## 2 Background
### The CEGRA Method  
The CEGRA method can use decision-making and information evaluation to incompletely explore the degree of association between two series, quantify the distance between observed and target values using discrete measures, conduct correlation analyses, establish models, and enable decision making between various schemes. CEGRA can effectively manage uncertainty, multivariate input, and discrete data in decision problems [25; 26; 27]. In this study, novel algorithms were designed to assess the decision making of an IAMS for the online manufacturing of sports shoes. The implementation of the algorithms used in this study are based on the CEGRA and grey situation decision-making system proposed by [PERSON] [11]. The steps in the algorithms are described as follows:  
_Step 1.1: Defining evaluations valuable for assessing online sports shoe online manufacturing IAMSs._  
Let (_EPF_, _X_) be the exploiting resource space when _EPF_ is equipped with mapping functions as follows:  
\[\text{EPF: Resources}\rightarrow\text{efficacies (v)} \tag{1}\]  
In the above, a general consensus among experts has been reached to establish a model, facilitating the ultimate goal of evaluating the performance of collaboration between technology companies according to CEGRA concepts. _Step 1.2: Categorizing data and evaluating performance._  
Data from the IAMSs can be categorized as data resources or resources possessing information, \(\theta(d)\), as shown below:  
\[\theta(d)=\{v(\bullet),value;s(\bullet),sign;p(\bullet),polarity;b(\bullet),background;c( \bullet),comotation\}\]  
Otherwise, only digits exist, without a resource. If \(dec_{o}(\bullet)\) is the entirety of complete decision-making information for event \(o\), the following data category is obtained:  
\[\text{Decision-making data}:\ \theta(d)\in dec_{o}(\bullet) \tag{2}\]  
_Step 1.3: Finding patterns based on grey resource theory._  
In grey resource theory, the situation, \(S_{ij}=(a_{i},\,b_{j})\), is defined as a pattern, where \(a_{i}\) is a decision maker and \(b_{j}\) is a resource. Hence, the effectiveness of \(a_{i}\) and \(b_{j}\) can expressed using \(S_{ij}\). _GM_ is defined as a mapping function, \(GM\big{(}S_{ij}\big{)}=R_{ij}\), where \(R_{ij}\) is the value of the mapping function, such that,  
\[S_{ij}=(a_{i},\,b_{j}),\,S_{ij}=S_{GM}\ \text{(grey modeling pattern)} \tag{3}\]Hence, if the mapping of an event, \(E_{ij}\), is defined as \(GM\Big{(}E_{ij}\Big{)}=R_{ij}\), the synthetic measured effectiveness value for this event is defined as:  
\[R_{i}^{\Sigma}=\frac{1}{n}{\sum_{j=1}^{n}R_{ij}} \tag{4}\]  
_Step 1.4: Contrasting cause-effect incidences and normalizing data._  
In the cause-effect space \(\Big{(}P_{aux},P_{eff}\Big{)}\), a cause-effect relationship can only be obtained between individuals with cause-effect incidences that are essential contrasts; no contrasts are required for irrelevant individuals. In this study, cause-effect incidence contrasting was defined as follows. First, let \(\alpha\) and \(\beta\) be the contrasted individuals, \(\alpha,\beta\in P\). \(\mu=\Delta=\alpha-\beta\) then represents a comparison gene, and for series comparison, \(\Delta=\Delta_{0i}(k)=\big{|}\chi_{0}(k)-\chi_{i}(k)\big{|}\), where \(\Delta\to 0\) denotes melt and \(\Delta\to\infty\) indicates irrelevance. The contrasted result, \(v\) is given as follows:  
\[v=\frac{\varsigma+\pi}{\mu+\pi}\,,\;\varsigma\to 0 \tag{5}\]  
If \((\alpha^{c},\beta)\) is considered a cause-effect related contrast, then \(\alpha>>\beta\) or \(\beta<<\alpha\Rightarrow\mu\to\infty\) if \(\alpha\) and \(\beta\) are irrelevant, such that \(v\to 0\). Alternatively, if \(\alpha\) and \(\beta\) are incident in terms of cause and effect, then \(\alpha\to\beta\Rightarrow\mu\to 0\Rightarrow v\to 1\).