Conclusion

From the previous three sections it can be concluded that if we use exclusively one of the three parts of the upper level objective function in (3.1) only the solution to the first part is possibly unique. When the prior OD-matrix D0 satisfies the traffic regime constraints, this OD-matrix is the unique solution to

f1. For the second part of the upper level objective function, it turns out to be difficult to draw general conclusions on the uniqueness of the solutions. But the given example in the Dogbone network shows that the solutions tof2are possibly non-unique. The same can be concluded for the solutions to the third part of the objective function; Also the solutions to f3 are possibly non-unique. Therefore in practice it is required to put always some weight on the first part of the upper level objective functionf1. However note that this will not always guarantee a unique solution.

Chapter 4

Solution method

In chapter 3 the matrix estimation problem for STAQ-squeezing with fixed route choice and no junction modelling as considered within this study has been formulated and discussed. In this chapter it is described in detail how this problem is solved, using the matrix estimation method for STAQ as developed by Brederode et al. [5].

Analysing the proposed matrix estimation method [5] resulted in some remarks, which are given in a gray text box. Those remarks are important in the further development of the given matrix estimation method.

4.1

General idea

Below the matrix estimation problem as considered within this study is shown1. Again the considered STAQ variant, STAQ-squeezing with fixed route choice and no junction modelling, is denoted by STAQ-squeezing∗_.

min D F(D) = minD w1f1(D, D 0_{) +w} 2f2(y,y˜) +w3f3(τ,τ˜) s.t. y, τ= STAQ-squeezing∗(D) D≥0 χj=        1 then P p∈Pj ψ_prsDrs Q ij∈_IJ˜_jp αij(D)≥Cj, 0 then P p∈Pj ψrs p Drs Q ij∈IJ˜jp αij(D)≤Cj, ∀j∈J .˜ (4.1)

It can be seen that the link flowsy and the path queueing delaysτ in the upper level minimization problem are depending on the OD-vectorD. The lower level assignment model STAQ-squeezing∗ can calculate for every possible OD-vector

D the flows ya on all links a and the path queueing delays τp on all paths p.

But considering STAQ, it is not possible to determine explicit relationshipsy(D) 1_{This problem has been defined in chapter 3. See equation (3.1) for the matrix estimation}

Dk _{= arg min}

D [w1f1(D, D0) +w2f2(y(D),y˜) +w3f3(τ(D),˜τ)]

s.t. nonnegativity and traffic regime constraints

y(D)≈A(Dk−1_)Dk−1₊δ A(D)D δD _D=Dk−1(D−D k−1₎ τ(D)≈ T 2 αˆ(D k−1₎−1₋~₁ STAQ-squeezing∗(Dk₎ Dk k=k+ 1 A(Dk), δA_δD(Dkk), ˆα(D k₎

Figure 4.1: Proposed solution approach for STAQ.

andτ(D). As such, the given matrix estimation problem for STAQ-squeezing∗ is a bi-level optimization problem. The upper level minimization problem is constrained by the lower level assignment problem. In chapter 2 it has been described, that to solve this bi-level matrix estimation problem, Brederode et al [5] propose a heuristic algorithm that iteratively assigns the OD-vector from the upper level into the lower level and then solves the upper level optimization problem using information from the lower level to approximate the relationship between the OD-demands and the link flows y(D) and the relationship between the OD-demands and the path queueing delaysτ(D). This solution approach is shown in figure 4.1. By approximating the link flows and path queueing delays in the upper level problem, less lower level assignments are required, which possibly speeds up the optimization process.

Remark 1: But note that the approximated link flows and path queueing delays might differ significantly from the actual link flows and path queueing delays as can be calculated by the lower level assignment model STAQ-squeezing∗. Therefore, it could be better to perform only one or a few steps within the upper level optimization problem (instead of solving it to optimality) before a new lower level assignment is performeda_.

a_{Another possibility is to put more weight on the prior part of the objective function.}

Recall from section 2.1 that the proposed solution approach as shown in figure 4.1 is adopted from the conventional matrix estimation method for traditional STA models [8]. However, within the conventional matrix estimation problem no path queueing delays2 _{and no traffic regime constraints are considered. And} more importantly, Brederode et al. [5] have introduced a different way of approx- imating the link flows in the upper level optimization problem. Not only the assignment matrix, but also the sensitivity of the assignment matrix to changes in the OD-demands is taken into account in the proposed first order Taylor approximation of the link flows.

In the next section the upper level approximation of the link flows is explained in detail. Subsequently in section 4.3 it is described how the required sensitivity of the assignment matrix to changes in the OD-demands is determined from the lower level assignment model STAQ-squeezing∗. Then in section 4.4 the calculation and the upper level approximation of the path queueing delays are discussed. And finally in section 4.5 it is explained how the traffic regime constraints are taken into account in the given upper level optimization problem.