THE MULTIPLE CHOICE PROBLEM WITH INTERACTIONS BETWEEN CRITERIA

ABSTRACT

An important problem in Multi-Criteria Decision Analysis arises when one must select at least two alternatives at the same time. This can be denoted as a multiple choice problem. In other words, instead of evaluating each of the alternatives separately, they must be combined into groups of n alternatives, where n = 2. When the multiple choice problem must be solved under multiple criteria, the result is a multi-criteria, multiple choice problem. In this paper, it is shown through examples how this problemcan be tackled on a bipolar scale. The Choquet integral is used in this paper to take care of interactions between criteria. A numerical application example is conducted using data from SEBRAE-RJ, a non-profit private organization that has the mission of promoting competitiveness, sustainable developmentand entrepreneurship in the state of Rio de Janeiro, Brazil. The paper closes with suggestions for future research.

Keywords:
Multi-Criteria Decision Analysis; bipolar Choquet integral; multiple choice problem; formation of portfolios; fuzzy logic

1 INTRODUCTION

The project portfolio management process involves different stages of decision making. At the end of those stages, projects that add value to organizations are selected and prioritized. Companies that work with multiple projects require a vision or integrated form of management that encompasses all of the projects from their portfolios. The methods that are used to form a portfolio of projects tend to emphasize the importance of uncertainty as a process variable. The term uncertainty usually refers to both the resources and the results that must be achieved (^{He & Zhou, 2011}; ^{Yu et al., 2012}). This statement is especially true in a multiple interacting criteria context.

The issue of dependency between criteria has been approached by different authors in Multi-Criteria Decision Analysis (^{Čančer, 2010}; ^{Dalalah et al., 2012}; ^{Öztürk, 2006}; ^{Roy, 2009}). However, that issue has been virtually untouched within the context of portfolio selection under multiple criteria.

Selecting a portfolio of projects is a problem that has been approached using Multi-Criteria Decision Analysis (^{Vetschera & Almeida, 2012}; ^{Anagnostopoulos & Mamanis, 2010}; ^{Carazo et al., 2010}). However, in many cases when portfolios must be selected under multiple criteria, the interactions among the criteria are not considered as much as they should be, and additive aggregation procedures are used. A mathematical model that has been quite useful for modeling the interactions between criteria is the Choquet integral (^{Choquet, 1953}). The objective of this paper is to show how the bipolar Choquet integral can be used for determining two or three combinations of choices for projects to be performed at the same time under multiple criteria, given that there are interactions between criteria. Ordering the two or three project combinations is indeed a multiple choice problem. A numerical application example is conducted using the data from (^{Gomes et al., 2009}).

2 THE CHOQUET INTEGRAL AS A MULTICRITERIA RANKING MODEL IN THE UNIPOLAR SCALE

The Choquet integral makes use of fuzzy measures. Those measures are very important forproblems that require reliability (in a sense that an element belongs to a set) and plausibility which is dual to reliability. The fuzzy measures also assign different degrees of importance to preferences and verify whether the criteria are met.

Following (^{Grabisch & Labreuche, 2010}), consider the set X = X₁ × X₂ ... × Xn of feasible alternatives. The decision maker has preferences with respect to X that are expressed by a binary relation of the type ≿.

Now consider the function that is given by x ≥ y → F(μ₁(x₁), ..., μn(xn)) ≥ F(μ₁(y₁), ..., μn(yn)), where F is the Choquet integral and μi : Xi → S, i = 1, ..., n are aggregation functions. S ⊂ R⁺ is a scale that represents the decision maker's preferences.

There are two types of scales: the first scale, the limited unipolar scale, applies when S = [0; 1], where zero means the absence of a property and 1 means the total certainty about the existence of such a property. In modeling, one can affirm the existence in Xi of two elements that have the notations Ui and Pi, where Ui is an element of Xi that represents the complete dissatisfaction of the decision maker and Pi represents his complete satisfaction, then μi(Ui) = 0 and μi(Pi) = 1. The second scale, the unlimited unipolar scale, applies when S = R⁺. Thisscale serves to represent the priorities and relative importance. For convenience, we use the notation μi(Si) = 1.

