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Chapter 4: Response Surface Methodology 4.1 Concepts and Terms 4.2 Classic Response Surface Designs for Second-Order Models 4.3 Steepest Ascent Method (Optional) 1 Chapter 4: Response Surface Methodology 4.1 Concepts and Terms 4.2 Classic Response Surface Designs for Second-Order Models 4.3 Steepest Ascent Method (Optional) 2 Objectives 3 Understand response surface methodology and its sequential nature. Distinguish among different kinds of optima. Illustrate different types of surfaces and relate those surfaces to model equations. Response Surface Methodology 4 Response surface methodology (RSM) uses various statistical, graphical, and mathematical techniques to develop, improve, or optimize a process. The most frequent applications of RSM are in the industrial area, where several predictor variables are used to predict some performance measure or quality characteristic of a product or process. Sequential Nature of RSM 5 You used designed experiments to determine what factors are important in determining the qualities of your product or the results of your process. Now you want to find factor settings or a factor region that optimize your response or responses. Finding these factor settings or factor region is an iterative process that utilizes designs discussed in the previous chapters and designs to be discussed in this chapter. New Purpose Requires New Designs At this point, you need more detailed information than the screening design could give you; optimum conditions are difficult to find using only screening designs. For this, you need more data in order to add new effects to your model. The design you select and the new data points you add depend on the model you want to fit. To estimate linear parameters, you could use a two-level fractional factorial design of resolution ≥ 3. cross-product parameters, you could use a fractional factorial design of resolution ≥ 5. quadratic parameters, data for at least 3 levels of each factor is required. 6 The Number of Levels Model adequacy requires testing (m+1) levels of a factor with order m polynomial effects. Linear effects, m = 1, require two factor levels. Quadratic effects, m = 2, require three factor levels. Effects greater than second-order, m ≥ 3, are unusual. You can avoid a large number of levels by restricting the factor ranges. 7 Center Points Center points provide a third level to support a quadratic model are positioned symmetrically between the ends of the test range {-1, 0, +1} are a common element in response surface designs are focused replication to estimate pure error add to screening designs to provide additional information. 8 9 4.01 Multiple Answer Poll Which of the following statements are true about center points? a. They are positioned halfway between the low and high settings. b. They are coded as 0. c. They are commonly found in response surface designs. d. They are used to estimate pure error. 10 4.01 Multiple Answer Poll – Correct Answers Which of the following statements are true about center points? a. They are positioned halfway between the low and high settings. b. They are coded as 0. c. They are commonly found in response surface designs. d. They are used to estimate pure error. 11 General Optimum Response You might have a general objective in mind for your response. More is better (maximum). Less is better (minimum). Target is better (range of response values). 12 Specific Acceptable Response Values You might have a specific objective for your goal. For example: Yield should be at least 90%. Impurity must be less than 5 mg/L. Final acidity is best at 4.5 0.2 pH. 13 Satisfy More Than One Goal You might have more than one response, and each response has a unique goal. Some responses have more demanding specifications than other responses. Some responses are more important than other responses. The best factor settings are a trade off of all of the response goals. 14 15 4.02 Multiple Choice Poll What is the most common goal for the processes at your business? a. To maximize b. To minimize c. To hit a target 16 Design Under Factor Constraints You might also have a factor with specifications, or a constraint. Such specifications or constraints might: Account for physical limitations. Minimize costs or safety risks. Manage inventory or control cost. You want to satisfy this constraint when you find the optimum settings. 17 The Response Surface and its Model Your response usually depends on more than one factor. The shape of the response as a function of two or more factors defines a surface. You want to explore the surface for the optimum response without testing every possible point. This true response model that describes the surface is usually unknown. A smooth interpolating function is usually used, and it generally includes quadratic and interaction effects. Y 0 i X i i j X i X j ii X 18 2 i RSM Models For Two Factors Screening y=0 + 1x1 + 2x2 + 12x1x2 + Steepest ascent y=0 + 1x1 + 2x2 + Optimization y=0 + 1x1 + 2x2 + 12x1x2 + 11x1 + 22x2 + 2 19 2 Good Design A good design is both effective and efficient. An effective design enables you to obtain sufficient data to fit an interpolating model that provides unbiased predictions with sufficient precision. An efficient design enables you to obtain the most precise estimates for a given budget on the number of runs. 20 Shape of the Response This demonstration illustrates the concepts discussed previously. 