Y=f(x), What does it stand for?

Y=f(x), can be used to determine cause and effect in a Lean Six Sigma project through data collection or to statistically assess performance and identify areas for improvement. This formula is known as the “breakthrough equation” in Lean Six Sigma. The Y=f(x) equation identifies the dependent output (or Y(s)) for a process that has different inputs (or X(s)).

Let’s break it down into its parts.

  • Y: The outcome, result, or set of results that you desire
  • X:  The inputs, factors, or whatever it takes to achieve the desired outcome (there may be more than one x).
  • F: The function or process that takes the inputs and turns them into the desired outcome.

An example

In a bottling process, the output (Y) could equal “Fill Height”. An input (x) that could affect the “Fill Height” (the Y) could be the five different filler lines used to fill the bottles. The function of email openings (f) is more effective when there are more coupons (x).

How does the DMAIC process and Y=f(x) work together?

DMAIC can be used to identify process input variables and process outcomes. The two together can make a powerful partnership. How does each of the DMAIC words work towards Y=f(x), then?

  • Define describes how to understand the outcome Y and how to measure it.
  • Measure allows you to prioritize the potential x’s, and measure the x and Y.
  • Analyze allows you to verify the relationship between x, Y and determine the most important x.
  • Improve aims to implement fixes to help Y and address x.
  • Control monitors x and y over time.

Let’s take a closer look at the DMAIC process

Y=f(x) – Define the Y

Determine whether the project has a defined business problem or process problem. This may be clear at times, but not always. If this is the case, you can get a clear understanding of the Y, such as the process problem in quantifiable terms within the contexts of project goals.

The project charter defines the problem that the project is designed to solve. To measure the progress of the project, you will need to establish the baseline for the project metrics. It is helpful to get data from customers.

The business plan will place the Y in the context of the overall organizational strategy, financial targets, customer fulfillment, and other relevant goals. The define phase of the process is completed once these goals have been identified.

Y=f(x) – Measure the Xs

Map your process. This will allow you to determine the causes of any Xs in your formula. You will see which Xs have the most influence on Y. Prioritization will be required to reduce the number of Xs.

Data will not help to reduce potential Xs, since the project is still in its beginning stages. To determine what is most important, brainstorming is necessary. Next, you will need to identify patterns that can be used to establish a baseline capability for the project.

Analyze to find the most important Xs

Verify and quantify the X/Y relationship. This phase can be completed using statistical and graphical tools. This is where the equation is very powerful. Know that we know the Y(s) and the Xs, now it is time to use qualitative tools (like the FMEA) and Quantitative tools (like hypothesis testing) to determine the function (or “f”) in the equation Y=f(x). This will help you understand which x’s are most important to solving the problem Y.

Improve and Optimize the Important Xs

What can you do to improve on what you have at this stage? You can do this by brainstorming ideas and coming up with creative solutions. You will need to reduce the number of solutions that you have come up with and focus only on those that are truly necessary.

You will need to rate each solution using criteria, such as how it improves Y or how it addresses x. There are also other criteria that you can use such as ease, cost, and so on. The best way to reduce this number is to determine the likelihood of these solutions failing and how to avoid them.

Create Controls to Sustain the Improvements

You can’t set aside the work you have done and the improvements that you made. They must be sustainable. You can now see the key points and plan for actions if the process does not follow the plan. The x’s are the leading indicators, or checkpoints, in a process management diagram. The Y is the last checkpoint or lagging indicator at the end of a cycle. You’re basically dealing with a project dashboard that measures the metrics you need to monitor to keep your project on track. This will display the performance of Y over time. All this information can be analyzed at the end of the project. These data will not only help you guide your project, but they will also serve as a precedent for future projects and help you evaluate their merits.