Random Utility Models for Trip Demand

Travel-Demand Models

4.2 Trip-based Demand Model Systems

4.2.1 Random Utility Models for Trip Demand

Regardless of the particular functional form used, each partial share in the previous structure can be modeled following a descriptive or a behavioral approach.

However, it is worthwhile to derive partial share model systems that are consis- tent with the general results of random utility theory presented in Chap. 3, where random utility models were introduced as a tool for representing choices from among a discrete set of alternatives(1, . . . , j, . . . , m). Recall that, in the preced- ing section, a trip was viewed as the result of choices over multiple dimensions. In the most general case, therefore, random utility models for travel demand consider alternatives that represent sequences of choices in all the trip dimensions consid- ered. In a four-step model, for example, an alternative might consist of making a particular numberxof trips, for purposes, in time periodh, in order to reach desti- nationd, by modem, and pathk. In this case the symbolj, which denoted a generic alternative in Chap. 3, is equivalent to a sequence[x, d, m, k].

This section proposes two methods for defining a partial share system of models consistent with the hypotheses underlying random utility models. The first method factors a random utility model over the whole sequence of travel choice dimensions into a product of multiple random utility models, each having the same functional form as the original model but involving only a subset of the choice dimensions. The results presented in Chap. 3 on the multinomial logit and hierarchical logit models can be applied for this approach: such models, as was seen, are particularly suited to this purpose.

By contrast, the second method directly specifies the system of partial shares using random utility models, and then imposes conditions that ensure a consistent behavioral interpretation.

The factoring procedure is first described for a situation involving choice in only two dimensions, destination d and mode m; the more general case is considered subsequently. To simplify notation, the user classi, origin zoneo, trip purpose s, and time periodhare taken as understood here and in the rest of this section. Let us assume that the systematic utility associated with a particular choice alternative pairdm,V_dm,¹⁰ may be broken down into a part V_d that depends on destination d, and a partV_m/d that, given the destination choiced, depends on modem. This assumption is consistent with the hypothesis stated above that choice dimensions are considered in sequence: destination choice is affected by mode choice, but the latter, for a given destination, depends only on the attributes of alternative modes and not on those of the destination. The termVdcould be a function of the attributes of the destination, regardless of the mode used to reach it. For shopping trips, for example, attributes might include the number of shops or area of display space; an elementary specification might be:

Vd=β1SHOPSd

10As noted, variables (systematic utility, EMPU, random residuals, etc.) are understood to depend on the origin zoneo, trip purposes, and time periodh; thus notations such asV_dm, V_m/d, and p[dm]are used instead ofV_dm/osh, V_m/oshd, andp[dm/osh], respectively.

The termV_m/d is instead a function of attributes of both the mode and the desti- nation, such as travel time and the monetary cost incurred in reachingdby modem fromo:

V_m/d=β₂T_m/d+β₃C_m/d

In conclusion, the perceived utility of alternativedmmay be expressed:

Udm=Vd+Vm/d+εdm (4.2.3)

Assuming that the residualsε_dm are i.i.d. Gumbel with parameterθ, the previ- ous chapter showed that the probability of choosing alternativedmis given by the multinomial logit model:

p[dm] = exp[(Vd+Vm/d)/θ]

d^′

m^′/d^′exp[(Vd^′+V_m^′_/d^′)/θ] (4.2.4) whered^′ andm^′are generic indexes and the sums are extended to all destinations and to all modes available for each destination for the user class in question.

Factoring (4.2.4) requires finding expressions for the probability of the mode choice given the destinationp[m/d], and of the destination choicep[d].

The probabilityp[m/d]may be obtained directly by applying the definition of the random utility model to (4.2.3):

p[m/d] =Pr[V_d+V_m/d+ε_dm> V_d+V_m′/d+ε_dm′]

=Pr[Vm/d+εdm> V_m^′_/d+ε_dm^′] ∀m^′=m

and from the assumptions made about the distribution of residuals, we again obtain the multinomial logit model:

p[m/d] = exp[Vm/d/θ]

m^′exp[Vm^′/d/θ] (4.2.5) The probabilityp[d]of choosing destinationd regardless of mode may be de- rived from the stability properties of Gumbel variables with respect to maximization.

