Arc-based train slot generation model

5.6.1 Arc-based train slot generation model

An arc-based train slot generation model for addressing the train timetabling problem is proposed based on the time-space network representation described in section 5.4. Notation used in developing this formulation is given next.

5.6 .1.1 Notation

K : set of routes. N p : set of terminals.

I k : set of candidate train slots operating on each route k ∈ K .

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x a : binary decision variable that indicates whether or not the arc a is used by the train timetable.

η k : suggested train frequency for each route k ∈ K .

c a : operational cost of the track a∈ A p .

δ set of conditions that cannot occur simultaneously on the same track. :

i ) : shipment delay cost of the train slot i ∈ I .

5.6.1.2 Model Formulation The arc-based train slot generation formulation is given in (1) through (7).

Min z ( x ) =

∑∑∑ c a x a +

subject to

x ∑∑ x

a = η , ∀ k ∈ K (2)

a ≤ 1 , ∀ i ∈ I ∀ k ∈ ∑ K (3)

a ≤ 1 , ∑ i ∀ ∈ I ∀ k ∈ K (5)

x a ≤ 1 ∑  , ∀C ∈ (6)

a ∈C

a ∈ {} 0 , 1 , ∀ a ∈ A ∀ i ∈ I (7)

Binary decision variable x a represents whether or not the track segment a associated with a certain period of time on the time-space network G p is included in the timetable. The objective given in equation (1) seeks to minimize the total operational cost of shipping

shipments along the provided train route for the given schedule within the corridor and to

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minimize the total shipment delay incurred along the corridor. Constraints (2) ensure that the total number of trains on a route that will be operated is equivalent to the number of train slots, k η , that may be operated for transporting the required shipments for the route.

Constraints (3) ensure that at most one track associated with a train is selected among those leaving the pseudo super-source terminal α . Constraints (4) are the mass balance constraints for each terminal. These constraints impose equality on the number of selected arcs associated with a train entering and leaving each arrival or departure terminal. Constraints (5) ensure that at most one track associated with a train is selected among those arriving at the pseudo super-sink terminal. Constraints (6) prevent two consecutive trains from running on the same track at the same time (or within the time of the minimum headway), while imposing the track capacity constraints. Note that  is defined as the set of conflicts between

any pairs of trains as shown in Figure 5-3 and C ∈  is defined as the subset of conflicts for a specific pair of trains. Binary integrality requirements for every arc are given in Constraints

The size of the formulation is a function of the number of the track segments in the time-space network (number of decision variables), the number of routes, the suggested train frequency for each route, the time length of the planning horizon, and the number of the train conflict constraints. One can expect the number of decision variables and constraints in the arc-based train slot generation problem formulation to be quite large for a real-world problem instance. The number of train slots will increase exponentially with the number of arcs p A of

p G ; thus, it is not likely that one will be able to create an exact, efficient solution technique to solve the arc-based train slot generation problem and an alternative formulation is

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proposed. The alternative formulation is based on concepts of multicommodity network flow models and a column generation-based technique is proposed for its solution. The formulation and solution approach are described in the following sections.