带注资的生产动态控制和优化

doi:10.16381/j.cnki.issn1003-207x.2022.2584

摘要/Abstract

摘要：

本文考虑控制企业的采购生产及资金的流动，使得股东未来的累积期望贴现收益达到最大。在一个生产周期结束时，若公司持有现金量超过给定边界，则超出部分作为红利分发给股东；若公司盈余不足以支付随后的最优生产成本，则允许股东注入资金满足生产需求。文中建立了一个马尔可夫决策模型，假设任意周期的单位生产成本和上一期的单位售价、需求量相关，并服从一个联合分布；本文提供了计算最优值函数和最优生产策略的方法，其中包括考察期时长有限时值函数满足的迭代关系式，以及考察期时长无限制时值函数满足的Bellman方程，并用压缩映射原理证明最优值函数是Bellman方程的唯一解。本文最后给出了数值算例，采用文中的方法计算最优值函数和最优策略并分析结果。

关键词: 注资策略, 边界分红策略, 最优生产策略, Bellman方程, 压缩映射

Abstract:

A common problem in the production and management of enterprises，especially retail enterprises， how to make reasonable purchasing and production plans to achieve the business objectives of enterprises， is studied in this paper. In addition to production，sales，inventory and other activities，paying dividends to shareholders and accepting capital injection from shareholders are also essential links of enterprises. It is of practical signiﬁcance to formulate purchasing and production strategies with the goal of maximizing shareholders’ interests.On the basis of previous relevant studies，it is considered that enterprises operate a class of products，similar to the multi-cycle newsboy model. An enterprise controls the purchase or production and capital ﬂow，adopts the barrier dividend strategy to pay dividends to shareholders at the end of each cycle，or accepts capital injection from shareholders when the amount of cash held is insuﬃcient to pay the optimal production cost in the next period. In addition，it is assumed that the unit production cost of any period，the unit selling price of the previous period and the quantity demanded obey a joint distribution. In this paper，a discrete time Markov decision model is proposed to maximize the future cumulative expected discounted returns of shareholders，the aim is to ﬁnd the optimal production strategy and the corresponding optimal value function. Under the condition that the assessment period is limited，the iterative relation satisﬁed by the optimal value function is given by the formula of total expectation. When the assessment period is unlimited，the Bellman equation satisﬁed by the value function is given，and the optimal value function is proved to be the only solution by the contraction mapping principle.Finally，a numerical example is given according to the method provided in this paper，and the results mainly provide the following information： 1） When the duration of the assessment period is limited，the optimal production strategy of each period which is far away from the expiration time basically remains unchanged. When other conditions remain unchanged，the optimal production of each period shows a monotonous trend with the increase of the initial inventory. When the inventory reaches a certain value，the production is 0，indicating that the inventory should be sold ﬁrst and no reproduction is needed. 2） When the duration of the assessment period is unlimited，the value function of a ﬁnite assessment period with suﬃcient length can approximate the optimal value function of the inﬁnite case. The length of the ﬁnite assessment period can be calculated when the error is given. 3） No matter the assessment period is ﬁnite or inﬁnite，the value of the optimal value function is aﬀected by the dividend barrier，and the decision maker can adjust the dividend barrier appropriately to achieve the optimal state.A decision method for enterprises is provided，and the research on enter prise decision making is enriched.

Key words: capital injection strategy, barrier dividend strategy, optimal production strategy, Bellman equation, contraction mapping

中图分类号:

C934

郑宽宽, 谭激扬, 吕志科, 牛龙龙. 带注资的生产动态控制和优化[J]. 中国管理科学, 2024, 32(11): 115-124.

Kuankuan Zheng, Jiyang Tan, Zhike Lv, Longlong Niu. Dynamic Control and Optimization of Production with Capital Injection[J]. Chinese Journal of Management Science, 2024, 32(11): 115-124.

图/表 9

图1

表1

表2

最优生产策略 ϕ 1 * ( p 1 , g )"

$p 1$ g	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	……
1.2	8	8	7	6	5	4	3	2	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	……
1	10	10	9	8	7	6	5	4	3	2	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	……
0.8	12	12	10	10	10	9	9	9	9	5	5	4	2	2	1	0	0	0	0	0	0	0	0	0	0	……
0.6	16	16	16	16	16	16	16	16	16	16	13	13	13	11	11	8	8	8	7	6	4	4	3	2	1	……

表2

表3

最优值函数 V 0 * ( u , g , p 1 )随 u的变化"

$u$	0	1	2	3	4	5
$V 0 * (u, 9,1.2)$	90.4834	91.4615	92.4996	93.4767	94.4560	95.4786
$V 0 * (u, 11,1)$	92.6786	93.6563	94.6847	95.6673	96.6447	97.6596
$V 0 * (u, 15,0.8)$	96.1290	97.1046	98.1151	99.0959	100.0731	101.0743
$V 0 * (u, 24,0.6)$	-	103.3881	104.3789	105.3806	106.3584	107.3439
$u$	6	7	8	9	10
$V 0 * (u, 9,1.2)$	96.4572	97.4376	98.4574	99.4374	100.4174
$V 0 * (u, 11,1)$	98.6407	99.6193	100.6322	101.6122	102.5922
$V 0 * (u, 15,0.8)$	102.0545	103.0331	104.0323	105.0123	106.7644
$V 0 * (u, 24,0.6)$	108.3382	109.3170	110.2995	111.2919	112.2719

