Let's take a look at our next example, which is the so-called EOQ model. That's considered as motivating example. Suppose there's accompany, this company is requiring a certain component and that has a constant consumption rate. For example, this particular airline company, this company needs 500 taillights per year. This is some products that you need to purchase from your component suppliers, and from time to time, it may break down so you need to replace and use a new one. Roughly in each year, you estimate roughly you need 500, so that's the case. There's something that you need to use and consume. You're going to purchase these products from a manufacturer and you will have a unit price which is 500. The thing is that we assume taillights are consumed somehow in a constant rate throughout the year. Think about your purchasing rice for your workers, or you are talking about something that you need to consume constantly. They are something like that. Then whenever we want to place an order, there is the ordering cost, let's say $5 regardless of the order quantity, so what's that? When you want to purchase something, you may need to initiate an ordering process. You need to have someone to sit down and take a look at the inventory, and then you say, okay, I need to purchase something. Then you give a call to your supplier, the supplier returns your call and you talk about whether I may buy spare parts and that and that takes some time. Then you need to print out a copy, have someone to sign, and then send that to your suppliers and ask for your suppliers to send the products to you. You need to take a look, you need to double-check and make sure that everything is done. You need to have invoice, you need to pay blah blah blah. All of these has something you need to pay and that has nothing to do with your order quantity. If you order one product, you need to pay that. If you order 1,000 products, you still pay $5. That's different from your purchasing cost. An ordering cost has nothing to do with your purchasing quantity. Also, there is something called holding cost. What's that? Your holding cost is something that you need to pay to store a product in your warehouse, for example, one month. Let's say storing that thing takes two cents and that help you to store your product in your warehouse for one month. You may want to ask, what's the point of paying two cents, to store your products? Obviously, your warehouse take some money. You need to have someone to take care of that warehouse, you need to pay the salary. There are utilities, you need to pay for electricity, you need to pay for waters. Probably, most importantly, whenever you pay $500 to get one unit of product, that product will always be there. It will not become two products, three products and so on. But if you put $500 into a bank, it will become 501, 502 gradually, because you get interest income. Whenever you put money in your bank, you get some money back. But if you put an inventory in your warehouse, you do not get anything back. That opportunity cost is also a certain source of holding cost. It doesn't really matter whether you are storing broccoli or whether you are storing books. What matters is as long as you need to pay to get this product, then you need to pay some holding cost, depending on the price, depending on interest rates, and so on. Having all of these, your company is asking about one question. We want to minimize the total cost, which is a sum of ordering, purchasing and holding cost, for a satisfying this particular demand. We want to ask, how much should we order in one particular order? When should we place in order? If we were to ask this question, maybe first you need to ask yourself, well, why do we want to have a small order? Why do we want to have a large order? The idea is very simple. Let's say, per year I need 500. All right. That's order once per year, so in every year I place an order for 500 units. If I do that, then each year I only need to pay $5 as my ordering cost. But then I'm going to have a lot of inventory, because the consumption rate is not so high. I would have so many money transformed into inventory, put into warehouse, and that actually have some holding cost. That may not be a good idea. Or on the other way, maybe I say okay, let me place 10 orders per year, then my inventory level would be not so high. Then I can save my holding cost. But then I need to pay $50 per year, as my ordering cost. That also may not be a good idea. There is a trade-off between ordering cost and holding cost. If I order a lot in an order, I'm going to save ordering cost, but then I need to pay a lot for holding cost. If my orders are small, I am going to save holding cost, but then I need to order frequently. Then that comes ordering cost. The whole problem is about when to order and how much to order, that's the ordering problem we are talking about. This question may be answered with the so-called EOQ model. EOQ model is the abbreviation of economic order quantity model. You may see that this is a model, helping us to make these order quantity decision. We look for the order quantity that is the most economic. Which means we want to look for a balance between the ordering cost and he holding cost. Technically, what we want to do is that we want to formulate a nonlinear program, where an optimal solution is our optimal order quantity. The key here, obviously, is that our objective function should somehow include all those costs we just mentioned. Once we have that, we may want to solve this problem and then get to an optimal solution. Later we will do that. But before that, let's take a look at those assumptions for the most basic EOQ models. We mentioned to you that you need to have some annual number of units you need. You need to have demand. Well, demand is somehow estimated, so that's deterministic. Then it has a constant consumption rate. Each day you use one dollar, one dollar, one dollar, one unit, one unit, one unit, and that's constant. Then you have a fixed ordering cost. The cost has nothing to do with your order quantity. We assume that no shortage is allowed. You really need to make sure that you always have units. Shortage is allowed and you don't need to buy anything. We assume the ordering lead time is zero, which means you don't need to worry about when to order. You don't need to worry about every time. Once you order, then you get a product immediately. Today you place an order, tomorrow you'll get it. Then the inventory holding cost is also constant. We don't say at the beginning of the year you have a higher interest rate, at the ending of the year, you have higher interest rate, we don't say that. We assume the inventory cost is constant. You may feel that these assumptions seems to be strong, but don't forget all the models that we try to use, is some abstraction from the world. As long as this model may capture some important trade-offs that we are talking about, as long as it helps us to understand, to deal with the real problems, as long as the model is useful, then it's a good model.
