模型不确定下的收益管理动态定价策略研究
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摘要
收益管理是一套通过控制库存或价格来为企业的产品或服务科学管理需求以使有限库存收益最大的管理理念和方法。传统的关于收益管理动态定价策略的研究大都利用随机模型来刻画不确定需求,并假定该概率模型是已知的,以此来对价格决策进行优化。但现实中决策者往往并不具有需求模型的完全信息,而根据不恰当的模型优化得到的价格可能得到错误的决策。本文放松了这一完全信息假设,研究了单个零售企业在模型不确定下如何对有限库存进行动态定价以使期望收益最大的问题。通过在价格优化时考虑到需求模型的不确定,可以使优化的价格具有更强的适用性。
本文首先研究了结构化模型不确定下,如何对有库存约束的易逝品制定价格策略的问题。利用贝叶斯方法在销售过程中对不确定参数进行学习,分别研究了连续需求学习和周期性需求学习的动态定价问题。在连续需求学习的动态定价问题中,本文将顾客到达过程构造为一个贝努利过程,利用贝叶斯方法对每个周期有顾客到达的概率进行学习,将该问题构造为一个随机动态规划模型,并分析了最优价格策略与最优值函数的结构性质。在周期性需求学习的动态定价问题中,利用乘式需求函数对需求进行建模,利用贝叶斯方法对随机变量分布中的不确定参数进行学习,将该问题构造为一个依赖于销售历史的随机动态规划模型,分析了最优值函数的性质。
接下来本文研究了非结构化模型不确定下,如何对有库存约束的易逝品制定鲁棒价格策略的问题。利用相对熵来刻画模型不确定,将定价问题构造为一个决策者与“自然”的二人零和非合作博弈,建立了基于相对熵约束和基于相对熵惩罚的鲁棒定价模型,证明在一定条件下这两个模型可以得到相同的价格策略。对于单周期定价问题,分析了最优价格的性质。对于多周期鲁棒定价问题,证明该问题可以通过动态规划求解价格策略,并分析了鲁棒动态定价问题与指数效用下的风险规避型动态定价问题的关系。
然后本文将上述单产品鲁棒动态定价模型扩展到多产品,研究了模型不确定下的多种相关易逝品的鲁棒动态定价问题。仍利用相对熵来刻画模型的不确定,分别建立了同一模型不确定水平下的动态定价模型和不同模型不确定水平下的动态定价模型。对于前者证明可以利用动态规划递归求解,但由于所谓的“维数灾难”难以计算最优价格策略,本文设计了结合神经网络、遗传算法和随机模拟的混合智能算法来计算最优价格策略;对于后者,发现难以利用动态规划来计算价格策略,本文提出了结合遗传算法和随机模拟的启发式算法来计算开环策略。数值算例验证了算法的有效性。
最后本文将易逝品鲁棒动态定价模型扩展到非易逝品,研究了非结构化模型不确定下有库存约束的非易逝品收益管理问题。首先利用随机最优控制理论对完全信息下的动态定价问题进行建模,分析了最优值函数和最优价格策略的结构性质。然后将相对熵过程的概念进行了扩展来刻画需求模型的不确定,并将鲁棒动态定价问题构造为一个二人零和随机微分博弈,给出了最优值函数满足的Hamilton-Jacobi-Isaacs (HJI)方程,然后通过验证定理证明了HJI方程的解就是动态定价问题的值函数。最后通过名义需求率为指数函数的算例对鲁棒价格策略和最优值函数进行了说明。
Revenue management is a collection of strategies and tactics firms use to scientifically manage demand for their products and services by controlling inventory or price to maximize their revenue. Traditional research on dynamic pricing strategy for revenue management uses stochastic model to represent the uncertain demand and assumes that the stochastic demand model is known so as to optimize the pricing decision based on this model. In practice, however, the decision-maker usually does not posses complete information about the demand model and the optimized pricing may be wrong when using incorrect model. This dissertation will relax this full information assumption and studies the dynamic pricing problem of a monopolist retailing firm with finite inventory in the presence of model uncertainty. By considering model uncertainty in optimization, we can make the pricing policy more applicable in practice.
