Concavity and monotonicity properties in a groundwater management model

We consider a discrete‐time groundwater model in which the cost of pumping takes a slightly different form to that which has been traditional in the research literature to date. This enables us to prove that (a) the optimal pumping quantity is nondecreasing in the ground water stock, (b) the stock level remaining after each period's pumping is also nondecreasing in the groundwater stock, (c) the optimal decision is determined by maximizing a concave function, and finally (d) the optimal pumping quantity is nonincreasing in the number of periods to go. We show that (a)–(c), while intuitive, do not hold under traditional modeling assumptions. We also explain the connections between our results and similar ones for some classic problems of operations research. © 2011 Wiley Periodicals, Inc. Naval Research Logistics 00: 000–000, 2011     

Shortest path interdiction problem with arc improvement recourse: A multiobjective approach

We consider the shortest path interdiction problem involving two agents, a leader and a follower, playing a Stackelberg game. The leader seeks to maximize the follower's minimum costs by interdicting certain arcs, thus increasing the travel time of those arcs. The follower may improve the network after the interdiction by lowering the costs of some arcs, subject to a cardinality budget restriction on arc improvements. The leader and the follower are both aware of all problem data, with the exception that the leader is unaware of the follower's improvement budget. The effectiveness of an interdiction action is given by the length of a shortest path after arc costs are adjusted by both the interdiction and improvement. We propose a multiobjective optimization model for this problem, with each objective corresponding to a different possible improvement budget value. We provide mathematical optimization techniques to generate a complete set of strategies that are Pareto‐optimal. Additionally, for the special case of series‐parallel graphs, we provide a dynamic‐programming algorithm for generating all Pareto‐optimal solutions.