سهمیه بندی موجودی پویا با سفارش معوق مخلوط شده و فروش از دست رفته

Customers may react differently when stockouts occur. In this paper we investigate the rationing policy for an inventory system with a mixture of demand classes of backorder type and lost sales type. Since the penalty cost of backorders varies with time, the priorities of demand classes also alter with time. This totally changes the problem structure compared with the classic rationing models. A dynamic rationing policy is studied in this paper by considering the dynamics of demand priorities. A Markov decision model is developed to obtain the optimal dynamic rationing levels for multiple demand classes. The results indicate that between the priority switching points, rationing levels often exhibit different patterns. For lost sales demand classes, the rationing levels always decrease as the remaining time approaches to zero. For backorder demand classes, the rationing levels increase in some parts due to declining of the priorities. The rationing levels of all demand classes finally decline to zero to reduce the inventory holding cost. The application of dynamic rationing is further extended from a single period model to a multi-period (S,T) model where unit cost has to be included. The optimal ordering policy is proved to be a myopic base stock policy and the dynamic rationing policy in the single period model can still be applied with modified time-independent penalty costs for lost sales classes. To overcome the computational complexity, a heuristic dynamic rationing policy is introduced. Due to its good outcome, implementing such a heuristic dynamic rationing policy can be a practical solution for inventory system with mixed backorders and lost sales, in order to enhance the system performance.

In the traditional inventory models, demand is considered to be homogeneous and often fulfilled with the first-come-first-server principle. Nowadays customers are often classified into different groups, for examples VIPs, premium and ordinary ones, who are considered to bring different values to companies. In a finite time horizon without production and replenishment opportunity, inventory is generally limited. Thus at certain time points, inventory is preferred to be reserved for the demand classes with high added value in order to maximise profits or avoid expensive penalty costs. Inventory rationing is one method to achieve these goals. Before implementing a rationing policy, we often need to define the priorities of demand classes. Most literature assigns priorities to demand classes according to their penalty costs which often have the same structure. This implies that customers are of the same type, i.e. they have similar behaviour in facing stockouts but the values of penalty cost vary. In practice customers may exhibit different purchasing behaviours, in particular with different reactions to stockouts. Some are willing to wait until demand is fulfilled, and the others walk away immediately or purchase from other sources. These two responses lead to backorders and lost sales, respectively. There are plenty of industrial cases with such kinds of mixed phenomenon. For example, e-commerce companies usually perform as suppliers for large business customers as well as serve individual consumers. When the inventory is out of stock, the individual consumers will likely switch their purchasing to other websites, while the large business customers will wait because of price discounts or constraints in the contracts. When we apply the rationing polices to different classes which react differently to stockouts, the decisions become more complicated. If there is a long remaining time till the end of period (or next replenishment), we may reject the customer demand of lost sales type in order to reserve inventory to the ones of backorder type which may have a high penalty cost due to a long waiting time. On the other hand, when the decision point is approaching the next replenishment, we may ask the customers of backorder type to wait for a short while and use the inventory to avoid further lost sales, because the penalty cost associated with backorders becomes less than that of lost sales. In summary, due to different dimensions of the penalty costs, we need to switch the protection focus and consequently adjust the amount of reserved inventory at different decision points along with time. In this paper, we therefore study the rationing policies in an inventory system with a mixture of backorders and lost sales. Since at least one demand class (backorder type) has a penalty cost changing over time, the importance of demand classes should also change along with time and consequently the demand priority may switch. In this circumstance, a dynamic rationing policy should be employed. Thus, in this paper we first develop a model for dynamic rationing in a single period. Such a system is studied because the pattern and advantages of dynamic rationing can be explored. We then apply dynamic rationing in a multi-period system and check its impact on ordering policy and cost. For the purpose of making the dynamic rationing policy easy to be implemented, we also introduce a simple algorithm with a heuristic policy for obtaining the rationing levels. The structure of this paper is as follows. In Section 2 we introduce two streams of literature related to our study. In Section 3, we develop a discrete Markov model for obtaining the optimal dynamic rationing levels in a single period. The dynamic rationing is then extended to a multi-period system in Section 4. Then a heuristic dynamic rationing policy is developed in Section 5. The conclusions are drawn in Section 6.

This paper studies an inventory system with a mixture of backorders and lost sales, which represent two different responses to stockouts in common practice. However, with our best knowledge, there is no study on inventory rationing with this research orientation. Most studies focus on the cases of having two/multiple backorders or two/multiple lost sales. Our study thus fills the research gap by formulating such a problem, deriving cost expressions and examining performance when rationing policies are applied. One major feature of such a problem is due to its changing priority among classes along with time. Since the rationing decisions are largely dependent on the remaining time and the priority gap (determined by the penalty costs during the remaining time), we formulate the problem by a Markov decision model. According to this model, we reveal that the optimal dynamic rationing levels are divided into several sections by switching time points where demand classes switch the priorities. In general, all rationing levels should decrease to zero as the remaining time becomes zero. However, for backorder type classes, the rationing levels could increase in some sections due to the opposite effects of priority gap and the remaining time. For lost sales type classes, their rationing levels always decrease as time approaches to the end of period. We further investigate dynamic rationing policy in a multi-period system. The unit cost has to be considered because the consideration of lost sales type demand influences the expected inventory status at the beginning of each period. We prove that the optimal ordering policy is a myopic base stock policy and the optimal rationing policy has the same principle as in a single period model but with modified time-independent penalty cost of lost sales classes. The numerical results show that the dynamic rationing improves the cost compared with non-rationing policy. However, with a mixture of backorders and lost sales, the improvement does not necessarily increase along with the period length T . Given an initial inventory level S , there exists a period length View the MathML sourceT^ where rationing policies have minor improvement compared with non-rationing. If T is close to View the MathML sourceT^, there is no need to implement rationing policies. Only when T is apart from View the MathML sourceT^, i.e. one class dominates in priority during most of the period, rationing policies should be suggested. Due to the calculation complexity in the Markov model, we also develop a heuristic dynamic policy which performs very close to the optimal one. The calculation is easy and can be realised in Excel, as we obtain the closed-form formulae for the approximated rationing levels. Such a development of heuristic policy also makes dynamic rationing possible to implement in practice. We have limited our study by zero lead time, which ignores the clearing mechanism for handling backorders. Common policies such as (r,Q) and (s,S) should be further studied. Including the clearing mechanism can definitely be one direction for future research. It should improve our understanding of the rationing policies, but it also raises challenges of more complicated analysis.