**3. A functional model for public portfolio optimization using genetic algorithms**

Let us consider a set Pr of public projects whose consequences can be estimated by the DM. These projects have been considered acceptable after some prior evaluation. That is, the DM would support all of them, given that enough funds are available and that no mutually exclusive projects are members of the set. However, projects are not, in general, mutually independent. In fact, they can be redundant or synergetic. Furthermore, they may establish conflicting priorities, or compete for material or human resources, which are indivisible, unique, or scarce.

For the sake of generality, let us consider a planning horizon partitioned in T adjacent time intervals. When T=1, this problem is known as the stationary budgeting problem (one budgeting cycle) (Chan et al., 2005). In non-stationary cases, there could be different levels of available funds for each period.

In its more general form, a portfolio is a finite set of pairs of projects and periods {(pi, t(pi))}, where pi∈ Pr and t(pi) ∈T denotes the period when pi starts. A portfolio is feasible whenever it satisfies financial and scheduling restrictions, including precedence, and it does not contain redundant or mutually exclusive projects. These restrictions may also be influenced by equity, efficiency, geographical distribution, and the priorities imposed by the DM. In particular, if only one budgeting cycle is considered, the portfolios are subsets of Pr.

The set of projects is partitioned in different areas, according to their knowledge domain, their social role, or their geographic zone of action. One project can only be assigned to one area. Such partition is usually due to the DM's interest for obtaining a balanced portfolio. Given a set of areas *A* = {*A*1, *A*2, …, *A*n}, the DM can set the minimum and maximum amounts of funding that will be assigned to projects belonging to area *Ai* ∈ *A*.

The general problem is to determine which projects should be supported, in what period should the support start, and the amount of funds that each project should receive, provided that the overall social benefit from the portfolio is maximised.

In order to have a formal problem statement, we should answer the following questions.


Public Portfolio Selection Combining Genetic Algorithms and Mathematical Decision Analysis 147

social values but rather the ambition of a certain group. Thus, even under the premise of ethical behavior, the SDM ⎯who is supposed to distribute resources according to social preferences⎯ can only act in response to his/her own preferences. The reasons for this are that either the SDM hardly knows the actual social preferences, or he/she pursues his/her own satisfaction ⎯according to his/her preferences⎯ in an honest attempt to achieve what he/she thinks is socially better. Unethical behavior or lack of information can cause the SDM's preferences to significantly deviate from the predominant social interests. In turn, this situation might trigger events such as social protests claiming to reduce the distance between the SDM's preferences and social interests. Therefore, solving a public projectportfolio selection problem is about finding the best solution from the SDM's perspective. This solution (under the premise of ethical behavior) should be close to the portfolio with

In order to maximize the portfolio's subjective return (that is, the return from the SDM perspective), we must build a value function that satisfies relation ≿portfolios. For a starting analogy, let us accept that each project's return can be expressed by a monetary value, in a similar way as cost-benefit analysis. If no synergy and no redundancy exist (or they can be neglected) among the projects, the overall portfolio's return can be calculated as

In Equation 1, *N* is the cardinality of *Pr*. The value of *xi* is set to 1 whenever the *i*-th project is

Let *M*i denote the funding requirements for the *i*-th project. Let *d* be an N-dimensional vector of real values. Each value, *d*i, of vector *d* is associated to the funding given to the *i-*th project. If a project is not supported, then the corresponding value in *d* associated to such project will be set to zero. With this, we can now formally define the problem of portfolio

**Problem definition 1**. Portfolio selection optimization can be obtained after maximizing *Rt*, subject to *d* ∈ *RF*, where *RF* is a feasible region determined by the available budget, constraints for the kind of projects allowed in the portfolio, social roles, and geographic zones. Problem 1 is a variant of the knapsack problem, which can be efficiently solved using 0-1 programming. Unfortunately, this definition is an unrealistic model for most social portfolio

1. For Equation 1 to be valid, the monetary value associated to each project's social impact must be known. Monetary values can be added to produce a meaningful figure. However, due to the existence of indirect as well as intangible effects on such projects, it is unrealistic to assume that such monetary equivalence can be defined for all projects. If we cannot guarantee that every *c*i in Equation 1 is a monetary value, then the

2. Most of the times, the decision is not about accepting or rejecting a project but rather

supported, otherwise *xi* = 0. Finally, *c*i is the return value of the *i*-th project.

*R*t = *x*<sup>1</sup> *c*1 + *x*<sup>2</sup> *c*2 + …+ *x*<sup>N</sup> *c*N (1)

the highest social return.

follows.

selection.

**3.1 A Functional model of the subjective return** 

selection problems due to the following issues.

expression becomes meaningless.

about the feasibility of assigning sufficient funds to it.

To achieve the goal of maximizing social return we need to formally define a real-valued function, *Vsocial*, that does not contravene the relation ≿social. The construction of such function is, however, problematical due to the following reasons.


The preference-indifference social relation is required to be transitive and complete over social states (premise *i*). However, due to the known limitations for constructing collective rational-preferences (e.g., Condorcet's Paradox, Arrow's Impossibility Theorem, and context-dependent preferences), (Bouyssou et. al., 2000; Tversky and Simonson, 1993; French, 1993), and to the difficulty in obtaining valid information about social preferences from the decision maker, premises *i* and *ii* are rarely fulfilled in real-world cases (Sen, 2000, 2008).

The success of public policies is measured in terms of their contribution to social equity and social "efficiency". A project's social impact should be an integrated assessment of such criteria. In the research literature, it is possible to find several methods that have been proposed for estimating a project contribution to social well-fare. Unfortunately, they all show serious limitations for handling intangible attributes. Furthermore, these methods' objectivity for measuring the contribution of each project or public policy is questionable. In any society, a wide variety of interests and ideologies can coexist. This human condition makes it complicated to reach a consensus on what an effective measure of social benefit should be. In turn, the absence of consensus leads to a lack of objectivity on any defined measure. This lack of objectivity is closely related to a nonexistent function of social preference and to the ambiguity of collective preferences as reported by Condorcet, Arrow, and Sen (Bouyssou et al., 2000; Sen, 2000, 2008).

While the social impact is objective, its assessment is highly subjective as it depends on the ideology, preferences and values of the person measuring the impact. This subjectivity, however, does not necessarily constitute a drawback as it is not arbitrary. In the end, decision making does not lack of subjective elements. The set of criteria upon which the decision making is based should strive to be objective. However, the assessment of the combined effect of such criteria, some of them in conflict with each other, is subjective in nature as it depends on the perception of the decision maker. The objectivity of decision making theory is not based on eliminating all subjective elements. Instead, it is based on creating a model that reflects the system of values of the decision maker.

In every decision problem it is necessary to identify the main actor whose values, priorities, and preferences, are to be satisfied. In this context (the problem of efficiently and effectively allocating public resources), we will call "supra-decision-maker" (SDM) to this single or collective actor. For the rest of the discussion, we drop the idea of modeling public returns from a social perspective in favor of modeling the SDM's preferences.

Focusing exclusively on the SDM's preferences is a pragmatic representation of the problem that raises ethical concerns. This is particularly true when the SDM is elected democratically and, as such, his/her decisions formally represent the preferences of the society. In real life, an SDM may possibly have a very personal interpretation of social welfare and subjective parameters to evaluate project returns that do not necessarily represent the generalized social values but rather the ambition of a certain group. Thus, even under the premise of ethical behavior, the SDM ⎯who is supposed to distribute resources according to social preferences⎯ can only act in response to his/her own preferences. The reasons for this are that either the SDM hardly knows the actual social preferences, or he/she pursues his/her own satisfaction ⎯according to his/her preferences⎯ in an honest attempt to achieve what he/she thinks is socially better. Unethical behavior or lack of information can cause the SDM's preferences to significantly deviate from the predominant social interests. In turn, this situation might trigger events such as social protests claiming to reduce the distance between the SDM's preferences and social interests. Therefore, solving a public projectportfolio selection problem is about finding the best solution from the SDM's perspective. This solution (under the premise of ethical behavior) should be close to the portfolio with the highest social return.

#### **3.1 A Functional model of the subjective return**

146 Bio-Inspired Computational Algorithms and Their Applications

To achieve the goal of maximizing social return we need to formally define a real-valued function, *Vsocial*, that does not contravene the relation ≿social. The construction of such

The preference-indifference social relation is required to be transitive and complete over social states (premise *i*). However, due to the known limitations for constructing collective rational-preferences (e.g., Condorcet's Paradox, Arrow's Impossibility Theorem, and context-dependent preferences), (Bouyssou et. al., 2000; Tversky and Simonson, 1993; French, 1993), and to the difficulty in obtaining valid information about social preferences from the decision maker, premises *i* and *ii* are rarely fulfilled in real-world cases (Sen, 2000,

The success of public policies is measured in terms of their contribution to social equity and social "efficiency". A project's social impact should be an integrated assessment of such criteria. In the research literature, it is possible to find several methods that have been proposed for estimating a project contribution to social well-fare. Unfortunately, they all show serious limitations for handling intangible attributes. Furthermore, these methods' objectivity for measuring the contribution of each project or public policy is questionable. In any society, a wide variety of interests and ideologies can coexist. This human condition makes it complicated to reach a consensus on what an effective measure of social benefit should be. In turn, the absence of consensus leads to a lack of objectivity on any defined measure. This lack of objectivity is closely related to a nonexistent function of social preference and to the ambiguity of collective preferences as reported by Condorcet, Arrow,

While the social impact is objective, its assessment is highly subjective as it depends on the ideology, preferences and values of the person measuring the impact. This subjectivity, however, does not necessarily constitute a drawback as it is not arbitrary. In the end, decision making does not lack of subjective elements. The set of criteria upon which the decision making is based should strive to be objective. However, the assessment of the combined effect of such criteria, some of them in conflict with each other, is subjective in nature as it depends on the perception of the decision maker. The objectivity of decision making theory is not based on eliminating all subjective elements. Instead, it is based on

In every decision problem it is necessary to identify the main actor whose values, priorities, and preferences, are to be satisfied. In this context (the problem of efficiently and effectively allocating public resources), we will call "supra-decision-maker" (SDM) to this single or collective actor. For the rest of the discussion, we drop the idea of modeling public returns

Focusing exclusively on the SDM's preferences is a pragmatic representation of the problem that raises ethical concerns. This is particularly true when the SDM is elected democratically and, as such, his/her decisions formally represent the preferences of the society. In real life, an SDM may possibly have a very personal interpretation of social welfare and subjective parameters to evaluate project returns that do not necessarily represent the generalized

creating a model that reflects the system of values of the decision maker.

from a social perspective in favor of modeling the SDM's preferences.

function is, however, problematical due to the following reasons.

i. A set of well defined social preferences must exist.

ii. This set of preferences must be revealed.

and Sen (Bouyssou et al., 2000; Sen, 2000, 2008).

2008).

In order to maximize the portfolio's subjective return (that is, the return from the SDM perspective), we must build a value function that satisfies relation ≿portfolios. For a starting analogy, let us accept that each project's return can be expressed by a monetary value, in a similar way as cost-benefit analysis. If no synergy and no redundancy exist (or they can be neglected) among the projects, the overall portfolio's return can be calculated as follows.

$$R\_{\ell} = \chi\_1 \cdot c\_1 + \chi\_2 \cdot c\_2 + \dots + \chi\_N \cdot c\_N \tag{1}$$

In Equation 1, *N* is the cardinality of *Pr*. The value of *xi* is set to 1 whenever the *i*-th project is supported, otherwise *xi* = 0. Finally, *c*i is the return value of the *i*-th project.

Let *M*i denote the funding requirements for the *i*-th project. Let *d* be an N-dimensional vector of real values. Each value, *d*i, of vector *d* is associated to the funding given to the *i-*th project. If a project is not supported, then the corresponding value in *d* associated to such project will be set to zero. With this, we can now formally define the problem of portfolio selection.

**Problem definition 1**. Portfolio selection optimization can be obtained after maximizing *Rt*, subject to *d* ∈ *RF*, where *RF* is a feasible region determined by the available budget, constraints for the kind of projects allowed in the portfolio, social roles, and geographic zones.

