Simultaneous competitive supply chain network design with continuous attractiveness variables

•We analyse the simultaneous competitive supply chain network design problem.•We discuss the effect of different modes on the problem.•A two-stage algorithm is proposed to solve the problem.•Bi-level programming is used in the proposed algorithm.

A two-stage algorithm is developed for the competitive supply chain network design problem in which two competitors simultaneously enter the market with no existing rival chain, shape their networks, and set attractiveness of their opened distribution centres to maximize their profits. The customer behaviour is based on the Huff gravity-based rule.The proposed algorithm is constructed based on the Lemke and Howson algorithm and variational inequality formulation with the help of bi-level programming, the modified projection method, and the possibility theory. We derive the equilibrium condition and establish the finite dimensional variational inequality formulation, and provide properties of the equilibrium patterns in terms of the results of existence and uniqueness. Finally, we generate some instances and use a real world study to discuss the effect of the different structures of the competitors, namely centralized, decentralized, cooperative, or unknown modes, on the equilibrium solution.