Optimal resource allocation in activity networks - stochastic environment

We treat the problem of optimally allocating a single resource under uncertainty to minimise the sum of resource cost and tardiness cost. We assume that the work content of an activity is the source of the 'internal' uncertainty - as opposed to the 'external' uncertainty. When the work content is known only in probability, we discuss the approach via stochastic programming and demonstrate its inadequacy. We treat the special case when the work content is exponentially distributed. This results in a continuous-time Markov chain with a single absorbing state. We establish convexity of the cost function and develop a policy iteration-like approach that achieves the optimum in finite number of steps. In case of arbitrary probability distribution of the work content, we develop a simulation-based optimisation method that incorporates sampling optimisation and variance reduction techniques.