Inventory control: EOQ and ABC

In short

Inventory control decides how much to order and when to order so that the total of ordering, carrying and shortage costs is least, while material is available when production or a customer needs it. The Economic Order Quantity (EOQ) gives the order size at which annual ordering cost and annual carrying cost balance; ABC analysis classifies items by annual consumption value so that control effort goes where the money is. Inventory is one of the heaviest-weighted topics in GATE PI/ME (EOQ, reorder level, shortage and discount models appear almost every year) and in university Industrial Engineering and Operations Management papers. In jobs, it is the daily work of stores, materials and supply-chain roles at Tata Motors, Maruti, Bajaj, L&T, Hindustan Unilever, Asian Paints, Flipkart and Reliance Retail, and the core of any SAP MM / ERP inventory module.

Explained

Why inventory exists and what it costs

• Functions: decouple successive stages, absorb demand and lead-time variability, allow economic lot sizes and transport loads, hedge against price rise, and smooth seasonal demand.
• Types by stage: raw material, work-in-process (WIP), finished goods, stores and spares, consumables, scrap.
• Types by purpose: cycle (lot-size) stock, safety (buffer) stock, pipeline (in-transit) stock, anticipation stock, decoupling stock, dead stock.
• Cost elements:
  • Ordering / set-up cost (Co) — ₹ per order: requisition, tendering, follow-up, inspection, GRN; for in-house production it is the set-up cost per batch.
  • Carrying / holding cost (Cc or Ch) — ₹ per unit per year: interest on capital (usually the largest part), storage, insurance, handling, obsolescence, deterioration, pilferage. Often quoted as Cc = i × C, where i is the carrying-cost fraction (typically 15–30% per year in Indian industry) and C the unit price.
  • Shortage / stockout cost (Cs) — backorder handling, lost contribution, lost goodwill, line stoppage.
  • Purchase cost — relevant only when price varies with quantity (discount models) or when GST input credit differs.
• Taxes: GST paid on purchases is input tax credit for a registered buyer, so the GST component is not normally included in the unit price used for carrying cost; cess and non-creditable duties are included.

EOQ: assumptions, derivation and sensitivity

• Assumptions of the basic (Wilson/Harris) model: demand is known, constant and continuous; lead time is constant; the entire order arrives in one instalment; no shortages; unit price and the two cost rates are constant; infinite replenishment rate; single item, no interaction.
• Derivation outline: average inventory = Q/2, number of orders = D/Q. Total annual cost TC(Q) = (D/Q)·Co + (Q/2)·Cc. Differentiate: dTC/dQ = −DCo/Q² + Cc/2 = 0 → Q\* = √(2DCo/Cc). Second derivative 2DCo/Q³ > 0, so it is a minimum.
• Key property: at Q\*, annual ordering cost = annual carrying cost = √(2DCoCc)/2 each, and minimum relevant cost = √(2DCoCc). Many exam questions are solved in one line using this.
• Flatness (robustness): TC is flat near the optimum. If the order size is k×Q\*, TC/TC\* = (k + 1/k)/2 — ordering 20% more than EOQ raises relevant cost by only about 1.7%. This justifies rounding EOQ to a pack size, truck load or standard bundle.
• Effect of parameters: Q\* ∝ √D — doubling demand raises EOQ only by 41% (hence inventory grows as the square root of throughput, the basis of warehouse consolidation). Q\* ∝ √Co — halving set-up time (SMED) cuts batch size by 29%, the economic logic behind JIT.
• Limitations: constant demand is unrealistic for seasonal or lumpy items; it ignores budget and space limits, multi-item constraints, perishability, price inflation and supplier capacity; costs Co and Cc are hard to estimate; it assumes independent demand, so dependent (assembly) items are planned by MRP instead.

