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“覃学苦练（69）：精读硕士论文《考虑原制造商双向公平关切的闭环供应链决策研究》公平中性集中决策求解”
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“Qin Xuekulian (69): Intensive reading of the master's thesis "Research on closed-loop supply chain decision-making considering the two-way fairness concerns of the original manufacturer" Fair neutral centralized decision-making solution”
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一、思维导图（Mind mapping）

本期推文围绕公平中性集中决策模型求解展开精读，该部分是第3章量化分析的核心，旨在通过明确决策逻辑、验证模型可行性、求解关键变量与界定边界与评估绩效的完整流程，构建供应链系统最优的基准。

This article focuses on a detailed analysis of the solution method for a fair and neutral centralized decision-making model. This section is the core of the quantitative analysis in Chapter 3, aiming to establish an optimal benchmark for supply chain systems by clearly defining the decision-making logic, verifying model feasibility, solving for key variables, defining boundaries, and evaluating performance through a complete process.

二、精读内容（Intensive reading content）

（一）集中决策的决策顺序（Decision-making order in centralized decision-making）

公平中性集中决策下，原制造商、零售商、第三方再制造商整合为单一决策主体（如“供应链联盟”），按共同协商与同步决策开展活动：

Under fair and neutral centralized decision-making, original manufacturers, retailers, and third-party remanufacturers are integrated into a single decision-making entity (such as a "supply chain alliance") and carry out activities based on joint consultation and synchronized decision-making:

首先，三方结合市场需求与产品成本，共同确定三种产品终端售价，因追求整体利润最大，无需考虑批发价、专利许可费等内部变量；

First, the three parties jointly determine the final selling prices of the three products based on market demand and product costs. Since the goal is to maximize overall profit, they do not need to consider internal factors such as wholesale prices or patent licensing fees.

其次，依据最优价格与消费者需求规律，同步确定生产量以匹配市场需求；最后，计算系统利润、消费者剩余及社会福利，完成决策闭环。这种决策无先后博弈与信息不对称，主体共享信息、协同行动，从全局出发决策，避免分散决策中领导者与随者博弈的效率损失。

Secondly, based on the optimal price and consumer demand laws, the production volume is determined simultaneously to match market demand; finally, the system profit, consumer surplus and Social welfare are calculated to complete the decision-making loop. This approach eliminates sequential bargaining and information asymmetry; all parties share information and act collaboratively, making decisions from a holistic perspective, thereby avoiding the efficiency losses inherent in decentralized decision-making where one party acts as a leader and the other as a follower.

（二） 最优价格求解（Optimal price calculation）

在验证模型可行性后，依据边际利润等于零的经济学原理，利润最大化的均衡条件求解最优价格。对于集中决策的系统利润函数，分别对三种产品的价格求一阶偏导，并令偏导数等于零，形成包含三个价格变量的方程组。

After verifying the feasibility of the model, the optimal price is determined by applying the economic principle that marginal profit equals zero, which is the equilibrium condition for profit maximization. For the system profit function under centralized decision-making, we take the first-order partial derivative with respect to the price of each of the three products and set the derivatives equal to zero, thereby forming a system of three equations with price variables.

通过联立求解该方程组，最终得到三种产品的最优定价：其定价逻辑均遵循平衡成本与市场需求的原则，既考虑产品自身的生产/再制造成本，也结合消费者对不同产品的支付意愿，确保价格既能覆盖成本，又能最大限度激发市场需求，实现成本、需求与利润的最优平衡，贴合现实中成本加成定价与市场需求调节的商业逻辑。

By solving this system of equations, we obtain the optimal pricing for the three products. This pricing strategy adheres to the principle of balancing costs and market demand, considering both the production/remanufacturing costs of the products and consumers' willingness to pay. This ensures that the price covers costs while maximizing market demand, achieving an optimal balance between cost, demand, and profit, thus aligning with the real-world business logic of cost-plus pricing and market demand adjustment.

（三） 最优生产量推导（Derivation of the optimal production quantity）

得到最优价格后，将其代入前文基于消费者效用推导的需求函数，即可计算出三种产品的最优生产量。需求函数刻画了价格差异对需求量分配的影响，因此当价格确定时，每种产品的市场需求量也随之确定。

Once the optimal price is determined, substituting it into the demand function derived from consumer utility allows us to calculate the optimal production quantity for each of the three products. The demand function describes how price differences affect the allocation of demand; therefore, when the price is fixed, the market demand for each product is also determined.

这一过程将价格决策转化为生产决策，形成价格、需求量与利润的完整逻辑链，为后续计算系统利润、消费者剩余等绩效指标提供了量化基础，使决策从价格层面延伸至运营层面。

This process transforms pricing decisions into production decisions, establishing a complete logical link between price, demand, and profit, and providing a quantitative basis for calculating performance indicators such as system profit and consumer surplus. This allows decision-making to extend from the pricing level to the operational level.

