Cross-entropy method for multiple attribute decision making under the triangular fuzzy neutrosophic sets

Abstract

Cloud computing is emerging as an advanced stage of enterprise information technology in a highly competitive society, and is now in a phase of rapid development. Enterprises using cloud computing can reduce the cost of building infrastructure and reap huge benefits. Due to the convenience of cloud computing, more and more enterprises are inclined to use cloud services to build their business systems. However, there are many different cloud service providers in the market, and enterprises need scientific decision-making tools to determine which cloud service provider to choose. The cloud service provider selection is viewed as the multi-attribute decision-making (MADM). In this paper, the cross-entropy method under triangular fuzzy neutrosophic sets (TFNSs) is built based on the traditional cross-entropy method. Firstly, the TFNSs is introduced. Then, combine the traditional fuzzy cross-entropy method with TFNSs information, the triangular fuzzy neutrosophic number cross-entropy (TFNN-CE) method is established for MADM under TFNSs. Finally, a numerical example for cloud service provider selection has been given and some comparisons is used to illustrate advantages of cross-entropy method with TFNSs.

Keywords

Multiple attribute decision making (MADM) problems triangular fuzzy neutrosophic sets (TFNSs)cross-entropy method cloud service provider selection

1. Introduction

In recent years, cloud computing [1], as an emerging network computing service model, has gained more and more widespread attention. Cloud computing integrates a large amount of computing, storage and software resources through virtualization, grid computing and other information technologies to form a virtual shared resource pool of huge scale and provide users with the required information services [2]. Cloud computing has changed the traditional information service model by integrating infrastructure, platform, software and other information resources to provide on-demand services for users like water, electricity, gas and other social resources [3]. Based on the advanced service concept of cloud computing, industries such as medical, finance, energy, e-government, education and research, telecommunication and manufacturing are all undergoing a major change in information technology, and through the deep integration with cloud computing technology, a new development model based on industrial characteristics is gradually formed. Among them, the manufacturing industry, as a typical physical application field, has fused with cloud computing technology to give rise to a new service-oriented networked manufacturing model of cloud manufacturing [4, 5, 6, 7, 8]. Cloud manufacturing integrates existing networked manufacturing technologies, cloud computing, Internet of Things, and high-performance computing to achieve unified and centralized intelligent management and operation of various manufacturing resources and provide readily accessible, on-demand, secure, and high-quality manufacturing services for the whole manufacturing life cycle [9, 10]. Cloud manufacturing unifies the two ideas of centralized use of decentralized resources and decentralized services [11, 12, 13]; at the same time, it presents the processes of demonstration, design, production and processing, experimentation, simulation, operation and management, and integration in the whole manufacturing life cycle to users in the form of services, which can realize on-demand services [14, 15, 16, 17]. This goal cannot be achieved without the support of virtualization technologies [18, 19, 20], which transform physical manufacturing resources into logical manufacturing resources through technologies such as the Internet of Things, information-physical systems, and virtualization of computing systems, enabling resources to serve users in an efficient, agile, reliable, secure, and convenient manner. Multi-granular classification and aggregation of resources will further improve the efficiency and performance of resource virtualization, sharing, and distribution in cloud manufacturing, and provide better services to users [21, 22, 23]. An important challenge in deploying and implementing cloud manufacturing model for enterprises or enterprise alliances is to choose the right cloud service provider, which is a common core issue for the integration of healthcare, finance, energy, e-government and other industries with cloud computing.

Multi-attribute decision making (MADM) occupies an important place in decision science [24, 25]. In the decision-making process, most of them need to consider multiple criteria and multiple choices, thus, multi-criteria decision analysis (MCDA) also plays an important part in the decision-making issues [26, 27, 28, 29]. According to the continuity and discrete nature of decision-making issues, MCDA is divided into multi-objective decision making (MODM) and MADM. The rise of cloud computing has intensified the competition in the cloud service market. Therefore, how to choose the best provider among the many cloud service providers offering functionally similar products to create the maximum benefit for enterprises has become a hot issue for current research in both academia and industry. The cloud service provider selection is viewed as the MADM [30, 31, 32, 33, 34, 35]. Compared with Characteristic Objects Method (COMET) [36, 37], Stable Preference Ordering Towards Ideal Solution (SPOTIS) method [38] and Data vARIability Assessment Technique for Order of Preference by Similarity to Ideal Solution (the DARIA-TOPSIS method) [39], cross-entropy [40] is an important topic in fuzzy set theory and it is an important tool to measure the degree of difference between two systems [41, 42, 43, 44, 45, 46]. Biswas, Pramanik [47] defined the triangular fuzzy neutrosophic numbers (TFNNs) which is easy to depict the uncertain information during the cloud service provider selection. The purpose of this defined work is to devise the cross-entropy method to study TFNN-MADM problems more effectively. In this paper, the cross-entropy method under triangular fuzzy neutrosophic sets (TFNSs) is built based on the traditional cross-entropy method. Firstly, the TFNSs is introduced. Then, combine the traditional fuzzy cross-entropy method with TFNSs information, the triangular fuzzy neutrosophic number cross-entropy (TFNN-CE) method is established for MADM under TFNSs. Finally, a numerical example for cloud service provider selection has been given and some comparisons is used to illustrate advantages of cross-entropy method with TFNSs.

