A problem cannot be merely solved by a graphical method when there are a large number of decision variables and constraints. Practical mathematical procedures exist to solve such complex problems. One such procedure is called the Simplex Method. Such procedures for the solution of the Linear Programming problem can be referred to in any book on operational Research. The idea behind this is to explain the concepts and applications of Linear Programming in the area of Production and Operations Management. The manual solution by Simplicity method may not be difficult when the number of variables and the number of constraints are one-digit numbers. Beyond that, the manual procedure may become laborious and cumbersome. Today there are computers and readymade Computer Software packages available to take care of this problem.
The assumptions underlying Linear programming are:
1. Objective Function and the constants are all linear relationships:
The corollaries of this assumption are that:
(a) We assume that there are no economies of scale or diseconomies of scale; six units of product Y require six times as much of raw material as required for one unit of Y.
(b) We assume that there are no interactions between the decision variables; the total raw material required for X and Y was a simple addition of individual requirements for X and Y.
2. There is only one objective function. In our product mix problems there was only one objective and that to maximize the profit. But, the case is not always so simple in practice. Often, for a particular decision, an organization may have a number of objectives with possibly some priorities between them, all of which need to be considered together. For instance, the production planning should be such that both the total costs of production as well as the time delays for delivery of the products need to be minimized. This type of a problem cannot be solved by simple Linear Programming. Linear Programming, therefore confines itself to a single objective or a situation where the multiple objectives need to be transformed/modified into a single objectives function.
Other Related Methods:
There are procedures by which a multiple objectives problem can be transformed to a single objective problem. This can be done by considering the trade-offs between different objectives or by ranking different objectives in terms of priorities and converting the problem to a series to a series of single objective problems. A procedure called Goal Programming can also be used in such a case.
The case where the decision variables cannot take fractional values, a related technique called Integer Programming can be used.
Application of Linear Programming:
Linear programming can be used effectively for production and operations Management situations. Usually the objective is to either maximize the profit or minimize the profit or minimize the total cost or the delay factor. There are always constraints or limitations on production capacity, the quality of the products and constraints on the sale ability of a product in a particular period of time. There may be different constraints related to the decision variables based at different time periods. For instance, in the months of harvest the labor availability is very low, or that in a company, absenteeism is high during the months of summer due to the wedding season. Many of the production problems can be formulated into Linear Programming problems. Thus it is a very useful technique for various planning and other decisions in production operations.
The basic work content in solving a linear programming problem is not in its solution per se, but in its formulation. Mathematical formulation of the problem is the first step in linear programming. Needless to say, if the formulation is wrong the solution would also be wrong.