Orbit charts, and why you should use them

What I propose to call “Orbit Chart” is rarely used in manufacturing today, and I think it should be. This chart tracks the path followed by a system or an object in a plane where the coordinates are two of its characteristics. In earlier applications, there were geographical coordinates; in current ones, they may be the numerator and denominator of a financial ratio like Return On Net Assets (RONA), metrics of productivity and quality, or physical characteristics, like the depth and diameter of a drilled hole.

Most data visualizations in manufacturing are limited to simple charts, that show how one parameter varies by category, in  bar graphs or continuously as a function of time in line plots. Charts that show more dimensions require more effort both to generate and to the read, but reveal information that you would not otherwise find.

For details, click on the following:

What is an “orbit chart”?

When you have an object moving in a plane, you can separately plot each coordinate against time, but the juxtaposition of the two plots would not show you the path followed in two dimensions. When you want to study a possible relationship between two parameters from a table of values where both are measured on the same objects, you usually start part generating a scatter plot, from which you try to infer some form of correlation between the two. When you do this, however, you lose the sequence information. When you consider two neighboring points on your scatter plot, you have no way to tell whether they are measurements on two units of product made consecutively or with many others in between.

On an orbit chart, you connect the points that succeed each other in the sequence, and label each point with its sequence number. As a result, what you plot is no longer a cloud of points, but a path followed by your object in the plane of your two parameters. On a road trip, the sequence of your locations at the end of each day are not independent: where you are tonight is where you were last night plus today’s increment. In machining with a tool that wears out, its condition after the 50th workpiece is what it was after the 49th, plus the effect of the 50th. The orbit chart is a visualization tool for this kind of phenomenon.

Let us assume that you are plotting the quality performance of a production line, as represented by its first-pass yield, against its productivity in terms of units/operator/shift. If you are practicing management whack-a-mole, you improve quality at the expense of productivity, by adding inspections and rework, or productivity at the expense of quality, by pressuring operators to cut corners. In this case, you can expect the orbit of your production line in the Quality versus Productivity plane to be a cycle, looking somewhat like a figure eight, and showing no real improvement.

Orbit chart of Quality versus Productivity Whack-a-Mole

Orbit chart of Quality versus Productivity Whack-a-Mole

On the other hand, if you are practicing Lean, quality and productivity improve together, resulting in a chart that does not loop.

Plotting an orbit, of course, is not always meaningful. In the following example, we show a scatterplot of two parameters measured on a sequence of independent events. The scatterplot lends itself to correlation and regression analysis of the two parameters, but tracing the path of the values in the sequence of points as on the right is meaningless.

The orbit chart is meaningless if the points are independent

The orbit chart is meaningless if the points are independent

Examples of orbit charts

The examples below are not from manufacturing but from military history, nuclear power plant maintenance, macro-economics, and ecosystems analysis. 25 years ago, radar charts were used in magazine like Britain’s The Economist, to compare parameters like inflation, unemployment and growth in multiple countries, but were unknown in factories, where they now are commonplace. Such may be the fate of orbit charts as well. The caveat is that, like radar charts, they are richer and more sophisticated than the usual charts  you find on performance boards, and may be difficult for operators to relate to. As a result, they may be more useful as an analytical tools for engineers and managers than as a communication tool on the shop floor. The only way to find out is to try.

Minard’s Russia campaign chart

Perhaps the best known example of an orbit chart is Minard’s map showing the path of Napoleon’s army in 1812 Russia, on the offensive in brown, and retreating in black, with the thickness of the line showing the size of the army. It is annotated with dates, and with a temperature chart below. It was drawn in 1869, and Edward Tufte brought to the attention of readers in The Visual Display of Quantitative Information, as the most eloquent summary ever written about this disastrous campaign, and the mother of all infographics. Please click on the picture if you would like to enlarge it and read the text, knowing that it is in French.Minard's Russia campaign chart Tufte calls it a “narrative of space and time.” I prefer to call it “orbit chart” because (1) it is a shorter name and (2) the x and y coordinates do not necessarily represent space, nor is the index of the points always time. In manufacturing, for example, x and y could be quality characteristics,  indexed by the serial numbers rather than time.

