## 4 Problem Statement and Formulation
### MILP Formulation
#### 4.2.2 Constraints  
To guarantee proper selection of sojourn locations and RPs and assigning the rest of sensor-nodes to sufficient RPs that can minimize the data delivery time, the following constrains must be met:  
1. _The maximum number of RPs on the trajectory should not exceed \(P_{max}\)._ \[|\Delta|\leq P_{max},\] (9)  
**(Note) The method for calculating the maximum number of RPs:** Since the trajectory of the mobile-sink is pre-defined to obtain the maximum number of RPs, we assume in an ideal case that all of the RPs are located beside the trajectory with dense distribution as illustrated in Figure 3a in which, as an example, the sensor-nodes \(A\), \(B\) and \(C\) are in the communication range of each other and all of them can contact with the mobile-sink directly. However, among these three sensor-nodes one of them should contact with the mobile-sink as RP and other two sensor-nodes should transfer their data to the selected RP. Otherwise, the mobile-sink should collect the data from all of RPs in the periphery of the trajectory that consumes a considerable amount of time in every tour. Thus, the maximum number of RPs can be obtained as follows. \[P_{max}=(\frac{D}{(2\times\delta_{d_{p}}+2\times\delta_{d_{Y}})-\xi}\times 2 )\times 2. \tag{10}\]  
such that \(\delta_{d_{p}}\) and \(\delta_{d_{Y}}\) are two consecutive sensor-nodes as depicted in Figure 3b,In Equation (10), \(D\) represents the total distance of the trajectory. By assuming dense distribution of the sensor-nodes in the periphery of the trajectory, the maximum number of RPs can be obtained by dividing the total distance \(D\) to the summation of communication diameter of the two consecutive sensor-nodes with subtracting from the intersected line that is represented by \(\xi\). The obtained value should be multiplied by 2 because in this part of the trajectory two RPs should be selected. And by considering the symmetry condition in another side of the trajectory, the obtained value should be multiplied by 2. The value of \(\xi\) is user-defined. Obviously, due to assuming dense distribution of the sensor-nodes in the whole trajectory, we need to consider two arbitrary consecutive sensor-nodes and the obtained value will be same for rest of the sensor-nodes. 2. _To optimize energy-consumption in communication between sensor-nodes, the solution should not contain energy-triangles._ For example, if sensor-node \(A\) can contact with sensor-node \(C\) directly, then sensor-node \(B\) cannot play hop-role between them because \(E_{AC}\leq E_{AB}+E_{BC}\). The following expressions illustrate the constraint formally. \[\forall i=1,...,n\;,\forall j=1,...,n\;\text{and}\;i\
eq j\text{:}\] \[F_{i}(s_{j})= \begin{cases}1&\text{if sensor\;}j\;\text{is in the communication range of sensor }i,\\ 0&\text{else.}\end{cases} \] (11) \[H_{i}(s_{j})= \begin{cases}1&\text{if sensor\;}i\;\text{contacts with sensor\;}j\;\text{directly,}\\ 0&\text{else.}\end{cases} \] (12) \[F_{i}(s_{j})\times H_{i}(s_{j})=1.\] (13)
3. _The selected RP should be in communication range of the sojourn location._ \(\forall i=1,...,z\;\text{and}\;\forall j=1,...,m\): \[J_{i}(s_{j})= \begin{cases}1&\text{if sensor\;}j\;\text{is in the communication range of sojourn location }i,\\ 0&\text{else.}\end{cases} \] (14) \[\Pi_{i}= \begin{cases}1&\text{if sensor\;}j\;\text{is RP,}\\ 0&\text{else.}\end{cases} \] (15) \[J_{i}(s_{j})\times\Pi_{i}=1.\] (16)
4. _Each sensor-node should send its data to only one RP for preventing data redundancy and extra energy-consumption in the network._  
Figure 3: (**a**) An example of dense distribution of the sensor-nodes in the periphery of the trajectory. (**b**) Two consecutive sensor-nodes that work as RP and their counterparts in other side of the trajectory. \(\forall i=1,...,n\) and \(\forall j=1,...,m\): \[A_{j}(s_{i})= \begin{cases}1&\text{if sensor $i$ is assigned to RP j,}\\ 0&\text{else.}\end{cases} \] (17) \[\sum_{k=1}^{n}A_{j}(s_{k})=1.\] (18)
5.