Abstract
The selection of third-party logistics (TPL) providers can greatly influence the development of a company, and it has therefore become a hot topic for research. The process of selecting TPL providers is complicated because it includes fuzzy information that can be depicted by single-valued neutrosophic numbers (SVNNs). This paper establishes a new method for selecting TPL providers under single-valued neutrosophic environments. This method simultaneously takes into account the interrelationships among criteria and the different priority levels of criteria. To construct the method, we first introduce Frank operations of SVNNs, considering that extant operations for SVNNs lack flexibility and robustness. Then, we develop the single-valued neutrosophic Frank normalized prioritized Bonferroni mean (SVNFNPBM) operator utilizing the proposed Frank operations. The SVNFNPBM operator considers the interrelationships among criteria and their distinct priority levels by combining the Bonferroni mean and the prioritized average operators. Subsequently, we construct a method for selecting TPL providers based on the proposed operator. Finally, we describe a numerical example of selecting a TPL provider and conduct a comparative analysis to verify the applicability and feasibility of the proposed method.
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1 Introduction
Many companies consider logistics seriously because of the great influence it exerts on economic development. Sometimes, companies choose to outsource logistics to third-party logistics (TPL) providers, mainly because TPL providers can render professional services at lower costs compared with internal logistics. The process of selecting TPL providers has attracted the attention of researchers [1, 2]. In essence, the problem of selecting a TPL provider involves multi-criteria decision-making (MCDM). Several criteria must be considered when selecting a TPL provider, including the cost of service, operational experience in the industry, customer satisfaction, and market reputation. In this respect, methods for selecting TPL providers can be described through MCDM models.
The process of selecting TPL providers involves fuzzy information that cannot be addressed by crisp values. Zadeh [3] developed fuzzy logic and fuzzy sets (FSs) to handle fuzzy information. However, FSs have some limits; for example, they cannot depict uncertain information. Numerous extensions of FSs have been proposed [4–6] to tackle the deficiencies of FSs. To address the issue that FSs cannot depict information uncertainty, Turksen [7] developed interval-valued FSs. Basing the work on FSs, Atanassov [8] introduced the notion of nonmembership and proposed intuitionistic fuzzy sets (IFSs). Subsequently, Atanassov and Gargov [9] presented interval-valued IFSs based on IFSs. Furthermore, Torra and Narukawa [10, 11] defined hesitant fuzzy sets (HFSs) to depict decision makers’ hesitancy, which cannot be expressed by FSs. Chen et al. [12] then developed interval-valued HFSs (IVHFSs). However, HFSs and IVHFSs are unable to deal with indeterminacy in decision-making processes. To overcome this problem, Smarandache [13, 14] introduced neutrosophic sets (NSs), which depict fuzzy information using the functions of truth, indeterminacy, and falsity. Notably, the function of indeterminacy is independent of the functions of truth and falsity [15]. Nevertheless, NSs are hard to apply in practical problems because the values of the truth, indeterminacy and falsity functions lie within ]0−, 1+[ [16]. Therefore, Wang et al. [17] presented single-valued neutrosophic sets (SVNSs), in which truth, indeterminacy, and falsity functions lie between zero and one. Since then, SVNSs have been studied [18, 19] and extended by combining them with other theories [20–24]. In addition, SVNSs and various other extensions of FSs have been applied in various fields, including decision-making [25–33], medical diagnosis [34–37], green product development [38], tourists’ restaurant selection [39], and cloud service selection [40].
The aggregation operator represents an important tool for constructing MCDM methods [41]. Numerous researchers [42, 43] have studied aggregation operators under all sorts of fuzzy environments, including single-valued neutrosophic environments. Most aggregation operators suppose that the elements integrated are mutually independent. However, criteria may be correlative in some practical problems, like the selection of TPL providers. In these situations, operational experience in the industry and market reputation may influence the cost of service, and customer satisfaction may both depend on and affect operational experience in the industry. In solving these kinds of practical problems, it is important to consider interrelationships among criteria, which can be accomplished by the strategy of the Bonferroni mean (BM) operator [44]. Researchers have studied the BM and weighted BM (WBM) operators in several fuzzy environments [45], including intuitionistic fuzzy environments [46–48], hesitant fuzzy environments [49], and neutrosophic environments [50]. However, Zhou and He [51] pointed out that WBM operators do not satisfy reducibility; to conquer this shortcoming, they developed the normalized WBM (NWBM) operator [51] for intuitionistic fuzzy environments. Subsequently, the NWBM operator has been applied in a few other fuzzy environments, like hesitant fuzzy environments [52] and single-valued neutrosophic environments [53].
In some practical MCDM problems, criteria have different priority levels. For instance, the four criteria involved in selecting TPL providers can have distinct degrees of priority. A company may attach more importance to customer satisfaction than the other three criteria. That company may set the cost of service as the second most important criterion, followed by operational experience in the industry. In this situation, market reputation represents the least significant criterion. When tackling this kind of problem, the distinct priorities of the criteria should be taken into consideration. This necessitates the introduction of the prioritized average (PA) operator [54], which considers criteria priority levels by assigning each criterion a weight calculated according to its priority. The higher the priority of a criterion is, the greater its weight will be. The PA operator has been applied in a variety of fuzzy environments, including intuitionistic fuzzy environments [55–57], hesitant fuzzy environments [58, 59], 2-tuple fuzzy environments [60, 61], and interval-valued neutrosophic environments [62].
