Alternatively, one can incorporate Equations 7b into Equation 7a. This new expression, which is shown as Equation 7c, is
one which should be minimized concurrently to maximizing objective function 4a or 4b. Unfortunately, it is not easy to
combine one expression that desires to maximize likelihood with another that desires to minimize link flow error, as
Lagrangian can only add equality constraints to a constrained objective function. A proposed solution to this problem
involves taking the partial derivatives of Equation 6c with respect to each of the trip cells that are to be estimated, as
demonstrated in Equation 8a. This yields as many equations as there are trip cells, as shown in Equation 8b.
Furthermore, setting these derivatives equal to 0 is equivalent to minimizing Equation 8a. However, while equation 7a
could not be added to the maximum likelihood objective function, the equalities in Equation 8b can. This produces an
unconstrained objective function that always yields a feasible solution, as is shown in Equation 9.