Slide3d Simplex Method

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Reference: Decision Maths 2 textbook,chapter 1, section 1

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Simplex Algorithm1. Construct system of equations from LP inequalities

1.1 rewrite objective formula to be equal to zero.1. remove the inequalities from the constraints b! a""ing slac#

variables. Place s!stem of equations in a tableau

$. %epeat the following until solution is optimal$.1 &"entif! the pivot column

'he pivot column is the column with the largest negative value inthe objective equation

$. &"entif! the pivot element using ratio test.Compute ratios of %(S to the correspon"ing entr! in the pivotcolumn'he minimum of these ratios "efines the pivot element.

$.$ )a#ing the pivotManipulate the system of equations so that* pivot element becomes 1* all remaining element in pivot column become +

$., Chec# if the solution is optimal.Solution is optimal if there is no negative entr! in objective row.

LP xample

0

160

320

480

640

800

960

0 160 32 0 480 640 800 960

x

y

)aximise P- x +./!

subject to x ! 1+++

x ! 10++

$x ! ,++

&nitial solution

P ! "

at 2+3 +4

)aximise P - x +./!

subject to x ! 1+++

x ! 10++

$x ! ,++

)aximise P

where P *x *+./! - +subject to x ! s1 - 1+++

x ! s - 10++

$x ! s$ - ,++Constraintequations

Constraintequations

5bjective equation5bjective equation

Slac# variablesSlac# variables

LP xample

2#""1++$+

1$""+1+1+

1"""++111+

"+++*+./*11

%(Ss$ss1!xP

%&MPL' ()*L)+

x ! ", y ! "3 P- +3s1

- 1+++3 s

- 10++3 s$

- ,++

&nitial solution:

1

,++1++$+

10+++1+1+

1+++++111+

++++*+./*11

%(Ss$ss1!xP

P&-( 1 .hoosin/ the pivot columnChoose the column withthe most negative number in objective row

pivot column

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,++6$

-/++,++1++$+

10++6

-70+10+++1+1+

1+++61

-1+++1+++++111+

++++*+./*11

%(Ss$ss1!xP

P&-( 1 .hoosin/ the pivot element

%atio test )in. of $ ratios gives 2 as pivot element

pivot ro0 pivot element

1

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,++1++$+

10+++1+1+

1+++++111+

++++*+./*11

%(Ss$ss1!xP

P&-( 1

pivot ro0 pivot element

1

8ext operation9ivi"e through the pivot row b! the pivot element

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,++1++$+

$"""$""$1"

1+++++111+

++++*+./*11

%(Ss$ss1!xP

P&-( 12

After "ivi"e through the pivot row b! the pivot element

8ext operation 5bjective row pivot row

70+++.0++.01+

$"""$"3"4"1

%(Ss$ss1!:P

P&-( 1After objective row pivot row

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8ext operation ;irst constraint row * pivot row

70+++.0++.01+

2$""3"$1"$""

70+++.0+*+.$+1

%(Ss$ss1!xP

P&-( 1 After first constraint row * pivot row

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8ext operation 'hir" constraint row < $ x pivot row1$"131$""$""

70+++.0++.01+

0++*+.01+.0++

70+++.0+*+.$+1

%(Ss$ss1!xP

P&-( 1After thir" constraint row < $ x pivot row

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10+1*1.0++.0++

70+++.0++.01+

0++*+.01+.0++

70+++.0+*+.$+1

%(Ss$ss1!xP

n5 of P&-( 13

1$"1*1.0++.0++

$"++.0++.01+

2$"+*+.01+.0++

$"++.0+*+.$+1

%(Ss$ss1!xP

n5 of P&-( 1 6e0 solution

P - 70+3 x ! $", y ! "3 s1 - 0+3 s - +3 s$ - 10+

P -0.3y +0.5s2 = 750

0.5y +s1 + = 250

x +0.5y +0.5s2 = 750

0.5y -1.5s2 +s3 = 150

P -0.3y +0.5s2 = 750

0.5y +s1 + = 250

x +0.5y +0.5s2 = 750

0.5y -1.5s2 +s3 = 150

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160

320

480

640

800

960

0 160 3 20 48 0 6 4 0 80 0 9 60

x

y

7raphical equivalent

)aximise P- x +./!

