How to End Up With a Negative Reduced Cost in Linear Programming

Having a variable with a negative reduced cost in your linear programming model indicates that increasing the value of that variable will have a negative effect on the objective function's value. The reduced cost of a variable refers to how much the coefficient of the variable will have to improve before it will have a positive value in the optimal solution. For a maximization problem, the coefficient will need to increase; for a minimization problem, the coefficient will need to decrease.

Instructions

    • 1

      Take the optimal solution for your linear programming model. For example, consider a car manufacturing plant with an optimal production plan involving an optimal value of 1,000 cars and 500 jeeps. Imagine one of the constraints being that the total number of cars has to be less than or equal to twice the amount of jeeps. Both vehicles have a positive net contribution. In this model, increasing or lowering the amount of cars or jeeps will decrease the manufacturing plant's profit. At this point both variables are basic, because they don't have an upper or lower bound. Both variables have a corresponding reduced cost of 0.

    • 2

      Modify the model by lowering the net profit contribution of one of the variables to a point where it's a nonbasic variable and its optimal value is at the lower bound of 0. For example, decrease the net contribution of jeeps to such a low number that the optimal solution involves only manufacturing cars. The reduced costs value of the jeeps variable may now have a negative value, such as -100 dollars per jeep.

    • 3

      Adjust the objective cost coefficient of the variable to make it active in your optimal solution. For example, increase the profit contribution of jeeps by at least 100 dollars. This will make jeeps a basic variable and it'll be profitable to manufacture jeeps once again.

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