Ch 3 Introduction to Linear Programming

By Kanchala Sudtachat

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

Ex1 Giapetto’s Woodcarving

Ex1 Giapetto’s Woodcarving

3.2 The Graphical Solution of Two-variable LP

Binding and Nonbinding Constraints

The Graphical Solution to Minimization Problems Ex 2 Dorian Auto

Ex 2 Dorian Auto

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.

Ex 3 Alternative or Multiple Optimal Solutions

Ex 3 Alternative or Multiple Optimal Solutions

Ex 4 Infeasible LP

Ex 5 Unbounded LP

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)

3.4 A Diet Problem

3.5 A Work-Scheduling Problem Ex 7 Post Office Problem

3.8 Blending Problems Ex 12: Oil Blending

3.9 Production Process Models Ex 13 Brute Production Process

Ex 14 Sailco Inventory