Inventory Control & EOQ Models

Inventory control (EOQ) is a high-yield GATE ME topic (~2 marks). The economic-order-quantity formula and its total-cost/reorder logic are heavily and predictably tested.

Basic EOQ model

Balances ordering cost (per order) against holding/carrying cost (per unit per year):

• EOQ: Q* = √(2·D·S/H), where D = annual demand, S = ordering cost per order, H = holding cost per unit per year.
• At the EOQ, ordering cost = holding cost (they are equal at the optimum). Minimum total cost (excluding purchase) = √(2·D·S·H).
• Number of orders = D/Q*; cycle time = Q*/D. Reorder point ROP = d·L (demand rate × lead time); with variability add safety stock.

Extensions

• EOQ is robust: total cost is flat near Q*, so small errors in Q barely raise cost.
• Production/EPQ model (finite replenishment rate p): Q* = √(2DS/H) × √(p/(p − d)) — larger than EOQ because stock builds while producing.
• Quantity-discount model: compare total cost (including purchase price) at the EOQ and at each price-break quantity.
• ABC analysis: classify items by annual value (A = few, high value → tight control).

Exam Tricks & Tips

• 🎯 EOQ = √(2DS/H) — the single most-tested IE formula; watch units (S per order, H per unit per year).
• 🎯 At EOQ, ordering cost = holding cost — a fast check and a common conceptual MCQ.
• 🎯 Minimum total (inventory) cost = √(2DSH) — derived by substituting EOQ back.
• 🎯 EPQ > EOQ because stock accumulates during production (factor √(p/(p−d))).
• 🎯 Reorder point ROP = demand × lead time (+ safety stock) — don't forget the safety stock under demand variability.
• ❌ Common mistake: mixing per-order and per-unit costs, or annual vs per-period demand — EOQ needs D annual, S per order, H per unit per year, all consistent.

Expected exam pattern

A 1-mark EOQ, number-of-orders, or ROP NAT, and a 2-mark total-cost, EPQ, or quantity-discount problem. The "ordering = holding at EOQ" fact and the √(2DSH) minimum cost are frequent points.

Quick recap

EOQ Q* = √(2DS/H); at optimum ordering cost = holding cost; min total cost = √(2DSH). Orders = D/Q*, cycle = Q*/D, ROP = d·L (+ safety stock). EPQ = EOQ·√(p/(p−d)) (finite production). Quantity discount: compare total cost at breaks. ABC by annual value.

Cover the answer, recall, then check. 11 cards on inventory control for GATE ME.

Q1. State the basic EOQ formula.
A1. Q* = √(2·D·S/H), where D = annual demand, S = ordering cost/order, H = holding cost/unit/year.

Q2. What is true about the two cost components at the EOQ?
A2. Annual ordering cost equals annual holding cost — they are balanced at the optimum quantity.

Q3. Minimum total inventory cost (excluding purchase)?
A3. √(2·D·S·H), obtained by substituting the EOQ back into the total-cost expression.

Q4. Number of orders per year and cycle time at EOQ?
A4. Number of orders = D/Q*; cycle time = Q*/D.

Q5. Define the reorder point.
A5. ROP = demand rate × lead time (d·L), plus safety stock when demand/lead time varies.

Q6. How does the EPQ (production) model differ from EOQ?
A6. Q* = √(2DS/H)·√(p/(p − d)) — larger than EOQ, because inventory builds up during finite-rate production.

Q7. Why is EOQ described as robust?
A7. The total-cost curve is flat near the optimum, so moderate errors in Q raise cost only slightly.

Q8. How is a quantity-discount decision made?
A8. Compare total annual cost (ordering + holding + purchase) at the EOQ and at each price-break quantity; pick the lowest.

Q9. What is ABC analysis?
A9. Classifying inventory by annual value: A (few items, high value, tight control), B (moderate), C (many, low value, loose control).

Q10. What is safety stock for?
A10. A buffer against demand and lead-time variability, preventing stockouts between reorder and receipt.

Q11. Effect of doubling annual demand on EOQ?
A11. EOQ rises by √2 (≈ 41%), since Q* ∝ √D.

