So let's take an example. Coyote Inc orders detonators from the Acme Company. For fans of the Road Runner you know what I'm talking about. So, every time Coyote orders from Acme, it must incur a cost of shipping which is charged at a flat rate of 75 per shipment. Coyote consumes these detonators at a steady rate of 2,000 per year. Each detonator costs Coyote $10 and Coyote has a capital cost of 20% per annum. Storage costs for the detonators are $2 per year. So, how much should Coyote order to minimize its annual cost? Let's look at the data that we've been given. Our demand is 2,000 per year. Our setup cost or ordering cost is $75 per order. Our carrying cost depends on the cost of storage plus the capital cost and so, that turns out to be the storage costs was $2, the carrying cost is per item $10 times 20 percent. So, 0.2 per year and so that gives us a total of $4. If you plug it into the EOQ formula, Q* turns out to be 273.86 or approximately 274. So, Coyote should order 274 detonators at a time. Now, when we think in terms of this order 274, notice that we took the 273.86 and rounded it up to 274. How much of our differences that make in terms of the cost? Well, the difference it makes is not that significant and the reason why it's not that significant is notice that the minimum point for the total cost curve occurs in the relatively flat part of the total cost curve. So, small changes in the order quantity do not affect the cost very much. In fact, if we were to quadruple the ordering cost or quadruple the demand, then the EOQ quantity only doubles. Okay? Similarly, quadrupling the holding cost only halves the EOQ. What this tells us is that the EOQ is relatively robust to small changes in both the input data and to small rounding that we can do to the final answer. Now, why does this matter? Well, practically some of these costs are somewhat difficult to assess. What should be included in the setup cost? Oftentimes, many of the costs that are involved in set-ups and ordering, are costs across multiple parts and then they are apportioned to individual parts based on accounting principles. But accounting principles don't necessarily tell you exactly what the cost is and for that specific part. So, those costs are somehow approximate. Not only that, despite what we may choose to calculate, suppliers often insist that you order minimum quantity. So that even if you decide that you wanted to order 274 of these detonators, the supplier might say the minimum order for detonators is 1,000 and that forces you then to order 1,000. It may also be possible that suppliers supply in package quantities and the package quantity might be 100. So, if I want to order 274, I have to order three packages of 100, which means I have to order 300. In all these cases, we have to make adjustments to the EOQ quantity that we've calculated, but the robustness of the EOQ model saves us in this case because the cost to not increase dramatically if we have to shift from the calculator or the optimum EOQ quantity q* that we calculate. A second practical consideration is that the EOQ model assumed that we have instantaneous replenishment. But instantaneous replenishment may be difficult in a manufacturing environment. You cannot make things instantaneously. It takes a certain amount of time to produce parts. So, there might be a rate of production which we will denote by R, in our manufacturing process. So, instead of having an instantaneous replenishment, we replenished at the rate of R. In that case, we modify the EOQ formula and create what's called the Economic Manufacturing Quantity formula. Sometimes people call it the Economic Production Quantity formula. So, it's either the EMQ or the EPQ. So, the EMQ looks very similar except for a couple of extra terms. So, instead of having EOQ which was square root of 2 times SD divided by H, we now have this extra term in the numerator where we multiply by the production rate R and then in the denominator we have a term R-T. So, the EMQ is square root of 2 times S times D times R divided by H, the holding cost times (R-D) which is the rate of production minus the demand rate. So, that's the EMQ. Let's take this example further. Coyote decides that instead of ordering from Acme, Coyote is going to manufacture their detonators in-house and they can do it at the rate of 5,000 per year. Now, to keep the data similar to our previous example, let's assume that the setup cost for this manufacturing remains $75 orders, and the consumption rate remains 2,000 and the holding cost also remains the same. We can now calculate the economic manufacturing quantity and it turns out if you plug in all those values into the formula, our economic manufacturing quantity turns out to be 354. Remember our economic order quantity when we're ordering from Acme, was 274. The economic manufacturing quantity is now larger is 354. But what does this mean? Well, let's assume that Coyote has manufacturing going on for 365 days a year and they have demand continuously during those 365 days of the year. So, the amount of time it takes to produce the economic quantity of 354 at the rate of 5,000 per year can be calculated as 354 divided by 5,000 to get it in years multiplied by 365 to convert it into days, and that turns out to be 25.81 days. To consume that same 354, Coyote requires 354 divided by 2,000 that many years multiplied by 365 to get the days or 64.52 days. So, Coyote produces 425.81 days and consumes the amount that is produced in 25.81 days over 64.52 days. Let's look at the inventory graph for this particular system. So, for 25.81 days, Coyote is producing detonators at the rate of 5,000 per year but during the same time, it's also consuming them at the rate of 2,000 per year. So therefore, the rate at which inventory is building up becomes 5,000-2,000 or 3,000 divided by 365 to find the rate of buildup per day. So, we get detonator inventory building up at the rate of 8.22 detonators per day and that's reflected by the increasing inventory in the first part of the triangle that represents the inventory for each cycle. So, inventory keeps building up, it reaches a maximum of 212.13 and then at that time productions stops. Coyote continues to consume inventory at the rate of 2,000 per year or 5.48 per day and so, inventory gets depleted until it drops down to 0, at which point Coyote resumes manufacturing again and starts rebuilding inventory. So, we still have a triangular shape for our inventory profile, but unlike the EOQ where inventory used to increase instantaneously to its maximum, it now builds up over a period of time. So, we've looked at two possible scenarios. One in which manufacturing or replenishment is occurring instantaneously and one in which replenishment or manufacturing occurs over a period of time. For each of these two scenarios, we looked at the costs associated with our ordering decisions. The ordering cost plus the inventory costs and came up with the optimum quantity that should be ordered to minimize the total cost of this particular decision.