表3

表4

最优值函数 V 0 * ( u , g , p 1 )随 g的变化"

$g$	0	1	2	3	4	5	6	7
$V 0 * (4, g, 1.2)$	-	-	-	-	-	-	91.0087	96.1625
$V 0 * (6, g, 1.0)$	-	-	-	-	-	92.6786	93.6563	94.6847
$V 0 * (8, g, 0.8)$	-	-	93.6321	94.4922	95.3226	96.1474	96.9537	97.7378
$V 0 * (10, g, 0.6)$	97.2150	97.9784	98.7196	99.4349	100.1324	100.8169	101.4909	101.4909
$g$	8	9	10	11	12	13	14	……
$V 0 * (4, g, 1.2)$	93.3081	94.4560	95.6026	96.6447	97.5911	98.4462	99.2717	……
$V 0 * (6, g, 1.0)$	95.6673	96.6447	97.6596	98.6407	99.5834	100.4349	101.2567	……
$V 0 * (8, g, 0.8)$	98.5011	99.2717	100.0731	100.8920	101.6213	102.4191	103.2154	……
$V 0 * (10, g, 0.6)$	102.1435	102.7783	103.3881	103.9607	104.5910	105.2120	105.7365	……

表4

表5

最优值函数 V 0 * ( u , g , p 1 )随 p 1的变化"

$p 1$	1.2	1.0	0.8	0.6
$V 0 * (6,15, p 1)$	102.0545	102.0545	102.0545	102.9678
$V 0 * (7,9, p 1)$	97.4376	97.6596	97.6596	-
$V 0 * (8,12, p 1)$	101.5715	101.5715	101.6213	103.2372
$V 0 * (9,5, p 1)$	94.6245	95.6673	95.6673	-
$V 0 * (10,0, p 1)$	89.6899	91.6324	93.9339	97.2150

表5

表6

各期的最优生产决策"

$k$	1	2	3	4	5	6	7	8	9	10
$ϕ k * (1.2,0)$	8	8	8	8	8	8	8	8	8	7
$ϕ k * (1.0,5)$	6	6	6	6	6	6	6	5	5	2
$ϕ k * (0.8,8)$	9	9	9	6	6	6	5	4	3	0
$ϕ k * (0.6,12)$	13	13	13	13	13	11	8	5	0	0

表6

表7

不同分红边界 b对应的最优值函数"

$b$	1	2	3	4	5
$V 0 * (1,23,1.0)$	61.6535	78.6145	78.9423	-	-
$V 0 * (2,25,0.6)$	*	85.5992	92.9882	99.5517	106.4432
$V 0 * (4,15,1.2$	*	*	*	85.1119	92.7129
$V 0 * (5,16,0.8)$	*	*	*	*	101.8918
$b$	6	7	8	9	10
$V 0 * (1,23,1.0)$	102.9689	102.8506	102.5169	101.7574	100.8060
$V 0 * (2,25,0.6)$	107.0991	106.6747	106.1683	105.3574	104.3789
$V 0 * (4,15,1.2$	101.0470	101.8460	101.7105	101.0303	100.0731
$V 0 * (5,16,0.8)$	103.7821	103.6673	103.4801	102.7901	101.8501

表7

表8

无限期情形下的最优值函数"

$(u, g, p 1)$	$V * (u, g, p 1)$
(0,10,1.2)	523.7008
(2,8,1.2)	523.3789
(4,7,1.0)	524.7476
(6,6,1.0)	525.7250
(8,3,0.8)	525.7012
(10,5,0.6)	532.8930

