# 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*.

<!--
Guidance: How to use this template
- Start with "When to use it": the first decision is which model fits (repeated orders or one season, known or
  uncertain demand). Most exam errors are model choice errors, not arithmetic.
- Rebuild Worked example A in a spreadsheet with the Excel formulas shown. Then fill every [ ] in Worked example B
  and solve Exercise C on paper before checking the answer key.
- Keep the costs per unit per year straight: the holding cost H is usually a percentage of the unit cost, per year.
- Waiting lines and service capacity are in mba/mba-queueing-card; process capacity and Little's Law are in
  mba/mba-process-analysis-card; the bullwhip effect and supply chain coordination are in mba/mba-operations-guide.
- Related templates: mba/mba-operations-guide, mba/mba-process-analysis-card, mba/mba-queueing-card,
  mba/mba-quality-card, mba/mba-formula-sheet, mba/mba-glossary, mba/mba-ratio-analysis-card,
  mba/mba-relevant-costs-card, mba/mba-error-log, mba/mba-concept-flashcards, finance/working-capital-review,
  academic/stats-distributions-sheet, academic/stats-probability-basics.
-->

## Document control

<!-- Guidance: Record the data source for demand, costs and lead times and the date of the data. Inventory parameters go stale quickly: a new supplier or a price change changes every answer on this card. -->

| Field | Value |
|---|---|
| Course or project | [Operations core, inventory sessions / client project] |
| Learner or analyst | [Your name] |
| Items analysed | [Northfield Trailhead 400 headlamp and Summit Parka (fictional) / your SKUs] |
| Demand data and period | [Point-of-sale data, YYYY-MM-DD to YYYY-MM-DD / forecast version] |
| Cost data | [Ordering cost study, holding cost rate approved by finance] |
| Tool | [Excel / Google Sheets / Python with scipy] |
| Reviewed by | [Study group / tutor / mentor] |
| Last updated | YYYY-MM-DD |
| Version | [0.1] |

## When to use it

<!-- Guidance: Pick the model before you calculate. Ask two questions: does the product sell over and over with replenishment, or in one season with no second chance? Is demand during the lead time known or uncertain? -->

**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.

| Situation | Model on this card | Key output |
|---|---|---|
| Steady demand, repeated orders, a fixed cost per order | EOQ (Harris 1913) | Order quantity $Q^*$ |
| Repeated orders, uncertain demand during the supplier lead time | Reorder point with safety stock (continuous review) | Reorder point ROP, safety stock |
| One selling season or one-off event, leftover units are sold off cheaply | Newsvendor (Arrow, Harris and Marschak 1951) | Order quantity at the critical ratio |
| Many items, limited management attention | ABC classification | Which items get tight control |
| Reading or benchmarking a company | Inventory turns and days of inventory (Little 1961) | Turns, days, cash tied up |

| Use when | Do not use alone when |
|---|---|
| Demand is roughly stable or can be described by a mean and a standard deviation | Demand has strong trend or seasonality within the planning horizon (forecast first, see academic/stats-time-series-arima) |
| Costs per order and per unit held can be estimated | Capacity, shelf space or budget limits bind (add constraints) |
| One item can be planned on its own | Items are ordered together from one supplier (joint replenishment) or share a truck |
| Lead time is known or its variation can be measured | The supplier is unreliable in ways that a standard deviation does not capture |

## The idea in plain language

<!-- Guidance: Read this section aloud before the formulas. If you can explain why ordering more often lowers holding cost but raises ordering cost, the EOQ formula will make sense. -->

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

<!-- Guidance: Check that D and H use the same time unit (per year) and that daily demand and lead time use the same unit (days). Write the unit next to every input. -->

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

$$
TC(Q) = \frac{D}{Q}S + \frac{Q}{2}H, \qquad Q^* = \sqrt{\frac{2DS}{H}}, \qquad TC(Q^*) = \sqrt{2DSH}
$$