According to (^{Sugeno, 1974}), the function u : 2N → R is a capacity if u(Φ) = 0. A capacity μ that satisfies μ(A) ≤ μ(B), A ⊆ B is a fuzzy measure. This fuzzy capacity is normalized if μ(N) = 1, where N is the set of natural numbers. The fuzzy capacity is additive if, for all disjoint sets A, B ⊆ N, one has μ(A ∪ B) = μ(A) + μ(B). It is symmetrical if, for all subsets A, B, we have |A| = |B| ⇒ μ(A) = μ(B).

The formal definition of the discrete Choquet integral in a unipolar scale can be defined as follows: Let f : N → R⁺ to be the Choquet integral f in relation to a capacity μ given by:

C_μ(f) = Σ(f_σ(i₎ - f_σ(i_{- 1)})μ({σ₁, ..., σn})

where σ is a permutation in N such that f_σ1 ≤...≤ f_σn and f_σ0 = 0.

To construct an example, assume that the scores of 4 students in 3 subjects are as shown in Table 1.

Table 1
Students' evaluations in 3 subjects.

Criterion	Alternative
	Student A	Student B	Student C	Student D
Subject 1	4	3	1	2
Subject 2	6	5	6	5
Subject 3	3	5	3	2

The dean of the school wants to give a full scholarship to a student by sticking to the following rule: every chosen student must be good in subjects 1, 2 and 3 (exactly in this order, that is subject 1 is more important than subject 2 and subject 2 is more important than subject 3) (i.e. Subject 1 > Subject 2 > Subject 3).

The ordering of these 4 students can be determined by using the Choquet integral as shown in Table 2. The steps below are then followed.

Table 2
Ranking obtained by using the Choquet integral in the unipolar scale.

Criterion	Alternatives
	Student A	Student B	Student C	Student D
Subject 1	4μ({1}) = 4 · 0.35 = 1.4	3μ({1}) = 3 · 0.35 = 1.05	μ({1}) = 0.35	2μ({1}) = (2) · 0.35 = 0.7
Subject 2	2μ({1, 2}) = (6 − 4) · 0.34 = 0.68	2μ({1, 2}) = (5 − 3) · 0.34 = 0.68	5μ({1, 2}) = (6 − 1) · 0.34 = 1.7	3μ({1, 2}) = (5 − 2) · 0.34 = 1.02
Subject 3	3μ({2, 3}) = 3 · 0.33 = 0.99	(5 − 5)μ({2, 3}) = 0	(6 − 3)μ({2, 3}) = (3) · 0.33 = 0.99	7μ({2, 3}) = (5 − 2) · 0.33 = 0.99
Choquet integral	1.4 + 0.68+ 0.99 = 3.07	1.05+ 0.68 + 0 = 1.73	0.35+ 1.7 + 0.99 = 3.04	0.7 + 1.02+ 0.99 = 2.71
Ordering	1	4	2	3

Step 1 - Determining the fuzzy measures

A fuzzy measure indicates the degree of evidence that an element belongs to a set. It was used a 2-additive model and Shapley-Schubik index to determine the fuzzy measures.

For example, considering three subjects in this order Subject 1 > Subject 2 > Subject 3. The fuzzy measures used in this example were:

μ({1, 2, 3}) = 1, μ(Φ) = 0, μ({1}) = 0.35, μ({2}) = 0,

μ({3}) = 0, μ({1, 2}) = 0.34, μ({1, 3}) = 0, μ({2, 3}) = 0.33.

Step 2: Calculating the Choquet integral

These calculations are performed by summing the values along each column. This sum gives the values of the Choquet integral. The ranking of the alternatives that are provided by the Choquet integral is then obtained by ordering these alternatives from the highest to the lowest values. The results are presented in Table 2.

The resulting order is: Student A > Student C > Student D > Student B. This arrangement means that Student A is preferable to the other students.

