16 Economic order quantity (EOQ) model
Inventory management, Economic order quantity, Supply chain, Sawtooth model, Condition of certainty
Inventory is money sitting around in a different form.—Rhonda Abrams
Introduction
The basic EOQ model considers one product, constant demand, certain order arrivals, and no stockouts. It balances 2 costs:
- Ordering cost: The fixed cost incurred each time an order is placed. Fewer orders reduce this cost.
- Holding cost: The cost of storing and maintaining inventory. Lower average inventory reduces this cost.
Frequent orders increase ordering cost. Large, infrequent orders increase holding cost. The economic order quantity (EOQ) minimizes their sum under the model assumptions.
Harris (1913) presents the model and its closed-form solution. Taft (1918) extends it to production settings through the economic production quantity (EPQ). Erlenkotter (1990) gives a historical account.
Problem statement: Determine the order quantity \(Q\) that minimizes ordering and holding costs over the planning horizon.
Assumptions:
- Demand is continuous, constant, and known.
- Replenishment or lead time is constant and known.
- The model excludes stockouts and safety stock.
- All demand is satisfied without backorders.
- Unit price is constant and independent of order quantity.
- The model excludes inventory in transit.
- Goods are purchased on a delivered price basis, with transportation costs included in the purchase price.
- The model contains one inventory item.
- The planning horizon is infinite.
- Capital is unconstrained.
EOQ formulation
Table 16.1 defines the EOQ variables.
| 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 cycle 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 \(A\frac{R}{Q}\).
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 \(Q\): \[ \begin{align} Q^2 & = \frac{2AR}{VW} \\ Q & = \sqrt{\frac{2AR}{VW}} = \sqrt{\frac{2AR}{S}} \end{align} \]
Example: Inventory cycles and EOQ costs
The initial example uses 120 units per order and annual demand of 3,600 units. In a 360-day year, inventory falls by 10 units per day. Figure 16.1 shows what happens over 30 days. Change the values below the figure to see how the cycle and its costs respond.
Change the example
The cost figure uses the same order quantity and demand as the sawtooth. It separates annual ordering and holding costs, then shows their sum. The minimum occurs at the economic order quantity in Figure 16.2.
Plot settings
Both figures assume steady, known demand and immediate replenishment when stock reaches zero. The cost figure omits purchase cost because unit price stays constant as order quantity changes. The model follows Harris (1913); Alnahhal et al. (2024) reviews later changes to its assumptions.
Adjustments to the EOQ model
EOQ extensions relax assumptions about costs, demand, supply, products, and transport. Alnahhal et al. (2024) classifies the requirements that change the classical formula.
These requirements include:
- Input parameters: Changes to cost estimates, including:
- 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: Stepwise or trending costs.
- Variants: Changes to demand, supply, quality, and product behavior:
- 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).
- Technology and environmental effects:
- 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: RFID, sensors, and digital twins change data accuracy, visibility, and process errors.
- Operating constraints:
- 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.
Variable holding costs
Holding costs can vary with storage duration, inventory level, or season. Alnahhal et al. (2024) identifies variable holding cost as a frequently studied EOQ adjustment.
Alfares and Ghaithan (2019) reviews EOQ models with 2 types of variable holding cost:
- Time-dependent holding costs: These increase with storage duration. Perishable goods may require more preservation effort over time, and pharmaceuticals may require refrigeration.
- Stock-dependent holding costs: These increase with inventory level through additional space, financing cost, spoilage, or obsolescence.
Both types often use linear functions. Most reviewed models use EOQ formulations and direct differential calculus to minimize total inventory cost.