Single-valued neutrosophic entropy and similarity measures to solve supplier selection problems

3.1 Single-valued neutrosophic entropy

Definition 3.1. Assume that α = 〈α₁, α₂, α₃〉 is a SVNV. A function E : SVNV ¯ → [ 0 , 1 ] is called a single-valued neutrosophic entropy on α, if it satisfies the following requirements:

E (α) =0 ⇔ α _t  = 0 or 1, t = 1, 2, 3;

α _t  ≤ β _t  ≤ 0.5, t = 1, 2, 3, or α _t  ≥ β _t  ≥ 0.5, t = 1, 2, 3 .

With the help of trigonometric functions, we establish an information measure formula for SVNV α as follows: e ( α ) = 1 3 ( 2 - 1 ) ∑ t = 1 3 ( sin α t - α t c + 1 4 π + cos α t - α t c + 1 4 π - 1 ) .(1)

Theorem 3.1. Assume that α = 〈α₁, α₂, α₃〉 is a SVNV, the information measure formula e (α), defined by Equation (1), satisfy the four requirements in Definition 3.1.

Proof. Let f ( x ) = 1 2 - 1 ( sin x 4 π + cos x 4 π - 1 ) , x ∈ [ 0 , 2 ] ,(2)

it is follows that f ′ ( x ) = df ( x ) dx = π 4 ( 2 - 1 ) ( cos x 4 π - sin x 4 π ) = 2 π 4 ( 2 - 1 ) cos x + 1 4 π , x ∈ [ 0 , 2 ] .(3)

It is easy to know that f′ (x) ≥0, x ∈ [0, 1] and f′ (x) ≤0, x ∈ [1, 2]. Thus, if x ∈ [0, 1], f (x) is monotonically increasing function; if x ∈ [1, 2], f (x) is monotonically decreasing function. In addition, we can obtain that 0 ≤ f (x) ≤1, and f min ( x ) = 0 ⇔ x = 0 or x = 2 ,(4) f max ( x ) = 1 ⇔ x = 1 .(5)

In what follows, we testify that e (α) satisfies the four requirements listed in Definition 3.1.

Assume that e (α) =0.

Since 0 ≤ α _t  ≤ 1, t = 1, 2, 3, then α t - α t c + 1 = α t - ( 1 - α t ) + 1 = 2 α t ∈ [ 0 , 2 ] . Therefore, from the above analysis, we have 0 ≤ e (α) ≤1. Therefore, e ( α ) = 0 ⇔ f ( α t - α t c + 1 ) = 0 , t = 1 , 2 , 3 . In addition, f ( α t - α t c + 1 ) = 0 , t = 1 , 2 , 3 if and only if α t - α t c + 1 = 2 α t = 0 , t = 1 , 2 , 3 or α t - α t c + 1 = 2 α t = 2 , t = 1 , 2 , 3 . It follows that e (α) =0 if and only if α _t  = 0 or 1, t = 1, 2, 3.

Assume that α _t  = 0 or 1, t = 1, 2, 3, then we have α t - α t c + 1 = 0 or 2 , t = 1 , 2 , 3 , one can obtain that f ( α t - α t c + 1 ) ) = 0 , t = 1 , 2 , 3 , thus e (α) =0.

Since α t - α t c + 1 = α t - ( 1 - α t ) + 1 = 2 α t · ∈ [ 0 , 2 ] , then according to the above analysis, one can obtain that e ( α ) = 1 ⇔ f ( α t - α t c + 1 ) ) = 0 , t = 1 , 2 , 3 ⇔ α t - α t c + 1 = 1 , t = 1 , 2 , 3 ⇔ α t = 0.5 , t = 1 , 2 , 3 . Therefore, e (α) =1 ⇔ 〈α₁, α₂, α₃〉 = 〈0.5, 0.5, 0.5〉.

e ( α c ) = 1 3 ( 2 - 1 ) ∑ t = 1 3 ( sin α t c - ( α t c ) c + 1 4 π + cos α t c - ( α t c ) c + 1 4 π - 1 ) = 1 3 ( 2 - 1 ) ∑ t = 1 3 ( sin α t c - α t + 1 4 π + cos α t c - α t + 1 4 π - 1 ) = 1 3 ( 2 - 1 ) ∑ t = 1 3 ( sin 2 - ( α t - α t c + 1 ) 4 π + cos 2 - ( α t - α t c + 1 ) 4 π - 1 ) = 1 3 ( 2 - 1 ) ∑ t = 1 3 ( cos α t - α t c + 1 4 π + sin α t - α t c + 1 4 π - 1 ) = e ( α ) .(6)

