替代竞争情境下关联需求固定产出比联产品加工决策研究

doi:10.16381/j.cnki.issn1003-207x.2022.2442

Abstract

Abstract:

The fixed-proportion co-production system （FPCS）， which produces two products in fixed proportions by processing a raw material， is widely observed in oil refining， agricultural product processing， chemical industry and so on. In practice， many manufacturers with FPCS are face the challenging problem of mismatches between the fixed-proportion yields and the independent uncertain demands for different petroleum products. In this paper， the production/inventory game of two manufacturers with FPCS is studied. The unmet demand of one manufacturer’s product can be met by the other manufacturer’s leftover stock of the same product， if available， and is lost otherwise. The manufacturers compete for the substitute demands by choosing their own purchase/processing quantities.The problem is formuated as a multi-product multi-period game in which the aggregate demands and the market shares for the two manufacturers’ products are uncertain and independent. It is shown that the production/inventory game is submodular and the response functions are contraction maps， so the existence of a unique Nash equilibrium is guaranteed. It is found that the best response function is strictly decreasing in the rival’s decisions and decreasing in the initial inventory levels of the two products. It is shown that the produce-up-to policy is the equilibrium policy. The impact of competition is examined by showing the equilibrium processing quantity and payoffs are increasing with substitution rates. Finally， we show that our work can be extended to the infinite-period case. The numerical studies are conducted to show the value of our model， and the impacts of some parameters on the equilibrium policies and the players’ payoffs. The managerial insights is also generated from the research findings：（1） Considering the influence of a fixed output ratio， in each period， a high initial inventory level of a certain co-product will put the manufacturer at a disadvantage in the competition in the markets of other co-products. A high initial inventory level of a certain co-product held by the competitor means that the manufacturer can obtain a larger share in the markets of other co-products.（2） An increase in the substitution rate of a certain co-product will lead to an increase in the total quantity of products in the markets of other co-products and a decrease in the substitution demand due to stock-outs， thus reducing the intensity of competition.（3） The manufacturer’s equilibrium processing quantity decreases as the initial inventory levels of its own and competitor’s co-products increase. The traditional produce-up-to strategy is not applicable in a competitive setting. The improved equilibrium strategy in the form of produce-up-to depends on the beginning inventory levels of various co-products of both the competitor and the manufacturers itself.These results have important implications for the competing strategies of the manufacturers with FPCS.Compared with the existing literature， we make the following contributions：（1） the inventory competition games between two manufacturers are studied for the multi-period （finite and infinite） cases， under the co-production system；（2） The concavity and submodularity of the payoff functions， and the existence and uniqueness of the equilibria of the games are shown；（3） For the multi-period （finite， infinite） cases， it is shown that the Nash equilibrium policies have a simple structure （the produce-up-to policy）.

Key words: co-production system, fixed yield proportions, substitution competition, processing game, multi-period

CLC Number:

F224

Shuang He,Liang Zhao,Jian Zhang, et al. Production/Inventory Competition Between Firms with Fixed-Proportions Co-production Systems and Aggregate Demands[J]. Chinese Journal of Management Science, 2025, 33(10): 76-85.

Figures/Tables 6

符号	定义
i	生产商的索引， $i = 1,2$
k	联产品的索引， $k = 1,2, 3, …, n$
$p i, k t$	第t周期生产商i产品k的销售价格
$θ k$	产品k的产出比例
$c i$	生产商i的单位采购和加工成本
$d k t$	第t周期产品k的市场总需求
$f d k t$ , $F d k t$	$d k t$ 的PDF和CDF
$α i, k$	生产商i产品k的需求替代率
$s i, k$	生产商i未售出产品k的单位残值
$l i, k$	生产商i产品k的单位缺货成本
$q i t$	第t周期生产商i的采购/加工数量
$I i, k t$	第t周期生产商i加工前产品k的库存水平
$y i, k t$	第t周期生产商i加工后产品k的库存水平
$γ i, k t$	第t周期生产商i产品k的市场份额
$g γ 1, k t$ , $G γ 1, k t$	$γ 1, k t$ 的PDF和CDF

