Web Reference: The minimum-cost flow problem (MCFP) is an optimization and decision problem to find the cheapest possible way of sending a certain amount of flow through a flow network. If the supplies, demands, and capacities of a minimum cost flow problem are all integral, then every basic feasible solution is integer valued. Therefore, the simplex method will provide an integer optimal solution. For problems with lower and upper bound constraints, also need to keep track of arcs with flows at lower bound and arcs with flows at upper bound (see Bertsekas’ book, section 5.3)
YouTube Excerpt: A detailed explanation on the Linear Programming formulation of MCNFP, followed by working on sample

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Minimum Cost Flow Algorithm Step by Step Example 1
Celebrity Minimum cost flow problem Net Worth
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Celebrity Min Cost Flow: Cycle Cancelling Algorithm and ILP Formulation Profile
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