Statistical Quality Control in Operations Management

Objec­tives and basic idea of SQC#

The main objec­tive of SQC is to mon­i­tor process vari­a­tion and keep the process in a state of sta­tis­ti­cal con­trol so that qual­ity stays within accept­able stan­dards. In prac­tice it answers four ques­tions:

• Is the process work­ing nor­mally?
• Are defects increas­ing?
• Has a spe­cial prob­lem appeared?
• Is cor­rec­tive action needed now?

The whole sub­ject rests on one idea: vari­a­tion always exists, but not all vari­a­tion is dan­ger­ous. SQC sep­a­rates nat­ural, expected vari­a­tion from unusual vari­a­tion that sig­nals a prob­lem. This makes qual­ity man­age­ment pre­ven­tive and sci­en­tific rather than reac­tive. The approach was devel­oped by Wal­ter She­whart at Bell Tele­phone Lab­o­ra­to­ries in the 1920s and was later pro­moted world­wide by W. Edwards Dem­ing.

Types of vari­a­tion in a process#

Com­mon cause (chance) vari­a­tion#

This is the nat­ural, ran­dom vari­a­tion built into the process. It comes from many small sources act­ing together, such as slight machine vibra­tion, minor dif­fer­ences in mate­r­ial, small dif­fer­ences in how a worker per­forms a task, and nor­mal room tem­per­a­ture changes. A process show­ing only com­mon-cause vari­a­tion is said to be in con­trol, and its out­put is pre­dictable. Reduc­ing com­mon-cause vari­a­tion usu­ally requires a man­age­ment deci­sion, such as buy­ing a bet­ter machine or chang­ing the method.

Spe­cial cause (assign­a­ble) vari­a­tion#

This is unusual vari­a­tion that can be traced to a spe­cific, iden­ti­fi­able cause, for exam­ple a machine break­down, a wrong set­ting, a defec­tive lot of raw mate­r­ial, a worker mis­take or a sud­den tem­per­a­ture change. It needs inves­ti­ga­tion and cor­rec­tive action. The main job of SQC is to sig­nal when a spe­cial cause is prob­a­bly present.

Main tools of SQC#

SQC is usu­ally divided into three broad areas:

• Descrip­tive sta­tis­tics: mean, range and stan­dard devi­a­tion, which sum­marise a set of mea­sure­ments.
• Sta­tis­ti­cal process con­trol (SPC): tak­ing sam­ples dur­ing pro­duc­tion and plot­ting them on con­trol charts to decide whether the process is in con­trol.
• Accep­tance sam­pling: inspect­ing a ran­dom sam­ple from a lot of incom­ing or fin­ished goods and accept­ing or reject­ing the whole lot on that basis.

Sup­port­ing tools include sam­pling plans, fre­quency dis­tri­b­u­tions, his­tograms, Pareto charts, check sheets, cause-and-effect dia­grams and scat­ter dia­grams. For most BBA and MBA exam­i­na­tions, the con­trol chart is the most impor­tant tool.

Process con­trol and prod­uct con­trol#

Process con­trol#

Process con­trol checks the process while pro­duc­tion is going on. Sam­ples are taken at inter­vals and plot­ted on con­trol charts so that prob­lems are cor­rected before many defec­tive units are made. It is pre­ven­tive.

Prod­uct con­trol#

Prod­uct con­trol checks the fin­ished or incom­ing prod­uct to decide whether a lot should be accepted or rejected. Accep­tance sam­pling is the main tech­nique. It is a judge­ment on out­put that already exists, so it detects rather than pre­vents defects.

What is a con­trol chart?#

A con­trol chart is a time-ordered graph of sam­ple results with three hor­i­zon­tal lines:

• Cen­tre line (CL): the process aver­age or expected value.
• Upper con­trol limit (UCL): the high­est value expected from com­mon-cause vari­a­tion alone.
• Lower con­trol limit (LCL): the low­est value expected from com­mon-cause vari­a­tion alone.

Most charts use three-sigma lim­its, that is, the cen­tre line plus or minus three stan­dard devi­a­tions of the plot­ted sta­tis­tic. If the process is sta­ble and the sta­tis­tic is roughly nor­mal, about 99.7 per cent of points should fall inside these lim­its, so a point out­side is strong evi­dence of a spe­cial cause.

Why con­trol charts are use­ful#

• They show at a glance whether the process is sta­ble.
• They give an early warn­ing before large num­bers of defec­tives are pro­duced.
• They stop oper­a­tors from over-adjust­ing a process that is actu­ally fine.
• They pro­vide a record for prob­lem solv­ing and con­tin­u­ous improve­ment.

