Bullwhip effect and complexity analysis in a multi-channel supply chain considering price game with discount sensitivity

This paper constructs a multi-channel supply chain that includes a manufacturer, a dual-channel retailer and an online retailer. The dual-channel retailer has a traditional channel and an online channel, while the manufacturer cooperates with the retailers and opens a direct channel simultaneously. The horizontal and historical price discount sensitivities are taken into account to establish the price-sensitive demand functions. A price game model with the heterogeneous expectations is proposed and analysed with the methods of the stability domain, the bifurcation diagram and the maximum Lyapunov exponent. The complexity and bullwhip effect of each channel are investigated with respect to the price adjustment speed and historical price discount sensitivity. The results show that a moderate or low price discount sensitivity keeps the system stable and a high price discount sensitivity brings the system into the twofold cycle state or chaotic state. It is interesting that if both online channels increase the adjustment speeds with similar strategy the system will enter chaos through the Neimark–Sacker bifurcation. The bullwhip effect is affected by the price adjustment speed and the price discount sensitivity, especially in the double-period or chaotic state, where the bullwhip of the overall supply chain is increased significantly. The results show that controlling price discount sensitivity is useful for supply chain node companies.