Reorder level, safety stock and review systems

• Lead time (L): time from raising the order to material being available for use.
• Reorder level (ROL) with certain demand: ROL = d × L (d = demand per day/week).
• With variability: ROL = d̄L + z·σ_L, where safety stock SS = z·σ_L; σ_L = σ_d√L for constant lead time, and σ_L = √(L̄·σ_d² + d̄²·σ_LT²) when lead time also varies (L̄ = mean lead time, σ_LT = standard deviation of lead time).
• Service level and z: 90% → 1.28, 95% → 1.65, 97.5% → 1.96, 99% → 2.33. Service level here is the probability of no stockout during a lead-time cycle (cycle service level), not the fraction of units supplied (fill rate).
• Q-system (continuous review / two-bin / fixed order quantity): order a fixed Q\* whenever stock falls to ROL. Needs perpetual records (now automatic with barcodes/ERP); used for A-class and critical items.
• P-system (periodic review / fixed order interval): review every T and order up to a target level M = d̄(T + L) + z·σ_(T+L). Simpler, allows combined orders from one supplier, but needs more safety stock because the exposure period is T + L. Suited to B and C items.
• Two-bin system: physical version of the Q-system — the second bin holds ROL quantity; when the first empties, reorder. Common for fasteners and consumables in Indian stores.
• s,S (min–max) system: order up to S whenever stock falls below s.
• Lead-time reduction, not safety stock, is the real lever: SS ∝ √L.

Extensions of EOQ examined in exams

• Economic Batch Quantity (EBQ) / finite replenishment rate: production rate p, demand rate d, p > d. Only (1 − d/p) of each unit produced is stocked. Q\* = √[2DCo / (Cc(1 − d/p))] — larger than EOQ, because average inventory is lower.
• Planned shortages (backordering allowed), shortage cost Cs per unit per year: Q\* = √[2DCo/Cc] × √[(Cc + Cs)/Cs] — larger than EOQ; maximum inventory = Q\*·Cs/(Cc + Cs).
• Quantity discounts (all-units): EOQ no longer minimises total cost because purchase cost changes. Procedure: (1) compute EOQ for each price break using that price's Cc; (2) if EOQ is feasible in its own bracket, keep it, else push it up to the lowest quantity in that bracket; (3) compute total cost including purchase cost DC + (D/Q)Co + (Q/2)Cc for each candidate; (4) choose the least. Incremental-discount problems are treated similarly but only the excess quantity gets the lower price.
• Multi-item constraints: with a budget or space limit, use a Lagrange multiplier — Q\* = √[2DCo/(Cc + 2θ·C)] — and adjust θ until the constraint binds.
• Single-period (newsvendor) model for perishables and one-shot buys: stock up to the quantity where P(demand ≤ Q\*) = Cu/(Cu + Co_over), with Cu = underage (shortage) cost and Co_over = overage cost per unit.
• Inflation/price-rise buying and buffer for imported items are handled as judgement adjustments over EOQ, not as the model itself.

ABC analysis and its companion classifications

• Principle (Pareto, 80:20): rank items by annual consumption value = annual usage × unit price, cumulate, and classify.
  • A: roughly top 10–20% of items, about 70–80% of value — tight control, frequent review, low safety stock, EOQ with accurate data, weekly reporting, senior-level approval, vendor development.
  • B: next 20–30% of items, about 15–20% of value — moderate control, periodic review.
  • C: remaining 50–70% of items, about 5–10% of value — simple control: two-bin, bulk orders, annual purchase, large safety stock (cheap insurance).
  • These percentage bands are conventional guidelines, not fixed rules; firms set their own cut-offs.
• Steps: list items → compute annual value → sort descending → cumulate value and item count → compute cumulative % → draw the Pareto/ABC curve → set cut-offs → assign control policies.
• Limitations: value alone ignores criticality (a ₹50 O-ring can stop a line), lead time, shelf life and availability; needs periodic re-classification as prices and usage change.
• Companion schemes (combine with ABC as a matrix — e.g. AV items get the strictest control):
  • VED — Vital, Essential, Desirable: by criticality to operations; standard for spares in Indian PSUs, railways and refineries.
  • FSN / FNSD — Fast, Slow, Non-moving (Dead): by movement rate; used to find obsolete stock.
  • SDE — Scarce, Difficult, Easy to obtain: by procurement difficulty.
  • HML — High, Medium, Low unit price (not annual value); used for security and verification norms.
  • XYZ — by value of stock lying in store on a date, as against ABC's consumption value; an AZ item (high consumption, low stock) is well managed, a CX item is over-stocked.
  • GOLF — Government, Ordinary, Local, Foreign source of supply.
  • SOS — Seasonal / Off-seasonal.
• Physical verification practice: ABC drives cycle counting — A items counted monthly or quarterly, B half-yearly, C once a year (annual physical verification), which also satisfies statutory stock-audit requirements under the Companies Act, 2013 and the auditor's inventory certification.

