Track capacity modification

5.6.3 Track capacity modification

Given the train slots generated through the use of the initial track capacity allocation (or the path-based train slot generation model) and the shipment delay at terminals estimated from the simulation platform, the track time capacity for each train slot will be adjusted to reduce the delay incurred at the terminals. The track time capacity usage adjustment on the

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train slots from the initial train timetable generated or from the previous train timetable produced by the path-based train slot generation model is addressed in this section.

The goal of the track capacity modification is to improve the shipment delay in the pick-up/drop-off terminal by adjusting the departure and arrival times of the train slots and to generate additional train slots based on the residual track capacity. The train slots from the previous timetable will be adjusted such that the shipment delays from scheduled departure times (referred to herein as shipment-at-terminal delay) in the pick-up/drop-off terminals can

be reduced. The timetable will be modified through a train slot adjustment method in which the trains are assumed to be able to travel within a certain allowable range of the speed. Additional train slots will be constructed between existing trains if residual track capacity remains. The resulting modified timetable, therefore, can be used as an input for the train slot generation model.

The train slot adjustment method has the goal of adjusting the train slots to accommodate more shipments that have been identified as waiting at the pick-up/drop-off terminals for the next available train. This is done by adjusting the train slot’s departure time at the terminal. A neighboring solution (i.e. an alternate timetable) to the current timetable is generated through a train departure adjustment procedure applied to every drop-off/pick-up terminal in which the train departure time can be adjusted within a neighborhood if there is significant delay incurred due to a train’s scheduled departure time (i.e. if the next scheduled departure is far from the shipment’s arrival time).

The delay to the shipment incurred at the terminal based on the current train timetable is computed. The amount of delay at the terminal is determined by the shipment’s arrival time,

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which is mainly affected by the predetermined distribution of the shipment generation rate and the train/truck arrival time at this terminal. The delay to the shipment is called the shipment-delay-at-terminal. This delay corresponds to a certain train and is defined as the time period starting when the shipment arrives at the terminal and ending when the shipment is loaded on the train. The shipment-delay-at-terminal may result from either the shipment process time at the terminal or the shipment waiting time for the next available train. In this work, it is assumed that the cause for the delay is unknown. Thus, it is assumed that half of the delay is caused by the shipment process activity at the terminal. For shipments arriving approximately at the same time and waiting for the same train, the shipment-delay-at- terminal for these shipments form a cluster-delay-at-terminal, defined as the shipment-delay- at-terminal times the number of shipments.

Through the use of the train departure adjustment procedure, the largest cluster-delay- at-terminal will be selected to be reduced by shifting the train’s departure time back to the mid-point in time of the range of activities that contribute to the cluster-delay-at-terminal. The new departure time from a terminal must be later than the train’s arrival time at that terminal. The departure and arrival times for each terminal visited by the train at points succeeding this terminal must be rescheduled by checking the track time capacity. To allow the train to be rescheduled successfully (i.e. without conflicts with other train slots), the train is allowed to travel on the track by varying the speed within a reasonable range so that the disturbance caused by rescheduling the trains can be minimized.

After finishing the adjustment to the departure and arrival times of the existing train schedules, additional train slots will be constructed by randomly creating train slots for every

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route based on the residual track capacity while ensuring that the track capacity is not violated. Each train slot will be constructed from its origin to its destination while keeping a minimum headway between trains on the same track.

Let k I denote the set of trains on route k ∈ K from the previous timetable. For each train k i ∈ I , let E

i = { f , ( f + 1 )..., ( g − 1 ), g } denote an ordered set of the terminals that train i will visit in sequence along the route in which

f is defined as the route’s origin and g is defined as the route’s destination. Train i’s arrival time and departure time are indicated as

α (m ) and β (m ) for each terminal m ∈ E i . Let L denote a set of terminals in which the train will load/unload shipments, change locomotives or process border crossing activities.

The train slot adjustment method is described as follows.

Step 1. (Choose a terminal)

Choose a terminal l l ∈ L . Let M represent the set of trains that will load/unload shipments at terminal l . Remove l from L . If L = ∅ , terminate.

Step 2. (Search neighborhood delay)

2.0 Choose one train l i ∈ M that departs the earliest at terminal l . Remove train i

from l M . If M = ∅ , return to Step 1.

2.1 Construct a neighborhood of train i’s departure time β (l ) . Let h i , j denote the headway between two consecutive trains, i and j. The neighborhood’s range is

expressed as follows.

