# Evaluation of Investment Policy Based on Multi-Attribute Decision-Making Using Interval Valued T-Spherical Fuzzy Aggregation Operators

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## Abstract

**:**

## 1. Introduction

- To introduce the idea of the IVTSFS that incorporates the impression of an event with four membership grades in terms of intervals with no limitations.
- To propose aggregation tools that can be applied to those problems where the IVFS, IVIFS, and IVPFS fail to be applied.
- To study a MADM problem, utilizing the aggregation operators of IVTSFSs where the selection of the best investment policy is carried out.
- To prove that the proposed aggregation operators will reduce to match other pre-existing aggregation operators, hence showing their superiority, by using some suitable constraints.

## 2. Preliminaries

**Definition**

**1.**

**Remark**

**1.**

- (1)
- q-ROPFS if we take${\u0219}^{l}\left(\u1d8d\right)={\u0219}^{u}\left(\u1d8d\right)$and${d}^{l}\left(\u1d8d\right)={d}^{u}\left(\u1d8d\right)$. Yager [6]
- (2)
- (3)
- PyFS if we take$q=2$,${\u0219}^{l}\left(\u1d8d\right)={\u0219}^{u}\left(\u1d8d\right)$and${d}^{l}\left(\u1d8d\right)={d}^{u}\left(\u1d8d\right)$. Yager [5]
- (4)
- (5)
- IFS if we take$q=1$,${\u0219}^{l}\left(\u1d8d\right)={\u0219}^{u}\left(\u1d8d\right)$and${d}^{l}\left(\u1d8d\right)={d}^{u}\left(\u1d8d\right)$. Atanassov [3]
- (6)
- (7)

**Definition**

**2.**

**Remark**

**2.**

- (1)
- (2)
- (3)
- (4)
- (5)
- (6)

**Definition**

**3.**

- (1)
- $\u0104\u2a01\mathcal{B}=\left(\sqrt{{\u0219}_{\u0104}^{2}+{\u0219}_{\mathcal{B}}^{2}-{\u0219}_{\u0104}^{2}.{\u0219}_{\mathcal{B}}^{2}},{d}_{\u0104}.{d}_{\mathcal{B}}\right)$.
- (2)
- $\u0104\u2a02\mathcal{B}=\left({\u0219}_{\u0104}.{\u0219}_{\mathcal{B}},\sqrt{{d}_{\u0104}^{2}+{d}_{\mathcal{B}}^{2}-{d}_{\u0104}^{2}.{d}_{\mathcal{B}}^{2}}\right)$
- (3)
- $\lambda .\u0104=\left(\sqrt{1-{\left(1-{\u0219}_{\u0104}^{2}\right)}^{\lambda}},{\left({d}_{\u0104}\right)}^{\lambda}\right)$.
- (4)
- ${\u0104}^{\lambda}=\left({\left({\u0219}_{\u0104}\right)}^{\lambda},\sqrt{1-{\left(1-{d}_{\u0104}^{2}\right)}^{\lambda}}\right)$.

## 3. The Significance of Interval-Valued Fuzzy Structures

## 4. Interval-Valued T-Spherical Fuzzy Set

**Definition**

**4.**

**Theorem**

**1.**

- (1)
- (2)
- IVSFS: if we consider$q=2$.
- (3)
- (4)
- (5)
- (6)
- (7)
- (8)
- (9)
- (10)
- (11)
- (12)
- (13)

