[MUSIC] Hi everyone, welcome back to Operations Research, this is Ling J Kun again, and is very happy to see you again. Probably you are not very happy to see me again, but anyway let's do it. So today is our week four and today we have again a new topic. We're going to talk about nonlinear programming. Okay, so what's this? We know that in many cases we need to deal with nonlinear situations, nonlinear formula. For example, when you make your product pricing decision, when you increase your price, you know that demand is going to decrease typically. So if that's the case, then when you multiply your price or your margin with your demand. That decision, that profit function is going to be nonlinear to your price, okay? It's going to be something like this. When your price becomes very large, that's going to create some problem to your demand. So if that's the case, then we're going to say we are facing a nonlinear function here. So if a function is nonlinear, then of course, your linear programming or integer programming thing is not going to be enough. So that's why we need nonlinear programming, so we will see examples like pricing, okay? In pricing problems we will give you formulation and then tell you what's the basic trade off between demand and price. We will also have examples about inventory, so what's the issue about inventory? In many cases, when you make your ordering decision, you need to tell your supplier how much you want to purchase today. If you purchase a lot, you're going to have a lot of inventory in your warehouse and then that creates inventory cost for you. But if you order just a little bit, then you need to order very frequently and each time you order there may be some fixed costs. For example, maybe in each order, you need to pay the delivery fee. And the last delivery fee may has nothing to do with the amount of your ordering quantity. Each time you need to pay $500 to your supplier, next time another 500. So that kind of fixed ordering cost is going to create some trade off between your production and ordering quality. If you order a lot, you save ordering cost, but then you have a lot of inventory cost. If you order just a little bit, you have a short inventory cost, but then the delivery fee is going to be very high. So that's again another trade off in inventory problem. Lastly, we will give you a problem about portfolio optimization. So what's this? This is basically an application in finance. So sometimes we have some money we want to do some investment and there are so many items we may invest our money in. For example, if you are just looking at the stocks, every time when you want to do stock investment. You have so many options, stock one, stock two, stock three. When you want to do this budget allocation problem, it seems to be a linear program. Because you allocate your limited resources to multiple options, right? But sometimes you care about just profit. But in this particular case, when you do financial decisions, you always also care about risk, and the risk in many cases may be measured by variability. Which may be measured by variance or a standard deviation, something you heard about from your statistics course. So when we are talking about variance, standard deviation, whatever, the formula itself by nature is nonlinear. So that's why we need a nonlinear programming. So for all of these, we will give you formulation, we will give you applications. We will tell you what's the inside and eventually everything is talking about balance. Okay, we want to find a balance between high margin or high demand, if this is a pricing problem. We want to find a balance between high ordering cost and high inventory costs. We want to find a balance between high expected return and high risk. Those kinds of things are looking for balance. We probably need to find the right way to adjust our allocation or adjust our investment. So that we find the best balance between all different trade offs. So balance will be the main issue in nonlinear programming, today, let's see how to do this.