Types of con­trol charts#

The chart cho­sen depends on the type of data being col­lected.

Vari­able data#

Vari­able data are mea­sured on a con­tin­u­ous scale, such as length in mil­lime­tres or weight in grams. X-bar and R charts are used together: the X-bar chart watches the process cen­tre, and the R chart watches its spread.

Attribute data#

Attribute data are counted: an item is either defec­tive or not, or the num­ber of defects on it is counted. Attribute charts are sim­pler to use because no mea­sur­ing instru­ment is needed, but they need larger sam­ples to give the same infor­ma­tion.

Worked exam­ple 1: X-bar and R chart#

Sup­pose a plant makes bot­tle caps with a tar­get diam­e­ter of 50 mm. Every hour an inspec­tor mea­sures 5 caps ($n = 5$). The means and ranges of 10 sam­ples are shown below.

Step 1: Grand mean. The sum of the ten means is 500.0 mm.

$$\displaystyle \bar{\bar{X}} = \frac{500.0}{10} = 50.00 \text{ mm}$$

Step 2: Aver­age range. The sum of the ten ranges is 9.0 mm.

$$\displaystyle \bar{R} = \frac{9.0}{10} = 0.90 \text{ mm}$$

Step 3: Con­trol chart fac­tors. From stan­dard tables for $n = 5$: $A_2 = 0.577$, $D_3 = 0$ and $D_4 = 2.114$.

Step 4: X-bar chart lim­its.

$$UCL_{\bar{X}} = \bar{\bar{X}} + A_2\bar{R} = 50.00 + 0.577 \times 0.90 = 50.00 + 0.519 = 50.52 \text{ mm}$$

$$LCL_{\bar{X}} = \bar{\bar{X}} - A_2\bar{R} = 50.00 - 0.519 = 49.48 \text{ mm}$$

Step 5: R chart lim­its.

$$UCL_R = D_4\bar{R} = 2.114 \times 0.90 = 1.90 \text{ mm}, \qquad LCL_R = D_3\bar{R} = 0$$

Step 6: Inter­pret. All ten means lie between 49.48 and 50.52 mm, and all ten ranges lie between 0 and 1.90 mm, so the process is in con­trol and these lim­its can be used for future mon­i­tor­ing. Now sup­pose sam­ple 11 gives a mean of 50.62 mm. It is above the UCL, so the oper­a­tor should stop and look for a spe­cial cause, such as a worn die or a changed machine set­ting.

Attribute charts: p-chart and c-chart#

The p-chart#

A p-chart mon­i­tors the pro­por­tion (frac­tion) of defec­tive items in a sam­ple. It is used when each item is classed sim­ply as defec­tive or non-defec­tive. For exam­ple, if 4 bulbs out of 100 are defec­tive, the pro­por­tion defec­tive is 0.04, or 4 per cent. The lim­its are:

$$\displaystyle UCL_p = \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}, \qquad LCL_p = \bar{p} - 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}$$

If the LCL comes out neg­a­tive, it is set to zero, because a pro­por­tion can­not be neg­a­tive.

The c-chart#

A c-chart mon­i­tors the num­ber of defects per unit when the unit or sam­ple size is con­stant, for exam­ple the num­ber of bub­bles, scratches and spots on one painted metal sheet. One sheet may have 3 defects, another 5, another 2. Because defect counts usu­ally fol­low the Pois­son dis­tri­b­u­tion, the stan­dard devi­a­tion is the square root of the mean:

$$UCL_c = \bar{c} + 3\sqrt{\bar{c}}, \qquad LCL_c = \bar{c} - 3\sqrt{\bar{c}}$$

For instance, if sheets aver­age $\bar{c} = 4$ defects, then $UCL_c = 4 + 3 \times 2 = 10$ and $LCL_c = 4 - 6 = -2$, which is set to 0.

Dif­fer­ence between p-chart and c-chart#

Worked exam­ple 2: p-chart#

Sup­pose an LED bulb fac­tory tests 10 sam­ples of 200 bulbs each ($n = 200$). The num­bers of defec­tive bulbs found are 6, 4, 8, 5, 7, 3, 9, 6, 5 and 7.

