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This paper proposes the application of multi-criteria analysis to the problem of allocating public funds to competing programs, projects, or policies, with a subjective approach applied to define the concept of highest portfolio social return. This portfolio corresponds to an attainable non-strictly outranked state of the social object under consideration. Its existence requires a decision-maker (DM) to establish a relational preference system of minimal consistency. As the number of feasible portfolios increases exponentially, the DM’s asymmetric preference relation should be computable to perform an exploration of the portfolio space. The complexity of many real situations requires evolutionary algorithms, but, in presence of many objectives, evolutionary algorithms are inefficient. We overcome this problem by using the extended non-outranked sorting genetic algorithm (NOSGA-II), which handles multi-criteria preferences through a robust model based on a binary fuzzy outranking relation expressing the truth value of the predicate “portfolio x is at least as good as portfolio y”. The DM is assumed to be capable of assigning the parameters for constructing the outranking relation. In case of collective decision-making, we first assume that the possible differing values of the group members are not strongly conflicting, so that a consensus can be achieved on the model’s parameters. Otherwise, we propose a method in which each member of the heterogeneous group gets his/her own best portfolio. These individual solutions are then aggregated in a group’s best acceptable portfolio, which maximizes a measure of group satisfaction and minimizes regret. The proposal is examined through two real size problems, in which good solutions are reached; the first example is useful to illustrate the case of social-action program selection; the second illustrates the case of basic research project portfolios.

A central and frequently contentious issue in public policy analysis is the allocation of funds to competing projects. Public financing for social projects is particularly scarce. Very often, the requested budget ostensibly overwhelms what can be granted; moreover, strategic, political and ideological criteria pervade the administrative decisions on such assignments (cf. [20] and [35]). For accomplishing normative criteria, such as those underlying prevalent public policies or governmental ideology, it is convenient to prioritize projects and construct project-portfolios according to rational principles (e.g., maximizing social benefits). According to [15], public projects may be characterized as follows: • The projects may be undoubtedly profitable, but their effects are indirect, perhaps only visible in the long term, and hard to quantify. • Aside from their potential economic contributions to social welfare, there are intangible benefits that should be considered to achieve an integral social view. • Equity, concerning impacts and the social conditions of the benefited individuals, should also be considered. Governmental decision-makers (DMs) and top managers of funding organizations must solve two related problems: (i) evaluation of particular projects; and (ii) creation of a project-portfolio. Although the strategic objective requires solving (ii), some kind of (i) is required as a premise. The most popular approach is cost-benefit analysis (e.g., [3]) though it has received important criticisms (cf. [4], [10], [14], [20] and [24]) such as: • the presence of numerous public goods for which no market price is available; • the difficulty of evaluating “benefits” and “costs” from a “social point of view”; • the difficulty of measuring some effects in well-defined units and, thus, to price them out; • the rationality of “social preferences” is still open to question, but if social preferences are ill-defined, the meaning of the net social present value of a project is far from obvious; • the controversial character of the social discounting rate, and moreover, the difficulty of assigning a suitable value to it; • the controversial assumption of a perfect capital market; • the overwhelming presence of uncertainty (technological changes, future prices, etc.); • the difficulty of considering irreversible effects; • the presence of effects that are highly complex, which may demand very long time periods; • the presence of effects which are very unevenly distributed among individuals and which can raise important equity concerns. A contending approach is multi-criteria analysis, which comprises a variety of techniques for exploring the preferences of concerned DMs as well as models for analyzing the complexity inherent to real decisions. Some of the broadly known multi-criteria approaches are MAUT (cf. [28]), AHP (cf. [39] and [40]), and outranking methods [5], [22] and [36]. Multi-criteria analysis constitutes a good option to overcome the limitations of the cost-benefit analysis because it may handle intangibles, ambiguous preferences, and veto conditions. Different multi-criteria methods have been proposed for addressing project evaluation and portfolio selection (e.g., [2], [12], [15], [16], [25], [31], [33] and [34]. Their advantages are documented in the specialized literature (e.g., [27], [30] and [46]). Multi-criteria analysis offers techniques for: – selecting the best project or a small set of equivalent “best” projects (called Pα problem) according to the known classification by Roy [37]; – sorting projects into several predefined categories (e.g., “good”, “bad”, “acceptable”) (Pβ problem); and – ranking projects from the best to the worst (Pγ problem). These methods are appropriate for evaluating and ranking projects, since multi-criteria information is aggregated at the project level. However, as far as portfolio selection is concerned, the usefulness of multi-criteria methods is questionable. They work on the set of projects, but portfolio selection is a problem of choosing among a set of feasible portfolios, hence multi-criteria information should be aggregated at the portfolio level. The point is that the decision should be made based on the best portfolio rather than on the best individual projects. It is simply not sufficient to compare one project with another. Instead, it is necessary to compare portfolios. The best projects do not necessarily compose the best portfolio. There are three basic ways of making comparisons at the portfolio level: First: Each portfolio value is calculated by aggregating the values of its component projects (cf. [14], [16] and [31]). The main difficulty is that projects’ values should be assessed in a ratio scale (cf. [16] and [31]), which may require an enormous effort from the DM, mainly when synergetic combinations of projects are considered (e.g., [6]). Additionally, such assessments become questionable, even meaningless, when the DM is not a single person but a heterogeneous group. Second: Each portfolio (let us denote it by C′) can be seen as a way to transform the current state of the social object under consideration (denoted by E0) into a different state E′. From a normative point of view, there should be a value function U agreeing with the DM’s preferences on the set of feasible states of the social object. If a cardinal value function U is assessed, the value of C′ can be measured by U(E′) − U(E0). Unfortunately, the practical value of this statement is strongly limited for several reasons (cf. [20]). The existence of such cardinal value functions is not guaranteed for real DMs (see [37] for a discussion of the practical limitations of decision actors). Moreover, even if the DM approached an ideal normative behavior, it would be extremely difficult, if not impossible, to specify the DM’s value function. The third basic approach is multi-objective portfolio optimization, in which each project produces certain effects on the social object. This effect, change, or benefit, can be described by a multi-dimensional vector. At a portfolio level, the whole change produced by a portfolio is a result of aggregating individual projects’ effects. If the aggregation function is computable, the finding of the best social portfolio can be modeled by a multi-objective optimization problem. Several applications, focusing on the allocation of public resources, have been proposed in this area (see [44] and [48] for documented surveys on this topic). In the portfolio space, multi-objective optimization algorithms search for a Pareto-acceptable solution of a vector optimization problem. The objectives represent either the benefits or the social costs of each supported project, in a way that can be quantified in some scale and aggregated considering the whole portfolio. Often, this may be formally described as a multi-objective knapsack problem. The research literature identifies 0–1 goal programming as a suitable approach for solving this kind of problem (cf. [1], [7], [13], [29], [41], [44] and [48]). In this view, the Pareto solution is obtained by assigning a goal to every objective (a priori modeling of the DM’s preferences). Nevertheless, the approach performs satisfactorily to an extent. Most of the proposals are aimed for specific applications, but it is difficult to apply them to highly complex scenarios. The human mind is limited to handling 5–8 distinct pieces of information simultaneously (cf. [32]), being thus unable to identify the best compromise solution when the DM should compare even a small subset of compromise solutions in problems with more than a few objectives. The conflicting point is that, for assessing portfolio social-consequences many attributes may be required. Hence, the cognitive limitations of the DM when dealing with more than four conflicting attributes become an important concern (cf. [32]). A prior model of the DM’s preferences can hardly offer a good solution. Moreover, the incorporation of these preferences while searching for a solution, or afterwards, becomes a difficult task when the size of the problem increases. From the discussion so far, it is evident that a more general formulation of the project-portfolio selection problem, able to contemplate diverse scenarios for public-funds assignments, is needed. For this purpose, it is necessary to consider: (a) a set of proper attributes describing social concerns; (b) interactions and synergies between projects; and (c) distribution of projects within the planning horizon. Items (b) and (c) have been scarcely investigated (cf. [6] and [7]); their modeling is intricate, involving problems that have only been solved by powerful heuristics and extensive computational effort [6]. Aside from the mathematical complexities conveyed by (b) and (c), the conflicting point is that the assessment of social portfolio selection problems may require