Abstract: This paper uses delay differential equation to describe the delay phenomenon of quality improvement investment on brand goodwill promotion, and the differential game models under five decision-making modes are constructed, which are non-cooperative game mode, cooperative game mode and partial cooperative game modes of cost sharing(one-way partial cooperative game of manufacturer participating in marketing, one-way partial cooperative game of retailer participating in production, two-way partial cooperative game of manufacturer participating in marketing and retailer participating in production). With the help of Hamilton maximum principle, the optimal quality improvement input strategy of manufacturer, the optimal marketing effort strategy of retailer and supply chain profit are obtained in five cases. Comparing the results of five game models, we find that: 1)Delay phenomenon can reduce manufacturers' enthusiasm for quality improvement investment, but it has no effect on retailers' marketing efforts. Under the influence of delay phenomenon, brand goodwill decays first and then improves. 2)Cooperative game is always the best choice for supply chain performance. Although three kinds of cost-sharing partial cooperative game contracts can not achieve complete coordination of supply chain, they can improve non-cooperative game. 3)Of the three kinds of cost-sharing contracts, two-way cooperative partial cooperative game is the best choice for supply chain, but its Pareto improvement effect increases with the delay time. The results will no longer be obvious.

Key words: delay differential equation, delay phenomenon, differential game, quality improvement input, marketing effort

摘要： 利用时滞微分方程刻画质量改进投入对品牌商誉提升的延迟现象,分别构建了制造商和零售商采取非合作博弈、合作博弈以及成本分担的部分合作博弈(制造商参与营销的单向部分合作博弈、零售商参与生产的单向部分合作博弈、制造商参与营销及零售商参与生产的双向部分合作博弈)五种决策模式下的微分博弈模型。借助哈密尔顿极大值原理,求解得到五种情形下的制造商最优质量改进投入策略和零售商的最优营销努力策略以及供应链利润。对比五种博弈模式下的结果发现:1)延时现象会降低制造商进行质量改进投入的积极性,但对零售商营销努力无影响;品牌商誉在延迟现象影响下出现先衰减后提升的演进规律;2)合作博弈对于供应链绩效总是最优的,三种成本分担的部分合作博弈契约虽不能实现供应链的完全协调,但可以对非合作博弈情形进行帕累托改进;3)对比两种单向部分合作博弈,在提高供应链利润方面,制造商参与营销的成本分担契约优于零售商参与生产的成本分担契约;4)三种成本分担契约中,双向合作的部分合作博弈是供应链的最优选择,但随着延迟时间增大,其帕累托改进效果将不再明显。

关键词: 时滞微分方程, 延迟现象, 微分博弈, 质量改进投入, 营销努力