Engineering & Technology · Manufacturing Processes & Process Selection

Process capability, resource use and manufacturing trade-offs

A process repeatedly produces holes with almost the same diameter, but the holes are consistently too large. Another makes many acceptable parts but wastes substantial energy on rejected units. Repeatability, conformance and resource efficiency are related questions that need different evidence.

Material Structure & Properties and Engineering Design Methods & Trade-offs supply deformation, properties, requirements and trade-offs. This chapter recalls volume, stress and percentage arithmetic before using them. The last lesson introduces mean and standard deviation in context. All numerical prices, dimensions, shrinkage factors, times, yields, distributions and resource quantities are original teaching assumptions. They are not supplier data, machine settings, qualified procedures or allowable service values.

Back to the chapter

After reading, you can

Distinguish stable process behaviour from specification compliance and compare resource use per accepted function.

Compare what is required with what the process does

Specifications express acceptable product limits. A process distribution describes output variation under a stated operating condition. Stability means behaviour is sufficiently consistent over time for that model; capability asks how the stable process compares with the specification. A short run of acceptable measurements does not by itself establish long-term stability or prove that future parts will pass.

For a numerical description, μ denotes the process mean and σ its standard deviation, a measure of spread in the same units as the dimension. Under an assumed normal model, μ ± 3σ describes a central spread convention. It is not a hard boundary that every individual must obey. Estimates also depend on sampling, independence, measurement quality and whether the assumed distribution is appropriate.

Keep spread and centring separate

Our original theoretical distributions have σ = 0.02 mm and specification limits 9.90–10.10 mm. The first has μ = 10.00 mm; its μ ± 3σ interval is 9.94–10.06 mm. Shift the mean to 10.08 mm without changing spread, and that interval becomes 10.02–10.14 mm. Much more of the distribution now extends above the upper limit.

Two conventional indices make this distinction visible. Cp = (upper − lower limit)/(6σ) measures specification width relative to spread. Cpk uses the smaller distance from the mean to a limit, divided by 3σ. These indices require an appropriate process and statistical model; a favourable index is not an individual-part acceptance rule or a substitute for evidence of stability.

Horizontal linear axis diameter 9.86–10.18 mm. Vertical relative probability density normalised to one at each peak; no observed histogram implied. Blue mean 10.00 mm; purple mean 10.08 mm; both standard deviation 0.02 mm. Orange specification limits at 9.90 and 10.10 mm. Cp = 1.666… for both, Cpk = 1.666… centred and 0.333… shifted. The shifted distribution has substantially more area beyond the upper limit. Model assumptions do not establish a real process as stable or capable.

Count resources for accepted output

Compare resource use for the same useful output, including rejected and reworked parts within a stated boundary. Energy per started part may look favourable while energy per accepted part is poor. A recovered material stream can reduce some losses but may still need sorting, transport and reprocessing. Identify which operations and resource categories are included.

Rate, flexibility, quality and cost can conflict. A faster operation may require more finishing; a route with little shaping scrap may use more energy or have lower yield. Compare alternatives using measured or clearly assumed data and test which assumptions change the decision. A single percentage called “efficiency” cannot describe all these consequences.

Use records that link material lot, relevant process state, inspections and dispositions. When the process changes, assess whether old capability evidence still applies. Neither a process family name nor one successful prototype establishes the performance of a continuing production system.

Check capability arithmetic and an energy denominator

For the constructed distributions, Cp = 0.20/(6 × 0.02) = 1.666… for both. Centred Cpk is 0.10/0.06 = 1.666…. After the mean shift, its nearest-limit distance is 10.10 − 10.08 = 0.02 mm, so Cpk = 0.02/0.06 = 0.333…. These are exact-model comparisons, not estimates from a factory sample.

For a separate invented batch, 100 parts enter; 92 pass first inspection, five more pass after rework and three are rejected. Final accepted output is 97, while first-pass yield is 92%. Suppose all included processing and rework together consumed 200 kWh.

Energy per started part is 2.00 kWh; per accepted part it is 200/97 ≈ 2.06 kWh. If upstream material production, transport or use are excluded, call this an included-process energy comparison. It is neither a complete lifecycle result nor evidence that the accepted parts provide equivalent service.

Try the reasoning

Why is unchanged Cp insufficient after a mean shift?
Show the answer and reasoning
Cp uses only specification width and spread. Cpk also responds to the smaller remaining distance from the mean to a limit.
Does μ ± 3σ guarantee that all output lies inside that interval?
Show the answer and reasoning
No. Under the normal model there are tails beyond it, and a real process may also violate the model assumptions.
Why divide this batch’s 200 kWh by 97 rather than 100 when comparing accepted output?
Show the answer and reasoning
The useful accepted count is 97. Dividing by all starts hides the resources consumed for parts that did not become accepted output.

What to carry forward

Judge capability with stable, relevant data and explicit model assumptions. Compare resources per equivalent accepted output across a clear boundary.

References

Teaching resources and research supporting this lesson’s explanations and examples.