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This paper proposes a primal-dual method for solving the minimum cost flow problem with an addi- tional linear constraint. Each branch of a given network has two dimensional distance associated with it. The first element is related to the cost, and the second to the coefficient of the additional constraint. The lexico-shortest route is defined as the route with the minimum distance in the lexicographical ordering. A loop with negative distance is called a lexico-negative loop. A lexico-negative loop such that the first element of its distance is negative, never appears. When there exists a lexico-negative loop such that the first element is zero, and that the second is negative, the current primal solution (flow) is improved by changing the flow around the loop. Otherwise, the lexico-shortest route exists and the current dual solution is jmproved. Our algorithm is a pure network algorithm in the meaning that we need not know what is the basis for the current solution. It is very suitable to the cases when degeneracies often occur.

THE LEXICO-SHORTEST ROUTE ALGORITHM FOR SOLVING THE MINIMUM COST FLOW PROBLEM WITH AN ADDITIONAL LINEAR CONSTRAINT