## 3 GNL Based SUE with Consideration of Wayfinding Instructions
### Formulation of SUE Model  
The stochastic user equilibrium (SUE) model relaxes the assumption of deterministic user equilibrium (DUE) that users have perfect knowledge of the network state and consistently choose optimal routes. Various choice models are employed in the formulation of SUE. The multinomial logit (MNL) model proposed by [PERSON] implies the assumption of independent and irrelevant alternatives (IIA) (i.e., different choice are uncorrelated), which has a limit on interpreting route similarities because of the overlap issue [18]. The nested logit partly solved this problem by constructing a hierarchical logit model that captures correlation features between different choices. [PERSON] further introduces the cross-nested logit (CNL) model, which contains a two-level nesting structure that allows an alternative to belong to more than one nest. The two-level nesting structure is shown in Figure 5.  
Figure 4: Revised incident matrix in the context of static wayfinding instructions. Figure 5: Illustration of the two-level structure of CNL model. To improve the generalization performance of CNL, the generalized nested logit (GNL) model proposed by [PERSON] and [PERSON] with a higher degree of flexibility is applied in this study and it is proven that GNL is applicable in large-scale networks [17]. [PERSON] and [PERSON] presented the mathematical formulation of a GNL-based SUE model as follows [18]:  
\[minZ=Z_{1}+Z_{2}+Z_{3} \tag{2}\]  
\[Z_{1}\ = \sum_{m}\int_{0}^{x_{m}}c_{m}(w)dw \tag{3}\]  
\[Z_{2}= \frac{1}{\theta}\sum_{rs}\sum_{m}\sum_{k}\mu_{m}f_{mkl}^{rs}\frac{f_{mk}^{ rs}}{\left(\alpha_{mk}^{rs}\right)^{1/\mu_{m}}} \tag{4}\]  
\[Z_{3}=\frac{1}{\theta}\sum_{rs}\sum_{m}(1-\mu_{m})(\sum_{k}f_{mk}^{rs})ln(\sum_ {k}f_{mk}^{rs}) \tag{5}\]  
subject to  
\[\sum_{k}f_{k}^{rs}=q^{rs},\forall r,s \tag{6}\]  
\[f_{mk}^{rs}\geq 0,\forall m,k,r,s. \tag{7}\]  
The expression \(f_{mk}^{rs}ln\frac{f_{mk}^{rs}}{\left(\alpha_{mk}^{rs}\right)^{1/\mu_{m}}}=0\) when \(f_{mk}^{rs}=0\) or \(\alpha_{mk}^{rs}=0\). Where \(\theta\) is the dispersion coefficient that describe the sensitivity of route choice behavior to changes in travel cost, and \(\alpha_{mk}^{rs}\) is the inclusion coefficients representing the proportion of route \(k\) belonging to nest (link) \(m\)[17], and they can be obtained by the following formula:  
\[\alpha_{mk}=\frac{L_{m}}{L_{k}}\delta_{mk}. \tag{8}\]  
\(L_{m}\) and \(L_{k}\) are length of link \(m\) and path \(k\) and \(\sum_{m}\alpha_{mk}=1\). In this study, we can rewrite inclusion coefficients \(\alpha_{mk}^{rsi}\) by applying revised incident matrix \(\mathbf{\Delta}_{is}\), proposed earlier, to capture the influence to network flow generated by static wayfinding instructions:  
Figure 4:0  
where \(\delta_{mki}\) is an element of revised incident matrix. \(\mu_{m}\) is a nesting parameter indicating the dissimilarity of nest \(m\) (\(\mu_{m}\in[0,1]\)). In the CNL model, all nesting parameters \(\mu_{m}\) are equal [17]. When \(\mu_{m}\) equals 1, the model collapses to MNL, and when \(\mu_{m}\to 0\), the model becomes probabilistic at the link level and deterministic at the nest level [14]. [PERSON] and [PERSON] provide the following formula for the nesting parameter \(\mu_{m}\)[17]. Figure 4:1  
where \(N_{m}\) is the number of paths passing through link \(m\). In this formulation, \(\mu_{m}\) can be calculated directly from a given transportation network. In this study, the well-known Bureau of Public Roads (BPR) function is applied to calculate travel time cost on link \(m\). The BPR function is widely used in traffic assignment and traffic flow analysis. [PERSON] introduced a congestion boundary estimation approach and a method for examining the biases caused by the positioning of loop detectors on the macroscopic fundamental diagram (MFD), while [PERSON] employed the BPR function to model SUE and test the convergence performance of different algorithms [22; 36; 37].