折扣下多阶段易腐品库存问题的无概率解决方法

Probability-free solutions to the multi-period perishable inventory problem under discount

考虑到供应商常采用折扣营销方式吸引零售商以提高产品的销售量,本文应用弱集成算法给出折扣下易腐品库存问题的无概率解决方法。通过集成静态专家意见,从连续和离散角度分别给出价值折扣和数量折扣下的在线订购策略。从理论上证明了所给在线订购策略的累积收益接近最优专家意见的累积收益,它们具有良好的竞争性能。数值算例进一步表明离散型在线订购策略的竞争性能更优,且不同初始概率分布对策略竞争性能的影响较小。

The classic perishable inventory problem assumes that the retailer should not be affected by upstream suppliers in the ordering process.However,real-world operation and management entail a relationship of competition and cooperation between retailers and suppliers.Suppliers therefore influence retailers’decision-making processes.Suppliers often use discount marketing,which is effective in attracting retailers and increasing the sales volume of products.This also allows retailers to order products at lower cost,creating a win-win scenario.Value discounts and quantity discounts are commonly used in discount marketing.Value discounts allow a retailer to enjoy a cash discount when the retailer’s total order value reaches a certain value,and quantity discounts refer to a price discount offered by the supplier for all products when the retailer’s single order quantity exceeds a fixed value.Most studies in the literature on inventory management assume that the demand or demand distribution is known;however,in reality,it is difficult to obtain accurate demand and demand distribution.Historical demand data are usually the only information that is available.Developments in theoretical computer science allow online learning algorithms to completely depart from the assumption of statistical information and rely only on historical data for decision-making.The Weak Aggregating Algorithm(WAA),an online learning algorithm,constructs online strategy by learning from different experts’advice and aggregating it;the online strategy can thus track the best expert advice.This assists online decision makers in effectively integrating various types of information and making decisions on such information.This study expands the research on the single-product,multi-period perishable inventory problem in the context of value discounts and quantity discounts without any assumption of statistical information.This article applies the WAA to learning from different experts’advice in studying the perishable inventory problem when value discounts and quantity discounts can be used.The online ordering strategies(probability-free solutions)are constructed as follows.First,at the beginning of every period,experts recommend their static advice for the specific perishable inventory decision problem;such expert advice is fixed and will not be changed throughout all periods.Second,in the case of value discounts or quantity discounts,the online decision maker applies the WAA to assign different degrees of trust to different experts’advice and then integrates the advice to make a decision.Last,the actual decision result of the decision problem is obtained,and cumulative gains are computed to evaluate the performance of the experts and the online decision maker.This paper constructs online ordering strategies in different situations.For real-valued order quantities,we aggregate experts’advice to propose an online order strategy that considers value discounts in detail.We analyze a more practical case in which the order quantity is an arbitrary integer by constructing an online ordering strategy under value discounts through learning from expert advice.We then approach the matter from continuous and discrete angles to propose two explicit online order strategies under quantity discounts.This paper further verifies the performance of the proposed online ordering strategies through theoretical analysis and the use of numerical examples.Competitive performance is measured as the difference between the cumulative gains from using the online ordering strategies and the cumulative gains from following the best expert advice.The theoretical results show that the cumulative gains of the given online ordering strategies are as large as those of the best expert advice and that they outperform their benchmark strategies.The numerical examples indicate that the cumulative gains achieved by the given online ordering strategies fluctuate slightly around those achieved by the optimal expert advice.The competitive performance of the discrete online ordering strategies outperforms that of the continuous online ordering strategies.The theoretical results are therefore effectively validated.The numerical examples also show that different initial probability distributions have little impact on strategies’competitive performance.In the case of value discounts,the online ordering strategy has better competitive performance for perishable products with a higher unit price,a lower unit cost,and higher value discounts.Regarding quantity discounts,the online ordering strategy performs better when the perishable products have a higher unit price,a lower unit cost,lower quantity discounts,and a lower discount threshold.A consideration of the actual situation leads to explicit ordering rules being proposed in this paper.These enable retailers to make ordering decisions quickly and effectively without information on future demand and thus to meet market demand with the optimal order quantity and maximize their profits.This paper thus provides guidance for perishable goods inventory management.