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

摘要/Abstract

摘要：

本文研究了产出比例固定情况下两个生产商的多周期联产品加工博弈问题。每个生产商加工一种原材料能够同时得到比例固定的多种联产品。不同生产商的同类联产品间存在替代竞争关系，当生产商的某种联产品缺货时部分未满足的顾客会转而购买另一生产商的同类产品。他们通过选择加工量来实现个人损益最大化。本文首先在总需求和市场份额均随机的情况下建立多周期加工博弈模型，分析发现生产商均衡加工量会随着自身和竞争对手联产品期初库存的增加而减少；均衡解和均衡损益值会随着各联产品替代率的增加而增加。其次，在竞争情境下改进produce-up-to策略，证明它是均衡加工策略。再次，在无限周期下证明改进的produce-up-to策略依然是均衡策略。最后，利用仿真算例分析竞争及其强度和产出比对联产品系统的影响，并将均衡加工策略与两种静态策略进行比较。

关键词: 联产品, 固定产出比, 替代竞争, 加工博弈, 多周期

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

中图分类号:

F224

何爽,赵亮,张剑, 等. 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[J]. 中国管理科学, 2025, 33(10): 76-85.

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.

图/表 6

表1

参数和变量"

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

表1

表2

Oi,k1,t和Oi,k2,t的表达式"

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)

表2

表3

替代竞争对系统的影响"

$M D$	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

表3

表4

产品1缺货替代率α1,1对系统的影响"

$M D 1$	$α 1,1$	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

表4

表5

产品2缺货替代率α1,2对系统的影响"

$M D$	$α 1,2$	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

表5

表6

三种策略均衡损益值比较"

$M D 1$	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

表6

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