Evaluation of customer satisfaction of “Door-to-Door” whole-process logistic service with interval-valued intuitionistic fuzzy information

Abstract

The evaluation of the customer satisfaction of “Door-to-Door” whole-process logistic service refers to a multiple attribute decision making problem. In this paper, we study on the multiple attribute decision making using interval-valued intuitionistic fuzzy numbers. Inspired by the idea of dependent aggregation, we design the dependent interval-valued intuitionistic fuzzy Hamacher weighted geometric (DIVIFHWG) operator, where the associated weights rely on the aggregated interval-valued intuitionistic fuzzy arguments and can lower the influence of unfair interval-valued intuitionistic fuzzy arguments on the aggregated results via allocating low weights to “false” and “biased” ones and then utilized these elements to design some methods for multiple attribute decision making with interval-valued intuitionistic fuzzy numbers. In the end, an example to evaluate the customer satisfaction of “Door-to-Door” whole-process logistic service is provided to test the proposed algorithm.

Keywords

Interval-valued intuitionistic fuzzy numbers operational laws dependent interval-valued intuitionistic fuzzy Hamacher weighted geometric (DIVIFHWG) operator customer satisfaction whole-process logistic service

1 Introduction

Atanassov [1, 2] provided the concept of intuitionistic fuzzy set(IFS) characterized by a membership function and a non-membership function. Xu [3] the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator and then provided an application of the proposed IFHG operator to multiple attribute group decision making with intuitionistic fuzzy information. Atanassov and Gargov [4, 5] then discussed the interval-valued intuitionistic fuzzy set (IVIFS). Xu [6] and Wei & Wang [7] developed interval-valued intuitionistic fuzzy weighted geometric (IVIFWG) operator, the interval-valued intuitionistic fuzzy ordered weighted geometric (IVIFOWG) operator and the interval-valued intuitionistic fuzzy hybrid geometric (IVIFHG) operator. Wei [8] investigated the dynamic intuitionistic fuzzy multiple attribute decision making problems and proposed geometric aggregation operators, for example, 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. Wang and Liu [9] further developed some new geometric aggregation operators, such as the intuitionistic fuzzy Einstein weighted geometric operator and the intuitionistic fuzzy Einstein ordered weighted geometric operator, which extend the weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator to provide the environment where the given arguments are intuitionistic fuzzy values. Based on the Hamacher operations on interval-valued intuitionistic fuzzy sets, such as Hamacher sum, Hamacher product, etc., and Liu [10] further develop the induced interval-valued intuitionistic fuzzy Hamacher geometric aggregating operators.

The evaluation of the customer satisfaction of “Door-to-Door” whole-process logistic service is the multiple attribute decision making problem. In this paper, the authors study on the multiple attribute decision making with interval-valued intuitionistic fuzzy numbers. Motivated by the ideal of dependent aggregation [12] and Hamacher operations [13], we develop the dependent interval-valued intuitionistic fuzzy Hamacher weighted geometric (DIVIFHWG) operator, in which the associated weights only depend on the aggregated interval-valued intuitionistic fuzzy arguments and is able to lower the affect of unfair interval-valued intuitionistic fuzzy arguments on the aggregated results via allocating lower weights to “false” and “biased” elements and then utilize them to design some methods for multiple attribute decision making with interval-valued intuitionistic fuzzy numbers. Finally, an illustrative example for evaluating the customer satisfaction of “Door-to-Door” whole-process logistic service is proposed to test the developed approach.

2 Preliminaries

Atanassov and Gargov [4, 5] further introduced the interval-valued intuitionistic fuzzy set (IVIFS), which is a generalization of the IFS.

Definition 1. [4, 5]. Assume that X is an universe of discourse, An IVIFS $\tilde{A}$ over X is an object having the form: $\tilde{A} = {〈 x, {\tilde{μ}}_{A} (x), {\tilde{ν}}_{A} (x) 〉 | x \in X}$ (1)

where ${\tilde{μ}}_{A} (x) \subset [0, 1]$ and ${\tilde{ν}}_{A} (x) \subset [0, 1]$ are interval numbers, and $0 \leq sup ({\tilde{μ}}_{A} (x)) + sup ({\tilde{ν}}_{A} (x)) \leq 1$ , ∀ x ∈ X. For convenience, let ${\tilde{μ}}_{A} (x) = [a, b]$ , ${\tilde{ν}}_{A} (x) = [c, d]$ , so $\tilde{A} = ([a, b], [c, d])$ .

