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Destination Choice: Mathematical Formulation

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This page is part of the Category Destination Choice Models.

The most common implementation of destination choice is the multinomial logit (MNL) model. Gravity models, which are commonly used in aggregate, trip-based models, can be shown to be a special case of destination choice. Another type of trip distribution model is the intervening opportunities model, which has fallen into disuse in North America. On the other hand, data-driven approaches are emerging, facilitated by the availability of origin/destination big data.

## Multinomial logit

### General specification

The following discussion assumes familiarity with the general formulation of MNL. The destination choice problem is presented with reference to an individual decision-maker, however the model is equally applicable to aggregate, zone-based formulations.This section describes the specification of the destination choice utility function in general terms. More in-depth discussion of the factors that influence destination choice and how to represent them in the MNL model can be found here.

Given a trip origin *i*, and decision-maker *m*, the utility of each destination *j* can be written as follows:

$${ U }_{ j|im }={ \beta }_{ m }\times { TravelImpedance }_{ ij }+\ln { \left( { Size }_{ jm } \right) } $$

In this formulation, the utility of a destination depends on (a) the impedance or spatial separation between the trip origin and the destination, and (b) the size or attractions at the destination. This is the simplest representation of destination choice utility. The impedance term is oftentimes referred to as the *qualitative* utility component, while the size or attraction term is referred to as the *quantitative* component.

Impedance can be measured by distance, auto travel time, or a generalized cost, among other possible measures of spatial separation. A convenient measure of impedance is the inclusive value, or logsum, of the mode choice model. The mode choice logsums are used when it is desirable to have sensitivity to multi-modal level of service in the destination choice model. The coefficient of the impedance variable(s) can be generic (i.e., the same for all decision-makers), or it can vary for certain types of travelers. For example, it is often found that women with pre-school children at home tend to choose work locations that are closer to home than other workers, all else equal. This is represented in the utility function by a more negative coefficient on distance impedance for women with pre-school children than the distance coefficient used for other workers.

The attraction variable is commonly referred to as the size term. It measures the activity opportunities at each destination. In the case of a work location model, the size term is typically employment. In the case of a school location model, the size term can be school enrollment. For many other trip purposes, the size term is typically a linear combination of different types of employment, for example:

$${ Size }_{ jm }={ \alpha }_{ 1 }\times { RetailEmp }\quad +\quad { \alpha }_{ 2 }{ \times ServiceEmp }\quad +\quad { \alpha }_{ 3 }\times { ProductionEmp }$$ The size term always enters the utility function in log form. The log formulation is necessary so that the choice probability of a destination is directly proportional to the number of opportunities at the destination. In other words, if the number of jobs at a destination doubles, all else equal, then the choice probability of this destination approximately doubles.

$${ Pr }_{ j|im }=\quad \frac { exp\left( { U }_{ j|im } \right) }{ \sum _{ k }^{ }{ exp\left( { U }_{ k|im } \right) } } =\frac { exp\left( { \beta }_{ m }\times Imp \right) }{ \sum _{ k }^{ }{ exp\left( { U }_{ k|im } \right) } } \times { S }_{ jm }$$

A corollary of the size term log specification is that the choice probabilities are invariant with respect to the scale of the size term. That is, the choice probabilities remain the same when the entire size term is multiplied by an arbitrary factor, f:

$${ Pr }_{ j }=\quad \frac { exp\left( { V }_{ j }+\ln { \left( { S }_{ j } \right) } \right) }{ \sum _{ }^{ }{ exp\left( { V }_{ k }+\ln { \left( { S }_{ k } \right) } \right) } } =\frac { exp\left( { V }_{ j } \right) \times { S }_{ j } }{ \sum _{ }^{ }{ exp\left( { V }_{ k } \right) \times { S }_{ k } } } =\frac { exp\left( { V }_{ j } \right) \times f{ S }_{ j } }{ \sum { exp\left( { V }_{ k } \right) \times f{ S }_{ k } } } $$

For this reason, by convention one of the variables in the size term is given a coefficient value of 1. Doing so is optional in model application, but necessary when estimating the model, since otherwise the estimation problem is undetermined.

The availability of opportunities for some types of trips is sometimes not well captured with employment variables in the size term. Consider for example trips to the beach, to open spaces like parks, or to secondary vacation homes. For these types of attractors, instead of a traditional size variable, it may be preferable to use an indicator or qualitative variable.

The representation of the impedance measure need not be linear in the parameters; in fact, it is common for the marginal disutility with respect to distance to decrease with distance, as shown in the figure below. However, special rules apply when the impedance function is the mode choice logsum. To ensure proper elasticities between mode choice and destination choice, the mode choice logsum coefficient must take values between zero and one. (refer to section on joint dest choice/mode choice).

- Example.jpg
Caption1

- === Alternative-specific constants ===
- === Agglomeration effects and competing destinations===
- ===Equilibrium constraints / Shadow pricing===
- ==Gravity models as special case of MNL==
- ==Data driven models==
- =References=