Design of Experiments in Manufacturing

Design of experiments (DOE) is a planned way to change several process inputs at the same time and measure how they affect a result. The NIST/SEMATECH e-Handbook of Statistical Methods describes experimental design as a plan laid out before the experiment, chosen so that the data support valid conclusions for the effort spent. In manufacturing, the inputs are settings such as speed, feed, temperature or pressure. The result is something you measure on the part: a dimension, surface roughness or the scrap rate.

DOE replaces the habit of changing one setting while holding all others. The Penn State Department of Statistics calls that one-factor-at-a-time method very inefficient in its STAT 503 course notes and recommends designs that vary several factors together, which also show how factors interact.

Terms to Settle Before the First Run

Penn State lists the planning steps in this order: state the problem, choose factors, levels and ranges, select the response, choose the design, conduct the experiment, analyze the data and draw conclusions. Write the first three down before anyone touches the machine. A response measured at a different spot on each part, or a tool changed halfway through, makes the analysis hard to interpret.

Choose the Design by Objective

NIST sorts designs by objective and number of factors in its design selection guide. A comparative experiment asks whether one factor matters. A screening experiment finds the few important factors among many. A response surface experiment maps interactions and curvature to find better settings.

The run count of a two-level full factorial doubles with every factor. A half fraction halves it again.

The saving has a price. In the half fraction for three factors, each main effect is confounded with a two-factor interaction, so its estimate is really the sum of both. NIST accepts that only when the interactions can be assumed small. The same guide advises planning somewhat fewer runs than the budget allows, keeping spare runs for center points and for repeating runs that went wrong.

A Four-Run Example With Two Factors

The numbers in this example are invented. A turning operation has two factors, cutting speed A and feed B, each at an approved low and high level. The response is surface roughness Ra. NIST lists such a two-level full factorial in standard order with the levels coded −1 and +1. That order is for bookkeeping; the runs themselves are executed in random order.

An effect is the mean response where its column shows +1, minus the mean where it shows −1.

• Speed: (3.0 + 2.0) ÷ 2 − (4.0 + 5.0) ÷ 2 = 2.5 − 4.5 = −2.0 µm
• Feed: (5.0 + 2.0) ÷ 2 − (4.0 + 3.0) ÷ 2 = 3.5 − 3.5 = 0 µm
• Interaction A×B: (4.0 + 2.0) ÷ 2 − (3.0 + 5.0) ÷ 2 = 3.0 − 4.0 = −1.0 µm

On average, feed looks irrelevant. Yet its effect changes direction with speed. At low speed, more feed raises Ra from 4.0 to 5.0 µm; at high speed, it lowers Ra from 3.0 to 2.0 µm. An engineer who varies one factor at a time from the starting point tries speed (4.0 to 3.0 µm, better) and feed (4.0 to 5.0 µm, worse), then keeps speed high and feed low. That route ends at 3.0 µm after three runs and never tries run 4. Moving feed again at the new, high speed would reach run 4, but only by chance: nothing in the one-at-a-time logic says that feed changes direction.

Replicate, Randomize and Block

One run per combination gives effects, but no estimate of their noise. Penn State calls replication the basis for estimating how precise the results are. Suppose a second replicate, run in random order, gives the values in the third column.

With two values per run, each run adds its difference squared divided by two to the error sum of squares: 0.02 + 0.02 + 0.005 + 0.045 = 0.09 µm² with four degrees of freedom. The mean square error is 0.0225 µm², so s = 0.15 µm. Penn State gives the variance of an effect in a 2^k design as σ² ÷ (2^(k−2) · n) in Lesson 6. With k = 2 and n = 2, that is 0.0225 ÷ 2, a standard error of 0.106 µm.

From the means, speed has an effect of −2.05 µm (t ≈ 19.3), the interaction −0.85 µm (t ≈ 8.0) and feed +0.10 µm (t ≈ 0.9). With four degrees of freedom, the two-sided 5 % limit of the t distribution is 2.78. Speed and the interaction stand out clearly; the feed main effect stays within the noise.

Draw the run order for each replicate by lot. If two material lots are in use, put each replicate on one lot. The lot difference then appears as a block effect instead of hiding in the factor effects. When a factor such as furnace temperature is slow to change, the order is restricted and needs a split-plot analysis, which Penn State covers in a later lesson. Record the actual order and every restriction on the trial sheet.

Confirm Before You Release a Setting

The best combination in the example is high speed with high feed, at a mean of 2.15 µm. Before it becomes the standard, run confirmation parts at this setting with normal operators, tools and material, and compare the results with the prediction. Check what the experiment did not measure, such as tool life, cycle time and chip control. After release, the setting belongs under statistical process control, and the Cpk calculator shows whether the process stays capable.

What SYMESTIC Contributes to an Experiment

Planning and analyzing the experiment stays with your statistics software. The Cloud MES SYMESTIC records what happens in production before and after the trial. Scrap and rework are recorded by reason, manually or automatically from the machine signal. For a numeric process parameter, the Process Stability widget shows intervention limits as well as Cp and Cpk per segment and period. Cycle time is displayed as the last and the average value. In series production, you can then check whether a released setting keeps its Cpk.

The page on quality control software shows these views, and the article on statistical quality control covers the methods behind them.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods: What is experimental design?
• NIST/SEMATECH e-Handbook of Statistical Methods: How do you select an experimental design?
• NIST/SEMATECH e-Handbook of Statistical Methods: Two-level full factorial designs
• NIST/SEMATECH e-Handbook of Statistical Methods: Confounding (aliasing)
• Penn State Department of Statistics: STAT 503 Design of Experiments, Lesson 1
• Penn State Department of Statistics: STAT 503 Design of Experiments, Lesson 6 (2^k factorial design)