Dual slack variables can be incorporated into the problem, thus allowing the constraint conditions to be expressed as equalities. Letting LX and LY represent the two slack variables, constraint Equations 9.11 and 9.12 become
4 Rules for constructing the dual linear programming problem from its related primal are provided in Appendix 9A, at the end of this chapter.
These slack variables are subtracted from the constraint equations, because greater-than-or-equal-to inequalities are involved. Using slack variables, the left-hand sides of the constraint conditions are thus decreased to equal the right-hand sides' profit contributions. Dual slack variables measure the excess of input value over output value for each product. Alternatively, dual slack variables measure the opportunity cost associated with producing X and Y. This can be seen by examining the two constraint equations. Solving constraint Equation 9.13 for LX, for example, provides
This expression states that LX is equal to the implicit cost of producing 1 unit of X minus the profit contribution provided by that product. The dual slack variable LX is a measure of the opportunity cost of producing product X. It compares the profit contribution of product X, $12, with the value to the firm of the resources necessary to produce it.
A zero value for LX indicates that the marginal value of resources required to produce 1 unit of X is exactly equal to the profit contribution received. This is similar to marginal cost being equal to marginal revenue at the profit-maximizing output level. A positive value for LX indicates that the resources used to produce X are more valuable, in terms of the profit contribution they can generate, when used to produce the other product Y. A positive value for LX measures the firm's opportunity cost (profit loss) associated with production of product X. The slack variable Ly is the opportunity cost of producing product Y. It will have a value of zero if the implicit value of resources used to produce 1 unit of Y exactly equals the $9 profit contribution provided by that product. A positive value for LY measures the opportunity loss in terms of the foregone profit contribution associated with product Y.
A firm would not choose to produce if the value of resources required were greater than the value of resulting output. It follows that a product with a positive slack variable (opportunity cost) is not included in the optimal production combination.
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