MBA Inventory Management Concept Card Template

Inventory Management Concept Card: [Your course or product line]

A concept card for the inventory models of the MBA operations core: the economic order quantity (EOQ) for cycle stock, the reorder point and safety stock for uncertain demand during a lead time, the newsvendor model for one selling season, ABC classification to decide where to spend management attention, and inventory turns to read inventory on a balance sheet. It is for MBA students preparing a case or an exam and for managers in retail, distribution, manufacturing and healthcare supply. One example is fully solved, one is half solved and one is left for you, and the self-test answers sit in a separate key. Source basis: Harris (1913) for EOQ, Arrow, Harris and Marschak (1951) for the newsvendor logic, Little (1961) for the link between inventory and time, and the inventory chapters common to the leading operations textbooks, all written up in original words with fictional data (Northfield Outdoor Co. is an invented company). Replace every [bracketed] value and every row marked Example.

Document control

When to use it

One-sentence rule: use these models whenever you must decide how much to order and when to order, and you can put a cost on holding too much and on having too little.

The idea in plain language

Inventory exists for reasons, and each reason has its own model.

Cycle stock exists because ordering in batches is cheaper than ordering every unit separately. Every order has a fixed cost (purchasing work, receiving, a truck), so you want few, large orders. But every unit sitting on a shelf costs money (capital, space, insurance, obsolescence), so you want small orders. The EOQ balances the two: at the best order size the annual ordering cost equals the annual holding cost, and the total cost curve is flat near the optimum, so being 20% off hardly matters.

Safety stock exists because demand during the supplier lead time is uncertain. You reorder when stock falls to the reorder point: expected demand during the lead time plus a cushion. The cushion grows with the uncertainty of lead-time demand and with the service level you choose. Variation in the lead time itself often matters more than variation in daily demand.

Seasonal stock for a single season cannot be replenished: the parka you design in February sells in November or not at all. The newsvendor model compares the cost of ordering one unit too many (you sell it at a loss) with the cost of ordering one unit too few (you lose its margin). If running out is more costly than being left over, you order above the mean forecast. The critical ratio tells you exactly how far above.

ABC classification exists because not every item deserves the same attention. A few items usually carry most of the annual spend; they get careful forecasting, frequent review and tight safety stocks.

Finally, inventory turns and days of inventory read inventory from the financial statements. By Little's Law, days of inventory is simply the average time a unit (or a dollar of cost) spends in stock.

Formulas and notation

EOQ (Harris 1913). Annual relevant cost and its minimum:

Reorder point with safety stock (continuous review, normal lead-time demand):

Fill rate (share of demand met from stock), using the standard normal loss function $G(z)$:

Newsvendor (Arrow, Harris and Marschak 1951 give the marginal argument; the textbook form is):

Expected lost sales $= \sigma\,G(z)$, expected sales $= \mu - \sigma\,G(z)$, expected leftover $= Q -$ expected sales.

Inventory turns (Little 1961 applied to stock):

Excel equivalents. $Q^*$: =SQRT(2*D*S/H). $z$: =NORM.S.INV(CSL). Newsvendor $Q^*$: =NORM.INV(Cu/(Cu+Co), mu, sigma). Loss function: =NORM.S.DIST(z,FALSE)-z*(1-NORM.S.DIST(z,TRUE)).

Assumptions and limits

The EOQ formula was first published by Harris (1913) for a factory deciding how many parts to make at once; the reprint in Operations Research (1990) makes the original accessible. The newsvendor idea of balancing the expected marginal cost of too much against too little appears in Arrow, Harris and Marschak (1951), the paper that started modern inventory theory.

How managers use it

• Order sizes and frequency. Buyers use EOQ as an anchor, then round to case packs, pallets or full trucks. Because the curve is flat, a 15% deviation from $Q^*$ usually costs only about 1% more.
• Service level policy. The cycle service level is a management choice, not a technical one: it should reflect the margin lost and the customer goodwill at risk when an item is missing. The newsvendor critical ratio gives a principled way to set it.
• Supplier negotiations. Safety stock depends heavily on lead-time variation (Worked example A2). A reliable supplier can be worth a higher price, and a shorter lead time cuts both safety stock and forecast error.
• Seasonal buys and markdowns. Apparel, outdoor gear, publishing and fresh food buyers use newsvendor logic to set initial buys and plan clearance. Pre-season orders above mean demand are rational when margins are high.
• Working capital. Turns and days of inventory feed the cash conversion cycle (finance/working-capital-review and mba/mba-ratio-analysis-card). Releasing inventory frees cash.
• Supply chain design. Pooling stock in one distribution centre, postponing final customisation and sharing point-of-sale data with suppliers all reduce the uncertainty that safety stock pays for; see the bullwhip effect in mba/mba-operations-guide.

