A method to solve strategy based decision making problems with logarithmic T-spherical fuzzy aggregation framework

Abstract

T-Spherical fuzzy set (TSFS) is an improved extension in fuzzy set (FS) theory that takes into account four angles of the human judgment under uncertainty about a phenomenon that is membership degree (MD), abstinence degree (AD), non-membership degree (NMD), and refusal degree (RD). The purpose of this manuscript is to introduce and investigate logarithmic aggregation operators (LAOs) in the layout of TSFSs after observing the shortcomings of the previously existing AOs. First, we introduce the notions of logarithmic operations for T-spherical fuzzy numbers (TSFNs) and investigate some of their characteristics. The study is extended to develop T-spherical fuzzy (TSF) logarithmic AOs using the TSF logarithmic operations. The main theory includes the logarithmic TSF weighted averaging (LTSFWA) operator, and logarithmic TSF weighted geometric (LTSFWG) operator along with the conception of ordered weighted and hybrid AOs. An investigation about the validity of the logarithmic TSF AOs is established by using the induction method and examples are solved to examine the practicality of newly developed operators. Additionally, an algorithm for solving the problem of best production choice is developed using TSF information and logarithmic TSF AOs. An illustrative example is solved based on the proposed algorithm where the impact of the associated parameters is examined. We also did a comparative analysis to examine the advantages of the logarithmic TSF AOs.

Keywords

T-Spherical fuzzy set logarithmic operations spherical fuzzy set multi-attribute decision making methods

1 Introduction

Fuzziness and vagueness consistently exist in decision-making, quite possibly the main issues are to speak to credit alternatives properly. As of late, many devices have been created for portraying and communicating fuzziness and ambiguity. To handle such kinds of issues, the frame of the fuzzy set (FS) was established by Zadeh [1]. The FS takes a function to describe the MD of the element to a specific set under certain conditions, where the ordinary sets fail. The range for assigning the MDs to the elements of FS is 0 to 1 wherein 0 corresponds to no inclusion and 1 corresponds to full inclusion of the elements. Sometimes, our opinion about a phenomenon is of two types, i.e., we can be in favor of an object to some extent and at the same time, we may have a negative opinion about the same object. Unfortunately, the FS describes such a situation but with no flexibility which leads Atanassov [2] to develop the frame of intuitionistic FS (IFS) where an MD is accompanied by an NMD to describe the affiliation of some certain element to some specific set with considerably greater flexibility compared to the notion of FS. The restriction in an IFS on NMD and MD is that 0 ≤ MD + NMD ≤ 1. Yager [3] relaxed the restriction that is imposed on duplets of information in intuitionistic fuzzy settings by developing the notion of Pythagorean FS (PyFS). Yager’s restriction on the duplet of information is that 0 ≤ MD² + NMD² ≤ 1. Despite, IFS and PyFS can portray uncertain information precisely but still there exist some duplets of information that cannot be represented by IFS either by PyFS. To deal with such issues, a more comprehensive and flexible fuzzy framework of q-rung orthopair fuzzy set (QROFS) was developed by Yager [4]. The QROFS consists of all those duplets for which 0 ≤ MD^q + NMD ^q ≤ 1 for $q \in ℤ^{+}$ . Some recent work on the theory and applications of these duplets can be found in [5 –8].

The duplets of information presented using the frame of IFS, PyFS, and QROFS only discuss the two perspectives of the vagueness while the extent of remaining abstains denoted by AD and the measure of not taking part in the evaluation denoted by RD are being ignored that leads to the loss of information. Observing this issue led Cuong [9] to define the concept of picture FS (PFS) that uses four grades i.e., MD, AD, NMD, and RD to evaluate the membership of an object to specific sets. Cuong’s PFS inherit the same restriction of Atanassov IFS i.e., 0 ≤ MD + AD + NMD ≤ 1. Due to this restriction, only finitely many triplets can be used to portray fuzzy information that led Mahmood et al. [10] to introduce the frames of spherical FS (SFS) and subsequently TSFS to synthesize information where the notion of PFS gets fail. The restraint of SFS and TSFS are defined as 0 ≤ MD² + AD² + NMD² ≤ 1, 0 ≤ MD^q + AD^q + NMD^q ≤ 1 respectively. The framework of TSFS is an advanced version of FS theory to tackle information with greater flexibility and less information loss due to the usage of four kinds of degrees i.e., NMD, AD, MD, and RD. A handsome amount of work in this regard is being done on the AOs and similarity measures of the TSFS by several authors such as Ullah et al. [11 –14]. Garg et al. [15, 16] investigated the conception of TSF power AOs for multi attribute decision making (MADM) purposes. For some other relevant studies, one is referred to [17 –21].

Several types of AOs can be used to deal with MADM problems. Among this research, the notion of LAOs is considered the prominent one. In literature, a huge amount of work is established on the theory and applications of the LAOs in the layouts of IFS, PyFSs, ROFSs, and PFSs, etc. A study of LAOs in the settings of IFSs is investigated by Li and Wei [22]. Applications of MADM algorithms based on LAOs of PyFSs are comprehensively studied by Garg [23]. The study of LAOs has been extended to picture fuzzy environments for the purpose to solve MADM problems by [24]. Garg and Rani [25] developed LAOs in complex intuitionistic fuzzy settings to deal with MADM problems. The framework of neutrosophic sets for the aggregation of information based on LAOs is used by Garg [26]. Zhou et al. [27] investigated some logarithmic proportional AOs in group decision-making. Zhou et al. [28] also discussed some generalized ordered weighted logarithmic harmonic AOs for MADM purposes. For some other relevant theories on aggregation, one is referred to [29 –34].

As previously discussed, the environments of IFS, PyFS, and QROFS lead to the loss of information whenever we use these frameworks for modeling human opinion. Similarly, the framework of PFS and SFS have their shortcomings. Because of this observation, it is evident that the theories presented in [22 –24] are limited whenever it comes to their applicability. Due to this reason, we aim to introduce the conception of LAOs in the environment of TSFS where the uncertain information is depicted using the MD along with AD, NMD, and RD with greater flexibility by using a parameter q. The advantages of the logarithmic function are discussed below.

1. The rate of change can be represented graphically, using the logarithmic functions as an operator.

2. The data value can be elevated for small inputs.

3. The logarithmic scale can be used as a statistical auxiliary [35].

4. Resolving exponential functions into linear functions, logarithmic operators can improve the understanding.

The key advantage of the application of logarithmic can be considered as the transformation of exponential function into the linear function. Since exponential functions are hard to understand, for the general people, thus, using the logarithmic application, simplification can be conducted. In a similar vein, plotting large values in graph paper is not possible using ordinary mathematical approaches. In this context, logarithm considers 10 as the base value of the function, and thereby large values can be plotted in the graph papers quite easily. But in this manuscript, we are more generalized the base of the logarithmic into δ with a rule that is $0 < δ \leq min {s, q \sqrt[q]{1 - i {(u)}^{q}}, \sqrt[q]{1 - d {(u)}^{q}}} < 1, δ \neq 1$ . Keeping the advantages of the LOLs, Garg [23] firstly utilized the principle of LOLs in the environment of PyFSs and discussed their application in the MADM technique. For the duration of the aggregation technique, the main technique is to characterize the operational laws. Considering these outstanding activities, he introduced some new Pythagorean fuzzy exponential aggregation operators. Another side from, another significant numerical activity is LOLs, which are very valuable in the field of aggregation procedure. As such, with the creating sound of the PyFS both all around and in scope, there is a need to develop some new operational laws and accumulation operators. The existing operators are defined only based on PyFSs, up to date no one proposed these operators based on SFS and TSFSs. By keeping the advantages of TSFS which contain the degree of truth, abstinence, and falsity with a rule that is the sum of the q-powers of all grades is limited to the unit interval. The summary of the investigated approaches is discussed below.

1. To investigate logarithmic operational laws based on TSFSs.

2. To use logarithmic operational laws, we explore the idea of LAOs namely LTSFWA and LTSFWG operators.

3. It is proved that the previous work becomes the special case of the currently proposed work.

4. A MADM procedure is developed and exemplified by using the presented TSF LAOs.

5. A comparative study of the TSF LAOs with previous work is set up to see the advantages of the newly established work.

6. To explore the LOLs and their properties.

7. To find the validity and proficiency of the explored works, we resolved some numerical examples by using the proposed operators.

