Okay, so now it's time to talk about our last topic, which is about shadow prices. So to introduce the idea of shadow prices, we need to, again, takes a look at an example. So let's, again, consider a product mix problem. So you have learned it for many, many times. Let's just go to the formulation directly. The formulation basically says that you are able to produce tables, and let's say now you may produce chairs. So you let x1 and x2 be the number of tables, number of chairs produced. And then once you produce table or chairs, you may earn some money, all right? And also, they're going to consume about woods and the labors. So they have their unit consumption, and you have six units of wood, six units of labors. And all the values, obviously, should be non-negative. So very quickly, if you are given this kind of unit revenues, you're going to see that your optimal solution is to spend all your resources to produce tables, okay? Because the tables are more cost-efficient, all right? So the profit lies here, okay, and the optimal solution is here. So when you want to produce three units of tables, you will have some amount of labors left over. But you don't care, because using all your resources to produce tables is your optimal solution. Okay, so in many cases, we want to ask some what-if questions, okay? You got that optimal solution because you were solving that problem. What if the problem is changed? What if the unit price of chairs is not $1, actually becomes $2? What if each table requires not just two units of wood, what if that's three units? What if you have actually more woods, instead of just six units? So these questions are important in practice. Because in practice, your parameters may change from time to time, or sometimes your estimation of parameters may be inaccurate. Some people say I need two units of wood, but that number may also be obtained with some estimation, right? So it sometimes may be inaccurate. You really need to look at those estimations, and there are some uncertainty that you need to consider. Finally, you may be looking for some ways to improve your business. So you may want to ask, well, what if I may buy more woods, does that help? What if I may buy more labor hours, does that help? What if I may spend a lot of money to do some investment to make the production process for table more efficient? Something like that, you need to estimate the impact of your potential improvements so that you may determine whether you really want to do that. So if you have a model, then that's really possible, because you may modify the model to see what's the outcome. So that's something we need to do. But for practical or realistic problems, what if questions may actually be hard? Because even if that's just a tiny modification of one parameter, your optimal solution may change, right? You sometimes may think about solving the new problem to get your new optimal solution, but your problem may actually be very complicated, very difficult to solve. You may have millions of variables, millions of constraints, and you may have a bunch of possible what-if questions to answer. So we really need some better ways. And the collection of tools to answer what-if questions is sensitivity analysis. So this is obviously required. So let's say there's a successful story. There was a company called Pacific Lumber Company, now it's called another name. So this company owns a lot of forests, okay? And if you want to manage forest, typically you consider them as your potential products, but you also consider that as your assets. You need to maintain sustainability so that in your future you still have this forest. They grow up, they make new forest, and you want to have a long-term plan for production and for a lot of things. So they have an OR team develops a 120 years forest ecosystem management plan to talk about how to use linear programming to optimize the operations for maximizing profitability in the long run. So obviously, your program would be quite complicated. And what's even more complicated is that we all agree that the environment may change, so every long term plan somehow realize some estimation. So once you have a set of estimations, there is no way for you to feel comfortable, because your estimation maybe uncertain, your estimation may be inaccurate. So there are all kinds of things that you need to estimate, and you need to take care of all those possibilities so that you may make the best decision. That's why in that particular case you really need sensitivity analysis. So coming back to our example here, pretty much what-if questions can always be answered by formulating and solving a new optimization problem from the very beginning. But as we mentioned, this may be too time-consuming. What we really want is the following. Basically, we have already solved our original problem. So the original problem, original optimal solution should be utilized, and in particular we should utilize the optimal tableau, okay? Because once we solve a problem, we don't just have a solution, we have that optimal basis, optimal tableau. So we would try to start from the original optimal basic feasible solutions, try to see whether we may do just a few more iterations to get to our new optimal solution. That's our plan. And in that case, sometimes we will see that duality provides a theoretical background. So we will see one particular example today. So here we just want to introduce just one type of the what-if question. I want to ask, what if I add one unit of a certain resource into my resource pool, okay? So why do we want to consider this is because this is typically the case. So consider the following scenario. Suppose one day, a salesperson enter your office and it says that they're going to provide you additional unit of woods at $1 per unit. So you may need to first think about, well, do you need more woods? If you recall your memory saying that okay, previously, our optimal solution is here, okay? We produce three tables with no chairs, and we stop because we are running out of woods. So if that's the case, you would agree that okay, additional woods seems to be useful. But of course, everything has a price. If that one additional unit of woods cost you $1,000, you are not going to pay. If that cost you just $0.001, I guess you will pay. But what if that's a number in between these extremes, okay? You must have a systematic way to determine whether you want to buy these additional woods at $1 per unit. In practice, you have multiple resources, you have millions of resources, some thousands of resources. And for each of them you somehow need to make this kind of judgment, this kind of decisions, right? That would be too complicated for you to deal with them one by one. We need to have some systematic way. That's something we will introduce for the remaining lectures for today's lecture.