Algorithm for optimal chance constrained knapsack problem with applications to multi-robot teaming

Motivated by applications in multirobot team selection, in this paper, we present a novel algorithm for computing optimal solution of chance-constrained 0-1 knapsack problem. In this variation of the knapsack problem, the objective function is deterministic but the weights of the items are stochastic and therefore the knapsack constraint is stochastic. We convert the chance-constrained knapsack problem to a two-dimensional discrete optimization problem on the variance-mean plane, where each point on the plane can be identified with an assignment of items to the knapsack. By exploiting the geometry of the non-convex feasible region of the chance-constrained knapsack problem in the variance-mean plane, we present a novel deterministic technique to find an optimal solution by solving a sequence of deterministic knapsack problems (called risk-averse knapsack problem). We apply our algorithm to a multirobot team selection problem to cover a given route, where the length of the route is much larger than the length each individual robot can fly and the length that an individual robot can fly is a random variable (with known mean and variance). We present simulation results on randomly generated data to demonstrate that our approach is scalable with both the number of robots and increasing uncertainty of the distance an individual robot can travel.