# Jointly Production and Correlated Maintenance Optimization for Parallel Leased Machines
## 1 Introduction  
In classical scheduling strategies, it is generally assumed that the system is available to produce at all times, which is wrong since machines are subject to failures and breakdowns. Finding the optimal production plan taking into account the maintenance strategy is then necessary to increase the economic returns of a firm and to handle backorders. In the frame of production and maintenance optimization, [PERSON] et al. [1] presented a simultaneous optimization model of production flow and preventive maintenance actions taking into account the interaction between both services. The authors considered a system composed by a single unit producing one type of product in order to satisfy a constant demand. They proposed a model for the maintenance strategy based on the level of the inventory and the age of the machine. They minimized firstly the total maintenance cost in order to obtain the optimal age of preventive maintenance actions and then established the expression of the total costs associated with the holding stock and unsatisfied demands. The combination between both costs allows minimizing a global cost function integrating simultaneously the maintenance, the inventory as well as the unsatisfied demands costs in order to obtain an optimal security stock. So many works have treated the joint optimization of production and maintenance strategies. Some of these works were devoted to study the case of restricted and simple systems composed by only one machine while others were interested in more complex systems composed by several machines whose number is fixed and known during the production horizon. Finding the optimal production and maintenance policies, by minimizing the total costs, becomes then more complex when the number of machines is variable. [PERSON] et al. [2] developed an optimization model for a production system composed by one machine producing with a constant cadence in order to satisfy the demand. Sincethe machine is prone to random failures, corrective and preventive maintenance actions are performed at periodically dates T, 2T.. They proposed the construction of a security stock S to meet the demand during the period of machine shutdown. The objective of this work is to determine simultaneously the optimal periodicity of preventive maintenance actions \(T^{\star}\) as well as the optimal stock level \(S^{\star}\) which minimize the average total cost of maintenance, inventory and unsatisfied demands. [PERSON] et al. [PERSON] et al. (2016) presented an analytical model for a production line composed by \(n\) machines which have an identical production rate to satisfy a constant demand. Simulation and genetic algorithms are the approaches used to determine the optimal preventive maintenance dates and stock levels. Some authors treated the coupling between production and maintenance considering new constraints like subcontracting ([PERSON] et al. (2016) and [PERSON] et al. (2016)). [PERSON] et al. (2016) developed new maintenance policies while coupling maintenance and production under subcontracting constraint. To satisfy the customer demand, subcontracting is sometimes needed in order to hedge against shortage. Concerning reliability, the authors considered a constant failure rate for the subcontractor's machine and an optimal maintenance planning for the principal machine by proposing an improved maintenance policy which takes into account the constraint of subcontracting. [PERSON] et al. (2016) considered an order-dependent failure and developed an iterative method to solve the problem for a single-machine system. The model developed by [PERSON] et al. [PERSON] et al. (2016) optimizes jointly the production and the maintenance schedules for a single machine system taking into account both the quality robustness and the solution robustness. An approach based on a three-phase heuristic is used in order to find the optimal solution. [PERSON] et al. [PERSON] et al.