Different formulations for nested Logit models
There are different formulations for hierarchical nested Logit models. ABM Nested Demand follows the formulation in Ben-Akiva 1985. Transformations are given below for alternative formulations.
For the sake of clarity, we will concentrate on the case destination choice - mode choice. The letters d and m stand for destination and mode.
ABM Nested Demand
The following applies:
where
Vdm |
Base utility |
lD |
Location choice lambda |
lM |
Mode choice lambda |
Ad |
Attraction factor of destination d |
Transformation according to Ben-Akiva
The following applies:
where
Vd |
Utility of destination d (systematic part) |
Vm |
Uility of mode m (systematic part) |
Vdm |
Utility of mode m in combination with destination d (systematic part) |
μD |
Scale parameter destination |
μM |
Scale parameter mode |
Transformation according to ABM Nested Demand
ABM Nested Demand |
Ben-Akiva |
Vdm |
Vd + Vm + Vdm |
lM |
μM |
lM |
μD |
AM |
1 |
Transformation according to the tour-based model
The following applies:
where
Vd |
Utility of destination d |
Vdm |
Mode choice utility |
θ |
LogSum parameter |
Ad |
Attraction factor of destination d |
Transformation according to ABM Nested Demand
ABM Nested Demand |
Tour-based model |
Vdm |
Vd / θ + Vdm |
lM |
1 |
lD |
θ |
Ad |
Ad |
Another variant
The following applies:
where
Vdm |
Utility of mode m in combination with destination d (systematic part) |
θ |
LogSum parameter |
Ad |
Attraction factor of destination d |
Transformation according to ABM Nested Demand
ABM Nested Demand |
Variant |
Vdm |
Vdm / θ |
lM |
1 |
lD |
θ |
Ad |
Ad |
Web TAG
TAG essentially describes the case mode choice - destination choice.
- ABM Nested Demand
For the mode choice - destination choice case, the ABM formulation is Nested Demand:
- Web TAG
The following applies:
where
Gdm |
Generalised cost |
θmode |
Mode parameter |
ldist |
Distribution parameter |
Bd |
Attraction factor |
Transformation according to ABM Nested Demand
ABM Nested Demand |
Web TAG |
Vdm |
Gdm |
lM |
θ...λdist |
lD |
-λdist |
Ad |
Bd |