Question 7 of 7: DNC/CNC Characteristics, Uses of Statistical Quality Control, and an X (Individuals) Control Chart

Question 7: DNC/CNC Characteristics, Uses of Statistical Quality Control, and an X (Individuals) Control Chart (20 marks: 5/5/10)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(i) Characteristics of DNC and CNC

CNC (Computer Numerical Control). Each machine tool has its own dedicated computer controller that stores, edits, and executes the part program directly at the machine; the machine is self-contained — it can hold multiple programs, allow point-of-use editing, run diagnostics, and operate without any connection to another computer. This is the standard configuration for essentially all modern NC machine tools.

DNC (Distributed/Direct Numerical Control). In its original "Direct" sense, DNC meant a central computer controlling several machines' motion directly and in real time, with no individual machine controller — now essentially obsolete. In its modern "Distributed" sense, DNC is a central computer or server that stores the master part-program library and distributes programs over a network to multiple CNC machines, each of which still executes locally through its own CNC controller; the central system typically also collects shop-floor production and machine-status data. Its practical advantages are centralized, version-controlled program management (eliminating manual tape/USB program transfer and the transcription errors that go with it) and integration with production monitoring/scheduling systems (an early form of what is now called a Manufacturing Execution System).

(ii) Uses of Statistical Quality Control in Manufacturing

Statistical quality control (SQC) is used to: monitor a process's stability over time via control charts, distinguishing routine common-cause (natural) variation from special-cause (assignable) variation that signals a real process change; give an early, proactive warning that a process is drifting out of control before it actually produces out-of-specification parts, rather than relying on inspecting parts after the fact; reduce inspection cost by replacing 100% inspection with statistically justified acceptance-sampling plans wherever appropriate; support process-capability studies (Cp/Cpk) that verify a process is inherently able to meet the design tolerance before it is put into full production; provide objective, quantitative data for continuous-improvement and root-cause-analysis efforts; and support supplier qualification and incoming-inspection decisions by giving an objective basis for accepting or rejecting a supplied lot or process.

(iii) X (Individuals) Control Chart

Given. Five individual process measurements (inches): $X_1=4.001$, $X_2=4.003$, $X_3=4.002$, $X_4=4.005$, $X_5=4.000$; control limits to be set at $\pm 3$ standard deviations. A later check sample of three measurements: $4.005$, $4.007$, $4.006$ inches.

Find. (a) The centre line, UCL, and LCL for an X (individuals) chart, and the chart itself. (b) Whether the check sample indicates the process has gone out of control.

Approach. Use the five given measurements to estimate the process mean $\bar{X}$ and standard deviation $s$, set 3-sigma limits on the individual values, then evaluate the check sample two ways — as individual points against those limits, and as a size-3 subgroup mean against control limits scaled for a subgroup of that size (a subgroup mean varies less than a single reading, so its own, tighter limits are the correct test for a shift in the process mean — flagged here as a check assumption since the source does not name a subgroup size for part (b)).

1. Process mean. $$\bar{X} = \dfrac{4.001+4.003+4.002+4.005+4.000}{5} = \dfrac{20.011}{5} = 4.0022\ \text{in.}$$
2. Sample standard deviation. Summing squared deviations from $\bar{X}$ and dividing by $n-1=4$, $$s = \sqrt{\dfrac{\sum (X_i-\bar{X})^2}{n-1}} = \sqrt{\dfrac{0.0000148}{4}} \approx 0.001924\ \text{in.}$$
3. Individuals-chart 3-sigma limits (part a). $$UCL = \bar{X}+3s = 4.0022+3(0.001924) \approx \boxed{4.0080\ \text{in.}}, \qquad LCL = \bar{X}-3s \approx \boxed{3.9964\ \text{in.}}$$
4. Check-sample mean (part b). $$\bar{X}_{\text{new}} = \dfrac{4.005+4.007+4.006}{3} = 4.006\ \text{in.}$$ All three individual readings (4.005, 4.006, 4.007) fall inside the individuals limits from Step 3, so no single reading alone signals an out-of-control point.
5. Subgroup-mean limits for $n=3$. A 3-reading subgroup mean has standard error $s/\sqrt{3}$, giving tighter limits centred on $\bar{X}$: $$UCL_{\bar{X}} = \bar{X}+3\dfrac{s}{\sqrt{3}} = 4.0022+3(0.001111) \approx \boxed{4.0055\ \text{in.}}, \qquad LCL_{\bar{X}} \approx 3.9989\ \text{in.}$$ Since $\bar{X}_{\text{new}}=4.006 > UCL_{\bar{X}}=4.0055$, the check sample's mean signals a real upward shift in the process mean.