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train scheduling by erfan

Journal article . Deadline 2020-03-19
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Optimization-based bound tightening has recognized as a practical approach to solve large-size instances of the MILP models (Gleixner et al., 2017). In this study, the preprocessing technique attempt to strengthen the basic mathematical model (TTP1) by introducing tight lower bounds. It refers to the methods for reducing the number of binary variables and the number of constraints used to model a given problem instance. Given the mathematical model presented in section 4, the problem is formulated as a MILP, in which thousands of constraints have to be satisfied, and a high number of variables take integer values. Due to the computational complexity of the problem, a variable fixing technique is proposed that has been applied to the mathematical model before the primary optimization model starts. The primary aim is to reduce the number of integer variables to cope with the complexity of the underlying mathematical model. Also, it is still challenging to find the optimal solutions for the basic MILP model because of the existence of big-M constraints, represented in equations (14), (15), and (16). It could be highly advantageous for MILP solvers to deal with a lower number of big-M constraints (Belotti et al., 2016). It is experimentally demonstrated that the existence of such constraints may result in increased computational complexity, making the basic optimization model impractical. Here, the partial variable fixing technique provides bounds for the decision variables associated with the train skip-stop pattern. In the result section, it will be shown that the performance of the partial fixing procedure for skip-stop variables, i.e., , has a direct influence on the problem complexity. Based on preliminary numerical experiments, the partial variable fixing procedure can reduce the number of redundant Big-M constraints.