# Jointly Production and Correlated Maintenance Optimization for Parallel Leased Machines
## 1 Introduction
(2016) considered the quality aspect of the system in order to integrate the lot-sizing problem to the preventive maintenance scheduling. A system composed by \(M\) defined machines was considered and the production and maintenance schedules were determined using a genetic algorithm. The system undergoes quality inspections which are accompanied with adjustment and calibration activities to confirm that the machine will start the next production interval in its normal state. If inspection detected that a shift has occurred (the machine is in a degraded state), all items produced on the related machine interval need to be checked, and defective items should be separated. [PERSON] et al. [PERSON] et al. (2016) studied a production system with a decreasing production capacity over time and determined its production and maintenance plans. [PERSON] et al. (2016) studied a multi-product single machine system, which is subject to random failures. They minimized the sum of the total production and maintenance costs related to inventory, backorder, production, set-up, preventive maintenance, and minimal repair under demand satisfaction and machine capacities constraints. Some works have dealt with the group maintenance scheduling optimization; we can mention [PERSON] et al. [PERSON] et al. (2016) which is one of the first works to deal with this type of problem. The authors considered a system composed by \(n\) independent identical machines which fail separately and developed a nomograph for all machines which allows determining the time for repair as well as the total maintenance cost. The group maintenance activity is performed when a certain time interval elapsed. [PERSON] et al. [PERSON] et al. (2016) considered two time-intervals (0, T) and (T, T + W) during which failures are either removed by minimal repairs or removed by replacement or are left idle. During the first interval (0, T), if a unit fails at age \(y\) it is either replaced by a new one with a probability \(p(y)\) or it undergoes a minimal repair with a probability \(1\)-\(p(y)\). During the second interval (T, T + W), no replacement is performed until the group maintenance; failures, when occurred, are either removed by minimal repairs or are left idle. The group maintenance strategy in this work is performed at time T + W or upon the \(k\)th idle whichever comes first. More recent works have also dealt with the group maintenance optimization problem; [PERSON] et al. [PERSON] et al. (2016) studied the impact of opportunistic maintenance on the effectiveness of condition-based maintenance. They considered a system composed by three components mounted in series and varied few elements such as the number of components, the length of the opportunistic zone and used a dynamic grouping in which group maintenance activity is carried out only for the non-failed components that are in the opportunistic zone. [PERSON] et al. [PERSON] et al. (2016) developed a group maintenance policy for parallel machines. Their model allows determining jointly the optimal maintenance frequencies, the optimal positions of the maintenance activities and the optimal job sequence. An algorithm in which the authors fixed the maintenance frequencies is provided in order to solve the problem. In the work of [PERSON] et al. [15], the authors used a genetic algorithm in order to develop an optimum group maintenance schedule of a water-distributed network. A judgment matrix based on the adjacent geographical distribution of the pipelines is used to control the searching space of the maintenance grouping optimization model. [PERSON] et al. [16] developed a dynamic maintenance grouping approach for a multi-components system using two optimization algorithms (genetic and multifit). The developed approach allows obtaining an optimal maintenance schedule taking into account the availability of repairmen constraints. [PERSON] et al.