Chapter 3 | Part 02: Dual Form of A LP Model | Faraday JR

Added: 2026-05-06

[00:00:00]Welcome to Faraday Junior YouTube channel.

[00:00:02]In this episode, we are going to look at the dual form of a linear programming model.

[00:00:09]All right. As I said earlier, every linear or every LP problem is having two forms.

[00:00:16]We have the first or the original form that is known as the primal and then we have the second second form that is known as the dual.

[00:00:31]So, normally all the problems we solved so far, we've been dealing with the primal.

[00:00:36]Yes, so we are going to look at the dual form or when you are given a linear program model in the primal form, how do you convert it to the primal form. But before then, let me talk this.

[00:00:49]Whenever you have a maximization model when you have a maximization model in the primal state, primal in the dual, you have it to be a minimization problem. And then when you have a minimization when you convert it to the dual, you have it to be a maximization.

[00:01:16]And then the number of variables in the dual form, number of variables in the dual form in the dual form is equal to the number of constraints in the primal.

[00:01:40]Yes.

[00:01:42]And there are other relationships between the There are more relationships between the uh primal and then the dual form.

[00:01:49]You can refer to your handbook or do your own research and then get more of them, but I think these are the two that I want to share with you.

[00:01:59]So, let's take an example.

[00:02:04]Let's take an example.

[00:02:06]So, we have this maximization problem and then we have to convert it to the dual form.

[00:02:15]So, what do we do?

[00:02:16]Uh you know, as I said earlier, I said the number of constraints in the or the the variables in the dual form is equal to the number of constraints we have in the primal form. So, this is the primal.

[00:02:31]How many constraints we have here? We have three constraints. One, two, three.

[00:02:35]Which means that in the dual form, we are supposed to have three decision variables or three variables.

[00:02:44]Yes, so let's quickly convert this to its dual form. So, we have X1 plus 2 X2 or equal to 10.

[00:02:58]This is one.

[00:02:59]Plus 6 X2 This is 2 X1 plus 4 X2 equal to 40.

[00:03:09]So, we do this Y1, Y2, Y3.

[00:03:14]Let us assign one variable to each of every constraint. So, Y1 for the first constraint, Y2 for the second constraint and then Y3 for the third constraint.

[00:03:23]So, this is how we are going to write our our constraint for the dual form. We go this way.

[00:03:30]So, here we are going to What's the coefficient of We use the coefficient The coefficient of this one, X1 here is one. So, 1 Y1.

[00:03:41]The coefficient of X1 here is 6. So, 6 Y2.

[00:03:46]Plus 6 Y2. The coefficient of X1 here is 8. So, plus 8 Y3.

[00:03:55]Then the coefficient of X1 in the objective function here is 4. So, we equate it to 4. So, we Sorry.

[00:04:04]It will be this way.

[00:04:09]Yes.

[00:04:10]4.

[00:04:12]Less than or equal to 4.

[00:04:15]Then we go to the Remember I said when we have a maximization problem in the primal state, its duality will give us a minimization minimization problem. So, this is supposed to be So, it will no longer be less than or equal to since it will be a minimization problem, it's going to give us greater than or equal to 4. So, let's do for the second constraint or this.

[00:04:50]So, we have to the coefficient of X2 here is 2. So, 2 Y1 plus this 6 here, Y2.

[00:05:02]So, 6 Y2 plus this 4 here, 4 Y3.

[00:05:07]4 Y3 greater than or equal to the coefficient of X2 in the maximization or the objective function is 5.

[00:05:14]So, 5. You realize that the number of variables you have here, the number of variables you have in the dual form is equal to the number of constraints we have here. We have how many variables?

[00:05:25]Y1, Y2, Y3, three variables. It's equal to the number of constraints we have here. We have one, two, three constraints. And the number of constraints we have in the dual form is equal to the number of variables we have in the primal state. You see in the primal state we have X1 and X2 only.

[00:05:40]Yes, so we have two variables here.

[00:05:43]Then that gave us two constraints here. And then we have three constraints here.

[00:05:52]And that gave us three variables here.

[00:05:55]Yes, so our objective function will now be min of ZD is equal to So, 10 Y1 10 Y1 plus 36 Y2 plus 40 Y3.

[00:06:17]So, this will be our answer for this question. And our non-negativity restriction will be Y1, Y2, Y3 all greater than or equal to zero. So, this is the dual form of the problem. This is the dual form for this problem. All right, let's take another example where we have where we have equal sign and then Yes.

[00:06:52]Then we have equal sign.

[00:06:54]So, when you have equal sign, what do you do?

[00:06:57]Or what do we do?

[00:07:00]This is a maximization problem.

[00:07:02]And then we have the equal sign in it.

[00:07:04]So, you will just go ahead and then convert it to the dual form just like how we did the previous one, but then you have an unrestricted variable in it. So, you let's go ahead and do it. So, you have 4 X1 plus 2 X2 24.

