Decision making method for evaluating logistics companies based on the ordered representation of the polygonal fuzziness 1

Abstract

A polygonal fuzzy numbers can describe fuzzy information by means of finite ordered real numbers. It not only overcomes the complexity of traditional fuzzy number operations, but also keeps some good properties of trapezoidal fuzzy numbers, and it can approximate general fuzzy numbers with arbitrary precision. In this paper, a weighted arithmetic average operator is defined by the ordered representation and its operations of the polygonal fuzzy numbers, and a new Euclidean distance for measuring the polygonal fuzzy numbers is given. Secondly, in view of cost attribute and benefit attributes, the polygonal fuzzy decision matrix is normalized, and the weighted Euclidean distance is used to solve the positive (negative) ideal solution and the relative closeness of the decision matrix, and then a new decision method is given. Finally, the effectiveness of the proposed decision-making method is illustrated by an example of the evaluation of logistics companies by shopping websites.

Keywords

Polygonal fuzzy number ordered representation Euclidean distance positive (negative) ideal solution multiple attribute decision making

1. Introduction

With the rapid development of China’s economy and the fierce competition of market operators, shopping websites are increasingly demanding for logistics companies. At present, these websites mainly conduct comprehensive evaluation of logistics companies through multiple attribute indicators, such as transport efficiency, transport cost, transport safety, service attitude and throughput, and through the evaluation results choose a satisfactory logistics company. In reality, some logistics companies are often limited by their transport capacity and scale, inevitably some indicators can not meet the requirements of shopping websites or consumers, so they may be mercilessly eliminated by the market. Nowadays, popular shopping websites mainly focus on optimizing the selection of indicators such as high transportation efficiency, low transportation cost, good security and good service attitude to win reputation, which is undoubtedly the fundamental guarantee for shopping websites to maximize profits. Actually, logistics evaluation mainly uses some relevant data and mathematical models to evaluate the whole logistics system, so as to establish a control system that is conducive to the reasonable planning of the logistics system, and can accurately reflect the internal rationalization status of the logistics system and establish a sustainable development business model.

In 2014, Kao and Liao [1] proposed an evaluation method for logistics transportation and its service system based on fuzzy set theory and objective programming. In 2016, J Q Wang et al. utilized the likelihood-based TODIM approach of fuzzy language information to study the evaluation system of logistics outsourcing in [2], and a likelihood function was established for multi-hesitant fuzzy linguistic term elements based on a generalized function of the possibility degree of real numbers. In 2017, Cagliano and Marco et al. studied the most important factors that influence the productivity of the urban fleet of a logistics service provider in [3], and through a regression analysis they proposed that three main levers are shown to have significant impacts on productivity. In 2018, Singh and Qunasekaran et al. [4] put forward the fuzzy AHP and fuzzy TOPSIS methods with the help of the third party logistics in cold chain management. In recent years, some work on evaluating logistics enterprise and decision-making has been gradually increased. See [5 –7]. These beneficial results provide some new evaluation techniques and methods for logistics companies from different perspectives.

In 2006, Wang [8] put forward an interval intuitionistic fuzzy decision-making method for incompletely determined multi-attribute information, and gave a multi-criteria programming method [9], the evidence reasoning method is extended to the multi-attribute decision making of intuitionistic trapezoidal or interval fuzzy numbers. See [10, 11]. In 2007, Xu first suggested the multi-attribute decision-making model with intuitionistic fuzzy number as attribute value in [12], a linear programming model was established, and proposed a multi-attribute group decision-making scheme based on a variety of aggregators. See [13 –16]. In 2009, Wei defined the expected value and score function of intuitionistic trapezoidal fuzzy numbers by weighted arithmetic average operator in [17], and suggested a new multi-attribute group decision-making method [18]. In 2010, Wan et al. [19] studied the group decision-making problem of multi-attribute information based on intuitionistic trapezoidal fuzzy numbers, and gave the multi-attribute decision-making method of interval intuitionistic trapezoidal fuzzy numbers. In 2014, Lan et al. [20] not only discussed the completeness of trapezoidal fuzzy number space, but also put forward a new practical decision methods. Although these results provided some new evaluating methods for logistics companies from different perspectives, the old trapezoid or interval intuitionistic fuzzy number is still used as a basic element in the description of fuzzy phenomena. It has to be said that it is a weak point.

The polygonal fuzzy numbers are different from intuitionistic fuzzy nmbers or Pythagorean fuzzy soft sets, it can approximate or represent the general fuzzy number by means of a finite number of ordered real numbers, and it may describe the fuzzy information according to the arbitrary accuracy. It is not only a generalization of triangular or trapezoidal fuzzy numbers, but also its operation is simple and clear and satisfies the closeness. In 2011, Wang [21] first studied the approximation performance of forward polygonal fuzzy neural network by using n- polygonal fuzzy numbers. In Ref. [22], the construction and approximation process of single input and single output polygonal fuzzy neural network were studied. Later, a FCM clustering algorithm and TOPSIS decision-making method of multi-attribute index information were proposed by using the ordered representation of polygonal fuzzy numbers. See [23, 24]. These results show many advantages of the polygonal fuzzy numbers from different aspects, and it also lays a theoretical foundation for the wide application of trapezoid or interval fuzzy numbers.

The main contents of this paper are as follows: In section 2, the ordered representation of polygonal fuzzy numbers, examples and their operation rules are given; In section 3. A new Euclidean distance of the polygonal fuzzy numbers is proposed, and the approximation accuracy of the polygonal fuzzy numbers is discussed. In section 4, the expression of multi-attribute information and information aggregation of logistics companies are described by the ordered representation, and the expression of linear function is given. In section 5, a new normalization method for polygonal decision matrix is given, and the relative closeness degrees are solved by the weighted Euclidean distance and positive (negative) ideal solutions. Finally, a new decision method is given. Finally, the effectiveness of the proposed method is illustrated by an example of evaluating logistics companies through shopping websites in section 6.

2. Polygonal fuzzy numbers and their ordered representation

Normally, the operations of general fuzzy numbers are not linear, it can only rely on the complex Zadeh expansion principle for arithmetic operations, even for triangular or trapezoidal fuzzy numbers is very difficult. This brings serious inconvenience to the further application of fuzzy numbers. In fact, trapezoidal fuzzy numbers are not only generalizations of triangular fuzzy numbers, but more importantly, it can approximate to a fuzzy number by superposition of several small trapezoids. In other words, a fuzzy number can intercept n small trapezoids according to different n values, and the finite superposition of these small trapezoids can approximate to express a fuzzy numbers, as shown in Fig. 1.

Fig. 1

Intercepting several small trapezoidal images on the membership function A (x).

According to Fig. 1, it is not difficult to see that the number of small trapezoids intercepted on a given fuzzy number A depends on the value of n. The larger the value of n, the more the number of small trapezoids and the more intersections they have. These small trapezoids have stronger ability to approximate fuzzy numbers, but their complexity also increases.

Definition 2.1. For a given fuzzy number $A \in F_{0} (ℝ)$ and natural number $n \in ℕ$ , if the closed interval [0,1] on the y- axis is divided into n small closed intervals, the subdivision points are x_i = i/n, i = 1, 2, ·· · , n. If there are 2n + 2 ordered real numbers ${a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot,$ $a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2}} \subset ℝ$ and satisfy $a_{0}^{1} ⩽ a_{1}^{1} ⩽ \cdot \cdot \cdot$ $⩽ a_{n}^{1} ⩽ a_{n}^{2} ⩽ \cdot \cdot \cdot ⩽ a_{1}^{2} ⩽ a_{0}^{2}$ , the membership function A (x) takes straight line segments on each $[a_{i - 1}^{1}, a_{i}^{1}]$ and $[a_{i}^{2}, a_{i - 1}^{2}]$ , i = 1, 2, ·· · , n, that is to say, for arbitrary $x \in ℝ$ , and its membership function is defined as $A (x) = {\begin{matrix} \frac{i - 1}{n} + \frac{x - a_{i - 1}^{1}}{n (a_{i}^{1} - a_{i - 1}^{1})}, & x \in [a_{i - 1}^{1}, a_{i}^{1}], i = 1, 2, \cdot \cdot \cdot, n \\ 1, & x \in [a_{n}^{1}, a_{n}^{2}] \\ \frac{i - 1}{n} + \frac{a_{i - 1}^{2} - x}{n (a_{i - 1}^{2} - a_{i}^{2})}, & x \in [a_{i}^{2}, a_{i - 1}^{2}], i = 1, 2, \cdot \cdot \cdot, n \\ 0, & Otherwise \end{matrix} .$

Then A is called an n-polygonal fuzzy number on $ℝ$ , or simply called a polygonal fuzzy number, abbreviated as $Z_{n} (A) = (a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2})$ , and the right-hand ordered real array $(a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot$ , $a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2})$ is called an ordered representation of A. See Fig. 2

Fig. 2

Images of the membership function of n-polygonal fuzzy number A.

According to Fig. 2, there are obviously support set Supp $A = [a_{0}^{1}, a_{0}^{2}]$ , Ker $A = [a_{n}^{1}, a_{n}^{2}]$ , and satisfies $A (a_{i}^{q}) = i / n$ , q = 1, 2 ; i = 1, 2, ·· · . n. Especially, when $a_{n}^{1} = a_{n}^{2}$ , the each components in the ordered representation can not be omitted, and 2n + 2 real numbers still need to be guaranteed. In addition, it is not difficult to see that the polygonal fuzzy numbers have similar properties with triangular or trapezoidal fuzzy numbers, and each n-polygonal fuzzy number A is uniquely determined by 2n + 2 ordered real numbers ${a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot$ , $a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2}}$ .

Example 1. For a given ordered representation A = (-4, - 3, 0, 1, 2, 4, 5, 7), try to determine the analytic expression of its corresponding membership function.

