You built the model.
Then what?

A model is a means, not a deliverable

Nobody ships a model. They ship a decision: a mill reduction schedule, a dryer operating window, a soft-sensor readout, a control-plan setpoint. The model is the tool that lets you answer a question cheaply — in silico — instead of running another 30 heats.

So the honest way to ask "then what?" is to ask what question are you trying to answer. Each question routes you to a different next step. Here's the whole menu on one plate; the rest of the lesson takes each item in turn.

Predict — and pick the right interval

The simplest use: plug new settings into the fitted equation and read off \(\hat{Y}\). For a yield-strength model that's predict(reduction=18%, C=0.06, Nb=0.035, coil_temp=560). The number itself is the easy part. The part people get wrong — and the part that changes decisions — is how much to trust it.

Decision rule: if you're setting policy about the process on average ("does grade X70 clear 355 MPa on average?"), the CI is your tool. If you're making a call about the next pipe or next heat ("will this heat pass?"), you owe the customer a PI. Quoting a CI where a PI belongs is how a process that looks capable on paper ships nonconforming product.

That lone +1 under the root is the entire difference between the two intervals. Drop it and you have a CI; keep it and you have a PI. As sample size grows the CI term shrinks toward zero, but the +1 never does — the PI can't get narrower than the process's own noise. The widget below lets you feel that.

Connection between them

Strip away the regression machinery and the same logic shows up in the simplest possible case — a plain sample mean, no model at all. For a mean \(\bar{X}\) with sample standard deviation \(S\) and \(t = t_{\alpha/2,\,n-1}\):

Confidence interval — for the process average

Prediction interval — for a single next observation

Notice the prediction interval carries \(\sqrt{1 + \frac{1}{n}}\) where the confidence interval only has \(\sqrt{\frac{1}{n}}\). Since \(1 + \frac{1}{n} > \frac{1}{n}\) for every sample size, the prediction interval is always wider. And as \(n\) grows the \(\frac{1}{n}\) term melts toward zero — the CI can shrink to a point, but the PI can't drop below \(\bar{X} \pm tS\), because that leading 1 is the process's own scatter. It is the exact same role the +1 plays inside the regression formula above.

Optimize — find the best settings

Prediction answers "what if I set X here." Optimization inverts it: where should X be so Y is best? Three flavors depending on the goal — maximize (throughput), minimize (defect rate, energy), or hit a target (yield strength inside a spec band, not just as high as possible).

Single response is the easy case

With one Y, walk the response surface to its peak/valley: contour and surface plots to see the landscape, then the optimizer to pin the coordinates. In Excel that's Solver on the fitted equation; in Python, scipy.optimize.minimize (negate to maximize); in Minitab, the Response Optimizer.

Real problems have competing responses

You almost never optimize one thing in isolation. On a roughing-mill reduction schedule you want strength up, toughness held, roll load and energy down, all at once — and the setting that maxes one degrades another. Two standard ways to handle it:

• Desirability functions (Derringer–Suich). Convert each response to a 0–1 desirability \(d_i\) (0 = unacceptable, 1 = ideal), then combine into one composite \(D=(d_1 d_2 \cdots d_k)^{1/k}\), optionally weighted by importance. The optimizer maximizes \(D\). Because it's a geometric mean, any \(d_i = 0\) forces \(D=0\) — one violated response kills the whole setting. That's the behavior you want.
• Overlaid contour plots. Draw each response's spec contour on the same factor axes; the region where all of them are simultaneously satisfied is your operating window. This is precisely the deliverable behind an "operating-window optimization" — you're not handing over a single point, you're handing over the feasible zone plus a recommended center.

Rank drivers · robustify · simulate

Confirm — reality gets a vote

The model is a hypothesis about the process. Before you roll a new schedule to production, you test the hypothesis: run confirmation trials at the predicted optimum and compare what actually happens to what the model promised.

Deploy & monitor — hold the gains

A confirmed optimum that isn't held drifts back to baseline within a quarter. Deployment is what turns a one-time improvement into a permanent one, and it's where the model often gets a second life as an operational asset.

• Lock the setpoints into the control plan. The optimized settings become documented operating parameters with tolerances — tightest on the high-leverage factors from item 3, loosest where you have slack.
• Put SPC on the vital few. Control charts on the key X's and the Y catch drift before it becomes scrap. This is the "hold" mechanism — signal vs noise, tampering vs missed shifts.
• Run the model as a soft sensor. If Y is slow, expensive, or destructive to measure (Charpy, tensile) and the X's are cheap and fast, the model predicts Y in near-real-time from routine inputs — an inferential measurement between physical tests.
• Score new data in a pipeline. Batch or streaming, the fitted model becomes a step in production: new heat → predict → flag if predicted Y approaches spec.
• Monitor for drift. Models decay as equipment wears, feedstock shifts, seasons change. Track the residuals (actual − predicted) on a control chart of their own; a trend or a widening spread is the model telling you it's going stale and needs a refit.

