Approaches to modeling train scheduling problems as job-shop problems with blocking constraints

Abstract

The problem of scheduling a set of trains traveling through a given railway network consisting of single tracks, sidings and stations is considered. For every train a fixed route and travel times, an earliest departure time at the origin and a desired arrival time at the destination are given. A feasible schedule has to be determined which minimizes total tardiness of all trains at their destinations. This train scheduling problem is modeled as a job-shop scheduling problem with blocking constraints, where jobs represent trains and machines constitute tracks or track sections. Four MIP formulations without time-indexed variables are developed based on two different transformation approaches of parallel tracks and two different types of decision variables leading to job-shop scheduling problems with or without routing flexibility. A computational study is made on hard instances with up to 20 jobs and 11 machines to compare the MIP models in terms of total tardiness values, formulation size and computation time.

Appendix: Notation

Sets and indices

${{\mathcal {J}}}$:=:

$\{J_i \mid i=1,2,\ldots ,n\}$ set of jobs (trains)

${\mathcal {L}}^k$:=:

$\{l \mid l=1,2,\ldots ,m_k \}$ set of parallel machine units (track lines) of machine $M_k$

${{\mathcal {M}}}$:=:

$\{M_k \mid k=1,2,\ldots ,m\}$ set of machines (track sections)

${{\mathcal {M}}}^i$:=:

$\{M_k \mid \exists ~rout_{ij} = k \}$ set of machines, on which job $J_i$ has to be processed

${\mathcal {O}}^{i}$:=:

$\{O_{ij} \mid j=1,2,\ldots ,n_i\}$ ordered set of operations of job $J_i$

$OpMa^{k}$:=:

$\{O_{ij} \mid rout_{ij} = k \}$ set of operations having to be processed on machine $M_k$

${\mathcal {R}}^k$:=:

$\{r \mid r=1,2,\ldots , R_k\}$ set of order positions on machine $M_k$

${\mathcal {R}}^{kl}$:=:

$\{r \mid r=1,2,\ldots , R_{kl}\}$ set of order positions on unit l of machine $M_k$

Parameters

$d_i$:=:

due time of job $J_i$, (desired arrival time of a train at its final destination)

$p_{ij}$:=:

processing time of operation $O_{ij}$, (running time of a train in a track section)

$r_i$:=:

release time of job $J_i$, (earliest departure time of a train at its starting station)

$rout_{ij}$:=:

index of the machine $M_k$, on which operation $O_{ij}$ has to be processed

Decision variables

$C_i$:=:

completion time of job $J_i$, (arrival time of a train at its final destination)

$T_i$:=:

tardiness of job $J_i$, (tardiness of a train at its final destination)

$s_{ij}$:=:

starting time of operation $O_{ij}$ of job $J_i$

$$\begin{aligned}&x_{ijkl}:=\left\{ \begin{array}{ll} 1 &{}\hbox { if operation }O_{ij}\hbox { is scheduled on unit }l\hbox { of machine }M_k \\ 0 &{}\hbox { otherwise}. \\ \end{array}\right. \\&x_{ijk}^r:=\left\{ \begin{array}{ll} 1 &{}\hbox { if operation }O_{ij}\hbox { is scheduled as the }r-\hbox {th operation }\\ &{}\hbox { on machine }M_k \\ 0 &{}\hbox { otherwise}. \\ \end{array}\right. \\&x_{ijkl}^r=\left\{ \begin{array}{ll} 1 &{}\hbox { if operation }O_{ij}\hbox { is scheduled as the }r-\hbox {th operation on unit }l \\ &{}\hbox { of machine }M_k \\ 0 &{}\hbox { otherwise}. \\ \end{array}\right. \\&x_{ik}^t:=\left\{ \begin{array}{ll} 1 &{}\hbox { if job }J_{i}\hbox { is processed on machine }M_k\hbox { in period }t\\ 0 &{}\hbox { otherwise}. \\ \end{array}\right. \\&y_{iji'j'k}:=\left\{ \begin{array}{ll} 1 &{}\hbox { if operation }O_{ij}\hbox { is scheduled before }O_{i'j'}\hbox { on machine }M_k \\ 0 &{}\hbox { otherwise}. \\ \end{array}\right. \\&y_{iji'j'kl}:=\left\{ \begin{array}{ll} 1 &{}\hbox { if operation }O_{ij}\hbox { is scheduled before }O_{i'j'}\hbox { on the }l\hbox {th unit }\\ &{}\hbox { of machine }M_k \\ 0 &{}\hbox { otherwise}. \\ \end{array}\right. \end{aligned}$$