We investigate a supply chain selling a mass customization (MC) product to the market. A decision model is built when the degree of customization, price and lead time of the MC product affect market demand. The optimal decisions of the MC supply chain are analyzed in non-cooperative decision-making and cooperative decision-making. Moreover, the sensitivity analysis of optimal customer order decoupling point (CODP) position is conducted. Our results show that compared with non-cooperative decision-making, the optimal CODP moves to a later position in cooperative decision-making. The relationships between the optimal CODP position and the price sensitivity coefficient are different in two different scenarios. The impact of the customization sensitivity coefficient on the optimal CODP position is opposite to the impact of the lead time sensitivity coefficient on the optimal CODP position. When the production cost coefficient or investment coefficient increases, the optimal CODP moves to an earlier position.