21 22 4.03 Quiz Match each surface type with its graph. A. B. C. D. 23 E. 1. 2. 3. 4. 5. Planar Surface Quadratic Surface Saddle Surface Ridge Surface Twisted Surface 4.03 Quiz – Correct Answer Match each surface type with its graph. A. B. C. D. E. 1-A, 2-B, 3-D, 4-C, 5-E 24 1. 2. 3. 4. 5. Planar Surface Quadratic Surface Saddle Surface Ridge Surface Twisted Surface 25 Chapter 4: Response Surface Methodology 4.1 Concepts and Terms 4.2 Classic Response Surface Designs for Second-Order Models 4.3 Steepest Ascent Method (Optional) 26 Objectives 27 Understand properties of Box-Behnken and Central Composite designs. Generate and analyze a Box-Behnken design. Generate and analyze a Central Composite design. Box-Behnken Design The Box-Behnken design incorporates three levels (coded –1, 0, +1) has points that are vertices of a polygon and equidistant from the center avoids extreme points (for example, [+1, +1, +1]) does not utilize a screening run is the smallest classic response surface design for fewer than five factors has uniform blocks: each level of each factor is in each block an equal number of times and the center points are evenly divided among the blocks. 28 Box-Behnken Design Example 29 The objective of an experiment is to reduce the unpleasant odor of a chemical product. The response variable is Odor, and it is believed that a secondorder model is required. The factors are temperature (Temp) with a range of 40-120, gas and liquid ratio (GL Ratio) with a range of .2-.7, and packing height (Height) with a range of 2-6. Box-Behnken Design Example 30 There are 3 continuous factors (Temp, GL Ratio, and Height). For a Box-Behnken design, there will be 15 runs: 12 to examine all possible combinations of low and high for each pair of factors (with the third factor at the center) and 3 center points. The plot shows the position of the design points. Box-Behnken Design This demonstration illustrates the concepts discussed previously. 31 32 4.04 Quiz Match each graph with its name. A. C. 33 B. 1. Surface Plot 2. Contour Plot 3. Prediction Profiler 4.04 Quiz – Correct Answer Match each graph with its name. A. C. B. 1. Surface Plot 2. Contour Plot 3. Prediction Profiler 1-B, 2-C, 3-A 34 Central Composite Design X X X X X X 35 X X The CCD design incorporates five levels (coded –α,-1, 0,+1,+α) shares screening runs has axial points (±α) for one factor and 0 for all other factors is the largest factorial design for 3 factors and the smallest for 5 or more factors has points that form a cube plus a star Central Composite Design Example 36 Recall the experiment to determine which factors are important to determine Seal Strength for a bread wrapper. The experimenter decides to run a CCD to understand the shape of the response surface and to find an optimum setting for these factors. The factors are % Polyethylene with a range of 85-95, Cooling Temperature with a range of 120-140, and Sealing Temperature with a range of 220-240. Central Composite Design Example 37 There are 3 continuous factors (% Polyethylene, Cooling Temperature, and Sealing Temperature). For a CCD design with uniform precision, there will be 20 runs: 8 from the 23 design, 6 axial points, and 6 center points. The plot shows the position of the design points. Central Composite Design This demonstration illustrates the concepts discussed previously. 38 39 4.05 Multiple Answer Poll Which of the following are properties of the Box-Behnken design? a. Avoids extreme design points b. Incorporates axial points c. Is rotatable or nearly rotatable d. Supports screening runs e. Is the smallest classic response surface design for fewer than 5 factors f. Is a spherical design 40 4.05 Multiple Answer Poll – Correct Answers Which of the following are properties of the Box-Behnken design? a. Avoids extreme design points b. Incorporates axial points c. Is rotatable or nearly rotatable d. Supports screening runs e. Is the smallest classic response surface design for fewer than 5 factors f. Is a spherical design 41 Exercise This exercise reinforces the concepts discussed previously. 42 43 4.06 Quiz The Prediction Profiler output from the exercise is below. Examine the solutions for the three variables. Which variable’s solution is problematic? Why? 44 4.06 Quiz – Correct Answer The Prediction Profiler output from the exercise is below. Examine the solutions for the three variables. Which variable’s solution is problematic? Why? Distance – the solution is problematic because distance cannot be negative. JMP gave this answer because there was no constraint on Distance. 