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6CHAPTER Statistical Process

SUPPLEMENT Control

DISCUSSION QUESTIONS

1. The Central Limit Theorem provides the basis for the calculation of required sample size.

2. The ultimate goal of the X and R-charts is to ascertain, by a sampling procedure, that the relevantparameter is kept within specific upper and lower bounds. The Xbar chart alone tells us only thatthe average or variable values are within the appropriate limits. The combination ofX and the R-charts allows one to determine that both the average and the deviations are within the limits.

3. X-charts: depict the variation in average value of a variable (weight or diameter, for example)

R-charts: depict the average range or deviation of a variablep-charts: depict the average value of an attribute (percent defective, for example)c-charts: depict the number of times an attribute (defect, for example) occurs

4. A process can be out of control because of assignable variation, which can be traced to specificcauses. Examples include such factors as:n Tool wearn A change in raw materialsn A change in working environment (temperature or humidity, for example)n Tired or poorly trained labor

5. Walter Shewhart brought his knowledge of statistics to bear on the problems of statistical samplingand quality control, providing the foundations of statistical quality control as we know it.

6. Natural variations are those variations that are inherent in the process and for which there is noidentifiable cause. These variations fall in a natural pattern.

Assignable causes are variations beyond those that can be expected to occur because of naturalvariation. These variations can be traced to a specific cause.

7. Occurrences (items) can fall outside of the expected range of the process and still be in controlbecause we expect that to happen in the tails of the distribution.

Also, patterns such as trends, too many points on one side of the mean, or strong fluctuationsin readings can cause a process to be out of control.

8. Producers risk: the risk of rejecting a good lotConsumers risk: the risk of accepting a defective lot

9. Type I error: when one rejects a hypothesis that is in fact true (i.e., conclude that a batch isunacceptable when it is not)Type II error: when one accepts a hypothesis that is in fact false (i.e., conclude that a batch isacceptable when it is not)

10. Cpk is the way we express process capability. It measures the difference between desired and an

actual dimension of products made in a process.

A Cpk of 1.0 means that the process variation is centered within the desired upper and lower

specifications.

Chapter 6 Supplement: Statistical Process Control 87

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11. A run of 5 implies that assignable variation is present.

12. The 5 steps used to establish X andR charts are:n Collect 2025 samples ofn = 4 orn = 5 from a stable process.

n Compute X and R , set limits (usually at 3 sigma). If the current process is not stable, use a

desired mean instead ofX.n

Graph the sample means and ranges and see if they fall outside the limits.n Look for points or patterns indicating the process is out of control.n Collect more samples, and revalidate limits if needed.

13. The desired mean is used when the mean of a process being observed is unknown or out of control.

14. A run test is used to help spot abnormalities in a control chart process. It is used if points are notindividually out of control, but form a pattern above or below the nominal line.

15. Managerial issues include:n Selecting places in their process that need SPCn Deciding which type of control charts best fitn Setting rules for workers to follow if certain points or patterns emerge

END-OF-SUPPLEMENT PROBLEMS

S6.1 n = 10, X= 75 , = 195. , z= 3

UCL

LCL

= + H K=

= FHIK=

75 3195

107685

75 3195

107315

..

..

S6.2 n = 5, X= 50 , = 018. , z= 3

UCL

LCL

= +

H K=

= FHIK=

50 3172

55230

50 3172

54770

..

..

S6.3 n = 5. From Table S6.1, A2 0 577= . , D4 2115= . , D3 0=

(a) UCL X A R

LCL X A R

X

X

= + = + =

= = =

2

2

50 0 577 4 52 308

50 0 577 4 47 692

. .

. .

(b) UCL D R

LCL D R

R

R

= = =

= = =

4

3

2115 4 8 456

0 4 0

. .

S6.4 n = 6 . From Table S6.1, A2 0 483= . , D4 2 004= . , D3 0=

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

46 0 483 2 46 966

46 0 483 2 45 034

2 004 2 4 008

0 2 0

. .

. .

. .

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S6.5 n = 10 . From Table S6.1, A2 0308= . , D4 1777= . , D3 0233= .