First, we study the limited inventory pricing problem for perishable products in the presence of structured model uncertainty. Bayesian method is applied to learn some uncertain parameters of the demand model during the selling process, and dynamic pricing problems with continuous-time demand learning and periodic demand learning are studied, respectively. For the dynamic pricing problem with continuous-time demand learning, customer arrival process is modeled as a Bernoulli process and the arrival probability in each period is updated by applying Bayesian theorem. The dynamic pricing problem is formulated as a stochastic dynamic programming. Structural properties for optimal pricicing policy and optimal value function are analyzed. For the dynamic pricing problem with periodic demand learning, the multiplicative demand function is applied to model the demand in each period and Bayesian method is used to learn the uncertain parameters in the distribution of random variable. The pricing problem is formulated as a history dependant stochastic dynamic programming. Stuctural properties for optimal value function are analyzed.
Next, we study the limited inventory pricing problem for perishable products with unstructured model uncertainty. Relative entropy is used to characterize the model uncertainty, the pricing problem is model as a two-person zero-sum game between decision-maker and“nature”and two robust pricing models based on relative entropy constraint and relative entropy penalty are built, respectively. For the single-period pricing problem, we analyze properties of the optimla price. For the multi-period robust pricing problem, we show that the optimal pricing policy can be computed recursively by dynamic programming. The relationship between robust pricing problem and risk-averse pricing problem with exponential utility is also analyzed.
Then, we extend the above single product robust dynamic pricing model to the multi-product case and study the pricing problem for multiple related perishable products with the consideration of model uncertainty. Relative entropy is used to model model uncertainty, and dynamic pricing models with single ambiguity level and multi-ambiguity levels are studied. For the former, we prove that it can be solved by dynamic programming recursively. The optimal pricing policy, however, is hard to compute because of the“curse of dimension”. We propose a hybrid intelligent algorithm that integrates neural network, generic algorithm and stochastic simulation to compute the optimal pricing policy; for the latter, dynamic programming cannot be applied to compute the pricing policy. Therefore, we propose a heuristic algorithm that integrates generic algorithm and stochastic simulation to compute the open-loop policy. Numerical experiments justify the effectiveness of the algorithms.
At last, we extend the idea of robust dynamic pricing model for perishable product to non-perishable product case, and study the revenue management problem for non-perishable products with limited inventory in the presence of unstructured model uncertainty. The nominal problem is modeled by stochastic optimal control theory and the structural properties of the optimal value function and the optimal pricing policy are studied. Then, the concept of relative entropy process is generalized to describe model uncertainty. The robust dynamic pricing problem is formulated as a two-person zero-sum stochastic differential game and the Hamilton-Jacobi-Isaacs equation the optimal value function satisfied is derived. A verification theorem is proved to show that the solutions the HJI equation is the value function of the dynamic pricing problem. The results are illustrated by an example with exponential nominal demand rate.
本文首先研究了结构化模型不确定下,如何对有库存约束的易逝品制定价格策略的问题。利用贝叶斯方法在销售过程中对不确定参数进行学习,分别研究了连续需求学习和周期性需求学习的动态定价问题。在连续需求学习的动态定价问题中,本文将顾客到达过程构造为一个贝努利过程,利用贝叶斯方法对每个周期有顾客到达的概率进行学习,将该问题构造为一个随机动态规划模型,并分析了最优价格策略与最优值函数的结构性质。在周期性需求学习的动态定价问题中,利用乘式需求函数对需求进行建模,利用贝叶斯方法对随机变量分布中的不确定参数进行学习,将该问题构造为一个依赖于销售历史的随机动态规划模型,分析了最优值函数的性质。
接下来本文研究了非结构化模型不确定下,如何对有库存约束的易逝品制定鲁棒价格策略的问题。利用相对熵来刻画模型不确定,将定价问题构造为一个决策者与“自然”的二人零和非合作博弈,建立了基于相对熵约束和基于相对熵惩罚的鲁棒定价模型,证明在一定条件下这两个模型可以得到相同的价格策略。对于单周期定价问题,分析了最优价格的性质。对于多周期鲁棒定价问题,证明该问题可以通过动态规划求解价格策略,并分析了鲁棒动态定价问题与指数效用下的风险规避型动态定价问题的关系。
然后本文将上述单产品鲁棒动态定价模型扩展到多产品,研究了模型不确定下的多种相关易逝品的鲁棒动态定价问题。仍利用相对熵来刻画模型的不确定,分别建立了同一模型不确定水平下的动态定价模型和不同模型不确定水平下的动态定价模型。对于前者证明可以利用动态规划递归求解,但由于所谓的“维数灾难”难以计算最优价格策略,本文设计了结合神经网络、遗传算法和随机模拟的混合智能算法来计算最优价格策略;对于后者,发现难以利用动态规划来计算价格策略,本文提出了结合遗传算法和随机模拟的启发式算法来计算开环策略。数值算例验证了算法的有效性。
最后本文将易逝品鲁棒动态定价模型扩展到非易逝品,研究了非结构化模型不确定下有库存约束的非易逝品收益管理问题。首先利用随机最优控制理论对完全信息下的动态定价问题进行建模,分析了最优值函数和最优价格策略的结构性质。然后将相对熵过程的概念进行了扩展来刻画需求模型的不确定,并将鲁棒动态定价问题构造为一个二人零和随机微分博弈,给出了最优值函数满足的Hamilton-Jacobi-Isaacs (HJI)方程,然后通过验证定理证明了HJI方程的解就是动态定价问题的值函数。最后通过名义需求率为指数函数的算例对鲁棒价格策略和最优值函数进行了说明。
