Properties of linear programming model

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Properties of linear programming model

The following properties of the linear programming model:
1. Relationship among decision variables must be linear in nature.
2. A model must have an objective function.
3. Resource constraints are essential.
4. A model must have a non-negativity constraint

Formulation of Linear Programming

Formulation of linear programming is the representation of problem situation in a mathematical form. It involves well defined decision variables, with an objective function and set of constraints.

Objective Function

The objective of the problem is identified and converted into a suitable objective function. The objective function represents the aim or goal of the system (i.e., decision variables) which has to be determined from the problem. Generally, the objective in most cases will be either to maximize resources or profits or, to minimize the cost or time.
For example, assume that a furniture manufacturer produces tables and chairs. If the manufacturer wants to maximize his profits, he has to determine the optimal quantity of tables and chairs to be produced.
Optimal production of tables
Profit from each table sold
Optimal production of chairs
Profit from each chair sold.
Total profit from tables = p1
Total profit from chairs = p2
The objective function is formulated as below,
Maximize Z or Zmax = p1 x1 + p2 x2


When the availability of resources are in surplus, there will be no problem in making decisions. But in real life, organizations normally have scarce resources within which the job has to be performed in the most effective way. Therefore, problem situations are within confined limits in which the optimal solution to the problem must be found.
Considering the previous example of furniture manufacturer, let w be the amount of wood available to produce tables and chairs. Each unit of table consumes w1 unit of wood and each unit of chair consumes w2 units of wood.
For the constraint of raw material availability, the mathematical expression is,
w1 x1 + w2 x2 w

Non-negativity Constraint

Negative values of physical quantities are impossible, like producing negative number of chairs, tables, etc., so it is necessary to include the element of non-negativity as a constraint i.e., x1, x2 0

General Linear Programming Model

A general representation of LP model is given as follows:
Maximize or Minimize, Z = p1 x1 + p2 x2 ………………pn xn
Subject to constraints,
w11 x1 + w12 x2 + ………………w1n xn or = or w1
w21 x1 + w22 x2 ………………w2n xn or = or w2
. . . .
. . . .
. . . .
wm1 x1 + wm2 x2 +………………wmn xn           or =       wm

Non-negativity constraint,

xi    o      (where i = 1,2,3 …..n)