β ( l ) − h i , i − 1 < neighborho od < β ( l ) + h i , i + 1 ,

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where i − 1 is indicated as the train that departs before train i and i + 1 is indicated as the train that departs after train i.

2.2 Search the maximum cluster delay indicated as d from the set of clusters defined as D i for train i in the neighborhood constructed from Step 2. Remove d from D i .

2.3 Let d τ denote a cluster delay

d ∈ D i corresponding to the train i. If the cluster delay occurred before β (l ) , go to step 3.0. Otherwise, go to 3.1.

Step 3. (Modify train departure time at a chosen terminal)

3.0 Force the train to depart earlier. Let µ = β ( l ) − α ( l ) . Change train i’s departure

d time at terminal d l . β ( l ) = β ( l ) − τ

i / 2 . If µ ≤ 0 , β ( l ) = β ( l ) − τ i / 2 + µ + ε , where ε is defined as a small amount of the time. Go to Step 4.

3.1 Force the train to depart later. Change train i’s departure time at terminal l . β d ( l ) = β ( l ) + τ

i / 2 + υ , where υ is defined as the time period between the new train departure time and the starting time of the cluster delay.

Step 4. (Adjust timetable on the train slot)

4.0 Let α (n ) and β (n ) indicate the departure and arrival times, respectively, for every terminal k n∈ E

j visited in order by train j ∈ I , k ∈ K , i ≠ j . Let train i and j be two consecutive trains departing/arriving at the same terminal. Based on the new

departure time obtained from Step 3, find the new arrival time α (m ) and departure time β (m ) for every m ∈ E i that will be visited in order by train i after terminal l by identifying the new movement arcs and arrival arc along the time-space

network, where no conflicts exist with train j: For train i and train j, let | Δ ( β ( m ), β ( n )) | , where m = , denote the difference n

between the two trains’ arrival times and let | Δ ( α ( m ), α ( n )) | , where m =, n denote the difference between the two trains’ departure times.

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(1) | Δ ( β ( m ), β ( n )) > | minimum departure headway

(2) | Δ ( α ( m ), α ( n )) > | minimum arrival headway

(3) α ( m ) > α ( n ) and β ( m + 1 ) < β ( n + 1 ) , where m + 1 = n + 1 . Note that terminal m+1 is visited by train i right after terminal m and terminal n+1 is

visited by train j after terminal n. If any of the arrival/departure times cannot be adjusted, return to Step 2.2;

otherwise, the train slot is successfully modified. Return to Step 2.0. Step 5. (Select a route for creating additional train slots)

5.0 Let K denote a set of routes and B denote a set of routes for which the train slots will be created. B= K

5.1 If B = ∅ , return to Step 5.0. Otherwise, select a route k ∈ B with the longest travel time. B = B \k { } . Construct the train slot i along route k beginning from the

origin, employing the available track capacity of p A in G. p

Step 6. (Construct a train slot along the time-space network)

6.0 Choose feasible departure arc ( α , q u ) where u ∈ E i , α = f .

Search the earliest available track time capacity between pseudo super-source node α and the origin terminal

f in the arc set p A . If a feasible train departure arc cannot be identified, return to Step 5.1.

q 6.1 Choose feasible movement arc r ( u , v ) , where v ∈ E

i , u ∈ E i , u ≠ v . Let q denote train i’s earliest feasible departure time at terminal u. Starting from

time q, search the earliest available track time capacity from terminal u to terminal v. If a feasible movement arc cannot be identified, return to Step 5.1. If

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the arrival terminal v is a destination terminal g, go to Step 6.3. Continue, otherwise.

6.2 Choose feasible process/siding arc q ( v , u ) where v ∈ E

i , u ∈ E i , u = v . Let r denote the train i’s arrival time at terminal v and q denote the time when the

train finishes its required activity at the terminal. Return to Step 6.1.

6.3 Choose feasible arrival arc r ( v , β ) where v ∈ E

The train i’s slot is obtained. Return to Step 5.1.

Many different combinations of train slots can be generated through the use of this method. The resulting set of potential feasible train slots provide necessary input to the column generation technique proposed in the next section for determining the train timetable to transport the shipment such that the operational cost and the delay are minimized. The technique can be repeated on different initial sets of potential feasible train slots and the best solution from these runs can be employed.