 self-sufficiency, assistant professor, department of civil engineering, watch on its technology, solapur. In today's session, we are going to see the duality of the linear programming problem. At the end of the session, learner will be able to determine the dual of the linear programming problem from the primal one. Now let us see what is linear programming problem and the dual. The linear programming problem is a mathematical modeling technique in which we see the objective functions containing of non-negative variables which are subject to the constraints of the particular equality or inequality equation form. This is an example which is showing you the objective function and the subject to the constraints. There are two constraints present in this particular problem, and the x1 and x2 are the two decision variables which are restricted by greater than or equal to 0. The dual of the linear programming problem is what? Every primal linear programming problem has an associated with another linear programming problem called as the dual one. Dual of the dual is the primal and vice versa. Here, the primal problem, if it is of maximized case, is dual will be of minimized case. And if the particular problem, if its primal is minimized case, its dual will be of maximized case. Now, to solve the particular problem, your particular LPP should be in the standard form. Now, first is what? All the variables which are present in the LPP should be of non-negative variables. Or you can say the decision making variables should be of non-negative. I am showing you the example, x1 and x2 are the variables in the LPP, then it should be restricted by positives and that is x1, x2 is greater than or equal to 0. Secondly, the second is what? For LPP of maximized case, all constraints should be of left-hand side should be less than or equal to right-hand side type. This is an example which you are showing that objective function maximized case, z is equal to 5x1 plus 2x2 subjected to the constraint 3x1 plus 5x2 less than or equal to 15. Second constraint, again it is restricted by the less than or equal to 10. The left-hand side is less than or equal to right-hand side, this is for the maximized case. And for the minimized case, the constraint should be of left-hand side should be greater than or equal to right-hand side type. And after observing this particular some necessary changes which are required for the conversion of the LPP in the standard form for the duality solving. If the variable in LPP is unrestricted by the sign, that is the particular variable is free one, that particular variable we have to replace it by the two new decision variables. In this particular problem in the particular we are observing that it is of maximized case and there are two constraints present in this and this particular problem in this the x1 is restricted by the particular sign but x2 is unrestricted by the sign. So to convert this particular problem in the standard form I have to replace this particular x2 by two new decision variables that is x3 and x4 and this is replaced by the difference between the two. Now, objective function maximized case that is equal to 5x1 plus 2 x2 minus x4 subject to the constraint 3x1 plus 5x 5 multiplied by x3 minus x4 less than or equal to 15. Similarly, for the second constraint also I will replace x2 by the new decision variables. Finally, I have to represent here the restriction for that particular old variable and the newly replaced variable in place of x2 that is why x1, x3, x4 are greater than or equal to 0. Now, the next necessary changes is when all the constraints are not in right direction as per the objective function cases then convert the constraints as per the objective function cases by multiplying it by minus 1. Objective function here is maximized case that is equal to 5x1 plus 2x2 and in this particular problem which are the subject to constraint the second constraint is not as per the case of the objective function as we have seen in earlier slide that is slide number 4 all the constraints should be of particular direction as per the case for the maximized case left hand side should be less than or equal to right hand side but we are observing that even though it is maximized case here left hand side is greater than or equal to right hand side. To convert it into the standard form I have to multiply it by minus 1. Now, after multiplying it by the minus 1 the newly obtained constraint is minus 5x1 minus 2x2 less than or equal to 10 and this is a standard form of constraint required for the particular LPP according to its case. Now, next necessary changes which is required is what if the constraints involve an equivalent equation then to obtain the inequality equation it is required to replace it by the pairs of inequality equations in the opposite direction. In this particular provision problem the second constraint is of equality equation up till now we have seen that all the problem of LPP having the constraints of inequality form but here in this particular problem we are observing that the second constraint is of equality to convert this second equation in the standard form we have to replace it by the two new equations this is two newly obtained equations that are 5x1 plus 2x2 greater than or equal to 10 and the second is 5x1 plus 2x2 less than or equal to 9. These are the two newly obtained variables which are in inequality form. Now, after replacing it again this is a second constraint which is not occurring to the case as in earlier in this slide we have seen that we have to convert the particular constraints according to the case means in slide number 4 we are saying that the particular if it is of maximize case left hand side should be of less than or equal to right hand side so it is not as per the requirement so to convert this particular second constraint in the standard form multiply again it by minus 1 now after multiplying it by minus 1 as shown here multiply by this greater than or equal to minus 1 into 10 the newly obtained will be minus 5x1 minus 2x2 greater than or equal to 10 and this constraint is a newly obtained constraint as per the standard this is a newly obtained LPP which can be solved last point which is required to see that if the primary LPP contain n non-negative variables and m set of constraints then in the dual LPP we should see that or we observe that m non-negative variables and the n set of constraints are present for example I am showing you if in the primary problem n non-negative variables are there then in the dual there will be n set of constraints means 3 variables it will convert into 3 constraints now point to remember is what after converting of the particular primary problem to the dual one this is an example it is of maximized case which is converted into the minimized case here in the maximized case we are seeing 2 variables but in the dual we are observing the 3 variables as we are saying said that in the primal if there are 2 variables there will be 3 you can say 2 constraints in the particular dual problem and if in the primal there are 3 constraints there will be 3 variables in the dual here x1 and x2 are there this here y1 y2 y3 means these 2 here in the 2 variables in the primal now it is converted into 3 variables in the dual one and here there are 3 constraints in the particular primal problem now here we are observing that there are 2 variables 2 constraints in the particular dual one this is the necessary changes which we should be there in the particular problem according to the case now let us select the correct answer for this particular msec hope so you have selected the correct answer which are showing here these are the references for the today's session thank you