Supply Chain & Logistics Management

Inventory Models: EOQ, Newsvendor, and Safety Stock

Module 3

Inventory keeps supply chains moving when demand, supply, and lead times do not line up, and this lesson works through the classic models that decide how much and when to order: EOQ, the newsvendor, and the Q and P review systems.

1. Introduction to Inventory Management

Inventory Planning and Managing Uncertainty: module overview infographic

Inventory is a stock of goods that buffers imperfect synchronization between demand and production. It sits at the intersection of operations, marketing, and finance.

Department	Core Interest	Operational Impact
Operations	Flow and stability	Process flow, service levels, disruptions
Marketing	Customer experience	Product availability and lost sales
Finance	Capital efficiency	Ties up cash; hits the balance sheet

Stage	Definition	Examples
Raw Materials and Purchased Items	Inputs waiting for production	Steel coils, fabric, packaging material
Work in Process (WIP)	Partially completed goods on the shop floor	A smartphone or vehicle mid-assembly
Finished Goods / Merchandise	Completed products ready for sale	Retail shelves, fulfillment centers, dark stores
MRO Supplies	Materials that support operations but do not enter the product	Tools, lubricants, spare parts, cleaning supplies
Pipeline Inventory / Goods in Transit	Capital tied up moving between stages	Goods on a truck, ship, or between supplier and warehouse

Function of Inventory	Purpose	Example
Meet Anticipated Demand	Satisfy steady, predictable average demand	Supermarket staples like milk, bread, rice
Smooth Production	Build in low-demand periods for high-demand peaks	Stocking fans before summer
Decouple Operations	Buffer between steps to prevent system-wide shutdowns	WIP buffers when a machine goes down
Reduce Stockout Risk	Safety stock against demand or lead-time uncertainty	Buffers against weather delays or supplier failures
Economies of Scale	Larger batches cut per-unit ordering and transport costs (cycle inventory)	Batch production, container-load shipping
Hedge Price Increases	Speculative early buying before price rises	Buying commodity metals ahead of hikes
Permit Operations	Production and transport are not instantaneous	Unavoidable WIP and pipeline inventory
Quantity Discounts	Larger orders secure lower unit costs	Bulk purchasing

Average inventory equals the average demand rate times the average time a unit spends in the system.

ƒLittle's Law

I = D \times T

Where: I is average inventory, D is the average demand or flow rate, and T is the average time a unit spends in the system.

Worked example. If units spend 10 days in the system and the demand rate is 5 units/day:

I = 5 \times 10 = 50 \text{ units}

Downsides and Metrics

Excess inventory ties up capital, requires space and handling, risks shrinkage, spoilage, obsolescence, and forces markdowns. The objective is satisfactory customer service at reasonable inventory cost.

Metric	Calculation	Interpretation
Inventory Turnover	$\text{Inventory Turnover} = \frac{\text{Annual COGS}}{\text{Average Inventory Investment}}$	High turns = efficient use, but pushing too high causes stockouts; low-margin groceries need high turns, high-margin items tolerate lower
Days of Inventory	Days sales can continue if replenishment stops	High = excess stock; low = stockout risk

ƒInventory turnover

\text{Inventory Turnover} = \frac{\text{Annual COGS}}{\text{Average Inventory Investment}}

Where: average inventory investment is the average value of stock held over the year, measured at cost.

Effective management requires a tracking system, a reliable forecast (with error), lead times and their variability, cost estimates, and a classification system.

Counting System	Definition	Advantages	Disadvantages
Periodic	Physical count at fixed intervals	Simpler, less tracking sophistication	No continuous visibility; needs extra buffers between reviews
Perpetual	Continuous electronic tracking (barcodes, POS)	Excellent visibility, fast reaction	High record-keeping cost; still needs physical verification for shrinkage

These environments align with the two classic review models: continuous review (Q system) and periodic review (P system).

2. Inventory Costs and ABC Classification

Cost Type	Definition	Characteristics	Notation
Purchase Cost	Amount paid per unit	Largest absolute cost, but does not drive frequency unless discounts apply	C
Holding (Carrying) Cost	Cost of storing inventory over time	Interest/cost of capital, rent, insurance, taxes, spoilage, shrinkage, obsolescence	h = I × C (I = carrying rate)
Ordering (Setup) Cost	Cost of placing and receiving an order or preparing a run	Fixed per order regardless of size	K
Shortage (Stockout) Cost	Penalty when demand exceeds stock	Lost sales, lost goodwill, backorder expediting; hardest to measure	Cu or penalty rates

ƒAnnual holding cost per unit

h = I \times C

Where: I is the annual carrying rate as a fraction of unit value and C is the purchase cost per unit.

