Hi there! In this post you will find overview of triangle method in ABC-analysis, its pros and cons, and advantage over classical methods.

Firstly, remember basics of Classical ABC-analysis.
The simplest method of classification can be called method of segmentation on three equal groups by quantity of positions or by value of result.

However, this method is only useful for even diminishing series (arithmetical progression). In practice, such distributions are rather exceptions. In examples below, you will see how irrational can be segmentation on equal groups.

Pareto rule create strong foundation for ABC-analysis – misbalance principle, which demonstrates not only asymmetric relation 20:80, but in addition allows using this asymmetry for more accurate classification.

Relative to ABC-analysis Pareto rule can be interpreted as: management 20% of positions allows to control whole system on 80%.
For example, examination of company products shown that 10% of goods creates 70% of expenses (group A), 20% – 20% of whole expenses (group B), and 70% of whole range of goods, that creates only 10% of expenses (group C). Thus, obviously, that company management must very carefully control income, moving and storage of goods from group A. Relative to group B control can be operational; relative to C – periodical.
Worth to note, that in different literature can be applied other percentage for segmentation, i.e. A – 15%, B – 20%, C – 65%. Such methods known as empirical or classical ABC-analysis.

To process a classical ABC-analysis we have to draw Pareto chart. Firstly, sort objects by descending result, then build diagram of cumulated total.

Then axis of diagram should be normalized by following formulas

On built Pareto Chart we can separate group A – usually it is 10% positions, group B – 20% positions, and C – 70%.
This is Classical ABC-analysis, based on Pareto rule. As already mentioned, in practice Classical Method gives more accurate results than division on equal groups, because Pareto rule empirically contained in structure of many business processes. However, rule 80:20 not always works and in such cases Classical method gives bad results. For common problem of classification, series of methods were developed, and one of them is Method of Triangle.

# Trap of Pareto

Let’s consider two simple sets of data and build Pareto Chart for them. But firstly, introduce definition of Pareto Point.
Definition. Pareto point – point on Pareto Chart with coordinated (Xp, Yp), for which works equality Xp+Yp=100%.
In table below offered two sets of data ordered by result value

 Position No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Set 1 610 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Set 2 500 500 500 500 500 500 500 500 24 24 24 24 24 24 24 24 24 24 24 24

Set 1 contains one position with very high value of result and nineteen positions with low value. Set 2 contains eight positions with high value of result and twelve positions with low value.

Pareto Charts and coordinates on Pareto Points shown on graphs below:

As an example, make segmentation of these sets on groups A, B and C using Classical Method.

 Position No A (10%) B (20%) C (70%) Set 1 610 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Set 2 500 500 500 500 500 500 500 500 24 24 24 24 24 24 24 24 24 24 24 24

From table we may see that in Set 1 group A contains 1 position with high value and 1 position with low value.
In Set 2 group C contains 2 values with high value and 12 positions with low value.

This analysis shows that:
• For some sets of objects Classical Method gives irrational segmentation (i.e. Set 2). Therefore, Classical ABC-analysis cannot be applied to any process.
• Pareto Chart visually shows level of position importance. Positions with high resulting values places on vertical (sharp) segment due to high contribution in total result. Positions with low resulting values – on horizontal (flat) part of chart.
• As closer Pareto Point placed to upper right corner of diagram quadrant, then segment of high values becomes more vertical, and horizontal part of chart – more flat. It means that part of high resulting values becomes smaller, in the same time part of “low positions” larger.

From this summary, we can make following conclusion: groups size depend on coordinates of Pareto Point, and rule 20:80 cannot be used literally. In practice, point of imbalance (Pareto Point) can have other coordinates and Classical ABC-analysis in such cases gives bad results.

Therefore, we need to find more advanced method of approximation in ABC-analysis.

# Method of triangle

In basis of this method lays fact that size of Pareto Groups depends on coordinates of Pareto Point. Desiring to receive formulas of dependence, we will consider all possible Pareto Charts going through Pareto Point (Xp, Yp). Notice, that set of all Pareto Points lays on line y=100%-x, which drawn on diagram by dashed line.

Two ultimate Pareto Charts going through point (Xp, Yp) have shape of two-section broken line, drawn on picture. Set of all Pareto Charts, going through (Xp, Yp) lays in area of two triangles built by these lines.

From picture we may see that on two-section broken lines part of positions with high resulting values vary from 0% to (Xp/Yp)*100%, and part of positions with low resulting values from 100% to (1-Xp/Yp)*100% correspondently.

These ultimate values can be used for determination of ABC boundaries. For this purpose we should draw two lines, parallel to diagonal going through Pareto Point (dashed line), through knees of two-section broken lines.
Points of crossing of these lines with Pareto Chart are desired boundaries of A, B and C groups.

Equations of lines can be expressed through Pareto Point (Xp, Yp):

Simplifying algorithm of boundaries search we can take middle points of segments DE and FG instead of looking for crossing with Pareto Chart. Then boundaries of ABC can be calculated by formulas:

Let’s check that formulas 5 and 6 satisfy to boundary conditions of Pareto Point:

• If Pareto Point tend to (0%, 100%), then by formulas 5 and 6 group A tend to 0%, B – to 0%, C – to 100% positions. It corresponds to goals of ABC analysis, where group with high resulting values get on almost vertical part of Pareto Chart, which projection on axis X tend to 0%, however group with low resulting values get on almost horizontal part of diagram and takes almost 100% values.
• If Pareto Point tend to (50%, 50%), then using formulas 5 and 6 group A tend to 0%, B – 100%, C – 0%. In this case Pareto Chart tend to diagonal line, and this corresponds to even distributed series. In this case all objects similar by resulting value and should go to one group. In addition, we cannon say is it bad group or good, thus better to take everything in group B. And method of triangle gives us such result.

Therefore, we can conclude that method of triangle makes narrow area of approximation and satisfies to boundary conditions of Pareto Point.

In the end, let’s consider how method of triangle works for distributions used in the beginning of post and compare results with simplest and Classical methods.

# Conclusion

In this post shown method of triangle for ABC-analysis, made comparison of this method with Classical and Simplest methods. We could see that Method of Triangle gives lower level of mistake among considered methods.

Original of article: http://quantresearch.ru/2012/metod-treugol-nika-v-avs-analize/

In next post I’ll show how to create calculation model in Excel for ABC-analysis with method of triangle.