## 7 An Illustrative Example of Project Portfolio Optimization
### Solution When All the DMs Accept the Interval Outranking Model and Its Assumptions  
Based on Assumption 3, the parameter values of the model (specified by the group members) are shown in Table 1 (See Appendix B). Weights and vetoes are the same as in [38]. Problem 11 was solved by each group member through the procedure explained in Section 4.2 and the evolutionary algorithm in Appendix C with the parameters given in Table 1 (c). The best compromise solution was identified by each DM (Assumption 2), as shown in Table 2. The conditions under which our computer experiments were carried out are the following: (1) the algorithms were implemented using the Java programming language and run on a computer with Intel Core i7-6700 HQ 2.6 GHz CPU, 8 GB of RAM, and Windows 10 Pro operating system; and (2) the solutions were obtained from 30 independent runs. The parameter values for the algorithms were tuned through a covering array experimental design [58]. The involved factors were population size, number of evaluations of the objective function vector and mutation probability, with levels of [50, 100, 200], [10,000, 100,000, 200,000] and [0.01, 0.02, 0.05], respectively. The results from the fine-tuning process indicated as the best configuration the following values: Population size = 100, number of evaluations = 100,000, and mutation probability = 0.02. The time required by the extended I-NOSGA to solve Problem 11 per group member was of 153 min and 56 s.  
Taking their best compromise as an ideal reference, the DMs selects their associated limiting boundary between \(C_{sat}\) and \(C_{dis}\). Each boundary is composed of three solutions that fulfill the conditions in Assumption 4. The boundaries are described in Table 2 in Appendix B. There is (\(\beta\),\(\lambda\))-preference favoring the best compromise in its comparison with the solutions in the limiting boundary (Condition _iii_ in Assumption 4), but there is no \(\alpha\)-dominance. Now, for identifying the best consensus, we should solve Problem 1. We use MOEA/D, whose brief description can be found in Appendix C. This algorithm found two solutions in the decision variable space, both fulfilling \(N_{sat}\) = 9 and \(N_{dis}\) = 0. In the objective space, such solutions are given in Table 3. The time consumed by MOEA/D to solve Problem 1 was 35 min and 59 s.  
There is no dissatisfaction with such solutions. All the DMs are satisfied except a single one who is neither satisfied nor dissatisfied. It is a high consensus level that may be accepted as the final collective decision. However, we suppose that the group moderator wishes to carry on a consensus reaching step to close even more the group judgments and beliefs. Suppose that, under the guidance from the moderator; the group members make closer criterion weights, veto thresholds, credibility thresholds \(\alpha\), \(\beta\), and \(\lambda\), and the required budget per project. We made a computer simulation of this consensus round as follows:The procedure in Algorithm 1 is applied for Parameter \(p\) in the set MP = {lower bound of \(w_{j}\) (\(j\) = 1,..., 9), upper bound of \(w_{j}\) (\(j\) = 1,..., 9), lower bound of \(v_{j}\) (\(j\) = 1,..., 9), upper bound of \(v_{j}\) (\(j\) = 1,..., 9), lower bound of \(\beta\), and upper bound of \(\beta\), \(\alpha\), \(\xi\)}. ```
For each \(p\) in MP  Let \(p_{i}\) be the value of Parameter \(p\) set by the \(i\)-th DM. Let \(p_{a}\) denote the average value of \(p\) on the set of DMs. Repeat from \(i\) = 1 to 10  Calculate \(d=p_{i}-p_{a}\)  If \(d>0\), update \(p_{i}\) as \(p_{i}-d/2\)  If \(d<0\), update \(p_{i}\) as \(p_{i}+1\)/\(2\)  If \(d=0\), then \(p_{i}\) keeps its value  End of Repeat  End of For
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**Algorithm 1**Consensus Round Simulation  
Applying Algorithm 1, the distance between the previous value of \(p\) from the \(i\)-th DM and the average value of such a parameter on the set of DMs is reduced by half.