THE REAL MATH

Linear programming involves choosing a course of action when the mathematical model of the problem contains only linear functions.

The maximization or minimization of some quantity is the objective in all linear programming problems.All LP problems have constraints that limit the degree to which the objective can be pursued.A feasible solution satisfies all the problem's constraints.An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing or smallest while mimizing.

The linear model consists of the following components:

A set of decision variables.

An objective function.

A set of constraints.

The linear programming model makes the following assumptions:

A Real Life Example

Omega Toys manufactures two toy models:

The decision variable are continuous

The parameters are know with certainty

The objective function and constraints exhibit constant return to scale

There are no interactions between the decision variables

1) Space Ray.

2) Zapper.

Resources are limited to

1000 pounds of special plastic.

40 hours of production time per week

Marketing requirement

Total production cannot exceed 700 dozens.

Number of dozens of Space Rays cannot exceed number of dozens of Zappers by more than 350.

Technological input

Space Rays requires 2 pounds of plastic and 3 minutes of labor per dozen.

Zappers requires 1 pound of plastic and 4 minutes of labor per dozen.

The desired production plan calls for:

To maximize the profits by producing Space Ray ($8 profit per dozen) and Zappers ($5 profit per dozen)

Use resources within the limitations.

Decisions variables:

X1 = Production level of Space Rays (in dozens per week).

X2 = Production level of Zappers (in dozens per week).

Objective Function:

Weekly profit, to be maximized

The Linear Programming Model is -

Max 8X1 + 5X2 (Weekly profit)

subject to

2X1 + 1X2 < = 1000 (Plastic)

3X1 + 4X2 < = 2400 (Production Time)

X1 + X2 < = 700 (Total production)

X1 - X2 < = 350 (Mix)

Xj> = 0, j = 1,2 (Nonnegativity)

This Assigment needs integrated research in order to validate and justify the points very effectively.