科學化決策的核心 – 用DOE找到最佳參數

As engineers, we often face numerous challenges in solving process-related issues or conducting experiments to determine which parameters yield the best process yield. However, with so many variables in a process, it can be difficult to identify the key factors, leading us to rely on experience and randomly select a few parameter sets for testing. After completing the experiments, we often lack confidence in the results, yet there's no time for more comprehensive testing. This problem arises from a lack of understanding of experimental design, causing us to repeatedly use a "single-factor, multiple-level" approach. The result is unreliable experiments that may not converge, and even if we do solve the process issue, it's often through guesswork.

DOE (Design of Experiments) is a statistical method used to plan, design, and analyze experiments to explore the effects of different factors (i.e., variables that influence a system or process) on outcomes. This approach is particularly useful in situations requiring multi-factor analysis, such as process optimization, new product development, or quality control. Through DOE, one can effectively identify the key factors and their interactions that impact a system, thereby maximizing the information gained from a limited number of experiments.

Basic Concepts

• Factors: Variables in an experiment that are controlled and altered, such as temperature, pressure, or concentration. These are the variables the experimenter aims to study in terms of their impact on the results.
• Levels: The different values or settings of a factor. For example, the factor of temperature may have three levels: low, medium, and high.
• Response: The outcome of the experiment, which can be measured as yield, quality characteristics, or other quantifiable data.
• Interactions: The combined effects of different factors on the response. The impact of one factor may change depending on the levels of other factors, which is known as interaction. For instance, increasing temperature might also raise concentration, thereby affecting the results.

Key Steps in Designing an Experiment

1. Define the Objective: Clearly outline the experiment's goal, which is typically to optimize one or more response variables. The objectives must be quantified, such as achieving a certain yield percentage or reaching a specific area measurement. For aspects that cannot be expressed with exact numbers, such as taste, appearance, or aroma—subjective judgments must still be quantified, for example, by using a 1-10 rating scale.
2. Select Factors and Levels: Determine the factors to be studied and their levels. This step is crucial as it influences the scope of the experiment and the reliability of the results. For example, as explained earlier, temperature is a factor in this experiment. By dividing the temperature into three groups—low, medium, and high—you create three levels for this factor.
3. Design the Experiment: Based on the selected factors and levels, structure the experiment. Common experimental design methods include full factorial design, fractional factorial design, response surface methodology (RSM), and Taguchi methods, each offering distinct advantages depending on the situation. Full factorial design is the most fundamental approach, serving as the foundation from which other methods are derived.
4. Conduct the Experiment: Execute the experiment according to the design and collect data. This step requires strict control of variables to ensure the accuracy of the experimental results.
5. Analyze the Data: This is the most critical stage of experimental design. Use statistical methods, such as regression analysis or analysis of variance (ANOVA), to analyze the collected data. The goal is to determine the impact of each factor and their interactions on the response, ultimately identifying the "optimal key factors."
6. Draw Conclusions and Optimize: Based on the analysis results, draw conclusions and offer optimization recommendations. For instance, the next steps may involve focusing solely on adjusting the key factors to optimize the process.

Application Scenarios

• Process Optimization: Identify how process parameters affect product quality and determine the optimal combination of parameters.
• New Product Development: Test various factors in the new product design to select the optimal design solution.
• Problem Solving: When issues arise in the process or product, use DOE to identify the root cause and propose improvement solutions.

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Case Examples

Let's go through an example to help you understand how to practically apply experimental design.

1. Define the Objective

Let's study how to brew the best-tasting coffee. We'll rate the flavor on a scale of 1 to 10, with 10 being the highest score.

2. Select Factors and Levels

For brewing coffee, we will select the following factors and their levels:

• Coffee Bean Roasting Time (Factor A): Short Roast (5 minutes) and Long Roast (10 minutes)
• Water Temperature (Factor B): Low Temperature (85°C) and High Temperature (95°C)
• Coffee Bean Grind Size (Factor C): Coarse Grind and Fine Grind

3. Design the Experiment

In this scenario, we'll use a full factorial design. You need to test all possible combinations of factors, resulting in 2^3 = 8 combinations:

Note: The number of runs in a full factorial experiment = Number of Levels^Number of Factors.

