# Jointly Production and Correlated Maintenance Optimization for Parallel Leased Machines
## 2 Problem Description and Industrial Context  
As mentioned above, in the present work we focus on a parallel manufacturing system composed by several identical machines. We referred, in this study, to a real case of a company specialized in metal and steel parts manufacturing. When the pieces ordered by customers are of a complex shape, the technology used by this firm is based on laser machining tools used to cut materials (steel in most cases) with extreme precision in order to obtain a high-quality product. Due to the expensiveness of laser tools, and since the company does not use this type of equipment regularly, the managers use to lease these machines in order to satisfy some customers' orders. Given the fluctuation of demand, the number of the leased machines does not remain constant and may vary from one period to another. Also, given the complexity of the technology used by these machines, maintenance activities are performed by specialists, generally from the lessor company. Leasing contracts are generally concluded for one year, or more, depending on the use of the equipment. The leasing fees are paid monthly at the debut of each period. Group maintenance activity for this kind of systems is then of a huge importance since it allows the lessee to avoid unavailability of repairmen, which is most frequently encountered in the case of an individual maintenance strategy. Inspired from [PERSON] et al. [21], and in order to take into account production planning while minimizing maintenance cost, we put emphasis on degradation function of each available machine according to its production rate. We consider the production system that has to satisfy a random demand \(d_{k}\) of one type of product. This demand fluctuates according to a normal distribution with mean and variance given respectively by \(\mu_{k}^{d}\) and \(\sigma_{d}^{2}\geq 0\). The number of machines to be leased depends on the customer's demand and varies from one production period to another. Processing time for each machine is also variable: in fact a machine can operate for a maximum duration in regular work time denoted by \(X_{\max}^{N}\), once exceeded this value, a machine can operate up to \(X_{\max}^{S}\) of extra time work (overtime). A periodic group maintenance strategy is performed to all machines in order to reduce the probability of failure. We put emphasis on the complexity of jointly determining production and maintenance plans for this kind of systems in which many variables are stochastic and strongly dependent. We do make the following assumptions:  
* All machines are considered new at the beginning of the time horizon. * All machines have the same production rate per time unit (same quantity of product for a same period of processing). * The number of machines per period is bounded as follows: \(m\leq M_{k}\leq M\). * Products are purchased by the customers through an inventory. The inventory level is calculated at the end of each production period. * Maintenance activities are not included in the lease contract. The lessor has to pay experts in order to perform preventive and corrective maintenance actions. * All machines are preventively maintained at every preventive maintenance interval and preventive maintenance activity renews the system (after each preventive maintenance action, the equipment is in \"as good as new\" state). If any machine fails within the preventive maintenance interval, corrective action with minimal repair is done and the machine is restored to the as-bad-as-old state. * Repair time (preventive activity) is constant and is the same for all machines. * Corrective maintenance activities' duration (minimal repair) is negligible compared to preventive maintenances. * Since production is off during preventive activities, the time taken for repair will be incremented to processing time of any machine that did not work for extra time, if any, or to the machine that worked the less otherwise in order to hedge against inventory shortage.