INTRODUCTION
The application of specific operations research techniques to determine the choice among several courses of action, so as to get an optimal value of the measures of effectiveness (objective or goal), requires to formulate (or construct) a mathematical model. Such a model helps to represent the essence of a system that is required for decision-analysis. The term formulation refers to the process of converting the verbal description and numerical data into mathematical expressions, which represents the relationship among relevant decision variables (or factors), objective and restrictions (constraints) on the use of scarce resources (such as labour, material, machine, time, warehouse space, capital, energy, etc.) to several competing activities (such as products, services, jobs, new equipment, projects, etc.) on the basis of a given criterion of optimality. The term scarce resources refers to resources that are not available in infinite quantity during the planning period. The criterion of optimality is generally either performance, return on investment, profit, cost, utility, time, distance and the like.
me, distance and the like. In 1947, during World War II, George B Dantzing while working with the US Air Force, developed LP model, primarily for solving military logistics problems. But now, it is extensively being used in all functional areas of management, airlines, agriculture, military operations, education, energy planning, pollution control, transportation planning and scheduling, research and development, health care systems, etc. Though these applications are diverse, all LP models have certain common properties and assumptions – that are essential for decision-makers to understand before their use.
STRUCTURE OF LINEAR PROGRAMMING MODEL
The general structure of an LP model consists of following three basic components (or parts). Decision variables (activities) The evaluation of various courses of action (alternatives) and select the best to arrive at the optimal value of objective function, is guided by the nature of objective function and availability of resources. For this, certain activities (also called decision variables) usually denoted by x1, x2, . . ., xn are conducted. The value of these variables (activities) represents the extent to which each of these is performed. For example, in a product-mix manufacturing problem, an LP model may be used to determine units of each of the products to be manufactured by using limited resources such as personnel, machinery, money, material, etc.
The value of certain variables may or may not be under the decision-maker’s control. If values are under the control of the decision-maker, then such variables are said to be controllable, otherwise they are said to be uncontrollable. These decision variables, usually interrelated in terms of consumption of resources, require simultaneous solutions. In an LP model all decision variables are continuous, controllable and nonnegative. That is, x1 ≥ 0, x2 ≥ 0, . . ., xn ≥ 0.
The objective function The objective function of each LP problem is expressed in terms of decision variables to optimize the criterion of optimality (also called measure-of-performance) such as profit, cost, revenue, distance etc. In its general form, it is represented as: Optimize (Maximize or Minimize)
Z = c1x1 + c2 x2 + . . . + cn xn,
where Z is the measure-of-performance variable, which is a function of x1, x2, . . ., xn. Quantities c1, c2, . . ., cn are parameters that represent the contribution of a unit of the respective variable x1, x2, . . ., xn to the measure-of-performance Z. The optimal value of the given objective function is obtained by the graphical method or simplex method.