## 3 GNL Based SUE with Consideration of Wayfinding Instructions
### Impact of Wayfinding Instructions in Network  
Many researchers developed storage structures and estimation methods to analyze the utilization of wayfinding instructions. [PERSON] introduced the conception of decision points where users make decision on route choices, and further provided a structure for storing wayfinding instruction information, which includes link guidance and destination names [33]. A consistency evaluation algorithm provided by [PERSON] calculates the degree of consistency along a path between the given origin-destination (OD) pair by combing network topology and guide signs [34]. The mathematical program for guide sign location given by [PERSON] optimizes the orientation accessibility as well as total costs [5]. However, extensive information of wayfinding instruction is required in above methods, and research examining the impact of static wayfinding instructions on traffic flow is still missing in literature. As illustrated in Figure 1, a typical static wayfinding instruction consists of two parts: (1) directions that provide link guidance; and (2) destinations that correspond to nodes in the network. According to the Manual on Uniform Traffic Control Devices (MUTCD), wayfinding signs direct users unfamiliar with local condition by directing them to specific destinations to prevent incorrect route choices and provide single-direction guidance in case of obscurity; hence, paths not guided ahead of this node will be unused assuming that travelers have imperfect information on travel time cost [35] (p. 268). Figure 2 shows impact on users' path choice behavior posed by static wayfinding instruction. Figure 1: Components of typical static wayfinding instruction. To demonstrate the impact of static wayfinding instructions at the network level, consider a traffic network denoted by \(G(N,L)\), where \(N\) is the set of nodes and \(L\) is the set of links. Let \(x_{m}\) and \(c_{m}\) represent the flow and cost on link \(m\in L\), where \(c_{m}\) is a function of \(x_{m}\) and can be written as \(c_{m}=c(x_{m})\). Similarly, \(f_{k}^{\prime\prime s}\) and \(c_{k}^{\prime s}\) represent the flow and cost, respectively, on path \(k\) between origin \(r\) and destination \(s\). Let \(q^{rs}\) denote the traffic demand between \(r\) and \(s\). The incident matrix is denoted by \(\mathbf{\Delta}\), with elements \(\delta_{mk}\), where \(\delta_{mk}\) is 1 if link \(m\) is a part of path \(k\), and \(\delta_{mk}\) is 0 otherwise. The matrix expression of the relationship between path flow and link flow can be written as follows:  
\[\mathbf{x}^{\mathbf{T}}=\mathbf{\Delta}\mathbf{f}^{\mathbf{T}} \tag{1}\]  
where \(\mathbf{f}=(\dots,\mathbf{f^{s}},\dots)\), and \(\mathbf{f^{s}}=(\dots,f_{k}^{rs},\dots)\) is a vector with elements representing the flow of each path connecting the OD pair \(r-s\). \(\mathbf{x}=(\dots,x_{a},\dots)\) denotes flows on each link. \(\mathbf{\Delta}\) is the incident matrix mentioned above. Figure 3 is a small example to demonstrate the above formulation. It can be seen that revising the incident matrix \(\mathbf{\Delta}\) is an efficient method for modeling the relationship between uses' link choices influenced by static wayfinding instructions and the path actually used. As discussed earlier, for users consistently following guided paths towards their destinations and freely selecting paths at decision points lacking required wayfinding instructions, this impact can be described as \(\delta_{nl}=0\) in the above presentation, where \(n\) is the link not guided and \(l\) is the path containing link \(n\), whose destination is informed by this wayfinding sign. The network with static wayfinding instructions can be modeled as follows: Let \(m\times m\) diagonal matrix \(\Lambda_{s}=\mathbf{diag}(i)\), with elements \(i_{1},\dots,i_{m}\) representing impact on link \(m\in L\) with destination \(s\). \(i_{j}=-1\) when link \(j\) is not guided and 1 otherwise.