8. The advantages, comparative analysis, and graphical expressions of the discovered theory are also discussed.

The rest of this manuscript is designed as in Section 2, we recall the idea of TSFS and some relevant theory. In section 3, we presented the logarithmic operational laws based on TSFSs, and some thorough investigation is carried out with the help of some results. In section 4, we carried out an investigation of the LAOs of TSFSs by introducing the notion of LTSFWA and LTSFWG operators. The study is extended to introduce the conception of logarithmic TSF ordered weighted averaging (LTSFOWA) and logarithmic TSF ordered weighted geometric (LTSFOWG) operators. An investigation about the characteristics of the TSF LAOs is also carried out. In Section 5, it is proved that the proposed LAOs are the generalization of the existing LAOs. In Section 6, a MADM procedure is developed by using the presented LAOs based on TSF information followed by a comprehensive example. The comparative analysis and advantages of the proposed approaches are discussed in Section 7 followed by a summary of the paper in Section 8.

2 Preliminaries

In this study, we recall the idea of TSFS and its operational laws. The idea of the TSF weighted averaging (TSFWA) operator and TSF ordered weighted averaging (TSFOWA) operator are also discussed in this section. Throughout this manuscript, the symbols i, s, and d express the AD, MD, and NMD where the universal set is denoted by X.

Definition 1: [10] A TSFS is demonstrated by: $B = {(u, (s (u), i (u), d (u))) : u \in X}$ (1) with a rule that 0 ≤ s^q (u) + i^q (u) + d^q (u) ≤ 1 for $q \in ℤ^{+}$ . The RD is given as $π (u) = \sqrt[q]{1 - s^{q} (u) - i^{q} (u) - d^{q} (u)}$ . The TSFNs are denoted and defined by: $B_{k} = (s_{k}, i_{k}, d_{k}), k = 1, 2, 3 \dots . . σ$ .

Definition 2: [10] For any three TSFNs $B = (s, i, d), B_{1} = (s_{1}, i_{1}, d_{1})$ , and $B_{2} = (s_{2}, i_{2}, d_{2})$ , we have $\begin{matrix} 1) B^{c} = (d, i, s) \\ 2) B_{1} \leq B_{2} iff s_{1} \leq s_{2}, i_{1} \geq i_{2} and d_{1} \geq d_{2} \\ 3) B_{1} = B_{2} iff B_{1} \leq B_{2} and B_{2} \leq B_{1} \\ 4) B_{1} \cap B_{2} = (min (s_{1}, s_{2}), min (i_{1}, i_{2}), max (d_{1}, d_{2})) \\ 5) B_{1} \cup B_{2} = (max (s_{1}, s_{2}), min (i_{1}, i_{2}), min (d_{1}, d_{2})) \\ 6) B_{1} \oplus B_{2} = (\sqrt[q]{s_{1}^{q} + s_{2}^{q} - s_{1}^{q} s_{2}^{q}}, i_{1} i_{2}, d_{1} d_{2}) \\ 7) B_{1} \otimes B_{2} = (s_{1} s_{2}, \sqrt[q]{i_{1}^{q} + i_{2}^{q} - i_{1}^{q} i_{2}^{q}}, \sqrt[q]{d_{1}^{q} + d_{2}^{q} - d_{1}^{q} d_{2}^{q}}) \\ 8) δ B_{1} = (\sqrt[q]{1 - {(1 - s_{1}^{q})}^{δ}}, i_{1}^{δ}, d_{1}^{δ}), δ > 0 \end{matrix}$ $\begin{matrix} 9) B_{1}^{δ} = (s_{1}^{δ}, \sqrt[q]{1 - {(1 - i_{1}^{q})}^{δ}}, \sqrt[q]{1 - {(1 - d_{1}^{q})}^{δ}}) \\ 10) δ^{B} = {\begin{matrix} (δ^{\sqrt[q]{1 - s^{q}}}, \sqrt[q]{1 - δ^{q i}}, \sqrt[q]{1 - δ^{q d}}) & δ \in (0, 1) \\ ({(\frac{1}{δ})}^{\sqrt[q]{1 - s^{q}}}, \sqrt[q]{1 - {(\frac{1}{δ})}^{q i},} \sqrt[q]{1 - {(\frac{1}{δ})}^{q d}}) & δ \geq 1 \end{matrix} \end{matrix}$

Definition 3: [10] For any TSFN $B = (s, i, d)$ , the score value can be computed as: $\begin{matrix} S (B) = s^{q} (u) - (d^{q} (u)) (π^{q} (u)), \\ S (B) \in [- 1, 1] \end{matrix}$ and the accuracy value can be computed as: $\begin{matrix} H (B) = (s^{q} (u) + i^{q} (u) + d^{q} (u)), \\ H (B) \in [0, 1] \end{matrix}$

To differentiate any number of TSFNs $B_{1} = (s_{1}, i_{1}, d_{1})$ and $B_{2} = (s_{2}, i_{2}, d_{2})$ , we use the following rules such that:

If $S (B_{1}) > S (B_{2})$ , then $B_{1} > B_{2}$

If $S (B_{1}) = S (B_{2})$ , then,

If $H (B_{1}) > H (B_{2})$ , then $B_{1} > B_{2}$

If $H (B_{1}) = H (B_{2})$ , then $B_{1} = B_{2}$

Furthermore, for a collection of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k})$ where k = 1, 2, 3 … σ, we define the following operational laws [14]:

1) A TSFWA operator is demonstrated by: $TSFWA (B_{1}, B_{2}, \dots, B_{σ}) = (\sqrt[q]{1 - \prod_{k = 1}^{σ} (1 - s_{k}^{q} (u))^{ω_{k}}}, \prod_{k = 1}^{σ} i_{k}^{ω_{k}}, \prod_{k = 1}^{σ} d_{k}^{ω_{k}})$ (2) where ω_k = (ω₁, ω₂, …, ω_σ) ^T express the weighted weight vector with a rule that is $\sum_{k = 1}^{σ} ω_{k} = 1$ .

2) A TSFOWA operator is demonstrated by: $TSFOWA (B_{1}, B_{2,} B_{3, \dots,} B_{σ}) = (\sqrt[q]{1 - \prod_{k = 1}^{σ} (1 - s_{σ (k)}^{q} (u))^{ω_{k}}}, \prod_{k = 1}^{σ} i_{σ (k)}^{ω_{k}}, \prod_{k = 1}^{σ} d_{σ (k)}^{ω_{k}})$ (3) where (σ (1) , σ (2) , …, σ (k)) express permutations of k = 1, 2, 3 …… σ, and ω_k = (ω₁, ω₂, . . , ω_σ) ^T express the weighted weight vector with a rule that is $\sum_{k = 1}^{σ} ω_{k} = 1$ .

3 T-spherical fuzzy logarithmic operational laws

The purpose of this study is to present some logarithmic operational laws (LOLs) using TSFNs based on these laws we explore the ideas of AOs for TSFNs. Suppose $B$ and δ express the TSFNs and real numbers such that δ ≠ 1, wherein log _δ0 and log ₁u is not investigated in positive numbers. We suppose that a TSFN $B = (s, i, d) \neq 0 = (0, 0, 0)$ and δ ≠ 1 throughout this manuscript.

Definition 4: For any TSFN $B = (s, i, d)$ , the LOLs are demonstrated by: ${log}_{δ} B = {\begin{matrix} u, \sqrt[q]{1 - {log}_{δ} s^{q} (u)}, \\ {log}_{δ} (\sqrt[q]{1 - i^{q} (u)}), {log}_{δ} (\sqrt[q]{1 - d^{q} (u)}) : u \in X \end{matrix}}$ (4)

It can be seen that the ${log}_{δ} B$ is also a TSFN. By the Definition 1 for all u ∈ X with a rule that 0 ≤ s^q (u) + i^q (u) + d^q (u) ≤ 1. If $0 < δ \leq min {s (u), \sqrt[q]{1 - i^{q} (u)}, \sqrt[q]{1 - d^{q} (u)}} \leq 1$ and δ ≠ 1, then:

The MD function is given by: $\sqrt[q]{1 - {({log}_{δ} s (u))}^{q}} : X \to [0, 1],$ $\forall u \in X \to {log}_{δ} (\sqrt[q]{1 - ({log}_{δ} s (u))^{q}}) \in [0, 1]$ (5)

The AD function is given by: ${log}_{δ} (\sqrt[q]{1 - i {(u)}^{q})}) : X \to [0, 1],$ $\forall u \in X \to {log}_{δ} (\sqrt[q]{1 - i {(u)}^{q}}) \in [0, 1]$ (6)

