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A nonlinear allocation problem

We consider the problem of deploying work force to tasks in a project network for which the time required to perform each task depends on the assignment of work force to the task, for the purpose of minimizing the time to complete the project. The rules governing the deployment of work force and the resulting changes in task times of our problem are discussed in the contexts of a) related work on project networks and b) more general allocation problems on polytopes. We prove that, for these problems, the obvious lower bound for project completion time is attainable.