IntroductionIn the previous lessons, we answered two questions: how much to order (the economic order quantity) and how much to keep in reserve (safety stock). That leaves a third question, one the warehouse manager faces every day: when to order?Inventory management offers two classic approaches. The first is the continuous review model, which we will call the Q system. The second is the periodic review model, or P system. Both use the concepts you already know—demand, lead time, safety stock and order quantity—but put them together differently. In this lesson, we will look at each system through formulas, charts and a numerical example comparing them for the same product.Notation used in this lesson (the same as in the lessons on economic order quantity and safety stock):Dt or d: average demand per unit of time (per day, per month...);tr or L: average replenishment lead time, expressed in the same time unit as demand;Es or ES: safety stock;Q: order quantity;Ip or T: time between inventory reviews in the P system;σ: standard deviation of demand per unit of time; z: number of standard deviations corresponding to the service level (1,28 for 90%, 1,65 for 95%, 2,06 for 98%).One rule applies throughout: the time units must match. If demand is measured in units per day, lead time and the review interval must be measured in days. Mixing monthly demand with a lead time in days is the most common calculation error in this topic.The Q system, or continuous review modelThe continuous review model appears in the literature under several names: continuous review model, reorder point policy, (Q, r) model, fixed order quantity system and two-bin system, among others. The idea is simple: set a fixed inventory level that triggers a new order of a predetermined quantity Q when it is reached. This level is the reorder point (PP), also called the replenishment point.The Q system does not necessarily require an economic order quantity. According to Buffa (1968), Q can be set using a rule of thumb, the supplier’s pack size or an economic order quantity model suited to the situation. Ordering in economical quantities is, of course, useful (see the lesson on economic order quantity).The reorder pointBecause demand varies, we cannot know in advance which day inventory will reach the reorder point. In the Q system, the time between orders varies: what stays fixed is the inventory level that triggers an order. That level is calculated as follows (Wu, 2000; Tubino, 2000):PP = Dt × tr + EsThe reorder point therefore accounts for two things: the inventory used while the order is on its way (Dt × tr) and the safety stock that covers above-average demand during that period. In the Q system, safety stock only needs to cover uncertainty during lead time, so we use the formula from the safety stock lesson:Es = z × σ × √trBuffa (1968) highlights an easily missed detail. When replenishment lead time is long and a quantity Q lasts less than tr, more than one order may be in transit at once. In that case, the reorder point must be compared with the inventory position, not the stock physically on hand:inventory position = stock on hand + outstanding orders (placed but not yet delivered) − unfilled customer ordersA new order is placed only when this position reaches PP. When lead time is short, the distinction matters little because an order arrives almost immediately.A 60-day simulation of an item with average demand of 40 units per day, order quantity Q = 480, lead time of 4 days, PP = 187 and ES = 27. Every order is the same size, but the time between orders varies (13, 13, 13 and 11 days) with demand.Average and maximum inventoryInventory rises and falls each day in the sawtooth pattern you saw in the economic order quantity lesson. To find average inventory, we could record the balance at the end of each day and take the average. There is a much simpler estimate: if Q is the order quantity and ES is safety stock, average inventory (EM) isEM = Q / 2 + ESand maximum inventory, just after a delivery (assuming average demand), is Emáx = Q + ES. Average inventory is the figure used to calculate holding costs and inventory turnover (see the lesson on inventory policy and turnover).Advantages and disadvantagesSome textbooks call continuous review a fixed-quantity, variable-interval system, which neatly describes how it works: order quantity stays fixed at Q, while the timing of each order depends on demand.The main advantage is near-real-time tracking of inventory levels. This makes it possible to order at the right time and hold less safety stock, since it only needs to cover lead time. It also gives accounting an accurate record of what has come in and gone out.The main drawback is the need for a reliable system that records every receipt and withdrawal: barcode scanners, an ERP system and physical counts to check accuracy. If the records are wrong, orders go out at the wrong time. That is why continuous review is not always worthwhile for every item.The two-bin system is a low-tech version of the Q system. Stock is kept in