# A-10

# Module A

# The Simplex Solution Method

variable into the basic solution. Naturally, we want to make as much money as possible, because the objective is to maximize profit. Therefore, we enter the variable that will give the greatest net increase in profit per unit. From Table A-7, we select variable x_{2 }as the entering basic variable because it has the greatest net increase in profit per unit, $50—the h i g h e s t p o s i t i v e v a l u e i n t h e r o w . c j - z j

## Table A-7

Selection of the Entering Basic Variable

c j

0 0

Basic Variables

s s_{2 }1

z j

c j - z j

40

50

0

0

Quantity

x_{1 }

x_{2 }

s_{1 }

s_{2 }

40

1

2

1

0

120

4

3

0

1

0

0

0

0

0

40

50

0

0

The pivot column is the column corresponding to the entering variable.

The x_{2 }column, highlighted in Table A-7, is referred to as the pivot column. (The opera- tions used to solve simultaneous equations are often referred to in mathematical terminol- ogy as pivot operations.)

The selection of the entering basic variable is also demonstrated by the graph in Figure A-2. At the origin nothing is produced. In the simplex method we move from one solution point to an adjacent point (i.e., one variable in the basic feasible solution is replaced with a variable that was previously zero). In Figure A-2 we can move along either the x_{1 }axis or the x_{2 }axis in order to seek a better solution. Because an increase in x_{2 }will result in a greater profit, we choose x_{2}.

## Figure A-2

Selection of which item to produce—the entering basic variable

x_{2 }

40

Produce mugs

30

20

A

10

B

0

10

C 20 30 Produce bowls

40

x_{1 }

The Leaving Basic Variable Since each basic feasible solution contains only two variables with nonzero values, one of the two basic variables present, s_{1 }or s_{2}, will have to leave the solution and become zero. Since we have decided to produce mugs (x_{2}), we want to produce as many as possible or, in other words, as many as our resources will allow. First, in the labor constraint we will use all