Inventory Planning and Managing Uncertainty

Inventory Foundations: Why Stock Exists and What It Costs

Inventory is a stock of goods held within a system. It ranges from screws and paperclips to machines, trucks and construction equipment, and matters in manufacturing, retail, hospitals, service operations and IT. Inventory bridges the mismatch between demand, supply and lead time; if production and demand were perfectly synchronised, little inventory would be needed.

It sits at the operations–marketing–finance intersection: operations needs flow, continuity and service; marketing needs availability and fewer lost sales; finance sees cash tied up on the balance sheet. Inventory reduces stockouts but can spoil, become obsolete or require markdowns. The objective is therefore the right service level at reasonable cost—not maximum inventory or minimum inventory.

Types and functions of inventory

Inventory serves eight major functions:

1. Anticipation stock meets expected/average customer demand (milk, bread, rice in a supermarket).
2. It smooths production when demand is seasonal but capacity cannot flex instantly (fans before summer; Diwali assortments).
3. It decouples operations, buffering process steps so a machine failure or late supplier does not stop everything.
4. Safety stock protects against uncertain demand and lead time—weather, supplier stockouts, wrong materials or quality problems.
5. Cycle inventory arises because orders/shipments are made in cost-efficient batches with fixed ordering/transport costs.
6. Speculative inventory hedges expected input-price increases.
7. It permits operations that take time: production and transport create unavoidable WIP and pipeline stock.
8. It captures quantity discounts from suppliers.

Little’s Law quantifies inventory in a flow system:

$I = R \times T$

where $I$ is average inventory, $R$ is average throughput/demand rate, and $T$ is average time in the system. At 5 units/day and 10 days in system, average inventory is $50$ units.

The fundamental trade-off and performance metrics

Too little inventory causes stockouts, lost sales and disrupted operations. Too much locks up capital, consumes warehouse/handling space, risks shrinkage, spoilage, pilferage, depreciation and obsolescence, and often forces markdowns.

The two core management questions are when to order and how much to order. Common metrics are:

$\text{Inventory turnover} = \frac{\text{Annual cost of goods sold}}{\text{Average inventory investment}}$

High turns often mean efficient inventory use, but are not automatically superior: low-margin grocery needs high turns, while high-margin items can tolerate lower turns; excessive turns can cause stockouts. Days of inventory on hand asks how many days sales can continue if replenishment stopped today—high days can signal excess, low days can signal stockout risk.

Effective management requires accurate on-hand/on-order records, a demand forecast plus forecast error, lead-time mean/variability, cost estimates and SKU classification. Forecasting and inventory are tightly linked: inventory is a response to forecast error and lead-time uncertainty.

Counting, tracking and review logic

These support two review policies:

• Continuous review / Q system: monitor continuously; at a reorder point, place a fixed quantity $Q$. Timing varies.
• Periodic review / P system: inspect every fixed time interval; order quantity varies to restore a target. Timing is fixed.

Key takeaways

• Inventory buffers real mismatches but is costly and risky to hold.
• Cycle stock is batch-driven; safety stock is uncertainty-driven.
• Little’s Law links inventory directly to flow rate and time.
• The correct service/cost balance—not maximum turnover—is the objective.
• Periodic tracking supports P logic; perpetual tracking makes Q logic feasible.

The four relevant cost buckets

The clean baseline for annual holding cost per unit is:

$h = iC$

where $h$ is currency per unit per year, $i$ is annual carrying/capital rate, and $C$ is unit purchase cost. If $C=\text{Rs. }180$ and $i=10\%$, then:

$h = 0.10 \times 180 = \text{Rs. }18\text{ per unit per year}$

Holding cost is broader than capital cost, but this expression shows why expensive items hurt to hold. For an order of $X$ units, purchase spending is $CX$ and the fixed order cost is $K$; do not confuse $C$ (unit purchase cost) with $K$ (fixed cost per order).

$\text{Total cost of an order of }X\text{ units}=K+CX$

ABC classification: focus effort where value is concentrated

A SKU (stock keeping unit) is a separately tracked/replenished product variant—for example, a shampoo’s size or fragrance. Managing every one of thousands of SKUs identically is wasteful. ABC analysis ranks items by annual monetary usage:

$\text{Annual dollar/rupee value} = \text{Annual demand} \times \text{Unit cost}$

Annual value is the starting criterion, but criticality, stockout risk, long lead time, obsolescence and regulatory need can override it. A low-value C bolt can still stop an assembly line; because its annual value is low, ordering more/earlier may be sensible.

