## 3 A Non-Real-Time DTN-Based Forwarding Algorithm
### A New Multi-Objective Mathematical Model
This decision variable models the amount of data that is sent through a link at any given time and it is used to maximize the amount of data flowing  
Figure 4: Problem graph. (**a**) Availability probability and data flow from node \(i\) to node \(j\); (**b**) Network graph with delivery probabilities varying through time. AP: Access Point. through the highest delivery probability paths and minimize the time used to get the data to their destination. Table 2 shows all parameters and sets used in the model. Figure 5 shows the network graph expanded through time. The multiobjective mathematical model maximizes the data flow through the paths with the highest delivery probabilities over time and at the same time it tries to minimize the delivery time for these messages in the network (see Equations (2) and (3). Constraints are described in Equations (4)-(7). Equations (4) and (5) are flow and balance constraints that require demands to be met, so all messages are delivered to their destination through one or several time steps. Equation (4) states that at time zero (0) what a node sends to other neighboring nodes and stores in its buffer should be equal to the offer of the node (\(b_{i}(t)\)) plus the messages received from other nodes. Equation (5) is a generalization of Equation (4) for times greater than zero. It takes into account the messages that are stored in a nodes' buffer from the previous time. Equations (6) and (7) are capacity constraints that require that for any given time, information flow over links and stored in buffers must be equal to or below these links and buffers' capacities. Finally, constraint in Equation (8) states that the decision variable \(x_{ij}(t)\) is a positive integer defined over all links for all paths. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Definition** \\ \hline \(N\) & Set of nodes \\ \(E\) & Set of edges \\ \(P\) & Set of candidate paths \\ \(T\) & Set of discrete time values \\ \(c_{ij}\) & Capacity of edge \((i,j)\in E\) \\ \(c_{ii}\) & Storage capacity of node \(i\) \\ \(a_{ij}(t)\) & Availability Probability of edge \((i,j)\in E\) at time \(t\). \(a_{ij}(t)=a_{ji}(t)\) \\ \(d_{ij}(t)\) & Delivery Probability of node \(i\) through node \(j\) at time \(t\) \\ \(b_{i}(t)\) & Set of offers or demands from node \(i\) at time \(t\) \\ \(t\) & discrete time value, where \(t\in T\) \\ \(\delta_{t}\) & Time interval duration \\ \(v\) & number of time values since last update \\ \(x_{ij}(t)\) & Decision variable. How much data flows through edge \((i,j)\in E\) at time \(t\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mathematical model parameters. Figure 5: Problem graph through time. Temporal links go from left to right and represent nodes’ buffers. Objective Functions  
\[\max\prod_{(i,j)\in P}d_{ij}(t)x_{ij}(t) \tag{2}\] \[\min\sum_{(i,j)\in P}\sum_{t\in T}x_{ij}(t)\delta_{t} \tag{3}\]  
Constraints  
\[\sum_{j\in N}x_{ij}(t)-\sum_{j\in N}x_{ji}(t)+x_{ii}(t)=b_{i}(t) \qquad\forall(i,j)\in E,i\
eq j,t=0 \tag{4}\] \[\sum_{j\in N}x_{ij}(t)-\sum_{j\in N}x_{ji}(t)+x_{ii}(t)-x_{ii}(t-1 )=b_{i}(t)\qquad\forall(i,j)\in E,i\
eq j,t\geq 1\] (5) \[x_{ij}(t)+x_{ji}(t)\leq c_{ij}\delta_{t}\qquad\forall(i,j)\in E,i\
eq j\] (6) \[x_{ii}(t)\leq c_{ii}\qquad\forall i\in N\] (7) \[x_{ij}(t)\in\mathbb{Z}_{\geq 0}\qquad\forall(i,j)\in E,p\in P \tag{8}\]