And the NMD function is given by: ${log}_{δ} (\sqrt[q]{1 - d {(u)}^{q}}) : X \to [0, 1],$ $\forall u \in X \to {log}_{δ} (\sqrt[q]{1 - d {(u)}^{q}}) \in [0, 1]$ (7)

Therefore, ${log}_{δ} B$ becomes a TSFS given by: ${log}_{δ} B = {\begin{matrix} (\begin{matrix} u, (\sqrt[q]{1 - ({log}_{δ} s {(u)}^{q}}), \\ {log}_{δ} (\sqrt[q]{1 - i {(u)}^{q}}), {log}_{δ} (\sqrt[q]{1 - d {(u)}^{q}}) \end{matrix}) : u \in X \\ 0 < δ \leq \min {s (u), \\ \sqrt[q]{1 - i {(u)}^{q}}, \sqrt[q]{1 - d {(u)}^{q}}} \leq 1, δ \neq 1 \end{matrix}}$ (8)

Definition 5: For any TSFN $B = (s, i, d)$ , the logarithmic TSFN (LTSFN) is demonstrated by: ${log}_{δ} B = {\begin{matrix} (\begin{matrix} \sqrt[q]{1 - ({log}_{δ} s {(u)}^{q},} \\ {log}_{δ} (\sqrt[q]{1 - i {(u)}^{q}}, \\ {log}_{δ} (\sqrt[q]{1 - d {(u)}^{q}} \end{matrix}); & 0 < δ \leq min {\begin{matrix} s, \sqrt[q]{1 - i {(u)}^{q},} \\ \sqrt[q]{1 - d {(u)}^{q}} \end{matrix}} < 1 \\ (\begin{matrix} \sqrt[q]{1 - ({log}_{\frac{1}{δ}} s {(u)}^{q}}, \\ {log}_{\frac{1}{δ}} (\sqrt[q]{1 - i {(u)}^{q}}, \\ {log}_{\frac{1}{δ}} (\sqrt[q]{1 - d {(u)}^{q}} \end{matrix}); & 0 < \frac{1}{δ} \leq min {\begin{matrix} s, \sqrt[q]{1 - i {(u)}^{q}}, \\ \sqrt[q]{1 - d {(u)}^{q}} \end{matrix}} < 1, δ \neq 1 \end{matrix}$ (9)

Theorem 1: For any TSFN $B = (s, i, d)$ , the value of ${log}_{δ} B$ is again a TSFN.

Proof: Allow TSFN $B = (s (u), i (u), d (u))$ satisfies s, i, d ∈ [0, 1]with a rule that 0 ≤ s^q + i^q + d^q ≤ 1. Then we discuss the following two steps:

Step 1: When 0 $< δ \leq min {s, \sqrt[q]{1 - i {(u)}^{q}}, \sqrt[q]{1 - d {(u)}^{q}}} < 1, δ \neq 1,$ with ${log}_{δ} B$ concerning δ . Thus, $0 \leq {log}_{δ} s, {log}_{δ} (\sqrt[q]{1 - i^{q}}, {log}_{δ} (\sqrt[q]{1 - d^{q}}) \leq 1$ . Hence $0 \leq \sqrt[q]{1 - {({log}_{δ} s)}^{q}} \leq 1$ , 0 $\leq {log}_{δ} (\sqrt[q]{1 - i^{q}}) \leq 1$ , $0 \leq {log}_{δ} (\sqrt[q]{1 - d^{q}}) \leq 1$ and $0 \leq 1 - ({log}_{δ} s)^{q} + ({log}_{δ} (\sqrt[q]{1 - i^{q}}))^{q} + {log}_{δ} (\sqrt[q]{1 - d^{q}}))^{q} \leq 1 .$ Therefore, ${log}_{δ} B$ is a TSFN $\forall q \in ℤ^{+}$ .

Step 2: After λ < 1 and $0 < \frac{1}{δ} < 1$ and $\frac{1}{δ} \leq min {s, \sqrt[q]{1 - i^{q}}, \sqrt[q]{1 - d^{q}}}$ , so it is easy to get that ${log}_{δ} B$ is a TSFN.

Example 1: For any TSFN $B = (0.7, 0.6, 0.5)$ and δ = 0.5, q = 3, we have ${log}_{δ} B = (\begin{matrix} \sqrt[3]{1 - {({log}_{0.5} (0.7))}^{3}}, \\ {log}_{0.2} (\sqrt[3]{1 - {(0.6)}^{3}}), {log}_{0.2} (\sqrt[3]{1 - {(0.5)}^{3}}) \end{matrix})$ $= (0.9523, 0.1171, 0.064)$

If δ = 6 then ${log}_{\frac{1}{δ}} B = (\begin{matrix} \sqrt[3]{1 - {({log}_{\frac{1}{6}} (0.7))}^{3}}, \\ {log}_{\frac{1}{6}} (\sqrt[3]{1 - {(0.6)}^{3}}), {log}_{\frac{1}{6}} (\sqrt[3]{1 - {(0.5)}^{3}}) \end{matrix})$ $= (0.9975, 0.0443, 0.0243)$

Theorem 2: For any TSFN $B = (s, i, d)$ , if 0 $< δ \leq min {s, \sqrt[q]{1 - i^{q}}, \sqrt[q]{1 - d^{q}}} \leq 1, δ \neq 1$ , then $\begin{matrix} 1) δ^{{log}_{δ} B} = B \\ 2) {log}_{δ} δ^{B} = B \end{matrix}$

Proof. 1) By using Definition 2, we obtain $δ^{{log}_{δ} B} = (\begin{matrix} δ^{\sqrt[q]{1 - (1 - {({log}_{δ} s)}^{q}}}, \\ \sqrt[q]{1 - δ^{q {log}_{δ} (\sqrt[q]{1 - i^{q}})},} \\ \sqrt[q]{1 - δ^{q {log}_{δ} (\sqrt[q]{1 - d^{q}})}} \end{matrix})$ $= (δ^{{log}_{δ} s}, \sqrt[q]{1 - (1 - i^{q}),} \sqrt[q]{1 - (1 - d^{q})})$ $= (s, i, d)$ $= B$

2) Again, by using Definition 2, we obtain: $\begin{matrix} {log}_{δ} δ^{B} = {log}_{δ} (δ^{\sqrt[q]{1 - s^{q}},} \sqrt[q]{1 - δ^{q i},} \sqrt[q]{1 - δ^{q d}}) \\ = (\begin{matrix} {(\sqrt[q]{1 - ({log}_{δ} δ^{\sqrt[q]{1 - s^{q}}}})}^{q}, \\ {log}_{δ} (\sqrt[q]{1 - (1 - δ^{q i})}), \\ {log}_{δ} (\sqrt[q]{1 - (1 - δ^{q d})}) \end{matrix}) \\ = (\sqrt[q]{1 - (1 - s^{q})}, i, d) \\ = (s, i, d) \\ = B \end{matrix}$

Theorem 3: For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = (1, 2)$ with $0 < δ \leq min {s_{k}, \sqrt[q]{1 - i_{k}^{q},} \sqrt[q]{1 - d_{k}^{q}}} \leq 1$ and δ ≠ 1. Then

${log}_{δ} B_{1} \oplus {log}_{δ} B_{2} = {log}_{δ} B_{2} \oplus {log}_{δ} B_{1}$

${log}_{δ} B_{1} \otimes {log}_{δ} B_{2} = {log}_{δ} B_{2} \otimes {log}_{δ} B_{1}$

Proof. Straightforward.

Theorem 4: For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = (1, 2)$ , with $0 < δ \leq min {s_{k}, \sqrt[q]{1 - i_{k}^{q},} \sqrt[q]{1 - d_{k}^{q}}} \leq 1$ and δ ≠ 1. Then

$({log}_{δ} B_{1} \oplus {log}_{δ} B_{2}) \oplus {log}_{δ} B_{3} = {log}_{δ} B_{1} \oplus ({log}_{δ} B_{2} \oplus {log}_{δ} B_{3})$

$({log}_{δ} B_{1} \otimes {log}_{δ} B_{2}) \otimes {log}_{δ} B_{3} = {log}_{δ} B_{1} \otimes ({log}_{δ} B_{2} \otimes {log}_{δ} B_{3})$

Proof. Straightforward.