ABC Classification

Firms with thousands of SKUs prioritize control based on annual monetary value (annual demand × unit cost).

Class	Fraction of Items	Share of Annual Value	Control Strategy
A	10% to 20%	60% to 70%	Close attention, tight control, frequent reviews, accurate forecasting, frequent cycle counting
B	Middle group	~20% to 30%	Moderate control between the extremes
C	50% to 60%	10% to 15%	Simple control, less frequent reviews, bulk ordering, larger safety margins

Cycle counting rotates physical counts: A items frequently, B moderately, C rarely.

Example: Item 8 alone is 10% of items but 52.7% of annual monetary value (Class A). Items 3, 6, and 1 are 30% of items and 40.8% of value (Class B). The remaining 6 items are 60% of items but only 6.5% of value (Class C).

3. Economic Order Quantity (EOQ)

The EOQ model (Ford Harris, 1915) analyzes cycle stock under deterministic conditions. Assumptions: single SKU; known deterministic annual demand; continuous constant demand rate; known constant lead time; instantaneous single-delivery replenishment; constant unit cost; constant holding cost; no shortages; no quantity discounts.

Inventory follows a sawtooth: jumps to Q on delivery, declines linearly to zero.

ƒTotal annual cost under EOQ

\mathrm{TC}(Q) = C \times D + K \times \frac{D}{Q} + h \times \frac{Q}{2}

Where: C is unit purchase cost, D is annual demand, K is the fixed cost per order, h is the annual holding cost per unit, and Q is the order quantity.

Purchase cost C × D is constant. Ordering cost K × D/Q falls as Q rises. Holding cost h × Q/2 rises linearly with Q (average cycle inventory Q/2). Setting dTC/dQ = 0:

ƒEconomic order quantity

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

Where: D is annual demand, K is the fixed ordering cost, and h is the annual holding cost per unit.

The second derivative is positive (a minimum). At Q*, annual holding cost exactly equals annual ordering cost.

Worked example. Given: D = 9,600 tires/year, K = $75 per order, h = $16 per tire per year, 288 working days.

Q^* = \sqrt{\frac{2 \times 9{,}600 \times 75}{16}} = 300 \text{ tires}

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

\text{Cycle length} = \frac{288}{32} = 9 \text{ workdays}

Calculation

\text{Annual holding} + \text{annual ordering} = 2{,}400 + 2{,}400 = 4{,}800

Answer: Q* = 300 tires, 32 orders per year, a 9-workday cycle, and $4,800 total annual holding + ordering cost.

Reorder Point (ROP) with Lead Time

The inventory replenishment cycle under uncertainty

L = 0: ROP = 0, orders arrive instantly.
L < cycle time T: ROP = daily demand × L.
L > T: pipeline orders exist; use leftover lead time L' = L mod T.

ƒReorder point under certainty

\mathrm{ROP} = d \times L

Where: d is average demand per day and L is the lead time in days, replaced by the leftover lead time L' = L mod T when the lead time exceeds one order cycle.

Worked example (L < T). With L = 5 days:

d = \frac{9{,}600}{288} = 33.33 \text{ tires/day}

\mathrm{ROP} = 33.33 \times 5 = 166.67 \approx 167 \text{ tires}

Worked example (L > T). For L = 15 days:

L' = 15 \bmod 9 = 6

\mathrm{ROP} = 33.33 \times 6 = 199.98 \approx 200 \text{ tires}

4. Newsvendor Model

When unsold products become obsolete or heavily marked down at period end, inventory cannot carry forward: each period is a one-shot decision.

Cost	Definition	Formula
Overage Cost (Co)	Net loss from one extra unsold unit	$C_o = C - S$ (C = purchase cost, S = salvage value)
Underage Cost (Cu)	Lost margin from being one unit short	$C_u = P - C$ (P = retail price)

ƒOverage cost

C_o = C - S

Where: C is the purchase cost per unit and S is the salvage value of an unsold unit.

ƒUnderage cost

C_u = P - C

Where: P is the retail price, so C_u is the margin lost on each unit of unmet demand, becoming P - C + g when a goodwill penalty g applies.

Worked example. Given: C = 30 rupees, P = 40, S = 10; demand equally likely 6, 7, 8, 9, or 10 (probability 0.2 each).

