Introduction to Linear Programming

A linear form is made a mathematical expression of the type a₁x₁ + a₂x₂ + .... + a_nx_n, where a₁, a₂, ..., a_n are constant and x₁, x₂ ... x_n are variable. The word Programming means the procedure of determining a particular program or plan of action. So Linear Programming (LP) is one of the highly significant optimization (maximization / minimization) techniques which are developed in the field of Operations Research (OR).

The methods applied for answering a linear programming problem are fundamentally simple problems; an answer can be achieved by a set of simultaneous equations. Though an exclusive solution for a set of simultaneous equations in n-variables (x₁, x₂ ... x_n), at least one of them is non-zero, can be obtained if there are precisely n relations. When the number of relations is more than or less than n then a unique solution does not exist but several trial solutions can be found.

In different practical conditions, the queries are seen in which the number of relations is not equal to the number of the number of variables and a lot of relations are in the form of inequalities (≤ or ≥) to maximize or minimize a linear function of the variables subject to like conditions. These problems are called as Linear Programming Problem (LPP).

General form of LPP

We develop a mathematical model for common problem of allocating resources to different activities. In particular, this model is to choose the values for x₁, x₂ ... x_n in order to maximize or minimize

Z = c₁x₁ + c₂x₂ +.............+c_nx_n  

subject to limitations or restrictions

a₁₁x₁ + a₁₂x₂ + .............+a₁nx_n (≤ or ≥) b₁

a₂₁x₁ + a₂₂x₂ + ...........+a₂nx_n (≤ or ≥) b₂

.

.

.

a_m1x₁ + a_m2x₂ + ..........+a_mnx_n (≤ or ≥) b_m

and

x₁ ≥ 0, x₂ ≥ 0,..., x_n ≥ 0

Where

Z = value of on the whole measure of performance

x_j = level of activity (for j = 1, 2, ..., n)

c_j = increase in Z that would result from each unit enhance in level of activity j

b_i = amount of resource i that is available for allocation to activities (for i = 1,2, ..., m)

a_ij = amount of resource i used by each unit of activity j

                              Data needed for LP model

• The level of activities x₁, x₂.........x_n are known as decision variables.
• The values of the c_j, b_i, a_ij (for i=1, 2 ... m and j=1, 2 ... n) are the input constants for the model. They are known as parameters of the model.
• The function being maximized or minimized Z = c₁x₁ + c₂x₂ +.... +c_nx_n is known as objective function.
• The restrictions are normally referred to as constraints. The constraint a_i1x₁ + a_i2x₂ ... a_inx_n are at times knownas functional constraint (L.H.S constraint). x_j ≥ 0 restrictions are referred to as non-negativity constraint.

Assumptions in LPP

1. Additivity

Sum of the resources utilized by diverse activities must be equivalent to the total amount of resources used by each activity for all resources collectively or individually.

1. Proportionality

The contribution or participation of each variable in the main function or its usage of the resources is directly proportional to the value of the variable that is if resource accessibility rises by some percentage, then the output will also rise by same percentage

1. Divisibility

The variables are not limited to integer values

1. Finiteness

Variables and constraints are limited in number.

1. Deterministic

Coefficients in the objective function and constraints are totally known and do not vary during the period under study in all the problems taken.

1. Optimality

In LPP, we establish the decision variables in order to optimize the objective function of the LPP.

1. The problem includes only one objective either profit maximization or cost minimization.

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