Inventory Management Review: Formulas, Concepts and an Integrated Exercise

30/10/2026 •0•0

An overview of inventory management covering key metrics, forecasting, costs, economic order quantity, safety stock, Q and P systems, ABC and XYZ analysis, and FIFO, with a complete exercise.

작성자:Leon

번역자Editorial

개정자Editorial

Introduction

This lesson closes the first part of the course, which focuses on inventory management. It introduces no new material. Instead, it brings together the concepts and formulas from lessons 3 to 12, shows how they fit together, and works through an exercise covering everything from sales history to the value of inventory on the balance sheet. Use it to review before an exam or as a quick reference. If a formula is unclear, return to the lesson indicated.

Inventory management addresses a series of questions, in order:

How much inventory do we want, and how do we know whether we have the right amount? Inventory policy and performance measures (lesson 3).
What kinds of materials do we have? Inventory types and classification (lesson 4).
How much will we use? Demand forecasting (lessons 5 and 6).
What does it cost to hold inventory or run out? Inventory costs (lesson 7).
How much should we order at a time? Economic order quantity (lesson 8).
How much should we keep in reserve? Lead time and safety stock (lesson 9).
When should we order? Continuous and periodic review (lesson 10).
Which items deserve the most attention? ABC and XYZ classification (lesson 11).
What is our inventory worth? Valuation using PEPS, UEPS and average cost (lesson 12).

Inventory policy and performance measures

An inventory policy is an agreement among the sales, production and finance teams on how to manage inventory. Sales and production want enough stock to avoid shortages; finance wants less stock so capital is not tied up. The fundamental trade-off is between responsiveness (more stock, faster service) and efficiency (less stock, lower costs). Performance measures show whether the policy is working:

Measure	Formula	What it measures
Service level	requests filled in full (or on time) ÷ total requests received	How effectively inventory meets demand
Inventory accuracy	items with accurate records ÷ total items counted	Reliability of item data and inventory records
Inventory turnover	demand (units issued) during the period ÷ average inventory during the period	How many times inventory turns over during the period
Days of supply	inventory ÷ demand per unit of time, or days in the period ÷ inventory turnover	How long inventory will last without replenishment
Return on capital (RC)	annual sales profit ÷ capital invested in inventory	The return on capital tied up in inventory

Turnover can be calculated in units for a single item or by value for all inventory (cost of goods sold ÷ average inventory at cost). High turnover and fewer days of supply indicate lean inventory; the reverse suggests capital is sitting idle. The two measures are inversely related: turnover of 12 times a year means inventory lasts about a month.

A company had cost of goods sold of R$ 120.000,00 for the year and average inventory of R$ 20.000,00. Using a 360-day year, what are its inventory turnover and days of supply?

Answer: Turnover of 6 times and 60 days of supply — turnover = 120.000 ÷ 20.000 = 6; days of supply = 360 ÷ 6 = 60 days. Reversing the division gives 0,17, which is not turnover: it is the fraction of the year that the inventory lasts.

Material types and classification

Classification groups materials with similar characteristics so managers can work with categories rather than thousands of individual items. A good classification system is comprehensive (covering physical, financial and accounting characteristics), flexible (allowing classifications to be combined) and practical (simple and objective). The most common classifications are:

By use: raw materials, work in process (intermediate goods), finished goods and auxiliary materials;
By accounting treatment: consumable or permanent materials. In the public sector, Portaria STN nº 448/2002 lists five criteria (a useful life of up to two years, fragility, perishability, incorporation into another item and transformation) that make a material consumable when applicable;
By hazard level and by perishability;
By consumption value: ABC classification;
By operational importance (criticality): XYZ classification.

