## 7 An Illustrative Example of Project Portfolio Optimization
### Solution When All the DMs Accept the Interval Dum Function Model and Its Assumptions  
The criterion weights in this model are in general different from the weights in an outranking model. Based on Assumption 6, for simplicity and without loss of generality, we set in this Section the same values provided by Table 2a. The model of Section 5 requires credibility thresholds \(\alpha\) and \(\xi\). These are set as in Table 2c. Specifications about projects and requirements are the same as in [[https://www.dropbox.com/s/t80u9](https://www.dropbox.com/s/t80u9) kbdcub6]([https://www.dropbox.com/s/t80u9](https://www.dropbox.com/s/t80u9) kbdcub6) jua/Projects%20 Description.pdf?dl=0 (accessed on 12 May 2021). Problem 11 was solved by each group member through the procedure explained in Section 5.2 and the evolutionary algorithm in Appendix C with the parameters given in Table 2a,c. According to Assumption 2, the best compromise solution for each DM was determined, as shown in Table 5. Under the same conditions defined in the previous experiments, the extended I-NOSGA algorithm required 37 min and 47 s per group member to solve Problem 11. With their best compromise as an ideal reference, the DMs identify their associated limiting boundary between \(C_{sat}\) and \(C_{dis}\). Each boundary is composed of three solutions that fulfill the conditions in Assumption 7. The boundaries are detailed in Table 7 in Appendix B. Once the limiting boundaries have been set, the evaluation of functions \(N_{sat}\) and \(N_{dis}\) is straightforward through Definitions 18-20. Using MOEA/D (see Appendix C), a single non-dominated solution of Problem 1 was found, with \(N_{sat}=6\), \(N_{dis}=0\). In the original nine-objective space, this solution is given in Table 6.  
As in Section 7.1, suppose that, under the guidance from the moderator, the group members make closer criterion weights, credibility thresholds \(\alpha\) and \(\xi\), and the required budget per project. Our computer simulation of this consensus round, similarly to Section 7.1, is performed through Algorithm 1. Here, the procedure in Algorithm 1 is applied for Parameter \(p\) in the set \(MP=\{\)lower bound of \(w_{j}\) (\(j=1,\,\ldots,\,9\)), upper bound of \(w_{j}\) (\(j=1,\,\ldots,\,9\)), \(\alpha,\,\xi\}\). Since the simulation algorithm is identical, the updated parameters are the same as in Table 4a,c. The interval budget required by projects is updated identically to the way followed in Section 7.1 (see the updated values in Table 5, Appendix B). Solving Problem 1, again, MOEA/D identified many solutions in the decision variable space, which satisfies the best consensus \(N_{sat}=10\) and \(N_{dis}=0\). To select a small subset of solutions, we use the \"min\" operator for conjunction as in Equation 20. The first ranked consensus solutions are provided in Table 7.  
The values \(\sigma_{Pt}(ZG_{ki},\,B_{i})\) from Equation 19 are shown in Table 8 (see Appendix B). These represent the level of satisfaction for each DM. **Remark 9**. : _The example has illustrated the importance of the assumptions on which the proposal is based, the way to build and use both model of multi-criteria preferences, the search for optimum consensus solutions, the handle of imperfect information through interval numbers, and the modeling of conservatism from individual DMs when they face this imperfect information. For simplicity and space consumption reasons, the example assumed that all the DMs face the same set of objective functions, and consider the same resource consumption per project and the same resource availability, although the DMs may have their own conservatism attitude concerning these issues; the ill-definition of proper collective judgments is modeled by interval numbers._