440. Perishables

440.1. EOQ for perishable items

Relax one dimension from basic EOQ: items deteriorate over time. Each unit decays at constant exponential rate — even sitting on the shelf untouched, fraction of stock is lost per unit time.

Examples: pharmaceuticals with shelf life, fresh produce, photographic film, blood products.

440.1.1. Setup

New variable beyond basic EOQ:

• = decay rate (fraction of stock lost per unit time)

Inventory dynamics with continuous demand and continuous decay:

The stock drops both from sales (rate ) and from spoilage (rate ). Solve this ODE with :

Cycle ends when :

The cycle is shorter than basic EOQ’s — decay shortens how long each order lasts.

440.1.2. Cost model — exact form

Per cycle:

• Order:
• Holding:

Compute the integral. Using :

After substituting and simplifying:

The exact TC per unit time is then — but is transcendental in , so there’s no clean closed-form . Use the small- approximation instead.

440.1.3. Small-decay approximation

For modest decay (), expand :

Decay loss per cycle = — quadratic in .

Annualizing: number of cycles per year is , so:

• Annual order cost:
• Annual holding cost: (basic EOQ approximation, leading order)
• Annual decay loss: units, valued at unit cost , costs

The decay term shows up as an extra holding-like cost: it scales with . Combine it with the standard holding term:

The decay rate effectively raises the holding cost from to .

440.1.4. Derive

Same calculus as basic EOQ with the inflated holding cost:

440.1.5. Final formulas (small- approximation)

Sanity check: as (no decay), and we recover basic EOQ exactly ✓. As (instant decay), — order infinitesimally small, infinitely often, so nothing decays.

The result is intuitive: decay acts like an extra carrying cost of per unit per unit time. This captures most of the effect when decay is moderate; for high , you’d need the exact transcendental form.