# A-6

# Module A

# The Simplex Solution Method

The initial simplex tableau for this model, with the various column and row headings, is shown in Table A-1.

Table A-1 The Simplex Tableau

c j

Basic Variables

## Quantity

____________________________ x_{1 }x_{2 }s_{1 }s 2

____________________________________________________________________

z j _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ c j - z j

The first step in filling in Table A-1 is to record the model variables along the second row from the top. The two decision variables are listed first, in order of their subscript magni- tude, followed by the slack variables, also listed in order of their subscript magnitude. This step produces the row with x_{1}, x_{2}, s_{1}, and s_{2 }in Table A-1.

The basic feasible solution in the initial simplex tableau is the origin

where all decision variables equal zero.

The next step is to determine a basic feasible solution. In other words, which two vari- ables will form the basic feasible solution and which will be assigned a value of zero? Instead of arbitrarily selecting a point (as we did with points A, B, and C in the previous section), the simplex method selects the origin as the initial basic feasible solution because the values of the decision variables at the origin are always known in all linear programming prob- lems. At that point x_{1 }= 0 and x_{2 }= 0; thus, the variables in the basic feasible solution are s_{1 }and s_{2}:

s 1 = 4 0 h r . 0 + 2 ( 0 ) + s 1 = 4 0 x 1 + 2 x 2 + s 1 = 4 0

and

s 2 = 1 2 0 l b . 4 ( 0 ) + 3 ( 0 ) + s 2 = 1 2 0 4 x 1 + 3 x 2 + s 2 = 1 2 0

At the initial basic feasible solution at the origin, only slack variables have a value greater than zero.

In other words, at the origin, where there is no production, all resources are slack, or unused. The variables s_{1 }and s_{2}, which form the initial basic feasible solution, are listed in Table A-2 under the column “Basic Variables,” and their respective values, 40 and 120, are listed under the column “Quantity.”

Table A-2 The Basic Feasible Solution

c j

Basic Variables

Quantity

____________________________ x_{1 }x_{2 }s_{1 }s

2

s_{1 }s

2

40 120

____________________________

j

z_{1 }c -z

j

____________________________