Spherical fuzzy sets and its representation of spherical fuzzy t-norms and t-conorms

Abstract

In this paper, we introduce spherical fuzzy sets (SFSs) which is an advanced tool of the fuzzy sets, intuitionistic fuzzy sets and picture fuzzy sets. We investigate the basic properties of SFSs and compare the idea SFSs with picture fuzzy sets. The spaces of spherical membership grades and picture membership grades are examined by graphically. Some useful operation such as spherical fuzzy t’-norms and spherical fuzzy t’-conorms are introduced. Also introduced spherical fuzzy negator and some classifications of spherical fuzzy t’-norms and spherical fuzzy t^′-conorms which are useful for development the aggregation operator to aggregate the spherical fuzzy information. Finally, a food circulation center evaluation problem is given as a practical application to demonstrate the usage and applicability of the proposed ranking approach.

Keywords

Spherical fuzzy sets spherical fuzzy t’-norm spherical fuzzy t’-conorm

1 Introduction

The complication of a system is growing every day in real life and to get the finest option from the set of possible one it is difficult for the decision makers. To attain single objective is difficult to summarize but not incredible. Many organizations found difficulties to set the motivation, goals and opinions complication. Thus, in organizational decisions, which simultaneously includes of numerous objectives, no matter takes an individual or by the committee. This reflection suggests that according to criteria solved optionally which restrict each decision maker to attain an ideal solution-optimum under each criteria involve in practical problem. Consequently, decision maker is more concentration to establishes more applicable and reliable technique to find the finest option. To handle the ambiguity and uncertainty data in decision making problems, the classical or crisp methods cannot be always effective. So, dealing with such uncertain situation Zadeh [21] in 1965 presented the idea of the fuzzy set. Zadeh assign membership grades to elements of a set in the interval [0,1] by offering the idea of fuzzy sets (FSs). Zadeh’s work in this direction is remarkable as many of the set theoretic properties of crisp cases were defined for fuzzy sets. FSs got the attentions of researchers and founds its applications in decision science, intelligence science, communications, engineering, computer sciences etc.

Atanassov’s [7] work on intuitionistic fuzzy set (IFSs) was quite remarkable as he extended the concept of FSs by assigning membership degree say P (x) along with non-membership degree say “N (x)” with condition that 0 ≤ P (x) + N (x) ≤1. Atanassov’s construction of IFSs is of exceptional reputation but decision makers are somehow restricted in assigning values due to the condition on P (x) and N (x). Sometimes sum of their membership degrees are superior then 1. In such situation, to attain reasonable outcome IFS fails. So, dealing with such situation Yager [19] in 2015, establish the concept of Pythagorean fuzzy sets (PyFS) by assigning membership degree say “P (x)” along with non-membership degree say “N (x)” with condition that 0 ≤ P² (x) + N² (x) ≤1.

In this paper we extend the concept of PyFS to Spherical fuzzy set by assigning neutral membership degree say “I (x)” along with positive and negative membership degrees say “P (x)” and “N (x)” with condition that 0 ≤ P² (x) + I² (x) + N² (x) ≤1.

Atanassov’s structure discourses only satisfaction and dissatisfaction degree of elements in a set which is quite insufficient as human nature has some sort of abstain and refusal issues too. Such hitches were considered by Cuong [8] in 2013, to proposed picture fuzzy sets (PFS) define as (P (x) , I (x) , N (x)) where the elements in triplet represent satisfaction, abstain and dissatisfaction degrees respectively, under the condition that 0 ≤ P (x) + I (x) + N (x) ≤1. This structure of Cuong is considerably more close to human nature than that of earlier concepts and is one of the richest research area now. Cuong [9] in 2014, established the relations, compositions and find the distance between picture fuzzy numbers, also he gives the concept of picture soft sets. Phong at all [15] in 2014, give the concept about the compositions of some picture fuzzy relations. Cuong [11] in 2015, introduced the fuzzy logic operators for picture fuzzy sets. Singh [16] in 2015, introduced the idea related to correlation coefficients for picture fuzzy sets. Cuong [10] in 2015, established the ideas like convex combination of PFNs, alpha-cuts of PFS, picture fuzzy relations. Cuong [12] in 2016, established the classification of representable t’-norm operators for PFSs. Son [17] in 2016, introduced the concept about generalized picture distance measure and give its applications. Wei [18] in 2017, introduced the cosine similarity measures for PFSs. Garg [14] in 2017, introduced the picture fuzzy aggregation to aggregate the picture fuzzy information’s.

Cuong’s construction of picture fuzzy sets is of exceptional reputation but decision makers are somehow restricted in assigning values due to the condition on P (x) , I (x) and N (x). Sometimes sum of their membership degrees are superior then 1. In such situation, to attain reasonable outcome PFS fails. To describe this situation, we take an example, for provision and in contradiction of the membership degrees. The alternatives are (1/5),(3/5) and (3/5) respectively. This gratifies the situation that their sum is superior then 1 and PFS fails to deal such type of data. Dealing with such kind of circumstances, Ashraf [1] in 2018 proposed new structure by defining spherical fuzzy sets (SFSs) which enlarge the space of membership degrees P (x) , I (x) and N (x) somehow bigger than that of picture fuzzy sets. In SFS, membership degrees are satisfying the condition 0 < P (x) + I (x) + N (x) <1. Hierarchy structure of spherical fuzzy set is shown in Fig. 1.

Fig.1

Hierarchy structure of spherical fuzzy set.

In this paper, we extend the notions of picture fuzzy t’-norm and picture fuzzy t’-conorm to the spherical fuzzy case. t’-norms and t’-conorms are used to define, intersection and union of spherical fuzzy sets and use to aggregate the spherical fuzzy information. After that, we generalize said representation theorems to spherical fuzzy connectives. Also, in this paper, we extend the TOPSIS method for spherical fuzzy numbers. Based on TOPSIS approach, a decision-making method has been established for ranking the alternatives by utilizing spherical fuzzy environment. The suggested technique has been demonstrated with an application to a food circulation center evaluation problem for viewing their effectiveness as well as reliability. The paper aims are: (a) to propose the idea of spherical fuzzy set (b) some debate on its basic operational characteristics, (c) define union, intersection based on t-norms and t- conorms, (d) establish spherical Fuzzy Negators and some classifications of spherical fuzzy t’-norm and spherical fuzzy t’-conorms, (e) extend the TOPSIS approach for SFNs and established a practical example that demonstrate the effectiveness as well as reliability of the suggested technique.

2 Preliminaries

Definition 1. [21] Let R ≠ φ be a universe set. Then the fuzzy set (FS) can be written as; $J = {〈 r, P_{j} (r) | r \in R 〉} .$

A FS in a set R is indicated by P_j : R → [0, 1]. The function P_j (r) indicate the positive membership degree of each r ∈ R.

Definition 2. [7] Let R ≠ φ be a universe set. Then the intuitionistic fuzzy set (IFS) can be written as; $J = {〈 r, P_{j} (r), N_{j} (r) | r \in R 〉} .$

An IFS in a set R is indicated by P_j : R → [0, 1] and N_j : R → [0, 1]. The functions P_j (r) and N_j (r) indicate the positive and the negative membership degrees of each r ∈ R, respectively. Furthermore P_j and N_j satisfy 0 ≤ P_j (r) + N_j (r) ≤1, for all r ∈ R.

Definition 3. [8] Let R ≠ φ be a universe set. Then the picture fuzzy set (PFS) can be written as; $J = {〈 r, P_{j} (r), I_{j} (r), N_{j} (r) | r \in R 〉} .$ where P_j : R → [0, 1], I_j : R → [0, 1] and N_j : R → [0, 1] are indicated the positive, neutral and negative membership degrees of each r ∈ R, respectively. Furthermore P_j, I_j and N_j satisfy 0 ≤ P_j (r) + I_j (r) + N_j (r) ≤1 for all r ∈ R.