Definition 2. [11] Assume that $\tilde{a} = ([a, b], [c, d])$ refers to an interval-valued intuitionistic fuzzy number, S of $\tilde{a}$ can be denoted: $S (\tilde{a}) = \frac{a - c + b - d + 2}{4}, S (\tilde{a}) \in [- 1, 1] .$ (2)

Definition 3. [11] Assume that $\tilde{a} = ([a, b], [c, d])$ be the interval-valued intuitionistic fuzzy number, its accuracy function H can be represented as follows [15]: $H (\tilde{a}) = \frac{a + b + c + d}{2}, H (\tilde{a}) \in [0, 1] .$ (3)

to estimate the level of accuracy of the interval-valued intuitionistic fuzzy value. In can be seen that, when the value of $H (\tilde{a})$ is higher, accuracy of the interval-valued intuitionistic fuzzy value $\tilde{a}$ is higher as well.

Liu [10] investigated the interval-valued intuitionistic fuzzy geometric aggregation operators with the help of the Hamacher operations.

Definition 4. [10] Assume that ${\tilde{a}}_{j} = ([a_{j}, b_{j}],$ [c_j, d_j]) (j = 1, 2, ⋯ , n) refers to a set of interval-valued intuitionistic fuzzy values, and then IVIFHWG : Qⁿ → Q is satisfied, if $\begin{matrix} {IVIFHWG}_{ω} ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \otimes_{j = 1}^{n} {({\tilde{a}}_{j})}^{ω_{j}} \\ = (\begin{matrix} [\frac{γ \prod_{j = 1}^{n} a_{j}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - a_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} a_{j}^{ω_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} b_{j}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - b_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} b_{j}^{ω_{j}}}], \end{matrix} \end{matrix}$ (4) $\begin{matrix} [\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{j})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - c_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{j})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - c_{j})}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{j})}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - d_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{j})}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - d_{j})}^{ω_{j}}}]) \end{matrix}$

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 IVIFHWG is named as the interval-valued intuitionistic fuzzy Hamacher weighted geometric (IVIFHWG) operator.

3 Dependent interval-valued intuitionistic fuzzy Hamacher weighted geometric (DIVIFHWG) operator fuzzy

Definition 5. Assume that ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ be a set of interval-valued intuitionistic fuzzy numbers, the average value of the score function of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ is computed as $s (\tilde{a}) = \frac{\sum_{j = 1}^{n} s ({\tilde{a}}_{j})}{n}$ (5)

Definition 6. Assume that ${\tilde{a}}_{1} = ([a_{1}, b_{1}], [c_{1}, d_{1}])$ and ${\tilde{a}}_{2} = ([a_{2}, b_{2}], [c_{2}, d_{2}])$ refers to 2 interval-valued intuitionistic fuzzy numbers, therefore, Hamming distance between ${\tilde{a}}_{1} = ([a_{1}, b_{1}], [c_{1}, d_{1}])$ and ${\tilde{a}}_{2} = ([a_{2}, b_{2}], [c_{2}, d_{2}])$ can be defined for the following equation [14]:

$\begin{matrix} d ({\tilde{a}}_{1}, {\tilde{a}}_{2}) \\ = \frac{| a_{1} - a_{2} | + | b_{1} - b_{2} | + | c_{1} - c_{2} | + | d_{1} - d_{2} |}{4} \end{matrix}$ (6)

Definition 7. Assume that ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ is a set of interval-valued intuitionistic fuzzy numbers, and $s (\tilde{a})$ the arithmetic average values of interval-valued intuitionistic fuzzy numbers, afterwards, this paper defines the standard deviation of values of these interval-valued intuitionistic fuzzy numbers as $σ = \sqrt{\frac{1}{n} \sum_{j = 1}^{n} d {(s ({\tilde{a}}_{j}), s (\tilde{a}))}^{2}}$ (7)

