Perishables

456. Perishables

456.1. EOQ for perishable items

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

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

456.1.1. Setup

New variable beyond basic EOQ:

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

Inventory dynamics with continuous demand $𝑑$ and continuous decay:

\frac{𝑑 𝐼}{𝑑 𝑡} = - 𝑑 - 𝜃 𝐼 (𝑡)

The stock drops both from sales (rate $𝑑$ ) and from spoilage (rate $𝜃 𝐼$ ). Solve this ODE with $𝐼 (0) = 𝑄$ :

𝐼 (𝑡) = (𝑄 + \frac{𝑑}{𝜃}) 𝑒^{- 𝜃 𝑡} - \frac{𝑑}{𝜃}

Cycle ends when $𝐼 (𝑇) = 0$ :

(𝑄 + \frac{𝑑}{𝜃}) 𝑒^{- 𝜃 𝑇} = \frac{𝑑}{𝜃} \to 𝑇 = (\frac{1}{𝜃}) ln (1 + \frac{𝑄 𝜃}{𝑑})

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

456.1.2. Cost model — exact form

Per cycle:

Order: $𝑆$
Holding: $ℎ \cdot \int_{0}^{𝑇} 𝐼 (𝑡) 𝑑 𝑡$

Compute the integral. Using $𝐼 (𝑡) = (𝑄 + 𝑑 / 𝜃) 𝑒^{- 𝜃 𝑡} - 𝑑 / 𝜃$ :

\int_{0}^{𝑇} 𝐼 (𝑡) 𝑑 𝑡 = - \frac{𝑄 + 𝑑 / 𝜃}{𝜃} \cdot (𝑒^{- 𝜃 𝑇} - 1) - 𝑑 𝑇 / 𝜃

After substituting $𝑒^{- 𝜃 𝑇} = 𝑑 / (𝑄 𝜃 + 𝑑)$ and simplifying:

\int_{0}^{𝑇} 𝐼 (𝑡) 𝑑 𝑡 = \frac{𝑄}{𝜃} - \frac{𝑑 𝑇}{𝜃}

The exact TC per unit time is then $𝑆 / 𝑇 + ℎ (𝑄 / (𝜃 𝑇) - 𝑑 / 𝜃)$ — but $𝑇$ is transcendental in $𝑄$ , so there’s no clean closed-form $𝑄^{*}$ . Use the small- $𝜃$ approximation instead.

456.1.3. Small-decay approximation

For modest decay ( $𝜃 𝑇 ≪ 1$ ), expand $𝑇$ :

𝑇 = (\frac{1}{𝜃}) ln (1 + 𝑄 \frac{𝜃}{𝑑}) \approx \frac{𝑄}{𝑑} - \frac{𝑄^{2} 𝜃}{2 𝑑^{2}} + 𝒪︀ (𝜃^{2})

Decay loss per cycle = $𝑄 - 𝑑 𝑇 \approx 𝑄^{2} 𝜃 / (2 𝑑)$ — quadratic in $𝑄$ .

Annualizing: number of cycles per year is $1 / 𝑇 \approx 𝑑 / 𝑄 + 𝜃 / 2$ , so:

Annual order cost: $𝑆 / 𝑇 \approx 𝑆 𝑑 / 𝑄$
Annual holding cost: $ℎ 𝑄 / 2$ (basic EOQ approximation, leading order)
Annual decay loss: $\approx 𝑄 𝜃 / 2$ units, valued at unit cost $𝑐$ , costs $𝑐 𝑄 𝜃 / 2$

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

TRC (𝑄) \approx 𝑆 (\frac{𝑑}{𝑄}) + (ℎ + 𝑐 𝜃) (\frac{𝑄}{2})

The decay rate effectively raises the holding cost from $ℎ$ to $ℎ + 𝑐 𝜃$ .

456.1.4. Derive $𝑄^{*}$

Same calculus as basic EOQ with the inflated holding cost:

𝑄^{*} \approx \sqrt{\frac{2 𝑆 𝑑}{ℎ + 𝑐 𝜃}}

456.1.5. Final formulas (small- $𝜃$ approximation)

𝑄^{*} \approx \sqrt{\frac{2 𝑆 𝑑}{ℎ + 𝑐 𝜃}} 𝑇^{*} \approx \frac{𝑄^{*}}{𝑑} {TRC}^{*} \approx \sqrt{2 𝑆 𝑑 (ℎ + 𝑐 𝜃)}

Sanity check: as $𝜃 \to 0$ (no decay), $ℎ + 𝑐 𝜃 \to ℎ$ and we recover basic EOQ exactly ✓. As $𝜃 \to \infty$ (instant decay), $𝑄^{*} \to 0$ — order infinitesimally small, infinitely often, so nothing decays.

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

Example

Given (shared EOQ params + a 5%/year decay rate):

Annual demand: $𝑑 = 12000$ units/year
Order cost: $𝑆$ = $50 / order
Holding cost: $ℎ$ = $2 / unit / year
Unit cost: $𝑐$ = $10
Decay rate: $𝜃 = 0.05$ /year (5% of stock spoils per year)

Step 1 — effective holding cost

ℎ_{eff} = ℎ + 𝑐 𝜃 = 2 + 10 \cdot 0.05 = 2 + 0.5 = 2.5

Decay raises the effective holding cost by 25% over the basic EOQ rate.

Step 2 — order quantity

𝑄^{*} \approx \sqrt{\frac{2 \cdot 50 \cdot 12000}{2.5}} = \sqrt{480000} \approx 693 units

Step 3 — cycle and cost

𝑇^{*} \approx \frac{𝑄^{*}}{𝑑} = \frac{693}{12000} \approx 0.058 years \approx 21 days

{TRC}^{*} \approx \sqrt{2 \cdot 50 \cdot 12000 \cdot 2.5} \approx 1732

$/year

Decay loss per year: $𝑄^{*} 𝜃 / 2 \approx 693 \cdot 0.05 / 2 \approx 17$ units (worth $170).

Compare to basic EOQ on the same params (ignoring decay):

Basic EOQ ( $𝜃 = 0$ ): $𝑄^{*} = 775$ , ${TRC}^{*} \approx$ $1549.
With decay ( $𝜃 = 0.05$ ): $𝑄^{*} \approx 693$ (smaller), ${TRC}^{*} \approx$ $1732 (12% higher).

Two effects, both pointing the right way:

Order smaller batches (693 vs 775) so less stock sits long enough to spoil.
Total cost rises — decay is a real cost you can’t optimize away, only minimize.

For higher $𝜃$ (e.g., fresh produce with $𝜃 \geq 1 /$ year), the small- $𝜃$ approximation breaks down and the exact transcendental form is needed; in practice, perishable inventory is more often modeled with a fixed shelf life $𝐿$ rather than continuous decay.