两种订货模式下订购和促销联合优化策略

本文基于CVaR决策准则,通过构建需求依赖于促销的一般需求模型,其中包括加法和乘法需求模型作为特例,考察了一次订货和允许紧急订货两种模式下风险厌恶零售商关于促销和库存的联合优化问题。讨论了紧急订购成本、风险厌恶以及市场需求变动对最优策略的影响,并对两种模式下零售商的最优策略和收益进行了比较分析,结果表明:在两种订货模式下,零售商的最优订购量和促销努力均随风险厌恶程度的增大而降低;紧急订货模式下的促销努力和实现的收益大于一次订货模式,且在加法需求模型下,紧急订货模式下的初次订购量小于一次订货模式的最优订购量。运用随机变量一阶和二阶交替随机占优的概念刻画了市场需求变动下零售商如何调整最优策略的充分条件。最后实施数值实验验证了理论分析结果。

Joint optimization of ordering and promoting in two ordering modes

Some factors,such as risk preferences,promotion behaviors,market demand fluctuations and ordering modes provided by upstream enterprises and so forth,can exert an important influence on ordering strategy of a retailer.To explore impacts of the factors on the retailer’s strategies,in this paper we built up a model on joint decision-making of a risk-averse newsvendor-like retailer upon ordering quantity and promotional effort,where market demand of the commodity operated by the retailer depends on promotional effort of the retailer.In particular,a generally promotion-dependent demand model was constructed,including additive demand model and multiplicative demand model as two special cases,so to make our discussion more generalized.Meanwhile,non-promotional case can be discussed as a special case in the generally promotion-dependent demand model.The CVaR(conditional value at risk)decision criterion was used to characterize risk aversion of the retailer.Under this way,the retailer’s risk aversion degree can be described by one unique coefficient.In addition,two ordering modes were discussed,respectively,in which one is one-time ordering mode(OOM)and the other emergency ordering mode(EOM).The retailer has only one ordering opportunity and cannot replenish the inventory in the former mode,and can exert initial ordering and replenish the inventory,if necessary,respectively at the beginning of and during the sales period in the latter mode.To begin with,we discussed the optimal strategies of the retailer in the two ordering modes,respectively.We characterized the first-order conditions jointly met by optimal ordering and promoting strategies.Based on that,we analyzed impacts of the retailer’s risk aversion and emergency order cost on the optimal strategies,by applying the implicit function theorem to the first order optimal conditions.The conclusion were drawn as follows.Firstly,the optimal ordering quantity decreases in risk aversion of the retailer,whether promotion efforts are exerted or not and regardless that emergency orders are allowed or not.Secondly,the retailer's promotion effort decreases in risk aversion in the two ordering modes.More accurately,the more the retailer is risk averse,the level of optimal promotional effort is smaller.Thirdly,the retailer's promotion effort decreases in emergency order cost.Fourthly,the ordering quantity increases in emergency order cost in the absence of sales effort and is uncertain in the presence of sales effort.Next,we implemented comparative analysis on the optimal strategies and the CVaRs of the retailer in the two modes.The optimal effort is bigger in EOM than in OOM,because the retailer has a replenishment opportunity in EOM.In particular,in additive demand model,the optimal initial ordering quantity is less in EOM than the ordering quantity in OOM.In the absence of promotional efforts,the order quantity in EOM is less than in OOM.However,the conclusion mentioned-above holds no longer valid in the presence of promotional efforts,but there exists an exception in the additive demand model.Furthermore,the CVaR realized by the retailer is bigger in EOM than that in OOM,owing to risk reduction caused by uncertainty in commodity demand,which,in turn,is because emergency ordering can be implemented,if necessary,after realization of the commodity demand.In the following,we explored impact of market demand fluctuations on optimal strategies of the retailer.In virtue of stochastic dominant concepts,especially first-order alternately stochastic dominant and second-order alternately stochastic dominant of two stochastic variables,we characterized the sufficient conditions how the retailer should adjust the strategies as the market demand varies.Firstly,in either OOM or EOM,a larger stochastic market demand leads to a greater promotion effort and a bigger order quantity.Secondly,in additive demand model or in case of no promotional efforts,if the stochastic market demand variables before and after a change are first-order alternately stochastic dominant each other,and the critical point of alternating stochastic dominance satisfies a specific condition,adjustment direction of the optimal order quantity in OOM is just the opposite to that in EOM.Some other results were also obtained.What’s more,we revealed mechanism behind some important results,and accordingly gave out managerial implication,which has enlightening significance to business practice in the real world.Finally,it exerted three sets of numerical experiments to verify the conclusions.The first set of numerical experiment was based on additive demand model,and the second was multiplicative demand model,on the assumption that market demand obeys negative exponential distribution.All results obtained by theoretical analyses were confirmed by the two experiments.The third set of experiment,assuming that market demand obeys uniformity distribution,was used to provide a counterexample in case of multiplicative demand model.The three sets of numerical experiments jointly showed that the initial order quantity in EOM is less than the order quantity in OOM with a few exceptions.