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# The Simplex Solution Method

is selected as the pivot row because it corresponds to the minimum positive ratio of 16. In selecting the pivot row, the -4 value for the x2 row was not considered because the mini- mum positive value or zero is selected. Selecting the x2 row would result in a negative quan- tity value for s1 in the fourth tableau, which is not feasible.

Table A-19 The Third Simplex Tableau

c j

3 6

x

2

x

1

Basic Variables

z j

z j - c j

6

Quantity

x1

8/5

0

24/5

1

168/5

6

0

3

0

0

x2

s1

s2

1

-2/5

1/5

0

3/10

-2/5

3

3/5

-9/5

0

3/5

-9/5

6

3

0

0

Quantity

x1

x2

s1

s2

8

4/3

1

0

-1/3

16

10/3

0

1

-4/3

24

4

3

0

-1

-2

0

0

-1

3 0

The fourth simplex tableau, with s1 replacing x1, is shown in Table A-20. Table A-20 is t h e o p t i m a l s i m p l e x t a b l e a u b e c a u s e t h e r o w c o n t a i n s n o p o s i t i v e v a l u e s . T h e o p t i - z j - mal solution is c j

s 2 = 0 e x t r a l b . o f p h o s p h a t e x 2 = 8 b a g s o f C r o p - q u i c k s 1 = 1 6 e x t r a l b . o f n i t r o g e n x 1 = 0 b a g s o f S u p e r - g r o Z = \$24, total cost of purchasing fertilizer

x s 2 1

Basic Variables

z j

z j - c j

Table A-20 Optimal Simplex Tableau

c j

# 1. Transform all Ú constraints to equations by subtracting a surplus variable and adding

an artificial variable. 2 . A s s i g n a v a l u e o f M t o e a c h a r t i f i c i a l v a r i a b l e i n t h e o b j e c t i v e f u n c t i o n . 3 . C h a n g e t h e r o w t o z j - c j . c j - z j c j

Although the fertilizer example model we just used included only Ú constraints, it is possible for a minimization problem to have and = constraints in addition to Ú con- straints. Similarly, it is possible for a maximization problem to have Ú and = constraints in addition to constraints. Problems that contain a combination of different types of inequality constraints are referred to as mixed constraint problems.

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