## 4 Bi-Objective Optimization Allocation Problem Model and Algorithm
### Industrial Robot-Monitoring Workflow Assignment Problem (IRMWAP) Formulation
\begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Definition and Description** \\ \hline memory\({}_{\text{i}}\) & The execution memory space of \(\text{t}_{\text{i}}\) \\ \hline pattern\({}_{\text{ij}}\) & The workflow pattern of \(\text{e}_{\text{ij}}\), including Sequence, Parallel Split, Synchronization, Exclusive Choice, Simple Merge \\ \hline data\({}_{\text{ij}}\) & The transfer data of \(\text{e}_{\text{ij}}\) \\ \hline comSpeed\({}_{\text{x}}\) & The computing speed of \(\text{v}_{\text{x}}\), cloud center \(>\) edge server \(>\) monitor \\ \hline latency\({}_{\text{x}}\) & The latency between monitor and \(\text{v}_{\text{x}}\) \\ \hline comCost\({}_{\text{x}}\) & The computing cost of \(\text{v}_{\text{x}}\), edge server \(>\) cloud center \(>\) monitor = 0 \\ \hline comCap\({}_{\text{x}}\) & The computing capacity of \(\text{v}_{\text{x}}\), cloud center \(>\) edge server \(>\) monitor \\ \hline bandwidth\({}_{\text{x}}^{\text{up}}\) & The upload bandwidth of \(\text{v}_{\text{x}}\), cloud center \(>\) edge server \(>\) monitor \\ \hline bandwidth\({}_{\text{x}}^{\text{down}}\) & The download bandwidth of \(\text{v}_{\text{x}}\), cloud center \(>\) edge server \(>\) monitor \\ \hline l\({}_{\text{xy}}\) & The communication latency of \(\text{f}_{\text{xy}}\), monitor to cloud center \(>\) monitor to edge server \\ \hline commCost\({}_{\text{xy}}\) & The communication cost of \(\text{f}_{\text{xy}}\), monitor to cloud center \(>\) monitor to edge server \\ \hline commCap\({}_{\text{xy}}\) & The communication capacity of \(\text{f}_{\text{xy}}\), cloud center to edge server \(>\) monitor to cloud center \(>\) monitor to edge server \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._The total optimization objective of cost includes communication cost \(\bar{F}_{\text{xiyiyi}}\) and computation cost \(\text{c}^{c}_{\text{xi}}\), which depends on the pricing strategy of the service provider and the usage of users. \[\text{C}=\sum\bar{F}_{\text{xiyi}}*\text{d}_{\text{xi}}*\text{d}_{\text{yi}}+ \sum\text{c}^{c}_{\text{xi}}*\text{d}_{\text{xi}}\]  
\[\bar{F}_{\text{xiyi}}^{c}=\text{data}_{\text{qi}}*\text{commCost}_{\text{xy}} \tag{2}\]  
\[c^{c}_{\text{xi}}=\frac{\text{instruction}_{\text{xi}}}{\text{comSpeed}_{\text {x}}^{c}}*\text{comCost}_{\text{x}}\]  
Each task can be allocated only one computing resource and each computing resource can perform multiple tasks in FIFO mode without exceeding its computing capacity. The data dependency between each pair of tasks cannot exceed the limit of communication capacity. \[\text{d}_{\text{xi}}=\left\{ \begin{array}{cc}1&\text{t}_{\text{i}}\text{ running in }\text{v}_{\text{x}},\\ 0&\text{others}.\end{array} \right. \tag{3}\]  
\[\sum_{\text{x}=1}^{\text{n}}\text{d}_{\text{xi}}=1 \tag{4}\]  
\[\sum\text{d}_{\text{xi}}*\text{instruction}_{\text{i}}\leq\text{comCap}_{ \text{x}} \tag{5}\]  
\[\sum\text{d}_{\text{xi}}*\text{d}_{\text{yi}}*\text{data}_{\text{ij}}\leq \text{commCap}_{\text{xy}} \tag{6}\]  
The above can be summarized as a Bi-objectives Generalized Quadratic Assignment Problem IRMWAP:  
\[\text{min}:\text{T},\text{C}\]  
\[\text{subject to}:\text{(3)},\text{(4)},\text{(5)},\text{(6)} \tag{7}\]  
Proof of Theorem 1.: From the above equation, it is easy to find that IRMWAP optimizes each objective in accordance with the definition of generalized quadratic assignment problem (GQAP), and GQAP has been proved to be an NP-hard problem [25], then IRMWAP is proved as an NP-hard problem.