All the aggregation operators described above are defined on the basis of algebraic product and sum. However, the algebraic operational laws lack flexibility and robustness. Frank operational laws [63] can be utilized to address this defect. Frank operational laws are generalizations of algebraic operational laws, but they are more flexible for involving a parameter. Researchers have studied Frank operational laws under fuzzy environments. For example, Zhang et al. [64] explored Frank operational laws in the context of IFSs and presented several intuitionistic fuzzy power aggregation operators along with the proposed laws. Zhang [65] further proposed Frank operational laws for interval-valued IFSs and defined two Frank aggregation operators for interval-valued IFSs. Similarly, Qin et al. [66] defined Frank operational laws for HFSs and developed several aggregation operators based on the proposed laws. However, Frank operational laws have barely been studied in the context of NSs.
SVNSs can depict fuzzy information in the process of selecting TPL providers as fully as possible. Criteria for selecting TPL providers are correlative and have different priorities. BM operators can consider the interrelationships among criteria, and PA operators can take into account the criteria’s distinct priorities. Furthermore, as mentioned above, Frank operational laws are flexible and robust. However, Frank operational laws for SVNSs have not been studied, and BM and PA operators based on Frank operational laws under single-valued neutrosophic environments have not been developed. In addition, extant methods for the selection of TPL providers cannot simultaneously take into consideration the interrelationships among criteria and their different priorities. To overcome these shortcomings, this paper puts forth a comprehensive method for selecting TPL providers. To achieve this goal, we first define Frank operational laws for SVNSs. Then, we develop a single-valued neutrosophic Frank normalized prioritized BM (SVNFNPBM) operator by combining PA and BM operators with the proposed Frank operational laws. Finally, we establish a method for selecting TPL providers based on the proposed SVNFNPBM operator.
The remainder of this paper is organized as follows. Section 2 reviews some basic concepts concerning SVNSs, Frank operations, and PA and BM operators. Section 3 defines Frank operations for single-valued neutrosophic numbers (SVNNs) and explores some properties of the proposed operations. Section 4 develops the SVNFNPBM operator and discusses its properties. Section 5 constructs a novel method for selecting TPL providers under single-valued neutrosophic environments. Section 6 describes a numerical example of selecting TPL providers and conducts a comparative analysis to demonstrate the applicability and feasibility of the proposed method. Section 6 also discusses the influence of parameters on the final ranking order. Finally, Sect. 7 outlines conclusions and suggests several directions for future research.
2 Preliminaries
This section reviews some basic concepts about SVNSs, Frank operations, and PA and BM operators, which will be utilized throughout the rest of the paper.
2.1 SVNSs
Definition 1
[13, 14] Let X be a finite set of points (objects), and let x denote a generic element in X. An NS A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x), and a falsity-membership function F A (x). For any arbitrary x in X, T A (x) ∊ ]0−, 1+[, I A (x) ∊ ]0−, 1+[, and F A (x) ∊ ]0−, 1+[. There is no constraint on the sum of T A (x), I A (x), and F A (x), that is, 0− ≤ T A (x) + I A (x) + F A (x) ≤ 3+.
However, NSs are difficult to apply in practical problems. To overcome this challenge, Wang et al. [17] defined SVNSs.
Definition 2
[17] Let X be a finite set of points (objects), and let x denote a generic element in X. An SVNS A in X is characterized by a truth-membership function T A (x), an indeterminacy-membership function I A (x), and a falsity-membership function F A (x). For any arbitrary x in X, T A (x) ∊ [0, 1], I A (x) ∊ [0, 1], and F A (x) ∊ [0, 1]. An SVNS A can be defined as:
An SVNS is a subclass of NS. The sum of T A (x), I A (x), and F A (x) lies between zero and three, that is, 0 ≤ T A (x) + I A (x) + F A (x) ≤ 3.
For convenience, we use x = (T x , I x , F x ) to represent an SVNN, which is an element in an SVNS.
Definition 3
[17] Let \(A = \left\{ {\langle x,T_{A} (x),I_{A} (x),F_{A} (x)\rangle |x \in X} \right\}\) be an SVNS. The complement A c of A is defined as \(A^{\text{c}} = \{ \langle x, F_{A} \left( x \right), 1 - I_{A} (x),T_{A} (x)\rangle |x \in X\} .\)
In order to compare two SVNNs, Şahin [42] defined a comparison method with the proposed score and accuracy functions.
Definition 4
[42] Let x = (T x , I x , F x ) be an SVNN. The score function sc of SVNN x can be defined as follows:
where sc(x) ∊ [−1, 1].
Definition 5
[42] Let x = (T x , I x , F x ) be an SVNN. The accuracy function l of SVNN x can be defined as follows:
where l(x) ∊ [−1, 1].
Definition 6
[42] Let x = (T x , I x , F x ) and y = (T y , I y , F y ) be two SVNNs. The comparison method between x and y can be defined as follows:
-
(1)
when sc(x) < sc(y), x ≺ y;
-
(2)
when sc(x) = sc(y) and l(x) < l(y), x ≺ y; and
-
(3)
when sc(x) = sc(y) and l(x) = l(y), x ∼ y.
Furthermore, the algebraic operations of SVNNs have been defined as follows.
Definition 7
[53] Let x = (T x , I x , F x ) and y = (T y , I y , F y ) be two SVNNs, and λ > 0. The algebraic operations can be defined as follows:
-
(1)
x ⊕ y = (T x + T y − T x T y , I x I y , F x F y );
-
(2)
x ⊗ y = (T x T y , I x + I y − I x I y , F x + F y − F x F y );
-
(3)
λx = (1 − (1 − T x )λ, I λ x , F λ x ); and
-
(4)
x λ = (T λ x , 1 − (1 − I x )λ, 1 − (1 − F x )λ).
