© 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.The strategy of the milk-run vehicle routing problem (VRP) is to design networks for material flow among the points in the supply chain while minimizing the total cost and the number of vehicles used. In addition to the theoretical VRP model, products may need to be delivered at specific time intervals and these restrictions affect the cost and the routes. A compromise milk-run route plan emerges by balancing the distance and the time window costs. Therefore, least total distance and least total time window costs should be considered as objectives for the multi objective milk-run modeling. In addition to the crisp model, the fuzzy multi-objective milk-run model should also be considered to increase the membership degrees of objectives by maximizing overall satisfaction degree. This paper contributes to the literature by modeling the multi-objective milk-run and fuzzy multi-objective milk-run optimization under time window constraints, and by applying the proposed approaches to produce suitable paths in the supply network. The validity of the model is demonstrated in an illustrative milk-run vehicle routing problem.