A sequential reduction algorithm for the large-scale fixed-charge network flow problems

The fixed-charge network flow problem (FCNFP) is widely used in industrial production and can be exactly solved by converting to mixed-integer linear programming (MILP). However, the long solving time of MILP solvers limits their applicability to large-scale problems. This paper proposes a sequential reduction algorithm called adaptive dynamic slope scaling procedure (ADSSP). ADSSP introduces an adaptive edge deletion strategy to improve solution quality and efficiency. Theoretical analysis proves the algorithm convergence and provides the best-case and worst-case time upper bounds of ADSSP. Numerical experiments show that ADSSP outperforms the previous linear programming-based iterative algorithms on different scales. In instances with hundreds of thousands of variables, ADSSP achieves a 2% improvement in the objective and takes only 3% solving time compared to the Cplex MILP Solver. The results demonstrate that ADSSP can significantly improve the solution quality with high efficiency for large-scale FCNFP. As a widely applicable optimization method, ADSSP can be a valuable tool for optimizing FCNFP and other similar problems.