3 Spherical fuzzy set and their basic operations

In this section, the idea of spherical fuzzy set and their operations are established. Here we explain that why we need spherical fuzzy sets? How it generalizes picture fuzzy sets? We made a comparison of spherical fuzzy set with that of picture fuzzy set also we examined their spaces graphically in Figs. 2 and 3.

Fig.2

Spherical fuzzy space (spherical representation).

Fig.3

Spherical fuzzy space (3D).

Definition 4. [1] Let R ≠ φ be a universe set. Then the spherical fuzzy set (SFS) can be written as; $J = {〈 r, P_{j} (r), I_{j} (r), N_{j} (r) | r \in R 〉} .$

Where P_j : R → [0, 1], I_j : R → [0, 1] and N_j : R → [0, 1] are indicated the positive, neutral and negative membership degrees of each r ∈ R, respectively. Furthermore P_j, I_j and N_j satisfy $0 \leq P_{j}^{2} (r) + I_{j}^{2} (r) + N_{j}^{2} (r) \leq 1$ for all r ∈ R.

Spherical fuzzy sets have its own importance in a circumstance where opinion is not only constrained to yes or no but there is some sort of abstinence or refusal. A good example of spherical fuzzy set could be decision making such as when four decision makers have four different categories of opinion about a candidate. Another example could be of voting where four types of voters occurs who vote in favor or vote against or refuse to vote or abstain. Spherical fuzzy set is a direct generalization of fuzzy set, intuitionistic fuzzy set and picture fuzzy set.

A question arises that why we need spherical fuzzy set or what are the boundaries of PFSs that leads us to spherical fuzzy sets? The main downside of PFSs is the restriction on it, i.e., $0 \leq P_{j} (r) + I_{j} (r) + N_{j} (r) \leq 1$

As this condition does not allows the decision makers to assign membership values of their own consent. The decision makers are somehow limitations in a specific domain. We consider an example P_j (r) =0.8, I_j (r) =0.5 and N_j (r) =0.3 which interrupts the condition that 0 ≤ P_j (r) + I_j (r) + N_j (r) ≤1 but if we take the square of these values such as, $P_{j}^{2} (r) = 0.64$ , $I_{j}^{2} (r) = 0.25$ and $N_{j}^{2} (r) = 0.09$ where the condition $0 \leq P_{j}^{2} (r) + I_{j}^{2} (r) + N_{j}^{2} (r) \leq 1$ is satisfies.

Another crucial thing to debate is that why we called it spherical fuzzy set? The answer to this question is very simple. Consider P_j (r), I_j (r) and N_j (r) represents the degrees of positive, neutral and negative memberships of a spherical fuzzy set respectively such that $0 \leq P_{j}^{2} (r) + I_{j}^{2} (r) + N_{j}^{2} (r) \leq 1$ . Now if we observe the given condition we may get the equation $P_{j}^{2} (r) + I_{j}^{2} (r) + N_{j}^{2} (r) \leq 1$ which represent the space occupied by a part of unit sphere as shown in figure (below). By a part of sphere, we mean that we consider the values of P_j (r), I_j (r) and N_j (r) in [0, 1]. The figure shown here is according to the inequality $P_{j} (r) \leq {(1 - I_{j}^{2} (r) + N_{j}^{2} (r))}^{\frac{1}{2}}$ .

Note that, if we put I_j (r) =0 in SPSs. Then SPSs reduced to Pythagorean fuzzy sets. Hence Spherical fuzzy sets are direct extension of Pythagorean fuzzy set and also the extension of picture fuzzy sets, also we seen the hierarchy structure (above) of the spherical fuzzy sets.

Let SFS (R) indicate the collection of all spherical fuzzy sets of R.

Now we assume F^* can be define as; $\begin{matrix} F^{*} = {f = (f_{1}, f_{2}, f_{3}) | f \in [0, 1]^{3}, \\ f_{1}^{2} + f_{2}^{2} + f_{3}^{2} \leq 1} . \end{matrix}$

If f ∈ F^* this means that f₁, f₂ and f₃ are thecomponents of f ∈ F^*. i.e. f = (f₁, f₂, f₃). Consider F^* be any set and order relation ≤_{F
^*}on F^* can be define as;

$\begin{matrix} f \leq_{F^{*}} j \Leftrightarrow (f_{1}, f_{2}, f_{3}) \leq_{F^{*}} (j_{1}, j_{2}, j_{3}) \\ \Leftrightarrow ((f_{1} < j_{1}) \land (f_{3} \geq j_{3})) \lor ((f_{1} = j_{1}) \\ \land (f_{3} > j_{3})) \lor \\ ((f_{1} = j_{1}) \land (f_{3} = j_{3}) \land (f_{2} \leq j_{2})), \end{matrix}$

for all f, j ∈ F^*. We define the projection mapping for components of f on F^* as; pl₁ (f) = f₁, pl₂ (f) = f₂ and pl₃ (f) = f₃, for all f ∈ F^*. We define infimum and supremum for each f, j ∈ F^*; $\begin{matrix} inf (f, j) = \\ {\begin{matrix} min (f, j) & if f \leq_{F^{*}} j or j \leq_{F^{*}} f \\ (\begin{matrix} f_{1} \land j_{1}, \\ \sqrt{1 - f_{1}^{2}} \land \sqrt{j_{1}^{2} - f_{3}^{2}} \lor j_{3}, \\ f_{3} \lor j_{3} \end{matrix}) & else \end{matrix} \end{matrix}$ $\begin{matrix} sup (f, j) = \\ {\begin{matrix} max (f, j) & if f \leq_{F^{*}} j or j \leq_{F^{*}} f \\ (f_{1} \lor j_{1}, 0, f_{3} \land j_{3}) & else \end{matrix} \end{matrix}$

Then (F^*, ≤ _{F
^*}) can be complete lattice, if φ ≠ A ⊆ F^*, we have inf L = (inf pl₁A, inf pl₂A, inf pl₃A), where $inf {pl}_{1} A = inf {f_{1} \in [0, 1] | \exists f = (f_{1}, f_{2}, f_{3}) \in A},$ $inf {pl}_{3} A = sup {f_{3} \in [0, 1] | \exists f = (f_{1}, f_{2}, f_{3}) \in A},$ $inf {pl}_{2} A = {\begin{matrix} inf {f_{2} : (inf {pl}_{1} A, f_{2}, inf {pl}_{3} A) \in A} \\ 1 - inf {pl}_{1} A - inf {pl}_{3} A, \\ if {(inf {pl}_{1} A, z, inf {pl}_{3} A) : each z \notin A \end{matrix}$ and sup L = (sup pl₁A, sup pl₂A, sup pl₃A), where $sup {pl}_{1} A = sup {f_{1} \in [0, 1] | \exists f = (f_{1}, f_{2}, f_{3}) \in A},$ $sup {pl}_{3} A = inf {f_{3} \in [0, 1] | \exists f = (f_{1}, f_{2}, f_{3}) \in A},$ $sup {pl}_{2} A = {\begin{matrix} sup {f_{2} : (sup {pl}_{1} A, f_{2}, sup {pl}_{3} A) \in A} \\ 0, if {(sup {pl}_{1} A, f_{2}, sup {pl}_{3} A) \notin A, \forall f_{2} \end{matrix}$

