برنامه ریزی یکپارچه تولید و تعمیر و نگهداری پیشگیرانه در سیستم های تولیدی رو به وخامت

The traditional production planning model based upon the famous linear programming formulation has been well documented. However, the integration of preventive maintenance planning in the same model is a recent problem. This paper proposes an extended linear programming model as a hybrid approach for computing the optimum production plan with minimum total cost. The dual objective problem of production planning and maintenance is treated into a mixed integer linear program. This program is not only considering cases of multi-lines, multi-periods and multi-items but also taking into account the deterioration of the lines. This deterioration is represented in the model as a reduction of production lines capacities in function of the time evolution. Maintenance operations are supposed to provide lines in an operational state as good as new, i.e. with a maximum capacity. Through the study of the models limits, it is shown that the proposed approach can deal with a broader range of problems than that of Aghezzaf and Najid (2008) [3]. An optimal relaxation technique based on the polyhedral theory is developed to improve the computational time and expand the limits of the proposed model. Also, a “Fix and Relax heuristic” is developed for complex problems. Their computation time and their difference are computed referring to the same lower bound and the same considerations as those presented by Aghezzaf and Najid. It is proved through more than 880 several simulations for each model with different capacities and different setup costs, that this approach can solve large size problems with moderate computational time and gap.

Within the field of production planning, the main goal is to meet the satisfaction of the demand for a few specific products. The most commonly used technique to solve these issues are based on a sound knowledge of the problem’s parameters, such as production cost, production order, etc. In fact, these parameters are subject to a number of uncertainties and rates of fluctuation during the production planning process. Therefore, some of these production plans are inexact or even irrelevent. In this case, the initial plan needs to be updated continually to consider the fluctuation of the quoted parameters. Then, this update becomes more and more crucial for a reactive planification [14]. The first authors to introduce how important is to consider the machines “reliability” as a parameter to elaborate a production plan are Older and Suri [11]. At first, the problem was treated and formulated as a stochastic process in which the failure moments as well as repairing ones were considered as a homogeneous Markov chain. Then, non-homogeneous Markov processes were elaborated in order to consider the machines capacities: Boukas [5] and Gharbi [7]. These papers are assuming that failure rate as well as control policy depends on machines life time. Thus it has been proved that machines tend to have more availability when a preventive maintenance plan is adopted. A number of studies had been done in these last two areas. Shapiro [15] and Pinedo [12] are overviewing all the relevent publications regarding the production planning issues. In the same context Sherif and Smith [16] and Dekker [6] came up with several studies on the subject of maintenance optimization models. However, almost all the previous published studies treat separately the two problems of production and maintenance scheduling. The two subjects were never integrated in the same model even if they were in direct correlation. Recently very few studies point the leverage of combining the two problems in a communal formulation. In this perspective, Graves and Lee [8] presented a scheduling model combining the two approaches and considering the case of only one machine and one maintenance operation on a time scale. The aim was to minimize weighted completion time of jobs. Lee and Chen [9] have extended the problem to multi machines. Our initiative belongs to this general context that justifies formulating an approach combining the production and the maintenance planning for a parallel machines system. This article comes in addition to and capitalises on the two previous publications of Aghezzaf and Najid [3] and [2], in which the objective function is to minimize the total cost of the two core subjects combined. Through their work not only the maintenance policy has been considered as periodical but also any machine failure would inherently cause a capacity fall. bf In extension to previous works, Najid et al. [10] have proposed a mixed-integer linear program in which the demand shortage and the reliability of the production line are taken into account. A Lagrangian-based heuristic procedure for the joint production planning and maintenance problem is proposed by Alaoui-Selsouli et al. [4], where the maintenance is considered flexible. Recently, Zhao et al. [18] have taken into account the order-dependent-failure (ODF) and proposed an iterative method to solve the problem on single machine. Qin et al. [13] have solve the production planning problem with finite and infinite horizon by considering a fuzzy system case. In this article, mathematical models are proposed for the simultaneous resolution of maintenance and production planning in a system composed of several production lines and several products on a finite horizon composed of periods of time. First, a separate study is proposed for production planning and maintenance planning. In this first part, all the assumptions and details of costs and capacities computation are presented. In Section 2.4, the optimal resolution proposed by Aghezzaf and Najid in [3] and the resulting algorithm PCPMAN are remembered and a more compact formulation, called the PCPMCYY model is proposed. Using simulations, it is shown that this reformulation (PCPMCYY) allows a gain in time of resolution. In Section 2.5, the objective is to reduce the computation time for the research of the optimal solution. To do this, it is shown with the “Theorem of integer solution optimization” that the relaxation of production variables does not influence the integrity of the optimal solution of the PCPMCYY model. Using simulations, and thanks to this relaxation, the computation time is reduced compared to the reformulation PCPMCYY. In the next section, a heuristic “Fix and Relax” is proposed. This heuristic is based on the principle of mobile time window studied by Absi and Kedad-Sidhoum [1] for the ”lot sizing” problem. Finally, in the last section, experimental simulations allow to compare the computation time and gaps between the lower bound between the Fix and Relax heuristic and the Lagrangian relaxation heuristic proposed by Aghezzaf and Najid in [3].

Through this publication, the model formulation of the linear programming in mixed numbers related to the dual problem of production and maintenance scheduling proposed by Aghezzaf and Najid was improved. A new formulation was derived which is more compact and able to tackle problems with more complexity. In the second part related to improving the model, a relax of the View the MathML sourcexitj and IitIit production variables was proposed, justified by the integer solution optimality theorem. The purpose of this was to reduce the time of computation for average size problems. Nevertheless, these exact methods are not relevant to tackle larger size problems. According to this reason, in the last section of this publication, the existing fix and relax heuristic was adapted for the lot-sizing problem to our case. Based on the realized simulations, it has been proved that variable observation window kind heuristics are showing excellent results compared to Lagrangian Relaxation. Further developments to the current publication could focus on the observation window size effect and also step size effect. Indeed, in this document, these two parameters are considered constants and the same all over the scheduling horizon. This choice is based on Absi and Kedad-Sidhoum’s [1] studies showing that an observation window size of three units of time with a single unit of time step shows good results. These studies, however, were run for lot-sizing problems. Also, another improvement could be considered: Adapting the Fix and Relax heuristic to a real time scheduling. Indeed, this approach would bring the advantage of being able to change data and parameters for the time scale following the observation window as opposed to other methods where elaborating production and maintenance plan can only be settled for the full observation horizon. Any unexpected change means a reschedule on the full horizon.