A K-Prototype Clustering Assisted Hybrid Heuristic Approach for Train Unit Scheduling

Abstract

This paper presents a K-Prototype assisted hybrid heuristic approach called SLIM+KP for solving large instances of the Train Unit Scheduling Optimization (TUSO) problem. TUSO is modelled as an Integer Multi-commodity Flow Problem (IMCFP) based on a Directed Acyclic Graph (DAG). When the problem size goes large, the exact solver is unable to solve it in reasonable time. Our method uses hybrid heuristics by iteratively solving reduced instances of the original problem where only a subset of the arcs in the DAG are heuristically chosen to be optimised by the same exact solver. K-Prototype is a clustering method for partitioning. It is an improvement of K-Means and K-Modes to handle clustering with the mixed data types. Our approach is designed such that the arcs of the DAG are clustered by K-prototype and each time only a small fraction of the arcs are selected to form the reduced instances. The capabilities of this framework were tested by realworld cases from UK train operating companies and compared with the results from running an exact integer solver. Preliminary results indicate the the proposed methodology achieves the same optimal solutions as the exact solver for small instances but within shorter time, and yields good solutions for instances that were intractable for the exact solver.

An auxiliary hybrid heuristic approach for objective function design evaluation

“…The real case is usually that the solutions with very similar or even the same objective function value have rather different structural properties. This phenomenon is also observed in experiments on Train Unit Scheduling Optimization (TUSO) Kwan, 2014, 2016b;Copado-Mendez et al, 2017;Lei et al, 2018), which is further explained in Section 4.1.…”

“…Like in crew scheduling, challenges in solving large scale instances also occur in the TUSO problem. In Copado-Mendez et al (2017), an HH approach named Size Limited Iterative Method (SLIM) is proposed, where arcs in a DAG represent train unit connection opportunities. Since the exact solver (RS-Opt) (Lin and Kwan, 2016b) is only able to solve small to medium sized instances in a reasonable time, SLIM will reduce problem instances by removing the majority of the arcs in the DAG while still ensuring the existance of feasible solutions and the quality of the series of solutions based on reduced instances will be improving until convergence.…”

“…CMSA and Extract-and-Augment share many common features and design philosophy, especially in the ways they use an exact solver to find out high quality components by solving reduced instances (Lin and Kwan, 2016b). A hybrid heuristic method named Size Limited Iterative Method (SLIM) has also been developed for larger instances (Copado-Mendez et al, 2017).…”

“…P HH will stop when it is believed that the best reduced instance so far G is a good approximation of the desirable G. Commonly used criteria are maximum time/iterations and/or none improvement on the objective function value for a certain number of iterations. The size of a region can be controlled through a parameter µ, logically, 0 < µ ≪ 1 to make sure |G| ≪ |G| such that P can deliver an "optimal" solution quickly (Copado-Mendez et al, 2017). On the other hand, the benchmark solution s * is obtained by considering the entire input G for P , i.e.…”

“…Mixed extraction strategies were used including only selecting the best solutions so far and selecting random solutions. A greedy method(Copado-Mendez et al, 2017) is employed to construct the initial feasible solution to start the process to ensure that each candidate objective function has the same start point.…”