Thanks, Michelle. We will just take a look at how to use Excel to solve linear programs. In there you'll also get some experience about what the solution would look like. Probably you will see some problems that many people have when they first take a look at linear programming. The thing is that we are going to produce something that is fractional. We see the examples saying that, the optimal solution is for each day to produce 884.2 desks, or we need to produce 189.5 tables. That seems to be not so reasonable because a desk is a desk, you cannot make 0.2 desks. Sometimes we may say, hey this linear program is not so realistic. The solution is not so realistic and cannot be executed. Well, sometimes it may be true, but actually in most cases is not. Why is that? Because if you really have a factory and you really need to produce one-half of table, what does that mean? That means today you do something and eventually, you complete a half of your table, and then tomorrow morning you start to do the second part and make it a whole complete table. Pretty much that means this 189.5 is a concept of average. It simply tells you that you need to open your facility today, tomorrow, the day after tomorrow, for all the days in average each day you produce 189.5 tables. That's a concept of average. If you take that into consideration, you actually realize, this is implementable. Another example is this personnel scheduling one. You may say, "Hey, how is it possible for me to assign 3.3 workers to start to work on Monday? How may we make it 53.3 workers to start to work on Wednesday? That's not realistic," I agree. In this case, you cannot take that average argument here. But the thing is that, so first, any model is used to help us to do decision-making, as long as it is helpful then it is useful. If we don't have this linear program, we have no way to start. But if we have this linear fractional solution this is a good point for us to start do some other adjustments. For example, you may just use a very trivial way to say, 3.3 cannot be executed, let's do it with three people. This one maybe let's do it with 53, this one, let's do it with 14, this one let's do it with 94, something like this. Once you do this adjustment to this rounding, you will get a feasible plan to be executed. You may also do some calculation here to see whether any constraint is satisfied, not satisfied. Probably, if you are unlucky, you are rounding will give you one or two constraints that are violated. But you also know that it's not going to be violated a lot, is going to be violated just a little bit. What does that mean? In practice, when you are thinking about, for Wednesday I need 150 workers. If you only have 149, is still okay, it's roughly you can take it. Basically, as long as you are using a model to solve real problems, there are always some accuracy problems. You'll always use model to approximate your real situation. If that's the case, when your solution is round up, round down a little bit. If you can take that accuracy loss, if you're satisfied with that, then your model is okay, then your model is useful. Basically, this is the end of our linear programming lecture. I added some notes about how to interpret your outcome and how to execute them in practice. Just a sentence to remind everybody, a model is a model. A model is to be used to approximate real situations in all cases. As long as the outcome of a model, which is the solution, as long as the solution is useful to help you make decisions, then it is useful. Of course, linear programming, well, it's just one kind of model. There are other kinds of models to be introduced in the next week and next, next week and so on. We will see if you really need integer outcomes, what to do, we have integer programming. We will see if you have non-linear constraints, non-linear objectives, what to do. We will have nonlinear programming. That will be the contents for the next week and so on, so see you next week.
