is a very important process for leadership and management. There are processes and techniques available for decision-making and improve the quality of the decision as well. This process involves numerous information, and the collected information need to be aggregated to find the desired result. Hence, aggregation operators are playing a vital role especially in the decision-making process. Triangular inequalities were protracted by the theoretical concept of triangular norms which are introduced from the scope of prospect metric [1]. To date, different operators have been used under different set environments. As the real-world problems contain uncertainty we need to use the concept of fuzzy and its extensions. Soft set theory is a general mathematical tool to handle with uncertainty of the real-world problems. Some of the operations on soft sets have been introduced and applied in various fields. The application of fuzzy soft set in decision-making problem has received more attention among the researchers. It has been applied in the field of engineering, economics and all the environmental areas as it deals with uncertainties successfully than other set environments by getting free from difficulties [16, 19].

Most of the objectives in real-time applications are communicated in linguistic terms, but a concise mathematical formula is not applicable in management decisions. If the objective of the decision is correctable, then the constraints of the decision may be flexible. Modeling decision-making process using soft set is more realistic and applied fruitfully to various problems. Representation of a soft set is described by the soft matrix and has many advantages in collecting the information and applying matrices. An algebraic structure of soft set theory has two types of soft sets like a soft set with a fixed set of parameters and with different sets of parameters with new operations, and this may be either similar or different [20,21,22,23,24,25,26,27, 52]. Soft set theory normally solves the problem using rough and fuzzy soft sets. Many of the conventional techniques for explicit modeling, logic, and calculations are crisp, precise and acceptable. But the data obtained from real-world problems are not always crisp. Because of this situation, one faces uncertainty in the real problems. The available theories, namely the theory of FSs, IFSs, vague sets, interval mathematics, rough sets are playing as the mathematical tool to handle the uncertainties. But still, all of these theories have their complication due to the insufficient parameterization mechanism of the mentioned theories. Hence, Molodtsov introduced the notion of soft theory to clear the impreciseness which is exempted from these kinds of difficulties.

A fuzzy set is the principle idea of fuzzy logic. It contributes a lot in dealing with uncertainties by including some impreciseness corresponding to the membership functions. Fuzzy sets are also called type-1 fuzzy sets (T1FSs) [17]. General additional operations in the interval [0, 1] are t norms and t conorms called triangular norms ever-present in the theory and applications of Type-2 FS (T2FSs). Generalization of T1FS to interval-valued FSs (IVFSs) can be done by generalizing the triangular norms to interval-based cases [18]. Many of the real-time problems are described by the flexibility of the constraint. Particularly in the decision-making process, this kind of flexibility could accompany workable solutions, where the aim and conditions identified by various parties convoluted in the decision-making are determined to one another and delighted to different degrees. This kind of soft constraints can be modeled by fuzzy sets.

Fuzzy set theory contributes to a methodology of representing and handling flexible or soft constraints. In many cases, there is imperfect knowledge of the data and hence, uncertainties should be taken care of in many dimensions and kinds. Choosing membership functions of linguistic terms is the essential one for dealing with uncertainty [5]. In T1FSs, the membership grade of the element is taken from the unit interval [0, 1] and is crisp. T2FSs are generally used for modeling imprecision and uncertainty in an enhanced way with the membership value itself is fuzzy. It is characterized by upper and lower membership functions. The interval between these membership functions represents the footprint of uncertainty, which is used to define the uncertainty level. In T2FSs, two membership functions (MFs) are available, namely primary and secondary membership functions. For all the values of the primary variable x on the universal set X, the function has membership rather than a characteristic value. Secondary membership function provides three-dimensional T2FS, where the third dimension produces a certain degree of freedom to deal with uncertainties. Also, this set can provide more parameters, so that more uncertainties can be handled [8, 9, 11, 42, 67, 68].

The generalized type-2 fuzzy set has more computational complexity for defuzzification and hence, it is necessary to use interval type-2 fuzzy sets. Interval type-2 fuzzy sets have been used in traffic control management, image processing, image extraction, control system, and pattern recognition. At the beginning stage of fuzzy sets, the analysis was made regarding the fact that the membership function of a conventional fuzzy set has no ambiguity connected with it which contradicts the word fuzzy though it implies lots of uncertainty. Hierarchical type-2 fuzzy logic control design has been proposed for autonomous mobile robots that express the efficiency of type-2 fuzzy sets than type-2 fuzzy sets. Due to the computational complexity of general type-2 fuzzy sets, interval type-2 fuzzy sets have been used. All of the results that are needed to implement an interval type-2 fuzzy set can be obtained using type-2 fuzzy mathematics. All of the results that are required to implement an interval type-2 fuzzy logic system can be acquired using type-1 fuzzy set. An interval type-2 fuzzy system has the benefit of direct and indirect approaches. In the direct method, rules are developed through the extraction of knowledge and in the latter case, through historical data. From this concept, it is concluded that interval type-2 fuzzy set is a hybrid approach of these two approaches.

A system with type-2 fuzzy sets allows modeling the uncertainties between the rules and parameters related to data analysis [23, 24, 43]. The concept of interval-valued fuzzy soft set or interval type-2 fuzzy soft set is the combination of interval type-2 fuzzy sets and soft set [25]. Generally, real-world problems are described by huge levels of numerical and linguistic uncertainties. Since the type-1 fuzzy system cannot handle these huge levels of uncertainties completely available in real-world applications, the type-2 fuzzy set has been used to overcome this issue and getting successful results [26].

Many aggregation operators have been introduced for aggregating the information so far. Different types of uncertainties are represented and solved by various methods. In the decision-making process, information is collected from several experts by preparing a survey using linguistic terms where uncertainty naturally exists as different words mean different things. This can be solved by T2 fuzziness, where the FSs have degrees of membership that themselves fuzzy rather than type-1 fuzzy sets [21, 58, 62].

If there are multiple numbers of inputs, then they can be accumulated into interval type-2 input MFs using the methodologies [31, 39]. T2FS is an expansion of T1FSs. The mathematical functions which are applied to combine the information are called aggregation operators. They play a vital aspect in the field of computational intelligence due to their capacity for combining pieces of linguistic information. Fuzzy logic is an engineering mechanism and is explained in a broad sense and is a multi-valued function [33, 34]. Some of the well-known fuzzy numbers are triangular, trapezoidal, right and left shoulder and piecewise linear functions. Fuzzy logic contributes to compositional calculation of degrees of truth. The set operations namely union, intersection, complement, and implication are working on the case of fuzzy using t-norm and t-conorm [15, 24] predicted for carbon monoxide concentration in mega-cities using an interval type-2 fuzzy expert system. Min [25] proposed the concept of similarity in soft set theory. Hagras and Wagner [26] analyzed the widespread of the type-2 fuzzy logic system in real-time applications. Zhang and Zhang [27] solved a decision-making problem using the type-2 fuzzy soft set. Borah et al. [28] examined some of the operations of fuzzy soft sets. Rajarajeswari and Dhanalakshmi [29] applied a similarity measure of fuzzy soft set based on distance in a decision-making problem. Basu et al. [30] determined the best alternative using various types of fuzzy soft set matrices. Wang et al. [31] proposed interval-valued intuitionistic fuzzy aggregation operators. Zhang and Zhang [32] applied a type-2 fuzzy soft set in a decision-making problem. Wang and Liu [27] proposed intuitionistic fuzzy aggregation operators using Einstein operations.

