Mathematical Model and Framework for Multi-Phase Project Optimization

Abstract

This research aims to assist investors of “real” tangible assets such as construction projects in making an optimal portfolio of phased and regular projects which will yield the best financial outcome calculated in terms of discounted cash flow of future anticipated revenues and costs. We use optimization techniques to find the optimal timing and phasing of a single project that has the potential of being decomposed into smaller sequential phases. Existence of uncertainties is inevitable especially in cases in which we are planning for long durations. In the presence of these uncertainties, full upfront commitment to large projects may jeopardize the rationality of investments and cause substantial economic risks. Breaking a big project into smaller stages (phases) and implementing a staged development is a potential mechanism to hedge the risk. Under this approach, by adding managerial flexibilities, we may choose to abandon a project at any time once the uncertain outcomes are not favorable. In addition to the benefits resulting from hedging unfavorable risks, phasing a project can transform a financially infeasible project into a feasible one due to less load on capital budgets during each time. Once some phases of a project are delayed and planned to be implemented sequentially, it is important to prepare the infrastructure required for their future development. Initially, we present a Mixed Integer Programming (MIP) model for the deterministic case with no uncertainties that considers interrelationships between phases of projects such as scheduling and costs (economy of scales) in addition to the initial infrastructural investment required for implementation of future phases. Pairing possible phases of a project and doing them in parallel is beneficial due to positive synergies between phases but on the downside requires larger capital investments. Unavailability of enough budgets to fully develop a profitable project will cause the investment to be carried out in different phases e.g. during times when the required capital for developing the next phase (or group of phases) is available. After, presenting the model for the deterministic case, we present a scenario-based multi-stage MIP model for the stochastic case. The source of uncertainty considered is future demand that is modeled using a trinomial lattice. We then present two methods for solving the stochastic problem and finding the value of the here and now decision variable (the size of the infrastructure/foundation). Finding the value of the here and now decision variable for all scenarios using a novel technique that does not require solving all the scenarios is the first method. The second method combines simulation and optimization to find good solutions for the here and now decision variable. Lastly, we present a MIP for the deterministic multi-project case. In this setting, projects could have multiple phases. The MIP will help the managers in making the project selection and scheduling decision simultaneously. It will also assist the managers in making appropriate decisions for the size of the infrastructure and the implementation schedule of the phases of each project. To solve this complex model, we present a pre-processing step that helps reduce the size of the problem and a heuristic that finds good solutions very fast.