3 THE CHOQUET INTEGRAL IN THE BIPOLAR SCALE

Using the same notation as in the previous section, for the bipolar Choquet we have the following (^{Grabisch & Labreuche, 2005}):

Let f : N → ^_Rn, A ⊆ ^_Rn be the Choquet integral of f with respect to the capacity μ given by

where σ is a permutation in N such that f_σ(1) ≤ ... ≤ f_σ(n₎ and f_σ(0) = 0.

According to (^{Greco & Figueira, 2003}), given a finite set J = {1, 2, ..., n}, a fuzzy measure μ is a function of the form: μ : 2J → [0, 1] such that μ(Φ) = 0, μ(J) = 1 (boundary conditions) and μ(C) = μ(D) if D ⊆ C, ∀ C, D ⊆ J (monotonicity condition).

Let P(J) be a set of pairs of subsets of

J : P(J) = {(C, D), C, D ⊆ J, C ∩ D = Φ}.

A bi-capacity μ in J is a function μ: P(J) → [0, 1] X [0, 1] such that μ(C, Φ) = (c, 0) and μ(Φ, D) = (0, d), c, d ∈ [0, 1]; μ(J, Φ) = (1, 0) and μ(Φ, J) = (0, 1) (boundary conditions). For each (C, D), (E, F) ∈ P(J) such that E ⊆ C, D ⊆ F, we have μ(C, D) = (c, d) and μ(E, F) = (e, f), e, f ∈ [0, 1] with c ≥ e and d ≥ f (monotonicity condition). We use the following notation: μ⁺(C, D) = c, μ^-(C, D) = d. A bi-capacity on the set J is a function : P(J) → [-1, 1] such that (Φ, Φ) = 0; (J, Φ) = 1 and (Φ, J) = -1 (boundary conditions). If E ⊆ C, D ⊆ F, then (C, D) ≥ (E, F) (monotonicity condition). From each bi-polar capacity μ in J, we can obtain a bi-capacity in J : (C, D) = μ+(C, D) - μ-(C, D), ∀ C, D ∈ P(J) (^{Greco & Figueira, 2003}).

For each x ∈ Rn : x+ = max{x, 0} is the positive part of x; for each x ∈ R : x- = max{-x, 0} is the negative part of x; for each x ∈ R : x+ = is the positive part of x(x₁, x₂, ..., x_n) ∈ ^_Rn; and x^- = is the negative part of x(x₁, x₂, ..., x_n) ∈ ^_Rn. Given x ∈ ^_Rn, we consider a permutation ( · ) of the elements of J such that |x₍₁₎| ≤ |x₍₂₎| ≤ ... ≤ |x_(j)| ≤ |x_(n)|. For each element j ∈ J, we have two subsets, C(j) = {i ∈ J : x_i ≥ |x_(j)|} and D(j) = {i ∈ J : -x_i ≥ |x_(j)|}. Considering a bi-capacity μ in J and a vector x ∈ ^_Rn, we can define the positive part of the bipolar Choquet integral as follows:

where J ≥ {j ∈ J/|xj| > 0}. In the same way, we can write the negative part of the bipolar Choquet integral as follows: Ch^-(x, μ) = Σj_∈J_> (|x₍j₎| - |x₍j_-1)|μ^-(C₍j₎, D₍j₎)). Therefore, the bipolar Choquet integral is ChB(x, μ) = C⁺(x, μ) + Ch^-(x, μ).

To illustrate the use of the bipolar Choquet integral, we now consider an example of the evaluation of apartments for rent based on three alternatives: near downtown, near a subway stationand low cost, which are given in Table 3. In this example, we have used a Likert scale, withwhich the opinions of experts varying from 1 (worst value) to 5 (best value). To select the best apartment, the client expresses his preferences as follows: (i) for an apartment near downtown, a low price is more important than being near the subway; therefore, apartment #1 is better than apartment #2; and (ii) for an apartment far from downtown, being near the subway station is more important than a low price; therefore, apartment #3 is better than apartment #4.

Table 3
Decision matrix for a bipolar example.