情况

表达式

(1) y 1, k t γ 1, k t > y 2, k t γ 2, k t

O 1, k 1, t = (p 1, k t + z 1, k) ∫ 0 y 2, k t / γ 2, k t (y 1, k t - γ 1, k t d k t) d F (d k t) + (p 1, k t + z 1, k) ∫ y 2, k t / γ 2, k t ρ 1 [y 1, k t - γ 1, k t d k t + α 1, k y 2, k t - α 1, k γ 2, k t d k t] d F (d k t) + l 1, k ∫ y 1, k t / γ 1, k t ∞ (γ 1, k t d k t - y 1, k t) d F (d k t), O 2, k 1, t = (p 2, k t + z 2, k) ∫ 0 y 2, k t / γ 2, k t [y 2, k t - γ 2, k t d k t] d F (d k t) + l 2, k ∫ y 2, k t / γ 2, k t ∞ (γ 2, k t d k t - y 2, k t) d F (d k t) .

(2) y 1, k t γ 1, k t ≤ y 2, k t γ 2, k t

O 1, k 2 = (p 1, k t + z 1, k) ∫ 0 y 1, k t / γ 1, k t (y 1, k t - γ 1, k t d k t) d F (d k t) + l 1, k ∫ y 1, k t / γ 1, k t ∞ (γ 1, k t d k t - y 1, k t) d F (d k t) O 2, k 2 = (p 2, k t + z 2, k) ∫ 0 y 1, k t / γ 1, k t [y 2, k t - γ 2, k t d k t] d F (d k t) + (p 2, k t + z 2, k) ∫ y 1, k t / γ 1, k t ρ 2 [y 2, k t - γ 2, k t d k t + α 2, k y 1, k t - α 2, k γ 1, k t d k t] d F (d k t) + l 2, k ∫ y 2, k t / γ 2, k t ∞ (γ 2, k d k t - y 2, k t) d F (d k t)

$M D$	Case	Sum	Payoff
0.9	Com.	138.68	15.92
0.9	Mon.	138.50	15.91
0.7	Com.	141.72	136.76
0.7	Mon.	137.78	132.96
0.5	Com.	130.57	243.51
0.5	Mon.	123.48	230.29
0.3	Com.	159.00	360.12
0.3	Mon.	146.38	331.54
0.1	Com.	181.44	410.97
0.1	Mon.	162.73	368.58

$M D 1$	$α 1,1$	Sum	Payoff
0.9	0.9	147.12	19.75
	0.7	142.84	18.54
	0.5	138.68	18.04
	0.3	142.53	17.10
	0.1	138.46	16.62
0.5	0.9	150.28	280.27
	0.7	143.42	267.46
	0.5	137.68	256.77
	0.3	119.68	223.20
	0.1	113.20	211.11
0.1	0.9	206.55	467.83
	0.7	194.51	440.56
	0.5	185.24	419.58
	0.3	172.89	391.61
	0.1	164.66	372.96

$M D$	$α 1,2$	Sum	Payoff
0.9	0.9	138.50	16.62
	0.7	138.52	16.63
	0.5	138.49	16.62
	0.3	138.46	16.62
	0.1	138.47	16.62
0.5	0.9	150.28	280.27
	0.7	143.42	267.46
	0.5	137.68	256.77
	0.3	130.96	244.22
	0.1	123.87	231.02
0.1	0.9	227.40	515.06
	0.7	214.14	485.04
	0.5	203.95	461.94
	0.3	190.35	431.14
	0.1	181.29	410.61

$M D 1$	Strategy	Q₁	Q₂	Q₃	Q₄	Q₅	Sum	Payoff
0.9	Max.	270.51	139.27	0	0	0	529.78	-575.35
	Myo.	182.82	99.54	0	0	0	342.36	-318.62
	Bas.	140.68	77.12	0	0	0	212.8	8.74
0.5	Max.	119.42	99.9	0	0	0	339.32	-302.19
	Myo.	104.02	69.31	47.36	11.19	0	206.88	-190.69
	Bas.	63.65	78.92	68.51	46.08	0	282.16	289.45
0.1	Max.	90.28	70.15	60.14	49.58	68.89	254.05	499.55
	Myo.	93.12	78	67.99	57.46	86.8	258.38	530.05
	Bas.	52.74	87.62	82.61	82.08	46.43	256.48	554

