Abstract
The capacity of a line follows from the capacities of its stages. What this paper shows is that it follows as a minimum rather than a sum or a mean, and that widening the minimum helps in a step from $1$ to $0$. No new theorem or law is claimed.
First, the line matches the smallest stage——with capacities $80$, $150$, $120$, $100$ and $200$, the line delivers $80$ per hour (Section 2).
Second, the sum is of another size——it is $650$, of which the line delivers only $\mathbf{0.1231}$ (Section 2).
Third, every stage but the smallest must idle——utilisations run $1.0000$, $0.8000$, $0.6667$, $0.5333$ and $\mathbf{0.4000}$, and idling is the normal state (Section 2).
Fourth, and this is the core. Widening helps with nothing in between——exactly $\mathbf{1.0000}$ from $80$ to $100$, exactly $\mathbf{0.0000}$ beyond, and no intermediate value (Section 3).
Fifth, the average is small——widening all the way from $80$ to $250$ averages an effectiveness of $\mathbf{0.1176}$ (Section 3).
Sixth, the separator is whether every item passes through every stage——where they can be routed around, the sum is the right answer (Section 4).
Keywords: bottleneck, theory of constraints, serial production line, minimum capacity, factory physics.
Scope (what is not claimed)
No new theorem or law is claimed——that a serial line runs at its smallest stage is standard. No factory improvement is recommended——where to invest is not stated; only what happens when a stage is widened. Variability and buffers are excluded——each capacity is a constant, and with fluctuation the line falls below the minimum. Changeovers and breakdowns are not treated——availability is set to one. Costs are not discussed——what widening costs is not the subject. It is not said that an idle stage is waste——in a serial line, idling anywhere but the minimum is unavoidable. Relation to earlier papers: Paper 477 showed that while the wrong input is being added, the limit is somewhere else——that paper is about locating the limit, and this one about the shape of the help once it is located: place there, step here. Paper 194 showed that the four means are one family with a single equality fence——the minimum is the far end of that family, the power mean as the exponent goes to $-\infty$, and the line’s capacity falls there. Paper 300 showed that sharing a root is decidable——“the line is the minimum” and “the help is a step” are separate roots. What is added is measuring the ratio to the sum as $\mathbf{0.1231}$, listing five utilisations to put a number on idling being normal, confirming the switch from $\mathbf{1.0000}$ to $\mathbf{0.0000}$ to $12$ digits on both sides of the boundary, measuring the average as $\mathbf{0.1176}$, and placing the separator on whether every item passes through every stage.
1. Notation and the Question
Setting: five stages in series, every item passing through all of them in turn. Writing $c_i$ for the capacity of stage $i$ in items per hour, the line delivers
$$C=\min_i c_i$$Observation 1.1 no sum is taken. Five capacities are brought together and not added——one is chosen and the rest discarded.
Observation 1.2 nor is it a mean. It is none of the four means of Paper 194 but sits at the end of that family——the minimum is the power mean with the exponent sent to $-\infty$. The line’s capacity falls to the very outside of the family.
Question: how much does the line gain when the narrowest stage is widened?
2. The Line Is the Minimum
Computation 2.1: five stage capacities.
| stage | capacity (per hour) | utilisation |
|---|---|---|
| machining | $80$ | $\mathbf{1.0000}$ |
| painting | $100$ | $0.8000$ |
| assembly | $120$ | $0.6667$ |
| inspection | $150$ | $0.5333$ |
| packing | $200$ | $\mathbf{0.4000}$ |
| sum | $650$ | — |
| line capacity | $\mathbf{80}$ | — |
| ratio to the sum | $\mathbf{0.1231}$ | — |
Observation 2.2 the line matches the smallest. At $80$——another size entirely from the sum of $650$, a ratio of $\mathbf{0.1231}$.
Observation 2.3 only the smallest stage is fully used. The other four fall below one——packing reaches only $\mathbf{0.4000}$, standing idle three-fifths of the time.
Observation 2.4 idling is the normal state. Raising packing’s utilisation adds nothing to the line——in a serial line, idling anywhere but the minimum is unavoidable, and driving every utilisation up produces only inventory.
Observation 2.5 the sum does mean something elsewhere. Were the five in parallel, with each item needing only one of them, the line would deliver $650$——the same five numbers give $80$ in series and $650$ in parallel: a difference of more than $8.1$-fold produced by the arrangement alone.
3. The Help Has Nothing In Between
Computation 3.1: widening machining alone.
| machining capacity | line capacity | effectiveness over the interval |
|---|---|---|
| $80$ | $80$ | — |
| $90$ | $90$ | $\mathbf{1.0000}$ |
| $100$ | $100$ | $\mathbf{1.0000}$ |
| $120$ | $100$ | $\mathbf{0.0000}$ |
| $150$ | $100$ | $\mathbf{0.0000}$ |
| $250$ | $100$ | $\mathbf{0.0000}$ |
Observation 3.2 this is the core. The effectiveness is either $\mathbf{1.0000}$ or $\mathbf{0.0000}$, never between——both confirmed to $12$ digits, on either side of the boundary.
Observation 3.3 the switch is at the second narrowest stage. It falls to zero the moment painting’s $100$ is passed——the limit has moved from machining to painting, and no further widening of machining gets past painting.
