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SETTING UP THE INITIAL SIMPLEX TABLEAU 217

Once we have the problem in tableau form, we then can obtain an initial Simplex
tableau that shows a basic feasible solution and then use the Simplex procedure to
search for improved solutions to this initial tableau. Given that the Simplex proce-
dure is iterative, we keep searching for improved solutions until we can no longer
find a better one. The current solution is then the optimal solution. To summarize,
the following steps are appropriate:
Step 1. Formulate the problem.
Step 2. Set up the problem in the standard form using slack and surplus variables
as appropriate.
Step 3. Set up the problem in tableau form.
Step 4. Set up the initial Simplex tableau.
Step 5. Search for improvements in the current solution shown in the tableau
until no further improvement can be made.

5.3 Setting Up the Initial Simplex Tableau

Clearly, the tableau we have developed simply sets out the LP problem formulation.
In order to proceed, we need to create a tableau that represents an initial basic,
feasible solution. This will allow us to begin the Simplex procedure and search for
the optimal solution. When an LP problem with all less-than-or-equal-to constraints
is written in the standard form, it is easy to find a basic feasible solution. We simply
set the decision variables equal to zero and solve for the values of the slack variables.
For the HighTech problem, we then have: x 1 ¼ 0, x 2 ¼ 0, s 1 ¼ 150, s 2 ¼ 20 and
s 3 ¼ 300 as the initial basic feasible solution (the slack variables simply take the
values of the right-hand side values of the constraints). Note that this solution puts
us at Point 1 in Figure 5.1. We now need an initial Simplex tableau that corresponds
to this point and to this basic, feasible, solution. This is not difficult to do. The initial
Simplex tableau for the HighTech problem is shown below:

Value
x 1 x 2 s 1 s 2 s 3
Basis 50 40 0 0 0
3 5 1 0 0 150
s 1
0 1 0 1 0 20
s 2
8 5 0 0 1 300
s 3

To practise setting up the We label the first column Basis to indicate the basic solution and we show
portion of the simplex the basic variables s 1 , s 2 , s 3 . From the tableau we can determine the values for
tableau corresponding to each of the basic variables: s 1 ¼ 150, s 2 ¼ 20, s 3 ¼ 300. Non-basic variables – those
the objective function
and constraints at this not appearing in the Basis list – automatically take zero values. Here, we have x 1
point, try Problem 4. and x 2 both equal to zero. Notice that for each basic variable, its corresponding
column in the tableau has a 1 in the only non-zero position. Such columns are
known as unit columns or unit vectors. Second, a row of the tableau is associated
with each basic variable. This row has a 1 in the unit column corresponding to the
basic variable. The value of each basic variable is then given by the b value in the

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