Unplanned versus planned downtime in nuclear power plants

I used the orbit chart below on p. 282 of Working with Machines, to compare the maintenance performance of the Japanese and French nuclear industries in the 1980s:

France versus Japan nuclear plant downtime stats 1979-1989

Jean-Pierre Mercier, from the French electrical utility EDF, published this chart to compare the evolution of nuclear reactor downtime in Japan and France. Each node on each of the orbits indicates performance in a given year, in terms of planned unavailability on the y-axis, and unplanned unavailability on the x-axis, and the sum of the two gives the total unavailability, so that diagonal lines indicate a constant total value. The orbits enable us to track year-by-year progress, and improvement is marked on the chart by movement towards the origin. The Japanese and the French orbits both show improvement over the years but are so different that they do not even intersect, which begs the question of why. What did the Japanese and French industries do so differently that it produced such radically different results? Once the chart prompts you to ask the question, it is easily answered:

  • France has one national utility company, with reactors of just two designs all made by the same supplier. This company reduced downtime by redesigning components or subsystems that failed in one reactor, and retrofitting the changes to all sites of the same design.
  • Japan has nine private utilities and reactors with many different designs, which made the French approach impossible. Instead, they overdid preventive maintenance in the beginning, and gradually improved it, eventually achieving almost the same performance as the French utility.

GINI index versus GDP in Brazil and the US, 1980-2011

This was back in 1990. Fast forward to 2013, and I am reading a new book by Alberto Cairo, called The Functional Art, about the design of information graphics and visualization. On p. XIX of the Introduction, I find the following chart of the orbit of Brazil’s economy in terms of GDP on the x-axis and the GINI index of inequality on the y-axis through five presidential administrations: Brazil GINI versus GDP chart As explained on Wikipedia, the GINI index is a cleverly defined ratio, which is 0 if every member of a society has an equal share of its wealth, and 100 if it is all in the hands of a single individual. Worldwide, among the countries for which data is available, Sweden’s GINI index of 23 is the lowest, while South Africa and Lesotho have the highest, at 62. The US has gone from a low of 38 in the late 1940 to 47.7  in 2010. The GDP is a better known metric, and is here shown as evaluated in constant US$ by institutions external to Brazil, like the World Bank and the IMF.

Assuming the underlying Brazilian economic statistics are credible, this chart tells quite a story, from the peak of inequality with low growth during hyperinflation under Sarney to sustained growth with steady reductions in inequality under Lula, the only pause in growth coming with the financial crisis of 2008. Would we see anything similar from plotting the same chart for the US economy? I tried, and the result is as follows: US-Gini-index-versus-GDP-1980-2011What does this chart tell us? The first obvious conclusion are:

  1. Regardless of economic circumstances or the political affiliation of the president, inequality has steadily increased in the US for over 30 years.
  2. We must always be wary of highly aggregated numbers. The  US census bureau warns us that the formula for calculating the GINI index was changed for 1993, and that before-and-after comparisons are therefore not meaningful. On the face of the charts, it appears that even the high point of 47.7 for the US is lower than the low point of 53.8 for Brazil, but we would have to assume that the numbers are calculated the same way, which is doubtful. The formula was changed in the US; it may be different in Brazil and, unbeknownst to us, it may have been changed as well along the way.
  3. We can see, at the Bush(43)/Obama juncture, that the crisis of 2008 had a bigger impact on the US than on Brazil.
  4. While still roughly one sixth  the size of the US economy, Brazil’s is growing faster. Back in 1980, it was less than one tenth.

Populations of predators and preys

For a long time, orbit charts have been used in population dynamics, with the x-axis being a prey population and the y-axis a predator population. Orbit charts can represent both theoretical models and actual data, when these are available. The following example, from a course taught at Portland State University,  shows a simulation starting at the bottom right-hand side with a large prey population and few predators. This stimulates population growth for the predator, which depletes the prey population. This leads to a food shortage for the predators, causing their population to collapse, which in turn gives preys the opportunity to multiply again…

Orbit chart - Predator-prey limit cycle

Predator-prey limit cycle

Eventually, the two populations spiral down not to a stable point, but to a repeating loop called limit cycle.

