7.3 Inventory Models, Economic Order Quantity (EOQ) & Safety Stock
Key Takeaways
- The classic Economic Order Quantity (EOQ) formula (EOQ = sqrt((2 * D * S) / H)) identifies the exact order lot size that minimizes the sum of annual ordering costs and annual inventory holding costs.
- Total Annual Inventory Cost comprises Ordering Costs ((D / Q) * S), Holding Costs ((Q / 2) * H), and Acquisition Spend (D * C); at optimal EOQ, annual ordering costs exactly equal annual holding costs.
- Evaluating supplier quantity discounts requires calculating EOQ across price tiers, verifying feasibility within volume break thresholds, and comparing Total Annual Costs (including raw purchase spend).
- Continuous Review (Q, R) systems monitor inventory continuously and order a fixed Q whenever the reorder point R is breached, protecting against stockouts during Lead Time (L), whereas Periodic Review (P, T) systems order variable quantities up to target T at fixed intervals, protecting across P + L.
- Reorder Points (ROP) incorporate buffer Safety Stock to absorb demand and lead time variability: ROP = (d * L) + SS, where SS = Z * sigma_d * sqrt(L) for variable demand and constant lead time.
7.3 Inventory Models, Economic Order Quantity (EOQ) & Safety Stock
Inventory is both an indispensable operational shock absorber and one of the largest capital investments on an enterprise balance sheet. Holding insufficient inventory triggers line stoppages, emergency freight premiums, lost customer goodwill, and lost sales; conversely, holding excessive inventory drains working capital, incurs severe carrying costs, and exposes the organization to obsolescence and spoilage.
Supply management professionals must deploy rigorous mathematical models to balance the economic trade-offs between ordering frequency, carrying costs, batch sizes, and stockout risk protection.
1. Strategic Functions and Cost Structures of Inventory
The Strategic Roles of Inventory:
- Cycle Stock (Lot Size Inventory): Inventory resulting from purchasing or manufacturing in batches rather than unit-by-unit, capturing economies of scale in ordering, transport, and production setup.
- Safety Stock (Buffer Inventory): Surplus inventory held to protect against stochastic variability in customer demand rate and supplier delivery lead time.
- Pipeline / Transit Inventory: Inventory currently in transit between supply chain nodes (I_transit = d * L_transit).
- Anticipation / Pre-Build Stock: Inventory accumulated in advance of known seasonal demand surges, plant shutdowns, or anticipated supplier price increases.
- Decoupling Inventory: Buffers placed between sequential manufacturing workstations to prevent an upstream machine breakdown from halting the entire assembly line.
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| THE THREE CORE INVENTORY COSTS |
| |
| [1. UNIT ACQUISITION COST (C)] |
| - Direct purchase price paid to supplier or direct manufacturing cost |
| - Annual Acquisition Spend = D × C |
| │ |
| [2. ORDERING / SETUP COST (S)] │ |
| - Fixed administrative expense incurred per purchase order placed |
| - Sourcing, PO generation, dock receiving, QA inspection, invoice match |
| - In manufacturing: machine teardown, re-tooling, calibration scrap |
| - Annual Ordering Cost = (D / Q) × S |
| │ |
| [3. HOLDING / CARRYING COST (H)] │ |
| - Annual cost to carry one unit in stock for one full year (H = i × C) |
| - Four Primary Components: |
| 1. Capital Cost (WACC / Opportunity Cost of money): 10% - 20% |
| 2. Storage Space Cost (Rent, utilities, warehouse labor): 3% - 8% |
| 3. Inventory Service Cost (Insurance, property taxes): 1% - 3% |
| 4. Inventory Risk Cost (Obsolescence, shrinkage, damage): 3% - 10% |
| - Typical Total Annual Carrying Rate (i): 15% to 35% of unit value |
| - Annual Holding Cost = (Q / 2) × H |
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2. The Classic Wilson Economic Order Quantity (EOQ) Model
Originally formulated by Ford Whitman Harris (1913) and popularized by R.H. Wilson, the Economic Order Quantity (EOQ) model determines the optimal order batch size (Q*) that minimizes the total annual cost of ordering and holding inventory.
