# Switching Control of Closed-Loop Supply Chain Systems with Markov Jump Parameters
## 1 Introduction  
With the development of productivity and the growing awareness of resource use and environmental protection, research on closed-loop supply chains has attracted more and more attention [1; 2; 3; 4]. In recent years, research on manufacturing and remanufacturing systems has attracted extensive attention. Industry practice shows that remanufacturing in closed-loop supply chains not only helps to reduce resource consumption but also helps to save costs and improve competitive advantages [5]. Because remanufacturing is profitable and environmentally efficient, many enterprises have established their own recycling and remanufacturing systems and voluntarily carried out remanufacturing activities, such as IBM, HP, BMW, Ford, Apple, Kodak, Xerox, Caterpillar, etc. [6; 7]. Therefore, consideration of the remanufacturing link in closed-loop supply chain systems [8; 9] is being paid more attention. Ref. [10] considers the supply chain model with a product remanufacturing link, Ref. [11] investigates the problem of effective channel design for closed-loop supply chain systems, and Ref. [12] determines productivity, remanufacturing rates, and disposal rates through a cost optimization approach. Ref. [13] applied the Pontryagin maximum value principle [14] and studied a linear model for optimizing production, remanufacturing, and abandonment strategies. In addition, a number of scholars have started to focus on closed-loop supply chain production-inventory control systems: Ref. [15] used optimal control ideas to study the dynamic capacity planning problem of closed-loop supply chain remanufacturing systems; [PERSON]. [16] applied the ideas of game theory to the network equilibrium problem of closed-loop supply chain systems; and [PERSON]. [17; 18] applied fuzzy logic ideas to study the product recovery strategy of closed-loop supply chains. It is worth noting that the above-mentioned literature mainly focuses on the static environment, while it is obvious that the static model is not sufficient to portray the dynamic characteristics of closed-loop supply chain systems such as demand fluctuation, production lead time, sales forecast, etc. Therefore, supply chain analysis based on a dynamic model has achieved some results [19; 20; 21; 22], among which, modeling the closed-loop supply chain as a kind of dynamic behavior of switching system has attracted great attention from many scholars [23; 24], and it is generally believed that a switching system with Markov jump parameters is an appropriate model to describe the closed-loop supply chain system. The robust production and maintenance scheduling problem is described in [19] as aminimax statistical control problem. The machine state process is modeled as a finite-state Markov chain whose generator depends on the rate of aging, productivity, and maintenance, modeling the demand rate as an unknown interference process. Ref. [25] studies the control problem of a closed-loop supply chain switching system with Markov jump parameters. Aiming at the uncertainty problem in the process of remanufacturing, based on the input lag control strategy, the Markov switching idea is applied to the controller design and performance analysis of the system. However, modeling the production-inventory model of a closed-loop supply chain system as a switching system with Markov jump parameters has rarely been reported [26], which is one of the main motivations for this study. The bullwhip effect is the phenomenon of demand fluctuation amplification in supply chain and is the most important performance indicator in supply chain operations [27; 28; 29]. The bullwhip effect exists in a supply chain and more extensive ERP, e-commerce, and other management systems, including modern logistics operation. It has important theoretical significance and wide application prospect to study bullwhip effect of a closed-loop supply chain system with a robust \(H_{\infty}\) control method. This paper studies the closed-loop supply chain production-inventory system of remanufacturing.