45 46 Chapter 4: Response Surface Methodology 4.1 Concepts and Terms 4.2 Classic Response Surface Designs for Second-Order Models 4.3 Steepest Ascent Method (Optional) 47 Objectives 48 Understand the procedures of the method of steepest ascent to find an optimum. Design a sequence of experiments using the method of steepest ascent. Stages to Find Optimum Three stages depict the usual sequence to study a process. 1. Screen important effects: this tells you what factors are important in the first region of optimization. 2. Optimize factor settings: this is only possible if your current region of investigation contains the optimum you seek. 3. Verify optimum conditions: this is done after an optimum is found. An intermediate step between steps 1 and 2 might be necessary to find the right region of optimization. 49 Eyes On the Optimum 50 Visualize the Response 51 Your screening experiment will sample some region in the possible factor space. This region is anchored in space by its center; the coded level for every factor at this point is zero. Without prior knowledge, the range of factor settings and the multitude of potential factors result in a large space to search. The complexity of your response goals might further reduce the likelihood that the initial region will contain the optimum. Graphs Are Useful Searching Tools Use graphs to visualize the response during your search. Prediction Profile plot (1D) is a slice of a contour or a surface plot. Contour plot (2D) is a topographical map of the response surface. Surface plot (3D) of the response shows the shape of the response from various angles. 52 Method of Steepest Ascent The method of steepest ascent is an iterative process that uses multifactor screening designs and center points is based on a search strategy (choose direction and step size) accommodates minimizing, maximizing, or targeting goals. 53 Linear Screening Models 54 The original design space does not contain the optimum. If you are not close to the optimum, then quadratic effects should be small: think of the side of a smooth mountain. A linear screening model provides the trajectory toward the goal. Steps in the Search Process 55 Screen for the important factors and assess the location of the optimum. Determine the direction and step size towards the optimum from the screening model. Take uniform steps, i for each factor xi, while the response improves. Continue stepping along until there is no further increase or until a decrease in the response is observed. Check the direction and model lack of fit. Determine the Direction Consider an example of a two-factor case without interaction. The linear model parameters define the path of steepest ascent. Y = 0 + 1X1 + 2X2 1 and 2 indicate the change in Y for a one-unit change in coded X. The vector {1, 2} points to the optimum from the origin. 56 Step Size 57 You want to take the largest steps possible to reach the optimum in the fewest steps, but too large of a step size might mean that you miss the optimum. Use engineering and scientific knowledge to weigh the possibilities, and take some risk. Replicate if necessary to detect a significant improvement. After a Stop When the response either decreases or no longer increases, it is either because the maximum response is in the vicinity or because it is not in the vicinity, but instead you are on a ridge or at a saddle point. Therefore you have to decide if you want to stay here and find the optimum or determine a new path and step size to continue to search. To decide, first design a two-level screening experiment and include center points (and shift the initial origin to the best stop point), fit a new first-order model, and test for a lack of fit. 58 Continue or Stay 59 Continue the search if there is no detectable or significant lack of fit. The maximum response is not in this region, so you need to determine the new direction and step size. Stay if the lack of fit test based on the center points suggests a quadratic effect in the current region. In this case, design an experiment to ensure model adequacy for a response surface model. A central composite design is a good choice for this sequence. Caveats for Steepest Ascent 60 The same principles apply if the goal is a minimum response but in an opposite sense. This is called the path of steepest descent. The goal might instead be a target. Stop this search when the range of the response includes the target. The method of steepest ascent is aggressive. Smaller adjustments can be used safely to maintain an optimum response. This method is known as evolutionary operation (EVOP). Path of Steepest Ascent for Yield Example 61 A chemical engineer wants to maximize the Yield of a process. Two factors have been determined as important: Reaction Time and Reaction Temperature. Currently the process is completed in about 35 minutes at a temperature of about 155 degrees. Method of Steepest Ascent This demonstration illustrates the concepts discussed previously. 62 63