A multi-stage non-deterministic model for project scheduling under resources constraints

Nowadays the general purpose of project scheduling is the development of decision-aids to help managers improving the planning, the programming and the control of their projects. This target is hard to achieve because projects have complex descriptions and because such decisions include multiple choices about the allocation of resources and the project calendar, which have to be improved in terms of several criteria such as total duration, net present value, risk of failing to meet the planned durations for important fractions of the project, etc. A review of the major approaches adopted to describe projects in terms of time and resources is presented in this paper showing that, unfortunately, too large deterministic models are usually obtained for projects with a realistic size. A new formulation of the general problem of project scheduling under resources constraints is proposed in this paper considering the project as a sequence of stages. According to this approach, the project can be studied as a sequence of decisions on resources allocation where each time interval between successive decisions (stage duration) is explicitly formulated as a decision variable. This formulation is very powerful and flexible as it avoids large amounts of binary variables and it allows progressive levels of detail concerning the discretization of the project schedule. Also, non-deterministic attributes can be introduced and the optimization problem can be easily solved for most common objective functions. Analytical solutions can also be deduced using the proposed model if specific hypotheses about the restrictions on durations and resources are assumed.