The structure of this paper is depicted as: the definition of TFNSs is introduced in Section 2. The cross-entropy method is built for TFNN-MADM in Section 3. A numerical example for cloud service provider selection is used to show the TFNNCE method and some comparisons are also conducted to illustrate more advantages of the cross-entropy method with TFNNs in Section 4. Section 5 gives some useful conclusions.

2. Preliminaries

Biswas, Pramanik [47] defined the TFNSs.

Definition 1 [47]. Let $X$ be a fix set, the TFNSs $Y Y$ is defined:

$\displaystyle YY=\{(x,YA(x),YB(x),YC_{Y}(x))|x\in X\}$ (1)

where $YA(x),YB(x),YC_{Y}(x)\in[{0,1}]$ represent the truth membership, indeterminacy membership and falsity membership which is depicted by triangular fuzzy numbers.

$\displaystyle YA(x)=({YA^{L}(x),YA^{M}(x),YA^{U}(x)}),0\leqslant YA^{L}(x)% \leqslant YA^{M}(x)\leqslant YA^{U}(x)\leqslant 1$ (2) $\displaystyle YB(x)=({YB^{L}(x),YB^{M}(x),YB^{U}(x)}),0\leqslant YB^{L}(x)% \leqslant YB^{M}(x)\leqslant YB^{U}(x)\leqslant 1$ (3) $\displaystyle YC(x)=({YC^{L}(x),YC^{M}(x),YC^{U}(x)}),0\leqslant YC^{L}(x)% \leqslant YC^{M}(x)\leqslant YC^{U}(x)\leqslant 1$ (4)

For convenience, we let $YY=\{{({YA^{L},YA^{M},YA^{U}}),({YB^{L},YB^{M},YB^{U}}),({YC^{L},YC^{M},YC^{U}% })}\}$ be a TFNN $0\leqslant YA^{U}+YB^{U}+YC^{U}\leqslant 3$ .

Definition 2 [47]. Assume there are three TFNNs $YY_{1}=\{({YA_{1}^{L},YA_{1}^{M},YA_{1}^{U}}),({YB_{1}^{L},YB_{1}^{M},YB_{1}^{% U}}),\linebreak({YC_{1}^{L},YC_{1}^{M},YC_{1}^{U}})\}$ , $YY_{2}=\{{({YA_{2}^{L},YA_{2}^{M},YA_{2}^{U}}),({YB_{2}^{L},YB_{2}^{M},YB_{2}^% {U}}),({YC_{2}^{L},YC_{2}^{M},YC_{2}^{U}})}\}$ and $YY=\{{({YA^{L},YA^{M},YA^{U}}),({YB^{L},YB^{M},YB^{U}}),({YC^{L},YC^{M},YC^{U}% })}\}$ , the operation laws of them are defined:

$\displaystyle(1)\,YY_{1}\oplus YY_{2}$ $\displaystyle=\left\{{\begin{array}[]{l}({YA_{1}^{L}+YA_{2}^{L}{-}YA_{1}^{L}YA% _{2}^{L},YA_{1}^{M}+YA_{2}^{M}{-}YA_{1}^{M}YA_{2}^{M},YA_{1}^{U}+YA_{2}^{U}{-}% YA_{1}^{U}YA_{2}^{U}}),\\ ({YB_{1}^{L}YB_{2}^{L},YB_{1}^{M}YB_{2}^{M},YB_{1}^{U}YB_{2}^{U}}),({YC_{1}^{L% }YC_{2}^{L},YC_{1}^{M}YC_{2}^{M},YC_{1}^{U}YC_{2}^{U}})\\ \end{array}}\right\};$ $\displaystyle(2)\;YY_{1}\otimes YY_{2}$ $\displaystyle=\left\{{\begin{array}[]{l}({YA_{1}^{L}YA_{2}^{L},YA_{1}^{M}YA_{2% }^{M},YA_{1}^{U}YA_{2}^{U}}),\\ ({YB_{1}^{L}+YB_{2}^{L}{-}YB_{1}^{L}YB_{2}^{L},YB_{1}^{M}+YB_{2}^{M}{-}YB_{1}^% {M}YB_{2}^{M},YB_{1}^{U}+YB_{2}^{U}{-}YB_{1}^{U}YB_{2}^{U}}),\\ ({YC_{1}^{L}+YC_{2}^{L}{-}YC_{1}^{L}YC_{2}^{L},YC_{1}^{M}+YC_{2}^{M}{-}YC_{1}^% {M}YC_{2}^{M},YC_{1}^{U}+YC_{2}^{U}{-}YC_{1}^{U}YC_{2}^{U}})\\ \end{array}}\right\};$ $\displaystyle(3)\;\lambda YY=\left\{{\begin{array}[]{l}({1-({1-YA^{L}})^{% \lambda},1-({1-YA^{M}})^{\lambda},1-({1-YA^{U}})^{\lambda}}),\\ ({({YB^{L}})^{\lambda},({YB^{M}})^{\lambda},({YB^{U}})^{\lambda}}),\\ ({({YC^{L}})^{\lambda},({YC^{M}})^{\lambda},({YC^{U}})^{\lambda}})\\ \end{array}}\right\},\lambda>0;$ $\displaystyle(4)\;YY^{\lambda}=\left\{{\begin{array}[]{l}({({YA^{L}})^{\lambda% },({YA^{M}})^{\lambda},({YA^{U}})^{\lambda}}),\\ ({1-({1-YB^{L}})^{\lambda},1-({1-YB^{M}})^{\lambda},1-({1-YB^{U}})^{\lambda}})% ,\\ ({1-({1-YC^{L}})^{\lambda},1-({1-YC^{M}})^{\lambda},1-({1-YC^{U}})^{\lambda}})% \\ \end{array}}\right\},\lambda>0.$

From Definition 2, it’s clear that the operation laws have the following properties.