**Definition**

**5.**

- (1)
- $\u0104\oplus \mathcal{B}=\left\{\left(\left[\begin{array}{c}\sqrt[q]{{\u0219}_{\u0104}^{l}{}^{q}\left(\u1d8d\right)+{\u0219}_{\mathcal{B}}^{l}{}^{q}\left(\u1d8d\right)-{\u0219}_{\u0104}^{l}{}^{q}\left(\u1d8d\right).{\u0219}_{\mathcal{B}}^{l}{}^{q}\left(\u1d8d\right)},\\ \sqrt[q]{{\u0219}_{\u0104}^{u}{}^{q}\left(\u1d8d\right)+{\u0219}_{\mathcal{B}}^{u}{}^{q}\left(\u1d8d\right)-{\u0219}_{\u0104}^{u}{}^{q}\left(\u1d8d\right).{\u0219}_{\mathcal{B}}^{u}{}^{q}\left(\u1d8d\right)}\end{array}\right],\left[{i}_{\u0104}^{l}\left(\u1d8d\right).{i}_{\mathcal{B}}^{l}\left(\u1d8d\right),{i}_{\u0104}^{u}\left(\u1d8d\right).{i}_{\mathcal{B}}^{u}\left(\u1d8d\right)\right],\left[{d}_{\u0104}^{l}\left(\u1d8d\right).{d}_{\mathcal{B}}^{l}\left(\u1d8d\right),{d}_{\u0104}^{u}\left(\u1d8d\right).{d}_{\mathcal{B}}^{u}\left(\u1d8d\right)\right]\right)\right\}$
- (2)
- $\u0104\otimes \mathcal{B}=\left\{\left(\left[{\u0219}_{\u0104}^{l}\left(\u1d8d\right).{\u0219}_{\mathcal{B}}^{l}\left(\u1d8d\right),{\u0219}_{\u0104}^{u}\left(\u1d8d\right).{\u0219}_{\mathcal{B}}^{u}\left(\u1d8d\right)\right],\left[{i}_{\u0104}^{l}\left(\u1d8d\right).{i}_{\mathcal{B}}^{l}\left(\u1d8d\right),{i}_{\u0104}^{u}\left(\u1d8d\right).{i}_{\mathcal{B}}^{u}\left(\u1d8d\right)\right],\left[\begin{array}{c}\sqrt[q]{{d}_{\u0104}^{l}{}^{q}\left(\u1d8d\right)+{d}_{\mathcal{B}}^{l}{}^{q}\left(\u1d8d\right)-{d}_{\u0104}^{l}{}^{q}\left(\u1d8d\right).{d}_{\mathcal{B}}^{l}{}^{q}\left(\u1d8d\right)},\\ \sqrt[q]{{d}_{\u0104}^{u}{}^{q}\left(\u1d8d\right)+{d}_{\mathcal{B}}^{u}{}^{q}\left(\u1d8d\right)-{d}_{\u0104}^{u}{}^{q}\left(\u1d8d\right).{d}_{\mathcal{B}}^{u}{}^{q}\left(\u1d8d\right)}\end{array}\right]\right)\right\}$
- (3)
- $\lambda \u0104=\left(\left[\sqrt[q]{1-{\left(1-{\u0219}_{\u0104}^{l}{}^{q}\right)}^{\lambda}},\sqrt[q]{1-{\left(1-{\u0219}_{\u0104}^{u}{}^{q}\right)}^{\lambda}}\right],\left[{\left({i}_{\u0104}^{l}\right)}^{\lambda},{\left({i}_{\u0104}^{u}\right)}^{\lambda}\right],\left[{\left({d}_{\u0104}^{l}\right)}^{\lambda},{\left({d}_{\u0104}^{u}\right)}^{\lambda}\right]\right)$
- (4)
- ${\u0104}^{\lambda}=\left(\left[{\left({\u0219}_{\u0104}^{l}\right)}^{\lambda},{\left({\u0219}_{\u0104}^{u}\right)}^{\lambda}\right],\left[{\left({i}_{\u0104}^{l}\right)}^{\lambda},{\left({i}_{\u0104}^{u}\right)}^{\lambda}\right],\left[\sqrt[q]{1-{\left(1-{d}_{\u0104}^{l}{}^{q}\right)}^{\lambda}},\sqrt[q]{1-{\left(1-{d}_{\mathcal{B}}^{u}{}^{q}\right)}^{\lambda}}\right]\right)$

**Remark**

**3.**

- (1)
- TSFSs: if we consider${\u0219}^{l}={\u0219}^{u},{i}^{l}={i}^{u},$and${d}^{l}={d}^{u}$.
- (2)
- IVSFSs: if we consider$q=2$.
- (3)
- SFSs: if we consider$q=2$and${\u0219}^{l}={\u0219}^{u},{i}^{l}={i}^{u}$and${d}^{l}={d}^{u}$.
- (4)
- IVPFSs: if we consider$q=1$.
- (5)
- (6)
- IVq-ROPFSs: if we consider${i}^{l}={i}^{u}=0$.
- (7)
- q-ROPFSs: if we consider${\u0219}^{l}={\u0219}^{u},{i}^{l}={i}^{u}$and${d}^{l}={d}^{u}$.
- (8)
- (9)
- (10)
- (11)
- (12)
- IVFSs: if we consider$q=1$and${i}^{l}={i}^{u}={d}^{l}={d}^{u}=0$.
- (13)
- FSs: if we consider$q=1$and${\u0219}^{l}={\u0219}^{u},{i}^{l}={i}^{u}=0={d}^{l}={d}^{u}$.

