THE MULTIPLE CHOICE PROBLEM WITH INTERACTIONS BETWEEN CRITERIA

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 problem can 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 nonprofit private organization that has the mission of promoting competitiveness, sustainable development and entrepreneurship in the state of Rio de Janeiro, Brazil. The paper closes with suggestions for future research.


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.

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: where σ is a permutation in N such that f σ 1 ≤ . . .≤ f σ n and f σ 0 = 0.
To construct an example, assume that the scores of 4 students in 3 subjects are as shown in Table 1.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.

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: 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: This arrangement means that Student A is preferable to the other students.

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): 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 : Let P(J ) be a set of pairs of subsets of ), e, f ∈ [0, 1] with c ≥ e and d ≥ f (monotonicity condition).We use the following notation: For each x ∈ R n : 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 : Considering a bi-capacity μ in J and a vector x ∈ R n , we can define the positive part of the bipolar Choquet integral as follows: where J ≥ { j ∈ J /|x j | > 0}.In the same way, we can write the negative part of the bipolar Choquet integral as follows: 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 station and low cost, which are given in Table 3.In this example, we have used a Likert scale, with which 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.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 2additive model and the Shapley-Chubik index [Grabisch & Labreuche (2010)].Those measures are presented below: 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.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.

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 and sustainable 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 C 1 = cost of project; C 2 = generated revenue/total cost of project; C 3 = degree of synergy in the use of SEBRAE-RJ's products in the project; C 4 = capacity to contribute to the sustainable development of the region; C 5 = capacity to interact with other sectors of the economy; C 6 = capacity to generate employment and income; C 7 = degree of adherence of the partnerships in the management as well as governance of the project; C 8 = chance of success; and C 9 = 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'.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.The membership values for line 1 of the decision matrix are presented in Table 6.This task is accomplished for the whole decision matrix, as shown in Table 7.In this last table m i is the membership value for line i of the decision matrix (i = 1, . . ., 7).
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.
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: We then consider the following order of criteria: Step

-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.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, C 1 (cost of project with mean 1193.88),C 2 (generated revenue/total cost of project with mean 0.17) and C 3 (degree of synergy in the use of SEBRAE-RJ's products in the project with mean 6.18).
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.
It can be observed that project P 3 is present in all combinations for criteria that are related with minimum cost.high generated revenue and high synergy.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 P 3 is present in all of them.Table 12 presents a two combination choice for the criteria C 7 (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 P 3 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.
In the case of three-project portfolio alternatives, the Choquet integral was calculated for all three portfolios combinations and all three criteria combinations that is, the MinMax operator was used for each.
Pesquisa Operacional, Vol.35(3), 2015 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.

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

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

Table 3 -
Decision matrix for a bipolar example.

Table 4 -
Rank ordering for a bipolar example.

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

Table 6 -
Membership values for of the decision matrix.

Table 7 -
Membership values for all data of decision matrix.

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

Table 11
presents a two combination choice for the criteria C 4 (capacity to contribute to the sustainable development of the region with mean 9.45), C 5 (capacity to interact with other sectors of the economy with mean 9.64) and C 6 (capacity to generate employment and income with mean 9.1).

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