[00:07:26]They are X1 plus 7 X2 28.

[00:07:33]2 X1 plus 3 X2 Sorry, this is supposed to be equal sign.

[00:07:41]This is equal to. Sorry.

[00:07:44]18.

[00:07:46]All right, so as we did earlier, Y1 for the first constraint, Y2 for the second constraint and Y3 for the third constraint. So, the dual form is going to give us As I said earlier, you write the equation this way by transposing the matrix.

[00:08:03]So, 4 Y1 because of this 4 and Y1.

[00:08:06]And then you have 1 here and Y2.

[00:08:09]So, plus Y2 plus 2 Y3.

[00:08:15]Y3.

[00:08:16]This is a maximization problem.

[00:08:18]And it is less than or equal to, so it should give us a minimization problem, so greater than or equal to.

[00:08:27]And that will be the coefficient of X1 in the objective function, 3.

[00:08:34]And let's go to the second constraint.

[00:08:35]We are going to write it this way. So, 2 Y1 plus 7 Y2 plus 3 Y3 less than or equal to this 7.

[00:08:49]All right.

[00:08:51]So, we are going to have our objective function to be ZD values.

[00:09:03]24 Y1 plus 28 Y2 plus 18 Y3.

[00:09:15]And our restrictions is going to be our non-negativity restriction is going to be Y1, Y3 greater than or equal to zero.

[00:09:29]And our Y2 because that was the variable assigned to the constraint with the equal sign, so that one is going to be unrestricted.

[00:09:39]Unrestricted.

[00:09:42]It's just as simple as that. So, that's the dual form of the linear program model given. So, whenever you have an equal sign, the variable you assigned to the constraint with the equal sign is going to be unrestricted.

[00:09:59]All right, so let's take another example.

[00:10:03]Let me take another example where we have a greater than or equal to constraint in a maximization problem or where we have mixed constraints.

[00:10:15]Let me take this out.

[00:10:18]So we have mixed constraints.

[00:10:21]As we have mixed constraints here and then an unrestricted variable here as well.

[00:10:25]So let's see how we are going to solve this problem.

[00:10:29]So you can see that this is a maximization problem. You know that all maximization problems have are supposed to have all maximization constraints are supposed to have less than or equal to constraints. But here we have greater than or equal to constraint. So what do we do?

[00:10:46]You have to multiply that particular constraint by -1. So this constraint you're going to multiply both sides, the right the right hand side and the left hand side by -1 so that we convert this greater than or equal to to less than or equal to. So let's quickly do that. So we write our constraints.

[00:11:03]We have x1 plus 4 x2 plus 2 x3 less than or equal to 16.

[00:11:12]We have 3 x2 plus x3 equal to 10.

[00:11:20]And then we have -6 x1 plus 2 x2 Sorry, minus.

[00:11:32]That should be minus because you are multiplying you are multiplying minus one throughout.

[00:11:37]So that should be minus.

[00:11:39]And then this sign this sign the greater than or equal to is going to change to less than or equal to and this one is also going to be minus 20. As simple as that. So we go ahead and then convert this to the dual form.

[00:11:54]Go ahead and then convert this to the dual form.

[00:11:58]So doing that we are going to assign our variables to the constraint y1, y2, y3.

[00:12:09]So as we do and so. So we have one here so 1 y1 plus here you don't have any so 0 y2 can just not write it.

[00:12:22]0 y2 and then -6 y3.

[00:12:26]So -6 y3.

[00:12:39]Then equal to 10.

[00:12:44]Yeah, then we move to the next one. The 10 here is a is a 0 to or is the coefficient of 10 in the coefficient of x1 in the objective function here.

[00:12:53]We move to the next one, x2.

[00:12:55]So 4 y1 plus 3 y2 minus 2 y3 all greater than or equal to the coefficient of x2 in the objective function that's 3.

[00:13:12]We move to the last one on this side.

[00:13:15]So 2 y1 plus plus y2 all equal to 8.

[00:13:34]You will understand why we have equal sign here.

[00:13:38]This is because we have x3 to be unrestricted.

[00:13:42]You know that when we have when we have an equal sign in the primal we have an equal sign in the primal and then we convert it to the dual the variable that is assigned to the constraint with the equal sign is given as unrestricted.

[00:14:02]So it's just the same idea that you do it.

[00:14:04]So this x3 means that the third constraint is having equal sign. Simple as that.

[00:14:10]So this one our non-negativity restriction is going to be this way and then we are going to have y2 to be unrestricted.

[00:14:25]This is because our second constraint here is having equal sign.

[00:14:29]So our second variable is unrestricted.

[00:14:32]This as we have the we have x3 to be unrestricted here and our third constraint in the dual form is also having equal sign. So it's just the same idea that you do it. So our >> [music]