Performance measures and Indian practice

• Inventory turnover ratio = annual cost of goods sold / average inventory at cost (times per year); days of inventory = 365 / turnover.
• Fill rate = units supplied from stock / units demanded; stockout frequency; inventory to sales ratio; carrying cost as % of turnover.
• Valuation: Indian companies value inventory under Ind AS 2 / AS 2 — lower of cost and net realisable value, with cost by FIFO or weighted average (LIFO is not permitted).
• GST: supplies are taxed at 5%, 18% or 40% since 22 September 2025 (with a nil/exempt category); stock transfers between a company's own GSTINs in different states are taxable supplies, which pushes firms toward fewer, larger warehouses — an EOQ-and-consolidation decision.
• Related tools in practice: SAP MM / Oracle / Tally for records; min–max and reorder-point automation; VMI (vendor-managed inventory) and consignment stock in auto plants around Pune, Chennai and Gurugram; IS 15910-type guidance and company stores manuals govern stores procedure — quote the specific standard only after checking its current number and edition, as several stores and codification standards have been revised or withdrawn.

Key points

• EOQ - order size that minimises ordering + carrying cost: Q\* = √(2DCo/Cc).
• Balance property - at EOQ, annual ordering cost exactly equals annual carrying cost.
• Minimum relevant cost - √(2DCoCc); purchase cost DC is excluded unless price varies with quantity.
• Square-root law - EOQ and hence cycle stock vary as √D and √Co; cost rises as √D too.
• Robustness - TC/TC\* = (k + 1/k)/2 for Q = kQ\*, so moderate deviation from EOQ is cheap.
• EBQ - finite production rate: Q\* = √[2DCo/(Cc(1 − d/p))] > EOQ.
• Backorder model - Q\* = EOQ × √[(Cc + Cs)/Cs]; allowing shortages increases the order size.
• Quantity discount - compare total cost including purchase price at each feasible candidate, not EOQ alone.
• Reorder level - ROL = d̄L + z·σ_L; safety stock = z·σ_L and scales with √L.
• Q-system vs P-system - fixed quantity at a variable time versus variable quantity at a fixed time; P needs more safety stock (exposure T + L).
• Carrying cost fraction - Cc = i × C, with i commonly 15–30% per year, interest being the largest element.
• ABC analysis - classification by annual consumption value; A few items carry most of the value.
• A/B/C bands - about 10–20% / 20–30% / 50–70% of items against roughly 75% / 20% / 5% of value (indicative).
• VED, FSN, SDE, HML, XYZ, GOLF - criticality, movement, availability, unit price, stock value, source.
• Cycle counting - A monthly/quarterly, B half-yearly, C annually, instead of one annual count for everything.
• Inventory turnover - COGS / average inventory; higher turnover means less capital locked, but too high risks stockouts.
• Ind AS 2 - inventory at lower of cost and net realisable value; FIFO or weighted average only, LIFO prohibited.
• EOQ's limits - independent, steady demand only; dependent demand is planned by MRP, and JIT attacks Co instead of accepting it.

Formulas and examples

Example 1 — Basic EOQ with all derived quantities. A Pune auto-component plant uses D = 24,000 bearings/year at C = ₹250 each. Co = ₹450 per order, carrying cost i = 20% per year.

