The design of a multi-stream plate-fin heat exchanger is a highly integrated task with multiple opposing objectives and many degrees of freedom. This work shows how it can be fully or partially automated by the combination of a detailed three-dimensional simulation model and an optimization routine. The desired task is formulated as the target of a multi-objective optimization and solved using a genetic algorithm. The workflow is presented using a cryogenic plate-fin heat exchanger with four process streams. The design is optimized towards high efficiency, low pressure drop, and low unit weight by adjusting the outer geometry, the inlet and outlet distributor configuration, and the detailed stream geometry. A detailed analysis of the Pareto-set gives a good overview of possible solutions, and the optimization routine can automatically find a feasible design with a reasonable tradeoff between the objectives. All elements of the framework are implemented in open source software. A highlight of this research is that a very comprehensive and detailed simulation model is employed in the optimization framework. Thus, the presented method can be easily adjusted to fit the needs of other engineering tasks.