Step 1: Aver­age pro­por­tion defec­tive. Total defec­tives = 60; total inspected = $10 \times 200 = 2{,}000$.

$$\displaystyle \bar{p} = \frac{60}{2000} = 0.030$$

Step 2: Stan­dard devi­a­tion of the pro­por­tion.

$$\displaystyle \sigma_p = \sqrt{\frac{0.03 \times 0.97}{200}} = \sqrt{0.0001455} = 0.01206$$

Step 3: Con­trol lim­its.

$$UCL_p = 0.030 + 3 \times 0.01206 = 0.0662$$

$$LCL_p = 0.030 - 0.0362 = -0.0062 \rightarrow 0$$

Step 4: Inter­pret. The sam­ple pro­por­tions range from $3/200 = 0.015$ to $9/200 = 0.045$. All lie inside 0 to 0.0662, so the defec­tive rate is sta­ble at about 3 per cent. The process is in con­trol, although man­age­ment may still decide that 3 per cent is too high and work to improve it.

In-con­trol and out-of-con­trol processes#

In-con­trol process#

A process is in con­trol when points fall ran­domly inside the lim­its with no unusual pat­tern. Only com­mon-cause vari­a­tion is present, and out­put is pre­dictable.

Out-of-con­trol process#

A process is likely to be out of con­trol when any of the fol­low­ing appears:

• One or more points fall out­side the UCL or LCL.
• A run of about seven or more con­sec­u­tive points lies on one side of the cen­tre line.
• A steady upward or down­ward trend of sev­eral points, often caused by tool wear.
• Repeat­ing cycles or points hug­ging one limit.

An out-of-con­trol sig­nal means "inves­ti­gate", not "the prod­uct is bad". The cause must be found and removed before the process is trusted again.

Con­trol lim­its ver­sus spec­i­fi­ca­tion lim­its#

A process can be in con­trol yet still pro­duce units out­side spec­i­fi­ca­tion if its nat­ural spread is too wide. The com­par­i­son of process spread with the spec­i­fi­ca­tion width is called process capa­bil­ity.

Role of sam­pling in SQC#

Check­ing every unit is slow and costly, and for destruc­tive tests (such as test­ing a fuse by blow­ing it) it is impos­si­ble. SQC there­fore relies on sam­ples. A good sam­ple is taken ran­domly, at reg­u­lar inter­vals, and in ratio­nal sub­groups (units made under sim­i­lar con­di­tions), so that vari­a­tion within a sam­ple reflects com­mon causes and vari­a­tion between sam­ples reveals spe­cial causes. In accep­tance sam­pling, a sam­pling plan states the sam­ple size and the max­i­mum num­ber of defec­tives allowed before a lot is rejected.

Why SQC is bet­ter than only final inspec­tion#

Final inspec­tion finds defects after the money has already been spent mak­ing them. It is also tir­ing and error-prone when vol­umes are high. SQC checks the process dur­ing pro­duc­tion, so prob­lems are fixed at the source, fewer defec­tives are made, and inspec­tion effort is reduced. In short, SQC builds qual­ity in rather than inspect­ing it in.

SQC in man­u­fac­tur­ing and ser­vices#

Man­u­fac­tur­ing: bot­tle fill vol­umes, shaft diam­e­ters, tablet weights, paint defects, bulb fail­ures and weld strength are all mon­i­tored with con­trol charts.

Ser­vices: banks track the pro­por­tion of trans­ac­tions with errors; hos­pi­tals mon­i­tor wait­ing times and infec­tion rates; call cen­tres chart aver­age han­dling time; air­lines track the pro­por­tion of flights delayed; restau­rants count cus­tomer com­plaints per day.

Advan­tages and lim­i­ta­tions#

Advan­tages of SQC#

• Early prob­lem detec­tion before many defec­tives are pro­duced.
• Bet­ter process con­trol and fewer unnec­es­sary adjust­ments.
• Reduced wastage of mate­r­ial, rework and scrap.
• Bet­ter con­sis­tency of out­put and cus­tomer sat­is­fac­tion.
• Bet­ter deci­sion mak­ing based on facts rather than opin­ion.
• Lower cost, because sam­pling is cheaper than 100 per cent inspec­tion.

Lim­i­ta­tions of SQC#

• It needs cor­rect, hon­est data; poor mea­sure­ment gives mis­lead­ing charts.
• It needs trained peo­ple who can cal­cu­late lim­its and read pat­terns.
• It does not solve prob­lems auto­mat­i­cally; it only sig­nals them.
• It may not cap­ture every qual­ity issue, such as design faults or rare defects missed by sam­pling.
• Sam­pling always car­ries some risk of a wrong deci­sion.

SQC com­pared with qual­ity con­trol#

Qual­ity con­trol is the broad func­tion of ensur­ing prod­ucts and ser­vices meet stan­dards. It includes inspec­tion, test­ing, sup­plier checks, train­ing and cor­rec­tive action. Sta­tis­ti­cal Qual­ity Con­trol is the part of qual­ity con­trol that uses sta­tis­ti­cal tools such as sam­pling and con­trol charts. All SQC is qual­ity con­trol, but not all qual­ity con­trol is sta­tis­ti­cal.