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

$$
ROP = \bar{d}\,L + SS, \qquad SS = z\,\sigma_{LTD}, \qquad z = \Phi^{-1}(\text{CSL})
$$

$$
\sigma_{LTD} = \sigma_d\sqrt{L} \ \text{(fixed lead time)}, \qquad
\sigma_{LTD} = \sqrt{L\,\sigma_d^2 + \bar{d}^2\,\sigma_L^2} \ \text{(variable lead time)}
$$

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

$$
G(z) = \varphi(z) - z\,\big(1 - \Phi(z)\big), \qquad \text{Fill rate} = 1 - \frac{\sigma_{LTD}\,G(z)}{Q}
$$

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

$$
C_u = p - c, \qquad C_o = c - s, \qquad F(Q^*) = \frac{C_u}{C_u + C_o}, \qquad Q^* = \mu + z\,\sigma \ \text{for normal demand}
$$

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

**Inventory turns** (Little 1961 applied to stock):

$$
\text{Turns} = \frac{\text{Cost of goods sold}}{\text{Average inventory at cost}}, \qquad
\text{Days of inventory} = \frac{365}{\text{Turns}}
$$

| Symbol | Meaning | Unit |
|---|---|---|
| $D$ | Annual demand | units per year |
| $S$ | Fixed cost per order (purchasing, receiving, transport fixed part) | currency per order |
| $H$ | Holding cost per unit per year, usually $h \times c$ | currency per unit per year |
| $h$, $c$ | Holding cost rate, unit cost | % per year, currency per unit |
| $Q$, $Q^*$ | Order quantity, optimal order quantity | units |
| $\bar{d}$, $\sigma_d$ | Mean and standard deviation of daily demand | units per day |
| $L$, $\sigma_L$ | Mean and standard deviation of the lead time | days |
| $\sigma_{LTD}$ | Standard deviation of demand during the lead time | units |
| CSL, $z$ | Cycle service level (probability of no stock-out in a cycle), its standard normal quantile | %, none |
| $\varphi$, $\Phi$, $G(z)$ | Standard normal density, distribution function, loss function | none |
| $p$, $s$ | Selling price, salvage value of a leftover unit | currency per unit |
| $C_u$, $C_o$ | Cost of one unit too few (underage), one unit too many (overage) | currency per unit |
| $\mu$, $\sigma$ | Mean and standard deviation of season demand | units |

**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

<!-- Guidance: Every model here rests on strong assumptions. Name the one that matters most for your item before you present a number. -->

| Model | Key assumptions | What breaks it | What to do |
|---|---|---|---|
| EOQ | Constant, known demand; fixed ordering cost; linear holding cost; no stock-outs; instant replenishment; no quantity discounts | Seasonal demand, truck or case-pack constraints, price breaks | Round to case packs (the curve is flat), check total cost including purchase price at each price break, plan seasonal items differently |
| Reorder point | Normal lead-time demand; independent days; continuous review | Lumpy or intermittent demand, correlated days, periodic review | Use empirical lead-time demand, other distributions, or the periodic review form with review period added to lead time |
| Safety stock with CSL | CSL counts cycles without a stock-out, not units served | Managers often mean fill rate when they say "service level" | Report both CSL and fill rate (Worked example A3) |
| Newsvendor | One period, known costs, demand distribution known, no reorder | Second production run possible, demand estimate biased, substitution between items | Model a reorder option, debias forecasts using past forecast errors, consider pooled or postponed products |
| ABC | Annual usage value captures importance | A cheap part that stops a production line, a critical drug | Add criticality as a second criterion (ABC-XYZ, VED in healthcare) |
| Turns | Average inventory is representative | Year-end inventory is window-dressed, seasonal peaks | Use monthly averages; compare within the same industry |

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

<!-- Guidance: Connect each output to a decision someone signs. Inventory policy is a trade-off between cash, cost and customer service, so finance, operations and sales will all care. -->