Criterion	Alternative
	Apartment # 1	Apartment # 2	Apartment # 3	Apartment # 4
Near downtown	5	5	2	2
Near subway station	4	5	5	4
Near subway station	3	4	4	3

Step 1 - Determining the fuzzy measures

Consider the following ordering of criteria: (i) low price > near subway station for an apartment near downtown; and (ii) near subway station > low price for an apartment far from downtown. This arrangement allows us to establish a relation between the fuzzy measures using a 2-additive model and the Shapley-Chubik index [(^{Grabisch & Labreuche, 2010})]. Those measuresare presented below:

μ({1, 2, 3}) = 1, μ(Φ) = 0, μ({1}) = 0.39, μ({2}) = 0,

μ({3}) = 0, μ({1, 2}) = 0.33, μ({1, 3}) = 0, μ({2, 3}) = 0.31

Step 2 - Calculating the Choquet integral

In Table 4, we present the rank ordering obtained by using the bipolar Choquet integral.

Computations are performed by determining the Min and Max values along each column. The MaxMin operator gives the values of the Choquet integral. The ranking of the alternatives that are provided by the Choquet integral is then obtained by ordering these alternatives from the highest to the lowest values. The results are presented in Table 4.

Table 4
Rank ordering for a bipolar example.

Criterion	Alternative
	Apartment # 1	Apartment # 2	Apartment # 3	Apartment # 4
Near downtown	5μ({1}) = 5 · 0.39 = 1.95	5μ({1}) = 5 · 0.39 = 1.95	3μ({1}) = 3 · 0.39 = 0.78	3μ({1}) = 3 · 0.39 = 0.78
Near subway station	μ({1, 2}) = (5 − 4) · 0.33 = 0.33	μ({1, 2}) = (5 − 5) · 0.33 = 0	μ({1, 2}) = (5 − 2) · 0.33 = 0.99	μ({1, 2}) = (4 − 2) · 0.33 = 0.66
Low price	(4 − 3)μ({2, 3}) = 1 · 0.31 = 0.31	(5 − 4)μ({2, 3}) = 1 · 0.31 = 0.31	(5 − 4)μ({2, 3}) = 1 · 0.31 = 0.31	(4 − 3)μ({2, 3}) = 1 · 0.31 = 0.31
Min operator	1.95 × 0.33× 0.31 = 0.2	1.95× 0 × 0.31 = 0	0.78× 0.99× 0.31 = 0.24	0.78× 0.66 × 0.31 = 0.16
Max operator	1.95 + 0.33+ 0.31 = 2.59	1.95+ 0 + 0.31 = 2.26	0.78+ 0.99+ 0.31 = 2.08	0.78+ 0.66 + 0.31 = 1.75
Choquet integral	2.59	2.26	2.08	1.75
Rank Ordering	1	2	3	4

By using the bipolar Choquet integral the logic and desired solution is obtained. This solution is the following: Apartment #1 > Apartment #2 and Apartment #3 > Apartment #4.

4 THE SEBRAE-RJ CASE STUDY

A numerical application example is conducted using data from (^{Gomes et al., 2009}). SEBRAE-RJ is a non-profit private organization that has the mission of promoting competitiveness andsustainable development and encouraging entrepreneurship in the state of Rio de Janeiro, Brazil. In conjunction with the Strategies and Guidelines area of that organization, nine criteria were defined to evaluate different development projects. These criteria were C₁ = cost of project; C₂ = generated revenue/total cost of project; C₃ = degree of synergy in the use of SEBRAE-RJ's products in the project; C₄ = capacity to contribute to the sustainable development of the region; C₅ = capacity to interact with other sectors of the economy; C₆ = capacity to generate employment and income; C₇ = degree of adherence of the partnerships in the management as well as governance of the project; C₈ = chance of success; and C₉ = degree of visibility that the project would bring to SEBRAE-RJ. The decision matrix is presented in Table 5. In Table 5, 'Cr' stands for 'Criterion'.

Table 5
Decision matrix for the SEBRAE-RJ Case Study.