Cagman and Deli [33] introduced products of FP-soft sets and applied in a decision-making problem and the same authors [34] proposed means of FP-soft sets and applied in a decision-making problem. Hernandez et al. [35] analyzed the role of t-norms on type-2 fuzzy sets. Bobillo and Straccia [36] introduced aggregation operators for fuzzy ontologies. Mondal and Roy [37] projected the theory of fuzzy soft matrix and its application in a decision-making process. **e et al. [38] applied the concept of fuzzy soft sets in medical diagnosis for gray relational analysis. Muthumeenakshi and Muralikrishna [39] examined SFPM analysis using a fuzzy soft set. Liang et al. [40] proposed new aggregation operators under triangular intuitionistic fuzzy numbers and applied in a decision-making problem. Qin et al. [41] proposed novel aggregation operators for triangular interval type-2 fuzzy set using Frank triangular norms. Rajarajeswari and Dhanalakshmi [42] introduced theoretical concepts of interval-valued intuitionistic fuzzy soft matrix. Chen et al. [43] analyzed dynamic decision-making using interval-valued triangular fuzzy soft set. Castillo et al. [44] commented on interval type-2 fuzzy sets and intuitionistic fuzzy sets. Sola et al. [45] described the relationship between interval type-2 fuzzy sets and generalization of interval-valued fuzzy sets. Alcantud [46] introduced a novel alternative approach and applied in a decision-making problem. Tripathy and Sooraj [47] solved a decision-making problem using interval-valued fuzzy soft sets. Sellappan et al. [48] evaluated the risk priority number in design failure mode and examined the effects using factor analysis. Hernandez et al. [49] proposed the model of type-2 fuzzy sets. Deli and Cagman [50] introduced two person fuzzy soft games and extended to n-person fuzzy soft games. Deli and Cagman [51] contributed a probabilistic equilibrium solution of soft games. Bajestani et al. [52] analyzed the role of interval type-2 fuzzy regression model with crisp inputs and type-2 fuzzy outputs for TAIEX forecasting. Sudharsan and Ezhilmaran [53] proposed a new aggregation operator for interval-valued intuitionistic fuzzy numbers and applied in a decision-making problem. Shenbagavalli et al. [54] determined attribute weights for fuzzy soft set and applied in a decision-making problem. Selvachandran and Sallah [55] introduced interval-valued complex fuzzy soft sets. Senthilkumar [56] solved a decision-making problem using a weighted fuzzy soft matrix. Nagarajan et al. [57] proposed the methodology for image extraction for the DICOM image using the type-2 fuzzy set. Anusuya and Nisha [58] introduced a type-2 fuzzy soft set with distance measure. Lathamaheswari et al. [59] made a review of applications of type-2 fuzzy logic in the field of biomedicine. Dinagar and Rajesh [60] introduced a new approach on the aggregation of interval-valued fuzzy soft matrix and applied in a multi-criteria decision-making problem. Qin et al. [61] proposed a new method for a decision-making problem under interval-valued intuitionistic fuzzy environment. Lathamaheswari et al. [62] made a review of applications of the type-2 fuzzy controller. Nagarajan et al. [63] applied the concept of type-2 fuzzy in image edge detection. Selvachandran and Singh [64] proposed the theoretical concepts of interval-valued complex fuzzy soft set and applied in a real-time application. Mahmooda et al. [65] introduced lattice ordered intuitionistic fuzzy soft sets. Arora and Garg [66] proposed robust aggregation operators for multi-criteria decision-making problem. Alcantud and Torrecillas [67] introduced the intertemporal choice of fuzzy soft sets. Smarandache [68] introduced extended versions of the soft set. Nagarajan et al. [69] analyzed traffic control management using interval type-2 fuzzy sets and interval neutrosophic sets. Nagarajan et al. [70] examined the stability of an intelligent system using a type-2 fuzzy controller. Shi and Fan [71] introduced fuzzy soft sets as L-fuzzy sets. Hassan and Al-Qudah [72] proposed fuzzy parameterized complex multi-fuzzy soft sets. Cakalli et al. [73] defined strong pre-continuity with fuzzy soft sets. Khali et al. [74] appraised interval-valued hesitant fuzzy soft sets and generalized trapezoidal fuzzy soft sets. Khan and Zhu [75] introduced a new methodology for parameter reduction of fuzzy soft set. Deli and Karaaslan [76] proposed bipolar FPSS-theory and applied in a decision-making problem. Deli [77] introduced convex and concave sets under soft sets and fuzzy soft sets environments. Nagarajan et al. [78] proposed a new aggregation operator using Frank triangular norms for interval-valued triangular fuzzy soft numbers and applied in a decision-making problem. Rahimi [79] applied a fuzzy soft set in patients’ prioritization.

From this literature review, to the best of our knowledge, there is no contribution of research for profit analysis in decision-making process using triangular interval type-2 fuzzy soft numbers under triangular interval type-2 fuzzy soft environment. Also, this is the first study to apply the decision-making process for analyzing profit using triangular interval type-2 fuzzy soft numbers.

Basic concepts

In this section, some of the basic concepts are presented for the better understanding of the work.

Definition 3.1 [39]

Let \( (M_{\alpha } )_{\alpha \in [0,1]} \) be a group of aggregation operators (AOs) which is non-decreasing. If A is an AO, then

$$ M_{\text{A}} :\bigcup\limits_{n \in N} {[0,\;1]^{n} } \to [0,\;1]. $$

Definition 3.2 [53]

Let \( \overline{M} \) be the triangular interval type-2 fuzzy set and is defined in Fig. 1.

Fig. 1
figure 1

Triangular interval type-2 fuzzy set

Full size image

The membership functions (MFs) are developed using the triangular fuzzy number in interval type-2 fuzzy set which is called TIT2FS. In IT2FS, upper and lower membership functions are represented by a triangular fuzzy number \( \overline{M} = \left\langle {\left[ {\underline{{l_{\text{M}} }} ,\;\overline{{l_{\text{M}} }} } \right],\;c_{M} ,\;\left[ {\underline{{r_{\text{M}} }} ,\overline{{r_{\text{M}} }} } \right]} \right\rangle \), and are defined by

$$ \begin{aligned} {\text{LMF}}_{{\overline{M} }} (x) = \left\{ {\begin{array}{*{20}l} {\frac{{x - \overline{{l_{M} }} }}{{c_{M} - \overline{{l_{M} }} }},} & {} & {\overline{{\quad l_{M} }} \le x < c_{M} } \\ {1,} & {} & {\quad x = c_{M} } \\ {\frac{{x - \underline{{r_{M} }} }}{{c_{M} - \underline{{r_{M} }} }},} & {} & {\quad c_{M} \le x < \underline{{r_{M} }} } \\ {0,} & {} & {\quad {\text{otherwise,}}} \\ \end{array} } \right. \hfill \\ \hfill \\ \end{aligned} $$
(1)
$$ {\text{UMF}}_{{\overline{M} }} (x) = \left\{ {\begin{array}{*{20}l} {\frac{{x - \underline{{l_{M} }} }}{{c_{M} - \underline{{l_{M} }} }},} & {} & {\underline{{l_{M} }} \le x < c_{M} } \\ {1,} & {} & {x = c_{M} } \\ {\frac{{x - \overline{{r_{M} }} }}{{c_{M} - \overline{{r_{M} }} }},} & {} & {c_{M} \le x < \overline{{r_{M} }} } \\ {0,} & {} & {\text{otherwise,}} \\ \end{array} } \right. $$
(2)