Holding too much inventory ties up cash; too little risks stockouts. Inventory models find the sweet spot. GATE reliably tests the Economic Order Quantity (EOQ) and its variants — among the most formula-clean, high-scoring Industrial Engineering topics.

Core concept: the EOQ balances ordering cost (favours large, infrequent orders) against holding cost (favours small, frequent orders) to minimise total inventory cost.

Deep explanation

Beginner — the basic EOQ

With annual demand D, ordering cost per order C_o, and holding cost per unit per year C_h:
EOQ Q* = √(2 D C_o/C_h).
At the EOQ, ordering cost = holding cost (the two curves cross at the minimum). Total minimum cost (excluding purchase) = √(2 D C_o C_h). Number of orders = D/Q*; cycle time = Q*/D.

Intermediate — reorder point and safety stock

• Reorder point (ROP) = demand during lead time = d × L (d = demand rate, L = lead time). Order when stock hits ROP.
• Safety stock buffers demand/lead-time variability: ROP = d̄L + z·σ_L, where z is the service-level factor. Higher service level → more safety stock.
• EOQ is robust: being off the optimum order size by ±20% raises total cost only slightly (a flat minimum).

Advanced — EOQ variants

• Production/EPQ model (finite replenishment rate p): Q* = √(2 D C_o/C_h) × √(p/(p − d)) — larger than EOQ because stock builds gradually.
• Quantity discounts: compute EOQ, then check discount price breaks; the optimum may be a break quantity even if it exceeds EOQ (compare total costs including purchase).
• Backorder model: allowing planned shortages (with shortage cost) increases the optimal order quantity.
• ABC analysis prioritises control: A-items (high value, tight control), B, C (low value, loose control).

Worked example

Annual demand D = 10,000 units, ordering cost C_o = 50 per order, holding cost C_h = 4 per unit per year. Find the EOQ and the number of orders per year.
Q* = √(2 D C_o/C_h) = √(2 × 10,000 × 50/4) = √(1,000,000/4) = √250,000 = 500 units.
Number of orders = D/Q* = 10,000/500 = 20 orders per year. (Check: ordering cost = 20×50 = 1000; holding = (500/2)×4 = 1000 — equal, confirming the EOQ.)

GATE relevance

EOQ (√(2DC_o/C_h)), the ordering = holding cost condition, reorder point/safety stock, and the EPQ and quantity-discount variants are staple Industrial Engineering questions. The EOQ formula and its properties (flat minimum, equal costs) are must-knows.

Exam tricks & shortcuts

• EOQ = √(2DC_o/C_h); at the optimum, annual ordering cost = annual holding cost — a quick check.
• Total minimum cost (excl. purchase) = √(2 D C_o C_h).
• EPQ > EOQ by the factor √(p/(p−d)) (finite production rate).
• Mnemonic: "Root of two-D-C-o over C-h."

Key formulas

• Economic order quantity: EOQ = √(2DS/H) (D = annual demand, S = ordering cost, H = holding cost/unit/yr).
• Total cost: TC = (D/Q)S + (Q/2)H + DC.
• Number of orders: D/EOQ; cycle time = EOQ/D.
• Reorder point: ROP = d·L (+ safety stock).
• EOQ with shortages / quantity discounts adjust the base model.

EOQ minimises total inventory cost by balancing ordering and holding costs; reorder point covers lead-time demand.

Worked Example

Problem: A product has an annual demand of 10000 units. Each order costs ₹100 to place, and the holding cost is ₹2 per unit per year. Find the economic order quantity (EOQ), the number of orders placed per year, and the total annual inventory cost (ordering + holding).

Solution:
Economic order quantity (Wilson formula):
EOQ = √(2·D·C_o/C_h) = √(2 × 10000 × 100/2) = √(2000000/2) = √1000000 = 1000 units.

Number of orders per year:
N = D/EOQ = 10000/1000 = 10 orders/year.

Total inventory cost = ordering cost + holding cost:
Ordering = (D/Q)·C_o = 10 × 100 = ₹1000.
Holding = (Q/2)·C_h = (1000/2) × 2 = ₹1000.
Total = 1000 + 1000 = ₹2000 per year.

Note that at the EOQ the ordering and holding costs are equal — a defining property of the optimum.

Answer: EOQ = 1000 units, 10 orders/year, total inventory cost = ₹2000/year.