Embed Size (px)
Citation preview
LINEAR PROGRAMMING
Example 1
0
160
320
480
640
800
960
0 160 320 480 640 800 960
x
yMaximise I = x + 0.8ysubject to x + y 1000
2x + y 15003x + 2y
2400
Initial solution:
I = 0
at (0, 0)
LINEAR PROGRAMMING
Example 1
Maximise I = x + 0.8ysubject to x + y 1000
2x + y 15003x + 2y 2400
Maximise Iwhere I - x - 0.8y = 0subject to x + y + s1 = 1000
2x + y + s2 = 1500
3x + 2y + s3 = 2400
I x y s1 s2 s3 RHS
1 -1 -0.8 0 0 0 0
0 1 1 1 0 01000
0 2 1 0 1 01500
0 3 2 0 0 12400
SIMPLEX TABLEAU
I = 0, x = 0, y = 0, s1 = 1000, s2 = 1500, s3 = 2400
Initial solution
I x y s1 s2 s3 RHS
1 -1 -0.8 0 0 0 0
0 1 1 1 0 0 1000
0 2 1 0 1 0 1500
0 3 2 0 0 1 2400
PIVOT 1
Choosing the pivot column
Most negative number in objective row
I x y s1 s2 s3 RHS
1 -1 -0.8 0 0 0 0
0 1 1 1 0 0 1000 1000/1
0 2 1 0 1 0 1500 1500/2
0 3 2 0 0 1 2400 2400/3
PIVOT 1
Choosing the pivot element
Ratio test: Min. of 3 ratios gives 2 as pivot element
I x y s1 s2 s3 RHS
1 -1 -0.8 0 0 0 0
0 1 1 1 0 0 1000
0 1 0.5 0 0.5 0 750
0 3 2 0 0 1 2400
PIVOT 1
Making the pivot
Divide through the pivot row by the pivot element
I x y s1 s2 s3 RHS
1 0 -0.3 0 0.5 0 750
0 1 1 1 0 0 1000
0 1 0.5 0 0.5 0 750
0 3 2 0 0 1 2400
PIVOT 1
Making the pivot
Objective row + pivot row
I x y s1 s2 s3 RHS
1 0 -0.3 0 0.5 0 750
0 0 0.5 1 -0.5 0 250
0 1 0.5 0 0.5 0 750
0 3 2 0 0 1 2400
PIVOT 1
Making the pivot
First constraint row - pivot row
I x y s1 s2 s3 RHS
1 0 -0.3 0 0.5 0 750
0 0 0.5 1 -0.5 0 250
0 1 0.5 0 0.5 0 750
0 0 0.5 0 -1.5 1 150
PIVOT 1
Making the pivot
Third constraint row – 3 x pivot row
I x y s1 s2 s3 RHS
1 0 -0.3 0 0.5 0 750
0 0 0.5 1 -0.5 0 250
0 1 0.5 0 0.5 0 750
0 0 0.5 0 -1.5 1 150
PIVOT 1
New solution
I = 750, x = 750, y = 0, s1 = 250, s2 = 0, s3 = 150
0
160
320
480
640
800
960
0 160 320 480 640 800 960
x
y
LINEAR PROGRAMMING
Example
Maximise I = x + 0.8ysubject to x + y 1000
2x + y 15003x + 2y
2400
Solution after pivot 1:
I = 750
at (750, 0)
I x y s1 s2 s3 RHS
1 0 -0.3 0 0.5 0 750
0 0 0.5 1 -0.5 0 250
0 1 0.5 0 0.5 0 750
0 0 0.5 0 -1.5 1 150
PIVOT 2
Most negative number in objective row
Choosing the pivot column
I x y s1 s2 s3 RHS
1 0 -0.3 0 0.5 0 750
0 0 0.5 1 -0.5 0 250 250/0.5
0 1 0.5 0 0.5 0 750 750/0.5
0 0 0.5 0 -1.5 1 150 150/0.5
PIVOT 2
Choosing the pivot element
Ratio test: Min. of 3 ratios gives 0.5 as pivot element
