# Project Scheduling with Continuously-Divisible, Doubly Constrained Resources

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## Abstract

This paper deals with a class of project scheduling problems concerning the allocation of continuously divisible resources under conditions in which both total usage at every moment and total consumption over the period of project duration are constrained. Typical examples of such resources, called doubly constrained, are money or energy, when the constraint on power or the rate of expenditure cannot be ignored as neither, of course, can. the constraint on resource consumption. Also manpower must often be considered as a doubly constrained resource. Mathematical models of project activities in which performing speeds are continuous functions of resource amounts are considered. The objective is a schedule which minimizes project duration. Thus, the problems considered are generalizations of both the classical project scheduling problem and the time-cost trade-off problem. The properties of optimal schedules are given for strictly concave, concave and convex activity models. On the basis of these properties, methods for finding optimal schedules are described for independent and dependent activities. We also consider the minimum resource consumption ensuring minimum project duration for a given level of resource usage, and the minimum level of resource usage ensuring minimum project duration for a given level of resource consumption. Possible generalizations of the presented results are indicated.

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