where \( \underline{{l_{M} }} ,\;\overline{{l_{M} }} ,\;c_{M} ,\;\underline{{r_{M} }} ,\;\overline{{r_{M} }} \) are the reference points on TIT2FS satisfying the condition \( 0 \le \underline{{l_{M} }} \le \overline{{l_{M} }} \le c_{M} \le \underline{{r_{M} }} \le \overline{{r_{M} }} \le 1 \). If we consider X as a set of real numbers, a TIT2FS in X is called TIT2FN. The FOU is the area between lower and upper membership functions in Fig. 1. If \( \,\underline{{l_{M} }} = \overline{{l_{M} }} ,\;\underline{{r_{M} }} = \overline{{r_{M} }} \), then \( {\text{UMF}}_{{\overline{M} }} (x) \) = \( {\text{LMF}}_{{\overline{M} }} (x) \) for all the values of \( x \) in \( X \), then the TIT2FS will become Type-1 case. Here, FOU is the footprint of Uncertainty.

Definition 3.3 [4]

Let \( \overline{M} = \left\langle {[\underline{{l_{M} }} ,\overline{{l_{M} }} ],c_{M} ,[\underline{{r_{M} }} ,\overline{{r_{M} }} ]\,} \right\rangle \) be a triangular interval type-2 fuzzy number (TIT2FN) in Fig. 1. The score function for TIT2FN is defined as follows:

$$ {\text{SF}}\left( {\overline{M} } \right) = \left( {\frac{{\underline{{l_{M} }} + \overline{{r_{M} }} }}{2} + 1 \times } \right)\frac{{\underline{{l_{M} }} + \overline{{l_{M} }} + \underline{{r_{M} }} + \overline{{r_{M} }} + 4c_{M} }}{8}. $$
(3)

Definition 3.4 [39]

Let \( P_{i} = \left\langle {\left[ {l_{{P_{i} }}^{ - } ,l_{{P_{i} }}^{ + } } \right],\;\;m_{{P_{i} }} ,\;\;\left[ {r_{{P_{i} }}^{ - } ,r_{{P_{i} }}^{ + } } \right]} \right\rangle ,\;\;i = 1,2,3, \ldots ,n \) be a set of triangular interval type-2 fuzzy soft numbers (TIT2FSNs) and if \( {\text{TIT}}2{\text{FSWA:}}\;\;\varOmega^{n} \to \varOmega \), then triangular interval type-2 fuzzy soft weighted arithmetic (TIT2FSWA) operator and triangular interval type-2 fuzzy soft weighted geometric (TIT2FSWG) operator are defined by

$$ {\text{TIT}}2{\text{FSWA}}_{\xi } \left( {P_{1} ,P_{2} , \ldots ,P_{n} } \right) = \xi_{1} \cdot P_{1} \oplus \xi_{2} \cdot P_{2} \oplus \cdots \oplus \xi_{n} \cdot P_{n} \quad \left( {{\text{TIT}}2{\text{FSWA}}} \right), $$
(4)
$$ {\text{TIT}}2{\text{FSWG}}_{\xi } \;\left( {P_{1} ,P_{2} , \ldots ,P_{n} } \right) = P_{1}^{{\xi_{1} }} \otimes P_{2}^{{\xi_{2} }} \otimes \cdots \otimes P_{n}^{{\xi_{n} }} \quad \left( {{\text{TIT}}2{\text{FSWG}}} \right), $$
(5)

where \( \xi = \left( {\xi_{1} ,\;\xi_{2} , \ldots ,\;\zeta_{n} } \right)^{\text{T}} \) is the weight vector and \( \xi_{i} \ge 0 \). If \( \xi = \left( {1/n,\;1/n, \ldots ,1/n} \right)^{\text{T}} \), then TIT2FSWA and TIT2FSWG operators become triangular interval type-2 fuzzy soft averaging operators.

Definition 3.5 [41]

Let \( U \) be the universe of discourse, \( \wp (U) \) be the power set of \( U \) and \( P \) a set of attributes. Then the pair \( (F,\;U) \), where \( F:P \to \wp (U) \) is called a soft set over \( U \). In other words, a soft set over \( U \) is a parameterized family of subsets of the universe. For \( \varepsilon \in P \), \( F(\varepsilon ) \) is regarded as the set of \( \varepsilon \)-approximate elements of the soft set \( (F,\;U) \).

Definition 3.6 [41]

Let \( U \) be the universe of discourse, \( \wp (U) \) be the power set of \( U \) and \( P \) a set of attributes. Then the pair \( (F,\;U) \), where \( F:P \to \wp (U) \) is called a soft set over \( U \). A triangular-valued fuzzy soft set is a parameterized family of triangular-valued fuzzy soft of \( U \) and therefore its inverse is the set of all triangular-valued fuzzy sets of \( U \) i.e., \( \wp (U) \). Hence triangular-valued fuzzy soft set is a special case of a fuzzy soft set.

Let \( A \subseteq E \), then \( (F,\;A) \) is a triangular interval type-2 fuzzy soft set of \( U \), where \( F \) is a map** given by \( F:A \to {\text{TIT}}2{\text{FS(}}U ) \). Here, \( {\text{TIT}}2{\text{FS}}(U) \) is the set of all triangular interval type-2 fuzzy sets of \( U \) and the membership degree that object \( x \) holds parameter \( e \), where \( x \in U \) and \( e \in A \), then \( F(e) = \left\{ {\left\langle {x,\left[ {\mu_{F(e)}^{ - } (x),\;\mu_{F(e)}^{ + } (x)} \right]} \right\rangle /x \in U} \right\} \). And for the triangular interval type-2 fuzzy soft set, \( F(e) = \left\{ {\left\langle {x,s_{F(e)} (x)} \right\rangle /x \in U} \right\} \), where \( s_{F(e)} (x) \) is the triangular interval type-2 fuzzy number of \( x \) in \( F(e) \).

Definition 3.7 [41]

If

$$ s = \left\{ {\begin{array}{*{20}c} {\left( {a^{ - } ,b,c^{ - } } \right)} \\ {\left( {a^{ + } ,b,c^{ + } } \right)} \\ \end{array} } \right.,\quad 0 \le a^{ - } \le a^{ + } \le b \le c^{ - } \le c^{ + } , $$

then \( s \) is called triangular interval type-2 fuzzy number.