Observation 3.4 the average is small. Widening by $170$, from $80$ to $250$, gains the line only $20$, an average effectiveness of $\mathbf{0.1176}$——above $0.1$ and below $0.2$; “about thirty per cent” set beforehand would have been wrong.
Observation 3.5 Paper 477 asks in a different place. There the subject was telling where the limit is——here it is the shape of the help at a limit already located: place there, step here.
Observation 3.6 a consequence of the step. There is no interval where widening half helps——estimating an improvement linearly understates it before the boundary and overstates it after. The error runs both ways.
4. The Separator
Observation 4.1 the separator is whether every item passes through every stage. If it must, the minimum; if it can be routed around, the sum——the same five numbers give $80$ and $650$. What decides is the arrangement, not the capacities.
Observation 4.2 a second separator is whether the capacities are constants. With fluctuating stages the line falls below the minimum——matching the minimum exactly requires no fluctuation and no starvation. Only that ideal side is examined here.
Observation 4.3 separate roots under Paper 300. “The line is the minimum” carries seriality——“the help is a step” carries the minimum moving to a different stage. With one stage only, the first survives and no step appears.
Observation 4.4 what is not said. An idle stage is not called waste——in a serial line idling is unavoidable, and removing it produces inventory.
5. Roll Call and Closing
- line capacity $\mathbf{80}$ per hour (Computation 2.1): the minimum.
- sum $650$ (Computation 2.1): a ratio of $\mathbf{0.1231}$.
- machining utilisation $\mathbf{1.0000}$ (Computation 2.1): only the minimum fills.
- packing utilisation $\mathbf{0.4000}$ (Computation 2.1): idle three-fifths.
- effectiveness $\mathbf{1.0000}$ from $80$ to $100$ (Computation 3.1): to $12$ digits.
- effectiveness $\mathbf{0.0000}$ beyond $100$ (Computation 3.1): the far side, to $12$ digits.
- average effectiveness $\mathbf{0.1176}$ (Observation 3.4): measured, then bracketed.
- in parallel, $650$ (Observation 2.5): arrangement alone, over $8.1$-fold.
- the separator is passage through every stage (Observation 4.1): minimum or sum.
- a second separator is constancy (Observation 4.2): fluctuation drops it lower.
Closing: The capacity of a line follows from the capacities of its stages. It follows, however, as a minimum rather than a sum or a mean——five stages of $80$, $100$, $120$, $150$ and $200$ sum to $650$, yet in series the line delivers $\mathbf{80}$ per hour, only $\mathbf{0.1231}$ of the sum. It is none of the four means of Paper 194 but the end of that family, the power mean with the exponent sent to $-\infty$. Consequently only the smallest stage reaches a utilisation of $\mathbf{1.0000}$, while packing sits at $\mathbf{0.4000}$ and stands idle three-fifths of the time. In a serial line this is unavoidable, and driving every utilisation up produces only inventory. And widening the narrowest stage helps with nothing in between——the effectiveness is exactly $\mathbf{1.0000}$ from $80$ to $100$ and exactly $\mathbf{0.0000}$ once painting’s $100$ is passed, taking no intermediate value, both confirmed to $12$ digits from either side of the boundary. The limit has moved from machining to painting, so widening by $170$ all the way to $250$ gains the line only $20$, an average effectiveness of $\mathbf{0.1176}$. “About thirty per cent” set beforehand would have been wrong. Because the help is a step, estimating an improvement linearly understates it before the boundary and overstates it after. The separator is whether every item passes through every stage——if it must, the minimum; if it can be routed around, the sum, so that the same five numbers give $\mathbf{80}$ and $650$. What decides is the arrangement, not the capacities. A second separator is whether the capacities are constants, since fluctuating stages drop the line below the minimum. Under Paper 300, “the line is the minimum” carries seriality while “the help is a step” carries the minimum moving elsewhere: with one stage only, no step appears. Placed beside Paper 477——that paper is about telling where the limit is, and this one about the shape of the help at a limit already located: place there, step here. To be honest——no factory improvement is recommended; where to invest was not stated, only what happens when a stage is widened. Capacities were taken as constants, changeovers and breakdowns were excluded with availability set to one, and the cost of widening was not discussed. An idle stage is not called waste either, since in a serial line idling is unavoidable.
References
- Buzacott, J. A., & Shanthikumar, J. G. (1993). Stochastic Models of Manufacturing Systems. Prentice Hall.
- Goldratt, E. M., & Cox, J. (1984). The Goal: A Process of Ongoing Improvement. North River Press.
- Hardy, G. H., Littlewood, J. E., & Polya, G. (1934). Inequalities. Cambridge University Press.
- Hopp, W. J., & Spearman, M. L. (2011). Factory Physics (3rd ed.). Waveland Press.
- Yamagishi, Y. (2026). The Four Means Are One Family, and There Is Only One Equality Fence [Paper 194]. Zenodo. 10.5281/zenodo.22106809
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). While the Wrong Input Is Being Added, the Limit Is Somewhere Else [Paper 477]. Zenodo. 10.5281/zenodo.22733686
Reference
Yamagishi, Y. (2026). Widening Helps Either Fully or Not at All, With Nothing In Between [Paper 550]. YOI: https://www.yamagishiyuuki.com/yoi/550
DOI: 10.5281/zenodo.23014187
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 550. This paper is a preprint and has not been peer reviewed.