2.2 Frank operations
Frank operations include the Frank product and Frank sum. The Frank product ⊗ F is a triangular norm, and the Frank sum ⊕ F is a triangular conorm.
Definition 8
[63] Let u and z be two real numbers. The Frank product ⊗ F and Frank sum ⊕ F between u and z are defined as follows:
where γ > 1.
Wang and He [67] discussed some properties of the Frank product and Frank sum:
Based on the limit theory, it can be proved that when γ → 1, u ⊕ F z → u + z − uz and u ⊗ F z → uz [67]. The Frank product and Frank sum reduce to the probabilistic product and probabilistic sum.
2.3 PA and BM operators
Yager [54] proposed the PA operator, which considers the priority levels of criteria. The PA operator is defined as follows.
Definition 9
[54] Let C = {C 1, C 2, …, C n } be a set of criteria, and a prioritization C 1 ≻ C 2 ≻ ··· ≻ C n exists among the criteria. The performance value of object y under criterion C i is denoted by C i (y) (i = 1, 2, …, n), and C i (y) ∊ [0, 1]. The PA operator is defined as follows:
where \(w_{i} = \frac{{H_{i} }}{{\sum\nolimits_{t = 1}^{n} {H_{t} } }},H_{i} = \prod\nolimits_{k = 1}^{i - 1} {C_{k} } (y)\,\,\left( {i \ge 2} \right)\), and H 1 = 1.
In some practical situations, criteria are interdependent. In order to consider the interrelationships among criteria, Bonferroni [44] developed the BM operator.
Definition 10
[44] Let p, q > 0 and b i (i = 1, 2, …, n) be a collection of nonnegative real numbers. The BM operator is defined as follows:
Considering the different relative importance of each element, Xu and Yager [68] further defined the WBM operator on the basis of the BM operator.
Definition 11
[68] Let p, q > 0 and b i (i = 1, 2, …, n) be a collection of nonnegative real numbers. w = (w 1, w 2, …, w n )T is the weight vector of b i (i = 1, 2, …, n), \(\sum_{i = 1}^{n} w_{i} = 1\), and w i ∊ [0, 1]. The WBM operator is defined as follows:
Nevertheless, the WBM operator does not have the reducibility. To address this defect, Zhou and He [51] proposed the NWBM operator as follows.
Definition 12
[51] Let p, q > 0 and b i (i = 1, 2, …, n) be a collection of nonnegative real numbers. w = (w 1, w 2, …, w n )T is the weight vector of b i (i = 1, 2, …, n), \(\sum_{i = 1}^{n} w_{i} = 1\), and w i ∊ [0, 1]. The NWBM operator is defined as
3 Frank operations of SVNNs
This section introduces Frank operations of SVNNs based on the algebraic operations of SVNNs described in Definition 7 and the Frank operations described in Definition 8. Furthermore, this section discusses some properties of the proposed Frank operations of SVNNs.
Definition 13
Let x = (T x , I x , F x ) and y = (T y , I y , F y ) be two SVNNs, γ > 1, and λ > 0. Frank operations of SVNNs are defined as follows:
-
(1)
\(x\, \oplus_{F} \,y = \left( {1 - \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{1 - T_{x} }} - 1} \right)\left( {\gamma^{{1 - T_{y} }} - 1} \right)}}{\gamma - 1}} \right),} \right. \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{I_{x} }} - 1} \right)\left( {\gamma^{{I_{y} }} - 1} \right)}}{\gamma - 1}} \right), \left. {\log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{F_{x} }} - 1} \right)\left( {\gamma^{{F_{y} }} - 1} \right)}}{\gamma - 1}} \right)} \right);\)
-
(2)
\(x\, \otimes_{F} \,y = \left( {\log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{T_{x} }} - 1} \right)\left( {\gamma^{{T_{y} }} - 1} \right)}}{\gamma - 1}} \right),} \right. 1 - \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{1 - I_{x} }} - 1} \right)\left( {\gamma^{{1 - I_{y} }} - 1} \right)}}{\gamma - 1}} \right), \,\left. {1 - \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{1 - F_{x} }} - 1} \right)\left( {\gamma^{{1 - F_{y} }} - 1} \right)}}{\gamma - 1}} \right)} \right);\)
-
(3)
\(\lambda \, \cdot_{F} \,x = \left( {1 - \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{1 - T_{x} }} - 1} \right)^{\lambda } }}{{\left( {\gamma - 1} \right)^{\lambda - 1} }}} \right),} \right. \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{I_{x} }} - 1} \right)^{\lambda } }}{{\left( {\gamma - 1} \right)^{\lambda - 1} }}} \right), \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{F_{x} }} - 1} \right)^{\lambda } }}{{\left( {\gamma - 1} \right)^{\lambda - 1} }}} \right);\)
and
-
(4)
\(x^{{ \wedge_{F} \lambda }} = \left( {\log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{T_{x} }} - 1} \right)^{\lambda } }}{{\left( {\gamma - 1} \right)^{\lambda - 1} }}} \right),} \right. 1 - \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{1 - I_{x} }} - 1} \right)^{\lambda } }}{{\left( {\gamma - 1} \right)^{\lambda - 1} }}} \right), 1 - \log_{\gamma } \left( {1 + \frac{{\left( {\gamma^{{1 - F_{x} }} - 1} \right)^{\lambda } }}{{\left( {\gamma - 1} \right)^{\lambda - 1} }}} \right).\)
Example 1
Let x = (0.4, 0.2, 0.3) and y = (0.6, 0.1, 0.2) be two SVNNs. Suppose γ = 2, then by the Frank operations listed in Definition 13, we have x ⊕ F y = (1 − log2 (1 + (21−0.4 − 1)(21−0.6 − 1)), log2 (1 + (20.2 − 1)(20.1 − 1)), log2 (1 + (20.3 − 1)(20.2 − 1))) = (0.7799, 0.0153, 0.0488), and x ⊗ F y = (log2(1 + (20.4 − 1)(20.6 − 1)), log2(1 + (21−0.2 − 1)(21−0.1 − 1)), log2 (1 + (21−0.3 − 1)(21−0.2 − 1))) = (0.2201, 0.7153, 0.0.5488).