$\displaystyle(1)\;YY_{1}\oplus YY_{2}=YY_{2}\oplus YY_{1},YY_{1}\otimes YY_{2}% =YY_{2}\otimes YY_{1},({({YY_{1}})^{\lambda_{1}}})^{\lambda_{2}}=({YY_{1}})^{% \lambda_{1}\lambda_{2}};$ (5) $\displaystyle(2)\;\lambda({YY_{1}\oplus YY_{2}})=\lambda YY_{1}\oplus\lambda YY% _{2},({YY_{1}\otimes YY_{2}})^{\lambda}=({YY_{1}})^{\lambda}\otimes({YY_{2}})^% {\lambda};$ (6) $\displaystyle(3)\;\lambda_{1}YY_{1}\oplus\lambda_{2}YY_{1}=({\lambda_{1}+% \lambda_{2}})YY_{1},({YY_{1}})^{\lambda_{1}}\otimes({YY_{1}})^{\lambda_{2}}=({% YY_{1}})^{({\lambda_{1}+\lambda_{2}})}.$ (7)

Definition 3 [47]. Let $YY=\{{({YA^{L},YA^{M},YA^{U}}),({YB^{L},YB^{M},YB^{U}}),({YC^{L},YC^{M},YC^{U}% })}\}$ be a TFNN, the score functions and accuracy functions of $Y Y$ is:

$\displaystyle SF({YY})=\frac{1}{12}\left[{\begin{array}[]{l}8+({YA^{L}+2YA^{M}% +YA^{U}})\\ -({YB^{L}+2YB^{M}+YB^{U}})\\ -({YC^{L}+2YC^{M}+YC^{U}})\\ \end{array}}\right],SF({YY})\in[{0,1}]$ (8) $\displaystyle AF({YY})=\frac{1}{4}[{({YA^{L}+2YA^{M}+YA^{U}})-({YB^{L}+2YB^{M}% +YB^{U}})}],AF({YY})\in[{-1,1}]$ (9)

For two TFNNs $YY_{1}$ and $YY_{2}$ , based on the Definition 3, then

(1) if $SF({YY_{1}})\prec SF({YY_{2}})$ , then $YY_{1}\prec YY_{2}$ ;

(2) if $SF({YY_{1}})\succ SF({YY_{2}})$ , then $YY_{1}\succ YY_{2}$ ;

(3) if $SF({YY_{1}})=SF({YY_{2}}),AF({YY_{1}})\prec AF({YY_{2}})$ , then $YY_{1}\prec YY_{2}$ ;

(4) if $SF({YY_{1}})=SF({YY_{2}}),AF({YY_{1}})\succ AF({YY_{2}})$ , then $YY_{1}\succ YY_{2}$ ;

(5) if $SF({YY_{1}})=SF({YY_{2}}),AF({YY_{1}})=AF({YY_{2}})$ , then $YY_{1}=YY_{2}$ .

Bhandari and Pal [40] defined the cross entropy.

Definition 7 [40]. Assume that $\alpha\alpha=({\alpha\alpha({x_{1}}),\alpha\alpha({x_{2}}),\ldots,\alpha\alpha% ({x_{n}})})$ and $\beta\beta=(\beta\beta({x_{1}}),\beta\beta({x_{2}}),\ldots,\linebreak\beta% \beta({x_{n}}))$ are two sets on the $({x_{1},x_{2},\ldots,x_{n}})$ . The cross-entropy of $\alpha\alpha$ from $\beta\beta$ is built:

$\displaystyle CE({\alpha\alpha,\alpha\beta})=\sum\limits_{j=1}^{n}{\left({% \alpha\alpha({x_{j}})\ln\frac{\alpha\alpha({x_{j}})}{\beta\beta({x_{j}})}+({1-% \alpha\alpha({x_{j}})})\ln\frac{1-\alpha\alpha({x_{j}})}{1-\beta\beta({x_{j}})% }}\right)}$ (10)

which indicates the discrimination degree of $\alpha\alpha$ from $\beta\beta$ .

Then, TFNN cross-entropy (TFNN-CE) shall be defined based on the modified fuzzy cross-entropy [40] and TFNNs [47].

