6 Economic Order Quantity (EOQ) Model
Keywords
Inventory Management, Economic Order Quantity, Supply Chain, Sawtooth Model, Condition of Certainty
Inventory is money sitting around in a different form.—Rhonda Abrams
6.1 Introduction
Picture a simple and predictable world: a single product, constant demand like clockwork, flawless order arrivals, and no stockouts. In this scenario, two costs compete:
- Ordering Cost: A fixed cost incurred each time an order is placed. Reducing the number of orders minimizes this cost.
- Holding Cost: The cost associated with storing and maintaining inventory. Lower average inventory levels reduce this cost.
Striking the right balance is key: ordering too frequently leads to high setup costs, while ordering too infrequently results in excessive holding costs. The Economic Order Quantity (EOQ) identifies the optimal order quantity that minimizes the total cost of inventory management.
This concept and its closed-form solution were first introduced in a seminal 1913 paper by Harris (1913). Subsequently, Taft (1918) expanded the model to production settings, leading to the Economic Production Quantity (EPQ) model. For a detailed historical perspective, refer to Erlenkotter (1990).
Problem Statement: We aim to determine the optimal order quantity \(Q\) for a single item that minimizes the total cost, which is the sum of ordering and holding costs over a planning horizon.
Assumptions:
- Demand occurs at a continuous, constant, and known rate, ensuring predictability in inventory needs.
- Replenishment or lead time is constant and known, allowing precise planning for order arrivals.
- Stockouts or safety stocks are not considered, assuming perfect alignment between supply and demand.
- All demand is fully satisfied without backorders, ensuring no unmet customer needs.
- The price or cost per unit remains constant and does not depend on the order quantity, excluding quantity discounts from the analysis.
- Inventory in transit is not considered, focusing solely on inventory held at the facility.
- Goods are purchased on a delivered price basis, with transportation costs included in the purchase price.
- The model focuses on a single inventory item, avoiding interactions or dependencies between multiple items.
- The planning horizon is infinite, assuming a long-term perspective for inventory decisions.
- Capital availability is unlimited, removing financial constraints from the decision-making process.
6.2 EOQ Formulation
Table 6.1 summarizes the key variables used in the EOQ model.
| Symbol | Description |
|---|---|
| \(R\) | Annual rate of demand (units) |
| \(Q\) | Quantity ordered (units) |
| \(\frac{1}{2}Q\) | Avg. number of units during order cycle |
| \(A\) | Cost of placing an order |
| \(V\) | Cost of one unit of inventory |
| \(W\) | Carrying cost/unit of inventory/year (%) |
| \(S\) | = \(VW\). Inventory carrying cost per unit per year |
| \(t\) | Order cycly time (days) |
| \(TAC\) | Total annual cost |
Considering a planning horizon of one year, the average inventory level is \(\frac{Q}{2}\) units, leading to an annual holding cost of \(\frac{1}{2}QVW\). The number of orders placed annually is \(\frac{R}{Q}\), resulting in an annual ordering cost of \(A\frac{R}{Q}\).
Hence, the total annual cost (TAC) is:
\[ TAC = \overbrace{\frac{1}{2}QVW}^{\substack{\text{Carrying}\\\text{costs}}} + \overbrace{A\frac{R}{Q}}^{\substack{\text{Ordering}\\\text{costs}}} \]
To find the optimal \(Q\):
- Differentiate the \(TAC\) function: \[ \frac{\partial(TAC)}{\partial Q} = \frac{VW}{2} - \frac{AR}{Q^2} \]
- Set \(\frac{\partial(TAC)}{\partial Q}=0\) and solve for \(Q\): \[ \begin{align} Q^2 & = \frac{2AR}{VW} \\ Q & = \sqrt{\frac{2AR}{VW}} = \sqrt{\frac{2AR}{S}} \end{align} \]
6.3 Example: Inventory Cycles (Sawtooth Model)
6.4 Example: EOQ Calculation
6.5 Adjustments to the EOQ Model
The EOQ model has been extended in various ways to accommodate more complex and realistic scenarios. Alnahhal et al. (2024) reviews EOQ models to provide insights into practical considerations and emerging trends affecting EOQ calculations. The study identifies the main requirements that can alter the classical EOQ formula and details the main factors within each requirement.
