Joint Replenishment

442. Joint Replenishment

442.1. Joint replenishment (multi-item EOQ with shared setup)

Relax one dimension from basic EOQ: number of items is no longer one. Multiple items share a single supplier (or a single delivery truck, or a single setup operation), so the order cost $𝑆$ is paid once per coordinated order — not separately per item.

Single supplier, multiple SKUs: order them all together on the same cadence, save on setup.

442.1.1. Setup

$𝑛$ items indexed $𝑖 = 1, \dots, 𝑛$ :

$𝐷_{𝑖}$ = annual demand for item $𝑖$
$ℎ_{𝑖}$ = holding cost per unit per year for item $𝑖$
Common setup cost: $𝑆$ (paid once when any order goes out, regardless of how many items)
Decision: a common cycle time $𝑇$ — every $𝑇$ years, order each item in the quantity needed for one cycle, $𝑄_{𝑖} = 𝐷_{𝑖} 𝑇$ .

This is the simplest joint-replenishment formulation. (Variants with item-specific minor setups $𝑠_{𝑖}$ in addition to a major $𝑆$ exist — see Roundy / Federgruen-Zheng — but we’ll keep it minimal here.)

442.1.2. Cost model

Per cycle:

One setup cost: $𝑆$
Holding for item $𝑖$ over the cycle: $ℎ_{𝑖} \cdot 𝑄_{𝑖} / 2 \cdot 𝑇 = ℎ_{𝑖} 𝐷_{𝑖} 𝑇 / 2 \cdot 𝑇$ . Wait — average inventory $𝑄_{𝑖} / 2$ , time per cycle $𝑇$ , so per-cycle holding cost is $ℎ_{𝑖} \cdot (𝑄_{𝑖} / 2) \cdot 𝑇 = ℎ_{𝑖} 𝐷_{𝑖} 𝑇^{2} / 2$ .

Per unit time, divide by $𝑇$ :

TC (𝑇) = \frac{𝑆}{𝑇} + \sum_{𝑖 = 1}^{𝑛} ℎ_{𝑖} \frac{𝐷_{𝑖} 𝑇}{2} = \frac{𝑆}{𝑇} + (\frac{𝑇}{2}) \sum_{𝑖 = 1}^{𝑛} ℎ_{𝑖} 𝐷_{𝑖}

The single decision $𝑇$ trades off setup cost (decreases with $𝑇$ ) against aggregate holding (increases with $𝑇$ ).

442.1.3. Derive $𝑇^{*}$

\frac{𝑑}{𝑑 𝑇} TC = - \frac{𝑆}{𝑇^{2}} + \frac{1}{2} \sum_{𝑖} ℎ_{𝑖} 𝐷_{𝑖} = 0

Solve for $𝑇^{*}$ :

𝑇^{{* 2}} = \frac{2 𝑆}{\sum_{𝑖} ℎ_{𝑖} 𝐷_{𝑖}}

𝑇^{*} = \sqrt{\frac{2 𝑆}{\sum_{𝑖} ℎ_{𝑖} 𝐷_{𝑖}}}

442.1.4. Final formulas

𝑇^{*} = \sqrt{\frac{2 𝑆}{\sum_{𝑖} ℎ_{𝑖} 𝐷_{𝑖}}} 𝑄_{𝑖}^{*} = 𝐷_{𝑖} 𝑇^{*} = 𝐷_{𝑖} \sqrt{\frac{2 𝑆}{\sum_{𝑗} ℎ_{𝑗} 𝐷_{𝑗}}} {TC}^{*} = \sqrt{2 𝑆 \sum_{𝑖} ℎ_{𝑖} 𝐷_{𝑖}}

Sanity check: for a single item ( $𝑛 = 1$ ), $𝑇^{*} = \sqrt{2 𝑆 / (ℎ_{1} 𝐷_{1})}$ , $𝑄_{1}^{*} = 𝐷_{1} 𝑇^{*} = \sqrt{2 𝑆 𝐷_{1} / ℎ_{1}}$ — exactly basic EOQ ✓.