Cr	Projects
	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11
C1	775.79	393.06	343.93	1572.24	327.55	884.38	2354.2	884.39	929.54	982.67	3684.94
C2	0.08	0.06	0.63	0.10	0.10	0.19	0.09	0.09	0.12	0.31	0.06
C3	4	6	4	6	6	8	10	4	10	6	4
C4	10	10	10	8	10	6	10	10	10	10	10
C5	10	8	10	10	10	10	10	8	10	10	10
C6	8	10	10	8	10	6	10	10	10	8	10
C7	10	8	10	10	8	10	10	8	10	8	10
C8	10	10	10	10	10	10	10	10	10	10	10
C9	10	10	10	10	10	10	10	10	10	10	10

By using the mean and standard deviation of each line of the decision matrix a Gaussian membership function can be utilized in order to minimize the spreading of the data (Oliveira et al., 2007). This approach is illustrated in Figure 1 for line 1 of the decision matrix.

Figure 1
Membership function adjusted to the data of line 1 of decision matrix.

The membership values for line 1 of the decision matrix are presented in Table 6.

Table 6
Membership values for of the decision matrix.

Project	Membership values
P1	0.93
P2	0.89
P3	0.89
P4	0.98
P5	0.89
P6	0.93
P7	0.01
P8	0.94
P9	0.94
P10	0.95
P11	0.97

This task is accomplished for the whole decision matrix, as shown in Table 7. In this last table mi is the membership value for line i of the decision matrix (i = 1, ..., 7).

Table 7
Membership values for all data of decision matrix.

Project	m1	m2	m3	m4	m5	m6	m7
P1	0.93	0.85	0.60	0.90	0.90	0.55	0.91
P2	0.89	0.79	0.99	0.90	0.10	0.66	0.45
P3	0.89	0.02	0.60	0.90	0.90	0.66	0.77
P4	0.98	0.91	0.99	0.50	0.90	0.55	0.77
P5	0.89	0.91	0.99	0.90	0.90	0.66	0.45
P6	0.93	0.99	0.70	0.02	0.90	0.01	0.77
P7	0.91	0.88	0.21	0.90	0.90	0.66	0.77
P8	0.94	0.91	0.60	0.90	0.10	0.66	0.45
P9	0.94	0.95	0.21	0.90	0.90	0.66	0.77
P10	0.95	0.68	0.99	0.90	0.90	0.55	0.45
P11	0.97	0.79	0.60	0.90	0.90	0.66	0.77

The Choquet integral was calculated for all two portfolio combinations and for all possible two criteria combinations. The MinMax operator was used for each two of them. Similarly, all three portfolio combinations for all possible three criteria combinations were taken. Since criteria 8 and 9 are irrelevant for the analysis as they lead to the same figures they were removed from Table 8.

Table 8
Fuzzified decision matrix for the SEBRAE-RJ case study.

Criterion	Projects
	P1	P2	P3	P4	P5	P6	P7	P8	P9	P10	P11
C1	0.93	0.89	0.89	0.98	0.89	0.93	0.91	0.94	0.94	0.95	0.97
C2	0.85	0.79	0.02	0.91	0.91	0.99	0.88	0.91	0.95	0.68	0.79
C3	0.60	0.99	0.60	0.99	0.99	0.70	0.21	0.60	0.21	0.99	0.60
C4	0.90	0.90	0.90	0.50	0.90	0.00	0.90	0.90	0.90	0.90	0.90
C5	0.90	0.10	0.90	0.90	0.90	0.90	0.90	0.10	0.90	0.90	0.90
C6	0.55	0.66	0.66	0.55	0.66	0.01	0.66	0.66	0.66	0.55	0.66
C7	0.77	0.45	0.77	0.77	0.45	0.77	0.77	0.45	0.77	0.45	0.77

By applying these values, we obtain a new decision matrix, which is presented in Table 8.

Step 1 - Determining the fuzzy measures

As fuzzy measures indicate the degree of evidence that an element belongs to a set, considering a 2-aditive model and the Shapley-Schubik index, the fuzzy measures used in this paper were:

μ({1, 2, 3, 4, 5, 6, 7, 8, 9}) = 1, μ(Φ) = 0, μ({1}) = 0.56, μ({2}) = 0,

μ({3}) = 0, μ({4}) = 0, μ({5}) = 0, μ({6}) = 0,

μ({7}) = 0, μ({8}) = 0, μ({9}) = 0, μ({1, 2}) = 0.54,

μ({1, 3}) = 0, μ({2, 3}) = 0.53, μ({3, 4}) = 0.49, μ({4, 5}) = 0.48,

μ({5, 6}) = 0.45, μ({6, 7}) = 0.44.

