## 3 Test of Bio-Inspired Algorithms on a Transportation Planning Problem
### A Biogeography-Based Optimization (BBO) Algorithm for the Problem  
We develop a new algorithm based on biogeography-based optimization (BBO) [62] for the problem. BBO is a relatively new evolutionary algorithm inspired by the science of biogeography. In the meta-heuristic, an individual solution is analogous to a habitat, the solution components are analogous to a set of suitability index variables (SIVs) and solution fitness is analogous to the habitat's suitable index (HSI). Central to the algorithm is the equilibrium theory, which indicates that high HSI habitats have a high species emigration rate and low HSI habitats have a high species immigration rate. To our best knowledge, there is no report in the literature that applies the BBO heuristic to transportation problems in disaster management. In our algorithm, each solution of the problem is encoded as a set of \(m\) sequences (permutations) \(\{\pi_{1},...,\pi_{i},...,\pi_{m}\}\), such that \(\bigcup_{i=1}^{m}C(\pi_{i})=\{1,...,n\}\) and \(C(\pi_{i})\cap C(\pi_{i^{\prime}})=\emptyset\) for any two different \(i\) and \(i^{\prime}\). The algorithm starts by initializing a population, \(P\), of random solutions. At each iteration, the solutions are sorted in increasing order of the objective value, and the immigration rate and emigration rate of the \(k\)-th solution is respectively calculated as:  
\[\lambda_{k} = I\frac{k}{|P|} \tag{9}\] \[\mu_{k} = E(1-\frac{k}{|P|}) \tag{10}\]  
where \(I\) and \(E\) are the maximum possible immigration and emigration rates, which are typically set to one. Then, the population evolved by migration from probably high fitness solutions to low fitness solutions and the random mutation on a part of solutions. The procedure continues, until the termination condition is satisfied. Algorithm 1 presents the framework of our BBO algorithm for the problem, where \(p\) is a predefined mutation rate and \(rand()\) returns a uniformly distributed random number in [0,1]. Let two solutions \(\mathbf{h}=\{\pi_{1},...,\pi_{i},...,\pi_{m}\}\) and \(\mathbf{h}^{\prime}=\{\pi^{\prime}_{1},...,\pi^{\prime}_{i},...,\pi^{\prime}_ {m}\}\); the migration from \(\mathbf{h}\) to \(\mathbf{h}^{\prime}\) at the \(i\)-th dimension is performed as follows:  
1. Let \(R_{i}\) be the set of tasks in \(\pi^{\prime}_{i}\), but not in \(\pi_{i}\). 2. Set \(\pi^{\prime}_{i}=\pi_{i}\) in \(\mathbf{h}^{\prime}\). 3. For each \(j\in R_{i}\), find the position, \(k\), such that \(j\) belongs to \(\pi_{k}\) of \(\mathbf{h}\), and then, insert \(j\) into \(\pi^{\prime}_{k}\) of \(\mathbf{h}^{\prime}\), such that the new \(\pi^{\prime}_{k}\) has the minimum length of completion time among all permutations of the tasks in \(\pi^{\prime}_{k}\). For example, suppose there are three sources and eight targets, \(\mathbf{h}=\{[5,3],[7,4],[2,6,8,1]\}\), \(\mathbf{h}^{\prime}=\{[5,4,1],[8,2],[7,3,6]\}\); when migrating the first dimension of \(\mathbf{h}\) to \(\mathbf{h}^{\prime}\), we have \(R_{1}=\{4,1\}\), and after migration \(\pi^{\prime}_{1}=[5,3]\); Task 4 is in \(\pi_{2}\) of \(\mathbf{h}\) and, thus, will be inserted into \(\pi^{\prime}_{2}=[8,2]\), such that \(\pi^{\prime}_{2}\) has the minimum length of completion time among all permutations of the set \(\{2,4,8\}\); similarly,Task 1 will be inserted into \(\pi_{3}^{\prime}\), such that it has the minimum length of completion time among all permutations of \(\{1,3,6,7\}\). The mutation operation of a solution \(\mathbf{h}=\{\pi_{1},...,\pi_{i},...,\pi_{m}\}\) at the \(i\)-th dimension is performed as follows: randomly select an \(i^{\prime}\) other than \(i\); drop a task, \(j\), from \(\pi_{i}\) and reinsert it into \(\pi_{i^{\prime}}\), such that the new \(\pi_{i^{\prime}}\) has the minimum length of the completion time. If a migration or mutation results in an infeasible solution, the operation is skipped on the current dimension and moves to the next dimension.