Units of F^* can be indicated by 1_{F
^*} = (1, 0, 0) and 0_{F
^*} = (0, 0, 1). Note that if f, j ∈ F^* then said to be incomparable w.r.t. ≤_{F
^*}if neither f ≤ _{F
^*}j nor j ≤ _{F
^*}f. Notation of incomparable is f||_{F
^*}j. Similarly, we can define lattice structure as, (F^*, ∧ , ∨), where each f, j∈ F^* ∃ f ∧ j and f ∨ j, note that $\begin{matrix} \begin{matrix} f \land j & = {min (f_{1}, j_{1}), min (f_{2}, j_{2}), max (f_{3}, j_{3})} \\ f \lor j & = {max (f_{1}, j_{1}), min (f_{2}, j_{2}), min (f_{3}, j_{3})} . \end{matrix} \end{matrix}$

Definition 5. Let R ≠ φ be a universe set. Then any two spherical fuzzy sets J₁, J₂ can be written as; $\begin{matrix} \begin{matrix} J_{1} & = {〈 r, P_{j_{1}} (r), I_{j_{1}} (r), N_{j_{1}} (r) | r \in R 〉} \\ J_{2} & = {〈 r, P_{j_{2}} (r), I_{j_{2}} (r), N_{j_{2}} (r) | r \in R 〉} \end{matrix} \end{matrix}$

The union of two spherical fuzzy sets J₁ and J₂ in universe set R can be written as $J_{1} \cup J_{2} = {\begin{matrix} (P_{j_{1}} (r), P_{j_{2}} (r)), \\ (I_{j_{1}} (r), I_{j_{2}} (r)), \\ (N_{j_{1}} (r), N_{j_{2}} (r)) \end{matrix}}$

The intersection of two spherical fuzzy sets J₁ and J₂ in universe set R can be written as $J_{1} \cap J_{2} = {\begin{matrix} min (P_{j_{1}} (r), P_{j_{2}} (r)), \\ min (I_{j_{1}} (r), I_{j_{2}} (r)), \\ max (N_{j_{1}} (r), N_{j_{2}} (r)) \end{matrix}}$

The complement of any spherical fuzzy set J₁ in universe set R can be written as $J_{1}^{c} = {N_{j_{1}} (r), I_{j_{1}} (r), P_{j_{1}} (r)} .$

3.1 Spherical fuzzy negators and spherical fuzzy norms

In this section, we established spherical fuzzy negators which are extended form of picture fuzzy negators and intuitionistic fuzzy negators. Also define the spherical fuzzy t’-norm and spherical fuzzy t’-conorm.

Definition 6. Suppose any decreasing mapping N : F^* → F^* is said to be a spherical fuzzy negator if its satisfies

N (0_{F
^*}) = 1_{F
^*}

N (1_{F
^*}) = 0_{F
^*}.

Note that if ∀f ∈ F^* we have N (N (f)) = f then given decreasing mapping N : F^* → F^* called involutive negator.

Definition 7. suppose any decreasing mapping N₀ : F^* → F^*, define as N₀ (f) = (f₃, 0, f₁), ∀f ∈ F^* be spherical fuzzy negator.

Suppose that we define f₄ = 1 - f₁ - f₂ - f₃. Then the involutive negator called standard negator, if mapping define as N_l (f) = (f₃, f₄, f₁), ∀f ∈ F^*.

Proposition 1. Let an involutive picture fuzzy negator be N. Then N (R₂) = R₂, N (0, 0, 0) = (0, 1, 0).

Proof. Suppose any involutive picture fuzzy negator be N and N (0, r₂, 0) ∉ R₂ Then N (0, r₂, 0) ∈ {(i, 0, 0) , (0, 0, k) , (i, 0, k) , (i, j, k) , (0, j, k) , (i, j,0)}, where i, j, k > 0.

Suppose that N (0, r₂, 0) = (i, 0, 0), where i > 0. Suppose f, f^/ ∈ F^* such that f, f^/ ≤ _{F
^*} (i, 0, 0) and f||_{F
^*}f^/. we know that N is decreasing and involutive, N (f), N (f^/) ≥ _{F
^*}N (i, 0, 0) = (0, r₂, 0)

Then pc₃N (f) = pc₃N (f^/) = 0 and N (f), N (f^/) are comparable w.r.t. ≤_{F
^*}.

Hence f, f^/ are comparable w.r.t. ≤_{F
^*}. Which is a contradiction.

Similarly, we gain the contradiction for remaining cases.

Thus, N (R₂) ⊂ R₂.

Since N is involutive, then we have N (R₂) = R₂.

Now suppose that N (0, 0, 0) = (0, α, 0), 0 ≤ α ≤ 1.

Since N is decreasing and involutive, then $N (0, 1, 0) = (0, β, 0) \leq_{F^{*}} N (0, α, 0) = (0, 0, 0),$

Then we attain β = 0 and N (0, 0, 0) = (0, 1, 0). □

Now, we define spherical fuzzy t’-norm and t’-conorm. These spherical fuzzy triangular norms help us to find the conjunction and disjunction operators for spherical fuzzy sets. These t’-norms are the extended form of picture fuzzy t’-norms and intuitionistic fuzzy t’-norms. Suppose for ∀f we define $\begin{matrix} I (f) & = {j \in F^{*} : j = (f_{1}, j_{2}, f_{3}), \\ 0 \leq j_{2} \leq 1 - (f_{1} + f_{3})} \end{matrix}$

Definition 8. A mapping T : (F^*) ² → F^* said to be a spherical fuzzy t’-norm if it satisfies

T (f, j) = T (j, f), ∀f, j ∈ F^* (commutativity)

T (f, T (j, l)) = T (T (f, j) , l), ∀f, j, l ∈ F^* (associativity)

T (f, j) ≤ _{F
^*}T (f, l), ∀f, j, l ∈ F^*, j ≤ _{F
^*}l, (monotonicity)

T (1_{F
^*}, f) ∈ I (f) , ∀ f ∈ F^* (boundary condition)

Definition 9. A mapping S : (F^*) ² → F^* said to be a spherical fuzzy t’-conorm if it satisfies

S (f, j) = S (j, f), ∀f, j ∈ F^* (commutativity)

S (f, S (j, l)) = S (S (f, j) , l), ∀f, j, l ∈ F^* (associativity)

S (f, j) ≤ _{F
^*}S (f, l), ∀f, j, l ∈ F^*, j ≤ _{F
^*}l (monotonicity)

T (0_{F
^*}, f) ∈ I (f) , ∀ f ∈ F^* (boundary condition)

In our next study, we indicate f ∧ j = min(f, j) = {min(f₁, j₁), min(f₂, j₂), max(f₃, j₃)}, f ∨ j = max(f, j) = {max(f₁, j₁), min(f₂, j₂), min(f₃, j₃)}, ∀f, j ∈ [0, 1].

Example 1. Some spherical fuzzy t’-norms are denoted and define as, ∀f, j ∈ F^* we have,

$T_{inf} (f, j) = inf (f, j) = {\begin{matrix} (\begin{matrix} f_{1} \land j_{1}, \\ \sqrt{1 - (f_{1}^{2} \land j_{1}^{2}) - (f_{3}^{2} \lor j_{3}^{2})}, \\ (f_{3} \lor j_{3}) \end{matrix}) & if f | |_{F^{*}} j \\ f \land j, & otherwise \end{matrix}$ .

T_min (f, j) = (f₁ ∧ j₁, f₂ ∧ j₂, f₃ ∨ j₃).

T (f, j) = (f₁ ∧ j₁, f₂j₂, f₃ ∨ j₃).

T (f, j) = (f₁j₁, f₂j₂, f₃ ∨ j₃).

