Throughput Math - Sortation

A sorter throughput spec with a clean number (“16,000 cartons per hour”) doesn’t tell you how it was derived, under what conditions it holds, or how fast it degrades when reality introduces contention. This page is the math behind those numbers.

The Fundamental Throughput Equation

Working in inches (practical for North American systems):

$$\text{CPM} = \frac{\text{FPM} \times 12}{\text{carton length (in)} + \text{gap (in)}}$$

To find throughput per hour from pitch and speed:

$$\text{Units/hr} = \frac{\text{FPM} \times 3600}{\text{pitch (in)}}$$

where pitch = carton length + gap (both in inches).

The 80% Nameplate Rule

Design at no more than 80% of nameplate FPM continuously. (MWPVL sortation primer)

Four compounding reasons:

1. Accumulation wave instability: Minor upstream delays create waves. At 100% nameplate, no speed margin exists to absorb them — the wave propagates downstream.
2. Gapping inefficiency: Real induction produces gaps varying ±15–20% around target. At 100% nameplate, cartons arriving slightly early or late miss their divert window.
3. Accelerated wear: Sliding shoe sorters at continuous 100% show significantly elevated shoe wear.
4. Surge margin: The 20% headroom absorbs the surge curve. For a Poisson-like carton arrival process with CV = 0.5, the 95th-percentile instantaneous demand: 10,000 CPH × (1 + 1.65 × 0.5) = 18,250 CPH instantaneous. The sorter nameplate must accommodate surge events, not just hourly averages.

Worked Example: The 10,000 CPH Case

Design requirement: 10,000 CPH. Average carton length: 24 in. Gap for sliding shoe sorter: 6 in. Pitch = 30 in.

Required speed: FPM = (10,000/60 × 30) / 12 = 417 FPM

A 450 FPM nameplate sorter provides 417 FPM with ~8% headroom above calculated requirement — thin but positive.

Applying the 80% rule: Design operating speed = 0.80 × 450 = 360 FPM.

Achievable throughput at 360 FPM: (360 × 12) / 30 = 144 CPM = 8,640 CPH

Gap: 10,000 CPH required vs. 8,640 CPH deliverable at 80% nameplate. You need either a faster sorter (≥550 FPM nameplate) or a second induction point.

Gap Requirements by Technology

Gap requirements are set by physics, not convention:

Technology	Minimum Gap	Nameplate FPM Range
Sliding shoe sorter	2–6 in (angle-dependent)	300–500 FPM
Cross-belt sorter	4–8 in	200–350 FPM
Pop-up wheel sorter	4 in	200–400 FPM
Tilt-tray sorter	Set by tray pitch	100–200 FPM

Technology and gap together determine the throughput ceiling: same 9-in carton, pop-up wheel vs. sliding shoe can produce a 52% throughput difference.

Merge Efficiency

When two lines merge into one, output is always less than the sum of inputs. This is a geometric constraint, not a configuration problem.

The theoretical merge efficiency ceiling:

$$\eta_{\text{merge}} = \frac{L}{L + g_{\min}}$$

For typical cartons (18 in) with a 3-in sliding shoe gap: η = 18/21 = 85.7%

In practice: real timing variability drops effective merge efficiency to 70–80% of combined inbound capacity.

Worked Example: 3-Way Merge

Three pick modules each feeding 50 CPM = 150 CPM combined.

Expected output: 150 × 0.75 = 112 CPM

If SAT requires 120 CPM, you are 8 CPM short. Options:

• Increase each module’s induction rate to at least 54 CPM (150 × 0.80 = 120 CPM)
• Add pre-merge accumulation (slug release can recover 5–8 percentage points of merge efficiency)
• Redesign the layout — a sequential merge (A into B, then AB into C) behaves differently than a 3-way simultaneous merge

Slug release: Accumulate 5–6 carton lengths upstream, release on a timed gate. Converts a random arrival process into a near-deterministic one. Recovers 5–8pp of merge efficiency without touching the sorter. Underused and undervalued.

Accumulation Conveyor Sizing

Accumulation buffers decouple upstream and downstream processes. Size for the 80th-percentile MTTR, not the mean — a buffer sized for average downtime is inadequate roughly half the time.

$$\text{Buffer length (ft)} = \text{Upstream rate (CPM)} \times \text{MTTR}{P{80}} \text{ (min)} \times \frac{\text{pitch (in)}}{12}$$

Worked Example

Upstream rate: 120 CPM. P80 MTTR for downstream station: 4 min. Pitch = 22 in.

Buffer cartons = 120 × 4 = 480 cartons Buffer length = 480 × 22/12 = 880 ft

880 feet of linear floor space. In practice: spiral conveyors compact this into 100–150 sq ft of footprint at the same capacity.

Zero-Pressure vs. Pressureless Accumulation

Type	How It Works	Use Case
Zero-pressure (ZPA)	Sensor-controlled zones; no contact between cartons	Fragile product, mixed carton sizes, high-value items
Pressureless	Continuous low-pressure zones; cartons touch	High-volume commodity items, uniform carton sizes

Zero-pressure is the default for general merchandise. Pressureless accumulation is appropriate when product can tolerate contact and cost sensitivity drives the layout.

Recirculation

Recirculation lanes handle no-reads and exception items. Rule of thumb: size recirculation capacity for 5–10% of peak induction rate. At 10,000 CPH sorter, a 7% recirculation rate consumes 700 CPH of effective capacity — reducing net throughput to ~9,300 CPH on the primary lanes.

Always include recirculation in the throughput model and in the FDS. A system with no documented recirculation behavior produces undefined behavior at exception events.

Source: 2.6-advanced-automation-design