Evaluation and research on the logistics efficiency of agricultural products with intuitionistic fuzzy information

Abstract

With respect to intuitionistic fuzzy multiple attribute decision making problems with completely unknown weight information, some operational laws of intuitionistic fuzzy numbers, score function and accuracy function of intuitionistic fuzzy numbers are introduced. To determine the attribute weights, a model based on the information entropy, by which the attribute weights can be determined, is established. We utilize the intuitionistic fuzzy Hamacher weighted averaging (IFHWA) operator to fuse the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. Finally, an illustrative example for evaluating the logistics efficiency of agricultural products is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Multiple attribute decision making intuitionistic fuzzy number intuitionistic fuzzy Hamacher weighted averaging (IFHWA) operator entropy logistics efficiency agricultural products

1. Introduction

In order to improve the accuracy of real-life decision-making, Zadeh [1] initially designed the fuzzy sets (FSs). Atanassov [2] designed the intuitionistic fuzzy sets (IFSs), which could be a generalization of FSs. He et al. [3] integrated the power averaging operators with IFSs and defined several intuitionistic fuzzy power interaction aggregation operators. Li and Wu [4] presented the intuitionistic fuzzy cross entropy distance and the GRA. Liang et al. [5] extended MABAC method to IFSs by utilizing the novel distance measures. Zhang and He [6] defined the extensions of intuitionistic fuzzy geometric interaction operators by using the t-norm and the corresponding t-conorm means. Chen et al. [7] developed the novel MCDM method based on the TOPSIS method and similarity measures in the context of IFSs. Khan et al. [8] put forward a novel similarity measure about IFNs depending on the distance measure of double sequence of bounded variation. Garg [9] developed some intuitionistic fuzzy averaging operators by taking the degrees of hesitation between the membership mathematical functions into consideration. Li et al. [10] developed a grey target decision making method in the form of IFNs on the basis of grey relational analysis [11]. Bao et al. [12] put forward the prospect theory and the evidential reasoning method under IFSs. Gupta et al. [13] extended the fuzzy entropy [14] to IFSs with axiomatic justification and proposed importance of parameter alpha. Gan and Luo [15] employed a hybrid method on the basis of DEMATEL and IFSs. Gupta et al. [16] modified the superiority and inferiority ranking (SIR) method and combined it under IFSs. Krishankumar et al. [17] developed IFSP (intuitionistic fuzzy set based PROMETHEE) which was a novel ranking method. Liu et al. [18] presented some novel intuitionistic fuzzy operators by extending the BM operator on the basis of the Dombi operations [19] and designed some MAGDM methods. Hao et al. [20] presented THE novel intuitionistic fuzzy MADM method depending on the decision theory. Gou et al. [21] pointed out a novel exponential operational law about IFNs and offered a method which was utilized to aggregate intuitionistic fuzzy information. Jin et al. [22] developed two group decision making (GDM) methods which could obtain the normalized intuitionistic fuzzy priority weights from the designed IFPRs on the basis of the order consistency and the multiplicative consistency. Luo and Wang [23] combined IFSs with VIKOR method relying on a novel distance measure which taking the IFSs into consideration. Wu et al. [24] gave the VIKOR method for financing risk assessment of rural tourism projects under IVIFSs. Lu and Wei [25] designed the TODIM method for performance appraisal on social-integration-based rural reconstruction under IVIFSs. Wu et al. [26] designed the algorithms for competiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators. Wu et al. [27] proposed some interval-valued intuitionistic fuzzy Dombi Heronian mean operators for evaluating the ecological value of forest ecological tourism demonstration areas.