$T (f, j) = (f_{1} j_{1}, f_{2} j_{2}, \sqrt{f_{3}^{2} + j_{3}^{2} - f_{3}^{2} j_{3}^{2}})$ .

$T (f, j) = (\begin{matrix} 0 \lor (\sqrt{f_{1}^{2} + j_{1}^{2} - 1}), \\ 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), \\ 1 \land (\sqrt{f_{3}^{2} + j_{3}^{2}}) \end{matrix})$ .

$T (f, j) = (\begin{matrix} 0 \lor (\sqrt{f_{1}^{2} + j_{1}^{2} - 1}), \\ 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), \\ \sqrt{f_{3}^{2} + j_{3}^{2} - f_{3}^{2} j_{3}^{2}} \end{matrix})$ .

$T (f, j) = (\begin{matrix} \frac{1}{2} (\sqrt{f_{1}^{2} + j_{1}^{2} - 1 + f_{1}^{2} j_{1}^{2}}) \lor 0, \\ \frac{1}{2} (\sqrt{f_{2}^{2} + j_{2}^{2} - 1 + f_{2}^{2} j_{2}^{2}}) \lor 0, \\ \sqrt{f_{3}^{2} + j_{3}^{2} - f_{3}^{2} j_{3}^{2}} \end{matrix})$ .

$T (f, j) = (\begin{matrix} f_{1} j_{1}, \\ 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), \\ \sqrt{f_{3}^{2} + j_{3}^{2} - f_{3}^{2} j_{3}^{2}} \end{matrix})$ .

$T (f, j) = (\begin{matrix} 0 \lor (\sqrt{f_{1}^{2} + j_{1}^{2} - 1}), \\ f_{2} j_{2}, \\ \sqrt{f_{3}^{2} + j_{3}^{2} - f_{3}^{2} j_{3}^{2}} \end{matrix})$ .

Example 2. Some spherical fuzzy t’-conorms are denoted and define as, ∀f, j ∈ F^* we have,

$S_{sup} (f, j) = sup (f, j) = {\begin{matrix} (f_{1} \lor j_{1}, 0, f_{3} \land j_{3}), & if f | |_{F^{*}} j \\ f \lor j, & otherwise \end{matrix}$ .

S_max (f, j) = (f₁ ∨ j₁, f₂ ∧ j₂, f₃ ∧ j₃).

S (f, j) = (f₁ ∨ j₁, f₂j₂, f₃ ∧ j₃).

S (f, j) = (f₁ ∨ j₁, f₂j₂, f₃j₃).

$S (f, j) = (\sqrt{f_{1}^{2} + j_{1}^{2} - f_{1}^{2} j_{1}^{2}}, f_{2} j_{2}, f_{3} j_{3})$ .

$S (f, j) = (\begin{matrix} 1 \land (\sqrt{f_{1}^{2} + j_{1}^{2}}), \\ 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), \\ 0 \lor (\sqrt{f_{3}^{2} + j_{3}^{2} - 1}) \end{matrix})$ .

$S (f, j) = (\begin{matrix} \sqrt{f_{1}^{2} + j_{1}^{2} - f_{1}^{2} j_{1}^{2}}, \\ 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), \\ 0 \lor (\sqrt{f_{3}^{2} + j_{3}^{2} - 1}) \end{matrix})$ .

$S (f, j) = (\begin{matrix} \sqrt{f_{1}^{2} + j_{1}^{2} - f_{1}^{2} j_{1}^{2}}, \\ \frac{1}{2} (\sqrt{f_{2}^{2} + j_{2}^{2} - 1 + f_{2}^{2} j_{2}^{2}}) \lor 0, \\ \frac{1}{2} (\sqrt{f_{3}^{2} + j_{3}^{2} - 1 + f_{3}^{2} j_{3}^{2}}) \lor 0 \end{matrix})$ .

$S (f, j) = (\begin{matrix} \sqrt{f_{1}^{2} + j_{1}^{2} - f_{1}^{2} j_{1}^{2}}, \\ 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), \\ f_{3} j_{3} \end{matrix})$ .

$S (f, j) = (\begin{matrix} \sqrt{f_{1}^{2} + j_{1}^{2} - f_{1}^{2} j_{1}^{2}}, \\ f_{2} j_{2}, \\ 0 \lor (\sqrt{f_{3}^{2} + j_{3}^{2} - 1}) \end{matrix})$ .

In 2015, notation of picture fuzzy t’-norms and picture fuzzy t’-conorms established by the Cuong et al. [cuo1] and investigated the condition where we obtain similar representation theorem as inpicture fuzzy information. In our study, further usage we assume F^* can be define as; $\begin{matrix} F^{*} = {f = (f_{1}, f_{2}, f_{3}) | f \in [0, 1]^{3}, \\ f_{1} + f_{2} + f_{3} \leq 1} . \end{matrix}$ suppose the order relation ≤_{F
^*} on F^* can be define as; $\begin{matrix} f \leq_{F^{*}} j \Leftrightarrow (f_{1}, f_{2}, f_{3}) \leq_{F^{*}} (j_{1}, j_{2}, j_{3}) \\ \Leftrightarrow (\begin{matrix} ((f_{1} < j_{1}) \land (f_{3} \geq j_{3})) \lor \\ ((f_{1} = j_{1}) \land (f_{3} > j_{3})) \lor \\ ((f_{1} = j_{1}) \land (f_{3} = j_{3}) \land (f_{2} \leq j_{2})) \end{matrix}), \end{matrix}$ for all f, j ∈ F^*. The projection mapping for components of f on F^* can be define as; pl₁ (f) = f₁, pl₂ (f) = f₂ and pl₃ (f) = f₃, for all f ∈ F^*. Units of F^* can be indicated by 1_{F
^*} = (1, 0, 0) and 0_{F
^*} = (0, 0, 1).

Definition 10. Suppose that T be the picture fuzzy t’-norm is said to be representable iff there exists two t’-norms t₁, t₂ and a t’-conorm s₃ on [0, 1] satisfy for each f, j ∈ F^*, $T (f, j) = (t_{1} (f_{1}, j_{1}), t_{2} (f_{2}, j_{2}), s_{3} (f_{3}, j_{3})) .$

Definition 11. Suppose that S be the picture fuzzy t’-conorm is said to be representable iff there exists two t’-norms t₁, t₂ and a t’-conorm s₃ on [0, 1] satisfy for each f, j ∈ F^*, $S (f, j) = (s_{3} (f_{1}, j_{1}), t_{2} (f_{2}, j_{2}), t_{1} (f_{3}, j_{3})) .$

Similarly, we can define,

Definition 12. Suppose that T be the spherical fuzzy t’-norm is said to be representable iff ∃ two t’-norms t₁, t₂ and a t’-conorm s₃ on [0, 1] satisfy for each f, j ∈ F^*, $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2})) .$

Definition 13. Suppose that S be the spherical fuzzy t’-conorm is said to be representable iff ∃ two t’-norms t₁, t₂ and a t’-conorm s₃ on [0, 1] satisfy for each f, j ∈ F^*, $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2})) .$

Proposition 2. Suppose that T (f, j) be a t^′-representable picture fuzzy t^′-norm T (f, j) = (t₁ (f₁, j₁) , t₂ (f₂, j₂) , s₃ (f₃, j₃)) for all f, j∈F^*, where t₁, t₂ are t^′-norms and s₃ is a t^′-conorm on [0, 1]. Then $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a spherical fuzzy t^′-norm.

