The non-equivalence between shadow prices and dual variables

Abstract

In the context of economic interpretation applied to linear programming problems, the concept of shadow price associated with the i-th constraint is explored, indicating the change in the objective function when the resource b_i of that constraint is modified by one unit. This impact reflects on the i-th dual variable w_i. In problems with non-degenerate primal optimal solutions, there's an established relationship between the shadow price of the i-th constraint and the i-th dual variable w_i. However, in scenarios with degenerate solutions, this relationship may be invalid. The purpose of this study is to conduct a detailed analysis of this dynamic and present two methodologies to compute the correct shadow prices in linear programming problems with degenerate primal optimal solutions. Using an example problem proposed by Strum (1969) that exhibits degenerate solutions, it demonstrates the non-equivalence between dual solution and shadow price. In the end, the correct shadow prices of the problem are determined by employing the strategies outlined in the article.