Suppose that α _t  ≤ β _t  ≤ 0.5, t = 1, 2, 3, while α t - α t c + 1 = 2 α t , β t - β t c + 1 = 2 β t , one can obtain that

Because f (x) is a monotonically increasing function of x ∈ [0, 1], thus f ( α t - α t c + 1 ) ≤ f ( β t - β t c + 1 ) , t = 1 , 2 , 3 ,(8)

it follows that e (α) ≤ e (β).

If α _t  ≥ β _t  ≥ 0.5, t = 1, 2, 3, we also can get that e (α) ≤ e (β). Thus, we complete the proof of Theorem 3.1. □

Definition 3.2. Assume that α = 〈α₁, α₂, α₃〉 is a SVNV, the information measure formula e (α), defined by Equation (1), is called the single-valued neutrosophic entropy of SVNV α.

Definition 3.3. Assume that α = 〈α₁, α₂, α₃〉 and β = 〈β₁, β₂, β₃〉 are two SVNVs. A real-valued function S : SVNV ¯ × SVNV ¯ → [ 0 , 1 ] is called a single-valued neutrosophic similarity measure of α and β, if it satisfies the following four properties:

S (α, β) =0 ⇔ α _t  - β _t  = -1 or 1, t = 1, 2, 3;

Similarly, based on the trigonometric functions, we construct the following information measure formula to measure the similarity level for SVNVs α = 〈α₁, α₂, α₃〉 and β = 〈β₁, β₂, β₃〉: s ( α , β ) = 1 3 ( 2 - 1 ) ∑ t = 1 3 ( 2 sin α t - β t + 2 4 π - 1 ) .(9)

Theorem 3.2. Assume that there are two SVNVs α = 〈α₁, α₂, α₃〉 and β = 〈β₁, β₂, β₃〉, the information measure formula s (α, β), defined by Eq. (9), satisfy the four requirements in Definition 3.3.

Proof. Let g ( x ) = 1 2 - 1 ( 2 sin x + 2 4 π - 1 ) , x ∈ [ - 1 , 1 ] ,(10)

then g ′ ( x ) = dg ( x ) dx = 2 π 4 ( 2 - 1 ) cos x + 2 4 π , x ∈ [ - 1 , 1 ] .(11)

Since g′ (x) ≥0, x ∈ [-1, 0] and g′ (x) ≤0, x ∈ [0, 1], thus, when x ∈ [-1, 0] ,  g (x) , is monotonically increasing function; when x ∈ [0, 1], g (x) is monotonically decreasing function. In addition, we also can obtain that 0 ≤ g (x) ≤1, and g min ( x ) = 0 ⇔ x = - 1 or x = 1 ,(12) g max ( x ) = 1 ⇔ x = 0 .(13)

In the following, we testify that s (α, β) satisfies the four properties listed in Definition 3.3.

Assume that s (α, β) =0.

As 0 ≤ α _t  ≤ β _t  ≤ 1, t = 1, 2, 3, it is obvious that -1 ≤ α _t  - β _t  ≤ 1, t = 1, 2, 3, thus 0 ≤ g (α _t  - β _t ) ≤1, t = 1, 2, 3,

which indicates that each term in the summation of s (α, β) is non-negative. If s (α, β) =0, then each term in the summation of s (α, β) is zero, i.e., g (α _t  - β _t ) =0, t = 1, 2, 3, thus α _t  - β _t  = -1 or 1, t = 1, 2, 3 .

Assume that α _t  - β _t  = -1 or 1, t = 1, 2, 3, then we can obtain that g (α _t  - β _t ) =0, t = 1, 2, 3, it follows that s (α, β) =0.

