Introduction to Linear Programming

BSAD 141

Dave Novak

Topics Covered

Discussion of Linear Programming Example handout

Linear Programming

Linear – a linear relationship exists between variables (like our regression example)

Program – a set of steps is followed to solve the model

The LP Model

1) Decision variables (x1, x2,… xi)Typically represent quantities of something What are the “optimal” values of x1, x2?

2) Objective Function (OF): the linear equation describing the mathematical relationship between the decision variables Only 1 OFThe expression we are maximizing or

minimizing

The LP Model 3) Constraints: linear equations expressing

values the decision variables (x1, x2) can or cannot take on

Restrictions or conditions that apply to the problemFor example, resource limitations like labor,

materials, production capacity, etc.Nonnegativity constraints

Discussion Consider the question of how many trucks

should be made by FordWhat is a problem definition or objective?What are some decision variablesWhat are some constraints?

• If profit increases based on the number of trucks sold, Ford would want to make an unlimited number of trucks…

• We have to impose realistic constraints

Irregular Problems May not always be a “best” solution

Multiple optimal solutionsInfeasible solution

• No values of (x1, x2) satisfy all constraints at the same time

Unbounded solution• Solution space for a Max problem is not bounded

When to use LP The decision involves minimizing or

maximizing something The decision involves questions about how

much or how many You have constraints on the decision

variablesLaborInput materialsProduction limitations

Example See the handout