• Cc = 0.20 × 250 = ₹50/unit/year
• Q\* = √(2 × 24,000 × 450 / 50) = √(21,600,000/50) = √432,000 ≈ 657 units (657.27)
• Orders/year = 24,000/657.27 ≈ 36.5; cycle time = 365/36.5 ≈ 10 days
• Ordering cost = 36.5 × 450 ≈ ₹16,432; carrying cost = (657.27/2) × 50 ≈ ₹16,432 — equal, as expected
• Minimum relevant cost = √(2 × 24,000 × 450 × 50) = √1,080,000,000 ≈ ₹32,863/year
• Total cost including purchase = 24,000 × 250 + 32,863 = ₹60,32,863/year
• If the supplier sells only in boxes of 800: k = 800/657.27 = 1.217 → TC ratio = (1.217 + 0.8217)/2 = 1.019, i.e. only 1.9% more (₹33,500). Accept the 800-box.

Example 2 — Economic Batch Quantity (in-house manufacture). A press shop makes a bracket at p = 500/day; the assembly line consumes d = 200/day, 250 working days → D = 50,000/year. Set-up cost ₹2,000 per batch; carrying cost ₹12/unit/year.

• 1 − d/p = 1 − 200/500 = 0.6
• Q\* = √[2 × 50,000 × 2,000 / (12 × 0.6)] = √(200,000,000/7.2) = √27,777,778 ≈ 5,270 units
• Maximum inventory = 5,270 × 0.6 = 3,162 units; average = 1,581
• Production run length = 5,270/500 ≈ 10.5 days; cycle = 5,270/200 ≈ 26.4 days
• Annual cost = (50,000/5,270) × 2,000 + 1,581 × 12 ≈ 18,975 + 18,972 ≈ ₹37,947/year
• Plain EOQ would give √(2 × 50,000 × 2,000/12) ≈ 4,082 units — EBQ is larger, as the model predicts.

Example 3 — Quantity discount (all-units). D = 10,000 units/year, Co = ₹400/order, i = 25% per year. Price: ₹100 for 1–1,999; ₹95 for 2,000–4,999; ₹90 for 5,000 and above.

• Least total cost is at Q = 5,000 units at ₹90, total ₹9,57,050/year — saving about ₹57,000 against the no-discount EOQ, even though carrying cost is eight times higher. This shows why purchase cost must be included in discount problems.

Example 4 — Reorder level, safety stock and service level. Average demand 120 units/day, standard deviation 30 units/day, lead time 9 days (constant), 95% cycle service level.

• σ_L = 30 × √9 = 90 units; z = 1.65 → SS = 1.65 × 90 ≈ 149 units
• ROL = 120 × 9 + 149 = 1,080 + 149 = 1,229 units
• If carrying cost is ₹40/unit/year, safety stock costs 149 × 40 ≈ ₹5,960/year for 95% service.
• Raise service to 99% (z = 2.33): SS = 210 units, ROL = 1,290, cost ≈ ₹8,400 — 41% more safety stock for 4 points of service.
• Cut lead time to 4 days instead: σ_L = 60, SS at 95% = 99 units, ROL = 480 + 99 = 579 — lead-time reduction beats buying service with stock.

Example 5 — ABC classification. Ten stores items (annual usage × unit price), sorted by annual value; total = ₹20,00,000.

• A = 2 items (20%) → 75% of value; B = 3 items (30%) → 21.5%; C = 5 items (50%) → 3.5%.
• Policy: P-4 and P-7 on weekly review, EOQ with the actual Co and Cc, vendor-managed or JIT delivery, monthly cycle count; B items on a periodic (P) system; C items bought once or twice a year on a two-bin system with generous buffer.
• If P-5 (₹2 gasket) is Vital in the VED sense, it becomes a CV item — low value but never allowed to run out, so it is held in large quantity despite being class C.

Where it is used

• Automotive and auto components (Tata Motors, Maruti Suzuki, Bajaj, Bosch, Sundram Fasteners): EOQ/EBQ for fasteners and castings, ABC-VED for spares, milk-run and VMI replenishment feeding line-side bins.
• FMCG and retail (HUL, ITC, Dabur, DMart, Reliance Retail, BigBasket): distributor and warehouse reorder points, fill-rate targets, FSN analysis to clear slow movers before expiry.
• Public sector and utilities (Indian Railways, NTPC, BPCL, SAIL, ONGC): stores and spares management is formally ABC + VED + SDE; insurance spares for critical equipment are stocked outside EOQ logic.
• Hospitals and pharma (AIIMS, Apollo, drug warehouses): ABC-VED matrix is the standard for drug store management; cold-chain and expiry drive single-period and FSN decisions.
• E-commerce and 3PL (Flipkart, Amazon India, Delhivery): SKU classification for slotting and fulfilment-centre placement; square-root law used to justify network consolidation.
• Job roles: materials/stores officer, supply-chain or inventory planner, SAP MM consultant, production planning engineer, procurement analyst; tested in GATE PI/ME, ESE, PSU materials-management interviews and MBA operations placements.

