4.5.1 Designs

Designs Introduction Designs in Lean Six Sigma refer to structured plans for collecting data so that cause–effect relationships can be understood reliably and efficiently. In the IASSC Black Belt context, this primarily means experimental designs used in the Improve phase to identify and optimize critical process factors. Good designs: - Align with a clear objective - Control or randomize nuisance influences - Use resources efficiently - Provide valid, interpretable results This article focuses on the core design concepts, types of designs, and practical decision rules needed to plan, run, and interpret experiments in a process improvement environment. --- Core Concepts in Experimental Designs Experimental Units, Factors, and Responses Every design rests on clear definitions. - Experimental unit: The item or entity that receives a treatment or condition (part, batch, transaction, customer, machine setup). - Factor: An input variable that is intentionally varied in an experiment to study its effect on the response (temperature, time, speed). - Level: A specific setting of a factor (e.g., 50°C and 60°C; “method A” and “method B”). - Response: The measured output of interest (cycle time, defect rate, yield, strength). - Treatment combination: A unique combination of factor levels applied to an experimental unit. To design an experiment: - Define the response clearly and how it will be measured. - Identify controllable factors you can set deliberately. - Recognize noise factors you cannot easily control but may be able to randomize or block. Main Effects and Interactions Designs aim to separate: - Main effect: The average change in the response when a factor changes level, ignoring other factors. - Interaction: When the effect of one factor depends on the level of another factor. Examples: - A temperature main effect: Increasing temperature from low to high reduces defects on average. - A temperature–time interaction: High temperature only reduces defects when time is also high; at low time, temperature has little effect. Understanding interactions is essential for robust improvement: - Large interactions can reverse conclusions about “best” settings. - Designs must be capable of detecting important interactions, not just main effects. Randomization, Replication, and Blocking These three principles shape reliable designs. - Randomization: Randomly order the experimental runs. - Protects against time-related and hidden trends. - Supports valid statistical inference (p-values, confidence intervals). - Replication: Repeating the same treatment combination. - Allows estimation of pure experimental error. - Increases precision of effect estimates. - Supports detection of lack of fit and nonlinearity (when combined with center points). - Blocking: Grouping similar experimental units or conditions into blocks to isolate nuisance variability. - Block: A set of runs conducted under similar nuisance conditions (shift, day, machine). - Blocking removes the block-to-block variation from the error term, improving sensitivity to factor effects. --- Screening Designs Purpose of Screening Screening designs are used early to: - Identify which factors among many are important. - Focus later, more detailed studies on the vital few factors. - Use resources efficiently when the number of potential factors is large. Assumptions in screening: - Only a small number of factors have meaningful effects. - Higher-order interactions (three-way and above) are negligible. - The primary need is to separate important from unimportant factors, not necessarily to build a full predictive model. Plackett–Burman Designs Plackett–Burman designs are resolution III two-level designs used when there are many factors and few runs. Key characteristics: - Two levels per factor (often coded -1 and +1). - Highly efficient: can screen many factors with relatively few runs. - Typically require a number of runs that is a multiple of 4 (e.g., 12, 16, 20). - Estimate main effects efficiently but confound them with some interactions. Strengths: - Very economical when factors are numerous. - Good for an initial pass to identify likely important factors. Limitations: - Main effects are aliased with two-factor interactions. - Not suitable when interactions are expected to be strong and must be interpreted. - Usually not used for final optimization. Use Plackett–Burman when: - There are many candidate factors. - Budget and time allow only a moderate number of runs. - You primarily want an initial shortlist of important factors for follow-up experiments. --- Two-Level Factorial Designs Full Factorial Designs A two-level full factorial design includes all possible combinations of factor levels. - For k factors, a full factorial has 2^k treatment combinations. - Each factor is at a low and high level. - All main effects and interactions up to the k-factor interaction can be estimated. Advantages: - Complete information about factor effects and interactions at the chosen levels. - Clear interpretation with no aliasing among factorial effects (before considering replications and blocks). Disadvantages: - Run count doubles with each added factor. - Becomes impractical as factor count grows. Use full factorial designs when: - The number of factors is modest (commonly 2–4, sometimes 5). - Understanding interactions is critical. - Adequate resources are available for all runs. Design Resolution Resolution describes the degree of