 Hello and welcome to the recession. In this session we discuss the perfect mathematical formulation of a linear programming problem. For this we will consider an example. We have this example of a company which manufactures toys of two types, type A and type B. Now So type A toys, 6 minutes are required for cutting and for type B toys, 8 minutes are required for cutting. Then for assembling toy A or toys of type A requires 12 minutes type B requires 6 minutes are immersed in the minutes available for cutting and then there are immersed in the minutes available and the profit is the toys of type A $30 for the toys of type B to find out the number of the company attains maximum profit. For this we consider two decision variables X and Y where X is the number of toys of type A manufactured by the company and Y is the number of toys of type B manufactured by the company. Now we have limitations on the times for cutting and assembling for the toys of two types A and B. So this means we would have two constraints, cutting constraint and assembling constraint. Now as we have 6 minutes are required for cutting of toys A manufactured by the companies taken as is the total time required by the toys of type A for cutting by the toys of type B for cutting and we have Y as the number of toys of type B. So 8 Y would be the total time taken by the toys of type B for cutting of at most 3 as 20 minutes available for cutting. So this means 6 X plus 8 Y would be less than equal to which would be equal to 180 plus 20 minutes or you can say 200 minutes. So 6 X plus 8 Y is less than equal to 200. So this is the constraint. Now from the table as you can see that we have 10 minutes available for assembling for the toys of type X would be the total time for assembling taken by the toys of type B for assembling. So 6 Y would be the total time taken by the toys of type B for assembling. 10 minutes are at most available for assembling. So 12 X plus 6 Y would be less than equal to plus 10 minutes or 6 Y is less than equal to 250 and so this is the we have the decision variables as X and Y which would be non-negative and Y greater than equal to 0 and these are the non-negative without the objective function for this problem. We are given that $60 profit would be attained from the toys of type A and $30 profit would be attained from the toys of type B. So would be equal to 60 X dollars manufactured by the company and $60 profit paid on manufacturing toys of type A and the same would be equal to 30 Y dollars. Now the total profit equal to plus 30 Y profit needs to be maximized. So we take Z be equal to plus 30 Y objective function for this problem. You see that this is a linear function X and Y which are the decision variables. So we now have the constraints and the objective function. Mathematically the given problem reduces to maximized equal to plus 30 Y plus 8 Y less than equal to 200 which is the cutting constraint then 12 X plus 6 Y is less than equal to 250 equal to 0. Y greater than equal to 0. So this is the linear programming problem written mathematically. So this is the mathematical formulation of the given linear programming problem in which we have to maximize this objective function subject to the constraints which are determined by a set of linear inequalities with variables as non-negative. So this concludes the session. Hope you have understood the concept of mathematical formulation of a linear programming problem.