- **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

<!-- Guidance: Use this diagram during a case: answer the questions in order and you will land on the right 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.

```mermaid
flowchart TD
  A["Inventory decision"] --> B{"Can you reorder during the selling period?"}
  B -->|"No: one season or event"| N["Newsvendor: order at the critical ratio"]
  B -->|"Yes: repeated replenishment"| C{"Is there a fixed cost per order?"}
  C -->|"Yes"| E["EOQ for the order size"]
  C -->|"No or negligible"| F["Order small and often, size set by packs and trucks"]
  E --> G{"Is demand during the lead time uncertain?"}
  F --> G
  G -->|"Yes"| H["Reorder point = mean lead-time demand + safety stock"]
  G -->|"No"| I["Reorder point = mean lead-time demand"]
  H --> J{"Many items to manage?"}
  I --> J
  N --> J
  J -->|"Yes"| K["ABC classification sets review frequency and service levels"]
  J -->|"No"| L["Monitor turns and days of inventory"]
  K --> L
```

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

<!-- Guidance: Rebuild each part in a spreadsheet with the Excel formulas shown. The numbers come from a re-runnable script; if yours differ, check the units of H and of the lead time first. -->

**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.**

| Quantity | Result | Excel |
|---|---|---|
| $Q^*$ | 1,046.42 units | `=SQRT(2*18250*150/5)` |
| Orders per year | 17.44 | `=18250/1046.42` |
| Days between orders | 20.93 | `=365/17.44` |
| Annual ordering cost | USD 2,616.06 | `=18250/1046.42*150` |
| Annual holding cost | USD 2,616.06 | `=1046.42/2*5` |
| Total relevant cost $TC(Q^*)$ | USD 5,232.11 | `=SQRT(2*18250*150*5)` |

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 $Q$)

| Order quantity *(Example)* | 500 | 800 | 1,200 | 1,500 | 2,000 |
|---|---|---|---|---|---|
| Total relevant cost (USD) | 6,725.00 | 5,421.88 | 5,281.25 | 5,575.00 | 6,368.75 |
| Above the minimum | 28.53% | 3.63% | 0.94% | 6.55% | 21.72% |

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

```vega-lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "description": "Northfield Trailhead 400 headlamp: annual ordering, holding and total relevant cost by order quantity (fictional, computed)",
  "width": 380,
  "height": 220,
  "data": {"values": [
    {"Q": 400, "cost": "Ordering", "usd": 6843.75}, {"Q": 400, "cost": "Holding", "usd": 1000.0}, {"Q": 400, "cost": "Total", "usd": 7843.75},
    {"Q": 600, "cost": "Ordering", "usd": 4562.5}, {"Q": 600, "cost": "Holding", "usd": 1500.0}, {"Q": 600, "cost": "Total", "usd": 6062.5},
    {"Q": 800, "cost": "Ordering", "usd": 3421.88}, {"Q": 800, "cost": "Holding", "usd": 2000.0}, {"Q": 800, "cost": "Total", "usd": 5421.88},
    {"Q": 1000, "cost": "Ordering", "usd": 2737.5}, {"Q": 1000, "cost": "Holding", "usd": 2500.0}, {"Q": 1000, "cost": "Total", "usd": 5237.5},
    {"Q": 1200, "cost": "Ordering", "usd": 2281.25}, {"Q": 1200, "cost": "Holding", "usd": 3000.0}, {"Q": 1200, "cost": "Total", "usd": 5281.25},
    {"Q": 1400, "cost": "Ordering", "usd": 1955.36}, {"Q": 1400, "cost": "Holding", "usd": 3500.0}, {"Q": 1400, "cost": "Total", "usd": 5455.36},
    {"Q": 1600, "cost": "Ordering", "usd": 1710.94}, {"Q": 1600, "cost": "Holding", "usd": 4000.0}, {"Q": 1600, "cost": "Total", "usd": 5710.94},
    {"Q": 1800, "cost": "Ordering", "usd": 1520.83}, {"Q": 1800, "cost": "Holding", "usd": 4500.0}, {"Q": 1800, "cost": "Total", "usd": 6020.83},
    {"Q": 2000, "cost": "Ordering", "usd": 1368.75}, {"Q": 2000, "cost": "Holding", "usd": 5000.0}, {"Q": 2000, "cost": "Total", "usd": 6368.75},
    {"Q": 2200, "cost": "Ordering", "usd": 1244.32}, {"Q": 2200, "cost": "Holding", "usd": 5500.0}, {"Q": 2200, "cost": "Total", "usd": 6744.32},
    {"Q": 2400, "cost": "Ordering", "usd": 1140.62}, {"Q": 2400, "cost": "Holding", "usd": 6000.0}, {"Q": 2400, "cost": "Total", "usd": 7140.62},
    {"Q": 2600, "cost": "Ordering", "usd": 1052.88}, {"Q": 2600, "cost": "Holding", "usd": 6500.0}, {"Q": 2600, "cost": "Total", "usd": 7552.88},
    {"Q": 2800, "cost": "Ordering", "usd": 977.68}, {"Q": 2800, "cost": "Holding", "usd": 7000.0}, {"Q": 2800, "cost": "Total", "usd": 7977.68}
  ]},
  "mark": {"type": "line", "point": true},
  "encoding": {
    "x": {"field": "Q", "type": "quantitative", "title": "Order quantity Q (units)"},
    "y": {"field": "usd", "type": "quantitative", "title": "Annual cost (USD)"},
    "color": {"field": "cost", "type": "nominal", "title": "Cost"}
  }
}
```