Worked ABC illustration

For 10 items with total annual value ₹75,910, sorting high-to-low showed item 8 alone (10% of items) contributed 52.7% of value: an A item. The next three items—3, 6 and 1 (30% of items)—contributed 40.8%: B items. The remaining six (60% of items) contributed only 6.5%: C items. Cutoffs are managerial choices, not natural laws; the purpose is differential control.

Cycle counting follows directly: count A items often, B sometimes and C rarely, instead of counting all stock once a year. Record error creates stockouts, excess ordering and operational disruption.

Key takeaways

• Inventory decisions trade holding, ordering and shortage costs; constant purchase cost often falls out of lot-size optimisation.
• $h=iC$ is a useful baseline but storage and deterioration can dominate in practice.
• ABC classifies by annual usage value, not simply unit price or volume.
• A items deserve precision; C items can use simpler controls, though critical C items still require availability protection.

Economic order quantity (EOQ)

Economic order quantity (EOQ) is the fixed batch size that balances annual ordering cost against annual holding cost for predictable demand. It is attributed to Ford Harris (1915).

The basic model assumes: one independently analysed item; known annual demand; continuous, constant demand rate; known constant lead time; each order received in one delivery; fixed $K$ per order; unit cost $C$; holding cost $h$ per unit-time; no shortages; no obsolescence; and no quantity discounts.

With annual demand $D$, order size $Q$, fixed ordering cost $K$, purchase cost $C$ and annual holding cost $h$:

$\text{Annual purchase cost}=CD$

$\text{Number of orders per year}=\frac{D}{Q}$

$\text{Annual ordering cost}=\frac{KD}{Q}$

Because stock falls linearly from $Q$ to $0$, average cycle inventory is:

$\bar{I}=\frac{Q}{2}$

$\text{Annual holding cost}=\frac{hQ}{2}$

Thus total annual cost is:

$TC(Q)=CD+\frac{KD}{Q}+\frac{hQ}{2}$

$CD$ is independent of $Q$, so minimise only ordering plus holding cost. Differentiating:

$\frac{dTC}{dQ}=-\frac{KD}{Q^2}+\frac{h}{2}=0$

$Q^*=\sqrt{\frac{2DK}{h}}$

The second derivative is positive for $Q>0$:

$\frac{d^2TC}{dQ^2}=\frac{2KD}{Q^3}>0$

so this is a minimum. At $Q^*$, annual ordering cost equals annual holding cost. Higher $K$ or $D$ raises $Q^*$; higher $h$ lowers it. $C$ does not appear directly when constant, but affects $Q^*$ indirectly if $h=iC$.

The inventory path is a sawtooth: delivery jumps stock to $Q$, then constant demand depletes it linearly. Large $Q$ means fewer orders but high average inventory; small $Q$ means many orders but low average inventory. EOQ is the compromise.

Worked EOQ: tire distributor

Inputs: $D=9{,}600$ tires/year, $K=75$ USD/order, $h=16$ USD/tire/year, and 288 operating days/year.

$Q^*=\sqrt{\frac{2(9{,}600)(75)}{16}}=\sqrt{90{,}000}=300\text{ tires}$

$\text{Orders/year}=\frac{9{,}600}{300}=32$

$\text{Cycle length}=\frac{Q^*}{D}=\frac{1}{32}\text{ year}=\frac{288}{32}=9\text{ workdays}$

$\text{Holding cost}=\frac{300}{2}(16)=2{,}400\text{ USD}$

$\text{Ordering cost}=\frac{9{,}600}{300}(75)=2{,}400\text{ USD}$

Annual relevant cost is $2{,}400+2{,}400=4{,}800$ USD (add $CD$ only if total purchase spending is required). The distributor orders 300 tires every 9 workdays.

When to order: deterministic reorder point (ROP)

With zero lead time, stock can reach zero before ordering, so $ROP=0$. With positive lead time $L$, order when stock covers expected demand while the shipment travels:

$ROP=dL$

where $d$ is demand per day and $L$ is lead time in days.

