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 linear-constrained problems, if an optimum exists.

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.

• BASIC CHARACTERISTICS:– Components– Assumptions

Components of the structure of a linear programming model

• Objective function: a mathematical expression of the goal– 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 functions

• Divisibility: noninteger values are acceptable

• 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 mathematically.

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…