We then consider the following order of criteria: C₁ > C₂ > C₃ > C₄ > C₅ > C₆ > C₇.

Step 2 - Choquet integral calculations

In Table 9, we present some calculations obtained by using the bipolar Choquet integral for projects 1 and 2 and for projects 1 and 3. The same has been done for all combinations of two projects.

Table 9
Some calculations for a two project selection of SEBRAE-RJ case study.

Criterion	Project # 1	Project # 2	Project # 1	Project # 3
C1	0.93μ({1}) = 0.93 · 0.56 = 0.52	0.89μ({1}) = 0.89 · 0.56 = 0.50	0.93μ({1}) = 0.93 · 0.56 = 0.52	0.89μ({1}) = 0.89 · 0.56 = 0.50
C2	(0.93 − 0.85)μ({1, 2}) = 0.14	(0.89 − 0.79)μ({1, 2}) = 0.19	(0.93 − 0.85)μ({1, 2}) = 0.14	(0.89− 0.02)μ({1, 2}) = 1.61
C3	(0.85 − 0.6)μ({2, 3}) = 0.13	(1 − 0.79)μ({2, 3}) = 0.11	(0.85 − 0.6)μ({2, 3}) = 0.13	(0.6 − 0.02)μ({2, 3}) = 0.31
C4	(0.9 − 0.6)μ({3, 4}) = 0.15	(1 − 0.9)μ({3, 4}) = 0.04	(0.9 − 0.6)μ({3, 4}) = 0.15	(0.9 − 0.6)μ({3, 4}) = 0.15
C5	(0.9 − 0.9)μ({4, 5}) = 0	(0.9 − 0.1)μ({4, 5}) = 0.38	(0.9 − 0.9)μ({4, 5}) = 0	(0.9 − 0.9)μ({4, 5}) = 0
C6	(0.9 − 0.55)μ({5, 6}) = 0.15	(0.66 − 0.1)μ({5, 6}) = 0.25	(0.9 − 0.55)μ({5, 6}) = 0.15	(0.9 − 0.66)μ({5, 6}) = 0.11
C7	(0.77 − 0.55)μ({6, 7}) = 0.09	(0.66 − 0.45)μ({6, 7}) = 0.09	(0.77 − 0.55)μ({6, 7}) = 0.09	(0.77− 0.66)μ({6, 7}) = 0.05
Min	Min(0.52, 0.14, 0.13, 0.15,0, 0.15, 0.09,0.50, 0.19, 0.11,0.04, 0.38, 0.25,0.09, 0.24, 0.24) = 0		Min(0.52,0.14, 0.13, 0.15,0, 0.15, 0.09,0.50, 1.61, 0.31,0.15, 0, 0.11, 0.05, 0.10, 0.10) = 0
Choquet integral (Max)	Max(0.52, 0.14, 0.13, 0.15,0, 0.15, 0.09,0.50, 0.19, 0.11,0.04, 0.38, 0.25,0.09, 0.24, 0.24) = 0.52		Max(0.52, 0.14, 0.13, 0.15,0, 0.15, 0.09,0.50, 1.61, 0.31,0.15, 0, 0.11, 0.05, 0.10, 0.10) = 1.61

These calculations are performed by using the Min and Max operators along considering the respective two columns projects. The MaxMin operator has been used to calculate the Choquet integral. The ranking of the alternatives that are provided by the Choquet integral is then obtained by ordering these alternatives from the highest to the lowest values.

The results are presented in Table 9.

A two-combination choice

The Choquet integral was calculated for all two portfolios combinations and all two criteria combinations that is, the MinMax operator was used for each two of them.

In Table 10 the ordering for two project combination portfolios is shown for the three most important criteria, C₁ (cost of project with mean 1193.88), C₂ (generated revenue/total cost of project with mean 0.17) and C₃ (degree of synergy in the use of SEBRAE-RJ's products in the project with mean 6.18).