subject to x ! 1+++

x ! 10++

$x ! ,++

Solution after

pivot 1P ! $"

at 270+3 +410+1*1.0++.0++

70+++.0++.01+

0++*+.01+.0++

70+++.0+*+.$+1

%(Ss$ss1!xP

P&-( 2

)ost negative number in objective row

.hoosin/ the pivot column3

10+6+.0-$++

10+1*1.0++.0++

70+6+.0-10++70+++.0++.01+

0+6+.0

-0++0++*+.01+.0++

70+++.0+*+.$+1

%(Ss$ss1!xP


%atio test )in. of $ ratios gives "$ as pivot element

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4""234"1""

70+++.0++.01+

0++*+.01+.0++

70+++.0+*+.$+1

%(Ss$ss1!xP

P&-( 2 Makin/ the pivot

9ivi"e through the pivot row b! the pivot element

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$++*$+1++

8#""93"#"""1

%(Ss$ss1!xP


5bjective row +.$ x pivot row

5

$++*$+1++

1""3111"""

/,++.=*+.,+++1

%(Ss$ss1!xP


;irst constraint row < +.0 x pivot row

5

$++*$+1++

9""312""1"

1++*111+++

/,++.=*+.,+++1

%(Ss$ss1!xP


Secon" constraint row < +.0 x pivot row

5

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$++*$+1++

=++*1++1+

1++*111+++

/,++.=*+.,+++1

%(Ss$ss1!xP

n5 of P&-( 25

4""*$+1++

9""*1++1+

1""*111+++

8#"+.=*+.,+++1

%(Ss$ss1!xP

n5 of P&-( 2 6e0 solutionP - /,+3 x ! 9"", y ! 4""3 s1 - 1++3 s - +3 s$ - +

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0

160

320

480

640

800

960

0 160 3 20 4 80 640 800 960

x

y

)aximise P- x +./!

subject to x ! 1+++

x ! 10++

$x ! ,++

Solution afterpivot

P ! 8#"

at 2=++3 $++4



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$++*$+1++

=++*1++1+

1++*111+++

/,++.=*+.,+++1

%(Ss$ss1!xP

P&-( 4 .hoosin/ the pivot column

)ost negative number in objective row

5

$++*$+1++

=++6=++*1++1+

1++611++*111+++

/,++.=*+.,+++1

%(Ss$ss1!xP


%atio test )in. of ratios gives 1 as pivot element

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$++*$+1++

=++*1++1+

1""3111"""

/,++.=*+.,+++1

%(Ss$ss1!xP


9ivi"e through the pivot row b! the pivot element

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1++*111+++

88""2""#""1

%(Ss$ss1!xP


5bjective row +., x pivot row

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#""1"32"1"

1++*111+++

//++.++.,++1

%(Ss$ss1!xP


Secon" constraint row < x pivot row

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9""31"41""

,++1+*+1+

1++*111+++

//++.++.,++1

%(Ss$ss1!xP


'hir" constraint row $ x pivot row

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=++*1+$1++

,++1+*+1+

1++*111+++

//++.++.,++1

%(Ss$ss1!xP

n5 of P&-( 47

9""*1+$1++

#""1+*+1+

1""*111+++

88"+.++.,++1

%(Ss$ss1!xP

n5 of P&-( 4 -ptimal solution

P - //+3 x ! #"", y ! 9""3 s1 - +3 s - 1++3 s$ - +

8o more negative entries in 5bjective %ow8o more negative entries in 5bjective %ow

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0

160

320

480

640

800

960

0 160 320 4 80 64 0 80 0 9 60

x

y

)aximise P- x +./!

subject to x ! 1+++

x ! 10++

$x ! ,++

5ptimal solution

after pivot $P ! 88"

at 2,++3 =++4


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$*variable LP Problem

> ?xample from 9 ?x 1A @$ 2page 14)aximise P - x1+!=z

Subject to

x $! ,z B $

=x =! z B /x3 !3 z +

Slide3d Simplex Method

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