In fact, it is obvious that the support set Supp A = [-4, 7], Ker A = [1, 2], and there are eight ordered real numbers in the given ordered representation. so that 2n + 2 =8 and the solution is n = 3. This means inserting two points λ₁ = 1/ - 2 and λ₂ = 2/ - 3 into the closed interval of y- axis [0, 1]. Therefore, the intersection coordinates of the membership function of 3- polygonal fuzzy number and the horizontal lines y = 1/ - 3 and y = 2/ - 3 can be expressed as $\begin{matrix} (- 4, 0), (- 3, \frac{1}{3}), (0, \frac{2}{3}), (1, 1), (2, 1), \\ (4, \frac{2}{3}), (5, \frac{1}{3}), (7, 0) . \end{matrix}$

Then the above eight coordinate points and its positions can easily determined in the plane coordinate system, and then connect the adjacent intersections with straight line segments in turn. Finally, we can obtain the membership function image of the obtained 3-polygonal fuzzy number A and its analytic formula as follow. See Fig. 3.

Fig. 3

Membership function image and its analytic expression of 3-polygonal fuzzy number A.

In addition, it is easy to obtain an ordered representation Z_n (A) of the n-polygonal fuzzy number by n-equidistant subdivision for the membership function A (x), so that the fuzzy number A approximated by finite ordered real numbers can be realized. But the better the approximation effect of Z_n (A) is, the greater the value of n, and satisfies $lim_{n \to \infty} D (Z_{n} (A), A) = 0 .$ In fact, the n-polygonal fuzzy number Z_n (A) is composed of several small trapezoids, in which the triangle can be regarded as a special trapezoid, and satisfies $A (a_{i}^{q}) = i / n$ , q = 1, 2 ; i = 1, 2, ·· · . n. Especially, when n = 1 ⇔ i = 1, then n = 1 means that the bottom ends and the center points are directly connected to form a triangle or trapezoid, that is, the Z₁ (A) degenerates into a triangular or trapezoidal fuzzy number. Hence, when n ⩾ 2, the ordered representation Z_n (A) can describe the fuzzy information more precisely.

Let $F_{0}^{tn} (ℝ)$ denote the set of all n-polygonal fuzzy numbers on $ℝ$ , and $F_{0}^{t n} (ℝ) \subset F_{0} (ℝ)$ . Next, For a given fuzzy number, its ordered representation will be solved by an example.

Example 2. Assuming that the membership function of the fuzzy number A $A (x) = {\begin{matrix} \sqrt{x + 1} - 1, & 0 ⩽ x < 3 \\ 1, & 3 ⩽ x ⩽ 4 \\ 3 - \sqrt{x}, & 4 < x ⩽ 9 \\ 0, & Otherwise \end{matrix} .$

When n = 2, 3, 4, try to solve the ordered representation of the corresponding n-polygonal fuzzy number A.

In fact, Supp A = [0, 9], when n = 2, only one subdivision point λ = 1/ - 2 on the y-axis. Let the left function $\sqrt{x + 1} - 1 = 1 / 2$ , then x₁ = 5/ - 4 is solved on [0, 3). Let right function $3 - \sqrt{x} = 1 / 2$ , then x₂ = 25/ - 4 is also solved on (4, 9]. So the ordered representation of 2-polygonal fuzzy numbers Z₂ (A) is $Z_{2} (A) = (0, \frac{5}{4}, 3, 4, \frac{25}{4}, 9) .$

Let n = 3, it has only two subdivision points λ₁ = 1/ - 3 and λ₂ = 2/ - 3 on the closed interval [0, 1] of y-axis. Let left function $\sqrt{x + 1} - 1 = 1 / 3$ , 2/ - 3 in turn, then the x₁ = 7/ - 9 and x₂ = 16/ - 9 can be solved on [0, 3), and satisfies x₁ < x₂. Let right function $3 - \sqrt{x} = 1 / 3, 2 / 3$ in turn, then x₃ = 49/ - 9 and x₄ = 64/ - 9 can also be solved on (4, 9], and satisfies x₂ < x₃ < x₄. so the ordered representation of 3-polygonal fuzzy number Z₃ (A) is as follows: $Z_{3} (A) = (0, \frac{7}{9}, \frac{16}{9}, 3, 4, \frac{49}{9}, \frac{64}{9}, 9) .$

Similarly, let n = 4, then there are three subdividing points λ₁ = 1/ - 4, λ₂ = 2/ - 4 and λ₃ = 3/ - 4. It is not difficult to obtain the ordered representation of Z₄ (A) as follows $Z_{4} (A) = (0, \frac{9}{16}, \frac{5}{4}, \frac{33}{16}, 3, 4, \frac{81}{16}, \frac{25}{4}, \frac{121}{16}, 9) .$

In particular, when n = 2, 3, the membership function image and the ordered representation of n-polygonal fuzzy number corresponding to A are shown in Figs. 4-5.

Fig. 4

Membership function images of A and Z₂ (A).

Fig. 5

Membership function images of A and Z₃ (A).

Next, we will give the operations of n-polygonal fuzzy numbers on $F_{0}^{t n} (ℝ)$ as follows.

Definition 2.2. For a given $n \in ℕ$ , let $A, B \in F_{0}^{t n} (ℝ)$ , and $A = (a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1}, a_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2})$ , $B = (b_{0}^{1}, b_{1}^{1}, \cdot \cdot \cdot$ , $b_{n}^{1}, b_{n}^{2}, \cdot \cdot \cdot, b_{1}^{2}, b_{0}^{2})$ . Define the addition and multiplication as

① $A + B = (a_{0}^{1} + b_{0}^{1}, a_{1}^{1} + b_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1} + b_{n}^{1}, a_{n}^{2} + b_{n}^{2}, \cdot \cdot \cdot, a_{1}^{2} + b_{1}^{2}, a_{0}^{2} + b_{0}^{2})$ ;

② $A \cdot B = (c_{0}^{1}, c_{1}^{1}, \cdot \cdot \cdot, c_{n}^{1}, c_{n}^{2}, \cdot \cdot \cdot, c_{1}^{2}, c_{0}^{2})$ , where $c_{i}^{1} = a_{i}^{1} b_{i}^{1} \land a_{i}^{1} b_{i}^{2} \land a_{i}^{2} b_{i}^{1} \land a_{i}^{2} b_{i}^{2}$ , $c_{i}^{2} = a_{i}^{1} b_{i}^{1}$ $\lor a_{i}^{1} b_{i}^{2} \lor a_{i}^{2} b_{i}^{1} \lor a_{i}^{2} b_{i}^{2}$ ,i = 0, 1, 2, ·· · , n. Especially,

$k \cdot A = {\begin{matrix} ({ka}_{0}^{1}, {ka}_{1}^{1}, \cdot \cdot \cdot, {ka}_{n}^{1}, {ka}_{n}^{2}, \cdot \cdot \cdot, {ka}_{1}^{2}, {ka}_{0}^{2}), k ⩾ 0 \\ ({ka}_{0}^{2}, {ka}_{1}^{2}, \cdot \cdot \cdot, {ka}_{n}^{2}, {ka}_{n}^{1}, \cdot \cdot \cdot, {ka}_{1}^{1}, {ka}_{0}^{1}), k < 0 \end{matrix} .$

Lemma 1. [25] Assuming $A, B \in F_{0} (ℝ)$ , for a given $n \in ℕ$ , and the polygonal operator $Z_{n} : F_{0} (ℝ) \to F_{0}^{t n} (ℝ)$ , then the addition and multiplication operations on fuzzy number space $F_{0} (ℝ)$ can be transformed into the expansion operations on $F_{0}^{t n} (ℝ)$ , i.e.,

① Z_n (A) = Z_n (B) ⇔ A_i/-n = B_i/-n ⇔ Z_n (A) _i/-n = Z_n (B) _i/-n, i = 0, 1, 2, ·· · , n .

② Z_n (A + B) = Z_n (A) + Z_n (B),

③ Z_n (A · B) = Z_n (A) · Z_n (B) and Z_n (k · A)) = k · Z_n (A),k > 0.

3. Metric and approximation of the polygonal fuzzy numbers

In 2011, it is proved that the fuzzy distance on the polygonal fuzzy number space $F_{0}^{t n} (ℝ)$ degenerates to $D (A, B) = \lor_{i = 0}^{n} (| a_{i}^{1} - b_{i}^{1} | \lor | a_{i}^{2} - b_{i}^{2} |)$ under the traditional fuzzy distance formula and Definition 2.2. See Ref. [21]. Although the distance is simple and intuitive, the accuracy of the metric is poor. Hence, a new metric (distance) E_n on $F_{0}^{t n} (ℝ)$ is introduced as follows.

For a given natural number $n \in ℕ$ , let the mapping $E_{n} : F_{0}^{t n} (ℝ) \times F_{0}^{t n} (ℝ) \to [0, + \infty)$ , and for arbitrary $A, B \in F_{0}^{t n} (ℝ)$ and $A = (a_{0}^{1}, a_{1}^{1}, \cdot \cdot \cdot, a_{n}^{1}, a_{n}^{2},$ $\cdot \cdot \cdot, a_{1}^{2}, a_{0}^{2})$ , $B = (b_{0}^{1}, b_{1}^{1}, \cdot \cdot \cdot, b_{n}^{1}, b_{n}^{2}, \cdot \cdot \cdot, b_{1}^{2}, b_{0}^{2})$ . Let

$\begin{matrix} E_{n} (A, B) \\ = \sqrt{\frac{1}{2 n + 2} \sum_{i = 0}^{n} ((a_{i}^{1} - b_{i}^{1})^{2} + (a_{i}^{2} - b_{i}^{2})^{2})} . \end{matrix}$ (1)

It is not difficult to verify that the mapping E_n constitutes a distance (metric) on $F_{0}^{t n} (ℝ)$ , we also call it an Euclidean distance.

Lemma 2. (Schwartz inequality) If {a₁, a₂, ·· · , a_n} and {b₁, b₂, ·· · , b_n} are two groups of real numbers given arbitrarily, then it must satisfy ${(\sum_{i = 1}^{n} a_{i} b_{i})}^{2} ⩽ \sum_{i = 1}^{n} a_{i}^{2} \cdot \sum_{i = 1}^{n} b_{i}^{2}$ .

Corollary 1. If a₁, a₂, b₁, b₂ > 0, then $\sqrt{a_{1} b_{1}} + \sqrt{a_{2} b_{2}} ⩽ \sqrt{(a_{1} + a_{2}) (b_{1} + b_{2})}$ .