**Theorem**

**2.**

- (1)
- $\u0104\u2a01\mathcal{B}=\mathcal{B}\u2a01\u0104$
- (2)
- $\u0104\u2a02\mathcal{B}=\mathcal{B}\u2a02\u0104$
- (3)
- $\lambda \left(\u0104\u2a01\mathcal{B}\right)=\lambda \u0104\u2a01\lambda \mathcal{B}$
- (4)
- ${\left(\u0104\u2a02\mathcal{B}\right)}^{\lambda}={\u0104}^{\lambda}\u2a02{\mathcal{B}}^{\lambda}$
- (5)
- ${\lambda}_{1}\u0104\u2a01{\lambda}_{2}\u0104=\left({\lambda}_{1}+{\lambda}_{2}\right)\u0104$
- (6)
- ${\u0104}^{{\lambda}_{1}}\u2a02{\u0104}^{{\lambda}_{2}}={\u0104}^{{\lambda}_{1}+{\lambda}_{2}}$
- (7)
- ${\left({\u0104}^{c}\right)}^{\lambda}={\left(\lambda \u0104\right)}^{c}$
- (8)
- $\lambda \left({\u0104}^{c}\right)={\left({\u0104}^{\lambda}\right)}^{c}$
- (9)
- ${\u0104}^{c}\u2a01{\mathcal{B}}^{c}={\left(\u0104\u2a02\mathcal{B}\right)}^{c}$
- (10)
- ${\u0104}^{c}\u2a02{\mathcal{B}}^{c}={\left(\u0104\u2a01\mathcal{B}\right)}^{c}$

**Proof.**

**Definition**

**6.**

**Remark**

**4.**

**Example**

**1.**

## 5. Averaging Aggregation Operators for Interval-Valued T-Spherical Fuzzy Sets

**Definition**

**7.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**(1) Idempotency:**

**Proof.**

**(2) Boundedness:**

**Proof.**

**(3) Monotonicity:**

**Proof.**

**Definition**

**8.**

**Theorem**

**5.**

**Proof.**

**Example**

**2.**

**Definition**

**9.**

**Theorem**

**6.**

**Proof.**

**Example**

**3.**

**Theorem**

**7.**

**Proof.**

## 6. Geometric Aggregation Operators for Interval-Valued T-Spherical Fuzzy Sets:

**Definition**

**10.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**(1) Idempotency:**

**(2) Boundedness:**

**(3) Monotonicity:**

**Definition**

**11.**

**Theorem**

**10.**

**Example**

**4.**

**Definition**

**12.**

**Theorem**

**11.**

**Proof.**

**Example**

**5.**

**Theorem**

**12.**

**Proof.**

## 7. Multi-Attribute Decision-Making Investment Planning

**Step****1:**- The decision makers assess the alternatives, given the attributes, and provide their information in the form of a decision matrix.
**Step****2:**- Apply the proposed aggregation tools to the decision matrix obtained in Step 1.
**Step****3:**- Compute the scores of the IVTSFNs obtained in Step 2.
**Step****4:**- Analyze the score values of the alternatives and rank them to obtain the best alternative.

**Example**

**6.**

**Step 1:**The evaluation of alternatives, ${\u0104}_{i}$, by the decision makers in the form of decision matrix ${D}_{ij}$.

**Step 2:**By applying the IVTSFWA operators to the decision matrix provided in Step 1, we get:

**Step 3:**This step involves the computation of the score values as such:

**Step 4:**In this step, the score values obtained in Step 3 are analyzed. Based on the score values, the ranking of the alternatives is given:

**Step 1:**We use the same evaluation of alternatives, ${\u0104}_{i}$, in the form of decision matrix ${D}_{ij}$ as above.

**Step 2:**By applying the IVTSFWG operators to the decision matrix provided in Step 1, we get: