# Public Project Portfolio Optimization under a Participatory Paradigm

Authors: Eduardo Fernandez; Rafael Olmedo
Journal: Applied Computational Intelligence and Soft Computing
 (2013)

---

## Abstract

A new democracy paradigm is emerging through participatory budgeting exercises, which can be defined as a public space in which the government and the society agree on how to adapt the priorities of the citizenship to the public policy agenda. Although these priorities have been identified and they are likely to be reflected in a ranking of public policy actions, there is still a challenge of solving a portfolio problem of public projects that should implement the agreed agenda. This work proposes two procedures for optimizing the portfolio of public actions with the information stemming from the citizen participatory exercise. The selection of the method depends on the information about preferences collected from the participatory group. When the information is sufficient, the method behaves as an instrument of legitimate democracy. The proposal performs very well in solving two real-size examples.


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## Body

## 1. Introduction

Even in the best scenarios, designing public policies is far from being an exact science, with quantitative determinations beyond all subjectivity. Without denying the objective content of the social interest, it is certain that the difficulty in apprehending it opens space to methods that seek to model the preferences of concrete individuals, able to express their preferences in a more or less consistent way.Up to now, democracy in the distribution of public resources has been fundamentally expressed in(i)
the action of groups empowered by the society to make budget decisions on its behalf (GESBD) (parliaments, communes, governing boards of public organizations), formed by members of the political class legitimized by the popular vote, but that respond to their personal and their party’s interests instead of to the will of the electorate,(ii)
the action ofGESBDs formed by officials and experts appointed by the executive power that rather than interests of the electorate, reflected only in a very indirect way, reflect policies already designed by the executive.(iii)
the attempts of participatory budgeting carried out at local level where the population’s priorities are directly heard by constituted authorities and are later reflected in the distribution of resources once the compatibility with the opinions of the political class has been achieved.Since its emergence in Porto Alegre, Brazil, participatory budgeting has spread to hundreds of Latin American cities and dozens of cities in other continents. “Participatory budgeting” can be defined as a public space in which the government and the society agree how to adapt the priorities of the citizenship to the public policy agenda. The utility of these participatory exercises is that the government obtains information about the priorities of the participating social sectors and thus can perhaps identify programs that are of consensual benefit.However, the challenge that is still not approached by “participatory budgeting” is how to translate priorities and policies stemming from the exercise into a system of concrete social action projects, each with well-defined resources and falling within a frame of an approved general budget. Although the priorities of the citizenship have been identified and are likely to be reflected in a ranking of public policy actions, it is still necessary to solve a portfolio problem of public projects that should implement the agreed agenda.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 is2N, where N is the number of projects. If synergetic projects and nonlinear and timing distributions effects are considered, the complexity of the resulting optimization model increases significantly. So far, the participatory budgeting exercises have been unable to deal with this kind of problem. Two questions arise. How to obtain the information about the preferences of the participants in the exercise? How to use it later to explore the space of portfolios and find the best solutions compatible with the participants’ satisfaction?Basic structures of collective decision making and information provided by participatory budgeting exercises are discussed in Section2. The way in which a public portfolio can be optimized is described in Section 3. Section 4 details the way in which a portfolio can be optimized when the group preference information comes from a project ranking. Section 5 presents a portfolio optimization method when the preference information is given as individual goals. These methods are illustrated by real size examples in Sections 4 and 5. Finally, we draw some conclusions in Section 6.


## 2. Some Background

### 2.1. Basic Structures for Collective Decision Making

Let us consider a decision-maker group (DMG) composed of M actors with heterogeneous value systems. This collective is in charge of building a public project portfolio. According to Marakas [1], we can distinguish three basic structures for collective work oriented to decision making.Figure1(a) covers the case in which the whole group is responsible for the decision, there is a complete symmetry between the different DMs, and the final decision is made according to previously established rules that define the way in which the group is “constituted.” Figure 1(b) corresponds to situations in which symmetry is lost, there exists a DMin charge of the decision (hereinafter referred to as group supra-decision maker (GSDM)), who bases it on the opinions of the collective that processes a lower level of decision; the members of this collective only interact with the GSDM. The structure in Figure 1(c) shows a complete interaction amongst all participants, but the main responsibility still falls on the GSDM whose final judgment will be sustained by the best possible consensus of the members of the collective. All of the three structures assume a certain level of collaboration, even though it is not necessarily exempted from contradictions. These three cases will be referred to as partner association, team, and committee, respectively.Basic structures for collective work.
(a)
Complete interaction and symmetry
(b)
A supra-decision maker interacting with the group
(c)
Complete interaction, no symmetry


### 2.2. Information Stemming from the Participatory Exercise and Its Exploitation

Participatory budgeting exercises preferably take the form of Figure1(c), with government representatives playing the role of GSDM. From this interaction, the GSDM can obtain the following preferential information from the actors representing the society:(A)
opinions about certain priorities that the public agenda should contain, accepted by a significant part of participants;(B)
individual preferences of each participant about the public policy actions requiring budgetary resources ordered in a ranking of priorities;(C)
if the public agenda success is measured through a set of indicators, the levels that each participant expects as a result of the budgeting exercise.Given the social heterogeneity and conflicting interests, usually it is not possible to achieve consensuses in individual preferences (case B) or individual goals (C).A portfolio is a subset of a setPr of projects or social actions which are under consideration. To distribute resources means to find a feasible portfolio that meets specific requirements. In practice, the GSDM is the one who solves the portfolio problem by using the obtained information. But the ethics of democracy (not only the electoral factor, but democracy as a government by and for the people) require the GSDM to adhere as much as possible to the preferential information stemming from the participatory exercises. It is noteworthy that in (A) and (B) social preferences are not collected over project portfolios, but over concrete actions. Since the budget distribution is a problem of project, policy, or public action portfolios, the issue is how to translate the information from participatory exercises into preferences over portfolios. In (C), the aspirations collected about the social status are likely to be contradictory among different participants, but social status indicators will vary as a result of the portfolio that is decided to be supported. Thus, the information in mode (C) collects preferences over portfolios or over the result of portfolios.Case (A) is predominant among real exercises of participatory budgeting. TheGSDM seeks to adapt his/her agenda previously conceived according to the opinions received and on that basis to make the distribution of budgetary resources. The mind of GSDM processes the information, partly modifies his/her own subjectivity, and decides the distribution of resources. The group representing the society is not the one solving the portfolio problem, but its opinions are taken into account.In case (B), the public action rankings given by the participants are an expression of their preferences over projects, not over portfolios. Let us suppose that a method for integrating individual rankings is applied in collective ordering as the Borda score or a procedure based on exploitation of collective fuzzy preference relations (e.g., [2–4]). With the resulting group ordering, the GSDM has more information about social preferences regarding different actions that are to be budgeted. The GSDM shall use that information to find the best portfolio.In the following section, we assume that theGSDM has at his/her disposal a ranking of priorities of public actions or projects stemming from the participatory exercise.


## 2.1. Basic Structures for Collective Decision Making

Let us consider a decision-maker group (DMG) composed of M actors with heterogeneous value systems. This collective is in charge of building a public project portfolio. According to Marakas [1], we can distinguish three basic structures for collective work oriented to decision making.Figure1(a) covers the case in which the whole group is responsible for the decision, there is a complete symmetry between the different DMs, and the final decision is made according to previously established rules that define the way in which the group is “constituted.” Figure 1(b) corresponds to situations in which symmetry is lost, there exists a DMin charge of the decision (hereinafter referred to as group supra-decision maker (GSDM)), who bases it on the opinions of the collective that processes a lower level of decision; the members of this collective only interact with the GSDM. The structure in Figure 1(c) shows a complete interaction amongst all participants, but the main responsibility still falls on the GSDM whose final judgment will be sustained by the best possible consensus of the members of the collective. All of the three structures assume a certain level of collaboration, even though it is not necessarily exempted from contradictions. These three cases will be referred to as partner association, team, and committee, respectively.Basic structures for collective work.
(a)
Complete interaction and symmetry
(b)
A supra-decision maker interacting with the group
(c)
Complete interaction, no symmetry


## 2.2. Information Stemming from the Participatory Exercise and Its Exploitation

Participatory budgeting exercises preferably take the form of Figure1(c), with government representatives playing the role of GSDM. From this interaction, the GSDM can obtain the following preferential information from the actors representing the society:(A)
opinions about certain priorities that the public agenda should contain, accepted by a significant part of participants;(B)
individual preferences of each participant about the public policy actions requiring budgetary resources ordered in a ranking of priorities;(C)
if the public agenda success is measured through a set of indicators, the levels that each participant expects as a result of the budgeting exercise.Given the social heterogeneity and conflicting interests, usually it is not possible to achieve consensuses in individual preferences (case B) or individual goals (C).A portfolio is a subset of a setPr of projects or social actions which are under consideration. To distribute resources means to find a feasible portfolio that meets specific requirements. In practice, the GSDM is the one who solves the portfolio problem by using the obtained information. But the ethics of democracy (not only the electoral factor, but democracy as a government by and for the people) require the GSDM to adhere as much as possible to the preferential information stemming from the participatory exercises. It is noteworthy that in (A) and (B) social preferences are not collected over project portfolios, but over concrete actions. Since the budget distribution is a problem of project, policy, or public action portfolios, the issue is how to translate the information from participatory exercises into preferences over portfolios. In (C), the aspirations collected about the social status are likely to be contradictory among different participants, but social status indicators will vary as a result of the portfolio that is decided to be supported. Thus, the information in mode (C) collects preferences over portfolios or over the result of portfolios.Case (A) is predominant among real exercises of participatory budgeting. TheGSDM seeks to adapt his/her agenda previously conceived according to the opinions received and on that basis to make the distribution of budgetary resources. The mind of GSDM processes the information, partly modifies his/her own subjectivity, and decides the distribution of resources. The group representing the society is not the one solving the portfolio problem, but its opinions are taken into account.In case (B), the public action rankings given by the participants are an expression of their preferences over projects, not over portfolios. Let us suppose that a method for integrating individual rankings is applied in collective ordering as the Borda score or a procedure based on exploitation of collective fuzzy preference relations (e.g., [2–4]). With the resulting group ordering, the GSDM has more information about social preferences regarding different actions that are to be budgeted. The GSDM shall use that information to find the best portfolio.In the following section, we assume that theGSDM has at his/her disposal a ranking of priorities of public actions or projects stemming from the participatory exercise.


## 3. Finding the Best Portfolio via Multiobjective Optimization

As stated by Fernandez et al. [5] and Fernandez-Gonzalez et al. [6], 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, however, 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 project-portfolio can be reduced to finding a method for assessing social project returns, or at least a comparative way to analyze alternative portfolio proposals. If we do not have reliable information about specific social project returns, the social impact of a given portfolio may be represented by a set of “proxy” variables measuring different criteria of portfolio quality. Let us denote by {v1,…,vK} such a set. Without loss of generality, we suppose that the GSDM’s preferences are increasing with values of vi.Assumption 1.
LetC* and C** be two feasible portfolios and V*=(v1,…,vK)* and V**=(v1,…,vK)** their respective image in the proxy objective space. Then, V* is Pareto-dominated by V** only if under the available information, C** is preferred to C* from the GSDM’s point of view.
LetC be the set of feasible portfolios. Feasibility is determined by budgetary constraints. Assuming that the problem is solved by and for the GSDM, the comparison of the social value of the two portfolios becomes meaningful. Let Ci and Cj be two feasible portfolios. Let us assume that AP is an asymmetric preference relation defined by the GSDM on a subset of C× C. This is a subjective relation based on the information that the GSDM has about the portfolios. If CiAPCj does not generate a loop in AP, then we can accept that Ci has a greater value for the GSDM than Cj. The necessary condition for CL to be the best portfolio with respect to AP is that there is no Ci∈C such that CiApCL. Furthermore, a sufficient condition for CL to be the best portfolio with respect to APisCLApCj for all Cj∈C.Proposition 2.
Under Assumption1 the best portfolio for the GSDM is a nondominated solution to the problem
(1)MaximizeC(v1,…,vK).The proof is very simple. Suppose thatCL is the best portfolio being a dominated solution to problem (1). If CL is dominated by CJ, then from Assumption 1, CJApCL. So, CL does not hold the necessary condition to be the best portfolio.The choosing of a set{v1,…,vK} depends on the available information about the portfolio social impact, or how it is perceived by the GSDM. At least we should distinguish the two cases: (i) when the preference information from the participants has been aggregated in a ranking of priorities (case (B) of Section  2.2); (ii) when the public agenda success is measured by a set of indicators, and the contribution of each project or public action to the improvement of these indicators is known (case (C) of Section  2.2).In the next section, we propose a way to build the set{v1,…,vK} and solve problem (1) in case (i). Case (ii) will be approached in Section 5.


## 4. Portfolio Selection When the Group Preference Information Comes from a Project (Action) Ranking

It is necessary to compare the quality of the possible portfolios in order to find the best one. This problem was firstly approached by Fernandez et al. [5], under the assumption “the portfolio impact on a decision-maker’s mind is determined by the number of supported projects and their particular rank.” We shall accept that if project a is clearly ranked better than b, then a is admitted to have “more social impact” than b. The GSDM should take this information from the ranking into account. The appropriateness of a portfolio is not only defined by the quality of the included projects, but also by the amount of contained projects. As in Fernandez et al. [5], the purpose will be to find the best portfolio by increasing the number of supported projects and controlling the possible disagreements with respect to the GSDM’s preferences, which as it is assumed, are incorporated in the input ranking. Some discrepancies may be acceptable between the information provided by the ranking and the decisions concerning the approval (hence, supporting) of projects, whenever this fact increases the number of projects in the portfolio. However, this inclusion should be controlled because the admission of unnecessary discrepancies is equivalent to underestimating the ranking information.Among different heuristics, we suggest one that is based on dividing the ranking in five categories; then, on this basis construct the specific preference relation. The categories are labeled (1) vanguard, (2) high-medium, (3) medium, (4) low-medium, and (5) rearguard. IfPr is the set of projects, let us define a “priority” function ρ:Pr→{1,2,3,4,5} whose image is the project label (for instance, ρ(a)=1⇔a is ranked in the vanguard).Through exercising such categorization, we may build up the preference relations according to the ranking, as explained below.Absolute preference (denoted bya≫b): a≫b⇔ρ(a)<ρ(b) and |ρ(a)-ρ(b)|≥2.Strict preference (a>b): a>b⇔((ρ(a)<ρ(b) and |ρ(a)-ρ(b)|=1) or (ρ(a)=ρ(b) and a is ranked better than b).Weak preference (a>~b): a>~b⇔aand b are ranked equal, but b needs more funds than a.Strict outranking (aSb): S=≫∪>∪ >~.Note thatS is an asymmetric relation expressing preference for one of the projects. Hence, a well-formed portfolio should be compatible with the information contained in S. Let C be a project portfolio that obtains support, with aSb and a∉C. In such a case, the inclusion of b into C would constitute a discrepancy with respect to the information contained in S. Thus, each portfolio may be associated a discrepancy set in the following way:
(2)D={(a,b)∈Pr×Prsuch thataSb,b∈C,a∉C}.D can be partitioned in(i)
a set of absolute discrepanciesDa={(a,b)∈Pr×Pr such that a≫b, b∈C, a∉C};(ii)
a set of strict discrepanciesDs={(a,b)∈Pr×Pr such that a>b, b∈C, a∉C};(iii)
a set of weak discrepanciesDw={(a,b)∈Pr×Pr such that a>~b, b∈C, a∉C}.Letna, ns, and nw denote the respective cardinality of the above sets. Considering also the number of supported projects in C (denoted by nPr), problem (1) can be transformed into
(3)MaxCnPr,Min(na,ns,nw)
and according to Proposition 2, the best portfolio should be selected by the GSDM among its nondominated solutions. It is him/her who will finally select the adequate compromise among the number of supported projects and the quality given by their rank. Compromise solutions will depend on the extent that the GSDM relies on the information provided by the rank ordering and facts such as pressures exerted on him/her to include high-ranked projects or to increase the number of approved projects, as well as his/her aversion to costly projects.


### 4.1. An Illustrative Example

As illustration, let us consider the problem of distributing 2.5 billion dollars among 100 projects, all of them deserving individual financing. Information about these projects (costs in million dollars and their ranks according to some preference ordering) is shown in Table1 (the data are fictitious).Table 1
Information about projects.
Position in ranking
Requirements (million dollars)
1
84.00
2
124.50
3
129.75
4
147.75
5
126.00
6
137.25
7
96.00
8
84.75
9
93.00
10
121.50
11
102.75
12
141.75
13
105.75
14
98.25
15
101.25
16
83.25
17
109.50
18
107.25
19
135.00
20
97.50
21
127.50
22
114.00
23
106.50
24
94.50
25
43.50
26
31.25
27
26.50
28
36.25
29
50.00
30
34.75
31
48.25
32
46.00
33
36.75
34
34.00
35
26.00
36
31.75
37
29.75
38
37.25
39
26.75
40
43.75
41
27.25
42
47.00
43
41.00
44
30.50
45
45.25
46
26.25
47
45.50
48
44.25
49
48.75
50
33.25
51
27.50
52
41.25
53
29.50
54
25.25
55
40.00
56
30.75
57
39.00
58
44.50
59
47.50
60
36.00
61
28.50
62
29.00
63
30.25
64
49.50
65
33.00
66
38.50
67
33.50
68
48.50
69
35.00
70
28.75
71
25.50
72
40.25
73
38.75
74
46.75
75
37.00
76
46.50
77
44.00
78
25.75
79
38.25
80
40.75
81
42.75
82
43.00
83
32.25
84
37.75
85
44.75
86
27.00
87
39.50
88
30.00
89
37.50
90
49.00
91
41.75
92
39.25
93
34.50
94
49.75
95
48.00
96
29.25
97
47.75
98
42.25
99
46.25
100
39.75
Total
5542.00Financing all projects would amount more than 5.5 billions, but the available funds are not enough to support them all. The traditional heuristics for solving this problem consider giving support by following the rank order until the available resources run out. If this method was used, the resources would reach up to project 22, implying a total expenditure of 2.468 billion dollars.Let us suppose that theGSDM is not averse to making contradictory decisions with respect to the ranking information, thus being prepared to accept some discrepancies in order to increase the number of supported projects. Hence, he/she wants to evaluate nondominated solutions to problem (3). However, this problem exhibits exponential complexity, which complicates its solution through classical techniques of multiobjective optimization even for a medium-size instances [5]. Since it is difficult to model GSDM preferences over its objective functions, it seems preferable to apply a method for building the Pareto optimal set, and from this result, the GSDM can select the best “compromise.” For this reason we choose a multiobjective evolutionary algorithm (MOEA). One advantage of MOEAs is that they handle efficiently exponential complexity of the search space. According to Fernandez et al. [5], several works report successful experiments applying MOEAs to large 0-1 multiobjective knapsack problems. Unlike classical multiobjective optimization techniques, MOEAs are not sensitive to mathematical properties of the objective functions or the shape of the feasible region; moreover, they can obtain many nondominated solutions in only one run [7]. Here, we apply the NSGA-II (nondominated sorting genetic Algorithm) developed by Deb et al. [8], which is the benchmark of evolutionary multiobjective optimization. The NSGA-II performance is fairly sensitive to the number of objective functions, but this is not a main drawback for solving problem (3). Its pseudocode is shown in Pseudocode 1 (adapted from [9]).Algorithm 1
Generate a sizen random population P, calculating the portfolio criteria.
Determine the different fronts assigning each individual a “rank” that is,  the front which it belongs to
and its crowding distance.
Execute the following as many times as generations
Generate an offspring population and calculate their criteria
Select the parents using binary tournament. Their crosses produce two descendants to
whom the mutation operator is applied
Combine the parents’ population and offspring population:
Evaluate theσ value in GDM × P. GDM is the group of M decision-makers
Calculate the satisfaction and dissatisfaction level for each individual:
If
σi(xG,xi*)≥λi and xG satisfies the i-DM restrictions; and all projects which i-DM
considers very important belong to the portfolio,
Then
The individuali is satisfied
If
σi(xG,xi*)<0.5 and σi(xi*,xG)≥λ, or aG doesn’t satisfy the i-DM restrictions or a
significant part of projects thati-DM considers very important is not in CG,
Then
The individuali is unsatisfied
Countnsat and ndis
Determine the different fronts
Select the new population so that the members of the first fronts belong to it, and if necessary,
execute the Crowding-Sort
Repeat the above-mentioned as another generation.pseudocode 1
PROCEDURENSGA-II (K′, Number_of_Generations)
Initialize PopulationP
Generate random population with sizeK′
Evaluate objective values
Assign rank (level) based on Pareto dominance
Generate Child PopulationQ with size K′
Perform Binary Tournament Selection
Perform Recombination and Mutation
FORI=1 to Number_of_Generations  DO
AssignP′=P∪Q
FOR each parent and child inP′ DO
Assign rank (level) based on Pareto dominance
Loop (inside) by adding solutions to the
next generation starting from the first front
untilK′ individual found; determine crowding
distance between points of each front
End FOR
Select points (elitist) of the lower front (with lower rank) and are outside
a crowding distance
ReplaceP by the K′ individual found
Generate Child PopulationQ with size K′
Perform Binary Tournament Selection
Perform Recombination and Mutation
End FOR
End PROCEDUREBinary encoding was used; a “1” in the individualmth allele means that the mth project belongs to this particular portfolio. One-point crossover and the standard mutation operator were implemented (cf. [10]). Binary tournament selection was performed as in [11].The approach for handling constraints is based on the principle that any individual satisfying the constraints is always better than any individual that does not. If the feasible individuals in the population are classified intok nondominated classes, then any unfeasible individual is relegated to class k+1. This implies that in binary tournaments any feasible individual will have priority over any nonfeasible individual.The parameters of the evolutionary search were: crossover probability = 1; mutation probability = 0.02; population size = 100 number of generations = 500.A representative sample of the Pareto front is shown in Table2. Solution 63 (nPr,na,ns,nw) = (22,0,0,0) corresponds to allocating funds according to the ranking. If the GSDM is willing to accept some minor discrepancies, he/she can select the solutions (nPr,na,ns,nw) = (25,0,16,0) or (24,0,13,0). Represented in the decision variable space, these solutions are shown in Figure 2.Table 2
Some Pareto solutions.
Solution
(nPr, na, ns, nw)
63
(22, 0, 0, 0)
91
(24, 0, 13, 0)
93
(25, 0, 16, 0)
100
(26, 0, 27, 0)
110
(26, 0, 27, 0)
112
(27, 0, 31, 0)
191
(28, 0, 44, 0)
114
(29, 0, 57, 0)
252
(30, 0, 70, 0)
48
(31, 0, 78, 0)
117
(31, 0, 78, 0)
107
(32, 0, 86, 0)
286
(33, 0, 103, 0)
198
(34, 3, 114, 0)
25
(34, 6, 112, 0)
123
(35, 6, 129, 0)
400
(36, 12, 156, 1)
154
(37, 16, 170, 0)
155
(38, 24, 182, 2)
30
(37, 25, 159, 1)Figure 2
Some acceptable solutions in the decision space.


## 4.1. An Illustrative Example

As illustration, let us consider the problem of distributing 2.5 billion dollars among 100 projects, all of them deserving individual financing. Information about these projects (costs in million dollars and their ranks according to some preference ordering) is shown in Table1 (the data are fictitious).Table 1
Information about projects.
Position in ranking
Requirements (million dollars)
1
84.00
2
124.50
3
129.75
4
147.75
5
126.00
6
137.25
7
96.00
8
84.75
9
93.00
10
121.50
11
102.75
12
141.75
13
105.75
14
98.25
15
101.25
16
83.25
17
109.50
18
107.25
19
135.00
20
97.50
21
127.50
22
114.00
23
106.50
24
94.50
25
43.50
26
31.25
27
26.50
28
36.25
29
50.00
30
34.75
31
48.25
32
46.00
33
36.75
34
34.00
35
26.00
36
31.75
37
29.75
38
37.25
39
26.75
40
43.75
41
27.25
42
47.00
43
41.00
44
30.50
45
45.25
46
26.25
47
45.50
48
44.25
49
48.75
50
33.25
51
27.50
52
41.25
53
29.50
54
25.25
55
40.00
56
30.75
57
39.00
58
44.50
59
47.50
60
36.00
61
28.50
62
29.00
63
30.25
64
49.50
65
33.00
66
38.50
67
33.50
68
48.50
69
35.00
70
28.75
71
25.50
72
40.25
73
38.75
74
46.75
75
37.00
76
46.50
77
44.00
78
25.75
79
38.25
80
40.75
81
42.75
82
43.00
83
32.25
84
37.75
85
44.75
86
27.00
87
39.50
88
30.00
89
37.50
90
49.00
91
41.75
92
39.25
93
34.50
94
49.75
95
48.00
96
29.25
97
47.75
98
42.25
99
46.25
100
39.75
Total
5542.00Financing all projects would amount more than 5.5 billions, but the available funds are not enough to support them all. The traditional heuristics for solving this problem consider giving support by following the rank order until the available resources run out. If this method was used, the resources would reach up to project 22, implying a total expenditure of 2.468 billion dollars.Let us suppose that theGSDM is not averse to making contradictory decisions with respect to the ranking information, thus being prepared to accept some discrepancies in order to increase the number of supported projects. Hence, he/she wants to evaluate nondominated solutions to problem (3). However, this problem exhibits exponential complexity, which complicates its solution through classical techniques of multiobjective optimization even for a medium-size instances [5]. Since it is difficult to model GSDM preferences over its objective functions, it seems preferable to apply a method for building the Pareto optimal set, and from this result, the GSDM can select the best “compromise.” For this reason we choose a multiobjective evolutionary algorithm (MOEA). One advantage of MOEAs is that they handle efficiently exponential complexity of the search space. According to Fernandez et al. [5], several works report successful experiments applying MOEAs to large 0-1 multiobjective knapsack problems. Unlike classical multiobjective optimization techniques, MOEAs are not sensitive to mathematical properties of the objective functions or the shape of the feasible region; moreover, they can obtain many nondominated solutions in only one run [7]. Here, we apply the NSGA-II (nondominated sorting genetic Algorithm) developed by Deb et al. [8], which is the benchmark of evolutionary multiobjective optimization. The NSGA-II performance is fairly sensitive to the number of objective functions, but this is not a main drawback for solving problem (3). Its pseudocode is shown in Pseudocode 1 (adapted from [9]).Algorithm 1
Generate a sizen random population P, calculating the portfolio criteria.
Determine the different fronts assigning each individual a “rank” that is,  the front which it belongs to
and its crowding distance.
Execute the following as many times as generations
Generate an offspring population and calculate their criteria
Select the parents using binary tournament. Their crosses produce two descendants to
whom the mutation operator is applied
Combine the parents’ population and offspring population:
Evaluate theσ value in GDM × P. GDM is the group of M decision-makers
Calculate the satisfaction and dissatisfaction level for each individual:
If
σi(xG,xi*)≥λi and xG satisfies the i-DM restrictions; and all projects which i-DM
considers very important belong to the portfolio,
Then
The individuali is satisfied
If
σi(xG,xi*)<0.5 and σi(xi*,xG)≥λ, or aG doesn’t satisfy the i-DM restrictions or a
significant part of projects thati-DM considers very important is not in CG,
Then
The individuali is unsatisfied
Countnsat and ndis
Determine the different fronts
Select the new population so that the members of the first fronts belong to it, and if necessary,
execute the Crowding-Sort
Repeat the above-mentioned as another generation.pseudocode 1
PROCEDURENSGA-II (K′, Number_of_Generations)
Initialize PopulationP
Generate random population with sizeK′
Evaluate objective values
Assign rank (level) based on Pareto dominance
Generate Child PopulationQ with size K′
Perform Binary Tournament Selection
Perform Recombination and Mutation
FORI=1 to Number_of_Generations  DO
AssignP′=P∪Q
FOR each parent and child inP′ DO
Assign rank (level) based on Pareto dominance
Loop (inside) by adding solutions to the
next generation starting from the first front
untilK′ individual found; determine crowding
distance between points of each front
End FOR
Select points (elitist) of the lower front (with lower rank) and are outside
a crowding distance
ReplaceP by the K′ individual found
Generate Child PopulationQ with size K′
Perform Binary Tournament Selection
Perform Recombination and Mutation
End FOR
End PROCEDUREBinary encoding was used; a “1” in the individualmth allele means that the mth project belongs to this particular portfolio. One-point crossover and the standard mutation operator were implemented (cf. [10]). Binary tournament selection was performed as in [11].The approach for handling constraints is based on the principle that any individual satisfying the constraints is always better than any individual that does not. If the feasible individuals in the population are classified intok nondominated classes, then any unfeasible individual is relegated to class k+1. This implies that in binary tournaments any feasible individual will have priority over any nonfeasible individual.The parameters of the evolutionary search were: crossover probability = 1; mutation probability = 0.02; population size = 100 number of generations = 500.A representative sample of the Pareto front is shown in Table2. Solution 63 (nPr,na,ns,nw) = (22,0,0,0) corresponds to allocating funds according to the ranking. If the GSDM is willing to accept some minor discrepancies, he/she can select the solutions (nPr,na,ns,nw) = (25,0,16,0) or (24,0,13,0). Represented in the decision variable space, these solutions are shown in Figure 2.Table 2
Some Pareto solutions.
Solution
(nPr, na, ns, nw)
63
(22, 0, 0, 0)
91
(24, 0, 13, 0)
93
(25, 0, 16, 0)
100
(26, 0, 27, 0)
110
(26, 0, 27, 0)
112
(27, 0, 31, 0)
191
(28, 0, 44, 0)
114
(29, 0, 57, 0)
252
(30, 0, 70, 0)
48
(31, 0, 78, 0)
117
(31, 0, 78, 0)
107
(32, 0, 86, 0)
286
(33, 0, 103, 0)
198
(34, 3, 114, 0)
25
(34, 6, 112, 0)
123
(35, 6, 129, 0)
400
(36, 12, 156, 1)
154
(37, 16, 170, 0)
155
(38, 24, 182, 2)
30
(37, 25, 159, 1)Figure 2
Some acceptable solutions in the decision space.


## 5. Selection of Public Project Portfolios When the Information about Preferences Is Given in terms of Goals

When an individual decision maker gives his/her particular project ranking, he/she is expressing preferences about projects, not about portfolios. In this section we are interested in the case when each member of a participatory group is able to express his/her preferences through a set of aspiration levels which reflect the desired state of the concerned social object. Since those levels result from the aggregated action of projects, such information reflects preferences about portfolios.The best portfolio for a heterogeneous participatory group should be the most acceptable agreement. Let us recall after Fernandez et al. [4] some ideas about collective agreement. When a heterogeneous group is making decisions, situations may arise that impede, even with the existence of the “constitution” rules, the determination of an acceptable agreement. These situations arise due to strong contradictions amongst important coalitions of its members, for example, when a solid opposition of a numerically significant minority exists and has to accept a preference that is supported by a weak majority of the group. To establish a collective agreement is only possible when unanimity or an appreciable level of consensus exists (cf. [4]). A definition of this last concept may be associated to the satisfying of the following two conditions.(a)
There is an important agreeing majority.(b)
There is no appreciable disagreeing minority.Fernandez et al. [4] alerted about the undesirable effects of the “majority dictatorship.” This could have a negative influence on the stability of the group, making it less governable. Such “dictatorship” could be acceptable as an organizational principle with simple applications and that bases its rationality in the idea that more individuals should have weight more than that of lesser individuals (such a judgment ignores the importance of disagreement intensity). “Majority dictatorship” performs well in collectives that are disciplined and stable, mostly if the group is fundamentally comfortable accepting the majority principle as a basis for its decisions, and/or mechanisms for motivation and coercion exist in order to support group stability over the potential inconformity of a minority. As acceptable as the organizing principle might be, one should not make the mistake of assigning an ontological value that is generated by its rationality for every organization. Rejecting “majority dictatorship,” the best agreement should maximize the group satisfaction and minimize the strong opposition. We need to define what a satisfactory/unsatisfactory portfolio is for each individual decision maker, followed by a way to find a good compromise between measures of satisfaction and unsatisfaction for the group.In order to make a formal definition of satisfaction/dissatisfaction, let us introduce several new assumptions.Assumption 2.
Each individual decision maker in the participatory event is able to set a goal vectorx*∈ℜK which contains his/her aspiration levels for the relevant criteria of the concerned social object.Assumption 3.
Each individual decision-maker in the participatory event is able to set a set of constraints which express budgetary limits and some particular preferences concerning classes of projects, social groups, or geographic regions. Let us denote byRFi the corresponding feasible region in the portfolio space.Assumption 4.
Each member of the participatory group can set a fuzzy preference relationσi(x,y) defined on the criterion space; σi values on [0,1] represent the degree of credibility of the statement “x is at least as good as y.”
In order to be computable from the multicriteria description ofx and y, σi(x,y) can be built by using outranking methods [12], as is shown in Section 5.2.Assumption 5.
Each one of the group coalitionsCoi is a sufficiently homogeneous group as to achieve a consensus on the fuzzy preference relation σi parameters, the constraint setting, the projects which are considered very important ones, and the goal vector xi*.Assumption 6.
TheGSDM can set a real value λ(0.5<λ<1) such that given (x,y) in the criterion space if σi(x,y)≥λ, the ith decision maker (or the ith coalition) may accept that “x is at least as good as y.”Definition 3.
Letxi* be the goal vector set by the ith decision-maker (or the ith coalition). Suppose that CG is a portfolio which is being considered by the group; xG is its image in the criterion space. The ith decision maker (or the ith coalition) is said to be satisfied by xG if and only if the following conditions hold:(i)
σi(xG,xi*)≥λ;(ii)
xG holds the constraints imposed by the ith decision-maker;(iii)
The projects which are considered very important by theith decision maker belong to CG.Definition 4.
Theith decision-maker (or the ith coalition) is said to be unsatisfied by xG if one of the following conditions hold:(i)
σi(xG,xi*)<0.5 and σi(xi*,xG)≥λ;(ii)
xG does not hold the constraints imposed by the ith decision maker;(iii)
most of the projects which are considered very important by theith decision maker do not belong to CG;LetRF be the union of the feasible regions RFi. Note that two integer functions nsat(CG) and ndis(CG) may be defined on RF. For each CG in RF with image xG  in the criterion space, there is a number nsat obtained by counting the number of individual decision-makers which are satisfied by xG. On the other hand, ndis is the result of counting the unsatisfied members. Note that ndis+nsat≤M. The difference M-(ndis+nsat) is the number of decision makers which do not support a favorable consensus but are not against it.The best agreement may be seen as the best compromise solution for the problem(4)Maximizensat,Minimizendis.CG∈RF.Group satisfaction and dissatisfaction are optimized by solving problem (4). Dominated solutions of (4) should not be good agreements. Only nondominated solutions of (4) satisfying a minimum level of satisfaction should be considered. At least the condition of “agreeing relative majority” nsat, >ndis is necessary. So, (4) can be transformed into
(5)Maximizensat,Minimizendiss.t.CG∈RFs.t.nsat,>ndis.For solving (5), any non-linear multiobjective optimization method able to generate the Pareto frontier is acceptable. Once the set D of nondominated solutions has been found, the GSDM should determine the “best” compromise. Such a most acceptable solution corresponds to the current stage of the group members’ preferences and beliefs.Unfeasibility of (5) should be a consequence of very conflicting value systems within the group. This is a tough situation which must be treated with care. The reasons for strong conflict should be determined. Further group discussions and the application of groupware techniques (e.g., [13, 14]) would be necessary to close divergent beliefs, preferences, and constraint settings in order to achieve feasible solutions.


### 5.1. An Evolutionary Algorithm to Solve Problem (5)

Solving problem (5) is not an easy task due to its size and the nonlinearity of its objective functions. We use evolutionary multiobjective optimization. Evolutionary algorithms are less sensitive to some mathematical properties of objective functions and constraints than the traditional optimization methods [9]. Additionally, in problem (5) an evolutionary algorithm helps to handle its complexity with respect to the number of applicant projects because computer efficiency of these methods is less sensitive to problem size in comparison to the traditional techniques.Algorithm 1 is a direct adaptation of theNSGA-II [11].In the binary tournament, two individuals are chosen randomly, the winner being the one with a smaller “rank.” If both are in the same front, the tournament is won by the one who has a bigger crowding distance.


### 5.2. Example: Resources Assignment for Social Action Programs (Projects)

This example is a fictitious decision making situation in which the participatory group chooses among 100 different social policies (projects), each with a direct social return. This is measured by using a nine-component vector (N1,N2,…,N9). Ni=nkj, the number of people belonging to the jth social category which receive the kth benefit level from that policy or project. In this example j=1,2,3 correspond to (extreme poverty, poverty, low-middle), and k = 1, 2, 3 to (high impact, middle impact, low impact). N1,N2, and N3 correspond to extreme poverty people; N7,N8, and N9 concern low-middle class.N
i
m denotes the value of Ni associated to the mth project. Let C be a portfolio (a subset of projects which receives financial support). The value of Ni for the whole portfolio is calculated as Ni(C)=z1Ni1+⋯+z100Ni100, where zm=1 if the mth project is supported, and zm=0 otherwise. The aim of this decision problem is to choose the “best” portfolio satisfying budget constraints. These are imposed by the type of project (educational, health, etc.), geographic region, and the whole portfolio. The total available budget was set as Total_budget=2.5 billion dollars. The constraints by class and region are given by0.3Total_budget ≤Budget_Class 1≤0.4 Total_budget,0.25Total_budget≤Budget_Class 2≤0.35 Total_budget,0.2Total_budget≤Budget_Class 3≤0.3 Total_budget,0.4Total_budget≤Budget_Region 1≤0.6 Total_budget,0.4Total_budget≤Budget_Region 2≤0.6 Total_budget.The degree of truthσ(x, y) of the statement “x is at least as good as y” was calculated as in the ELECTRE-III method (cf. [12]); that is,
(6)σ(x,y)=c(x,y)N(d(x,y)),
where c(x,y) denotes the degree of truth of the concordance predicate given by (7) and N(d(x,y)) denotes the degree of truth of the nondiscordance predicate
(7)c(x,y)=∑j∈Cx,ywj,
where Cx,y is the concordance coalition and w’s denote “weights” (w1+w2+⋯+wn=1).LetDx,y be the discordance coalition with “x is at least as good as y.” The intensity of discordance is measured in comparison with a veto threshold vtj, which is the maximum difference yj-xj compatible with σ(x,y)>0. The model for the nondiscordance predicate is based on (8) (cf. [15]) shown below:
(8)N(d(x,y))=minj∈Dx,y[I-dj(x,y)],dj(x,y)={1⟺∇j≥νtj,(∇j-uj)(νtj-uj)⟺uj<∇j<νtj,0⟺∇j≤uj,
where ∇j=yj-xj and uj is a discordance threshold (see Figure 3).Figure 3
Partial discordance relationdj(x,y).Let us admit that the participatory group is composed of five coalitions with conflicting value systems. CoalitionsCo1,Co2,…,Co5 are composed by 10, 9, 8, 7, 6 members, respectively (M=40). Each value system is characterized by a different vector of weights as shown in Table 3. According to Assumption 5, a particular best portfolio was obtained for each value system. Those solutions are shown in Table 4.Table 3
Different value systems expressed by criterion weights.
Coalition
w
1
w
2
w
3
w
4
w
5
w
6
w
7
w
8
w
9
Co
1
0.23
0.14
0.11
0.14
0.11
0.07
0.09
0.07
0.04
Co
2
0.14
0.23
0.07
0.11
0.14
0.09
0.07
0.11
0.04
Co
3
0.11
0.09
0.14
0.11
0.23
0.14
0.07
0.04
0.07
Co
4
0.07
0.11
0.23
0.04
0.14
0.09
0.14
0.07
0.11
Co
5
0.04
0.07
0.23
0.11
0.14
0.09
0.14
0.07
0.11Table 4
Best portfolios from the five coalitions (in thousands).
Coalition
N
1
N
2
N
3
N
4
N
5
N
6
N
7
N
8
N
9
Co
1
820
620
675
1095
1005
810
1086
660
630
Co
2
820
615
690
1140
1020
720
1050
630
690
Co
3
810
795
485
1095
1020
735
876
546
858
Co
4
820
670
585
1230
900
660
846
666
714
Co
5
750
780
585
1035
1005
795
924
672
756In order to apply Condition (iii) in Definition3, we proceeded as follows. For each coalition, the set of projects in its particular best portfolio was ranked according to their contribution to the most important criterion  (for instance, Criterion 1 for Co1, Criterion 2 for Co2, Criterion 5 for Co3, etc.). After that, the best ranked 10 projects were chosen as “very important projects” for the corresponding coalition. λ was set to 0.67. The models of σj were built by using the weights given in Table 3; the indifference thresholds were calculated as a measure of the error evaluating each objective, which was assessed as the 15% of its maximum value on the set of projects; the veto threshold for the ith criterion was settled as 0.5*(MaxNi-MinNi); operations Max and Min act on a population.We used the evolutionary algorithm of Section5.2 for solving problem (5). Binary encoding was chosen; a “1” in the individual mth allele means that the mth project belongs to this particular portfolio. Other parameters of the evolutionary search were crossover probability = 1; mutation probability = 0.02; population size = 100.Several nondominated fronts of the lastNSGA-II’s population in the objective space of problem (5) are shown in Figure 4. In this space, a single nondominated solution nsat=5 and ndis=0 was obtained; that is, the five coalitions are considered satisfied. After 20 runs, 500 generations per run, 14 nondominated points in the portfolio space were detected, which correspond to the solution (5,0). Their values in the original nine-dimensional criterion space are pointed out in Table 5.Table 5
Best solutionsxGk* of problem (5) on the original criterion space (values in thousands).
Portfolio
N
1
N
2
N
3
N
4
N
5
N
6
N
7
N
8
N
9
1
820
685
690
1110
1005
810
936
666
750
2
820
690
690
1095
1020
810
954
666
696
3
820
685
690
1110
1020
810
936
678
720
4
820
695
690
1095
1005
810
936
666
726
5
820
670
695
1110
1005
810
936
672
720
6
820
665
710
1110
1005
810
936
690
720
7
820
680
690
1110
1020
810
954
666
720
8
820
670
685
1110
1005
795
936
672
720
9
820
685
695
1110
1005
810
936
690
726
10
820
670
695
1095
1020
810
1002
666
672
11
820
665
690
1125
1005
750
936
666
750
12
820
670
690
1110
1005
810
936
672
720
13
820
675
695
1095
1005
810
984
666
702
14
820
675
690
1110
1005
810
936
666
696Figure 4
SeveralNSGA-II’s nondominated fronts of problem (5).Any of these portfolios is unanimously satisfactory for the group, but only one can be selected. When there are several solutions of problem (5) in the portfolio space, the GSDM should apply a voting rule or additional criteria for choosing a single one.Taking into account the indifference thresholds, solutions in rows fifth, eighth, twelfth, and fourteenth are strictly outranked byxG1*, so they should not be chosen as the best ones. Thus, the set shown in Table 5 is reduced to 10 solutions. On this set we propose to evaluate the fuzzy predicate “for all j, the jth group member considers that xGk* is at least as good as his/her particular goal vector xj*.” It is the conjunction of the fuzzy predicates “the jth group member considers that xGk* is at least as good as his/her particular goal vector xj*.” In [16] some desirable properties of the geometric mean as conjunction operator in the framework of the so-called “compensatory fuzzy logic” are discussed. Using the geometric mean for conjunction, the degree of truth of the first predicate can be calculated as
(9)μsat(xGK)=(∏j=1,2,…,Mσj(xGKxj*))1/M,
where σj denotes the fuzzy outranking function of the jth member and M is set to 40. The values of μsat (xGk) are provided by Table 6.Table 6
Values ofμsat(xGk).
x
G
k
μ
sat
(
x
G
k
)
x
G
1
0.7967
x
G
2
0.8510
x
G
3
0.8693
x
G
4
0.7818
x
G
5
0.7818
x
G
6
0.8693
x
G
7
0.7818
x
G
8
0.8415
x
G
9
0.7691
x
G
10
0.7654Solutions with greater values ofμsat should be more satisfactory. However, μsat(z1)>μsat (z2) may be not a sufficient condition to affirm that group satisfaction is greater with z1. This can be approached by the concept of the fuzzy strict order, which can be used in order to compare truth values of statements modeled through predicates. A fuzzy strict order is a predicate o: U2→[0,1] (U denotes the universe) that meets both of the following conditions [16, 17].(A1)
o(x,y)=n[o(y,x)](generalized fuzzy reciprocity) (n denotes the negation operator).(B1)
Ifo(x,y)≥0.5 and o(y,z)≥0.5, theno(x,z)≥max(o(x,y),o(y,z))(max-max fuzzy transitivity or strong stochastic transitivity).With the definition provided by the above (A1) and (B1), the function
(10)o(x,y)=0.5[C(x)-C(y)]+0.5
with n(x)=1-x is a strict order over the universe of the predicate C(cf. [18]), which has been already used successfully (cf. [17]). The predicate o(μsat(z1), μsat(z2)) allows measuring “how more satisfactory is z1 than z2 for the group.” o(μsat(z1), μsat(z2)) higher than 0.5 indicates a preference favoring z1. If o(μsat(z1),μsat(z2))≈0.5, indifference can be accepted.SolutionsxG3 and xG7 maximize μsat. Using (10) we obtain o(xG3,xG7)=0.50 and o(xG3,xG2)≈0.51, o(xG2,xG3)≈0.49.xG3 or xG7 can be chosen as the best solution by the GSDM. Figure 5 shows these solutions in the decision variable space.Figure 5
Best solutions in the decision variable space.


## 5.1. An Evolutionary Algorithm to Solve Problem (5)

Solving problem (5) is not an easy task due to its size and the nonlinearity of its objective functions. We use evolutionary multiobjective optimization. Evolutionary algorithms are less sensitive to some mathematical properties of objective functions and constraints than the traditional optimization methods [9]. Additionally, in problem (5) an evolutionary algorithm helps to handle its complexity with respect to the number of applicant projects because computer efficiency of these methods is less sensitive to problem size in comparison to the traditional techniques.Algorithm 1 is a direct adaptation of theNSGA-II [11].In the binary tournament, two individuals are chosen randomly, the winner being the one with a smaller “rank.” If both are in the same front, the tournament is won by the one who has a bigger crowding distance.


## 5.2. Example: Resources Assignment for Social Action Programs (Projects)

This example is a fictitious decision making situation in which the participatory group chooses among 100 different social policies (projects), each with a direct social return. This is measured by using a nine-component vector (N1,N2,…,N9). Ni=nkj, the number of people belonging to the jth social category which receive the kth benefit level from that policy or project. In this example j=1,2,3 correspond to (extreme poverty, poverty, low-middle), and k = 1, 2, 3 to (high impact, middle impact, low impact). N1,N2, and N3 correspond to extreme poverty people; N7,N8, and N9 concern low-middle class.N
i
m denotes the value of Ni associated to the mth project. Let C be a portfolio (a subset of projects which receives financial support). The value of Ni for the whole portfolio is calculated as Ni(C)=z1Ni1+⋯+z100Ni100, where zm=1 if the mth project is supported, and zm=0 otherwise. The aim of this decision problem is to choose the “best” portfolio satisfying budget constraints. These are imposed by the type of project (educational, health, etc.), geographic region, and the whole portfolio. The total available budget was set as Total_budget=2.5 billion dollars. The constraints by class and region are given by0.3Total_budget ≤Budget_Class 1≤0.4 Total_budget,0.25Total_budget≤Budget_Class 2≤0.35 Total_budget,0.2Total_budget≤Budget_Class 3≤0.3 Total_budget,0.4Total_budget≤Budget_Region 1≤0.6 Total_budget,0.4Total_budget≤Budget_Region 2≤0.6 Total_budget.The degree of truthσ(x, y) of the statement “x is at least as good as y” was calculated as in the ELECTRE-III method (cf. [12]); that is,
(6)σ(x,y)=c(x,y)N(d(x,y)),
where c(x,y) denotes the degree of truth of the concordance predicate given by (7) and N(d(x,y)) denotes the degree of truth of the nondiscordance predicate
(7)c(x,y)=∑j∈Cx,ywj,
where Cx,y is the concordance coalition and w’s denote “weights” (w1+w2+⋯+wn=1).LetDx,y be the discordance coalition with “x is at least as good as y.” The intensity of discordance is measured in comparison with a veto threshold vtj, which is the maximum difference yj-xj compatible with σ(x,y)>0. The model for the nondiscordance predicate is based on (8) (cf. [15]) shown below:
(8)N(d(x,y))=minj∈Dx,y[I-dj(x,y)],dj(x,y)={1⟺∇j≥νtj,(∇j-uj)(νtj-uj)⟺uj<∇j<νtj,0⟺∇j≤uj,
where ∇j=yj-xj and uj is a discordance threshold (see Figure 3).Figure 3
Partial discordance relationdj(x,y).Let us admit that the participatory group is composed of five coalitions with conflicting value systems. CoalitionsCo1,Co2,…,Co5 are composed by 10, 9, 8, 7, 6 members, respectively (M=40). Each value system is characterized by a different vector of weights as shown in Table 3. According to Assumption 5, a particular best portfolio was obtained for each value system. Those solutions are shown in Table 4.Table 3
Different value systems expressed by criterion weights.
Coalition
w
1
w
2
w
3
w
4
w
5
w
6
w
7
w
8
w
9
Co
1
0.23
0.14
0.11
0.14
0.11
0.07
0.09
0.07
0.04
Co
2
0.14
0.23
0.07
0.11
0.14
0.09
0.07
0.11
0.04
Co
3
0.11
0.09
0.14
0.11
0.23
0.14
0.07
0.04
0.07
Co
4
0.07
0.11
0.23
0.04
0.14
0.09
0.14
0.07
0.11
Co
5
0.04
0.07
0.23
0.11
0.14
0.09
0.14
0.07
0.11Table 4
Best portfolios from the five coalitions (in thousands).
Coalition
N
1
N
2
N
3
N
4
N
5
N
6
N
7
N
8
N
9
Co
1
820
620
675
1095
1005
810
1086
660
630
Co
2
820
615
690
1140
1020
720
1050
630
690
Co
3
810
795
485
1095
1020
735
876
546
858
Co
4
820
670
585
1230
900
660
846
666
714
Co
5
750
780
585
1035
1005
795
924
672
756In order to apply Condition (iii) in Definition3, we proceeded as follows. For each coalition, the set of projects in its particular best portfolio was ranked according to their contribution to the most important criterion  (for instance, Criterion 1 for Co1, Criterion 2 for Co2, Criterion 5 for Co3, etc.). After that, the best ranked 10 projects were chosen as “very important projects” for the corresponding coalition. λ was set to 0.67. The models of σj were built by using the weights given in Table 3; the indifference thresholds were calculated as a measure of the error evaluating each objective, which was assessed as the 15% of its maximum value on the set of projects; the veto threshold for the ith criterion was settled as 0.5*(MaxNi-MinNi); operations Max and Min act on a population.We used the evolutionary algorithm of Section5.2 for solving problem (5). Binary encoding was chosen; a “1” in the individual mth allele means that the mth project belongs to this particular portfolio. Other parameters of the evolutionary search were crossover probability = 1; mutation probability = 0.02; population size = 100.Several nondominated fronts of the lastNSGA-II’s population in the objective space of problem (5) are shown in Figure 4. In this space, a single nondominated solution nsat=5 and ndis=0 was obtained; that is, the five coalitions are considered satisfied. After 20 runs, 500 generations per run, 14 nondominated points in the portfolio space were detected, which correspond to the solution (5,0). Their values in the original nine-dimensional criterion space are pointed out in Table 5.Table 5
Best solutionsxGk* of problem (5) on the original criterion space (values in thousands).
Portfolio
N
1
N
2
N
3
N
4
N
5
N
6
N
7
N
8
N
9
1
820
685
690
1110
1005
810
936
666
750
2
820
690
690
1095
1020
810
954
666
696
3
820
685
690
1110
1020
810
936
678
720
4
820
695
690
1095
1005
810
936
666
726
5
820
670
695
1110
1005
810
936
672
720
6
820
665
710
1110
1005
810
936
690
720
7
820
680
690
1110
1020
810
954
666
720
8
820
670
685
1110
1005
795
936
672
720
9
820
685
695
1110
1005
810
936
690
726
10
820
670
695
1095
1020
810
1002
666
672
11
820
665
690
1125
1005
750
936
666
750
12
820
670
690
1110
1005
810
936
672
720
13
820
675
695
1095
1005
810
984
666
702
14
820
675
690
1110
1005
810
936
666
696Figure 4
SeveralNSGA-II’s nondominated fronts of problem (5).Any of these portfolios is unanimously satisfactory for the group, but only one can be selected. When there are several solutions of problem (5) in the portfolio space, the GSDM should apply a voting rule or additional criteria for choosing a single one.Taking into account the indifference thresholds, solutions in rows fifth, eighth, twelfth, and fourteenth are strictly outranked byxG1*, so they should not be chosen as the best ones. Thus, the set shown in Table 5 is reduced to 10 solutions. On this set we propose to evaluate the fuzzy predicate “for all j, the jth group member considers that xGk* is at least as good as his/her particular goal vector xj*.” It is the conjunction of the fuzzy predicates “the jth group member considers that xGk* is at least as good as his/her particular goal vector xj*.” In [16] some desirable properties of the geometric mean as conjunction operator in the framework of the so-called “compensatory fuzzy logic” are discussed. Using the geometric mean for conjunction, the degree of truth of the first predicate can be calculated as
(9)μsat(xGK)=(∏j=1,2,…,Mσj(xGKxj*))1/M,
where σj denotes the fuzzy outranking function of the jth member and M is set to 40. The values of μsat (xGk) are provided by Table 6.Table 6
Values ofμsat(xGk).
x
G
k
μ
sat
(
x
G
k
)
x
G
1
0.7967
x
G
2
0.8510
x
G
3
0.8693
x
G
4
0.7818
x
G
5
0.7818
x
G
6
0.8693
x
G
7
0.7818
x
G
8
0.8415
x
G
9
0.7691
x
G
10
0.7654Solutions with greater values ofμsat should be more satisfactory. However, μsat(z1)>μsat (z2) may be not a sufficient condition to affirm that group satisfaction is greater with z1. This can be approached by the concept of the fuzzy strict order, which can be used in order to compare truth values of statements modeled through predicates. A fuzzy strict order is a predicate o: U2→[0,1] (U denotes the universe) that meets both of the following conditions [16, 17].(A1)
o(x,y)=n[o(y,x)](generalized fuzzy reciprocity) (n denotes the negation operator).(B1)
Ifo(x,y)≥0.5 and o(y,z)≥0.5, theno(x,z)≥max(o(x,y),o(y,z))(max-max fuzzy transitivity or strong stochastic transitivity).With the definition provided by the above (A1) and (B1), the function
(10)o(x,y)=0.5[C(x)-C(y)]+0.5
with n(x)=1-x is a strict order over the universe of the predicate C(cf. [18]), which has been already used successfully (cf. [17]). The predicate o(μsat(z1), μsat(z2)) allows measuring “how more satisfactory is z1 than z2 for the group.” o(μsat(z1), μsat(z2)) higher than 0.5 indicates a preference favoring z1. If o(μsat(z1),μsat(z2))≈0.5, indifference can be accepted.SolutionsxG3 and xG7 maximize μsat. Using (10) we obtain o(xG3,xG7)=0.50 and o(xG3,xG2)≈0.51, o(xG2,xG3)≈0.49.xG3 or xG7 can be chosen as the best solution by the GSDM. Figure 5 shows these solutions in the decision variable space.Figure 5
Best solutions in the decision variable space.


## 6. Concluding Remarks

Participatory budgeting construction exercises can be seen as a work of a committee, in which the executive plays the role of the group supra-decision-maker and the participants are representatives (formal or not) of the society. There are three main ways how theGSDM can collect information from the participatory group:(A)
as opinions about certain priorities that the public agenda should contain;(B)
as individual preferences of each participant about the public policy actions requiring budgetary resources, ordered in a ranking of priorities;(C)
as levels of individual aspirations with respect to the indicators that reflect the status of the social object.The ethics of democracy requires from theGSDM to adhere as much as possible to the preferential information stemming from the participatory exercises. Since the budget distribution is a problem of the public project (actions) portfolio, the mode (A) offers very little preferential information, and the GSDM thus preserves huge freedom for his/her decisions. The mode (B) offers more preferential information; here the GSDM still has the solution of the portfolio problem on his/her hands, but the democracy ethics force him/her to seek solutions that are compatible with the ranking of priorities that stems from the participatory exercise.There are two main situations in which aGSDM would find revenues in this approach: (1) he/she is averse to projects whose costs exceed the average considerably; and (2) he/she is strongly interested in increasing the number of supported projects. The procedure presented in Section 4 allows searching for reasonable compromises between the group priorities and the GSDM’s interest in increasing the amount of public agenda actions.The preferential information obtained from mode (C) is sufficient to allow solving a portfolio problem by thoroughly exploring the space of solutions in order to maximize the satisfaction of the social representation. To the best of our knowledge, no other approach has been proposed for maximizing group satisfaction and minimizing dissatisfaction in portfolio optimization. The method described in Section5 transforms a group multiobjective portfolio optimization problem into a group choice problem on a decision set composed of a relatively small set of alternative portfolios. This set contains the possible acceptable consensuses in the portfolio space. Once such a set has been identified, other well-known techniques can be used to reach the final solution. The main advantages are that (a) the search for the best agreement is not limited to portions of the Pareto frontiers and (b) no voting rule is used by the optimization algorithm; no, to some extent, arbitrary way of handling collective preferences is needed. Very good performance of the whole proposal is illustrated by a real-size example.The role ofGSDM becomes rather that of a group-work facilitator. If the collective is formed as a representative sample of the social structure, the solutions obtained by our method reflect the social preference, and the procedure may become a paradigm of the best forms of democracy.



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*Source: 891781-2013-01-30.xml*