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DUALITY 269

Using Equation (6.10), let us restrict our interest to the interpretation of the primal
objective function. With x 1 and x 2 as the number of units of the Deskpro and the
UltraPortable that are assembled respectively, we have:
0 10 1 0 10 1
value Number of value Number of Total
@ per unit of A@ units of A þ @ per unit of A@ units of A ¼ value of
Deskpro Deskpro UltraPortable UltraPortable production
From Equation (6.11), we see that the coefficients of the dual objective function
(150, 20 and 300) can be interpreted as the number of units of resources available.
Thus, because the primal and dual objective functions are equal at optimality, we
have:
0 1 0 1 0 1
Units of Units of Units of
@ resource A u 1 þ @ resource A u 2 þ @ resource A u 3 ¼ Total value
1 2 3 of production
Thus, we see that the dual variables must carry the interpretations of being the value
per unit of resource. For the HighTech problem,
u 1 ¼ value per hour of assembly time
u 2 ¼ value per unit of the UltraPortable display
u 3 ¼ value per square metre of warehouse space
Have we attempted to identify the value of these resources previously? Recall that in
Section 6.1, when we considered sensitivity analysis of the right-hand sides, we
identified the value of an additional unit of each resource. These values were called
dual prices and are helpful to the decision maker in determining whether additional
units of the resources should be made available.
The analysis in Section 6.1 led to the following dual prices for the resources in the
HighTech problem.

Value per Additional Unit
Resource (dual price)
Assembly time E2.80
Portable display E0.00
Warehouse space E5.20

The dual variables are the Let us now return to the optimal solution for the HighTech dual problem. The
shadow prices, but in a values of the dual variables at the optimal solution are u 1 ¼ 2.80, u 2 ¼ 0 and
maximization problem,
they also equal the dual u 3 ¼ 5.20. For this maximization problem, the values of the dual variables and the
prices. For a dual prices are the same. For a minimization problem, the dual prices and the dual
minimization problem, variables are the same in absolute value but have opposite signs. So, the optimal
the dual prices are the values of the dual variables identify the dual prices of each additional resource or
negative of the dual
variables. input unit at the optimal solution.
In light of the preceding discussion, the following interpretation of the primal and
dual problems can be made when the primal is a product-mix problem.

Primal Problem Given a per-unit value of each product, determine how much of
each should be produced to maximize the value of the total production. Constraints
require the amount of each resource used to be less than or equal to the amount
available.

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