Page 21 - DCOM303_DMGT504_OPERATION_RESEARCH

Operations Research

Notes Linear Programming decisions are made obviously under certainty conditions i.e., when the
existing situation and the variables are known. The results obtained will be either optimal or
nearly optimal. It even helps in cross verification of the results obtained through the process of
mere intuition and the one arrived at with the use of Linear Programming technique while an
optimum solution is being anticipated.
The general Linear Programming Problem calls for optimizing (maximizing/minimizing) a
linear function for variables called the ‘objective function’ subject to a set of linear equations
and/or inequalities called the ‘constraints or restrictions.’

2.1 Basic Terminology

The word ‘linear’ is used to describe the relationship among two or more variables which are
directly or precisely proportional.
Programming’ means the decisions which are taken systematically by adopting alternative
courses of action.

Basic Requirements and their Relationships

1. Decision Variables and their Relationships: The decision variable refers to any candidate
(person, service, projects, jobs, tasks) competing with other decision variables for limited
resources. These variables are usually interrelated in terms of utilization of resources and
need simultaneous solutions, i.e., the relationship among these variables should be linear.
2. Objective Function: The Linear Programming Problem must have a well defined objective
function to optimize the results. For instance, minimization of cost or maximization of
profits. It should be expressed as linear function of decision variables (Z = X + X , where
1 2
Z represents the objective, i.e., minimization/maximization, X and X are the decision
1 2
variables directly affecting the Z value).
3. Constraints: There would be limitations on resources which are to be allocated among
various competing activities. These must be capable of being expressed as linear equalities
or inequalities in terms of decision variables.

4. Alternative Courses of Action: There must be presence of alternative solutions for the
purpose of choosing the best or optimum one.
5. Non-Negativity Restrictions: All variables must assume non-negative values. If any of
the variable is unrestricted in sign, a tool can be employed which will enforce the negativity
without changing the original information of a problem.
6. Linearity and Divisibility: All relationships (objective function and constraints) must
exhibit linearity i.e., relationship among decision variables must be directly proportional.
It is assumed that decision variables are continuous, i.e., fractional values of variables
must be permissible in obtaining the optimum solution.
7. Deterministic: In Linear Programming it is assumed that all model coefficients are
completely known. For example: profit per unit.

2.2 Application of Linear Programming

LP is a widely used technique of OR in almost every decision of a business and management.
However, Linear Programming is exclusively used in the following areas:
1. Production Management

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