IntroductionIn the lesson on inventory costs, we saw the trade-off: large orders reduce ordering costs but increase holding costs; small orders do the opposite. One of the best-known ways to balance these costs is to calculate an optimal order quantity for each item: the economic order quantity (EOQ).EOQ is the quantity that minimizes total inventory costs when ordered each time. The formula dates back more than a century: it is generally attributed to Ford W. Harris, who published it in 1913. It is also known as the Wilson formula, after R. H. Wilson, the consultant who popularized it among businesses. It remains built into inventory management systems and ERPs today.The model's assumptionsTo arrive at a simple formula, the model makes several assumptions about inventory:annual demand (D) is known, and usage is constant: the same quantity leaves inventory every day;inventory is always replenished by the same quantity (Q);replenishment is instantaneous and occurs exactly when inventory reaches zero;there are no stockouts: all demand is met;the cost per order (Cp) and the annual holding cost per unit (Cm) are fixed, and the price does not change with the quantity purchased (we will relax this last assumption in the section on discounts).Under these assumptions, inventory follows the familiar sawtooth pattern: an order of Q arrives, inventory falls steadily to zero, another order of Q arrives, and the cycle repeats. If daily usage is d and the time between replenishments is t, then Q = d × t.No real business has perfectly constant demand or instantaneous delivery. The model is still useful, however, because total cost is relatively insensitive to errors in the inputs, as we will see. Later lessons address the less realistic assumptions: lead time and demand uncertainty are accounted for through safety stock and the reorder point.The economic order quantity formulaTo meet annual demand D with orders of Q units, a company places D ÷ Q orders per year. Because inventory falls steadily from Q to zero, average inventory is Q ÷ 2. Therefore:Annual ordering cost = (D ÷ Q) × CpAnnual holding cost = (Q ÷ 2) × CmCm can be expressed in two ways, so pay close attention:as a cost per unit: "holding one unit costs R$ 2 per year" → Cm = R$ 2;as a percentage of the price: "the holding cost is 20% per year" and the item costs R$ 5 → Cm = 20% × 5 = R$ 1 per unit per year.Inventory cost is the sum of the two:Inventory cost = (Q ÷ 2) × Cm + (D ÷ Q) × CpHolding costs rise as order size grows, ordering costs fall, and their sum is lowest at the EOQThe chart shows what happens as Q changes: the holding-cost line rises, the ordering-cost curve falls, and their sum reaches a minimum. In this model, the minimum occurs exactly where the two costs are equal:(Q ÷ 2) × Cm = (D ÷ Q) × CpSolving for Q gives us the economic order quantity formula:EOQ = √(2 × D × Cp ÷ Cm)(If you know calculus, you can check: differentiate cost with respect to Q and set the result to zero to get the same expression.)Two figures derived from EOQ are often used alongside it:Number of orders per year = D ÷ EOQTime between orders = EOQ ÷ D (in years; multiply by 365, or by the number of working days, to get the interval in days). This interval underpins the periodic review system, covered in another lesson in this series.Watch the units: D, Cp and Cm must use the same time period. If demand is monthly and Cm is annual, convert one before calculating.Worked exampleA company buys panel components for R$ 5,00 per unit. It expects to use 4.000 units over the next year. Placing an order with its supplier costs R$ 30,00, and holding one component in inventory costs R$ 1,50 per year. Find: a) the economic order quantity; b) the number of orders per year; c) the total cost at the economic order quantity.a) EOQ = √(2 × 4.000 × 30 ÷ 1,5) = √160.000 = 400 unitsb) Number of orders = 4.000 ÷ 400 = 10 orders per year (approximately one every 36,5 days)c) Adding the purchase cost to the inventory costs:TC = D × P + (Q ÷ 2) × Cm + (D ÷ Q) × Cp = 4.000 × 5 + 200 × 1,5 + 10 × 30 = 20.000 + 300 + 300 = R$ 20.600At the EOQ, the holding cost (R$ 300) equals the ordering cost (R$ 300), just as the model predicts. This equality is a useful way to check your calculations.A company uses 7.200 units of an item per year. Each order costs R$ 100, and holding one unit in inventory costs R$ 4 per year. How many orders should it place each year if it follows the economic order quantity?Answer: 12 orders — EOQ = √(2 × 7.200 × 100 ÷ 4) = √360.000 = 600 units, and 7.200 ÷ 600 = 12 orders per year. 600 is the order quantity, not the number of orders.Total cost is fairly flat near the EOQA common objection is: "Ordering and holding costs are estimates, so the EOQ must be wrong." It will be, to some degree. But the total-cost curve is fairly flat near its minimum. Here are the inventory costs from the previous example at other order quantities:Order quantity QOrdering (D ÷ Q × 30)Holding (Q ÷ 2 × 1,5)Inventory costAbove minimum200R$ 600R$ 150R$ 750+25%300R$ 400R$ 225R$ 625+4%400 (EOQ)R$ 300R$ 300R$ 600—500R$ 240R$ 375R$ 615+2,5%600R$ 200R$ 450R$ 650+8%800R$ 150R$ 600R$ 750+25%Choosing an order quantity 25% above or below the EOQ (300 or 500) raises costs by only 2,5% to 4%. That has two practical implications:you can round the EOQ to fit the supplier's pack size, pallet size or order multiple without much concern;the EOQ is robust to estimation errors: because the formula uses a square root, if the estimated Cp is twice its actual value, the calculated order quantity is only 41% larger (√2 ≈ 1,41), and the cost is only slightly above the minimum.Annual demand for an item quadruples, while ordering cost and holding cost per unit stay the same. What happens to the economic order quantity?Answer: It doubles — demand is under the square root: √4 = 2. The number of orders per year also doubles (4 times the demand divided by 2 times the order quantity). It would quadruple only if EOQ were directly proportional to demand, which it is not.Quantity discountsSuppliers often offer a discount on orders above a minimum quantity. The price then depends on order size, so the fixed-price assumption no longer holds. The discount reduces the purchase cost, the largest part of total cost, but requires larger orders, increasing holding costs.There are more sophisticated formulas for this situation, but the simplest reliable approach is to calculate total cost, including purchase cost, for each price tier and choose the lowest:For each price tier, calculate the EOQ using the Cm for that price (when Cm is a percentage, it changes with the price).If the calculated EOQ is below the tier's minimum quantity, use that minimum (the smallest order eligible for the discount). If it falls within the tier, use the EOQ itself.Calculate total cost TC = D × P + (Q ÷ 2) × Cm + (D ÷ Q) × Cp for each candidate.Choose the lowest total cost.Worked exampleA company buys a product for resale at R$ 5,00 per unit. Its supplier offers the following prices: orders of 1.000 to 1.999 units cost R$ 4,80 per unit; orders of 2.000 units or more cost R$ 4,75 per unit. Each order costs R$ 49,00, annual holding cost is 20%, and expected demand is 5.000 units per year. What is the most economical order quantity?R$ 5,00 price tier (Cm = 20% × 5 = R$ 1,00): EOQ = √(2 × 5.000 × 49 ÷ 1) = √490.000 = 700 units, within the tier.R$ 4,80 price tier (Cm = R$ 0,96): the EOQ would be √(490.000 ÷ 0,96) ≈ 714, below the 1.000-unit minimum. Use Q = 1.000.R$ 4,75 price tier (Cm = R$ 0,95): the EOQ would be ≈ 718, below the 2.000-unit minimum. Use Q = 2.000.QPurchase costHolding costOrdering costTotal cost700 at R$ 5,00R$ 25.000350 × 1,00 = R$ 3505.000 ÷ 700 × 49 = R$ 350R$ 25.7001.000 at R$ 4,80R$ 24.000500 × 0,96 = R$ 4805 × 49 = R$ 245R$ 24.7252.000 at R$ 4,75R$ 