# A Cloud Based Disaster Management System
## 6 Optimized Rescue Plan Development
This group of objective functions can be defined as:  
\[\text{minimize}\ \sum_{k=1}^{m}\sum_{i=1}^{n+m}\sum_{j=1}^{n+m}x_{ijk}\text{C} \big{(}T_{i},T_{j}\big{)} \tag{2}\]  
subject to:  
\[\sum_{k=1}^{m}\sum_{i=1}^{n+m}x_{ijk}=1;\ \forall j=1,\ldots,n+m \tag{3}\]  
\[\sum_{k=1}^{m}\sum_{j=1}^{n+m}x_{ijk}=1;\ \forall i=1,\ldots,n+m \tag{4}\]  
\[\sum_{i=1}^{n+m}x_{ijk}=1;\ \forall k=1,\ldots,m \tag{5}\]\[\sum\limits_{j=1}^{n+m}x_{ijk}=1;\ \forall k=1,\ldots,m \tag{6}\]  
\[x_{ijk}\in\{0,1\};\ \forall i,j=1,\ldots,n+m;and\ k=1,\ldots,m \tag{7}\]  
Equations (3) and (4) guarantee that each robot visits each node only once. Equations (5) and (6) ensure that each robot begins and returns to its specific depot. Finally, restrictions brought about by Equation (7) show that the decision variables are twofold. The second group of objective function covers those that reduce the maximum cost for all robots, such as reducing the maximum mission time or reducing the maximum tour length. It is possible to model this group of objective functions as:  
\[\text{minimize}\ \max\limits_{k\in 1\ldots m}\left(\sum\limits_{i=1}^{n+m} \sum\limits_{j=1}^{n+m}x_{ijk}C\left(T_{i},T_{j}\right)\right) \tag{8}\]  
Subject to the same restrictions expressed within Equations (3)-(7). The third group of objective functions relates to balancing the workload between the robots, balancing the lengths of the tour, balancing mission time, as well as the number of targets allocated. It is possible to model this group of objective functions as:  
\[ \begin{array}{c}minimize\ \sum\limits_{k=1}^{m}\left[C_{k}-C_{avg}\right]\\ C_{k}=\left(\sum\limits_{i=1}^{n+m}\sum\limits_{j=1}^{n+m}x_{ijk}C\left(T_{i}, T_{j}\right)\right)\ K\in[1,m]\\ C_{avg}=\frac{\sum\limits_{k=1}^{n+m}\sum\limits_{i=1}^{n+m}\sum\limits_{j=1}^ {n+m}x_{ijk}C\left(T_{i},T_{j}\right)}{m}\end{array}  \tag{9}\]  
where \(C_{k}\) is the tour cost for the robot \(k\) (cost can be distance, energy, time, etc.). \(C_{avg}\) signifies the average cost of all tours. In our application model, it was assumed each robot possessed global knowledge of the targets within the disaster site requires a visit as well as the fire location. In addition, each robot is able to compute an estimation cost between its present location and the target to be visited, such as Euclidean distance, energy and time spent, subject to circumstance. The proposed AHP-MTSP solution put forth is weighted, meaning that we assigned the objectives to be optimized with different weights. We defined the global cost as the sum of the weighted costs of the different objective functions under consideration. Formally, let \(w=\left(w_{1}\cdots w_{p}\right)\) be a weight vector, where \(0<w_{i}<1\ \forall i=1,\ldots p\) and \(\sum\limits_{i=1}^{p}w_{i}=1\). The problem then consists of minimizing the function:  
\[ \begin{array}{c}\text{minimize}\ \text{g}(x|W)=\sum\limits_{i=1}^{p}w_{i}f_{i}(x)\\ \text{subject to}\ :\ x\in\ \Omega\end{array}  \tag{10}\]  
where \(\Omega\) is the decision space and \(f_{i}()\) is an objective function. To construct the weight vector \(W\), we used the AHP technique [66]. AHP is a decision-making approach with multi-criteria that can be used to solve complex decision problems [66]. The relevant data was obtained by means of a series of pairwise comparisons, which are used in terms of each individual decision criterion to obtain the weights of the targets and the relative performance measures of the alternatives. Figure 6 shows an overview of the proposed AHP-MTSP approach. First, the user defines his preference and priority of each objective function in a comparison matrix. This comparison matrix is then introduced to AHP to generate a weight vector. This weight vector is then used to compute the global cost, following Equation (10). Three different approaches were then executed, as shown in Figure 6. Finally, the best solutions for these three approaches were selected.