Demand forecasting

Every inventory decision starts with a forecast. The methods covered in lessons 5 and 6, from simplest to most advanced, are:

Method	Formula	When to use it
Last period	next period's forecast = last period's demand	Stable demand with no trend
n-period moving average	sum of demand over the last n periods ÷ n	Stable demand with fluctuations; a larger n produces a smoother forecast that responds more slowly
Weighted moving average	Σ (D_i × W_i) ÷ Σ W_i	When recent data should carry more weight
Exponential smoothing	D̄_t+1 = α × D_t + (1 − α) × D̄_t	A straightforward update each period; α ranges from 0 to 1 (a high α responds quickly, while a low α smooths fluctuations)
Least squares (linear regression)	Y = b + a × X, where a = [N × ΣXY − ΣX × ΣY] ÷ [N × ΣX² − (ΣX)²] and b = (ΣY − a × ΣX) ÷ N	Demand with an upward or downward trend

For seasonal demand, a period's seasonal index is its average demand divided by overall average demand: an index of 1,3 for December means sales in December are 30% above average.

A product's sales forecast for May was 200 units, and actual sales were 220. Using exponential smoothing with α = 0,3, what is the forecast for June?

Answer: 206 units — 0,3 × 220 + 0,7 × 200 = 66 + 140 = 206. Reversing the weights gives 214. With α = 0,3, the forecast moves only 30% of the way from the previous forecast toward actual sales.

Inventory costs and economic order quantity

Inventory costs fall into three categories (lesson 7):

Ordering cost (c_p): everything spent to place and receive an order (purchasing staff, communication, receiving and inspection). It rises with the number of orders;
Holding cost (c_m): the cost of capital, storage, services (insurance, taxes) and risks (obsolescence, losses, theft). It rises with average inventory and is usually expressed in R$ per unit per year or as an annual percentage of the item's value;
Stockout cost: lost sales, penalties, production stoppages and dissatisfied customers.

For an item with demand D, unit price c and order quantity Q, the total annual cost is (lesson 8):

CT = D × c + (Q / 2) × c_m + (D / Q) × c_p

The order quantity that minimizes this cost is the economic order quantity, where holding and ordering costs are equal:

LEC = √(2 × D × c_p ÷ c_m)

From this, you can calculate the number of orders per year (D ÷ Q) and the interval between orders (Q ÷ D). Take care: D and c_m must use the same time period (usually a year). If c_m is given as a percentage, multiply it by the item's price. With quantity discounts, calculate the full total cost (including D × c, which changes with the price) at the economic order quantity and at the minimum quantity for each discount tier, then choose the lowest-cost option.

A company uses 7.200 units of an item each year, spends R$ 50,00 per order and pays R$ 2,00 per unit per year to hold it in inventory. What is the economic order quantity, and how many orders does it place each year?

Answer: 600 units and 12 orders — LEC = √(2 × 7.200 × 50 ÷ 2) = √360.000 = 600; orders = 7.200 ÷ 600 = 12. The 360.000 option is the value inside the square root, before taking the root.

Safety stock and replenishment

The time between placing an order and receiving it is the lead time. During that time, demand may rise or delivery may be delayed. Safety stock covers this uncertainty (lesson 9). With normally distributed demand, a daily standard deviation of σ_D and lead time TR:

ES = z × σ_D × √TR

If lead time also varies (mean t, standard deviation σ_t) and average demand is D:

ES = z × √(σ_d² × t + σ_t² × D²)

Service level	60%	90%	95%	98%	99%
z (number of standard deviations)	0,25	1,28	1,65	2,06	2,33

Once safety stock is set, choose a replenishment system (lesson 10):

	Q system (continuous review)	P system (periodic review)
What stays fixed	Quantity (Q, usually the economic order quantity)	Interval (I_p = Q ÷ D)
Order trigger	Inventory reaches PP = D_t × t_r + E_s	Scheduled review date
How much to order	Q	E_máx − E_a − outstanding orders, where E_máx = D_t × (t_r + I_p) + E_s
Safety stock	z × σ × √L	z × σ × √(T + L)
Average inventory	Q ÷ 2 + ES	≈ D × I_p ÷ 2 + ES

Notice the square root: the standard deviation of demand grows with the square root of time, not time itself. Quadrupling the lead time doubles the safety stock.

An item has average demand of 100 units per day, a standard deviation of 20 units per day and a fixed lead time of 9 days. At a 95% service level (z = 1,65), what is its reorder point under the Q system?

Answer: 999 units — ES = 1,65 × 20 × √9 = 99 and PP = 100 × 9 + 99 = 999. Leaving out the square root gives ES = 297 and a reorder point of 1.197; leaving out safety stock gives 900.