Choosing the model

Two questions decide most inventory problems: can you reorder during the selling period, and is there a fixed cost per order? Answer them first, then add safety stock if lead-time demand is uncertain and an ABC policy if you manage many items.

Worked example A (fully solved): Northfield Outdoor Co.

Situation (Example, fictional). Northfield Outdoor Co. sells the Trailhead 400 headlamp through its stores and web shop. Annual demand is $D$ = 18,250 units (50 a day over 365 days). Each purchase order costs $S$ = USD 150 (buyer time, receiving, inbound freight fixed fee). The unit cost is $c$ = USD 20 and finance sets a holding cost rate of $h$ = 25% a year, so $H$ = USD 5.00 per unit per year.

A1. Economic order quantity.

At the optimum the two cost lines cross: ordering cost equals holding cost. The purchase cost itself (18,250 x 20 = USD 365,000) does not depend on $Q$ and is left out unless there are quantity discounts.

How flat is the curve? (=18250/Q*150+Q/2*5 for each $Q$)

If the supplier ships in cartons of 1,200, ordering 1,200 costs only 0.94% more than the theoretical optimum.

A2. Reorder point and safety stock. Daily demand averages $\bar{d}$ = 50 with $\sigma_d$ = 15 units; days are independent. The supplier lead time is $L$ = 9 days. Northfield wants a 95% cycle service level.

Now suppose the lead time varies with $\sigma_L$ = 2 days (customs delays):

Two days of lead-time variation multiply the safety stock by 2.44. Most of $\sigma_{LTD}^2$ (10,000 of 12,025) comes from the lead time, not from daily demand: a more reliable supplier is worth up to USD 531.77 a year on this one item in holding cost alone.

A3. Cycle service level versus fill rate. With the fixed 9-day lead time, $Q^*$ = 1,046.42 and $z$ = 1.6449:

A 95% CSL means a stock-out in about one cycle in twenty, but because each cycle sells about 1,046 units and the shortfall is small, customers find the headlamp in stock 99.9% of the time. Say which measure you mean.

A4. Newsvendor: Summit Parka limited edition. Northfield commissions a limited-edition parka for one winter season. Price $p$ = USD 240, cost $c$ = USD 110, leftovers go to the outlet channel at $s$ = USD 70. Season demand is forecast as normal with $\mu$ = 3,000 and $\sigma$ = 900. There is no second production run.

Compare three order quantities (expected values from the loss function; a 400,000-draw simulation agrees within USD 120):

Expected profit = 240 x sales + 70 x leftover - 110 x $Q$. Ordering the mean leaves about USD 14,000 of expected profit on the table because a lost sale costs more than three times a leftover unit. The scipy calculation used is norm.ppf(130/170, 3000, 900).

A5. ABC classification. Annual usage value (units x unit cost) of ten Northfield store items, sorted:

Policy used here (a choice, not a law): A up to 65% of cumulative value, B up to 90%, C the rest. Excel: share =C2/SUM($C$2:$C$11), cumulative =SUM($C$2:C2)/SUM($C$2:$C$11), class =IF(E2<=0.65,"A",IF(E2<=0.9,"B","C")). Three items (30% of SKUs) carry 64.6% of value; five items (50%) carry 12.5%. A items get weekly review and item-level forecasts; C items get simple two-bin rules. The method applies Pareto's observation that a few items carry most of the value to inventory control.

A6. Inventory turns. Northfield's cost of goods sold is USD 186.0m; inventory at cost was USD 44.0m at the start of the year and USD 49.0m at the end.

By Little's Law, a dollar of cost spends on average 91.25 days on Northfield's shelves.

Worked example B (half solved): water filter and guidebook

B1. EOQ and reorder point (Example, fictional). Northfield's backcountry water filter sells 7,300 units a year (20 a day). Ordering cost $S$ = USD 90, unit cost $c$ = USD 32, holding rate 25%. Daily demand SD $\sigma_d$ = 6; the lead time is a fixed 16 days; target CSL 98%.

1. $H$ = [ ] per unit per year (=32*0.25).
2. $Q^*$ = $\sqrt{2 \times 7{,}300 \times 90 / [\ ]}$ = [ ] units (=SQRT(2*7300*90/8)).
3. Orders per year = [ ]; total relevant cost $TC(Q^*)$ = USD [ ].
4. $z$ = NORM.S.INV(0.98) = [ ]; $\sigma_{LTD}$ = 6 x $\sqrt{16}$ = [ ]; safety stock = [ ]; ROP = 20 x 16 + [ ] = [ ].

B2. Newsvendor (Example, fictional). Northfield prints a regional trail guidebook once a year. Price USD 28, print cost USD 9, unsold copies are pulped for USD 2. Demand is normal with mean 800 and SD 150.