Definition 8. Assume that ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ be a set of interval-valued intuitionistic fuzzy numbers, that is,

$\begin{matrix} sim (s ({\tilde{a}}_{j}), s (\tilde{a})) & = & 1 - \frac{d (s ({\tilde{a}}_{j}), s (\tilde{a}))}{\sum_{j = 1}^{n} d (s ({\tilde{a}}_{j}), s (\tilde{a}))}, \\ j & = & 1, 2, \dots, n . \end{matrix}$ (8)

similarity between the jth largest interval-valued intuitionistic fuzzy values $s ({\tilde{a}}_{j})$ and the mean $s (\tilde{a})$ .

In real-life situations, the interval-valued intuitionistic fuzzy numbers ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ usually take the form of a collection of n preference values provided by n different individuals. Some individuals are able to allocate high or unduly low preference values to their preferred or repugnant objects. Hence, we need to allocate low weights to these “false” or “biased” opinions. The closer a preference value is to the mid one(s), the larger the weight we obtain. Therefore, utilizing Equation 8, IVIFHWG operator weights are defined as follows.

$\begin{matrix} w_{j} & = & \frac{sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}{\sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}, \\ j & = & 1, 2, \dots, n \end{matrix}$ (9)

We can see that w_j ≥ 0, j = 1, 2, ⋯ , n and $\sum_{j = 1}^{n} w_{j} = 1$ .

Especially, if $s ({\tilde{a}}_{j}) = s (\tilde{a})$ , for all i, j = 1, 2, ⋯ , n, then using Equation 9, w_j = $\frac{1}{n}$ is satisfied for all j = 1, 2, ⋯ , n.

Utilizing Equation 4, the following equation is satisfied. $\begin{matrix} IVIFHWG ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \otimes_{j = 1}^{n} ({\tilde{a}}_{j}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}) \end{matrix}$

$\begin{matrix} = (\frac{γ \prod_{j = 1}^{n} {(a_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - a_{j}))}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} + (γ - 1) \prod_{j = 1}^{n} {(a_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}, \\ \frac{γ \prod_{j = 1}^{n} {(b_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - b_{j}))}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} + (γ - 1) \prod_{j = 1}^{n} {(b_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}} \\ [\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} - \prod_{j = 1}^{n} {(1 - c_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - c_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} - \prod_{j = 1}^{n} {(1 - d_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - d_{j})}^{sim (s ({\tilde{a}}_{j}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{j}), s (\tilde{a}))}}]) \\ \frac{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} - \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(1 + b_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} + \prod_{j = 1}^{n} {(1 - b_{σ (j)})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}], \\ [\frac{2 \prod_{j = 1}^{n} c_{j}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(2 - c_{j})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} + \prod_{j = 1}^{n} c_{j}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}, \\ \frac{2 \prod_{j = 1}^{n} d_{j}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}{\prod_{j = 1}^{n} {(2 - d_{j})}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))} + \prod_{j = 1}^{n} d_{j}^{sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{σ (j)}), s (\tilde{a}))}}]) \end{matrix}$ (10) We call (10) a dependent interval-valued intuitionistic fuzzy Hamacher weighted geometric (DIVIFHWG) operator.

Inspired by paper [12], the following theorem can be obtained:

Theorem 1.Assume that ${\tilde{a}}_{j} = ([a_{j}, b_{j}], [c_{j}, d_{j}])$ be a set of interval-valued intuitionistic fuzzy numbers, and suppose that $s (\tilde{a})$ the average value of the score function of ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ . If $sim (s ({\tilde{a}}_{i}), s (\tilde{a})) \geq sim (s ({\tilde{a}}_{j}), s (\tilde{a}))$ , thenw_i ≥ w_j.