Moreover, the Frank operations of SVNNs in Definition 13 have the following properties.
Theorem 1
Let x, y, and z be three SVNNs, and λ 1, λ 2 > 0. The Frank operations of SVNNs satisfy the following properties:
-
(FP1) x ⊕ F y = y ⊕ F x;
-
(FP2) x ⊗ F y = y ⊗ F x;
-
(FP3) λ 1 · F (x ⊕ F y) = λ 1 · F x ⊕ F λ 1 · F y;
-
(FP4) \(\left( {x\, \otimes_{F} y} \right)^{{ \wedge_{F} \lambda_{1} }} = x^{{ \wedge_{F} \lambda_{1} }} \otimes_{F} y^{{ \wedge_{F} \lambda_{1} }}\);
-
(FP5) λ 1 · F x ⊕ F λ 2 · F x = (λ 1 + λ 2) · F x;
-
(FP6) \(x^{{ \wedge_{F} \lambda_{1} }} \, \otimes_{F} \,x^{{ \wedge_{F} \lambda_{2} }} = x^{{ \wedge_{F} \left( {\lambda_{1} + \lambda_{2} } \right)}}\);
-
(FP7) \(\lambda_{1} \cdot_{F} \left( {\lambda_{2} \cdot_{F} x} \right) = \lambda_{2} \cdot_{F} \left( {\lambda_{1} \cdot_{F} x} \right) = \left( {\lambda_{1} \lambda_{2} } \right) \cdot_{F} x\); and
-
(FP8) \(\left( {x \oplus_{F} y} \right) \oplus_{F} z = x \oplus_{F} \left( {y \oplus_{F} z} \right).\)
Proof
Assume that x = (T x , I x , F x ), y = (T y , I y , F y ), and z = (T z , I z , F z ).
(FP1) Based on operation (1) in Definition 13, it is true that
Therefore, x ⊕ F y = y ⊕ F x holds.
(FP2) Based on operation (2) in Definition 13, it follows that
Hence, x ⊗ F y = y ⊗ F x is proven to be right.
(FP3) Based on operations (1) and (3) in Definition 13,
and
Therefore, we have
and
Hence, λ 1 · F (x ⊕ F y) = λ 1 · F x ⊕ F λ 1 · F y.
(FP4) Based on operations (2) and (4) in Definition 13,
and
Therefore,
and
This makes it clear that \(\left( {x\, \otimes_{F} \,y} \right)^{{ \wedge_{F} \lambda_{1} }} = x^{{ \wedge_{F} \lambda_{1} }} \, \otimes_{F} \,y^{{ \wedge_{F} \lambda_{1} }} .\)
(FP5) Based on operations (1) and (3) in Definition 13, we have
and
Therefore,
Hence, λ 1 · F x ⊕ F λ 2 · F x = (λ 1 + λ 2) · F x holds.
(FP6) Based on operations (2) and (4) in Definition 13,
and
Hence,
Therefore, \(x^{{ \wedge_{F} \lambda_{1} }} \, \otimes_{F} \,x^{{ \wedge_{F} \lambda_{2} }} = x^{{ \wedge_{F} \left( {\lambda_{1} + \lambda_{2} } \right)}}\) is true.
(FP7) Based on operation (3) in Definition 13,
and
Therefore,
and
Hence, λ 1 · F (λ 2 · F x) = λ 2 · F (λ 1 · F x) = (λ 1 λ 2) · F x.
(FP8) Based on operation (1) in Definition 13,
and
Therefore, we have
and
Hence, (x ⊕ F y) ⊕ F z = x ⊕ F (y ⊕ F z) is true.
Thus, Theorem 1 holds.
4 SVNFNPBM operator
This section presents the definition of the single-valued neutrosophic Frank BM (SVNFBM) operator. Subsequently, we define the SVNFNPBM operator on the basis of the PA and NWBM operators. The SVNFNPBM operator simultaneously considers the priority levels of aggregated elements and the interrelationships among these elements by integrating the PA and WBM operators. Then, we discuss several properties of the SVNFNPBM operator.
Definition 14
Let p, q > 0, and x i = (T i , I i , F i ) (i = 1, 2, …, n) be a set of SVNNs. The SVNFBM operator is defined as follows:
This paper develops the SVNFNPBM operator based on the PA and NWBM operators as defined in Eqs. (7) and (10). The SVNFNPBM operator is defined as follows.
Definition 15
Let p, q > 0, and C = {C 1, C 2, …, C n } be a set of criteria, such that a prioritization C 1 ≻ C 2 ≻ ··· ≻ C n exists among the criteria. The performance value of object x under criterion C i is denoted by SVNN x i = (T i , I i , F i ) (i = 1, 2, …, n). The SVNFNPBM operator is defined as follows:
where \(w_{i} = \frac{{H_{i} }}{{\sum\nolimits_{t = 1}^{n} {H_{t} } }},H_{t} = \mathop \prod \nolimits_{k = 1}^{t - 1} \,sc\left( {x_{k} } \right)\,\left( {t \ge 2} \right)\), H 1 = 1, and sc(x k ) is the score value of SVNN x k obtained by Eq. (1).
