Post on 15-Dec-2015
What is Linear programming?
A mathematical program has objective and constraints Linear programming: An optimization problem whose objective function and constraints are linear.
So, what is a linear relationship? (1) the function is a sum of terms (2) each term of the function has at most one decision variable (multiplied by a constant). Example:
Linear?
1 2 1 2( , ) 100 200f x x x x
1 2 3 1 2 3( , , ) 4 5 7 400g x x x x x x
)5.020)(2()( pppf
1 2 3 1 2 3
2 1
31 2 3 1 2 2
1 2 2
2 2
(1) ( , , ) 4 7 10 7
(2) 3 14.5 30
(3) ( , , ) 10 2.5
(4) 7 100
(5) ( ) 1/
f x x x x x x
x x
f x x x x x x
x x x
f x x
Linear Function?
Implications of linear relationships1. Constant contribution of every decision variable.
2. The contribution of each decision variable is additive.
Because of these (and some other) reasons, computers can solve big problems fast if they are linear programs
What is Linear programming?
Step 1: Identify Decision Variables:
Step 2: Determine Objective function:
Step 3: Determine Constraints:
Step 4: Sign Restrictions:
LP “four”mulation
Things to keep in mind:
(1)Objective and constraints must be linear functions
(2)Decision variables are continuous (non-integer), and in solution may take on fractional values
(3)Coefficients must be deterministic constants
LP formulation
3.3 Special Cases We encounter three types of LPs that do not
have unique optimal solutions.1. Some LPs have an infinite number of
optimal solutions (alternative or multiple optimal solutions).
2. Some LPs have no feasible solutions (infeasible LPs)
3. Some LPs are unbounded: There are points in the feasible region with arbitrarily large (in a max problem) z-values.
Every LP with two variables (4 cases) Case 1: The LP has a unique optimal solution. Case 2: The LP has alternative or multiple
optimal solution Case 3: The LP is infeasible: the feasible
region contains no points. Case 4: The LP is unbounded: There are points
in the feasible region with arbitrarily large z- values (max problem or arbitrarily small z-values (min problem)