## 2 Related Literature
The authors propose a TI MILP to minimize simultaneously carbon footprint and TEC with machines having different consumption levels. Their study suggests that a trade-off between electricity costs and CO\({}_{2}\) emissions appears when the energy providers are coal-based. As in [7; 12], the assumption on time-dependent CO\({}_{2}\) emissions and electricity costs holds for this paper. OAS is a particular scheduling problem where the decision covers the selection of a subset of orders, among \(n\), and their sequencing in a capacity-constrained production system. Typically, this entails a fixed time frame to complete orders and an associated cost-driven event where the company fails to produce within the time-window. The solution space of OAS problem extends classical scheduling one, as jobs can be accepted or not. Indeed, at worst the number of possible solutions in OAS problem is \(\sum_{k=1}^{n}k!\), where all the \(k\)-permutations of \(n\) without repetition are considered, whereas only \(n!\) solutions form the solution space in classical scheduling problems. In the literature, for both single- or multi-machine systems, a variety of configurations of problems are considered such as sequence-dependent setup times [9; 28; 29], preemption [11] or resource constraints [10; 30]. As for our research, the immediate related works are those presented in [9; 28; 29]. A comprehensive survey [31] presents an overview on OAS problems, while in [32], a focus is made on scheduling problem with rejection. OAS involves mainly economic-related criteria, primarily embodied by the maximization of the total profit generated by the orders. Service level [33], percentage of accepted orders or utilization can also be maximized in OAS problems. In addition, cost penalties can be integrated in the objective function when tardiness or order rejection occur. For instance, [PERSON] et al. [9] maximize the total profit of accepted orders minus their possible tardiness penalties. As in scheduling problem, OAS solving methods involve exact and heuristic approaches. MILP [9; 28; 29], dynamic programming [34] and branching methods [35] have been employed for various OAS problems. In the meantime, as these problems are mostly NP-hard, a wide range of metaheuristics from local search to EA have been utilized and have shown very robust performances [36; 37; 38]. Besides, reports in the literature describe an order-based and a time-based FR heuristics applied to an OAS problem under resource constraints [30] that both achieve a tighter gap for large instances. A recent work of [PERSON] et [PERSON] [39] proposes a two-phase matheuristic that exploit a time-indexed model. First, they assign orders to time segments using the relaxed version of their model and generate a schedule subsequently; the solution is then improved by a VNS. This process is repeated until the termination criterion is met. However, energy aspects are not considered in their work. Literature on OAS considering resource constraints and/or energy aspects is very sparse. [PERSON] [10] tackles an resource-constrained OAS problem with the objective to maximize profit with rejection penalties using an EA and a priority rule heuristic. [PERSON] et al. [40] maximize the net revenue of a parallel machines system with order acceptance and a global constraint on the energy consumed by machines and their launch budget. In their work, a comparative analysis between diverse variable neighbor search algorithms and a dynamic programming approach is conducted. Considering electricity tariffs, to the best of our knowledge, three papers have been reported [12; 13; 41]. These papers follow up the works in [12; 13] that both investigate the OAS problem under TOU and CO2 emissions periods and sequence-dependent setup times with a disjunctive formulation in [12] and an ATI model in [13]. Moreover, this paper contributes to the comprehension of the OAS under energy aspects by introducing approached solving methods that improve the existing results.