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Among the decisions related to the milk supply chain, those related to the supply of raw milk from farms to the dairy factories are highly important. In this paper, a two-stage scenario-based possibilistic model is developed for designing a milk supply chain network from farms to the dairy factory in the form of location-routing problem. The milk which is collected by collection center (CC) vehicles or directly is delivered by farmers to CCs. The occurrence of disruption is considered in the form of probable scenarios. A given percentage of capacity of CCs and some of the existing routes might be unavailable under each disruption scenario. A possibilistic programming method is used to cope with epistemic uncertainty in parameters (cost, demand, and milk produced). Because of the mathematical model's high complexity in large sizes, a Lagrangian relaxation algorithm is also devised. The proposed model helps to make optimal decisions in the milk collection process from farms to factories according to existing constraints. The numerical results show the efficiency of the solution approach.