Definition 8. Assume that $YY_{1}=\{{({YA_{1}^{L},YA_{1}^{M},YA_{1}^{U}}),({YB_{1}^{L},YB_{1}^{M},YB_{1}^% {U}}),({YC_{1}^{L},YC_{1}^{M},YC_{1}^{U}})}\}$ and $YY_{2}=\{{({YA_{2}^{L},YA_{2}^{M},YA_{2}^{U}}),({YB_{2}^{L},YB_{2}^{M},YB_{2}^% {U}}),({YC_{2}^{L},YC_{2}^{M},YC_{2}^{U}})}\}$ . The TFNN-CE is built between $YY_{1}$ and $YY_{2}$ :

$\displaystyle\text{TFNN-CE}({YY_{1},YY_{2}})=\left({\begin{array}[]{l}\left({% \frac{YA_{1}^{L}+2YA_{1}^{M}+YA_{1}^{U}}{4}}\right)\ln\frac{\frac{YA_{1}^{L}+2% YA_{1}^{M}+YA_{1}^{U}}{4}}{\frac{YA_{2}^{L}+2YA_{2}^{M}+YA_{2}^{U}}{4}}\\ +\left({1-\frac{YA_{1}^{L}+2YA_{1}^{M}+YA_{1}^{U}}{4}}\right)\ln\frac{1-\frac{% YA_{1}^{L}+2YA_{1}^{M}+YA_{1}^{U}}{4}}{1-\frac{YA_{2}^{L}+2YA_{2}^{M}+YA_{2}^{% U}}{4}}\\ \end{array}}\right)+\left({\begin{array}[]{l}\left({\frac{YB_{1}^{L}+2YB_{1}^{% M}+YB_{1}^{U}}{4}}\right)\ln\frac{\frac{YB_{1}^{L}+2YB_{1}^{M}+YB_{1}^{U}}{4}}% {\frac{YB_{2}^{L}+2YB_{2}^{M}+YB_{2}^{U}}{4}}\\ +\left({1-\frac{YB_{1}^{L}+2YB_{1}^{M}+YB_{1}^{U}}{4}}\right)\ln\frac{1-\frac{% YB_{1}^{L}+2YB_{1}^{M}+YB_{1}^{U}}{4}}{1-\frac{YB_{2}^{L}+2YB_{2}^{M}+YB_{2}^{% U}}{4}}\\ \end{array}}\right)+\left({\begin{array}[]{l}\left({\frac{YC_{1}^{L}+2YC_{1}^{% M}+YC_{1}^{U}}{4}}\right)\ln\frac{\frac{YC_{1}^{L}+2YC_{1}^{M}+YC_{1}^{U}}{4}}% {\frac{YC_{2}^{L}+2YC_{2}^{M}+YC_{2}^{U}}{4}}\\ +\left({1-\frac{YC_{1}^{L}+2YC_{1}^{M}+YC_{1}^{U}}{4}}\right)\ln\frac{1-\frac{% YC_{1}^{L}+2YC_{1}^{M}+YC_{1}^{U}}{4}}{1-\frac{YC_{2}^{L}+2YC_{2}^{M}+YC_{2}^{% U}}{4}}\\ \end{array}}\right)$ (11)

which indicates the discrimination degree of $YY_{1}$ and $YY_{2}$ .

According to defined Shannon’s inequality [48], one could easily verify that $\text{TFNN-CE}({YY_{1},YY_{2}})\geqslant 0$ , and $\text{TFNN-CE}({YY_{1},YY_{2}})=0$ if and only if $({YA_{1}^{L},YA_{1}^{M},YA_{1}^{U}})=({YA_{2}^{L},YA_{2}^{M},YA_{2}^{U}}),(YB_% {1}^{L},\linebreak YB_{1}^{M},YB_{1}^{U})=({YB_{2}^{L},YB_{2}^{M},YB_{2}^{U}})% ,({YC_{1}^{L},YC_{1}^{M},YC_{1}^{U}})=({YC_{2}^{L},YC_{2}^{M},YC_{2}^{U}})$ .

3. Cross-entropy method for MADM issue with TFNNs

In this section, the cross-entropy method is defined for TFNN-MADM. Suppose there are $m$ alternatives $\{YB_{1},YB_{2},\ldots,YB_{m}\}$ , $n$ attributes $\{{YG_{1},YG_{2},\ldots,YG_{n}}\}$ and the steps of cross-entropy method is defined for TFNN-MADM are defined.

Step 1. Build the TFNNmatrix $YY=[{YY_{ij}}]_{m\times n}$ :

$\displaystyle\begin{array}[]{l}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;% \;\;\;\;\;\;\;\;\;\;\;\;{\begin{array}[]{*{20}c}{\;\;\;\;\;\;\;\;\;YG_{1}}&\;% \;{YG_{2}}&\;\ldots&\;{YG_{n}}\\ \end{array}}\\ YY=[{YY_{ij}}]_{m\times n}={\begin{array}[]{*{20}c}{YB_{1}}\\ {YB_{2}}\\ \vdots\\ {YB_{m}}\\ \end{array}}\left[{{\begin{array}[]{*{20}c}{YY_{11}}&{YY_{12}}&\ldots&{YY_{1n}% }\\ {YY_{21}}&{YY_{22}}&\ldots&{YY_{2n}}\\ \vdots&\vdots&\vdots&\vdots\\ {YY_{m1}}&{YY_{m2}}&\ldots&{YY_{mn}}\\ \end{array}}}\right]\\ \end{array}$ (12)

where $YY_{ij}=\{{({({YA_{ij}^{L}}),({YA_{ij}^{M}}),({YA_{ij}^{U}})}),({({YB_{ij}^{L}% }),({YB_{ij}^{M}}),({YB_{ij}^{U}})}),({({YC_{ij}^{L}}),({YC_{ij}^{M}}),({YC_{% ij}^{U}})})}\}$ .