Proposition 3. Suppose that S (f, j) be a t^′-representable picture fuzzy t^′-conorm S (f, j) = (s₃ (f₁, j₁) , t₂ (f₂, j₂) , t₁ (f₃, j₃)) for all f, j∈F^*, where t₁, t₂ are t^′-norms and s₃ is a t^′-conorm on [0, 1]. Then $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2})$ , $t_{2} (f_{2}^{2}, j_{2}^{2})$ , $t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a spherical fuzzy t^′-conorm.

Proposition 4. Suppose that T (f, j) be a t^′-representable picture fuzzy t^′-norm T (f, j) = (t₁ (f₁, j₁) , t₂ (f₂, j₂) , s₃ (f₃, j₃)) for all f, j ∈ F^*, where t₁, t₂ are t^′-norms and s₃ is a t^′-conorm on [0, 1], then satisfies for all f, j ∈ F^*, $0 \leq t_{1} (f_{1}^{2}, j_{1}^{2}) + t_{2} (f_{2}^{2}, j_{2}^{2}) + s_{3} (f_{3}^{2}, j_{3}^{2}) \leq 1 .$

Then $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2},$ $j_{3}^{2}))$ for all f, j ∈ F^* is a t’-representable spherical fuzzy t’-norm.

Proposition 5. Suppose that S (f, j) be a t^′-representable picture fuzzy t^′-conorm S (f, j) = (s₃ (f₁, j₁) , t₂ (f₂, j₂) , t₁ (f₃, j₃)) for all f, j ∈ F^*, where t₁, t₂ are t^′-norms and s₃ is a t^′-conorm on [0, 1], then satisfies for all f, j ∈ F^*, $0 \leq s_{3} (f_{1}^{2}, j_{1}^{2}) + t_{2} (f_{2}^{2}, j_{2}^{2}) + t_{1} (f_{3}^{2}, j_{3}^{2}) \leq 1 .$

Then $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a t’-representable spherical fuzzy t’-conorm.

3.2 Properties of spherical fuzzy t’-norm and t’-conorm

Definition 14. Suppose that $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2})$ , $t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a spherical fuzzy t’-norm is called Archimedean spherical fuzzy t’-norm iff it satisfies, $for each f \in F^{*} ∖ {0_{F^{*}}, 1_{F^{*}}}, T (f, f) <_{F^{*}} f .$

Definition 15. Suppose that $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2})$ , $t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a spherical fuzzy t’-norm is called nilpotent spherical fuzzy t’-norm iff it satisfies, $for each f, j \in F^{*} ∖ {0_{F^{*}}}, T (f, j) = 0_{F^{*}} .$

Definition 16. Suppose that $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2})$ , $t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a spherical fuzzy t’-norm is called strict spherical fuzzy t’-norm iff it satisfies, $for each f, j \in F^{*} ∖ {0_{F^{*}}}, T (f, j) \neq 0_{F^{*}} .$

Definition 17. Suppose that $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2})$ , $t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a spherical fuzzy t’-conorm is called Archimedean spherical fuzzy t’-conorm iff it satisfies, $for each f \in F^{*} ∖ {0_{F^{*}}, 1_{F^{*}}}, S (f, f) >_{F^{*}} f .$

Definition 18. Suppose that $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2})$ , $t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a spherical fuzzy t’-conorm is called nilpotent spherical fuzzy t’-conorm iff it satisfies, $for each f, j \in F^{*} ∖ {1_{F^{*}}}, S (f, j) = 1_{F^{*}} .$

Definition 19. Suppose that $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2})$ , $t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a spherical fuzzy t’-conorm is called strict spherical fuzzy t’-conorm iff it satisfies, $for each f, j \in F^{*} ∖ {1_{F^{*}}}, S (f, j) \neq 1_{F^{*}} .$

Proposition 6. Suppose that T^* be the nilpotent spherical fuzzy t^′-norm and T^** be the strict spherical fuzzy t^′-norm. Then we attain, $T^{*} \cap T^{* *} = φ .$

Proposition 7. Suppose that S^* be the nilpotent spherical fuzzy t^′-conorm and S^** be the strict spherical fuzzy t^′-conorm. Then we attain, $S^{*} \cap S^{* *} = φ .$

Proposition 8. Suppose that $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2})$ , $t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a spherical fuzzy t^′-norm, where if t₁, t₂ and s₃ are Archimedean. Then T is said to be Archimedean spherical fuzzy t^′-norm.

Proposition 9. Suppose that $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2})$ , $t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* is a spherical fuzzy t^′-conorm, where if t₁, t₂ and s₃ are Archimedean. Then S is said to be Archimedean spherical fuzzy t^′-conorm.

3.3 Some classification of spherical fuzzy t’-norm

In this section, we define some subclasses representable spherical fuzzy t’-norm as:

Definition 20. Suppose that T be a spherical fuzzy t’-norm is called strict-strict-strict (Δ_sss) iff $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ and t₂ are strict t’-norm and s₃ is strict t’-conorm on [0, 1].

Example 3. $T (f, j) = (f_{1} j_{1}, f_{2} j_{2}, \sqrt{f_{3}^{2} + j_{3}^{2} - f_{3}^{2} j_{3}^{2}})$ for all f, j ∈ F^*.

Definition 21. Suppose that T be a spherical fuzzy t’-norm is called nilpotent-nilpotent-nilpotent (Δ_nnn) iff $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ and t₂ are nilpotent t’-norm and s₃ is nilpotent t’-conorm on [0, 1].

Example 4. Nilpotent-nilpotent-nilpotent spherical fuzzy t’-norm have following examples.

$T (f, j) = (\begin{matrix} 0 \lor (\sqrt{f_{1}^{2} + j_{1}^{2} - 1}), \\ 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), \\ 1 \land (\sqrt{f_{3}^{2} + j_{3}^{2}}) \end{matrix})$ .

$T (f, j) = (\begin{matrix} 0 \lor (\sqrt[2 α]{f_{1}^{2 α} + j_{1}^{2 α} - 1}), \\ 0 \lor (\sqrt[2 β]{f_{2}^{2 β} + j_{2}^{2 β} - 1}), \\ 1 \land (\sqrt[2 γ]{f_{3}^{2 γ} + j_{3}^{2 γ}}) \end{matrix})$ , α, β, γ ≥ 1.

Definition 22. Suppose that T be a spherical fuzzy t’-norm is called nilpotent-nilpotent-strict (Δ_nns) iff $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ and t₂ are nilpotent t’-norm and s₃ is strict t’-conorm on [0, 1].

Example 5. Nilpotent-nilpotent-strict spherical fuzzy t’-norm have following examples.

$T (f, j) = (\begin{matrix} 0 \lor (\sqrt{f_{1}^{2} + j_{1}^{2} - 1}), \\ 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), \\ \sqrt{f_{3}^{2} + j_{3}^{2} - f_{3}^{2} j_{3}^{2}} \end{matrix})$ .

$T (f, j) = (\begin{matrix} \frac{1}{2} (\sqrt{f_{1}^{2} + j_{1}^{2} - 1 + f_{1}^{2} j_{1}^{2}}) \lor 0, \\ \frac{1}{2} (\sqrt{f_{2}^{2} + j_{2}^{2} - 1 + f_{2}^{2} j_{2}^{2}}) \lor 0, \\ \sqrt[2 α]{f_{3}^{2 α} + j_{3}^{2 α} - f_{3}^{2 α} j_{3}^{2 α}} \end{matrix})$ , α ≥ 1.

$T (f, j) = (\begin{matrix} 0 \lor (\sqrt[2 α]{f_{1}^{2 α} + j_{1}^{2 α} - 1}), \\ 0 \lor (\sqrt[2 β]{f_{2}^{2 β} + j_{2}^{2 β} - 1}), \\ \sqrt[2 γ]{f_{3}^{2 γ} + j_{3}^{2 γ} - f_{3}^{2 γ} j_{3}^{2 γ}} \end{matrix})$ , α, β, γ ≥ 1.

