小宇分享(三):《上下游联合减排与低碳宣传的微分博弈模型》

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摘要:在协同控制的集中式决策下,假设存在以最大化供应链整体利益为目标的中心决策者,其决策变量为E_M(t)和E_R(t)。虽然,实际运营中对于企业外部供应链系统很难存在以供应链整体利益最大化为目标的中心决策者,但是,可以以集中决策时的最优决策为标杆研究契约协调的效果

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“小宇分享(三):

“精读《上下游联合减排与低碳宣传的微分博弈模型》的协同控制的集中式决策”

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"Xiao Yu's sharing (Ⅲ)

"In-depth reading of the "Differential Game Model of Upstream and Downstream Joint Emission Reduction and Low-Carbon Promotion" on centralized decision-making for collaborative control"

一、思维导图(Mind mapping)

二、精读内容(Intensive reading content)

集中式决策下供应链系统的决策问题(Decision-making problems of the supply chain system under centralized decision-making):

在协同控制的集中式决策下,假设存在以最大化供应链整体利益为目标的中心决策者,其决策变量为E_M(t)和E_R(t)。虽然,实际运营中对于企业外部供应链系统很难存在以供应链整体利益最大化为目标的中心决策者,但是,可以以集中决策时的最优决策为标杆研究契约协调的效果。因此,首先对集中式决策进行分析(用上标c表示集中式决策)。

Under the centralized decision-making model of collaborative control, we assume the existence of a central decision-maker whose goal is to maximize the overall benefits of the supply chain. The decision variables are E_M(t) and E_R(t). Although, in actual operations, it is difficult for a central decision-maker to maximize the overall benefits of the supply chain in an enterprise's external supply chain system, the optimal decision under centralized decision-making can be used as a benchmark to study the effectiveness of contract coordination. Therefore, we first analyze centralized decision-making (using the superscript c to indicate centralized decision-making).

由假设6可知,在无限时间区间内供应链系统利润以贴现因子ρ进行贴现之后,得到整个供应链系统的决策问题如图所示:

Assumption 6 shows that after discounting the profit of the supply chain system with the discount factor ρ in an infinite time interval, the decision problem of the entire supply chain system is as shown in the figure:

集中式决策情况下的均衡结果(Equilibrium results under centralized decision-making):

为了得到问题的反馈均衡策略,采用Hamil-ton-Jacobi-Bellman方程(以下简称HJB方程)进行求解。由于动态参数条件下求解困难,所以假设模型中的参数都是与时间无关的常数。另外,为书写方便下文不再列出时间。求解的均衡结果如下图所示:

To determine the feedback equilibrium strategy for this problem, we used the Hamilton-Jacobi-Bellman equation (HJB equation). Because dynamic parameters are difficult to solve, we assume that all parameters in the model are constant and independent of time. For ease of explanation, time is omitted below. The resulting equilibrium is shown in the figure below:

证明过程(Proof process):

采用HJB方程进行求解,可知t时刻供应链系统的最优利润值函数如下图所示:

Solving the HJB equation, we can see that the optimal profit function of the supply chain system at time t is shown in the figure below:

所以t时刻的供应链系统的最优利润值函数转化为J_c^* s_c(τ,t) =e^{-ρt}* V_c* s_c(τ)如下图所示:

Therefore, the optimal profit function of the supply chain system at time t is transformed into J_c^* s_c(τ,t) =e^{-ρt}* V_c* s_c(τ) as shown in the figure below:

此时,供应链系统的最优控制问题满足如下HJB方程

At this time, the optimal control problem of the supply chain system satisfies the following HJB equation

计算关于E_M,E_R的海赛矩阵。可知,海赛阵半负定,目标函数为凹函数,如下图所示:

Calculate the Hessian matrix with respect to E_M and E_R. It can be seen that the Hessian matrix is negative semi-definite and the objective function is a concave function, as shown in the following figure:

所以关于变量E_M,E_R可以达到最大值,且最大值在偏导数等于0的点取得。因此对HJB方程分别关于E_M,E_R求偏导并联立求方程组可得:

Therefore, the variables E_M and E_R can reach their maximum values, and the maximum values are obtained at the point where the partial derivatives are equal to 0. Therefore, taking the partial derivatives of the HJB equation with respect to E_M and E_R and solving the equation system together yields:

根据所得HJB方程微分方程的特点,推测关于τ的线形最优值函数是HJB方程的解.于是,设函数V_c* s_c(τ)的具体表达式为V_cs_c(τ)= a_3τ+ b_3,将所设的表达式代入HJB方程可以得:

Based on the characteristics of the resulting HJB differential equation, we infer that the linear optimal value function with respect to τ is the solution to the HJB equation. Therefore, let the specific expression for the function V_c*s_c(τ) be V_cs_c(τ) = a_3τ + b_3. Substituting this expression into the HJB equation yields:

整理上面所得的式子,对比等式两端的同类项系数,可得关于a_3,b_3的约束方程组。可解得a_3,b_3,如下图所示:

By sorting out the equations above and comparing the coefficients of the like terms on both sides of the equation, we can obtain the constraint equations for a_3 and b_3. We can solve for a_3 and b_3 as shown in the figure below:

将 a*_3 ,b*_3 代入HJB方程,可以得到函数Vcs_c (τ)的表达式,再将式Vcs_c (τ)及其一阶导数代入式E_M,E_R可以得到供应链系统的均衡解,将所得均衡解代入状态方程,根据状态方程的边界条件τ(0)=τ0≥0,可得制造商单位产品减排量的最优轨迹,将式 Vcs_c (τ)代入式可以得到供应链系统的利润最优值函数。过程如下图所示:

Substituting a*_3 and b*_3 into the HJB equation yields the expression for the function Vcs_c (τ). Substituting Vcs_c (τ) and its first-order derivative into E_M and E_R yields the equilibrium solution for the supply chain system. Substituting this equilibrium solution into the state equation, and based on the boundary condition τ(0)=τ0≥0, yields the optimal trajectory for the manufacturer's unit product emissions reduction. Substituting Vcs_c(τ)intothe equation yields the optimal profit function for the supply chain system. The process is illustrated in the figure below:

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翻译:谷歌翻译

参考文献:[1]徐春秋,赵道致,原白云,等.上下游联合减排与低碳宣传的微分博弈模型[J].管理科学学报,2016,19(02):53-65.

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来源:LearningYard学苑

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