ИЗВЕСТИЯ РАИ. ТЕОРИЯ И СИСТЕМЫ УПРАВЛЕНИЯ, 2008, № 1, с. 29-36

СИСТЕМНЫЙ АНАЛИЗ И ИССЛЕДОВАНИЕ ОПЕРАЦИЙ

УДК 62.40

AN INTERACTIVE ALGORITHM FOR PORTFOLIO BI-CRITERIA OPTIMIZATION OF R&D PROJECTS IN PUBLIC ORGANIZATIONS*

© 2008 г. I. Litvinchev, F. López

Computing Center Russian Academy of Sciences, Moscow 119333, Vavilov, 40 Nuevo Leon State University, Department of Mechanical and Electrical Engineering, Pedro de Alba

s/n 66450, SN. Garza, NL, Mexico Received July 20, 2007

Abstract - In this paper we propose an interactive algorithm for the selection of portfolios of research and development (R&D) projects in public organizations based on a bi-criteria optimization model, the need for such a model arises when the decision maker (DM) doesn't trust enough on the portfolio quality measure. This algorithm efficiently exploit the structure and nature of the problem to support the DM. An interesting proposal is also the representation of a portfolio as a set of "rules of support/rejection"; in this way the DM can not only valuate the portfolio by its numerical measures but compare against his/her beliefs in a way that is more natural for him, which also allows for supporting with more arguments the solution obtained so far. Rough set methodology is employed for rule discovering.

Keywords: Decision Support Systems, Portfolio Optimization, R&D projects

INTRODUCTION

Projects selection is a problem present in most organizations that manage or invest funds in R&D, because there are more projects to support than funds to distribute among them. Projects selection is also an important problem top and middle managers must face in large public organizations (like government institution, universities, foundations, etc.) that are involved in funding or supporting R&D in some way. There are two subproblems strongly related with project selection: a) evaluation of individual projects, and b) building a portfolio of projects that maximized impact .

First, it is necessary to find projects with a positive potential (evaluation problem a)) and then decide what subset of these will be finally funded and the amount of funds assigned to each projects in portfolio. Funds are distributed based on organization's policies and in such a way that impact is maximized (portfolio problem b)) [1].

Most important differences between the portfolio problem of R&D projects in public organization and the classical portfolio problem in private organizations are:

• In public organizations subjective considerations are of great importance and must be taken into account; otherwise the solution could be of poor quality or not taken into consideration by the decision maker (DM). For example, sometimes there are policies that require the inclusion in portfolio of certain kind of projects (belonging to different areas). Funding such projects could reduce total impact of portfolio,

* The work was partially supported by RFBR (06-01-81020-Bel_a), CONAYT(61343, 61903) and PAICYT (CA1526-07).

but his inclusion is justified by diversity or other considerations [1].

• Public organizations are also interested in rising R&D. Then, a non explicit objective will be to support most projects as possible, but under the model's constraints.

Based on the above exposed distinctive characteristics about portfolio's optimization of R&D projects, a fairly builded portfolio with a high impact should hold an acceptable tradeoff between its quality and the quantity of funded projects. Therefore, it is convenient to have at hand a method for portfolio optimization that takes into account the principle cited above. The rest of the paper is devoted to discuss an interactive algorithm for portfolio optimization of R&D projects in public organizations.

After discussing some background in the next section, the mathematical model is described in Section 2. In Section 3 an interactive algorithm is introduced for solving the decision problem related to the proposed multobjective model and some procedures are presented in order to determine upper and lower bounds for the objectives. In Section 4 we present an original approach to represent portfolios as a set of decision rules over two categories (acceptance and rejection). In Section 5 all the procedures described are illustrated using a numerical example. Finally, some conclusions are presented.

1. BACKGROUND

Traditionally solution methods for portfolio optimization of R&D projects in public organizations are derived from a cost-benefit approach or are based on heuristics

like the one employed by the National Science Foundation. Critics to these approaches can be found in [2-5].