Step 2. Normalize the TFNNmatrix $NY=[{NY_{ij}}]_{m\times n}$ .

For benefit attributes:

$\displaystyle NY_{ij}=\left\{{\begin{array}[]{l}({({NYA_{ij}^{L}}),({NYA_{ij}^% {M}}),({NYA_{ij}^{U}})}),\\ ({({NYB_{ij}^{L}}),({NYB_{ij}^{M}}),({NYB_{ij}^{U}})}),\\ ({({NYC_{ij}^{L}}),({NYC_{ij}^{M}}),({NYC_{ij}^{U}})})\\ \end{array}}\right\}=\left\{{\begin{array}[]{l}({({YA_{ij}^{L}}),({YA_{ij}^{M}% }),({YA_{ij}^{U}})}),\\ ({({YB_{ij}^{L}}),({YB_{ij}^{M}}),({YB_{ij}^{U}})}),\\ ({({YC_{ij}^{L}}),({YC_{ij}^{M}}),({YC_{ij}^{U}})})\\ \end{array}}\right\}$ (13)

For cost attributes:

$\displaystyle NY_{ij}=\left\{{\begin{array}[]{l}({({NYA_{ij}^{L}}),({NYA_{ij}^% {M}}),({NYA_{ij}^{U}})}),\\ ({({NYB_{ij}^{L}}),({NYB_{ij}^{M}}),({NYB_{ij}^{U}})}),\\ ({({NYC_{ij}^{L}}),({NYC_{ij}^{M}}),({NYC_{ij}^{U}})})\\ \end{array}}\right\}=\left\{{\begin{array}[]{l}({({YC_{ij}^{L}}),({YC_{ij}^{M}% }),({YC_{ij}^{U}})}),\\ ({({YB_{ij}^{L}}),({YB_{ij}^{M}}),({YB_{ij}^{U}})}),\\ ({({YA_{ij}^{L}}),({YA_{ij}^{M}}),({YA_{ij}^{U}})})\\ \end{array}}\right\}$ (14)

Step 3. Define the TFNN positive ideal solution (TFNNPIS) $NY^{+}$ and $NY_{i}$ ;

$\displaystyle NY^{+}=\left\{{\begin{array}[]{l}({({NYA_{j}^{L}})^{+},({NYA_{j}% ^{M}})^{+},({NYA_{j}^{U}})^{+}}),\\ ({({NYB_{j}^{L}})^{+},({NYB_{j}^{M}})^{+},({NYB_{j}^{U}})^{+}}),\\ ({({NYC_{j}^{L}})^{+},({NYC_{j}^{M}})^{+},({NYC_{j}^{U}})^{+}})\\ \end{array}}\right\}$ (15) $\displaystyle SF\left\{{\begin{array}[]{l}({({NA_{j}^{L}})^{+},({NA_{j}^{M}})^% {+},({NA_{j}^{U}})^{+}}),\\ ({({NB_{j}^{L}})^{+},({NB_{j}^{M}})^{+},({NB_{j}^{U}})^{+}}),\\ ({({NC_{j}^{L}})^{+},({NC_{j}^{M}})^{+},({NC_{j}^{U}})^{+}})\\ \end{array}}\right\}$ (16) $\displaystyle\quad=\mathop{\max}\limits_{i}SF\left\{{\begin{array}[]{l}({({NYA% _{ij}^{L}}),({NYA_{ij}^{M}}),({NYA_{ij}^{U}})}),\\ ({({NYB_{ij}^{L}}),({NYB_{ij}^{M}}),({NYB_{ij}^{U}})}),\\ ({({NYC_{ij}^{L}}),({NYC_{ij}^{M}}),({NYC_{ij}^{U}})})\\ \end{array}}\right\}$ $\displaystyle NY_{i}=\left\{{\begin{array}[]{l}({({NYA_{ij}^{L}}),({NYA_{ij}^{% M}}),({NYA_{ij}^{U}})}),\\ ({({NYB_{ij}^{L}}),({NYB_{ij}^{M}}),({NYB_{ij}^{U}})}),\\ ({({NYC_{ij}^{L}}),({NYC_{ij}^{M}}),({NYC_{ij}^{U}})})\\ \end{array}}\right\}$ (17)

Step 4. Compute the TFNN cross-entropy (TFNNCE) between $NY^{+}$ and $NY_{i}$ as:

$\displaystyle\textit{TFNN-CE}({NY_{i},NY^{+}})=\sum\limits_{j=1}^{n}\omega_{j}% \left({\begin{array}[]{l}\left({\frac{\textit{NYA}_{ij}^{L}+2\textit{NYA}_{ij}% ^{M}+\textit{NYA}_{ij}^{U}}{4}}\right)\ln\frac{\frac{\textit{NYA}_{ij}^{L}+2% \textit{NYA}_{ij}^{M}+\textit{NYA}_{ij}^{U}}{4}}{\frac{({\textit{NYA}_{j}^{L}}% )^{+}+2({\textit{NYA}_{j}^{M}})^{+}+({\textit{NYA}_{j}^{U}})^{+}}{4}}\\ +\left({1-\frac{\textit{NYA}_{ij}^{L}+2\textit{NYA}_{ij}^{M}+\textit{NYA}_{ij}% ^{U}}{4}}\right)\ln\frac{1-\frac{\textit{NYA}_{ij}^{L}+2\textit{NYA}_{ij}^{M}+% \textit{NYA}_{ij}^{U}}{4}}{1-\frac{({\textit{NYA}_{j}^{L}})^{+}+2({\textit{NYA% }_{j}^{M}})^{+}+({\textit{NYA}_{j}^{U}})^{+}}{4}}\\ \end{array}}\right)+\sum\limits_{j=1}^{n}{\omega_{j}\left({\begin{array}[]{l}% \left({\frac{\textit{NYB}_{ij}^{L}+2\textit{NYB}_{ij}^{M}+\textit{NYB}_{ij}^{U% }}{4}}\right)\ln\frac{\frac{\textit{NYB}_{ij}^{L}+2\textit{NYB}_{ij}^{M}+% \textit{NYB}_{ij}^{U}}{4}}{\frac{({\textit{NYB}_{j}^{L}})^{+}+2({\textit{NYB}_% {j}^{M}})^{+}+({\textit{NYB}_{j}^{U}})^{+}}{4}}\\ +\left({1-\frac{\textit{NYB}_{ij}^{L}+2\textit{NYB}_{ij}^{M}+\textit{NYB}_{ij}% ^{U}}{4}}\right)\ln\frac{1-\frac{\textit{NYB}_{ij}^{L}+2\textit{NYB}_{ij}^{M}+% \textit{NYB}_{ij}^{U}}{4}}{1-\frac{({\textit{NYB}_{j}^{L}})^{+}+2({\textit{NYB% }_{j}^{M}})^{+}+({\textit{NYB}_{j}^{U}})^{+}}{4}}\\ \end{array}}\right)}+\sum\limits_{j=1}^{n}{\omega_{j}\left({\begin{array}[]{l}% \left({\frac{\textit{NYC}_{ij}^{L}+2\textit{NYC}_{ij}^{M}+\textit{NYC}_{ij}^{U% }}{4}}\right)\ln\frac{\frac{\textit{NYC}_{ij}^{L}+2\textit{NYC}_{ij}^{M}+% \textit{NYC}_{ij}^{U}}{4}}{\frac{({\textit{NYC}_{j}^{L}})^{+}+2({\textit{NYC}_% {j}^{M}})^{+}+({\textit{NYC}_{j}^{U}})^{+}}{4}}\\ +\left({1-\frac{\textit{NYC}_{ij}^{L}+2\textit{NYC}_{ij}^{M}+\textit{NYC}_{ij}% ^{U}}{4}}\right)\ln\frac{1-\frac{\textit{NYC}_{ij}^{L}+2\textit{NYC}_{ij}^{M}+% \textit{NYC}_{ij}^{U}}{4}}{1-\frac{({\textit{NYC}_{j}^{L}})^{+}+2({\textit{NYC% }_{j}^{M}})^{+}+({\textit{NYC}_{j}^{U}})^{+}}{4}}\\ \end{array}}\right)}$ (18)

Step 5. According to $\textit{TFNN-CE}({NY_{i},NY^{+}})$ , the smaller value of $\textit{TFNN-CE}({NY_{i},NY^{+}})$ is, the better alternative is.

4. The numerical example and comparative analysis

4.1 Numerical example

In recent years, driven by the Internet technology, cloud computing has been developed and applied rapidly. With the help of Internet technology, cloud computing can realize the sharing of software and hardware resources and make the resources available to other computers and devices. Some studies have shown that cloud computing will become the future trend of information technology and an important part of social utility infrastructure. Cloud computing is a service that is network-based, can be provided to users on demand, and achieves the separation of software and hardware infrastructure, which facilitates the use of services for enterprises and avoids the duplication of infrastructure. Cloud computing enables users to access services according to their needs and in a more flexible way through cloud services. In this new paradigm, users can enjoy services from any device and location, enabling a change in the way society works and business models. Due to the perceived convenience of cloud services, small and medium-sized enterprises have started to prefer using cloud services to build their business systems. With the widespread adoption of cloud services, a large number of cloud service providers have flooded into the market, and they offer cloud services with the same or similar functions. Therefore, how to choose the right product among many cloud service providers and select the best cloud service provider has become a hot issue that is currently studied by both academia and industry. Decision-making is widely present in the daily life of society, and the problem of cloud service provider selection can be summarized as the application of decision science in social practice. Therefore, the study of decision-making is of great importance. However, the cloud service provider selection can be attributed to the MADM problem. In this given section we provide a real numerical example for cloud service provider selection through cross-entropy method for TFNN-MADM. Assume that five cloud service provider $YB_{i}({i=1,2,3,4,5})$ to be chosen and four attributes to be assessed these urban and rural residents’ basic pension insurance systems: ⟀ YG ${}_{1}$ is cloud service features; ⟁ PY ${}_{2}$ is the supplier strength; ⟂ YG ${}_{3}$ is supplier service level; ⟃ YG ${}_{4}$ is supplier service quality. The five possible cloud service providers $YB_{i}({i=1,2,3,4,5})$ are to be evaluated by TFNNs under four attributes (attribute’s weight is (0.2, 0.3, 0.4, 0.1)).

