First in first out: the record says oldest, the yard newest

Type the receipts and the picks in the order they happened, and set how many lots a lane will hold. Each receipt takes a lane permanently and a pick takes whatever is on top of the lowest lane still holding something, so the order lots leave in is not the order they arrived in. The page counts the pairs that come out reversed, names the lot the rule puts on top next, and shows the count falling to zero at a height of one.

Live result

At a height of 2 the rule sends 2 pairs out in reverse of the order they arrived in. The lot it puts on top next is A.

The two order rails

Try a record

Height limit2lots

How many lots one lane will hold. Above eight this page stops, because the binding question stops being height.

Edit the event record

Events

Event 1

Event kind

Lot name

Your own label for this lot. Spaces at the ends are trimmed, so two lots cannot look identical.

Event 2

Event kind

Lot name

Your own label for this lot. Spaces at the ends are trimmed, so two lots cannot look identical.

Event 3

Event kind

Lot name

Your own label for this lot. Spaces at the ends are trimmed, so two lots cannot look identical.

Event 4

Event kind

Lot name

Your own label for this lot. Spaces at the ends are trimmed, so two lots cannot look identical.

Event 5

Event kind

Lot name

Your own label for this lot. Spaces at the ends are trimmed, so two lots cannot look identical.

Event 6

Event kind

Every lot

1

Lot: A
Lane: 1
Position in the lane: 1
Left in the replay: yes
Leaves at: 2

2

Lot: B
Lane: 1
Position in the lane: 2
Left in the replay: no
Leaves at: 1

3

Lot: C
Lane: 2
Position in the lane: 1
Left in the replay: yes
Leaves at: 4

4

Lot: D
Lane: 2
Position in the lane: 2
Left in the replay: yes
Leaves at: 3

5

Lot: E
Lane: 3
Position in the lane: 1
Left in the replay: yes
Leaves at: 5

Every lot
Received	Lot	Lane	Position in the lane	Left in the replay	Leaves at
1	A	1	1	yes	2
2	B	1	2	no	1
3	C	2	1	yes	4
4	D	2	2	yes	3
5	E	3	1	yes	5

The two orders

Height limit

Value: 2

Lanes in use

Value: 3

Pairs that leave in reverse

Value: 2

Lot on top next

Value: A

Outgoing order

Value: B, A, D, C, E

Receipt order

Value: A, B, C, D, E

The two orders
The two orders	Value
Height limit	2
Lanes in use	3
Pairs that leave in reverse	2
Lot on top next	A
Outgoing order	B, A, D, C, E
Receipt order	A, B, C, D, E

How many lots a lane will hold

lots

Fromlots

Tolots

In steps oflots

It does not check the values in between.

No other integer in the range leaves how many pairs come out reversed at 2. Every admissible integer in this range was checked.

The answer	2
Values sampled	1 lot to 8 lots, 8 values, step 1 lot
Holds from	2 lots
Holds to	2 lots
Nearest value that changes it	1 lot and 3 lots
The answer there	0 / 4

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.

Here is the maths

The rule used on this page

Receipts are numbered in the order you type them, starting at one on this page and at zero in the calculation. Each receipt is assigned permanently to a lane, which is its number divided by the height limit and rounded down. A receipt is placed on top of whatever its lane already holds. A pick takes the top lot from the lowest numbered lane that still holds something. A gap made by a pick is never backfilled and no lot is ever moved between lanes. That rule is the page's, it is deliberately small, and it is not taken from either source.

\ell_i=\left\lfloor\frac{i}{h}\right\rfloor

where

\ell_i

is the lane number in the calculation,

i

the receipt number starting at zero, and

h

the height limit.

A worked example

At height 2, receipts A and B enter lane 1, C and D enter lane 2, and E enters lane 3.
The pick takes B from the top of the lowest occupied lane. A is then the lot on top next.
Draining the rule gives B, A, D, C, E.
The pairs B with A and D with C leave in reverse, so the count is 2.