Definition 23. Suppose that T be a spherical fuzzy t’-norm is called strict-nilpotent-strict (Δ_sns) iff $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ is strict t’-norm, t₂ is nilpotent t’-norm and s₃ is strict t’-conorm on [0, 1].

Example 6. $T (f, j) = (f_{1} j_{1}, 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1})$ , $\sqrt{f_{3}^{2} + j_{3}^{2} - f_{3}^{2} j_{3}^{2}})$ .

Definition 24. Suppose that T be a spherical fuzzy t’-norm is called nilpotent-strict-strict (Δ_nss) iff $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ is nilpotent t’-norm, t₂ is strict t’-norm and s₃ is strict t’-conorm on [0, 1].

Example 7. $T (f, j) = (0 \lor (\sqrt{f_{1}^{2} + j_{1}^{2} - 1}), f_{2} j_{2}$ , $\sqrt{f_{3}^{2} + j_{3}^{2} - f_{3}^{2} j_{3}^{2}})$ .

Proposition 10. strict-strict-nilpotent spherical fuzzyt^′-norm can be written as, $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2})$ , $t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ is strict t^′-norm, t₂ is strict t^′-norm and s₃ is nilpotent t^′-conorm on [0, 1].

Any spherical fuzzy t’-norm cannot be represented in the form of strict-strict-nilpotent spherical fuzzy t’-norm.

Proposition 11. Suppose that T be a spherical fuzzy t^′-norm is called strict t^′-norm iff $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* and T belongs to one of subclasses that are Δ_sss, Δ_nns, Δ_nss, Δ_sns.

Proposition 12. Suppose that T be a spherical fuzzy t^′-norm is called nilpotent t^′-norm iff $T (f, j) = (t_{1} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), s_{3} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* and T belongs to subclass that is Δ_nnn.

3.4 Some classification of spherical fuzzyt’-conorm

In this section, we define some subclasses representable spherical fuzzy t’-conorm as:

Definition 25. Suppose that S be a spherical fuzzy t’-conorm is called strict-strict-strict (∇ _sss) iff $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ and t₂ are strict t’-norm and s₃ is strict t’-conorm on [0, 1].

Example 8. $S (f, j) = (\sqrt{f_{1}^{2} + j_{1}^{2} - f_{1}^{2} j_{1}^{2}}, f_{2} j_{2}, f_{3} j_{3})$ for all f, j ∈ F^*.

Definition 26. Suppose that S be a spherical fuzzy t’-conorm is called nilpotent-nilpotent-nilpotent (∇ _nnn) iff $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ and t₂ are nilpotent t’-norm and s₃ is nilpotent t’-conorm on [0, 1].

Example 9. Nilpotent-nilpotent-nilpotent spherical fuzzy t’-conorm have following examples.

$S (f, j) = (\begin{matrix} 1 \land (\sqrt{f_{1}^{2} + j_{1}^{2}}), \\ 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), \\ 0 \lor (\sqrt{f_{3}^{2} + j_{3}^{2} - 1}) \end{matrix})$ .

$S (f, j) = (\begin{matrix} 1 \land (\sqrt[2 α]{f_{1}^{2 α} + j_{1}^{2 α}}), \\ 0 \lor (\sqrt[2 β]{f_{2}^{2 β} + j_{2}^{2 β} - 1}), \\ 0 \lor (\sqrt[2 γ]{f_{3}^{2 γ} + j_{3}^{2 γ} - 1}) \end{matrix})$ , α, β, γ ≥ 1.

Definition 27. Suppose that S be a spherical fuzzy t’-conorm is called strict-nilpotent-nilpotent (∇ _snn) iff $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ and t₂ are nilpotent t’-norm and s₃ is strict t’-conorm on [0, 1].

Example 10. Strict-nilpotent-nilpotent spherical fuzzy t’-conorm have following examples.

$S (f, j) = (\begin{matrix} \sqrt{f_{1}^{2} + j_{1}^{2} - f_{1}^{2} j_{1}^{2}}, \\ 0 \lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), \\ 0 \lor (\sqrt{f_{3}^{2} + j_{3}^{2} - 1}) \end{matrix})$ .

$S (f, j) = (\begin{matrix} \sqrt[2 α]{f_{1}^{2 α} + j_{1}^{2 α} - f_{1}^{2 α} j_{1}^{2 α}}, \\ \frac{1}{2} (\sqrt{f_{2}^{2} + j_{2}^{2} - 1 + f_{2}^{2} j_{2}^{2}}) \lor 0, \\ \frac{1}{2} (\sqrt{f_{3}^{2} + j_{3}^{2} - 1 + f_{3}^{2} j_{3}^{2}}) \lor 0 \end{matrix})$ , α ≥ 1.

Definition 28. Suppose that S be a spherical fuzzy t’-conorm is called strict-nilpotent-strict (∇ _sns) iff $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ is strict t’-norm, t₂ is nilpotent t’-norm and s₃ is strict t’-conorm on [0, 1].

Example 11. $S (f, j) = (\sqrt{f_{1}^{2} + j_{1}^{2} - f_{1}^{2} j_{1}^{2}}, 0$ $\lor (\sqrt{f_{2}^{2} + j_{2}^{2} - 1}), f_{3} j_{3})$ .

Definition 29. Suppose that S be a spherical fuzzy t’-conorm is called strict-strict-nilpotent (∇ _ssn) iff $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ is nilpotent t’-norm, t₂ is strict t’-norm and s₃ is strict t’-conorm on [0, 1].

Example 12. $S (f, j) = (\sqrt{f_{1}^{2} + j_{1}^{2} - f_{1}^{2} j_{1}^{2}}, f_{2} j_{2}, 0$ $\lor (\sqrt{f_{3}^{2} + j_{3}^{2} - 1}))$ .

Proposition 13. Nilpotent-strict-strict spherical fuzzy t^′-conorm can be written as, $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2})$ , $t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^*, where t₁ is strict t^′-norm, t₂ is strict t^′-norm and s₃ is nilpotent t^′-conorm on [0, 1].

Any spherical fuzzy t’-conorm cannot be represent in the form of nilpotent-strict-strict spherical fuzzy t’-conorm.

Proposition 14. Suppose that S be a spherical fuzzy t^′-conorm is called strict t^′-conorm iff $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* and S belongs to one of subclasses that are ∇_sss, ∇ _snn, ∇ _ssn, ∇ _sns.

Proposition 15. Suppose that S be a spherical fuzzy t^′-conorm is called nilpotent t^′-conorm iff $S (f, j) = (s_{3} (f_{1}^{2}, j_{1}^{2}), t_{2} (f_{2}^{2}, j_{2}^{2}), t_{1} (f_{3}^{2}, j_{3}^{2}))$ for all f, j ∈ F^* and S belongs to subclass that is ∇_nnn.

