The Minimum Cost Flow Problem and The Network Simplex Solution Method

The Minimum Cost Flow (MCF) Problem is to send flow from a set of supply nodes, through the arcs of a network, to a set of demand nodes, at minimum total cost, and without violating the lower and upper bounds on flows through the arcs. The MCF framework is particularly broad, and may be used to model a number of more specialised network problems, including Assignment, Transportation and Transshipment problems, the Shortest Path Problem, and the Maximum Flow problem. The Network Simplex Method (NSM) is an adaption of the bounded variable primal simplex algorithm, specifically for the MCF problem. The basis is represented as a rooted spanning tree of the underlying network, in which variables are represented by arcs. The method iterates towards an optimal solution by exchanging basic and non-basic arcs. At each iteration, an entering arc is selected by some pricing strategy, and an arc to leave the basis is ascertained. The construction of a new basis tree is called the pivot. There are many strategies for selecting the entering arc, and these determine the speed of solution.

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