Properties of Linear Programming Model
The following properties of the linear programming model:
1. A 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 nonnegativity constraint.
Formulation of Linear Programming
Formulation of linear programming is the representation of problem situation in a mathematical form. It involves welldefined 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 maximise resources or profits or, to minimise the cost or time.
For example, assume that a furniture manufacturer produces tables and chairs. If the manufacturer wants to maximise his profits, he has to determine the optimal quantity of tables and chairs to be produced.
Let

x_{1}

=

Optimal production of tables


p_{1}

=

Profit from each table sold


x_{2}

=

Optimal production of chairs


p_{2}

=

Profit from each chair sold.

Hence,


Total profit from tables = p_{1}

x_{1}



Total profit from chairs = p_{2}

x_{2}

The objective function is formulated as below,
Maximize Z or Zmax = p1 x1 + p2 x2
Constraints
When the availability of resources is in surplus, there will be no problem in making decisions. But in real life, organisations 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 a furniture manufacturer, let w be the amount of wood available to produce tables and chairs. Each unit of table consumes a 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
In addition to raw material, if other resources such as labour, machinery and time are also considered as constraint equations
Nonnegativity 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 nonnegativity as a constraint i.e., x_{1}, x_{2} 0