Many real-world applications—such as optimizing traffic flow, power distribution,      communication networks, and financial transactions—frequently rely on network flow models. These problems often involve conflicting objectives, such as maximizing flow while minimizing cost or egress time. An effective strategy is to pose such problems as multi-objective optimization models that provide a set of optimal solutions and allow trade-offs between objectives. In this study, we focus on maximizing flow from a single source to multiple sinks in a multi-sink network. The problem is modeled as a multicriteria optimization problem, and an algorithm based on the ϵ-constraint method is proposed. The results are compared with solutions obtained using the weighted sum method.