Parker Hannifin’s “North-by-Northwest” chart

Parker Hannifin is a diversified manufacturing company that included the following orbit chart is in its 2012 annual report:

Orbit chart -- Parker Hannifin North-by-Northwest chart

Given the type of publication, the axes are unlabeled. Internally, this chart is also generated by division and monitored by General Managers. The black straight line represent constant RONA, and the desired movement is upwards and orthogonal to it, hence the nickname of “North-by-Northwest chart” given to it by managers. When it was first introduced a decade ago, it was not immediately understood, but it has taken root in the organization. I would connect the dots and annotate it as follows:

Orbit chart -- Parker Hannifin North-by-Northwest chart annotated

We can also see on this chart that, while the RONA improvements of 2010 and 2011 involved movement along both axes, in 2012, it was only a reduction in Net Assets/Sales, which is no doubt meaningful to someone familiar with the company’s operations.

Orbit charts for spacecraft

The term “orbit” of course is from astronomy, and I found a great example of an orbit chart drawn by NASA’s George Resteck of the path of the Pioneer 10 and 11 probes sent to explore the solar system in the early 1970s. As we can see on the chart, Pioneer 10 “only” managed to fly by Jupiter; Pioneer 11 flew by both Jupiter and Saturn, but crossed the other planets’ orbits far from where they were.
Pioneer 10 and 11 flight path
Only Voyager 2, launched 5 years later, managed to get close to all big four planets: Jupiter, Saturn, Uranus, and Neptune. The closest I could find to a similar chart for Voyagers 1 and 2 is as follows:

Voyager 1 and 2 flight paths

It is the same general idea, but with less details particularly on planet positions over time, and less information on scale. You can also find animations of their paths, but Resteck’s chart for Pioneer 10 and 11 still gives you the most information at a glance.

Recovery from crisis at Toyota versus GM

The following orbits of Toyota and GM profitability as a function of number of vehicles produced in recovery from crisis were included in a previous post:

GM sales and profits chart

Toyota sales and profits chart 2001-2011

The first shows GM through the growth of the twenties and the great depression; the second, Toyota through its 2001-2008 boom, followed by the financial crisis, the mass recalls of 2010, and the Fukushima earthquake and Thailand floods of 2011. It also shows how the economics of the auto industry changed in 80 years. In good times, today’s mature automobile industry yields profit margins that are barely 1/3 of what they used to be, on volumes that are many times higher. In the worst year of the great depression, 1932, GM made only 28% as many vehicles as in 1929. If the worst of the current crisis was in 2009-2010, Toyota’s drop in volume, while similar in absolute terms to GM’s in the great depression, was much smaller in relative terms, at barely 15% off from the 2008 peak.

The Toyota chart further shows three distinct periods:

  1. From 2001 to 2004, profit margins and volume rose together, suggesting that Toyota was enjoying some form of economies of scale.
  2. From 2004 to 2008, volume kept rising rapidly, but profit margins were flat. Toyota was criticized during that time for pursuing faster growth than it could manage.
  3. From 2008, the dominant effect is the financial crisis and recovery. with the 2010 recalls further reducing volume.

Janet Yellen’s orbit chart of inflation versus unemployment

David J (Joe) Armstrong pointed out to me an article from the New York Times on 10/9/2013 about Janet Yellen’s analysis of inflation and unemployment in the US since the Great Depression, using orbit charts and animations. The overall chart is as follows:

Inflation versus unemployment in US 19690-2013

See the articles for animations and explanations of the different segments.

Lora Cecere’s orbit charts of  inventory turns versus operating margins for Colgate, P&G, Walmart, and Target

David J (Joe) Armstrong also pointed out to me an article from Forbes Magazine on 12/13/2013 about supply chain collaboration, using orbit charts to show lack of progress. The charts are as follows:

Colgate and P&G inventory turns versus operatings margins

walmart versus target inventory turns versus operating margins

Read the article for Lora Cecere’s analysis of the significance of these charts. My first question about them is whether these companies compute the plotted parameters in the same way. Only if we can be sure that they do, can we venture some conclusions from the charts. No units are given on the operating margin axis. Given that it is a ratio, I assume “0.06” means 6%.

