## 2 Preference Discovery in Semi-Structured Decision Making  
A key aspect of semi-structured decision making is that preferences are not initially known to the decision makers themselves but are _discovered_ through an iterative process in which the decision makers gain decisive insights on their preferences, goals, and constraints. In this _preference discovery_ process, decision makers progressively enhance their understanding of what drives optimal decisions until a satisfactory solution is generated. The decision makers often start by \"generating\" an initial solution (let us say \(\mathbf{x}^{(1)}\)) whose _quantifiable_ performance measures are evaluated analytically with a closed-form expression, a simulation, a system dynamics model, etc. (see [22] for a review of model-driven DSSs). Based on this initial solution, the decision makers reflect on their preferences (which we denote as \(\pi\) in the following), understanding which aspects (metrics) of the solution align with their values and which do not. The decision makers then identify qualitative factors that were not initially considered but are important for a more comprehensive solution. These new insights are used to generate a new solution that hopefully reflects better their preferences. The new solution is then assessed with regard to quantifiable metrics. This process is repeated, with a new solution \(\mathbf{x}^{(h)}\) at each cycle \(h\), embedding a more refined understanding of what is important to the decision makers. This iterative process helps in managing the complexity of real-world decisions by breaking down the problem and progressively building up a more complete solution. Figure 1 describes this iterative process. When the decision is combinatorial (or continuous) in nature (see, e.g., [23; 24]), feasible solutions cannot be enumerated because they are infinite or huge in number. In this case, the structured part of the decision-making process leads to an _optimization problem_ in which only core quantitative factors are considered. Under these hypotheses, the semi-structured decision can be seen as the solution of an _optimization problem involving decision maker preferences_:  
\[(\mathcal{P}) \min\ z=f^{\pi}(\mathbf{x})\] (1) s.t. \[g_{i}(\mathbf{x})=0,\quad i=1,\ldots,m \tag{2}\] \[h_{r}^{\pi}(\mathbf{x})=0,\quad r=1,\ldots,q_{\pi} \tag{3}\]  
In this formulation, both the objective function \(f^{\pi}(\mathbf{x})\) (assumed to be minimized) and constraints \(h_{r}^{\pi}(\mathbf{x})=0\) depend on the preferences \(\pi\) of the decision makers and acknowledge the role of human tacit knowledge and contextual factors that affect the decision-making process. More formally, at the initial iteration (\(h=0\)), no preferences (or some initially known preferences) are considered (which is denoted as \(\pi_{0}\) in the following) and only the structured component of the decision-making process is accounted for. Hence, the following optimization problem is solved  
\[(\mathcal{P}^{(0)}) \min\ z=f^{\pi_{0}}(\mathbf{x})\] (4) s.t. \[g_{i}(\mathbf{x})=0,\quad i=1,\ldots,m \tag{5}\]  
where \(f^{\pi_{0}}(\mathbf{x})\) represents an approximation of \(f^{\pi}(\mathbf{x})\), in which no preferences are taken into account, and the constraints are independent of preferences. Once a solution \(\mathbf{x}^{(0)}\) of \(P^{(0)}\) is obtained, it is analyzed by the decision makers and evaluated in light of their preferences. Whenever this solution does not consider some of the preferences, the decision makers devise a refined approximation \(f^{\pi_{1}}(\mathbf{x})\) of the \"true\" objective function \(f^{\pi}(\mathbf{x})\), and possibly an expanded set of constraints \(h^{\pi_{1}}_{r}(\mathbf{x}),r=1,\ldots,q_{\pi_{1}}\). Then, the new approximation is solved, its solution analyzed, and so on. We now report the model at a generic iteration \(h=1,\ldots\) for future reference in this article:  
Figure 1: Preference discovery in semi-structured decision making. \[(\mathcal{P}^{(h)}) \min\ z=f^{\pi_{h}}(\mathbf{x})\] (6) s.t.