References 39

[1]	Wu D S， Zhang B F， Baron O. A trade credit model with asymmetric competing retailers［J］. Production and Operations Management， 2019， 28（1）： 206-231.
[2]	Mason J B， Wilkinson J B. Supermarket product availability and the consumer response［J］. Consumer Satisfaction， Dissatisfaction and Complaining Behavior， 1977， 24（3）： 153-158.
[3]	Emmelhainz M A， Stock J R， Emmelhainz L W. Guest commentary： consumer responses to retail stockouts［J］. Journal of Retailing， 1991， 67（2）： 138-147.
[4]	Yigal G， Ashish T， Kai W. Co-production models with random functionality yields［J］. IIE Transactions， 1996， 28（5）： 391-403.
[5]	Hsu A， Bassok Y. Random yield and random demand in a production system with downward substitution［J］. Operations Research， 1999， 47（2）： 277-290.
[6]	Tomlin B， Wang Y. Pricing and operational recourse in coproduction systems［J］. Management Science， 2008， 54（3）： 522-537.
[7]	Chen Y J， Tomlin B， Wang Y. Coproduct technologies： product line design and process innovation［J］. Management Science， 2013， 59（12）： 2772-2789.
[8]	Chen Y J， Tomlin B， Wang Y. Dual coproduct technologies： implications for process development and adoption［J］. Manufacturing & Service Operations Management， 2017， 19（4）： 692-712.
[9]	Lu T， Chen Y J， Tomlin B， et. al . Selling co-products through a distributor： the impact on product line design［J］. Production & Operations Management， 2019， 28（4）： 1010-1032.
[10]	Boyabatl O. Supply management in multi-product firms with fixed proportions technology［J］. Management Science， 2015， 61（12）： 3013-3031.
[11]	Sunar N， Plambeck E. Allocating emissions among co-products： implications for procurement and climate policy［J］. Manufacturing & Service Operations Management， 2016， 18（3）： 414-428.
[12]	Zhang P， Xu X， Shi V， et al. Simultaneous inventory competition and transshipment between retailers［J］. International Journal of Production Economics， 2020， 230：1-14.
[13]	周品，徐和，陈鹏宇，等. 随机需求环境下联产品系统价格与产量决策研究［J］. 管理工程学报， 2020， 34（2）： 156-160.
	Zhou P， Xu H， Chen P Y， et. al . Co-product’s joint pricing and quantity decisions with stochastic demand and fixed proportion technology［J］. Journal of Industrial Engineering/Engineering Management， 2020， 34（2）： 156-160.
[14]	Boyabatli O， Nguyen J， Wang T. Capacity management in agricultural commodity processing and application in the palm industry［J］. Manufacturing & Service Operations Management， 2017， 19（4）： 551-567.
[15]	Liu H， Zhang J， Cheng T C E， et al. Optimal production-inventory policy for the multi-period fixed proportions co-production system［J］. European Journal of Operational Research， 2020， 280（2）： 469-478.
[16]	Lippman S A， McCardle K F. The competitive newsboy［J］. Operations Research， 1997， 45（1）： 54-65.
[17]	Netessine S， Rudi N. Centralized and competitive inventory models with demand substitution［J］. Operations Research， 2003， 51（2）： 329-335.
[18]	Li M， Petruzzi N C， Zhang J. Overconfident competing newsvendors［J］. Management Science， 2016， 63（8）： 2637-2646.
[19]	Wu M， Zhu S X， Bai T. A loss averse competitive newsvendor problem with anchoring［J］. Omega， 2018， 81： 99-111.
[20]	Wu D， Zhang B， Baron O. A trade credit model with asymmetric competing retailers［J］. Production and Operations Management， 2019， 28（1）： 206-231.
[21]	Silbermayr L. A review of non-cooperative newsvendor games with horizontal inventory interactions ［J］. Omega， 2019， 92： 102-148.
[22]	苏兵，徐渝，陈金亮，等. 单周期二人博弈可替代产品库存模型的应用研究［J］. 中国管理科学， 2005， 13（1）： 53-59.
