## 1 Introduction  
With the increasing importance of oil, the development of effective operations of an oil refinery has attracted wide attention. If an oil refinery is operated properly, one can reduce production and inventory cost [1], and may even increase profit $10 per ton of products or more [2]. A refinery is operated hierarchically with three levels: production planning, production scheduling, and process control. With mature planning techniques [3, 4], optimal production plans can be obtained by using linear programming-based commercial software [5]. Meanwhile, the efficiency of process control is improved by the installation of advanced control systems [6]. Production scheduling is at the middle level and should fulfill planning demands and instruct process control. A schedule defines and sequences detailed operation activities for the whole scheduling horizon, which makes the problem highly challenging [7]. Because of the extreme complexity of short-term scheduling, we find no effective techniques and software tools to deal with it. This scheduling job is still done manually in practice. To bridge the gap between the production planning and process control, the development of effective short-term scheduling techniques is of paramount importance. Mathematical programming models are applied to solve the crude oil scheduling problem. They can be categorized as discrete-time and continuous-time ones. The models in [8, 9, 10, 11] are representative of the former. They are formulated by dividing the scheduling horizon into a number of time intervals with uniform time durations. An event, such as the start or end of an operation, should happen at the boundary of a time interval. Such a model has a major defect in that the time interval must be sufficiently small to describe the scheduling problem with acceptable accuracy. As a result, a model for real-life applications usually contains a computationally infeasible number of binary variables and constraints. To overcome this disadvantage, some researchers choose continuous-time models [12; 13; 14]. Although their use reduces the number of discrete variables, nonlinear constraints are introduced such that the problem is extremely difficult to solve. Moreover, to make the problem solvable, most discrete-time and continuous-time models make special assumptions that, unfortunately, lead to a low-quality or unrealistic solution for real-world cases [15]. With a very detailed schedule being required, the scheduling problem of crude oil operations becomes incredibly difficult [7]. To reduce the computational complexity, this problem is studied in the viewpoint of control theory by extending the Petri net models applied in discrete manufacturing systems [16; 17; 18; 19; 20]. By this approach, a two-level operational architecture for the short-term scheduling problem of crude oil operations is proposed to decompose the problem into two sub-problems [21; 22; 23; 24]. At the upper level, a refining schedule is specified for distillers without involving detailed operations to optimize some objectives, such as maximizing production rate. It determines the types, amount, processing rate and sequence of crude oil to be processed for each distiller. At the lower level, a detailed operation schedule is provided, such as charging tank charging/discharging and oil transportation via a pipeline, to realize a refining schedule. Based on this architecture, to ensure the feasibility of an obtained schedule, the dynamic behavior of crude oil operations is modeled by a hybrid Petri net (PN) and schedulability conditions are derived for different cases [21; 22; 25; 26; 27; 28]. With the schedulability conditions as constraints, an optimal and realizable refining schedule can be efficiently found [29; 30], while a detailed schedule is obtained recursively. Note that these studies are conducted under the assumption that there are enough charging tanks in the system.