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This paper is addressing a new class of on-demand transport problems oriented toward customers. A mixed-integer linear programming model is proposed with new effective constraints that contribute to enhancing the quality of service. An exact resolution has been achieved, leading to lower bounds of the solution space of real cases of on-demand transport problems. To overcome the exponential computational time of the exact resolution, an evolutionary descent method is developed. It relies on a new operator for perturbing the search. The comparative results between the new method and the branch and bound show low gaps for almost all the instances tested with lower execution times. The results of the evolutionary descent method are also compared with the results of two different heuristics, namely a Tabu Search and an Evolutionary Local Search. Our evolutionary method demonstrates its effectiveness through competitive and promising results. [ABSTRACT FROM AUTHOR]