Linear programming

Linear programming
Linear programming…
• …is a quantitative management tool to
obtain optimal solutions to problems that
involve restrictions and limitations (called
constrained optimization problems).
• …consists of a sequence of steps that
lead to an optimal solution to linearconstrained problems, if an optimum
Typical areas of problems
Determining optimal schedules
Establishing locations
Identifying optimal worker-job assignments
Determining optimal diet plans
Identifying optimal mix of products in a
factory (!!!)
• etc.
Linear programming models
• …are mathematical representations of
constrained optimization problems.
– Components
– Assumptions
Components of the structure of a
linear programming model
• Objective function: a mathematical expression of the
– e. g. maximization of profits
• Decision variables: choices available in terms of
amounts (quantities)
• Constraints: limitations restricting the available
alternatives; define the set of feasible combinations of
decision variables (feasible solutions space).
– Greater than or equal to
– Less than or equal to
– Equal to
• Parameters. Fixed values in the model
Assumptions of the linear
programming model
• Linearity: the impact of decision variables
is linear in constraints and the objective
• Divisibility: noninteger values are
• Certainty: values of parameters are known
and constant
• Nonnegativity: negative values of decision
variables are not accepted
Model formulation
The procesess of assembling information
about a problem into a model.
• This way the problem became solved
1. Identifying decision variables (e.g.
quantity of a product)
2. Identifying constraints
3. Solve the problem.
Graphical linear programming
1. Set up the objective function and the
constraints into mathematical format.
2. Plot the constraints.
3. Identify the feasible solution space.
4. Plot the objective function.
5. Determine the optimum solution.
1. Sliding the line of the objective function away from
the origin to the farthes/closest point of the feasible
solution space.
2. Enumeration approach.
Lets solve some exercises…