ABC and XYZ classification and inventory valuation

ABC classification (lesson 11): rank items by consumption value (quantity × unit cost), calculate the cumulative percentage and divide them into classes. Typically, a small number of A items (10% to 20%) account for 65% to 80% of the value, while the majority of items (C) account for 5% to 10%. A items receive tight controls and continuous review; C items receive simpler controls.

XYZ classification (lesson 11): based on criticality. X = low (a shortage has little impact, and the item is easy to replace); Y = medium (a substitute is available, but at some cost); Z = highest (essential, with no substitute). ABC measures value; XYZ measures the consequences of a shortage. An item can be both C and Z. In texts from other countries, XYZ may instead refer to demand variability: check which meaning is intended.

Valuation (lesson 12): CMV = opening inventory + purchases − closing inventory.

PEPS: the oldest costs are assigned to items sold first; closing inventory carries the most recent costs;
UEPS: the most recent costs are assigned to items sold first; closing inventory carries the oldest costs;
Weighted average cost: moving (recalculated with each receipt) or periodic (calculated once at the end of the period).

When prices are rising, PEPS produces the lowest CMV and the highest profit; UEPS produces the opposite result. In Brazil, CPC 16 (R1) permits PEPS, weighted average cost and, for items that are not interchangeable, specific identification. The Income Tax Regulations (Decreto nº 9.580/2018, art. 307) allow average cost or the cost of the most recently acquired goods. UEPS is not permitted, either for financial reporting or for tax purposes.

Which statement about ABC and XYZ classification and inventory valuation is correct?

Answer: An inexpensive part that is class C under ABC can be class Z if running out would stop production — ABC measures consumption value, while XYZ measures criticality, so the classifications are independent. Unit cost alone does not make an item class A, XYZ does not measure quantity, and UEPS is not permitted in Brazil.

Worked exercise bringing it all together

A quick-print shop wants to organize its stock of A4 paper (measured in reams). The data:

sales over the last six months, in reams: 1.010, 980, 1.020, 990, 1.000 and 1.010;
the print shop operates 300 days a year;
cost per ream: R$ 25,00; cost per order: R$ 75,00; holding cost: 20% of the item's value per year;
standard deviation of daily demand: 8 reams; supplier lead time: 5 days; target service level: 95% (z = 1,65).

Step 1: Forecasting. Demand shows no trend, so a moving average is enough. The three-month moving average is (990 + 1.000 + 1.010) ÷ 3 = 1.000 reams per month. (Exponential smoothing with α = 0,2, starting with a forecast equal to the first month's sales, gives 1.004, almost the same result.) Annual demand: D = 12 × 1.000 = 12.000 reams; daily demand: d = 12.000 ÷ 300 = 40 reams per day.

Step 2: Economic order quantity. c_m = 20% × 25 = R$ 5,00 per ream per year.

LEC = √(2 × 12.000 × 75 ÷ 5) = √360.000 = 600 reams.

Number of orders = 12.000 ÷ 600 = 20 per year; average time between orders = 600 ÷ 40 = 15 working days.

CT = 12.000 × 25 + (600 ÷ 2) × 5 + 20 × 75 = 300.000 + 1.500 + 1.500 = R$ 303.000,00 per year. At the economic order quantity, holding costs (R$ 1.500) equal ordering costs (R$ 1.500).

Holding costs, ordering costs and their combined total for the A4 paper in this exercise. The total reaches its minimum at 600 reams, where the two cost curves intersect.

Step 3: Safety stock and reorder point (Q system).

ES = 1,65 × 8 × √5 = 29,5, rounded to 30 reams.

PP = 40 × 5 + 30 = 230 reams. When stock falls to 230, order a batch of 600.

Average inventory = 600 ÷ 2 + 30 = 330 reams.

Step 4: Inventory metrics. Annual turnover = 12.000 ÷ 330 = 36,4 times. Inventory coverage = 330 ÷ 40 = 8,25 working days.

Step 5: What if inventory were reviewed periodically? With a review every 15 days:

ES = 1,65 × 8 × √(15 + 5) = 59,0, rounded to 60 reams (roughly twice as much).