C_o = 30 - 10 = 20

C_u = 40 - 30 = 10

Since Co > Cu, overstocking mistakes cost twice as much as understocking, biasing the order below the demand mean of 8. Building the profit matrix over all demand-quantity pairs, expected profit peaks at Q* = 7 copies with expected profit 64 rupees (Q = 6 gives 60, Q = 8 gives 62, Q = 9 gives 54, Q = 10 gives 40).

Answer: Q* = 7 copies, expected profit 64 rupees.

Derivation and the Critical Fractile

The (Q+1)-th unit sells if D > Q (gain Cu) and stays unsold if D ≤ Q (loss Co). Setting the expected payoff to zero using F(Q) = P(D ≤ Q):

ƒCritical fractile (newsvendor optimum)

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

Where: F is the cumulative distribution of demand, so Q* is the smallest quantity whose probability of covering demand reaches the critical ratio.

This is the optimal Cycle Service Level: the probability of not stocking out. With a goodwill penalty g, Cu becomes P - C + g.

Discrete application (Hemant)

\text{Critical ratio} = \frac{10}{30} = 0.33

F(6) = 0.2, F(7) = 0.4; the smallest Q with F(Q) ≥ 0.33 is Q* = 7.

Worked example. Given: C = $0.25, P = $0.75, S = $0.10; weekly demand normal with μ = 11.73, σ = 4.74.

C_o = 0.15

C_u = 0.50

\text{Critical ratio} = \frac{0.50}{0.65} = 0.77

Z for 0.77 ≈ 0.74.

Q^* = \mu + Z \times \sigma = 11.73 + 0.74 \times 4.74 = 15.24

Answer: Q* = 15.24, rounded up to 16 copies to meet the target service level.

5. The Q Model (Continuous Review)

Manages longer shelf-life products in multi-period replenishment with demand uncertainty.

ƒInventory position

\text{Inventory Position} = \text{On-Hand} + \text{On-Order (Pipeline)} - \text{Backorders}

Where: on-order is stock already ordered but still in the pipeline, and backorders are unfilled customer demand.

Monitoring inventory position instead of on-hand prevents duplicate ordering while pipeline deliveries are in transit.

Safety stock is the buffer above expected demand that reduces stockout risk. In continuous review it protects only during the lead time L (the vulnerable protection period).

Q* comes from the EOQ formula using expected annual demand.

ƒReorder point under demand uncertainty

\mathrm{ROP} = \text{Expected Demand During Lead Time (DDLT)} + \text{Safety Stock}

Where: DDLT is the mean demand over the lead time and safety stock is the buffer covering variability during that protection period.

ƒSafety stock (continuous review)

\text{Safety Stock} = Z \times \sigma_{\mathrm{DDLT}}

Where: Z is the service-level factor from the standard normal table and sigma DDLT is the standard deviation of demand over the lead time.

ƒStandard deviation of lead time demand

\sigma_{\mathrm{DDLT}} = \sigma_{\text{weekly}} \times \sqrt{L}

Where: L is the lead time in weeks, assuming independent weekly demand.

Worked example. Given: weekly demand normal (μ = 10, σ = 8), K = $45, h = $12/unit/year, L = 3 weeks, 70% service level.

D = 10 \times 52 = 520 \text{ units/year}

Q^* = \sqrt{\frac{2 \times 520 \times 45}{12}} \approx 62.5, \text{ so } 63 \text{ units}

\mathrm{DDLT} = 10 \times 3 = 30

\sigma_{\mathrm{DDLT}} = 8 \times \sqrt{3} \approx 13.86

With Z(70%) = 0.55:

\text{Safety Stock} = 0.55 \times 13.86 = 7.62

\mathrm{ROP} = 30 + 7.62 = 37.62 \approx 38 \text{ units}

Answer: order 63 units whenever the inventory position falls to 38 units.

6. The P Model (Periodic Review)

The P model is triggered by a fixed time interval T rather than an inventory threshold. The order-up-to level S is fixed, and the order quantity varies each review.

ƒP model order quantity

q = S - \text{Inventory Position}

Where: S is the order-up-to level and the inventory position is measured at the moment of review.

Because inventory is not monitored continuously, a stockout can go unnoticed until the next review, so the system must be protected for the entire T + L window. Since T + L > L, periodic review requires higher safety stock than continuous review for the same service level.

