Solution evaluation via objective function

Line blocking with vehicle interchange uses an objective function for evaluating the quality of a solution. The objective function measures solution properties where there is room for improvement. It comprises the objective function of line blocking without vehicle interchange (Construction of the graph) as a component.

The following solution properties are evaluated:

  • Costs: Objective function of line blocking without vehicle interchange. This especially comprises the number of vehicles per vehicle combination as well as the service km and empty km and service times and empty times, as well as the layovers within and outside of a depot
  • Number of vehicle units: Exceeding the predefined number of available vehicle units
  • Consideration of volumes: Too low capacities (total or seats) regarding the OD demand
  • Line purity, local definition: Number of transfers between the different lines
  • Line purity, global definition: Number of different lines in a block
  • Number of vehicle combinations per line: Number of different vehicle combinations used on the same line
  • Regularity: Number of vehicle journey sections, who's vehicle journey section occurrence lies in at least two different blocks or blocking days
  • Difference from reference solution (only when a reference solution has been specified): Difference to this reference solution in form of deviating transfers between vehicle journey sections
  • State of charge: How often, how much and how long is the state of charge with respect to a user-defined line blocking item type less than the minimum required state of charge.

The following function is used as an objective function (OF), which is based on the comparison between the calculated and the estimated values for each of the objective function components:

where

ci

Influential factor (Procedure parameters) for the indicator i, where ∑ci > 0

ofi

Objective function component for indicator i according to the upper list

comparisoni

Comparison value for indicator i on a comparable scale

The individual component properties are apply as follows:

Component

Calculation

Cost

The objective function component costs evaluate the solution according to the same criteria as the line blocking without vehicle interchange.

It therefore applies

of Costs = ∑fe • Costs(e) (fe = Flow on edge e)

For the comparison value, vehicle combinations are randomly selected and thus a solution calculated. The costs are then used as a comparison value.

Number of vehicle units

The sum is calculated over all vehicle units, across the number of used but not available vehicles per vehicle unit

The comparison value comparisonvehicle is 1. Note: This is how you achieve very strong penalization, because this criterion must apply "hard" if it is used at all.

Consideration of volumes

For the calculation, for each vehicle journey item i first the difference between its volume and its capacity (cap) is calculated as follows

The capacity sums up from the selected capacity (seats or total) of all vehicle combinations of all vehicle journey sections that service this vehicle journey item.

Due to the current parameter settings, a vehicle journey's value results from the particular formula below:

using option Average volume

using option Peak volume

The value is the total calculated for all vehicle journeys:

The value of this objective function component, which applies for the random solution, used for cost estimation, is used as a comparison value.

Line purity, local definition

The number of line changes between successive vehicle journeys in the block are measured. The benchmark is the number of occurrences of vehicle journey sections in total (therefore the number of all changeovers between successive vehicle journey block items).

Line purity, global definition

The number is calculated minus 1 of the line per block, summed up over all blocks.

The comparison value is the number minus 1 of the lines per partition, summed up over all partitions.

Regularity

Dispersion of the occurrences of vehicle journey sections is measured for different blocks or optionally for different blocking days.

The following applies:

ofregularity = Sum of vehicle journey sections |{Blocks / blocking days which contain the VJS}| - 1

Comparisonregularity = (Sum of occurrences of the VJS in the line blocking time interval – 1)

Distance to starting solution

The number of changeovers from vehicle journey section to vehicle journey section, which differ from the comparison solution, is measured.

The comparison value is the number of all changeovers from vehicle journey section to vehicle journey section in the comparison solution.

Table 226: Objective function components for line blocking with vehicle interchange

Note: Objective function components, which are not relevant to the specific planning task, can be switched off by setting the respective coefficient to 0. This is recommended, because optimization up until the solutions, considering the hidden properties, is thus suppressed. Finding good solutions regarding the remaining criteria is accelerated accordingly.

Note: Solutions that would be optimal with a different weighting of the objective function components (Pareto-Optima) and that are found during optimization can be saved. This helps you to estimate the optimization scope and the corresponding weights.