Revenue management is a collection of strategies and tactics firms use to scientifically manage demand for their products and services by controlling inventory or price to maximize their revenue. Traditional research on dynamic pricing strategy for revenue management uses stochastic model to represent the uncertain demand and assumes that the stochastic demand model is known so as to optimize the pricing decision based on this model. In practice, however, the decision-maker usually does not posses complete information about the demand model and the optimized pricing may be wrong when using incorrect model. This dissertation will relax this full information assumption and studies the dynamic pricing problem of a monopolist retailing firm with finite inventory in the presence of model uncertainty. By considering model uncertainty in optimization, we can make the pricing policy more applicable in practice.
First, we study the limited inventory pricing problem for perishable products in the presence of structured model uncertainty. Bayesian method is applied to learn some uncertain parameters of the demand model during the selling process, and dynamic pricing problems with continuous-time demand learning and periodic demand learning are studied, respectively. For the dynamic pricing problem with continuous-time demand learning, customer arrival process is modeled as a Bernoulli process and the arrival probability in each period is updated by applying Bayesian theorem. The dynamic pricing problem is formulated as a stochastic dynamic programming. Structural properties for optimal pricicing policy and optimal value function are analyzed. For the dynamic pricing problem with periodic demand learning, the multiplicative demand function is applied to model the demand in each period and Bayesian method is used to learn the uncertain parameters in the distribution of random variable. The pricing problem is formulated as a history dependant stochastic dynamic programming. Stuctural properties for optimal value function are analyzed.
Next, we study the limited inventory pricing problem for perishable products with unstructured model uncertainty. Relative entropy is used to characterize the model uncertainty, the pricing problem is model as a two-person zero-sum game between decision-maker and“nature”and two robust pricing models based on relative entropy constraint and relative entropy penalty are built, respectively. For the single-period pricing problem, we analyze properties of the optimla price. For the multi-period robust pricing problem, we show that the optimal pricing policy can be computed recursively by dynamic programming. The relationship between robust pricing problem and risk-averse pricing problem with exponential utility is also analyzed.
Then, we extend the above single product robust dynamic pricing model to the multi-product case and study the pricing problem for multiple related perishable products with the consideration of model uncertainty. Relative entropy is used to model model uncertainty, and dynamic pricing models with single ambiguity level and multi-ambiguity levels are studied. For the former, we prove that it can be solved by dynamic programming recursively. The optimal pricing policy, however, is hard to compute because of the“curse of dimension”. We propose a hybrid intelligent algorithm that integrates neural network, generic algorithm and stochastic simulation to compute the optimal pricing policy; for the latter, dynamic programming cannot be applied to compute the pricing policy. Therefore, we propose a heuristic algorithm that integrates generic algorithm and stochastic simulation to compute the open-loop policy. Numerical experiments justify the effectiveness of the algorithms.
At last, we extend the idea of robust dynamic pricing model for perishable product to non-perishable product case, and study the revenue management problem for non-perishable products with limited inventory in the presence of unstructured model uncertainty. The nominal problem is modeled by stochastic optimal control theory and the structural properties of the optimal value function and the optimal pricing policy are studied. Then, the concept of relative entropy process is generalized to describe model uncertainty. The robust dynamic pricing problem is formulated as a two-person zero-sum stochastic differential game and the Hamilton-Jacobi-Isaacs equation the optimal value function satisfied is derived. A verification theorem is proved to show that the solutions the HJI equation is the value function of the dynamic pricing problem. The results are illustrated by an example with exponential nominal demand rate.
引文
②资料来源: Sullivan L.美国零售业青睐优化定价软件.《信息周刊》网络版.沈樱立译. 2006-2-9. http://www.informationweek.com.cn/iarticle/11071.html
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