Suppose \( s = \left[ {(a^{ - } ,\;a^{ + } );\;b;\;(c^{ - } ,\;c^{ + } )} \right] \) and \( t = [(k^{ - } ,\;k^{ + } );\;l;\;(m^{ - } ,\;m^{ + } )] \) are the triangular interval type-2 fuzzy numbers, then

$$ \begin{aligned} s \cap t & = \left[ \hbox{min} (a^{ - } ,\;k^{ - } ),\;\;\hbox{min} (a^{ + } ,\;k^{ + } );\;\;\hbox{min} (b,\;l); \right.\\ &\quad \left.\hbox{min} (c^{ - } ,\;m^{ - } ),\;\;\hbox{min} (c^{ + } ,\;m^{ + } ) \right], \end{aligned} $$
(6)
$$ \begin{aligned} s \cup t & = \left[ \hbox{max} (a^{ - } ,\;k^{ - } ),\;\;\hbox{max} (a^{ + } ,\;k^{ + } );\;\;\hbox{max} (b,\;l); \right. \\ &\quad \left. \hbox{max} (c^{ - } ,\;m^{ - } ),\;\;\hbox{max} (c^{ + } ,\;m^{ + } ) \right], \end{aligned} $$
(7)
$$ \begin{aligned} r\,s & = \left[ \left( {1 - (1 - a^{ - } )^{r} , \;\; 1 - (1 - a^{ + } )^{r} } \right); \;\; 1 - (1 - b)^{r} ; \right. \\ &\quad \left. \left( {1 - (1 - c^{ - } )^{r} ,\;\;1 - (1 - c^{ + } )^{r} } \right) \right],\quad r > 0, \end{aligned} $$
(8)

And the complementary set of \( t \) is defined as

$$ t^{c} = \left[ {\left( {1 - m^{ + } ,\;1 - m^{ - } } \right);\;1 - l;\;\left( {1 - k^{ + } ,\;1 - k^{ - } } \right)} \right]. $$
(9)

Proposed methodology

In this section, a new aggregation operator namely triangular interval type-2 fuzzy soft weighted arithmetic (TIT2FSWA) operator is proposed with their desired mathematical properties in detail.

Definition 4.1

(Aggregation Operators for Triangular Interval Type-2 Fuzzy Soft Numbers (TIT2FSNs))

Let \( P_{{k_{ij} }} = \left\langle {\left( {r_{ij}^{ - } ,\;r_{ij}^{ + } } \right),\;s_{ij} ,\;\left( {t_{ij}^{ - } ,\;t_{ij}^{ + } } \right)} \right\rangle ,\;\;i = 1,2, \ldots ,n\,{\text{and}}\,j = 1,2, \ldots ,m \) be the TIT2FSNs and \( \chi_{j} ,\,\,\xi_{i} \) be the weight vectors of the parameters \( k_{j} \)’s and experts \( x_{i} \), respectively, then \( {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{nm} }} } \right) = \mathop \oplus \nolimits_{j = 1}^{m} \chi_{j} \left( {\mathop \oplus \nolimits_{i = 1}^{n} \xi_{i} F_{{k_{ij} }} } \right) \) and satisfying the following conditions: \( \sum\nolimits_{j = 1}^{m} {\chi_{j} = 1} \) and \( \sum\nolimits_{i = 1}^{n} {\xi_{i} = 1} \), \( \chi_{j} > 0,\;\xi_{i} > 0 \).

We used the notations for \( \prod\nolimits_{j} { = \wp_{j} } \) and \( \prod\nolimits_{i} { = {\mathbb{P}}_{i} } \) throughout the paper.

Theorem 4.2

Let \( P_{{k_{ij} }} = \left\langle {\left( {r_{ij}^{ - } ,r_{ij}^{ + } } \right),\;s_{ij} ,\left( {t_{ij}^{ - } ,t_{ij}^{ + } } \right)} \right\rangle , \) \( i = 1,2, \ldots ,n\,\,{\text{and}}\,\,j = 1,2, \ldots ,m \) be the set of TIT2FSNs, the aggregated value using TIT2FSWA operator is also TIT2FSN and is given by

$$ \begin{aligned} & {\text{TIT}}2{\text{FSWA}}\left( {P_{{e_{11} }} ,\;P_{{e_{12} }} , \ldots ,P_{{e_{nm} }} } \right) \hfill \\ &\quad = \left\langle {\left[ {\left( {1 - \left( {\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - k_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right),\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - k_{ij}^{ + } } \right)^{{^{{\xi_{i} }} }} } \right)^{{\chi_{j} }} } \right)} \right)} \right]} \right., \\ &\qquad\quad 1 - \wp_{j} \left( {\left( {{{\mathbb{P}}}_{i} \left( {1 - b_{ij} } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right), \hfill \\ &\qquad \left. {\left. {\left( {1 - \left( {\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - c_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right),1 - \left( {\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - c_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right)} \right]} \right\rangle . \hfill \\ \end{aligned} $$
(10)

Proof

For \( n = 1,\xi_{1} = 1 \):

Using the operational law,

$$ \begin{aligned} c \cdot P & = \left\langle \left[ {1 - (1 - r^{ - } )^{c} ,1 - (1 - r^{ + } )^{c} } \right],1 - (1 - s)^{c} , \right. \\ &\quad \left. \left[ {1 - (1 - t^{ - } )^{c} ,1 - (1 - t^{ + } )^{c} } \right] \right\rangle , \end{aligned} $$
$$ \begin{aligned} & {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{1m} }} } \right) \hfill \\ &\quad = \mathop \oplus \limits_{j = 1}^{m} \chi_{j} P_{{k_{1j} }} \hfill \\ & \quad = \left\langle \left[ \left( {1 - \left( {\wp_{j} \left( {1 - r_{1j}^{ - } } \right)^{{\chi_{j} }} } \right), 1 - \left( {\wp_{j} \left( {1 - r_{1j}^{ + } } \right)^{{\chi_{j} }} } \right)} \right) \right], \right.\\ &\qquad \left. 1 - \left( {\wp_{j} \left( {1 - s_{1j} } \right)^{{\chi_{j} }} } \right), \right.\\ &\qquad \left. \left. \left( 1 - \left( {\wp_{j} \left( {1 - t_{1j}^{ - } } \right)^{{\chi_{j} }} } \right),1 - \left( {\wp_{j} \left( {1 - t_{1j}^{ + } } \right)^{{\chi_{j} }} } \right) \right)\right] \right\rangle . \hfill \\ \end{aligned} $$

For \( m = 1,\;\xi_{1} = 1 \)

$$ \begin{aligned} & {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{21} }} , \ldots ,P_{{k_{n1} }} } \right) = \mathop \oplus \limits_{i = 1}^{n} \xi_{i} P_{{k_{i1} }} \hfill \\ &\quad = \left\langle \left[ {\left( {1 - \left( {\wp_{i} \left( {1 - r_{i1}^{ - } } \right)^{{_{{\xi_{i} }} }} } \right),1 - \left( {\wp_{i} \left( {1 - r_{i1}^{ + } } \right)^{{_{{\xi_{i} }} }} } \right)} \right)} \right] , \right. \hfill \\ &\qquad \left. 1 - \left( {\wp_{i} \left( {1 - s_{i1} } \right)^{{\xi_{i} }} } \right), \left( 1 - \left( {\wp_{i} \left( {1 - t_{i1}^{ - } } \right)^{{\xi_{i} }} } \right), \right. \right. \\ &\qquad \left.\left. \left.1 - \left( {\wp_{j} \left( {1 - t_{i1}^{ + } } \right)^{{\xi_{i} }} } \right) \right)\right] \right\rangle . \hfill \\ \end{aligned} $$

Hence, the result is true for \( n = 1,\;m = 1 \).

