Path-based train slot generation model

5.6.2 Path-based train slot generation model

The train scheduling problem is reformulated as an integral multicommodity network flow problem that relies on a train slot representation of the track capacity of each route. Differently from the arc-based train slot generation model, each decision variable in this model is a path-based variable representing whether or not the train slot is selected in the timetable. The train slot representation is constructed based on the time-space network described in section 5.3. Solution of the model can provide a train timetable for a given planning period for which demand is known.

Notation used in developing this formulation is given as follows.

5.6 .2.1 Notation

T : set of days in the planning horizon. K : set of routes. L : set of shipments loading/unloading terminals.

I k : set of candidate train slots operating on the route k ∈ K . y binary decision variable that indicates whether or not the train slot k

i ∈ I is operated.

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

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u lt : maximum number of train slots which can transport shipments generated at unloading/unloading terminal l ∈ L at time t ∈ T .

i : operational cost for each train slot i ∈ I .

δ k lt ( y i ) : binary indicator that represents whether or not the train slot i ∈ I visits the terminal l at time t . ρ k ( µ

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

5.6.2.2 Model Formulation The formulation given in (8) through (11) is referred to herein as the path-based train

slot generation formulation.

Min z ( x ) = ∑∑ c i y i + ∑∑ ρ ( µ i ) (8)

subject to

i = η , ∀ k ∈ K (9)

δ ( y ) y ≤ u , ∀ l ∈ L ∀ ∑∑ t lt i i lt ∈ T (10)

i ∈ {} 0 , 1 , ∀ i ∈ I ∀ k ∈ K (11)

The objective given in equation (8) seeks to minimize the total delay incurred along the corridor and total operational costs required to transport the shipments within the corridor. Constraints (9) ensure that the total number of train slots employed along the corridor on each route must be operated to satisfy the suggested train frequency k η on route k. Constraints (10)

force the number of train slots in which these train slots pass the terminal at a given time to

be no larger than the number of train slots necessary to transport the shipments at terminal l at time t. Binary integral requirements of the decision variables are given in constraints (11).

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UMD and UOB

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Classification:

Unclassidied

Sixth Framework Programme

Thus, a train slot-based binary multicommodity network flow formulation is provided with simple constraint structure.

i , where i ∈ I , represents a potential feasible train slot of route k ∈ K . These train slots are generated to ensure that even if all train slots are operated, there will be no conflicts incurred within any track segment. A set of potential feasible train slots is required as input for the train slot generation formulation. This input will be generated primarily based on the initial timetable created by the initial track capacity allocation technique. Given the initial timetable and the estimated delay generated from the simulation platform, a set of potential feasible train slots will be created through the use of a track capacity modification heuristic developed as part of this work to be employed as the input to the train slot generation formulation. The near optimal train timetable for the selected set of potential feasible train slots will be determined by a column generation-based technique described in the section 5.6.4. The column generation technique can be run many times over many sets of train slots and the solution with the set of train slots with the best solution value is chosen.

In the train slot generation model, each binary decision variable k y