03-10-2012, 11:26 AM
Linear Programming
Linear Programming.pdf (Size: 620.95 KB / Downloads: 132)
Definition And Characteristics Of Linear Programming
Linear Programming is that branch of mathematical programming which is designed to solve optimization problems where all the constraints as will as the objectives are expressed as Linear function .It was developed by George B. Denting in 1947. Its earlier application was solely related to the activities of the second’ World War. However soon its importance was recognized and it came to occupy a prominent place in the industry and trade.
Linear Programming is a technique for making decisions under certainty i.e.; when all the courses of options available to an organisation are known & the objective of the firm along with its constraints are quantified. That course of action is chosen out of all possible alternatives which yields the optimal results. Linear Programming can also be used as a verification and checking mechanism to ascertain the accuracy and the reliability of the decisions which are taken solely on the basis of manager's experience-without the aid of a mathematical model.
Some of the definitions of Linear Programming are as follows:
"Linear Programming is a method of planning and operation involved in
the construction of a model of a real-life situation having the following elements:
(a) Variables which denote the available choices and
(b) the related mathematical expressions which relate the variables to the
controlling conditions, reflect clearly the criteria to be employed for measuring the benefits flowing out of each course of action and providing an accurate measurement of the organization’s objective. The method maybe so devised' as to ensure the selection of the best alternative out of a large number of alternative available to the organization
Linear Programming is the analysis of problems in which a Linear function of a number of variables is to be optimized (maximized or minimized) when whose variables are subject to a number of constraints in the mathematical near inequalities.
From the above definitions, it is clear that:
(i) Linear Programming is an optimization technique, where the underlying objective is either to maximize the profits or to minim is the Cosp.
(ii) It deals with the problem of allocation of finite limited resources amongst different competiting activities in the most optimal manner.
(iil) It generates solutions based on the feature and characteristics of the actual problem or situation. Hence the scope of linear programming is very wide as it finds application in such diverse fields as marketing, production, finance & personnel etc.
(iv) Linear Programming has be-en highly successful in solving the following types of problems :
(a) Product-mix problems
(b) Investment planning problems
© Blending strategy formulations and
(d) Marketing & Distribution management.
(v) Even though Linear Programming has wide & diverse’ applications, yet all LP problems have the following properties in common:
Constant Value of objective & Constraint Equations. Before a Linear Programming technique could be applied to a given situation, the values or the coefficients of the objective function as well as the constraint equations must be completely known. Further, Linear Programming assumes these values to be constant over a period of time. In other words, if the values were to change during the period of study, the technique of LP would loose its effectiveness and may fail to provide optimal solutions to the problem.
However, in real life practical situations often it is not possible to determine the coefficients of objective function and the constraints equations with absolute certainty. These variables in fact may, lie on a probability distribution curve and hence at best, only the Iikelil1ood of their occurrence can be predicted. Mover over, often the value’s change due to extremely as well as internal factors during the period of study. Due to this, the actual applicability of Linear
Programming tools may be restricted.
Applications Of Linear Programming Techniques In Indian Economy
In a third world developing country like India, the various factors of productions such as skilled labour, capital and raw material etc. are very precious and scarce. The policy planner is, therefore faced with the problem of scarce resource allocation to meet the various competing demands and numerous conflicting objectives. The traditional and conventional methods can no longer be applied in the changed circumstances for solving this problem and are hence fast losing their importance in the current economy. Hence, the planners in our country are continuously and constantly in search of highly objective and result oriented techniques for sound and proper decision making which can be effective at all levels of economic planning. Nonprogrammed decisions consist of capacity expansion, plant location, product line diversification, expansion, renovation and modernization etc. On the other hand, the programmed decisions consist of budgeting, replacement, procurement, transportation and maintenance etc.
In These modern times, a number of new and better methods ,techniques and tools have been developed by the economists all over the globe. All these findings form the basis of operations research. Some of these well-known operations research techniques have been successfully applied in Indian situations, such as: business forecasting, inventory models - deterministic and probabilistic, Linear Programming.Goal programming, integer programming and dynamic programming etc.
Main Application Areas Of Linear Programming
In the last few decades since 1960s, no other mathematical tool or technique has had such a profound impact on the management's decision making criterion as Linear Programming well and truly it is one of the most important decision making tools of the last century which has transformed the way in which decisions are made and businesses are conducted. Starting with the Second World War till the Y -2K problem in computer applications, it has covered a great distance.