Theorem 5. For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) k = (1, 2)$ , with $0 < δ \leq min {s_{k}, \sqrt[q]{1 - i_{k}^{q},} \sqrt[q]{1 - d_{k}^{q}}} \leq 1$ and δ ≠ 1, p, p₁, p₂ > 0. Then

p (log _λ α₁ ⊕ log _λ α₂) = p log _λ α₁ ⊕ p log _λ α₂

(log _λ α₁ ⊗ log _λ α₂) ^p = (log _λ α₁) ^p ⊗ (log _λ α₂) ^p

p₁ log _λ α₁ ⊕ p₂ log _λ α₁ = (p₁ + p₂) log _λ α₁

(log _λ α₁) ^p
₁ ⊗ (log _λ α₁) ^p
₂ = (log _λ α₁) ^p₁+p₂

((log _λ α₁) ^p
₁) ^p
₂ = (log _λ α₁) ^{p
₁
p
₂}

Proof. Trivial.

Theorem 6: For any TSFN (s, i, d), if $0 < δ_{1} \leq δ_{2} \leq \min {s, \sqrt[q]{1 - i^{q}}, \sqrt[q]{1 - d^{q}}} \leq 1, δ_{1}, δ_{2} \neq 1$ , then ${log}_{δ_{1}} B \geq {log}_{δ_{2}} B$ and ${log}_{δ_{1}} B \leq {log}_{δ_{2}} B$ for $0 < \frac{1}{δ_{2}} \leq \frac{1}{δ_{1}} \leq \min {s, \sqrt[q]{1 - i^{q}}, \sqrt[q]{1 - d^{q}}} \leq 1, δ_{1}, δ_{2} \neq 1$ .

Proof. Based on Definition 2, we obtain: $\begin{matrix} {log}_{δ_{1}} B = (\sqrt[q]{1 - {({log}_{δ_{1}} s_{1})}^{q}}, {log}_{δ_{1}} (\sqrt[q]{1 - i^{q}}), {log}_{δ_{1}} (\sqrt[q]{1 - d^{q}})) \\ {log}_{δ_{2}} B = (\sqrt[q]{1 - {({log}_{δ_{1}} s_{2})}^{q}}, {log}_{δ_{2}} (\sqrt[q]{1 - i^{q}}), {log}_{δ_{2}} (\sqrt[q]{1 - d^{q}})) \end{matrix}$

If $0 < δ_{1} \leq δ_{2} \leq \min {s, \sqrt[q]{1 - i^{q}}, \sqrt[q]{1 - d^{q}}} \leq 1, δ_{1}, δ_{2} \neq 1$ , then $1 - {({log}_{δ_{1}} s_{1})}^{q} \geq 1 - {({log}_{δ_{1}} s_{2})}^{q}, {log}_{δ_{1}} (\sqrt[q]{1 - i^{q}}) \leq {log}_{δ_{2}} (\sqrt[q]{1 - i^{q}})$ and ${log}_{δ_{1}} (\sqrt[q]{1 - d^{q}}) \leq {log}_{δ_{2}} (\sqrt[q]{1 - d^{q}})$

By using δ₁, δ₂ > 1 and δ₁ ≤ δ₂ then we get $0 < \frac{1}{δ_{2}} \leq \frac{1}{δ_{1}} \leq \min {s, \sqrt[q]{1 - i^{q}}, \sqrt[q]{1 - d^{q}}} \leq 1, δ_{1}, δ_{2} \neq 1,$ so, by hypothesis, we know that ${log}_{δ_{1}} B \leq {log}_{δ_{2}} B$ .

Theorem 7: For any two TSFNs $B_{1} = (s_{1}, i_{1}, d_{1})$ and $B_{2} = (s_{2}, i_{2}, d_{2})$ , if s₁ ≤ s₂, i₁ ≥ i₂ and d₁ ≥ d₂ $B_{1} \leq B_{2}, 0 < δ \leq \min {s_{k}, \sqrt[q]{1 - i_{k}^{q}}, \sqrt[q]{1 - d_{k}^{q}}} \leq 1, δ \neq 1,$ then ${log}_{δ} B_{1} \leq {log}_{δ} B_{2}$ .

Proof. Straightforward.

4 TSF logarithmic aggregation operators

This section aims to introduce the novel concepts of the LTSFWA and LTSFWG operators using the LOLs of TSFNs discussed in the previous section. Throughout this paper, we denote the weight vector by ω_k = (ω₁, ω₂ … , ω_σ) ^T such that ω_k > 0 and $\sum_{k = 1}^{σ} ω_{k} = 1$ .

Definition 6. For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = 1, 2, \dots, σ)$ with $0 < δ_{1} \leq \min {s, \sqrt[q]{1 - i^{q}}, \sqrt[q]{1 - d^{q}}} \leq 1, δ_{1} \neq 1$ , the LTSFW operator LTSFWA :⊖ ^σ → ⊖ is demonstrated by: $LTSFWA (B_{1}, B_{2}, \dots, B_{σ}) = ω_{1} {log}_{δ_{1}} B_{1} \oplus ω_{2} {log}_{δ_{2}} B_{2} \oplus \dots ω_{σ} {log}_{δ_{σ}} B_{σ}$ (10)

Theorem 8: For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = 1, 2, \dots, σ)$ , by using Definition 2 and Definition 6, we obtain: $\begin{matrix} LTSFWA (B_{1}, B_{2}, \dots, B_{σ}) = \\ {\begin{matrix} (\begin{matrix} \sqrt[q]{1 - \prod_{k = 1}^{σ} {(\log_{δ_{k}} s_{k})}^{q ω_{k}}}, \\ \prod_{k = 1}^{σ} {(\log_{δ_{k}} (\sqrt[q]{1 - i_{k}^{q}}))}^{ω_{k}}, \\ \prod_{k = 1}^{σ} {(\log_{δ_{k}} (\sqrt[q]{1 - d_{k}^{q}}))}^{ω_{k}} \end{matrix}); 0 < δ_{k} \leq \\ min {s_{k}, \sqrt[q]{1 - i_{k}^{q}}, \sqrt[q]{1 - d_{k}^{q}}} \leq 1, δ_{k} \neq 1 \\ (\begin{matrix} \sqrt[q]{1 - \prod_{k = 1}^{σ} {(\log_{\frac{1}{δ_{k}}} s_{k})}^{q ω_{k}}}, \\ \prod_{k = 1}^{σ} {(\log_{\frac{1}{δ_{k}}} (\sqrt[q]{1 - i_{k}^{q}}))}^{ω_{k}}, \\ \prod_{k = 1}^{σ} {(\log_{\frac{1}{δ_{k}}} (\sqrt[q]{1 - d_{k}^{q}}))}^{ω_{k}} \end{matrix}); 0 < \frac{1}{δ_{k}} \leq \\ min {s_{k}, \sqrt[q]{1 - i_{k}^{q}}, \sqrt[q]{1 - d_{k}^{q}}} \leq 1, δ_{k} \neq 1 \end{matrix} \end{matrix}$ (11)

Proof. Trivial.

Remark 1. If $δ_{1} = δ_{2} = \dots = δ_{σ} = δ,; 0 < δ \leq \min {s_{k}, \sqrt[q]{1 - i_{k}^{q}}, \sqrt[q]{1 - d_{k}^{q}}} \leq 1, δ \neq 1$ , then LTSFWA operator is: $LTSFWA (B_{1}, B_{2}, \dots, B_{σ}) = (\begin{matrix} \sqrt[q]{1 - \prod_{k = 1}^{σ} {({log}_{δ} s_{k})}^{q ω_{k}}}, \\ , \prod_{k = 1}^{σ} {({log}_{δ} (\sqrt[q]{1 - i_{k}^{q}}))}^{ω_{k}}, \\ , \prod_{k = 1}^{σ} {({log}_{δ} (\sqrt[q]{1 - d_{k}^{q}}))}^{ω_{k}} \end{matrix})$ (12)