The key point is that, in 12 years, none of these companies moved towards the best scenario of a high number of inventory turns and a high operating margin. In the case of Walmart versus Target, they seemed to have had “orthogonal” strategies, with Walmart moving up and down in inventory turns without any large gains in margin, while Target was moving back and forth in margin without major changes in inventory turns.

I think there are many factors that these charts don’t show, such as the impact of the competitive environment and the financial crisis. While neither manufacturers like P&G and Colgate nor retailers like Walmart or Target have shown much progress, the impact of inventory on their operations is quite different. Overall, the manufacturing companies buy materials and generate their margins from selling finished goods; they may have some items on consignment, but only a minority.

By contrast, if I understand how very large retailers work, their income does not come from markups on products but from the float in accounts payable. They pay suppliers a month or two after collecting revenues from selling to consumers, which adds up to  a very large float, and the companies’ profits come from the income generated by this float in financial markets. In essence, their inventory is on consignment and their holding costs are nothing like the manufacturers’. They still have the costs of operating warehouses, but their own money is not tied up in materials paid upfront. On the other hand, their margins are sensitive to the health of the financial markets.

How to generate orbit charts in Excel

No matter how great orbit charts may be, not many people in manufacturing will use them unless they are easy to generate with Excel. Generating the orbit itself is not a problem. With Excel 2007 or 2010, all you have to do is, under the Insert tab, select Scatter and click one of the two options for Scatter with lines and markers.  Then you can use the various formatting options to refine the axes, gridlines, etc. Most of the charts in this post were generated this way.

Selecting scatter chart with straight lines and markers on Excel

Selecting scatter chart with straight lines and markers in Excel

As you can see in the following example, Excel does not mind the curve looping and spiraling:

Looping and spiraling chart generated with Excel

Looping and spiraling chart generated with Excel

Labeling the points on the charts is trickier. If there are few enough, you can manually add text boxes on the chart, which is what I did, but it would not work for thousands of points. In fact, for any large number of points, your only options are:

  1. Labeling every n-th point. 
  2. Making labels pop up next to a point when you hover on it or click on it.

The labeling option in Excel charts will display the numeric values for x and y or the name of the data series next to each point, which doesn’t tell you anything you don’t already know. You want to label each point with the value of its index in the data table so that, if it is time, you can know when each point was generated. Excel won’t do it, but Microsoft Support provides a Macro with which you can, with the result as follows:

Orbit chart with date labels affixed by Excel macro

Orbit chart with date labels affixed by Excel macro

In  Alberto Cairo’s chart, the successive presidencies of Brazil were marked by a different color. In Excel, you could achieve this effect by have having a separate data series for each presidency, which you could then color as you wish.

Orbit charts for multidimensional data

Visualizing two dimensions of the evolution over time of a group of machines or the output of a production line is an improvement over plotting just one. But what if, instead of two, you have fifty or even ten characteristics of interest?

You cannot see a point with 10 coordinates, but you can use dimensionality reduction techniques to work around this problem. Principal Component Analysis, for example, projects these multidimensional points onto a plane so that the projections contain most of the variability of the full multidimensional cloud. Linda E. Kavraki provides the following illustration of the concept:

a) A data set given as 3-dimensional points. b) The three orthogonal Principal Components (PCs) for the data, ordered by variance. c) The projection of the data set into the first two PCs, discarding the third one.

a) A data set given as 3-dimensional points. b) The three orthogonal Principal Components for the data, ordered by variance. c) The projection of the data set into the first two Principal Components

The coordinates on this plane are two uncorrelated linear combinations of the full set of coordinates called first and second principal components. Then you can plot the orbit of your population in this plane. Technically, it is straightforward, because you will easily find software packages to perform Principal Component Analysis. Minitab does it, and so does the XLSTAT add-in to Excel.