SQC and con­tin­u­ous improve­ment#

Once spe­cial causes are removed and the process is sta­ble, man­agers can work on reduc­ing com­mon-cause vari­a­tion, which nar­rows the con­trol lim­its over time. SQC there­fore sup­ports the Plan-Do-Check-Act cycle, Total Qual­ity Man­age­ment and Six Sigma pro­grammes. Each improve­ment is con­firmed by the chart: if the new lim­its are tighter, the change worked.

Key terms#

Sta­tis­ti­cal Qual­ity Con­trol

The use of sta­tis­ti­cal meth­ods to mon­i­tor, con­trol and improve qual­ity.

Com­mon cause vari­a­tion

Nat­ural, ran­dom vari­a­tion inher­ent in a sta­ble process.

Spe­cial cause vari­a­tion

Unusual vari­a­tion from an iden­ti­fi­able source that needs cor­rec­tion.

Con­trol chart

A time-ordered plot of sam­ple results with a cen­tre line and upper and lower con­trol lim­its.

X-bar chart

A vari­ables chart that tracks the sam­ple mean, show­ing the process cen­tre.

R chart

A vari­ables chart that tracks the sam­ple range, show­ing process spread.

p-chart

An attrib­utes chart for the pro­por­tion of defec­tive items in a sam­ple.

c-chart

An attrib­utes chart for the num­ber of defects per unit of con­stant size.

Accep­tance sam­pling

Accept­ing or reject­ing a whole lot on the basis of a ran­dom sam­ple.

Spec­i­fi­ca­tion lim­its

Tol­er­ances set by design or the cus­tomer for indi­vid­ual units.

Com­mon ques­tions#

What is the main dif­fer­ence between com­mon and spe­cial causes of vari­a­tion?#

Com­mon causes are many small, ran­dom sources that are always present in a sta­ble process. Spe­cial causes are spe­cific, iden­ti­fi­able events such as a bro­ken tool or a bad batch of mate­r­ial. Con­trol charts are designed to sig­nal spe­cial causes so that they can be removed.

When should a p-chart be used instead of a c-chart?#

Use a p-chart when each item is judged as defec­tive or non-defec­tive and you want the frac­tion defec­tive. Use a c-chart when you count the num­ber of defects on a unit of con­stant size, since one unit may carry sev­eral defects.

Why are X-bar and R charts used together?#

The X-bar chart shows whether the process aver­age has shifted, while the R chart shows whether vari­abil­ity has changed. A process can keep the right aver­age but become more erratic, which only the R chart will reveal.

Does a point inside the con­trol lim­its mean the prod­uct meets spec­i­fi­ca­tion?#

Not nec­es­sar­ily. Con­trol lim­its describe process behav­iour and are cal­cu­lated from data, while spec­i­fi­ca­tion lim­its come from the cus­tomer. A sta­ble process with too much nat­ural spread can still pro­duce out-of-spec­i­fi­ca­tion units.

Why is the lower con­trol limit some­times set to zero?#

For pro­por­tions and defect counts, the for­mula can give a neg­a­tive value when the aver­age is small. Since a neg­a­tive pro­por­tion or count is impos­si­ble, the LCL is set to zero.

Is SQC use­ful in ser­vice organ­i­sa­tions?#

Yes. Banks, hos­pi­tals, air­lines and call cen­tres chart error rates, wait­ing times, delays and com­plaints in exactly the same way that fac­to­ries chart dimen­sions and defects.

Ref­er­ences#

1. Mont­gomery, D. C. Intro­duc­tion to Sta­tis­ti­cal Qual­ity Con­trol. Wiley.
2. She­whart, W. A. (1931) Eco­nomic Con­trol of Qual­ity of Man­u­fac­tured Prod­uct. D. Van Nos­trand Com­pany.
3. Dem­ing, W. E. (1986) Out of the Cri­sis. MIT Press.
4. Steven­son, W. J. Oper­a­tions Man­age­ment. McGraw-Hill Edu­ca­tion.
5. Heizer, J., Ren­der, B. and Mun­son, C. Oper­a­tions Man­age­ment: Sus­tain­abil­ity and Sup­ply Chain Man­age­ment. Pear­son.

Related read­ing#

• Qual­ity Man­age­ment in Oper­a­tions Man­age­ment
• Main­te­nance Man­age­ment in Oper­a­tions Man­age­ment
• Sequenc­ing in Oper­a­tions Man­age­ment