Risk assessment of supply chain finance with intuitionistic fuzzy information

Abstract

Supply chain finance is a new financing model that makes the industry chain as an organic whole chain to develop financing services. Its purpose is to combine with financial institutions, companies and third-party logistics companies to achieve win-win situation. The supply chain is designed to maximize the financial value. The supply chain finance business in our country is still in its early stages. Conducting the research on risk assessment and control of the supply chain finance business has an important significance for the promotion of the development of our country supply chain finance business. The paper investigates the dynamic multiple attribute decision making problems, in which the decision information, provided by decision makers at different periods, is expressed in intuitionistic fuzzy numbers. We first develop one new aggregation operators called dynamic intuitionistic fuzzy Hamacher weighted averaging (DIFHWA) operator. Moreover, a procedure based on the DIFHWA and IFHWA operators is developed to solve the dynamic multiple attribute decision making problems where all the decision information about attribute values takes the form of intuitionistic fuzzy numbers collected at different periods. Finally, an illustrative example for risk assessment of supply chain finance is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Dynamic multiple attribute decision making dynamic intuitionistic fuzzy Hamacher weighted averaging (DIFHWA) operator intuitionistic fuzzy Hamacher weighted averaging (IFHWA) operator risk assessment supply chain finance

1 Introduction

Many vendors have less market power than the focal firm. Thus, they usually sold their product in open account to keep long-term relationship with the focal firm, namely, they supplied working capital for the supply chain. So they need finance form banks or other finance institutions [1]. At the same time, they are always SMEs, which are start-up firms, and lack of capital accumulation. They cannot put enough collateral, which is hard information to get banking, to bank to get loan. In the meantime, they are lack of good reputations, which are soft information for bank to supply credit for them. Thus, finance becomes the bottleneck of the vendor’s development [2, 3]. Not only the vendors can’t further expand production, but also affect the stability and competitiveness of the supply chain. They remain SMEs, and supply working capital for the supply chain. In other words, these vendors fall into a vicious cycle of credit: supply working capital-no cash-no collateral-not loans-not develop –small firms-supply working capital, etc. The research about financial innovation to finance the vendors, which are at the established information constraints and asset constraints, is an interesting topic for bank to ensure their earnings, for SMEs to get loans to help their further development, and for keeping supply chain stability and competitiveness [5, 6]. Though now there are a lot of literature the supply chain finance, there are five aspects topics worthy of studying: First, how to model the bank credit behavior for vendors based on supply chain background? Second, how to build theoretical model which study the relation between the supply chain characteristics (such as focal firm Capacity, vendor’s capacity and supply chain governance) and bank credit for vendor? We will study it in what way? Third, on the basis of the theoretical model, the need to further demonstrate relationship between the focal firm Capacity, vendors capacity and supply chain governance and the bank credit level for vendors, if the relationship cannot be observed directly, but which is a mediator variable that can be inferred from bank credit condition (such as transparency) and ultimate behavioral outcomes (including credit lines, time and trust) [7, 8]. Is this variable the bank cognition about vendor? Fourth, whether the model that the focal firm involved in the vendor’s financing is a moderator variable? How does the model that the focal firm involved in the vendor’s financing adjust the strength of the relationship between banks cognition about vendors with the focal firm capacity, vendor’s capacity and supply chain governance level, etc.? How the models that the focal firm involved in the vendor’s financing adjust the strength of the relationship between banks cognition about vendors with the level of bank credit? How the model that the focal firm involved in the vendor’s financing adjust the strength of the relationship between the focal firm capacity, the strength of the relationship between vendors and supply chain governance capacity with the level of vendor’s credit? Fifth, whether the banking mode, including the requirements of bank about information which is from soft to hard, that is, from hard to easy to get loan, is a moderator? How does it adjust the strength of the relationship between banks cognitions about vendor’s credit line, credit time and credit strict condition [9 –12]?

