[MUSIC] Hi everyone, welcome back to operations research. So we're going to continue our discussions about theory for this particular week 3. We're going to talk about something more about sensitivity analysis and the simplex method. So again, this week is dedicated to theories about linear programming. So that's say, we have a very typical linear program. Suppose I want to maximize 3x1+2x2 subject to x1+2x2 must be less than or equal to 8, or 2x1+4x2 must be less than equal to 10, something like that. So suppose your variables are all non negative and so on and so on. Okay, so somehow, when you have a linear program like this. In many cases, you may want to ask some sensitivity analysis questions. For example, last time we mentioned about what's going to happen if the right hand side value is changed by one, all right? So, for example, that say, this is actually 11. From 10 to 11, we know how to evaluate the impact of this change is through shadow price. Okay, so today we want to talk about something else, for example, in some cases, you're going to have additional variables. Okay, so let me write it down additional variables, okay. So that somehow means your resources are still there. But now you're going to have additional variables like this, okay? So if you want to, for example, make some efforts to produce your third product, the new product that you just invented. You're going to consume some resources. But maybe that's going to be a popular product, and you're going to earn a lot of money. Okay, so this is your new product, x3. So if this is the case now, you have a new linear program to solve, right? You have a new column or new variable. The thing is that if you already have 1000 products with 1000 of constraints. If you add one additional columns resolving the whole problem maybe too time consuming. We're going to tell you if you already have the solution for the original program. Now, when you have one additional column, you want one additional variable. How would you go from the original optimal solution to do some a little bit more simplex iterations to get to your new optimal solution? We're going to see how. Another thing is that, maybe you're going to have additional constraints, okay? So, what does that mean? Sometimes it's going to be the case that you're going to face new restrictions. So that say, you will now say that, okay, 3x1+x2+x3 cannot be greater than or equal to let's say 5. So this may happen, for example, you're producing a lot of products, but sometimes regulation changes. The law changes, or this market changes so that you face new constraints. Then again, in this particular case, we want to ask if we have solved our original problem and get our original optimal solution. May we do something else? Do a little bit more to go from the original optimal solution to a new one. If you are able to do that, is going to make your decision speed faster. So again, this is possible. But the interesting thing is that here we need a very special technique which is called the dual simplex method. So it's going to combine your understanding about the simplest method and something about duality that you learned from the previous week. Combining both of them, we are able to deal with this additional constraint problem. And by having this, we're going to give you a very quick understanding about when we are using branch and bound to solve interior programs. We're going to solve a lot of linear programs, right? And we're going to tell you with dual simplex how we may make it faster. By doing that, we're going to move to integer programs. And in the next, next week, we will talk about theories about integer programming.