E_máx = 40 × (5 + 15) + 60 = 860 reams. If there are 250 reams on hand at the review and none on order, order 860 − 250 = 610 reams.

Step 6: ABC and XYZ. Annual consumption of five items at the print shop:

Item	Annual consumption	Unit cost	Annual consumption value	Cumulative %	ABC	XYZ
A4 paper (ream)	12.000	R$ 25,00	R$ 300.000	68,2%	A	Z
Toner	400	R$ 180,00	R$ 72.000	84,5%	B	Z
Pen	30.000	R$ 1,50	R$ 45.000	94,8%	B	X
Stapler	500	R$ 30,00	R$ 15.000	98,2%	C	Y
Paper clips (box)	2.000	R$ 4,00	R$ 8.000	100,0%	C	X

With cutoffs at 80% (A) and 95% (B), paper is the only A item, accounting for 68% of the value. The print shop cannot print without paper or toner, so both are Z items. The takeaway: A4 paper is both A and Z, making it the item that warrants continuous monitoring, the calculated safety stock and supplier negotiations. Pens, classified B and X, can be reviewed periodically.

Step 7: Inventory valuation. In one month, the print shop started with 100 reams at R$ 24,00 each, bought 600 at R$ 25,00 and then 600 at R$ 26,00, and used 1.000 reams. That leaves 300. Total cost of goods available: 2.400 + 15.000 + 15.600 = R$ 33.000,00.

FIFO: the remaining 300 are from the last batch: ending inventory = 300 × 26 = R$ 7.800,00; cost of goods sold = 33.000 − 7.800 = R$ 25.200,00.
Periodic weighted-average cost: 33.000 ÷ 1.300 = R$ 25,3846 per ream; ending inventory = R$ 7.615,38; cost of goods sold = R$ 25.384,62.
LIFO (for comparison only; not permitted in Brazil): the oldest units remain, 100 × 24 + 200 × 25 = R$ 7.400,00; cost of goods sold = R$ 25.600,00.

When prices are rising, FIFO produces the lowest cost of goods sold. The print shop can use either FIFO or weighted-average cost for its financial statements and taxes.

What comes next

That wraps up Part 1. Part 2 of the course covers where materials come from and where they go: purchasing (including in the public sector), stockrooms, warehousing and transportation. Many of the tools from this part return there: economic order quantity informs purchasing negotiations, ABC analysis guides supplier evaluation and inventory counts, and lead time is largely a transportation issue.

Exercises

Several of these exercises come from the professor's Logistics exams at CEFET/RJ (2018); questions with ambiguous information have been revised.

Easy

1) (CEFET/RJ exam, 2018) An inventory count covered 16.915 items: 4.910 in class A, 9.125 in class B and 2.880 in class C. Discrepancies were found in 268 class A items, 438 class B items and 55 class C items. Calculate inventory accuracy.

2) (CEFET/RJ exam, 2018) A building supply store's cost of goods sold was R$ 26.000,00 last year, and its average inventory at cost was R$ 2.470,00. Calculate inventory turnover and coverage, using a 365-day year.

3) (CEFET/RJ exam, 2018) Eletrônica Estrela uses 3.000 units of an item per year. Each order costs R$ 60,00, and holding one unit in inventory costs R$ 1,00 per year. How many units should it order at a time to minimize total inventory costs?

4) (CEFET/RJ exam, 2018) A company uses exponential smoothing with α = 0,30. Its forecast for the first week of July was 800 units, and actual demand was 850. What is the forecast for the second week?

5) (CEFET/RJ exam, 2018) A supermarket sells an average of 514 boxes per day, with a daily standard deviation of 73 boxes. The supplier delivers in 1 day. For a 90% service level (z = 1,28), how many boxes should be held as safety stock?

6) (DPE-AM civil service exam, cited in the 2018 CEFET/RJ exam) XYZ classification of materials is based on:
a) the item's location in the organization's and its suppliers' inventories;
b) the material's financial importance, based on its highest demand value;
c) the distinction between consumable and durable materials;
d) the item's perishability, with an emphasis on storage periods;
e) how critical or indispensable the item is to the organization's operations.

