With changes to include partial trip data, the dynamic optimization problem solved by Analyst Drive is given by

Equation Set 9

with the new variables V_obs and V_scale representing observed partial trip volume and the scale adjusted partial trip volumes respectively. The user should see the section on partial trip data for further explanation of these variables and the partial trip cost function term in general. In the new case of estimating warm up period matrices, the variable A_w will not be calculated and instead warm up periods will be included in the calculation of the A matrix.

As in the static case, A is the route choice probability matrix, the design variable X is the OD matrix, X₀ is the initial X matrix, and b is a vector of observed counts. A_t and b_t correspond to turning count versions of the route choice probability matrix and observed counts respectively, and the variables A_w and X_w represent the variables A and X during the warm up period. X_c, b_c and b_tc are diagonal matrices containing confidence values for their respective terms. The matrix product AX gives the simulated volume, and the count vector b is adjusted by the warmup volume A_wX_w to produce a vector c, i.e. c = b - A_wX_w. The boundary constraints are treated with a hybrid penalty / reduced gradient method. In this manner penalty terms are added to create an augmented cost function (Note that Equation 10 uses simplified notation, not explicating confidence terms and the part-trip data component)

Equation 10

where β is a scaling factor and B gives a discrete boundary penalty function in the same manner as Equation 7.