The aggregation comes through the single $\sqrt{\sum_{𝑖} ℎ_{𝑖} 𝐷_{𝑖}}$ — items with small $ℎ_{𝑖} 𝐷_{𝑖}$ contribute little to the optimal cycle, so they’re ordered alongside the big- $ℎ_{𝑖} 𝐷_{𝑖}$ items at no extra setup cost.

442.1.5. Why is this cheaper than independent EOQs?

Order each item independently with its own EOQ: each pays $𝑆$ per order, total cost is $\sum_{𝑖} \sqrt{2 𝑆 𝐷_{𝑖} ℎ_{𝑖}}$ .

Joint replenishment: $\sqrt{2 𝑆 \sum_{𝑖} ℎ_{𝑖} 𝐷_{𝑖}}$ .

By Cauchy-Schwarz / Jensen, $\sqrt{\sum_{𝑖} 𝑎_{𝑖}^{2}} \leq \sum_{𝑖} 𝑎_{𝑖}$ , so:

\sqrt{2 𝑆 \sum_{𝑖} ℎ_{𝑖} 𝐷_{𝑖}} \leq \sum_{𝑖} \sqrt{2 𝑆 ℎ_{𝑖} 𝐷_{𝑖}}

Joint always wins. The savings grow with the number of items and the unevenness of $ℎ_{𝑖} 𝐷_{𝑖}$ across them.

Example

Given (3 items sharing a single supplier with shared setup):

Common setup: $𝑆$ = $50 / order
Item 1: $𝐷_{1} = 12000$ , $ℎ_{1}$ = $2
Item 2: $𝐷_{2} = 6000$ , $ℎ_{2}$ = $3
Item 3: $𝐷_{3} = 3000$ , $ℎ_{3}$ = $4

Step 1 — aggregate holding rate

\sum_{𝑖} ℎ_{𝑖} 𝐷_{𝑖} = 2 \cdot 12000 + 3 \cdot 6000 + 4 \cdot 3000 = 24000 + 18000 + 12000 = 54000

Step 2 — common cycle time

𝑇^{*} = \sqrt{\frac{2 \cdot 50}{54000}} = \sqrt{\frac{100}{54000}} \approx 0.0430 years \approx 16 days

Step 3 — order quantities

𝑄_{1}^{*} = 𝐷_{1} 𝑇^{*} = 12000 \cdot 0.043 \approx 516 units

𝑄_{2}^{*} = 𝐷_{2} 𝑇^{*} = 6000 \cdot 0.043 \approx 258 units

𝑄_{3}^{*} = 𝐷_{3} 𝑇^{*} = 3000 \cdot 0.043 \approx 129 units

Step 4 — total cost

{TC}^{*} = \sqrt{2 \cdot 50 \cdot 54000} = \sqrt{5400000} \approx 2324

$/year

Compare to independent EOQs (each item ordered separately with full $𝑆$ ):

Item 1: $𝑄_{1} = \sqrt{2 \cdot 50 \cdot 12000 / 2} = 775$ , ${TC}_{1} = \sqrt{2 \cdot 50 \cdot 12000 \cdot 2} \approx 1549$
Item 2: $𝑄_{2} = \sqrt{2 \cdot 50 \cdot 6000 / 3} \approx 447$ , ${TC}_{2} = \sqrt{2 \cdot 50 \cdot 6000 \cdot 3} \approx 1342$
Item 3: $𝑄_{3} = \sqrt{2 \cdot 50 \cdot 3000 / 4} \approx 274$ , ${TC}_{3} = \sqrt{2 \cdot 50 \cdot 3000 \cdot 4} \approx 1095$
Total independent: $1549 + $1342 + $1095 = $3986.

Joint replenishment saves $1662/year (42% lower) by sharing the setup across items. The savings scale with the number of items — at 10 SKUs of similar size, savings would approach 70%.