**Dual SoIution
as Dual Prices**

First, consider the dual variables u_{1}
and u_{2}. u_{1} has a value of $8, which means an hour
of time of machine 1 has a value of $8. How would the optimal solution
change if there were an additional hour of machine 1 time available? Since
the original solution was to produce 4 units of A (x_{1} = 4) and
none of B, the additional machine 1 time would be used to produce more
units of A. Since one unit of product A takes 2 hours on machine 1, an
additional hour will enable us to produce one half of a unit of A. The
resulting change in profit is one half of $16, or $8. (Note that there
is sufficient excess machine 2 capacity for this production.)

The actual cost of renting additional capacity of machine 1 may be greater or less than $8. If it is less (say $6), then the company should consider renting some additional capacity of machine 1. This is because for each additional hour of available capacity rented, the return is $8, whereas the cost is $6, resulting in an incremental gain of $2.

The value of u_{2} is zero, which means the
"opportunity cost" of an hour of time on machine 2 is zero. This is consistent
with the fact that machine 2 has idle hours following the optimum schedule
of production and making more time on machine 2 available would not increase
profit.

Dual prices are also called shadow prices or marginal values.

Dual prices can be interpreted as the "cost" of a
constraint. We can say that a unit of slack of machine l, which has only
four hours available, "costs" at a rate of $8 per hour (u_{1} has
a value of $8 per unit). It would be worth $8 per hour in increased profitability
to obtain an additional hour on machine 1. Thus, the dual price measures
the value or worth of relaxing a constraint by acquiring an additional
unit of that factor of production.

Now consider the dual slack variables, u_{3}
and u_{4}. The dual slack variables measure the opportunity loss
involved in production of the corresponding primal variable. Since u_{3}
is the slack variable in the first dual constraint, it corresponds to x_{1}
in the primal. Similarly, u_{4} corresponds to x_{2}. u_{4}
has a value of $18; this means that the cost of forcing a unit of x_{2}
into the solution is $18. U_{3} has a value of zero, which means
that there is no opportunity loss involved in producing x_{1}.
Thus, x_{1} is being produced in positive quantity in the optimal
solution, whereas x_{2} is not being produced. If there is a positive
opportunity loss associated with a variable, it will have the value of
zero in the optimal solution; conversely, if the opportunity loss associated
with a variable is zero, the variable will take on some positive value
in the optimal solution. We use the term-reduced costs to refer to the
dual solution variables interpreted in this fashion.

Note that the values of four for u_{1} and
u_{4} are the same (except for the sign) as the
values for x_{3} and x_{2}. This is not a coincidence.
The u values of the dual solution are uniquely the
values of corresponding variables in the primal solution. By corresponding
variables, it is meant that dual ordinary variables (u_{1} and
u_{2}) are associated with primal slack variables (x_{3}
and x_{4}), and dual slack variables (u_{3} and u_{4})
are associated with primal ordinary variables (x_{1} and x_{2}).
These associations make sense, since the dual ordinary variable u_{1}
measures the opportunity cost per hour of using machine 1, and the primal
slack variable x_{3} represents the amount of unused capacity of
machine 1. Similarly, the
values of the dual give the values of the corresponding variables in the
primal solution. Thus, the simplex table for the primal solution provides
both the primal solution and (through the
values) the values of the dual variables.