Consider the result is true for \( m = q_{1} + 1,\;n = q_{2} \) and \( m = q_{1} ,\;n = q_{2} + 1 \)

Now,

$$ \begin{aligned} & \mathop \oplus \limits_{j = 1}^{{q_{1} + 1}} \chi_{j} \left( {\mathop \oplus \limits_{i = 1}^{{q_{2} }} \xi_{i} P_{{k_{ij} }} } \right) \hfill \\ &\quad = \left\langle {\left[ {\left( {1 - \wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right),\;\left( {1 - \wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right]} \right.,\\ &\quad \left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - b_{ij} } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right), \hfill \\ &\quad \left. {\left. {\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right),\;\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right]} \right\rangle . \hfill \\ \end{aligned} $$

And

$$ \begin{aligned} & \mathop \oplus \limits_{j = 1}^{{q_{1} }} \chi_{j} \left( {\mathop \oplus \limits_{i = 1}^{{q_{2} + 1}} \xi_{i} P_{{k_{ij} }} } \right) \hfill \\ &\quad = \left\langle {\left[ {\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right),\;\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right]} \right., \\ &\quad \left. { {\left[ {\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right),\,\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right]} } \right\rangle . \hfill \\ \end{aligned} $$

Therefore, it holds for \( m = q_{1} + 1,\;n = q_{2} + 1 \).

Hence by method of induction, the result is true for all the values of \( m,\;n \ge 1 \). Since, \( 0 \le r_{ij}^{ - } \le 1 \),

$$ \begin{aligned} \Leftrightarrow 0 \le {{\mathbb{P}}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} \le 1 \Leftrightarrow 0 \le \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\psi_{j} }} \le 1, \hfill \\ \Leftrightarrow 0 \le \left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right) \le 1. \hfill \\ \end{aligned} $$

Similarly,

$$ \begin{aligned} 0 \le \left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right) \le 1, \hfill \\ 0 \le \left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - s_{ij} } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right) \le 1, \hfill \\ 0 \le \left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right) \le 1, \hfill \\ 0 \le \left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right) \le 1. \hfill \\ \end{aligned} $$

For \( m = 1 \),

$$ \begin{aligned} & {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,\;P_{{k_{21} }} , \ldots ,P_{{k_{n1} }} } \right) \hfill \\ &\quad = \left\langle {\left[ {\left( {1 - {\mathbb{P}}_{i} \left( {1 - r_{i1}^{ - } } \right)^{{\xi_{i} }} } \right),\;\left( {1 - {\mathbb{P}}_{i} \left( {1 - r_{i1}^{ + } } \right)^{{\xi_{i} }} } \right)} \right]} \right.,\\ &\quad \left( {1 - {\mathbb{P}}_{i} \left( {1 - b_{i1} } \right)^{{\xi_{i} }} } \right), \hfill \\ &\quad \left. {\left. {\left( {1 - {\mathbb{P}}_{i} \left( {1 - t_{i1}^{ - } } \right)^{{\xi_{i} }} } \right),\;\left( {1 - {\mathbb{P}}_{i} \left( {1 - t_{i1}^{ + } } \right)^{{\xi_{i} }} } \right)} \right]} \right\rangle . \hfill \\ \end{aligned} $$

\( \Rightarrow \) the aggregation operator defined under TIT2FS environment is considered as a special case of the proposed operator.

Theorem 4.3

If \( P_{{k_{ij} }} = P_{k} = \left\langle {\left( {r_{{}}^{ - } ,r_{{}}^{ + } } \right),s,\left( {t_{{}}^{ - } ,t_{{}}^{ + } } \right)} \right\rangle ,\;i = 1,2, \ldots ,n\,\,{\text{and}}\,\,j = 1,2, \ldots ,m \), then

$$ {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{nm} }} } \right) = P_{k} . $$
(11)

Proof

Since all \( P_{{k_{ij} }} = P_{k} = \left\langle {\left( {r_{{}}^{ - } ,r_{{}}^{ + } } \right),s,\left( {t_{{}}^{ - } ,t_{{}}^{ + } } \right)} \right\rangle , \)

$$ \begin{aligned} & {\text{TIT2FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{nm} }} } \right) \hfill \\ &\quad = \left\langle {\left[ {\left( {1 - \left( {\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - k_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right),\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - k_{ij}^{ + } } \right)^{{^{{\xi_{i} }} }} } \right)^{{\chi_{j} }} } \right)} \right)} \right]} \right.,\\ &\quad 1 - \wp_{j} \left( {\left( {{\mathbb{P}}_{i} \left( {1 - b_{ij} } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right), \hfill \\ &\quad \left. {\left[ {1 - \left( {\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - c_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right),1 - \left( {\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - c_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right]} \right\rangle \hfill \\ &\quad = \left\langle {\left[ {1 - \left( { \left( {\left( {1 - r_{ij}^{ - } } \right)^{{\sum\limits_{i = 1}^{n} {\xi_{i} } }} } \right)^{{\sum\limits_{i = 1}^{n} {\chi_{j} } }} } \right),1 - \left( { \left( {\left( {1 - r_{ij}^{ + } } \right)^{{\sum\limits_{i = 1}^{n} {\xi_{i} } }} } \right)^{{\sum\limits_{i = 1}^{n} {\chi_{j} } }} } \right)} \right]} \right.,\\ &\qquad 1 - \left( {\left( { \left( {1 - s_{ij} } \right)^{{\sum\limits_{i = 1}^{n} {\xi_{i} } }} } \right)^{{\sum\limits_{j = 1}^{m} {\chi_{j} } }} } \right),\left. \left[ 1 - \left( { \left( {\left( {1 - t_{ij}^{ - } } \right)^{{\sum\limits_{i = 1}^{n} {\xi_{i} } }} } \right)^{{\sum\limits_{i = 1}^{n} {\chi_{j} } }} } \right), \right.\right. \\ &\quad \left.\left. 1 - \left( { \left( {\left( {1 - t_{ij}^{ + } } \right)^{{\sum\limits_{i = 1}^{n} {\xi_{i} } }} } \right)^{{\sum\limits_{i = 1}^{n} {\chi_{j} } }} } \right) \right] \right\rangle \hfill \\ &\quad = \left\langle \left[ {1 - \left( {1 - r_{{}}^{ - } } \right),1 - \left( {1 - r_{{}}^{ + } } \right)} \right], \right. \\ &\quad \left. 1 - \left( {1 - s} \right),\left[ {1 - \left( {1 - t_{{}}^{ - } } \right),1 - \left( {1 - t_{{}}^{ + } } \right)} \right] \right\rangle \\ &\quad = \left\langle {\left( {r_{{}}^{ - } ,r_{{}}^{ + } } \right),s,\left( {t_{{}}^{ - } ,t_{{}}^{ + } } \right)} \right\rangle = P_{k} . \hfill \\ \end{aligned} $$

Hence the proof.