two bins. When the first is empty, an order is placed; the second holds exactly the reorder-point quantity.A bakery sells an average of 30 bags of flour per day. Its supplier delivers in 5 days, and the bakery keeps 40 bags as safety stock. Under the Q system, at what inventory level should it place an order?Answer: 190 bags — PP = 30 × 5 + 40 = 190. The most common mistake is to leave out safety stock and answer 150, which covers only consumption during lead time.The P system, or periodic review modelThe periodic review model also goes by several names: periodic review model, periodic order model, fixed reorder cycle system and, in inventory management notation, an (R, S) or (T, S) policy. Inventory is reviewed every R (or T) units of time, and enough is ordered to bring it up to level S. In short, the time between reviews is fixed, while the order quantity varies with consumption since the previous review (Buffa, 1968). The quantity ordered is therefore usually different from the optimal order quantity: it is whatever is needed to cover demand until the next order arrives.The P system’s main advantage is flexibility in choosing when to review inventory. According to Novaes and Alvarenga (1994), reviews take place at fixed intervals, removing the need for the continuous monitoring required by the Q system. Companies can schedule reviews so that orders for several products from the same supplier fall on the same date, simplifying purchasing and making it possible to obtain shipping discounts (Novaes and Alvarenga, 1994; Tubino, 2000). For this reason, Novaes and Alvarenga consider periodic review the system most widely used by companies.The review intervalThe interval can follow any convenient schedule (every Monday, the 1st of each month, a sales representative’s visit) (Tubino, 2000). It can also be calculated from the economic order quantity to bring it close to the optimum (Buffa, 1968): if the economic order quantity is Q and demand is D, the interval at which each order averages one economic order quantity isIp = Q / DFor example, an economic order quantity of 480 units and demand of 40 units per day give Ip = 480 / 40 = 12 days.The maximum inventory level (order-up-to level) and order quantityAt each review, the business calculates how much it needs to order to reach the maximum inventory level, also called the order-up-to level. This is safety stock plus expected demand over the review interval and lead time. An order placed today must last until the following order arrives: that order will not be placed for another Ip units of time and will arrive tr units of time after that.Emáx = Dt × (tr + Ip) + EsThe quantity ordered at each review takes the current stock balance (Ea) into account:Q = Emáx − EaOne caution: any earlier order still in transit must also be deducted (Q = Emáx − Ea − outstanding orders), for the same reason that the Q system uses inventory position. If the result is zero or negative, nothing is ordered in that cycle.Notice in the chart that physical stock never reaches the order-up-to level. That level represents the inventory position immediately after an order is placed; by the time the goods arrive, tr days of demand have already been met. On average, physical inventory in the P system is approximatelyEM ≈ Dt × Ip / 2 + EsThe same item and demand as in the Q-system chart, now reviewed every 12 days, with a lead time of 4 days, an order-up-to level of 693 and ES = 53. The review interval stays the same, but each order is a different size (563, 415, 429, 481 and 507 units).Safety stock in the P systemBecause the P system checks inventory only at set intervals, it introduces another source of uncertainty. After a review, the next order will not be placed for Ip units of time and will not arrive until tr units of time later. Safety stock must cover demand variability over the entire review interval plus lead time, not just during lead time:Es = z × σ × √(T + L)Here, T is the time between reviews, L is lead time, σ is the standard deviation of demand per unit of time and z is the number of standard deviations corresponding to the desired service level. This is the P system’s main disadvantage: at the same service level, it requires more safety stock than the Q system.An item with σ = 10 units per day and a lead time of 4 days switches from continuous review to periodic review every 12 days, while keeping the same service level. What happens to its safety stock?Answer: It doubles because the factor changes from √4 to √16 — in the Q system, ES covers only lead time (√4 = 2); in the P system, it covers the review interval plus lead time (√(12 + 4) = √16 = 4). Because variability grows with the square root of time, rather than time itself, ES doubles instead of quadrupling.Worked example: the same item under both systemsA distributor sells an average of 40 units of a product per day, with a standard deviation of 8 units per day. The supplier delivers in 4 days, and the calculated economic order quantity is 480 units. The company wants a 95% service level (z = 1,65).Q