Proof. According to Lemma 2, where n = 2, $\sqrt{a_{i}}$ and $\sqrt{b_{i}}$ are used to replace a_i and b_i(i = 1, 2), respectively, the conclusion is obtained immediately.

Next, we will verify that the operator E_n really constitutes a metric on the space $F_{t}^{n} (ℝ)$ according to Formula (1).

In fact, the non-negativity and symmetry of E_n (A, B) are evident, so it only need to prove that E_n satisfies three-point inequality, i.e., for any $A, B, C \in F_{0}^{t n} (ℝ)$ , then E_n (A, B) ⩽ E_n (A, C) + E_n (C, B) . Since the coefficient 1/ - (2n + 2) under the square root of Formula (1) does not affect the proof of three-point inequality. For simplicity, let $C = (c_{0}^{1}, c_{1}^{1}, \cdot \cdot \cdot, c_{n}^{1}, c_{n}^{2}, \cdot \cdot \cdot, c_{1}^{2}, c_{0}^{2})$ , it only need to prove the following conclusions, i.e., $\begin{matrix} \sqrt{\sum_{i = 0}^{n} ((a_{i}^{1} - b_{i}^{1})^{2} + (a_{i}^{2} - b_{i}^{2})^{2})} \\ ⩽ \sqrt{\sum_{i = 0}^{n} ((a_{i}^{1} - c_{i}^{1})^{2} + (a_{i}^{2} - c_{i}^{2})^{2})} \\ + \sqrt{\sum_{i = 0}^{n} ((c_{i}^{1} - b_{i}^{1})^{2} + (c_{i}^{2} - b_{i}^{2})^{2})} . \end{matrix}$

Actually, as $| a_{i}^{1} - b_{i}^{1} | ⩽ | a_{i}^{1} - c_{i}^{1} | + | c_{i}^{1} - b_{i}^{1} |$ , i = 0, 1, 2, ·· · , n, by Lemma 2 (Schwartz inequality) it is not difficult to obtain $\begin{matrix} \sum_{i = 0}^{n} (a_{i}^{1} - b_{i}^{1})^{2} \\ ⩽ \sum_{i = 0}^{n} (a_{i}^{1} - c_{i}^{1})^{2} + \sum_{i = 0}^{n} (c_{i}^{1} - b_{i}^{1})^{2} \\ + 2 \sum_{i = 0}^{n} | a_{i}^{1} - c_{i}^{1} | \cdot | c_{i}^{1} - b_{i}^{1} | \\ ⩽ \sum_{i = 0}^{n} (a_{i}^{1} - c_{i}^{1})^{2} + \sum_{i = 0}^{n} (c_{i}^{1} - b_{i}^{1})^{2} \\ + 2 \sqrt{\sum_{i = 0}^{n} (a_{i}^{1} - c_{i}^{1})^{2} \cdot \sum_{i = 0}^{n} (c_{i}^{1} - b_{i}^{1})^{2}} \\ = {(\sqrt{\sum_{i = 0}^{n} (a_{i}^{1} - c_{i}^{1})^{2}} + \sqrt{\sum_{i = 0}^{n} (c_{i}^{1} - b_{i}^{1})^{2}})}^{2} . \end{matrix}$

Similarly, under the condition $| a_{i}^{2} - b_{i}^{2} | ⩽ | a_{i}^{2} - c_{i}^{2} | + | c_{i}^{2} - b_{i}^{2} |$ , i = 0, 1, 2, ·· · , n, we can also obtain $\begin{matrix} \sum_{i = 0}^{n} (a_{i}^{2} - b_{i}^{2})^{2} \\ ⩽ {(\sqrt{\sum_{i = 0}^{n} (a_{i}^{2} - c_{i}^{2})^{2}} + \sqrt{\sum_{i = 0}^{n} (c_{i}^{2} - b_{i}^{2})^{2}})}^{2} . \end{matrix}$

Let $\sum_{i = 0}^{n} (a_{i}^{1} - c_{i}^{1})^{2} = λ_{1}$ , $\sum_{i = 0}^{n} (a_{i}^{2} - c_{i}^{2})^{2} = λ_{2}$ , $\sum_{i = 0}^{n} (c_{i}^{1} - b_{i}^{1})^{2} = β_{1}$ , $\sum_{i = 0}^{n} (c_{i}^{2} - b_{i}^{2})^{2} = β_{2}$ . By Corollary 1, it is not hard to get

$\begin{matrix} \sqrt{\sum_{i = 0}^{n} ((a_{i}^{1} - b_{i}^{1})^{2} + (a_{i}^{2} - b_{i}^{2})^{2})} \\ = \sqrt{\sum_{i = 0}^{n} (a_{i}^{1} - b_{i}^{1})^{2} + \sum_{i = 0}^{n} (a_{i}^{2} - b_{i}^{2})^{2}} \\ ⩽ \sqrt{(\sqrt{λ_{1}} + \sqrt{β_{1}})^{2} + (\sqrt{λ_{2}} + \sqrt{β_{2}})^{2}} \\ = \sqrt{λ_{1} + λ_{2} + β_{1} + β_{2} + 2 (\sqrt{λ_{1} β_{1}} + \sqrt{λ_{2} β_{2}})} \\ ⩽ \sqrt{(λ_{1} + λ_{2}) + (β_{1} + β_{2}) + 2 (\sqrt{(λ_{1} + λ_{2}) (β_{1} + β_{2})}} \\ = \sqrt{λ_{1} + λ_{2}} + \sqrt{β_{1} + β_{2}} \\ = \sqrt{\sum_{i = 0}^{n} ((a_{i}^{1} - c_{i}^{1})^{2} + (a_{i}^{2} - c_{i}^{2})^{2})} \\ + \sqrt{\sum_{i = 0}^{n} ((c_{i}^{1} - b_{i}^{1})^{2} + (c_{i}^{2} - b_{i}^{2})^{2})} . \end{matrix}$

That is to say, E_n (A, B) ⩽ E_n (A, C) + E_n (C, B). Then E_n is an Euclidean metric on $F_{t}^{n} (ℝ)$ .

In addition, since the polygonal fuzzy number Z_n (A) can approximate to a general fuzzy number A, people are more concerned about how to determine the minimum n value to make the approximation accuracy ɛ > 0, and satisfies D (Z_n (A) , A) < ɛ. Actually, as a special fuzzy number, the polygonal fuzzy number must satisfy the traditional fuzzy distance formula $D (A, B) = \underset{α \in (0, 1]}{\lor} d_{H} (A_{α}, B_{α})$ of P. Diamond, where d_H is a Hausdorff metric. Especially, when $A_{α} = [a_{α}^{1}, a_{α}^{2}]$ and $B_{α} = [b_{α}^{1}, b_{α}^{2}]$ are abounded closed intervals for arbitrary α ∈ (0, 1], it is easy to obtain $D (A, B) = \underset{α \in (0, 1]}{\lor} (| a_{α}^{1} - b_{α}^{1} | \lor | a_{α}^{2} - b_{α}^{2} |)$

Next, for a given precision ɛ > 0, we will use an example to find a smaller n value so that D (Z_n (A) , A) < ɛ. In fact, if we do equidistant subdivision on the closed interval [0, δ]: δ/ - n ⩽ 2δ/ - n < ·· · < iδ/ - n < ·· · < (n - 1) δ/ - n < δ, then there is i ∈ {1, 2, ·· · , n} makes (i - 1) δ/ - n ⩽ α ⩽ i1ptδ/ - n and A_i1ptδ/-n ⊂ A_α ⊂ A_(i-1)δ/-n = Z_n (A) _{(i-1) 1ptδ/-n} for arbitrary α ∈ (0, δ]. According to the three-point inequality of Hausdorff metric d_H, then the following conclusion can be obtained that

$d_{H} (Z_{n} (A)_{α}, A_{α}) ⩽ d_{H} (Z_{n} (A)_{α}, Z_{n} (A)_{\frac{i - 1}{n}}) + d_{H} (Z_{n} (A)_{\frac{i - 1}{n}}, A_{\frac{i - 1}{n}}) + d_{H} (A_{\frac{i - 1}{n}}, A_{α})$ $⩽ d_{H} (A_{\frac{i}{n}}, A_{\frac{i - 1}{n}}) + d_{H} (A_{\frac{i - 1}{n}}, A_{α}) .$ (2)

It should be noted that formula $D (A, B) = \underset{α \in (0, 1]}{\lor} (| a_{α}^{1} - b_{α}^{1} | \lor | a_{α}^{2} - b_{α}^{2} |)$ can only be used to describe the distance between A and Z_n (A), but the above Formula (1) can not be used.

Suppose the membership function of fuzzy number A is $A (x) = {\begin{matrix} \sqrt{6 x + 4} - 1, - 1 / 2 ⩽ x < 0 \\ 1, 0 ⩽ x ⩽ 1 / 2 \\ 1 - \sqrt{6 x - 3}, 1 / 2 < x ⩽ 2 / 3 \\ 0, \end{matrix}$ , for arbitrary α ∈ (0, 1], then it is easy to obtain the cut set $\begin{matrix} A_{α} = & [\frac{1}{6} (1 + α)^{2} - \frac{2}{3}, \frac{1}{6} (1 - α)^{2} + \frac{1}{2}], \\ A_{\frac{i - 1}{n}} = & [\frac{1}{6} (1 + \frac{i - 1}{n})^{2} - \frac{2}{3}, \frac{1}{6} (1 - \frac{i - 1}{n})^{2} + \frac{1}{2}], \\ i = 1, 2, \cdot \cdot \cdot, n, \end{matrix}$ and satisfies $α - \frac{i - 1}{n} ⩽ \frac{1}{n}$ , $2 + α + \frac{i - 1}{n} ⩽ 2 + \frac{i}{n} + \frac{i - 1}{n} ⩽ 4$ ,i = 1, 2, ·· · , n. According to Formula (2), there must be $\begin{matrix} d_{H} (Z_{n} (A)_{α}, A_{α}) \\ ⩽ d_{H} (A_{\frac{i}{n}}, A_{\frac{i - 1}{n}}) + d_{H} (A_{\frac{i - 1}{n}}, A_{α}) \\ = \frac{1}{6} ((1 + \frac{i}{n})^{2} - (1 + \frac{i - 1}{n})^{2}) \\ \lor \frac{1}{6} ((1 - \frac{i - 1}{n})^{2} - (1 - \frac{i}{n})^{2}) \\ + \frac{1}{6} ((1 + α)^{2} - (1 + \frac{i - 1}{n})^{2}) \\ \lor \frac{1}{6} ((1 - \frac{i - 1}{n})^{2} - (1 - α)^{2}) \\ = \frac{1}{6 n} (2 + \frac{i}{n} + \frac{i - 1}{n}) \lor \frac{1}{6 n} (2 - \frac{i - 1}{n} - \frac{i}{n}) \\ + \frac{1}{6} ((α - \frac{i - 1}{n}) (2 + α + \frac{i - 1}{n})) \\ \lor \frac{1}{6} ((α - \frac{i - 1}{n}) (2 - α - \frac{i - 1}{n})) \\ = \frac{1}{6 n} (2 + \frac{i}{n} + \frac{i - 1}{n}) \\ + \frac{1}{6} ((α - \frac{i - 1}{n}) (2 + α + \frac{i - 1}{n})) \\ ⩽ \frac{4}{6 n} + \frac{4}{6 n} = \frac{4}{3 n} . \end{matrix}$