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

60 0 308 3 60 924

60 0 308 3 59 076

1777 3 5 331

0 223 3 0 669

. .

. .

. .

. .

Hour X R Hour X R Hour X R1 3.25 0.71 9 3.02 0.71 17 2.86 1.432 3.10 1.18 10 2.85 1.33 18 2.74 1.293 3.22 1.43 11 2.83 1.17 19 3.41 1.614 3.39 1.26 12 2.97 0.40 20 2.89 1.095 3.07 1.17 13 3.11 0.85 21 2.65 1.086 2.86 0.32 14 2.83 1.31 22 3.28 0.467 3.05 0.53 15 3.12 1.06 23 2.94 1.588 2.65 1.13 16 2.84 0.50 24 2.64 0.97

AverageX=

2.982, Average R = 1.02375, n = 4 . From Table S6.1, A2

0 729= . , D4

2 282= . ,D3 0 0= . .

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

2 982 0 729 1024 3 728

2 982 0 729 1024 2 236

2 282 1024 2 336

0 1024 0

. . . .

. . . .

. . .

.

The smallest sample mean is 2.64, the largest 3.39. Both are well within the control limits.Similarly, the largest sample range is 1.61, also well within the control limits. We can concludethat the process is presently within control. However, the first five values for the mean are abovethe expected mean; this may be the indication of a problem in the early stages of the process.

2.000

Sample

5 2510 2015

2.50

3.00

3.50

4.00

Control Chart

LCL

UCL

X


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0.000

Sample

5 2510 2015

0.50

1.00

1.50

2.00

Control Chart R

LCLUCL

2.50

S6.7 Sample X R Sample X R Sample X R

1 63.5 2.0 10 63.5 1.3 19 63.8 1.32 63.6 1.0 11 63.3 1.8 20 63.5 1.63 63.7 1.7 12 63.2 1.0 21 63.9 1.04 63.9 0.9 13 63.6 1.8 22 63.2 1.85 63.4 1.2 14 63.3 1.5 23 63.3 1.76 63.0 1.6 15 63.4 1.7 24 64.0 2.07 63.2 1.8 16 63.4 1.4 25 63.4 1.58 63.3 1.3 17 63.5 1.19 63.7 1.6 18 63.6 1.8

X= 6349. , R = 15. , n = 4 . From Table S6.1, A2 0 729= . , D4 2282= . , D3 0 0= . .

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

63 49 0 729 15 64 58

63 49 0 729 15 62 40

2 282 15 3 423

0 15 0

. . . .

. . . .

. . .

.

The process is in control.

60.000

Sample

5 2510 2015

61.00

62.00

63.00

64.00

Control Chart

LCLUCL

65.00

30

X


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Sample

0.000 5 2510 2015

1.00

2.00

3.00

4.00

Control Chart R

LCLUCL

30

S6.8 Time Box 1 Box 2 Box 3 Box 4 Average Range

9 AM 9.8 10.4 9.9 10.3 10.10 0.6010 AM 10.1 10.2 9.9 9.8 10.00 0.4011 AM 9.9 10.5 10.3 10.1 10.20 0.6012 PM 9.7 9.8 10.3 10.2 10.00 0.60

1 PM 9.7 10.1 9.9 9.9 9.90 0.40Average 10.04 0.52

n = 4 . From Table S6.1, A2 0 729= . , D4 2282= . , D3 0 0= . .

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

10 04 0 729 0 52 10 42

10 04 0 729 0 52 9 66

2 282 0 52 1187

0 0 52 0

. . . .

. . . .

. . .

.

The smallest sample mean is 9.9, the largest 10.2. Both are well within the control limits.Similarly, the largest sample range is 0.6, also well within the control limits. Hence, we canconclude that the process is presently within control.

One step the QC department might take would be to increase the sample size to provide aclearer indication as to both control limits andwhether or not the process is in control.

S6.9 X= 1990. , R = 0 34. , n = 4 , A2 0 729= . , D4 2282= .

(a) UCL

LCL

= + =

= =

( )

( )

19 90 0 729 0 34 2015

19 90 0 729 0 34 19 65

. . . .

. . . .

(b) UCL

LCL

= =

=

( )2 282 0 34 0 78

0

. . .

