Unchanged region: values you checked, not a range you proved

Sweep one assumption and watch one answer. Where it holds, the values line up into a region that reads like a guarantee. It is not one: narrow the step and a third answer appears.

Live result

Your list returns 2 answers, and the same range read at the finest step this panel carries returns 3.

step in your list300
first change, your list800 to 1,100
finest step this panel reads25

A wider step can hide a change that a narrower one finds.

Try a step

The quantity you are assuming

units

Fromunits

Tounits

In steps ofunits

It does not check the values in between.

Across the values sampled, which quote is cheaper stays the rate only quote from 200 units, the lowest value sampled to 800 units. The nearest value where it changes is 1,100 units, where it becomes the fee and rate quote.

The answer	the rate only quote
Values sampled	200 units to 1,700 units, 6 values, step 300 units
Holds from	200 units, lowest sampled
Holds to	800 units
Nearest value that changes it	1,100 units
The answer there	the fee and rate quote

What an unchanged region does and does not prove is set out in a short lesson.

The sweep samples only the values you entered. Between samples the answer is not checked, so read the boundary as observed, not proven.

The same range, at the shortest step this panel reads

The shaded span connects checked values with the same reading as at 800. It does not prove that the reading holds between checked values.

the rate only quote	200 to 925
too close to call	950 to 1,050
the fee and rate quote	1,075 to 1,700

What the region is a fact about

A sweep evaluates a model at listed values and reports exactly what came back.

The region between those values is a shape, not a measurement of every value it covers.

Its width follows the distance between listed values, so a wide region can mean stability or a thin list.

A narrower range around a boundary makes the boundary claim stronger.

The subject changes and the instrument does not. The first sweep moved a quantity along a straight line of money. This one moves a delay along a timetable of two departures, where the answer can only be one of a few names.

A delay meets a finite departure schedule

This second example keeps every time fixed except the entered delay. It selects the earliest one of the two stated departures that is at or after the ready time. It does not infer another departure when both have passed.

The path is ready at 6. With a delay of 0 it takes the departure at 6. Delays from 1 to 4 take the departure at 10. Above 4, the entered schedule is exhausted.

The times held fixed

First departure plus first travel	$0 + 5$
Transfer time	1
The complete downstream schedule	6 \| 10
Downstream travel	3
Number of entered departures	2

The example describes a missed connection as later waiting, following the mechanism discussed in Missed Connections and Later Waiting, Transit Cooperative Research Program, 1997. The source describes the mechanism and does not validate these invented times.

Entered first leg delay

time units

Fromtime units

Totime units

In steps oftime units

It does not check the values in between.

Across the values sampled, which departure the path selects stays the departure at 10 from 1 time unit to 4 time units. The nearest value where it changes is 0 time units, where it becomes the departure at 6.

The answer	the departure at 10
Values sampled	0 time units to 5 time units, 6 values, step 1 time unit
Holds from	1 time unit
Holds to	4 time units
Nearest value that changes it	0 time units
The answer there	the departure at 6

What an unchanged region does and does not prove is set out in a short lesson.

Every admissible integer in this range was checked. Values outside the range remain unchecked.

The same panel, on three live pages

Document workback
Unlocks the correction allowance for the selected document.
Reorder point
Unlocks the spread in .
Service level
Unlocks the cost of one stockout while watching where the sits.

The same panel appears again on the counterexample shelf, where the assumption being opened belongs to a rule somebody else stated.

Here is the maths

The formula, in words

The whole model is one subtraction. The quote that carries a fee starts 900 higher and gains 0.90 a unit on the other one, so the gap between them is 900 less 0.90 a unit and it falls in a straight line.
The crossing is where that gap is zero. 900 divided by 0.90 is 1,000 units, and that is the only quantity at which the two quotes cost the same.
The band is where the gap is smaller than the tolerance. Two quotations that differ by 50 on a job of this size differ by less than either of them rounds by, so the page names that condition rather than picking a winner inside it.
A sweep is a list and a comparison. Build the list of values, evaluate the model at each one, compare each answer with the answer at the value you started from, and report the run either side of it that matched.

\Delta = 900 - 0.90 \times Q

where

\Delta

is the difference and

Q

the quantity in units.

Q_* = \frac{900}{0.90} = 1{,}000

where

Q_*

is the crossing.too close to call

\lvert\Delta\rvert \le 50

band

850 \le 0.90 \times Q \le 950

A worked example

At 800 units the quote with a fee costs $900 + 3.20 \times 800 = 3{,}460$ and the quote without costs $4.10 \times 800 = 3{,}280$ . The gap is 180 in favour of the quote without a fee.

The sweep on this page starts at 200, ends at 1,700 and steps by 300, so it evaluates six values: 200, 500, 800, 1,100, 1,400 and 1,700. The gap at those six is $720$ , $450$ , $180$ , $-90$ , $-360$ and $-630$ . Three positive, three negative, and the sign turns over between 800 and 1,100.

At a step of 25 the same range gives sixty one values, and five of them return a third answer: 950, 975, 1,000, 1,025 and 1,050, where the gap is $45$ , $22.50$ , $0$ , $-22.50$ and $-45$ , all inside the tolerance of 50. The panel now reports that the answer holds from 200 to 925 and changes at 950.

Solving instead of sweeping puts the band at 944.4 to 1,055.6 units, so at whole quantities it runs from 945 to 1,055, a hundred and eleven of them. The six value list caught none. The sixty one value list caught five. Neither list found the ends of the band, because neither list was asked about them.

What this does not model

It does not model a quotation. A real one carries validity, minimum quantities, surcharges, currency, credit terms and a delivery date, and any of them can decide the comparison before the arithmetic starts.
It does not choose the tolerance. 50 is stated by this page because two estimates of this size do not separate below it. On your own numbers that figure is a judgement and it moves the band.
It does not know the quantity. The whole page exists because the quantity is assumed rather than known, which is what makes it an assumption and worth sweeping.
The longer list is not the answer either. Sixty one values is sixty one values. Its region has the same standing as the six value one, and it was drawn by the same function.
A sweep says nothing about a model's fitness. It reports where a model's own answer turns over. Whether the model was the right one to ask is a different question and this page does not touch it.

Check it yourself

Set the step to 25 and read the region, then keep the step at 25 and set the bounds to 700 and 1,000. The second sweep evaluates thirteen values, and it puts the change in the same place, between 925 and 950, on a range a fifth as wide with a fifth as many values. Narrowing the range is what makes a boundary claim stronger. Sixty one values had already put the change there and could not make it any tighter.

The two quotes on this page are invented for it and describe no real offer. The page compares two lines of arithmetic and nothing else: it does not price anything, it does not say which quote anyone should take, and it does not model what a real quotation carries. A sweep, on this page and everywhere else on this site, reports the values it evaluated. It has no view about the values between them.