This paper proposes an algorithm for solving the minimum cost project scheduling problerm with an additional linear constraint whose right hand side is a parameter. Instead of putting the additional constraint, we add it to the objective function, where it is multiplied by a parameter. First, we get an optimal solution when a parameter is zero, and next, increase it to infinity. To do so, we iterate to find two dimensional flow on the given arrow diagram. The bounds for each arrow flow are determined by the current solution. The first elements are related to the costs, and the second to the coefficients in the additional constraint. A lexico-bounded flow is defined as the flow such that each arrow flow is within the bounds in the lexicographical ordering. When a lexico-bounded flow exists, the parameter is increased. Otherwise, the function of the additional constraint is increased. Our algorithm is almost dual to the lexico-shortest route algorithm for the minimum cost flow problem with an additional linear constraint. For example, a loop is replaced by a cutset. Our algorithm is also applicable to a project scheduling problem with two objectives by combining them with a parameter.
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