Thanks Mitchell again. Here we just use Microsoft Excel to solve integer programs. You solve some problems. For example, this person [inaudible] , personnel scheduling problem, you solve these screenshots again. Here a very good thing is that with the help of integer program, we are able to specifically set all these variables to be integers. That's how you see the outcome as four people, 52 people and so on and so on. You may see that, if you recall what we did in a previous lecture, we have linear programs. We have something like 3.33 persons and so on. If you do run a rundown by yourself, actually it takes a lot of time for you to get this optimal integer solution. You actually really need the help with integer programming to give you this optimal integer solution. From this perspective, integer programming can be, you may say it's very useful. But I need to remind you one thing is that typically when we are really solving problems, if this is a quantity like the number of people, the number of workers, the number of products to produce, if this is the quantity, we typically do not assign it to be integer variables. Why is that? First, you also see from Mitchell's lecture, if you set variables to be integer, the solution process, the computational time is going to increase a lot. In this particular seven variable example, you already need a lot of time to solve this problem, a few seconds. In practice, you are not going to solve the problem with seven variables. You may solve problems with 7,000 variables, 7 million variables, all are possible. If you set that 7,000 variables to be integer, basically no one can solve this problem. You'll probably recall that when last week, this week we'll have a lot of examples. As long as you have a lot of products, a lot of facilities, a lot of markets, a lot of this and that, you'll have a lot of variables. If they are all continuous variables, linear programming can solve them in one shot. But if you set many of them to be integer variables is going to take you a lot of time. Basically you cannot solve it. If we cannot solve a model, then the model becomes useless. When this variable is not so critical, when it is not necessary to set it to be integer, don't do that. For example here, coming back to this example. Last time we told you that, well, if you use linear programming, that's pretty much fine. If this is 3.33, rounded up, rounded down it may not be as good as this integer one, but it's not going to be too bad because there is running up and running down idea is not going to lose accuracy too much. Especially in practice, if you are a company that tries to use operations research to help you to do decision making. It must be the case that you are province scale is really large. If you only have three or four or five persons to schedule, you're not going to use mathematical tools. You will do it by yourself. Only if you have hundreds of people's, thousands of peoples to assign, then you need mathematical tools, then you need operations research. Where you have thousands, hundreds of persons, millions of products, rounding up or down by one or two does not really create any trouble. This is one thing to remind you. A similar thing happens in this second example. Here we have two sets of variables. According to my previous announcement here, when you are talking about this, the number of books to be delivered. This quantity is not going to be set to be integers. I will always set them to be continuous variables. You also saw that Mitchell did not set them to be integer variables, even if we thought we'd know a book is, the book is should not be 1.5 of a book. When we are using this model, we do not set it to integers because that's going to create a lot of troubles in computation. But here, there is one thing that you really need to set it to be binary or to be integer, that's the facility construction decision. Mitchell really set it to binary. Why is that? Because this is a facility construction decision. You either build it or not, you really cannot make it 0.3, 0.7, if your model says, "Hey, lets Build 70 percent of this facility, that does not make any sense. If this is really too far from the practice, then you really need to restrict your model's behavior, then you set it to binary. Pretty much we have a conclusion whether to set a variable to be integer, binary, or continuous depends on how critical that integer constraint is. For theory construction, it is critical, set it to binary. Quantity, shipment, production quantity, person for schedule, that's not critical, set it to continuous. I understand that this may take a little bit of exercise to really get you to understand when to set it to binary when to set it to continuous when it should be integer, when it's not. If you need a very simple rule, I have a rule here. If you think that thing is quantity, then set it to continuous. If you think that thing is an assignment, then set it to binary. This is a very simple way for you to make a judgment about whether you should set a variable to be continuous or to binary. You may ask, "Well, when should we set it to integer?" In my personal experience, there are very few situations that you need to set a variable to be integer but not binary. Typically, when we are using integer programming, we just use continuous and the binary variables. We do not use integer variables so often. This is one suggestion for you, if you are beginner in this particular field, when you have a quantity do it as continuous, when you have assignment, do it as binary. That's pretty much what I want to conclude in this particular part. We now are very good at formulating all kinds of problems. But still there is one thing that we don't know how to deal with. We don't know how to deal with non-linear problems. In practice, there are just so many situations that are non-linear. When you increase your price, it is going to affect your demand, and then your profit is typically non-linear to your decision, the price. There are all kinds of situations with non-linear formula. Next week we will talk about non-linear programming. That's the end of today's lecture. See you next time.
