An $$\alpha $$ α -risk appetite cost minimizing model for multi-commodity capacitated p-hub median problem with time windows and uncertain flows

The uncertain flow is a common factor that leads to risks in hub-and-spoke systems. Confronted with risks, it cannot be ignored that the decision maker has different risk-bearing capacities, which influence decision-making. This paper presents an $$\alpha $$ α -risk appetite to characterize the risk-bearing capacity and employs the concept of belief degree to construct it. A multi-commodity capacitated p-hub median model is built with time windows, uncertain flows and the $$\alpha $$ α -risk appetite objective function. In this model, the uncertain flows are depicted as uncertain variables, which possess an empirical uncertainty distribution even with little data. Uncertainty theory has the advantage in converting the proposed uncertain model into a deterministic equivalent form. While the deterministic model is still a mixed 0–1 integer programming problem and has challenges in theory and practice for addressing large-scale problems. Based on the difficulty, we design a hybrid genetic algorithm, where a new principle of cis-position assignment is proposed and inserted into this algorithm to accelerate the solving process. By comparing the proposed model with a deterministic benchmark model, the computational results demonstrate that taking the $$\alpha $$ α -risk appetite and uncertainty into account could avoid the losses of neglecting risks. The observations of cost and hub location reflect the model’s sensitivity about risk appetite levels. Finally, the effectiveness of the proposed algorithm is verified by two data sets, the Civil Aeronautics Board data and the Australian Post data.