Therefore, according to the definition of the above-mentioned fuzzy distance D, it is obtained immediately that $D (Z_{n} (A), A) = \underset{α \in (0, 1]}{\lor} d_{H} (Z_{n} (A)_{α}, A_{α}) ⩽ \frac{4}{3 n} .$

Let ɛ = 0.1, if $D (Z_{n} (A), A) ⩽ \frac{4}{3 n} < ɛ = 0.1$ , then $n ⩾ \frac{4}{3 ɛ} = \frac{4}{0.3} \approx 13.33$ . So we can take n = 14.

Note 1. In fact, n = 14 means that the ordered representation of Z₁₄ (A) consists of 2n + 2 =30 ordered real numbers, and when n ⩾ 14, the Z_n (A) can approximate to A and satisfy the approximation accuracy ɛ = 0.1. If ɛ = 0.01, that means n ⩾ 134, then Z_n (A) satisfies the approximation accuracy, and the ordered representation of Z₁₃₄ (A) will consist of 270 ordered real numbers. This fact is easy to accept in theory, but in practice it can only be realized by means of computer and its MATLAB software programming. Actually, the Formula (3) derived from Hausdorff distance is relatively rough, and the error estimates obtained are naturally more rough. In practical problems, for a given accuracy ɛ > 0, the required n value will be much smaller than the above estimated value. Ordinary circumstances, the minimum n value can be determined by computer simulation experiment, and when the required precision is not high, the natural number n within 10 can meet the requirement.

4. Representation and aggregation of multi-attribute information

It is well known that triangular fuzzy numbers can be expressed by three ordered real numbers, trapezoidal fuzzy numbers can be expressed by four ordered real numbers, and the polygonal fuzzy numbers can not only be expressed by six, eight or even 2n + 2 ordered real numbers, but also can approximate the given general fuzzy numbers with arbitrary precision, and can guarantee the closeness of the arithmetic operations and the excellent properties of trapezoidal fuzzy numbers, so as to ensure less loss of useful information. Usually, when shopping websites evaluate logistics companies synthetically, they are often disturbed by internal or external factors. If they use simple triangular or trapezoidal fuzzy numbers to express fuzzy information, they often lose some useful information, which leads to unreliable evaluation results or inaccurate evaluation. However, broken-line fuzzy numbers can make up for these shortcomings. Especially, the description for multi-attribute information by the polygonal fuzzy number is more comprehensive and specific. In fact, for a given natural number n, it is not difficult to transform the general fuzzy number into an ordered representation of n-polygonal fuzzy number (See Example 2), and then transform the complex fuzzy number operation which originally relied on the Zadeh extension principle to a simple linear operation (Definition 2.2). See [22 , 25]. Consequently, it is more practical and in line with the operators’ vital interests to evaluate the multi-attribute index system of logistics companies by using the ordered representation of polygonal fuzzy numbers.

Usually, a shopping website platform evaluates logistics companies synthetically through multiple indicators such as transportation cost, transportation efficiency, transportation reliability, service attitude and commodity throughput, and then evaluates and chooses satisfactory logistics companies. Actually, the total volume of goods transported by each logistics company is limited by the company’s own size and operational capacity. For example, if there are too few goods transported every day, the transportation cost is low, but the profit is small, so logistics companies and shopping websites are certainly not satisfied. If there are too many goods transported every day, the transportation cost will increase naturally, and even the logistics companies can not afford to exceed their own carrying capacity, which will lead to the decline of transportation efficiency. This situation is certainly unsatisfactory for shopping websites. Therefore, under the premise of not expanding reproduction, only the appropriate transportation cost of logistics companies can make logistics companies satisfied. However, how to make the transportation capacity of logistics companies “appropriate” is not a simple problem.

For this reason, we can quantitatively study this problem by means of mathematical modeling. For example, we use smooth Gaussian fuzzy number to describe the satisfaction degree of shopping websites to logistics companies’ transportation costs, and assume that road transportation costs mainly include vehicle attrition, toll collection, fuel consumption and labor cost. If a shopping website uses the Gaussian fuzzy number G to evaluate the logistics cost of a logistics company, and the Gaussian membership function is $G (x) = e^{- \frac{(x - 4)^{2}}{6}}, 0 ⩽ x ⩽ 8 .$

Here, the x-axis represents the transportation cost of the logistics company; the y-axis represents the satisfaction degree of the shopping website. When the transportation cost is less than RMB 40,000/day, the satisfaction degree of logistics company increases with the increase of transportation cost; when the transportation cost reaches RMB 40,000/day, the satisfaction degree reaches the highest level; when the daily transportation cost exceeds RMB 40,000/day, the satisfaction degree decreases with the increase of transportation cost; when the transportation cost is RMB 80,000/day, the satisfaction degree is G (8) = G (0) = e^-8/-3 ≈ 0.0695; when the transportation cost exceeds RMB 80,000/day, the satisfaction degree is 0.

For simplicity, let n = 2, 3, 4, according to Example 1 the ordered representation of Gaussian membership function G (x) = e^{-(x-4)²/- 6} can be obtained as follows: ${\begin{matrix} Z_{2} (G) = (0, 4 - \sqrt{6 ln 2}, 4, 4, 4 + \sqrt{6 ln 2}, 8) \approx (0, 1.9607, 4, 4, 6.0393, 8) \\ Z_{3} (G) = (0, 4 - \sqrt{6 ln 3}, 4 - \sqrt{6 ln \frac{3}{2}}, 4, 4, 4 + \sqrt{6 ln \frac{3}{2}}, 4 + \sqrt{6 ln 3}, 8) \\ \approx (0, 1.4326, 2.4402, 4, 4, 5.5597, 6.5674, 8) \\ Z_{4} (G) = (0, 4 - 2 \sqrt{3 ln 2}, 4 - \sqrt{6 ln 2}, 4 - \sqrt{6 ln \frac{4}{3}}, 4, 4, 4 + \sqrt{6 ln \frac{4}{3}}, 4 + \sqrt{6 ln 2}, 4 + 2 \sqrt{3 ln 2}, 8) \\ \approx (0, 1.1159, 1.9607, 2.6862, 4, 4, 5.3138, 6.0393, 6.8840, 8) \end{matrix},$

The images of the membership function Z_n (G) (x) corresponding to the n-polygonal fuzzy number can also be obtained as shown in Figs. 6–8.

Fig. 6

Images of G (x) and Z₂ (G) (x).

Fig. 7

Images of G (x) and Z₃ (G) (x).

Fig. 8

Images of G (x) and Z₄ (G) (x).

From Figs. 6– 8, it is easy to see that with the increase of n value, the ability of Z_n (G) (x) to approximate the Gauss function G (x) is stronger, but its complexity is also increased. Therefore, the index information can not only reduce the loss of useful information based on the ordered representation of fuzzy numbers, but also enhance the comprehensiveness and accuracy of describing fuzzy information. This is because the ordered representation of a fuzzy number can be 2n + 2 ordered real numbers. To describe the information of an index, the traditional triangular or trapezoidal fuzzy numbers can only be represented by 3 or 4 ordered real numbers, and they are only special cases of the polygonal fuzzy numbers.

Let’s take Fig. 7 as an example (n = 3) to illustrate that the ordered representation Z₃ (G) describes the satisfaction degree of shopping websites with transportation costs. The ordered representation can be abbreviated to Z₃ (G) = (0, 1.43, 2.44, 4, 4, 5.56, 6.57, 8), which indicates the satisfaction degree of shopping websites with transportation costs. For example, when the daily transportation cost variable x of logistics companies is [1.43, 2.44], the satisfaction function is Z₃ (G) (x) =0.3307x - 0.1386. In particular, its satisfaction is Z₃ (G) (2) =0.5228 when x = 2. Similarly, if the transportation cost x is graded continuously according to the arrow trend: 0 → 1.43 → 2.44 → 4 =4 → 5.56 → 6 . 57 → 8 (RMB 10.000), the shopping network will be satisfied, and the satisfaction of stations to transport costs can be expressed by a linear function Z₃ (G) (x), i.e. $\begin{matrix} Z_{3} (G) (x) \\ = {\begin{matrix} 0.1841 x + 0.0695, 0 ⩽ x < 1.43 \\ 0.3307 x - 0.1386, 1.43 ⩽ x < 2.44 \\ 0.2193 x + 0.1453, 2.44 ⩽ x ⩽ 4 \\ - 0.2193 x + 1.8547, 4 ⩽ x ⩽ 5.56 \\ - 0.3308 x + 2.5059, 5.56 < x ⩽ 6.57 \\ - 0.1845 x + 1.5455, 6.57 < x ⩽ 8 \end{matrix} . \end{matrix}$

In fact, it is very easy to obtain an analytic expression of a linear function Z₃ (G) (x) based on the two point equation. in which the ordered representation Z₃ (G) and the linear function Z₃ (G) (x) are only two different forms, and they are mutually unique. Obviously, the ordered representation is much simpler than the linear function. Hence, the satisfaction of transportation cost described by the ordered representation Z₃ (G) is not only closer to the smooth Gaussian membership function G (x), but also more accurate than the triangular or trapezoidal fuzzy number. Of course, in practical problems we do not have to divide the points equally along the y-axis points $\frac{1}{3}$ and $\frac{2}{3}$ , and also transportation cost along the x-axis.

