Next we will talk about modeling, which as we always mention, is the heart of operations research. I would like to use this particular case to illustrate the ideas of the three levels of modeling. We will first do conceptual modeling and then mathematical modeling, and then eventually we are able to create computer models that may really generate solutions. Actually, all of this has been mentioned in the past. But hopefully after this case study, after you have learned so much things in this case study, we would make all these ideas as concrete as possible. We always try to solve a problem by creating three levels one by one. As long as you are trying to solve a problem with Operations Research, you always do this. The first thing is to create a conceptual model. Originally you have a business problem or you have a real problem. In that problem, there are all kinds of information, all kinds of people telling you a lot of things. Most of them are useless in sometimes, or I should say most of them are not in the core of the problem that you are interested in. In that case, the conceptual model is a thing that use words to describe problems. You need to cut down those useless information, remove them, and just focus on the required information, relevant information. You need to first act as a domain expert or try to talk with domain experts to see what are really inside your problems, and try to use words to describe them. For example, your uneasy problem. Our goal is to minimize total cost, from that description. We are facing some situations that we are having trade-off between operating costs and the service cost, and that's a description, and so on. Our descriptions should be able to be understood by a manager. The manager should be a typical manager, that is not so technical. Maybe have no knowledge in operations research, maybe has not used mathematics for 20 years. But still our description should make sense to the manager, letting the manager understand, and letting the manager agree that this is indeed the problem that they are trying to solve. If the manager wants to maximize profit, don't try to maximize revenues. That's two different things. Your problem description or problem definition need to be understood and agreed by the manager. Typically if you describe things in words, words are not so precise. The description in conceptual model may not be precise enough for us to write a computer program, and that's why we need a mathematical model. A mathematical model consists of decision variables, constraints and objective function. That's pretty much all the thing that we have told you in our previous course, Operations Research 1. You are able to write programs for linear programming, non-linear programming, integer program, and so on. Those mathematical terms that follows formulas are hard to be understood by a traditional manager. But you really need that, so that when you hand in the mathematical model to an engineer, that engineer will feel that the definitions are precise enough to build a computer model. If you have some experience in writing computer programs, you really know that we require a specs, our specification for writing a computer program. Someone needs to tell us what's the objective for our programs. Someone need to tell us, what's the input, what are the meanings, what the output that we are expected and so on. We need a spec to write a computer program. In this our studies mathematical model is the spec. You don't want to just give rough words to engineers. They will complain to you that those descriptions are not precise enough. There must be someone that is able to translate those vague business problem descriptions to precise specifications. That's the role played by an operations researchers. Finally, once we have a mathematical model, we may hand in that mathematical model to a computer scientists, to people who are able to write a program, and the computer model is a concrete program in Python, in C++ in Excel in whatever. That program may be executed and really generate a solution. In this case you have a program, you input data, and then that program gives you output, which is a suggested print for your planning problem. That would be hard to be built if you don't have a mathematical model. You need a spec so that some people may write a program for it. Your computer program can be an Excel Solver, can be a Python program invoking Gurobi, can be an implementation of a heuristic algorithm in C++, in Java, in whatever. Whatever computer programs you are talking about, I would need to emphasize that what's the most difficult and the most important thing is to create the spec. As long as you have a spec, all those people who are taking introductory computer programming course will be able to create a relevant computer models according to the computer languages that you are using. But writing a mathematical model is hard. It takes some training. That's why we have the first course. You'll need to have those mathematical backgrounds, plus, you need to know what's the real business problems that the manager is concerned about. There really need to be a bridge between the business people and engineering people. Business people only understand a conceptual model. Those engineers, they are only talking to computer models. You are really the person to bridge these two kinds people by writing a correct and accurate and appropriate mathematical model. That's why we always say mathematical modeling is the heart of Operations Research. Later we will show you the whole process.