Learning Outcomes
• Mahasiswa dapat menghitung
pemecahan masalah/kasus model
PL dengan menggunakan metode
grafik..
Outline Materi:
• Contoh-contoh pemecahan kasus PL
dengan menggunakan metode grafik..
Contoh-contoh Kasus
Optimal
Solution
y
Example 1:
4
Maximize
x+y
Subject to: x + 2 y  2
x3
3
Feasible
Region
2
y4
x0 y0
1
0
These LP animations were
created by Keely Crowston.
x
0
1
2
3
Contoh-contoh Kasus
y
Example 2:
4
Minimize **
Multiple
Optimal
Solutions!
x-y
Subject to: 1/3 x + y  4
-2 x + 2 y  4
3
2
Feasible
Region
x3
x0 y0
1
0
0
1
2
3
x
Contoh-contoh Kasus
y
Example 3:
40
Minimize
x + 1/3 y
Subject to: x + y  20
-2 x + 5 y  150
30
Feasible
Region
20
x5
x 0 y 0
10
x
Optimal
Solution
0
0
10
20
30
40
Do We Notice Anything From These 3
Examples?
Extreme point
y
y
y
4
4
40
3
3
30
2
2
20
1
1
10
0
0
1
2
3
x
0
0
1
2
3
x
0
x
0
10
20
30
40
A Fundamental Point
y
y
y
4
4
40
3
3
30
2
2
20
1
1
10
0
0
1
2
3
x
0
0
1
2
3
x
0
x
0
10
20
30
If an optimal solution exists, there is
always a corner point optimal solution!
40
Graphing 2-Dimensional LPs
Second
Corner pt.
Example 1:
Optimal
Solution
y
4
Maximize
x+y
Subject to: x + 2 y  2
x3
3
Feasible
Region
2
y4
x 0 y 0
1
Initial
0
Corner pt.
x
0
1
2
3
And We Can Extend this to Higher
Dimensions
Linear Programs in higher
dimensions
minimize
subject to
z=
7x1 + x2 + 5x3
x1
- x2
+ 3x3 >= 10
5x1
+ 2x2 -
x1,
x2,
x3
x3 >= 6

0
What happens at (1,2,3)?
What does it tell us about z* = optimal value of z?
The Dual
maximize
subject to
z’ =
10y1 + 6y2
y1
+ 5y2 <= 7
-y1
+ 2y2 <= 1
3y1 –
y1,
y2
y2 <= 5

0
What is the dual of a dual?
Every feasible solution of the dual gives a lower bound on z*
The Primal
minimize
z=
subject to
7x1 + x2 + 5x3
x1
- x2
+ 3x3 >= 10
5x1
+ 2x2 -
x1,
x2,
x3
x3 >= 6

0
Every feasible solution of the primal is an upper bound on
the solution to the dual.
LP Geometry
• Forms
a n dimensional polyhedron
• Is convex : If z1 and z2 are two feasible
solutions then λz1+ (1- λ)z2 is also feasible.
• Extreme points can not be written as a
convex combination of two feasible points.
Barrier Algorithms
Simplex solution path
Barrier central path
o Predictor
o Corrector
Optimum
Interior Point Methods
y
objective
LP optimum
feasible solutions
x
Linear Program
y
objective
optimum of
LP relaxation
IP optimum
feasible
solutions =
rounding down optimum
of LP relaxation
x
Integer Program