**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.

| Quantity | Result | Excel |
|---|---|---|
| $z$ for 95% | 1.6449 | `=NORM.S.INV(0.95)` |
| $\sigma_{LTD}$ (fixed lead time) | 45.00 units | `=15*SQRT(9)` |
| Safety stock | 74.02 units | `=NORM.S.INV(0.95)*15*SQRT(9)` |
| Reorder point | 524.02, so reorder at 525 units | `=50*9+74.02` |
| Annual holding cost of safety stock | USD 370.09 | `=74.02*5` |

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

| Quantity | Result | Excel |
|---|---|---|
| $\sigma_{LTD}$ (variable lead time) | 109.66 units | `=SQRT(9*15^2+50^2*2^2)` |
| Safety stock | 180.37 units | `=NORM.S.INV(0.95)*109.66` |
| Reorder point | 630.37, so 631 units | `=50*9+180.37` |
| Annual holding cost of safety stock | USD 901.86 | `=180.37*5` |

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:

| CSL | $z$ | Safety stock | Expected units short per cycle | Fill rate | Excel (units short) |
|---|---|---|---|---|---|
| 90% | 1.2816 | 57.67 | 2.13 | 99.796% | `=45*(NORM.S.DIST(1.2816,FALSE)-1.2816*(1-NORM.S.DIST(1.2816,TRUE)))` |
| 95% | 1.6449 | 74.02 | 0.94 | 99.910% | `=45*(NORM.S.DIST(1.6449,FALSE)-1.6449*(1-NORM.S.DIST(1.6449,TRUE)))` |
| 99% | 2.3263 | 104.69 | 0.15 | 99.985% | `=45*(NORM.S.DIST(2.3263,FALSE)-2.3263*(1-NORM.S.DIST(2.3263,TRUE)))` |

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.

| Quantity | Result | Excel |
|---|---|---|
| $C_u$ (lost margin per unit short) | USD 130 | `=240-110` |
| $C_o$ (loss per unit left over) | USD 40 | `=110-70` |
| Critical ratio | 0.7647 | `=130/(130+40)` |
| $z$ | 0.7215 | `=NORM.S.INV(0.7647)` |
| $Q^*$ | 3,649.37, so order 3,650 | `=NORM.INV(130/170,3000,900)` |