These requirements include:
- Input Parameters: Adjustments to better reflect real-world costs, such as:
- Inventory Holding Costs: Varying by storage technology and item class.
- Variable Holding Costs: Changes due to storage duration, inventory levels, or seasonal fluctuations
- Ordering Costs: Comprising fixed and variable components.
- Variable Ordering Costs: Following steps or trends rather than being constant.
- Variants: Adjustments for real-world complexities, such as:
- Obsolescence and Depreciation: Items losing value or becoming unusable over time.
- Variable Demand and Seasonality: Changes in demand patterns over time.
- Variable Prices: Fluctuations in purchase prices and holding rates.
- Stochastic Supplier Capacity: Random availability due to outages, quotas, or yield losses.
- Imperfect Quality: Defect rates, rework, and returns affecting effective yield and cost.
- Growing Items: Inventories that gain weight or value in storage (e.g., livestock or cultured goods).
- Trends:
- Environmental Effects: Incorporating carbon and other externalities into cost considerations.
- New Products with No Forecast of Demand: Using robust, fuzzy, or learning policies for items with little demand history.
- Effect of Industry 4.0: Leveraging technologies like RFID, sensors, and digital twins to improve visibility and reduce errors.
- Practical Considerations:
- Size of Logistics Unit: Constraining orders to cases, layers, pallets, or totes.
- Subgrouping of Products: Grouping items by value, space, risk, or temperature for representative parameters and differentiated policies.
- Simplified Models: Using tractable formulas or regression surrogates when data are scarce.
- Supply Chain Disruptions: Adapting EOQ strategies to handle shocks such as pandemics or conflicts.
- Shipment Size: Considering freight economics and preferred shipment sizes by mode and lane.
6.5.1 Variable Holding Costs
In practical scenarios, holding costs often vary due to factors such as storage duration, inventory levels, or seasonal fluctuations. Alnahhal et al. (2024) shows that this is one of the most studied EOQ adjustments in the literature.
Alfares and Ghaithan (2019) provides an extensive review of EOQ models incorporating variable holding costs, categorizing them into two primary types:
- Time-Dependent Holding Costs: These are modeled as increasing functions of storage duration, reflecting the need for advanced preservation technologies to maintain item value over time. For instance, perishable goods may incur higher costs the longer they are stored and pharmaceuticals may require refrigeration.
- Stock-Dependent Holding Costs: These account for the higher expenses associated with larger inventory levels, such as additional storage space, increased financing costs, and greater risks of spoilage and obsolescence.
Linear functional forms are the most commonly assumed representation for both types of variable holding costs. Most models in this domain are EOQ-based, with direct differential calculus frequently employed to derive optimal solutions. The primary objective across these models remains the minimization of total inventory costs.
“Abstract: Inventory management is crucial for companies to minimize unnecessary costs associated with overstocking or understocking items. Utilizing the economic order quantity (EOQ) to minimize total costs is a key decision in inventory management, particularly in achieving a sustainable supply chain. The classical EOQ formula is rarely applicable in practice. For example, suppliers may enforce a minimum order quantity (MOQ) that is much larger than the EOQ. Some conditions such as imperfect quality and growing items represent variants of EOQ. Moreover, some requirements, such as the reduction of CO2 emissions, can alter the formula. Moreover, disruptions in the supply chain, such as COVID-19, can affect the formula. This study investigates which requirements must be considered during the calculation of the EOQ. Based on a literature review, 18 requirements that could alter the EOQ formula were identified. The level of coverage for these requirements has been tracked in the literature. Research gaps were presented to be investigated in future research. The analysis revealed that, despite their importance, at least 11 requirements have seldom been explored in the literature. Among these, topics such as EOQ in Industry 4.0, practical EOQ, and resilient EOQ have been identified as promising areas for future research.” (Alnahhal et al., 2024, p. 1) (pdf)
Input parameters. Getting holding and ordering costs right is foundational, since the EOQ you compute is only as good as those inputs. In practice, holding cost varies by storage technology and item class, and ordering cost is often partly fixed and partly variable. Example: a warehouse that upgrades a freezer zone for pharmaceuticals will see higher energy and depreciation components in its holding rate than for ambient goods, which can shift the optimal order size even if demand is unchanged. Similarly, if a supplier begins to charge a per-shipment handling fee plus a per-pallet fee, treating ordering cost as a single constant will bias the EOQ.