Problem 1 is a variant of the knapsack problem, which can be efficiently solved using 0-1 programming. Unfortunately, this definition is an unrealistic model for most social portfolio selection problems due to the following issues.


Public Portfolio Selection Combining Genetic Algorithms and Mathematical Decision Analysis 149

In Equation 2, *c*i represents the subjective value of the *i*-th project. Equations 1 and 2 are formally equivalent. However, the resulting value of *V* only makes sense if there is a process

Before we proceed to the description of the rest of the assumptions, we need to introduce

**Definition 3**: An elementary portfolio is a portfolio that contains only projects of the same category. It will be expressed in the form of a C-dimensional vector, where C is the number of discrete categories. Each dimension is associated to one particular category. The value in each dimension corresponds to the number of projects in the associated category. Consequently, the C-dimensional vector of an elementary portfolio with n projects will have

**Assumption 5:** The SDM can define a complete relation ≿ on the set of elementary portfolios. That is, for any pair of elementary portfolios, P and Q, one and only one of the

**Assumption 6 (Essentiality):** Given two elementary portfolios, P and Q, defined over the same category. Let P = **(**0, 0, …, n, 0, …, 0) and Q = (0, 0,…, m, 0, …, 0). P is preferred to Q if

From the set of discrete categories, let C1 be the lowest category, CL be the highest, and Cj a

**Assumption 7 (Archimedean):** For any category Cj, there is always an integer value *n* such that the SDM would prefer a portfolio composed of *n* projects in the C1 category to any

**Assumption 8 (Continuity):** If an elementary portfolio P = (x, 0, …, 0,…, 0) is preferred to an elementary portfolio Q = (0,…, 1, 0,…, 0), defined over category j for 1 < j ≤ L, there is always a pair of integers values n and m (n > m) such that an elementary portfolio with n projects of the lowest category is indifferent to another elementary portfolio with m projects of the *j*-th

Assumption 5 characterizes the normative claim of the functional approach for decisionmaking. Assumption 6 is a consequence of Assumption 4 (additivity) combined with the premise that all projects satisfy minimal acceptability requirements. Assumption 7 is a consequence of both essentiality and the non-bounded character of the set of natural numbers. Assumption 8 simulates the way in which a person balances a scale using a set of

Let us say that c1 is a number representing the subjective value of the projects belonging to the lower category C1. Similarly, let us use cj to represent the value of projects in category Cj. Now, suppose that the elementary portfolios P (containing n projects in C1) and Q (integrated by m projects in Cj) are indifferent. That is, P and Q have the same V value. If we

combine Assumption 8 with Equation 2, we obtain the following expression.

to assign meaningful values to *c*i.

the concept of elementary portfolio.

the form (0, 0, …, n, 0, …, 0).

following propositions is true.

an only if n > m.

category.

category preferred to C1.

• Portfolio P is preferred to portfolio Q • Portfolio Q is preferred to portfolio P • Portfolios P and Q are indifferent.

portfolio composed of a single project in the Cj category.

two types of weights whose values are relative primes.


The functional normative approach presented in Section 2 is used to address the first issue on this list. Here, we present a new approach based on the work of Fernandez and Navarro (2002), Navarro (2005), Fernandez and Navarro (2005), and Fernandez et al. (2009). Addressing issues 2 to 5 on the list above requires using a heuristic search and optimization methods.

This new approach is constructed upon the following assumptions.

**Assumption 1**: Every project has an associated value subjectively assigned by the SDM. This value increases along with the project's impact.

**Assumption 2**: This subjective value reflects the priority that the SDM assigns to the project. Each project is assigned to a category from a set of classes sorted in increasing order of preference. These categories can be expressed qualitatively (e.g., {poor, fair, good, very good, excellent}) or numerically in a monotonically increasing scale of preferences.

**Assumption 3**: Projects assigned to the same category have about the same subjective value to the SDM. Therefore, the granularity of the discrete scale must be sufficiently fine so that no two projects are assigned to the same class if the SMD can establish a strict preference between them.

**Assumption 4 (Additivity)**: The sum of the subjective values of the projects belonging to a portfolio is an ordinal-valued function that satisfies relation ≿portfolios.

Fernandez et al. (2009) rationalize this last assumption by considering that each project is a lottery. A portfolio is, in consequence, a "giant" lottery being played by a risk-neutral SDM. Under this scenario, the subjective value of projects and portfolios corresponds to their certainty equivalent value.

Under Assumption 4, the interaction between projects cannot be modeled. Synergy and redundancy in the set of projects are characteristics that require special consideration that will be introduced later.

Under Assumptions 1 and 4, the SDM assess a subjective value to portfolio given by the following equation.

$$V = \mathbf{x}\_1 \mathbf{c}\_1 + \mathbf{x}\_2 \mathbf{c}\_2 + \dots + \mathbf{x}\_N \mathbf{c}\_N \tag{2}$$

In Equation 2, *c*i represents the subjective value of the *i*-th project. Equations 1 and 2 are formally equivalent. However, the resulting value of *V* only makes sense if there is a process to assign meaningful values to *c*i.

Before we proceed to the description of the rest of the assumptions, we need to introduce the concept of elementary portfolio.

**Definition 3**: An elementary portfolio is a portfolio that contains only projects of the same category. It will be expressed in the form of a C-dimensional vector, where C is the number of discrete categories. Each dimension is associated to one particular category. The value in each dimension corresponds to the number of projects in the associated category. Consequently, the C-dimensional vector of an elementary portfolio with n projects will have the form (0, 0, …, n, 0, …, 0).

**Assumption 5:** The SDM can define a complete relation ≿ on the set of elementary portfolios. That is, for any pair of elementary portfolios, P and Q, one and only one of the following propositions is true.


148 Bio-Inspired Computational Algorithms and Their Applications

3. The effects of synergy between projects can be significant on the portfolio social return. Therefore, they must be modeled. For instance consider the following two projects, one for building a hospital and the other for building a road that will enhance access to such hospital. Both of such projects have, individually, an undeniable positive impact.

5. It is possible that for a pair of projects (i and j) *c*i >> *c*j and *M*i >> *M*j, the solution to this problem indicates that project i should not be supported (*x*i = 0) whereas project j is supported (*x*j = 1). The SDM might not agree to this solution, as it fails to support a high-impact project while it provides funds to a much less important project.

The functional normative approach presented in Section 2 is used to address the first issue on this list. Here, we present a new approach based on the work of Fernandez and Navarro (2002), Navarro (2005), Fernandez and Navarro (2005), and Fernandez et al. (2009). Addressing issues 2 to 5 on the list above requires using a heuristic search and optimization

**Assumption 1**: Every project has an associated value subjectively assigned by the SDM. This

**Assumption 2**: This subjective value reflects the priority that the SDM assigns to the project. Each project is assigned to a category from a set of classes sorted in increasing order of preference. These categories can be expressed qualitatively (e.g., {poor, fair, good, very good, excellent}) or numerically in a monotonically increasing scale of

**Assumption 3**: Projects assigned to the same category have about the same subjective value to the SDM. Therefore, the granularity of the discrete scale must be sufficiently fine so that no two projects are assigned to the same class if the SMD can establish a strict preference

**Assumption 4 (Additivity)**: The sum of the subjective values of the projects belonging to a

Fernandez et al. (2009) rationalize this last assumption by considering that each project is a lottery. A portfolio is, in consequence, a "giant" lottery being played by a risk-neutral SDM. Under this scenario, the subjective value of projects and portfolios corresponds to their

Under Assumption 4, the interaction between projects cannot be modeled. Synergy and redundancy in the set of projects are characteristics that require special consideration that

Under Assumptions 1 and 4, the SDM assess a subjective value to portfolio given by the

*V*= *x*<sup>1</sup> *c*1 + *x*<sup>2</sup> *c*2 + …+ *x*<sup>N</sup> *c*N (2)

4. Time dependences between projects are not considered by Problem definition 1.

Furthermore, such situation will be difficult to explain to the public opinion.

However their combined social impact is superior.