In the process of MADM with intuitionistic fuzzy information [28, 29, 30], sometimes, the attribute values take the form of intuitionistic fuzzy numbers, and the information about attribute weights is incompletely known or completely unknown because of time pressure, lack of knowledge or data, and the expert’s limited expertise about the problem domain. All of the above methods, however, will be unsuitable for dealing with such situations. Therefore, it is necessary to pay attention to this issue. The aim of this paper is to develop another method, based on the information entropy method, to overcome this limitation. The remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to intuitionistic fuzzy sets. In Section 3 we introduce the MADM problem with intuitionistic fuzzy information, in which the information about attribute weights is completely unknown, and the attribute values take the form of intuitionistic fuzzy numbers. To determine the attribute weights, a model based on the information entropy method, by which the attribute weights can be determined, is established. We utilize the intuitionistic fuzzy Hamacher weighted averaging (IFHWA) operator to aggregate the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. In Section 4, an illustrative example for evaluating the logistics efficiency of agricultural products is pointed out. In Section 5 we conclude the paper and give some remarks.

2. Preliminaries

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

Definition 1 [1]. Let $X$ be a universe of discourse, then a fuzzy set is defined as:

$\displaystyle A=\{\langle x,\mu_{A}(x)\rangle|{x\in X}\}$ (1)

Which is characterized by a membership function $\mu_{A}:X\to[{0,1}]$ , where $\mu_{A}(x)$ denotes the degree of membership of the element $x$ to the set $A$ [3].

Atanassov [2] extended the fuzzy set to the IFS, shown as follows:

Definition 2 [31]. An IFS $A$ in $X$ is given by

$\displaystyle A=\{\langle x,\mu_{A}(x),\nu_{A}(x)\rangle|{x\in X}\}$ (2)

Where $\mu_{A}:X\to[{0,1}]$ and $\nu_{A}:X\to[{0,1}]$ , with the condition $0\leqslant\mu_{A}(x)+\nu_{A}(x)\leqslant 1$ , $\forall x\in X$ . The numbers $\mu_{A}(x)$ and $\nu_{A}(x)$ represent, respectively, the membership degree and non-membership degree of the element $x$ to the set $A$ .

Definition 3 [31]. For each IFS $A$ in $X$ , if

$\displaystyle\pi_{A}(x)=1-\mu_{A}(x)-\nu_{A}(x),\forall x\in X.$ (3)

Then $\pi_{A}(x)$ is called the degree of indeterminacy of $x$ to $A$ [1, 2].

Definition 4 [32]. Let $I_{1}=({\mu_{1},\nu_{1}})$ and $I_{2}=({\mu_{2},\nu_{2}})$ be IFNs, the score and accuracy functions of $I_{1}$ and $I_{2}$ can be expressed:

$\displaystyle S({I_{1}})=\mu_{1}+\mu_{1}({1-\mu_{1}-\nu_{1}}),$ (4) $\displaystyle S({I_{2}})=\mu_{2}+\mu_{2}({1-\mu_{2}-\nu_{2}})$ $\displaystyle H({I_{1}})=\mu_{1}+\nu_{1},H({I_{2}})=\mu_{2}+\nu_{2}$ (5)

For two IFNs $I_{1}$ and $I_{2}$ , regarding on the Definition 3, then

(1)

$\text{if }s({I_{1}})<s({I_{2}}),\text{ then }I_{1}<I_{2}$ ;

(2)

$\text{if }s({I_{1}})>s({I_{2}}),\text{ then }I_{1}>I_{2}$ ;

(3)

$\text{if }s({I_{1}})=s({I_{2}}),h({I_{1}})<h({I_{2}}),\text{ then }I_{1}<I_{2}$ ;

(4)

$\text{if }s({I_{1}})=s({I_{2}}),h({I_{1}})>h({I_{2}}),\text{ then }I_{1}>I_{2}$ ;

(5)

$\text{if }s({I_{1}})=s({I_{2}}),h({I_{1}})=h({I_{2}}),\text{ then }I_{1}=I_{2}$ .

