Section: New Results
Reliability and maintenance of systems
Participants : Kondo H. Adjallah, Valerio Boschian, Mohamed Dahane, Sofiene Dellagi, Sophie Hennequin, Nidhal Rezg.
Generally, this theme concerns the optimization of integrated maintenance policies for manufacturing systems. New integrated maintenance policies are developed and optimized in order to prove its performance according to the traditional policies existing.
Nowadays, the hard competition between the enterprises brings us to revise the currently adopted strategies of production and maintenance. In fact, the satisfaction of the client in time became a difficult spot since demands are random. In this context, subcontracting is defined as the procurement of products or services from external sources. It is justified by many reasons like cost reduction, production flexibility improvement, skill/resource shortage or proximity to markets. In this context our works deal with problems of management of subcontracting services for subcontractor enterprises. We study the constraint of subcontracting under a combined approach of maintenance management and production control for production system under a supplier - customer relationship with a "principal costumer". In order to increase the exploitation of the production capacity this system provides subcontracting services to another customer called "contractor", under a subcontractor - contractor relationship. In [14] , we considered the profitability of subcontracting activities for subcontractor companies. The subcontracting imposes periods of unavailability of the production unit in order to periodically perform subcontracting tasks. We analytically show the conditions under which subcontracting is profitable, based on a given policies of maintenance, production control and assignment to subcontracting. We discuss profitability of two cases of subcontracting constraints: occasional and long-term relationship. In the first case we integrate the subcontracting task in the subcontractor plan without changing the decision variables values (maintenance interval and stock level). The second case imposes a new optimization of decision variables. For this case, we investigated the problem of the unforeseen extension of the subcontracting duration and its impact on the generated costs of maintenance, inventory and shortage. In the same context, we study analytically the importance of the beginning instant of subcontracting tasks in [13] , [31] . This study proves that the optimal instant to allocate machine to subcontracting is exactly the moment when the capacity of the stock is reached.
Dealing with the development of new integrated maintenance policies under the subcontracting constraint, another subcontractor aspect is studied. We started by considering two subcontractors which have different service cost and availability rate. The strategy consists at relaying on one of the two subcontractors and switching to the other at certain dates. This strategy is justified and optimized analytically in order to determine the optimal preventive maintenance date for the manufacturing system and the optimal switching dates between the two subcontractors [15] . Furthermore, we proposed a prospect related to this study. The goal is the development of an optimal switching strategy between several subcontractors. We considered the building of a safety stock contrarily to the just in time strategy.
Considering only one subcontractor which satisfies a part of the demand in order to take the possibility of building a safety stock, a new research direction is explored. The objective is to manage simultaneously the integrated maintenance-production plan by determining the optimal safety stock level and the optimal preventive maintenance dates while minimizing the average total costs (production, maintenance, inventory and demand loss) [54] .
More then, it's easy to see that the manufacturing system degradation evolve according to the production rate, Thus, for a given randomly demand, we established an optimal production plan which minimizes the average total holding and production costs. Using this optimal production plan and its influence on the manufacturing system failure rate, an optimal maintenance scheduling which minimizes the average maintenance cost has been established [34] . We have solved analytically the problem using the linear quadratic Gaussian (LQG) control theory [55] [35] .
Another study deals with the combination between production and maintenance plan for a manufacturing system satisfying a random demand subjected to random failures of the manufacturing system. The aim of this study is to establish an economical production/maintenance plan minimizing the average total cost and to illustrate the significant influence of the production rate on the manufacturing system degradation [33] . Moreover, [53] considered an extended version of the problem by taking into account the demand rejection [53] . Two studies investigating new intelligent integrated maintenance and production or service strategies, dealing with complex reliability problems are presented in [51] .
In the same context of the dependence between production and failure rates, we considered the problem of production control when production rates depend on the failure rate. The objective is to determine the production planning over a finite horizon minimizing the generated costs (inventory, production and maintenance) [32] .
Another research work uses the prognosis concept to develop a set of maintenance policies which integrate the schedule of maintenance missions performed by navy ship [8] . The prognosis result is based on the evaluation of the degradation law, i.e. by taking into account the variations of the environmental and operational conditions. The aim is to determine the optimal business plan (choice and scheduling missions) combined to an optimal maintenance plan. To model the failure law, we established a relationship between the times between failure (from feedback) and risk factors of each navy mission. Also, we have developed two preventive maintenance policies based on a dynamic failure law for a finite planning horizon. The first preventive maintenance policy is sporadic the aim is to determine the number of maintenance activities to perform and the choice of the mission that must be followed by preventive activity, in order to minimize maintenance costs . The optimal solutions are obtained by the extended great deluge algorithm [45] . The second preventive maintenance policy is periodic. With a numerical procedure, we established the optimal number of maintenance actions [23] [44] . In a recent research work, based on a finite time horizon, we have extended this work by considering a production system, which must satisfy a set of requests. The objective is to determine the best production plan to minimize both the holding costs and maintenance costs [60] .
Another research program focuses on the development of models considering maintenance imperfections. Indeed, most preventive maintenance models assume that the system is restored to as good as new at each maintenance actions and consider the intervention time as negligible. Hence, the system may not be restored to as good as new immediately after the completion of maintenance action. Our approach is based on a fuzzy logic model which allows taking into account imperfections. These later are essentially due to technician's experience, the level of complexity of the restoration, and the time taken by maintenance actions. After a maintenance, the machine returns to an age between as good as new and as bad as old. Fuzzy logic is preferred over crisp logic because it is relatively easy to implement in this situation considering that the human factor is hardly interpreted by analytical methods because of its unpredictable nature. Simulation-based optimization is used to have a more reactive and accurate tool for parishioners. By taken into account the impact of the imperfections due to human factors, the period for the preventive maintenance, which minimizes the expected cost rate per unit of time or maximizes the availability of the system, is evaluated by a simulation-based optimization [18] .
Uncertainty in a machine reliability is commonly present, but not always completely modeled. In general, statistical laws like Weibull, normal or exponential probability distribution are used, but they only model randomness of the reliability and not the fuzziness. Additionally, the systems parameters like the costs and durations of maintenance, are used to be taken like real constants without taking into account the unsharpness and imprecision of these parameters. In our work, the concept of fuzzy random probability distribution is also introduced to the maintenance model to cover in a wider level the uncertainty of the reliability of the machines. This method gives a better representation of the randomness as well as the fuzziness of the reliability. The systems parameters are modeled not as real constants but as fuzzy constant numbers to introduce their unsharpness. Finally the goal is to minimize the total cost of maintenance per unit time, by finding the optimal age at which preventive maintenance must be performed in a fuzzy environment [28] .
For maintenance decision support, we are working on data and information collection, in terms of production systems' health information chain modeling and data analysis, and in terms diagnosis and prognosis tasks scheduling and optimization [20] . These investigations aim at equipments', maintainability and reliability data quality enhancement, health monitoring, risks evaluation, maintenance actions decidability studies and lifecycle management [26] , [27] . This is a new topic of investigation that was recently initiated to address e-maintenance problems arising from the implementation of advanced Information and Communication Technologies for the maintenance decision support.