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| THE EOQ COST TRADE-OFF CURVE |
| |
| Annual Cost ($) |
| ▲ |
| │ / Total Cost (TC) |
| │ | /─── |
| │ | * Minimum Total Cost / |
| │ | ▼ / Annual Holding Cost |
| │ | *──────────────────* (Q / 2) * H |
| │ | / / |
| │ | / / |
| │ | / / |
| │ | / / |
| │ | / / |
| │ | / / |
| │ | / / |
| │ | // |
| │ | *◄────── Parity Point: |
| │ | / | (Annual Ordering = Annual Holding) |
| │ / | | |
| │ / | | Annual Ordering Cost |
| │ / | | (D / Q) * S |
| │ / | | |
| +──────────────────/─────────│──────│──────────────────────────────► |
| 0 Q* (EOQ) Order Size|
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Fundamental Assumptions of the Basic EOQ Model:
- Demand rate (D) is continuous, constant, known, and independent.
- Lead time (L) is constant, known, and fixed.
- The entire order quantity (Q) arrives in a single batch (instantaneous receipt).
- Unit purchase price (C) is fixed (no volume discounts).
- Ordering cost (S) and holding cost rate (i) are constant and known.
- No stockouts, backorders, or shortages are permitted.
Mathematical Formulation & Derivation:
Total Annual Inventory Cost (TC):
TC = Annual Ordering Cost + Annual Holding Cost + Annual Purchase Cost
TC = ((D / Q) * S) + ((Q / 2) * H) + (D * C)
To find the minimizing quantity Q*, take the first derivative with respect to Q and set to zero:
d(TC)/dQ = -(D * S) / Q^2 + H / 2 = 0
(D * S) / Q^2 = H / 2 ===> Q^2 = (2 * D * S) / H
EOQ = Q* = sqrt((2 * D * S) / H) = sqrt((2 * D * S) / (i * C))
Where:
- D = Annual demand (units/year)
- S = Fixed cost per order ($/order)
- H = Annual holding cost per unit ($/unit/year) = i * C
- i = Annual inventory carrying charge (%/year)
- C = Unit purchase price ($/unit)
- Q = Order quantity (units/lot)
[!IMPORTANT] The EOQ Parity Theorem: At the exact EOQ (Q*), Annual Ordering Cost equals Annual Holding Cost: (D / Q*) * S = (Q* / 2) * H. If a supply manager discovers that annual ordering costs exceed holding costs, order size Q is currently smaller than EOQ; if holding costs exceed ordering costs, order size Q is larger than EOQ.
3. Quantity Discount Evaluation Procedure
Suppliers frequently offer tiered price breaks to incentivize large order volumes. Evaluating quantity discounts requires balancing unit price savings against higher inventory holding costs.
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| QUANTITY DISCOUNT EVALUATION ALGORITHM |
| |
| [Step 1: Calculate EOQ for Lowest Unit Price (Highest Discount Bracket)] |
| │ |
| v |
| [Step 2: Feasibility Check] |
| Is calculated EOQ ≥ Minimum Quantity required for that discount? |
| │ │ |
| YES NO |
| │ │ |
| v v |
| [Optimal Quantity Found] [Step 3: Test Higher Brackets] |
| Calculate Total Cost (TC) at EOQ Set Q = Price Break Minimum |
| This is the lowest cost solution. Calculate TC at Break Quantity |
| and compare with feasible EOQ |
| at higher price tiers |
| │ |
| v |
| [Step 4: Select Minimum TC] |
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Total Cost Comparison Formula for Discounts:
TC(Q) = ((D / Q) * S) + ((Q / 2) * i * C) + (D * C)
Because unit price C varies across tiers, the annual purchase acquisition spend (D * C) changes and MUST be included in the total cost comparison.