many attributes; and multi-objective meta-heuristics do not work properly in problems with many dimensions. Hence, the way of finding the best social portfolio via multi-objective optimization has been, up to now, strongly limited. However, some recent advances in evolutionary multiobjective optimization should be tested in the framework of public portfolio selection. The main objectives of this paper are the following: • to make an operational characterization of the best social portfolio; • to prove the equivalence between the best portfolio and the best compromise solution of an associated multi-objective optimization problem; • to apply the recently proposed NOSGA II method (cf. [19]) to social portfolio selection problems. The rest of this paper is structured as follows: a characterization of the best social portfolio in terms of a DM’s asymmetric preference relation is presented in Section 2. Under very general premises a computer characterization of the best portfolio is given in Section 3 by using the relational system of preferences proposed in [19]. On this background, the search for the best portfolio is reduced to solving a three-objective optimization problem in Section 4. Section 5 illustrates the application of the proposal to social action program’s resources allocation, and basic research portfolio selection. The NOSGA-II algorithm is used for solving the associated three-objective optimization problem. In Section 6 the proposal is extended to situations in which the entity in charge of making decisions is a heterogeneous group involving such conflicting values that it cannot even agree on preferences with minimal consistency. A didactic example is also provided in this section. Finally, concluding remarks are presented in Section 7.

The main objectives of this paper have been accomplished. An operational characterization of the best social portfolio has been achieved, and its equivalence with the best compromise solution of an associated multi-objective optimization problem has been proved. The extension of the NOSGA-II method to solving social portfolio selection problem has provided good results. There are many previous papers suggesting 0–1 multi-objective programming to solve public portfolio selection problems, but most of them are specific applications which cannot be extended to highly complex scenarios. Some merits of this paper can be in: (i) establishing the relevance of the subjective approach; (ii) providing a general multi-objective formulation and justifying its validity under very general premises. Through the subjective approach to preference modeling in multi-criteria decision analysis, the concept of best public-project portfolio has been unambiguously defined. To the best of our knowledge, this is another merit of this research. Thus, the optimization of a public-project portfolio can be achieved without an aggregation of return measurements in a single value; for this purpose, the following is required: – a description of every project’s return in terms of a consistent family of criteria; – a method to integrate the contributions of the projects composing the portfolio; – a relational system of the DM’s preferences satisfying minimal consistence requirements; and – a fuzzy outranking relation defined on the criterion space. The number of criteria introduced to describe the social object is not a problem whenever a good measure of the credibility on the statement “portfolio Cx is at least as preferred as Cy”, is available. The ELECTRE methodology renders suitable models for that credibility measure. The capacity to handle non-cardinal and qualitative information, imprecision in objective values and threshold effects is another advantage of our proposal. In some examples solved by the NOSGA-II algorithm the results are quite satisfactory. They suggest that our proposal can obtain good results in problems with hundreds of projects (or social programs) described by many criteria. In fact, the approach of building a value-function based portfolio is clearly outranked by our proposal in numerous random instances with 150 and 500 projects. However, the closeness of the so obtained solution to the “actual” best portfolio depends on the precision of the preference model as a characterization of the true SDM’s preference relational-system. Cases with several budgeting cycles are not exponentially more complicated, being hence approachable by this method. Cases with interacting projects may be solved by NOSGA-II, but with an exponential increment of computer effort. Some important assumptions of our method are not fulfilled when a very heterogeneous group is in charge of solving the portfolio problem. The parameters of a group fuzzy outranking relations could not be comprehensively settled. Perhaps other merit of this paper is to propose a method for searching the best group portfolio in the feasible portfolio space. This method is based on: (a) construction of a measure of group satisfaction; (b) construction of a measure of group non-satisfaction; (c) each member of the group gets his/her own best portfolio; (d) these individual solutions are then aggregated in a group’s best acceptable portfolio, which maximizes the measure of group satisfaction and minimizes regret. This approach could be useful for increasing social participation in decision-making on allocation of public resources, a tool of participatory democracy.