Example 2. Let $B_{1} = (0.4, 0.01, 0.5), B_{2} = (0.3, 0.03, 0.6)$ , and $B_{3} = (0.5, 0.05, 0.7)$ be three TSFSNs and ω = (0.1, 0.2, 0.4) ^T is the weight vector of them. Suppose δ₁ = δ₂ = δ₃ = 0.2 and q = 3then $LTSFWA (B_{1}, B_{2}, B_{3}) = (\begin{matrix} \sqrt[q]{1 - \prod_{k = 1}^{3} {({log}_{δ_{k}} s_{k})}^{q ω_{k}}}, \\ \prod_{k = 1}^{3} {({log}_{δ_{k}} (\sqrt[q]{1 - i_{k}^{q}}))}^{ω_{k}}, \\ \prod_{k = 1}^{3} {({log}_{δ_{k}} (\sqrt[q]{1 - d_{k}^{q}}))}^{ω_{k}} \end{matrix})$ $= (\begin{matrix} \sqrt[3]{\begin{matrix} 1 - {({log}_{0.2} (0.4))}^{0.3} \\ \times ({log}_{0.2} (0.3))^{0.6} \times {({log}_{0.2} (0.5))}^{0.12} \end{matrix}}, \\ {log}_{0.2} {(\sqrt[3]{1 - {(0.01)}^{3}})}^{0.1} \times \\ {log}_{0.2} {(\sqrt[3]{1 - {(0.03)}^{3}})}^{0.2} \times {log}_{0.2} {(\sqrt[3]{1 - {(0.05)}^{3}})}^{0.4}, \\ {log}_{0.2} {(\sqrt[3]{1 - {(0.5)}^{3}})}^{0.1} \times {log}_{0.2} {(\sqrt[3]{1 - {(0.6)}^{3}})}^{0.2} \\ {log}_{0.2} {(\sqrt[3]{1 - {(0.7)}^{3}})}^{0.4} \end{matrix})$ $= (0.8855, 0.0000076, 0.1447)$

For $δ_{1} = δ_{2} = \dots = δ_{σ} = δ, 0 < δ \leq \min_{k} {s_{(k)}, \sqrt[q]{1 - i_{(k)}^{q}}, \sqrt[q]{1 - d_{(k)}^{q}}} \leq 1, δ_{k} \neq 1$ and ω_k be the weight vector of TSFN $B_{k}$ such that ω_k > 0 and $\sum_{k = 1}^{σ} ω_{k} = 1$ .

Property 1. For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = 1, 2, \dots, σ)$ , if $B_{k} = B (k = 1, 2, \dots, σ)$ , we have $LTSFWA (B_{1}, B_{2}, \dots, B_{σ}) = {log}_{δ} B$ (13)

Proof. Straightforward.

Property 2. For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = 1, 2, \dots . σ)$ , if $B^{-} = (\min_{k} {s_{k}}, \max_{k} {i_{k}}, \max_{k} {d_{k}})$ and $B^{+} = (\max_{k} {s_{k}}, \min_{k} {i_{k}}, \min_{k} {d_{k}}),$ then ${log}_{δ} B^{-} \leq LTSFWA (B_{1}, B_{2}, \dots, B_{σ}) \leq {log}_{δ} B^{+}$ (14)

Proof. Straightforward.

Property 3. For any two families of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k})$ and $B_{k}^{*} = (s_{k}^{*}, i_{k}^{*}, d_{k}^{*})$ (k = 1, 2 … , σ), if $s_{k} \leq s_{k}^{*}, i_{k} \leq i_{k}^{*}$ and $d_{k} \leq d_{k}^{*}$ , then

$\begin{matrix} LTSFWA (B_{1}, B_{2}, \dots, B_{σ}) \\ \leq LTSFWA (B_{1}^{*}, B_{2}^{*}, \dots, B_{σ}^{*}) \end{matrix}$ (15)

Proof. Straightforward.

Definition 7. For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = 1, 2, \dots, σ)$ , with $0 < δ_{1} \leq \min {s, \sqrt[q]{1 - i^{q}}, \sqrt[q]{1 - d^{q}}} \leq 1, δ_{1} \neq 1$ , then the LTSFOWA operatorLTSFOWA :⊖ ^σ → ⊖ is demonstrated by:

$\begin{matrix} LTSFOWA (B_{1}, B_{2}, \dots, B_{σ}) \\ = (ω_{1} {log}_{δ_{O (1)}} B_{O (1)}) \oplus (ω_{2} {log}_{δ_{O (2)}} B_{O (2)}) \\ \oplus \dots \oplus (ω_{O} {log}_{δ_{O (σ)}} B_{O (σ)}) \end{matrix}$ (16) where O represent the permutation of $(1, 2, \dots, σ) \forall B_{O (k - 1)} \geq B_{O (k)}$ where k = 2, 3, …, σ.

Theorem 9. For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = 1, 2, \dots, σ)$ , then by using Definition 2 and Definition 7, we obtain: $\begin{matrix} LTSFOWA (B_{1}, B_{2}, \dots, B_{σ}) = \\ {\begin{matrix} (\begin{matrix} \sqrt[q]{1 - \prod_{k = 1}^{σ} {(\log_{δ_{o (k)}} s_{o (k)})}^{q ω_{k}}}, \\ \prod_{k = 1}^{σ} {(\log_{δ_{o (k)}} (\sqrt[q]{1 - i_{o (k)}^{q}}))}^{ω_{k}}, \\ \prod_{k = 1}^{σ} {(\log_{δ_{o (k)}} (\sqrt[q]{1 - d_{o (k)}^{q}}))}^{ω_{k}} \end{matrix}); 0 < δ_{o (k)} \leq \\ min {s_{o (k)}, \sqrt[q]{1 - i_{o (k)}^{q}}, \sqrt[q]{1 - d_{o (k)}^{q}}} \leq 1, δ_{k} \neq 1 \\ (\begin{matrix} \sqrt[q]{1 - \prod_{k = 1}^{σ} {(\log_{\frac{1}{δ_{o (k)}}} s_{o (k)})}^{q ω_{k}}}, \\ \prod_{k = 1}^{σ} {(\log_{\frac{1}{δ_{o (k)}}} (\sqrt[q]{1 - i_{o (k)}^{q}}))}^{ω_{k}}, \\ \prod_{k = 1}^{σ} {(\log_{\frac{1}{δ_{o (k)}}} (\sqrt[q]{1 - d_{o (k)}^{q}}))}^{ω_{k}} \end{matrix}); 0 < \frac{1}{δ_{o (k)}} \leq \\ min {s_{o (k)}, \sqrt[q]{1 - i_{o (k)}^{q}}, \sqrt[q]{1 - d_{o (k)}^{q}}} \leq 1, δ_{k} \neq 1 \end{matrix} \end{matrix}$ (17)

Proof: Straightforward.

Definition 8: For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = 1, 2, \dots, σ)$ ,with $0 < δ_{1} \leq \min {s, \sqrt[q]{1 - i^{q}}, \sqrt[q]{1 - d^{q}}} \leq 1, δ_{1} \neq 1$ , then the LTSFWG operatorLTSFWG :⊖ ^σ → ⊖ is demonstrated by:

$\begin{matrix} LTSFWG (B_{1}, B_{2}, \dots, B_{σ}) \\ = {({log}_{δ_{1}} B_{1})}^{ω_{1}} \otimes {({log}_{δ_{2}} B_{2})}^{ω_{2}} \\ \otimes \dots \otimes {({log}_{δ_{σ}} B_{σ})}^{ω_{σ}} \end{matrix}$ (18)

Theorem 10: For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = 1, 2, \dots, σ)$ , then by using Definition 2 and Definition 8, we obtain: $\begin{matrix} LTSFWG (B_{1}, B_{2}, \dots, B_{σ}) = \\ {\begin{matrix} (\begin{matrix} \prod_{k = 1}^{σ} \sqrt[q]{{(1 - {(\log_{δ_{k}} s_{k})}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(\log_{δ_{k}} (\sqrt[q]{1 - i_{k}^{q}}))}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(\log_{δ_{k}} (\sqrt[q]{1 - d_{k}^{q}}))}^{q})}^{ω_{k}}} \end{matrix}); 0 < δ_{k} \leq min_{k} {s_{k}, \sqrt[q]{1 - i_{k}^{q}}, \sqrt[q]{1 - d_{k}^{q}}} \leq 1, δ_{k} \neq 1 \\ \begin{matrix} (\begin{matrix} \prod_{k = 1}^{σ} \sqrt[q]{{(1 - {(\log_{\frac{1}{δ_{k}}} s_{k})}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(\log_{\frac{1}{δ_{k}}} (\sqrt[q]{1 - i_{k}^{q}}))}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(\log_{\frac{1}{δ_{k}}} (\sqrt[q]{1 - d_{k}^{q}}))}^{q})}^{ω_{k}}} \end{matrix}); 0 < \frac{1}{δ_{k}} \leq min_{k} {s_{k}, \sqrt[q]{1 - i_{k}^{q}}, \sqrt[q]{1 - d_{k}^{q}}} \leq 1, δ_{k} \neq 1 \end{matrix} \end{matrix} \end{matrix}$ (19)

Proof: Trivial.