ƒProtection period (periodic review)

\text{Protection Period} = T + L

Where: T is the fixed review interval and L is the replenishment lead time.

ƒSafety stock (periodic review)

\text{Safety Stock} = Z \times \sigma_D \times \sqrt{T + L}

Where: sigma D is the standard deviation of demand per period and Z is the service-level factor.

ƒOrder-up-to level

S = \mu_D \times (T + L) + Z \times \sigma_D \times \sqrt{T + L}

Where: mu D is mean demand per period, so S covers expected demand over the whole protection period plus the safety buffer.

Worked example. Given: daily demand normal (μ = 60, σ = 7), L = 6 days, K = 10 INR, h = 0.5 INR/unit/year, 365 days; the current policy orders 1,200 units at IP = 360.

\mathrm{DDLT} = 360

\sigma_{\mathrm{DDLT}} = 7 \times \sqrt{6} = 17.46

The current ROP of 360 gives

Z = \frac{360 - 360}{17.46} = 0

a 50% cycle service level. For 95% (Z = 1.65):

D = 21{,}900

Q^* = \sqrt{\frac{2 \times 21{,}900 \times 10}{0.5}} = 936 \text{ units}

\text{Safety Stock} = 1.65 \times 17.46 = 28.8

\mathrm{ROP} = 388.8 \approx 388 \text{ units}

Answer: Q* = 936 units with ROP = 388 units for a 95% service level.

Worked example. Given: daily demand (μ = 10, σ = 3), T = 30 days, L = 14 days, 98% service (Z = 2.05), current inventory 150.

S = 10 \times 44 + 2.05 \times 3 \times \sqrt{44} = 440 + 6.15 \times 6.63 \approx 481 \text{ units}

q = 481 - 150 = 331 \text{ units}

Answer: order 331 units this period.

7. Pooling Strategies

Pooling combines demand streams, product variants, or delays commitment so statistical variability partially cancels, reducing safety stock for the same service level.

Strategy	Definition	Examples	Trade-offs
Location Pooling	Consolidate N decentralized sites into centralized hubs	Central warehouse serving multiple regions	Complexity, higher transport costs, slower response
Product Pooling	Consolidate variants into a universal design	O'Neill wetsuits merged surfer and diver SKUs	Reduced differentiation, lost price segmentation
Lead-Time Pooling (Postponement)	Delay final differentiation until demand is realized	Paint tinting at retail, Benetton dyeing after knitting	Needs modular designs, higher up-front costs

ƒDecentralized safety stock

\text{Decentralized safety stock} = N \times Z \times \sigma

Where: N is the number of separate stocking locations, Z the service-level factor, and sigma the demand standard deviation at one location.

ƒSquare root law (centralized safety stock)

\text{Centralized safety stock} = \sqrt{N} \times Z \times \sigma

Centralization cuts safety stock by a factor of √N (independent demands).

Correlation effect: for two locations the pooled variability depends on how their demands move together.

ƒPooled standard deviation with correlation

\sigma_{\text{total}} = \sigma \times \sqrt{2 \times (1 + \rho)}

Where: rho is the correlation between the two locations' demands, so rho = 0 recovers the square root law and rho = 1 removes the pooling benefit entirely.

Positive correlation (demands move together) increases variance and erodes pooling benefits; negative correlation offsets demand and maximizes benefits.

Memory hook: Pooling mnemonic "LPL": Location, Product, Lead-time. Benefits shrink as correlation ρ rises; √N is the ceiling with independent demands.

8. Exam Essentials

Feature	Continuous Review (Q)	Periodic Review (P)
Trigger	Inventory hits ROP	Time interval T elapses
Order Quantity	Fixed (Q)	Variable (q = S - IP)
Protection Period	L	T + L
Safety Stock	Lower	Higher (longer protection)
Tracking Cost	Higher (perpetual system)	Lower (review-time checks)

Feature	Newsvendor	EOQ
Time Horizon	Single period, one-shot	Multi-period cycle
Perishability	High (obsolete after period)	Low (carries forward)
Core Trade-off	Overage (Co) vs. Underage (Cu)	Ordering (K) vs. Holding (h)
Safety Stock Logic	Embedded in critical fractile	Added explicitly to ROP

Must-know: SKU; $\text{Inventory Position} = \text{On-Hand} + \text{On-Order} - \text{Backorders}$ ; $C_o = C - S$ ; $C_u = P - C$ ; critical fractile $\frac{C_u}{C_u + C_o}$ ; cycle service level; postponement; square root law.

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