(c) The ranges are ok; the means are not in control.

S6.10 X= 10, R = 3 3.

(a) standard deviation = 1.36, x = =136 5 0 61. .


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(b) Using x

UCL

LCL

= + =

= =

( )

( )

10 3 0 61 1183

10 3 061 817

. .

. .

Using A2 0 577= .

UCL

LCL

= + =

= =

( )

( )

10 33 0 577 1190

10 33 0 577 810

. . .

. . .

(c) UCL

LCL

= =

= =

( )

( )

2115 33 6 98

0 33 0

. . .

.

(d) Yes, both mean and range charts indicate process is in control.

S6.11 RDesired = 3 5. , XDesired = 50 , n = 6

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

3

4

50 0 483 35 5169

50 0 483 3 5 48 31

2 004 3 5 7 014

0 3 5 0

. . .

. . .

. . .

.

The smallest sample range is 1, the largest 6. Both are well within the control limits.The smallest average is 47, the largest 57. Both are outside the proper control limits.

Therefore, although the range is within limits, the average is outside limits, and apparentlyincreasing. Immediate action is needed to correct the problem and get the average within thecontrol limits again.

S6.12 Sample Number Sample Range Sample Mean

1 1.10 462 1.31 453 0.91 464 1.10 475 1.21 486 0.82 477 0.86 508 1.11 499 1.12 51

10 0.99 5211 0.86 5012 1.20 52

R = 1049. , X= 48583. , n = 10 . Use X= 47 , R = 10. , n = 10

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

47 0 308 1 47 308

47 0 308 1 46 692

1777 1 1777

0 223 1 0 223

. .

. .

. .

. .

The smallest sample range is 0.82, the largest 1.31. Both are well within the control limits.Almost all the averages are outside the control limits. Therefore the process is out of control.


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While the range is within limits, the average is outside limits, and apparently increasing.Immediate action is needed to correct the problem and get the average within the control limitsagain.

44.000

Sample

2 104 86

46.00

48.00

50.00

52.00

Control Chart

LCL

UCL

54.00

12 14

X

0.000

Sample

2 104 86

0.50

1.00

1.50

2.00Control Chart R

LCL

UCL

12 14

S6.13 0.51

0.505

0.495

0.49

drill bit (largest)

drill bit (smallest)

0 505 0 49 0 015. . . = , 0 015 0 00017 88. . = holes within standard0 495 0 49 0 005. . . = , 0 005 0 00017 29. . = holes within standardAny one drill bit should produce at least 29 holes that meet tolerance, but no more than 88 holes

before being replaced.

S6.14 UCL pp p

n

LCL pp p

n

p

p

= +( )

= ( )

31

31


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n = 100

Percent Defective (p) 1 p p p n1 ( ) LCLp UCLp

0.02 0.98 0.014 0.0 0.0620.04 0.96 0.020 0.0 0.1000.06 0.94 0.024 0.0 0.1320.08 0.92 0.027 0.0 0.161

0.10 0.90 0.030 0.01 0.190

S6.15 UCL pp p

n

LCL pp p

n

p

p

= +( )

= ( )

31

31

n = 200

Percent Defective (p) 1 p p p n1 ( ) LCLp UCLp

0.01 0.99 0.0070 0.0 0.03100.02 0.98 0.0099 0.0 0.04970.03 0.97 0.0121 0.0 0.06630.04 0.96 0.0139 0.0 0.08170.05 0.95 0.0154 0.0038 0.09620.06 0.94 0.0168 0.0096 0.11040.07 0.93 0.0180 0.0160 0.12400.08 0.92 0.0192 0.0224 0.13760.09 0.91 0.0202 0.0294 0.15060.10 0.90 0.0212 0.0364 0.1636

0.000

Percent defective

0.02 0.10.04 0.080.06

0.05

0.1

0.15

0.2Control Limits for Percent Defective

LCL

UCL

S6.16

EMBED EQUATION

UCL pp p

n

LCL pp p

n

UCL

LCL

p

p

p

p

.. .

.

.. .

.