Variants. Real systems rarely meet the classical assumptions, so models incorporate effects like variable or seasonal demand, imperfect quality, deterioration, price dynamics, or stochastic supplier capacity. Each alters the cost trade-off and therefore the order size. Example: for a fresh-food SKU with 3 percent daily spoilage, you would prefer smaller, more frequent orders than the classic EOQ suggests. For a component with occasional defective lots, the effective yield and rework costs make the economic lot larger or smaller depending on inspection and return policies.
Trends. Sustainability goals and Industry 4.0 technologies are changing EOQ practice. Adding carbon cost to transport and storage can justify larger, less frequent replenishments on efficient modes, while sensors, RFID, and better data can shorten lead times and reduce safety buffers, which often favors smaller lots. Example: a firm that starts pricing CO₂ at €100 per ton and shifts from ad-hoc trucking to weekly milk runs by rail might increase lot size, whereas the same firm installing RFID at goods-in can cut counting errors and shorten cycle counts, which can decrease lot size for high-value items.
Practical considerations. Operations constraints determine what you can actually order: unit loads, shipment size steps, subgrouping items for shared parameters, simplified formulas for quick use, and resilience to disruptions. Example: if outbound handling requires full pallets of 56 cases, the actionable policy is “order in multiples of 56” and possibly “fill a 33-palette trailer,” not the exact continuous EOQ. During a disruption like a supplier outage, a contingency policy may intentionally over-order ahead of risk windows, trading higher holding cost for service continuity.
Inventory holding costs for different storage systems. Holding cost is not one number. It depends on space, handling, technology, risk, and capital tied up, and differs across ambient racks, cold rooms, AS/RS, and outside yards. A pallet of vaccines in a freezer has higher energy, depreciation, and risk components than paper towels in ambient racking, so the optimal lot $Q^*$ shifts even if demand is unchanged. Good EOQ practice therefore estimates item-specific holding rates by storage class.
Variable holding cost over time. Holding cost often changes with time or stock level due to inflation, seasonality of utilities, congestion, or aging. Models handle time-dependent, stock-dependent, or mixed forms, which alter the cost curve and usually reduce $Q^*$ when late-period costs are higher. Example: electricity surcharges in summer make chilled storage more expensive from June to August, so smaller, more frequent orders are preferable then.
Estimating ordering costs. Many firms either guess a flat fee or ignore ordering cost, yet it has fixed and variable parts: admin labor, approvals, transport booking, receiving, inspection, and per-unit fees. Underestimating it inflates order frequency and workload. Example: if receiving adds a per-pallet inspection fee, a single “€40 per order” assumption understates true cost for large shipments.
Variable ordering costs. Ordering cost often follows steps or trends rather than a constant. Learning effects can reduce it with experience, carrier fees add per-unit components, and workload peaks can raise it. Example: a 3PL charges a booking fee plus a per-pallet handling fee, creating a stepwise cost that makes some batch sizes clearly superior.
Obsolescence and depreciation. Items lose value or become unusable over time, adding an extra cost term that penalizes large lots. Electronics, fashion, and dated packaging are typical. Example: a label redesign scheduled in three months makes over-ordering today risky, so the economic lot shrinks relative to the classical EOQ.
