Modeling diminishing returns in project resource planning

This paper presents a series of approaches that offer a variety of options for modeling project scheduling in situations involving diminishing returns from increasing allocations of resources. The models allow the analyst to balance activity durations and costs to attain an optimal allocation of resources and activity times. A base example is utilized throughout the development to illustrate the approaches. A linear approximation model is developed, followed by a non-linear model, and finally a discrete multi-modal model. A key feature of all of these models is that the relationship between the duration of an activity and the resources allocated to it are incorporated directly into the constraints of the models. The paper concludes with several suggestions of areas for future research, including potential approaches for the solution of large, operational problems.