（四） 模型可行性验证（Model feasibility verification）

求解的首要步骤是从数学上验证系统利润函数存在唯一最优解，这是后续推导的前提。研究通过构建系统利润函数关于三种产品价格的三阶Hessian矩阵。

The first step in finding the solution is to mathematically verify that the system's profit function has a unique optimal solution; this is a prerequisite for subsequent derivations. The study achieves this by constructing the third-order Hessian matrix of the system's profit function with respect to the prices of the three products.

分析其负定性：一阶顺序主子式为负、二阶顺序主子式为正、三阶顺序主子式为负，证明系统利润函数是三种产品价格的联合凹函数。这一结论意味着，存在且仅存在一组价格组合，能使供应链系统利润达到最大值，不会出现“多个最优解”或“利润无上限”的情况，为模型的科学性与后续求解的有效性提供了数学支撑，确保推导结果具有明确的经济意义。

Analyzing its negative definiteness: the first-order principal minor is negative, the second-order principal minor is positive, and the third-order principal minor is negative, which proves that the system's profit function is a jointly concave function of the three product prices. This conclusion implies that there exists a unique optimal price combination that maximizes the supply chain system's profit, thus avoiding scenarios of multiple optimal solutions or unbounded profit. This provides mathematical support for the scientific validity of the model and the effectiveness of subsequent solutions, ensuring that the derived results have clear economic significance.

（五） 系统利润与约束条件（System profitability and constraints）

1. 系统最优利润计算（System optimal profit calculation）

将最优价格与最优生产量代入系统利润函数，可得到集中决策下供应链的最大系统利润。该利润值是供应链整体效率最优的核心标志，后续分散决策的系统利润需与之对比，以衡量分散决策因双重边际化产生的效率损失，同时也是设计协调契约时需达到的理想目标，为判断契约是否有效的提供量化标准。

By substituting the optimal price and optimal production quantity into the system profit function, we can obtain the maximum system profit under centralized decision-making. This profit value serves as the core indicator of the optimal overall efficiency of the supply chain. The system profit under decentralized decision-making should be compared with this value to measure the efficiency loss caused by the double marginalization effect. Furthermore, it represents the ideal target to be achieved when designing coordination contracts, providing a quantitative standard for evaluating the effectiveness of such contracts.

2. 成本约束条件界定（Defining cost constraints）

为确保求解结果符合现实商业逻辑（如生产量非负、企业有利可图），研究还推导了产品成本的约束范围。这些约束条件的核心作用是界定模型的适用场景，排除成本不合理导致决策无意义的情况，使模型结论更具现实针对性，而非单纯的理论推导。

To ensure that the solution results align with real-world business logic (such as non-negative production quantities and profitable operations), the study also derived constraints for product costs. The core purpose of these constraints is to define the applicable scenarios for the model, eliminating situations where unreasonable costs lead to meaningless decisions, thus making the model's conclusions more practically relevant, rather than merely a theoretical derivation.

（六） 消费者剩余与社会福利计算（Calculating consumer surplus and social welfare）

除核心的价格、生产量、利润外，求解过程还将最优决策变量代入消费者剩余与社会福利函数，得到集中决策下的具体数值。其中，消费者剩余通过积分计算不同产品购买群体的效用差值总和，反映消费者从购买中获得的额外利益；

In addition to the core elements of price, production quantity, and profit, the solution process also substitutes the optimal decision variables into the consumer surplus and social welfare functions to obtain specific numerical values under centralized decision-making. Specifically, consumer surplus is calculated by integrating the sum of utility differences across different groups of product buyers, reflecting the additional benefits consumers derive from their purchases.

社会福利则为系统利润与消费者剩余之和，体现供应链对社会整体价值的贡献。这两个指标的求解突破了仅关注企业利润的局限，从消费者利益和社会价值维度完善了集中决策的绩效评估体系，为后续对比分散决策、分析公平关切行为的影响提供了更全面的判断依据。

Social welfare is defined as the sum of system profit and consumer surplus, reflecting the overall value that the supply chain contributes to society. By incorporating these two indicators, the evaluation system for centralized decision-making goes beyond solely focusing on enterprise profits, and enhances the performance assessment framework by considering both consumer interests and social value. This provides a more comprehensive basis for comparing decentralized decision-making approaches and analyzing the impact of fairness considerations.

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翻译：谷歌翻译

参考资料：百度百科、Chat GPT

参考文献：闫毓敏.考虑原制造商双向公平关切的闭环供应链决策研究[D].湖南大学,2021.

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