4. Continuous Review (Q, R) vs. Periodic Review (P, T) Systems
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| CONTINUOUS REVIEW VS. PERIODIC REVIEW |
| |
| CONTINUOUS REVIEW (Q, R SYSTEM) PERIODIC REVIEW (P, T SYSTEM) |
| - Fixed Order Quantity (Q) - Fixed Review Interval (P) |
| - Variable Order Timing - Variable Order Quantity (Q) |
| - Continuously monitors Inventory - Audits inventory every P days |
| Position: IP = (OH + OO - Backorders) (e.g., every 2 weeks) |
| - Order Q triggered when IP ≤ ROP - Orders Q = Target T - IP |
| - Protection Horizon: Lead Time (L) - Protection Horizon: (P + L) |
| - Requires lower safety stock - Requires higher safety stock |
| - Best for Class A high-value items - Best for Class C cheap items |
| and automated ERP/RFID systems and multi-item joint freight |
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| CONTINUOUS REVIEW (Q, R) INVENTORY POSITION CYCLE |
| |
| Inventory Position |
| ▲ |
| │ Order Arrives (Q) |
| │ *───────────* |
| │ │/ / |
| │ │ / / |
| ROP ┼───┼──*───────────*─────────────────── Reorder Point |
| │ │ / Lead / Lead |
| │ │ / Time (L) / Time (L) |
| SS ┼───┼─────*───────────*──────────────── Safety Stock |
| │ │ │ │ |
| +───┴─────┴───────────┴────────────────► |
| 0 Order Placed Order Placed Time |
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5. Reorder Point (ROP) & Safety Stock (SS) Mathematics
The Reorder Point represents the inventory level that triggers a replenishment order of size Q.
1. Constant Demand Rate and Constant Lead Time:
When daily demand (d) and lead time (L) are perfectly deterministic:
ROP = d * L
2. Variable Demand Rate and Constant Lead Time:
When daily demand fluctuates with standard deviation sigma_d, but lead time L is constant:
ROP = (d * L) + SS
SS = Z * sigma_L = Z * sigma_d * sqrt(L)
Where:
- d = Average daily demand
- L = Supplier lead time (in days)
- Z = Standard normal distribution score corresponding to the desired cycle service level
- sigma_d = Standard deviation of daily demand
- sigma_L = sigma_d * sqrt(L) = Standard deviation of demand during lead time
3. Variable Lead Time and Constant Demand Rate:
When daily demand d is constant, but supplier lead time varies with standard deviation sigma_L:
SS = Z * d * sigma_L
4. Variable Demand Rate AND Variable Lead Time (Combined Stochastic Uncertainty):
When both demand and lead time are independent random variables:
SS = Z * sqrt((L * sigma_d^2) + (d^2 * sigma_L^2))
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| STANDARD SERVICE LEVELS & Z-SCORES (CPSM) |
| |
| CYCLE SERVICE LEVEL (%) Z-SCORE (NORMAL VARIATE) |
| ----------------------- ------------------------ |
| 90.0% 1.28 |
| 95.0% 1.645 |
| 97.5% 1.96 |
| 98.0% 2.05 |
| 99.0% 2.33 |
| 99.9% 3.09 |
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[!NOTE] The Exponential Law of Service Level Costs: Increasing customer service level from 90% (Z = 1.28) to 95% (Z = 1.645) requires a 28.5% increase in safety stock. However, increasing service level from 95% to 99.9% (Z = 3.09) requires an 87.8% increase in safety stock. Supply managers must weigh the carrying cost of extreme buffers against the commercial cost of a stockout.