3.5 Distance between SFSs

Definition 30. Let R ≠ φ be a universe set and any two spherical fuzzy sets J₁, J₂. Then normalized Hamming distance d_NHD (J₁, J₂) is given as for all r ∈ R, $\begin{matrix} d_{NHD} (J_{1}, J_{2}) = \frac{1}{n} \sum_{l = 1}^{n} (| P_{j_{1}} (r) - P_{j_{2}} (r) | \\ + | I_{j_{1}} (r) - I_{j_{2}} (r) | + | N_{j_{1}} (r) - N_{j_{2}} (r) |) . \end{matrix}$

Definition 31. Let R ≠ φ be a universe set and any two spherical fuzzy sets J₁, J₂. Then normalized Euclidean distance d_NED (J₁, J₂) is given as for all r ∈ R,

$d_{NED} (J_{1}, J_{2}) = \sqrt{\frac{1}{n} \sum_{l = 1}^{n} ((P_{j_{1}} (r) - P_{j_{2}} (r))^{2} + (I_{j_{1}} (r) - I_{j_{2}} (r))^{2} + (N_{j_{1}} (r) - N_{j_{2}} (r))^{2})} .$

4 MADM problems by using TOPSIS method for SFNs

In this section, method for solving the MADM problems by utilizing TOPSIS method [22] for the spherical fuzzy information is established. Suppose that any finite collection C ={ c₁, c₂, …, c_m } of m alternatives and any finite collection G ={ g₁, g₂, …, g_n } of n attributes. Let $J_{a} = [J_{a_{jk}}] = [{〈 r, P_{a_{jk}} (r), I_{a_{jk}} (r), N_{a_{jk}} (r) 〉 | r \in R}]_{m \times n}$ $\begin{matrix} g_{1} & g_{2} & \dots & g_{n} \\ c_{1} & 〈 P_{a_{11}}, I_{a_{11}}, N_{a_{11}} 〉 & 〈 P_{a_{12}}, I_{a_{12}}, N_{a_{12}} 〉 & \dots & 〈 P_{a_{1 n}}, I_{a_{1 n}}, N_{a_{1 n}} 〉 \\ c_{2} & 〈 P_{a_{21}}, I_{a_{21}}, N_{a_{21}} 〉 & 〈 P_{a_{22}}, I_{a_{22}}, N_{a_{22}} 〉 & \dots & 〈 P_{a_{2 n}}, I_{a_{2 n}}, N_{a_{2 n}} 〉 \\ . & . & . & . & . \\ . & . & . & . & . \\ . & . & . & . & . \\ c_{m} & 〈 P_{a_{m 1}}, I_{a_{m 1}}, N_{a_{m 1}} 〉 & 〈 P_{a_{m 2}}, I_{a_{m 2}}, N_{a_{m 2}} 〉 & \dots & 〈 P_{a_{mn}}, I_{a_{mn}}, N_{a_{mn}} 〉 \end{matrix}$ be the decision matrix, where {〈r, P_{a
_jk} (r), I_{a
_jk} (r), N_{a
_jk} (r) 〉|r ∈ R} are the collection of SFNs.

If weight vector τ ={ τ₁, τ₂, …, τ_n } of attribute, with τ_l ≥ 0 and $\sum_{l = 1}^{n} τ_{l} = 1$ . Then, technique of handling the MADM problems are listed below:

Step 1. Normalized the given Decision Matrix.

In extensively, there are attributes which have two types (1) benefit criteria (2) worse criteria.

If given criteria is worse type then we use given below equation to modify the worse criteria into benefit criteria, $J^{c} = {〈 r, N_{a_{jk}} (r), I_{a_{jk}} (r), P_{a_{jk}} (r) 〉 | r \in R} .$

Where J^c is the complement of J. If given criteria is benefit type then no need to be normalized. P_{a
_jk}

Step 2. We use given below equations to calculate the positive ideal solution (PIS) J⁺ and negative ideal solution (NIS) J^-. $\begin{matrix} J^{+} & = & {j_{1}^{+}, j_{2}^{+}, \dots, j_{n}^{+}} \\ = & {max_{j} P_{a_{jk}}, min_{j} I_{a_{jk}}, min_{j} N_{a_{jk}} | k = 1, 2, \dots, n} . \end{matrix}$ $\begin{matrix} J^{-} & = & {j_{1}^{-}, j_{2}^{-}, \dots, j_{n}^{-}} \\ = & {min_{j} P_{a_{jk}}, min_{j} I_{a_{jk}}, max_{j} N_{a_{jk}} | k = 1, 2, \dots, n} \end{matrix}$ 5ptStep 3. In this step we use distance formulas to calculate the d_NHD (J_l, J_k) and d_NED (J_l, J_k) from PIS to each alternatives a_i (i = 1, 2, …, m) and similarly find distance between NIS to each alternatives a_i (i = 1, 2, …, m) respectively.

Step 4. In this step we use given below equations to find the closeness relation among each alternatives a_i (i = 1, 2, …, m). $\begin{matrix} l_{NHD} (J_{k}) = \frac{d_{NHD} (J_{k}, J^{-})}{d_{NHD} (J_{k}, J^{-}) + d_{NHD} (J_{k}, J^{+})} \\ l_{NED} (J_{k}) = \frac{d_{NED} (J_{k}, J^{-})}{d_{NED} (J_{k}, J^{-}) + d_{NED} (J_{k}, J^{+})} \end{matrix}$

Step 5. Rank all the alternatives a_i (i = 1, 2, …, m) and select the best one(s) in accordance with a_i (i = 1, 2, …, m). If any alternative has the highest l_NHD (J_k) value, then it is the finest alternative for decision maker, and similar by using l_NED (J_k) value, respectively.

4.1 An application to a Food circulation center evaluation problem

In this section, the proposed ranking method is applied to deal with circulation center evaluation problem. Consider a committee of decision-maker to perform the evaluation and to select the most suitable circulation center, among the three circulation centers F₁, F₂, F₃ and F₄. The decision maker evaluates the circulation center according to four attributes, which are given as follows:

Cost of transportation (C₁),

the load of capacity (C₂),

the satisfy demand with minimum delay (C₃), and

the quality (C₄):

According to the suitability ratings of three alternatives, F₁, F₂, F₃ and F₄ under four attributes C₁, C₂, C₃ and C₄ can be obtained as shown in Table 1.

Table 1
Food circulation center information

C ₁ C ₂ C ₃ C ₄

F ₁ 〈0.3, 0.8, 0.5〉〈0.8, 0.4, 0.3〉〈0.4, 0.5, 0.7〉 (0.4, 0.3, 0.4)

F ₂ 〈0.2, 0.6, 0.7〉〈0.3, 0.9, 0.1〉〈0.5, 0.3, 0.7〉 (0.6, 0.1, 0.6)

F ₃ 〈0.4, 0.8, 0.4〉〈0.5, 0.8, 0.2〉〈0.2, 0.3, 0.7〉 (0.5, 0.3, 0.4)

F ₄ 〈0.5, 0.3, 0.8〉〈0.6, 0.6, 0.3〉〈0.3, 0.6, 0.5〉 (0.4, 0.4, 0.4)

	C ₁	C ₂	C ₃	C ₄
F ₁	〈0.3, 0.8, 0.5〉	〈0.8, 0.4, 0.3〉	〈0.4, 0.5, 0.7〉	(0.4, 0.3, 0.4)
F ₂	〈0.2, 0.6, 0.7〉	〈0.3, 0.9, 0.1〉	〈0.5, 0.3, 0.7〉	(0.6, 0.1, 0.6)
F ₃	〈0.4, 0.8, 0.4〉	〈0.5, 0.8, 0.2〉	〈0.2, 0.3, 0.7〉	(0.5, 0.3, 0.4)
F ₄	〈0.5, 0.3, 0.8〉	〈0.6, 0.6, 0.3〉	〈0.3, 0.6, 0.5〉	(0.4, 0.4, 0.4)

Step 1. Since C₁, C₃ are cost-type criteria and C₂, C₄ are benefit-type criteria. So, we have need to normalize the decision matrices. Normalized decision matrix are shown in Table 2.