Indeed, ifU_d^∗stands for the utility associated with destinationdby the most suitable mode, then:

U_d^∗=V_d+max

m^′

(V_m′/d+ε_dm′) (4.2.6) and, by the stability property,U_d^∗is again Gumbel distributed with expected value

E U_d^∗

=E

Vd+max

m^′ (V_m^′_/d+ε_dm^′)

=Vd+θln

m^′

exp[Vdm^′/θ]

=V_d+θ Y_d (4.2.7)

178 4 Travel-Demand Models whereθis, once again, the parameter associated with random variableU_d^∗andY_dis the logsum variable introduced in Sect. 3.3.1. This allows (4.2.6) to be expressed as U_d^∗=V_d+θ Y_d+ε_d^∗ (4.2.8) whereε_d^∗is still a Gumbel random variableG(0, θ )with zero mean and parameterθ.

Using the random utility model definition (3.3.6), the probability of choosing destinationdmay be calculated by replacingU_jwithU_d^∗, and a logit model is once again obtained:

p[d] = exp[(Vd/θ )+Y_d]

d^′exp[(Vd^′/θ )+Y_d′] (4.2.9) Finally, it is easy to verify that the product ofp[m/d]andp[d], expressed re- spectively by (4.2.5) and (4.2.9), again givesp[dm], expressed by (4.2.4).

A different partial share model may be obtained by using a hierarchical logit model. In this case, the elementary alternatives (dm) are grouped by destination:

groupI_d thus contains pairs(d, m^′)for all the available mode alternativesm^′ that serve destinationd. In this case (see Sect. 3.3.2), it is assumed that the random residualε_dmfollows a Gumbel distribution with parameterθ_oand that can be broken down into the sum of two random variablesη_dandτ_m/d:

Udm=Vdm+εdm=Vd+Vm/d+ηd+τm/d (4.2.10) As shown in Sect. 3.3.2, the decomposition ofε_dm into the two components intro- duces a covariance between the residuals of alternativesdmanddm^′:

Cov(εdm, ε_dm^′)=Var(ηd)=(π²/6).

θ_o²−θ_d²

(4.2.11) whereθoandθdare the parameters of Gumbel distributions associated, respectively, with the root node and with all the intermediate decision nodes.

The behavioral interpretation of (4.2.11) is that the decision-maker perceives in a similar fashion the destination/mode alternatives that have the same destination but not those that have the same mode. Figure4.3shows schematically the two utility function structures corresponding to (4.2.3) and (4.2.10).

By applying the results of Sect. 3.3.2, the probability of choosing modemcon- ditional on destinationd is again provided by a multinomial logit model, the ex- pression for which may be obtained by substitutingj =m, k=d, andθ=θd in expression (3.3.12):

p[m/d] = exp[Vm/d/θ_d]

m^′exp[Vm^′/d/θd] (4.2.12) which is the same as (4.2.5) except for parameterθ.

By the same token, the destination choice probability may be obtained by (3.3.17):

p[d] = exp[Vd/θ_o+δY_d]

d^′exp[Vd^′/θ_o+δY_d′] (4.2.13)

Fig. 4.3 Example of alternative utility function structures corresponding to a logit and hierarchical logit specification of a model for two destinations and two modes

where

δ=θ_d/θ_o (4.2.14)

The probability of choosing pair dm may thus be obtained from (4.2.12) and (4.2.13) as

p[dm] =p[d] ·p[m/d] = exp[V_d/θ_o+δY_d]

d^′exp[Vd^′/θ_o+δY_d′]

· exp[Vm/d/θ_d]

m^′exp[V_m^′_/d/θ_d] (4.2.15) Note that the difference between the multinomial logit (4.2.4) and hierarchical logit models (4.2.15) lies in the value of the parameter δ defined in (4.2.14). As stated in Sect. 3.3.2, this parameter may take values between 0 and 1; forδ=1 the hierarchical logit model coincides with the logit.