Linear Programming.pdf (Size: 620.95 KB / Downloads: 132)
Definition And Characteristics Of Linear Programming
Linear Programming is that branch of mathematical programming which is designed to solve optimization problems where all the constraints as will as the objectives are expressed as Linear function .It was developed by George B. Denting in 1947. Its earlier application was solely related to the activities of the second’ World War. However soon its importance was recognized and it came to occupy a prominent place in the industry and trade.
Linear Programming is a technique for making decisions under certainty i.e.; when all the courses of options available to an organisation are known & the objective of the firm along with its constraints are quantified. That course of action is chosen out of all possible alternatives which yields the optimal results. Linear Programming can also be used as a verification and checking mechanism to ascertain the accuracy and the reliability of the decisions which are taken solely on the basis of manager's experience-without the aid of a mathematical model.
Some of the definitions of Linear Programming are as follows:
"Linear Programming is a method of planning and operation involved in
the construction of a model of a real-life situation having the following elements:
(a) Variables which denote the available choices and
(b) the related mathematical expressions which relate the variables to the
controlling conditions, reflect clearly the criteria to be employed for measuring the benefits flowing out of each course of action and providing an accurate measurement of the organization’s objective. The method maybe so devised' as to ensure the selection of the best alternative out of a large number of alternative available to the organization
Linear Programming is the analysis of problems in which a Linear function of a number of variables is to be optimized (maximized or minimized) when whose variables are subject to a number of constraints in the mathematical near inequalities.
From the above definitions, it is clear that:
(i) Linear Programming is an optimization technique, where the underlying objective is either to maximize the profits or to minim is the Cosp.
(ii) It deals with the problem of allocation of finite limited resources amongst different competiting activities in the most optimal manner.
(iil) It generates solutions based on the feature and characteristics of the actual problem or situation. Hence the scope of linear programming is very wide as it finds application in such diverse fields as marketing, production, finance & personnel etc.
(iv) Linear Programming has be-en highly successful in solving the following types of problems :
(a) Product-mix problems
(b) Investment planning problems
© Blending strategy formulations and
(d) Marketing & Distribution management.
(v) Even though Linear Programming has wide & diverse’ applications, yet all LP problems have the following properties in common:
Constant Value of objective & Constraint Equations. Before a Linear Programming technique could be applied to a given situation, the values or the coefficients of the objective function as well as the constraint equations must be completely known. Further, Linear Programming assumes these values to be constant over a period of time. In other words, if the values were to change during the period of study, the technique of LP would loose its effectiveness and may fail to provide optimal solutions to the problem.
However, in real life practical situations often it is not possible to determine the coefficients of objective function and the constraints equations with absolute certainty. These variables in fact may, lie on a probability distribution curve and hence at best, only the Iikelil1ood of their occurrence can be predicted. Mover over, often the value’s change due to extremely as well as internal factors during the period of study. Due to this, the actual applicability of Linear
Programming tools may be restricted.
Applications Of Linear Programming Techniques In Indian Economy
In a third world developing country like India, the various factors of productions such as skilled labour, capital and raw material etc. are very precious and scarce. The policy planner is, therefore faced with the problem of scarce resource allocation to meet the various competing demands and numerous conflicting objectives. The traditional and conventional methods can no longer be applied in the changed circumstances for solving this problem and are hence fast losing their importance in the current economy. Hence, the planners in our country are continuously and constantly in search of highly objective and result oriented techniques for sound and proper decision making which can be effective at all levels of economic planning. Nonprogrammed decisions consist of capacity expansion, plant location, product line diversification, expansion, renovation and modernization etc. On the other hand, the programmed decisions consist of budgeting, replacement, procurement, transportation and maintenance etc.
In These modern times, a number of new and better methods ,techniques and tools have been developed by the economists all over the globe. All these findings form the basis of operations research. Some of these well-known operations research techniques have been successfully applied in Indian situations, such as: business forecasting, inventory models - deterministic and probabilistic, Linear Programming.Goal programming, integer programming and dynamic programming etc.
Main Application Areas Of Linear Programming
In the last few decades since 1960s, no other mathematical tool or technique has had such a profound impact on the management's decision making criterion as Linear Programming well and truly it is one of the most important decision making tools of the last century which has transformed the way in which decisions are made and businesses are conducted. Starting with the Second World War till the Y -2K problem in computer applications, it has covered a great distance.