Interdiction Games on Markovian PERT Networks

Abstract

I n a stochastic interdiction game a proliferator aims to minimize the expected duration of a nuclear weapons development project, and an interdictor endeavors to maximize the project duration by delaying some of the project tasks. We formulate static and dynamic versions of the interdictor's decision problem where the interdiction plan is either precommitted or adapts to new information revealed over time, respectively. The static model gives rise to a stochastic program, whereas the dynamic model is formalized as a multiple optimal stopping problem in continuous time and with decision-dependent information. Under a memoryless probabilistic model for the task durations, we prove that the static model reduces to a mixed-integer linear program, whereas the dynamic model reduces to a finite Markov decision process in discrete time that can be solved via efficient value iteration. We then generalize the dynamic model to account for uncertainty in the outcomes of the interdiction actions. We also discuss a crashing game where the proliferator can use limited resources to expedite tasks so as to counterbalance the interdictor's efforts. The resulting problem can be formulated as a robust Markov decision process.

Scheduling Markovian PERT networks to maximize the net present value: New results

“…Let m := max P ∈P |E(P )| denote the maximum number of tasks that can be active in parallel, then it is not difficult to show that |P| is bounded above by m i=0 n i ; see [4]. Indeed, P is completely determined by E(P ) and n i bounds the number of states having |E(P )| = i.…”

A Decomposition Approach for Stochastic Shortest-Path Network Interdiction with Goal Threshold

“…Shortest-path network interdiction problems can be extended for various application scenarios [23]. Wei et al [24] introduced a threshold on the capacity of a network and proposed algorithms based on decomposition and duality formulation to optimize the interdictor's resource consumption.…”

Timely exposure of a secret project: Which activities to monitor?

“…Harney et al (2006) describe the tasks necessary to produce "a first small batch of nuclear weapons" together with their durations and precedence constraints. The project-which has also served as an example in (Brown et al, 2009;Gutin, Kuhn, & Wiesemann, 2014;Pinker et al, 2013)-consists of 124 nondummy tasks that can be grouped into five main parts as listed in Table 5. The table's third and fourth column give each part's start and end time (in weeks) under the late-start schedule with a deadline equal to the project's minimum makespan of 468 weeks.…”