23.7501.000 × 0,95 = R$ 9502,5 × 49 = R$ 122,50R$ 24.822,50The best choice is to order 1.000 units at a time. The second discount cuts the purchase cost by a further R$ 250, but it means holding twice as much inventory, and the extra holding cost wipes out the savings.An item has annual demand of 1.200 units, an ordering cost of R$ 50 and an annual holding cost of 20%. Its price is R$ 10, but falls to R$ 9,50 for orders of 500 units or more. The EOQ at full price is about 245 units, with a total cost of about R$ 12.490. What should the company do?Answer: Order 500 units at R$ 9,50, for a total cost of R$ 11.995 — purchase cost 1.200 × 9,50 = R$ 11.400; ordering cost 1.200 ÷ 500 × 50 = R$ 120; holding cost 250 × 1,90 = R$ 475; total R$ 11.995, less than R$ 12.490. Ordering 251 at R$ 9,50 is not possible: orders below 500 units do not qualify for the discount.The model's limitsEOQ is a starting point, not a rule to follow regardless of circumstances. Watch for cases where its assumptions do not match reality:Lead time. The model assumes an order arrives the moment it is placed. In practice, delivery takes time, so the order must be placed before inventory runs out. See the lesson on lead time and safety stock.Uncertain demand. When demand varies, EOQ can still determine order size, but safety stock is needed to prevent stockouts.In-house production. When an item is made in-house and enters inventory gradually while it is being used, a variation called the economic production quantity accounts for the production rate.Physical and financial constraints. Warehouse space, product shelf life, supplier pack sizes and available cash may rule out the calculated order quantity. Because costs are fairly flat near the optimum, adjusting the quantity to meet these constraints usually costs little.ExercisesEasy1) (Logistics Exam, CEFET/RJ, 2018) Eletrônica Estrela uses 3.000 units of an item per year. The ordering cost is R$ 60,00 per order, and the cost of holding each unit in inventory is R$ 1,00 per year. To minimize total inventory cost, how many units should it order each time?2) A retailer has annual demand of 15.000 units for an item. The annual holding cost is R$ 0,80 per unit, and the cost of placing an order is R$ 35,00. Determine the economic order quantity.3) An item has annual demand of 2.400 units, an ordering cost of R$ 75,00 and an annual holding cost of R$ 4,00 per unit. Calculate the economic order quantity, the number of orders per year and the interval between orders in days (using a 365-day year).4) An item costs R$ 25,00. The annual holding-cost rate is 20%, annual demand is 5.000 units, and each order costs R$ 40,00. What is the economic order quantity?5) A company automated its purchasing process, cutting the cost of each order to one-quarter of its previous level. Demand and holding costs are unchanged. What happens to the economic order quantity?Advanced6) VendeMais sells a product with annual demand of 40.000 units. Placing a purchase order costs R$ 30,00. The annual holding cost is R$ 0,30 per unit. There is also a fixed cost of R$ 50,00 per year for this item, regardless of order size. Calculate:a) the total annual inventory cost for orders of 2.500 units, excluding purchase costs;b) the economic order quantity.7) An electronics manufacturer uses 600 printed circuit boards a year and pays R$ 20,00 per order. The annual holding cost is R$ 0,10 per unit, based on average inventory. There is also an annual storage cost of R$ 0,025 per unit, based on the space needed for maximum inventory. Each board costs R$ 1,00. Calculate:a) the economic order quantity;b) the total cost, including the purchase cost;c) the number of orders placed per year.8) An appliance manufacturer expects to sell 4.500 refrigerators a month next year. Each refrigerator uses a compressor purchased for R$ 18,00 per unit using the economic order quantity. Holding a