Definition 5 [33]. Let $a_{j}=({\mu_{j},\nu_{j}})({j=1,2,\cdots,n})$ be a collection of IFNs, and let IFHWA: $Q^{n}\to Q$ , if

$\displaystyle\text{IFHWA}_{\omega}({\tilde{a}_{1},\tilde{a}_{2},\cdots,\tilde{% a}_{n}})=\mathop{\oplus_{\varepsilon}}\limits_{j=1}^{n}({\omega_{j}\tilde{a}_{% j}})=\left(\frac{\begin{array}[]{c}\prod\limits_{j=1}^{n}({1+({\gamma-1})\mu_{% j}})^{\omega_{j}}\\ -\prod\limits_{j=1}^{n}{({1-\mu_{j}})^{\omega_{j}}}\\ \end{array}}{\begin{array}[]{c}\prod\limits_{j=1}^{n}({1+({\gamma-1})\mu_{j}})% ^{\omega_{j}}\\ +({\gamma-1})\prod\limits_{j=1}^{n}{({1-\mu_{j}})^{\omega_{j}}}\end{array}},% \right.\left.{\frac{\gamma\prod\limits_{j=1}^{n}{\nu_{j}^{\omega_{j}}}}{\begin% {array}[]{c}\prod\limits_{j=1}^{n}{({1+({\gamma-1})({1-\nu_{j}})})^{\omega_{j}% }}\\ +({\gamma-1})\prod\limits_{j=1}^{n}{\nu_{j}^{\omega_{j}}}\\ \end{array}}}\right)$ (6)

where $\omega=({\omega_{1},\omega_{2},\cdots,\omega_{n}})^{T}$ be the weight vector of $\tilde{a}_{j}({j=1,2,\cdots,n})$ , and $\omega_{j}>0$ , $\sum\limits_{j=1}^{n}{\omega_{j}}=1$ , then IFWA is called the intuitionistic fuzzy Hamacher weighted averaging (IFHWA) operator.

3. Method for intuitionistic fuzzy multiple attribute decision making problems with entropy weight

The following assumptions or notations are used to represent the intuitionistic fuzzy MADM problems with entropy weight information. The alternatives are known. Let $A=\{{A_{1},A_{2},\cdots,A_{m}}\}$ be a discrete set of alternatives. The attributes are known. Let $G=\{{G_{1},G_{2},\cdots,G_{n}}\}$ be a set of attributes; The information about attribute weights is incompletely known. Let $w=({w_{1},w_{2},\cdots,w_{n}})\in H$ be the weight vector of attributes, where $w_{j}\geqslant 0,j=1,2,\cdots,n$ , $\sum\nolimits_{j=1}^{n}{w_{j}}=1$ .

Suppose that $\tilde{R}=({\tilde{r}_{ij}})_{m\times n}=({\mu_{ij},\nu_{ij}})_{m\times n}$ is the intuitionistic fuzzy decision matrix, where $\mu_{ij}$ indicates the degree that the alternative $A_{i}$ satisfies the attribute $G_{j}$ given by the decision maker, $\nu_{ij}$ indicates the degree that the alternative $A_{i}$ doesn’t satisfy the attribute $G_{j}$ given by the decision maker, $\mu_{ij}\subset[{0,1}]$ , $\nu_{ij}\subset[{0,1}]$ , $\mu_{ij}+\nu_{ij}\leqslant 1$ , $i=1,2,\cdots,m$ , $j=1,2,\cdots,n$ .

Definition 6. Let $\tilde{R}=({\tilde{r}_{ij}})_{m\times n}=({\mu_{ij},\nu_{ij}})_{m\times n}$ be an intuitionistic fuzzy decision matrix, $\tilde{r}_{i}=(\tilde{r}_{i1},\tilde{r}_{i2},\linebreak\cdots,\tilde{r}_{in})$ be the vector of attribute values corresponding to the alternative $A_{i}i=1,2,\cdots,m$ , then we call

