Question 1 of 6: Economic Order Quantity and the Design of an Inventory Control System

Question 1: Economic Order Quantity and the Design of an Inventory Control System (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A purchased part bought against a steady annual requirement, with the ordering cost quoted per purchase order and the carrying charge quoted as a percentage of the money tied up in stock.

Find. The approximate economic order quantity for the part, and then, in essay form, how a working inventory control system copes with uncertainty, what the form of its information feedback changes, and why it matters whether the system produces to stock or to order.

Approach. Part (a) is the classical square-root lot size: express the carrying charge as dollars per unit per year, then balance the annual ordering cost against the annual carrying cost; parts (b) to (d) are discussed afterwards in the light of what that model assumes and where it breaks down.

1. Part (a) — convert the carrying charge into dollars per unit per year. The charge is quoted as a fraction of the value held, so it must be multiplied by the unit price before it can be used in a lot-size formula: $$H = i\,c = 0.20 \times 10 = 2.00\ \text{per unit per year}$$ Carrying one part in stock for a full year therefore ties up two dollars of interest and storage.
2. Set the total variable cost and minimise it. Ordering D/Q times a year at S per order and carrying an average of Q/2 units gives $$TVC(Q) = \frac{D}{Q}S + \frac{Q}{2}H$$ Differentiating with respect to Q and setting the derivative to zero gives the square-root lot size.
3. Evaluate the economic order quantity. Substituting the given data, $$Q^{*} = \sqrt{\frac{2DS}{H}} = \sqrt{\frac{2(30{,}000)(20)}{2.00}} = \sqrt{600{,}000} = 774.6$$ $$\boxed{Q^{*} \approx 775\ \text{parts per order}}$$ which is the answer the question asks for.
4. Check the balance and report the operating consequences. At the optimum the two cost components must be equal, and they are: $$\frac{D}{Q^{*}}S = \frac{30{,}000}{774.6}(20) = 774.60 \qquad \frac{Q^{*}}{2}H = \frac{774.6}{2}(2.00) = 774.60$$ so the total variable cost is $\sqrt{2DSH} = \sqrt{2(30{,}000)(20)(2.00)}$, that is 1,549.19 dollars a year, on 38.7 orders a year or about one order every 6.5 working days.
5. Confirm that the answer only needs to be approximate. The total-cost curve is very flat near its minimum: ordering 700 at a time costs 1,557.14 and ordering 900 at a time costs 1,566.67, which are only 0.5 and 1.1 per cent above the optimum. That is why the question asks for the approximate quantity — rounding 775 to a convenient carton, pallet or price-break quantity costs almost nothing, and is normally the right operating decision.

Part (b) — how practical systems absorb uncertainty. The square-root model above assumes demand, lead time and yield are all known constants; no real system enjoys that. Practical systems therefore keep the lot-size calculation but attach to it a second, separate decision about when to order, and it is that decision which carries the protection. The device is safety stock: the reorder point is set at the expected demand over the lead time plus a buffer sized from the standard deviation of demand during lead time, so the buffer is chosen against a stated service level rather than guessed. Where the lead time itself is variable, its variance is combined with the demand variance, since a late delivery and a heavy week do the same damage. Where the process is the supplier, uncertain yield is met the same way, by inflating the order quantity by the expected scrap fraction so that the good output, not the gross output, meets the requirement. Around these numerical devices sit three procedural ones: periodic review of the parameters so that a drifting demand rate is caught, expediting rules for the exceptions that the statistics deliberately do not cover, and ABC classification, which spends the analytical effort and the tight control where the money is and leaves the C items on generous buffers and simple two-bin rules.

Part (c) — what the type of information feedback changes. The feedback loop is what converts a formula into a control system, and its type decides both the responsiveness and the running cost of the system. A perpetual (continuous-review) system posts every issue and receipt as it happens, so the stock position is known at all times and an order can be released the instant the reorder point is pierced; the buffer then has to cover only the lead time. A periodic-review system looks at the stock only every T periods, so a stock-out can begin unseen at any point in the review interval and the buffer must cover the lead time plus the review interval — measurably more inventory for the same service level, bought in exchange for cheaper record keeping and the convenience of ordering many items from one supplier on one day. Two-bin and other visual systems are the limiting case: the feedback is physical rather than clerical, it costs almost nothing and it is entirely adequate for low-value items. The choice is therefore an economic one between the cost of information and the cost of the inventory that ignorance requires, and it is properly made item by item rather than for the plant as a whole.

Part (d) — the value of classifying by whether the system produces for inventory. The distinction matters because it decides which variable absorbs the uncertainty, and therefore which model is legitimate. A produce-to-stock system meets demand from a finished-goods buffer, so the customer sees a service level and the inventory absorbs the variability; its control problem is exactly the one solved above, and its performance measures are fill rate and turns. A produce-to-order system holds no finished stock, so the variability is absorbed by the order backlog instead: the customer sees a delivery lead time, the control problem becomes scheduling and capacity planning rather than lot sizing, and the performance measures are delivery promise and shop-floor utilisation. Classifying the system first thus tells the analyst where to look for the cost, prevents the common error of applying a finished-goods reorder-point model to a jobbing shop, and identifies the assemble-to-order middle ground, where the customer decoupling point is deliberately placed at the component level so that standard parts are made to stock and only the final configuration waits for the order.