Supply chain finance risk is an uncertainty of loss in the process of financial institutions such as banks provide financial services of supply chain. Faced with more uncertainty, the supply chain financial risk has strong complexity and concealment, and it brings huge losses and damage to all participants in the supply chain finance business by making use of the correlation and vulnerability of the supply chain and financial system, making the real earnings of supply chain finance business and expected returns to produce larger deviation. More and more complex financial risk has become a key problem which restricts the development of the supply chain finance, how to accurately identify, scientifically monitor, effectively prevent and control complex supply chain finance risk is a realistic requirement and urgent task to ease the financing difficulty of medium and small enterprises, expand the business space of banks, develop the competitiveness of third industry. As a relatively new research field, the current literature research on supply chain financial risk management mostly focuses on the concept and qualitative description of value. In the aspect of theoretical research, a theory system on supply chain financial risk comprehensive management has not been formed. In the aspect of practical application, scientific quantitative measure model tools and effective riskmanagement techniques and methods are scanty. Because of this, based on the era and realistic background of current supply chain finance development, according to the inherent requirement and the research logic of comprehensive risk management, in this paper, we investigate the dynamic multiple attribute decision making problems, in which the decision information, provided by decision makers at different periods, is expressed in intuitionistic fuzzy numbers. We first develop one new aggregation operators called dynamic intuitionistic fuzzy Hamacher weighted averaging (DIFHWA) operator. Moreover, a procedure based on the DIFHWA and IFHWA operators is developed to solve the dynamic multiple attribute decision making problems where all the decision information about attribute values takes the form of intuitionistic fuzzy numbers collected at different periods. Finally, an illustrative example for risk assessment of supply chain finance is given to verify the developed approach and to demonstrate its practicality and effectiveness.

2 Preliminaries

In the following, we introduce some basic concepts related to intuitionistic fuzzy sets.

Definition 1. [13] Let X be a universe of discourse, then a fuzzy set is defined as: $A = {〈 x, μ_{A} (x) 〉 | x \in X}$ (1) which is characterized by a membership function μ_A : X → [0, 1], where μ_A (x) denotes the degree of membership of the element x to the set A.

Atanassov [14, 15] extended the fuzzy set to the IFS, shown as follows:

Definition 2. [14, 15] An IFS A in X is given by $A = {〈 x, μ_{A} (x), ν_{A} (x) 〉 | x \in X}$ (2)

where μ_A : X → [0, 1] and ν_A : X → [0, 1], with the condition 0 ≤ μ_A (x) + ν_A (x) ≤ 1, ∀ x ∈ X . The numbers μ_A (x) and ν_A (x) represent, respectively, the membership degree and non- membership degree of the element x to the set A [1, 2].

Definition 3. [14, 15] For each IFS A in X, if $π_{A} (x) = 1 - μ_{A} (x) - ν_{A} (x), \forall x \in X .$ (3)

Then π_A (x) is called the degree of indeterminacy of x to A [1, 2].

Definition 4. [16] Let $\tilde{a} = (μ, ν)$ be an intuitionistic fuzzy number, a score function S of an intuitionistic fuzzy value can be represented as follows[44]: $S (\tilde{a}) = μ - ν, S (\tilde{a}) \in [- 1, 1] .$ (4)

Definition 5. [17] Let $\tilde{a} = (μ, ν)$ be an intuitionistic fuzzy number, a accuracy function H of an intuitionistic fuzzy value can be represented as follows [45]: $H (\tilde{a}) = μ + ν, H (\tilde{a}) \in [0, 1] .$ (5) to evaluate the degree of accuracy of the intuitionistic fuzzy value $\tilde{a} = (μ, ν)$ , where $H (\tilde{a}) \in [0, 1]$ . The larger the value of $H (\tilde{a})$ , the more the degree of accuracy of the intuitionistic fuzzy value $\tilde{a}$ .

Similar to the relation between mean and variance in statistics. Based on the score function S and the accracy function H, in the following, Xu [18] give an order relation between two intuitionistic fuzzyvalues.

