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Table A-6

The Complete Initial Simplex Tableau

The variable with the largest positive cj - zj value is the entering variable.

The Simplex Method

A-9

N o w t h e r o w i s c o m p u t e d b y s u b t r a c t i n g t h e r o w v a l u e s f r o m t h e ( t o p ) r o w v a l u e s . F o r e x a m p l e , i n t h e c o l u m n t h e r o w v a l u e i s c o m p u t e d a s T h i s v a l u e a s w e l l a s o t h e r v a l u e s a r e s h o w n i n T a b l e A - 6 , w h i c h i s t h e c o m p l e t e i n i - ( i . e . , t h e Z v a l u e ) i s g i v tial simplex tableau with all values filled in. This tableau represents the solution at the ori- gin, where x1 = 0, x2 = 0, s1 = 40, and s2 = 120. The profit represented by this solution e n 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 0 i n T a b l e A - 6 . T h i s z j c j - z j 4 0 - 0 = 4 0 . c j - z j x 1 solution is obviously not optimal because no profit is being made. Thus, we want to move to a solution point that will give a better solution. In other words, we want to produce either some bowls (x1) or some mugs (x2). One of the nonbasic variables (i.e., variables not in the present basic feasible solution) will enter the solution and become basic. c j z j c j - z j

c j

0 0

Basic Variables

s s2 1

z j

c j - z j

40

50

0

0

Quantity

x1

x2

s1

s2

40

1

2

1

0

120

4

3

0

1

0

0

0

0

0

40

50

0

0

The Entering Nonbasic Variable As an example, suppose the pottery company decides to produce some bowls. With this deci- sion x1 will become a basic variable. For every unit of x1 (i.e., each bowl) produced, profit will be increased by $40 because that is the profit contribution of a bowl. However, when a bowl (x1) is produced, some previously unused resources will be used. For example, if

x1 = 1

then

s 1 = 3 9 h r . o f l a b o r 1 + 2 ( 0 ) + s 1 = 4 0 x 1 + 2 x 2 + s 1 = 4 0 h r . o f l a b o r

and

s 2 = 1 1 6 l b . o f c l a y 4 ( 1 ) + 3 ( 0 ) + s 2 = 1 2 0 4 x 1 + 3 x 2 + s 2 = 1 2 0 l b . o f c l a y

In the labor constraint we see that with the production of one bowl, the amount of slack, or unused, labor is decreased by 1 hour. In the clay constraint the amount of slack is decreased by 4 pounds. Substituting these increases (for x1) and decreases (for slack) into the objective function gives

c j Z = 40(1) + 50(0) + 0(-1) + 0(-4) Z = $40 z j

T h e f i r s t p a r t o f t h i s o b j e c t i v e f u n c t i o n r e l a t i o n s h i p r e p r e s e n t s t h e v a l u e s i n t h e r o w ; t h e s e c o n d p a r t r e p r e s e n t s t h e v a l u e s i n t h e r o w . T h e f u n c t i o n e x p r e s s e s t h e f a c t t h a t t o I n produce some bowls, we must give up some of the profit already earned from the items they replace. In this case the production of bowls replaced only slack, so no profit was lost. g e n e r a l , t h e r o w v a l u e s r e p r e s e n t t h e n e t i n c r e a s e p e r u n i t o f e n t e r i n g a n o n b a s i c c j - z j z j c j

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