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In this paper, we consider a Markovian model for a two-echelon inventory/return system. The system consists of a supply plant with infinite capacity and a central warehouse for inventory and returns. There are also a number of local warehouses which are also capable of re-manufacturing of returns to products. To obtain a high service level of handling inventory and returns, lateral transshipment of demands is allowed among the local warehouses. The objective is to minimize the total expected operating cost by choosing the maximum inventory level at the local warehouses. We present some theoretical results for the proposed model. Numerical examples are also given to demonstrate the performance of the system.

The efficiency of product/service delivery is one of the major concerns in many industries including re-manufacturing industry. Since customers are usually scattered over a large regional area, a network of locations (local warehouses) for inventory of products and handling of returns is necessary to maintain a high service level. In our study, Lateral Transshipment (LT) is allowed among the local warehouses to enhance the service level. LTs are also very practical in many organizations having multiple locations linked by computers. Substantial savings can be realized by the sharing of inventory in the local warehouses (Robinson, 1990). A number of research publications have been appeared in this area. Kukreja et al. (2001) developed a single-echelon and multi-location inventory model for slow moving and consumable products. Aggarwal and Moinzadeh (1994) studied the emergency replenishments for a two-echelon model with deterministic lead time. Ching (1997) considered a multi-location inventory system where the process of LT is modeled by Markov-modulated Poisson Process (MMPP). Both numerical algorithm and analytic approximation were developed to solve the steady state probability distribution of the system (Ching, 1997, Ching et al., 1997 and Ching et al., 2013). Alfredsson and Verrijdt (1999) considered a two-echelon inventory system for service parts with emergency supply options in terms of LT and direct delivery. Olsson (2009) proposed a single-echelon, two-location system with LT and derived the optimal policy by using stochastic dynamic programming. Paterson et al. (2012) proposed a quasi-myopic approach to the development of a strongly performing enhanced reactive transshipment policy. Service levels are improved while the aggregate costs incurred in managing the system can be significantly reduced. Tiacci and Saetta (2011a) proposed a ‘preventive’ transshipment policy, which means that transshipment occurs if the inventory level of a warehouse is lower than a predetermined threshold. Tiacci and Saetta (2011b) then extended the policy by considering stock redistribution among warehouses. Minner and Silver (2005) compared two transshipment policies for two warehouses analytically: (i) never transship and (ii) always transship when shortage occurs at one location and stock available at another. Hochmuth and Köchel (2012) suggested using simulation optimization for the control problem of multi-location inventory systems with LT. Summerfield and Dror (2012) discussed several two-stage decentralized inventory problems using the same framework. Graphical taxonomy trees were depicted andstochastic programming models were also applied. Readers are referred to Paterson et al. (2011) for a recent review on LT. Recently, studies on multi-echelon inventory systems with returns are getting more attention. Hoadley and Heyman (1977) considered a one-period multi-echelon model that allows LT between stocking points at the same multi-echelon level. Returns are handled at the first echelon and the optimal initial stock level at the first echelon was obtained. Lee (1987) considered a similar situation with continuous monitoring on inventory levels. Korugan and Gupta (1998) considered a two-echelon inventory system with return flows, where demand and return rates are mutually independent. Min et al. (2006) proposed a nonlinear mixed-integer programming model and a genetic algorithm that can solve the multi-echelon reverse logistics problem involving product returns. Mitra (2009) considered a two-echelon inventory system with returns. The system consists of one depot and one distributor only. Both deterministic and stochastic demand rates were considered. He and Zhao (2012) proposed a multi-echelon supply chain with both demand and supply uncertainty. Unsold products can be returned from retailers to manufacturer. Vercraene et al. (2014) studied the coordination of manufacturing, remanufacturing and returns acceptance in a hybrid manufacturing/remanufacturing system and proposed several heuristic control rules. Reviews on multi-echelon inventory research with returns can be found in Dekker et al. (2004) and Fleischmann and Minner (2003). However, most of the literature considered handling returns at the upper echelons but few considered handling returns at the lower echelons. One possible approach to study multi-echelon system is using Markovian model. Gross et al. (1993) obtained the steady-state probabilities for large multi-echelon reparable item inventory systems without considering LT. Saetta et al. (2012) presented an analytical approach for Markovian performance analysis of a serial multi-echelon supply chain without considering returns. Some recent studies on applying Markovian model in inventory control problems are Liu et al. (in press) and Sebnem Ahiska et al. (2013). We remark that none of the studies considered a Markovian multi-echelon model for handling both returns and LT. In this paper, we propose an inventory/returns model based on the frameworks and analyses in Alfredsson and Verrijdt (1999) and Lee (1987). The model of the system consists of a supply plant with infinite capacity, a central warehouse and a number of local warehouses with re-manufacturing capacity. Here we consider a queueing model for a two-echelon inventory system. Queueing model is a useful tool for many inventory models and manufacturing systems that can assist with long-run decision, see for instance Buzacott and Shanthikumar (1993), Ching et al., 1997, Ching et al., 2003 and Ching et al., 2013 and Ching (2001). The rest of the paper is organized as follows. In Section 2, we propose a two-echelon inventory model. In Section 3, we consider a Markovian queueing model for the local warehouses. In Section 4, we consider an aggregated model for the central warehouse. We then give the expected operating cost function in Section 5 and a numerical example is given in Section 6. Finally, concluding remarks are given in Section 7 to conclude the paper.

In this paper, we present a two-echelon inventory/return system. The system consists of a supply plant with infinite capacity and a central warehouse for inventory. There is also a number of local warehouses having capacities for re-manufacturing of products. Lateral transshipment of demands is allowed among the local warehouses. We develop a Markovian queueing model for the local warehouses. We then propose an aggregated inventory model for the central warehouse and the local warehouses. Fast algorithm for solving the steady state probability distribution of the inventory level is also developed. We obtain the optimal inventory level at the local warehouses. Numerical example and sensitivity analyses are also provided. The followings are some possible research issues which may be addressed in the future. Firstly, the current work can be extended by considering a general multi-echelon inventory system instead of a two-echelon system. Secondly, we only consider constant and common demand and return rates for each local warehouse. All the local warehouses are identical in terms of cost structure and capacity. It is interesting to investigate a system consisting of non-identical local warehouses. Thirdly, the demand is satisfied on the basis of first-come-first-served. This constraint can be further relaxed. Finally in our current model, the excess demand is lost. A system which is capable of handling backlog may be considered in future research.