	Su B， Xu Y， Chen J L， et. al . Research on the substitutable product inventory model and its application［J］. Chinese Journal of Management Science， 2005， 13（1）： 53-59.
[23]	陈敬贤，王国华，梁樑．顾客退货影响的多零售商库存博弈［J］.系统工程学报， 2013， 28（1）： 101-108.
	Chen J X， Wang G H， Liang L. Inventory game among multi retailers under customer returns’ impact ［J］. Journal of Systems Engineering， 2013， 28（1）： 101-108.
[24]	曹国昭，齐二石．替代品竞争环境下损失厌恶报童问题研究［J］.管理学报， 2013， 10（6）： 898-904.
	Cao G Z， Qi E S. Loss-averse newsvendor problem based on competition between substitutable product ［J］. Chinese Journal of Management， 2013， 10（6）： 898-904.
[25]	叶涛锋，达庆利.需求替代情形下反应能力的价值分析［J］.中国管理科学， 2012， 20（3）： 175-184.
	Ye T F， Da Q L. On the value of reactive capacity under demand substitution［J］. Chinese Journal of Management Science， 2012， 20（3）： 175-184.
[26]	关旭，马士华，应丹丰.基于库存分担策略的装配系统准时供货模型研究［J］.中国管理科学， 2013， 21（4）： 44-52.
	Guan X， Ma S H， Ying D F. On-time delivery model in a ‘n to 1’ assembly system with inventory sharing policy［J］. Chinese Journal of Management Science， 2013， 21（4）： 44-52.
[27]	邵婧. 两层级分散决策供应链库存转运问题研究［J］. 中国管理科学， 2016， 24（1）： 76-81.
	Shao J. Impact of transshipment on a decentralized supply chain［J］. Chinese Journal of Management Science， 2016， 24（1）： 76-81.
[28]	Avsar Z M， Baykal‐Gursoy M. Inventory control under substitutable demand： a stochastic game application［J］. Naval Research Logistics， 2002， 49（4）： 359-375.
[29]	Nagarajan M， Rajagopalan S. A multiperiod model of inventory competition［J］. Operations Research， 2009， 57（3）： 785-790.
[30]	Olsen T L， Parker R P. Inventory management under market size dynamics［J］. Management Science， 2008， 54（10）： 1805-1821.
[31]	Olsen T L， Parker R P. On Markov equilibria in dynamic inventory competition［J］.Operations Research， 2014. 62（2）： 332-344.
[32]	张剑，张菊亮. 供应不确定条件下多周期库存博弈［J］. 中国管理科学， 2018， 26（4）： 67-77.
	Zhang J， Zhang J L. Multi-period inventory competition under yield uncertainty［J］. Chinese Journal of Management Science， 2018， 26（4）： 67-77.
[33]	Zhang J， He S， Zhang J， et al. Purchase and retrieval competition for seasonal produce［J］. Naval Research Logistics （NRL）， 2020， 67（3）： 161-184.
[34]	张剑，何爽，师璞杰，等.替代竞争情境下考虑期初库存水平和存储能力约束的多周期订货博弈问题研究［J］.中国管理科学， 2024， 32（1）： 319-324.
	Zhang J， He S， Shi P J， et. al . Multi-period inventory substitute competition under the initial inventory level and the capacity constraints［J］. Chinese Journal of Management Science， 2024， 32（1）： 319-324.
[35]	Cachon G P， Netessine S. Game theory in supply chain analysis ［M］//Simchi-Levi D， Wu S D， Shen Z J M. Handbook of Quantitative Supply Chain Analysis. New York： Springer US， 2004： 13-65.
[36]	Glasserman P. Perturbation analysis of production networks ［M］//David D， Yao. Stochastic Modeling and Analysis of Manufacturing Systems.New York： Springer NY， 1994.
[37]	Topkis D M. Supermodularity and complementarity ［M］. New Jersey： Princeton University Press， 1998.
[38]	Parlar M. Game theoretic analysis of the substitutable product inventory problem with random demands［J］. Naval Research Logistics， 1988. 35： 397-409.
[39]	Bertsekas D P， Shreve S E. Stochastic optimal control： The discrete time case［M］. Mathematics in Science and Engineering. Belmont， MA： Academic Press， 1978.

Production/Inventory Competition Between Firms with Fixed-Proportions Co-production Systems and Aggregate Demands

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