Table 2

Normalized Food circulation center information

	C ₁	C ₂	C ₃	C ₄
F ₁	〈0.5, 0.8, 0.3〉	〈0.8, 0.4, 0.3〉	〈0.7, 0.5, 0.4〉	(0.4, 0.3, 0.4)
F ₂	〈0.7, 0.6, 0.2〉	〈0.3, 0.9, 0.1〉	〈0.7, 0.3, 0.5〉	(0.6, 0.1, 0.6)
F ₃	〈0.4, 0.8, 0.4〉	〈0.5, 0.8, 0.2〉	〈0.7, 0.3, 0.2〉	(0.5, 0.3, 0.4)
F ₄	〈0.8, 0.3, 0.5〉	〈0.6, 0.6, 0.3〉	〈0.5, 0.6, 0.3〉	(0.4, 0.4, 0.4)

Step 2. Now, we find the PIS and NIS respectively. $\begin{matrix} J^{+} = {〈 0.8, 0.3, 0.2 〉, 〈 0.8, 0.4, 0.1 〉, \\ 〈 0.7, 0.3, 0.2 〉, 〈 0.6, 0.1, 0.4 〉}, \end{matrix}$ and $\begin{matrix} J^{-} = {〈 0.4, 0.3, 0.5 〉, 〈 0.3, 0.4, 0.3 〉, \\ 〈 0.5, 0.3, 0.5 〉, 〈 0.4, 0.1, 0.6 〉} . \end{matrix}$

Step 3. Now we fine the distance between PIS to each alternatives a_i (i = 1, 2, …, m). $\begin{matrix} Normalized Hamming Distance & Value \\ d_{NHD} (J_{1}, J^{+}) & 0.475 \\ d_{NHD} (J_{2}, J^{+}) & 0.475 \\ d_{NHD} (J_{2}, J^{+}) & 0.550 \\ d_{NHD} (J_{4}, J^{+}) & 0.500 \end{matrix}$ and similarly $\begin{matrix} Normalized Euclidean Distance & Value \\ d_{NED} (J_{1}, J^{+}) & 0.370 \\ d_{NED} (J_{2}, J^{+}) & 0.427 \\ d_{NED} (J_{2}, J^{+}) & 0.435 \\ d_{NED} (J_{4}, J^{+}) & 0.346 \end{matrix}$

Now we find the distance between NIS to each alternatives a_i (i = 1, 2, …, m). $\begin{matrix} Normalized Hamming Distance & Value \\ d_{NHD} (J_{1}, J^{-}) & 0.525 \\ d_{NHD} (J_{2}, J^{-}) & 0.500 \\ d_{NHD} (J_{2}, J^{-}) & 0.575 \\ d_{NHD} (J_{4}, J^{-}) & 0.475 \end{matrix}$ and similarly $\begin{matrix} Normalized EuclideanDistance & Value \\ d_{NED} (J_{1}, J^{-}) & 0.424 \\ d_{NED} (J_{2}, J^{-}) & 0.400 \\ d_{NED} (J_{2}, J^{-}) & 0.415 \\ d_{NED} (J_{4}, J^{-}) & 0.370 \end{matrix}$

Step 4. Now we find the closeness relation among each alternatives a_i (i = 1, 2, …, m). $\begin{matrix} l_{NHD} (J_{1}) = \frac{d_{NHD} (J_{1}, J^{-})}{d_{NHD} (J_{1}, J^{-}) + d_{NHD} (J_{1}, J^{+})} & \frac{0.525}{0.525 + 0.475} = 0.527 \\ l_{NHD} (J_{2}) = \frac{d_{NHD} (J_{2}, J^{-})}{d_{NHD} (J_{2}, J^{-}) + d_{NHD} (J_{2}, J^{+})} & \frac{0.500}{0.500 + 0.475} = 0.512 \\ l_{NHD} (J_{3}) = \frac{d_{NHD} (J_{3}, J^{-})}{d_{NHD} (J_{3}, J^{-}) + d_{NHD} (J_{3}, J^{+})} & \frac{0.575}{0.575 + 0.550} = 0.511 \\ l_{NHD} (J_{4}) = \frac{d_{NHD} (J_{4}, J^{-})}{d_{NHD} (J_{4}, J^{-}) + d_{NHD} (J_{4}, J^{+})} & \frac{0.475}{0.475 + 0.500} = 0.487 \end{matrix}$ and $\begin{matrix} l_{NED} (J_{1}) = \frac{d_{NED} (J_{1}, J^{-})}{d_{NED} (J_{1}, J^{-}) + d_{NED} (J_{1}, J^{+})} & \frac{0.424}{0.424 + 0.370} = 0.534 \\ l_{NED} (J_{2}) = \frac{d_{NED} (J_{2}, J^{-})}{d_{NED} (J_{2}, J^{-}) + d_{NED} (J_{2}, J^{+})} & \frac{0.400}{0.400 + 0.427} = 0.483 \\ l_{NED} (J_{3}) = \frac{d_{NED} (J_{3}, J^{-})}{d_{NED} (J_{3}, J^{-}) + d_{NED} (J_{3}, J^{+})} & \frac{0.415}{0.415 + 0.435} = 0.488 \\ l_{NED} (J_{4}) = \frac{d_{NED} (J_{4}, J^{-})}{d_{NED} (J_{4}, J^{-}) + d_{NED} (J_{4}, J^{+})} & \frac{0.370}{0.370 + 0.346} = 0.516 \end{matrix}$

Step 5. Now rank the all the alternative by descending order, tabular representation is given below, $\begin{matrix} Calculation result of each food circulation center \\ \begin{matrix} Normalized Hamming Distance \\ distance & l_{NHD} (J_{1}) > l_{NHD} (J_{2}) > l_{NHD} (J_{3}) > l_{NHD} (J_{4}) \\ Ranking & F_{1} > F_{2} > F_{3} > F_{4} \\ Normalized Euclidean Distance \\ distance & l_{NED} (J_{1}) > l_{NED} (J_{4}) > l_{NED} (J_{3}) > l_{NED} (J_{2}) \\ Ranking & F_{1} > F_{4} > F_{3} > F_{2} \end{matrix} \end{matrix}$

Hence F₁ food circulation center is our finest alternative.

5 Conclusion

In this paper, we construct spherical fuzzy sets and examined how spherical fuzzy sets are advanced tools of the fuzzy sets, intuitionistic fuzzy sets and picture fuzzy sets. Spherical fuzzy set is the direct extension of Pythagorean fuzzy set, we seen that how we put neutral membership, I_j (r) =0 in SPSs to reduce in Pythagorean fuzzy sets. Also seen that how SFSs is extension of picture fuzzy set by taking squares of the membership degrees we obtain the spherical fuzzy sets. In this paper, spaces of spherical membership grades and pictorial membership grades are examined by graphically. We developed spherical fuzzy t’-norms Ť, spherical fuzzy t’-conorms Š and introduced some classifications of spherical fuzzy triangular norms and conorms which are useful for establish the aggregation operator to aggregate the spherical fuzzy information. According to the importance of evaluating options and ranking fuzzy quantities in the design of algorithms that are used for solving fuzzy linear optimization problems such as simplex-based algorithms, an approach for ranking is proposed in this paper; to illustrate the usage, applicability, and advantages of the proposed ranking approach, it has been applied for evaluating the circulation center of food as an applicable problem. We have outlined a practical example about of an arrangement of drive frameworks to check the created approach and to show its common uses and adaptability as compare to conventional methods. A lot much work needs to be done on ideas because of its usefulness in practical problems. Some directions are graphs of spherical fuzzy sets, aggregation operators on these sets which will enable us to deal with different kind of practical challenges.

Footnotes

Acknowledgments

The authors 2,3,5 extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups programs under grant numberR,G.P1/76-40.

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