## 3 Problem Description  
The OAS with sequence-dependent setup times, release date under TOU costs and taxed carbon periods is investigated in this paper. Parameters and notations are detailed in this section (Table 1). Each order \(j=1,\ldots,n\) is completely defined by its processing time \(p_{j}\) release date \(r_{j}\), due date \(d_{j}\), deadline \(d_{j}\), revenue \(e_{j}\), power consumption \(\Omega_{j}\) and tardiness penalties \(w_{j}\). In addition, a sequence-dependent setup time \(s_{ij}\) is defined between any pair of orders \(i\) and \(j\). A dummy order \(0\) is introduced in order to start the sequence. Each of its properties are set to zero except its setup time \(s_{0j}\) between any order \(j\). An order \(j\) is accepted when it is sequenced in the span ranging from its release date \(r_{j}\) to its deadline \(\bar{d}_{j}\) and rejected otherwise. A tardiness penalty \(w_{j}\) is subtracted to an order \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Units** \\ \hline \(n\) & Number of orders & \\ \(T\) & Planning horizon & min \\ \(p_{j}\) & Processing time of order \(j=0,\ldots,n\) & min \\ \(r_{j}\) & Release date of order \(j=0,\ldots,n\) & min \\ \(d_{j}\) & Due date of order \(j=0,\ldots,n\) & min \\ \(d_{j}\) & Deadline of order \(j=0,\ldots,n\) & min \\ \(e_{j}\) & Revenue generated by order \(j=0,\ldots,n\) & \$ \\ \(\Omega_{j}\) & Power consumption of order \(j=1,\ldots,n\) & kWh \\ \(w_{j}\) & Tardiness penalties of order \(j=0,\ldots,n\) & min \\ \(s_{ij}\) & Setup-times between order \(i=0,\ldots,n\) and order \(j=1,\ldots,n\) & min \\ \(m\) & Number of TOU intervals & \\ \(b_{k}\) & Starting time of TOU interval \(k=1,\ldots,m\) & min \\ \(EC_{k}\) & Electricity cost of TOU interval \(k=1,\ldots,m\) & \$/kWh \\ \(h\) & Number of CO\({}_{2}\) emission intervals & \\ \(g_{l}\) & Starting time of CO\({}_{2}\) interval \(l=1,\ldots,h\) & min \\ \(q_{l}\) & Amount of CO\({}_{2}\) emission per kWh in interval \(l=1,\ldots,h\) & kg/kWh \\ \(Tax\) & Tax per kg of CO\({}_{2}\) emitted & \$/kg \\ \(c_{jl}\) & Energy cost of order \(j=1,\ldots,n\) at time \(t=r_{j},\ldots,\bar{d}_{j}\) & \$ \\ \(f_{ji}\) & Profit of order \(j=1,\ldots,n\) at time \(t=r_{j},\ldots,\bar{d}_{j}\) & \$ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters nomenclature. revenue \(e_{j}\) for each time unit beyond its due date \(d_{j}\). In Figure 1, \(w_{j}\) represents the slope of the revenue decay between \(d_{j}\) and \(d_{j}\). Moreover, in the original work, the planning horizon is divided into intervals with fluctuating TOU tariffs and CO\({}_{2}\) emissions. Each TOU interval \(k=1,\ldots,m\) is characterized by a starting time \(b_{k}\) and an electricity cost \(EC_{k}\). Each CO\({}_{2}\) emissions interval \(l=1,\ldots,h\) is determined by a starting time \(g_{l}\) and an amount of CO\({}_{2}\) per kg and a _Tax_ per emitted kg of carbon. As in [7,12], by assumption, CO\({}_{2}\) emissions are time-dependent, i.e., the emitted amount fluctuates over the day as the employed power sources are coal-based during off-peak hours and gas-based during mid-peak and on-peak hours. In this problem, the objective is to maximize the revenue minus tardiness penalties and energy costs. For simplification reasons, the energy costs can be calculated at each time-slots rather than at each intervals, especially as TOU and CO\({}_{2}\) emissions intervals partition differently the horizon. The energy cost \(c_{ji}\) for any order \(j=1,\ldots,n\) at any period \(t=r_{j},\ldots,\bar{d_{j}}\) is thus computed with the formula given in Equation (1). \[c_{ji}=\frac{\Omega_{j}}{60}(\sum_{k=1}^{m}EC_{k}\bar{1}_{b_{k-1}\leq t<b_{k}}+ Tax\sum_{l=1}^{h}q_{l}\bar{1}_{g_{l-1}\leq t<g_{l}}) \tag{1}\]  
The energy cost of each time period \(t\) and for each order \(j\) corresponds to the sum of the respective TOU and CO\({}_{2}\) taxed emissions costs of the examined period \(t\) multiplied by the order's energy consumption expressed into minutes.