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268 CHAPTER 6 SIMPLEX-BASED SENSITIVITY ANALYSIS AND DUALITY

The final tableau for the original HighTech Industries problem is shown here.

x 1 x 2 s 1 s 2 s 3
Basis c B 50 40 0 0 0

x 2 40 0 1 0.32 0 0.12 12
s 2 0 0 0 0.32 1 0.12 8
x 1 50 1 0 0.20 0 0.20 30
50 40 2.80 0 5.20 1 980
z j
0 0 2.80 0 5.20
c j – z j

The optimal solution to the primal problem is x 1 ¼ 30, x 2 ¼ 12, s 1 ¼ 0, s 2 ¼ 8 and
s 3 ¼ 0. The optimal value of the objective function is 1980.
What observation can we make about the relationship between the optimal value
of the objective function in the primal and the optimal value in the dual for the
HighTech problem? The optimal value of the objective function is the same (1980)
for both. This relationship is true for all primal and dual linear programming
problems and is stated as property 1.
Property 1
If the dual problem has an optimal solution, the primal problem has an
optimal solution and vice versa. Furthermore, the values of the optimal
solutions to the dual and primal problems are equal.
This property tells us that if we solved only the dual problem, we would know that
HighTech could make a maximum of E1980.

Interpretation of the Dual Variables
Before making further observations about the relationship between the primal and
the dual solutions, let us consider the meaning or interpretation of the dual variables
u 1 , u 2 and u 3 . Remember that in setting up the dual problem, each dual variable is
associated with one of the constraints in the primal. Specifically, u 1 is associated with
the assembly time constraint, u 2 with the UltraPortable display constraint and u 3
with the warehouse space constraint.
To understand and interpret these dual variables, let us return to property 1 of
the primal–dual relationship, which stated that the objective function values for the
primal and dual problems must be equal. At the optimal solution, the primal
objective function results in:

50x 1 þ 40x 2 ¼ 1980 (6:10)

while the dual objective function is:

150u 1 þ 20u 2 þ 300u 3 ¼ 1980 (6:11)

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