| We treat the problem of optimally allocating resources of limited availability under uncertainty to the various activities of a project to minimize a certain economic objective composed of resource cost and tardiness cost. Traditional project scheduling methods assume that the uncertainty resides in the duration of the activities. Our research assumes that the work content (or "effort") of an activity is the source of uncertainty and the duration is the result of the amount of resource allocated to the activity, which then becomes the decision variable. The functional relationship between the work content ( w), the resource allocation (x), and the duration of the activity (y) is arbitrary, though we assume that the relationship obeys the "power law." In other words, y = f(w, xgamma), where the exponent gamma is some constant.; As preliminary, we first treat the problem assuming that the work content is known deterministically. We develop two new models, a nonlinear programming model, which can be used when resource availabilities are continuous, and an integer program that handles the case when resource availabilities are discrete. When the work content is known only in probability, we first 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 a finite number of steps. In case of arbitrary probability distribution of the work content, we develop a simulation-cum optimization method that incorporates sampling optimization and variance reduction techniques, and which can be used for the purposes of estimation of total project cost, resource consumption levels, etc. |
