Optimising a coal rail network under capacity constraints

Abstract

This research deals with an innovative methodology for optimising the coal train scheduling problem. Based on our previously published work, generic solution techniques are developed by utilising a “toolbox” of standard well-solved standard scheduling problems. According to our analysis, the coal train scheduling problem can be basically modelled a Blocking Parallel-Machine Job-Shop Scheduling (BPMJSS) problem with some minor constraints. To construct the feasible train schedules, an innovative constructive algorithm called the SLEK algorithm is proposed. To optimise the train schedule, a three-stage hybrid algorithm called the SLEK-BIH-TS algorithm is developed based on the definition of a sophisticated neighbourhood structure under the mechanism of the Best-Insertion-Heuristic (BIH) algorithm and Tabu Search (TS) metaheuristic algorithm. A case study is performed for optimising a complex real-world coal rail system in Australia. A method to calculate the lower bound of the makespan is proposed to evaluate results. The results indicate that the proposed methodology is promising to find the optimal or near-optimal feasible train timetables of a coal rail system under network and terminal capacity constraints.

Appendix: Proof for Proposition 1

The following computational experiment aims to validate Proposition 1 given in Sect. 3. For a block $B_{l} = (\pi (u_{l - 1} ),\pi (u_{l - 1} + 1), \ldots ,\pi (u_{l} ))$, ∀l = 1, 2,…,k, where it is assumed that there are k blocks in total on the critical sequence $\Uppi$, we consider a subset of I-Moves $W_{l} (\Uppi ) = \{ (a,b) \in V:a,b \in \{ u_{l - 1} + 1, \ldots ,u_{l} - 1\} \}$ which are performed insider the block B _l . Let $W(\Uppi ) = \cup_{l = 1}^{k} W_{l} (\Uppi )$ and we have observed an important property in designing the neighbourhood structure for the BPMJSS problem. For any permutation sequence $\Uppi \prime \in N(W(\Uppi ),\Uppi )$, it holds $C_{\max } (\Uppi \prime ) \ge C_{\max } (\Uppi )$ for BPMJSS.

The data of a 10-job 19-machine (10-train 19-section) BPMJSS numerical example are given in the Tables 4, 5, 6, and 7.

Table 4 The sectional running times of a BPMJSS example

Table 5 The section (machine) sequence of each train (job)

Table 6 Then number of units for each section (machine)

Table 7 Computational experiments for validating proposition 1

Assume that the initial order of jobs for this BPMJSS case is:

$$ \Uppi = J_{0} - J_{1} - J_{2} - J_{3} - J_{4} - J_{5} - J_{6} - J_{7} - J_{8} - J_{9} .$$

After applying the SLEK constructive algorithm, the feasible BPMJSS schedule is obtained with the makespan of 190.84. The critical sequence P on the bottleneck machine is:

$$ {\rm P} = (J_{5} - J_{0} - J_{1} - J_{2} - J_{3} - J_{4} - J_{6} - J_{7} - J_{8} - J_{9} ) .$$

Thus, there are three blocks (k = 3) on the critical sequence P:

$$ {\rm P} = (B_{1} ,B_{2} ,B_{3} ) $$

$B_{1} = (\pi (0))$, $B_{2} = (\pi (1),\pi (2),\pi (3),\pi (4),\pi (5))$ and $B_{3} = (\pi (6),\pi (7),\pi (8),\pi (9))$;$B_{1} = (J_{5} )$, $B_{2} = (J_{0} ,J_{1} ,J_{2} ,J_{3} ,J_{4} )$ and $B_{3} = (J_{6} ,J_{7} ,J_{8} ,J_{9} )$.

Now we can define the set of I-moves

$$ W(\Uppi ) = \cup_{l = 1}^{k} W_{l} (\Uppi ) $$

as follows:$W_{1} (\Uppi ) = \phi$;

$$ W_{2} (\Uppi ) = \{ (2,3),(2,4),(3,4)\} $$

$$ W_{3} (\Uppi ) = \{ (7,8)\} $$

In terms of computational experiments, the makespans of the neighbours defined by the moves $W(\Uppi ) = \cup_{l = 1}^{k} W_{l} (\Uppi )$ are presented in Table 7.

From Table 7, it is proved that any neighbouring BPMJSS solution defined by $N(W(\Uppi ),\Uppi )$ cannot improve the solution quality, namely, $C_{\max } (\Uppi \prime ) \ge C_{\max } (\Uppi )$, $\forall \Uppi \prime \in N(W(\Uppi ),\Uppi )$.