Table 10
A two combination choice based on the first three most important criteria C₁, C₂ and C₃.

A two combination choice	C1	C2	C3	Observations
P3 − P7	343.93 and 2354.2 mean = 1349.07	0.63 and 0.092 mean = 0.36	4 and 10 mean = 7	low cost; high generated revenue; high synergy
P3 − P5	343.93 and 327.55 mean = 335.74	0.63 and 0.10 mean = 0.36	4 and 6 mean = 5	low cost; high generated revenue; medium synergy
P3 − P6	775.79 and 343.92 mean = 559.86	0.08 and 0.63 mean = 0.36	4 and 8 mean = 6	low cost; high generated revenue; medium synergy
P3 − P9	343.93 and 929.54 mean = 636.74	0.63 and 0.122 mean = 0.375	4 and 10 mean = 7	low cost; high generated revenue; medium synergy
P3 − P8	343.93 and 884.39 mean = 1349.07	0.63 and 0.096 mean = 0.362	4 and 4 mean = 4	low cost; high generated revenue; low synergy
P2 − P3	393.06 and 343.93 mean = 368.5	0.061 and 0.09 mean = 0.345	6 and 4 mean = 5	low cost; high generated revenue; low synergy
P3 − P4	343.93 and 1572.24 mean = 958.09	0.63 and 0.1 mean = 0.36	4 and 6 mean = 5	low cost; high generated revenue; low synergy
P1 − P3	343.93 and 343.93 mean = 614.16	0.63 and 0.192 mean = 0.41	4 and 4 mean = 4	low cost; high generated revenue; low synergy
P3 − P10	343.93 and 982.67 mean = 663.3	0.63 and 0.305 mean = 0.47	4 and 6 mean = 5	low cost; high generated revenue; low synergy
P3 − P11	343.93 and 3684.94 mean = 2014.44	0.63 and 0.064 mean = 0.346	4 and 4 mean = 4	low cost; high generated revenue; low synergy

In this paper values higher that mean values are considered as high; values near mean values are considered as mean; and values lower that mean values are considered as low. This holds for Tables 10, 11 and 12.

Table 11
A two combination choice based on criteria C₄, C₅ and C₆.

A two-combination choice	C4	C5	C6	Observations
P3 - P5	10 and 10 mean = 10	10 and 10 mean = 10	10 and 10 mean = 10	high capacity to contribute to the sustainable development of the region; high capacity to interact with other sectors of the economy; high capacity to generate employment and income
P3 - P7	10 and 10 mean = 10	10 and 10 mean = 10	10 and 10 mean = 10	high capacity to contribute to the sustainable development of the region; high capacity to interact with other sectors of the economy; high capacity to generate employment and income
P3 - P9	10 and 10 mean = 10	10 and 10 mean = 10	10 and 10 mean = 10	high capacity to contribute to the sustainable development of the region; high capacity to interact with other sectors of the economy; high capacity to generate employment and income
P3 - P11	10 and 10 mean = 10	10 and 10 mean = 10	10 and 10 mean = 10	high capacity to contribute to the sustainable development of the region; high capacity to interact with other sectors of the economy; high capacity to generate employment and income
P1 - P3	10 and 10 mean = 10	10 and 10 mean = 10	8 and 10 mean = 9	high capacity to contribute to the sustainable development of the region; high capacity to interact with other sectors of the economy; medium capacity to generate employment and income

Table 12
A two combination choice based on criteria C₇.

A two-combination choice	C7	Observations
P1 - P3	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income
P3 - P4	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income
P3 - P6	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income
P3 - P7	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income
P3 - P9	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income
P3 - P11	10 and 10 mean = 10	high degree of adherence of the partnerships in the management as well as governance of the project; high chance of success; high degree of visibility that the project would bring to SEBRAE-RJ; capacity to generate employment and income

It can be observed that project P₃ is present in all combinations for criteria that are related with minimum cost. high generated revenue and high synergy.

Table 11 presents a two combination choice for the criteria C₄ (capacity to contribute to the sustainable development of the region with mean 9.45), C₅ (capacity to interact with other sectors of the economy with mean 9.64) and C₆ (capacity to generate employment and income with mean 9.1).