Theorem 4.4

If \( P_{{k_{ij} }}^{ - } = \left\langle \left[ \min_{j} \min_{i} \left\{ {r_{ij}^{ - } } \right\},\min_{j} \min_{i}\right.\right.\break \left.\left. \left\{ {r_{ij}^{ + } } \right\}\, \right], \min_{j} \min_{i} \left\{ {s_{ij}^{{}} } \right\} \right.,\left. \left[ {\min_{j} \min_{i} \left\{ {t_{ij}^{ - } } \right\},\min_{j} \min_{i} \left\{ {t_{ij}^{ + } } \right\}\,} \right] \right\rangle \)and \( P_{{k_{ij} }}^{ + } = \left\langle {\left[ {\max_{j} \max_{i} \left\{ {r_{ij}^{ - } } \right\},\max_{j} \max_{i} \left\{ {r_{ij}^{ + } } \right\}\,} \right]} \right.,\max_{j} \max_{i} \left\{ {s_{ij}^{{}} } \right\}, \)\( \left. {\left[ {\max_{j} \max_{i} \left\{ {t_{ij}^{ - } } \right\},\;\max_{j} \max_{i} \left\{ {t_{ij}^{ + } } \right\}} \right]} \right\rangle \), then

$$ P_{{k_{ij} }}^{ - } \le {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{nm} }} } \right) \le P_{{k_{ij} }}^{ + } . $$
(12)

Proof

Since

$$ P_{{k_{ij} }} = \left\langle {\left( {r_{ij}^{ - } ,r_{ij}^{ + } } \right),\;s_{ij} ,\;\left( {t_{ij}^{ - } ,t_{ij}^{ + } } \right)} \right\rangle ,\;\mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} \le r_{ij}^{ - } \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\}, $$
$$ \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ + } } \right\} \le r_{ij}^{ + } \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ + } } \right\},\mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {s_{ij}^{{}} } \right\} \le s_{ij}^{{}} \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {s_{ij}^{{}} } \right\}, $$
$$ \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {t_{ij}^{ - } } \right\} \le t_{ij}^{ - } \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {t_{ij}^{ - } } \right\},\mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {t_{ij}^{ + } } \right\} \le t_{ij}^{ + } \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {t_{ij}^{ + } } \right\}. $$

Consider,

$$ \begin{aligned} & \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} \le r_{ij}^{ - } \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} \hfill \\ &\quad \Leftrightarrow 1 - \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} \le 1 - r_{ij}^{ - } \le 1 - \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} \hfill \\ &\quad \Leftrightarrow 1 - \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\}^{{\xi_{i} }} \le \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} \\ &\quad \le 1 - \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\}^{{\xi_{i} }} \hfill \\ &\quad \Leftrightarrow \left( {1 - \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\}} \right)^{{\sum\limits_{j = 1}^{m} {\chi_{j} } }} \le {{\mathbb{P}}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} \\ &\qquad \le \left( {1 - \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\}} \right)^{{\sum\limits_{j = 1}^{m} {\chi_{j} } }} \hfill \\ &\quad \Leftrightarrow \left( {1 - \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\}} \right) \le \wp_{j} \left( {{{\mathbb{P}}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }}\\ &\qquad \le \left( {1 - \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\}} \right) \hfill \\ &\quad \Leftrightarrow \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} \le 1 - \wp_{j} \left( {{{\mathbb{P}}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} \\ &\qquad \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\}. \hfill \\ \end{aligned} $$

Similarly,

$$ \begin{aligned} & \Leftrightarrow \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ + } } \right\} \le 1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ + } } \right\} \hfill \\ &\Leftrightarrow \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {s_{ij}^{{}} } \right\} \le 1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - s_{ij}^{{}} } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {s_{ij}^{{}} } \right\} \hfill \\ & \Leftrightarrow \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {t_{ij}^{ - } } \right\} \le 1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {t_{ij}^{ - } } \right\} \hfill \\ & \Leftrightarrow \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {t_{ij}^{ + } } \right\} \le 1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {t_{ij}^{ + } } \right\}. \hfill \\ \end{aligned} $$

Let \( \beta = {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{nm} }} } \right) = \left\langle {\left( {r_{\beta }^{ - } ,r_{\beta }^{ + } } \right),s,\left( {t_{\beta }^{ - } ,t_{\beta }^{ + } } \right)} \right\rangle . \)

Then, from the above results,

$$ \begin{aligned} & \Leftrightarrow \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} \le r_{\beta }^{ - } \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\}\\ & \Leftrightarrow \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ + } } \right\} \le r_{\beta }^{ + } \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ + } } \right\} \hfill \\ & \Leftrightarrow \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {s_{ij}^{{}} } \right\} \le s_{\beta }^{{}} \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {s_{ij}^{{}} } \right\} \\ &\Leftrightarrow \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {t_{ij}^{ - } } \right\} \le t_{\beta }^{ - } \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {t_{ij}^{ - } } \right\} \hfill \\ & \Leftrightarrow \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {t_{ij}^{ + } } \right\} \le t_{\beta }^{ + } \le \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {t_{ij}^{ + } } \right\}. \hfill \\ \end{aligned} $$

Thus, by the definition of a score function

$$ \begin{aligned} & S\left( \beta \right) = \left( {\frac{{r_{ij}^{ - } + t_{ij}^{ + } }}{2} + 1} \right) \times \frac{{r_{ij}^{ - } + r_{ij}^{ + } + t_{ij}^{ - } + t_{ij}^{ + } + 4s_{ij} }}{8} \hfill \\ &\quad \le \left( {\frac{{\mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} + \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {t_{ij}^{ + } } \right\}}}{2} + 1} \right) \hfill \\ &\qquad \times \frac{{\mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} + \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {r_{ij}^{ + } } \right\} + \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {t_{ij}^{ - } } \right\} + \mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {t_{ij}^{ + } } \right\} + 4\mathop {\hbox{max} }\limits_{j} \mathop {\hbox{max} }\limits_{i} \left\{ {s_{ij} } \right\}}}{8} = S\left( {P_{{k_{ij} }}^{ + } } \right) \hfill \\ &\quad S\left( \beta \right) = \left( {\frac{{r_{ij}^{ - } + t_{ij}^{ + } }}{2} + 1} \right) \times \frac{{r_{ij}^{ - } + r_{ij}^{ + } + t_{ij}^{ - } + t_{ij}^{ + } + 4s_{ij} }}{8} \ge \left( {\frac{{\mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} + \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {t_{ij}^{ + } } \right\}}}{2} + 1} \right) \hfill \\ \hfill \\ &\qquad \times \frac{{\mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ - } } \right\} + \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {r_{ij}^{ + } } \right\} + \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {t_{ij}^{ - } } \right\} + \mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {t_{ij}^{ + } } \right\} + 4\mathop {\hbox{min} }\limits_{j} \mathop {\hbox{min} }\limits_{i} \left\{ {s_{ij} } \right\}}}{8} \hfill \\ &\quad = S\left( {P_{{k_{ij} }}^{ - } } \right). \hfill \\ \end{aligned} $$
(13)

If \( S\left( {P_{{k_{ij} }}^{{}} } \right) \le S\left( {P_{{k_{ij} }}^{ + } } \right)\,\& S\left( {P_{{k_{ij} }}^{{}} } \right) \ge S\left( {P_{{k_{ij} }}^{ - } } \right) \), then by the comparison law between TIT2FSNs, we get

$$ P_{{k_{ij} }}^{ - } \le {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{nm} }} } \right) \le P_{{k_{ij} }}^{ + } . $$

Hence the theorem.

