 Hello friends, myself Mr. Sanjeev B. Naik working as an assistant professor in Mechanical Engineering Department, Walton District of Technology, Singapore. In this video, I am explaining the linear programming technique, which is used in managerial decision making in the industries or in the business. At the end of this session, learners will be able to know the various applications of linear programming and also will be able to formulate linear programming problem. So, linear programming is one of the most versatile and powerful technique provided by operations research in managerial decision making process for best possible allocation of limited resources. So, we know that every industry or business is having the resources which are very limited. So, it is always expected to have a very best possible utilization of those resources to satisfy the overall objective of the business or the industry and hence linear programming deals with the optimization of the objective function. So, requirement is to formulate the linear programming model, which essentially consists of a set of linear equations known as constraints equations for conditions imposed by the limited resources and whereas a linear equation for objective function. Now, the applications of linear programming is very versatile and very vast. It can be said that every department of an industry can be applied with one or other form of the linear programming. So, in production department, we know that the number of products are produced by utilizing common resources. Whereas, the resources available are limited, however the profit contributed by the product is different and that is why it is always interesting to know the optimal product mix by allocating available resources in such a way that the total profit is maximized and that is why it becomes very versatile application in production management. Similarly, in finance management for allocating the limited budget for various activities involved for the industry or business to achieve overall objective. Similarly, in marketing management to decide upon the marketing policies, linear programming technique helps in research and development, in personal management, in transportation. So, almost all the department, they can solve their problem of allocation of resources by making use of one or other form of linear programming. Now, pause this video for a while and think on the industrial resources. So, just list out all the industrial resources that you know where you can imagine they are imposing the constraints on objective. The requirement to use the linear programming technique is to formulate the linear programming model. So, the basic components of linear programming model are the decision variables, objective function and the constraints. So, in formulation of linear programming problem, one has to identify the decision variables of the problem. As I said that, if the manufacturing department is producing number of products, so the quantity of products to be produced by using limited resources, so that the profit is maximized. In this case, the quantity of product to be produced is the decision variable, so that that will satisfy the objective of maximization of profit and that is why first we have to identify the decision variables and we have to formulate the objective function having relationship with the decision variables as a linear equation and also it is required to formulate the constraint equations which are imposed by the limited resources on decision variables also must be formulated as linear equations. Now it should be ensured that the decision variables can never be negative, either they are 0 or more than 0 and that is what is called as non-negativity restriction, it should be inherently applied to the model. One can explain the formulation of LPP by considering this example. So let me consider an example where a company manufactures two products A and B which are contributing profit of Rs. 40 and Rs. 30 per unit respectively, that means product A is contributing Rs. 40 per unit and product B is contributing Rs. 30 per unit whereas both of them require certain resource, what the requirement resource is grinding machine and polishing machine. So these are the resources which have been required which are common to both the products. The requirement of these resources are for product A to manufacture one unit it requires four hours of grinding time and two hours of polishing machine time and similarly product B requires two hours of grinding time and three hours of polishing time and that is why availability of these resources are limited, per week these are available as a unit 40 and 60 hours respectively. So by utilizing these limited available time of the both the machines, now it is required to decide the number of units of product A and B to be produced to maximize the profit. So that is what an objective to solve this problem. So this is the way we have to identify the objective and objective function and decision variables and the constants. So for this problem we can formulate the LPP by identifying the decision variables as I said the number of units to be produced of product A, let it be reaction and X2 is the number of units of product B to be produced so as to maximize the total profit and that is why X1 and X2 become the decision variables. Now with the help of these variables we have to formulate what is known as objective function. So as we know that the objective function of this problem is to maximize the profit and the contribution of this profit is by product A and B and as we know that the variable that we are using is the quantity X1 for product A and quantity X2 for product B and that is why the total profit as objective function maximized Z is obtained as a linear equation as a 40 X1 plus 30 X2. So as we know that 42 rupees is the profit contribution for A and 30 rupees for B and that is why it gives us 40 X1 plus 30 X2 as the total profit which is been obtained and objective is to maximize it. But this quantity X1 and X2 depend upon the constraints of utilization of the resources. So two resources provide two constraints which are imposed on decision variables. So we know that the first constraint is grinding machine for which the availability is only 40 hours, per week it is available 40 hours. That means we can use equal to 40 hours or less than 40 hours. So total time consumed by both the products on grinding machine is restricted to maximum 40 or less than 40. And then the contribution of requirement of the machine hours for product A is 4X1 because 4 hours per unit for product A is the requirement. So total requirement is 4 into X1. Similarly for product B the requirement is 2 hours per unit so it is 2 into X2. So total requirement is 4X1 plus 2X2 and that must be restricted to 40 hours either less or equal. Similarly we can see the another constraint which is a polishing machine. So maximum availability is 60 so we can use less or equal to 60. In this case the equation which can be developed as a linear 2X1 plus 3X2 as a 2 hours per unit so 2 into X1 for product A 3 hours per unit for product B 3 into X2 must be less or equal to 60. So this is the way the another linear equation having the sign less or equal should be developed for polishing machine constraints. So two constraints are imposed on the decision variables. So one has to satisfy these two constraints to decide value of X1 and X2 and then we can calculate what is the maximum profit which can be derived. And then inherently we know that we should assure to this model that X1 and X2 is always greater than 0. So either we produce the product or is not produced and that is why either its value will be 0 or more than 0 and that will be the requirement of non-negativity restriction. So this is the way the given problem can be converted into appropriate LP model. So these are my references. So books I am referring over here. Thank you.