systemSafety stock: ES = 1,65 × 8 × √4 = 26,4, rounded up: 27 units;Reorder point: PP = 40 × 4 + 27 = 187 units;Average inventory: EM = 480 / 2 + 27 = 267 units;Average time between orders: 480 / 40 = 12 days (though this varies in practice).System P, with the review interval derived from the economic order quantityInterval: Ip = 480 / 40 = 12 days;Safety stock: ES = 1,65 × 8 × √(12 + 4) = 52,8, rounded up: 53 units;Target inventory level: Emáx = 40 × (4 + 12) + 53 = 693 units;If the review finds 230 units in stock and none in transit: Q = 693 − 230 = 463 units;Average inventory: EM ≈ 40 × 12 / 2 + 53 = 293 units.System QSystem PWhat stays fixedOrder quantity (480)Review interval (12 days)What variesWhen an order is placedOrder quantityWhat triggers an orderInventory reaches the reorder point (187)The scheduled review date arrivesSafety stock27 (covers 4 days)53 (covers 16 days)Average inventory267293Monitoring requiredContinuous (every withdrawal recorded)Only on the review dateIn this example, system P holds an average of 26 more units, nearly 10% more inventory. That is the cost of checking stock only every 12 days.In the example above, the stockroom clerk finds 180 units in stock during the review on day 24, with another 100 units from an earlier order still in transit. How many units should be ordered?Answer: 413 units — Q = 693 − 180 − 100 = 413. Forgetting the order in transit gives an answer of 513 and leaves inventory 100 units above the target.Which system should you use?Periodic review is widely used because it is simple: monitoring inventory takes little effort. Continuous review provides more control and requires less safety stock, provided the company records every inventory movement.Many companies use both. Class A items (see the lesson on ABC and XYZ classification), or items with highly uncertain demand, are managed through continuous review. For these items, every extra unit ties up cash, while every stockout affects customers. Low-value items or those with predictable demand can be managed through periodic review.In general, when the necessary technology is available, continuous review costs less and provides better control. Even so, system P is preferable in some situations:products that need a physical inspection to check their condition, such as food and other perishables: the periodic review provides an opportunity to remove anything that has expired;suppliers that offer discounts or lower shipping costs when several items are ordered on the same day: combining orders cuts ordering and transportation costs;suppliers with fixed visiting or delivery days, as is common in retail and the public sector.Which of the following situations is the clearest case for system P (periodic review)?Answer: A store that buys 30 items from the same wholesaler, which delivers only on Tuesdays and offers free shipping on combined orders — the fixed interval aligns with the delivery day, and combining orders saves on shipping. The other three describe critical or expensive items with continuous tracking, for which continuous review is more effective.ReferencesROSA, MAYERLE and GONÇALVES. Controle de estoque por revisão contínua e revisão periódica: uma análise comparativa utilizando simulação. Produção, v. 20, n. 4, p. 626-638, 2010. The basis for this lesson; the citations of Buffa (1968), Wu (2000), Tubino (2000), and Novaes and Alvarenga (1994) come from this article.NOVAES, A. G.; ALVARENGA, A. C. Logística aplicada: suprimento e distribuição física. São Paulo: Pioneira, 1994.TUBINO, D. F. Manual de planejamento e controle da produção. 2nd ed. São Paulo: Atlas, 2000.ExercisesEasy1) (Logistics exam, CEFET/RJ, 2018) Manoel sells 20 savory snacks a day. Whenever his stock falls to 60 snacks, he orders 80 more from his supplier, Dona Maria. She takes 2 days to make and deliver them.Identify the replenishment model, sketch the inventory-level graph, and calculate the applicable variables: demand (D), order quantity (Q), lead time (L), safety stock (ES), replenishment interval (IR), reorder point (PR), average inventory (EM), and target inventory level (EA).2) Average demand for product Y is 50 units a day, and each purchase order is for 400 units. The supplier takes 3 days to deliver an order. The company holds safety stock equal to 2 days of demand.Identify the replenishment model, sketch the inventory-level graph, and calculate the applicable variables: demand (D), order quantity (Q), lead time (L), safety stock (ES), replenishment interval (IR), reorder point (PR), average inventory (EM), and target inventory level (EA).3) An item's economic order quantity is 900 units, and average demand is 30 units a day. If the company adopts periodic review with an interval derived from the economic order quantity, how often should it review its stock?4) Under system P, an item's target inventory level is 600 units. At the review, there are 220 units on the shelf and 150 units from an earlier order in transit. How many units should be ordered?5) Under system Q, an item's reorder point is 350 