Variable demand and seasonality. When demand changes with time or price, the average-rate assumption breaks. Seasonal or trend models favor phase-aligned ordering and often smaller lots near peaks when stock turns faster. Example: sunscreen demand spikes in summer, so a retailer staggers inbound lots before the peak and avoids carrying peak inventory through low-sales months.
Variable prices. Purchase prices and holding rates move with energy, commodities, or supplier policies, and can correlate with demand. This shifts both the unit value in holding cost and the purchasing bill, changing $Q^*$. Example: during a resin shortage, both price and demand for sanitizer bottles rose together, calling for a resilient policy rather than a discount-only model.
Stochastic supplier capacity. Supply may be capped or randomly available due to outages, quotas, or yield losses. Policies must hedge with safety stock or pre-buy windows, sometimes increasing order size despite higher holding cost. Example: a fab supplier with random monthly capacity forces customers to place larger, earlier orders to secure share.
Imperfect quality. Defect rates, rework, and returns change the effective yield and cost. If inspection is after receipt, more units must be ordered to achieve a good-units target, and the optimal lot can increase or decrease depending on costs of screening and backorders. Example: 5 percent defect rate with cheap rework may still justify larger, less frequent orders to amortize setup.
Growing items. Some inventories gain weight or value in storage, like livestock or cultured goods, which flips the holding logic. Optimal policy balances growth benefits against mortality and care costs. Example: a poultry farmer times chick purchases and slaughter age to minimize total feed, housing, mortality, and ordering costs.
Environmental effects. Adding carbon and other externalities changes the cost trade-off and sometimes the transport mode. Firms may accept larger, less frequent shipments on efficient modes to cut emissions or redesign milk runs, while greener warehousing can reduce holding rates. Example: pricing CO₂ pushes a shift from frequent truckloads to consolidated rail shipments with a different EOQ.
New products with no forecast of demand. With little history, demand distributions are unknown, so robust, fuzzy, or learning policies are used. Early lots are intentionally small to buy information and avoid obsolete stock, then scale as signal quality improves. Example: a hardware startup pilots micro-lots and updates order size from customer feedback before national launch.
Effect of Industry 4.0. RFID, sensors, and digital twins improve visibility, shorten effective lead times, and reduce errors, which typically lowers safety stock and lot size, while automation can lower ordering and receiving costs. Example: RFID at goods-in cuts counting time and shrinkage, enabling more frequent replenishment with smaller batches.
Size of logistics unit. Real orders are constrained to cases, layers, pallets, or totes. Handling benefits of unit loads compete with higher average inventory. Policies therefore choose multiples of the unit load near the theoretical $Q^*$. Example: if a pallet holds 56 cases, the actionable rule is to order the nearest multiple of 56 rather than the exact continuous optimum.
Subgrouping of products. Estimating unique parameters for thousands of SKUs is impractical. Grouping by value, space, risk, or temperature allows representative parameters and differentiated policies. Example: ABC families each receive calibrated holding and ordering costs, so A-items get tighter, smaller lots while C-items use coarser rules.
Simplified models. Practitioners need tractable formulas or regression surrogates when data are scarce. Simplified EOQ variants approximate optimality within small error and support fast what-ifs. Example: a regression-based estimator predicts $Q^*$ from demand and lead-time proxies when full cost decomposition is not available.
Supply chain disruptions. Shocks such as pandemics or conflicts change demand, lead times, and capacity. Resilient EOQ blends pre-buy, safety stock, and emergency sourcing to trade holding cost for service. Example: a pharmacy increased lot sizes during COVID-19 spike periods to buffer uncertain supply and demand swings.
Shipment size. Freight economics impose preferred shipment sizes by mode and lane, so the cost minimum is often at discrete breakpoints. Integrating transport, capital in transit, and inventory costs yields a shipment-aware lot size. Example: a contract minimizes annual cost by choosing half-truck weekly rather than quarter-truck twice weekly once in-transit capital is included.