In this scenario, since the number of factors and levels is relatively small, using a full factorial design won't result in too many experimental runs. However, if there were more factors, a common approach would be to use the Taguchi method, setting up a Taguchi orthogonal array to further reduce the number of experimental runs.

4. Conduct the Experiment

You brew these 8 cups of coffee in sequence and then rate them based on aroma, taste, and other factors. Let's assume you received the following scores (out of 10):

5. Analyze the Data

After completing the experiment, let's organize the results. We can interpret and analyze the data using two types of charts: the "Main Effects Plot" and the "Interaction Plot."

Main Effects Plot

The Main Effects Plot displays the average effect of each factor on the response variable independently, helping to understand which factor has the most significant impact on the outcome without considering interactions. This plot is particularly useful when you want to identify whether a factor has a significant individual effect on the response. For example, if changing the levels of a factor noticeably alters the result, that factor is likely a primary influencing factor.

In this case, it's clear that water temperature has the most significant impact on the results, so it can be temporarily identified as the optimal key factor.

Interaction Plot

The Interaction Plot shows how two or more factors jointly affect the response variable. If the lines are parallel, it indicates no interaction, meaning the factors do not influence each other. However, if the lines tend to intersect, it suggests there is an interaction, with more pronounced crossing indicating a stronger interaction. This plot is essential when you need to understand how different factors influence each other and how these interactions collectively determine the response. It helps identify factors that may have little impact individually but significantly affect the outcome when combined in specific ways.

The three charts on the right display the interactions in the coffee flavor optimization example.

• Interaction between Roasting Time and Water Temperature (A × B): The chart shows that with a short roasting time (5 minutes), increasing the water temperature (from 85°C to 95°C) significantly improves the coffee flavor score (from 5 to 7). However, with a long roasting time (10 minutes), the effect of raising the water temperature is even more pronounced, with the score increasing from 4 to 7, and even 9. This indicates a significant interaction between water temperature and flavor, depending on the roasting time.
• Interaction between Roasting Time and Grind Size (A × C): The chart shows that the interaction between roasting time and grind size is relatively minor. Fine grinding yields higher flavor scores than coarse grinding at both roasting times, but the difference is not significant.
• Interaction between Water Temperature and Grind Size (B × C): In this chart, we see that at a lower water temperature (85°C), the impact of grind size on flavor scores is minimal. However, when the water temperature rises to 95°C, fine grinding significantly improves the flavor score (from 7 to 9), whereas coarse grinding is less effective.

Finally, after synthesizing the results, we observe that water temperature (Factor B) shows a significant impact under various conditions, particularly across different roasting times and grind sizes. The change in water temperature leads to the most noticeable variation in flavor scores, aligning with the main effects plot where water temperature had the greatest influence. Therefore, water temperature can be considered the optimal key factor in this example.

6. Draw Conclusions and Optimize

Using this method, we define the optimal key factor as "water temperature," rather than grind size or roasting time. This means that when optimizing coffee flavor, the primary focus should be on adjusting the water temperature first, followed by fine-tuning the roasting time and grind size of the coffee beans.

Through the method of experimental design, you can more efficiently narrow down variables to identify key factors, rather than relying solely on experience. This approach allows for faster and more accurate reduction in the number of experiments needed and provides a more principled and credible explanation to anyone involved.

This example involved three factors with two levels each, which is relatively straightforward and simple. However, if there were more factors and levels, the experimental results would more clearly highlight differences and convergence effects. I encourage you to try experimenting with more complex scenarios to see the powerful impact of this approach for yourself!

In summary, DOE is a powerful tool that enables engineers and scientists to systematically explore relationships between factors in complex multi-factor scenarios, achieving optimal experimental outcomes. Making good use of this approach can significantly enhance your work efficiency.

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