## 3 Model and Application
### Application of the Model to a Case Study  
The investment hierarchy model described in Section 3.1 was applied to a concrete case of an intervention program (PdI) developed by a water manager in Southern Italy. The research group responsible for the creation of this paper guided the implementation of the model, with the collaboration of the company involved in the experimentation. The utility in question is responsible for managing the integrated water service of 20 municipalities in the Campania region, which hosts a population of approximately 118,000 inhabitants over a total area of 677 km\({}^{2}\). The analysis of the PdA made it possible to obtain detailed information on the population served by the distribution and sewage services in the various municipalities. The specific data broken down by municipality have not been reported in this paper for reasons of synthesis but can be requested directly from the authors. In general, the water distribution service covers 97% of the resident population, while access to the sewage system is only available for around 82% of the population. Furthermore, the responsible body only partially manages the wastewater treatment service. Figure 2: Proposed model: hierarchization of aMi design alternatives with respect to time. The utility involved in the experiment has demonstrated its willingness to allow the implementation of the proposed model. It is important to underline that the collaboration between the scientific community and the utilities involved in the experimentation is essential to ensure the success of the PdIs. Once the optimal design alternatives are selected \(\mathrm{a_{Mi}}\) (an operation which, in the case in question, was performed using the AHP methodology and considering the indications and new criteria introduced in Subphase 2\({}_{3}\) of the proposed original protocol), the ARERA also requires the operator to define a time schedule for the identified interventions. In this regard, the model provides for the definition of a ranking that indicates the chronological priorities to be respected in the investments to be made. To identify a shared solution between the public aims set out in the Area Plan and the entrepreneurial aims of the utility, the model envisages a new application of the AHP, in which the goal is the temporal hierarchy of the project proposal's optimal terms \(\mathrm{a_{Mi}}\). The criteria for defining the ranking of the alternatives and the relative weights (unlike the previous phase, the criteria are now no longer equivalent) vary according to the following two scenarios:  
* Scenario 1 is representative of the strategic intervention priorities defined in the PdA. Specifically, the vector of weights is drawn up by submitting the judgment on the temporal urgency of the resolution of the different critical issues to the EGA, along with a pairwise comparison procedure;
* Scenario 2 is instead representative of the operator's business goals. The operator tends to invest in segments that ensure greater and more immediate profitability. In this scenario, therefore, the judgment on the weight vector is required of the utility to ensure that its interests are taken into account. Depending on the scenario, therefore, the EGA or the utility compare, according to their points of view, the critical areas, the subareas and finally the critical issues themselves (a reference was made to the classification scheme of the critical issues suggested by the ARERA). The comparative hierarchy (defined on a theoretical/conceptual level in Figure A1 of Appendix A) for this specific application of the AHP methodology is illustrated in Figure 3.  
The [CT] criticalities identified, grouped into areas and subareas, are summarized in Table 1. As anticipated, in Section 3.1, each criticality \(\mathrm{c_{i}}\) of the set {CT} was identified following a phase of consultation between the utility and EGA (even if the standard formally attributes this responsibility to the EGA).