$\displaystyle\tilde{r}_{i}=({\mu_{ij},\nu_{ij}})=\text{IFHWA}_{w}({\tilde{r}_{% i1},\tilde{r}_{i2},\cdots,\tilde{r}_{in}})=\left({\frac{\begin{array}[]{c}% \prod\limits_{j=1}^{n}({1+({\gamma-1})\mu_{j}})^{\omega_{j}}\\ -\prod\limits_{j=1}^{n}{({1-\mu_{j}})^{\omega_{j}}}\end{array}}{\begin{array}[% ]{c}\prod\limits_{j=1}^{n}({1+({\gamma-1})\mu_{j}})^{\omega_{j}}\\ +({\gamma-1})\prod\limits_{j=1}^{n}{({1-\mu_{j}})^{\omega_{j}}}\end{array}},}% \right.\left.{\frac{\gamma\prod\limits_{j=1}^{n}{\nu_{j}^{\omega_{j}}}}{\begin% {array}[]{c}\prod\limits_{j=1}^{n}{({1+({\gamma-1})({1-\nu_{j}})})^{\omega_{j}% }}\\ +({\gamma-1})\prod\limits_{j=1}^{n}{\nu_{j}^{\omega_{j}}}\end{array}}}\right),% i=1,2,\cdots,m.$ (7)

the overall value of the alternative $A_{i}$ , where $w=({w_{1},w_{2},\cdots,w_{n}})^{T}$ is the weight vector of attributes.

In the situation where the information about attribute weights is completely known, i.e., each attribute weight can be provided by the expert with crisp numerical value, we can weight each attribute value and aggregate all the weighted attribute values corresponding to each alternative into an overall one by using Eq. (7). Based on the overall attribute values $\tilde{r}_{i}$ of the alternatives $A_{i}({i=1,2,\cdots,m})$ , we can rank all these alternatives and then select the most desirable one(s). The greater $\tilde{r}_{i}$ , the better the alternative $A_{i}$ will be.

Entropy [34] was one of the concepts in thermodynamics originally and then Shannon first introduced the concept of information entropy in connection with communication theory. In these disciplines entropy is applied as a measure of disorder, unevenness of distribution and the degree of dependency or complexity of a system. Information entropy is an ideal measure of uncertainty and it can measure the quality of effective information. In the intuitionistic fuzzy MADM problems which have $m$ alternatives and $n$ attributes, the $j t h$ attribute’s entropy is defined as follows:

$\displaystyle H_{j}=-k\sum\limits_{i=1}^{m}{f_{ij}}\ln f_{ij},i=1,2,\cdots,m,$ $\displaystyle j=1,2,\cdots,n.$

where

$\displaystyle f_{ij}=\frac{H({\tilde{r}_{ij}})}{\sum\limits_{i=1}^{m}{H({% \tilde{r}_{ij}})}},k=\frac{1}{\ln m}.$ (8)

Assume that if $f_{ij}=0$ , then $f_{ij}\ln f_{ij}=0$ .

Then, the $j t h$ attribute’s entropy is defined as follows:

$\displaystyle w_{j}=\frac{1-H_{j}}{n-\sum\limits_{j=1}^{n}{H_{j}}}.$ (9)

Based on the above models, we develop a practical method for solving the MADM problems, in which the information about attribute weights is completely unknown, and the attribute values take the form of intuitionistic fuzzy information. The method involves the following steps:

Step 1.

Let $\tilde{R}=({\tilde{r}_{ij}})_{m\times n}$ be the intuitionistic fuzzy decision matrix, where $\tilde{r}_{ij}=(\mu_{ij},\linebreak\nu_{ij})$ , which is an attribute value, given by an expert, for the alternative $A_{i}\in A$ with respect to the attribute $G_{j}\in G$ , $w=(w_{1},\linebreak w_{2},\cdots,w_{n})$ be the weight vector of attributes, where $w_{j}\in[{0,1}],j=1,2,\cdots,n,\linebreak\sum\nolimits_{j=1}^{n}{w_{j}}=1$ .

Step 2.

Determine the entropy weight of each attribute according to Eqs (8) and (9).

