In the original or primal problem statement, the goal is to maximize profits, and the (primal) objective function is
The dual problem goal is to minimize implicit values or shadow prices for the firm’s resources. Defining VA, VB, and VC as the shadow prices for inputs A, B, and C, respectively, and π* as the total implicit value of the firm’s fixed resources, the dual objective function (the dual) is
Because the firm has 32 units of A, the total implicit value of input Ais 32 times A’s shadow price, or 32VA. If VA, or input A’s shadow price, is found to be $1.50 when the dual equations are solved, the implicit value of A is $48 (= 32 _ $1.50). Inputs B and C are handled in the same way.
Dual Constraints
In the primal problem, the constraints stated that the total units of each input used to produce X and Y must be equal to or less than the available quantity of input. In the dual, the constraints state that the total value of inputs used to produce one unit of X or one unit of Y must not be less than the profit contribution provided by a unit of these products. In other words, the shadow prices of A, B, and C times the amount of each of the inputs needed to produce a unit of X or Y must be equal to or greater than the unit profit contribution of X or of Y. Because resources have value only when used to produce output, they can never have an implicit value, or opportunity cost, that is less than the value of output.
In the example, unit profit is defined as the excess of price over variable cost, price and variable cost are both constant, and profit per unit for X is $12 and for Y is $9. As shown in Table, each unit of X requires 4 units of A, 1 unit of B, and 0 units of C. The total implicit value of resources used to produce X is 4VA + 1VB. The constraint requiring that the implicit cost of producing X be equal to or greater than the profit contribution of X is
Because 2 units of A, 1 unit of B, and 3 units of C are required to produce each unit of Y, the second dual constraint is
Because the firm produces only two products, the dual problem has only two constraint equations.
Dual Slack Variables
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 become
AND
These slack variables are subtracted from the constraint equations, because greaterthanorequalto inequalities are involved. Using slack variables, the lefthand sides of the constraint conditions are thus decreased to equal the righthand 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 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.
Azero 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 profitmaximizing output level. Apositive 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 Yexactly 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.
Afirm 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.
Solving the Dual Problem
The dual programming problem can be solved with the same algebraic technique that was employed to obtain the primal solution. In this case, the dual problem is
subject to
and
where
Because there are only two constraints in this programming problem, the maximum number of nonzerovalued variables at any corner solution is two. One can proceed with the solution by setting three of the variables equal to zero and solving the constraint equations for the values of the remaining two. By comparing the value of the objective function at each feasible solution, the point at which the function is minimized can be determined. This is the dual solution. To illustrate the process, first set VA = VB = VC = 0, and solve for LX and LY:
Because LX and LY cannot be negative, this solution is outside the feasible set. The values just obtained are inserted into Table as solution 1. All other solution values can be calculated in a similar manner and used to complete Table. It is apparent from the table that not all solutions lie within the feasible space. Only solutions 5, 7, 9, and 10 meet the nonnegativity requirement while also providing a number of nonzerovalued variables that are exactly equal to the number of constraints. These four solutions coincide with the corners of the dual problem’s feasible space.
At solution 10, the total implicit value of inputs A, B, and C is minimized. Solution 10 is the optimum solution, where the total implicit value of employed resources exactly equals the $108 maximum profit primal solution. Thus, optimal solutions to primal and dual objective functions are identical.
At the optimal solution, the shadow price for input C is zero, VC = 0. Because shadow price measures the marginal value of an input, a zero shadow price implies that the resource in question has a zero marginal value to the firm. Adding another unit of this input adds nothing to the firm’s maximum obtainable profit. A zero shadow price for input C is consistent with the primal solution that input C is not a binding constraint. Excess capacity exists in C, so additional units of C would not increase production of either X or Y. The shadow price for input A of $1.50 implies that this fixed resource imposes a binding constraint. If an additional unit of
Solutions for the Dual Programming Problem
A is added, the firm can increase total profit by $1.50. It would increase profits to buy additional units of input A at any price less than $1.50 per unit, at least up until the point at which A is no longer a binding constraint. This assumes that the cost of input A is currently fixed. If those costs are variable, the firm would be willing to pay $1.50 above the current price of input A to eliminate this constraint. Because availability of B also imposes an effective constraint, the firm can also afford to pay up to $6 for a marginal unit of B.
Finally, both dual slack variables equal zero at the optimal solution. This means that the implicit value of resources required to produce a single unit of X or Y is exactly equal to the profit contribution provided. The opportunity cost of producing X and Y is zero, meaning that the resources required for their production are not more valuable in some alternative use. This is consistent with the primal solution, because both X and Y are produced at the optimal solution. Any product with a positive opportunity cost is suboptimal and would not be produced.
Using the Dual Solution to Solve the Primal
The dual solution does not indicate optimal amounts of Xand Y. It does, however, provide all the information necessary to determine the optimum output mix. The dual solution shows that input C does not impose a binding constraint on output of X and Y. Further, it demonstrates that π = π* = $108 at the optimum output of X and Y. The dual solution also offers evidence on the value of primal constraint slack variables. To see this, recall the three constraints in the primal problem:
The dual solution indicates that the constraints on Aand B are binding, because both inputs have positive shadow prices, and only resources that are fully utilized have a nonzero marginal value. ccordingly, the slack variables SA and SB equal zero, and the binding primal constraints can be rewritten as
and
With two equations and only two unknowns, this system can be solved for QX and QY. Multiplying the second constraint by two and subtracting from the first provides
and
These values of QX and QY, found after learning from the dual which constraints were binding, are identical to the values found by solving the primal problem directly. Having obtained the value for QY, it is possible to substitute value for QY in constraint C and solve for the amount of slack in that resource:
These relations, which allow one to solve either the primal or the dual specification of a linear programming problem and then quickly obtain the solution to the other, can be generalized by the two following expressions:
Equation states that if an ordinary variable in the primal problem takes on a nonzero value in the optimal solution to that problem, its related dual slack variable must be zero. Only if a particular Qi is zero valued in the solution to the primal can its related dual slack variable, Li, take on a nonzero value. A similar relation exists between the slack variables in the primal problem and their related ordinary variables in the dual, as indicated by Equation 9.16. If the primal slack variable is nonzero valued, then the related dual variable will be zero valued, and vice versa.


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