Is my pareto steep enough?

The Pareto principle, commonly referred to as the 80/20 rule, suggests that a small percentage of contributors often accounts for a large portion of the results. In practical applications like warehouse solution design, it’s crucial to determine whether the Pareto curve is steep (dominated by a few contributors) or relatively flat (more evenly distributed outcomes).

While experienced analysts can often eyeball a Pareto curve and judge its steepness, this approach is prone to subjectivity and may be difficult for beginners. This article introduces a mathematical method to objectively quantify the steepness, offering confidence and consistency in interpretation.

Understanding the Pareto Curve, A Pareto curve is a graph that:

1. X-axis: Represents the cumulative percentage of contributors: SKUs

2. Y-axis: Represents the cumulative percentage of outcomes: sales, orderlines

3. The data is arranged in descending order of contribution to ensure the curve reflects the most impactful contributors first.

A steep curve implies that a small count of SKUs contribute to majority of quantity or orderlines, while a flat curve suggests all SKUs contribute quantity or orderlines equally (need to check both parameters separately).

Why Visual Inspection Falls Short?

An experienced analyst may quickly gauge whether a Pareto curve is steep, moderate, or flat by visual inspection. However:

1. Subjectivity: What appears steep to one expert may not to another.

2. No Numerical Proof: Eyeballing offers no quantifiable measure to back decisions.

3. Challenging for Rookies: Without experience, interpreting curves visually can be overwhelming.

This is where a mathematical approach becomes invaluable. By calculating the area outside the Pareto curve and using it as a measure of steepness, analysts can rely on consistent and objective analysis.

Quantifying Steepness Mathematically

1. The Total Reference Area: The area under the pareto curve is 1 when SKUs and quantity or orderlines are normalized in percentage form till 100%.

2. Area Under the Pareto Curve (AUC): The area under the Pareto curve (AUC) can be calculated using numerical integration techniques like the trapezoidal rule. This area represents the cumulative contribution of outcomes across contributors.

3. Area Outside the Pareto Curve (AOC): The area outside the curve is the difference between the total reference area and the AUC.

4. Area Outside = 1 - AUC

5. Steepness Index: The steepness index is derived from the area outside the curve. It quantifies the concentration of outcomes:

• Steepness Index = 1 - Area Outside

• Steep Curve: High steepness index (close to 0).

• Flat Curve: Low steepness index (close to 0.5).

The steepness index reaches a maximum of 0.5. When quantities or orderlines are evenly distributed across SKUs, the curve forms a diagonal and cannot fall below this line, as the values are arranged in descending order. Since the SKUs and quantities/orderlines are normalized to 100%, the maximum area under the curve is 1. Hence, the maximum available area outside the curve ranges from 0 to 0.5.