Exercises: Periodic Timetabling for Networks ARRIVAL/ Matheon Fall - - PDF document

Exercises: Periodic Timetabling for Networks

ARRIVAL/Matheon Fall School 2006 Thursday, September 28, 2006, Dabendorf (near Berlin), Germany

Exercise 1 Consider the ICE International line between Amsterdam CS and Frank- furt (Main) Hbf. According to the present timetable, the one-way trip takes 3 h 53 min. We assume the minimum turnaround time in both end- points to be 60 minutes. Furthermore, we assume the line to be operated every two hours.

• 1. Compute two timetables for the ICE International which require a

different number of trains!

• 2. Can you set up a Pesp-model with the four arrival and departure

events in Amsterdam and Frankfurt, which respects the trip times, the minimum turnaround times, and in which precisely those peri-

• dic timetables are feasible which can be operated with the smaller

number of trains that you determined in Part 1?

• 3. Introduce into your Pesp-model the station Cologne Hbf with ar-

rival and departure events for both directions. Set the minimum dwell time to three minutes (Frankfurt-Cologne 75 min, Cologne- Amsterdam 155 min) and allow for an additional dwell time of up to five minutes, in order to enable connections. Without cutting

• ff any timetable that was feasible with respect to the Pesp-model

that you developed in Part 2, can you ensure in your Pesp-model that precisely those periodic timetables remain feasible which can be

• perated with the smaller number of trains that you determined in

Part 1?

• 4. Would your answer to Part 3 be different, when allowing for a linear
• bjective function over the arcs and seeking for an optimum periodic

timetable? Hint: Exploit the cycle periodicity property!

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Exercise 2 Consider the hourly Brandenburg RegionalExpress lines RE3 and RE5:

• Elsterwerda/Senftenberg – Dabendorf – Berlin Lichterfelde Ost –

Berlin Gesundbrunnen – Stralsund/Schwedt

• Lutherstadt Wittenberg/Falkenberg – Berlin Lichterfelde Ost – Berlin

Gesundbrunnen – Rostock/Stralsund. In principle, the northern branches of these two lines could be inter- changed without having too severe impacts with respect to direct trav- ellers. Provide an excerpt of the PESP model for Brandenburg northbound regional traffic in the station Berlin Lichterfelde Ost in which the assign- ment of southern branches to northern branches is not fixed a priori.

• 1. What are the four relevant events at Berlin Lichterfelde Ost and

which lines/branches are associated with them?

• 2. Introduce arcs which ensure that the two lines are more or less equally

spaced over time, i.e. that a cyclic distance of 25 minutes is accepted, but 24 minutes are not!

• 3. Introduce line assignment arcs between the four vertices that ensure

that a node potential vectors π satisfies all your constraints, if and

• nly if for every arrival event there exists exactly one departure event

that occurrs between one and three minutes after the arrival event! Hint: Make use of disjunctive constraints, which involve parallel arcs, and profit from the “more or less equally spacing” that you ensure by having solved Part 2.

• Exercise 3

You are given an instance (D, ℓ, u) of the Periodic Event Scheduling

• Problem. Moreover, you are given a vector x that is defined on the arcs
• f D such that

xa ∈ [ℓa, ua], ∀a ∈ A, and which satisfies the cycle periodicity property for every oriented circuit

• f D.
• 1. Can you derive from x a node potential vector π such that

(πj − πi − ℓa) mod T ≤ ua − ℓa, ∀a = (i, j) ∈ A?

• 2. If so, how?
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Exercise 4 Consider the following directed graph D.

1 2 3 4 5 6 7 8

• 1. Find a non-integral cycle basis B of D.
• 2. Which oriented circuits of D are an integer linear combinations of

the circuits of B? Consider the following vector x that is defined on the arcs of A: xT := (20, 20, 20, 20, 15, 15, 15, 15).

• 3. For which oriented circuits of D does the vector x satisfy the cycle

periodicity property?

• Exercise 5∗

Consider any orientation of the generalized Petersen-graph P11,4. Determine a weight function w on the edges of P11,4 such that the unique minimum cycle basis of your weighted version of P11,4 is not integral. Hint: You may argue easily over all the circuits of P11,4 by observing that the bold edges of P11,4 form a cut. Moreover, there exists a weight function with weights in {4, 5, 12}.

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