Optimal Resource Allocation in Activity Networks under Stochastic Conditions

We treat the problem of optimally allocating a single resource under uncertainty to the various activities of a project to minimize a certain economic objective composed of resource utilization cost and tardiness cost. Traditional project scheduling methods assume that the uncertainty resides in the duration of the activities. Our research differs from the traditional view in that it assumes that the work content (or “effort”) of an activity is the source of the ‘internal’ uncertainty — as opposed to the ‘external’ uncertainty — and the duration is the result of the intensity of the resource allocated to the activity, which then becomes the decision variable. The functional relationship between the work content, the resource allocation, and the duration of the activity is arbitrary, though we take it to be hyperbolic. When the work content is known only in probability, we discuss the approach via stochastic programming and demonstrate its inadequacy to respond to the various questions raised in this domain. Our analysis treats the special case when the work content is exponentially distributed, which may be viewed as an ‘upper bound’ distribution on most probability distributions suggested in this field. 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. † Corresponding author.