Theorem 2
Let p, q > 0, and let C = {C 1, C 2, …, C n } be a set of criteria, such that a prioritization C 1 ≻ C 2 ≻ ··· ≻ C n exists among the criteria. The performance value of object x under criterion C i is denoted by SVNN x i = (T i , I i , F i ) (i = 1, 2, …, n). The aggregated value of x i (i = 1, 2, …, n) by SVNFNPBM operator in Eq. (12) is still an SVNN, and
where \(w_{i} = \frac{{H_{i} }}{{\sum\nolimits_{t = 1}^{n} {H_{t} } }}, H_{t} = \mathop \prod \nolimits_{k = 1}^{t - 1} \,sc\left( {x_{k} } \right) \, \left( {t \ge 2} \right)\), H 1 = 1, and sc(x k ) is the score value of SVNN x k obtained by Eq. (1).
Theorem 2 can be proved by mathematical induction as follows.
Proof
(1) Firstly, the following equation can be proved.
(a) When n = 2, based on operations (2) and (4) in Definition 13, we can determine that
and
Therefore, based on operation (3) in Definition 13, we can determine that
and
Then,
In other words, when n = 2, Eq. (14) holds.
(b) Assuming Eq. (14) holds for n = k, we can obtain the following equation:
When n = k + 1, based on property (FP8) in Theorem 1, we have
The following equation can be proven to calculate the value of \(\mathop {\mathop { \oplus_{F} }\limits_{i,j = 1,}^{k + 1} }\limits_{i \ne j} \left( {\frac{{w_{i} w_{j} }}{{1 - w_{i} }} \cdot_{F} \left( {\left( {x_{i} } \right)^{{ \wedge_{F} p}} \otimes_{F} \left( {x_{j} } \right)^{{ \wedge_{F} q}} } \right)} \right)\):
Equation (16) can be proven using mathematical induction.
(i) When k = 2, based on operations (2) and (4) in Definition 13, we can determine that
and
Therefore, based on operation (3) in Definition 13, we can determine that
and
Then,
in other words, when k = 2, Eq. (16) holds.
(ii) Assuming Eq. (16) holds for k = h, we can determine that
Moreover, it is true that
Based on operations (2) and (4) in Definition 13,
Hence, it is clear that
Hence, it is clear that
in other words, it is true that Eq. (16) holds for k = h + 1.
Therefore, Eq. (16) is true for all k.
The following equation can be proven in a similar fashion.
Therefore, based on Eqs. (15), (16), and (17), we can determine that
in other words, it is true that Eq. (14) holds for k = h + 1.
Therefore, Eq. (14) is also true for all k.
(2) According to Eq. (14) and operation (4) in Definition 13,
In addition, the following inequalities are correct:
and
which meet the requirements of an SVNN.
Therefore, Theorem 2 holds.
Example 2
Let x 1 = (0.4, 0.2, 0.3) and x 2 = (0.6, 0.1, 0.2) be two SVNNs, and let p = q = 1 and C 1 ≻ C 2. Based on Eq. (1),
Therefore,
and
Suppose that γ = 2, then, according to Eq. (13),
The following section investigates some additional properties of the SVNFNPBM operator.
Theorem 3
(Reducibility) Let C 1, C 2, …, C n have the same priority level; that is, \(w_{i} = \frac{1}{n}\) (i = 1, 2, …, n). Then, SVNFNPBMp,q (x 1, x 2, …, x n ) = SVNFBMp,q (x 1, x 2, …, x n ).
Proof
When \(w_{i} = \frac{1}{n}\) (i = 1, 2, …, n), based on Eq. (12), we can determine that
Based on the property (FP3) in Theorem 1,
Therefore,
Theorem 4
(Idempotency) Let all SVNNs x i (i = 1, 2, …, n) be equal, i.e., x i = x. Then, IVNBMp,q(x 1, x 2, …, x n ) = x.
Proof
Since x i = x (i = 1, 2, …, n),
Based on the property (FP6) in Theorem 1,
By applying Eq. (12) and property (FP5) in Theorem 1, we can determine that
5 Method for selecting TPL providers
This section establishes a novel method for selecting TPL providers using SVNNs.
Assume that there are m alternative TPL providers P = {P 1, P 2, …, P m } and n criteria C = {C 1, C 2, …, C n }. The criteria are correlative, the priority levels of the n criteria are different, and the alternatives will be assessed by e decision makers. If e = 1, selecting TPL provider is an MCDM problem. Let U = (a ij ) m×n be a single-valued neutrosophic decision matrix where \(a_{ij} = \left\langle {T_{{a_{ij} }} ,I_{{a_{ij} }} ,F_{{a_{ij} }} } \right\rangle\) is an evaluation value denoted by an SVNN. Moreover, \(T_{{a_{ij} }}\) in a ij indicates the degree of truth-membership to which the provider P i satisfies criterion C j , \(I_{{a_{ij} }}\) in a ij indicates the degree of indeterminacy-membership to which the provider P i satisfies criterion C j, and \(F_{{a_{ij} }}\) in a ij indicates the degree of falsity-membership to which the provider P i satisfies criterion C j . If e > 1, the selection of the TPL provider becomes a group decision-making problem. Let U k = (a k ij ) m×n be the single-valued neutrosophic decision matrix provided by the k-th (k = 1, 2, …, e) decision maker, and \(a_{ij}^{k} = \left\langle {T_{{a_{ij}^{k} }} ,I_{{a_{ij}^{k} }} ,F_{{a_{ij}^{k} }} } \right\rangle\). Then, \(\omega { = }\left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{e} } \right)^{T}\) will be the weight vector of the decision makers.