= +( )

= ( )

= +

=

=

=

31

31

0 015 30 015 0 985

5000 0313

0 015 30 015 0 985

5000 0013 or zero


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S6.17

EMBED EQUATION

UCL pp p

n

LCL pp p

n

UCL

LCL

p

p

p

p

.. .

.

.. .

.

= +( )

= ( )

= +

=

=

=

31

31

0 035 30 035 0 965

5000 0597

0 035 30 035 0 965

5000 0103

S6.18

EMBED EQUATION

p

p

p

UCL p

LCL p

= =

= + = + =

= = = =

( )

( )

( )

0 035 0 965

1000 0184

3 0 035 3 0 0184 0 0901

3 0 035 3 0 0184 0 0201 0

. ..

. . .

. . .

Increased control limits by more than 50%. No, sample size should not be changed, as quality willbe seriously affected.

S6.19

EMBED EQUATION

UCL pp p

n

LCL pp p

n

UCL

LCL

p

p

p

p

.. .

.

.. .

.

= +( )

= ( )

= +

=

=

=

31

31

0 011 30 011 0 989

10000 0209

0 011 30 011 0 989

10000 0011

S6.20 n = 200 , p = =( )50 10 200 0 025.

UCL p p pn

LCL pp p

n

UCL

LCL

p

p

p

p

.. .

.

.. .

.

= + ( )

= ( )

= +

=

=

=

3 1

31

0 025 30 025 0 975

2000 0581

0 025 30 025 0 975

2000 0081 or zero

The highest percent defective is .04; therefore the process is in control.

S6.21 Number Number NumberDay Defective Day Defective Day Defective

1 6 8 3 15 42 5 9 6 16 53 6 10 3 17 64 4 11 7 18 55 3 12 5 19 46 4 13 4 20 37 5 14 3 21 7


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pp

N

p p

n

i= = =

=( )

=

=

( )

98

211000 0467

1 0 0467 0 9533

1000 0211

.

. ..

For a 3p-chart, the upper control level is given by:

UCL p

LCL

= + = + =

=

3 0 0467 3 0 0211 011

0

. . .

0.000

Sample

5 2510 2015

0.05

0.10

0.15

0.20

Chartp

LCL

UCL

The process is in control.

S6.22 Average blemishes/table = =2000 100 20 . Using a normal approximation to the Poisson distribu-tion:

c

UCL c

LCL c

c

c

=

= =

= + = + =

= = =

20

20 4 472

3 20 3 4 472 33 4 33

3 20 3 4 472 6 6 7

.

. .

. .

or blemishes

or blemishes

Yes42 blemishes is considerably above the upper control limit.

S6.23 c

UCL c c

LCL c c

=

= + = + =

= = =

6

3 6 3 6 1335

3 6 3 6 135 or 0

.

.

Ten complaints are within the control limits, so this many complaints would not be consideredunusual.

S6.24 Cpk =

NM QP = =NM( )( ) ( )( )min of or8135 8 00

3 0 04

8 00 7 865

3 0 04

0135

0121125

0135

0121125

. .

.,

. .

.

.

.. ,

.

.. . Therefore Cpk = 1125. .

The process is centered and will produce within the specified tolerance.


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S6.25 Cpk =

NM QP = =NM( )( ) ( )( )min of or41 4

3 01

4 3 9

3 01

01

0 30 33

01

0 30 33

.

.,

.

.

.

.. ,

.

.. . Therefore Cpk = 0 33. . The process

will not produce within the specified tolerance.

S6.26 Cpk =

NM QP NM( )( ) ( )( )min of or16 5 16

3 1

16 15 5

3 1

0 5

3

0 5

3

.,

. .,

.. Therefore Cpk = 0166. .

S6.27 Time Box 1 Box 2 Box 3 Box 4 Average

9 AM 9.8 10.4 9.9 10.3 10.110 AM 10.1 10.2 9.9 9.8 10.011 AM 9.9 10.5 10.3 10.1 10.212 PM 9.7 9.8 10.3 10.2 10.0

1 PM 9.7 10.1 9.9 9.9 9.9Average = 10.04

Std. Dev. = 0.11

101 10

3 0110 3

.

..

