# An ANN-Based Approach for Real-Time Scheduling in Cloud Manufacturing
## 1 Introduction
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Description** \\ \hline \(b_{j}^{(c)}\) & The processing start time of \(\mathbb{T}_{j}^{(c)}\). \\ \(c\) & Label of MP (\(c\in\mathcal{C}\)). \\ \(d_{j}^{(c)}\) & The ideal due time of \(\mathbb{T}_{j}^{(c)}\), its upper and lower bound are \(\overline{d}_{j}^{(c)}\) and \(\underline{d}_{j}^{(c)}\). \\ \(e_{j}^{(c)}\) & The finish time of \(\mathbb{T}_{j}^{(c)}\). \\ \(\delta_{j}^{(c)}\) & The ready time of \(\mathbb{T}_{j}^{(c)}\). \\ \(i\) & Label of MP (\(i\in\mathcal{I}_{k}\)). \\ \(j\) & Label of MT (\(j\in\mathcal{J}_{c}\)). \\ \(k\) & Label of MS (\(k\in\mathcal{K}\)). \\ \(l_{j}^{(c)}\) & The logistics duration of \(\mathbb{T}_{j}^{(c)}\). \\ \(p_{j}^{(c)}\) & The processing duration of \(\mathbb{T}_{j}^{(c)}\). \\ \(p_{j}^{(c)}\) & The required resource capacity of \(\mathbb{T}_{j}^{(c)}\). \\ \(x_{ij}^{(c)}\) & 0–1 decision variable to indicate if \(\mathbb{T}_{j}^{(c)}\) will be allocated to \(\mathbb{R}_{j}^{(k)}\) or not. \\ \(\mathbf{y}(j,c,i,k)\) & The actual completion status of \(\mathbb{T}_{j}^{(c)}\) after processed in \(\mathbb{R}_{i}^{(k)}\). \\ \(\mathbf{\hat{y}}(j,c,i,k)\) & The predicted completion status of \(\mathbb{T}_{j}^{(c)}\) after processed in \(\mathbb{R}_{i}^{(k)}\). \\ \(B_{c}\) & The publish time of \(\mathbb{P}_{c}\). \\ \(C_{i}^{(k)}\) & The unit cost of \(\mathbb{R}_{i}^{(k)}\). \\ \(D_{i}^{(k)}(\mathbb{T}_{j}^{(c)})\) & The preference function of \(\mathbb{T}_{j}^{(c)}\) allocated on \(\mathbb{R}_{i}^{(k)}\). \\ \(L_{c,i}^{(k)}\) & The logistics duration between \(\mathbb{P}_{c}\) and \(\mathbb{S}_{i}\). \\ \(L_{c}\) & The logistics duration vector of \(\mathbb{P}_{c}\), each element corresponds to a type-matched MR. \\ \(\mathcal{M}_{i}^{(k)}(t)\) & The in-processing set of MT in \(\mathbb{R}_{i}^{(k)}\) at time \(t\). \\ \(\mathbb{P}_{c}\) & MP labeled in \(c\). \\ \(p_{i}^{(k)}\) & The service quality of \(\mathbb{R}_{i}^{(k)}\). \\ \(\mathcal{P}_{j}^{(c)}\) & The immediate predecessor set of \(\mathbb{T}_{j}^{(c)}\). \\ \(\mathcal{Q}_{i}^{(i)}(t)\) & The processing pending queue of MTs in \(\mathbb{R}_{i}^{(k)}\) at time \(t\). \\ \(\mathbb{R}_{j}^{(k)}\) & MR labeled in \((i,k)\). \\ \(\mathbb{S}_{k}\) & MS in type \(k\). \\ \(\mathcal{S}_{j}^{(c)}\) & The immediate successor set of \(\mathbb{T}_{j}^{(c)}\). \\ \(\mathbb{T}_{j}^{(c)}\) & MT labeled in \((j,c)\). \\ \(V_{i}^{(k)}(t)\) & The maximum available capacity of \(\mathbb{R}_{i}^{(k)}\) at time \(t\), its upper bound is \(\overline{V}_{i}^{(k)}\). \\ \(\mathcal{Z}_{j}^{(c)}\) & The object value of \(\mathbb{P}_{c}\), \(x\in\{d,m,u\}\) represents [make-span, cost, quality]. \\ \(\kappa_{j}^{(c)}\) & The required service type of \(\mathbb{T}_{j}^{(c)}\). \\ \(\mu_{i,x}\) & The mean value of MTs in queue, \(x\in\{g,p,l,v,\underline{d},\overline{d}\}\) represents the feature mark. \\ \(\sigma_{i,x}^{(c,l)}\) & The standard deviation value of MTs in queue, \(x\in\{g,p,l,v,\underline{d},\overline{d}\}\) represents the feature mark. \\ \(\varphi_{i,n}^{(c,l)}\) & The probability for due time value located in interval \(n\) of \(\mathbb{T}_{j}^{(c)}\) if \(\mathbb{R}_{i}^{(k)}\) is allocated (\(n=0,1,2\)). \\ \(\mathbb{I}_{\text{predicate}}\) & Indicator function (\(\mathbb{I}_{\text{predicate}}=1\) if the predicate is true, else 0). \\ \(|\mathcal{S}|\) & The element count of set \(\mathcal{S}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nomenclature. Providers arrange MTs using professional software and deliver the completed MTs to demanders. Therefore, the scheduling problem in the CMfg environment is as follows: how to reasonably allocate the emerging MTs to the type-matched MRs in real-time under the constraints of task precedence, resource occupation, and logistics duration while reducing the service costs, improving the service quality, and shortening the make-span for each MP? Since the scheduling environment of CMfg is larger and more dynamic than that of the traditional workshop [17; 18], it is much more difficult to find the optimal solution.