Economic Order Quantity (EOQ & EPQ) Engine

The Mathematical Foundation of Lot Sizing

Originally developed by Ford Whitman Harris in 1913 and popularized by R. H. Wilson, the Economic Order Quantity (EOQ) model remains the foundational pillar of modern inventory control, material requirements planning (MRP), and warehouse optimization. At its core, the model resolves a fundamental commercial conflict:

• Ordering Costs (S): Every purchase order triggers administrative labor, requisition processing, invoice reconciliation, shipping dock receiving, and inbound inspection. To minimize annual ordering overhead, purchasing agents naturally favor placing massive, infrequent orders.
• Inventory Holding Costs (H): Storing surplus stock consumes capital interest, insurance premiums, warehouse footprint leasing, climate control energy, security, obsolescence risk, and shrinkage. To minimize holding overhead, warehouse supervisors prefer ordering in lean, frequent micro-batches.

By formulating Total Relevant Annual Cost (TC) as a function of lot size Q:

Taking the first derivative with respect to Q, setting it to zero (first-order condition for unconstrained optimization), and solving yields the classic square-root formula:

Because the second derivative d²TC/dQ² = 2DS/Q³ > 0 for all positive Q, the cost function is strictly convex, guaranteeing that Q* is a global minimum.

Advanced Models: EPQ, Backorders & Price Breaks

While the textbook EOQ assumes instantaneous replenishment, real-world manufacturing and wholesale procurement require specialized adjustments:

1. Economic Production Quantity (EPQ / Non-Instantaneous Receipt)

When an item is produced internally on a production line or delivered continuously over time at daily rate P, units are consumed simultaneously at demand rate D (P > D). As a result, the maximum warehouse inventory accumulation never reaches batch size Q, but rather peaks at I_max = Q × (1 - D/P). This dampens holding cost, yielding a larger production run size:

2. Quantity Discount Decision Algorithm

Suppliers frequently offer volume price breaks. Because unit price C decreases at higher tiers, total purchase cost D × C must be included in the comparison. The step-by-step algorithm executed by our engine is:

1. Calculate raw EOQ for each price tier using that tier's holding cost H_j = i × C_j.
2. Check feasibility: If raw EOQ falls within the tier's range [Q_min, Q_max], it is feasible. If raw EOQ is below Q_min, round up to Q_min. If raw EOQ exceeds Q_max, the tier is infeasible.
3. Compute Total Annual Cost TC(Q) = (D/Q)S + (Q/2)H_j + D × C_j for each feasible Q.
4. Select the batch size yielding the absolute lowest total annual cost.

Worked Numerical Example

Notice that at the exact economic optimum, Annual Ordering Cost ($707.10) virtually equals Annual Holding Cost ($707.00). This parity is an inherent mathematical property of the classic EOQ model.

The Robustness of EOQ: Why Being "Close Enough" Works

One of the most valuable insights in operations research is the flatness of the EOQ total cost curve around Q*. Even if actual batch size deviates significantly from Q*, the percentage cost penalty is surprisingly minimal. Analytically, if actual lot size is Q = k × Q*, the ratio of total relevant costs is:

For instance, ordering 20% too much (k = 1.20) or 20% too little (k = 0.80) results in a mere 1.67% penalty in total inventory management costs. This mathematical robustness allows procurement teams to round batch sizes to full carton quantities, pallet multiples, or container loads without financial ruin.

Frequently Asked Questions (Answer-First)