What a reversed pair is

Two lots form a reversed pair when their order in the full outgoing sequence is the reverse of their order in the receipt sequence. The number this page reports is a count of pairs, not a count of lots that came out in the wrong place and not a count of moves anybody would have to make. In the worked example four lots are involved across two pairs, and the answer is two.

Nothing here counts a relocation

The rule never moves a lot to reach another one, so there is no relocation to count and this page reports none.

The two answers move independently

The pair count and the lot the rule puts on top next are different answers about the same record. On six receipts with a pick after the third and another after the fifth, heights three and four both give six reversed pairs while the next lot changes. The sweep watches the count and the finding names both.

Check it yourself

Set the height to 6 and leave the sweep at 1 to 8 in steps of 1. The answer holds from 5 through 8, because five receipts already fit in one lane and raising the ceiling of a lane that is not full changes nothing. Then set the height to 1. The count is zero, and it is zero only there.

Counting the crossings on the chart

Every place two connectors cross is one reversed pair, and the marked crossings stop at twelve. Where three connectors meet at one point the marks sit on top of each other, so read the number from the table rather than from the drawing.

The calculation domain

The calculation accepts a height limit from 1 to 8 and 1 to 100 events with lot names up to 80 characters. This page offers 4 to 14 events and shorter names because the chart draws one cell for every receipt.

Boundary

It replays the events you typed under the one lane rule stated on this page. It does not determine how a real yard is laid out, whether a lot can be reached, or what an inventory record ought to assume.

Source anchors

Auburn University, an open dissertation on block stacked warehouses. United States Government Accountability Office, inventory system requirements. The first describes lots placed on one another on a floor and what that does to reaching them. The second sets out first in first out as an assumption an inventory record makes rather than a description of a yard. The rule on this page is the page's own and every event on it comes from your own typing.

Calculation version

1.0.0

The whole record prints. Your browser can save it as a PDF instead of printing it.

This page will never prescribe an order for taking lots out, count the moves anybody would have to make, lay out a floor, or call the lot it names next the lot that should be issued. It describes what its own stated rule produces from the events you typed. It does not determine that a lot is accessible, safe, conforming, released or within its shelf life.

Stack order record

First in first out: the record says oldest, the yard newest

Accepted inputs

Height limit: 2
Event 1: Receipt A
Event 2: Receipt B
Event 3: Receipt C
Event 4: Receipt D
Event 5: Receipt E
Event 6: Pick

All results

Height limit: 2
Lanes in use: 3
Pairs that leave in reverse: 2
Lot on top next: A
Lots already picked in the replay: B
Outgoing order: B, A, D, C, E
Receipt order: A, B, C, D, E
Lot record 1: A, 1, 1, 2, yes
Lot record 2: B, 1, 2, 1, no
Lot record 3: C, 2, 1, 4, yes
Lot record 4: D, 2, 2, 3, yes
Lot record 5: E, 3, 1, 5, yes

Sweep of How many lots a lane will hold

The assumption: How many lots a lane will hold
Values sampled: 1 to 8, 8
The answer: 2
Holds across: 2 to 2 lots
Nearest change: 1 lots, 0 and 3 lots, 4
Reading the boundary: The sweep samples only the values you entered. Between samples the answer is not checked, so read the boundary as observed, not proven.

Record fields

Calculation version: 1.0.0
Boundary: It replays the events you typed under the one lane rule stated on this page. It does not determine how a real yard is laid out, whether a lot can be reached, or what an inventory record ought to assume.
Calculation trace: Number every receipt in typed order and assign it permanently to a lane from the height limit. Replay every pick from the top of the lowest numbered occupied lane. Drain the same rule to form the full outgoing order. Count every pair whose outgoing positions reverse its receipt positions. Name the top lot in the lowest numbered lane that remains occupied.
Source anchor: Auburn University, an open dissertation on block stacked warehouses. United States Government Accountability Office, inventory system requirements.