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This paper explores flexible service policies for an (r, Q) Markov inventory system with two classes of customers, ordinary and prioritized customers. When the on-hand inventory drops to pre-determined safety level r, arrival ordinary customers receive service at probability p. Firstly, the inventory level state transitions equation is set up. The steady-state probability distribution and the system's performance measures which are used for the inventory control are derived. Next, a long-run average inventory cost function is established and a mixed integer optimization model is set up. And, an improved genetic algorithm for the optimum control policies is presented. Finally, the optimal inventory control polices and the sensitivities are investigated through numerical experiments.

Inventory systems usually satisfy demands from more than one types of customer, each of which may possess respective characteristics, such as affordable price and quality of services. Typically, the variable demands of different customers result in various service priority. The type of priority inventory demand is classically categorized into booked orders and unscheduled orders. Booked orders, which are from long-term contracts and have much higher shortage lost, must be satisfied preferentially, whereas unscheduled orders, which are from the stochastic demand, may bear lower shortage cost and can be lost. The real-life situations and extensive implications drive us to consider the inventory system with two demand classes. The practices of inventory management face multiple classes of demands. Veinott (1965) considered a critical level policy for solving the problem of several demand classes in inventory systems. Nahmias and Demmy (1981), Dekker and Kleijn (1998) and Deshpande et al. (2003) also studied inventory control problems with different classes of customer. Hung et al. (2012) consider the dynamic rationing problem for inventory systems with multiple demand classes and general demand processes. They assume that back orders are allowed. The aim is to find the threshold values for this dynamic rationing policy. In the lost sales case, an important issue in the inventory systems is the inventory control policies of optimal inventory (Ha, 1997 and Dekker et al., 2002). Melchiors et al. (2000) derived a continuous review inventory system with lost sales and two demand classes. They proposed a formula for the total expected cost and presented a numerical procedure for optimization. But, they could not prove convexity of the cost function. Sapna Isotupa (2006) analyzed a similar model using exponentially distributed lead-times, and then established a long-run expected cost function. He proved that cost function is pseudo-convex in both parameters s/r and Q. Other scholars such as Berman and Kim (1999), Berman and Sapna (2001) and Schwarz et al. (2006) examined queueing-inventory systems over the last two decades. They investigated the behavior of service systems with an attached inventory. They defined a Markovian system process and then used classical optimization methods to find the optimal control policy of the inventory ( Krishnamoorthy et al., 2006 and Manuel et al., 2008). Recently, Zhao and Lin, 2011 investigated a queueing-inventory system with two classes of customers and found a priority service rule to minimize the long-run expected waiting cost. Ioannidis (2011) propose a simple threshold type policy for a two-class system in which the production, service, and back-ordering decisions are integrated. He proposed a simple threshold type heuristic policy for the joint control of inventories and backorders. Even though there are some models in the literature that incorporate two or multiple classes of demands considering possible lost sales for rejecting ordinary customers' demands, there is a lack of studies using more flexible service polices. This paper presents flexible service policies for an (r , Q ) Markov inventory system with two classes of customers. Three major differences from the literature are outlined here. Firstly, our paper introduces a priority parameter pp, which is different from the previous papers by Sapna Isotupa (2006) and Zhao and Lin (2011) on Markov inventory systems with two classes of customers. The parameter p(0≤p≤1)p(0≤p≤1) is used for controlling the application of priority. When ordinary customers arrive, the system makes a decision whether or not to offer service; when prioritized customers arrive, they are served in priority. If p=1p=1, there is no priority service in the inventory system. If p=0p=0, there is a strict priority service in the inventory system. In this case, our model is the same as Sapna Isotupa (2006). If 0<p<10<p<1, when the on-hand inventory drops to the safety level rr, arrival ordinary customers will receive service at probability pp. Secondly, our paper establishes a mixed integer optimization model with integer variables (r, Q) and real variable (p). We adopt a real coded genetic algorithm genetic named MI-LXPM for solving integer and mixed integer constrained optimization problems. Lastly, we conduct eight numerical experiments for investigating the sensitivities of system parameters and reveal more management insights than the literature. We will describe our Markov inventory model in Section 2 and derive the steady-state performance measures in Section 3. In Section 4, we will first establish a long-run average inventory cost function and then prove that the cost function is pseudo-convex about rr and QQ for fixed parameter pp. A mixed integer optimization model will be established and the MI-LXPM algorithm will be presented in Section 5. This will be followed by some numerical experiments that investigate the service discipline with different cost in Section 6. The paper will be concluded in Section 7.

This paper investigated a Markov inventory system with two classes of demands, (r,Q)(r,Q) replenishment policy, and an additional flexible service discipline. We introduce a parameter pp to distinguish the service policy. We formulate the inventory level states process into a Markov process and derive the steady state probability distribution of the inventory levels. Several system performance measures are also obtained for the inventory control, which include the average inventory, the reorder rate, the shortage cost and the average inventory cost. We establish a mix integer optimization model for the inventory control and use an efficient MI-LXPM algorithm for the optimum solutions. Several numerical experiments were conducted and presented to test the system parameters. The results show that the parameter p can distinguish the service policy. The paper contributes to the literature by introducing an additional flexible service discipline into the Markov inventory system that endues the inventory management polices to be much more flexible. This research can be generalized to the situation where demands arrival is a compound Poisson process. In many cases, each customer demands more than one unit item. In the case of batch demand, the context becomes much more complex and much more challenging. The demand fluctuation will be much bigger under the batch demands than that under a single demand. The new inventory control policies are expected to propose for future work.