The challenge is making sense of the orbit chart. When you just plot the projection of your cloud of points onto the first two principal components, you may notice a small clump of points off to the side and identify them as outliers. But, when the points are generated over time, following the orbit may not tell you much because the coordinates are linear combinations of the original coordinates with no obvious meaning. The first principal component could well be three times the length of the ship minus half the captain’s age.

Metrics in Lean – Chart junk in performance boards and presentations

Manufacturing professionals who read Edward Tufte‘s books on visualization may be stunned to discover that their 3D pie charts, stacked bar charts, and green safety crosses are chart junk. These charts are common, both on shop floor performance boards and in management presentations. But they are information-poor, and their decorative elements distract, confuse, and occasionally mislead.  The purpose of plotting is not to dazzle, but to discover patterns, understand the underlying phenomena, and communicate with people whose livelihood is affected by these patterns.

For details, see:

Pabulum pies

Following is an example of what Microsoft Support considers to be a pie chart with a “spiffy, professional look,” and a few comments about it.
This chart not only uses buckets of ink to display five data points, it also violates other good design principles:
  1. The legends are remote from their objects, forcing the reader to look in two places to understand each item.
  2. The chart begs a question that it could answer but doesn’t: percentages of what total amount?
  3. The shadow and the light reflection convey no information.
  4. The 3D effect makes the 21% slice appear larger than it actually is.
If we put the legends on their objects, add the total amount as a subtitle, get rid of the vinyl-cushion look, and take a top view so that the wedge sizes are not distorted, we get the following:
Microsoft pie chart example - improved

One feature that now stands out is that the wedges are ordered by decreasing size, except for the last two: “Desserts” is larger than “Beverages” but comes after. It is the example Microsoft uses for training and they don’t explain the sequence. While this chart is an improvement over the previous version, but do we actually need it? A picture may be worth a 1,000 words but, if what you have to say fits in 10 words, a picture may be overkill. Compare the pie chart with the following table of the data in it:

Microsoft pie chart example - data table

If you compare a pie chart with a sorted table of the data in it, you see that the chart uses orders of magnitude more ink in multiple colors than the table, without telling you anything that isn’t already obvious from the table. In Edward Tufte’s book, it makes it the kind of chart junk that you should banish from your materials.

3D pie charts are actually worse, because they are not only useless but misleading in that the perspective distorts the apparent size of the wedges. This is an illustration of another of Tufte’s principles, that a graphic display should not have more dimensions than the data. In 3D pie charts, the height of the pie is meaningless. The only circumstance in which showing pie thicknesses would be useful if when showing several pie charts side by side, and both pie diameter and thickness would represent parameters of each pie.

Stacks astray

Vertical bar charts, also known as column charts, usually serve to show the evolution of a quantity for which data is available for a sequence of periods. It is used when there are few periods and interpolation between periods does not make sense. If you plot daily sales over a year in a vertical bar chart, the columns will be so densely packed that you will only see the line formed by their tops anyway, and you might as well just use a line plot.
Bar charts - Daily sales as bar versus line plot
On the other hand, if you are plotting monthly sales, you use vertical bars, because interpolation between two points does not give you meaningful intermediate numbers. If you plotted temperature readings taken at fixed intervals, for example on a hospital patient, you would interpolate to estimate the temperature between readings, and therefore you use a line plot rather than vertical bars.
Bar chart of monthly sales

Bar chart of monthly sales

There is nothing objectionable to the ordinary, vertical bar chart. It is simple to produce and easy to read, and provides information that is not obvious in a table of numbers. Stacked bar charts, however, are another matter. They attempt to show the evolution over multiple periods of both an aggregate quantity and its breakdown in multiple categories, and do a poor job of it, especially when a spurious 3rd dimension is added for decoration.
We can retun to the Microsoft support web site for a tutorial on stacked bar charts. The building-block look of the Microsoft example may be appropriate for an audience of preschoolers.