1. $C_u$ = [ ], $C_o$ = [ ], critical ratio = [ ].
2. $z$ = [ ]; $Q^*$ = 800 + [ ] x 150 = [ ] copies (=NORM.INV(19/26,800,150)).
3. Is the order above or below mean demand, and why?

Exercise C (unsolved): trail race event tees and a reorder decision

Northfield sponsors a trail race and sells event T-shirts only on race weekend (Example, fictional). Price USD 35, cost USD 12, leftovers sold to a discount buyer for USD 4. Demand is normal with mean 1,200 and SD 300.

1. Which model applies, and why not EOQ?
2. Compute $C_u$, $C_o$, the critical ratio, $z$ and the optimal order quantity.
3. Compute expected sales, expected leftover and expected profit at that quantity. (Use the rounded order quantity; $G(z)$ for $z$ = 0.65 is about 0.1554.)
4. The discount buyer withdraws and leftovers are worth nothing. What is the new order quantity?
5. Separately, plain Northfield logo tees sell all year: 4,800 a year, ordering cost USD 60, holding cost USD 3 per unit per year. Compute the EOQ. If annual demand quadruples, by what factor does the EOQ change?

Connections to other cards

• mba/mba-process-analysis-card: Little's Law links inventory, flow rate and flow time; batch sizes from EOQ affect process flow time.
• mba/mba-queueing-card: safety capacity in services plays the role that safety stock plays in goods.
• mba/mba-operations-guide: the bullwhip effect (Lee, Padmanabhan and Whang 1997), lean and just-in-time inventory, supply chain design.
• mba/mba-quality-card: defects and returns add inventory and reduce effective supply.
• mba/mba-relevant-costs-card: only costs that change with the decision belong in $S$ and $H$.
• mba/mba-ratio-analysis-card and finance/working-capital-review: turns and days of inventory in the cash conversion cycle.
• academic/stats-distributions-sheet: the normal distribution, quantiles and the loss function.

Common mistakes and misconceptions

• Mixing time units. Using an annual $H$ with monthly $D$, or daily $\sigma_d$ with a lead time in weeks. Convert first.
• Putting the purchase price into the EOQ trade-off. Without quantity discounts the purchase cost is the same for any $Q$; it enters only through $H = h \times c$.
• Treating EOQ as precise. The cost curve is flat; rounding to a case pack is fine. Spending days refining $S$ to the cent is wasted effort.
• Scaling safety stock with $L$ instead of $\sqrt{L}$. With independent days, the SD of lead-time demand grows with the square root of the lead time.
• Ignoring lead-time variation. In Worked example A2 it more than doubles safety stock.
• Confusing cycle service level with fill rate. A 95% CSL is not "95% of customers served"; the fill rate is usually much higher.
• Ordering the mean in a newsvendor problem. The mean is optimal only when $C_u = C_o$. High-margin, low-salvage products justify ordering above the mean.
• Getting $C_o$ wrong. The overage cost is cost minus salvage, not the full unit cost, unless leftovers are worth nothing.
• Using a biased forecast as $\mu$. If past forecasts were 15% too high, correct $\mu$ and use the SD of forecast errors, not of raw demand, as $\sigma$.
• ABC by unit price alone. Classification uses annual usage value (volume x cost), and critical low-value items need a separate flag.
• Comparing turns across industries. A grocer and a jeweller have very different normal turns; benchmark within the industry.

Self-test

1. $D$ = 10,000 units a year, $S$ = USD 50, $H$ = USD 4. Compute $Q^*$ and $TC(Q^*)$.
2. In question 1, the holding cost doubles. Does $Q^*$ halve? By what factor does it change?
3. Why is the total cost curve flat near the EOQ, and what practical freedom does that give a buyer?
4. Daily demand has mean 40 and SD 10; lead time is a fixed 4 days. Compute safety stock and the reorder point for a 90% CSL ($z$ = 1.2816).
5. Lead time doubles from 4 to 8 days, everything else equal. By what factor does safety stock change?
6. Explain in one sentence the difference between cycle service level and fill rate.
7. A newsvendor item has $p$ = 50, $c$ = 20, $s$ = 5, demand normal with mean 500 and SD 100. Compute the critical ratio and the optimal order.
8. In question 7, should the order be above or below 500 if $s$ rises to 15? Explain without computing.
9. A company has cost of goods sold of USD 120m and average inventory of USD 30m. Compute turns and days of inventory.
10. Interleaving: name the model. (a) A hospital orders flu vaccine once before the season. (b) A distributor reorders printer paper every few weeks from a supplier with a fixed delivery fee. (c) A retailer wants to know which 20 of its 2,000 items need weekly forecast reviews.