Theorem 4.5

If \( P_{k} = \left\langle {\left( {r_{{}}^{ - } ,r_{{}}^{ + } } \right),s,\left( {t_{{}}^{ - } ,t_{{}}^{ + } } \right)} \right\rangle \) is TIT2FSN, then

$$ \begin{aligned} {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} \oplus P_{k} ,P_{{k_{12} }} \oplus P_{k} , \ldots ,P_{{k_{nm} }} \oplus P_{k} } \right) \hfill \\ = {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{nm} }} } \right) \oplus P_{k} . \hfill \\ \end{aligned} $$
(14)

Proof

Since \( P_{k} \) and \( P_{{k_{ij} }}^{{}} \) are TIT2FSNs,

$$ \begin{aligned} & P_{k} \oplus P_{{k_{ij} }}^{{}} = \left\langle {\left[ {1 - \left( {1 - r^{ - } } \right)\left( {1 - r_{ij}^{ - } } \right),1 - \left( {1 - r^{ + } } \right)\left( {1 - r_{ij}^{ + } } \right)} \right]} \right., \\ & \quad 1 - \left( {1 - s} \right)\left( {1 - r_{ij}^{{}} } \right),\left. \left[ 1 - \left( {1 - t^{ - } } \right)\left( {1 - t_{ij}^{ - } } \right),\right. \right.\\ &\quad \left. \left. 1 - \left( {1 - t^{ + } } \right)\left( {1 - t_{ij}^{ + } } \right) \right] \right\rangle , \hfill \\ \end{aligned} $$
$$ \begin{aligned} & {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} \oplus P_{k} ,P_{{k_{12} }} \oplus P_{k} , \ldots ,P_{{k_{nm} }} \oplus P_{k} } \right) \hfill \\ &\quad \mathop \oplus \limits_{j = 1}^{m} \chi_{j} \left( {\mathop \oplus \limits_{i = 1}^{n} \xi_{i} \left( {P_{{k_{ij} }} \oplus P_{k} } \right)} \right) \hfill \\ &\quad = \left\langle {\left[ {\left( {1 - \wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} \left( {1 - r_{{}}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right.,} \right.\left. {\left( {1 - \wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ + } } \right)^{{\xi_{i} }} \left( {1 - r_{{}}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right],\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - s_{ij} } \right)^{{\xi_{i} }} \left( {1 - s} \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right), \hfill \\ &\quad\qquad \left[ {\left( {1 - \wp_{j} \left( {{{\mathbb{P}}}_{i} \left( {1 - t_{ij}^{ - } } \right)^{{\xi_{i} }} \left( {1 - t_{{}}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right.,\left. {\left. {\,\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ + } } \right)^{{\xi_{i} }} \left( {1 - t_{{}}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right]} \right\rangle \hfill \\ &\quad = \left\langle {\left[ {\left( {1 - \left( {1 - r_{{}}^{ - } } \right)\wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right.,} \right.\left. {\left( {1 - \left( {1 - r_{{}}^{ + } } \right)\wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right],\left( {1 - \left( {1 - s} \right)\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - s_{ij} } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right), \hfill \\ &\quad\qquad \left[ {\left( {1 - \left( {1 - t_{{}}^{ - } } \right)\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right.,\left. {\left. {\,\left( {1 - \left( {1 - t_{{}}^{ + } } \right)\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right]} \right\rangle \hfill \\ &\quad = \left\langle {\left[ {\left( {1 - \wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right.,} \right.\left. {\left( {1 - \wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right],\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - s_{ij} } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right), \hfill \\ &\qquad\quad \left[ {\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right.,\left. {\left. {\,\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right]} \right\rangle \hfill \\ &\quad \oplus \left\langle {\left( {r_{{}}^{ - } ,r_{{}}^{ + } } \right),s,\left( {t_{{}}^{ - } ,t_{{}}^{ + } } \right)} \right\rangle \hfill \\ &\quad = {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{nm} }} } \right) \oplus P_{k} . \hfill \\ \end{aligned} $$

Hence the theorem.

Theorem 4.6

For any real number \( c > 0 \), we have

$$ {\text{TIT}}2{\text{FSWA}}\left( {c \cdot P_{{k_{11} }} ,c \cdot P_{{k_{12} }} , \ldots ,c \cdot P_{{k_{nm} }} } \right) = c \cdot {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{nm} }} } \right). $$
(15)

Proof

By the operational law of TIT2FSN,

$$ \begin{aligned} c \cdot P_{{k_{ij} }} & = \left\langle \left[ {1 - \left( {1 - r_{ij}^{ - } } \right)^{c} ,1 - \left( {1 - r_{ij}^{ + } } \right)^{c} } \right], \right. \\ &\quad \left. 1 - \left( {1 - s_{ij}^{{}} } \right)^{c} ,\left[ {1 - \left( {1 - t_{ij}^{ - } } \right)^{c} ,1 - \left( {1 - t_{ij}^{ + } } \right)^{c} } \right] \right\rangle , \end{aligned} $$
$$ \begin{aligned} & {\text{TIT}}2{\text{FSWA}}\left( {c \cdot_{{k_{11} }} ,c \cdot P_{{k_{12} }} , \ldots ,c \cdot P_{{k_{nm} }} } \right) \hfill \\ &\quad = \left\langle {\left[ {\left( {1 - \wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{c \cdot \xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right.,} \right.\left. {\left( {1 - \wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ + } } \right)^{{c \cdot \xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right],\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - s_{ij} } \right)^{{c \cdot \xi_{i} }} } \right)^{{\chi_{j} }} } \right), \hfill \\ &\quad\qquad \left[ {\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ - } } \right)^{{c \cdot \xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right.,\left. {\left. {\,\left( {1 - \wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ + } } \right)^{{c \cdot \xi_{i} }} } \right)^{{\chi_{j} }} } \right)} \right]} \right\rangle \hfill \\ &\quad = \left\langle {\left[ {\left( {1 - \left\{ {\wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right\}^{c} } \right)} \right.,} \right.\left. {\left( {1 - \left\{ {\wp_{j\,} \left( {{\mathbb{P}}_{i} \left( {1 - r_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right\}^{c} } \right)} \right],\left( {1 - \left\{ {\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - s_{ij} } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right\}^{c} } \right), \hfill \\ &\qquad\quad \left[ {\left( {1 - \left\{ {\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ - } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right\}^{c} } \right)} \right.,\left. {\left. {\,\left( {1 - \left\{ {\wp_{j} \left( {{\mathbb{P}}_{i} \left( {1 - t_{ij}^{ + } } \right)^{{\xi_{i} }} } \right)^{{\chi_{j} }} } \right\}^{c} } \right)} \right]} \right\rangle \hfill \\ &\quad = c\, \cdot {\text{TIT}}2{\text{FSWA}}\left( {P_{{k_{11} }} ,P_{{k_{12} }} , \ldots ,P_{{k_{nm} }} } \right). \hfill \\ &\quad \end{aligned} $$

Hence the theorem.

