## 2 Background
### The TOPSIS Method  
The TOPSIS method for multiple attribute group decision making was developed by [PERSON] and [PERSON] [12]. It has subsequently been used to evaluate the purchase of business intelligence systems, product quality improvements, flow control in a manufacturing system, and intelligent home energy management in smart grids [30; 31; 32; 33].  
In this paper, a combination of the TOPSIS method as introduced by [PERSON] and [PERSON] [12], and the implementation described by [PERSON] et al. [34] is applied for evaluation. The calculation steps are summarized as follows:  
_Step 2.1: Establishing an IAMS evaluation decision matrix._  
The IAMS decision matrix, \(S\), is defined as below,  
\[S=\begin{array}{c}a_{1}\\ a_{2}\\ \vdots\\ a_{i}\\ \vdots\\ a_{m}\end{array}\begin{array}{c}\begin{pmatrix}b_{1}&b_{2}&\cdots&b_{j}& \cdots&b_{n}\\ \end{pmatrix}\\ x_{11}&x_{12}&\cdots&x_{1j}&\cdots&x_{1n}\\ x_{21}&x_{22}&\cdots&x_{2j}&\cdots&x_{2n}\\ \vdots&\vdots&\cdots&\vdots&\vdots&\vdots\\ x_{i1}&x_{22}&\vdots&x_{ij}&\vdots&x_{in}\\ \vdots&\vdots&\cdots&\vdots&\vdots&\vdots\\ x_{m1}&x_{m2}&\cdots&x_{mj}&\cdots&x_{mm}\\ \end{array} \tag{13}\]  
where \(a_{i}\) denotes the evaluated online manufacturing IAMSs, \(b_{j}\) represents the criterion evaluated in _the online manufacturing IAMSs, \(i=1\), \(2\),..., \(m\), \(j=1\), \(2\),..., \(n\), and \(x_{ij}\)_ indicates the performance rating of an evaluated online manufacturing IAMS \(a_{i}\) with respect to criterion \(b_{j}\).  
_Step 2.2: Data Normalization._  
Data is transformed to a normalized scale as follows:  
\[\text{for benefit criteria}_{ij}=\frac{x_{ij}}{\sqrt{\sum\limits_{i=1}^{m}x_{ ij}^{2}}} \tag{14}\]\[\text{for cost criteria }r_{ij}=\frac{1/x_{ij}}{\sqrt{\sum\limits_{i=1}^{m}1/x_{ij}^{2}}} \text{for }i=1,\ 2,\ \ldots,\ m;j=1,\ 2,\ \ldots,\ n \tag{15}\]  
_Step 2.3: Establishing a weighted normalization matrix_  
In the TOPSIS framework, the weighted normalized performance matrix is defined as  
\[V=\left[\begin{array}{cccccc}v_{11}&v_{12}&\cdots&v_{1j}&\cdots&v_{1n}\\ v_{21}&v_{22}&\cdots&v_{2j}&\cdots&v_{2n}\\ \vdots&\vdots&\cdots&\vdots&\vdots&\vdots\\ v_{i1}&v_{i2}&\vdots&v_{ij}&\vdots&v_{in}\\ \vdots&\vdots&\cdots&\vdots&\vdots&\vdots\\ v_{m1}&v_{m2}&\cdots&v_{mj}&\cdots&v_{mn}\end{array}\right] \tag{16}\]  
\(v_{ij}=w_{j}\times x_{ij}\), for \(i=1,\ 2,\...,\ m;\ j=1,\ 2,\...,\ n\),  
where \(w_{j}\) denotes the weight of criterion \(j\).  
_Step 2.4: Calculating the separation measures_  
To calculate the separation measures, which, in this study, characterize an IAMS's proximity to optimal performance, the ideal solution is first calculated as follows  
\[d^{+}=(v_{1}^{+},\ v_{2}^{+},\ \cdots,\ v_{n}^{+}),\ \text{where}\ v_{j}^{+}= \max\limits_{i}v_{ij} \tag{17}\]  
\[d^{-}=(v_{1}^{-},\ v_{2}^{-},\ \cdots,\ v_{n}^{-}),\ \text{where}\ v_{j}^{-}= \min\limits_{i}v_{ij} \tag{18}\]  
The distance between the ideal solution and the negative ideal solution for each alternative is then calculated as  
\[d_{i}^{+}=\sqrt{\sum\limits_{j=1}^{n}\left(v_{ij}-v_{j}^{+}\right)^{2}}\quad i =1,2,\ \ldots,\ m, \tag{19}\]  
\[d_{i}^{-}=\sqrt{\sum\limits_{j=1}^{n}\left(v_{ij}-v_{j}^{-}\right)^{2}}\quad i =1,2,\ \ldots,\ m, \tag{20}\]  
_Step 2.5: Calculating the relative closeness coefficient and rank the center preference order_  
The relative closeness to the ideal solution of each online manufacturing IAMSs can be calculated as:  
\[a_{i}^{*}=\frac{d_{i}^{-}}{d_{i}^{+}+d_{i}^{-}},\ \text{for}\ i=1,2,\ \ldots,\ m \tag{21}\]