P = The model satisfied the Luce s axiom of independence of irrelevant alternatives (IIA) which can be stated as

Transcription

1 1.4 Multinomial logit model The multinomial logit model calculates the probability of choosing mode. The multinomial logit model is of the following form and the probability of using mode I, p is given by U i = Utility of mode i U r = utility of mode r (r = 1,2,3 I,...,m) A term relating to the subject mode I appear as the numerator and the summation of the similar terms corresponding to all competing modes in placed in the denominator. Disaggregate Behavioral models are i) Policy oriented ii) Geographically transferable iii) Data efficient The model satisfied the Luce s axiom of independence of irrelevant alternatives (IIA) which can be stated as Where any two alternatives have a non-zero probability of being chosen, the ratio of one probability over the other is unaffected by the pressure or absence of any additional alternatives in the choice set. Example: Car vs. blue bus and red bus Say car is competing with blue bus in a traffic scenario. In this case the probability of selection of car and blue bus is equal i.e in this case the ratio of their probabilities comes out to be 1.0. Now another bus is introduced as a competitor in the new situation the probability of selection of either of the buses or the car becomes 0.33 and the ratio of previous two competitor modes again remains 1.0.

2 Initially, this property was considered as advantages of the model, as it allows treating the new alternative problem neatly. But same property is perceived as disadvantages in case the correlated alternatives are present and multinomial logit model fails in that condition. This means that in the example taken above blue bus and red bus belong to the same segment and therefore competes with each other. These together compete with the car mode. In that case the probability of choosing a car or bus mod should be equal i.e Further within bus category the probability of choosing either a blue bus or a red bus should again be equal i.e On the whole, the probability of choosing either bus will become 0.50*0.50=0.25 and that of car will remain This will satisfy the probability i.e. the sum of probabilities should be equal to Calibration of multinomial logit model The general equation of multinomial logit model is of the following form and the probability of using model i & p is given by (i) U i = Utility of mode i U r = Utility of mode r (r = 1, 2, 3, - - -,I, m) The general form of this equation resembles the fractional term employed by the gravity model of trip distribution. A term relating to the subject mode appears as the numerator and the summation of the similar term corresponding to all competing modes is placed in the denominator this specification ensure that all trip have been estimated to occurs on a specific interchange are assigned to the available modes; That is the following trip balance equation is satisfied (ii)

3 Equation (ii) would still be satisfied by writing the proportion attracted by each mode as For reasons that lie beyond the scope of this book the logistic transformation of the utilities [Eq.-(i)] is preferred Case Studies:- A calibration study result in the following utility equation U K = a K X X X X 4 X 1 = Access plus egress time, in min X 2 = Waiting time, in min X 3 = Line, haul time, in min X 4 = Out of pocket costs, in Rs K = Subject Mode The trip distribution forecast for a particular interchange was a target year volume of Q ij = 5000 Person trip per day. During the target year trip maker on this particular interchange will have the choice between the private automobile (A) and a local bus system (B). The target year service attributes of the two competing modes have been estimated to be Table 4.2 Attribute X1 X2 X3 X4 Automobile Local bus

4 Assuming that the calibrated mode specific constant are 0.00 for the automobile mode (i.e. base mode) and for the bus mode, apply the logit model to estimate the target year market share of the two modes and resulting fare box revenue of the bus system. Solution: The utility equation yield U (A) = (0.025*5) - (0.032*0) - (0.015*20) - (0.002*100) = And U (B) = According to the logit equation (1) P (A) = 0.71 & P (B) = 0.29 Therefore the market share of each mode is Q ij (A) = (0.71*5000) = 3550 trips/day Q ij (B) = (0.29*5000) = 1450 trips/day The fare box revenue estimated is (1450 trips/day)*(rs: 0.50/trip) = Rs: 725 per day. Discussion: The terms of the utility function used in this example are negative quantities they represent cost (i.e. disutility) components. The more negative quantities is, the less attractive the mode will be. Because of the exponential transformation of the utilities, the market shares are not directly proportional to the magnitude of utility. Division of the numerator and denominator of equation (i) by e U(A) results in the following form P (B) = & P (A) =

5 Where U* is the difference in the utilities of two modes.