Definition 9: For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = 1, 2, \dots, σ)$ , with $0 < δ_{1} \leq \min {s, \sqrt[q]{1 - i^{q}}, \sqrt[q]{1 - d^{q}}} \leq 1, δ_{1} \neq 1$ , then the LTSFOWG operator LTSFOWG :⊖ ^σ → ⊖ is demonstrated by:

$\begin{matrix} LTSFOWG (B_{1}, B_{2}, \dots, B_{σ}) \\ = {({log}_{δ_{O (1)}} B_{O (1)})}^{ω_{1}} \otimes {({log}_{δ_{O (2)}} B_{O (2)})}^{ω_{2}} \\ \otimes \dots \otimes {({log}_{δ_{O (σ)}} B_{O (σ)})}^{ω_{σ}} \end{matrix}$ (20) where is O the permutation of (1, 2, …, σ) such that $B_{O (k - 1)} \geq B_{O (k)}$ and $0 < δ_{O (k)} \leq \min {s_{O (k)}, \sqrt[q]{1 - i_{O (k)}^{q}}, \sqrt[q]{1 - d_{O (k)}^{q}}} \leq 1, δ_{O (k)} \neq 1$ fork = 2, 3 … σ.

Theorem 11: For any family of TSFNs $B_{k} = (s_{k}, i_{k}, d_{k}) (k = 1, 2, \dots, σ)$ , then by using Definition 2 and Definition 9, we obtain: $\begin{matrix} LTSFOWG (B_{1}, B_{2}, \dots, B_{σ}) = \\ {\begin{matrix} (\begin{matrix} \prod_{k = 1}^{σ} \sqrt[q]{{(1 - {(\log_{δ_{o (k)}} s_{k})}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(\log_{δ_{o (k)}} (\sqrt[q]{1 - i_{o (k)}^{q}}))}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(\log_{δ_{o (k)}} (\sqrt[q]{1 - d_{k}^{q}}))}^{q})}^{ω_{k}}} \end{matrix}); 0 < δ_{o (k)} \leq min_{k} {s_{k}, \sqrt[q]{1 - i_{o (k)}^{q}}, \sqrt[q]{1 - d_{o (k)}^{q}}} \leq 1, δ_{k} \neq 1 \\ \begin{matrix} (\begin{matrix} \prod_{k = 1}^{σ} \sqrt[q]{{(1 - {(\log_{\frac{1}{δ_{o (k)}}} s_{k})}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(\log_{\frac{1}{δ_{o (k)}}} (\sqrt[q]{1 - i_{o (k)}^{q}}))}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(\log_{\frac{1}{δ_{o (k)}}} (\sqrt[q]{1 - d_{o (k)}^{q}}))}^{q})}^{ω_{k}}} \end{matrix}); 0 < \frac{1}{δ_{o (k)}} \leq min_{k} {s_{k}, \sqrt[q]{1 - i_{o (k)}^{q}}, \sqrt[q]{1 - d_{o (k)}^{q}}} \leq 1, δ_{k} \neq 1 \end{matrix} \end{matrix} \end{matrix}$ (21)

Proof: Straightforward.

Remark 2: If $δ_{1} = δ_{2} = \dots . = δ_{σ} = δ, 0 < δ \leq \min_{k} {s_{O (k)}, \sqrt[q]{1 - i_{O (k)}^{q}}, \sqrt[q]{1 - d_{O (k)}^{q}}} \leq 1$ . Then LTSFOWA and LTSFOWG operators are having the following shape: $LTSFOWA (B_{1} . B_{1}, B_{1}) = (\begin{matrix} \sqrt[q]{1 - \prod_{k = 1}^{σ} {({log}_{λ} s_{O (k)})}^{{q ω}_{k}}}, \\ \prod_{k = 1}^{σ} {({log}_{λ} (\sqrt[q]{1 - i_{O (k)}^{q}}))}^{ω_{k}}, \\ \prod_{k = 1}^{σ} {({log}_{λ} (\sqrt[q]{1 - d_{O (k)}^{q}}))}^{ω_{k}} \end{matrix})$ $LTSFOWG (B_{1} . B_{1}, B_{1}) = (\begin{matrix} \prod_{k = 1}^{σ} \sqrt[q]{{(1 - {({log}_{δ} s_{O (k)})}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {({log}_{δ} (\sqrt[q]{1 - i_{O (k)}^{q}}))}^{q})}^{ω_{k}},} \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {({log}_{δ} (\sqrt[q]{1 - d_{O (k)}^{q}}))}^{q})}^{ω_{k}}} \end{matrix})$

5 Consequences

In this section, we discuss some special cases of LAOs of TSFS because of some limitations on the proposed LTSFWA and LTSFWG operators.

Firstly, the logarithmic AOs for TSFSs is written as: $LTSFWA (B_{1}, B_{2}, \dots, B_{σ}) = (\begin{matrix} \sqrt[q]{1 - \prod_{k = 1}^{σ} {({log}_{δ} s_{k})}^{q ω_{k}}}, \\ \prod_{k = 1}^{σ} {({log}_{δ} (\sqrt[q]{1 - i_{k}^{q}}))}^{ω_{k}}, \\ \prod_{k = 1}^{σ} {({log}_{δ} (\sqrt[q]{1 - d_{k}^{q}}))}^{ω_{k}} \end{matrix})$ $LTSFWG (B_{1}, B_{2}, \dots B_{σ}) = (\begin{matrix} \prod_{k = 1}^{σ} \sqrt[q]{{(1 - {({log}_{δ_{k}} s_{k})}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(({log}_{δ_{k}} (\sqrt[q]{1 - i_{k}^{q}}))}^{q})}^{ω_{k}},} \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(({log}_{δ_{k}} (\sqrt[q]{1 - d_{k}^{q}}))}^{q})}^{ω_{k}}} \end{matrix})$

1) The LTSFWA and LTSFWG operators reduce to logarithmic spherical fuzzy weighted averaging (LSFWA) and weighted geometric (LSFWG) operators if we take q = 2 $LSFWA (B_{1}, B_{2}, \dots, B_{σ}) = (\begin{matrix} \sqrt{1 - \prod_{k = 1}^{σ} {({log}_{δ} s_{k})}^{2 ω_{k}}}, \\ \prod_{k = 1}^{σ} {({log}_{δ} (\sqrt{1 - i_{k}^{2}}))}^{ω_{k}}, \\ \prod_{k = 1}^{σ} {({log}_{δ} (\sqrt{1 - d_{k}^{2}}))}^{ω_{k}} \end{matrix})$ $LSFWG (B_{1}, B_{2}, \dots, B_{σ}) = (\begin{matrix} \prod_{k = 1}^{σ} \sqrt{{(1 - {({log}_{δ_{k}} s_{k})}^{2})}^{ω_{k}}}, \\ \sqrt{1 - \prod_{k = 1}^{σ} {(1 - {({log}_{δ_{k}} (\sqrt{1 - i_{k}^{2}}))}^{2})}^{ω_{k}}} \\ \sqrt{1 - \prod_{k = 1}^{σ} {(1 - {({log}_{δ_{k}} (\sqrt{1 - d_{k}^{2}}))}^{2})}^{ω_{k}}} \end{matrix})$

2) The LTSFWA and LTSFWG operators reduce to logarithmic picture fuzzy weighted averaging (LPFWA) and weighted geometric (LPFWG) operators if we take q = 1 $LPFWA (B_{1}, B_{2}, \dots, B_{σ}) = (\begin{matrix} 1 - \prod_{k = 1}^{σ} {({log}_{δ} s_{k})}^{ω_{k}}, \\ \prod_{k = 1}^{σ} {({log}_{δ} (1 - s_{k}))}^{ω_{k}}, \\ \prod_{k = 1}^{σ} {({log}_{δ} (1 - d_{k}))}^{ω_{k}} \end{matrix})$ $LPFWG (B_{1}, B_{2}, \dots, B_{σ}) = (\begin{matrix} \prod_{k = 1}^{σ} {(1 - ({log}_{δ_{k}} s_{k}))}^{ω_{k}}, \\ (1 - \prod_{k = 1}^{σ} {(1 - ({log}_{δ_{k}} (1 - i_{k}))}^{ω_{k}}), \\ (1 - \prod_{k = 1}^{σ} {(1 - ({log}_{δ_{k}} (1 - d_{k}))}^{ω_{k}}) \end{matrix})$

