# A-12

# Module A

# The Simplex Solution Method

The pivot row is the row corresponding to the leaving variable.

The leaving basic variable is the variable that corresponds to the minimum nonnegative quotient, which in this case is 20. (Note that a value of zero would qualify as the minimum quotient and would be the choice for the leaving variable.) Therefore, s_{1 }is the leaving variable. (At point A in Figure A-3, s_{1 }equals zero because all the labor is used to make the 20 mugs.) The s_{1 }row, highlighted in Table A-8, is also referred to as the pivot row.

s s_{2 }

1

40 120

1 4

2 3

1 0

0 1

z

j

0

0

0

0

0

c -z

j

j

40

50

0

0

40

50

0

0

x_{1 }

x_{2 }

s_{1 }

s_{2 }

Quantity

0 0

Basic Variables

## Table A-8

Pivot Column, Pivot Row, and Pivot Number

c j

The pivot number is the number at the intersection of the pivot column and row.

The value of 2 at the intersection of the pivot row and the pivot column is called the pivot number. The pivot number, row, and column are all instrumental in developing the next tableau. We are now ready to proceed to the second simplex tableau and a better solution.

Developing a New Tableau Table A-9 shows the second simplex tableau with the new basic feasible solution variables o f a n d a n d t h e i r c o r r e s p o n d i n g v a l u e s . c j s 2 x 2

## Table A-9

c The Basic Variables and j V a l u e s f o r t h e S e c o n d Simplex Tableau

c j

50 0

Basic

40

50

0

0

Variables

Quantity

x_{1 }

x_{2 }

s_{1 }

s_{2 }

x s_{2 }2

z j

c j - z j

Computing the new tableau pivot row values.

The various row values in the second tableau are computed using several simplex for- mulas. First, the x_{2 }row, called the new tableau pivot row, is computed by dividing every value in the pivot row of the first (old) tableau by the pivot number. The formula for these computations is

new tableau pivot row values =

old tableau pivot row values pivot number

Computing all remaining row values.

The new row values are shown in Table A-10. To compute all remaining row values (in this case there is only one other row), another formula is used.