From the above analysis and Equation (12), one can obtain that

s ( α , β ) = 1 ⇔ g ( α t - β t ) = 1 , t = 1 , 2 , 3 ⇔ α t - β t = 0 , t = 1 , 2 , 3 ⇔ 〈 α 1 , α 2 , α 3 〉 = 〈 β 1 , β 2 , β 3 〉(14)

It is known that for ∀x ∈ R, we have sin ( π 2 - x ) = sin ( π 2 + x ) , then one can obtain that

s ( α , β ) = 1 3 ( 2 - 1 ) ∑ t = 1 3 ( 2 sin α t - β t + 2 4 π - 1 ) = 1 3 ( 2 - 1 ) ∑ t = 1 3 ( 2 sin ( π 2 + α t - β t 4 π ) - 1 ) = 1 3 ( 2 - 1 ) ∑ t = 1 3 ( 2 sin ( π 2 - α t - β t 4 π ) - 1 ) = 1 3 ( 2 - 1 ) ∑ t = 1 3 ( 2 sin β t - α t + 2 4 π - 1 ) = s ( β , α )(15)

Assume that 0 ≤ α _t  ≤ β _t  ≤ γ _t  ≤ 1, t = 1, 2, 3, then -1 ≤ - γ _t  ≤ - β _t  ≤ - α _t  ≤ 0, t = 1, 2, 3, thus -1 ≤ α _t  - γ _t  ≤ β _t  - γ _t  ≤ 0, t = 1, 2, 3, -1 ≤ α _t  - γ _t  ≤ α _t  - β _t  ≤ 0, t = 1, 2, 3 .

Because g (x) is a monotonically increasing function of x ∈ [-1, 0], thus g ( α t - γ t ) ≤ g ( α t - β t ) , t = 1 , 2 , 3 ,(16) g ( α t - γ t ) ≤ g ( β t - γ t ) , t = 1 , 2 , 3 ,(17)

it follows that S ( α , γ ) ≤ S ( α , β ) , S ( α , γ ) ≤ S ( β , γ ) .

If 1 ≥ α _t  ≥ β _t  ≥ γ _t  ≥ 0, t = 1, 2, 3, then -1 ≤ - α _t  ≤ - β _t  ≤ - γ _t  ≤ 0, t = 1, 2, 3, thus     1 ≥ α _t  - γ _t  ≥ β _t  - γ _t  ≥ 0, t = 1, 2, 3,     1 ≥ α _t  - γ _t  ≥ α _t  - β _t  ≥ 0, t = 1, 2, 3 .

Because g (x) is a monotonically decreasing function of x ∈ [0, 1], thus g ( α t - γ t ) ≤ g ( α t - β t ) , t = 1 , 2 , 3 ,(18) g ( α t - γ t ) ≤ g ( β t - γ t ) , t = 1 , 2 , 3 ,(19)

it follows that

S (α, γ) ≤ S (α, β) , S (α, γ) ≤ S (β, γ) .

This completes the proof of Theorem 3.2. □

Definition 3.4. Assume that there are two SVNVs α = 〈α₁, α₂, α₃〉 and β = 〈β₁, β₂, β₃〉, the information measure formula s (α, β), defined by Equation (9), is called the single-valued neutrosophic similarity measure between α and β.

In this subsection, the relationship between the single-valued neutrosophic entropy and similarity measure is discussed in details, and we also get that Eq. (1) and Eq. (9) can be transformed by each other.

Theorem 3.3. Assume that α = 〈α₁, α₂, α₃〉 is a SVNV, then we have E (α) = S (α, α ^c ).

Proof. In what follows, in order to verify S (α, α ^c ) is a single-valued neutrosophic entropy, i.e., S (α, α ^c ) satisfy four properties in Definition 3.1.

If E (α) =0, then S (α, α ^c ) =0. From Definition 3.3, we have α t - α t c = - 1 or 1 , t = 1 , 2 , 3 , i.e., 2α _t  - 1 = -1 or 1, t = 1, 2, 3, thus α _t  = 0 or 1, t = 1, 2, 3.

Assume that α _t  = 0 or 1, t = 1, 2, 3, then α t - α t c = -1 or 1, t = 1, 2, 3. According to Definition 3.3, we can get that S (α, α ^c ) =0, that is E (α) =0.

