# Jointly Production and Correlated Maintenance Optimization for Parallel Leased Machines
## 4 Optimization Approach
### Gravity Center Approach  
In the literature, the GCA is frequently used in logistics systems in order to study the location selection of single distribution center [24; 25]. It is used to minimize the sum of depot operating cost and routing costs [26].  
We adopt this approach in the following part and apply it to our manufacturing system in order to find an optimal preventive maintenance plan. Our methodology consists of determining an optimal maintenance interval \(T_{n}\) at which we can group several preventive maintenances so that \(\sum\limits_{i=1}^{M_{k}}A_{i}.T_{n}T_{i}=\overrightarrow{0}\). In order to find the value of \(T_{n}\), we use the following expression used to find the center-of-gravity of different points:  
\[X_{n}=\frac{\sum\limits_{i=1}^{M_{k}}p_{i}/d_{i}}{\sum\limits_{i=1}^{M_{k}}p_{ i}/d_{i}} \tag{10}\]  
with \(p_{i}\) is the weight factor assigned to each machine \(i\) and it corresponds in our case study to the average number of failures for each machine \(i\); \(x_{i}\) is the maintenance interval of each machine for an individual maintenance strategy; and \(d_{i}\) is the distance (in time unit) between the group maintenance location (interval) and the individual maintenance location for each machine \(i\). \(d_{i}\) can be described as follows:  
\[d_{i}=\left[\left(T_{n}-T_{i}\right)^{2}\right]^{1/2} \tag{11}\]  
The choice of the average number of failures as the weight factor is justified by the fact that the degradation of each machine \(i\) is strongly correlated to its production rate (the quantity produced): more the machine works (\(p_{ik}\) is important), more its degradation is high (average number of failures). So, it would be preferable to bring its maintenance closer to an optimal new date (or to bring it further otherwise). The expression of \(T_{n}\) is as follows:  
\[T_{n}=\frac{\sum\limits_{i=1}^{M_{k}}A_{i}.T_{i}/d_{i}}{\sum\limits_{i=1}^{M_{ k}}A_{i}/d_{i}} \tag{12}\]where the average number or failures for each machine \(i\) can be expressed as follows:  
\[\begin{array}{c}A_{i}=\sum\limits_{j=0}^{N-1}\left[\sum\limits_{i=In(j\times \frac{T_{n}}{M})+1}^{\left[n((j+1))\times\frac{T_{n}}{M}\right]}\int\limits_{0}^ {A}\lambda_{ij}(t)+\int\limits_{0}^{(j+1)\times T_{i}-In((j+1)\times\frac{T_{ n}}{M})\times A\Omega}\lambda_{In((j+1)\times\frac{T_{n}}{M})+1j}(t)dt\\ +\int\limits_{0}^{(In((j+1)\times\frac{T_{n}}{M})+1)\times A\Omega-(j+1)\times T _{i}}\frac{n_{In((j+1)\times\frac{T_{n}}{M})+1}}{U_{\max}}\times\lambda_{n}(t) dt\right]\end{array} \tag{13}\]  
In order to find the optimal value of \(T_{n}\), we find at first a value \(T_{n}^{0}\) using the following expression:  
\[T_{n}^{0}=\frac{\frac{M_{n}}{M}A_{i}T_{i}}{\sum\limits_{i=1}^{A}A_{i}} \tag{14}\]  
Once found, \(T^{0}\) is replaced in Formula (11) in order to calculate \(T_{n}\). Then, a new value \(T_{n}^{1}\) is found which will be replaced in the same formula in order to find another value \(T_{n}^{2}\), and so on, repeated until the two identical results show up within iterations. The value found corresponds to the optimal interval of group maintenance activity for the leased system.  
Figure 2 describes the difference between individual and group preventive maintenance strategies for parallel leased machines system