In Analyst Drive, the static estimation procedure corresponds to the optimization problem given by Equation Set 4.

Equation Set 4

where A is the route choice probability matrix, the design variable X is the OD matrix to be estimated, X₀ is the initial OD matrix, b is a vector of observed counts, b_c is a diagonal matrix which contains the observed count confidence values and X_c is a diagonal matrix that gives the OD matrix confidence values. The matrix product AX gives the simulated volume. T_ic and T_jc are diagonal matrices that give the Trip Ends confidence values. TE_i and TE_j gives the inital OD matrix trip end totals and XE_i and XE_j the estimated trip end totals for I and J. ω, τ₁ and τ_j are weighting factors for their respective terms. Constraints X_lower and X_upper are lower and upper limits for the estimated OD matrix.

Analyst Drive also can perform an estimation on multiple matrices using a single set of counts. There are two ways to estimate multiple matrices with a single count set in Analyst Drive, the first simply preprocesses the count set according to initial volume proportions from the input matrix. The second method is more theoretically sound as it makes no assumptions of the validity in pressuming the proportions of the estimated matrix should correspond to those of the input matrix. The minimization problem for this method is the same as that of Equation Set 4 but replaces the simulated volume with a sum over the OD matrices to be estimated, uses a cumulative prior matrix term and does not use trip ends. This results in the optimization Equation Set 5.

Equation Set 5

Here, i indexes over the matrices and N is the total number of matrices corresponding to the count set b.

For this optimization problem, boundary constraints are treated as penalty terms that are added to create an augmented cost function

Equation 6

where β is a scaling factor and B gives a discrete boundary penalty function of the form

Equation 7

It is important to note that numerically, β is a function of the iteration number as it is multiplied by a scalar factor δ at each iteration. The discrete equation to represent this relationship is

Equation 8

where k is the current iteration number. This has the effect of allowing the optimization procedure to find better solutions early in the process which may have values that are out of bounds, but eventually enforces the boundary constraints through increasing the penalty for straying out of bounds at every iteration. The Analyst Drive program accepts an initial value for β as well as the scalar multiplier δ as parameter inputs. If a problem converges in very few iterations (e.g. those with few OD variables and/or counts), the initial penalty value β should be set very high so to enforce the boundary conditions from the first iteration.