# FORMULATION OF LINEAR PROGRAMMING - Quantitative Techniques for management

## Formulation of Linear Programming Problem examples

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.

Let x1= Optimal production of tables
P1= Profit from each table sold
X2= Optimal production of chairs
P2= Profit from each chair sold.

Hence, Total profit from tables = p1 x1

Total profit from chairs = p2 x2

The objective function is formulated as below,
Maximize Z or Z max = 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, 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

In addition to raw material, if other resources such as labor, machinery and time are also considered as constraint equations.

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

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