It can be observed that most of combinations have high capacity to contribute to the sustainable development of the region, high capacity to interact with other sectors of the economy and high capacity to generate employment and income. It also can be seen that project P₃ is present in all of them. Table 12 presents a two combination choice for the criteria C₇ (degree of adherence of the partnerships) in the management as well as governance of the project with mean 9.3. In Table 12 criteria 8 and 9 were not considered since they were found to be redundant.

It can be observed that all combinations have high degree of adherence of the partnerships in the management as well as governance of the project, high chance of success and high degree of visibility that the project would bring to SEBRAE-RJ.

It also can be seen that project P₃ is present in all of them. In conclusion, we now reach the ordering by the bipolar Choquet integral for alternative two-project portfolios, as shown Table 13 presents the results obtained by using the bipolar Choquet integral for two-combination portfolios. Table 14 presents the results obtained by using the bipolar Choquet integral for three-project portfolio alternatives with similar calculations.

Table 13
Ordering by the Choquet bipolar Integral for two-choice projects 2.

Two-combination Portfolios	Results obtained using the bipolar Choquet Integral
P3 - P7	1
P3 - P9	2
P3 - P6	3
P3 - P9	4
P3 - P11	5

Table 14
Ordering obtained using the bipolar Choquet Integral for three-combination portfolios.

Three-combination Portfolios	Results obtained using the bipolar Choquet Integral
P1 - P2 - P3	1
P1 - P3 - P6	2
P1 - P3 - P8	3
P1 - P3 - P9	4
P1 - P3 - P10	5
P1 - P3 - P11	6
P2 - P3 - P4	7
P2 - P3 - P7	8

In the case of three-project portfolio alternatives, the Choquet integral was calculated for allthree portfolios combinations and all three criteria combinations that is, the MinMax operator was used for each.

It can be observed that all projects are selected to compose a three-combination choice ofportfolio for all nine criteria and the project P₃ is present in all.

5 CONCLUSIONS

We have concluded that the bipolar Choquet integral is adequate for solving multiple choice problems. As an application example, we have used the bipolar Choquet integral to show how SEBRAE-RJ can determine which two or three project combination choices should be formed. All of the selected two- and three-project combination portfolios do not cost too much and lead to high generated revenues/total project cost, high capacity to contribute to the sustainable development of the region and high capacity to create employment and income. For a two-choice problem, projects P₂, P₄, P₅, P₈ and P₁₀ are not selected, and for a three-choice problem, all projects are selected. In essence, we have shown that the use of the bipolar Choquet integral could allow forming a portfolio of projects by considering the measures of interactions among criteria. However, one must keep in mind the limitations that are related to the use of the Choquet integral, such as the requirement to have the aggregation (e.g., utility) functions fixed a priori (^{Bouyssou et al., 2012}). Nevertheless, when the bipolar Choquet integral can be used, the approach presented in this paper can be extended for n = 3 by induction.

An algorithm for the solution of a generic multiple choice problem based on the bipolarChoquet integral can be a generalization of a two portfolio choice. The steps to be followed should then be following: (1) determine the fuzzy measures to indicate the degree of evidence that an element belongs to a set, considering a 2-additive model and the Shapley-Schubik index; (2) select a two portfolio alternative and calculate the bipolar Choquet integral for all two portfolio combinations and all two criteria combinations by using the MinMax operator for each combination; (3) for a three- project portfolio alternatives, the Choquet integral has to be calculated for all three portfolio combinations and all three criteria combinations that is, the MinMax operator has to be used for each of them; (4) and the same calculations can carried out for more than three projects selection. All these operations can be performed, as they were in this paper, by using a MATLAB program.

For future research, it is recommended to design and run detailed sensitivity analyses on using other types of membership functions and alternative values for the parameters.

ACKNOWLEDGMENTS

The research leading to this article was partially supported by CNPq through projects 305732/2012-9 and 303363/2014-2.

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Publication Dates

• Publication in this collection
  Sep-Dec 2015

History

• Received
  14 May 2014
• Accepted
  12 Sept 2015