confounding in a design. - Resolution III: - Main effects are aliased with two-factor interactions. - Two-factor interactions are aliased with other two-factor interactions. - Adequate for initial screening when interactions are assumed small. - Resolution IV: - Main effects are not aliased with two-factor interactions. - Two-factor interactions may be aliased with each other. - Allows unbiased estimation of main effects, even if two-factor interactions exist. - Resolution V: - Main effects are not aliased with two- or higher-order interactions. - Two-factor interactions are not aliased with each other but may be aliased with three-factor interactions. - Suitable when two-factor interactions are important to estimate reliably. Higher resolution is generally better but requires more runs. Choose resolution based on: - Need to interpret interactions. - Expected complexity of the system. - Available resources. --- Fractional Factorial Designs Motivation and Basic Idea When full factorial designs are too large, fractional factorial designs use only a fraction of the full treatment combinations to reduce run count. - For k factors, a 2^(k-p) design uses 1/2^p of the full factorial runs. - Example: - 2^(5-1) = 2^4 = 16 runs instead of 32 for 5 factors. - Fractions are constructed systematically using generators that define how some factors are combinations of others. Benefits: - Substantially fewer runs than full factorial. - Allow estimation of main effects and some interactions when resolution is adequate. - Highly useful for screening and early-stage optimization. Trade-off: - Some effects become aliased (confounded) with others. - Interpretation requires understanding the alias structure. Aliasing and Defining Relation In a fractional factorial: - Aliasing: Two or more effects (e.g., A, BC) are statistically indistinguishable; changing one combination is equivalent to changing the other pattern in the observed data. - Generator: A relation used to define the fraction, such as D = ABC. - Defining relation: All words obtained by multiplying generators together, including the identity term I. This defines which effects are aliased. Example (2^(4-1) design with generator D = ABC): - Defining relation: I = ABCD. - Aliasing: - A = BCD - B = ACD - C = ABD - D = ABC - AB = CD - AC = BD - AD = BC Interpretation: - A main effect is confounded with one three-factor interaction. - Two-factor interactions are confounded with other two-factor interactions. This is a resolution IV design: - Main effects are clear of two-factor interactions. - Two-factor interactions may be aliased with each other. Choosing Fractions and Resolution When selecting a fractional factorial: - Decide the highest-order interactions likely to be important. - Choose a resolution that does not confound the effects you care most about. General guidelines: - Use resolution III for preliminary screening with many factors and strong simplicity assumptions. - Use resolution IV when you must estimate main effects clearly but can treat two-factor interactions cautiously. - Use resolution V when two-factor interactions are critical and three-factor interactions are negligible. If initial fractional factorial results are ambiguous: - Augment the design with additional runs to de-alias specific effects. - Move to a higher-resolution design or full factorial for key factors. --- Center Points and Curvature Purpose of Center Points Center points are runs where all quantitative factors are set at mid-level values between low and high. Roles of center points: - Provide an estimate of pure error (when replicated). - Enable a test for curvature (nonlinearity) in the response across the factor range. Interpretation: - If center point response is significantly different from the average of the corner points, curvature is present. - Curvature indicates that a two-level linear model may be inadequate and that a more complex design (e.g., response surface) is needed. When to include center points: - When factors are continuous and ranges are wide. - When you suspect nonlinear relationships. - When planning a sequential strategy: screen first, then refine with response surface methods if curvature is found. --- Response Surface Designs When to Use Response Surface Designs Response surface designs refine knowledge of critical factors after screening: - Objective shifts from “which factors matter” to “what are the best settings.” - Nonlinear relationships and interactions are expected. - Optimization within a chosen factor region is required. Response surface designs use at least three levels per factor to model curvature. Common goals: - Fit a second-order (quadratic) model. - Locate optimum or near-optimal factor settings. - Explore the shape of the response surface (ridges, valleys, plateaus). Central Composite Designs A central composite design (CCD) is a standard response surface design building on a two-level factorial or fractional factorial core. Components: - Factorial points: The 2^k or 2^(k-p) points at low and high levels of each factor. - Axial (star) points: Points that extend along each factor axis beyond the low and high levels, enabling curvature estimation. - Center points: Multiple runs at the center to estimate pure error and test for curvature. Key parameters: - Alpha (α): Distance of axial points from the center (in coded units). - Chosen to achieve desired properties such as rotatability or orthogonality. - Coded units: Transform original factor scales so that low = -1, center = 0, high = +1, and star points at ±α. Types of central composite designs: - Central Composite Circumscribed (CCC): - Star points lie outside the factorial cube. - Common when extended factor ranges are feasible. - Central Composite Inscribed (CCI): - Star points define the limits of the region. - Factorial points lie inside; used when physical limits constrain extremes. - Central Composite Face-Centered (CCF): - Star points coincide with the factorial faces (α = 1). - No factor levels beyond the low and high; easier to run, somewhat less rotatable. Uses: - Fit quadratic models with moderate run counts. - Understand both main effects, interactions, and curvature. - Find stationary points (optima, saddle points) and explore predicted responses across the region. --- Blocking and Restrictions in Designs Blocking in Factorial and Response Surface Designs Blocking is often necessary when all runs cannot be performed under equivalent conditions. Examples of blocking factors: - Day or shift - Machine or production line - Operator groups - Raw material lots Design considerations: - Assign runs to blocks so that blocking factors do not confound critical treatment effects. - Modify design generators or use block generators to maintain design resolution while accommodating blocks. - Analyze data with block terms in the model to remove block-to-block variability from the error term. Blocking trade-offs: - Reduces unexplained error and increases sensitivity to factor effects. - May increase aliasing if block effects confound with some interactions. - Must be pre-planned in the design stage; cannot be fully corrected post-hoc. Hard-to-Change Factors and Split-Plot Logic Some factors are difficult or costly to change between runs (e.g., oven, machine, environment). These create practical constraints: - Whole-plot factors: Hard-to-change, adjusted infrequently. - Sub-plot factors: Easy-to-change, can vary run to run within a whole plot. Implications: - Designs with such constraints behave like split-plot structures, with different error terms for whole-plot and sub-plot factors. - Precision for hard-to-change factor effects is often lower than for easy-to-change factors. - Randomization is restricted: whole plots are randomized as units; runs within a whole plot are randomized for sub-plot factors. Planning actions: - Identify hard-to-change factors early. - Choose or adapt designs that accommodate restricted randomization. - Recognize that analysis must account for this structure when estimating effects and their precision. --- Practical Design Strategy Sequential Experimentation Effective design work rarely relies on a single experiment. Instead, use a sequence: - Start with a screening design: - Use Plackett–Burman or fractional factorial to identify important factors. - Include center points when quantitative factors suggest possible curvature. - Refine with follow-up experiments: - Resolve key aliases by adding runs or shifting to a higher-resolution design. - Drop clearly unimportant factors to simplify subsequent studies. - Optimize with response surface designs: - Use a central composite variant to model curvature. - Determine optimal factor settings according to performance and robustness. Benefits of a sequential approach: - Reduces risk of over-investing early in complex designs. - Allows learning to guide later decisions. - Enhances interpretability and robustness of final recommendations. Design Selection Criteria When selecting a design, consider: - Objective: - Screening: many factors, few runs, main effects focus. - Optimization: fewer factors, detailed response modeling. - Number of factors: - Many factors favor screening and fractional designs. - Few factors may allow full factorial or direct response surface designs. - Interactions and curvature: - If interactions are expected to be important, avoid low-resolution designs. - If curvature is suspected, plan for center points and response surface stages. - Resource constraints: - Available runs, time, and cost. - Practical limits on factor ranges and combinations. - Operational constraints: - Need for blocking or shift-based scheduling. - Hard-to-change factors and restricted randomization. --- Summary Designs in Lean Six Sigma provide structured plans for experiments that uncover how process factors influence critical responses. Key knowledge includes: - Clear definitions of experimental units, factors, levels, and responses. - Understanding main effects and interactions, and how designs allow them to be estimated. - Using randomization, replication, and blocking to control variation and support valid inference. - Applying screening designs (such as Plackett–Burman and fractional factorials) to identify important factors efficiently. - Choosing appropriate design resolution and understanding aliasing in fractional factorials. - Using center points to detect curvature and support model adequacy checks. - Employing response surface designs, especially central composite designs, to model curvature and optimize factor settings. - Accounting for blocking and hard-to-change factors in design structure and analysis. - Adopting a sequential, resource-conscious experimentation strategy to move from screening to optimization. Mastery of these design principles enables effective, efficient experimentation that leads to reliable process improvement and optimized performance.