Model Development: The problem can be formulated as a Travelling Salesperson Problem $($TSP$).$ In this problem, the sites to be visited by the rover are the cities in the TSP$.$ The objective is to find the shortest possible route that visits each site exactly once and returns to the origin site $($base$).$ The distances between the sites are given in the problem statement.

Dynamic Programming Implementation: A deterministic dynamic programming algorithm can be developed to solve this problem. The algorithm starts by calculating the shortest paths between all pairs of sites. Then, it iteratively builds up the optimal solutions to subproblems, which are the shortest paths that visit a subset of sites and end at a particular site. The final solution is the shortest path that visits all sites and returns to the base. The algorithm ensures that each site is visited exactly once and that the total distance traveled by the rover is minimized.

Optimization Strategy: The route of the rover can be further optimized by considering factors such as energy consumption, time constraints, and the number of samples collected. For example, the rover could prioritize visiting sites that are closer to the base or that have a higher probability of containing valuable samples. The optimization strategy should be designed to balance these different objectives.

Simulation and Analysis: The performance of the proposed algorithm can be evaluated by simulating the rover's exploration route and analyzing various metrics, such as the total distance covered, energy consumption, time taken, and the number of samples collected. This can be done using a software such as Python, Matlab, or Java.

Deliverables:

A comprehensive report detailing the problem formulation, algorithm design, and implementation steps.

Visualization of the optimized route and comparison with alternative strategies.

Analysis of the simulation results and conclusions about the efficiency and effectiveness of the proposed solution.