Proposed algorithm

In this section, an efficiency of the proposed operators has been proved by the practical example for a decision-making problem. Consider a set of \( d \) different alternatives, \( V = \left\{ {V_{1} ,\;V_{2} , \ldots ,\;V_{d} } \right\} \) which will be evaluated by the set of \( n \) experts \( w_{1} ,\;w_{2} , \ldots ,\;w_{n} \) under the parameters \( K = \left\{ {k_{1} ,\;k_{2} , \ldots ,\;k_{m} } \right\} \) and the weight vectors are \( \chi = \left( {\chi_{1} ,\;\chi_{2} , \ldots ,\;\chi_{n} } \right)^{\text{T}} \) and \( \xi = \left( {\xi_{1} ,\;\xi_{2} , \ldots ,\;\xi_{m} } \right)^{\text{T}} \), respectively. Also,\( \xi_{i} \in (0,1] \), \( \chi_{j} \in (0,1] \), \( \sum\nolimits_{j = 1}^{m} {\chi_{j} } = 1 \) and \( \sum\nolimits_{i = 1}^{n} {\xi_{i} } = 1 \). Here, the experts/decision makers give their preferences values in terms of TIT2FSNs \( P_{{k_{ij} }} = \left\langle {\left( {r_{ij}^{ - } ,r_{ij}^{ + } } \right),s_{ij} ,\left( {t_{ij}^{ - } ,t_{ij}^{ + } } \right)} \right\rangle ,i = 1,2, \ldots ,n\,{\text{and}}\,j = 1,2, \ldots ,m \). The overall collective decision matrix is represented by \( S = \left( {P_{{k_{ij} }} } \right)_{n \times m} \). According to the expert’s preference values, the aggregated value of TIT2FSN \( \gamma_{h} ,\,\,\left( {h = 1,2, \ldots ,d} \right) \) is calculated for the alternatives using the proposed aggregation operator. The score function of the aggregated TIT2FSN \( P_{{h_{d} }} \left( {h = 1,\;2, \ldots ,\;d} \right) \) is used to rank the alternatives.

The above acknowledged methodology has been summarized as follows.

Step 1: Gather the information associated with all the alternatives under various criteria/parameters in the form of Triangular Interval Type-2 Fuzzy Soft matrix (TIT2FSM) \( S \) is defined by,

$$ S = \left\langle {\left( {r_{ij}^{ - } ,r_{ij}^{ + } } \right),s_{ij} ,\left( {t_{ij}^{ - } ,t_{ij}^{ + } } \right)} \right\rangle_{n \times m} . $$

Step 2: Normalize the matrix \( S \) by converting the rating values of the cost parameters into benefit type by applying the following formula.

$$ R_{ij} = \left\{ {\begin{array}{l} {P_{{k_{ij} }} ,\quad {\text{for}}\,{\text{benefit}}\,{\text{type}}\,{\text{parameters}}} \\ {P_{{k_{ij} }}^{c} ,\quad {\text{for}}\,{\text{cost}}\,{\text{type}}\,{\text{parameters,}}} \\ \end{array} } \right. $$

where \( P_{{k_{ij} }}^{c} = \left\langle {\left( {1 - t_{ij}^{ - } ,1 - t_{ij}^{ + } } \right),1 - s_{ij} ,\left( {1 - r_{ij}^{ - } ,1 - r_{ij}^{ + } } \right)} \right\rangle \) is the complement of \( P_{{k_{ij} }} \).

If all the parameters are of the same type, then normalization is not necessary.

Step 3: Aggregate the TIT2FSNs \( P_{{k_{ij} }} \) for all the alternatives \( V = \left\{ {V_{1} ,\;V_{2} , \ldots ,\;V_{d} } \right\} \) into the collective decision matrix \( \gamma_{h} \) using proposed TIT2FSWA operator.

Step 4: Calculate the score value of all the alternatives

Step 5: Using score value of the alternatives, select the best one.

Step 6: End.

Application for financial gain analysis using the proposed algorithm

Consider a decision-making problem for profit analysis with different alternatives. The board of four experts \( d_{1} ,\;d_{2} ,\;d_{3} ,\;d_{4} \) whose weight vector is \( \xi = \left( {0.3,\;0.2,\;0.1,\;0.4} \right)^{\text{T}} \) will award their preference values \( V_{1} ,\;V_{2} ,\;V_{3} ,\;V_{4} \) under some criteria/parameters \( K \) = {“High production (\( k_{1} \)), Number of clients (\( k_{2} \)), Number of products (\( k_{3} \)), Duration (\( k_{4} \))} with the weight vector \( \chi = \left( {0.35,\;0.25,\;0.25,\;0.15} \right)^{\text{T}} . \) Using the proposed algorithm, profit analysis is determined as follows.

Step 1: The four experts \( d_{i} \) will measure profit for four alternatives in terms of TIT2FSNs. Parameters and their values of rating are summarized as below (Tables 1, 2, 3, 4).

Table 1 Triangular interval type-2 fuzzy soft matrix for \( V_{1} \)
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Table 2 Triangular interval type-2 fuzzy soft matrix for \( V_{2} \)
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Table 3 Triangular interval type-2 fuzzy soft matrix for \( V_{3} \)
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Table 4 Triangular interval type-2 fuzzy soft matrix for \( V_{4} \)
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Step 2: Here all the parameters are in the same type, there is no necessary for normalization.

Step 3: The different opinions of the experts for each alternative are aggregated using Eq. (10).

The aggregated value of the four alternatives is,

$$ \gamma_{1} = \left\langle {\left( {0.4354,0.5422} \right),0.6271,\left( {0.6971,0.8066} \right)} \right\rangle , $$
$$ \gamma_{2} = \left\langle {\left( {0.4009,0.5044} \right),0.6019,\left( {0.6965,0.8082} \right)} \right\rangle , $$
$$ \gamma_{3} = \left\langle {\left( {0.3480,0.4018} \right),0.5130,\left( {0.6445,0.7565} \right)} \right\rangle , $$
$$ \gamma_{4} = \left\langle {\left( {0.4448,0.5481} \right),0.6090,\left( {0.6518,0.7660} \right)} \right\rangle . $$

Step 4: Using Eq. (13), the obtained score values of the four alternatives are

$$ {\text{SV}}\left( {\gamma_{1} } \right) = 1.0126, $$
$$ {\text{SV}}\left( {\gamma_{2} } \right) = 0.9663, $$
$$ {\text{SV}}\left( {\gamma_{3} } \right) = 0.8156, $$
$$ SV\left( {\gamma_{4} } \right) = 0.9726, $$

and the ranking is \( {\text{SV}}\left( {\gamma_{1} } \right) > {\text{SV}}\left( {\gamma_{2} } \right) > {\text{SV}}\left( {\gamma_{4} } \right) > {\text{SV}}\left( {\gamma_{3} } \right) \), the symbol ‘\( > \)’ represents, superior to.

Step 5: Hence, high production is the best alternative for getting more profit.

Comparative analysis

This section provides a comparative study of the proposed method with the existing method. A comparison of the results between existing and new techniques is shown in Table 5.

Table 5 Comparative analysis
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It is observed that high production and duration have more possibilities for profit analysis. These results overlap the proposed result. Hence, the proposed methodology can be utilized using the concepts of the triangular interval type-2 fuzzy soft set to solve the decision-making problem suitably in comparison with the existing methods.

Conclusion

Aggregation operator plays a vital role in decision-making as there is a need for aggregating multi attributes to decide the best one. Though many aggregation operators are available, a new aggregation operator namely a triangular interval type-2 fuzzy soft weighted arithmetic (TIT2FSWA) operator has been proposed with its desired mathematical properties in detail. Since it is the combination of fuzzy soft set and triangular interval type-2 fuzzy set, more uncertainties due to the usage of linguistic terms can be addressed well. Also, profit analysis has been done by choosing the best alternative among the alternatives among the different alternatives using the proposed aggregation operator and it is found that the first alternative high production is the best alternative for getting more profit. The present work may be extended under hyper soft set, whole hyper soft set and plithogenic hyper soft set environments.