compressor in inventory costs R$ 1,536 per year, and placing a purchase order costs R$ 200,00.a) What is the order quantity for compressors?b) How many orders will be placed per year?9) A company currently uses the economic order quantity. Annual demand is 1.800 units, the economic order quantity is 600 units, the annual holding cost is R$ 2,50 per unit, and the ordering cost is R$ 250,00. The company is considering a new policy: deliveries directly to the production line in batches of 100 units, packed in containers costing R$ 35,00 each (which would become the cost per delivery). Which option is better for the company?10) A supermarket buys rice in 5 kg packages and expects to sell 25.000 kg a year. The supplier accepts only three order sizes: 10.000 kg at R$ 7,00 per package; 15.000 kg at R$ 6,50 per package; or 20.000 kg at R$ 6,00 per package. Processing a rice order costs R$ 3.500,00, while storage and capital costs together amount to 50% of the price per year. Which order size should the supermarket choose?Answer Key1) LEC = √(2 × 3.000 × 60 ÷ 1) = √360.000 = 600 units.2) LEC = √(2 × 15.000 × 35 ÷ 0,80) = √1.312.500 ≈ 1.146 units.3) LEC = √(2 × 2.400 × 75 ÷ 4) = √90.000 = 300 units. Orders per year: 2.400 ÷ 300 = 8. Interval: 365 ÷ 8 ≈ 45,6 days (or 300 ÷ 2.400 = 0,125 year × 365).4) First, calculate Cm: 20% × 25 = R$ 5,00 per unit per year. LEC = √(2 × 5.000 × 40 ÷ 5) = √80.000 ≈ 283 units.5) The economic order quantity is halved. Cp is under the square root: multiplying it by 1/4 multiplies LEC by √(1/4) = 1/2. The company will place twice as many orders for smaller quantities, halving its average inventory.6) a) Ordering: 40.000 ÷ 2.500 = 16 × 30 = R$ 480; holding: 2.500 ÷ 2 = 1.250 × 0,30 = R$ 375; fixed costs: R$ 50. Total: R$ 905.b) LEC = √(2 × 40.000 × 30 ÷ 0,30) = √8.000.000 ≈ 2.828 units. The fixed R$ 50 is excluded from the formula because it does not change with order size.7) There are two inventory costs: R$ 0,10 based on average inventory (Q ÷ 2) and R$ 0,025 based on maximum inventory (Q). The inventory cost is (Q ÷ 2) × 0,10 + Q × 0,025 + (600 ÷ Q) × 20. This is equivalent to using Cm = 0,10 + 2 × 0,025 = 0,15 in the formula.a) LEC = √(2 × 600 × 20 ÷ 0,15) = √160.000 = 400 units.b) Purchases: 600 × 1,00 = R$ 600; holding: 200 × 0,10 = R$ 20; storage: 400 × 0,025 = R$ 10; ordering: 1,5 × 20 = R$ 30. Total: R$ 660.c) 600 ÷ 400 = 1,5 orders per year (3 orders every 2 years).8) First, calculate annual demand: 4.500 × 12 = 54.000 compressors (Cm is an annual cost).a) LEC = √(2 × 54.000 × 200 ÷ 1,536) = √14.062.500 = 3.750 compressors.b) 54.000 ÷ 3.750 = 14,4 orders per year.The R$ 18,00 purchase price is not used in the formula because Cm is already given as a monetary amount.9) Current policy: 1.800 ÷ 600 = 3 orders × 250 = R$ 750; holding: 600 ÷ 2 = 300 × 2,50 = R$ 750; R$ 1.500 per year.New policy: 1.800 ÷ 100 = 18 deliveries × 35 = R$ 630; holding: 50 × 2,50 = R$ 125; R$ 755 per year.Deliveries of 100 units are the better option, at nearly half the cost. Cutting the cost per delivery from R$ 250 to R$ 35 changes the optimal order quantity, allowing the company to hold much less inventory.10) Demand in packages: 25.000 ÷ 5 = 5.000 per year. The available order sizes are 2.000, 3.000 and 4.000 packages (10, 15 and 20 thousand kg), with Cm equal to 50% of the price for each option:2.000 at R$ 7,00 (Cm = 3,50): 5.000 × 7 = 35.000 + 2,5 orders × 3.500 = 8.750 + 1.000 × 3,50 = 3.500; total: R$ 47.250;3.000 at R$ 6,50 (Cm = 3,25): 32.500 + 1,667 × 3.500 = 5.833,33 + 1.500 × 3,25 = 4.875; total: R$ 43.208,33;4.000 at R$ 6,00 (Cm = 3,00): 30.000 + 1,25 × 3.500 = 4.375 + 2.000 × 3,00 = 6.000; total: R$ 40.375.The lowest-cost option is to order 4.000 packages (20.000 kg) at a time. The purchase discount (R$ 5.000 a year compared with the full price) more than offsets the additional holding cost.
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