Okay, so that was our general introduction to integer programming formulation, okay? So we mentioned about selecting variables, selecting constraints, or fixed charge problems. Actually, there are more, but if we only talk about general things, not so interesting. So let's jump into the first field of applications, facility location problems. So one very typical managerial decision is to build facilities. And you need to decide where to build your facilities, how many facilities to build, if you are opening convenience stores, restaurants, whatever you need to decide where to open these facilities. If you want to build warehouse, you want to build distribution centers again, you need to choose locations. Factory you need to choose locations, power stations, fire stations, police stations, even though these are public facilities, you still need to choose locations, right? So very typically you need to consider demand. Maybe you also consider the cost. But anyway that seems to be an optimization problem, right? So a similar question is where to locate a scarce resource. So this may not be some buildings to make, but maybe you need to determine where to put some important things. For example, if you have a limited amount or number of fire engines or ambulances, you need to somehow distribute them in your area, you don't want to put all of them in one building, right? You need to somehow distribute them, how to do that. If you have police officers, they need to work Around to make sure that everything is fine, how to distribute them. If you have some ice cream machines, you want to install them into your convenience stores, but you cannot do it in any store then you need to decide where to do that. So these are all facility location problems in general, even though sometimes you are not building facilities. But as long as you are allocating some important things and that takes some cost and then that has an impact on your performance, then it is a facility location problem. So we will focus on discrete facility location problems which means there are a finite set of candidate locations. And among them we want to choose a subset of locations to build our facilities. So in general, there are some different things to keep in mind. There must be some demand nodes, and then there must be some potential locations in almost all applications, okay? We build facilities at locations to serve the demands. For example, distribution centers is make to ship products to retail stores. In that case, our retail stores or Lowe's retail stores, they are our demand nodes or we build fire stations to cover cities, towns, villages, so that we may get there in time. So there must be some locations to choose from and the sound demand nodes. Facility location problems are typically categorized based on their objective functions. So there are all kinds of facility location problems, but you may solve them with different purpose. So we will give you some introduction to three types of facility location problems, set covering, maximum covering, fixed charge location problems. The first one is that you need to cover all your demands, you need to take care of everyone. So you'll want to save your cost as much as possible. The second thing is that the number of facilities you may build is fixed, is given you want to cover as many persons as possible. So similar but not the same, they both want to cover something, but the objectives are somewhat different. Finally, you are not restricted in the number of facilities to build but building facilities now has some cost and covering demand is also not required. You don't need to take care of anyone, but covering people, having some benefits. So you need to find some trade-off between building facility and covering persons. You need to having some balance between the cost and the benefits. So later not see how to do any of them.