Hard

7) (CEFET/RJ exam, 2018) A manufacturer sold the following quantities of product AM this year: January 4.100, February 3.800, March 3.800, April 4.000, May 4.100, June 4.200 and July 4.900.
a) What is the August forecast using the last-period method?
b) What is the August forecast using a 3-month moving average?
c) Without knowing actual August sales yet, and using the August forecast from part b in their place, what would the September forecast be using a 3-month moving average?
d) For another product, January demand has averaged 270 units, and average annual demand is 5.400 units. Calculate January's seasonal index and the January forecast for a year with projected annual demand of 6.000 units.

8) (CEFET/RJ exam, 2018) A store buys a product for R$ 5,00 per unit. The supplier offers discounts: R$ 4,80 per unit for orders of 1.000 to 1.999 units, and R$ 4,75 per unit for orders of 2.000 units or more. Each order costs R$ 49,00, annual holding costs are 20% of the item's price, and annual demand is 5.000 units. What is the most economical order quantity?

9) (CEFET/RJ exam, 2018) A distributor's daily demand for peppermint candies follows a normal distribution with a mean of 5.000 kg and a standard deviation of 350 kg. The manufacturer's delivery time is also normally distributed, with a mean of 5 days and a standard deviation of 2 days. Calculate safety stock for a 95% service level (z = 1,65). Which source of uncertainty has the greater effect on the result?

10) (CEFET/RJ exam, 2018) Identify the replenishment system used in each situation and calculate the requested values.
a) Seu Manoel sells 20 savory snacks per day. Whenever stock falls to 60 snacks, he orders 80 from Dona Maria, who takes 2 days to make and deliver them. Calculate demand during lead time, implied safety stock, average time between orders and average inventory.
b) Every December, a public school checks its pen inventory and places a replenishment order that arrives in February (2 months later). Consumption is 100 pens per month, safety stock is 200 pens, and at the last review there were 180 pens on hand and no outstanding orders. Calculate maximum inventory (target stock), the order quantity and approximate average inventory.

11) A company began the month with 200 units at R$ 10,00 each. On day 5, it bought 300 units at R$ 12,00; on day 10, it issued 400 units; on day 15, it bought 200 units at R$ 13,00; and on day 20, it issued 150 units. Calculate cost of goods sold and ending inventory under FIFO and moving weighted-average cost, and explain why FIFO produces the lower cost of goods sold.

Answer key

1) Total discrepancies = 268 + 438 + 55 = 761 items. Correct items = 16.915 − 761 = 16.154. Accuracy = 16.154 ÷ 16.915 = 95,5%. By class: A = (4.910 − 268) ÷ 4.910 = 94,5%; B = (9.125 − 438) ÷ 9.125 = 95,2%; C = (2.880 − 55) ÷ 2.880 = 98,1%. Class A, the most important, has the lowest accuracy: cycle counts should cover it more frequently.

2) Turnover = 26.000 ÷ 2.470 = 10,5 times per year. Coverage = 365 ÷ 10,53 = 34,7 days (or, directly, 2.470 ÷ 26.000 × 365 = 34,7 days).

3) LEC = √(2 × 3.000 × 60 ÷ 1) = √360.000 = 600 units per order, giving 3.000 ÷ 600 = 5 orders per year.

4) Forecast = 0,30 × 850 + 0,70 × 800 = 255 + 560 = 815 units.

5) With a lead time of 1 day, √1 = 1: ES = 1,28 × 73 × 1 = 93,4, rounded up to 94 boxes. Mean demand (514) does not enter the safety stock calculation; it would enter the reorder point (514 × 1 + 94 = 608).

6) Answer e. XYZ measures criticality (the impact of a stockout); financial importance (b) is measured by ABC, while consumable versus durable (c) is an accounting classification.

7) a) Last-period method: the August forecast equals July sales, 4.900.
b) 3-month moving average: (4.100 + 4.200 + 4.900) ÷ 3 = 13.200 ÷ 3 = 4.400.
c) For September, the last three values become June, July and the August forecast: (4.200 + 4.900 + 4.400) ÷ 3 = 13.500 ÷ 3 = 4.500.
d) Average monthly demand = 5.400 ÷ 12 = 450. January index = 270 ÷ 450 = 0,6 (January sales are 40% below average). Forecast: (6.000 ÷ 12) × 0,6 = 500 × 0,6 = 300 units.