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

| Order quantity *(Example)* | In-stock probability | Expected sales | Expected leftover | Expected lost sales | Expected profit (USD) |
|---|---|---|---|---|---|
| 3,000 (the mean) | 50.00% | 2,640.95 | 359.05 | 359.05 | 328,961.83 |
| 3,650 (critical ratio) | 76.49% | 2,876.18 | 773.82 | 123.82 | 342,950.40 |
| 4,000 | 86.67% | 2,939.59 | 1,060.41 | 60.41 | 339,729.80 |

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:

| SKU | Item *(Example)* | Annual usage value (USD) | Share | Cumulative share | Class |
|---|---|---|---|---|---|
| NF-101 | Alpine tent 2P | 412,000 | 24.8% | 24.8% | A |
| NF-104 | Headlamp Trailhead 400 | 365,000 | 22.0% | 46.8% | A |
| NF-102 | Down sleeping bag | 296,000 | 17.8% | 64.6% | A |
| NF-103 | Trail running shoe | 238,000 | 14.3% | 79.0% | B |
| NF-106 | Merino base layer | 141,000 | 8.5% | 87.5% | B |
| NF-107 | Camp stove | 77,000 | 4.6% | 92.1% | C |
| NF-105 | Water filter | 58,000 | 3.5% | 95.6% | C |
| NF-108 | Fuel canister | 39,000 | 2.3% | 98.0% | C |
| NF-109 | Carabiner set | 22,000 | 1.3% | 99.3% | C |
| NF-110 | Repair kit | 12,000 | 0.7% | 100.0% | C |
| Total | | 1,660,000 | 100.0% | | |

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.

| Quantity | Result | Excel |
|---|---|---|
| Average inventory | USD 46.5m | `=(44+49)/2` |
| Turns | 4.00 | `=186/46.5` |
| Days of inventory | 91.25 days | `=365/4` |
| Inventory at 5 turns | USD 37.2m | `=186/5` |
| Cash released | USD 9.3m | `=46.5-37.2` |
| Annual holding cost saved at 25% | USD 2.325m | `=9.3*0.25` |

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

<!-- Guidance: Fill each [ ] with a number and an Excel formula, then check the answer key. Notice which inputs differ from Worked example A and predict the direction of each change before you compute. -->

**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

<!-- Guidance: Solve on paper first, then confirm with Excel. Write the model you chose and why before you compute anything. -->

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

<!-- Guidance: Interleave: after this card, solve one problem from a neighbouring card and decide which model fits before you compute. -->

- **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

<!-- Guidance: Read this list before an exam. Copy any mistake you make into mba/mba-error-log with the correct reasoning. -->

- **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

<!-- Guidance: Answer from memory, in writing, before scrolling to the answer key. Return after two days and after a week. Mark misses in mba/mba-error-log. -->

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

<!-- Guidance: This section is yours. Rate your confidence honestly and plan the next review. -->

| Item | My entry |
|---|---|
| Confidence with inventory models (1 = lost, 5 = could teach it) | [ ] |
| An item in my own work or life that fits the newsvendor model | [ ] |
| Questions for my study group or professor | [ ] |
| Mistakes logged in mba/mba-error-log | [Date and short description] |
| Next review date (spacing: +2 days, +1 week, +1 month) | YYYY-MM-DD |
| Next card to study | [mba/mba-queueing-card / mba/mba-quality-card] |

## Sources and further reading

<!-- Guidance: Cite the primary work for each model. Textbooks are named for further reading only; nothing on this card is copied from them. -->

- 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

<!-- Guidance: Check your answers only after writing your own. For each miss, write down why your first answer was wrong. -->

**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

<!-- Guidance: Add a row each time you change demand data, costs or policy choices. -->

| Version | Date | Author | Change summary |
|---|---|---|---|
| 0.1 | YYYY-MM-DD | [Your name] | First draft from the template |
| 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced Northfield data with own SKUs, CSL policy set with finance |

## 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.