Okay, so now let's formulate this problem, so again, we need to list our parameters that use these notations, capital D is our annual demand measured as unit. For example, 500 per unit per year, K is the unit ordering cost, for example, $5 per unit, h is the unit holding cost per year. For example, if you need to pay $0.20 per month for one unit, then per year, you multiply it by 12. Finally, p is the unit purchasing cost, for example, $500 per tail light. So these are the numbers that we already know. Once we have that, then now we want to make a decision on our ordering quantity, order quantity. So q is the number of units we want to order in a particular order, all right? Our decision is going to be made so that we may minimize the annual total cost. So for all our calculations or derivations below, we will use one year as our time unit. You actually may use one month, one century, whatever to be your time unit but don't forget to make it consistent, throughout all the numbers, okay? And then, in this case, actually, as long as we fix the lens of our time interval, you're capital D can be considered as some kind of demand rate, okay? 500 units per year, well that's some kind of demand rate, so the key for formulating and your key problem, is to consider to understand, what's going to happen at your warehouse, what's going to happen at your inventory? So to formulate this problem, we need to understand how the inventory level is affected by our decision. And inventory level, briefly speaking is the number of inventory we have on hand. So because there's no ordering the time, what we need to do is to place an order when we see no inventory, okay? So at the end of each day, you take a look at your warehouse, if you still have something you don't need to do anything. And if you can see that, my God, there's no product there, then you place an order tomorrow morning, you get it right? So, you place an order when you have no inventory, as inventory is consumed at a constant rate, or inventory level would be something like this. So throughout the year, every day your inventory is going to decrease, decrease, decrease, decrease. And once you hit zero, you're going to place an order and then you get replenishment and then your inventory level would go up. Because in the morning, some people knock your door and give you 50 units, all right? And then, day by day Use goes down, goes down, goes down, and then go up, goes down, goes down, goes down and then go up and so on and so on. We're going to say that if you order q units per day, then your inventory level would have a shape in this particular format, okay? Every time you have 50 units and then it gradually go down and once you hit 0, it goes up to 50 units again, you're going to have this triangular inventory level shape. So once we have that the same situation we know it will repeat until the end of the world, all right? So we will not will ask in average, what's the number of units we are storing in our warehouse? Well, in average, our inventory level would be q over 2 to agree, for example, if you order 50 units In each order then now you have 50, 49, 48, 47, up to zero is equivalent to having 25 units throughout the year. And by equivalent, I mean the amount you need to pay is actually equivalent to that one okay? Or you may think in that way. Basically we are talking about inventory cost, total inventory cost. So when you have q units, there is an inventory cost, when you have q minus o 1 unit, there's another inventory cost and so on, so on. Inventory cost is pretty much proportional to the amount of inventory you have, so if you want to calculate the inventory cost, that would be, you need to get all the areas under your curve. And that's pretty much q over 2 times your time lens, okay? If you are talking about 1 year, then you put 1 here, that's how we express our inventory cost, you either look at this graph and understand that in average it's just like everyday we have q over 2. Or you do some integral, in the integration and you get the area below the curve you are going to get the same thing. So now we are ready to do our formulation, once I choose my ordering quantity, my annual holding cost would be hq over 2, okay? For one year, the length of the time period would be 1, and my inventory level is q over 2 in average, right? So q over two is my inventory level in average, and I need to pay edge for that for each year. And then I have annual purchasing cost, for each year I need to buy Capital D units, and whether I buy it in 10 orders or 1 orders it doesn't really matter, because I just need to pay pD to complete this task. And then my annual ordering cost would be here, I need to first calculate the number of orders in a year, that will be d over q, all right? For d over q, obviously lateral because if I need 500 units per year, and if I order 50 units in each order, I need to place 10 orders and that's 10 is this number, if I need to place that many orders, each order cost me k dollars, then that product would be my annual ordering cost. So collectively nonlinear program for optimizing the ordering decision would be this one. I'm going to choose q such that I may minimise the sum of setup, I mean, ordering cost, purchasing cost and inventory cost. You may clearly see that, well, if I increase q, I'm going to make the inventory cost higher, if I purchase a lot and put them into my warehouse, I need to pay a lot of holding cost. But once I do that, I would have a reduction on ordering cost, if I decrease q, I'm going to pay more about setup cost, but then inventory cost would be saved. So that's some kind of trade off and this formulation hopefully would help us find an optimal solution if we know how to solve it, right? So pD is actually a constant, it has nothing to do with your quantity. So in many cases, a more relevant objective function would be just this one, all right? We only want to choose the q to find the best trade off between the setup cost and the holding cost. So that's the end for formulating the EOQ problem, so the interesting thing is that we certainly have some nice way to solve the EOQ problem. So for this particular model, there is a way to find an optimal solution, either analytically or computationally, we're not going to introduce those things here. You just need to try to believe in me, so that this can be solved and then, you may think this is a toy model, actually its not. EOQ model maybe one of the most useful, most widely used inventory model in practice, why is that? Because, first, it's simple indeed, but it captures very important trade off that you need to consider in practice, or they're more or less, they create some different costs and you need to find the best policy. You should not order too much, you should not order too few, this model is actually going to help you, so that's about the EOQ.
