The issue discussed in this paper is a bi-level problem in which two rivals compete in attracting customers and maximizing their profits which means that competitors competing for market share must compete in the centers that are going to be located in the near future. In this paper, a nonlinear model presented in the literature considering customer preferences is linearized. Customer behavior means that the customer patronizes the most attractive (most comfort) location that he/she wants to be served among the locations of the first-level decision maker (Leader) and the second-level decision maker (Follower). Four types of exact algorithms have been introduced in this paper which include three types of full enumeration procedures and a developed branch-and-bound procedure. Moreover, a clustering-based algorithm has been presented that can provide a good approximation (a good lower bound) to the mentioned binary problem. For this purpose, the numerical results obtained are compared with the results of the full enumeration, heuristic and the branch-and-bound procedure.

The hybrid cardiovascular modeling approach integrates an in vitro experiment with a computational lumped‐parameter simulation, enabling direct physical testing of medical devices in the context of closed‐loop physiology. The interface between the in vitro and computational domains is essential for properly capturing the dynamic interactions of the two. To this end, we developed an iterative algorithm capable of coupling an in vitro experiment containing multiple branches to a lumped‐parameter physiology simulation. This algorithm identifies the unique flow waveform solution for each branch of the experiment using an iterative Broyden's approach. For the purpose of algorithm testing, we first used mathematical surrogates to represent the in vitro experiments and demonstrated five scenarios where the in vitro surrogates are coupled to the computational physiology of a Fontan patient. This testing approach allows validation of the coupling result accuracy as the mathematical surrogates can be directly integrated into the computational simulation to obtain the “true solution” of the coupled system. Our algorithm successfully identified the solution flow waveforms in all test scenarios with results matching the true solutions with high accuracy. In all test cases, the number of iterations to achieve the desired convergence criteria was less than 130. To emulate realistic in vitro experiments in which noise contaminates the measurements, we perturbed the surrogate models by adding random noise. The convergence tolerance achievable with the coupling algorithm remained below the magnitudes of the added noise in all cases. Finally, we used this algorithm to couple a physical experiment to the computational physiology model to demonstrate its real‐world applicability.