Problem formulation

The formulation of the possession scheduling problem is shown in section 4.3.1. The explanation on the formulated formulas are shown in section 4.3.2. The aspects of the formulas, are shown in section 4.3.3.

Mathematical formulation

For RQ1 and RQ2, the problem formulation is the same. The difference between RQ1 and RQ2 is the content of matrix 𝐶𝑖𝑗,𝑟

max 𝑓(𝑥) = ∑ ∑ 𝑥𝑖,𝑤 𝑤 ∈ 𝑊 𝑖 ∈ 𝐼 4.1 s.t. ∑ 𝑥𝑖,𝑤 𝑤 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 4.2 𝑥𝑖,𝑤 + 𝑥𝑗,𝑤 ≤ 2 − 𝐶𝑖,𝑗,𝑟 ∀𝑖, 𝑗, 𝑟, 𝑤 ∈ 𝐼, 𝐽, 𝑅, 𝑊 4.3 𝑒𝑖,𝑤∗ 𝑥𝑖,𝑤 ≤ 1∀𝑖, 𝑤 ∈ 𝐼, 𝑊 4.4 ∑ 𝑥𝑖,v≤ 1 𝑣=𝑤+𝑚 𝑣=𝑤 ∀𝑖, 𝑣 ∈ 𝐼, 𝑉 4.5

In RQ3, there are some small adaptions in the objectives and constraints:

• Objective function has to minimize the number of possessions during an event. • All demand for possessions have to be planned.

• It is allowed that more than one possession are planned only during the holidays.

min 𝑓(𝑥) = ∑ ∑ 𝑒𝑖,𝑤∗ 𝑥𝑖,𝑤 𝑤 ∈ 𝑊 𝑖 ∈ 𝐼 4.6 s.t. ∑ 𝑥𝑖,𝑤 𝑤 = 𝑑𝑖 ∀𝑖 ∈ 𝐼 4.7 𝑥𝑖,𝑤 + 𝑥𝑗,𝑤 ≤ 2 − 𝐶𝑖,𝑗,𝑟 ∀𝑖, 𝑗, 𝑟, 𝑤 ∈ 𝐼, 𝐽, 𝑅, 𝑊 4.8 ∑ ℎ𝑖,𝑣∗ 𝑥𝑖,𝑣≤ 1 𝑣=𝑤+𝑚 𝑣=𝑤 ∀𝑖, 𝑣 ∈ 𝐼, 𝑉 4.9

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Explanation on problem formulation constraints

This section explains the mathematical formulation of the objective function and the mathematical formulation of the constraints. The numbers refer to the above described formuleas.

4.1 – Objective function

The objective of the research is to plan all possessions from the set of required possessions. Then, the number of planned possessions have to be maximized, under the constraints which can be found in the corridor book.

4.2 – Constraint: Demand have to be satisfied

When there is no possession planned, it is not required to create capacity to work on the tracks. As constraint is formulated: the number of planned possessions on a corridor part cannot exceed the number of required possessions.

4.3 – Constraint: Combinations of possessions in a weekend have to be allowed

As a result of a planned possession, freight trains have to be diverted when operation is required and passengers have to be transported by buses or have to be diverted. To reduce the disruption for passengers, some combinations of possessions on the railway network infrastructure are not allowed. In section 3.4 it is described why and which combinations are not allowed for every type of combination-constraint. When a combination for a constraint is not allowed, this is indicated by matrix 𝐶𝑖,𝑗,𝑟 which is explained in section 3.The value of 𝐶𝐴,𝐵,𝑟

means that there is any conflict between corridor part A and corridor part B (=1) or not (=0) for constraint r=1. Depening on the combination-constraint, the matrix 𝐶𝑖,𝑗,𝑟 divers. The following

scenario is not allowed:

𝒙𝑨,𝒘 𝒙𝑩,𝒘 𝑪𝑨,𝑩,𝟏 𝒙𝑨,𝒘+ 𝒙𝑩,𝒘≤ 𝟐 − 𝑪𝑨,𝑩,𝟏

1 1 1 1 + 1 ≤ 1

The other combinations are allowed:

𝒙𝑨,𝒘 𝒙𝑩,𝒘 𝑪𝑨,𝑩,1 𝒙𝑨,𝒘+ 𝒙𝑩,𝒘≤ 𝟐 − 𝑪𝑨,𝑩,𝟏 1 0 1 1 + 0 ≤ 1 0 1 1 0 + 1 ≤ 1 0 0 1 0 + 0 ≤ 1 1 1 0 1 + 1 ≤ 2 0 1 0 0 + 1 ≤ 2 1 0 0 1 + 0 ≤ 2 0 0 0 0 + 0 ≤ 2

4.4 – Constraint: Possession cannot be planned in the same weekend as an event.

The combination of a planned possession in the same weekend as a claimed weekend of an event by the railway undertakings is not allowed. Therefore the multiplication 𝐸𝑖,𝑤∗ 𝑥𝑖,𝑤 should

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4.5 – Constraint: Minimum interval

The interval between two possessions should be 3 weekends. For all combinations of 4 successive weekends of the year, a possession can only be planned on 1 of that weekends.

4.6 – Objective function

Different from RQ1 and 2, the problem formulation is adapted. In RQ3 the focus is on the number of possessions in the same weekend as a event should be minimized, when all required possessions have to be maximized. RQ1 will indicate that a possession planning without conflicts with the constraints is not possible, RQ3 indicates which possessions have to be planned in conflict with the constraints.

4.7 – Constraint: Demand have to be satisfied

All possessions have to be planned, thus the sum of all planned possessions is equal to the required possessions

4.8 – Constraint: Combinations of possessions in a weekend have to be allowed See 4.3

4.9 – Constraint: Interval between possessions on the same part of the network is at least 3 weekends.

See 4.5

Aspects of the mathematical problem

The used software package should be deal with the following aspects: • Input:

o All constraints or the corridor book should be able to implement in a genetic form.

o Script should be deal with:

▪ adapted input: Whether there is a conflict or not

▪ size: When an corridor part is added or removed, or weekend is added or removed.

• Model:

o Value of all X is binary: 0 or 1.

o Objective function and constraints are linear.

o Find the global maximum or minimum within a maximum time period.

o Able to find the global maximum or minimum by equality constraints and inequality constraints.

• Output:

o Possession planning

o Function value

o Corridor parts that cannot be planned under the constraints of the corridor book,

▪ Indicated by a table with the required possessions, planned possessions and the number of not-planned possessions.

▪ Indicated by a map with planned of possessions and not-planned possessions

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