Taking into consideration those criticisms, an integral methodology was developed in [6] based on multi-criteria decision analysis and utility theory. This methodology takes into account the uncertainty of the individual projects success and propose a quality measure of portfolio which permits to compare different portfolios, not only individual projects as in most heuristic approaches to fund R&D projects It was proposed in [6] to build the portfolio of R&D projects in public organizations as a solution of an optimization problem where the objective function is the portfolio quality measure.

The most important criticism to this proposal lies in the fact that even if the portfolio quality measure is theoretically correct, it can be of less or none application value in practice. This happens because the projects'weights (which are given by the decision maker), represent the certainty equivalent of some lotteries. Then, it is questionable that the DM can give accurate information about the certainty equivalent only based on projects evaluation.

This criticism could be solved by adding as an objective the maximization of the quantity of funded projects. Then the tradeoff between both objectives (portfolio's quality and quantity of funded projects) depends on how much the DM trust the quality measure (how confident he/she is about those certainty equivalents). If the DM trusts in the quality measure, this should be reflected in his/her decisions. If not, the DM must base his decisions on the quantity of supported projects. In any other case, the DM's preferences are represented by an intermediate tradeoff between this two extreme positions, which implies just to model and solve a multiobjective problem which considers maximization of portfolio's quality and quantity of funded projects. It is worth noting that both objectives are in conflict to each other.

A multiobjective interactive algorithm for the capital investment problem was developed in [7]. This algorithm takes into account nadir and ideal points for each function and only consider inclusion or exclusion of projects in portfolio.

In [8, 9] it was presented an ant colony based heuristic to search for efficient portfolios based on a multiob-jective model, but these results are limited to at most 30 projects. In [10] they compare some metaheuristics and present an improvement of the P-ACO procedure proposed in [8, 9], but they also were focused on design optimization procedures for searching efficient solutions. They test instances with 36 projects and 6 objective at most and recognized that developing an interactive procedure is challenging. The conclusion was made that when the number of non dominated solutions is too high, as is the case in most real problems, then it is perhaps best to embed the generation of non dominated solutions in an interactive procedure where the preferences of the DM are used to guide the optimization process and restrict the search area to promises ones instead of search on the whole space of efficient non dominated solutions.

In this paper we present a multiobjective model for the portfolio optimization of R&D projects in public organizations, considering two objectives: portfolio's quality and the number of funded projects. We address the solution process as a decision support problem, developing an interactive algorithm to solve it. This follows in a way different from the proposed in [11], exploiting the problem's structure to derive bounds o reference points to help the DM. Moreover, in the quest for a portfolio with an acceptable tradeoff between its quality and funded projects quantity, the DM must compare portfolios, not projects. This is a difficult task because there is much information related to a portfolio. Then, a challenge to any interactive comparison procedure is how to structure portfolio information in such a way that doesn't overkill the DM capacity of concentrate simultaneously on different aspects to achieve his/her goal. The interactive algorithm proposed in this paper addresses this issue and only a few portion of portfolio's information is presented to the DM in a drill down fashion interface, where the DM chooses the detail level of portfolio information to be rendered at any time. The upper level (less detailed) shows the two numerical measures: quality and quantity of funded projects, the lowest level (most detailed) shows funds assigned to each project.

Finally, we present a portfolio as a set of "rules of support/rejection". In this way the DM can not only valuate the portfolio by its numerical measures but compare against his/her beliefs in a more natural way and has arguments to support or explain his decisions.

2. MATHEMATICAL MODEL

The main parameters are: N (N > 1) - total quantity of projects; K (K > 1) - total quantity of areas in which the projects can be classified; Mj - quantity of money that fulfill completely the requirements of project j; mj -quantity of money that in case of not been fulfill there will be serious reasons to think that project j is not adequately funded; PG - total amount of funds available to

distribute among projects; P+ - maximum amount of funding to distribute among projects belonging to area i; P- - minimum amount of funding or distribute among projects belonging to area i. Decision vari

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