The cross-entropy method is defined for TFNN-MADM to select the best cloud service provider.

Step 1. Build the TFNN matrix $YY=[{YY_{ij}}]_{m\times n}$ (See Table 1).

Table 1
The TFNN information

	YG ${}_{1}$	YG ${}_{2}$
YB ${}_{1}$	$\left\{{\begin{array}[]{l}({0.47,0.51,0.84}),\\ ({0.24,0.55,0.66}),\\ ({0.29,0.42,0.72})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.62,0.75,0.94}),\\ ({0.47,0.52,0.62}),\\ ({0.58,0.66,0.71})\\ \end{array}}\right\}$
YB ${}_{2}$	$\left\{{\begin{array}[]{l}({0.42,0.66,0.98}),\\ ({0.52,0.66,0.89}),\\ ({0.42,0.56,0.68})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.53,0.67,0.79}),\\ ({0.32,0.55,0.62}),\\ ({0.45,0.67,0.84})\\ \end{array}}\right\}$
YB ${}_{3}$	$\left\{{\begin{array}[]{l}({0.32,0.51,0.75}),\\ ({0.38,0.45,0.58}),\\ ({0.29,0.47,0.66})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.52,0.68,0.82}),\\ ({0.63,0.71,0.81}),\\ ({0.43,0.56,0.72})\\ \end{array}}\right\}$
YB ${}_{4}$	$\left\{{\begin{array}[]{l}({0.58,0.64,0.82}),\\ ({0.56,0.92,0.81}),\\ ({0.28,0.48,0.79})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.49,0.64,0.75}),\\ ({0.58,0.79,0.93}),\\ ({0.48,0.64,0.88})\\ \end{array}}\right\}$
YB ${}_{5}$	$\left\{{\begin{array}[]{l}({0.55,0.67,0.72}),\\ ({0.37,0.48,0.59}),\\ ({0.28,0.34,0.46})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.64,0.81,0.72}),\\ ({0.49,0.55,0.67}),\\ ({0.38,0.44,0.59})\\ \end{array}}\right\}$
	YG ${}_{3}$	YG ${}_{4}$
YB ${}_{1}$	$\left\{{\begin{array}[]{l}({0.32,0.51,0.75}),\\ ({0.65,0.56,0.67}),\\ ({0.25,0.48,0.59})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.61,0.82,0.94}),\\ ({0.22,0.37,0.59}),\\ ({0.26,0.35,0.48})\\ \end{array}}\right\}$
YB ${}_{2}$	$\left\{{\begin{array}[]{l}({0.41,0.53,0.62}),\\ ({0.35,0.52,0.74}),\\ ({0.46,0.51,0.68})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.56,0.60,0.81}),\\ ({0.44,0.75,0.97}),\\ ({0.52,0.73,0.81})\\ \end{array}}\right\}$
YB ${}_{3}$	$\left\{{\begin{array}[]{l}({0.27,0.42,0.65}),\\ ({0.32,0.56,0.87}),\\ ({0.62,0.91,0.33})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.51,0.82,0.36}),\\ ({0.46,0.64,0.97}),\\ ({0.72,0.91,0.83})\\ \end{array}}\right\}$
YB ${}_{4}$	$\left\{{\begin{array}[]{l}({0.33,0.96,0.42}),\\ ({0.53,0.64,0.75}),\\ ({0.51,0.65,0.82})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.42,0.56,0.86}),\\ ({0.45,0.81,0.63}),\\ ({0.38,0.53,0.67})\\ \end{array}}\right\}$
YB ${}_{5}$	$\left\{{\begin{array}[]{l}({0.72,0.81,0.32}),\\ ({0.56,0.65,0.82}),\\ ({0.42,0.54,0.64})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.61,0.73,0.85}),\\ ({0.28,0.54,0.68}),\\ ({0.37,0.44,0.75})\\ \end{array}}\right\}$

Step 2. Normalize the matrix $YY=[{YY_{ij}}]_{5\times 4}$ to $NY=[{NY_{ij}}]_{5\times 4}$ , for all the attributes are benefit, the normalization is not really needed.

Step 3. Determine the TFNNPIS (See Table 2).

Table 2

The TFNNPIS and TFNNNIS

	YG ${}_{1}$	YG ${}_{2}$
TFNNPIS	$\left\{{\begin{array}[]{l}({0.58,0.64,0.82}),\\ ({0.56,0.92,0.81}),\\ ({0.28,0.48,0.79})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.64,0.81,0.72}),\\ ({0.49,0.55,0.67}),\\ ({0.38,0.44,0.59})\\ \end{array}}\right\}$
	YG ${}_{3}$	YG ${}_{4}$
TFNNPIS	$\left\{{\begin{array}[]{l}({0.72,0.81,0.32}),\\ ({0.56,0.65,0.82}),\\ ({0.42,0.54,0.64})\\ \end{array}}\right\}$	$\left\{{\begin{array}[]{l}({0.61,0.73,0.85}),\\ ({0.28,0.54,0.68}),\\ ({0.37,0.44,0.75})\\ \end{array}}\right\}$

Step 4. Compute the TFNN-CE (See Table 3).