=

( )( )and

10 9 9

3 0110 3

=

( )( )

.

..

As 0.3 is less than 1, the process will not produce within the specified tolerance. This makes for aninteresting discussion of the control chart for problem S6.8.

S6.28 X Range

Upper Control Limit 61.13136 41.6232Center Line (ave) 49.776 19.68Lower Control Limit 38.42064 0.00

Recent DataSample

Hour 1 2 3 4 5 X R

26 48 52 39 57 61 51.4 2227 45 53 48 46 66 51.6 21

28 63 49 50 45 53 52.0 1829 47 70 45 52 61 57.0 2530 45 38 46 54 52 47.0 16

(a) Yes, the process appears to be under control. Samples 2630 stayed within the boundaries ofthe upper and lower control limits for both X andR charts.

(b) The observed lifetimes have a mean of approximately 50 hours, which supports the claimmade by Ward Battery Corp. However, the variance from the mean needs to be controlledand reduced. Lifetimes should deviate from the mean by no more than 5 hours (10% of thevariance).


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CASE STUDIES

BAYFIELD MUD COMPANY

1. The first thing that must be done is to develop quality control limits for the sample means. This canbe done as follows. Because the process appears to be unstable, we can use the desired mean as thenominal line. Desired x = 50 0. , s = 12. (from past results of Wet-Land Drilling),

= = = =s n 12 6 12 2 45 0 489. . . . . At a 99.73% confidence interval Z= 3 :

UCL X

LCL X

X

X

= + = + = + =

= = =

3 50 3 0 489 50 1467 5147

3 50 147 48 53

. . .

. .

Now that we have appropriate control limits, these must be applied to the samples taken on theindividual shifts:

Day Shift*

Time Ave Low High Ave Low High Ave Low High

6:00 49.6 48.7 50.7 48.6 47.4 52.0 48.4 45.0 49.07:00 50.2 49.1 51.2 50.0 49.2 52.2 48.8 44.8 49.78:00 50.6 49.6 51.4 49.8 49.0 52.4 49.6 48.0 51.89:00 50.8 50.2 51.8 50.3 49.4 51.7 50.0 48.1 52.7

10:00 49.9 49.2 52.3 50.2 49.6 51.8 51.0 48.1 55.211:00 50.3 48.6 51.7 50.0 49.0 52.3 50.4 49.5 54.112:00 48.6 46.2 50.4 50.0 48.8 52.4 50.0 48.7 50.91:00 49.0 46.4 50.0 50.1 49.4 53.6 48.9 47.6 51.2

Evening ShiftTime Ave Low High Ave Low High Ave Low High

2:00 49.0 46.0 50.6 49.7 48.6 51.0 49.8 48.4 51.03:00 49.8 48.2 50.8 48.4 47.2 51.7 49.8 48.8 50.84:00 50.3 49.2 52.7 47.2 45.3 50.9 50.0 49.1 50.65:00 51.4 50.0 55.3 46.8 44.1 49.0 47.8 45.2 51.26:00 51.6 49.2 54.7 46.8 41.0 51.2 46.4 44.0 49.77:00 51.8 50.0 55.6 50.0 46.2 51.7 46.5 44.4 50.08:00 51.0 48.6 53.2 47.4 44.0 48.7 47.2 46.6 48.99:00 50.5 49.4 52.4 47.0 44.2 48.9 48.4 47.2 49.5

Night ShiftTime Ave Low High Ave Low High Ave Low High

10:00 49.2 46.1 50.7 47.2 46.6 50.2 49.2 48.1 50.711:00 49.0 46.3 50.8 48.6 47.0 50.0 48.4 47.0 50.8

12:00 48.4 45.4 50.2 49.8 48.2 50.4 47.2 46.4 49.21:00 47.6 44.3 49.7 49.6 48.4 51.7 47.4 46.8 49.02:00 47.4 44.1 49.6 50.0 49.0 52.2 48.8 47.2 51.43:00 48.2 45.2 49.0 50.0 49.2 50.0 49.6 49.0 50.64:00 48.0 45.5 49.1 47.2 46.3 50.5 51.0 50.5 51.55:00 48.4 47.1 49.6 47.0 44.1 49.7 50.5 50.0 51.9

* Bold-faced type indicates a sample outside the quality control limits.

