## 4 Mathematical Formulation and Optimization Procedure  
A multi-objective problem (MOP) can be stated as follows [9]:  
\[\text{minimize}/\text{maximize}\ \mathbf{y}=\mathbf{f}(\mathbf{x})= \left(f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),\ldots,f_{k}(\mathbf{x})\right), \tag{38}\] \[\text{subject to}\ \mathbf{g}(\mathbf{x})=\left(g_{1}(\mathbf{x}),g_{2}( \mathbf{x}),\ldots,g_{m}(\mathbf{x})\right)\leq\mathbf{L}, \tag{39}\]  
where \(\mathbf{f}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}\) is the objective function, \(g_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}\), \(i=1,\ldots,m\), are the constraints, \(\mathbf{L}=(l_{1},l_{2},\ldots,l_{m})\), \(\mathbf{x}=(x_{1},x_{2},\ldots,x_{n})\in X\) is the decision vector (or vector of decision variables), and \(\mathbf{y}=(y_{1},y_{2},\ldots,y_{k})\in Y\) is the objective vector. We denote by \(X\) and \(Y=\mathbf{f}(X)\), the decision space and the objective space, respectively.  
The optimization of the maintenance interval in terms of cost and benefit can be expressed as a MOP. The formulation of both variables has been presented in Section 3. It can be observed that both models depend on \(w_{c}\) and \(M_{min}\) which act as decision variables and are encoded in the decision vector \(\mathbf{x}\). In our problem, the problem is bi-objective (\(n=2\)) \(\mathbf{f}(\mathbf{x})=\left(C(\mathbf{x}),P(\mathbf{x})\right)\), and the aim is to minimize the cost function \(C(\mathbf{x})\) and to maximize the profit function \(P(\mathbf{x})\). In this optimization approach, maintenance managers' and plant managers' purposes are considered jointly: maintenance managers usually try to minimize costs related to equipment inefficiencies (\(C(\mathbf{x})\)), whereas plant managers aim at maximizing the profitability of the plant (\(P(\mathbf{x})\)). It is widely known that multi-objective optimization should be complementary with respect to measures taken into consideration. In this case, \(C(\mathbf{x})\) and \(P(\mathbf{x})\) are defined as complementary, since \(\Psi\) is defined as a constant value, so the cost terms of \(C(\mathbf{x})\) do not have an influence on it. In a real system, \(\Psi\) is a consequence of the sale price and all the productive and logistic costs, including the ones formulated in the equipment model. However, if the model had to consider \(C(\mathbf{x})\) to obtain \(\Psi\), other costs, such as logistics, management and raw material costs should be considered. It is worth remarking that the same joint \(C(\mathbf{x})\) and \(P(\mathbf{x})\) optimization approach was applied to the optimization of preventive maintenance in [9]. The cost function \(C(\mathbf{x})\) is defined as the sum of the maintenance costs, costs related to production speed loss and quality costs. It is calculated using Equations (18), (19), (20), (22), (31) and (35):  
\[C(\mathbf{x})=C_{cmt}+C_{pm}+C_{cm}+C_{u}+C_{sl}+C_{q}. \tag{40}\]  
The profit \(P(\mathbf{x})\) is calculated using (37). The vector of constraints, \(\mathbf{g}(\mathbf{x})\), is given by  
\[\mathbf{g}(\mathbf{x})=\left(C(\mathbf{x}),P(\mathbf{x}),U(\mathbf{x})\right), \tag{41}\]  
which means that \(m=3\), and \(\mathbf{L}=\left(C_{i},P_{i},U_{i}\right)\), \(C_{i}\), \(P_{i}\) and \(U_{i}\) being the cost, the profit