As is well known that normal distribution has been widely used. it has been seen that events are the aggregation of many smaller. Xu et al. discussed a normal distribution approach to estimate the weight of DOUWA operator [12]. Inspired by this idea, we should provide another algorithm to obtain DIVIFHWG weights: $w_{j} = \frac{1}{\sqrt{2 π} σ} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}, j = 2, \dots, n .$ (11)

where $s (\tilde{a})$ and σ are the arithmetic mean and the standard deviation of these interval-valued intuitionistic fuzzy arguments variables ${\tilde{a}}_{j} (j = 1, 2, \dots, n)$ , respectively.

Consider that w_j ≥ 0, j = 1, 2, ⋯ , n and $\sum_{j = 1}^{n} w_{j} = 1$ , then by (11), we have

$\begin{matrix} w_{j} & = & \frac{\frac{1}{\sqrt{2 π} σ} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}{\sum_{j = 1}^{n} \frac{1}{\sqrt{2 π} σ} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} \\ = & \frac{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}{\sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}, j = 1, 2, \dots, n . \end{matrix}$ (12)

then by (4), we have

$\begin{matrix} DIVIFHWG ({\tilde{a}}_{1}, {\tilde{a}}_{2}, \dots, {\tilde{a}}_{n}) \\ = \otimes_{j = 1}^{n} ({\tilde{a}}_{σ (j)}^{e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{σ (j)}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}) \\ = (\frac{γ \prod_{j = 1}^{n} {(a_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - a_{j}))}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} + (γ - 1) \prod_{j = 1}^{n} {(a_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}, \\ \frac{γ \prod_{j = 1}^{n} {(b_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - b_{j}))}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} + (γ - 1) \prod_{j = 1}^{n} {(b_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}} \\ [\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} - \prod_{j = 1}^{n} {(1 - c_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} + (γ - 1) \prod_{j = 1}^{n} {(1 - c_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} - \prod_{j = 1}^{n} {(1 - d_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) d_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}} + (γ - 1) \prod_{j = 1}^{n} {(1 - d_{j})}^{e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}} / \sum_{j = 1}^{n} e^{- \frac{{(d (s ({\tilde{a}}_{j}), s (\tilde{a})))}^{2}}{2 σ^{2}}}}}]) \end{matrix}$ (13)

Through Equations 9 and 12, we can find that all related weights of the DIVIFHWG operator rely on the aggregated interval-valued intuitionistic fuzzy numbers.

4 Model for multiple attribute decision making with interval-valued intuitionistic fuzzy information

In this section, we can utilize DIVIFHWG operator to the multiple attribute decision making problems with interval-valued intuitionistic fuzzy numbers. Assume that A ={ A₁, A₂, ⋯ , A_m } is a discrete set of alternatives, and G ={ G₁, G₂, ⋯ , G_n } be the set of attributes. The information about attribute weights can be find. Suppose that $\tilde{R} = {({\tilde{r}}_{ij})}_{m \times n} = ([a_{ij}, b_{ij}],$ [c_ij, d_ij]) _m×n is the interval-valued intuitionistic fuzzy decision matrix, [a_ij, b_ij] ⊂ [0, 1], [c_ij, d_ij] ⊂ [0, 1], b_ij + d_ij ≤ 1, i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n.

Next, we exploit the DIVIFHWG operator to multiple attribute decision making with interval-valued intuitionistic fuzzy information. The proposed algorithm is made up of the following phases:

By using the Equations 9 and 12 to determine the weight information of the attribute.

Exploiting the decision information given in matrix $\tilde{R}$ , and the DIVIFHWG operator