The following section proposes a method to rank TPL provider(s) and select the perfect one based upon the SVNFNPBM operator.
-
Step 1: Normalize the decision matrix.
Two kinds of different criteria can exist in problems of selecting TPL providers: benefit criteria and cost criteria. To unify all criteria, the decision matrix must be normalized. If e = 1, the normalized decision matrix N = (g ij ) m×n can be obtained by
$$g_{ij} = \left\{ {\begin{array}{*{20}l} {a_{ij}, } \hfill & {\quad {\text{if}}\,{\kern 1pt} C_{j} \,{\kern 1pt} {\text{is}}\,{\text{a}}\,{\text{benefit}}\,{\text{criterion},}} \hfill \\ {a_{ij}^{c} ,} \hfill & {\quad {\text{else},}} \hfill \\ \end{array} } \right.$$(18)where a c ij is the complement of a ij , and a c ij can be calculated according to Definition 3.
If e > 1, the decision matrices U k (k = 1, 2, …, e) can be normalized as N k = (g k ij ) m×n by
$$g_{ij}^{k} = \left\{ {\begin{array}{*{20}l} {a_{ij}^{k} ,} \hfill & {{\text{if}}{\kern 1pt} \,C_{j} \,{\text{is}}{\kern 1pt} \,{\text{a}}\,{\text{benefit}}\,{\text{criterion},}} \hfill \\ {a_{ij}^{kc}, } \hfill & {\text{else},} \hfill \\ \end{array} } \right.$$(19)where \(g_{ij}^{k} = \langle T_{{g_{ij}^{k} }} ,I_{{g_{ij}^{k} }} ,F_{{g_{ij}^{k} }} \rangle ,\) a kc ij is the complement of a k ij , and a kc ij can be calculated according to Definition 3.
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Step 2: Obtain the comprehensive decision matrix.
If e = 1, the decision matrix does not need to be integrated, and the comprehensive decision matrix is N* = (b ij ) m×n = N. If e > 1, based on the Frank weighted average operator of SVNSs, the comprehensive decision matrix N* = (b ij ) m×n can be obtained by
$$\begin{aligned} b_{ij} & {\text{ = SVNFWA}}\left( {g_{ij}^{1} ,g_{ij}^{2} , \ldots ,g_{ij}^{e} } \right) \\ & = \left( {1 - \log_{\gamma } \left( {1 + \frac{{\prod\limits_{k = 1}^{e} {\left( {\gamma^{{1 - T_{{g_{ij}^{k} }} }} - 1} \right)^{{\omega_{k} }} } }}{{\left( {\gamma - 1} \right)\prod\limits_{k = 1}^{e} {\left( {\gamma - 1} \right)^{{\omega_{k} - 1}} } }}} \right),} \right. \\ & \quad \log_{\gamma } \left( {1 + \frac{{\prod\limits_{k = 1}^{e} {\left( {\gamma^{{I_{{g_{ij}^{k} }} }} - 1} \right)^{{\omega_{k} }} } }}{{\left( {\gamma - 1} \right)\prod\limits_{k = 1}^{e} {\left( {\gamma - 1} \right)^{{\omega_{k} - 1}} } }}} \right), \\ & \quad \left. {\log_{\gamma } \left( {1 + \frac{{\prod\limits_{k = 1}^{e} {\left( {\gamma^{{F_{{g_{ij}^{k} }} }} - 1} \right)^{{\omega_{k} }} } }}{{\left( {\gamma - 1} \right)\prod\limits_{k = 1}^{e} {\left( {\gamma - 1} \right)^{{\omega_{k} - 1}} } }}} \right)} \right). \\ \end{aligned}$$(20) -
Step 3: Obtain the score value sc ij of each b ij (i = 1, 2, …, m; j = 1, 2, …, n).
Using Eq. (1), we can obtain the score value sc ij of each b ij .
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Step 4: Obtain the value of H ij (i = 1, 2, …, m; j = 1, 2, …, n).
H ij can be calculated based on the score value sc ij of each b ij using the following equation:
$$H_{ij} = \prod\limits_{t = 1}^{j - 1} {sc_{it} } ,\quad {\kern 1pt} H_{i1} = 1.$$(21) -
Step 5: Obtain the weight w ij of each b ij (i = 1, 2, …, m; j = 1, 2, …, n).
w ij can be calculated on the basis of H ij using the following equation:
$$w_{ij} = \frac{{H_{ij} }}{{\sum\nolimits_{t = 1}^{n} {H_{it} } }}.$$(22) -
Step 6: Obtain the overall performance value r i (i = 1, 2, …, m) of TPL provider P i .
The overall performance value r i can be obtained using the SVNFNPBM operator in Eq. (13).
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Step 7: Obtain the score value sc i of each r i (i = 1, 2, …, m).
Using the score function of the SVNN in Eq. (1), we can obtain the score value sc i of each r i .
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Step 8: Obtain the accuracy value l i of each r i (i = 1, 2, …, m).
Based on the accuracy function of the SVNN in Eq. (2), we can obtain the accuracy value l i of each r i .
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Step 9: Rank TPL providers according to the comparison method of SVNNs.
Using the comparison method presented in Definition 6, we can obtain the ranking order of alternative TPL providers.
