Abstract: This paper investigates a multi-node, multi-level and interactively connected supply chain network structure consisting of manufacturers, retailers and consumer markets. The nodal enterprises in the network compete with each other and depend on each other because of their benefit objectives. A change in any nodal enterprises in a certain link may cause one in the whole supply chain network. The supply chain network equilibrium is based on the nodal enterprises to maximize their own interests, and the interests of other members are considered, so that the overall supply chain network can be optimized by all-staff collaboration. This method can take the internal and external factors that affect the supply chain network equilibrium into consideration in this supply chain network, and provide a scientific basis for exploring the optimal decision of nodal enterprises under the intervention of internal and external forces. It is of great theoretical significance and practical value to study supply chain network from the perspective of global equilibrium optimization.
In recent years, both in application and in academic research, the topic of supply chain network equilibrium problem has been of great interest, so as to arouse a strong research interest among many scholars. The research area involves the model, analysis and computation for this problem. In this paper, the research aims to establish a nonlinear complementarity model for the supply chain equilibrium problem. This work is different from previous research in that it focuses on two conditions in which a manufacturer cannot sell all of its products, and both manufacturers and retailers use all free gift with purchase to promote sales. Based on this, optimizing behaviors of the various decision-makers can be modeled by KKT conditions, the equilibrium conditions of the manufacturers, the retailers and the consumer markets are derived respectively, and a supply chain equilibrium nonlinear complementarity model with all free gift with purchase and unbalanced production-marketing is presented. Some properties of the equilibrium pattern in terms of existence, boundedness and global uniqueness are provided. In particular, the (strict) monotonicity of the functions involved in the model is shown. For this reason, the algorithm proposed by CHEN et al. (2021) is used to solve this model, and the global convergence and R-linear convergence rate of the algorithm are given.
Finally, we illustrate the model through several numerical examples, in which the equilibrium prices and product shipments are computed under a variety of scenarios, and the variation characteristics on the equilibrium price and product shipment pattern are investigated by a change in relevant parameters such as the number of the product given away, retailer’s management cost and so on. The numerical results indicate that both the equilibrium price and the product shipment quantity increase with an increase in the quantity of gifts given away by manufacturers to retailers and by retailers to consumer markets within a certain number of gifts. The validity and reliability of the model are verified.
However, some factors affecting the supply chain network equilibrium, such as uncertain market demand, time-varying selling prices of retailers and carbon emission policies, are not considered in this paper. It will be the future research direction to further explore the network equilibrium of perishable low-carbon supply chain network equilibrium with random demand and all free gift with purchase. In addition, this work is supported by the Natural Science Foundation of China (12071250), and Shandong Provincial Natural Science Foundation (ZR2021MA088). Particularly, the authors gratefully acknowledge the valuable comments of the editor and the anonymous reviewers.

Key words: supply chain equilibrium, complementarity model, the cost of the product given away, unbalanced production-marketing

摘要： 本文旨在建立一个由制造商、零售商和消费市场组成的供应链均衡非线性互补模型。与以往研究不同,本文关注的是有些制造商不可能销售所有的产品给零售商,并且制造商和零售商均采用全量买赠促销。在此基础上,优化了各决策者的行为,利用KKT条件,分别推导了制造商、零售商和消费市场的均衡条件,建立了具有全量赠送成本、产销不平衡的供应链均衡非线性互补模型。探讨了该模型解的存在性、有界性和全局唯一性等性质,尤其给出了该模型所含函数的(严格)单调性。正因如此,可应用CHEN等(2021)提出的算法求解该模型,并给出算法的全局收敛性和R-线性收敛率。通过数值实验,制造商给零售商、零售商给消费市场馈赠数量函数不同的条件下,分别给出了该问题的均衡价格和产品交易模式,验证了模型的有效性和可靠性。

关键词: 供应链均衡, 互补模型, 赠送产品成本, 产销不平衡