Definition 6. [18] Let ${\tilde{a}}_{1} = (μ_{1}, ν_{1})$ and ${\tilde{a}}_{2} = (μ_{2}, ν_{2})$ be two intuitionistic fuzzy values, $s ({\tilde{a}}_{1}) = μ_{1} - ν_{1}$ and $s ({\tilde{a}}_{2}) = μ_{2} - ν_{2}$ be the scores of $\tilde{a}$ and $\tilde{b}$ , respectively, and let $H ({\tilde{a}}_{1}) = μ_{1} + ν_{1}$ and $H ({\tilde{a}}_{2}) = μ_{2} + ν_{2}$ be the accuracy degrees of $\tilde{a}$ and $\tilde{b}$ , respectively, then if $S (\tilde{a}) < S (\tilde{b})$ , then $\tilde{a}$ is smaller than $\tilde{b}$ , denoted by $\tilde{a} < \tilde{b}$ ; if $S (\tilde{a}) = S (\tilde{b})$ , then

if $H (\tilde{a}) = H (\tilde{b})$ , then $\tilde{a}$ and $\tilde{b}$ represent the same information, denoted by $\tilde{a} = \tilde{b}$ ; (2) if $H (\tilde{a}) < H (\tilde{b})$ , $\tilde{a}$ is smaller than $\tilde{b}$ , denoted by $\tilde{a} < \tilde{b}$ .

3 DIFHWA operator

Xu and Yager [19] investigated the dynamic intuitionistic fuzzy multiple attribute decision making problems and developed some aggregation operators such as the dynamic intuitionistic fuzzy weighted averaging (DIFWA) operator and uncertain dynamic intuitionistic fuzzy weighted averaging (UDIFWA) operator to aggregate dynamic or uncertain dynamic intuitionistic fuzzy information. Moreover, based on the DIFWA and UDIFWA operators respectively, they have developed two procedures for solving the dynamic intuitionistic fuzzy multiple attribute decision making problems where all the attribute values are expressed in intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers. Wei [20] investigated the dynamic intuitionistic fuzzy multiple attribute decision making problems where all the attribute values are expressed in intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers and proposed some geometric aggregation operators such as the dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator and uncertain dynamic intuitionistic fuzzy weighted geometric (UDIFWG) operator to aggregate dynamic or uncertain dynamic intuitionistic fuzzy information. He & Teng [21] investigated the dynamic hybrid multiple attribute decision making problems, in which the decision information, provided by decision makers at different periods, is expressed in intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers, respectively. They first develop two new aggregation operators called dynamic intuitionistic fuzzy Einstein weighted geometric (DIFEWG) operator and dynamic interval-valued intuitionistic fuzzy Einstein weighted geometric (DIVIFEWG) operator. Moreover, a procedure based on the DIFEWG and DIVIFEWG operators is developed to solve the dynamic hybrid multiple attribute decision making problems where all the decision information about attribute values takes the form of intuitionistic fuzzy numbers or interval-valued intuitionistic fuzzy numbers collected at different periods. The intuitionistic fuzzy set has received more and more attention since its appearance [22 –35].

In order to aggregate the intuitionistic fuzzy information, Huang [36] developed the intuitionistic fuzzy Hamacher weighted averaging (IFHWA)operator.

Definition 7. Let ${\tilde{a}}_{j} = (μ_{j}, ν_{j}) (j = 1, 2, \dots, n)$ be a collection of Intuitionistic fuzzy numbers, then their aggregated value by using the IFHWA operator isalso an IFN, and $\begin{matrix} {IFHWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \oplus_{j = 1}^{n} (ω_{j} {\tilde{a}}_{j}) \\ = (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{j})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - μ_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{j})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - μ_{j})}^{ω_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} ν_{j}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - ν_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} ν_{j}^{ω_{j}}}) \end{matrix}$ (6)

where ω = (ω₁, ω₂, …, ω_n) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

If γ = 2, intuitionistic fuzzy Hamacher weighted averaging (IFHWA) operator reduces to the intuitionistic fuzzy Einstein weighted averaging (IFEWA) operator proposed by Wang & Liu [37].

Definition 8. [37] Let ${\tilde{a}}_{j} = (μ_{j}, ν_{j}) (j = 1, 2, \dots, n)$ be a collection of intuitionistic fuzzy values, and let IFEWA: Qⁿ → Q, if

$\begin{matrix} {IFEWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\oplus_{ɛ}}_{j = 1}^{n} (ω_{j} {\tilde{a}}_{j}) \\ = (\frac{\prod_{j = 1}^{n} {(1 + μ_{j})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - μ_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + μ_{j})}^{ω_{j}} + \prod_{j = 1}^{n} {(1 - μ_{j})}^{ω_{j}}}, \\ \frac{2 \prod_{j = 1}^{n} ν_{j}^{ω_{j}}}{\prod_{j = 1}^{n} {(2 - ν_{j})}^{ω_{j}} + \prod_{j = 1}^{n} ν_{j}^{ω_{j}}}) \end{matrix}$ (7) where ω = (ω₁, ω₂, …, ω_n) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ , then IFEWA is called the intuitionistic fuzzy Einstein weighted averaging (IFEWA) operator.

