First pass yield and rolled throughput yield describe different scopes of process performance. Confusing those scopes can make a line appear efficient even when substantial rework occurs between material release and final acceptance.
First pass yield measures the proportion of units that complete a process step correctly on the first attempt. A unit that requires correction, adjustment, repair, rerouting or repeated processing fails first pass at that step, even if it eventually becomes acceptable.
Rolled throughput yield measures the proportion expected to complete every step correctly on the first attempt. For a process with multiple stations, RTY is the product of the individual station yields:
RTY = Y1 × Y2 × ... × Yn
The distinction depends on where measurement occurs. A station-level measure describes one operation. A final-gate measure describes the output accepted at the end. RTY describes first-pass performance across the complete sequence.
What multiplication does over a process sequence
High station yields can produce a much lower RTY when the process contains many steps. Each additional operation creates another opportunity for a first-pass failure, and multiplication carries that effect through the full sequence.
The table shows RTY for uniform station yields as process steps accumulate.
| Steps | 99% each | 98% each | 95% each | 90% each |
|---|---|---|---|---|
| 1 | 99.0% | 98.0% | 95.0% | 90.0% |
| 5 | 95.1% | 90.4% | 77.4% | 59.0% |
| 10 | 90.4% | 81.7% | 59.9% | 34.9% |
| 15 | 86.0% | 73.9% | 46.3% | 20.6% |
| 20 | 81.8% | 66.8% | 35.8% | 12.2% |
| 25 | 77.8% | 60.3% | 27.7% | 7.2% |
Figure 1 plots the same calculation across the full range of process lengths. The marked point on the 95% curve shows the cumulative result clearly: 10 steps operating at 95% FPY produce 59.9% RTY.
The individual stations can remain within their established yield expectations while the sequence generates frequent first-pass failures. Line length therefore matters whenever yield is used to describe overall operating performance. A station yield cannot be extended to the entire process without accounting for every other station.
This arithmetic also explains why a line-level yield requires consistent definitions. Each station must count first-pass success using the same unit basis and the same treatment of rework. If one operation records accepted output after correction while another records initial attempts, multiplying the values produces a result with mixed meanings.
Why final-inspection yield can overstate line performance
Final inspection yield answers a useful question: what proportion of output passed the final acceptance gate? That measure supports release decisions, outgoing quality control and customer protection.
It does not establish how efficiently the preceding operations produced that output. Material may have been adjusted, reprinted, relaminated, reinspected, refinished or repacked before reaching the final gate. When those events remain outside the reported yield, the final number includes the results of successful rework without counting the rework itself.
RTY assigns each first-pass failure to the process sequence. A corrected unit can still become conforming output, while its earlier failure continues to affect RTY. This treatment separates product acceptance from process execution.
The practical consequence is that two lines can report the same final-gate yield while requiring different amounts of intervention. Final inspection alone records the accepted result. RTY records the probability of achieving that result through first-pass success at every station.
Worked example: seven-station converting line
Consider a converting line with die cutting, printing, laminating, slitting, inspection, finishing and packing. Every station operates at or above 96% FPY.
| Station | FPY | Cumulative yield |
|---|---|---|
| Die cut | 98.5% | 98.5% |
| 97.0% | 95.5% | |
| Laminate | 99.5% | 95.1% |
| Slit | 98.0% | 93.2% |
| Inspect | 96.0% | 89.4% |
| Finish | 97.5% | 87.2% |
| Pack | 99.5% | 86.8% |
| Line measure | Result |
|---|---|
| RTY of the line | 86.8% |
| Final gate, Pack alone | 99.5% |
| Overstatement | 12.7 points |
DPU = -ln(RTY) |
0.142 |
Check: exp(-DPU) |
86.8% |
Figure 2 places each station FPY beside the cumulative yield through that point. The pale columns show that every station has a high individual yield. The cumulative line declines from 98.5% after die cutting to 86.8% after packing.
The dashed final-gate rule remains at 99.5%, which is the FPY reported for Pack alone. Its distance from the line RTY is 12.7 points. That gap represents the performance difference concealed when the final station’s yield is reported as the yield of the complete line.
Pack at 99.5% describes Pack. It also describes the immediate final gate when that station controls release. The 86.8% RTY describes the probability that a unit passes all seven stations on its first attempt.
The example also connects RTY to defects per unit under a Poisson assumption:
DPU = -ln(RTY)
Using the line RTY gives a DPU of 0.142. The inverse check returns the original RTY:
RTY = exp(-DPU)
The result is 86.8%. This relationship is useful when process reporting includes both yield and defect-rate measures, provided the Poisson assumption fits the application and the underlying counts use compatible definitions.
How to identify the yield being reported
The fastest test is to ask where the measurement was taken and which attempts were included.
A value measured at one operation is station FPY when it counts units completed correctly on their first attempt through that operation. It describes local execution and supports improvement work at that station.
A value measured at the last acceptance gate is final-inspection yield. It describes the proportion accepted there, including units that may have received corrective processing earlier in the sequence.
A value calculated across every operation in order is RTY. It requires the first-pass yield for each included station and multiplies those yields through the process sequence.
The reporting label should identify the scope directly. “Pack FPY” identifies a station. “Final acceptance yield” identifies an outgoing gate. “Line RTY” identifies the full sequence. A generic label such as “overall yield” leaves the measurement point and rework treatment undefined.
The underlying count provides a second check. If the numerator includes units that passed after correction, the measure includes recovered output. If each station records only units that passed on the first attempt, the station values can support an RTY calculation.
Using the measures together
A complete operating view can include station FPY, line RTY and final-inspection yield. Each measure supports a separate management question.
Station FPY identifies where first-pass failures occur. Line RTY quantifies their combined effect across the process. Final-inspection yield confirms the condition of output at release.
The difference between final-gate yield and RTY indicates how much process performance disappears from view when reporting begins only at final acceptance. Investigation then moves to the station records: rework quantities, repeat operations, defect categories and disposition paths.
The arithmetic can be completed with the FPY and RTY calculator by entering the yield from each process step.
Yield reporting becomes technically clear when the measurement point, process scope and treatment of rework appear with the value. Those definitions allow plant management, operations and quality personnel to interpret the same number consistently.