Freight transport modal choice models used in the context of studies covering large inter-regional or international areas are generally difficult to set up because of lack of precise or even simply relevant data. Transportation costs (C) and transit times (T) are for instance often among the only figures that can be gathered.
C and T being alternative specific, a conditional logit can be used, that computes the probability Pr to choose mode m among n alternatives. It can be written as:
Pr_m=(exp(αC_m+βT_m+δ_m))/(∑_(j=1)^n〖exp(αC_j+βT_j+δ_j)〗)
However, C and T are correlated by nature. Consequently, the sign of the α or β estimators can be of the “wrong” sign (both must be negative). In order to avoid this problem, a Box-Cox transformation (https://en.wikipedia.org/wiki/Power_transform) is often applied to the C and T independent variables.
A lot has been written on the how-to find the lambda value that best fits the data, but the problem here is not of the same nature. Indeed, one has to identify the lambda’s (one for C, another for T) that (for instance) maximize the likelihood of the logit model, with the additional constraint that the signs of the estimators for C and T must be of the expected sign.
Obviously, one can test all the possible values of lambda. However, this can rapidly be time consuming or even not practicable. Indeed, if one tests all the values between -2 and +2 with a step of 0.1, 41 values must be tested for one lambda. If the utility function contains 2 explanatory variables (which is the case in the example given above), 41^2 = 1681 logits must be solved. If a third variable is added, one would have to compute 68981 logits.
What would be the best approach to solve this problem (identify “valid” good lambda’s in an acceptable computing time)?