6. Comprehensive Worked Numerical Application
Sourcing & Inventory Scenario:
Titan Precision Bearings manages the procurement of specialized ceramic ball bearings (Part #CB-440) for high-speed industrial turbines. Sourcing and inventory data have been audited as follows:
- Annual Demand (D): 12,000 units/year
- Operating Year: 250 business days/year
- Average Daily Demand (d): 12,000 / 250 = 48 units/day
- Standard Deviation of Daily Demand (sigma_d): 10 units/day
- Supplier Delivery Lead Time (L): 9 business days (Constant)
- Order / Setup Cost (S): $150.00 per order
- Unit Purchase Price (C): $50.00 per unit
- Annual Inventory Carrying Rate (i): 20% per year
- Desired Cycle Service Level: 95% (Z = 1.645)
Step 1: Calculate Annual Unit Holding Cost (H)
H = i * C = 0.20 * $50.00 = $10.00 per unit per year
Step 2: Calculate Economic Order Quantity (EOQ)
EOQ = sqrt((2 * D * S) / H) = sqrt((2 * 12,000 * 150) / 10) = sqrt(3,600,000 / 10) = sqrt(360,000) = 600 units
Step 3: Calculate Annual Inventory Operating Costs at EOQ
-
Annual Ordering Cost: Annual Ordering Cost = (D / Q) * S = (12,000 / 600) * $150 = 20 orders/year * $150 = $3,000.00
-
Annual Holding Cost (Cycle Stock): Annual Holding Cost = (Q / 2) * H = (600 / 2) * $10 = 300 units * $10 = $3,000.00
-
Total Annual Inventory Management Cost (TC_inventory): TC_inventory = Ordering Cost + Holding Cost = $3,000 + $3,000 = $6,000.00
(Notice that at EOQ, Annual Ordering Cost $3,000 exactly equals Annual Holding Cost $3,000, validating the Wilson parity theorem).
-
Total Annual Acquisition Spend: Annual Spend = D * C = 12,000 * $50 = $600,000.00
-
Total Annual System Cost: TC_total = $6,000 + $600,000 = $606,000.00
Step 4: Calculate Safety Stock (SS)
With variable demand and constant lead time (L = 9 days, sigma_d = 10, Z = 1.645):
SS = Z * sigma_d * sqrt(L)
SS = 1.645 * 10 * sqrt(9) = 1.645 * 10 * 3 = 1.645 * 30 = 49.35 ≈ 50 units
Annual Holding Cost of Safety Stock: SS * H = 50 * $10 = $500.00 / year.
Step 5: Calculate Reorder Point (ROP)
ROP = (d * L) + SS
ROP = (48 units/day * 9 days) + 50 units = 432 + 50 = 482 units
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| SUMMARY OF OPERATIONAL POLICY |
| |
| PARAMETER CALCULATED VALUE OPERATIONAL DIRECTIVE |
| -------------------------- ------------------ ------------------------ |
| Order Quantity (EOQ) 600 units Order in lots of 600 |
| Orders Placed per Year 20 orders Place an order every 12.5 days
| Reorder Point (ROP) 482 units Trigger order when IP ≤ 482
| Safety Stock Buffer 50 units Protect against 95% surges|
| Total Inventory Policy Cost $6,500 / year $3k order + $3.5k holding |
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A supply manager calculates the Economic Order Quantity (EOQ) for an industrial valve as 800 units based on annual demand of 16,000 units, order setup costs of $200 per order, and annual holding costs of $10 per unit. When auditing current accounting operations, the manager discovers that the purchasing department has actually been ordering in lots of 400 units. Which of the following cost conditions currently exists within the enterprise?
A component distributor offers a manufacturing buyer a price discount schedule: $20.00 per unit for orders under 1,000 units, and $18.00 per unit for orders of 1,000 units or more. Annual demand is 5,000 units, order cost is $45 per order, and the annual inventory holding rate is 20% of unit price. What is the optimal order quantity?
A distribution center experiences daily demand for a medical supply component that is normally distributed with an average daily demand of 100 units and a standard deviation of 15 units. Supplier delivery lead time is constant at 16 days. If executive management establishes a 98% cycle service level target (Z = 2.05), what is the required safety stock and reorder point (ROP)?