If γ = 1, intuitionistic fuzzy Hamacher weighted averaging (IFHWA) operator reduces to the intuitionistic fuzzy weighted averaging (IFWA) operator proposed by Xu [18].

Definition 9. Let ${\tilde{a}}_{j} = (μ_{j}, ν_{j}) (j = 1, 2, \dots, n)$ be a collection of intuitionistic fuzzy values, and let IFWA: Qⁿ → Q, if

$\begin{matrix} {IFWA}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) = {\oplus_{ɛ}}_{j = 1}^{n} (ω_{j} {\tilde{a}}_{j}) \\ = (1 - \prod_{j = 1}^{n} {(1 - μ_{j})}^{ω_{j}}, \prod_{j = 1}^{n} ν_{j}^{ω_{j}}) \end{matrix}$ (8) where ω = (ω₁, ω₂, …, ω_n) ^T be the weight vector of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ , then IFWA is called the Intuitionistic fuzzy weighted averaging (IFWA) operator.

However, the IFHWA operator can only be utilized to aggregate the intuitionistic fuzzy information with time independent arguments. If time is taken into account, for example, the intuitionistic fuzzy information may be collected at different periods, then, it’s unsuitable for dealing with such situations.

Definition 10. [19] Let t be a time variable, then we call α (t) = (μ_α(t), ν_α(t)) an intuitionistic fuzzy variable, where ³⁵ $μ_{α (t)} \in [0, 1], ν_{α (t)} \in [0, 1], μ_{α (t)} + ν_{α (t)} \leq 1 .$

For an intuitionistic fuzzy variable α (t) = (μ_α(t), ν_α(t)), if t = t₁, t₂, …, t_p, then a (t₁) , a (t₂) , …, a (t_p) indicate p intuitionistic fuzzy numbers collected at p different periods. In the following, we shall propose the dynamic intuitionistic fuzzy Hamacher weighted averaging (DIFHWA) operator.

Definition 11. Let ${\tilde{a}}_{t_{k}} = (μ_{t_{k}}, ν_{t_{k}}) (k = 1, 2, \dots, p)$ be a collection of intuitionistic fuzzy numbers at p different periods t_k (k = 1, 2, …, p), and let DIFHWA: Qⁿ → Q, if

$\begin{matrix} {DIFHWA}_{η (t)} ({\tilde{a}}_{t_{1}}, {\tilde{a}}_{t_{2}}, \dots, {\tilde{a}}_{t_{p}}) \\ = \oplus_{k = 1}^{p} (η (t_{k}) {\tilde{a}}_{t_{k}}) \end{matrix}$ (9) where η (t) = (η (t₁) , η (t₂) , …, η (t_p)) ^T be the weight vector of the periods t_k (k = 1, 2, …, p), and η (t_k) ∈ [0, 1] , $\sum_{k = 1}^{p} η (t_{k}) = 1$ . Then, DIFHWA is called the dynamic intuitionistic fuzzy Hamacher weighted averaging (DIFHWA) operator.

By Definition 11, (9) can be rewritten as follows:

$\begin{matrix} {DIFHWA}_{η (t)} ({\tilde{a}}_{t_{1}}, {\tilde{a}}_{t_{2}}, \dots, {\tilde{a}}_{t_{p}}) = \oplus_{k = 1}^{p} (η (t_{k}) {\tilde{a}}_{t_{k}}) \\ = (\frac{\prod_{k = 1}^{p} {(1 + (γ - 1) μ_{t_{k}})}^{η (t_{k})} - \prod_{k = 1}^{p} {(1 - μ_{t_{k}})}^{η (t_{k})}}{\prod_{k = 1}^{p} {(1 + (γ - 1) μ_{t_{k}})}^{η (t_{k})} + (γ - 1) \prod_{k = 1}^{p} {(1 - μ_{t_{k}})}^{η (t_{k})}}, \frac{γ \prod_{k = 1}^{p} {(ν_{t_{k}})}^{η (t_{k})}}{\prod_{k = 1}^{p} {(1 + (γ - 1) (1 - ν_{t_{k}}))}^{η (t_{k})} + (γ - 1) \prod_{k = 1}^{p} {(ν_{t_{k}})}^{η (t_{k})}}) \end{matrix}$ (10)

For DIFHWA operator, to determine the weight vector η (t) = (η (t₁) , η (t₂) , …, η (t_p)) ^T of the periods t_k (k = 1, 2, …, p) is a very important step.