\begin{matrix} {\tilde{r}}_{i} = ([a_{i}, b_{i}], [c_{i}, d_{i}]) = DIVIFHWG ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) \\ = \oplus_{j = 1}^{n} ({({\tilde{a}}_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}) \\ = ([\frac{2 \prod_{j = 1}^{n} a_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}{\prod_{j = 1}^{n} {(2 - a_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} + \prod_{j = 1}^{n} a_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}, \\ \frac{2 \prod_{j = 1}^{n} b_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}{\prod_{j = 1}^{n} {(2 - b_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} + \prod_{j = 1}^{n} b_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}], \\ [\frac{\prod_{j = 1}^{n} {(1 + c_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} - \prod_{j = 1}^{n} {(1 - c_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}{\prod_{j = 1}^{n} {(1 + c_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} + \prod_{j = 1}^{n} {(1 - c_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}, \\ \frac{\prod_{j = 1}^{n} {(1 + d_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} - \prod_{j = 1}^{n} {(1 - d_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}{\prod_{j = 1}^{n} {(1 + d_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} + \prod_{j = 1}^{n} {(1 - d_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}]) \\ [\frac{2 \prod_{j = 1}^{n} c_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}{\prod_{j = 1}^{n} {(2 - c_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} + \prod_{j = 1}^{n} c_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}, \\ \frac{2 \prod_{j = 1}^{n} d_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}{\prod_{j = 1}^{n} {(2 - d_{ij})}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))} + \prod_{j = 1}^{n} d_{ij}^{sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i})) / \sum_{j = 1}^{n} sim (s ({\tilde{a}}_{ij}), s ({\tilde{a}}_{i}))}}]) \end{matrix}

Compute scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ of the collective overall values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ to rank all the alternatives A_i (i = 1, 2, ⋯ , m) and then choose the best one(s).

Rank all the alternatives A_i (i = 1, 2, ⋯ , m) and choose the best one(s) according to $S ({\tilde{r}}_{i})$ and $H ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ .

Algorithm End.

5 Numerical example

Along with the development of the network and the information technology, the global economy has presented the new development trend. The new economy has rapidly developed and the market environment is complicated, logistics service enterprise’ survival and development are difficult to predict. Pay attention to the logistics service enterprise’ dynamic environment to keep sustainable competitive advantage under the new economic era, to win the competitive advantage of the network resources and social capital, and strengthen logistics service enterprise’ competitiveness, it has become the realistic problem to be solved urgently for the academic circles and business community. In fact, because for a long time is affected by planned economy system influence and restrict market reputation mechanism, which leads China’s logistics service enterprises existing in logistics services overall development level is low, professional logistics service ability problems such as weak. In order to adapt to the development of China’s logistics service industry energetically, improving the logistics service enterprises’ dynamic capabilities to enhance sustainable competitive advantage and strengthen the competitiveness of the enterprises, many existed papers based on the theory of enterprise social capital, organization learning, enterprises’ dynamic capabilities, logistics service capabilities, as the research foundation, Using the theoretical study, empirical study and case analysis combination of methods, Learning from the introduction of the latest idea for knowledge view, Systematic analysis of the action mechanism and influence relation for the logistics service enterprises’ dynamic capabilities on social capital, organizational learning From the “structure-relationships-cognitive” three dimensional systematic. Preliminary established the theoretical framework and policy framework for the study of social capital, organization learning on logistics service enterprises’ dynamic ability influence, It has the extremely important practical significance and theoretical value to provide a theoretical basis and decision-making reference for the vigorously development of logistics service industry, and enhance the logistics service enterprises’ dynamic capabilities. This part provides a numerical example to testify the customer satisfaction of “Door-to-Door” whole-process logistic service with interval-valued intuitionistic fuzzy information. There are five possible logistic service enterprises A_i (i = 1, 2, 3, 4, 5) for 4 attributes G_j (j = 1, 2, 3, 4). The 4 attributes include: ding172 G₁ is the paid service; ding173 G₂ is the information service; ding174 G₃ is the goods receiving; ding175 G₄ is the complaint handling. The 5 possible logistic service enterprises A_i (i = 1, 2, 3, 4, 5) are designed evaluate the interval-valued intuitionistic fuzzy information.

