by Philip Mayfield
Quantum XL's Latitude Plots are a visual depiction of the relationship between the expected variation and the latitude window. If used properly, Latitude Plots are an exceptional communication technique, but can be confusing to the uninitiated. This article introduces Latitude Plots and explains how to read them.
To serve as an example, we will use the stack of two parts.
The sum of the heights of Part A and Part B equals the total height of the stack. The equation for the stack height is therefore Stack Height = Part A + Part B. If we know that Part A ~ N(23,22) (normally distributed with a mean = 23 and a standard deviation = 2), Part B ~ N(7,12), and that the stack's Lower Specification Limit (LSL) is 25cm with an Upper Specification Limit (USL) equal to 35cm, then we can easily create this model in Quantum XL.
Running the Monte Carlo simulation for this model produces the following Histogram.
The vertical red lines represent the spec limits (LSL = 25, USL = 35), and the area in red represents simulations out of specification, sometimes referred to as defects. Thus far, this has been "cookie cutter" Monte Carlo model building and simulation. Let's make a Latitude Plot of this by selecting "Quantum XL" - "Latitude Plot".
Below is the Latitude Plot for this example. The plot consists of the following items.
Red Box = Expected Variation Region
Area Between LSL and USL Lines = Latitude Window
Blue Dots = Simulations in specification
Red Dots = Simulations out of specification
Each of these items is explained in more detail below.
The first item of interest are the variables which are plotted on the X and Y Axes. The X Axis is the dimension of Part B and the Y Axis is the dimension of Part A. The output (stack height) is not plotted on either axis. In the center of the Latitude Plot is a red box. This red box represents the expected variation and is the area of combined 99% variation in both Part A and Part B. In the Latitude Plot below, the 99% interval has annotations added in red.
Part A is Normally distributed with a mean of 23 and a standard deviation of 2; therefore, 99% of the Part As will fall in the range of 17.85 to 28.15. Part B ~ N(7,12) and so 99% of Part Bs will fall in the range of 4.24 to 9.58. The red box that is created by connecting the four points is the expected variation region for Part A and Part B. If you happen to be a statistician, you might expect to see an ellipse instead of a square. We chose to use a square as it is easier for the user to understand and communicate.
The Latitude Plot also includes two diagonal lines that pass through the square. These lines represent the specification limits for the stack height. In the graph below, markings have been added to show direction. Remember that the LSL = 25 and the USL = 35. If a combination of Part A and Part B falls above the pink line, then the total stack height will be greater than 35. If a combination of Part A and Part B falls below the brown line, then the total stack height will be less than 25. The area between the pink and brown lines indicate the area where a product can be manufactured in specification and is sometimes called the "latitude window".
I should point out that these are the lines for this specific model. Other models can have more or fewer lines that can be non-linear and run in a variety of directions.
The dots that are located in the latitude plot demonstrate the observed values of 1,000 Monte Carlo simulations. If an individual simulation is in specification, between the LSL and the USL, then the dot is colored blue. Simulations out of specification are colored red. The first item of note is that there are simulations out of the expected variation region (red box). This can occur as the red box represents the range for 99% of the data; therefore, we expect 1% of the data to fall outside the red box.
In the Latitude Plot below, two of the simulated values have arrows pointing to them. The top red dot is a Monte Carlo simulation that happens to have Part A = 8.8 and Part B = 29.4. This point results in a stack height of 8.8 + 29.4 = 38.2 which is greater than the USL of 35; therefore, the dot is red. In the lower right corner of the plot is a dot with Part A = 9.9 and Part B = 20.7. The stack height for this simulation would be 9.9 + 20.7 = 30.6 which is between the LSL and USL, and so the dot is blue.
Probably the easiest way to view a Latitude Plot is to show one with the resulting histogram. In the Latitude Plot below, the area below the brown line represents combinations of parts that are less than 25cm tall. These parts show on the histogram as the small portion of the tail to the left of the LSL. The area above the pink line corresponds to the area in red on the right side of the histogram.
At the start of this article, I said that the purpose of the Latitude Plot is to visually depict the relationship between the expected variation and the latitude window. For this stack, the expected variation region is a little larger than we would like when compared to the operating window. We do have points that are outside of specification generating significant defects. We have several options on how to handle this problem, which vary from widening the specification limits (usually not a good idea), reducing the part variation, sorting the parts (also usually not a good idea), or redesigning the product to handle additional variation.
This was a very simple example of Latitude Plots. In the next article, we will explore an example with more than one output.