## Achieving symmetry and obscurity

##### Dec 17, 2022

The following diagram found in an article on a logistics problem absorbed me for the larger part of an hour:

I haven't seen this chart form before, and it looks cute.

Quickly, I realize this to be one of those charts that require a big box "How to read me". The only hint comes in the chart title: the chart concerns combinations of planning problems. The planning problems are listed on the left. If you want to give it a go, try now before continuing with this blog post.

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It took me and a coworker together to unpack this chart. Here's one way to read it:

Assume I want to know what other problems the problem of "workforce allocation" is associated with. I'd go to the workforce allocation row, then scan both up and down the diagonals. Going up, I see that the authors found one (1) paper that discusses workforce allocation together with workforce level, two (2) papers that feature workforce allocation together with storage location assignment, etc. while going down, I see that workforce allocation is paired with batching in two papers and with order consolidation & sorting in one paper.

You may recognize the underlying data as a type of correlation matrix, which is commonly shown as an upper or lower triangular matrix. Indeed, the same data can be found in a different presentation in the same paper:

All the numbers are the same. What happened was the designer transformed the upper triangular matrix into an inverted (isoceles) triangle, then turned it aside. The row labels are preserved, while the column labels are dropped. Then, the row labels are snapped to cover the space which was formerly the empty lower triangular matrix.

A gain in symmetry, a loss in clarity.

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Why is this cute, symmetric arrangement so much harder to read? It's out of step with the reader's cognitive path. The reader first picks a planning problem, then scans up and down looking for the correct pair.

Compare this to the matrix view: the reader picks a pair of problems, then finds the single cell that gives the number of articles.

One *could* borrow the reading strategy from the matrix, and proceed like this:

The reason why this cognition path doesn't come naturally is that there is only one set of labels on this triangular chart, compared to two sets in the common matrix format. It's unusual to have to pick out two items simultaneously from a single axis.

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In the end, even though I like the idea of inducing symmetry, I am not convinced by the result.

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The color scheme for the cells is also baffling. According to the legend, the dark color indicates research that solves a pair of problems in an integrated way while the light color is used when the researchers only analyze the interactions between the two problems.

What's odd is that each cell (pair of problems) is designated a single color. Since we expect researchers to take the different approaches to solving a given pair of problems, we deduce that the designated color represents the most frequent approach. What then does the number inside each cell represent? It can be the number of papers applying the color-coded solution approach, or it can be the total number of papers regardless of the solution approach.

P.S. [12-18-2022] See comments below for other examples of the triangular chart.

It reminds me of a house of quality that has fallen over 8-).

https://www.whatissixsigma.net/house-of-quality-qfd/

Posted by: John Rouillard | Dec 17, 2022 at 10:09 AM

JR: Thanks for the link. I was trying in vain to find other instances of this chart form, and you found one!

Posted by: Kaiser | Dec 18, 2022 at 02:57 PM

I found it familiar, perhaps because it reminded me of the distance charts I might have seen on a maps as a child. Here's an example:

http://www.iraqwho.com/pic/ditances.gif

[ed: link may take a while to load]

Posted by: Michiel Duvekot | Dec 18, 2022 at 04:20 PM

MD: That's a great example. In this example, they did not make it symmetric, and allowed the text labels to go diagonal. The most important thing is to cue readers to pick out two labels.

This works well if the distances are symmetric e.g. euclidean distances. I'm wondering how to adapt it for practical distances - it's likely that one direction takes more time to travel than the other direction, depending on size of city, and time of day.

Posted by: Kaiser | Dec 18, 2022 at 06:06 PM