The menu, mapped to DMAIC

None of this is new machinery — it's the back half of the roadmap you already run. Model-building closes out Analyze; everything after it lives in Improve and Control.

Stat ▸ Regression ▸ Regression ▸ Fit Regression Model · Best Subsets · Predict · Factorial Plots · Contour Plot · Surface Plot · Overlaid Contour Plot · Response Optimizer

Once you've run Fit Regression Model (or a DOE analysis), Minitab stores the equation and every option below operates on it. The three you'll reach for after fitting map straight onto the menu you already know — Predict, Factorial Plots, and Response Optimizer — and they cover items 1, 3, and 2 of the menu above.

Response      Goal        Lower   Target   Upper   Weight  Import
Strength      Maximize    355     420        —        1       1
Toughness     Target       80     120      160        1       1
Roll load     Minimize      —       —      9.5        1       1

Solution
Reduction = 16.4%   Carbon = 0.058%

Composite Desirability  D = 0.87
Predicted:  Strength 401 MPa   Toughness 118 J   Roll load 9.1 MN

Data ▸ What-If Analysis ▸ Scenario Manager · Goal Seek · Data Table  |  Data ▸ Analyze ▸ Solver

Excel does the same four jobs without a stats package — the trick is that your model has to live in a cell as a formula so these tools can push values through it. Once your fitted equation is in a cell referencing input cells, all four What-If tools are just different ways to interrogate it. Solver (a separate add-in in the Analyze group) is the heavy one.

Objective (Energy $/t)   Min      →   19.4
Variable  Reduction   =   15.1%
Variable  Carbon      =   0.056%

Constraints
Strength   401 MPa   ≥ 355   ✓ satisfied
Toughness  121 J     ≥ 100   ✓ satisfied
Roll load  8.8 MN    ≤ 9.5   ✓ satisfied

Which tool is which, across platforms

Statistics Implementation

The interactive consoles above are teaching models. When you sit down with real output, you drive the same four jobs — predict, bound, optimize, and check capability — through actual Excel functions and Minitab menus. This section is the lookup table. The worked numbers use the lesson's toy model, strength = 290 + 4.6·R + 320·(C−0.05), with residual spread s ≈ 11 MPa.

Excel functions

Array note: modern Excel spills LINEST automatically; in older versions enter it as an array with Ctrl+Shift+Enter. The optimisation and What-If tools (Solver, Goal Seek, two-variable Data Table) are covered in the In Excel section above.

Excel use cases

• Point prediction: put the fitted equation =290+4.6*B1+320*(B2-0.05) in a cell with reduction in B1 and carbon in B2. At R = 16.4%, C = 0.058% it returns about 368 MPa.
• Prediction interval: with s from STEYX (≈ 11) and t = T.INV.2T(0.05, n−2), a single-heat 95% PI is roughly fit ± t·√(s² + SE_fit²). Quote the lower bound when you promise a heat clears 355 MPa.
• Confidence interval: same t but only SE_fit (drop the s² term) — the narrower band for the average heat, not a single one.
• Predicted capability: after a Monte-Carlo Data Table, compute =(mean−355)/(3*sd) for Cpk and =1−NORM.DIST(355, mean, sd, TRUE) for the fraction clearing spec.

Minitab navigation

• Fit the model: Stat → Regression → Regression → Fit Regression Model. From a designed experiment use Stat → DOE → Factorial → Analyze Factorial Design or Stat → DOE → Response Surface → Analyze Response Surface Design.
• Predict with CI & PI (item 1): Stat → Regression → Regression → Predict. Enter the factor settings; the table returns Fit, SE Fit, 95% CI, and 95% PI side by side. DOE alternative: Stat → DOE → Response Surface → Predict. Choose Predict (not a fitted-line plot) when you need the numeric interval table for specific settings.
• Optimize (item 2): Stat → DOE → Response Surface → Response Optimizer (or the Factorial equivalent) for multi-response desirability. For a feasible operating window rather than one point, use Stat → DOE → Response Surface → Overlaid Contour Plot.
• Rank drivers & interactions (item 3): Stat → DOE → Factorial → Factorial Plots (Main Effects and Interaction) and the Pareto of standardized effects from the analysis dialog.
• Capability from the prediction (item 5): Stat → Quality Tools → Capability Analysis → Normal.
• Monitor residuals (item 7): Stat → Control Charts → Variables Charts for Individuals → I-MR on the actual−predicted residuals.

Traps between "model built" and "value delivered"