objective.functions: Construct objective functions

In travelr: Functions for Travel Demand Modeling

Description

An objective function provides a measure of least cost used in evaluating equilibrium conditions during highway assignment. This help section describes strategies for building objective functions.

Usage

build.general.objective.function(aset, ff.vol, ff.cost, tol = 1e-04, max.depth = 14)
cost.integrator(cost.function,ff.vol,ff.cost,cong.vol,cong.cost,tol=1e-8,max.depth=14)

Arguments

aset	An assignment.set with a cost.function
ff.vol	Free flow (or lower bound) volumes (vector or matrix depending on type of cost function)
ff.cost	Link costs corresponding to free-flow (or lower bound) volumes
tol	Convergence tolerance for cost function integration using adaptive quadrature
max.depth	Maximum allowed recursion level during cost integration (a maximum of 2^max.depth+1 calls to the cost function will be made)
cost.function	The cost function (volume/delay) that relates link volume to link cost
cong.vol	Congested (or upper bound) volumes
cong.cost	Link costs corresponding to congested (or upper bound) volumes)

Details

Objective functions are called from within the assignment algorithms (highway.assign like this:

objective.function(volume)

Parameter:

volume

volumes for which the objective function is to be computed; see below

The objective function returns a numeric scalar value that is used to assess equilibrium (which is presumed to be reached when the objective function has been minimized).

It is important to note that the objective function is not directly minimized by any of the equilibrium assignment methods currently implemented in travelr: the actual search direction toward a “better” solution is built from a linear approximation based on the cost function. The objective function is used simply to evaluate the quality of the current solution. The ability to change that strategy at will is, of course, one of the beauties of open software. Anyone who has a deeper understanding of these issues is welcome to suggest or implement improvements to the travelr algorithms.

The helper function build.general.objective.function will construct a usable objective.function from an arbitrary cost.function. The free-flow volumes and costs are ordinarily computed by new.assignment.set and are available as the ff.flow and ff.cost elements, respectively.

The resulting objective function uses cost.integrator to perform numerical integration. The cost function is integrated from free-flow to congested volumes. While the resulting objective function is reasonably accurate and will work with almost any cost function, it will always be faster (and possibly much more accurate) to build an analytic objective function (see the helper function build.BPR.objective.function for an example of how to do that for the common BPR functional form.

The cost.integrator function is also available as a general-purpose R utility which will evaluate integrals for functions taking a vector of parameters. In the case of a vector value, the function to be integrated is presumed to be the summation of the cost.function evaluated for each of the vector values. This is considerably more flexible than the standard integrate function, but also a very much slower since the integration will be performed repeatedly during the equilibrium line search. The cost.integrator function uses adaptive quadrature and Simpson's rule for approximating definite integrals.

It is possible to write objective functions that ignore certain assignment classes (for example, if loading a certain class always at free-flow conditions). It is nevertheless usually more efficient in those cases to construct multiple assignment sets and load the classes separately.

Helper functions are also available to construct cost (and objective) functions in the classic form known as the BPR formula (BPR stands for “Bureau of Public Roads”, a precursor of the Federal Highway Administration).

Value

The objective.function required by highway.assign returns a numeric vector or data.frame of the same shape as volume

build.general.objective.function returns a function that can be used as an objective function in equilibrium highway assighment

cost.integrator returns a (scalar) numeric value that results from integrating the supplied function between lower and upper bounds

Author(s)

Jeremy Raw

See Also

assignment.set for a description of how the objective function is supplied.

bpr.functions for BPR helper functions.

highway.assign for how objective functions are used.

travelr documentation built on May 2, 2019, 5:17 p.m.