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One-way car-sharing systems are transportation systems that allow customers to rent cars at stations scattered around the city, use them for a short journey, and return them at any station. The maximum customers' satisfaction problem concerns the task of assigning the cars, initially located at given stations, to maximize the number of satisfied customers. We consider the problem with two stations where each customer has exactly two demands in opposite directions between both stations, and a customer is satisfied only if both their demands are fulfilled. For solving this problem, we propose mixed-integer programming (MIP) models and matheuristics based on local search. We created a benchmark of instances used to test the exact and heuristic approaches. Additionally, we proposed a preprocessing procedure to reduce the size of the instance. Our MIP models can solve to optimality 85% of the proposed instances with 1000 customers in 10 minutes, with an average gap smaller than 0.1% for all these instances. For larger instances (2500 and 5000 customers), except for some particular cases, they presented an average gap smaller than 0.8%. Also, our local-based matheuristics presented small average gaps which are better than the MIP models in some larger instances.