Interview questions

• Q: What exactly does EOQ minimise, and what does it leave out? A: It minimises annual ordering plus carrying cost; purchase cost is left out because it is DC regardless of order size — unless price depends on quantity, when it must be included.
• Q: At EOQ, what is the relation between ordering and carrying cost? A: They are exactly equal, each √(2DCoCc)/2, so the total relevant cost is √(2DCoCc) — a quick check on any EOQ answer.
• Q: If annual demand quadruples, what happens to EOQ and to total relevant cost? A: Both double, since each varies as √D — inventory does not grow in proportion to sales.
• Q: Why is EBQ larger than EOQ? A: Because stock builds up only at rate (p − d) during the run, so average inventory per unit ordered is lower and a bigger batch is affordable.
• Q: Your supplier insists on a minimum order 30% above EOQ. How much does that cost? A: TC ratio = (1.3 + 1/1.3)/2 ≈ 1.035, so about 3.5% extra on ordering plus carrying cost — usually worth accepting for a price or freight concession.
• Q: How does JIT square with EOQ? A: JIT does not reject EOQ; it attacks the set-up/ordering cost Co (SMED, local suppliers, blanket orders), and since Q\* ∝ √Co, small lots become economic.
• Q: Difference between ABC and VED, and why use both? A: ABC ranks by annual consumption value, VED by criticality to operations; a cheap but vital gasket is C by ABC and V by VED, so the matrix prevents a low-value item from stopping a line.
• Q: How do you decide safety stock? A: From demand and lead-time variability and a chosen service level: SS = z·σ_L; cost it against the shortage cost rather than fixing it as "one month's usage".
• Q: When would you use a periodic review system over a reorder-point system? A: When many items come from one supplier and can be ordered together, when transactions are not recorded continuously, or for low-value C items where simplicity matters more than optimality.
• Q: Which inventory valuation methods are allowed in India? A: Under Ind AS 2 / AS 2, inventory is at lower of cost and net realisable value, with cost by FIFO or weighted average; LIFO is not permitted.
• Q: A plant reports inventory turnover of 4 and a competitor 12. Is 12 always better? A: Usually it means less locked capital, but very high turnover with poor fill rate or frequent expediting shows understocking — read turnover together with service level and stockout cost.
• Q: Why does EOQ fail for an assembly component? A: Its demand is dependent and lumpy, driven by the parent's production schedule, so lot sizing is done inside MRP against the master schedule rather than by a constant-demand formula.

Common mistakes

• Mixing units: monthly demand with annual carrying cost, or forgetting that Cc = i × C must be per unit per year.
• Including purchase cost DC in a plain EOQ comparison (harmless but confusing), yet omitting it in a quantity-discount problem — where it decides the answer.
• In discount problems, taking the lowest-price EOQ even when it is below that bracket's minimum quantity, instead of pushing Q up to the break point.
• Using √(2DCo/Cc) for a finite production rate, or forgetting the (1 − d/p) factor in EBQ and in maximum-inventory calculations.
• Setting reorder level = d·L only, with no safety stock, then blaming the formula for stockouts; or computing σ_L as σ_d·L instead of σ_d·√L.
• Classifying ABC by unit price (that is HML) or by stock on hand (that is XYZ) instead of annual consumption value.
• Treating A/B/C as fixed at 10/20/70% of items and never re-classifying after prices, usage or product mix change.

Related topics

MRP and ERP in manufacturing · Forecasting and aggregate planning · Production planning and control · Operations research basics · Supply chain and logistics management · Lean manufacturing and JIT