6 Numerical example
6.1 The steps of the proposed method
This subsection uses a numerical example of selecting TPL providers with SVNNs to verify the applicability of the proposed approach.
The following example of selecting TPL providers is adapted from Ref. [69].
An electronic commerce retailer intends to find an appropriate TPL provider with whom to carry out long-term cooperation. Preliminary selection identifies four alternative providers {P 1, P 2, P 3, P 4}. These TPL providers are assessed by experts with respect to four criteria: (1) the cost of service (C 1); (2) operational experience in the industry (C 2); (3) customer satisfaction (C 3); and (4) market reputation (C 4). C 2, C 3, and C 4 are benefit criteria, while C 1 is a cost criterion. The electronic commerce retailer identifies the priority relationship of the four criteria as C 3 ≻ C 1 ≻ C 4 ≻ C 2. Moreover, these four criteria are correlative. For the sake of reflecting reality in more detail and obtaining fuzzy and uncertain information, the evaluation values provided by one expert are transformed into SVNNs. Table 1 lists the transformed decision information.
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Step 1: Normalize the decision matrix.
Since C 2, C 3, and C 4 are benefit criteria, while C 1 is a cost criterion, the decision matrix needs to be normalized using Eq. (18). Table 2 presents the normalized decision information.
Table 2 Normalized decision information -
Step 2: Obtain the comprehensive decision matrix.
Since e = 1, the decision matrix does not need to be integrated, and the comprehensive decision matrix is the same matrix presented in Table 2.
-
Step 3: Obtain the score value sc ij of each b ij (i = 1, 2, …, m; j = 1, 2, …, n).
Using Eq. (1), the score value sc ij of each b ij can be obtained. Table 3 shows the corresponding score values sc ij (i = 1, 2, …, m; j = 1, 2, …, n).
Table 3 The score values -
Step 4: Obtain the value of H ij (i = 1, 2, …, m; j = 1, 2, …, n).
Using Eq. (21), the value of H ij can be obtained. Table 4 gives the corresponding values.
Table 4 The value of H ij -
Step 5: Obtain the weight w ij of each b ij (i = 1, 2, …, m; j = 1, 2, …, n).
Using Eq. (22), the weight w ij of each b ij can be obtained. Table 5 depicts the corresponding weights w ij (i = 1, 2, …, m; j = 1, 2, …, n).
Table 5 The corresponding weight matrix -
Step 6: Obtain the overall performance value r i (i = 1, 2, …, m) of provider P i .
According to the SVNFNPBM operator in Eq. (13), supposing that γ = 2 [66] and p = q = 1 [53], the overall performance value r i of each provider P i can be identified as follows: r 1 = (0.5394, 0.1167, 0.2255), r 2 = (0.589, 0.1331, 0.2096), r 3 = (0.4789, 0.1388, 0.2549), and r 4 = (0.4856, 0.2182, 0.2059).
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Step 7: Obtain the score value sc i of each r i (i = 1, 2, …, m).
According to Eq. (1), the score value sc i of each r i can be identified as sc 1 = 0.5403, sc 2 = 0.5566, sc 3 = 0.5732, and sc 4 = 0.4216.
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Step 8: Obtain the accuracy value l i of each r i (i = 1, 2, …, m).
Using Eq. (2), the accuracy value l i of each r i can be identified as l 1 = 0.2865, l 2 = 0.3525, l 3 = 0.1871, and l 4 = 0.2124.
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Step 9: Rank the TPL providers using the comparison method for SVNNs.
Since sc 3 > sc 2 > sc 1 > sc 4, according to the comparison method in Definition 6, we can rank the TPL providers as P 3 ≻ P 2 ≻ P 1 ≻ P 4. The most desirable TPL provider is P 3.
6.2 Influence of parameters
To investigate the influence of various parameters, this subsection calculates the ranking results with varying values for p, q, and γ. Two parts are included in this subsection. The first part discusses the impact of p and q on the ranking order, and the second one discusses the influence of γ.
To study the impact of p and q, in the first part, we determine and compare the ranking results using different pairs of p and q. The pairs of p and q can be distributed into three classes. The value of p is smaller than that of q in the first class, the value of p equals that of q in the second class, while the value of p is greater than that of q in the third class. In addition to changing the relationships between p and q, we also explore the effect of p and q on the ranking order of the four TPL providers by increasing the difference between p and q. Table 6 shows the significant pairs of p and q and their respective final ranking orders according to the proposed method.
As Table 6 shows, these three classes may produce different ranking orders. Furthermore, different ranking orders may exist within the same class in the presence of distinct differences between p and q. The first class yields two different rankings. When the value of p is smaller than 0.01, TPL provider P 3 performs better than TPL provider P 1. Meanwhile, the company prefers P 1 to P 3 when p is greater than 0.1. In the second class, the ranking orders are the same, and P 1 is consistently preferred to P 3. The third class also produces the same ranking order, which is different from the order in the second class. In these three classes, the best TPL provider is always P 2 and the worst is always P 4.
These results suggest that the values of p and q can influence the ranking order of TPL providers. From this perspective, the proposed method for selecting TPL providers is a flexible one.
To investigate the influence of γ, in the second part, we determine and compare the ranking orders using different values of γ. Table 7 presents the significant values of γ and the respective ranking orders according to the proposed method.