If γ = 2, dynamic intuitionistic fuzzy Hamacher weighted averaging (DIFHWA) operator reduces to the dynamic intuitionistic fuzzy Einstein weighted averaging (DIFEWA) operator.

Definition 12. Let ${\tilde{a}}_{t_{k}} = (μ_{t_{k}}, ν_{t_{k}}) (k = 1, 2, \dots, p)$ be a collection of intuitionistic fuzzy numbers at p different periods t_k (k = 1, 2, …, p), and let DIFEWA: Qⁿ → Q, if

$\begin{matrix} {DIFEWA}_{η (t)} ({\tilde{a}}_{t_{1}}, {\tilde{a}}_{t_{2}}, \dots, {\tilde{a}}_{t_{p}}) = \oplus_{k = 1}^{p} (η (t_{k}) {\tilde{a}}_{t_{k}}) \\ = (\frac{\prod_{k = 1}^{p} {(1 + μ_{t_{k}})}^{η (t_{k})} - \prod_{j = 1}^{n} {(1 - μ_{t_{k}})}^{η (t_{k})}}{\prod_{k = 1}^{p} {(1 + μ_{t_{k}})}^{η (t_{k})} + \prod_{k = 1}^{p} {(1 - μ_{t_{k}})}^{η (t_{k})}}, \\ \frac{2 \prod_{k = 1}^{p} {(ν_{t_{k}})}^{η (t_{k})}}{\prod_{k = 1}^{p} {(2 - ν_{t_{k}})}^{η (t_{k})} + \prod_{k = 1}^{p} {(ν_{t_{k}})}^{η (t_{k})}}) \end{matrix}$ (11) where η (t) = (η (t₁) , η (t₂) , …, η (t_p)) ^T be the weight vector of the periods t_k (k = 1, 2, …, p), and η (t_k) ∈ [0, 1] , $\sum_{k = 1}^{p} η (t_{k}) = 1$ , then DIFEWA is called the dynamic intuitionistic fuzzy Einstein weighted averaging (DIFEWA) operator.

If γ = 1, dynamic intuitionistic fuzzy Hamacher weighted averaging (DIFHWA) operator reduces to the dynamic intuitionistic fuzzy weighted averaging (DIFWA) operator proposed by Xu & Yager [19].

Definition 13. Let ${\tilde{a}}_{t_{k}} = (μ_{t_{k}}, ν_{t_{k}}) (k = 1, 2, \dots, p)$ be a collection of intuitionistic fuzzy numbers at p different periods t_k (k = 1, 2, …, p), and let DIFWA: Qⁿ → Q, if

$\begin{matrix} {DIFWA}_{η (t)} ({\tilde{a}}_{t_{1}}, {\tilde{a}}_{t_{2}}, \dots, {\tilde{a}}_{t_{p}}) = \oplus_{j = 1}^{n} (η (t_{k}) {\tilde{a}}_{t_{k}}) \\ = (1 - \prod_{j = 1}^{n} {(1 - μ_{t_{k}})}^{η (t_{k})}, \prod_{j = 1}^{n} {(ν_{t_{k}})}^{η (t_{k})}) \end{matrix}$ (12)

where η (t) = (η (t₁) , η (t₂) , …, η (t_p)) ^T be the weight vector of the periods t_k (k = 1, 2, …, p), and η (t_k) ∈ [0, 1] , $\sum_{k = 1}^{p} η (t_{k}) = 1$ , then DIFWA is called the dynamic intuitionistic fuzzy weighted averaging (IFWA) operator