For the tire example, daily demand is $d=9{,}600/288=33.33$ tires/day. With $L=5$ days:

$ROP=(33.33)(5)=166.67\approx167\text{ tires}$

Order at 167 tires; demand over the next five days reduces stock to about zero when the replenishment arrives. The 5-day lead time is less than the 9-day order cycle.

If $L=15$ days, it exceeds the 9-day cycle and multiple orders are in transit. Under the source’s on-hand treatment, the residual unfilled portion is $L\bmod T$, where $T$ is the order-cycle time:

$L'=L\bmod T=15\bmod9=6\text{ days}$

$ROP=dL'=(33.33)(6)\approx200\text{ tires}$

The key is tracking pipeline orders; later continuous-review practice defines the ROP by inventory position, not shelf stock alone.

Key takeaways

• EOQ answers how much: balance $KD/Q$ with $hQ/2$.
• At EOQ, annual ordering and holding costs are equal.
• Deterministic ROP answers when: expected demand during lead time.
• Long lead times require explicit tracking of multiple outstanding/pipeline orders.

When leftovers do not economically carry forward

The newsvendor / single-period / one-shot model applies when a decision is made before uncertain demand is known and leftover stock becomes obsolete or must be deeply marked down: monthly magazines, seasonal fashion or perishable products. Unlike EOQ, too much today cannot simply flow into the next equivalent cycle.

The trade-off is overage (one unit too many) versus underage (one unit too few), not long-run ordering versus holding.

For purchase cost $C$, selling price $P$ and salvage value $S$:

$C_o=C-S \qquad \text{(overage cost)}$

$C_u=P-C \qquad \text{(underage cost)}$

If a lost sale also damages future business, include goodwill penalty $g$:

$C_u=P-C+g$

$g$ is difficult to estimate because it is the present value of lost future margin and negative word of mouth.

Hemant Patel’s monthly-magazine calculation

Hemant pays ₹30, sells at ₹40, and salvages unsold monthly copies at ₹10. Demand is equally likely to be 6, 7, 8, 9 or 10 copies ($0.2$ each).

$C_u=40-30=\text{Rs. }10,\qquad C_o=30-10=\text{Rs. }20$

The larger overage cost suggests a conservative quantity below the middle demand (8). The full profit table confirms this. If demand is at least $Q$, profit is $10Q$; if demand is below $Q$, profit is $10D-20(Q-D)$.

Since each scenario probability is $0.2$, expected profit is the average of the column. Maximum expected profit is ₹64 at $Q=7$ copies.

Critical-fractile rule

Consider whether to buy the $(Q+1)$th unit. It sells if $D>Q$, gaining $P-C$, and remains if $D\le Q$, losing $C-S$. Add units while expected incremental payoff is positive; stop at zero:

$\Pr(D>Q)(P-C)+\Pr(D\le Q)(S-C)=0$

Using $F(Q)=\Pr(D\le Q)$ gives:

$[1-F(Q)]C_u=F(Q)C_o$

$F(Q^*)=\frac{C_u}{C_u+C_o}$

$Q^*=F^{-1}\left(\frac{C_u}{C_u+C_o}\right)$

The ratio is the critical fractile. It is also the optimal cycle service level—the probability demand is fully met/no stockout in the period.

Expected-payoff warm-up

The marginal decision follows ordinary expected-payoff logic. A ₹100 lottery ticket has a 25% chance of paying ₹200 and a 75% chance of returning a ₹50 consolation. Its expected net payoff is:

$0.25(200-100)+0.75(50-100)=25-37.5=-\text{Rs. }12.5$

It is a losing bet on average. The newsvendor’s next-unit decision uses the same two-outcome expected-payoff logic: sale versus leftover.

For Hemant:

$\frac{C_u}{C_u+C_o}=\frac{10}{10+20}=0.33$

CDF values for demand 6–10 are $0.2,0.4,0.6,0.8,1.0$. Choose the smallest quantity with CDF at least $0.33$: $Q^*=7$, agreeing with the expected-profit table.

Continuous-demand newsvendor: Mac’s weekly magazine

Mac pays 0.25 USD, sells at 0.75 USD and returns unsold copies for 0.10 USD. Weekly demand is normal with $\mu=11.73$, $\sigma=4.74$.

