10.3 EOQ, Reorder Point, and Safety Stock

Economic order quantity (EOQ), reorder point (ROP), and safety stock form the quantitative core of independent-demand inventory planning. Domain VI expects you to compute EOQ, interpret what it assumes, set ROP from lead-time demand plus safety stock, and reason about service levels — including how quantity discounts change the replenishment decision.

EOQ — The Basic Model

Assumptions (classical): steady known demand, constant lead time, instantaneous replenishment (or lead time handled separately via ROP), fixed order cost S, holding cost H per unit per year, no quantity discounts, no stockouts planned.

EOQ = Q = √(2DS / H)*

where:

• D = annual demand (units/year)
• S = cost per order (or setup)
• H = annual holding cost per unit (often i × C)

Worked example 1 — full EOQ

A distributor sells 10,000 widgets/year. Each order costs $32. Unit cost is $20; carrying rate is 25%/year.

1. Compute H: H = 0.25 × 20 = $5/unit/year
2. EOQ: Q = √(2 × 10,000 × 32 / 5) = √(128,000) ≈ 357.8* → round to 358 (or policy-friendly 360)
3. Orders/year ≈ 10,000 / 358 ≈ 27.9
4. Annual ordering cost ≈ 27.9 × 32 ≈ $893
5. Average cycle inventory ≈ 358 / 2 = 179
6. Annual carrying cost ≈ 179 × 5 ≈ $895

Ordering ≈ carrying at the optimum — a useful exam check.

Time between orders ≈ 365 / 27.9 ≈ 13.1 days.

Worked example 2 — sensitivity

Using the same D and S, suppose unit cost rises to $32 (still 25% carrying):

• H = 0.25 × 32 = $8
• Q = √(2 × 10,000 × 32 / 8) = √(80,000) ≈ 283*

Higher holding cost → smaller EOQ. If order cost rises to $72 with original H = 5:

• Q = √(2 × 10,000 × 72 / 5) = √(288,000) ≈ 537*

Higher order cost → larger EOQ. Remember the square-root dampening: quadrupling demand does not quadruple EOQ — it doubles EOQ.

Average inventory and total relevant cost

TRC(Q) = (D / Q) × S + (Q / 2) × H

At Q*, TRC is minimized. For example 1:

• TRC ≈ 893 + 895 ≈ $1,788/year (excluding purchase cost D × C, which is fixed without discounts)

Reorder Point (Continuous Review)

When inventory position hits the reorder point, place an order for Q (often EOQ).

ROP = dL + SS

where:

• d = demand rate during lead time (same units)
• L = lead time
• SS = safety stock

If demand is daily and lead time in days:

dL = d × L

Worked example — ROP without safety stock

Average demand = 40 units/day; lead time = 7 days; SS = 0.

• ROP = 40 × 7 = 280 units

With SS = 65:

• ROP = 280 + 65 = 345 units

Inventory position reminder

Inventory position = on-hand + on-order − backorders (allocations may also be subtracted, depending on system rules). ROP is compared to position, not merely to on-hand — otherwise you double-order while a replenishment is already inbound.

Safety Stock and Service Level

Safety stock buffers uncertainty in demand and/or lead time. A common normal-demand model for continuous review:

SS = z × σ_L

where z is the service factor from the standard normal distribution and σ_L is the standard deviation of demand over lead time.

If demand periods are independent and lead time is constant:

σ_L = σ_d × √L

Worked example — safety stock

Daily demand σ = 12 units; L = 9 days; target cycle service level ≈ 95% (z ≈ 1.65).

• σ_L = 12 × √9 = 12 × 3 = 36
• SS = 1.65 × 36 ≈ 59.4 → 59 units

If average demand is 50/day:

• ROP = 50 × 9 + 59 = 450 + 59 = 509 units

Service level vocabulary

Illustrative z values (approximate): 84% → 1.00; 90% → 1.28; 95% → 1.65; 97.5% → 1.96; 99% → 2.33. You will not always be asked for a z table lookup, but you must know directionality: raising service level raises safety stock nonlinearly in the tail.

Lead-time uncertainty (concept)

If lead time also varies, demand-during-lead-time variance increases further (combined demand and lead-time uncertainty). Exam takeaway: longer and more variable lead times inflate SS; shrinking mean and variance of lead time is often cheaper than stacking warehouse inventory forever.

Quantity Discounts — Overview

Suppliers may offer price breaks (all-units or incremental). For all-units discounts, unit cost C falls when order quantity reaches a breakpoint, which also changes H = iC.

Procedure (exam-friendly):

1. Compute EOQ at each applicable price (using that price's H).
2. If EOQ is feasible in that price's quantity range, it is a candidate.
3. If EOQ is below the range, use the minimum quantity needed to get that price as a candidate.
4. Compute total annual cost = purchase + ordering + holding for each candidate; choose the minimum.

TAC = (D × C) + (D / Q) × S + (Q / 2) × (i × C)

Worked discount comparison

D = 8,000/year; S = $100; i = 20%.

At $10: H = 0.2 × 10 = $2; EOQ = √(2×8000×100/2) = √800,000 ≈ 894 (feasible in 1–999).

• TAC₁₀ = 8,000×10 + (8,000/894)×100 + (894/2)×2 ≈ 80,000 + 895 + 894 ≈ $81,789

At $9.60: H = 1.92; EOQ = √(2×8000×100/1.92) ≈ √833,333 ≈ 913 — not feasible (needs ≥1,000). Use Q = 1,000.

• TAC₉.₆₀ = 8,000×9.60 + (8,000/1,000)×100 + (1,000/2)×1.92 = 76,800 + 800 + 960 = $78,560

At $9.25: H = 1.85; EOQ = √(2×8000×100/1.85) ≈ √864,865 ≈ 930 — not feasible. Use Q = 3,000.

• TAC₉.₂₅ = 8,000×9.25 + (8,000/3,000)×100 + (3,000/2)×1.85 = 74,000 + 267 + 2,775 = $77,042

Best among these candidates: order 3,000 at $9.25 (lowest TAC), even though EOQ at that price is far below 3,000. Blindly using the unrestricted EOQ at the lowest price would be invalid; blindly taking the lowest price without TAC can also fail when holding cost explodes — here the discount still wins, but you prove it with total cost.

Periodic Review Contrast (Brief)

Continuous review triggers on ROP. Periodic review orders up to a target level every T periods. Order-up-to level covers demand over T + L plus safety stock for that protection interval. Longer review intervals need more safety stock — another Domain VI tradeoff between administrative simplicity and inventory investment.

Exam Focus

Carry a calculator mindset: compute H, then EOQ, then verify ordering ≈ holding. For ROP, separate known lead-time demand from safety stock. For discounts, evaluate total annual cost at feasible quantities, not unit price alone. Those three habits unlock most quantitative inventory items on CPIM.