8) Calculate the economic order quantity for each price tier (c_m = 20% of the price), then adjust it to the tier's minimum order quantity:
a) R$ 5,00: LEC = √(2 × 5.000 × 49 ÷ 1,00) = 700, within the tier. CT = 5.000 × 5,00 + (700 ÷ 2) × 1,00 + (5.000 ÷ 700) × 49 = 25.000 + 350 + 350 = R$ 25.700,00.
b) R$ 4,80: LEC = √(2 × 5.000 × 49 ÷ 0,96) = 714, below the minimum of 1.000, so use 1.000. CT = 24.000 + 500 × 0,96 + 5 × 49 = 24.000 + 480 + 245 = R$ 24.725,00.
c) R$ 4,75: LEC = 718, below the minimum, so use 2.000. CT = 23.750 + 1.000 × 0,95 + 2,5 × 49 = 23.750 + 950 + 122,50 = R$ 24.822,50.
The lowest total cost comes from ordering 1.000 units at R$ 4,80. The larger discount does not pay off: it saves R$ 250 on the purchase price, but the extra inventory adds R$ 470 in holding costs, while fewer orders save only R$ 122,50.

9) ES = z × √(σ_d² × t + σ_t² × D²) = 1,65 × √(350² × 5 + 2² × 5.000²) = 1,65 × √(612.500 + 100.000.000) = 1,65 × √100.612.500 = 1,65 × 10.030,6 = approximately 16.550 kg. Lead-time uncertainty dominates: its contribution (100.000.000) is more than 160 times the contribution from demand (612.500). Reducing lead-time variability would cut safety stock far more than improving demand forecasts.

10) a) Q system (continuous review): an order is triggered when inventory reaches 60, and the order quantity is always 80. Demand during lead time = 20 × 2 = 40; safety stock = reorder point − demand during lead time = 60 − 40 = 20 savory snacks; average time between orders = Q ÷ D = 80 ÷ 20 = 4 days; average inventory = 80 ÷ 2 + 20 = 60 savory snacks.
b) P system (periodic review): orders are triggered on scheduled review dates. Review interval = 12 months; lead time = 2 months. Target inventory = D × (IR + L) + safety stock = 100 × (12 + 2) + 200 = 1.600 pens. Order quantity = 1.600 − 180 = 1.420 pens. Average inventory ≈ D × IR ÷ 2 + safety stock = 100 × 12 ÷ 2 + 200 = 800 pens.

11) Total inventory available during the month = 200 × 10 + 300 × 12 + 200 × 13 = 2.000 + 3.600 + 2.600 = R$ 8.200,00; 550 units are sold, leaving 150.
a) FIFO: the 400 units sold on day 10 comprise all 200 units at R$ 10 and 200 at R$ 12 = 2.000 + 2.400 = R$ 4.400; 100 units at R$ 12 remain. The 150 units sold on day 20 comprise those 100 units at R$ 12 and 50 at R$ 13 = 1.200 + 650 = R$ 1.850. Cost of goods sold = 4.400 + 1.850 = R$ 6.250,00; ending inventory = 150 × 13 = R$ 1.950,00.
b) Moving average cost: after the purchase on day 5, (2.000 + 3.600) ÷ 500 = R$ 11,20; cost of the 400 units sold on day 10 = 400 × 11,20 = R$ 4.480, leaving 100 units at R$ 11,20 = R$ 1.120. After the purchase on day 15, (1.120 + 2.600) ÷ 300 = R$ 12,40; cost of the 150 units sold on day 20 = 150 × 12,40 = R$ 1.860. Cost of goods sold = 4.480 + 1.860 = R$ 6.340,00; ending inventory = 150 × 12,40 = R$ 1.860,00.
Check: under both methods, cost of goods sold + ending inventory = R$ 8.200,00. When prices are rising, FIFO assigns the older, cheaper costs to goods sold and leaves the more recent costs in inventory, resulting in lower cost of goods sold and higher ending inventory.

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