## 3 Model and Application  
### The Proposed Model: The AHP for the Temporal Hierarchy of Optimal Investments in the Water Sector  
The temporal hierarchization model proposed in this work represents the third and final phase of a complex original protocol to support the public decision maker, aimed at the process of allocating resources to investments in the water sector in a territory [20]. The purpose of the innovative protocol, which meets the requirements imposed by the ARERA for the selection of project alternatives, is to build an architecture capable of rationalizing the design choices for investments in water infrastructures of the Area Plan (PdA). As mentioned, this document constitutes the main means of technical-economic and financial planning at a territorial level, defining the status of the service as well as establishing the objectives to aim for, the technical and organizational standards, the investments to be made in the respect for available resources, the tariff impact and the organizational and management model. The area government bodies (EGA) draw up the PdA. The latter are bodies identified by the regions for each optimal territorial area (ATO), a supra-municipal geographical area in which the IWSM is organized. All municipalities falling within the ATO are compulsarily participating in the EGAs. The exercise of the responsibilities of the municipalities themselves in the field of water resource management, including the planning of water infrastructures, is transferred to the EGAs. The PdA drawn up by the EGAs consists of four parts: (i) a reconnaissance of the infrastructures; (ii) a program of interventions (PdI); (iii) a management and organizational model; (iv) an economic/financial plan [8]. The proposed original protocol, of which the model presented therein is an integral part, can be a useful tool for drafting the PdI. In fact, in the PdI, the EGA highlights the investments that the water manager must make in the four-year planning period to respond to emerging needs in the area of competence. The needs of the entire ATO are represented in the PdI through a set of criticalities {CA}. This set is, in turn, deduced from a broader set of criticalities {C} defined by the national authority (ARERA), which groups the individual criticalities c\({}_{\rm i}\) into areas and sub-areas. If the reference ATO is not managed by a single utility, but by several water operators who deal with different portions of the territory T, the subset {CA} will have to be analyzed based on the specificities of each operator. In this way, it will be possible to identify a subset of {CT} criticalities within {CA}, for which coherent and adequate investments will be required. For each criticality c\({}_{\rm i}\) of the sub-set {CT}, thanks to the collaboration with the operator, the EGA will have to identify the project a\({}_{\rm j}\) to be included in the PdI to decrease the impact of the criticality c\({}_{i}\), starting with a multiplicity of possible design solutions a\({}_{i}\). The proposed original protocol, which is substantiated in three phases (summarized in Figure 1), aims to facilitate the EGA in carrying out the operations just described. Figure 1: Flow chart representative of the logic of the original general protocol designed. Phase 3, which represents the model under study in this work, is highlighted with a black rectangle. The first two phases of the operative protocol were defined and tested in a previous study, to which reference is made for further details [20]. These phases are summarized as follows:  
* _Phase 1--The selection and measurement of critical issues_: First of all, the EGA and the operator must define the criticalities \(c_{i}\) of the subset {CT} by deducing them from those already exposed by the EGA in the subset {CA}, which, as mentioned, can be deduced from the set {C} defined by the AREA (Subphase 1\({}_{1}\)).