表8

参考文献 22

1	李兰兰，诸克军，郭海湘.多周期报童模型在煤炭物资库存管理中的应用［J］.运筹与管理，2010，19（5）：167-170+183.
	Li L L， Zhu K J， Guo H X. Application of multi-period newsboy extended model in coal material inventory management［J］. Operations Research and Management Science， 2010， 19（5）： 167-170+183.
2	Kim G， Wu K， Huang E. Optimal inventory control in a multi-period newsvendor problem with nonstationary demand［J］. Advanced Engineering Informatics， 2015， 29（1）： 139-145.
3	Chen X A， Wang Z Z， Yuan H S. Optimal pricing for selling to a static multi-period newsvendor［J］. Operations Research Letters， 2017， 45（5）： 415-420.
4	Ullah M， Khan I， Sarkar B. Dynamic pricing in a multi-period newsvendor under stochastic price-dependent demand［J］. Mathematics， 2019， 7（6）： 520-534.
5	Yang S L， Shi C V， Zhao X. Optimal ordering and pricing decisions for a target oriented newsvendor［J］. Omega， 2011， 39（1）： 110-115.
6	Pasandideh S H R， Keshavarz M. A multi objective model for determining ordering strategy within different constraints［J］. International Journal of Mathematics in Operational Research， 2015， 7（1）： 52-68.
7	Ota M， Srinivasan S， Nandakumar C D. Optimal order quantity by maximising expected utility for the newsboy model［J］. International Journal of Procurement Management， 2019， 12（4）： 410-424.
8	Zhou P， Ruland W. Dividend payout and future earnings growth［J］. Financial Analysts Journal， 2006， 62（3）： 58-69.
9	Tan J Y， Yang X Q. Optimal dividend strategy in compound binomial model with bounded dividend rates［J］. Acta Mathematicae Applicatae Sinica， English Series， 2014， 30（4）： 859-870.
10	陈名芹，刘星.基于资金供应链价值视角的最优现金持有方案研究［J］.中国管理科学，2015，23（8）：92-101.
	Chen M Q， Liu X. Optimal cash holding structure： A perspective on the value of capital supply chain［J］. Chinese Journal of Management Science， 2015， 23（8）： 92-101.
11	陈树敏，曾燕，谷爱玲.R&D企业最优技术投资与分红策略研究［J］.系统工程理论与实践，2019，39（6）：1394-1406.
	Chen S M， Zeng Y， Gu A L. Optimal technology investment-dividend strategy for a R&D firm［J］. Systems Engineering-Theory & Practice， 2019， 39（6）： 1394-1406.
12	徐港，王晨，谭激扬.期望红利现值近似计算的随机算子方法和扩散方法［J］.湘潭大学学报（自然科学版），2022，44（5）：1-9.
	Xu G， Wang C， Tan J Y. Stochastic operator method and diﬀusion method for approximate calculation of the expected present value of dividends［J］. Journal of Xiangtan University（Natural Science Edition）， 2022， 44（5）： 1-9.
13	李岩，刘国欣.经典风险模型分红控制过程的最优停止问题［J］.应用数学学报，2010，33（6）：1123-1132.
	Li Y， Liu G X. Optimal stopping problem of dividend control process in classical risk model［J］. Acta Mathematicae Applicatae Sinica， English Series， 2010， 33（6）： 1123-1132.
14	Tan J Y， Ma Y H， Zhang H J， et al. Optimal control strategies for dividend payments and capital injections in compound Markov binomial risk model with penalties for deficits［J］. Communications in Statistics-Theory and Methods， 2017， 46（10）： 5072-5092.
15	李鹏，周明，孟辉. 脉冲和正则控制下的最优注资：一种混合策略［J］.中国科学：数学，2018，48（4）：565-578.
	Li P， Zhou M， Meng H. Optimal stochastic impulse and regular control for capital injections： A hybrid strategy［J］. Science China（Mathematics）， 2018， 48（4）： 565-578.
16	Zhuo J， Yang H L， Yin G. A numerical approach to optimal dividend policies with capital injections and transaction costs［J］. Acta Mathematicae Applicatae Sinica， English Series， 2017， 33（1）： 221-238.
17	Lindensjö K， Lindskog F. Optimal dividends and capital injection under dividend restrictions［J］. Mathematical Methods of Operations Research，2020，92（3）： 461-487.
18	Sana S S. Price sensitive demand with random sales price： A newsboy problem［J］. International Journal of Systems Science， 2012， 43（3）： 491-498.
19	Roy A， Sana S S， Chaudhuri K. A joint venturing of single supplier and single retailer under variable price， promotional eﬀort and service level［J］. Pacific Science Review B： Humanities and Social Sciences， 2015， 1（1）： 8-14.
20	许民利，李展.需求依赖于价格情境下基于Copula-CVaR的报童决策［J］.控制与决策，2014， 29（6）： 1083-1090.
	Xu M L， Li Z. Newsvendor decision based on Copula-CVaR with price-dependent demand［J］. Control and Decision， 2014， 29（6）： 1083-1090.
21	孟志青，郑敏超.零售价波动销售策略下的报童模型研究［J］.系统工程理论与实践，2020，40（10）： 2648-2657.
	Meng Z Q， Zheng M C. Research on newsvendor model under retail price fluctuation sales strategy［J］. Systems Engineering-Theory & Practice， 2020， 40（10）： 2648-2657.
22	程其襄，张奠宙，魏国强，等.实变函数与泛函分析基础［M］.北京：高等教育出版社，2010.
	Cheng Q X， Zhang D Z， Wei G Q， et al. Real variable functions and fundamentals of functional analysis［M］. Beijing： Higher Education Press， 2010.

[1]	曾燕, 李仲飞, 朱书尚, 伍慧玲. 基于CRRA效用准则的资产负债管理[J]. 中国管理科学, 2014, 22(10): 1-8.
[2]	沈铁松, 熊中楷. 基于新产品的供应链最优生产策略[J]. 中国管理科学, 2004, (2): 55-59.
[3]	刘夏清, 李林, 杨招军. 指数效用下企业的风险投资策略[J]. 中国管理科学, 2003, (2): 66-69.