4 Dynamic intuitionistic fuzzy multiple attribute decision making problem

The following assumptions or notations are used to represent the dynamic multiple attribute decision making with intuitionistic fuzzy information. The alternatives are known. Let A ={ A₁, A₂, …, A_m } be a discrete set of alternatives. The attributes are known. Let G ={ G₁, G₂, …, G_n } be a set of attributes, w = (w₁, w₂, …, w_n) ^T, where w_j ≥ 0, j = 1, 2, …, n, $\sum_{j = 1}^{n} w_{j} = 1$ . There are p different periods t_k (k = 1, 2, …, p), whose weight vector is η (t) = (η (t₁) , η (t₂) , …, η (t_p)) ^T, where η (t_k) ∈ [0, 1] , k = 1, 2, …, p, $\sum_{k = 1}^{p} η (t_{k}) = 1$ . Suppose that $\tilde{R} (t_{k}) = {({\tilde{r}}_{ij} (t_{k}))}_{m \times n} = {(μ_{ij} (t_{k}), ν_{ij} (t_{k}))}_{m \times n}$ is the intuitionistic fuzzy decision matrix at periods t_k (k = 1, 2, …, p), where μ_ij (t_k) indicates the degree that the alternative A_i satisfies the attribute G_j at periods t_k (k = 1, 2, …, p), ν_ij (t_k) indicates the degree that the alternative A_i doesn’t satisfy the attribute G_j at periods t_k (k = 1, 2, …, p), such that $\begin{matrix} μ_{ij} (t_{k}) \in [0, 1], ν_{ij} (t_{k}) \in [0, 1], \\ μ_{ij} (t_{k}) + ν_{ij} (t_{k}) \leq 1, i = 1, 2, \dots, m, \\ j = 1, 2, \dots, n . \end{matrix}$

Based on the above decision information, we develop a practical method to rank and select the most desirable alternative(s). The method involves the following steps:

Step 1. Utilize the DIFHWA operator:

$\begin{matrix} {\tilde{r}}_{ij} = (μ_{ij}, ν_{ij}) {= DIFHWA}_{η (t)} ({\tilde{r}}_{ij} (t_{1}), {\tilde{r}}_{ij} (t_{2}), \dots, {\tilde{r}}_{ij} (t_{p})) \\ = (\frac{\prod_{k = 1}^{p} {(1 + (γ - 1) μ_{ij} (t_{k}))}^{η (t_{k})} - \prod_{k = 1}^{p} {(1 - μ_{ij} (t_{k}))}^{η (t_{k})}}{\prod_{k = 1}^{p} {(1 + (γ - 1) μ_{ij} (t_{k}))}^{η (t_{k})} + (γ - 1) \prod_{k = 1}^{p} {(1 - μ_{ij} (t_{k}))}^{η (t_{k})}}, \frac{γ \prod_{k = 1}^{p} {(ν_{ij} (t_{k}))}^{η (t_{k})}}{\prod_{k = 1}^{p} {(1 + (γ - 1) (1 - ν_{ij} (t_{k})))}^{η (t_{k})} + (γ - 1) \prod_{k = 1}^{p} {(ν_{ij} (t_{k}))}^{η (t_{k})}}) \end{matrix}$ (13)

To aggregate all the intuitionistic fuzzy decision matrices $\tilde{R} (t_{k}) = {({\tilde{r}}_{ij} (t_{k}))}_{m \times n} = (μ_{ij} (t_{k}), ν_{ij} (t_{k}))_{m \times n}$ (k = 1, 2, …, p) into a complexintuitionistic fuzzy decision matrix $\tilde{R} = ({\tilde{r}}_{ij})_{m \times n} = (μ_{ij}, ν_{ij})_{m \times n}$ .

Step 2. Utilize the IFHWA operator:

$\begin{matrix} {\tilde{r}}_{i} & = & (μ_{i}, ν_{i}) {= IFHWA}_{w} ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) \\ = & (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{ij})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - μ_{ij})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) μ_{ij})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - μ_{ij})}^{ω_{j}}}, \frac{γ \prod_{j = 1}^{n} {(ν_{ij})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - ν_{ij}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(ν_{ij})}^{ω_{j}}}) \end{matrix}$ (14)

to obtain the overall values ${\tilde{r}}_{i} = (μ_{i}, ν_{i})$ of the alternatives A_i (i = 1, 2, …, m).

Step 3. Calculate the $S ({\tilde{r}}_{i}), H ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ of the overall intuitionistic fuzzy values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ to rank all the alternatives A_i (i = 1, 2, …, m) and then to select the best one(s).

