﻿ 基于碳排放限额和低碳销售努力的博弈模型分析及控制
 控制与决策  2020, Vol. 35 Issue (2): 357-366 0

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SI Feng-shan, WANG Jing, DAI Dao-ming. Analysis and control of game model based on carbon emission quota and low carbon sales efforts[J]. Control and Decision, 2020, 35(2): 357-366. DOI: 10.13195/j.kzyjc.2018.0722.
[复制英文]

### 文章历史

Analysis and control of game model based on carbon emission quota and low carbon sales efforts
SI Feng-shan , WANG Jing , DAI Dao-ming
School of Management Science and Engineering, Anhui University of Finance and Economics, Bengbu 233030, China
Abstract: Considering the carbon emission quota and low carbon sales efforts in the dual-channel supply chain, the single period static game model and the multi-period dynamic game model are constructed to explore the optimal decision and the system stability respectively. In the static model, the optimal decision analytic formulas under the centralized decision and the decentralized decision are given. In the dynamic model, the local asymptotic stability conditions of the game system are explored, and the effect of delay parameters on the stability of the system is discussed. In addition, the instability system is effectively controlled. The study shows that the increase of carbon emission quota can increase the profit of the supply chain system; with the increase of low carbon sales efforts, the profits of the supply chain system and manufacturer will be reduced; the unreasonable delay parameter will lead to instability of the system.
Keywords: carbon emission quota    low carbon sales efforts    differential price game    Hopf bifurcations    stability    control
0 引言

1 问题描述与假设

1) 制造商和零售商为有限理性, 在Stackelberg博弈中, 制造商是领导者, 零售商是追随者.

2) 制造商为了降低单位产品的碳排放量, 需要支付一定的碳减排技术研发成本.这里仅考虑一次性资金投入的情况, 如果单位产品的碳减排量为e, 则制造商需要承担的成本为kme2/2, km为碳减排成本系数.制造商的其他成本均忽略.

3) 零售商的低碳销售成本也仅考虑一次性投入的状况.若零售商的低碳销售努力程度为s, 则需要承担的成本为krs2/2, 其中kr为低碳销售努力的成本系数.零售商的其他成本均忽略.

4) 为了确保制造商和零售商能正常获利, 设定pr>w, pm>w>c.

5)制造商单位产品的碳减排量不高于单位产品的碳排放量, 即ee0.

6) 消费者具有低碳偏好, 所以消费者对零售商的低碳销售努力颇为敏感.

2 单周期静态博弈模型

 (1)
 (2)

2.1 集中决策

 (3)

 (4)

2.2 分散决策

 (5)
 (6)

 (7)
 (8)

 (9)
 (10)
 (11)

 (12)
 (13)

 (14)
 (15)

3 静态模型仿真

1) 关于命题2, 集中决策下低碳销售努力s对最优直销价pm.c*和最优零售价pr.c*的影响如图 1所示.

 图 1 最优直销价和最优零售价的变化趋势

 图 2 供应链最优系统利润的变化趋势

2) 关于命题4, 分散决策下直销价pm和批发价w对最优零售价pr.d*和最优销售努力sd*的影响如图 3所示.

 图 3 最优零售价和最优低碳销售努力的变化趋势

3) 集中决策和分散决策下的最优策略比较分析如表 1所示.

4 多周期时滞动态博弈模型

 (16)

 (17)

 (18)
5 Hopf分岔的存在性和局部渐近稳定性

Δ3=(e-e0)pe.模型(18)在该均衡点处的特征方程式为

 (19)

5.1 τ=0的情形

 (20)

5.2 τ>0的情形

 (21)

 (22)

 (23)

 (24)

f(ω)=ω 12+N10ω 10+N8ω 8+N6ω 6+N4ω 4+N2ω 2+N0.因, 故式(24)至少有一个正根.不失一般性, 假设式(24)有12个正根, 即为ωk(k=1, 2, ..., 12).对于每一个正根ωk都存在一系列的{τk(j)|k=1, 2, ..., 12;j=0, 1, ...}与之对应, 即

τ0=min{τk(j)|k=1, 2, ..., 12;j=0, 1, ...}=min{τk(0)|k=1, 2, ..., 12}=τ _k0(0), 此时ω0=ωk0.于是

 (25)

 (26)

 (27)

6 动态模型数值仿真

 (28)
6.1 时滞参数对系统稳定性的影响

 图 4 系统状态的变化趋势

6.2 碳减排量和低碳销售努力对价格和利润的影响

 图 5 价格和利润的变化趋势

 图 6 制造商利润和供应链系统利润的变化趋势

 图 7 制造商利润的变化趋势

7 稳定性控制

 图 8 当τ=0.33时, 系统(28)的吸引子

 (29)

τ =0.33时, 系统(29)关于μ的分岔图如图 9所示.

 图 9 系统(29)关于μ的分岔图

 图 10 当μ =0.07时, 系统(29)的吸引子
 图 11 当μ =0.08时, 系统(29)的吸引子

8 结论

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