$\begin{matrix} \tilde{R} = [\begin{matrix} ([0.4, 0.5], [0.3, 0.4]) & ([0.4, 0.6], [0.2, 0.4]) \\ ([0.6, 0.7], [0.2, 0.3]) & ([0.6, 0.7], [0.2, 0.3]) \\ ([0.3, 0.6], [0.3, 0.4]) & ([0.5, 0.6], [0.3, 0.4]) \\ ([0.7, 0.8], [0.1, 0.2]) & ([0.6, 0.7], [0.1, 0.3]) \\ ([0.3, 0.4], [0.2, 0.3]) & ([0.3, 0.5], [0.1, 0.3]) \end{matrix} \\ \begin{matrix} ([0.1, 0.3], [0.5, 0.6]) & ([0.3, 0.4], [0.3, 0.5]) \\ ([0.4, 0.7], [0.1, 0.2]) & ([0.5, 0.6], [0.1, 0.3]) \\ ([0.5, 0.6], [0.1, 0.3]) & ([0.4, 0.5], [0.2, 0.4]) \\ ([0.3, 0.4], [0.1, 0.2]) & ([0.3, 0.7], [0.1, 0.2]) \\ ([0.2, 0.5], [0.4, 0.5]) & ([0.3, 0.4], [0.5, 0.6]) \end{matrix}] \end{matrix}$

Next, we utilize the DIVIFHWG operator to multiple attribute decision making for evaluating the customer satisfaction of “Door-to-Door” whole-process logistic service.

(1) If we use the DIVIFHWG operator to decision making with interval-valued intuitionistic fuzzy information, we obtain the weight of DIVIFHWG operator is: $w = (0.16, 0 . 34, 0 . 28, 0 . 22) .$

(2) Exploit the DIVIFHWG operator, we achieve the overall values ${\tilde{r}}_{i}$ of the logistic service enterprises A_i (i = 1, 2, ⋯ , 5). $\begin{matrix} {\tilde{r}}_{1} & = & ([0.3548, 0.3780], [0.3764, 0.4591]) \\ {\tilde{r}}_{2} & = & ([0.5178, 0.5871], [0.1889, 0.2035]) \\ {\tilde{r}}_{3} & = & ([0.3801, 0.5409], [0.2951, 0.2456]) \\ {\tilde{r}}_{4} & = & ([0.5012, 0.6307], [0.1859, 0.1668]) \\ {\tilde{r}}_{5} & = & ([0.3482, 0.3569], [0.3166, 0.4014]) \end{matrix}$ (3) Compute the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, 5)$ of the interval-valued intuitionistic fuzzy values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5)$ $\begin{matrix} S ({\tilde{r}}_{1}) & = & - 0.0513, S ({\tilde{r}}_{2}) = 0.3562 \\ S ({\tilde{r}}_{3}) & = & 0.1902, S ({\tilde{r}}_{4}) = 0.3896 \\ S ({\tilde{r}}_{5}) & = & - 0.0065 \end{matrix}$ (4) Rank all the logistic service enterprises A_i (i = 1, 2, 3, 4, 5) according to the scores $S ({\tilde{r}}_{i})$ of the values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5) : A_{4} ≻ A_{2} ≻ A_{3} ≻ A_{5} ≻ A_{1}$ , and then the most suitable logistic service enterprise is represented as A₄.

6 Conclusion

The evaluation of the customer satisfaction of “Door-to-Door” whole-process logistic service refers to the multiple attribute decision making problems. We study on the multiple attribute decision making with interval-valued intuitionistic fuzzy numbers. Inspired by the idea of dependent aggregation, we propose a dependent interval-valued intuitionistic fuzzy Hamacher weighted geometric (DIVIFHWG) operator. In the end, an example to test the customer satisfaction of “Door-to-Door” whole-process logistic service is given to test the proposed algorithm and to illustrate its practicality and effectiveness. In the following study, we consider to extend the proposed algorithm to other fields [15 –22].

Footnotes

Acknowledgments

The work was supported by the National Natural Science Foundation of China under Grant No. 71173218, Key Project of National Social Science Foundation of China under Grant No. 12AGL003, Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20130095110002, Guangxi Philosophy and Social Science Fund Project under Grant No. 13BGL011, the New Century Teaching Reform Project in Guangxi under Grant Nos. 2012JGB118 & 2015JGB167 and Guangxi Development Research Center of Humanities and Social Science-Scientific Research Project “Outstanding Youth Characteristic Research Group” Pearl River-Xijiang River Intelligent Economic Belt Research Group.

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