Now we are ready to create our mathematical model. This is not an easy thing, and almost always we start by defining sets, indexes, and the variables. We know there are some entities in our problem. They are customers, they are candidate locations, and that there are scale levels. These are some things that we need to worry about in our problem. That's a I, J and the K, they are a set of customers, locations and scale levels. Later, we will see how they may be used. Very quickly, we would say, small i, small j, and the small k. Each of them are indexes. They represent a customer of facility location, or a facility scale level. Throughout our model, whenever you see small i, you know, that's the customer, small j, that's a facility or a location, and a small k, that means a scale level. Then we are ready to define our decision variables. As we always mention, you always define your decision variables before you define anything else, before you define parameters, before you define objectives. Because whenever you are trying to tell anybody, once you are in your model, you always tell them know what's the decision that your model is going to make for those decision-makers. Our model is going to make a suggestion on whether to build a facility, in which scale level. X, j, k would be one if a facility is built as location j with scale level k or zero otherwise. In this case, x, j, k certainly has j, and the k with location, in the scale level as the indexes. Because for each location j, you will choose one out of those possible scale levels. Xjk is our main decision variables. Then we also need to do customer assignments. Y i, j is one. If customer i is served by site j, which means facility j. Then finally, w, j is the number of engineers that are allocated in location j. Once we have this set of decision variables, we have covered all those decisions that we need to do. We need to deal with models, and we need to use the model to determine facility construction, to determine customer assignment, and engineer allocation. Now we are ready to describe more information that we need for building our models. That say f, j, k is the annual office rent for building a facility at location j with scale level k. You may expect and as different location, the typical office rent is different. If you need a larger space, then you need to pay more. That's why here, your f, has two indexes. The office rent is different from location to location, from scale level to scale levels. You have h, i which is the annual number of services that are needed for customer i. For different customers, this number may be different for post offices, maybe you need to go there very frequently. But maybe for 711, you don't need to go there very frequently. For Taipei post office, maybe it's very busy, so you need to go there for a lot of times. But maybe for one entitle, you don't need to visit there many times. That number may be different from customer to customer. Between a facility and a customer, there is a distance, and that distance also has something to do with the route that you need to go through. There must be a different cost between locations. That cost is per service cost for you to send one engineer, to travel between facility and the customers. D, i, j is that cost for sending one engineer to do the service between facility j, and the customer i. You have a cost c j. C j is the cost for hosting one engineer in location j. Again, that costs may be different from locations to locations. M, j, k. M, j, k is the maximum number of engineers that may be hosted in facility j, with scale level k. Scale level k, is for example, there may be several different scale levels. That says scale level 0, means you don't build a facility there. Scale level 1, 2, 3, 4 is become bigger and bigger, but for Taipei, scale level 1 may be having a different space that is different from the scale level 1, in Taejong. Or maybe in Taejong you only have two possible scale levels. Maybe in Taejong you have, three, maybe in Taipei, you have four, and so on and so on. Maybe in scale level 1, in Taipei means 100 ping or something like that, and maybe in Taejong, that means 80 ping and so on. Somehow in any particular location j, as long as you choose one scale level, then you know what's the maximum number of engineers that may be hosted there. That are some concrete numbers that you need to know so that later you may determine whether your engineer allocation, is feasible. Whenever you want to allocate engineers there's a reason. Is because that an engineer, must provide some services. But each engineer only have eight hours per day and each engineer need to take some holidays in a year. There is a maximum number of services that an engineer may complete in a year. That's go with small s. That number is pretty much the same throughout Taiwan. If that's the case, then there is a number that all these companies is going to estimate. Because that directly tells the company how many people we need to hire for these particular technical service. S is roughly, let's say 1,000 or 1,000 and the 500 and so on. That's the number of services that an engineer may complete roughly in the year. Then finally, there is a parameter called a, i, j is what? This is a small a, i, j. A, i, j is a binary parameter, or don't forget, is a binary parameter instead of variable. That parameter is saying that if a, i, j is one, that means location j in a customer i, they are close enough. What does that mean? Suppose you have a post office here, and then suppose you have a facility there, may be the distance is just too far from each other so that you don't have enough time to go there by sending of engineer from facility j. Because almost always when you sign a contract with, for example, 711, or when you sign a contract with the post office, they will say that, after I call you in two hours, you need to send someone into my stores, something like that. You have just two hours, four hours constraints. If you really have a two hour constraint, there's no way that a Taipei a facility may serve a post office in Gaussian. In that case, we would state that pair of a, i, j to be zero, if a facility j is not a close enough to the customer i. If that's the case, we said a, i, j to be zero, so that even if your facility j is open, it cannot be used to serve customer i. That's all the parameters we need.
