MATH 527 Deterministic OR
Graphical Solution Method for Linear Programs
MATH 327 - Mathematical Modeling
2
4 12 20
10
20
30
6025 21 xx
MATH 327 - Mathematical Modeling
3
4 12 20
10
20
30
6025 21 xx
MATH 327 - Mathematical Modeling
4
4 12 20
10
20
30
6025 21 xx
MATH 327 - Mathematical Modeling
5
4 12 20
10
20
30
6025 21 xx
2021 xx
MATH 327 - Mathematical Modeling
6
4 12 20
10
20
30
6025 21 xx
2021 xx
MATH 327 - Mathematical Modeling
7
4 12 20
10
20
30
6025 21 xx
2021 xx
MATH 327 - Mathematical Modeling
8
4 12 20
10
20
30
6025 21 xx
2021 xx
01 x
MATH 327 - Mathematical Modeling
9
4 12 20
10
20
30
6025 21 xx
2021 xx
01 x
02 x
MATH 327 - Mathematical Modeling
10
4 12 20
10
20
30
6025 21 xx
2021 xx
01 x
02 x
Feasible region
The feasible region is a polygon!!
MATH 327 - Mathematical Modeling
11
How do we find the optimal solution?? We must graph the
isoprofit line.– Straight line– All points on the line
have the same objective value
– When problem is minimization, called an isocost line.
How??– Choose any point in
the feasible region– Find its objective
value (or z-value)– Graph the line
objective function = z-
value.
MATH 327 - Mathematical Modeling
12
4 12 20
10
20
30
Isoprofit linez = 300
MATH 327 - Mathematical Modeling
13
4 12 20
10
20
30
Isoprofit line
MATH 327 - Mathematical Modeling
14
4 12 20
10
20
30
Isoprofit line
MATH 327 - Mathematical Modeling
15
4 12 20
10
20
30
Isoprofit line
MATH 327 - Mathematical Modeling
16
4 12 20
10
20
30
Isoprofit line
MATH 327 - Mathematical Modeling
17
4 12 20
10
20
30
Isoprofit linez = 433 1/3
optimal solution: (20/3, 40/3)z = 433 1/3
MATH 327 - Mathematical Modeling
18
Binding vs. Nonbinding
A constraint is binding if the optimal solution satisfies that constraint at equality (left-hand side = right-hand side). Otherwise, it is nonbinding.
Binding constraints keep us from finding better solutions!!
MATH 327 - Mathematical Modeling
19
4 12 20
10
20
30
optimal solution: (20/3, 40/3)z = 433 1/3
MATH 327 - Mathematical Modeling
20
4 12 20
10
20
30
optimal solution: (20/3, 40/3)z = 433 1/3
binding
MATH 327 - Mathematical Modeling
21
4 12 20
10
20
30
optimal solution: (20/3, 40/3)z = 433 1/3
binding
binding
MATH 327 - Mathematical Modeling
22
Convex Sets
A set of points S is a convex set if the line segment joining any two points in S lies entirely in S
ConvexNonconvex
MATH 327 - Mathematical Modeling
23
Extreme Points
A point P in a convex set S is an extreme point if, for any line segment containing P which lies entirely in S, P is an endpoint of that segment.
A
B
C
D
MATH 327 - Mathematical Modeling
24
Extreme Points
A point P in a convex set S is an extreme point if, for any line segment containing P which lies entirely in S, P is an endpoint of that segment.
A
B
C
D
C and D are extreme pointsA and B are not
MATH 327 - Mathematical Modeling
25
Interesting Facts
The extreme points of a polygon are the corner points.
The feasible region for any linear program will be a convex set.
MATH 327 - Mathematical Modeling
26
Interesting Facts
The feasible region will have a finite number of extreme points
Extreme points are the intersections of constraints (including nonnegativity)
Any linear program that has an optimal solution has an extreme point that is optimal!!
What are the implications?
MATH 327 - Mathematical Modeling
27
2 6 10
4
8
12
1826 21 xx
MATH 327 - Mathematical Modeling
28
2 6 10
4
8
12
1826 21 xx
MATH 327 - Mathematical Modeling
29
2 6 10
4
8
12
1826 21 xx
MATH 327 - Mathematical Modeling
30
2 6 10
4
8
12
1826 21 xx
3053 21 xx
MATH 327 - Mathematical Modeling
31
2 6 10
4
8
12
1826 21 xx
3053 21 xx
MATH 327 - Mathematical Modeling
32
2 6 10
4
8
12
1826 21 xx
3053 21 xx
MATH 327 - Mathematical Modeling
33
2 6 10
4
8
12
1826 21 xx
3053 21 xx
01 x
MATH 327 - Mathematical Modeling
34
2 6 10
4
8
12
1826 21 xx
3053 21 xx
01 x02 x
MATH 327 - Mathematical Modeling
35
2 6 10
4
8
12
1826 21 xx
3053 21 xx
01 x02 x
Feasible Region
MATH 327 - Mathematical Modeling
36
2 6 10
4
8
12
Isocost linez = 54
MATH 327 - Mathematical Modeling
37
2 6 10
4
8
12
Isocost line
MATH 327 - Mathematical Modeling
38
2 6 10
4
8
12
Isocost line
MATH 327 - Mathematical Modeling
39
2 6 10
4
8
12
Isocost line
MATH 327 - Mathematical Modeling
40
2 6 10
4
8
12
Isocost linez = 36 1/4
optimal solution: (5/4, 21/4)z = 36 1/4
MATH 327 - Mathematical Modeling
41
Special Cases
So far, our models have had– One optimal solution– A finite objective value
Does this always happen?
What if it doesn’t?
MATH 327 - Mathematical Modeling
42
Special Case # 1: Unbounded Linear Programs If maximizing: there are points in the
feasible region with arbitrarily large objective values.
If minimizing: there are points in the feasible region with arbitrarily small objective values.
MATH 327 - Mathematical Modeling
43
Special Case #1: Unbounded Linear Programs
maximization minimization
MATH 327 - Mathematical Modeling
44
CAUTION!!!
There is a difference between an unbounded linear program and an unbounded feasible region!!!
MATH 327 - Mathematical Modeling
45
Special Case #2: Infinite Number of Optimal Solutions When isoprofit/isocost lie intersects an
entire line segment corresponding to a binding constraint
Occurs when isoprofit/isocost line is parallel to one of the binding constraints
MATH 327 - Mathematical Modeling
46
Special Case #2: Infinite Number of Optimal Solutions
MATH 327 - Mathematical Modeling
47
Special Case # 3: Infeasible Linear Program Feasible Region is empty
MATH 327 - Mathematical Modeling
48
Every Linear Program
Has a unique optimal solution, or…..
Has infinite optimal solutions, or…..
Is unbounded, or…..
Is infeasible.