Step 4. Rank all the alternatives A_i (i = 1, 2, …, m) and select the best one(s) in accordance with $S ({\tilde{r}}_{i})$ and $H ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ .

Step 5. End.

5 Numerical example

It is unanimously recognized by the theorists in the world that Small and Medium-sized Enterprises (SMEs) will act as the leading role in the economic growth of the 21st century. SMEs play an irreplaceable and vital role in advancing rapid progress of national economy, alleviating employment pressure as well as promoting market prosperity and social stability. However, due to a series of reasons such as small scale, low labor productivity on the whole, low information transparency and so on, SMEs are at a disadvantage in fierce market competition, which is prominently reflected in financing difficulties. The difficulty in financing is the bottleneck restricting the development of SMEs, making impossible their sustainable and sound development. Stable development of supply chain requires SMEs to strengthen financing capacity and reduce financing cost through innovation of financial service in the context of supply chain. In this section, an illustrative example for risk assessment of supply chain finance is given to verify the developed approach with intuitionistic fuzzy information. Let us suppose there is an investment company, which wants to invest a sum of money in the best option. There is a panel with five possible Small and Medium-sized Enterprises to invest the money. The investment company must take a decision according to the following four attributes: ding172 G₁ is the industry environmental risk; ding173 G₂ is the enterprise credit; ding174 G₃ is the finance assets; ding175 G₄ is the supply chain operation. The five possible Small and Medium-sized Enterprises A_i (i = 1, 2, 3, 4, 5) are to be evaluated using the intuitionistic fuzzy information by the decision maker under the above four attributes at the periods t_k (k = 1, 2, 3), as listed in the following matrix. $\begin{matrix} \tilde{R} (t_{1}) \\ = [\begin{matrix} (0.4, 0.5) & (0.5, 0.4) & (0.2, 0.7) & (0.1, 0.8) \\ (0.6, 0.4) & (0.6, 0.3) & (0.6, 0.3) & (0.3, 0.6) \\ (0.5, 0.5) & (0.4, 0.5) & (0.4, 0.4) & (0.5, 0.4) \\ (0.7, 0.2) & (0.5, 0.4) & (0.2, 0.5) & (0.1, 0.5) \\ (0.5, 0.3) & (0.3, 0.4) & (0.6, 0.2) & (0.4, 0.4) \end{matrix}] \\ \tilde{R} (t_{2}) \\ = [\begin{matrix} (0.2, 0.7) & (0.1, 0.8) & (0.4, 0.5) & (0.5, 0.4) \\ (0.6, 0.3) & (0.3, 0.6) & (0.6, 0.4) & (0.6, 0.3) \\ (0.4, 0.4) & (0.5, 0.4) & (0.5, 0.5) & (0.4, 0.5) \\ (0.2, 0.5) & (0.1, 0.7) & (0.7, 0.2) & (0.5, 0.4) \\ (0.6, 0.2) & (0.4, 0.4) & (0.5, 0.3) & (0.3, 0.4) \end{matrix}] \\ \tilde{R} (t_{3}) \\ = [\begin{matrix} (0.5, 0.4) & (0.6, 0.3) & (0.3, 0.6) & (0.2, 0.7) \\ (0.7, 0.3) & (0.7, 0.2) & (0.7, 0.2) & (0.4, 0.5) \\ (0.6, 0.4) & (0.5, 0.4) & (0.5, 0.3) & (0.6, 0.3) \\ (0.8, 0.1) & (0.6, 0.3) & (0.3, 0.4) & (0.2, 0.6) \\ (0.6, 0.2) & (0.4, 0.3) & (0.7, 0.1) & (0.5, 0.3) \end{matrix}] \end{matrix}$

Let η (t) = (0.2, 0.3, 0.5) ^T be weight vector of the periods t_k (k = 1, 2, 3), and w = (0.35, 0.15, 0.20, 0.3) ^T be weight vector of the attributes G_j (j = 1, 2, 3, 4).

Then, we utilize the proposed procedure to get the most desirable Small and Medium-sized Enterprise.