$C_o=0.25-0.10=0.15,\qquad C_u=0.75-0.25=0.50$

$\text{Critical fractile}=\frac{0.50}{0.50+0.15}=0.77$

The standard-normal quantile for $0.77$ is $z\approx0.74$:

$Q^*=\mu+z\sigma=11.73+(0.74)(4.74)=15.24$

Order 15 or 16 copies; 16 ensures meeting at least the target fractile under an upward rounding convention. In Excel, use the inverse normal CDF directly with probability, mean and standard deviation.

Key takeaways

• Newsvendor is for one-shot/perishable decisions, not recurring EOQ cycles.
• Overage cost is $C-S$; underage cost is $P-C$ plus any goodwill penalty.
• The critical fractile selects the quantity where marginal expected gain from another unit becomes zero.
• Higher overage relative to underage means a lower service level and more conservative order quantity.

Inventory position and safety stock

With positive lead time, decisions should use inventory position:

$IP=\text{On-hand inventory}+\text{On-order inventory}-\text{Backorders}$

It prevents double ordering when stock is already in transit. With zero lead time/no backorders, $IP$ equals on-hand inventory.

Safety stock is inventory above expected demand that reduces stockout risk. In a normal single-period context, $Q=\mu+z\sigma$ and $z\sigma$ is that buffer; in replenishment systems, the relevant uncertainty window is the protection period.

Q model: continuous review, fixed quantity

The Q model continuously monitors $IP$, assumes backorders rather than lost sales, and orders the same fixed $Q$ when $IP$ reaches $ROP$. Thus $Q$ is fixed and timing varies. EOQ can set $Q$ from expected annual demand, while safety stock sets when to order.

In continuous review, the only vulnerable interval is the replenishment lead time $L$: after placing an order, no intervention can make it arrive earlier. With independent demand and fixed lead time:

$\mu_{DLT}=\mu_dL$

$\sigma_{DLT}=\sigma_d\sqrt{L}$

$SS=z\sigma_{DLT}$

$ROP=\mu_{DLT}+z\sigma_{DLT}$

Here $z$ corresponds to the chosen cycle service level (probability of no stockout during lead time). If both demand and lead time are uncertain, demand during lead time is a random sum; closed-form mean/standard-deviation expressions exist but are outside this module’s scope.

Q-model worked example: appliance store

Weekly demand is normal with mean 10 and standard deviation 8; $K=45$ USD/order, $h=12$ USD/unit/year, $L=3$ weeks, 52 weeks/year, and target cycle service level 70%.

Expected annual demand:

$D=(10)(52)=520$

Fixed EOQ:

$Q^*=\sqrt{\frac{2(520)(45)}{12}}=\sqrt{3900}=62.5\approx63\text{ units}$

Lead-time demand parameters:

$\mu_{DLT}=(10)(3)=30$

$\sigma_{DLT}=8\sqrt3=13.86$

For 70% service, $z\approx0.55$. Therefore:

$SS=(0.55)(13.86)=7.62$

$ROP=30+7.62=37.62\approx38\text{ units}$

Order 63 units whenever inventory position hits 38.

P model: periodic review, variable quantity

The P model reviews every fixed $T$ days/weeks and orders enough to raise inventory position to an order-up-to level $S$:

$Q_{\text{order}}=S-IP$

Timing is fixed, order quantity varies. It is attractive when continuous monitoring is costly/impractical, multiple SKUs can be ordered on a common supplier calendar, or procurement has weekly/monthly rhythm.

Its protection period is longer than Q model’s:

$\text{Protection period}=T+L$

The system can stock out just after review, remain unseen until the next review, then wait lead time again. For independent daily normal demand:

$SS=z\sigma_d\sqrt{T+L}$

$S=\mu_d(T+L)+z\sigma_d\sqrt{T+L}$

Thus P systems generally need more safety/average inventory than Q systems at the same service level, but are administratively simpler and enable order consolidation.

Continuous-review calculation: current versus 95% target

Daily demand: $\mu_d=60$, $\sigma_d=7$; $L=6$ days; $K=\text{Rs. }10$; $h=\text{Rs. }0.5$/unit/year; 365 days/year. Current policy: $Q=1{,}200$, $ROP=360$.

