## 3 Traffic Multi-Maps Generation Based on Disjoint Path Flows
### TWM Weights Generation Based on Dj-ksp  
The TWM generation function \(\Pi\) needs to be modified as \(\Pi_{DjkSP}\) (19) to create weight maps that encourage usage of the selected Dj-kSP routes \(\{r_{i}^{d}\}\). To achieve this, we create the traffic groups \(Q\) according to the different Dj-kSP \(\{r_{i}^{d}\}\) and generate a TWM map \(\left[\mu_{i}^{d}\right]\). The use of a specific map with promoted edge weights induces the creation of the sub-flows \(\{x_{i}^{d}\}\). \[\Pi_{DjkSP}:\Theta,[\Gamma_{m}],\Phi,\{r_{i}^{d}\},Y\rightarrow\left[\mu_{i} ^{d}\right] \tag{19}\] \[\mu_{i}^{d}=\{\left[\beta_{e}^{d,i}\right],\Gamma_{m}\}\]  
Each map \(\mu_{m}\) is obtained by applying a transformation function \(Y\) to the weight of the edges, which are part of the Dj-kSP \(r_{i}^{d}\). The most basic transformation function \(Y_{Lin}\) is the linear scaling by a constant factor \(\alpha^{e}\) to the original edge weight to decrease its weight and force the edges to be selected during the trip routing process. \[Y_{Lin}:\Theta,r_{i}^{d},\alpha^{e}\rightarrow\left[\beta_{e}^{d,i}\right]= \left[w_{e}\right]*\alpha^{e} \tag{20}\]  
The linear transformation does not assure that the selected path flow \(r_{i}^{d}\) strictly has a path with lower cost than the best shortest path, unless an \(\alpha\) factor is carefully selected. The \(Y_{Stretch}\) strategy is defined in (21) to solve it, where each map \(\mu_{m}\) uses a different \(\alpha_{m}\) depending on the global scaling factor \(\alpha^{e}\) and the relative cost influence (stretch) of the path flow \(r_{i}^{d}\) over the total path flows \(\{r_{i}^{d}\}\) available for the same flow. It is calculated using the free-flow total costs \(ffc_{i}^{d}\). \[Y_{Stretch}:\Theta,r_{i}^{d},\left[ffc_{i}^{d}\right],\alpha^{e}\rightarrow \left[\beta_{e}^{d,i}\right]=\left[\omega_{e}\right]*\alpha_{m},\forall e\in \{r_{i}^{d}\} \tag{21}\] \[\alpha_{m}=\alpha^{e}*\frac{min(ffc_{i}^{d})}{ffc_{m}^{d}}\]  
For instance, if we consider the flows \(\{x_{1},x_{2},x_{3}\}\) with the corresponding path flows \(\{\{r_{11},r_{12},r_{13},r_{14}\},\{r_{21},r_{22},r_{23},r_{24}\},\{r_{31},r_ {32}\}\}\), Table 2 shows the \(\alpha_{m}\) values for the path flows according to their path cost (free-flow cost) and an \(\alpha^{e}=0.5\) factor. Using TWM for vehicle routing expands the concept of routing paths to the routing area. In urban traffic networks, a particular routing path may have multiple small alternatives through the closest crossings and streets, which minimally impacts global path costs. They may provide efficient solutions when an edge becomes congested or suffers an incident. We can also lighten these minor path variations around the nodes \(\{\eta_{x}\}_{i}^{d}\) contained in the path flow \(r_{i}^{d}\) so that the routing agent can consider them as a suitable alternative while searching for the best path considering the edge occupancy. The mechanism to generate the TWM for the routing area around the path flow \(r_{i}^{d}\) is straightforward and described by (22):  
\[Y_{Strictch\_Area}:\Theta,r_{i}^{d},c_{i}^{d},\pi^{c},R,\alpha^{1} \rightarrow\left[\theta_{e}^{dj}\right]=[\omega_{e}]*\alpha_{m}^{*} \tag{22}\] \[\alpha_{m}^{*}=\alpha_{m},\forall e\in\{r_{i}^{d}\}\] \[\alpha_{m}^{*}=\alpha_{m}*\alpha^{R},\forall e\in\Theta,\ dist( \eta_{a},\eta_{b})\in[1,R],\] \[\eta_{a}\in\{r_{i}^{d}\},\eta_{b}\in\epsilon\]  
First, we select all the edges \(\{e_{e}\}_{i}^{d}\) connecting the neighbor nodes at a distance \(R\) (proximity radius) from the \(\{\eta_{x}\}_{i}^{d}\) nodes of the path flow, and then a new \(\alpha_{m}^{*}\) factor is applied to them considering a new \(\alpha^{R}\) penalty. Figure 3 illustrates this method with a synthetic traffic network and two completely disjoint path-flows. The distance between two nodes \(dist(\eta_{a},\eta_{b})\) may be evaluated in diverse ways: as the number of hops (edges) that separate them or as the cost function between them. For simplicity, we will consider the number of hops between them.