As Table 7 shows, the score values on an individual criterion differ as the value of γ varies. The bigger the value of γ, the higher the score for an individual criterion. However, the ranking orders remain the same as the corresponding values of γ differ; in other words, the value of γ does not influence the ranking order of TPL providers. The worst provider is always P 4, and the best is always P 2. Decision makers can determine an appropriate value for γ based on their own preferences. In practice, the value of γ can be calculated using regression analysis and previous data provided by the decision makers.
6.3 Comparative analysis
This subsection compares the proposed method with four extant methods to verify its feasibility. Sahin [42] developed the first two methods, which make use of the weighted arithmetic average (WAA) operator and the weighted geometric average (WGA) operator, respectively, to aggregate values under every criterion. Furthermore, the two methods proposed by Sahin [42] rank alternatives using a score function that is identical to that in Eq. (1). Ye [70] detailed the third method, which was constructed with the proposed cross-entropy method for SVNSs. Liu and Wang [53] proposed the fourth method, which utilizes the SVNNWBM operator developed based on the algebraic operational laws. Table 8 lists the ranking orders according to these five methods.
Table 8 demonstrates that the proposed method produces a different ranking order from those obtained by the four extant methods. According to the method using WAA [42], the best TPL provider is P 2, while the other four methods identify P 3. The worst TPL provider is P 4 across the five methods. The method using WGA [42], the method using cross-entropy [70], and the method using SVNNWBM [53] produce the same ranking orders. Meanwhile, the method using WAA [42] and the proposed method produce two different ranking orders, which are also distinct from the ranking order obtained by the other three methods.
The reasons why these differences exist in Table 8 are as follows. The differences exist between the ranking order of the proposed method and those of the first three methods because of three possible reasons. The first one is that the first three methods do not take into account the interrelationships among criteria, while the proposed method considers these interrelationships using the BM operator. The second one lies in the criteria’s priority levels. The proposed method considers the criteria’s different priorities, while the first three methods assume that the criteria all have the same priority levels. The last one can be traced back to the fact that the proposed method is constructed based on Frank operations, while the first three methods use algebraic operational laws. Therefore, it is reasonable that the ranking orders of the first three methods differ from that of the proposed method. Meanwhile, we discuss two main reasons for the dissimilarities existing between the ranking orders respectively obtained by the method using the SVNNWBM [53] and the proposed method, even though both methods consider the interrelationships among criteria. One main reason is that different weight vectors of criteria are involved in two methods. The weights for the criteria in the method using the SVNNWBM [53] are subjectively provided by decision makers lacking of objectivity. Conversely, the proposed method takes the criteria priorities into account by objectively obtaining the weight vector of the criteria through the weight method using the PA operator. And the other main reason lies in the difference of operational laws. The proposed method utilizes Frank operational laws, while the SVNNWBM [53] method makes use of algebraic operational laws. As discussed in Sect. 1, Frank operational laws are more robust than algebraic operational laws.
Generally speaking, the proposed method can be successfully utilized to select TPL providers. By combining the BM operator and the PA operator, this method takes into consideration the interrelationships among criteria and their varying priority levels. In addition, the proposed method is more robust than extant methods based on algebraic operational laws. Compared with extant methods, the proposed method using the SVNFNPBM operator is more suitable for tackling practical problems in which criteria are independent and have different priority levels. The ranking order of the proposed method can identify a TPL provider that is more in line with decision makers’ preferences than the providers selected by other methods.
7 Conclusion
SVNNs can comprehensively depict fuzzy information involved in the process of selecting TPL providers. This paper established a method based on the proposed Frank operations for selecting TPL providers under single-valued neutrosophic environments. The method took into account the interrelationships among criteria and the different priority levels of criteria. To do that, we developed an SVNFNPBM operator by combining the NWBM and PA operators. Moreover, we demonstrated the applicability and feasibility of the proposed method using a numerical example and a comparative analysis. In addition, this paper discussed the influence of the all involved parameters on ranking order.
The major contributions of this paper can be summarized as follows. First of all, this paper introduces Frank operations of SVNNs that are more flexible and robust than the extant algebraic operations of SVNNs. In addition, this paper presents and proves some properties of the proposed Frank operations. Secondly, this paper proposes a novel single-valued neutrosophic aggregation operator that combines the BM and PA operators to consider simultaneously the interrelationships among criteria and their distinct priority levels. The proposed aggregation operator is based on Frank operations, while most of the previous single-valued neutrosophic aggregation operators are based on algebraic operations. Last but not least, this paper constructs a decision-making method for the selection of TPL providers based on the proposed aggregation operator. The new method is then proven to be useful in selecting a TPL provider. The proposed MCDM method using the SVNFNPBM operator is more effective than extant methods in addressing situations with independent criteria that have different priority levels, which occur in most practical problems. In other words, the proposed method using the SVNFNPBM operator has a higher practical value than extant methods.
Several directions for future research may be promising. First, the aggregation operator proposed in this paper can be introduced into interval-valued neutrosophic environments. Second, applications of the proposed MCDM method can be explored to tackle practical problems in other areas, such as selecting hotels, cloud services, or renewable energy sources. The common feature of these practical problems is that multiple criteria involved are interdependent and have different priority levels. Third, the complexity of the proposed method can be improved with the help of computer technology. In the future, we will devote ourselves to reducing the complexity of the method as well as increasing accuracy.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (Nos. 71501192 and 71571193). The authors also would like to express appreciation to the anonymous reviewers and editors for their helpful comments that improved the paper.
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Ji, P., Wang, Jq. & Zhang, Hy. Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in selecting third-party logistics providers. Neural Comput & Applic 30, 799–823 (2018). https://doi.org/10.1007/s00521-016-2660-6
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DOI: https://doi.org/10.1007/s00521-016-2660-6