For example, we choose the Gauss membership function $G (x) = exp (- \frac{(x - 5)^{2}}{25})$ , let n = 3, and the transportation cost trend 0 → 2 →4 → 5 =5 → 6 →8 → 10 is regarded as an ordered representation Z₃ (G) = (0, 2, 4, 5, 5, 6, 8, 10), then the non-equidistant subdivision points on the y-axis can reversely be calculated by the Gaussian function G (x). See Figs. 9-10 below.

Fig. 9

Non-equidistant subdivision determined by the ordered representation on when n = 2.

Fig. 10

Non-equidistant subdivision determined by the ordered representation on when n = 3.

Because the ordered representation of a polygonal fuzzy number can describe in more detail the satisfaction degree of shopping websites with multi-index information such as transportation cost, transportation efficiency and service attitude of some logistics companies. For this reason, the aggregation method of multiple weighted arithmetic average operators is proposed below.

Definition 4.1. For a given $n \in ℕ$ , let polygonal fuzzy numbers $A_{j} \in F_{0}^{t n} (ℝ)$ , j = 1, 2, ·· · , l, and the ordered representations $A_{j} = (a_{0}^{1} (j), a_{1}^{1} (j), \cdot \cdot \cdot, a_{n}^{1} (j), a_{n}^{2} (j), \cdot \cdot \cdot, a_{1}^{2} (j), a_{0}^{2} (j))$ , the weight of each A_j is ω_j ∈ [0, 1], and satisfies $\sum_{j = 1}^{l} ω_{j} = 1$ . Let $W (A_{1}, A_{2}, \cdot \cdot \cdot, A_{l}) = \sum_{j = 1}^{l} ω_{j} A_{j}$ , then W is called a weighted arithmetic average operator of polygonal fuzzy numbers.

According to the addition and multiplication operations of polygonal fuzzy numbers given in Definition 2.1, it is not difficult to obtain the value of weighted arithmetic average operator after aggregation, which can be expressed as

$\begin{matrix} W (A_{1}, A_{2}, \cdot \cdot \cdot, A_{l}) \\ = (\sum_{j = 1}^{l} ω_{j} a_{0}^{1} (j), \sum_{j = 1}^{l} ω_{j} a_{1}^{1} (j), \cdot \cdot \cdot, \\ \sum_{j = 1}^{l} ω_{j} a_{n}^{1} (j), \sum_{j = 1}^{l} ω_{j} a_{n}^{2} (j), \cdot \cdot \cdot, \\ \sum_{j = 1}^{l} ω_{j} a_{1}^{2} (j), \sum_{j = 1}^{l} ω_{j} a_{0}^{2} (j)), \end{matrix}$ (3) where the each component element $\sum_{j = 1}^{l} ω_{j} a_{i}^{q} (j)$ at the right end of Formula (3) keeps increasing in order. Hence, the aggregation value $W (A_{1}, A_{2}, \cdot \cdot \cdot, A_{l}) \in F_{0}^{t n} (ℝ)$ .

Example 3. Let n = 3, given three ordered representations $A_{1} = (- 3, - \frac{5}{2}, - 1, - \frac{1}{2}, \frac{1}{2}, 3, \frac{7}{2}, 6)$ , $A_{2} = (- 2, - \frac{3}{2}, 1, 2, \frac{7}{2}, 4, 6, 7)$ and $A_{3} = (- \frac{3}{2}, - \frac{4}{3}, - \frac{5}{6}, 0, 2, \frac{28}{9}, \frac{34}{9}, 4)$ , and the corresponding weight are ω₁ = 0.3, ω₂ = 0.5 and ω₃ = 0.2.

Obviously, according to Formula (3), the aggregation value W (A₁, A₂, A₃) of three ordered representations can be directly calculated as $\begin{matrix} W (A_{1}, A_{2}, A_{3}) \\ = (\sum_{j = 1}^{3} ω_{j} a_{0}^{1} (j), \sum_{j = 1}^{3} ω_{j} a_{1}^{1} (j), \sum_{j = 1}^{3} ω_{j} a_{n}^{1} (j), \\ \sum_{j = 1}^{3} ω_{j} a_{n}^{2} (j), \sum_{j = 1}^{3} ω_{j} a_{1}^{2} (j), \sum_{j = 1}^{3} ω_{j} a_{0}^{2} (j)) \\ = (- \frac{11}{5}, - \frac{53}{30}, \frac{8}{15}, \frac{17}{20}, \frac{23}{10}, \frac{317}{90}, \frac{173}{36}, \frac{61}{10}) . \end{matrix}$

5. Decision-making method

At present, logistics companies are the key departments of commodity circulation in some developed countries, and the transportation cost is a core index of the whole logistics activity. Hence, how to reduce the transportation cost is the most important task for logistics companies. Of course in the increasingly competitive market economy, how to maximize the company’s profits and efficiency are the most concerned issue of all entrepreneurs. In addition, for consumers, how to aggregate multiple evaluation index values of logistics companies into a comprehensive evaluation value through a suitable mathematical model is a most concerned issue. Especially for some large-scale decision-making problems, multiple decision makers are often required to participate, which is also a popular decision-making method at present.

Let {A₁, A₂, ·· · , A_1ptm} be a set of all alternative schemes for evaluating a logistics company on a shopping website, in which each scheme A_i has l attributes indexes to be evaluated, and the attribute index set is {p₁, p₂, ·· · , p_l}, and the corresponding weight vector of the attribute set is {ω₁, ω₂, ·· · , ω_l}. Without loss of generality, let the evaluation index value of each alternative scheme A_i (i = 1, 2, ·· · , m) for the attribute p_j (j = 1, 2, ·· · , l) may be expressed in the form of the polygonal fuzzy number as follows:

$\begin{matrix} M_{ij} = & (m_{0 i}^{1} (p_{j}), m_{1 i}^{1} (p_{j}), \cdot \cdot \cdot, m_{n i}^{1} (p_{j}), m_{n i}^{2} (p_{j}), \\ \cdot \cdot \cdot, m_{1 i}^{2} (p_{j}), m_{0 i}^{2} (p_{j})), \end{matrix}$ (4) where the each component element on the right satisfies $\begin{matrix} 0 & ⩽ m_{0 i}^{1} (p_{j}) ⩽ m_{1 i}^{1} (p_{j}) ⩽ \cdot \cdot \cdot ⩽ m_{ni}^{1} (p_{j}) \\ ⩽ m_{ni}^{2} (p_{j}) ⩽ \cdot \cdot \cdot ⩽ m_{1 i}^{2} (p_{j}) ⩽ m_{0 i}^{2} (p_{j}) . \end{matrix}$

It is not difficult to obtain the polygonal fuzzy decision matrix M by the transformation criteria of some given linguistic variables and n-polygonal fuzzy numbers, and this concrete decision matrix M = (M_ij (p_j)) _m ×l is expressed as

$M = (\begin{matrix} M_{11} (p_{1}) & M_{12} (p_{2}) & \cdot \cdot \cdot & M_{1 j} (p_{j}) & \cdot \cdot \cdot & M_{1 l} (p_{l}) \\ M_{21} (p_{1}) & M_{22} (p_{2}) & \cdot \cdot \cdot & M_{2 j} (p_{j}) & \cdot \cdot \cdot & M_{2 l} (p_{l}) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ M_{i 1} (p_{1}) & M_{i 2} (p_{2}) & \cdot \cdot \cdot & M_{ij} (p_{j}) & \cdot \cdot \cdot & M_{il} (p_{l}) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ M_{m 1} (p_{1}) & M_{m 2} (p_{2}) & \dots & M_{mj} (p_{j}) & \dots & M_{m l} (p_{l}) \end{matrix}),$

where each attribute p_j occupies the weight ω_j ∈ [0, 1], and it satisfies $\sum_{j = 1}^{l} ω_{j} = 1$ , and the each element M_ij (p_j) is expressed as Formula (4).

Generally speaking, the attributes of schemes are divided into cost attributes and benefit attributes, in which the larger the value of benefit attributes, the higher the performance value obtained; the smaller the value of cost attributes, the higher the performance value.

Next, a new decision-making method from the alternative schemes according to the attribute value of each scheme will be given as follows:

Step 1. Normalized decision information. To eliminate the potential impact of different physical dimensions on decision results, the following Formulas (5)-(6) can be used to normalize the polygonal fuzzy decision matrix (M_ij (p_j)) _m ×l to (s_ij) _m ×l, where the representative element s_ij (n-polygonal fuzzy number) is expressed as $\begin{matrix} s_{ij} = & (s_{0 i}^{1} (p_{j}), s_{1 i}^{1} (p_{j}), \cdot \cdot \cdot, s_{ni}^{1} (p_{j}), s_{ni}^{2} (p_{j}), \cdot \cdot \cdot, \\ s_{1 i}^{2} (p_{j}), s_{0 i}^{2} (p_{j})) . \end{matrix}$

For the cost attribute, let $s_{ki}^{q} (p_{j}) = {\begin{matrix} \frac{{max}_{1 ⩽ i ⩽ l} m_{0 i}^{2} (p_{j}) - m_{ki}^{2} (p_{j})}{{max}_{1 ⩽ i ⩽ l} m_{0 i}^{2} (p_{j}) - {min}_{1 ⩽ i ⩽ l} m_{0 i}^{1} (p_{j})}, \\ q = 1; k = 0, 1, 2, \cdot \cdot \cdot, n \\ \frac{{max}_{1 ⩽ i ⩽ l} m_{0 i}^{2} (p_{j}) - m_{ki}^{1} (p_{j})}{{max}_{1 ⩽ i ⩽ l} m_{0 i}^{2} (p_{j}) - {min}_{1 ⩽ j ⩽ l} m_{0 i}^{1} (p_{j})}, \\ q = 2; k = n, n - 1, \cdot \cdot \cdot, 2, 1, 0 \end{matrix}$ (5)