3) The LTSFWA and LTSFWG operators reduce to logarithmic q-rung orthopair fuzzy weighted averaging (LQROFWA) and geometric (LQROFWG) operators if we neglect the AD. $LQROFWA (B_{1}, B_{2}, \dots, B_{σ}) = (q \sqrt[q]{1 - \prod_{k = 1}^{σ} {({log}_{δ} s_{k})}^{q ω_{k}}}, \prod_{k = 1}^{σ} {({log}_{δ} (\sqrt{1 - d_{k}^{2}}))}^{ω_{k}})$ $LQROFWG (B_{1}, B_{2}, \dots B_{σ}) = (\begin{matrix} \prod_{k = 1}^{σ} \sqrt[q]{{(1 - {({log}_{δ_{k}} s_{k})}^{q})}^{ω_{k}}}, \\ \sqrt[q]{1 - \prod_{k = 1}^{σ} {(1 - {(({log}_{δ_{k}} (\sqrt[q]{1 - d_{k}^{q}}))}^{q})}^{ω_{k}}} \end{matrix})$

4) The LTSFWA and LTSFWG operators reduce to logarithmic Pythagorean fuzzy weighted averaging (LPyFWA) and geometric (LPyFWG) operators if we neglect the AD and take q = 2 $LPyFWA (B_{1}, B_{2}, \dots, B_{σ}) = (\begin{matrix} \sqrt{1 - \prod_{k = 1}^{σ} {({log}_{δ} s_{k})}^{2 ω_{k}}}, \\ \prod_{k = 1}^{σ} {({log}_{δ} (\sqrt{1 - d_{k}^{2}}))}^{ω_{k}} \end{matrix})$ $LPyFWG (B_{1}, B_{2}, \dots B_{σ}) = (\begin{matrix} \prod_{k = 1}^{σ} \sqrt{{(1 - {({log}_{δ_{k}} s_{k})}^{2})}^{ω_{k}}}, \\ \sqrt{1 - \prod_{k = 1}^{σ} {(1 - {(({log}_{δ_{k}} (\sqrt{1 - d_{k}^{2}}))}^{2})}^{ω_{k}}} \end{matrix})$

5) The LTSFWA and LTSFWG operators reduce to logarithmic intuitionistic fuzzy weighted averaging (LIFWA) and geometric (LIFWG) operators if we neglect the AD and we take q = 1 $LIFWA (B_{1}, B_{2}, \dots, B_{σ}) = (\begin{matrix} 1 - \prod_{k = 1}^{σ} {({log}_{δ} s_{k})}^{ω_{k}}, \\ \prod_{k = 1}^{σ} {({log}_{δ} (1 - d_{k}))}^{ω_{k}} \end{matrix})$ $LIFWG (B_{1}, B_{2}, \dots B_{σ}) = (\begin{matrix} \prod_{k = 1}^{σ} {(1 - ({log}_{δ_{k}} s_{k}))}^{ω_{k}}, \\ (1 - \prod_{k = 1}^{σ} {(1 - ({log}_{δ_{k}} (1 - d_{k}))}^{ω_{k}}) \end{matrix})$ \begin twocolumn\end twocolumn

6 MADM Procedure by Using Investigated Operators

In this section, we aim to utilize the MADM procedure using TSF information. As discussed earlier, the decision making process involves human opinion and a human opinion is based on the favor, abstinence, against, and refusal degree that can be modeled with four types of degrees i.e. MD, AD, NMD, and RD. Thus representing information using a TSFS is to cover all possible aspects of human opinion.

In this section, we aim to develop a MADM procedure based on LAOs. For this, we choose the family of alternatives such that $B = {B_{1}, B_{2}, \dots . B_{m}}$ and their attributes $E = {E_{1}, E_{2}, \dots . . E_{n}}$ based on which the alternatives are to be ranked. ω_k = (ω₁, ω₂, …, ω_σ) ^T is used as a weight vector of the TSF information. In the process of MADM, we describe the alternatives $B = {B_{1}, B_{2}, \dots . B_{m}}$ using TSFNs under the observation of selected attributes. A brief algorithm is proposed as follows:

Step 1: Develop the decision matrix, where MDs, NMDs, and ADs are used to describe the finite alternatives $B = {B_{1}, B_{2} \dots B_{m}}$ under the attributes $E = {E_{1}, E_{2}, \dots E_{n}}$ . A common value of q is obtained for which every triplet in the decision matrix represents a TSFN.

Step 2: By using the LTSFWA and LTSFWG operators to aggregate the TSF information provided in the decision matrix.

Step 3: At this stage, we aim to rank the alternatives $B = {B_{1}, B_{2}, \dots B_{m}}$ using the score values of the aggregated information.

Step 4: Rank all the alternatives by examining their scores and choose the optimum one.

A flowchart elaborating all the MADM steps is given in Fig. 1 as below:

Fig. 1

Flowchart of the MADM algorithm.

Example 3: An example based on MADM is discussed by Garg [23] where strategic decision-making is analyzed in the case of the production of different products. We adapt the same example and discuss it in the environment of TSFSs. In it, expect that an organization is going to make a new product to expand its business. The organization aims to achieve maximum profit goal and for that, the intended product must be useful for a large number of customers. There can be more than 2 to 3 levels of customers for whom the product is supposed to be designed.

$B_{1}$ : Lower-class-oriented product.

$B_{2}$ : Middle-class-oriented product.

$B_{3}$ : Upper-class-oriented product.

$B_{4}$ : Product that is designed for the lower and middle class.

$B_{5}$ : Product that is designed for the middle and upper class.

The above-discussed strategies are considered as alternatives. Further, in any business, there can be more than one types of benefits from a specific product and can be categorized as:

$E_{1} :$ Profits in the short run.

$E_{2} :$ Profit in long run.

$E_{3} :$ Profit in a specific period.

$E_{4} :$ Average profit and image of the business.

$E_{5} :$ Other factors.

The five strategies are examined based on the five attributes as discussed above observing a weight vector defined by (0.3, 0.2, 0.1, 0.25, 0.15) ^Tand following the steps of the MADM algorithm defined earlier the stepwise calculations are as follows:

Step 1: In this step, we develop the decision matrix where every item is in the form of TSFN provided by the decision-making panel of the organization. The decision matrix is given as follows:

Step 2: By using the LTSFWA and LTSFWG operators, we aggregate the information of the decision matrix, which are discussed in Table 2, such that:

Table 2

Aggregated values by using the LTSFWA and LTSFWG operators

	LTSFWA	LTSFWG
$B_{1}$	(0.0368, 0.0174, 0.0054)	(0.0494, 0.0872, 0.1527)
$B_{2}$	(0.0872, 0.0064, 0.0022)	(0.0174, 0.1440, 0.2092)
$B_{3}$	(0.1440, 0.0022, 0.0006)	(0.0064, 0.2092, 0.2860)
$B_{4}$	(0.2092, 0.0006, 0.0001)	(0.0022, 0.2860, 0.3793)
$B_{5}$	(0.0872, 0.0174, 0.0006)	(0.0174, 0.0872, 0.2860)

Step 3: By using the score function of TSFNs defined in Definition 3 we find the score values of the aggregated information in Step 2.

Step 4: We Rank all the alternatives, and the ranking of the alternatives can be described as: $B_{4} \geq B_{3} \geq B_{5} \geq B_{2} \geq B_{1}$ (22) $B_{1} \geq B_{2} \geq B_{4} \geq B_{3} \geq B_{5}$ (23)

Table 3

Score values of the aggregated values of Table 2

Methods	Score Values
LTSFWA	$S (B_{1}) = 0.0316, S (B_{2}) = 0.0853, S (B_{3}) = 0.1435, S (B_{4}) = 0.2091, S (B_{5}) = 0.0867$
LTSFWG	$S (B_{1}) = - 0.0591, S (B_{2}) = - 0.1143, S (B_{3}) = - 0.1362, S (B_{4}) = - 0.1240, S (B_{5}) = - 0.1569$

Table 1

Decision matrix based on TSFNs

	$E_{1}$	$E_{2}$	$E_{3}$	$E_{4}$	$E_{5}$
$B_{1}$	(0.9, 0.8, 0.7)	(0.91, 0.81, 0.71)	(0.92, 0.82, 0.72)	(0.93, 0.83, 0.73)	(0.94, 0.84, 0.64)
$B_{2}$	(0.8, 0.7, 0.6)	(0.81, 0.71, 0.61)	(0.82, 0.72, 0.62)	(0.83, 0.73, 0.63)	(0.84, 0.74, 0.64)
$B_{3}$	(0.7, 0.6, 0.5)	(0.71, 0.61, 0.51)	(0.72, 0.62, 0.52)	(0.73, 0.63, 0.53)	(0.74, 0.64, 0.54)
$B_{4}$	(0.6, 0.5, 0.4)	(0.61, 0.51, 0.41)	(0.62, 0.52, 0.42)	(0.63, 0.53, 0.43)	(0.64, 0.54, 0.44)
$B_{5}$	(0.8, 0.8, 0.5)	(0.81, 0.81, 0.51)	(0.82, 0.82, 0.52)	(0.83, 0.83, 0.53)	(0.84, 0.84, 0.54)

The best option is $E_{4}$ and $E_{1}$ by using both AOs. To examine the reliability and effectiveness of the AOs, we choose the TSF types of information, and by using the investigated operators to resolve them.

