Using Control Charts to Keep an Eye on Variability

Operations Management

Dr. Ron Lembke

Goal of Control Charts See if process is “in control”

Process should show random values No trends or unlikely patterns

Visual representation much easier to interpret Tables of data – any patterns? Spot trends, unlikely patterns easily

NFL Control Chart?

Control Charts

UCL

LCL

avg

Values

Sample Number

Definitions of Out of Control1. No points outside control limits

2. Same number above & below center line

3. Points seem to fall randomly above and below center line

4. Most are near the center line, only a few are close to control limits

1. 8 Consecutive pts on one side of centerline

2. 2 of 3 points in outer third

3. 4 of 5 in outer two-thirds region

Control Charts

Normal Too Low Too high

5 above, or below Run of 5 Extreme variability

Control Charts

UCL

LCL

avg

1σ

2σ

2σ

1σ

Control Charts

2 out of 3 in the outer third

Out of Control Point? Is there an “assignable cause?”

Or day-to-day variability?

If not usual variability, GET IT OUT Remove data point from data set, and recalculate

control limits

If it is regular, day-to-day variability, LEAVE IT IN Include it when calculating control limits

Attributes vs. VariablesAttributes: Good / bad, works / doesn’t count % bad (P chart) count # defects / item (C chart)

Variables: measure length, weight, temperature (x-bar

chart) measure variability in length (R chart)

p Chart Control Limits

# Defective Items in Sample i

# Samples

Sample iSize

z = 2 for 95.5% limits z = 3 for 99.7% limitsp = avg defect raten = avg sample sizesp = sample std dev

pp szpUCL

pp szpLCL

n ni

i1

k

k

p X i

i1

k

ni

i1

k

n

ppsp

)1(

p Chart ExampleYou’re manager of a 1,700 room hotel. For 7 days, you collect data on the readiness of all of the rooms that someone checked out of. Is the process in control (use z = 3)?

© 1995 Corel Corp.

p Chart Hotel Data# Rooms No. Not Proportion

Day n Ready p

1 1,300 130 130/1,300 =.1002 800 90 .1133 400 21 .0534 350 25 .0715 300 18 .066 400 12 .037 600 30 .05

p Chart Control Limits

079.0150,4

326

150,4

30...90130

1

1

k

ii

k

ii

n

Xp

8.5927

150,4

7

600...80013001

k

nn

k

ii

068.7/)05.0...113.010.0( p

p Chart Solution

8.592,079.0 np

111.0*3079.0CL pszp

0457.0LCL,1123.0UCL

0333.0079.0

0111.0

8.592

079.01079.01sp

n

pp

Hotel Room Readiness P-Bar

1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

UCL

Actual

LCL

R Chart Type of variables control chart

Interval or ratio scaled numerical data

Shows sample ranges over time Difference between smallest & largest values

in inspection sample

Monitors variability in process Example: Weigh samples of coffee &

compute ranges of samples; Plot

Why do we need 2 charts?Consistent, but the average is in the wrong place

UCL

LCL

UCL

LCL

X-Bar Chart R Chart

The average works out ok, but way too much variability between points

X-Bar Chart R Chart

UCL

LCL

UCL

LCL

You’re manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control?

Hotel Example

Hotel DataDay Delivery Time

1 7.30 4.20 6.10 3.455.552 4.60 8.70 7.60 4.437.623 5.98 2.92 6.20 4.205.104 7.20 5.10 5.19 6.804.215 4.00 4.50 5.50 1.894.466 10.10 8.10 6.50 5.066.947 6.77 5.08 5.90 6.909.30

R &X Chart Hotel Data

SampleDay Delivery TimeMean Range

1 7.30 4.20 6.10 3.45 5.555.32 7.30 + 4.20 + 6.10 + 3.45 + 5.55

5Sample Mean =

R &X Chart Hotel Data

SampleDay Delivery TimeMean Range

1 7.30 4.20 6.10 3.45 5.555.32 3.85

7.30 - 3.45Sample Range =

Largest Smallest

R &X Chart Hotel Data

SampleDay Delivery TimeMean Range

1 7.30 4.20 6.10 3.45 5.555.32 3.85

2 4.60 8.70 7.60 4.43 7.626.59 4.27

3 5.98 2.92 6.20 4.20 5.104.88 3.28

4 7.20 5.10 5.19 6.80 4.215.70 2.99

5 4.00 4.50 5.50 1.89 4.464.07 3.61

6 10.10 8.10 6.50 5.06 6.947.34 5.04

7 6.77 5.08 5.90 6.90 9.306.79 4.22

R Chart Control Limits

UCL D R

LCL D R

R

R

k

R

R

ii

k

4

3

1

Sample Range at Time i

# Samples

Table 10.3, p.433

Control Chart Limits, p.161

n A2 D3 D4

2 1.88 0 3.278

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 2.00

7 0.42 0.08 1.92

R Chart Control Limits

894.37

22.4...27.485.31

k

RR

k

ii

0894.3*0*

232.8894.3*11.2*

3

4

RDLCL

RDUCL

R

R

10.3 Table from , 43 DD

R Chart Solution

1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

UCLRangeLCL

X Chart Control Limits

k

RR

k

XX

RAXUCL

k

ii

k

ii

X

11

2

Sample Range at Time i

# Samples

Sample Mean at Time i

X Chart Control LimitsA2 from Table 10-3

k

RR

k

XX

RAXLCL

RAXUCL

k

ii

k

ii

X

X

11

2

2

Control Chart Factors, p. 161

n A2 D3 D4

2 1.88 0 3.278

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 2.00

7 0.42 0.08 1.92

R &X Chart Hotel Data

SampleDay Delivery TimeMean Range

1 7.30 4.20 6.10 3.45 5.555.32 3.85

2 4.60 8.70 7.60 4.43 7.626.59 4.27

3 5.98 2.92 6.20 4.20 5.104.88 3.28

4 7.20 5.10 5.19 6.80 4.215.70 2.99

5 4.00 4.50 5.50 1.89 4.464.07 3.61

6 10.10 8.10 6.50 5.06 6.947.34 5.04

7 6.77 5.08 5.90 6.90 9.306.79 4.22

X Chart Control Limits

894.37

22.4...27.485.3

813.57

79.6...59.632.5

1

1

k

RR

k

XX

k

ii

k

ii

566.3894.3*58.0813.5*

060.8894.3*58.0813.5*

2

2

RAXLCL

RAXUCL

X

X

X Chart Solution*

1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

UCLMeanLCL

Summary Overview of “In Control” Attribute vs Continuous Control Charts P Charts X-bar and R charts