## 1 Introduction
[PERSON] and [PERSON] [14] have presented the case for using a carbon tax to control carbon emissions, and have illustrated the implications for the UK manufacturing. In addition, they have computed a total abatement cost curve corresponding to different levels of emissions' curbing to see whether there are critical points at which costs rise quickly. [PERSON] et al., [15] have developed a mathematical programming model of a flow shop-scheduling problem that considers peak power load, energy consumption, and associated carbon emission in addition to cycle time. The objective is to determine the optimal scheduling that increases the energy consumption and the carbon emissions. [PERSON] and [PERSON] [16] have proposed an optimal design for a manufacturing/remanufacturing system that sorts worn products into three quality levels. The authors have determined optimal production decisions regarding new and remanufactured products while considering carbon tax policy. As in a real case, the authors have considered that the carbon emissions from producing remanufactured products is different from those of new ones. Indeed, the benefit of the remanufacturing is that the worn products are recovered and reused. In addition, the carbon emission from producing remanufactured products is lower than that for producing new ones. More recently, [PERSON] et al., [17] have considered a manufacturer who produces new parts in the first period and makes new and remanufactured parts in the second period under carbon tax policy, where the tax price differs over the two periods. The authors have determined optimal manufacturing and remanufactured plans that reduce the carbon emissions when the carbon tax varies. [PERSON] et al., [18] have considered a supply chain network constrained by a stringent mandatory carbon cap, the purpose of which is to examine how stringent carbon regulations and operational decision modes jointly influence the profitability and emissions control of the system. The authors have shown that only when the mandatory cap is set in intermediate level rather than excessively mild or tight can it be effective to balance system profitability and emission control well. Indeed, the mandatory carbon policy is very efficient to curb the carbon emissions; however, due to its sternness, this policy drives away the investors. Some European countries are periodically applying the mandatory carbon regulation. For example, the French government applies the mandatory carbon policy on transport and manufacturing activities for a defined period when a high pollution rate is revealed. To the best of our knowledge, there is no work in the literature which deals with a periodic mandatory carbon policy. In this paper, we will combine the carbon tax and mandatory carbon regulation. Indeed, we will consider the carbon tax as a permanent policy to reduce the overall emissions, and the mandatory carbon as a periodic regulation when the emissions exceed a tolerated threshold. One of the processes to curb carbon emissions in the industrial domain is remanufacturing; Remanufacturing, or refurbing, is considered as a great industrial process, within which worn-out products are restored to new products, providing benefits both environmentally and economically, while at the same time enhancing their image as environmentally responsible, since products are reused instead of being discarded, including curbing carbon emissions [19]. It is supposed that remanufactured products are profitable, at 45%-65% of the price of a new product. Furthermore, they are beneficial under emission regulation [20]. In general, the remanufacturing is a series of industrial processes: disassembly of the worn product on parts, restoring of the reusable parts, replacement of the parts if necessary, and reassembly of the parts [21]. Most of the published papers in the literature have assumed that remanufactured products have the same quality as new ones [22; 23; 24; 25]. However, in practice, the quality of new products is mostly higher than for remanufactured ones.