Decision Modeling/Linear Programming

1. Discuss why and how you would use a liner programming model for a project of your choice, either from your own work or as a hypothetical situation. Be sure that you stae your situation first, before you develpp the LP model

Linear programming is a modeling technique that is used to help managers make logical and informed decisions. All date and input factors are known with certainty. Linear program models are developed in three different steps:

* Formulation

* Solution

* Interpretation

The formulation step deals with displaying the problem in a mathematical form. Once that is developed the solution stage solves the problem and finds the variable values. During the interpretation stage the sensitivity analysis gives managers the opportunity to answer hypothetical questions regarding the solutions that are generated.

There are four basic assumptions of linear programming and they are as follows:

* Certainty

* Proportionality

* Additivity

* Divisibility

Linear programming is the development of modeling and solution procedures which employ mathematical techniques to optimize the goals and objectives of the decision-maker. Programming problems determine the optimal allocation of scarce resources to meet certain objectives. Linear Programming Problems are mathematical programming problems where all of the relationships amongst the variables are linear.

Components of a LP Formulation are as follows:

* Decision Variables

* Objective Function

* Constraints

* Non-negativity Conditions

Decision variables represent unknown quantities. The solutions for these terms are what we would like to optimize. Objective function states the goal of the decision-maker. There are two types of objectives:

* Maximization

* Minimization

Constraints put limitations on the possible solutions of the problem. The availability of scarce resources may be expressed as equations or inequalities which rule out certain combinations of variable values as feasible solutions. Non-negativity conditions are special constraints which require all variables to be either zero or positive.

Linear Programming Example-Problem 1

Chad's Pottery Barn has enough clay to make 24 small vases or 6 large vases. He has only enough of a special glazing compound to glaze 16 of the small vases or 8 of the large vases. Let X1 = the number of small vases and X2 = the number of large vases.

The smaller vases sell for $3 each, and the larger vases would bring $9 each.

Formulation:

Objective function: Maximize 3X1 + 9X2

Subject to: Clay constraint: 1X1 + 4X2  24

1(0) + 4x=24

X=6

(0, 6)

1X+ 4(0)=24

Y=24

(24, 0)

Glaze constraint: 1X1 + 2X2  16

1(0) + 2X=16

X=8

(8, 0)

1X+2(0)=16

Y=16

(0,16)

X1 @ $3.00 X2 @ $9.00 Income

A 0 0 0

B 0 6 $54

C 8 4 $60*

D 16 0 $48

Evaluating all possible corner points that might be the optimal solution, the optimum income