## 1 Introduction  
Nowadays, the rapid changes of the global economic conditions, demand for a high-quality product at a low price, the requirement for a varied assortment, and that of reducing delivery time have imposed even more constraints on manufacturers to remain competitive. Moreover, customers are becoming more exigent, while production systems are getting more complex and subject to multiple uncertainties [1]. Therefore, the need to develop new industrial strategies, including maintenance, production, and quality, have become crucial to reduce the costs and satisfy customers' multiple requirements. Production planning optimization and maintenance strategy have always been a great challenge for industrial companies. The use of a new role of maintenance function to better support value creation, by contributing the economic dimension and considering the environmental and social aspects, are the new trends by [PERSON] et al. [2]. Indeed, they presented a new approach to transform and solve the indicator merge of initial maintenance to a new synthetic index, taking into account the interactions between criteria, in order to measure the maintenance sustainability level and to improve the results of economic, social, and environmental factors of the system of maintenance. However, the product or service quality decreases when the process is affected in terms of time, total cost, and constraints, or influencing the quality of the final product or service of customers. In this paper, we will use the six-sigma method to increase the products and processes quality, thereby guaranteeing products meet customer requirements and ensure economic sustainability. The main objective is to reduce the process variation and the related high deficiency rate by using, for example, the control chart based on the \"statistical process control\" (SPC) methodology. In this context, the six-sigma method will employ the Define, Measure, Analyze, Improve, and Control (DMAIC) methodologies [3]. The optimum plans of production and maintenance, including inventory, delivery, delay penalty, and quality, create a ranked decision-making procedure. Production system failure and malfunction are among the causes of non-compliant items and profit loss consequences. Therefore, the combination of statistical process control of quality, maintenance strategy, and optimal production planning will be an unfailing solution that minimizes costs and maximizes production system reliability. In this context, different works in the literature allowed the coordination of two of the fundamental functions: production, maintenance, and quality. It began with the integration of production and maintenance, then production with quality or maintenance with quality, and recently, most comprehensive trends have jointly dealt with all three factors. According to the service level requirement, this work's uniqueness determines the ideal combination of production, maintenance, and delivery amounts based on the control chart that identify the state of process (stable state or critical state) and the correlation between production and maintenance. Different authors studied various issues of production and maintenance. Firstly, we start by discussing the literature works of the production and maintenance problem. In this context, [PERSON] et al. [4] were one of the earliest authors to improve production in their paper \"Automatic transfer lines with buffer stock\". They considered the problem of integrated production to maintenance policies by analyzing the buffer stocks in improving process productivity. [PERSON] et al. [5], in their work \"Impact of delivery time on optimal production, delivery, and maintenance planning,\" decided at the same time the ideal production, delivery, and maintenance accounting delivery time, machine failure, and random demand, in addition to the withdrawal right.