An improved Lagrangian relaxation approach to scheduling steelmaking-continuous casting process

- We study a mixed integer mathematical model for the scheduling in the steelmaking-continuous casting process.
- The concave-convex procedure is introduced to deal with the subproblems of the relaxed problem.
- The convergence of the concave-convex procedure is established under some appropriate assumptions.
- An improved conditional surrogate subgradient algorithm is proposed to solve the Lagrangian dual problem.
- A simple heuristic algorithm is designed to construct a feasible schedule by adjusting the solutions of the relaxed problem.

In the steelmaking continuous-casting (SCC) process, scheduling problem is a key issue for the iron and steel production. To improve the productivity and reduce material consumption, optimal models and approaches are the most useful tools for production scheduling problems. In this paper, we firstly develop a mixed integer nonlinear mathematical model for the SCC scheduling problem. Due to its combinatorial nature and complex practical constraints, it is extremely difficult to cope with this problem. In order to obtain a near-optimal schedule in a reasonable computational time, Lagrangian relaxation approach is developed to solve this SCC scheduling problem by relaxing some complicated constraints. Owing to the existence of the nonseparability coming from the product of two binary variables, it is still hard to deal with this relaxed problem. By making use of their characteristics, the subproblems of the relaxed problem can be converted into different difference of convex (DC) programming problems, which can be solved effectively by using the concave-convex procedure. Under some reasonable assumptions, the convergence of the concave-convex procedure can be established. Furthermore, we introduce an improved conditional surrogate subgradient algorithm to solve the Lagrangian dual (LD) problem and analyze its convergence under some appropriate assumptions. In addition, we present a simple heuristic algorithm to construct a feasible schedule by adjusting the solutions of the relaxed problem. Lastly, some numerical results are reported to illustrate the efficiency and effectiveness of the proposed method.