By utilizing Definition 3.1 and Definition 3.3, one can obtain that

E ( α ) = 1 ⇔ S ( α , α c ) = 1 ⇔ 〈 α 1 , α 2 , α 3 〉 = 〈 α 1 c , α 2 c , α 3 c 〉 ⇔ 〈 α 1 , α 2 , α 3 〉 = 〈 1 - α 1 , 1 - α 2 , 1 - α 3 〉 ⇔ α t = 1 - α t , t = 1 , 2 , 3 ⇔ α t = 0.5 , t = 1 , 2 , 3 ⇔ 〈 α 1 , α 2 , α 3 〉 = 〈 0.5 , 0.5 , 0.5 〉(20)

According to Definition 2.2, we have α t c = 1 - α _t , t = 1, 2, 3, then ( α t c ) c = ( 1 - α t ) c = 1 - ( 1 -α _t ) = α _t , t = 1, 2, 3, that is (α ^c )  ^c ) = α. Therefore, E (α ^c ) = S (α ^c , (α ^c )  ^c ) = S (α ^c , α) = S (α, α ^c ) = E (α).

With the same reason, if α _t  ≥ β _t  ≥ 0.5, t = 1, 2, 3, we can also prove that E (α)≤E (β). This completes the proof of Theorem 3.3. □

The following corollary can be obtained in accordance with Theorem 3.3.

Corollary 3.1. Assume that α = 〈α₁, α₂, α₃〉 is a SVNV, then we have e (α) = s (α, α ^c ).

5.1 Numerical example

In recent years, in order to achieve success in competitions and keep customers satisfied, companies pay more attention to supply chain management. The supply chain is a complex optimization selection problem involving a large number of participants. One of the goals for the complete supply chain is to find an optimal supplier from the set alternative suppliers. Under the pressure of global competition, enterprises are more inclined to improve the ability of supply chain management to enhance the competitive advantages. Supplier selection as one of the important parts of supply chain management, aims at evaluating and improving organizational ability of a supplier in analyzing, identifying and treating competition risks. Selecting a desirable supplier will protect supply chain from the risks arising from emergency events [37].

Considering the supplier selection problems. An investment company wants to invest a sum of money in the best option. There are five suppliers can be selected: car supplier (x₁), food supplier (x₂), computer supplier (x₃), game supplier (x₄), arms supplier (x₅). The DMs evaluate these suppliers by means of four main attributes, i.e., C₁: risk, C₂: the cost of investment, C₃: environmental impact and C₄: rate of return. The DMs utilize SVNVs α ij = 〈 α 1 ij , α 2 ij , α 3 ij 〉 to express the evaluation information of five suppliers x _i  (i = 1, 2, 3, 4, 5) with respect to the above four attributes C _j  (j = 1, 2, 3, 4), and then we can establish a single-valued neutrosophic decision matrix R = ( α ij ) 5 × 4 , which is shown in Table 1.

Now, we select the best supplier by using the proposed MADM method, which is described by the following decision steps:

Step 1. By using Equation (21), we convert all the cost attribute values to the benefit attribute values, and then we obtain the normalized single-valued neutrosophic decision matrix ℵ = (χ _ij ) _5×4. Table 2 lists the normalized decision information.

Step 2. By utilizing model (22) and Equation (24), we derive the attribute weights: w 1 = 0.1923 , w 2 = 0.3208 , w 3 = 0.1260 , w 4 = 0.3609 .

Step 3. Calculate the similarity measures from supplier x _i to X^* and X_* by applying Equations (25 and 26): S 1 + = 0.4011 , S 2 + = 0.4617 , S 3 + = 0.5407 , S 4 + = 0.3220 , S 5 + = 0.5794 , S 1 - = 0.3894 , S 2 - = 0.3533 , S 3 - = 0.2915 , S 4 - = 0.4619 , S 5 - = 0.2308 .

Step 4. Computer the closeness degree Γ _i  (i = 1, 2, 3, 4, 5) as follows: Γ 1 = 0.5074 , Γ 2 = 0.5665 , Γ 3 = 0.6497 , Γ 4 = 0.4108 , Γ 5 = 0.7151 .

Step 5. As Γ₅ > Γ₃ > Γ₂ > Γ₁ > Γ₄, then the ranking order of the five suppliers is x₅ ≻ x₃ ≻ x₂ ≻x₁ ≻ x₄. Thus, x₅ is the best choice among the five suppliers.

In the following, in order to validate the effectiveness of the proposed MADM method, we compare our proposed model with previous methods in the existing literature, and then we highlight the advantages of the proposed method.