Another oddity of the Microsoft example is that it does not follow the convention by which vertical bars are used when the categories on the x-axis represent consecutive time buckets. When they are not ordered categories, you usually prefer horizontal bars, not only because we are used to the x-axis representing time in performance charts but also because this chart has horizontal category labels, readable without tilting your head.

Based on the preceding section, the first question we might ask is whether we need graphics at all, and the answer is yes. When we take a look at the data in table format, even though there are only 16 points, no pattern is immediately apparent:

Stacked bar Microsoft example source table

If we forget the spurious 3rd dimension and the building-block look, and toggle the axes so that the x-axis represents time, we get the following:

Stacked bar Microsoft example 3D removed

From this chart, it is obvious that total sales collapsed from the 2nd to the 4th quarter; it was not obvious from either the table of number or the 3D chart Microsoft presented as a model. Even in this form, however, the stacked bar chart is ineffective at answering the immediate follow-up question of whether the decline in sales was more pronounced in some regions than others. For example, if we try to isolate the Northeast on this chart, we find bars floating at different heights. On the other hand, the answers become visible if you de-stack the bars, as in the following:

Stacked bar Microsoft example unstacked

For example, you can see that sales actually grew from the 1st to the 2nd quarter in both Eastern regions, while declining throughout the year everywhere else. Yes, it takes up more space than the stacked bar, but is that really a concern when, even in a simple example like this one, it lets you see better?

Safety crosses to bear

The safety performance of a plant matters, and not only to the potential victims of accidents. Injuries must be rare events, occur with ever decreasing frequency, and have both immediate countermeasures and permanent solutions to prevent recurrence. In light of this, what kind of graphic summaries would be appropriate to represent the safety performance of a whole plant, or of a shop within it?

A common metric is the number of consecutive days without a lost-time accident. It is a relevant measure, but imperfect in that its use has been known to lead thoughtless managers to blame victims for hurting their department’s performance and pressure them not to report injuries. It is also necessary to show detailed information about each accident, and to categorize injuries, for example, in terms of whether they affected hands, shoulders, feet, etc. and where exactly they occurred.

In light of these considerations, what is industry using? In the US, the National Safety Council uses a green cross in its logo and awards a Green Cross for Safety Medal to one company each year. That makes the green cross a symbol of safety, and it has motivated some to make it the basis of a safety performance tracking chart. You start with a cross shape subdivided into rectangles numbered 1 through 31 and, on each day of the month, you place a magnet on the corresponding spot, which is green if nothing happened, yellow if a minor incident happened, and red for a lost time accident.

It is difficult to think of the use of a chart in the form of a symbol of safety as anything but a gimmick. The green cross has no connection with any requirements we can think of for a chart of safety performance. It does not make it particularly easy to count consecutive days without incident, nor does it bear any information about the nature or location of the accidents. The green cross shape evokes safety but has nothing to do with the key questions about employee safety.

To visualize a sequence of rare events, a technique that comes to mind is the timeline. You arrange events around a centerline that shows the passage of time, as in the following example that summarizes 10 years of the history of the iPhone.

Timeline 10 years of iPhone history

Safety-related events in a factory do not need a 10-year timeline, but possibly a six-month timeline along the following model:

Timeline of sports events january-june 2010

Note that the time between events is immediately visible, and that each event has some explanation and photographic documentation. For location information, you can pin injury locations on an outline of a human body and a map of the shop floor. While these may not be pleasant charts to consider, they are a means of starting a conversation in the team on what the safety issues are and the means of preventing injuries.

There are, of course, other safety issues, like repetitive stress, that are not associated with discrete events and do not appear on these charts, but they do not appear on the green crosses either.

Recommendation on performance board design

Performance boards come in all sorts of shapes, as in the following examples:

As a performance board for a shop floor team, I recommend the following template:

Performance board template

Performance board template

This template has one column per dimension of performance and one row for each type of information, as follows:

  1. The top row is for the latest news, what happened this shift or last.
  2. The second row is for historical trends in aggregate performance.
  3. The third row is for a breakdown of the aggregate into its constituent parts, such as the most common injuries, defect categories, most frequently made products, or the employee skills matrix.
  4. The bottom is about actions or projects in progress in each of these areas.