# Quantum Computing Approaches for Mission Covering Optimization
## 3 MCO Scenarios
### Scenario 1: Primary and Secondary Resources  
The first scenario involves two categories of resources: primary and secondary. Primary resources are ready to be used for mission completion. Secondary resources are allocated when primary resources are not able to perform their duty. Primary resources that cannot be allocated to missions are removed from the set of primary resources. For example, the mission covering optimization solution shown in Figure 1 is composed of three missions requiring three, two and two resources, respectively, for mission completions. Real-world applications are aeronautical missions and operations with pilots as resources. This particular example is used throughout the paper. Suppose in the planning phase that seven pilots were allocated to cover the three missions; they are on-duty (in the primary resource set), and three other pilots are on-call (in the secondary resource set). Suppose at a certain point in time, two of the pilots became sick, and they were removed from the resource group. This is an emergency, an unforeseen situation that requires re-running an optimization algorithm to cover the missions, including secondary resources. The pilots are pulled from the set of five pilots on-duty, ready to cover the missions and two pilots from the secondary resource set should also be included. The challenge is to find the best allocation of pilots to missions using all the pilots on-duty and two pilots on-call. The goal of our effort is to use quantum computing to solve this problem and analyze results from different quantum computing hardware implementations. The following outlines all the rules in this specific scenario:  
1. The set of missions in the problem is \(\mathbf{M}=\{m_{1},\ldots,m_{N_{m}-1},U\}\). 2. The set of resources is \(\mathbf{R}=\{r_{1},\ldots,r_{N_{r}}\}\). 3. There only exists one qualification, which is the ability to fly a plane, \(\mathbf{Q}=\{q_{1}\}\). 4 Resources can have a capability of 1 or 2. Therefore, the capability function is then \(\mathbb{C}\colon\mathbf{R}\times\mathbf{Q}\to\{1,2\}\). This is a way to represent how ready the resource is to be allocated to a mission. It can be thought of as an ordering for the allocation of resources; resources with capability 2 should be allocated before ones with 1. Resources that have capability 2 are referred to as primary resources, which are assigned to missions first. Resources that have capability 1 are referred to as secondary resources, which are assigned to missions only if primary resources become unavailable. 5. \(\texttt{REQ}_{M}\) is the mission's requirement. For example: * \(\texttt{REQ}_{M}(m_{1},q_{1})=3\). * \(\texttt{REQ}_{M}(m_{2},q_{1})=2\). * \(\texttt{REQ}_{M}(m_{3},q_{1})=2\). 6. There are no resource requirements. Therefore, the resource requirement function is \[\texttt{REQ}_{R}\colon\mathbf{R}\times\mathbf{Q}\to\{0\}.\]
7. \(f_{obj}\) is the function used to score the different problem solutions in terms of cost The mission cost includes two parts: * The mission cost represents how many of the mission's requirements are not met. It is a penalty introduced every time a mission is not able to accomplish its goal due to a lack of resources. * The precedence cost measures how well each solution allocates resources with higher capabilities before others to missions. In the specific example, it means that it is desirable that primary pilots are allocated to missions before secondary pilots. As discussed previously, MCO covers a broad spectrum of problems distinguishable only by its objective function. In the scenario described here, the objective function reflects the cost, which is the sum of the mission cost and precedence cost. All solutions are measured in terms of the cost in order to find the optimal solution. This scenario can be described by using a matrix arrangement of Boolean variables, shown in Figure 1. Each row of the table in the figure represents a mission, and each column represents a resource.