units. There are 120 units physically in stock, 300 units from an order in transit, and 20 units owed on overdue customer orders. Should another order be placed now?Hard6) (Logistics exam, CEFET/RJ, 2018) Every December, public school X checks its stock of pens and orders more. The order arrives in February. The school uses 100 pens a month and holds 200 pens as safety stock. At the last check, it had 180 pens in stock.Identify the replenishment model, sketch the inventory-level graph, and calculate the applicable variables: demand (D), order quantity (Q), lead time (L), safety stock (ES), replenishment interval (IR), reorder point (PR), average inventory (EM), and target inventory level (EA).7) Using the economic order quantity, a distributor has determined that the ideal interval for replenishing deodorant at its supermarket customers is 20 days. It delivers 5 days after each review. Supermarket A has 200 deodorants in stock and none in transit. Its safety stock is 100, and demand is 2 units a day.Identify the replenishment model, sketch the inventory-level graph, and calculate the applicable variables: demand (D), order quantity (Q), lead time (L), safety stock (ES), replenishment interval (IR), reorder point (PR), average inventory (EM), and target inventory level (EA).8) An item has average demand of 25 units a day, a standard deviation of 6 units a day, and a lead time of 9 days. Its stock is reviewed every 7 days. For a 90% service level (z = 1,28), calculate the safety stock and target inventory level under system P. Then calculate the safety stock and reorder point if the item switches to system Q.9) An item has demand of 900 units a month (a 30-day month), a standard deviation in daily demand of 8 units, and a lead time of 6 days. The company wants a 95% service level.a) Under system P, with reviews every 15 days, calculate the safety stock and target inventory level. Today's review finds 250 units in stock and none in transit: how many should be ordered?b) Under system Q, calculate the safety stock and reorder point.10) In the lesson's worked example (average inventory of 267 units under system Q and 293 under system P), the item costs R$ 50, and the annual holding-cost rate is 24%. Managing the item through continuous review would cost the company an additional R$ 500 a year for barcode scanning and stock audits. Which system costs less for this item?Answers1) System Q (the same quantity is ordered whenever stock reaches a set level). D = 20 per day; Q = 80; L = 2 days; PR = 60. Demand during lead time = 20 × 2 = 40, so ES = 60 − 40 = 20. EM = 80 ÷ 2 + 20 = 60. Maximum inventory = 80 + 20 = 100. Average time between orders = 80 ÷ 20 = 4 days.2) System Q. D = 50 per day; Q = 400; L = 3 days; ES = 2 × 50 = 100. PR = 50 × 3 + 100 = 250. EM = 400 ÷ 2 + 100 = 300. Maximum inventory = 400 + 100 = 500. Average time between orders = 400 ÷ 50 = 8 days.3) Ip = Q ÷ D = 900 ÷ 30 = 30 days.4) Q = Emáx − Ea − units in transit = 600 − 220 − 150 = 230 units.5) Inventory position = 120 + 300 − 20 = 400 units. Since 400 is above the reorder point of 350, no order should be placed now. Looking only at physical stock (120) would prompt an unnecessary order and leave too much inventory when both shipments arrive.6) System P (review on a fixed date, once a year). D = 100 per month; IR = 12 months; L = 2 months (from December to February); ES = 200.EA = 100 × (12 + 2) + 200 = 1.400 + 200 = 1.600 pens.Q = 1.600 − 180 = 1.420 pens.EM ≈ 100 × 12 ÷ 2 + 200 = 800 pens.Note that the 180 pens will not cover the 2-month wait (200 pens): at average consumption, the school will run short by about 20 pens before delivery. Its inventory was already below the safety-stock level at the review.7) System P. IR = 20 days; L = 5 days; D = 2 per day; ES = 100; current stock = 200.EA = 2 × (20 + 5) + 100 = 150 units.Q = 150 − 200 = −50, meaning supermarket A does not need to order this cycle: it already has more than the target inventory level. EM ≈ 2 × 20 ÷ 2 + 100 = 120 units under normal conditions.8) System P: ES = 1,28 × 6 × √(7 + 9) = 1,28 × 6 × 4 = 30,72 → 31; EA = 25 × (9 + 7) + 31 = 400 + 31 = 431.System Q: ES = 1,28 × 6 × √9 = 1,28 × 6 × 3 = 23,04 → 24; PP = 25 × 9 + 24 = 225 + 24 = 249.9) First, calculate daily demand, since the standard deviation and time periods are given in days: 900 ÷ 30 = 30 units per day.a) ES = 1,65 × 8 × √(15 + 6) = 1,65 × 8 × 4,583 = 60,5 → 61; EA = 30 × (6 + 15) + 61 = 630 + 61 = 691; order quantity = 691 − 250 = 441 units.b) ES = 1,65 × 8 × √6 = 1,65 × 8 × 2,449 = 32,3 → 33; PP = 30 × 6 + 33 = 180 + 33 = 213.10) Cm = 24% × 50 = R$ 12 per unit per year. System P holds an average of 293 − 267 = 26 more units: 26 × 12 = R$ 312 per year in additional holding costs. System Q would cost R$ 500 more per year to monitor. For this item, system P is cheaper (by R$ 188 per year). For a more expensive item or one with less predictable demand, the result could be reversed. That is why class A items are usually managed with continuous review.
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