By using the single-valued neutrosophic Frank normalized prioritized Bonferroni mean (SVNFNP-BM) operator, Ji et al. [38] developed an approach to deal with the decision-making problems. Utilize Ji et al. [38]’s approach to address the aforementioned problem, the main steps are as follows:

Step 1. See Step 1, then we can obtain the normalized single-valued neutrosophic decision matrix ℵ = (χ _ij ) _5×4.

Step 2. By using Equations (21 and 22) in [38], we obtain the corresponding weights w _ij  (i = 1, 2, 3, 4, 5, j = 1, 23, 4), which are shown in Table 3.

Step 3. Utilizing the SVNFNPBM operator (i.e., Eq. (13)) in [38], the overall single-valued neutrosophic values of the suppliers are obtained as follows (let γ = 10, p = q = 2): χ₁ = 〈0.5275, 0.1634, 0.2618〉, χ₂ = 〈0.5799, 0.1240, 0.2445〉, χ₃ = 〈0.6104, 0.1277, 0.2350〉, χ₄ = 〈0.4819, 0.1546, 0.2311〉, χ₅ = 〈0.6722, 0.1259, 0.2181〉 .

Step 4. According to Definition 4 in [35], we determine the score value sc (χ _i ) (i = 1, 2, 3, 4, 5) of each χ _i  (i = 1, 2, 3, 4, 5) as follows: sc ( χ 1 ) = 0.4694 , sc ( χ 2 ) = 0.5437 , sc ( χ 3 ) = 0.5600 , sc ( χ 4 ) = 0.4708 , sc ( χ 5 ) = 0.6012 .

Step 5. Because sc (χ₅)> sc (χ₃) > sc (χ₂) > sc (χ₄) > sc (χ₁), then the ranking order of the five suppliers is x₅ ≻ x₃ ≻ x₂ ≻ x₄ ≻ x₁, and the most desirable supplier is x₅.

In Ref. [35], based on the cotangent function, Ye proposed a weighted cotangent similarity measure for single-valued neutrosophic sets (SVNSs) and developed a new MADM method. By using the weighted cotangent similarity measure in [35], the optimal supplier can be selected as follows:

First, based on the original single-valued neutrosophic decision matrix R = ( α ij ) 5 × 4 , one can obtain the normalized single-valued neutrosophic decision matrix ℵ = (χ _ij ) _5×4 and attribute weights (See Step 1 and Step 2).

Then, we utilize the following weighted cotangent similarity measure [35]: WCot ( χ i , X * ) = ∑ j = 1 4 w j cot [ π 4 + π 4 max ( | χ 1 ij - χ 1 j * | , | χ 2 ij - χ 2 j * | , | χ 3 ij - χ 3 j * | ) ] ,(28)

to determine the similarity measure from supplier χ _i to ideal supplier X^*: WCot ( χ 1 , X * ) = 0.2015 , WCot ( χ 2 , X * ) = 0.2727 , WCot ( χ 3 , X * ) = 0.3126 , WCot ( χ 4 , X * ) = 0.3899 , WCot ( χ 5 , X * ) = 0.5054 .

As WCot (χ₅, X^*) > WCot (χ₄, X^*) > WCot (χ₃, X^*) > WCot (χ₂, X^*) > WCot (χ₁, X^*), then the ranking order of five suppliers is x₅≻ x₄ ≻ x₃ ≻ x₂ ≻ x₁, and the most desirable supplier is x₅.

According to above analysis, the ranking of the four suppliers and the most desirable supplier can be summarized in Table 4.

From the above numerical example and comparison with other methods, the proposed single-valued neutrosophic MADM method has the following characteristics:

Compared with the method developed by Ji et al. [38], although the method in this paper generates the same desirable supplier with Ji et al. [38]’s approach, our method and Ji et al. [38]’s approach obtain the different ranking of the suppliers. In fact, according to the original normalized single-valued neutrosophic decision matrix ℵ = (χ _ij ) _5×4, one can obtain that χ₁₁> χ₄₁, χ₁₂ > χ₄₂, χ₁₃ > χ₄₃, χ₁₄ < χ₄₄, then the supplier x₁ is superior to x₄. From the decision-making results in Table 4, our method can derive the correct ranking of supplier x₁ and x₄. In addition, Ji et al. [38]’s approach derives the corresponding weights of attributes by using score functions, which easy leads to weights of attributes are unreasonable, because the score functions are negative in some situations. As a consequence, our model is much more reasonable and scientific than Ji et al. [38]’s approach.