 Hello and welcome to this session. In this session we discuss the iso profit method of solving linear programming problems. Iso profit method is the method of maximizing the profit function. Now when we are given an objective function z equal to a x plus b y if an arbitrarily chosen fixed value k is assigned to the objective function z that is z equal to a x plus b y is equal to k where this k is sum arbitrarily chosen fixed value. Then this z equal to a x plus b y equal to k represents a straight line and every point on this line gives the same value of z. Straight line is called iso profit line. We know that iso means the same and for different values of z we obtain different lines. So for different values of z a family of iso profit lines which are equal to each other. Let us now consider an example of a linear programming problem to be solved by the iso profit method. The problem is to maximize the profit function given by z equal to 50 x plus 15 y subject to the constraints given by x plus y less than equal to 60, 5 x plus y less than equal to 100 and the non-negative constraints x greater than equal to 0 and y greater than equal to 0. Let us try solving this problem by the iso profit method. For this our first step is to find the feasible region as we given linear programming problem. Now to find the feasible region we need to graph these constraints and the common region determined by these constraints would get as the feasible region. After graphing the constraints we get this shaded region that is the common region determined by all the constraints and this shaded region is the feasible region that is the region O C A D O is the feasible region. Now in the next step we assign a convenient value to z and draw that line. We have the objective function or you can say the profit function z equal to 50 x plus 15 y and we give a constant value to the subjective function z. So let 50 x plus 15 y be equal to 150 which is the common multiple of both 50 and 15 or you can say 10 x plus 3 y is equal to 30. Let us now graph this straight line 10 x plus 3 y equal to 30. Let this equation be equation 1. We have graphed this equation and we get the straight line P 1 Q 1. So the line P 1 Q 1 represents the equation 1. Now as the objective function is of maximization type so we will draw lines parallel to this line 1. Next we would take the objective function minus 50 x plus 15 y equal to a constant value say 300 or you can say this equation is 10 x plus 3 y is equal to 60. Let this be equation 2 and now we will graph this equation 2 to obtain a straight line parallel to the line P 1 Q 1. So we have obtained the straight line P 2 Q 2 which represents equation 2 that is line P 2 Q 2 represents the equation 2 and this line is drawn parallel to the line P 1 Q 1. Now we will find a line which is farthest from the origin since the problem is of the maximization type and that line should have at least one point common to the feasible region. That means we have to draw some more lines parallel to the line P 1 Q 1 and from those parallel lines we would find the farthest line from the origin and it should also have a point common to the feasible region. As you can see we get this line P 4 Q 4. This line P 4 Q 4 is parallel to the line P 1 Q 1 is farthest from the origin with the point that is this point which coordinates 10 50 and then to the feasible region. As you can see this point A is common to the feasible region and the point A lies on the line P 4 Q 4 which is the line farthest from the origin and so we say the point A which coordinates 10 50 the maximum value of the objective function Z. Hence the solution is X equal to 10 and Y equal to 50 and value of Z is given Y to do the values of X and Y as 10 and 50 respectively in the objective function Z. So optimal value of Z is equal to 50 into 10 into 50 and this is equal to 500 plus 750 which gives us the value of Z as 1250. The optimal value of the objective function Z is equal to 1250. Optimal value or you can say the maximum value of Z. In this session we have understood the iso profit method of solving a linear programming problem.