Quality Control Management

Quality is the degree to which a product or service meets its specification and the needs of the customer ("fitness for use"): performance, reliability, durability, safety, conformance and appearance.

• Quality control (QC): the operational techniques and activities that check that products conform to specification, and take action on defects (inspection, testing, control charts). It is product oriented.
• Quality assurance (QA): all the planned and systematic activities (procedures, audits, training, documentation) that give confidence that the quality requirements will be met. It is process oriented.
• Total quality control (TQC): Feigenbaum's concept in which quality is built into every function of the organisation (design, purchasing, production, sales, service) and the aim is to give customer satisfaction at the lowest cost.
• Total quality management (TQM): a management philosophy where all employees are involved in continuous improvement of the quality of all products, processes and services to meet or exceed customer expectations.

Principles of TQM

1. Customer focus (internal and external customers).
2. Leadership and commitment of top management.
3. Involvement of all employees and teamwork.
4. Process approach and continuous improvement (Kaizen, PDCA cycle).
5. Decisions based on facts and data (SQC).
6. Supplier partnership.
7. Education and training.

Use of a control chart

A control chart is a plot of a sample statistic against time with a centre line and upper and lower control limits (UCL, LCL) at $\pm 3\sigma$. Points inside the limits show only chance variation; a point outside (or a trend) shows an assignable cause that has to be found and removed. $\bar X$ chart monitors the process average and $R$ chart monitors the spread.

Calculation (all ten samples)

$$\bar{\bar X} = \frac{500.21}{10} = 50.021\ \text{mm}, \qquad \bar R = \frac{0.65}{10} = 0.065\ \text{mm}$$

$\bar X$ chart:

$$UCL = \bar{\bar X} + A_2\bar R = 50.021 + 0.577(0.065) = 50.0585$$ $$LCL = 50.021 - 0.0375 = 49.9835$$

$R$ chart:

$$UCL_R = D_4\bar R = 2.114(0.065) = 0.1374, \qquad LCL_R = D_3\bar R = 0$$

Interpretation

All ranges (0.04 to 0.09) are below 0.1374, so the variation within samples is stable. In the $\bar X$ chart, sample 8 has $\bar X = 50.12 > 50.0585$ and is out of control; all others lie between 49.9835 and 50.0585.

Revised limits (without sample 8)

$$\bar{\bar X} = \frac{450.09}{9} = 50.010, \qquad \bar R = \frac{0.58}{9} = 0.0644$$ $$UCL = 50.010 + 0.577(0.0644) = 50.0472, \qquad LCL = 50.010 - 0.0372 = 49.9728$$ $$UCL_R = 2.114(0.0644) = 0.1362, \qquad LCL_R = 0$$

The remaining nine samples (49.99 to 50.04) lie within the revised limits, so the process is now in statistical control. The estimated process standard deviation is $\sigma = \bar R/d_2 = 0.0644/2.326 = 0.0277$ mm.

Answer: initial $\bar X$ limits 49.9835 to 50.0585 mm, $R$ limit 0 to 0.1374 mm (sample 8 out); revised $\bar X$ limits 49.9728 to 50.0472 mm, $R$ limit 0 to 0.1362 mm.

(a) Acceptance sampling

In acceptance sampling a random sample is drawn from a lot and inspected; the lot is accepted or rejected from the result. It is used when 100 % inspection is too costly, too slow or destructive.

• Single sampling plan: sample size $n$ and acceptance number $c$. If the number of defectives found is $\le c$, the lot is accepted; otherwise it is rejected. Simple and easy to administer.
• Double sampling plan: a first small sample $n_1$ is taken. If defectives $\le c_1$ accept; if $\ge c_2$ reject; if in between, take a second sample $n_2$ and decide on the combined count. It needs on average less inspection and gives a second chance to good lots.

The OC (operating characteristic) curve shows the probability of accepting a lot against its fraction defective. Producer's risk $\alpha$ (rejecting a good lot at AQL) and consumer's risk $\beta$ (accepting a bad lot at LTPD) are fixed by choice of $n$ and $c$.

(b) Process capability

Specification limits: $USL = 25.06$, $LSL = 24.94$ (tolerance $= 0.12$ mm).

$$C_p = \frac{USL - LSL}{6\sigma} = \frac{0.12}{6(0.018)} = 1.11$$ $$C_{pu} = \frac{USL - \mu}{3\sigma} = \frac{25.06-25.01}{0.054} = 0.926, \qquad C_{pl} = \frac{\mu - LSL}{3\sigma} = \frac{25.01-24.94}{0.054} = 1.296$$ $$C_{pk} = \min(C_{pu}, C_{pl}) = 0.926$$

Fraction outside the limits:

$$z_{upper} = \frac{25.06 - 25.01}{0.018} = +2.78, \qquad z_{lower} = \frac{24.94 - 25.01}{0.018} = -3.89$$

• Above $USL$: $P(z > 2.78) = 0.00274$
• Below $LSL$: $P(z < -3.89) = 0.00005$
• Total $\approx 0.00279$, i.e. 0.28 % (about 2800 parts per million)

Comment: $C_p > 1$ means the process spread fits the tolerance, but $C_{pk} < C_p$ shows that the process is off-centre (mean 0.01 mm above the middle of the tolerance). Re-centring on 25.00 mm would give $C_{pk} = C_p = 1.11$ and a reject rate of about 0.09 %. A $C_{pk}$ of 1.33 or more is usually required.

Answer: $C_p$ = 1.11, $C_{pk}$ = 0.926; about 0.28 % of shafts out of limits.