an optimal npv project scheduling with fixed work content and payment on milestones

project scheduling   net present value   we consider a project scheduling problem with permitted tardiness and discrete time/resource trade-offs under maximum net present value objective. in this problem, a project consists of a set of sequential phases such that each phase contains one or more sub-projects including activities interrelated by finish-start-type precedence relations with a time lag of zero, which require one or more renewable resources. there is also a set of unconstrained renewable resources. for each activity, instead of a fixed duration and known resource requirements, a total work content respect to each renewable resource is given which essentially indicates how much work has to be performed on it. this work content can be performed in different modes, i.e. with different durations and resource requirements as long as the required work content is met. based on the cost of resources units and resource requirements of each activity, there is a corresponding cash flow for the activity. each phase is ended with a milestone that corresponds to the phase income. we prove that the mode corresponding to the minimum possible duration of each activity is the optimal mode in this problem. we also present a simple optima scheduling procedure to determine the finish time of each activity .