$\mu_{DLT}=60(6)=360$

$\sigma_{DLT}=7\sqrt6=17.46$

$z_{\text{current}}=\frac{360-360}{17.46}=0$

Current cycle service level is 50%. For a 95% target, $z=1.65$:

$Q^*=\sqrt{\frac{2(60)(365)(10)}{0.5}}\approx936\text{ units}$

$ROP=360+(1.65)(17.46)=388.2\text{ units}$

Order 936 whenever $IP$ reaches about 388 to achieve 95% probability of no lead-time stockout.

Periodic-review calculation

Daily demand is normal with $\mu_d=10$, $\sigma_d=3$; $T=30$ days; $L=14$ days; service level 98% ($z\approx2.05$); current on-hand inventory is 150.

$S=(10)(30+14)+(2.05)(3)\sqrt{30+14}=480.8\approx481$

$Q_{\text{order}}=480.8-150=330.8\approx331\text{ units}$

At the next review, quantity will differ because it is always the gap between current $IP$ and $S$.

Select a high service level if backorder penalties, lost goodwill, contract fines or expediting are high; lower service can be appropriate if backorders are acceptable. There is no universal best policy—fit depends on monitoring cost, order coordination, product and stockout pain.

Key takeaways

• $IP$, not on-hand inventory alone, drives continuous-review ordering with pipeline stock.
• Q model protects uncertainty during $L$; P model protects $T+L$.
• Q: fixed quantity/variable time. P: fixed time/variable quantity.
• Higher service means higher safety stock and holding cost; lower service accepts more stockout/backorder risk.

Pooling and Postponement: Reducing the Uncertainty to Buffer

Pooling combines demand streams, variants or product commitment so variability partially cancels. It does not change average demand; it reduces the standard deviation that drives safety stock.

Location pooling and the square-root effect

For $N$ identical, independent regions, each with mean demand $\mu$, standard deviation $\sigma$ and the same service factor $z$:

$\text{Decentralised inventory}=N\mu+Nz\sigma$

$\text{Centralised inventory}=N\mu+\sqrt{N}z\sigma$

Mean demand grows linearly either way. The safety-stock term grows linearly across independent separate warehouses but only with $\sqrt N$ in a central pool. Centralisation also lowers facility overhead, but can increase transport cost, response time and operational complexity; real networks balance inventory efficiency against proximity/speed.

Correlation changes the benefit. For two equal-demand locations:

$\operatorname{Var}(X+Y)=2\sigma^2[1+\rho_{XY}]$

$\sigma_{X+Y}=\sigma\sqrt{2(1+\rho_{XY})}$

Positive correlation means demands rise/fall together, raising pooled variability and reducing pooling benefit. Negative correlation lets high demand in one region offset low demand in another, increasing the benefit.

Product pooling and delayed differentiation

Product pooling reduces variant-level uncertainty through a common design. O’Neill could stock separate surfer-logo and diver-logo wetsuits, requiring safety stock for two uncertain SKUs, or one generic wetsuit serving both segments. Similar examples are universal power adaptors, common automobile platforms and common components/packaging. The trade-off is less specialised functionality, differentiation and price-segmentation opportunity.

Lead-time pooling / postponement / delayed differentiation delays final commitment until demand information improves. Paint stores hold base paint plus pigments and mix the final shade at sale; Benetton’s dye-after-knitting approach retains generic garments longer. Modular designs let firms hold common modules and postpone final configuration (printers, electronics, configurable products).

Postponement works best when variety is important, total demand is more predictable than the split across variants, late customisation is quick/cheap, and differences are cosmetic (colour, label, packaging). Across location, product and lead-time pooling, the common rule is: reduce variability at the point where inventory is held.

Key takeaways

• Pooling cuts safety stock by reducing demand variability, not mean demand.
• Independent location pooling produces the square-root safety-stock advantage; positive correlation erodes it.
• Product pooling exchanges lower mismatch risk for less differentiation.
• Postponement keeps stock generic until better demand information arrives and needs suitable modular/process design.
• Strong supply chains combine policy levers (EOQ, Q/P safety stock) with design levers (pooling/postponement).