# Jointly Production and Correlated Maintenance Optimization for Parallel Leased Machines
## 1 Introduction
[17] studied the case of a multi-components system and proposed an age-based group maintenance policy that provides an optimal group maintenance time \(T^{*}\) by minimizing the average long-run maintenance cost. Concerning joint optimization of production and group maintenance policies, [PERSON] et al. [18] considered a series system of machines processing different types of jobs. They used a genetic algorithm in order to find the optimal group preventive maintenance policy (the optimal preventive maintenance interval) as well as the assignment of jobs on machines. Results are compared to an individual preventive maintenance policy in order to show the effectiveness of the used approach. In the work of [PERSON] et al. [19], the author considered a manufacturing system composed by a given number of cells consisting of a given number of machines which have to satisfy a random demand. The impact of preventive maintenance on the manufacturing system's performance is investigated. Static and dynamic environments were also taken into account in order to study the performance of the manufacturing system. The author used a multi-agents architecture to illustrate the scheduling problem. A discrete event simulation using Arena platform was used in order to simulate the manufacturing environment. The results found by the author show that changes and working time uncertainty have a significant effect on the performance of the manufacturing system since they lead to major benefits when a preventive maintenance policy is used. [PERSON] et al. [20] proposed a model for a group preventive maintenance optimization using the failure effects analysis. The authors considered a workshop composed by several machines which have two types of state: either good or failure state. For each state, they determined the average of failure effect using the Monte-Carlo simulation taking into account the different considered variables. The authors have considered a non-negligeable time for repair which leads to production losses. All aforementioned works in the frame of group maintenance strategy considered either a series system where failed equipment leads to a whole system breakdown or a fixed number of components (machines) in order to solve this kind of problem. However, and in the case of leased equipment, the number of machines to use can vary from a period to another according to the customers' demand. Also, few works have studied the case of a parallel machines system, in which, even if a machine fails, the system continues to perform its production process, unlike the case of a series system where a failed machine leads to the whole system breakdown. The problem becomes harder to resolve when the number of machines and the customer's demands are dynamic from a production period to another and depend on each other. This paper shows that it has a novelty and originality relative to this type of problem since it investigates the case study of a production system composed by a variable number \(M_{k}\) of identical parallel machines under lease contract. \(M_{k}\) can vary from a period to another according to a random customer's demand. Each machine has a unique processing time in regular work time; if any machine achieves certain duration in regular work time, then it can perform an extra work time in order to satisfy the random demand. The degradation rate of these machines is correlated with the production during a finite horizon. The other added value of this study consists of the establishment of a new method of correlated group preventive maintenance policy based on the gravity center approach (GCA). We put emphasis on the originality of this work since it is characterized by the use of new approaches and an analytical study taking into consideration the influence of the production rates on the system's degradation degree. Minimizing the cost of production and correlated preventive maintenance provides the optimal number of machines to be leased, the quantity to produce by each one and the optimal interval of group preventive maintenance.