# Coordination in Emergency Response System Design:
## 4 Two-Resource Maximal Covering Model  
In this section, we expand the maximal covering problem formulation to capture the presence of two emergency response resources that need to be coordinated. We consider each node on the urban transport network a candidate site for an ambulance and/or fire station. We do not explicitly model the spatial distribution of population density, which amounts to assigning the same priority for covering each node. The response times for each node are calculated for the dispatch of a vehicle from the closest station via the shortest route at an average vehicle speed of 60 km/h. Please see the Abbreviations section at the end of the manuscript for our notation. Naturally, \(o_{i}=1\) for \(i\in EAS\) and \(p_{i}=1\) for \(i\in EFS\). In order to capture the coordination between ambulances and fire stations, we define the auxiliary variable as follows:  
\[x_{ij}=o_{i}\times p_{j} \tag{1}\]  
That is, \(x_{ij}=1\) if the ambulance station \(i\) and the fire station \(j\) are established simultaneously; 0, otherwise. Since \(x_{ij}\) is nonlinear in terms of the decision variables, we will include its linearization in the model below. We now define two parameters that indicate the nodes that will be moved from the poor and adequate response time areas to the preferred response time areas, respectively. The value of \(A_{ijk}\) indicates the nodes that will move from  
Figure 1: Categorization of nodes according to the response time of the fire station, \(RTFS\), and the response time of the ambulance station, \(RTAS\): (**a**) the target lines; (**b**) response time areas. the poor area to the preferred area by the co-establishment of an ambulance station at node \(i\) and a fire station at node \(j\). That is, \(A_{ijk}\): a binary parameter, and each node \(k\in A^{Poor}\), where  
\[A_{ijk}=\left\{ \begin{array}{ll}1&\text{when }x_{ij}=1,\left|DH_{ik}-DF_{jk} \right|\leq\overline{WT},\ DH_{ik}\leq\overline{RTAS},\ DF_{jk}\leq\overline{ RTFS}\\ 0&\text{otherwise}\end{array} \right. \tag{2}\]  
Similarly, \(B_{ijl}\) keeps track of the nodes that move from the adequate area to the preferred area by the simultaneous establishment of an ambulance station at node \(i\) and a fire station at node \(j\). That is,  
\(B_{ijl}\): a binary parameter, and for each node \(l\in A^{Adap}\), where  
\[B_{ijl}=\left\{ \begin{array}{ll}1&\text{when }x_{ij}=1,\left|DH_{il}-DF_{jl} \right|\leq\overline{WT},\ DH_{il}\leq\overline{RTAS},\ DF_{jl}\leq\overline{ RTFS}\\ 0&\text{otherwise}\end{array} \right. \tag{3}\]  
_Mathematical Model_  
\[\text{max}f=\sum_{k\in A^{Poor}}y_{k}+\sum_{l\in A^{Adap}}z_{l} \tag{4}\]  
subject to  
\[\sum_{i\in A\otimes UAAS}o_{i}\leq NA \tag{5}\]  
\[\sum_{j\in EF\otimes AFS}p_{j}\leq NF \tag{6}\]  
\[x_{ij}\leq o_{i} \tag{7}\]  
\[x_{ij}\leq p_{j} \tag{8}\]  
\[x_{ij}\geq o_{i}+p_{j}-1 \tag{9}\]  
\[\sum_{i\in A\otimes UAAS}\sum_{j\in A\otimes UAAS}A_{ijk}x_{ij}\geq y_{k},\ \ \forall k\in A^{Poor} \tag{10}\]  
\[\sum_{i\in A\otimes UAAS}\sum_{j\in EAS\otimes UAAS}B_{ijl}x_{ij}\geq z_{l},\ \ \forall l\in A^{Adap} \tag{11}\]  
\[x_{ij}\in\{0,1\},\ \ \forall i\in EAS\cup AAS,\ \forall j\in EFS\cup AFS \tag{12}\]  
\[o_{i}\in\{0,1\},\ \forall i\in AAS \tag{13}\]  
\[\sum_{i\in A\otimes UAAS}o_{i}\leq NA \tag{5}\]0  
\[\sum_{i\in A\otimes UAAS}o_{i}\leq NA \tag{5}\]1  
\[\sum_{i\in A\otimes UAAS}o_{i}\leq NA \tag{5}\]2  
The objective function (4) maximizes the number of nodes in the preferred area after the establishment of the new ambulance and fire stations. It is possible to prioritize the number of nodes in the poor response time area by using a weight for the first term. Constraints (5) and (6) ensure the total number of ambulance stations and fire stations remain within the prespecified limits. Constraints (7)-(9) linearize (1) and ensure that only the established ambulance stations and fire stations can be selected to serve the transportation network. Constraint (10) is a logical constraint for the nodes originally in the poor response time area.