The maximum capacity path problem (MCPP) is a classical combinatorial optimization problem that seeks to find a path with the maximum capacity in a network. In this paper, we consider a fuzzy extension of the MCPP, where the capacities are given as arbitrary fuzzy numbers. Unlike previous approaches that rely on ranking functions or specific orderings, we formulate the fuzzy MCPP as a bi-objective path-finding problem, where one objective is to maximize the nominal capacity and the other is to optimize the reliability value of the path. We propose an efficient algorithm that can find a Pareto optimal path for any aggregation function between the two objectives. We also analyze the special case where the network is acyclic and show that the algorithm can be specialized to run in strongly polynomial time. Furthermore, we present an application of the fuzzy MCPP to the field of optimal control, where we use a discretization algorithm to transform a continuous routing problem into a discrete one and solve it using the proposed algorithm as a subroutine. To implement this application in practice, we run the algorithm on an old real-world project in Iran called Iranrud. Moreover, we report some computational results on grid networks with different sizes that illustrate the performance of the proposed algorithm.