Table 3

The TFNN-CE

Alternatives	YG ${}_{4}$
$\textit{TFNN-CE}({YB_{1},\textit{TFNNPIS}})$	0.4021
$\textit{TFNN-CE}({YB_{2},\textit{TFNNPIS}})$	0.4326
$\textit{TFNN-CE}({YB_{3},\textit{TFNNPIS}})$	0.2678
$\textit{TFNN-CE}({YB_{4},\textit{TFNNPIS}})$	0.3089
$\textit{TFNN-CE}({YB_{5},\textit{TFNNPIS}})$	0.3657

Step 5. According to the $\textit{TFNN-CE}({YB_{i},\textit{TFNNPIS}})$ , the order is: $YB_{3}>YB_{4}>YB_{5}>YB_{1}>YB_{2}$ and $YB_{3}$ is the best alternative.

4.2 Compare TFNNCE with some existing methods

Then, the TFNN-CE method is compared with TFNNWA operator [47], TFNNWG operator [47], single-valued triangular Neutrosophic Dombi prioritized normalized Bonferroni mean (SVTNDPNBM) operator [49], TFNN-MABAC method [50], TFNN-VIKOR method [51]. The ranking order is shown in Table 4.

Table 4
Order through different methods

	Order
TFNNWA operator	$YB_{3}>YB_{4}>YB_{5}>YB_{1}>YB_{2}$
TFNNWG operator [47]	$YB_{3}>YB_{4}>YB_{5}>YB_{1}>YB_{2}$
SVTNDPNBM operator [49]	$YB_{3}>YB_{4}>YB_{1}>YB_{5}>YB_{2}$
TFNN-MABAC method [50]	$YB_{3}>YB_{4}>YB_{5}>YB_{1}>YB_{2}$
TFNN-VIKOR method [51]	$YB_{3}>YB_{4}>YB_{5}>YB_{1}>YB_{2}$
TFNN-CE method	$YB_{3}>YB_{4}>YB_{5}>YB_{1}>YB_{2}$

Comparing the results with TFNNWA operator [47], TFNNWG operator [47], single-valued triangular Neutrosophic Dombi prioritized normalized Bonferroni mean (SVTNDPNBM) operator [49], TFNN-MABAC method [50], TFNN-VIKOR method [51], the order results are slightly different, however, the best and worst alternative is same. Thus, the TFNN-CE is effective and reasonable.

5. Conclusion

There is a wide variety of cloud service providers on the market today. For enterprises, choosing the best provider and finding the best partner is crucial, which will directly affect the growth of the enterprise. Therefore, it is necessary to build a system to facilitate Therefore, it is necessary to build a system to facilitate the selection process and improve the efficiency of decision making. The system should have the ability to manage supplier information. Such as the system should have the function to manage the supplier information, such as some supplier information changes to be able to change the supplier information, found potential and good reputation of the supplier to be able to add to the system, and the supplier information should be changed. suppliers with potential and good reputation should be able to be added to the system, and if some suppliers are found to fail to meet the requirements of the enterprise, they can be deleted. information. In addition, the administrator user should have the authority to modify user information. In addition, the administrator user should have the authority to modify the user information, including the personnel adjustment of the evaluation experts. The system should include supplier information. In summary, the system should include the management of supplier information, the adjustment of evaluation experts and the calculation of the comprehensive evaluation value of the cloud service. The system should include the management of vendor information, adjustment of evaluation experts, and calculation of comprehensive evaluation value of cloud service providers. The evaluation metrics of cloud service providers are often not single, but based on multiple evaluation metrics together to determine the final evaluation results. The process of cloud service provider selection often involves multiple decision experts, i.e., the selection problem of cloud service providers is a multi-attribute decision problem. For this kind of problem, multi-attribute decision theory provides a reasonable and reliable solution. The multi-attribute group decision theory provides a reasonable and reliable solution for such problems. In this paper, the cross-entropy method under triangular fuzzy neutrosophic sets (TFNSs) is built based on the traditional cross-entropy method. Firstly, the TFNSs is introduced. Then, combine the traditional fuzzy cross-entropy method with TFNSs information, the cross-entropy method is established for MADM under TFNSs. Finally, a numerical example for cloud service provider selection has been given and some comparisons is used to illustrate advantages of cross-entropy method with TFNSs.

In the future, the new method can be applied to solve supply of chain management, failure mode and effect analysis, system control, etc. In addition, considering that the superiority of cross-entropy, it can be extended to other fuzzy sets, such as Trapezoidal neutrosophic set.

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Cross-entropy method for multiple attribute decision making under the triangular fuzzy neutrosophic sets

Abstract

Keywords

1. Introduction

2. Preliminaries

4.1 Numerical example

Table 1 The TFNN information

Table 4 Order through different methods

References

Table 1
The TFNN information

Table 4
Order through different methods