Moreover, to find the proficiency and validity of the presented approaches, we compare the investigated operators with some existing operators which are discussed in the form of the next section with the help of Example 1. To find the reliability and accuracy of the explored operators, the information related to existing operators is discussed as follows: Garg et al. [16] explored the improved interactive AOs by using TSFSs. Liu et al. [20] developed the Einstein hybrid AOs by using TSFSs.

Hamacher AOs by using TSFSs were presented by Ullah et al. [11]. Then the comparative analysis of the investigated operators with some existing operators [16 , 11] is discussed in the form of Table 4.

Table 4

Comparative analysis of the proposed operators with some existing operators

Method	Score Values	Ranking Values
Garg et al. [16]	$S (E_{1}) = - 0.5185, S (E_{2}) = - 0.6304,$ $S (E_{3}) = - 0.6839,$ $S (E_{4}) = - 0.7334, S (E_{5}) = - 0.8374$	$E_{1} \geq E_{2} \geq E_{3} \geq E_{4} \geq E_{5}$
	$S (E_{1}) = 0.4193, S (E_{2}) = 0.7551,$ $S (E_{3}) = 0.9056,$ $S (E_{4}) = 0.9680, S (E_{5}) = 0.7200$	$E_{4} \geq E_{3} \geq E_{2} \geq E_{5} \geq E_{1}$
Liu et al. [20]	$S (E_{1}) = - 0.5499, S (E_{2}) = - 0.8100,$ $S (E_{3}) = - 0.9282,$ $S (E_{4}) = - 0.9765, S (E_{5}) = - 0.7485$	$E_{1} \geq E_{5} \geq E_{2} \geq E_{3} \geq E_{4}$
	$S (E_{1}) = 0.04998, S (E_{2}) = 0.3100,$ $S (E_{3}) = 0.4282,$ $S (E_{4}) = 0.4764, S (E_{5}) = 0.2486$	$E_{1} \geq E_{4} \geq E_{3} \geq E_{2} \geq E_{1}$
Ullah et al. [11]	$S (E_{1}) = - 0.6083, S (E_{2}) = - 0.8527,$ $S (E_{3}) = - 0.9510,$ $S (E_{4}) = - 0.9861, S (E_{5}) = - 0.7955$	$E_{1} \geq E_{5} \geq E_{2} \geq E_{3} \geq E_{4}$
	$S (E_{1}) = 0.1083, S (E_{2}) = 0.3527,$ $S (E_{3}) = 0.4510,$ $S (E_{4}) = 0.4861, S (E_{5}) = 0.2955$	$E_{4} \geq E_{3} \geq E_{2} \geq E_{5} \geq E_{1}$
Proposed Methods	$S (E_{1}) = 0.0316, S (E_{2}) = 0.0853,$ $S (E_{3}) = 0.1435,$ $S (E_{4}) = 0.2091, S (E_{5}) = 0.0867$	$E_{4} \geq E_{3} \geq E_{5} \geq E_{2} \geq E_{1}$
	$S (E_{1}) = - 0.0591, S (E_{2}) = - 0.1143,$ $S (E_{3}) = - 0.1362,$ $S (E_{4}) = - 0.1240, S (E_{5}) = - 0.1569$	$E_{1} \geq E_{2} \geq E_{4} \geq E_{3} \geq E_{5}$

From the above analysis, we obtain the same ranking results by using the existing operators and proposed operators. The obtained ranking is followed as: $E_{1} \geq E_{2} \geq E_{4} \geq E_{3} \geq E_{5}$ and the best option is $E_{1}$ . For ease of analysis, we portray the above-discussed results in the following Fig. 2.

Fig. 2

Graphical expressions of the information of the Table 4.

It can be seen that Fig. 2 contains three existing operators and one explored operator whose values are discussed in the form of Fig. 2. Each existing idea is containing five alternatives which are shown in different forms of colors. Therefore, the investigated operators based on TSFSs are extensively useful and more valid as compared to the existing ideas [16 , 11].

7 Discussion

LAOs are among the list of aggregation operators where the aggregation of fuzzy information involves logarithmic calculations. The LAO is successfully applied in various fuzzy frameworks including IFS, PyFSs, and PFSs. We proposed the idea of LAOs in the frame of TSFSs where uncertain information is expressed using four grades consisting of MD, NMD, AD, and RD and reduces information loss as a human opinion has exactly four types of opinion about an event. We mainly proposed two types of LAOs for TSFSs including LTSFWA and LTSFWG operators for aggregation purposes. It is observed that the proposed LAOs of TSFSs are the improved version of the existing LAOs. In section 5, it is proved that the LTAFWA and LTSFWG operators have generalized nature compare to the existing LAOs where it is shown that how the existing LAOs becomes the special cases of the TSFLAOs.

The MADM method based on LAOs of TSFSs provides the decision-maker an opportunity to describe the information using an MD, NMD, AD, and an RD. The LAOs of TSFs takes into account all these four aspects to aggregate the indefinite information and hence reducing the risk of information loss which is more likely to occur in the case of LAOs of IFSs [22], PyFS [23], and PFSs [24]. The variable parameter q is responsible for providing the decision maker a flexible ground for assigning the degrees which are very unlike in the case of existing LAOs. Based on the analysis of Section 5, we also conclude that the LAOs of TSFSs can be applied to the MADM problems discussed in [22 –24] however none of the existing LAOs can handle the data used in Example 3 which shows the superiority of the LTSFWA and LTSFWG operators.

8 Conclusion

In this paper, we comprehensively examined the notion of LAOs based on four types of membership grades to deal with uncertain and vague information. First, we introduced the idea of the LTSFWA operator and investigated some properties followed by the notion of the LTSFWG operator and its characteristics. A study of the generalization of the developed LTSFWA and LTSFWG operators because of some certain restrictions is established where it is shown that previously existing LAOs became special cases of the proposed operators. A MADM algorithm is discussed given LAOs of TSFSs followed by a comprehensive example where the effectiveness of the LAOs of the TSFSs is demonstrated. To strengthen the study, we set up a numerical comparative study of the current work with previously established AOs.

Advantages of the TSFLAOs

The advantages of TSFLAOs are given as follows:

1. TSFS takes four kinds of membership degrees to express a human opinion or uncertain information.

2. TSFS provides full flexibility for assigning the MD, AD, NMD, and RD.

3. Information loss using TSFSs becomes lesser than by using other fuzzy frameworks.

Disadvantages of the Previous Work

The disadvantages of the previously defined LAOs are discussed as follows:

1. The LAOs of IFSs [22] and PyFSs [23] discussed only two aspects of human opinion and provide very little flexibility hence leading to information loss.

2. The LAOs of PFSs [24] discussed four aspects of human opinion but proved to be very strict in assigning the MD, AD, NMD, and RD due to the restraint as discussed in [9].

In near future, we aim to investigate the LAOs with interval-valued information [12] and in the environment of complex TSFSs [36] for MADM purposes. We also aim to use the notion of LAOs of TSFSs in several methods of MADM including TOPSIS method [37], VIKOR method [38], MULTIMOORA method [39], AHP method [40], and TODIM method [41] to study its applicability in the environment of TSFSs. All these proposed method can be used in the assessments of projects, in the evaluation of investment policies, in performance evaluation, in information retrieval etc.

9 Data availability

The data used in this article are artificial and hypothetical, and anyone can use these data before prior permission by just citing this article.

10 Conflicts of interest

The authors declare that they have no conflicts of interest.

Footnotes

Acknowledgments

This work is supported by the Major Fundamental Research Funds for the Provincial Universities of Zhejiang (SJWZ2020002), Longyuan Construction Financial Research Project of Ningbo University (LYYB2002), Statistical Scientific Key Research Project of China (2021LZ33) and Statistical Scientific Key Research Project of Zhejiang (21TJZZ25).

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