My notes and next steps

Sources and further reading

• Harris, F. W. (1913). How many parts to make at once. Factory, The Magazine of Management, 10(2), 135-136, 152. Reprinted in Operations Research (1990), 38(6), 947-950.
• Arrow, K. J., Harris, T., and Marschak, J. (1951). Optimal inventory policy. Econometrica, 19(3), 250-272.
• Little, J. D. C. (1961). A proof for the queuing formula: L = λW. Operations Research, 9(3), 383-387.
• Lee, H. L., Padmanabhan, V., and Whang, S. (1997). Information distortion in a supply chain: the bullwhip effect. Management Science, 43(4), 546-558.
• Fisher, M. L. (1997). What is the right supply chain for your product? Harvard Business Review, March-April 1997.
• Textbooks (further reading, by name only): Cachon and Terwiesch, Matching Supply with Demand: An Introduction to Operations Management; Silver, Pyke and Thomas, Inventory and Production Management in Supply Chains; Chopra and Meindl, Supply Chain Management: Strategy, Planning, and Operation.
• Open resource: MIT OpenCourseWare, 15.761 Introduction to Operations Management (Spring 2013), https://ocw.mit.edu/courses/15-761-introduction-to-operations-management-spring-2013/.

Answer key

Worked example B. B1: (1) $H$ = USD 8.00. (2) $Q^*$ = 405.28 units. (3) 18.01 orders a year (=7300/405.28); $TC(Q^*)$ = USD 3,242.22 (=SQRT(2*7300*90*8)). (4) $z$ = 2.0537; $\sigma_{LTD}$ = 24.00; safety stock = 49.29 (=NORM.S.INV(0.98)*24); ROP = 320 + 49.29 = 369.29, so reorder at 370. B2: (1) $C_u$ = 28 - 9 = 19, $C_o$ = 9 - 2 = 7, critical ratio = 19 / 26 = 0.7308. (2) $z$ = 0.6151; $Q^*$ = 892.27, so print 893 copies. (3) Above the mean, because a missed sale (USD 19) costs much more than an unsold copy (USD 7).

Exercise C. (1) Newsvendor: one selling event, no replenishment, uncertain demand; EOQ assumes repeated orders and known demand. (2) $C_u$ = 23, $C_o$ = 8, critical ratio = 23 / 31 = 0.7419, $z$ = 0.6493, $Q^*$ = 1,394.80, so order 1,395 (=NORM.INV(23/31,1200,300)). (3) At 1,395 units ($z$ = 0.65, $G$ = 0.1554), expected lost sales = 300 x 0.1554 = 46.61, expected sales = 1,153.39, expected leftover = 241.61, expected profit = USD 24,595.04. (4) With $s$ = 0, $C_o$ = 12, critical ratio = 23 / 35 = 0.6571 and $Q^*$ = 1,321.40, so order 1,322. (5) EOQ = $\sqrt{2 \times 4{,}800 \times 60 / 3}$ = 438.18 tees (=SQRT(2*4800*60/3)); with demand four times higher the EOQ doubles to 876.36, because $Q^*$ grows with the square root of $D$.

Self-test.

1. $Q^*$ = $\sqrt{2 \times 10{,}000 \times 50 / 4}$ = 500 units; $TC$ = $\sqrt{2 \times 10{,}000 \times 50 \times 4}$ = USD 2,000.
2. No. $Q^*$ falls by a factor of $1/\sqrt{2}$, to 353.55 units.
3. Near the optimum, the rise in one cost line is almost offset by the fall in the other, so total cost changes slowly. Buyers can round to case packs, pallets or truckloads at little cost.
4. $\sigma_{LTD}$ = 10 x 2 = 20; safety stock = 1.2816 x 20 = 25.63; ROP = 160 + 25.63 = 185.63, so 186.
5. By $\sqrt{2}$ = 1.414 (from 25.63 to 36.25 units).
6. CSL is the probability of no stock-out during a replenishment cycle; fill rate is the share of demand served from stock.
7. $C_u$ = 30, $C_o$ = 15, critical ratio = 0.6667, $z$ = 0.4307, $Q^*$ = 543.07, so order 544.
8. Above 500, and higher than before: a higher salvage value lowers $C_o$, so the critical ratio rises.
9. Turns = 120 / 30 = 4.0; days of inventory = 365 / 4 = 91.25.
10. (a) Newsvendor. (b) EOQ with a reorder point. (c) ABC classification.

Revision history

Standard and licence note

Based on the published inventory theory literature, chiefly Harris (1913), Arrow, Harris and Marschak (1951) and Little (1961); all text, data and examples are original and fictional, and textbooks are named as further reading only. This template is not affiliated with or endorsed by any business school, publisher or course. Check your course materials for the notation your professor uses. Template licence: MIT.