Extension of the results to choices involving more than two dimensions is im- mediate. For example, the factored multinomial logit model for the sequence of choices[d, m, k]becomes:

p[dmk] = exp[Vd/θ+Y_d]

d^′exp[Vd^′/θ+Y_d′]· exp[Vm/d/θ+Y_m/d]

m^′exp[Vm^′/d/θ+Y_m′/d]

· exp[Vk/dm/θ]

k^′exp[Vk^′/dm/θ] (4.2.16)

where the logsum variables are defined as Y_d=ln

m^′

exp[Vm^′/d/θ+Y_m^′_/d] =ln

m^′

k^′

exp

(V_m^′_/d+V_k^′_/dm^′)/θ Ym/d=ln

k^′

exp[Vk^′/dm/θ] (4.2.17)

180 4 Travel-Demand Models The hierarchical logit model for these three choice dimensions takes the form:

p[dmk] = exp[V_d/θ_d+δ_dY_d]

d^′exp[Vd^′/θ_d+δ_dY_d^′]· exp[Vm/d/θm+δmYm/d]

m^′exp[Vm^′/d/θ_m+δ_mY_m^′_/d]

· exp[Vk/dm/θ_k]

k^′exp[V_k^′_/dm/θk] (4.2.18)

where

δ_d=θ_m

θ_d; δ_m= θ_k θ_m with

θ_d> θ_m> θ_k δd, δm<1 and the inclusive variablesY have the expressions:

Yd=ln

m^′

exp[V_m^′_/d/θm+δmY_m^′_/d] (4.2.19) Y_m/d=ln

k^′

exp[Vk^′/dm/θ_k] (4.2.20)

It is possible to define a form of factoring that is “weaker” than the one discussed here for logit and hierarchical logit models. In this second approach, the models that express the different steps of a partial step structure such as (4.2.2) are ran- dom utility models having different functional forms, such as logit for mode choice and probit for path choice. Therefore, the models corresponding to the sequence of partial choices cannot be obtained by factoring a single model that represents the choice of a compound alternative[d, m, k]. In this case, to maintain an inter- pretation consistent with the behavioral assumptions of random utility models, it is necessary for the model of each choice dimension to include an Expected Maximum Perceived Utility (EMPU) variable that reflects choice dimensions that are lower in the decision hierarchy.

For example, if in (4.2.10) we suppose thatτ_m/d is distributed jointly as multi- variate normal, the probability of choosing modemin (4.2.12) will be given by a probit model, and the utility of destination choice is:

U_d^∗=Vd+max

m^′ (V_m^′_/d+τ_m^′_/d)+ηd=U_d^∗=Vd+sd(V_m^′_/d)+τ_d^∗+ηd

wheresd is the EMPU that reflects mode choice. Moreover, if we assume that the sum of random variablesτ_d^∗andη_d is a Gumbel random variableG(0, θ )with zero mean and parameterθ, the destination choice model is a multinomial logit:

p[d] = exp[(Vd+sd(V_m^′_/d)/θ )]

d^′exp[(Vd^′+s_d′(V_m′/d^′)/θ )]

This approach may be extended to all choice dimensions by deriving partial share models analogous to those given by (4.2.16) and (4.2.18)

p[d, m, k] =p[d](Vd,sd)·p[m/d](Vm/d,sm/d)·p[k/dm](Vk/dm)

where the EMPU are expressed as:

sm/d=E max

k^′ (V_k^′_/dm+τ_k^′_/dm) s_d=E max

m^′

(V_m^′_/d+s_m^′_/d+τ_m^′_/d)

and the models that represent the various steps may have any functional form pro- vided that they can be obtained from the assumptions of random utility models.