“Stock-Dependent Holding Cost Many inventory models assume the holding cost to be a function of the inventory level or monetary value. This assumption is due to several real-life factors that make larger stock disproportionally more expensive to store. For example, additional storage space must be obtained by adding (possibly renting) extra storage facilities at a higher cost. Moreover, larger inventory imposes the need to invest greater amounts of capital at higher financing costs.” (Alfares and Ghaithan, 2019, p. 1747) (pdf)
“Time-Dependent Holding Cost In several inventory models, the unit holding cost is assumed to be an increasing function of storage time. In many realistic situations, this is due to the fact that the value of the stored items can decrease as they stay longer in the inventory. Therefore, longer storage time is more costly, because it requires more advanced technology in order to preserve the value of the stored items. Lee and Dye [7], for example, considered the holding cost (preservation technology) as a decision variable that affects the deterioration rate. By investing more in storage equipment and facilities, the retailer can slow down the deterioration of stored products.” (Alfares and Ghaithan, 2019, p. 1739) (pdf)
“Most of the papers with a variable holding cost (78%) presented EOQ-based inventory models, while a smaller number of them (22%) presented EPQ-based models. Most of these models (81%) assumed the holding cost to be a function of the storage time duration, while a smaller number of models (29%) considered it to be a function of the stock level. From the time-dependent holding cost models, the linear functional form is the most common assumption (49%). The most popular solution technique is to use direct differential calculus to derive formulas for the optimum solution. Minimizing the total cost is the dominant objective, which is pursued by twothirds of the selected papers. In addition, all the reviewed papers presented single-objective models, so far ignoring all multi-objective approaches. Finally, the great majority of reviewed papers (93%) validated their models through numerical examples rather than applied case studies.” (Alfares and Ghaithan, 2019, p. 1752) (pdf)
“Due to the existence of several compelling factors, holding cost variability is both a necessary and a realistic assumption in many practical inventory situations. Quite often, the holding cost increases with time because extended storage durations require more sophisticated and expensive storage facilities. For example, the long-term storage of food and pharmaceuticals requires refrigeration and special arrangements are needed to avoid spoilage. Similarly, volatile and flammable liquids need additional care and more stringent safety conditions for long-term storage. The unit holding cost also often increases with the inventory level, i.e., the quantity of stored items. The greater the amount of stored items, the longer the average time spent in inventory, and consequently the higher the chance of spoilage and obsolescence. Moreover, maintaining higher quantities in inventory requires additional (possibly rented) warehouses with correspondingly higher holding costs. In addition, buying and storing larger quantities require greater amounts of capital that are usually borrowed at higher financing costs. Other logical reasons for assuming variable holding costs are stated in the papers covered in this review.” (Alfares and Ghaithan, 2019, p. 1738) (pdf)
6.6 References
Alfares, Hesham K., and Ahmed M. Ghaithan. 2019. “EOQ and EPQ Production-Inventory Models with Variable Holding Cost: State-of-the-Art Review.” Arabian Journal for Science and Engineering 44 (3): 1737–55. https://doi.org/10.1007/s13369-018-3593-4.
Alnahhal, Mohammed, Batin Latif Aylak, Muataz Al Hazza, and Ahmad Sakhrieh. 2024. “Economic Order Quantity: A State-of-the-Art in the Era of Uncertain Supply Chains.” Sustainability 16 (14): 5965. https://doi.org/10.3390/su16145965.
Erlenkotter, Donald. 1990. “Ford Whitman Harris and the Economic Order Quantity Model.” Operations Research 38 (6): 937–46.
Harris, Ford Whitman. 1913. “How Many Parts to Make at Once.” Factory, The Magazine of Management.
Taft, E. W. 1918. “The Most Economical Production Lot.” Iron Age 101 (18): 1410–12.