Terminology and Requirements of Linear Programming           

The word linear is used to describe the relationship among too or more variables which are directly proportional. For example, if doubling (or tripling) the production of a production of product will exactly double (or triple) the profit and required resources, then it is a linear relationship. The word 'programming' means planning of activities in a 'optimal' if it maximizes or minimizes some measure or criterion of effectiveness such as profit cost or sales.

            Regardless of the way one defines linear programming certain basic requirements which are given below are necessary before the technique can be employed for optimization problems.

1.                  Decision variables and their relationship.  The decision (activity) variables refer to candidates (products, services, projects etc.) that are competition with one another for sharing the given limited resources. These variables are usually inter-related in terms of utilization of resources and need simultaneous solutions. The relationship among these variables should be linear.

2.                  Well defined objective function. A linear programming problem must have a clearly defined objective function to optimize which may be either to maximize contribution but utilizing available resources or it may be to produce at the lowest possible cost by using a limited amount of productive factors. It should be expressed as linear function of decision variables.

3.                  Presence of constraints or restrictions. There must be limitations on resources (like production capacity manpower time machines markets etc) 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 restrictions. There must be alternative courses of action. For example, it must be possible to make a selection between various combinations of the productive factors such as men machines materials markets, etc.

5.                  Non negative restrictions. All decision variables must assume non-negative values as negative values of physical quantities is an impossible situation. If any of the variables is unrestricted in sign a trick (discussed later in the chapter) can be employed which enforces non-negativity changing the original information of the problem.

6.                  Linearity. The basic requirements of a linear programming problem is that both the objective and constrains must be expressed in terms of linear equations or inequalities. It is well known that if the number of machines in a plant is increased the production in the plant also proportionately increases. Such a relationship, giving corresponding increment in one variable for every increment in other is called linear and can be graphically represented in the form of a straight line.

7.                  Finiteness. There must be finite number of activities and constraints otherwise an optimal solution cannot be computed.

8.                  Additivity. It means that sum of the resources used by different activities must be equal to the total quantity of resources used by each activity for all the resources individually and collectively. In other words, interaction among the activities of the resources does not exist.

9.                  Divisibility. This implies that solutions need not be in whole numbers (integers). Instead they are divisible and may take any fractional value. If a fraction of a product cannot be produced (like one fourth of a bus), an integer programming problem exists.

0.              Deterministic. We assume that conditions of certainty exist i.e., the coefficients in the objections function and constraints are completely known (deterministic) and do not change during the period being studied, e.g., profit per unit of each product amounts of resources available are fixed during the planning period.

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