## 6 Results
### TWM Assignment and Routing Strategies  
Different scenarios are compared to check the effectiveness and quality of the solution proposed by TWM(Dj-kSP):  
1. Free-Flow and All-Or-Nothing to obtain the minimum and maximum assignment values as discussed in previous sections.
2. The user equilibrium estimated solutions are provided by methods such as the cumulative, the successive averages, the Frank-Wolfe [5], and Wei [19]. Although they offer approximate theoretical values, they give us an insight into the actual quality of the solution offered by other strategies.
3. Ideal routing solutions for MTTS in hypothetical scenarios that do not use TWM and, thus, no adherence to them is considered. They are included for :  
Figure 6: DJ-kSP obtained (in red) for flow \(x_{N2,N3}\) with a disjoint level ratio of \(\phi=60\) for \(K=10\) different disjoint level policies (maxStretch, maxWeight, minStretch, minWeight) (created with Matlab R23A [71]).  
* Usage of Dj-kSP-based TWM, based on different adherenes \(\psi\) in the routing agents. \(\psi\)-TWM(f) When the vehicles use the TWM maps with a \(\psi\) adherence ratio and a random distribution (uniform or Fibonacci) for assignment. \(\psi\)-TWM(opt) When the vehicles use the TWM maps with a \(\psi\) adherence ratio and an optimal assignment with UCTV algorithm. It is our main objective.  
As pointed out before, some variations cannot be covered extensively in this paper, so we have selected the most relevant ones to illustrate how these parameters affect the TWM routing efficiency.  
UCTV Parameterization  
For the optimal assignment solutions, the UCTV genetic algorithm uses a \(\aleph_{1}\) chromosome with \(numPF\) genes, which will return a collection of \(numPF\) values. They correspond to the volume of vehicles of each sub-flow which will receive the corresponding TWM map. \(numPF\) depends on the disjoint level criteria and edge-discard criterion used, and of course, on the redundancy of feasible paths in the physical topology of the traffic network.  
The UCTV-GA uses an initial population of \(k_{1}*numPF\) individuals and \(k_{2}*numPF\) generations. Using \(k_{1}=5\), the algorithm's convergence to the mean value is quickly achieved in all scenarios in \(k_{2}\leqslant 10\) generations. These values have been selected after multiple experiments using high values and analyzing how the algorithm converges. Gene values in the chromosomes are \([0,1]\) normalized. Selection applies a stochastic uniform strategy in every evolutionary step using an elite count of 5% of immutable individuals and selecting random individuals from uniformly distributed population buckets arranged linearly. Crossover is limited to a \(k_{3}\) fraction of the genes, applying a \([0,1]\) normalized random scattered function. The crossover fraction applied is \(k_{3}=0.8\). Mutation uses a zero-mean Gaussian operator with a 1-normalized scaling factor that shrinks its amplitude at every iteration of the algorithm by a factor or \(\frac{S}{G_{max}}\) where \(g\) is the current generation of the algorithm and \(G_{max}\) is the total number of generations.  
Figure 8 shows UCTV genetic algorithm execution for the TWM(Dj-kSP) assignment, with 60% disjoint level, \"Edge-costs\" disjoint level policy, 6-kSP and minWeight edge discard. This scenario returns a \(numPF=21\).  
Figure 7: Fibonacci-based cumulative probability function for 10 path flows.