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TABLEAU FORM: THE GENERAL CASE 235

Now a 1 can be selected as the basic variable for this row, and its value is given by
the right-hand side. Once we have created tableau form by adding an artificial
variable to each equality constraint, the Simplex method proceeds exactly as
before.

Eliminating Negative Right-Hand Side Values

One of the properties of the tableau form of a linear programme is that the values
on the right-hand sides of the constraints have to be nonnegative. In formulating a
linear programming problem, we may find one or more of the constraints have
negative right-hand side values. To see how this situation might happen, suppose
that the management of HighTech has specified that the number of units of the
UltraPortable model, x 2 , has to be less than or equal to the number of units of the
Deskpro model, x 1 , after setting aside five units of the Deskpro for internal company
use. We could formulate this constraint as:

x 2 x 1 5 (5:13)

Subtracting x 1 from both sides of the inequality places both variables on the left-
hand side of the inequality. So,

x 1 þ x 2 5 (5:14)

Because this constraint has a negative right-hand side value, we can develop an
equivalent constraint with a nonnegative right-hand side value by multiplying both
sides of the constraint by 1. In doing so, we recognize that multiplying an inequal-
ity constraint by 1 changes the direction of the inequality.
So, to convert inequality (5.14) to an equivalent constraint with a nonnegative
right-hand side value, we multiply by 1 to obtain:

x 1 x 2 5 (5:15)

We now have an acceptable nonnegative right-hand side value. Tableau form for this
constraint can now be obtained by subtracting a surplus variable and adding an
artificial variable.
For a greater-than-or-equal-to constraint, multiplying by 1 creates an equivalent
less-than-or-equal-to constraint. For example, suppose we had the following greater-
than-or-equal-to constraint:

6x 1 þ 3x 2 4x 3 20
Multiplying by 1 to obtain an equivalent constraint with a nonnegative right-hand
side value leads to the following less-than-or-equal-to constraint:
6x 1 3x 2 þ 4x 2 20
Tableau form can be created for this constraint by adding a slack variable.
For an equality constraint with a negative right-hand side value, we simply multiply
by 1 to obtain an equivalent constraint with a nonnegative right-hand side value. An
artificial variable can then be added to create the tableau form.

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