Research Article A Bicriteria Approach Identifying Nondominated Portfolios Javier Pereira, 1 Broderick Crawford, 2,3 Fernando Paredes, 1 and Ricardo Soto 2,4 1 Escuela de Ingenier´ ıa Industrial, Universidad Diego Portales, 8370179 Santiago, Chile 2 Pontifcia Universidad Cat´ olica de Valpara´ ıso, 2362807 Valpara´ ıso, Chile 3 Universidad Finis Terrae, 7500000 Santiago, Chile 4 Universidad Aut´ onoma de Chile, 7500000 Santiago, Chile Correspondence should be addressed to Broderick Crawford; broderick.crawford@ucv.cl Received 27 February 2014; Accepted 18 June 2014; Published 3 July 2014 Academic Editor: Mohammad Khodabakhshi Copyright © 2014 Javier Pereira et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We explore a portfolio constructive model, formulated in terms of satisfaction of a given set of technical requirements, with the minimum number of projects and minimum redundancy. An algorithm issued from robust portfolio modeling is adapted to a vector model, modifying the dominance condition as convenient, in order to fnd the set of nondominated portfolios, as solutions of a bicriteria integer linear programming problem. In order to improve the former algorithm, a process fnding an optimal solution of a monocriteria version of this problem is proposed, which is further used as a frst feasible solution aiding to fnd nondominated solutions more rapidly. Next, a sorting process is applied on the input data or information matrix, which is intended to prune nonfeasible solutions early in the constructive algorithm. Numerical examples show that the optimization and sorting processes both improve computational efciency of the original algorithm. Teir limits are also shown on certain complex instances. 1. Introduction Markowitz provided one of the frst comprehensive theoreti- cal frameworks for the portfolio selection problem [1]. In his proposal, each portfolio is evaluated in terms of the expected return and risk. Ten, the efcient set, or efcient frontier, corresponds to all portfolios with the largest expected return, given a level of risk. From this framework, the expected return is usually evaluated as the weighted sum of the expected return from each project in the portfolio, while the risk value is evaluated by the variance of the portfolio. Tus, the investor may select the portfolios in the efcient frontier that best match her/his needs. In portfolio selection two vectors are defned [2]. First, the investment proportion vector corresponds to the proportion of money that the investor accepts to invest on each member of a set of securities (projects). Te criteria vector, instead, contains the values of measures evaluating the portfolio. In this sense, an efcient portfolio, in terms of the frst vector, is a nondominated portfolio, in the sense of the second one. A multicriteria portfolio selection problem supposes a criteria vector with three or more criteria [3], which is expected to be more difcult in terms of computing of nondominated portfolios. However, below, we show that depending on what it is to be taken into account as an evaluation measure, even with two criteria the generation of portfolios is a hard combinatorial problem. Since the portfolio selection problem intrinsically incor- porates business criteria, budget restrictions, and returns volatility [4], in the literature the problem is formulated as the maximization of the expected return, under uncertainty of returns. When the multicriteria version is considered, several utility functions need to be maximized, subject to constrains defning the feasible portfolios [1, 4, 5]. In this context, nondominated portfolios may be computed using multiobjective algorithms [6], evolutionary methods for mul- tiobjective models [7, 8], or preference programming [9]. In this article, we focus on the portfolio generation process, when business, budget, or even volatility information is poor. Several stringent situations obligate to split the project portfolio selection process into at least two phases: technical and business concerns. We place ourselves in the frst phase and formulate our problem in terms of satisfaction of a given set of technical requirements, with the minimum number Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 957108, 8 pages http://dx.doi.org/10.1155/2014/957108