This new approach is constructed upon the following assumptions.

portfolio is an ordinal-valued function that satisfies relation ≿portfolios.

value increases along with the project's impact.

methods.

preferences.

between them.

certainty equivalent value.

will be introduced later.

following equation.

**Assumption 6 (Essentiality):** Given two elementary portfolios, P and Q, defined over the same category. Let P = **(**0, 0, …, n, 0, …, 0) and Q = (0, 0,…, m, 0, …, 0). P is preferred to Q if an only if n > m.

From the set of discrete categories, let C1 be the lowest category, CL be the highest, and Cj a category preferred to C1.

**Assumption 7 (Archimedean):** For any category Cj, there is always an integer value *n* such that the SDM would prefer a portfolio composed of *n* projects in the C1 category to any portfolio composed of a single project in the Cj category.

**Assumption 8 (Continuity):** If an elementary portfolio P = (x, 0, …, 0,…, 0) is preferred to an elementary portfolio Q = (0,…, 1, 0,…, 0), defined over category j for 1 < j ≤ L, there is always a pair of integers values n and m (n > m) such that an elementary portfolio with n projects of the lowest category is indifferent to another elementary portfolio with m projects of the *j*-th category.

Assumption 5 characterizes the normative claim of the functional approach for decisionmaking. Assumption 6 is a consequence of Assumption 4 (additivity) combined with the premise that all projects satisfy minimal acceptability requirements. Assumption 7 is a consequence of both essentiality and the non-bounded character of the set of natural numbers. Assumption 8 simulates the way in which a person balances a scale using a set of two types of weights whose values are relative primes.

Let us say that c1 is a number representing the subjective value of the projects belonging to the lower category C1. Similarly, let us use cj to represent the value of projects in category Cj. Now, suppose that the elementary portfolios P (containing n projects in C1) and Q (integrated by m projects in Cj) are indifferent. That is, P and Q have the same V value. If we combine Assumption 8 with Equation 2, we obtain the following expression.

Public Portfolio Selection Combining Genetic Algorithms and Mathematical Decision Analysis 151

function of the set of supported projects. When a non-fuzzy model includes the binary indicator function of a crisp set, the fuzzy generalization provided by classical "fuzzy technology" is made substituting this function with a membership function expressing "the degree of membership" to the more general fuzzy set. In this way, Equation 5 becomes

Equation 6 was proposed by Fernandez and Navarro (2002) as a measure of a portfolio's

Redundancy between projects can be addressed using constraints. For every pair of

Let S = {S1, S2, …, Sk} be the set of coalitions of synergetic projects. In a model like the one represented by Equation 5, each of these coalitions should be treated as an (additional) individual project. As a result, each coalition has an associated cost (i.e., the sum of the costs of the individual projects in the coalition), and an evaluation. This evaluation should be better than the evaluation of any of the projects in the coalition. Let us assume that coalitions Si and Sj become projects PN+i and PN+j, respectively. If Si is a subset of Sj, then it does not make sense to include them both in a portfolio. Therefore, PN+i and PN+j must be considered redundant projects. Furthermore, if project pn is a member of Si, then the pair (pn, pN+i) is

Suppose that a feasible region of portfolios, RF, is defined by constraints on the total budget and on the distribution of projects by area. In addition, the SDM could include further

• The particular budget distribution of the portfolio could be very difficult to justify. Let us suppose that the SDM asserts that "project pj is much better than project pi". In consequence, any portfolio in which μi is greater than μj could be unacceptable. This implies the existence of some veto situations that can be modeled with the following constraint. For every project pi and pj, being si, and sj their corresponding evaluations, if (si – sj ) ≥ vs, then (μi (di) - μj (dj)) must be greater than (or equal to) 0, where vs is a veto

Let us use R'F, R'F ⊂ RF, to denote the set of values for the decision variables that make every portfolio acceptable. All the veto constraints are satisfied in R'F and there are no redundant projects in the portfolios belonging to this region. The optimization problem can now be

**Problem definition 2**. An optimal portfolio can be selected by maximizing U = Σik f(μik(dik)) wik, subject to *d* ∈ R'F , where dik indicates the financial support assigned to the k-th project

redundant projects, (pi, pj), i < j, condition μi(di) × μj(dj) = 0 should be enforced.

also redundant (since the value of pn is included in the value of pN+i).

constraints on the portfolios due to following reasons.

