Okay, so this is pretty much the end of the whole three courses. And now I'm going to give you some preview for the future because we don't have the next course. At least as for today, on the end of August, 2020 currently we are going to stop recording this set of all our courses, so it should be no course four. But there are still a lot of different things to learn. So maybe you still want to learn something more advanced here. I'm going to give you some key words. Maybe it's going to help you. So actually, we have learned a lot. We have OR one, two and three, right. If you have went through all of them with us, well, I cannot count, but right, that's roughly 20 different lectures. And in each lecture, we actually told you a lot of things. So I would say the whole collection of all these three courses is roughly not is larger or is more advanced than typical undergraduate courses. So the theory part is not as difficult as a graduate level course, but not so many undergraduate course. Maybe like this. So if you are with us for the whole three courses. Actually, you did a very good job. But of course, there are still many different things to study. When I was undergraduate student and the master student in and to you, I thought I already learned a lot from operations research. But then I go to Berkeley to study for my PhD in operations research. Then I basically relearned everything because they are just too many things to learn. So many of the following subject may itself form a single semester course. Okay for all these subjects like, for example, linear programming we only used roughly three weeks to talk about it. Application algorithms theory, right. Three to four weeks. But actually, it can be a one semester course in teacher programming, of course, can be a one semester course or even two semesters. Graphs and the network flow can be one semester. Convex programming. Nonlinear programming. So when I was in Berkeley, I took a course. I mean two courses about integer programming. One is called integer programming and the combinatorial optimization. The other one is called computational optimization. Basically, they are talking about integer programming. I took a course called graphs and the network flow. Basically, the whole semester is talking about network flow and the part of integer programming. I took a course from the W department about convex programming. I took a course for the whole semester talking about nonlinear programming. At that graduate level we don't have a course talking about linear programming, but in Berkeley is an undergraduate required course. So I served as the TA and the taught linear programming when I was a PhD student for many semesters. So all of these actually have much more things to be covered. But we don't have time to talk about it in this undergraduate course. So don't forget, there are still so many different things. And also there are even more that we don't even have any time to mention in this course, especially in this course. Basically, we ignore everything about uncertainty. So we assume we have all the information, or we can estimate all this information and we use the estimated information to make decision making, so that works in many cases. But still there are some cases that you cannot ignore those uncertainty. So in that case we need stochastic models or probabilistic models. There are courses or fields like stochastic processes, stochastic programming, robust programming, dynamic programming. We don't have time to talk about them in these collection of all our courses, but still, they are very important. When I was in Berkeley, there are two courses for two semesters talking about stochastic processes. There is a course in one semester talks about dynamic programming. Both are somewhat, very difficult. And the stochastic programming and robust programming exists in different courses on the idea may be covered and used in many different kinds of optimization courses. So when you want to go deeper, one specific way is to incorporate uncertainty. Incorporate probabilistic events into your optimization model. Obviously leads to go in to make a lot of things difficult, but you're underlying deterministic optimization skills may still help. There are also many related fields. Okay, so for example, there may be courses talking about scheduling, there maybe courses talking about algorithms. Typically from the computer science perspective, there may be courses talking about theory of computation, and there are today many machine learning courses. If you have a solid foundation in operations research. Taking all these courses is going to either makes you feel comfortable because you have the solid foundation or gives you different but related perspectives. That gives you an even bigger picture. And today optimization is not just used in computer science, in engineering or in business. It is used everywhere in particular. If you ask, those economists, ask those social scientists, in many cases they would say they need to add game theory into optimization. Or I should say this optimization tools in the game theory. They should be combined together so that we can model environment with multiple decision makers. In this collection, of course, is we talk about models with one decision maker, but in practice, your profit, your objective value may be affected by other people's decision. Okay, you may have competition among retailers. You may have competition among hotel, different hotels, or you may have upstream suppliers downstream customers. They don't really rely on you. They work with you, but you need to somehow induce them to do something that you asked. You want them to do? That's game theory. Okay, so in many cases, if you want to expand the application region for optimization. You need game theory so that you may talk about supply, chain management, competition, pricing and the many, many, many things else. That's also the thing that many operations researchers are doing today. So all those things are very interesting. If you want to go further to study them, don't forget to get more mathematical training before you really want to do that. Calculus obviously is needed. Or maybe you even want to learn analysis, real analysis or in some university is called advanced calculus. Okay, so it's no longer just differentiation, integration, whatever there are going to be to talk about deeper things. Okay, so I'm going to stop because that's too much. So if you really want to go deeper in the theory of operations research, don't forget to learn more about calculus and analysis. Obviously, discrete mathematics would be useful, especially if you like the algorithm met algorithm part more. Obviously, linear algebra is always used, so there are more linear algebra theory that if you know them, everything looks better for you. And finally, if you want to deal with those uncertain things, probability is of course needed. Okay, so all these things are something that you should learn. You should prepare yourself before you go deeper into operations research. So for me is almost the end of the semester of the course, so that means, say a little bit about myself. For me, operations research is a belief. So what does that mean? We believe the world can somehow be explained. Be estimated, be understood with operations research because, basically, we believe all decision problems are optimization problems. There are several things that you want to choose one front. You are optimizing some kind of objective functions. You want to find a direction to someplace you are doing some kind of optimization problems. It's just that sometimes you are unfamiliar with your objective functions. Sometimes you have different constraints and so on and so on. But basically everything is optimization somehow, and we believe many real world optimization problems may be modeled. Not every problem, but many problems may be modeled. And once we learn operations research, we can distinguish between models that we may solve and the models that we are unable to solve. Okay, so there are so many problems. Some may be modeled and for these models, some may be solved then for this part, at least for this part, we believe we should utilize operations research to help us do decision making to help us do better decision making. Because these models may be solved, we may get solutions, and the solutions may suggest us what to do. We don't need to really listen to the solutions without any change. We may take the solution, get an understanding about it, get an understanding about why it comes to this kind and do adjustment with if we need. When you face a really complicated situation, you really need suggestions. That's how operations research may help you. That might believe so. No field is perfect. No tool is perfect operations research, of course, is also not perfect. But there are always the situations that it may be used. For example, George Box are very famous. Scaler mentioned the least this sentence old models are wrong, but some are useful. So the idea is here. When you have a practical situation, you have a mathematical model. We always know that your mathematical model is not 100% the same as the real situation. No real situation can be 100% modeled. But your model as long as it is correct. As long as it is close enough to the real situation, it is useful. As long as it is useful it gives you some suggestions. Then we want to use it. That's my principle. And if you forget all those applicability or whatever, OR itself is so interesting, right. Because you have so many interesting theories. You have so many amazing coincidence that, My God. Primal and dual lake coincident LP duality and the LaGrange and duality lake coincident linear programming and interior programming with total unit modularity lake coincident. Those coincidence looks so surprising, but also not surprising because, you know there are underlying theories. Those underlying theories guarantees the things to happen. They are not coincidence. So that's something I feel. Wow, that's so interesting. That's so amazing. That's why I want to do research in this field. That's why I want to offer courses in operations research to also teach you or to give you this kind of amazing thing. So I don't know whether you feel amazing or you just want to finish the course as soon as possible. But anyway, that's all I want to say. Okay, so that's probably the end of the semester. The end of this course, the end of this serious, the end of this specialization whatever. That's the end of my lectures. So hopefully you enjoyed it. And if possible, see you next time you're in the future, bye.