For the benefit attribute, let $s_{ki}^{q} (p_{j}) = {\begin{matrix} \frac{m_{ki}^{1} (p_{j}) - {min}_{1 ⩽ i ⩽ l} m_{0 i}^{1} (p_{j})}{{max}_{1 ⩽ i ⩽ l} m_{0 i}^{2} (p_{j}) - {min}_{1 ⩽ i ⩽ l} m_{0 i}^{1} (p_{j})}, \\ q = 1; k = 0, 1, 2, \cdot \cdot \cdot, n \\ \frac{m_{ki}^{2} (p_{j}) - {min}_{1 ⩽ j ⩽ l} m_{0 i}^{1} (p_{j})}{{max}_{1 ⩽ i ⩽ l} m_{0 i}^{2} (p_{j}) - {min}_{1 ⩽ i ⩽ l} m_{0 i}^{1} (p_{j})}, \\ q = 2; k = n, n - 1, \cdot \cdot \cdot, 2, 1, 0 \end{matrix}$ (6)

Obviously, the denominator $max_{1 ⩽ i ⩽ l} m_{0 i}^{2} (p_{j}) - min_{1 ⩽ i ⩽ l} m_{0 i}^{1} (p_{j})$ (greater than 0) in the above expression is relatively fixed, then the value of each component element $s_{ki}^{q} (p_{j})$ depends mainly on its numerator part, and it can be concluded that each component $s_{ki}^{q} (p_{j})$ still satisfies $\begin{matrix} 0 & ⩽ s_{0 i}^{1} (p_{j}) ⩽ s_{1 i}^{1} (p_{j}) ⩽ \cdot \cdot \cdot ⩽ s_{ni}^{1} (p_{j}) ⩽ s_{ni}^{2} (p_{j}) \\ ⩽ \cdot \cdot \cdot ⩽ s_{1 i}^{2} (p_{j}) ⩽ s_{0 i}^{2} (p_{j}) . \end{matrix}$

That is to say, the each representative element s_ij of the normalized polygonal fuzzy decision matrix (s_ij) _m ×l is still an n-polygonal fuzzy number.

Step 2. According to Formula (3), all elements of the line i of the polygonal decision matrix (s_ij) _m ×l (some ordered representations) s_ij (j = 1, 2, ·· · , l) are aggregated into a comprehensive evaluation scheme, which is expressed as $S_{i} = (s_{i 1}, s_{i 2}, \cdot \cdot \cdot, s_{il}), i = 1, 2, \cdot \cdot \cdot, m,$ where each component element s_ij is an n-polygonal fuzzy number, and it can be expressed as $\begin{matrix} s_{ij} = & (s_{0 i}^{1} (p_{j}), s_{1 i}^{1} (p_{j}), \cdot \cdot \cdot, s_{ni}^{1} (p_{j}), s_{ni}^{2} (p_{j}), \cdot \cdot \cdot, \\ s_{1 i}^{2} (p_{j}), s_{0 i}^{2} (p_{j})) . \end{matrix}$

Step 3. Calculating the positive (or negative) ideal schemes $G_{j}^{+}$ (or $G_{j}^{-}$ ) of the polygonal decision matrix (s_ij) _m ×l and their attribute matrices, respectively. Let

${\begin{matrix} G_{j}^{+} = (max_{1 ⩽ i ⩽ m} s_{0 i}^{1} (p_{j}), max_{1 ⩽ i ⩽ m} s_{1 i}^{1} (p_{j}), \cdot \cdot \cdot, max_{1 ⩽ i ⩽ m} s_{ni}^{1} (p_{j}), max_{1 ⩽ i ⩽ m} s_{ni}^{2} (p_{j}), \cdot \cdot \cdot, max_{1 ⩽ i ⩽ m} s_{1 i}^{2} (p_{j}), max_{1 ⩽ i ⩽ m} s_{0 i}^{2} (p_{j})) \\ G_{j}^{-} = (min_{1 ⩽ i ⩽ m} s_{0 i}^{1} (p_{j}), min_{1 ⩽ i ⩽ m} s_{1 i}^{1} (p_{j}), \cdot \cdot \cdot, min_{1 ⩽ i ⩽ m} s_{ni}^{1} (p_{j}), min_{1 ⩽ i ⩽ m} s_{ni}^{2} (p_{j}), \cdot \cdot \cdot, min_{1 ⩽ i ⩽ m} s_{1 i}^{2} (p_{j}), min_{1 ⩽ i ⩽ m} s_{0 i}^{2} (p_{j})) \end{matrix};$

where j = 1, 2, ·· · , l. Let $G^{+} = (G_{1}^{+}, G_{2}^{+}, \cdot \cdot \cdot,$ $G_{l}^{+})^{T}$ , $G^{-} = (G_{1}^{-}, G_{2}^{-}, \cdot \cdot \cdot, G_{l}^{-})^{T}$ , then G⁺ and G^- are called the positive or negative ideal solution of the polygonal decision matrix (s_ij) _m ×l, respectively. The corresponding matrices G⁺ and G^- are the attribute matrix.

Step 4. Calculating the weighted Euclidean distances $D_{i}^{+} (S_{i}, G^{+})$ and $D_{i}^{-} (S_{i}, G^{-})$ between each comprehensive evaluation scheme S_i and the positive (or negative) ideal solution $G_{j}^{+}$ (or $G_{j}^{-}$ ), and $D_{i}^{+} (S_{i}, G^{+}) = \sum_{j = 1}^{l} ω_{j} E_{3} (s_{ij}, G_{j}^{+})$ and $D_{i}^{-} (S_{i}, G^{-}) = \sum_{j = 1}^{l} ω_{j} E_{3} (s_{ij}, G_{j}^{-})$ , i = 1, 2, ·· · , m, where the ω_j is the weight of the attribute p_j.

Step 5. Let $D_{i}^{*} = D_{i}^{+} (S_{i}, G^{+}) / (D_{i}^{+} (S_{i}, G^{+}) + D_{i}^{-} (S_{i}, G^{-})), i = 1, 2, \cdot \cdot \cdot, m$ , then $D_{i}^{*}$ is called a relative closeness degree of the comprehensive scheme S_i, and according to the criterion that the smaller the value of $D_{i}^{+} (S_{i}, G^{+})$ or the larger the value of $D_{i}^{-} (S_{i}, G^{-})$ , then the better the corresponding scheme is. That is to say, a new ranking method is given by $D_{i}^{*}$ .

In fact, the relative closeness degree determined in Step 5 is used as an evaluation criterion to test the advantages and disadvantages of the alternative schemes. That is, the closer the alternative schemes are to the positive ideal solution, the farther they are from the negative ideal solution, the better the alternative schemes are. In other words, the greater the relative closeness degree $D_{i}^{*}$ , the better the alternative schemes are. On the contrary, the smaller the relative closeness degree $D_{i}^{*}$ , the worse the alternative schemes are. Therefore, it is effective to decide the alternatives by ranking the relative closeness degree $D_{i}^{*}$ .

Next, the validity of the proposed method will be illustrated by multi-attribute evaluation of logistics companies through shopping websites.

6. Example analysis

A shopping website intends to conduct a comprehensive evaluation of logistics companies in order to select the most satisfactory partners. After primary selection, there are three alternative logistics companies A_i (i = 1, 2, 3), and the alternative set is {A₁, A₂, A₃}. the website intends to evaluate the above alternative logistics companies on the basis of four attribute indicators information {p₁, p₂, p₃, p₄}, and use the p₁ to express transportation cost, p₂- transportation efficiency, p₃- service attitude and p₄- commodity throughput. Assuming that {p₁, p₂, p₃} is a benefit attribute, and p₄ is a cost attribute, the weights of the four attributes indexes are ω₁ = 0.22, ω₂ = 0.36, ω₃ = 0.18 and ω₄ = 0.24, and the given parameters are n = m = 3 and l = 4. Try to choose the best logistics company to distribute goods through the proposed method.

Suppose that the shopping website intends to evaluate the satisfaction of logistics companies in terms of transportation cost, transportation efficiency, service attitude and commodity throughput. According to Section 4, the method of an ordered representation of transportation cost is given by introducing the Gauss function, we can similarly obtain the ordered representation of other three indicators information such as transportation efficiency, service attitude and commodity throughput. For simplicity, when n = 3, only the conversion criteria between the fuzzy linguistic variables and the ordered representation of transportation cost of logistics companies will be directly given (others are omitted). See the following Table 1 for specific conversion data.

Table 1
Conversion criteria between fuzzy linguistic variables and ordered representations of transportation cost

Fuzzy linguistic variables The ordered representations of logistics transportation costs/day

Unit: RMB 10,000/day

Very poor Δ₁ = (0.1, 0.5, 1.1, 1.3, 1.7, 1.9, 2.5, 3.0)

Poor Δ₂ = (0.3, 0.9, 1.1, 1.5, 1.8, 2.2, 3.0, 3.8)

Relatively poor Δ₃ = (1.5, 1.8, 2.3, 2.7, 3.0, 3.4, 3.9, 4.2)

Medium Δ₄ = (2.1, 2.8, 3.1, 3.4, 3.8, 4.1, 4.4, 5.1)

Better Δ₅ = (4.5, 4.7, 5.2, 5.5, 5.9, 6.2, 6.7, 6.9)

Good Δ₆ = (6.1, 6.8, 7.2, 7.7, 8.0, 8.5, 8.9, 9.6)

Very good Δ₇ = (7.9, 8.2, 8.4, 8.7, 9.0, 9.3, 9.5, 9.8)

Fuzzy linguistic variables	The ordered representations of logistics transportation costs/day
Very poor	Δ₁ = (0.1, 0.5, 1.1, 1.3, 1.7, 1.9, 2.5, 3.0)
Poor	Δ₂ = (0.3, 0.9, 1.1, 1.5, 1.8, 2.2, 3.0, 3.8)
Relatively poor	Δ₃ = (1.5, 1.8, 2.3, 2.7, 3.0, 3.4, 3.9, 4.2)
Medium	Δ₄ = (2.1, 2.8, 3.1, 3.4, 3.8, 4.1, 4.4, 5.1)
Better	Δ₅ = (4.5, 4.7, 5.2, 5.5, 5.9, 6.2, 6.7, 6.9)
Good	Δ₆ = (6.1, 6.8, 7.2, 7.7, 8.0, 8.5, 8.9, 9.6)
Very good	Δ₇ = (7.9, 8.2, 8.4, 8.7, 9.0, 9.3, 9.5, 9.8)

