Proceedings of the 2005 Systems and Information Engineering Design Symposium Ellen J. Bass, ed. ABSTRACT The planning process of public and private companies re- lies on optimal project selection and scheduling and the ef- ficient allocation of scarce resources. This process is com- plicated due in part to the fact that project investment must consider multiple criteria, project cash flows are uncertain, and there are several operational business and technical constraints. The proposed mixed-integer programming model assists the planning manager/analyst by choosing from a bank of projects in which projects to invest and when to invest. The model maximizes the sum of net pre- sent values of the chosen projects while minimizing their variance. The model satisfies simultaneously a set of precedence relations among projects; early and tardy pro- ject starting dates; exogenous budget limits; and endoge- nous project cash flow generation. Finally, by quantifying the opportunity cost, the model shows how arbitrary pro- ject selection and sequencing can reflect non-desirable so- lutions for the company and the society. 1 INTRODUCTION The planning process for companies is complex due to the great amount of investment projects, the interrelation be- tween them (Vonortas and Hertzfeld, 1998; Childs, Ott, and Triantis, 1998), the multiple criteria that can be rele- vant when evaluating each alternative (Benjamin, 1985; Ehie et. al 1990), and the constraints inherent to the corpo- rate operation (resources, time, regulation, among others). The mathematical models proposed in the project selection literature facilitate the decision-making process avoiding the use of subjective criteria that may generate certain states susceptible of improvement or sub-optimal solutions with harmful consequences for the company or the society. Since the pioneer work of Lorie & Savage (1955), the project selection problem has attracted several researchers. Many techniques have been applied to the project selection problem: linear programming (Benhard, 1969; Freeland and Rosenblatt 1978; Myers, 1972), multiobjective linear programming (Ringuest and Graves 1989; Ringuest and Graves, 1990), integer programming (Beged-Dov, 1965), goal programming (Benjamin 1985; Mukherjee and Bera, 1995), and evolutionary algorithms (Medaglia, 2003) among others. Benli and Yavuz (2002) have addressed timing and sequencing in project selection problems using zero-one programming. Their model provides starting dates for the projects, while maximizing the net present value (NPV). Gupta, Kyparisis and Ip (1992) and Kyparisis, Gupta and Ip (1995) use the same criteria in order to select and se- quence projects. Multiple criteria have been used to quantify the pro- jects’ performance both in selection and sequencing prob- lems. The most frequently used criteria are NPV (Gupta, Kyparisis and Ip 1992; Kyparisis, Gupta and Ip, 1995; Weingartner, 1967) and risk (Kangari and Boyer, 1981; Orman and Duggan, 1999; Stone 1973). Several research- ers have handled risk using the traditional Markovitz met- hodology (Markowitz, 1952). To the best of our knowledge, the proposed model is a novel addition to the existing project selection literature. It combines the project selection and sequencing decisions, while considering risk and profitability as optimization cri- teria. The uncertainty present in the forecasts of the pro- jects’ cash flows is the source of the NPV’s variance. The model provides an intertemporal risk diversification by in- cluding NPV covariance terms for all projects and all start- ing dates. The proposed model is an extension of our ex- perience in one of Colombia’s largest water and sewage companies, where a deterministic model was successfully designed and implemented (Medaglia et. al, 2005). This article is divided into four sections. Section 2 de- fines the portfolio’s expected return and variance when us- ing forecasts. Section 3 contains the formulation of the proposed mixed-integer programming model. In Section 4, we provide computational experiments using a set of sam- ple projects. In this Section, we also perform a sensitivity analysis and show how to construct an efficient frontier. We conclude in Section 5. 2 PROFITABILITY AND RISK UNDER THE FORECAST APPROACH The portfolio’s profitability is measured by the net present value (NPV). Let P be the set of investment projects to be considered. Let T be the planning horizon (no investments TOWARDS A MODEL FOR SELECTION AND SCHEDULING OF RISKY PROJECTS Jorge A. Sefair Industrial Engineering Department School of Economics Universidad de los Andes Bogotá, Colombia Andrés L. Medaglia Industrial Engineering Department Universidad de los Andes Bogotá, Colombia