Step 3.

Utilize the weight vector $w=(w_{1},w_{2},\cdots,\linebreak w_{n})$ and by Eq. (7), we obtain the overall values $\tilde{r}_{i}$ of the alternative $A_{i}(i=1,2,\cdots,\linebreak m)$ .

Step 4.

Calculate the scores $S({\tilde{r}_{i}})$ of the overall intuitionistic fuzzy preference value $\tilde{r}_{i}(i=1,2,\linebreak\cdots,m)$ to rank all the alternatives $A_{i}(i=1,\linebreak 2,\cdots,m)$ and then to select the best one(s) (if there is no difference between two scores $S({\tilde{r}_{i}})$ and $S({\tilde{r}_{j}})$ , then we need to calculate the accuracy degrees $H({\tilde{r}_{i}})$ and $H({\tilde{r}_{j}})$ of the overall intuitionistic fuzzy preference value $\tilde{r}_{i}$ and $\tilde{r}_{j}$ , respectively, and then rank the alternatives $A_{i}$ and $A_{j}$ in accordance with the accuracy degrees $H({\tilde{r}_{i}})$ and $H({\tilde{r}_{j}})$ .

Step 5.

Rank all the alternatives $A_{i}(i=1,2,\cdots,\linebreak m)$ and select the best one(s) in accordance with $S({\tilde{r}_{i}})$ and $H({\tilde{r}_{i}})({i=1,2,\cdots,m})$ .

Step 6.

End.

Table 1

Evaluation results of these methods

Methods	Order	The optimal alternative	The worst alternative
IFWA operator [35]	$A_{2}\succ A_{5}\succ A_{3}\succ A_{4}\succ A_{1}$	$A_{2}$	$A_{1}$
IFWG operator [36]	$A_{2}\succ A_{5}\succ A_{3}\succ A_{4}\succ A_{1}$	$A_{2}$	$A_{1}$
The MABAC method [37]	$A_{2}\succ A_{5}\succ A_{4}\succ A_{3}\succ A_{1}$	$A_{2}$	$A_{1}$
The GRA method [38]	$A_{2}\succ A_{5}\succ A_{3}\succ A_{4}\succ A_{1}$	$A_{2}$	$A_{1}$
The IFMSM operator [39]	$A_{2}\succ A_{5}\succ A_{3}\succ A_{4}\succ A_{1}$	$A_{2}$	$A_{1}$
The developed method	$A_{2}\succ A_{5}\succ A_{3}\succ A_{4}\succ A_{1}$	$A_{2}$	$A_{1}$

4. Illustrative example

Along with the improvement of economic level and people’s life quality, fresh agricultural products have gradually become one of the most common consumer goods in daily life. However, it is not suitable to store these fresh agricultural products at room temperature for a long time, due to their perishable properties. They are extremely sensitive to the external environment such as humidity, temperature. Since the development of logistics in China starts late, there is a lack in research focusing on establishing a supply chain network for fresh agricultural products. Current concerns that have been arisen include inefficient of track and trace system, inadequate logistics infrastructure, and susceptibility of fresh agricultural products to breaking during transportation, which may lead to severe and irreparable damage. Thus, developing an effective supply chain network for fresh agricultural products is vital to establish advantageous transport and delivery terms, reduce the risk of damage to goods, lower the logistics cost of enterprises, and improve the logistics efficiency. Despite the fact that more and more district has rich agricultural products resources, the occurrence of some adverse factors, such as high resources depletion rate during transportation, leading to dissatisfaction of the terminal customers and increasing unnecessary cost of the enterprises. In this chapter, an empirical application of evaluating the logistics efficiency of agricultural products will be provided by making use of proposed method. In order to evaluate the logistics efficiency of agricultural products, the expert group must take a decision according to the following four attributes: ⟀ G ${}_{1}$ is the product quality; ⟁ G ${}_{2}$ is the relationship closeness; ⟂ G ${}_{3}$ is the delivery performance; ⟃ G ${}_{4}$ is the price. The five possible logistics companies $A_{i}({i=1,2,3,4,5})$ are to be evaluated using the intuitionistic fuzzy information by the decision maker under the above four attributes, as listed in the following matrix.