According to the conversion criteria of the above-mentioned fuzzy linguistic variables, four attribute indices p_j (j = 1, 2, 3, 4) of three logistics companies {A₁, A₂, A₃} are evaluated by random selection of group customers by experts. The evaluation results are expressed as the initial polygonal fuzzy decision matrix (M_ij) _3 ×4 according to the arithmetic operations of the polygonal fuzzy number. Then the polygonal decision matrix (M_ij) _3 ×4 is normalized by Steps 1-2, its main process can be expressed as follows: $\begin{matrix} p_{1} p_{2} p_{3} p_{4} \begin{matrix} A_{1} \\ A_{2} \\ A_{3} \end{matrix} | (\begin{matrix} M_{11} & M_{12} & M_{13} & M_{14} \\ M_{21} & M_{22} & M_{23} & M_{23} \\ M_{31} & M_{32} & M_{33} & M_{34} \end{matrix}) \\ \underset{\to}{Normalization} (\begin{matrix} s_{11} & s_{12} & s_{13} & s_{14} \\ s_{21} & s_{22} & s_{23} & s_{24} \\ s_{31} & s_{32} & s_{33} & s_{34} \end{matrix}) = (s_{ij})_{3 \times 4}, \end{matrix}$ where each element s_ij of the normalized weighted decision matrix (s_ij) _3 ×4 is calculated according to Formulas (5)-(6) in the Step 1 of the algorithm, and all of their elements (3-polygonal fuzzy number) can be expressed specifically as

$\begin{matrix} {\begin{matrix} s_{11} = (0.01, 0.12, 0.18, 0.3, 0.34, 0.35, 0.72, 0.78) \\ s_{21} = (0.06, 0.13, 0.21, 0.3, 0.36, 0.39, 0.68, 0.76) \\ s_{31} = (0 . 12, 0.18, 0.36, 0.48, 0.67, 0.87, 0.94, 1) \end{matrix}, \\ {\begin{matrix} s_{12} = (0.01, 0.20, 0.24, 0.35, 0.47, 0.59, 0.66, 0.69) \\ s_{22} = (0.36, 0.57, 0.62, 0.74, 0.81, 0.93, 0.96, 1.00) \\ s_{32} = (0.21, 0.34, 0.48, 0.76, 0.79, 0.85, 0.89, 0.95) \end{matrix}, \\ {\begin{matrix} s_{13} = (0.66, 0.71, 0.76, 0.84, 0.92, 0.95, 0.98, 1 . 00) \\ s_{23} = (0.01, 0.13, 0.17, 0.23, 0.25, 0.31, 0.38, 0.46) \\ s_{33} = (0.68, 0.72, 0.76, 0.84, 0.92, 0.93, 0.98, 1 . 00) \end{matrix}, \\ {\begin{matrix} s_{14} = (0, 0.12, 0.14, 0.19, 0.21, 0.26, 0.33, 0.37) \\ s_{24} = (0.73, 0.80, 0.82, 0.87, 0.91, 0.93, 0.95, 1) \\ s_{34} = (0.82, 0.84, 0.88, 0.91, 0.93, 0.96, 0.98, 1) \end{matrix} . \end{matrix}$

According to Step 3, the positive ideal solution ${G′}^{+} = (G_{1}^{+}, G_{2}^{+}, G_{3}^{+}, G_{4}^{+})^{T}$ and the negative ideal solution $G^{-} = (G_{1}^{-}, G_{2}^{-}, G_{3}^{-}, G_{4}^{-})^{T}$ of the polygonal matrix (s_ij) _3×4 can be calculated, where the component elements $G_{j}^{+}$ and $G_{j}^{-}$ are calculated as shown in Table 2.

Table 2

Calculating results of positive (negative) ideal solution G⁺ (G^-) of the matrix (s_ij) _3×4

G ⁺	G ^-
$G_{1}^{+} = (0.03, 0.34, 0.39, 0.41, 0.71, 0.88, 0.94, 1)$	$G_{1}^{-} = (0.03, 0.23, 0.30, 0.39, 0.45, 0.57, 0.72, 0.79)$
$G_{2}^{+} = (0.47, 0.55, 0.62, 0.74, 0.82, 0.94, 0.98, 1)$	$G_{2}^{-} = (0.01, 0.20, 0.26, 0.37, 0.47, 0.59, 0.64, 0.68)$
$G_{3}^{+} = (0.68, 0.70, 0.76, 0.85, 0.93, 0.94, 0.99, 1)$	$G_{3}^{-} = (0.01, 0.13, 0.16, 0.22, 0.25, 0.30, 0.39, 0.42)$
$G_{4}^{+} = (0.83, 0.85, 0.89, 0.92, 0.94, 0.97, 0.99, 1)$	$G_{4}^{-} = (0.01, 0.11, 0.14, 0.19, 0.21, 0.26, 0.33, 0.36)$

This moment, if the calculation is based on the method given in Ref. [20], a new transformation criteria between fuzzy linguistic variables and trapezoidal fuzzy numbers need be given. That is to say, the elements 2-3 and 6-7 should be removed in the original ordered representation, such as, Δ₁ = (1, 5, 11 – mdash mdash, 13, 17, 19, 25 mdash mdash mdash, 30) modify them into Δ₁ = (1, 13, 17, 30), other original ordered representation Δ₂ to Δ₇ are also modified in a similar approach. Therefore, the normalized trapezoidal decision matrix $(s_{ij}^{'})_{3 \times 4}$ is obtained as follows:

$(s_{ij}^{'})_{3 \times 4} = (\begin{matrix} (0.00, 0.21, 0.24, 0.55) & (0.01, 0.23, 0.31, 0.45) & (0.19, 0.24, 0.27, 0.29) & (0.00, 0.25, 0.27, 0.48) \\ (0.04, 0.20, 0.24, 0.51) & (0.12, 0.25, 0.28, 0.34) & (0.01, 0.24, 0.26, 0.48) & (0.21, 0.25, 0.26, 0.28) \\ (0.05, 0.21, 0.29, 0.44) & (0.08, 0.28, 0.29, 0.35) & (0.19, 0.24, 0.28, 0.29) & (0.22, 0.25, 0.26, 0.27) \end{matrix}) .$

From this, the component elements ${G′}_{j}^{+}$ and ${G′}_{j}^{-}$ of the positive ideal solution scheme ${G′}^{+} = ({G′}_{1}^{+}, {G′}_{2}^{+}, {G′}_{3}^{+}, {G′}_{4}^{+})^{T}$ and negative ideal solution scheme ${G′}^{-} = ({G′}_{1}^{-}, {G′}_{2}^{-}, {G′}_{3}^{-}, {G′}_{4}^{-})^{T}$ of the trapezoidal decision matrix $(s_{ij}^{'})_{3 \times 4}$ are calculated as shown in Table 3.

Table 3

Calculating results of positive (negative) ideal solution G⁺ (G^–) of the matrix $(s_{ij}^{'})_{3 \times 4}$

${G^{'}}^{+}$	${G^{'}}^{-}$
${G^{'}}_{1}^{+}$ = (0.03, 0.38, 0.74, 0.89)	${G^{'}}_{1}^{+}$ = (0.02, 0.37, 0.46, 0.82)
${G^{'}}_{1}^{+}$ = (0.45, 0.70, 0.85, 0.93)	${G^{'}}_{1}^{+}$ = (0.01, 0.33, 0.51, 0.73)
${G^{'}}_{1}^{+}$ = (0.63, 0.79, 0.92, 0.97)	${G^{'}}_{1}^{+}$ = (0.03, 0.20, 0.27, 0.46)
${G^{'}}_{1}^{+}$ = (0.78, 0.90, 0.95, 0.98)	${G^{'}}_{1}^{+}$ = (0.00, 0.22, 0.34, 0.43)

According to Step 4, the comprehensive evaluation S_i = (s_i1, s_i2, s_i3) and $S_{i}^{'} = (s_{i 1}^{'}, s_{i 2}^{'}, s_{i 3}^{'})$ of 3-polygonal decision matrix (s_ij) _3×4 and the trapezoidal decision matrix $(s_{ij}^{'})_{3 \times 4}$ in Ref. [20] are calculated, respectively. Then the weighted Euclidean distances $D_{i}^{+} (S_{i}, G^{+})$ and $D_{i}^{-} (S_{i}, G^{-})$ based on the positive (or negative) ideal solutions G⁺ (or G^-) can be obtained. Here, the weights are calculated by ω = (0.22, 0.36, 0.18, 0.24), and then the comparison value of the relative closeness degrees $D_{i}^{*}$ and ${D′}_{i}^{*}$ are obtained by Step 5 as shown in Table 4.

Table 4

Comparison of two weighted Euclidean distances and their relative closeness degrees

I = 1,2,3	$D_{i}^{+} (S_{i}, G^{+})$	$D_{i}^{-} (S_{i}, G^{-})$	$D_{i}^{*}$	$D_{i′}^{+} (S_{i}^{'}, G^{+})$	$D_{i′}^{-} (S_{i}^{'}, G^{-})$	${D′}_{i}^{*}$
A ₁	$D_{1}^{+} (S_{1}, G^{+}) = 4.93$	$D_{1}^{-} (S_{1}, G^{-}) = 2.84$	$D_{1}^{*} = 0.63$	$D_{1′}^{+} (S_{1}^{'}, G^{+}) = 3.52$	$D_{1′}^{-} (S_{1}^{'}, G^{-}) = 4.66$	0.43
A ₂	$D_{2}^{+} (S_{2}, G^{+}) = 5.52$	$D_{2}^{-} (S_{2}, G^{-}) = 2.57$	$D_{2}^{*} = 0.68$	$D_{2′}^{+} (S_{2}^{'}, G^{+}) = 3.84$	$D_{2′}^{-} (S_{2}^{'}, G^{-}) = 3.02$	0.56
A ₃	$D_{3}^{+} (S_{3}, G^{+}) = 5.44$	$D_{3}^{-} (S_{3}, G^{-}) = 2.13$	$D_{3}^{*} = 0.72$	$D_{3′}^{+} (S_{3}^{'}, G^{+}) = 4.32$	$D_{3′}^{-} (S_{3}^{'}, G^{-}) = 3.39$	0.56

By comparing the data in Table 4, it is obvious that the relative closeness degree of the proposed method satisfies 0.72 > 0.68 > 0.63, that is to say, ranking of three logistics companies is $D_{3}^{*} > D_{2}^{*} > D_{1}^{*}$ according to relative closeness degree. Hence, the evaluation results of shopping websites for the three logistics companies are that A₃ is better than A₂, and A₂ is better than A₁. Of course, the third logistics company A₃ is the most satisfactory distribution company.

