## 2 The Proposed Methodology
### System Model and Decision Design  
An urban street lighting system is typically widespread, with a huge number of lighting points and an extensive power and communication and control network. Consequently, it is generally composed of a set of subsystems, which reproduce the lighting conditions in the street/road segments pertaining to adjacent urban zones. All the lighting units related to a given subsystem are supplied by a power substation and are automatically regulated by a control unit.  
The actual decision making is preceded by a decision design preparatory phase that comprises the analysis of the street lighting current conditions as well as the definition and characterization of appropriate retrofitting interventions, in the compliance with the norms and standards. This phase is performed by the decision maker in conjunction with technical experts on a zone by zone basis.  
First, the total energy savings achieved implementing the candidate refurbishment interventions plan is identified as the optimization criterion, since commonly used for assessing the energy performance of road lighting [20]. Subsequently, an on-site audit activity is performed. For each zone lighting subsystem, a diagnosis activity is conducted to evaluate the characteristics of interest, i.e., the classification and enumerations of deployed lighting units, and their parameters. The completion of the diagnosis phase allows estimating of the ex-ante status of the whole street lighting system.  
Successively, the assessment of appropriate refurbishment interventions is conducted. This activity implies an individual feasibility analysis in all the zone lighting subsystems, since each generally shows unique features, thus requiring customized retrofit measures. The outcome of such an activity is the set of possible interventions to be implemented in the given existing street lighting system, denoted as \(J=\{1,\ldots,j,\ldots,J\}\). We assume that the street lighting system satisfies the minimal technical requirements (e.g., photometric features) and all the retrofit measures to be implemented do not hinder the attainment of these technical requirements, while improving energy savings. Subsequently, each retrofit measure is characterized by three parameters: the application potential, the cost, and its payoff.  
The application potential of an intervention is defined as the metric concerning the maximum extension of the intervention implementation. For instance, in case of replacement of lighting units, the application potential consists in the number of units to be replaced, measured in pieces. Denoting the application potential of the \(j\)th action as \(u_{j}\), we collect the application potentials of all the actions \(j=1,\ldots,J\) in a column vector \(\mathbf{u}=\big{(}u_{1},\ldots,u_{j},\ldots,u_{J}\big{)}\).  
The cost of each intervention is modeled in accordance with a linear pricing model. Therefore, denoting as \(c_{j}\) the current unitary cost of the \(j\)th action, we collect all the unitary costs in a column vector \(\mathbf{c}=\big{(}c_{1},\ldots,c_{j},\ldots,c_{J}\big{)}\).  
The payoff of the generic intervention is defined as the per-period energy saving that the application of the specific action is expected to produce on the street lighting system. The payoff of each action is also modeled in accordance with a linear model. Therefore, denoting as \(p_{j}\) the payoff of the \(j\)th action, we collect all the current unitary payoffs in a column vector \(\mathbf{p}=\big{(}p_{1},\ldots,p_{j},\ldots,p_{J}\big{)}\). For instance, in case of the replacement of the aforementioned lighting units, the yearly energy consumption reduction per replaced lighting unit (measured in kWh/year) is assessed as \(p_{j}=\gamma-\gamma^{\prime}\), where \(\gamma\) and \(\gamma^{\prime}\), are the nominal energy consumptions of lighting units of a given type before and after the luminaire replacement, respectively.