$\displaystyle\tilde{R}=\left[{{\begin{array}[]{*{20}c}{({0.7,0.3})}&{({0.2,0.8% })}&{({0.2,0.3})}&{({0.8,0.2})}\\ {({0.6,0.4})}&{({0.3,0.5})}&{({0.5,0.4})}&{({0.4,0.6})}\\ {({0.4,0.5})}&{({0.4,0.4})}&{({0.3,0.5})}&{({0.6,0.2})}\\ {({0.6,0.3})}&{({0.3,0.6})}&{({0.4,0.2})}&{({0.5,0.3})}\\ {({0.5,0.3})}&{({0.5,0.4})}&{({0.5,0.4})}&{({0.4,0.2})}\\ \end{array}}}\right]$

Procedure for evaluating the logistics efficiency of agricultural products contains the following steps.

Step 1.
According to Eqs (8) and (9), we get the weight vector of attributes:

$\displaystyle w=({{0.3057}\,\,{0.1180}\,\,{0.2444}\,\,{0.3319}})^{T}$
Step 2.
Utilize the weight vector $w=(w_{1},w_{2},\cdots,\linebreak w_{n})$ and by Eq. (7), we obtain the overall values $\tilde{r}_{i}$ of the logistics companies $A_{i}\linebreak({i=1,2,\cdots,m})$ .

$\displaystyle\tilde{r}_{1}=({0.3523,0.5443}),$ $\displaystyle\tilde{r}_{2}=(0.5726,0.3495),$ $\displaystyle\tilde{r}_{3}=({0.4386,0.4549}),$ $\displaystyle\tilde{r}_{4}=({0.4534,0.3872}),$ $\displaystyle\tilde{r}_{5}=({0.4663,0.3013})$
Step 3.
calculate the scores $S({\tilde{r}_{i}})$ of the overall intuitionistic fuzzy preference values $\tilde{r}_{i}(i\linebreak=1,2,\cdots,m)$

$\displaystyle S({\tilde{r}_{1}})=-0.1920,S({\tilde{r}_{2}})=0.2230,$ $\displaystyle S({\tilde{r}_{3}})=-0.0163,S({\tilde{r}_{4}})=0.0662,$ $\displaystyle S({\tilde{r}_{5}})=0.1651$
Step 4.
Rank all the logistics companies $A_{i}(i=1,\linebreak 2,3,4,5)$ in accordance with the scores $S({\tilde{r}_{i}})({i=1,2,\cdots,5})$ of the overall intuitionistic fuzzy preference values $\tilde{r}_{i}(i=1,\linebreak 2,\cdots,m)$ : $A_{2}\succ A_{5}\succ A_{3}\succ A_{4}\succ A_{1}$ , and thus the most desirable logistics companies is $A_{2}$ .

In this part, our developed method is made comparison with some other methods to illustrate its superiority. Eventually, the results of other methods are depicted in Table 1.
5. Conclusion

In this paper, we have investigated the problem of MADM with completely unknown information on attribute weights to which the attribute values are given in terms of intuitionistic fuzzy numbers. To determine the attribute weights, a model based on the information entropy, by which the attribute weights can be determined, is established. We utilize the intuitionistic fuzzy Hamacher weighted averaging (IFHWA) operator to aggregate the intuitionistic fuzzy information corresponding to each alternative, and then rank the alternatives and select the most desirable one(s) according to the score function and accuracy function. Finally, an illustrative example for evaluating the logistics efficiency of agricultural products is given. In the future, we shall continue working in the application of the intuitionistic fuzzy multiple attribute decision-making to other domains [40, 41, 42, 43, 44, 45].

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