Finding Consistent Solutions

One method for tackling the problem of multiple equilibria is already implemented in OmniTRANS. Ideally we would like to find the same equilibrium every time we do a traffic assignment, regardless of the initial state of the network. This is not the case with asymmetric cost functions, as shown by Muijlwijk [19]. This is particularly problematic for variant studies in which consultants compare solutions of two networks that differ slightly. We want to be sure that all differences between the solutions in the variant study are the result of the changes in the network and not of different equilibria calculated for the variants. So we try to push the assignment algorithm to the same equilibrium by gradually introducing the turn delays throughout the iterations of the algorithm. The idea is that during the first iterations the turn delays are omitted and the algorithm calculates a solution that is close to the user equilibrium without junction modeling. Then the turn delays are introduced gradually, so that a UE with junction modeling that is ’close’ to the UE without junction modeling is found. To achieve this, we introduce a vector

µ= [µ1, µ2, . . . , µn], wherenis the maximum number of iterations and 0≤µi ≤1,

µi ≤µi+1 for 1≤i≤n−1 andµn= 1. We then change the objective function in

the i-th iteration to zi(f) = X a∈Al Z fa 0 ca(ω) dω+µi X a∈At Z fa 0 c0_a(ω) dω (4.3.3) Herec0_a is one of the approximation functions described in previous sections, since the integral of the actual turn delay function does not exist. The values in µ

are project and user specific and can be chosen by the user based on his or her preferences.

Chapter 5

Results

This chapter contains some results generated by the prototype of TAPAS made in OmniTRANS. All solutions calculated by TAPAS are compared with solutions calculated by FW and VA, the algorithms currently available in OmniTRANS. Section 5.1 contains a description of the prototype, with explanations of some of the design choices made. Section 5.2 shows some results of TAPAS, FW and VA without junction delays. In section 5.3 results of the three algorithms with junction delays incorporated are treated.

5.1

Prototype

A prototype was made in the OmniTRANS software package. Due to time con- straints only the essential parts of TAPAS described in chapter 3 were implemented and both the branch shift and proportionality procedures were skipped. This does not impair the ability of the prototype to converge. Proportionality is an auxil- iary method that produces more realistic path flow solutions but is not needed for convergence. Branch shifts are needed for guaranteed speedy convergence, but can be replaced by using the PAS that least violates the flow effectiveness condition. As for junction modeling, i.e. for dealing with asymmetric costs the methods de- scribed in section 4.3 were not implemented. Instead a simpler bisection method is implemented that iteratively approaches the step size for which the difference in cost between both segments of the PAS is minimal. The method is outlined in algorithm 4. As always we determine the maximal allowed flow shiftfmax and try

to determine the fraction λ of fmax that is optimal. For this procedure we define

δ(cλ) as the absolute cost difference between segment s1 and s2 if we would shift the fraction λ of the maximal allowed flow shift fmax. In the iteration scheme we

store λlow and λhigh, that are initialized at 0 and 1 respectively, as the current

lower and upper bound of the step size we want to perform. These bounds are then tightened by calculating δ(cλ) for two points halfway betweenλlow and λhigh

and then choosing new upper and lower bounds based on the cost differences. While this method is less precise than a direct calculation using for instance the diagonalization method, it does fit well in the iteration scheme of TAPAS where flow is shifted on a PAS multiple times in order to reach equilibrium. The main drawback of this method is that segment costs need to be calculated many times in each PAS shift, so calculation time of the algorithm goes up dramatically com- pared to TAPAS without junctions and therefore this method can only be used for demonstration purposes.

Algorithm 4: Procedure for finding the step size of flow shifts on a PAS with junctions

determine maximal allowed flow shift fmax

seti= 0 setλlow = 0 setλhigh = 1 setλ1 = 1/3 setλ2 = 2/3 while i <20do

calculate δ(cλlow), δ(cλhigh), δ(cλ1) and δ(cλ2)

determine δmin = min{δ(cλlow), δ(cλhigh), δ(cλ1), δ(cλ2)}

if δmin =δ(cλlow) then

λhigh =λ1 end

else if δmin =δ(cλ1)then

λhigh =λ2 end

else if δmin =δ(cλ2)then

λlow =λ1 end

else

λlow =λ2 end

λ1 =λlow+ 1/3(λhigh −λlow)

λ2 =λlow+ 2/3(λhigh −λlow)

end

Chapter 5. Results 51

Figure 5.1: Convergence of algorithms in OmniTRANS without junction de-

lays on the Amsterdam network