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## The Simplex Solution Method

This completes the process of filling in the initial simplex tableau. The remaining values i n t h e a n d r o w s , a s w e l l a s s u b s e q u e n t t a b l e a u v a l u e s , a r e c o m p u t e d m a t h e m a t i - c j - z cally using simplex formulas. j z j

The following list summarizes the steps of the simplex method (for a maximization model) that have been presented so far:

1. 2.

3.

First, transform all inequalities to equations by adding slack variables. Develop a simplex tableau with the number of columns equaling the number of vari- ables plus three, and the number of rows equaling the number of constraints plus four. Set up table headings that list the model decision variables and slack variables.

4.

5.

6.

Insert the initial basic feasible solution, which are the slack variables and their quan- tity values. A s s i g n v a l u e s f o r t h e m o d e l v a r i a b l e s i n t h e t o p r o w a n d t h e b a s i c f e a s i b l e s o l u t i o n c variables on the left side. Insert the model constraint coefficients into the body of the table. j

C o m p u t i n g t h e z j a n d c j - z j R o w s

The zj row values are computed by m u l t i p l y i n g t h e c o l u m n v a l u e s c by the variable column values and summing. j

So far the simplex tableau has been set up using values taken directly from the model. From t h i s p o i n t o n t h e v a l u e s a r e d e t e r m i n e d b y c o m p u t a t i o n . F i r s t , t h e v a l u e s i n t h e r o w a r e c o m p u t e d b y m u l t i p l y i n g e a c h c o l u m n v a l u e ( o n t h e l e f t s i d e ) b y e a c h c o l u m n v a l u e u n d e r Q u a n t i t y , a n d a n d t h e n s u m m i n g e a c h o f t h e s e s e t s o f v a l u e s . T h e z j s 2 x 1 , x 2 , s 1 , values are shown in Table A-5. c j z j

## Table A-5

T h e S i m p l e x T a b l e a u w i t h z Row Values j

c j

0 0

Basic Variables

Quantity

40 x1

50 x2

0 s1

0 s2

s s2

1

40 120

1 4

2 3

1 0

0 1

j

j

z c -z

j

0

0

0

0

0

F o r e x a m p l e , t h e v a l u e i n t h e r o w u n d e r t h e q u a n t i t y c o l u m n i s f o u n d a s f o l l o w s : z j

Q u a n t i t y c j

• 0

* 40 = 0

• 0

* 120 = 0 zq = 0

T h e v a l u e i n t h e r o w u n d e r t h e c o l u m n i s f o u n d s i m i l a r l y : x 1 z j

c j

x1

The simplex method works by moving from one solution (extreme) point to an adjacent point until it locates the best solution.

0*1=0 0*4=0 z j = 0

A l l t h e o t h e r r o w v a l u e s f o r t h i s t a b l e a u w i l l b e z e r o w h e n t h e y a r e c o m p u t e d u s i n g t h i s z formula. j

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