In fact, the weighted Euclidean distances $D_{i}^{+} (S_{i}, G^{+})$ obtained from the polygonal decision matrix is generally larger than the trapezoidal weighted distances ${D^{'}}_{i}^{+} ({S^{'}}_{i}, G^{+})$ , and $D_{i}^{-} (S_{i}, G^{-})$ is generally smaller than the trapezoidal weighted distance ${D^{'}}_{i}^{-} ({S^{'}}_{i}, G^{-})$ . As ${D^{'}}_{2}^{*} = {D^{'}}_{2}^{*} = 0.56$ , it is impossible to decide who is better fo r A₂ and A₃ according to Ref. [13], which is mainly due to the loss of information leading to the reduction of accuracy. For example, the ${D^{'}}_{2}^{*}$ ≈ 0.559767 and ${D^{'}}_{3}^{*}$ ≈ 0.560311 can be calculated by reserving six decimal places, the error is 0.000544, and ${D^{'}}_{3}^{*}$ is slightly larger than ${D^{'}}_{2}^{*}$ on the surface, but in practice, ${D^{'}}_{3}^{*}$ is slightly smaller than ${D^{'}}_{2}^{*}$ due to the interference of multiple link errors. Therefore, when the absolute error requirement is not high, the trapezoidal weighted distance proposed in [20] is difficult to make decisions on the schemes A₂ and A₃. However, if according to the proposed method in this paper, the weighted Euclidean distance can be obtained $D_{2}^{*} \approx 0.682324$ and $D_{3}^{*} \approx 0.718626$ , the error is 0.036302, and its relative error increases 66 times. Then $D_{3}^{*} > D_{2}^{*}$ , i.e., scheme A₃ is better than A₂.

At present, China’s logistics enterprises and e-commerce are becoming more and more developed and leading in the world. As the proportion of logistics cost in the total expenditure of an enterprise is second only to the cost of goods sold, reducing the logistics cost can make the enterprise occupy a larger market share in the market, so as to improve the competitiveness and sustainable development of the enterprise. Therefore, it is a well-known key problem to reduce the logistics transportation cost by scientific and reasonable logistics mode, control the distribution and packaging of goods. In this sense, the proposed decision method is a comprehensive evaluation of multi-source index information of logistics companies, it is more accurate than the decision-making method based on the triangular or trapezoidal fuzzy number. This is mainly due to the fact that polygonal fuzzy numbers can more accurately describe the fuzziness of objective things.

7. Conclusions

In fact, some important indexes to evaluate logistics companies are composed of transportation cost, transportation management fee, order processing fee, warehouse management fee, receiving and sending return fee, etc. In order to make logistics companies obtain the maximum profit and good evaluation, each of these processes must be closely coordinated and sustainable development. Otherwise, if the logistics cost is too high, the quality of goods can not be guaranteed, and the evaluation of after-sales service is poor, the logistics companies will It will make it difficult for logistics companies to survive and develop. Therefore, logistics providers should improve the logistics transportation system by adjusting the external transportation cost, reducing the logistics cost, expanding the sales volume and reducing the transportation cost, so as to make our own logistics companies occupy a place among many competitors. In this paper, a new Euclidean distance is proposed based on the orderly representation of polygonal fuzzy numbers, and a new normalization process is implemented for the polygonal decision matrix. Logistics companies are evaluated by solving the positive (negative) ideal solution and relative closeness of the decision matrix. Here, the evaluation and decision-making analysis of the important index of logistics transportation cost are mainly focused on, as for the evaluation of other indexes can be handled by adopting the similar method. Of course, the weight used in this method is still given, but in many practical problems, the weight is not known. Therefore, how to determine the weight more accurately is also one of the key issues to solve practical problems, which we need to continue to explore in depth next.

References

Kao

C.P.

, Liao

C.N.

, An evaluation approach to logistics service using fuzzy theory, quality function development and goal programming, Computers & Industrial Engineering 68 (2014), 54–64.

Wang

, Wang

J.Q.

, Zhang

H.Y.

, A likelihood-based TODIM approach based on multi-hesitant fuzzy linguistic information for evaluation in logistics outsourcing, Computers & Industrial Engineering 99 (2016), 287–299.

Cagliano

A.C.

, Marco

A.D.

, Mangano

, et al., Levers of logistics service providers’ efficiency in urban distribution, Operations Management Research 10(3-4) (2017), 104–117.

Singh

R.K.

, Gunasekaran

, Kumar

, Third party logistics (3PL) selection for cold chain management: A fuzzy AHP and fuzzy TOPSIS approach, Annals of Operations Research 267(1-2) (2018), 531–553.

Zhang

M.D.

, Huang

G.Q.

, Xu

S.X.

, et al., Optimization based transportation service trading in B2B e-commerce logistics, Journal of Intelligent Manufacturing 30(7) (2019), 2603–2619.

Bai

C.G.

, Sarkis

, Integrating and extending data and decision tools for sustainable third-party reverse logistics provider selection, Computers & Operations Research 110 (2019), 188–207.

Govindan

, Agarwal

, Darbari

J.D.

, An integrated decision making model for the selection of sustainable forward and reverse logistic providers, Annals of Operations Research 273(1-2) (2019), 607–650.

Wang

J.Q.

, Multi-criteria interval intuitionistic fuzzy decision making approach with incomplete certain information, Control and Decision 21(11) (2006), 1253–1256.

Wang

J.Q.

, Zhang

, Programming method of multi-criteria decision-making based on intuitionistic fuzzy number with incomplete certain information, Control and Decision 23(10) (2008), 1145–1148.

10.

Wang

J.Q.

, Zhang

, Multi-criteria decision-making method with incomplete certain information based on intuitionistic trapezoidal fuzzy number, Control and Decision 24(2) (2009), 226–230.

11.

Wei

G.W.

, An interval intuitionistic fuzzy number multiple attribute decision making method with incomplete weight information, Statistics and Decision 5(8) (2008), 208–211.

12.

Z.S.

, Approaches to multiple attribute decision making with intuitionistic fuzzy preference information, System Engineering Theory & Practice 27(11) (2007), 62–71.

13.

Z.S.

, Models for multiple attribute decision making with intuitionistic fuzzy information, International Journal of Uncertainty Fuzziness and Knowledge Based Systems 15(3) (2007), 285–297.

14.

Z.S.

, Yager

R.R.

, Some geometric aggregation operators based on intuitionistic fuzzy sets, International Journal of General System 35(4) (2006), 417–433.

15.

S.Z.

, Multi-person multi-attribute decision making models under intuitionistic fuzzy environment, Fuzzy Opimization Decision Making 6(3) (2007), 221–236.

16.

Z.S.

, Intuitionistic preference relations and their application in group decision making, Information Science 177(11) (2007), 2363–2379.

17.

Wei

G.W.

, Some geometric aggregation functions and their application to dynamic multiple attribute decision making in the intuitionistic fuzzy setting, International Journal of Uncertainty Fuzziness Knowledge Based Systems 17(2) (2009), 179–196.

18.

Wei

G.W.

, Some arithmetic aggregation operators with intuitionistic trapezoidal fuzzy numbers and their application to group decision making, Journal of Computer 5(3) (2010), 345–351.

19.

Wan

S.P.

, Dong

J.Y.

, Method of intuitionistic trapezoidal fuzzy number for multi-attribute group decision, Control and Decision 25(5) (2010), 773–776.

20.

Lan

, Fan

J.L.

, New complete metric on trapezoidal fuzzy numbers and its application to multi-criteria decision-making, Chinese Journal of Engineering Mathematics 27(6) (2010), 1001–1008.

21.

Wang

G.J.

, Li

X.P.

, Universal approximation of polygonal fuzzy neural networks in sense of K-integral norms, Science China. Information Sciences 54(11) (2011), 2307–2323.

22.

X.P.

, Li

, The structure and realization of a polygonal fuzzy neural network, International Journal of Machine Learning and Cybernetics 7(3) (2016), 375–389.

23.

Wang

G.J.

, Duan

, TOPSIS approach for multi-attribute decision making problems based on}-intuitionistic polygonal fuzzy sets description, Computers & Industrial Engineering 124(10) (2018), 573–581.

24.

Duan

, Wang

G.J.

, A FCM clustering algorithm based on polygonal fuzzy numbers to describe multiple attribute index information, Systems Engineering Theory & Practice 36(12) (2016), 3220–3228.

25.

Wang

G.J.

, Polygonal fuzzy neural network and fuzzy system approximation, Beijing: Science Press 2017.

Fuzzy linguistic variables	The ordered representations of logistics transportation costs/day
	Unit: RMB 10,000/day
Very poor	Δ₁ = (0.1, 0.5, 1.1, 1.3, 1.7, 1.9, 2.5, 3.0)
Poor	Δ₂ = (0.3, 0.9, 1.1, 1.5, 1.8, 2.2, 3.0, 3.8)
Relatively poor	Δ₃ = (1.5, 1.8, 2.3, 2.7, 3.0, 3.4, 3.9, 4.2)
Medium	Δ₄ = (2.1, 2.8, 3.1, 3.4, 3.8, 4.1, 4.4, 5.1)
Better	Δ₅ = (4.5, 4.7, 5.2, 5.5, 5.9, 6.2, 6.7, 6.9)
Good	Δ₆ = (6.1, 6.8, 7.2, 7.7, 8.0, 8.5, 8.9, 9.6)
Very good	Δ₇ = (7.9, 8.2, 8.4, 8.7, 9.0, 9.3, 9.5, 9.8)