Step 1. Utilize the DIFWG operator to aggregate all the intuitionistic fuzzy decision matrices $\tilde{R} (t_{k})$ into a complex intuitionistic fuzzy decision matrix $\tilde{R}$ as follows: $\tilde{R} = [\begin{matrix} (0.37, 0.36) & (0.51, 0.21) & (0.38, 0.37) & (0.47, 0.29) \\ (0.48, 0.30) & (0.54, 0.22) & (0.55, 0.20) & (0.63, 0.21) \\ (0.37, 0.40) & (0.29, 0.44) & (0.43, 0.29) & (0.27, 0.44) \\ (0.50, 0.33) & (0.60, 0.19) & (0.32, 0.35) & (0.52, 0.15) \\ (0.50, 0.26) & (0.56, 0.23) & (0.35, 0.30) & (0.55, 0.31) \end{matrix}]$

Step 2. Utilize the IFHWA operator to obtain the overall values ${\tilde{r}}_{i} = (μ_{i}, ν_{i})$ of the Small and Medium-sized Enterprises A_i (i = 1, 2, …, m). $\begin{matrix} {\tilde{r}}_{1} & = & (0.42, 0.32), {\tilde{r}}_{2} = (0.56, 0.23) \\ {\tilde{r}}_{3} & = & (0.33, 0.39), {\tilde{r}}_{4} = (0.45, 0.25) \\ {\tilde{r}}_{5} & = & (0.47, 0.29) \end{matrix}$

Step 3. Calculate the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ of the overall intuitionistic fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ to rank all the Small and Medium-sized Enterprises A_i (i = 1, 2, 3, 4, 5). $\begin{matrix} S ({\tilde{r}}_{1}) & = & 0.099, S ({\tilde{r}}_{2}) = 0.336 \\ S ({\tilde{r}}_{3}) & = & - 0.061, S ({\tilde{r}}_{4}) = 0.198 \\ S ({\tilde{r}}_{5}) & = & 0.179 \end{matrix}$

Step 4. Rank all the Small and Medium-sized Enterprises A_i (i = 1, 2, 3, 4, 5) and select the best one(s) in accordance with $S ({\tilde{r}}_{i})$ and $H ({\tilde{r}}_{i}) (i = 1, 2, 3, 4, 5)$ : A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃, and thus the most desirable Small and Medium-sized Enterprise is A₂. Thus, the Small and Medium-sized Enterprise A₂ is worth to invest the money with low risk of supply chain finance.

6 Conclusion

In the context of the supply chain, it has become an essential requirement for the stability of supply chain to enhance the financing ability and reduce the financing cost for small and medium enterprises through the innovative financial products. As a system innovation, Supply-chain Finance collects variety resources of business, finance, financial products to achieve the optimization welfare of the parties, which provides a solution of financing and technology bottlenecks for SMEs, which brings new profit models for the bank and explores business models innovation for developing third-party intermediary companies. Supply-chain Finance based on the real economy of supply chain, providing a range of financing products on the relationship between the upstream and downstream of supply chain, which can reduce financing costs for small and medium enterprises. Supply-chain Finance achieves the leap from static to dynamic of small and medium enterprises production study, to achieves the leap from the physical security to the property control of supply chain in risk prevention method, achieves the leap from the control of large enterprises financing to the attention small and medium enterprises financing. This financing model solves the problem of financing small and medium enterprises. In this paper, we investigate the dynamic multiple attribute decision making problems, in which the decision information, provided by decision makers at different periods, is expressed in intuitionistic fuzzy numbers. We first develop one new aggregation operators called dynamic intuitionistic fuzzy Hamacher weighted averaging (DIFHWA) operator. Moreover, a procedure based on the DIFHWA and IFHWA operators is developed to solve the dynamic multiple attribute decision making problems where all the decision information about attribute values takes the form of intuitionistic fuzzy numbers collected at different periods. Finally, an illustrative example for risk assessment of supply chain finance is given to verify the developed approach and to demonstrate its practicality and effectiveness. In the future, we shall continue our studies with other decision making models and information aggregating operators [38 –52].

Footnotes

Acknowledgments

The work was supported by the Subjects of Humanities and Social Sciences in Universities of Jiangxi Province (Project number: JJ1521), Social science planning project of Nanchang city in 12th Five-Year (Project number: JJ201511) and Social science planning project of Jiangxi Province (Project number: 13YJQ01).

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