01 · Move from screening to local optimization
Move from screening to local optimization
Response surface methods use designed experiments and local models to understand how continuous settings affect a response. After identifying important warehouse inputs, a team might optimize wave release spacing and batch size. The goal is a useful feasible operating region, with constraints on error rate, staffing, congestion, and throughput, rather than merely the smallest fitted cycle time.
RSM is sequential. A first order experiment can indicate an improving direction. Follow that direction with planned confirmation steps until progress slows or curvature becomes important, then collect a design capable of estimating a quadratic model. Do not fit a elaborate surface to too few convenient observations or extrapolate a local quadratic across the entire operation.
02 · Fit a quadratic with identifiable terms
Fit a quadratic with identifiable terms
For two coded factors, a quadratic model is ŷ = b0 + b1x + b2z + b12xz + b11x² + b22z². The coefficients describe local intercept, linear slopes, interaction, and pure curvature. Coding maps real settings into a useful centered scale and avoids comparing coefficients measured in incompatible units.
A two level factorial with centers can detect combined curvature but cannot distinguish each separate square term. Central composite designs add axial points and centers to factorial information. Box–Behnken designs provide another quadratic design structure for suitable numbers of factors. Select geometry based on safe settings, prediction precision, blocking, and feasibility. Some axial points in a conventional central composite design may exceed practical limits, requiring a different geometry or a custom design.
03 · Worked warehouse surface
Worked warehouse surface
Let x = (batch size−20)/10 and z = (release spacing−5)/2, with batch size in orders and spacing in minutes. Fictional observations at four corners (−1,−1), (−1,+1), (+1,−1), (+1,+1) are 36, 30, 26, and 24 minutes. At axial points (−1,0), (+1,0), (0,−1), (0,+1), times are 31, 23, 28, and 24. Three independently run centers (0,0) give 23, 24, and 25.
Least squares on these eleven points fits ŷ = 24−4x−2z+xz+3x²+2z² exactly for the eight noncenter points and at the center mean. Residuals occur at the centers: −1, 0, and +1. SSE = 2 with eleven minus six = five residual degrees of freedom, so residual mean square is 0.4. The center replicates estimate pure error as 2/2 = 1, while lack of fit is zero with three degrees of freedom. The neat fit is constructed for teaching; zero observed lack of fit is not proof the real process is exactly quadratic.
04 · Solve and classify the stationary point
Solve and classify the stationary point
Set partial derivatives to zero: −4+6x+z = 0 and −2+x+4z = 0. Solving gives x = 14/23 = 0.6087 and z = 8/23 = 0.3478. These correspond to about 26.09 orders and 5.70 minutes. The fitted response is 22.4348 minutes. Actual batch size is discrete, so assess feasible nearby settings rather than implementing 26.09 orders.
The Hessian matrix of second derivatives is [[6,1],[1,4]]. Its eigenvalues are positive, approximately 6.414 and 3.586, so the stationary point is a local minimum of the fitted surface. Negative eigenvalues would indicate a maximum; mixed signs a saddle. A stationary point is not inherently an optimum. It must also be feasible, within the credible model region, and acceptable for every operational response.
05 · Respect constraints and uncertainty
Respect constraints and uncertainty
If the allowed batch size is at most twenty four, x must be at most 0.4 and the unconstrained point is infeasible. On that boundary, minimizing over z gives z = (2−x)/4 = 0.4. Thus a boundary candidate is twenty four orders and 5.8 minutes, with fitted time 22.56. Check all feasible boundaries and relevant corners, and evaluate discrete settings and other constraints before calling a point best.
At the unconstrained fitted point, the ordinary fixed point mean prediction standard error from this design is about 0.3172. Using five degrees of freedom gives a nominal 95% interval of approximately 21.619 to 23.250 minutes. This conditional interval does not include uncertainty from selecting the optimum after fitting, and it does not describe a single future run. Uncertainty in the location can be substantial even when predicted responses across a flat basin are similar.
06 · Validate the surface and the chosen region
Validate the surface and the chosen region
Plot residuals against predictions, run order, and inputs. Examine curvature, unequal spread, leverage, and process drift. Independent replicates allow pure error to be separated from lack of fit when the design supports it. A nonsignificant lack of fit test with limited data does not prove adequacy. Avoid removing quadratic terms solely to improve individual p values when they are needed to represent credible curvature.
Confirm the chosen feasible setting with new independent runs and compare results to the prediction and customer threshold. Also test nearby settings to understand robustness to normal execution error. A broad acceptable basin can be preferable to a slightly better sharp point that is difficult to maintain. For multiple responses, display the feasible intersection of time, errors, and throughput constraints or use a transparent tradeoff rule. A desirability score depends on chosen weights and limits, so explain them.
07 · Judge quality and use through DMAIC
Judge quality and use through DMAIC
A strong application follows screening with an identifiable curvature design, classifies the fitted stationary point, checks constraints and diagnostics, and confirms an operating region. A weak application treats every quadratic solution as a minimum, accepts an infeasible point, or chooses settings beyond the experimental range. Another weak use reports many decimal places while ignoring discrete batch sizes and input control limitations.
Analyze establishes mechanisms and local model form. Improve chooses feasible settings with acceptable uncertainty and safeguards. Control defines the window, required input precision, monitoring responses, and conditions for reoptimization. Preserve the actual and coded design, quadratic equation, constraint set, diagnostic evidence, and confirmation results. The surface describes a local process under stated conditions, not a permanent law for every warehouse.
08 · Read geometry and model sensitivity
Read geometry and model sensitivity
The design plot marks the eight corner and axial settings plus the center, where three independent observations were made. Distinct input locations and replication provide different kinds of information. Locations help identify coefficients and lack of fit; repeats help estimate pure error. More repeats at one center cannot replace a missing direction needed to estimate separate curvature.
In the fictional surface, the optimum is inside the studied square, but its practical precision should not be inferred from several decimal places. Small changes in estimated linear or quadratic coefficients can move the stationary point. A bootstrap that resamples appropriate experimental units, or a model based uncertainty analysis, can explore this sensitivity. Respect the design and randomization structure when doing so.
If curvature is weak, the predicted basin may be broad and the optimizer location unstable. Choose a feasible stable region using cost and operational constraints rather than insisting on the exact estimated coordinate. If curvature is negative in one direction, the stationary point is a saddle and boundary search is essential. Always confirm predictions at implementable settings with new runs, especially when a constrained or rounded solution differs from the analytical stationary point.
Connections
Related methods and phase use
Placed under Analyze because of the outcome it supports. Use the method in other phases when its purpose and assumptions fit the work; the phase list is a guide to selection, not a compulsory sequence.
Sources
Professional foundations and further reading
The explanation and fictional teaching examples are written for LeanTrainingLab. Linked primary sources provide professional foundations; example values and local conventions are stated in the guide. The accepted toolset draws on Six Sigma+Lean Toolset, second edition (Lunau, editor; Springer, 2013). No book text or examples are reproduced here.