We've talked about inventories as being a vital part of meeting demand for both services and products. However, inventory is bring costs with them and since they bring costs with them, it's important to figure out what's the right amount of inventory to have so that they serve the purpose for which we are keeping them and yet we can keep costs as low as possible. So, let's look at a few different ways that we can decide on how much inventory to keep in our system and let's start out with transaction costs. What happens when we are keeping inventory for transactional efficiency? When we keep inventory for transactional efficiency, we are trying to balance the cost of ordering versus the cost of keeping that inventory or carrying that inventory. So accordingly, let's think in terms of what are those costs? So, the transaction costs for each order or each time we decided to set up might be and there's the cost of ordering which might be the cost of the person doing the ordering or the cost that we charge for every transaction in our computer system. Once that order is placed and it's shipped to us, there might be costs of transporting it back to us and then there's the cost of receiving the goods in our warehouse. So, ordering costs, receiving costs, fixed transportation costs, are all costs that become part of the setup cost or ordering cost and we usually denote that by the capital letter S. Now, there are other costs involved. For example, once I order a batch of an item, that batch will get slowly depleted. During the time that it is being consumed to, I have investments in this particular inventory. I have to store this particular inventory and there are costs associated with that storage as well. So, there are two costs that I need to be worried about. I have to worry about the storage costs, the cost of actually physically keeping this inventory in storage, and then there's the cost of the working capital that's tied up in this inventory. Together, we call this the holding cost. The amount of holding cost are depends on how much inventory I'm carrying and so, this is a particular cost that's dependent on the decision that I make of how much I choose to carry. So, I have two costs now. I have the cost of the transaction itself, the set-up cost, and then I have the holding cost of carrying inventory. Those two costs will depend on the amount that I choose to order at any given time. So, how often should I order and how much should I order at a time? Now, if I decide to order too often, then I will pay a lot of ordering costs or setup costs. However, because I'm ordering frequently, I don't have to order as much at a time and so my inventory carrying costs will be lower. On the other hand, if I want to lower my inventory cost, I can choose to order less frequently and so, I pay more inventory carrying costs and I pay less ordering costs. So, if you look at the figure, the small triangles represent the case when I order frequently and as you can notice that I order frequently so I have a lot of orders and my inventories are lower. When I order less frequently, the dotted red lines then I have lot fewer orders but a lot more inventory. So the question is, how do I balance these two costs? So, let's look at what happens to my costs for each of those individually. Now, if I choose to look at the ordering cost and I look at this on an annual basis, then the number of orders that I have to place for ordering depends on the order quantity that I have. So, if I take the annual demand, divided by the order quantity that tells me the number of orders that I will have in that particular year. That multiplied by the setup cost will then tell me the total cost of actually ordering. The inventory on the other hand, depends on the average inventory that I'm carrying. If I have an order quantity of Q, then the average inventory, if you look at our graph previously, starts of at a maximum of Q and ends up at a minimum of zero, so, the average is cube divided by two. That multiplied by the holding cost becomes Q divided by two times H and that becomes the annual cost of inventory. The total cost of having this inventory system then, is the cost of ordering plus the cost of inventory and so I add those two formula together to get the total cost of inventory. Now, as you can notice both those costs depend on the quantity that I'm ordering Q. So what happens as Q increases? So, if I look at the cost of ordering, if Q increases, then the cost of ordering which is shown by the red graph keeps dropping rapidly as the order quantity increases. It's a hyperbolic graph and it asymptotically goes to zero as the order quantity goes to infinity. The inventory carrying cost on the other hand, increases linearly so as Q increases, it increases as a straight line. When you add the two together, you get a total cost curve which drops rapidly initially then seems to flatten off and then gradually increases. The minimum total cost can be obtained where the inventory carrying cost curve and the ordering cost curve intersect. That gives me the location where I get the minimum total cost and if I'm trying to minimize total cost, that should be my order quantity. So, mathematically, I can solve this by finding out what's called the Economic Order Quantity or EOQ. So, the EOQ is the quantity that minimizes the total annual cost and it's given by Q star, we use Q superscript star to denote the economic order quantity. It's given by the square root of two times the demand, multiplied by the setup cost, divided by the holding cost. Now, if I choose to use a economic order quantity, then my total on annual costs which are the lowest possible total annual cost are given by square root of two times the demand multiplied by the setup cost, multiplied by the holding cost. Since the minimum is obtained where the two costs are equal, the annual ordering cost and the annual inventory costs, each of those is then half of the total annual cost and they're given by the formulae as shown there, which is square root of D times S, times H, divided by two.