Abstract: It is essential and important for a company to have a reasonable purchasing and production plan. Most relevant literature to this problem is the newsboy modeling one, which is a classic problem in operations research and management. It is noteworthy that the optimization objective of most relevant literature is to maximize the expected profit or minimize the cost. In addition, how to pay dividends to shareholders is also one of the problems considered by producers in the process of production and operation of enterprises. More common dividend methods are barrier dividend strategy, threshold dividend strategy and so on, and many studies have discussed them around the optimization of dividends.
This study extends and explores the newsboy modeling problem from a new perspective based on existing research. In this study, we assume that the firm produces and sells a class of products, similar to the multi-period newsboy model, and assume that the producer uses the barrier dividend strategy to pay dividends to shareholders. The main model used in this study is the discrete Markov decision model and the objective is to find the production strategy that maximizes the expected discounted dividend. Specifically, we consider two scenarios of production models. First, assuming that a firm produces, sells and stores a class of products for a given amount of principal, the market demand in different periods is a non-negative, independent and identically distributed discrete random variable. During any production cycle, the decision maker may face an inventory or supply shortage problem. We assume that the inventory charge for the current surplus product is paid at the end of each period, and the barrier dividend strategy is used to pay dividends to shareholders. In this paper, based on the above assumptions, we construct an iterative formula for maximizing the expected discounted dividend using the full-expectation formula. Under the condition that the length of the examination period is limited, Python software is used to carry out several iterations to obtain the optimal production strategy and the optimal value function in a limited period of time. In addition, we further consider a case where the length of the inspection period is unbounded until production is terminated when the producer's holdings of the prior are insufficient to cover the production and inventory costs in the next period. Similarly, the Bellman equation maximizing the expected discounted dividend is developed. The existence and uniqueness of the optimal value function, i.e., the optimal value function is the immovable point of Bellman equation, can be proved by the principle of compression mapping. Secondly, this study assumes that the unit cost in each period is correlated with the unit selling price and the market demand in the previous period as random variables, and the three follow a joint distribution and they are independent and identically distributed random vectors. We establish the iterative formulation in finite time and Bellman equation in the case of no time constraints, and obtain the optimal value function and the optimal production strategy in such a multivariate situation.
In conclusion, for a dividend optimization problem similar to a multi-period newsboy model, this study provides a method for finding an optimal solution which is useful in obtaining the expected discounted dividend that maximizes the expected discounted dividend and its corresponding optimal production schedule. However, there are some limitations in this study, for example, only a simple form of Markov decision-making problem is discussed in this paper, and more meaningful conclusions can be obtained if more complex conditions such as capital injection, investment, and Markov environmental process are further incorporated in the model.

Key words: optimal production strategy, barrier dividend strategy, Bellman equation, fixed point theorem

摘要： 本文探讨了生产经营过程中决策者非常关心的一个问题—如何制定合理的采购和生产策略。在边界分红策略下,本研究考虑控制企业的采购和生产,建立了一个马氏决策模型,目的是寻找使得考察期内期望贴现红利最大的生产策略及对应的最优值函数。本文探讨了两种情形:(1)仅市场需求量是随机变量的单变量情形;(2)任意生产周期的单位成本与前期的市场需求及售价具有相关关系且服从一个联合分布的多变量情形。在每种情形下,本文给出了最优生产策略满足的条件,以及考察期有限长条件下值函数满足的迭代关系式与考察期长无限制条件下的值函数满足的Bellman方程,并运用压缩映射原理证明时长无限制时的最优值函数是Bellman方程的唯一解。

关键词: 最优生产策略, 边界分红策略, Bellman方程, 不动点原理