• A possible redundancy exists between projects.

defined as follows.

belonging to the i-th category.

**3.4 A Genetic algorithm for optimizing public portfolio subjective value** 

threshold. In the following they will be called veto constraints.

U = Σik wik μik (6)

Equation 6 shown below.

**3.3 Synergy and redundancy** 

subjective value.

n c1= m cj ⇔ cj= (n/m)c1

If V is a value function, then every proportional function is also a value function satisfying the same preferences. Therefore, we can arbitrarily set c1=1 to obtain Equation 3 below.

$$\mathbf{c} \equiv \mathbf{n}/\mathbf{m} \tag{3}$$

In consequence, Equation 2 can now be re-stated as follows.

$$\mathbf{U} = \Sigma\_{i,k} \mathbf{w}\_{ik} \mathbf{x}\_{ik} \tag{4}$$

In Equation 4, the variable *j* is used to index categories, whereas variable *k* indexes projects. The value of w1k is set to 1, and wjk= n/mj, where mj denotes the cardinality of an elementary portfolio defined over category Cj. Additionally, factors wik might be interpreted as importance factors. These weights express the importance given by the SDM to projects within certain category. Therefore, they should be calculated from the SDM's preferences, expressed while solving the indifference equations between elementary portfolios, as stated by Assumption 8 and according to Equation 3. A weight must be calculated for every category. If the cardinality of the set of categories is too large, the resolution of such categories can be reduced to simplify the model. A temporary set of weights is obtained using these coarse categories. By interpolation on such set, the values of the original (finer resolution) set can be obtained.

#### **3.2 Fuzziness of requirements**

Another important issue is the imprecise estimation of the monetary resources required by each project. If *d*k are the funds assigned to the k-th project, then there is an interval [mk, Mk] for *d*k where the SDM is uncertain about whether or not the project is being adequately supported. Therefore, the proposition "the k-th project is adequately supported" may be seen as a fuzzy statement. If we consider that the set of projects with adequate funds is fuzzy, then the SDM can define a membership function μk(dk) representing the degree of truth of the previous proposition. This is a monotonically increasing function on the interval [mk ,Mk], such that μk(Mk) = 1, μk(mk) > 0, and μk(dk) = 0 when dk < mk.

The subjective value assigned by the SDM to the k-th project is based on the belief that the project receives the necessary funding for its operation. When dk<mk the SDM is certain that the project is not sufficiently funded. When mk ≤ dk < Mk, the SDM hesitates about the truth of that statement. This uncertainty affects the subjective value of the project, because it reduces the feasible impact of the project, which had been subjectively estimated under the premise that funding was sufficient. The reduction of the project's subjective value can be modeled by the product of the original value and a feasibility factor f. This factor is a monotonically increasing function with μk as an argument such that f(0) = 0 and f(1) = 1. Equation 5 below, is generated by introducing this factor into Equation 4, and assuming that f(μik) >0 ⇔ xik=1.

$$\mathbf{U} = \Sigma\_{\rm ik} \,\mathrm{f}(\mathfrak{mu}\_{\rm ik}) \text{ } \mathbf{w}\_{\rm ik} \tag{5}$$

The simplest definition of the feasibility factor is to make f(μik) = μik. This is equivalent to a fuzzy generalization of Equation 4. In such case, xik can be considered as the indicator function of the set of supported projects. When a non-fuzzy model includes the binary indicator function of a crisp set, the fuzzy generalization provided by classical "fuzzy technology" is made substituting this function with a membership function expressing "the degree of membership" to the more general fuzzy set. In this way, Equation 5 becomes Equation 6 shown below.

$$\mathbf{U} = \Sigma\_{\text{ik}} \mathbf{w}\_{\text{ik}} \boldsymbol{\mu}\_{\text{ik}} \tag{6}$$

Equation 6 was proposed by Fernandez and Navarro (2002) as a measure of a portfolio's subjective value.
