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36 I N N O VAT I O N S I N T R AV E L D E M A N D M O D E L I N G , V O L U M E 2 The only nontrivial transformation that requires I I explanation is the aggregation of non-transit- ) exp (Vi = exp (C ) × P (i ) = exp (C ) exponentiated utility, for which the following notation is i =1 i =1 Wn introduced: N I W (5) = exp (Vin ) i = 1, 2, . . . , I = choice alternatives; n =1 i =1 n = 1, 2, . . . , N = individual records to be aggre- gated within the group; Vin = known individual utilities; By combining Equations 2 and 5, it is possible to obtain Pn(i) = known individual probabilities; the expression for aggregate exponentiated utilities that Wn = person trips (1, for individual tours; party size, would exactly reproduce the target market shares and for joint tours); user benefits: Vi = unknown aggregate utilities; and W = total person trips for the group. ) exp (Vi = P ( i ) × exp (C ) Wn The purpose of utility aggregation is to find a utility I N W expression that will exactly replicate (a) aggregate mode = P (i ) × exp (Vin ) (6) shares and (b) total composite utility across all individ- n =1 i =1 ual records and consequently replicate a UB calculation. The aggregate mode shares for the group of records can be readily calculated as follows: Equation 6 has a simple intuitive interpretation. The aggregate exponentiated utility of each mode is propor- N tional to a product of two factors. The first one is equal W n × Pn ( i ) (1) to the aggregate mode share and reflects the improve- P (i ) = n =1 ment of each mode in comparison with the other (com- W peting) modes. The second one is equal to the weighted The first condition (replication of aggregate shares) geometric average of the individual (exponentiated) log leads to the following expression: sums; this component is sensitive to the overall improve- ment of all modes. Vi = ln P ( i ) + C (2) This aggregation calculation should be implemented separately for the base scenario and the build scenario where C denotes a utility scale constant that has to be and for each group of aggregated records. In the same determined. way that this aggregation can be applied for mode utili- The second condition (replication of the composite ties, it can be applied for the composite nontransit utility utility) leads to the following expression (for simplicity it as well as it can be generalized for any nested structure is assumed that the choice model is a simple multinomial by using a full mode choice log sum for each record LSn. logit model and the composite utility is calculated as a The aggregate exponentiated nontransit utility can be simple one-level log sum): calculated as follows: I N I P ( SOV ) + P ( HOV ) W × ln exp (Vi ) = Wn × ln exp (Vin ) (3) exp (VNT = ) i =1 n =1 i =1 + P ( NM ) + P ( SB) N Wn (7) Equation 4 results from an equivalent transformation × n =1 exp ( LSn ) W of Equation 3: I N Wn I where ) exp (Vi = exp × ln exp (Vin ) SOV = single-occupancy vehicle, i =1 n =1 W i =1 Wn HOV = high-occupancy vechicle, (4) N I W NM = nonmotorized vehicle, and = exp (Vin ) SB = school bus. n =1 i =1 The described methodology of UB calculation has By substituting the expression for aggregate utilities from been applied for the analysis of the recent LRT project in Equation 2 to Equation 4, the necessary formula for the Columbus, Ohio, and approved by FTA. The results are utility scale C is obtained: provided in a companion paper.