I x y s1 s2 s3 RHS
1 0 -0.3 0 0.5 0 750
0 0 0.5 1 -0.5 0 250
0 1 0.5 0 0.5 0 750
0 0 1 0 -3 2 300
PIVOT 2
Making the pivot
Divide through the pivot row by the pivot element
I x y s1 s2 s3 RHS
1 0 0 0 -0.4 0.6 840
0 0 0.5 1 -0.5 0 250
0 1 0.5 0 0.5 0 750
0 0 1 0 -3 2 300
PIVOT 2
Making the pivot
Objective row + 0.3 x pivot row
I x y s1 s2 s3 RHS
1 0 0 0 -0.4 0.6 840
0 0 0 1 1 -1 100
0 1 0.5 0 0.5 0 750
0 0 1 0 -3 2 300
PIVOT 2
Making the pivot
First constraint row – 0.5 x pivot row
I x y s1 s2 s3 RHS
1 0 0 0 -0.4 0.6 840
0 0 0 1 1 -1 100
0 1 0 0 2 -1 600
0 0 1 0 -3 2 300
PIVOT 2
Making the pivot
Second constraint row – 0.5 x pivot row
I x y s1 s2 s3 RHS
1 0 0 0 -0.4 0.6 840
0 0 0 1 1 -1 100
0 1 0 0 2 -1 600
0 0 1 0 -3 2 300
PIVOT 2
New solution
I = 840, x = 600, y = 300, s1 = 100, s2 = 0, s3 = 0
0
160
320
480
640
800
960
0 160 320 480 640 800 960
x
y
LINEAR PROGRAMMING
Example
Maximise I = x + 0.8ysubject to x + y 1000
2x + y 15003x + 2y
2400
Solution after pivot 2:
I = 840
at (600, 300)
I x y s1 s2 s3 RHS
1 0 0 0 -0.4 0.6 840
0 0 0 1 1 -1 100
0 1 0 0 2 -1 600
0 0 1 0 -3 2 300
PIVOT 3
Choosing the pivot column
Most negative number in objective row
I x y s1 s2 s3 RHS
1 0 0 0 -0.4 0.6 840
0 0 0 1 1 -1 100 100/1
0 1 0 0 2 -1 600 600/2
0 0 1 0 -3 2 300
PIVOT 3
Choosing the pivot element
Ratio test: Min. of 2 ratios gives 1 as pivot element
I x y s1 s2 s3 RHS
1 0 0 0 -0.4 0.6 840
0 0 0 1 1 -1 100
0 1 0 0 2 -1 600
0 0 1 0 -3 2 300
PIVOT 3
Making the pivot
Divide through the pivot row by the pivot element
I x y s1 s2 s3 RHS
1 0 0 0.4 0 0.2 880
0 0 0 1 1 -1 100
0 1 0 0 2 -1 600
0 0 1 0 -3 2 300
PIVOT 3
Making the pivot
Objective row + 0.4 x pivot row
I x y s1 s2 s3 RHS
1 0 0 0.4 0 0.2 880
0 0 0 1 1 -1 100
0 1 0 -2 0 1 400
0 0 1 0 -3 2 300
PIVOT 3
Making the pivot
Second constraint row – 2 x pivot row
I x y s1 s2 s3 RHS
1 0 0 0.4 0 0.2 880
0 0 0 1 1 -1 100
0 1 0 -2 0 1 400
0 0 1 3 0 -1 600
PIVOT 3
Making the pivot
Third constraint row + 3 x pivot row
I x y s1 s2 s3 RHS
1 0 0 0.4 0 0.2 880
0 0 0 1 1 -1 100
0 1 0 -2 0 1 400
0 0 1 3 0 -1 600
PIVOT 3
Optimal solution
I = 880, x = 400, y = 600, s1 = 0, s2 = 100, s3 = 0
0
160
320
480
640
800
960
0 160 320 480 640 800 960
x
y
LINEAR PROGRAMMING
Example
Maximise I = x + 0.8ysubject to x + y 1000
2x + y 15003x + 2y
2400
Optimal solution after pivot 3:
I = 880
at (400, 600)
