**1. Introduction**

138 Bio-Inspired Computational Algorithms and Their Applications

As a future research direction, the same analyses can be carried out for different problem domains, and with different structural parameter settings, and even the interaction between

[1] [Alander, 1992] Alander. J*"On optimal populations size of genetic algorithms"* Proc

[2] [Bäck, 1996] Bäck T, *"Evolutionary Algorithms in theory and Practice"*, Oxford University

[3] [Berry Michael, 2004] M. Berry , Survey of Text Mining – Clustering and Retrieval,

[4] [Berry Michael, et al,2008] M. Berry, Malu Castellano Editors:"*Survey of Text Mining II*",

[5] [Castillo,Fernandéz,León,2008] "*Information Retrieval with Cluter Genetic*" IADIS Data

[6] [Castillo,Fernandéz,León,2009] "*Feature Reduction for Clustering with NZIPF*" IADIS e-

[7] [Goldberd D,1989] *Genetic algorithms in search, optimization and machine learning*. Addison

[8] [Holland J.H, 1975] *Adaptation in Natural and Artificial Systems* University of Michigan

[9] [Michalewicz, 1999] Michalewicz Z.*"Genetic Algorithms + Data Structures = Evolution"*.

[10] [Olson David, 2008] Olson D. *"Advanced Data Mining Techniques"*, Springer 2008

[11] [Pao M.L, 1976] Pao M.*"Automatic indexing based on Goffman transition of word* 

[12] [Reeves CR, 1993] *Modern Heuristic Techniques for Combitational* Problems, Wiley, New

[13] [Schaffer et al,1989] Shaffer, et al, "*A study of control parameters performance on GA for* 

annual meeting 1977, volume 14:40th annual meeting, Chicago.

*occurrences"*. In American society for Information Science. Meeting (40th: 1977:ChicagoII). Information Management in the 1980's: proceedings of the ASIS

the numerical and structural parameters could be investigated.

**6. References** 

CompEuro 1992.

Press, 1996.

Springer 2004.

Springer,2008.

Mining, 2008.

Society 2009.

Springer-1999.

York, 1993.

Wesley M.A. 1989.

Press, Ann Arbor 1975.

ISBN:978-3-540-76916-3

*function optimization", 1989.* 

A central and frequently contentious issue in public policy analysis is the allocation of funds to competing projects. Public resources for financing social projects are particularly scarce. Very often, the cumulative budget being requested ostensibly overwhelms what can be granted. Moreover, strategic, political and ideological criteria pervade the administrative decisions on such assignments (Peterson, 2005). To satisfy these normative criteria, that underlie either prevalent public policies or governmental ideology, it is obviously convenient both to prioritize projects and to construct project-portfolios according to rational principles (e.g., maximizing social benefits). Fernandez et al. (2009a) assert that public projects may be characterized as follows.


Admittedly, the main difficulty for characterizing the "best public project portfolio" is finding a mechanism to appropriately define, evaluate, and compare social returns. Regardless of the varying definitions of the concept of social return, we can assert the tautological value of the following proposition.

**Proposition 1**: *Given two social projects, A and B, with similar costs and budgets, A should be preferred to B if A has a better social return.*

Ignoring, for a moment, the difficulties for defining the social return of a project portfolio, given two portfolios, C and D, with equivalent budgets, C should be preferred to D if and only if C has a better social return. Thus, the problem of searching for the best projectportfolio can be reduced to finding a method for assessing social-project returns, or at least a comparative way to analyze alternative portfolio proposals.

The most commonly used method to examine the efficiency impacts of public policies is "cost-benefit" analysis (e.g. Boardman, 1996). Under this approach, the assumed consequences of a project are "translated" into equivalent monetary units where positive

Public Portfolio Selection Combining Genetic Algorithms and Mathematical Decision Analysis 141

rigid process that has been questioned by several authors (e.g. Gabriel et al., 2006, Fernandez et al., 2009 a,b). According to our perspective, the decision on which projects should receive financing must be made based on the best portfolio, rather than on the best individual projects. Therefore, it is insufficient to compare projects to one another. Instead, it is essential to compare portfolios. Selecting a portfolio based on individual projects' ranking guarantees that the set of the best projects will be supported. However, this set of projects does not necessarily equals the best portfolio. In fact, these two sets might be disjoint. Under this scenario, it is reasonable to reject a relatively good (in terms of its social impact) but expensive project if it requires disproportionate funding (Fernandez et al. 2009 a,b).

Mavrotas et al. (2008) argue that, when the portfolio is optimized, good projects can be outranked by combinations of low-cost projects with negligible impact. However, this is not

• The decision maker can define his/her preferences over the set of feasible portfolios (by

• The nature of the decision maker should be defined. It must be clear that this entity can address social interest problems in a legit way. In addition, the following questions should be answered. Is the decision-maker a single person? Or is it a collective with homogeneous preferences such that these can be captured by a decision model? Or is it, instead, a heterogeneous group with conflicting preferences? How is social interest

• A computable model of the DM's preferences on the social impacts of portfolios is

• Portfolio selection is an optimization problem with exponential complexity. The set of possible portfolios is the power set of the projects applying for funding. The cardinality of the set of portfolios is 2N, where N is the number of projects. The complexity of this problem increases significantly if we consider that each project can be assigned a support level. That is, projects can be partially supported. Under these conditions, the optimization problem is not only about identifying which projects constitute the best portfolio but also about defining the level of support for each of

• If effects of synergetic projects or temporal dependencies between them are considered,

The first issue is related to the concepts of social preferences, collective decision, democracy, and equity. The second issue, on the other hand, constitutes mathematical decision analysis' main area of influence. These capabilities for building preference models that incorporate different criteria and perspectives is what makes these techniques useful (albeit with some

the complexity of the resulting optimization model increases significantly.

limitations) for constructing multidimensional models of conflicting preferences.

• The decision maker prefers the portfolio composed of more projects with lower costs. In order to solve the selection problem over the set of feasible portfolios, the following issues

α

problem

Therefore, obtaining the best portfolio is, we argue, equivalent to solving the *P*

a real shortcoming whenever the following conditions are satisfied.

using some quality measure, or even by intuition)

defined over the set of all feasible portfolios.

• Each project is individually acceptable

reflected on the decision model?

should be addressed.

required.

these projects.

consequences are considered "benefits" and negative consequences are considered "loses" or "costs". The temporal distribution of costs and benefits, modeled as net-cash-flows and adjusted by applying a "social discount rate", allows computing the net present-value of individual projects. A positive net present-value indicates that a project should be approved whenever enough resources are available (Fernandez et al., 2009a). Therefore, the net present-value of a particular project can be used to estimate its social return. As a consequence, the social impact of a project portfolio can be computed as the sum of the netpresent-value of all the projects in the portfolio. The best portfolio can then be found by maximizing the aggregated social return (portfolio net-present-social benefit) using 0-1 mathematical programming (e.g. Davis and Mc Keoun, 1986).

This cost-benefit approach is inadequate for managing the complex multidimensionality of the combined outcome of many projects, especially when it is necessary to assess intangibles that have no well-defined market values. In extreme cases, this approach favors unacceptable practices (either socially or morally) such as pricing irreversible ecological damages, or even human life. Aside from ethical concerns, setting a price to intangibles for which a market value is highly controversial can hardly be considered a good practice. For a detailed analysis on this issue, the reader is referred to the works by French (1993), Dorfman (1996), and Bouyssou et al. (2000).

Despite this drawback, cost-benefit analysis is the preferred method for evaluating social projects (Abdullah and Chandra, 1999). Besides, not using this approach for modeling the multi-attribute impacts of projects leave us with no other method for solving portfolio problems with single objective 0-1 programming. A contending approach to cost-benefit is multi-criteria analysis. This approach encompasses a variety of techniques for exploring the preferences of the Decision Makers (DM), as well as models for analyzing the complexity inherent to real decisions (Fernandez et al., 2009a). Some of the most broadly known multi-criteria approaches are MAUT (cf. Keeney and Raiffa, 1976), AHP (cf. Saaty, 2000, 2005), and outranking methods (Roy, 1990; Figueira et al., 2005; Brans and Mareschal, 2005).

Multi-criteria analysis represents a good alternative to overcome the limitations of costbenefit analysis as it can handle intangibles, ambiguous preferences, and veto conditions. Different multi-criteria methods have been proposed for addressing project evaluation and portfolio selection (e.g. Santhanam and Kyparisis, 1995 ; Badri et al., 2001 ; Fandel and Gal, 2001 ; Lee and Kim, 2001 ; Gabriel et al., 2006; Duarte and Reis, 2006; Bertolini et al., 2006; Mavrotas et al., 2006; Sugrue et al., 2006; Liesio et al., 2007 ; Mavrotas et al., 2008; Fernandez et al., 2009a,b). The advantages of these methods are well documented in the research literature and the reader is referred to Kaplan y Ranjithan (2007) and to Liesio et al. (2007) for an in-depth study on the topic.

Multi-criteria analysis offers techniques for selecting the best project or a small set of equivalent "best" projects (this is known as the *P*α problem, according to the known classification by Roy (1996)), classifying projects into several predefined categories (e.g. "good", "bad", "acceptable"), known as the *P*β problem, and ranking projects according to the preferences or priorities given by the decision maker (the *P*γproblem).

Given a set of ranked projects, funding resources may be allocated following the priorities implicit in the ranking until no resources are left (e.g. Martino, 1995). This is a simple but rigid process that has been questioned by several authors (e.g. Gabriel et al., 2006, Fernandez et al., 2009 a,b). According to our perspective, the decision on which projects should receive financing must be made based on the best portfolio, rather than on the best individual projects. Therefore, it is insufficient to compare projects to one another. Instead, it is essential to compare portfolios. Selecting a portfolio based on individual projects' ranking guarantees that the set of the best projects will be supported. However, this set of projects does not necessarily equals the best portfolio. In fact, these two sets might be disjoint. Under this scenario, it is reasonable to reject a relatively good (in terms of its social impact) but expensive project if it requires disproportionate funding (Fernandez et al. 2009 a,b). Therefore, obtaining the best portfolio is, we argue, equivalent to solving the *P*α problem defined over the set of all feasible portfolios.

Mavrotas et al. (2008) argue that, when the portfolio is optimized, good projects can be outranked by combinations of low-cost projects with negligible impact. However, this is not a real shortcoming whenever the following conditions are satisfied.

• Each project is individually acceptable

140 Bio-Inspired Computational Algorithms and Their Applications

consequences are considered "benefits" and negative consequences are considered "loses" or "costs". The temporal distribution of costs and benefits, modeled as net-cash-flows and adjusted by applying a "social discount rate", allows computing the net present-value of individual projects. A positive net present-value indicates that a project should be approved whenever enough resources are available (Fernandez et al., 2009a). Therefore, the net present-value of a particular project can be used to estimate its social return. As a consequence, the social impact of a project portfolio can be computed as the sum of the netpresent-value of all the projects in the portfolio. The best portfolio can then be found by maximizing the aggregated social return (portfolio net-present-social benefit) using 0-1

This cost-benefit approach is inadequate for managing the complex multidimensionality of the combined outcome of many projects, especially when it is necessary to assess intangibles that have no well-defined market values. In extreme cases, this approach favors unacceptable practices (either socially or morally) such as pricing irreversible ecological damages, or even human life. Aside from ethical concerns, setting a price to intangibles for which a market value is highly controversial can hardly be considered a good practice. For a detailed analysis on this issue, the reader is referred to the works by French (1993), Dorfman

Despite this drawback, cost-benefit analysis is the preferred method for evaluating social projects (Abdullah and Chandra, 1999). Besides, not using this approach for modeling the multi-attribute impacts of projects leave us with no other method for solving portfolio problems with single objective 0-1 programming. A contending approach to cost-benefit is multi-criteria analysis. This approach encompasses a variety of techniques for exploring the preferences of the Decision Makers (DM), as well as models for analyzing the complexity inherent to real decisions (Fernandez et al., 2009a). Some of the most broadly known multi-criteria approaches are MAUT (cf. Keeney and Raiffa, 1976), AHP (cf. Saaty, 2000, 2005), and outranking methods (Roy, 1990; Figueira et al., 2005; Brans and

Multi-criteria analysis represents a good alternative to overcome the limitations of costbenefit analysis as it can handle intangibles, ambiguous preferences, and veto conditions. Different multi-criteria methods have been proposed for addressing project evaluation and portfolio selection (e.g. Santhanam and Kyparisis, 1995 ; Badri et al., 2001 ; Fandel and Gal, 2001 ; Lee and Kim, 2001 ; Gabriel et al., 2006; Duarte and Reis, 2006; Bertolini et al., 2006; Mavrotas et al., 2006; Sugrue et al., 2006; Liesio et al., 2007 ; Mavrotas et al., 2008; Fernandez et al., 2009a,b). The advantages of these methods are well documented in the research literature and the reader is referred to Kaplan y Ranjithan (2007) and to Liesio et al. (2007)

Multi-criteria analysis offers techniques for selecting the best project or a small set of

classification by Roy (1996)), classifying projects into several predefined categories (e.g.

Given a set of ranked projects, funding resources may be allocated following the priorities implicit in the ranking until no resources are left (e.g. Martino, 1995). This is a simple but

β

α

problem, according to the known

problem, and ranking projects according to

problem).

γ

mathematical programming (e.g. Davis and Mc Keoun, 1986).

(1996), and Bouyssou et al. (2000).

for an in-depth study on the topic.

equivalent "best" projects (this is known as the *P*

the preferences or priorities given by the decision maker (the *P*

"good", "bad", "acceptable"), known as the *P*

Mareschal, 2005).


In order to solve the selection problem over the set of feasible portfolios, the following issues should be addressed.


The first issue is related to the concepts of social preferences, collective decision, democracy, and equity. The second issue, on the other hand, constitutes mathematical decision analysis' main area of influence. These capabilities for building preference models that incorporate different criteria and perspectives is what makes these techniques useful (albeit with some limitations) for constructing multidimensional models of conflicting preferences.

Public Portfolio Selection Combining Genetic Algorithms and Mathematical Decision Analysis 143

known for its representation of preferences as a fuzzy outranking relation. In this work,

When using the functional model, also known as the functional-normative approach (e.g. French, 1993), the Decision Maker must establish a weak preference relation, known as the *at least as good as* relation and represented by the symbol ≿.This relation is a weak order (a complete and transitive relation) on the decision set *A*. The statement "*a* is at least as good as *b*" (*a* ≿ *b*) is considered a logical predicate with truth values in the set {*False*, *True*}. If *a* ≿ *b* is false then *b* ≿ *a* must be true, implying a strict preference in favor of *b* over *a*. Given the transitivity of this relation, if the DM simultaneously considers that predicates *a* ≿ *b* and *b* ≿ *c* are true, then, the predicate *a* ≿ *c* is also set to true. This approach does not consider the situation where both predicates, *a* ≿ *b* and *b* ≿ *a*, are false, a condition known as incomparability. Because of this, the functional model requires the DM to have an unlimited

The relation ≿ can be defined over any set whose elements may be compared to each other and, as a result of such comparison, be subject to preferences. Of particular interest is the situation where the decision maker considers risky events and where the consequences of the actions are not deterministic but rather probabilistic. To formally describe this situation,

**Definition 1**. A lottery is a 2N-tuple of the form (p1, x1; p2, x2;… pN, xN), where xi ∈ ℜ represents the consequence of a decision, pi is the probability of such consequence , and the

Given that the relation ≿ is complete and transitive, it can be proven that a real-valued function V can be defined over the decision set A (V: A → ℜ), such that for all *a, b* ∈ A, V(*a*) ≥ V(*b*) ⇔ *a* ≿ *b*. This function is known as a value or utility function in risky cases (French, 1993). If the decision is being made over a set of lotteries, the existence of a utility function *U* can be proven such that *Ū*(*L1*) ≥ *Ū*(*L2*) ⇔ *L1* ≿ *L2*, where *L1* and *L2* are two lotteries from the decision set and *Ū* is the expected value of the utility function (French,

The value, or utility, function represents a well formed aggregation model of preferences. This model is constructed around the set of axioms that define the rational behavior of the decision maker. In consequence, it constitutes a formal construct of an ideal behavior. The task of the analyst is to conciliate the real versus the ideal behavior of the decision maker when constructing this model. Once the model has been created, we have a formal problem definition. This is a selection problem that is solved by maximizing either *V* or *Ū* over the set of feasible alternatives. From this, a ranking can be obtained by simply sorting the values of these functions. By dividing the range of these values into M contiguous intervals, discrete ordered categories can be defined for labeling the objects in the decision set A (for instance, Excellent, Very Good, Good, Fair, and Poor). These categories are considered as equivalence

When building a functional model, compatibility with the DM's preferences must be guaranteed. The usual approach is to start with a mathematical formulation that captures the essential characteristics of the problem. Parameters are later added to the model in a way that they reflect the known preferences of the decision maker. Hence, every time the

however, we will focus on the functional approach only.

let us introduce the concept of lottery at this point.

power of discrimination.

sum of all probabilities equals 1.

classes to which the objects are assigned to.

1993).

The DM´s preferences on portfolios (or their social impacts) can be modeled from different perspectives, using different methods, and to achieve different goals. Selecting one of these options depends on who the DM is (e.g., a single person or a heterogeneous group), as well as on how much effort this DM is willing to invest in searching for the solution to the problem. Therefore, the information about the impact and quality of the projects that constitute a portfolio can be obtained from the DM using one of several available alternatives. This requires us to consider different modeling strategies and, in consequence, different approaches for finding the solution to this problem. We should note that the DM's preferences can be modelled using different and varying perspectives; ranging from the normative approach that requires consistency, rationality, and cardinal information, to a totally relaxed approach requiring only ordinal information. The chosen model will depend on the amount of time and effort the decision maker is willing to invest during the modelling process, and on the available information on the preferences. Here, we are interested in constructing a functional-normative model of the DM's preferences on the set of portfolios.

Evolutionary algorithms are powerful tools for handling the complexity of the problem (third and fourth issues listed above). Compared with conventional mathematical programming, evolutionary algorithms are less sensitive to the shape of the feasible region, the number of decision variables, and the mathematical properties of the objective function (e.g., continuity, convexity, differentiability, and local extremes). Besides, all these issues are not easily addressed using mathematical programming techniques (Coello, 1999). While evolutionary algorithms are not more time-efficient than mathematical programming, they are often more effective, generally achieving satisfactory solutions to problems that cannot be addressed by conventional methods (Coello et al., 2002).

Evolutionary algorithms provide the necessary instruments for handling both the mathematical complexity of the model and the exponential complexity of the problem. In addition, mathematical decision analysis methods are the main tools for modelling the DM´s preferences on projects and portfolios, as well as for constructing the optimization model that will be used to find the best portfolio.

The rest of this chapter is organized as follows. An overview of the functional-normative approach to decision making, as well as its use as support for solving selection, ranking and evaluation problems is considered in Section 2. In Section 3, we study the public portfolio selection problem where a project's impact is characterized by a project evaluation, and the DM uses a normative approach to find the optimal portfolio (i.e., the case where maximal preferential information is provided). In the same section we also describe an evolutionary algorithm for solving the optimization problem. An illustrative example is provided in Section 4. Finally, some conclusions are presented in Section 5.

### **2. An outline of the functional approach for constructing a global preference model**

Mathematical decision analysis provides two main approaches for constructing a global preference model using the information provided by an actor involved in a decision-making process. The first of these approaches is a functional model based on the normative axiom of perfect and transitive comparability. The second approach is a relational model better 142 Bio-Inspired Computational Algorithms and Their Applications

The DM´s preferences on portfolios (or their social impacts) can be modeled from different perspectives, using different methods, and to achieve different goals. Selecting one of these options depends on who the DM is (e.g., a single person or a heterogeneous group), as well as on how much effort this DM is willing to invest in searching for the solution to the problem. Therefore, the information about the impact and quality of the projects that constitute a portfolio can be obtained from the DM using one of several available alternatives. This requires us to consider different modeling strategies and, in consequence, different approaches for finding the solution to this problem. We should note that the DM's preferences can be modelled using different and varying perspectives; ranging from the normative approach that requires consistency, rationality, and cardinal information, to a totally relaxed approach requiring only ordinal information. The chosen model will depend on the amount of time and effort the decision maker is willing to invest during the modelling process, and on the available information on the preferences. Here, we are interested in constructing a functional-normative model of the DM's preferences on the set

Evolutionary algorithms are powerful tools for handling the complexity of the problem (third and fourth issues listed above). Compared with conventional mathematical programming, evolutionary algorithms are less sensitive to the shape of the feasible region, the number of decision variables, and the mathematical properties of the objective function (e.g., continuity, convexity, differentiability, and local extremes). Besides, all these issues are not easily addressed using mathematical programming techniques (Coello, 1999). While evolutionary algorithms are not more time-efficient than mathematical programming, they are often more effective, generally achieving satisfactory solutions to problems that cannot

Evolutionary algorithms provide the necessary instruments for handling both the mathematical complexity of the model and the exponential complexity of the problem. In addition, mathematical decision analysis methods are the main tools for modelling the DM´s preferences on projects and portfolios, as well as for constructing the optimization model

The rest of this chapter is organized as follows. An overview of the functional-normative approach to decision making, as well as its use as support for solving selection, ranking and evaluation problems is considered in Section 2. In Section 3, we study the public portfolio selection problem where a project's impact is characterized by a project evaluation, and the DM uses a normative approach to find the optimal portfolio (i.e., the case where maximal preferential information is provided). In the same section we also describe an evolutionary algorithm for solving the optimization problem. An illustrative example is provided in

**2. An outline of the functional approach for constructing a global preference** 

Mathematical decision analysis provides two main approaches for constructing a global preference model using the information provided by an actor involved in a decision-making process. The first of these approaches is a functional model based on the normative axiom of perfect and transitive comparability. The second approach is a relational model better

be addressed by conventional methods (Coello et al., 2002).

Section 4. Finally, some conclusions are presented in Section 5.

that will be used to find the best portfolio.

of portfolios.

**model** 

known for its representation of preferences as a fuzzy outranking relation. In this work, however, we will focus on the functional approach only.

When using the functional model, also known as the functional-normative approach (e.g. French, 1993), the Decision Maker must establish a weak preference relation, known as the *at least as good as* relation and represented by the symbol ≿.This relation is a weak order (a complete and transitive relation) on the decision set *A*. The statement "*a* is at least as good as *b*" (*a* ≿ *b*) is considered a logical predicate with truth values in the set {*False*, *True*}. If *a* ≿ *b* is false then *b* ≿ *a* must be true, implying a strict preference in favor of *b* over *a*. Given the transitivity of this relation, if the DM simultaneously considers that predicates *a* ≿ *b* and *b* ≿ *c* are true, then, the predicate *a* ≿ *c* is also set to true. This approach does not consider the situation where both predicates, *a* ≿ *b* and *b* ≿ *a*, are false, a condition known as incomparability. Because of this, the functional model requires the DM to have an unlimited power of discrimination.

The relation ≿ can be defined over any set whose elements may be compared to each other and, as a result of such comparison, be subject to preferences. Of particular interest is the situation where the decision maker considers risky events and where the consequences of the actions are not deterministic but rather probabilistic. To formally describe this situation, let us introduce the concept of lottery at this point.

**Definition 1**. A lottery is a 2N-tuple of the form (p1, x1; p2, x2;… pN, xN), where xi ∈ ℜ represents the consequence of a decision, pi is the probability of such consequence , and the sum of all probabilities equals 1.

Given that the relation ≿ is complete and transitive, it can be proven that a real-valued function V can be defined over the decision set A (V: A → ℜ), such that for all *a, b* ∈ A, V(*a*) ≥ V(*b*) ⇔ *a* ≿ *b*. This function is known as a value or utility function in risky cases (French, 1993). If the decision is being made over a set of lotteries, the existence of a utility function *U* can be proven such that *Ū*(*L1*) ≥ *Ū*(*L2*) ⇔ *L1* ≿ *L2*, where *L1* and *L2* are two lotteries from the decision set and *Ū* is the expected value of the utility function (French, 1993).

The value, or utility, function represents a well formed aggregation model of preferences. This model is constructed around the set of axioms that define the rational behavior of the decision maker. In consequence, it constitutes a formal construct of an ideal behavior. The task of the analyst is to conciliate the real versus the ideal behavior of the decision maker when constructing this model. Once the model has been created, we have a formal problem definition. This is a selection problem that is solved by maximizing either *V* or *Ū* over the set of feasible alternatives. From this, a ranking can be obtained by simply sorting the values of these functions. By dividing the range of these values into M contiguous intervals, discrete ordered categories can be defined for labeling the objects in the decision set A (for instance, Excellent, Very Good, Good, Fair, and Poor). These categories are considered as equivalence classes to which the objects are assigned to.

When building a functional model, compatibility with the DM's preferences must be guaranteed. The usual approach is to start with a mathematical formulation that captures the essential characteristics of the problem. Parameters are later added to the model in a way that they reflect the known preferences of the decision maker. Hence, every time the

Public Portfolio Selection Combining Genetic Algorithms and Mathematical Decision Analysis 145

• When decisions are made by a collective, the transitivity of the preference relation

• In most cases, the DM does not have the time to refine the model until a precise utility

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,

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

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

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

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

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

• Under what conditions can the return of a portfolio be effectively maximized?

• How can objective and subjective criteria be incorporated for optimizing project-

particular, if only one budgeting cycle is considered, the portfolios are subsets of Pr.

amounts of funding that will be assigned to projects belonging to area *Ai* ∈ *A*.

• How can the return of a public project-portfolio be formally defined?

that the overall social benefit from the portfolio is maximised.

• What methods can be used to select the best portfolio?

Now, we provide a list of drawbacks we have identified on the functional approach.

• It cannot precisely model threshold effects, nor can it use imprecise information.

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

• It cannot incorporate ordinal or qualitative information. • In real life, DM's do not exactly follow a rational behavior.

cannot be guaranteed.

function is obtained.

**algorithms** 

unique, or scarce.

available funds for each period.

portfolio returns?

DM indicates a preference for object *a* over object *b*, the model (i.e., the value function V) must satisfy condition V(*a*) > V(*b*). Otherwise, the model should satisfy condition V(*a*) = V(*b*), indicating that the DM has no preference of *a* over *b*, nor has the DM a preference of *b*  over *a*. This situation is known as indifference on the pair (*a*, *b*). If V is an elemental function, these preference/indifference statements on the objects become mathematical expressions that yield the values of V's parameters. To achieve this, usually the DM provides the truth values of several statements between pairs of decision alternatives (ai, bi). Then, the model's parameter values are obtained from the set of conditions V(ai) = V(bi). Finally, the value and utility functions are generally expressed in either additive or product forms, and, in the most simple cases, as weighted-sum functions.

The expected gain in a lottery is the average of the observed gains in the lottery's history. If the DM plays this lottery a sufficiently large number of times, the resulting gain should be close the lottery's expected gain. However, it is not realistic to assume that a DM will face (play) the same decision problem several times as decision problems are, most of the times, unique and unrepeatable. Therefore it is essential to model the DM's behavior towards risk. Persons react differently when facing risky situations. In real life, a DM could be risk prone, risk averse, or even risk neutral. Personal behavior for confronting risk is obviously a subjective characteristic depending on all of the following.


All these aspects are closely related. While the first of them is completely subjective, the remaining three have evident objective features.

The ability for modeling the decision maker's behavior when facing risk is one of the most interesting properties of the functional approach. At this point, it is necessary to introduce the concept of certainty equivalence in a lottery.

**Definition 2**. Certainty equivalence is the "prize" that makes an individual indifferent between choosing to participate in a lottery or to receive the prize with certainty.

A risk averse DM will assign a lottery a certainty equivalence value lower than the expected value of the lottery. A risk prone DM, on the other hand, will assign the lottery a certainty equivalence value larger than the lottery's expected gain. We say a DM is risk neutral when the certainty equivalence value assigned to a lottery matches the lottery's expected gain. This behavior of the DM yields quite interesting properties on the utility function. For instance, it can be proven that a risk averse utility function is concave, a risk prone utility function is convex, and a risk neutral function is linear.

Let us conclude this section by summarizing both the advantages and disadvantages of the functional approach. We start by listing the main advantages of the functional approach.


Now, we provide a list of drawbacks we have identified on the functional approach.

• It cannot incorporate ordinal or qualitative information.

144 Bio-Inspired Computational Algorithms and Their Applications

DM indicates a preference for object *a* over object *b*, the model (i.e., the value function V) must satisfy condition V(*a*) > V(*b*). Otherwise, the model should satisfy condition V(*a*) = V(*b*), indicating that the DM has no preference of *a* over *b*, nor has the DM a preference of *b*  over *a*. This situation is known as indifference on the pair (*a*, *b*). If V is an elemental function, these preference/indifference statements on the objects become mathematical expressions that yield the values of V's parameters. To achieve this, usually the DM provides the truth values of several statements between pairs of decision alternatives (ai, bi). Then, the model's parameter values are obtained from the set of conditions V(ai) = V(bi). Finally, the value and utility functions are generally expressed in either additive or product forms, and, in the

The expected gain in a lottery is the average of the observed gains in the lottery's history. If the DM plays this lottery a sufficiently large number of times, the resulting gain should be close the lottery's expected gain. However, it is not realistic to assume that a DM will face (play) the same decision problem several times as decision problems are, most of the times, unique and unrepeatable. Therefore it is essential to model the DM's behavior towards risk. Persons react differently when facing risky situations. In real life, a DM could be risk prone, risk averse, or even risk neutral. Personal behavior for confronting risk is obviously a

• The specific situation of the DM as this determines the impact of failing or succeeding.

All these aspects are closely related. While the first of them is completely subjective, the

The ability for modeling the decision maker's behavior when facing risk is one of the most interesting properties of the functional approach. At this point, it is necessary to introduce

**Definition 2**. Certainty equivalence is the "prize" that makes an individual indifferent

A risk averse DM will assign a lottery a certainty equivalence value lower than the expected value of the lottery. A risk prone DM, on the other hand, will assign the lottery a certainty equivalence value larger than the lottery's expected gain. We say a DM is risk neutral when the certainty equivalence value assigned to a lottery matches the lottery's expected gain. This behavior of the DM yields quite interesting properties on the utility function. For instance, it can be proven that a risk averse utility function is concave, a risk prone utility

Let us conclude this section by summarizing both the advantages and disadvantages of the functional approach. We start by listing the main advantages of the functional

• Once the model exists, obtaining its prescription is a straight forward process.

between choosing to participate in a lottery or to receive the prize with certainty.

• The amount of the gain or loses that will result from making a decision.

most simple cases, as weighted-sum functions.

• The DM's personality

approach.

subjective characteristic depending on all of the following.

• The relationship of the DM with these gains and loses.

remaining three have evident objective features.

the concept of certainty equivalence in a lottery.

function is convex, and a risk neutral function is linear.

• It can model the DM's behavior towards risk.

• It is a formal and elegant model of rational decision making.