(a) Day shift (6:00 AM2:00 PM):


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Number of means within control limits

Total number of means=

23

2496%

(b) Evening shift (2:00 PM10:00 PM):


Total number of means

= 12

24

50%

(c) Night shift (10:00 PM6:00 AM):


Total number of means=

12

2450%

As is now evident, none of the shifts meet the control specifications. Bag weight monitoringneeds improvement on all shifts. The problem is much more acute on the evening and nightshifts staffed by the more recent hires.

Note also, that the number of samples indicating a short weight is much greater thanthe number indicating excess weight.

With regard to the range, 99.73% of the individual bag weights should lie within 3

of the mean. This would represent a range of 6, or 7.2. Only one of the ranges defined bythe difference between the highest and lowest bag weights in each sample exceeds this range.It would appear, then, that the problem is not due to abnormal deviations between thehighest and lowest bag weights, but rather to poor adjustments of the bag weight-feedercausing assignable variations in average bag weights.

The proper procedure is to establish mean and range charts to guide the bag packers.The foreman would then be alerted when sample weights deviate from mean and rangecontrol limits. The immediate problem, however, must be corrected by additional bag weightmonitoring and weight-feeder adjustments. Short-run declines in bag output may benecessary to achieve acceptable bag weights.

44.01

Sample

5 219 1713

46.0

48.0

50.0

52.0 LCLUCL

54.0Control Chart-XBayfield Case:

3 197 1511 23

Day

Eve

Night


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SPC AT THE GAZETTE

1. The overall fraction of errors (p) and the control limits are developed as follows:

p

sp p

n

=

=

=

=( )

=

=

Total number of errorsNumber of samples Sample size

12030 100

0 04

1 0 04 0 96

1000 0196

.

. ..

Then the control limits are given (for a 95% confidence interval; 95% = 1.96) by:

UCL p s

LCL p s

= + = + =

= = =

196 0 04 196 0 0196 0 0784

196 0 04 196 0 0196 0 0016

. . . . .

. . . . .

Errors in Fraction of Errors in Fraction of Sample Sample Errors (n/100)* Sample Sample Errors (n/100)

1 2 0.02 16 2 0.022 4 0.04 17 3 0.033 10 0.10 18 7 0.074 4 0.04 19 3 0.035 1 0.01 20 2 0.026 1 0.01 21 3 0.037 13 0.13** 22 7 0.078 9 0.09 23 4 0.049 11 0.11** 24 3 0.03

10 0 0.00 25 2 0.0211 3 0.03 26 2 0.0212 4 0.04 27 0 0.0013 2 0.02 28 1 0.0114 2 0.02 29 3 0.03

15 8 0.08 30 4 0.04* Bold-faced entries indicate sample fractions outside the quality control limits.** Indicates sample fractions outside the industry standard quality control limits.

Both the table presented above, and the control chart indicate that the quality requirements of theGazette are more stringent than those of the industry as a whole. In five instances, the fraction oferrors exceeds the firms upper control limit; in two cases, the industrys upper control limit isexceeded. An investigation, leading to corrective action, is clearly warranted.


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0.00

Sample

0.04

0.08

0.12

0 5 2510 2015 30

0.02

0.06

0.10

0.14

Firm LCLFirm UCL

Ind LCLInd UCL

UCL

LCL

Chartp

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GREEN RIVER CHEMICAL CO.

This is a very straightforward case. Running software to analyze the data will generate the X-chart as

UCL

LCL

X

X

: .

Nominal: .

: .

6113

4978

3842


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and the range chart as

UCL

LCL

R

R

:

Nominal:

: .00

41.62

19.68

0

Next, students need to take the means and ranges for the five additional samples.

Date Mean Range

April 6 52 147 57 258 47 169 51.4 22

10 51.6 21

The mean and the ranges are all well within the control limits for this week. There is, however, anoticeable change in the original data at time 13, where the range suddenly dropped. It then goes back upat time 16. The data were generated by students in class, and changes in the process were made at theaforementioned times. The control chart identifies that these changes took place.


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