Operations Management

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# Operations Management - PowerPoint PPT Presentation

Operations Management. Supplement 6 – Statistical Process Control. PowerPoint presentation to accompany Heizer/Render Principles of Operations Management, 7e Operations Management, 9e . Statistical Process Control (SPC). Variability is inherent in every process

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Presentation Transcript

Operations Management

Supplement 6 –

Statistical Process Control

PowerPoint presentation to accompany

Heizer/Render

Principles of Operations Management, 7e

Operations Management, 9e

Statistical Process Control (SPC)
• Variability is inherent in every process
• Natural or common causes
• Special or assignable causes
• Provides a statistical signal when assignable causes are present
• Detect and eliminate assignable causes of variation
Natural Variations
• Also called common causes
• Affect virtually all production processes
• Expected amount of variation
• Output measures follow a probability distribution
• For any distribution there is a measure of central tendency and dispersion
• If the distribution of outputs falls within acceptable limits, the process is said to be “in control”
Assignable Variations
• Also called special causes of variation
• Generally this is some change in the process
• Variations that can be traced to a specific reason
• The objective is to discover when assignable causes are present
• Incorporate the good causes

Each of these represents one sample of five boxes of cereal

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Weight

Samples

To measure the process, we take samples and analyze the sample statistics following these steps

(a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight

Figure S6.1

The solid line represents the distribution

Frequency

Weight

Samples

To measure the process, we take samples and analyze the sample statistics following these steps

(b) After enough samples are taken from a stable process, they form a pattern called a distribution

Figure S6.1

Central tendency

Variation

Shape

Frequency

Weight

Weight

Weight

Samples

To measure the process, we take samples and analyze the sample statistics following these steps

(c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape

Figure S6.1

Prediction

Frequency

Time

Weight

Samples

To measure the process, we take samples and analyze the sample statistics following these steps

(d) If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable

Figure S6.1

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Prediction

Frequency

Time

Weight

Samples

To measure the process, we take samples and analyze the sample statistics following these steps

(e) If assignable causes are present, the process output is not stable over time and is not predicable

Figure S6.1

Control Charts

Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes

Frequency

Upper control limit

Lower control limit

(b) In statistical control but not capable of producing within control limits

(c) Out of control

Size

(weight, length, speed, etc.)

Process Control

Figure S6.2

Types of Data

Variables

Attributes

• Characteristics that can take any real value
• May be in whole or in fractional numbers
• Continuous random variables
• Defect-related characteristics
• Classify products as either good or bad or count defects
• Categorical or discrete random variables

The mean of the sampling distribution (x) will be the same as the population mean m

x = m

s

n

The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n

sx =

Central Limit Theorem

Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve

Three population distributions

Mean of sample means = x

Beta

Standard deviation of the sample means

s

n

Normal

= sx =

Uniform

| | | | | | |

-3sx -2sx -1sx x +1sx +2sx +3sx

95.45% fall within ± 2sx

99.73% of all x

fall within ± 3sx

Population and Sampling Distributions

Distribution of sample means

Figure S6.3

For variables that have continuous dimensions

• Weight, speed, length, strength, etc.
• x-charts are to control the central tendency of the process
• R-charts are to control the dispersion of the process
• These two charts must be used together
Control Charts for Variables

For x-Charts when we know s

Upper control limit (UCL) = x + zsx

Lower control limit (LCL) = x - zsx

where x = mean of the sample means or a target value set for the process

z = number of normal standard deviations

sx = standard deviation of the sample means

= s/ n

s = population standard deviation

n = sample size

Setting Chart Limits

Hour 1

Sample Weight of Number Oat Flakes

1 17

2 13

3 16

4 18

5 17

6 16

7 15

8 17

9 16

Mean 16.1

s = 1

Hour Mean Hour Mean

1 16.1 7 15.2

2 16.8 8 16.4

3 15.5 9 16.3

4 16.5 10 14.8

5 16.5 11 14.2

6 16.4 12 17.3

n = 9

UCLx = x + zsx= 16 + 3(1/3) = 17 ozs

LCLx = x - zsx = 16 - 3(1/3) = 15 ozs

Setting Control Limits

For 99.73% control limits, z = 3

Variation due to assignable causes

Out of control

17 = UCL

Variation due to natural causes

16 = Mean

15 = LCL

Variation due to assignable causes

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1 2 3 4 5 6 7 8 9 10 11 12

Out of control

Sample number

Setting Control Limits

Control Chart for sample of 9 boxes

For x-Charts when we don’t know s

Upper control limit (UCL) = x + A2R

Lower control limit (LCL) = x - A2R

where R = average range of the samples

A2 = control chart factor found in Table S6.1

x = mean of the sample means

Setting Chart Limits

Sample Size Mean Factor Upper Range Lower Range

n A2D4D3

2 1.880 3.268 0

3 1.023 2.574 0

4 .729 2.282 0

5 .577 2.115 0

6 .483 2.004 0

7 .419 1.924 0.076

8 .373 1.864 0.136

9 .337 1.816 0.184

10 .308 1.777 0.223

12 .266 1.716 0.284

Control Chart Factors

Table S6.1

Process average x = 12 ounces

Average range R = .25

Sample size n = 5

Setting Control Limits

Process average x = 12 ounces

Average range R = .25

Sample size n = 5

UCLx = x + A2R

= 12 + (.577)(.25)

= 12 + .144

= 12.144 ounces

From Table S6.1

Setting Control Limits

Process average x = 12 ounces

Average range R = .25

Sample size n = 5

UCLx = x + A2R

= 12 + (.577)(.25)

= 12 + .144

= 12.144 ounces

UCL = 12.144

Mean = 12

LCLx = x - A2R

= 12 - .144

= 11.857 ounces

LCL = 11.857

Setting Control Limits
R – Chart
• Type of variables control chart
• Shows sample ranges over time
• Difference between smallest and largest values in sample
• Monitors process variability
• Independent from process mean

Upper control limit (UCLR) = D4R

Lower control limit (LCLR) = D3R

where

R = average range of the samples

D3 and D4 = control chart factors from Table S6.1

Setting Chart Limits

For R-Charts

Average range R = 5.3 pounds

Sample size n = 5

From Table S6.1 D4= 2.115, D3 = 0

UCLR = D4R

= (2.115)(5.3)

= 11.2 pounds

UCL = 11.2

Mean = 5.3

LCLR = D3R

= (0)(5.3)

= 0 pounds

LCL = 0

Setting Control Limits

(a)

These sampling distributions result in the charts below

(Sampling mean is shifting upward but range is consistent)

UCL

(x-chart detects shift in central tendency)

x-chart

LCL

UCL

(R-chart does not detect change in mean)

R-chart

LCL

Mean and Range Charts

Figure S6.5

(b)

These sampling distributions result in the charts below

(Sampling mean is constant but dispersion is increasing)

UCL

(x-chart does not detect the increase in dispersion)

x-chart

LCL

UCL

(R-chart detects increase in dispersion)

R-chart

LCL

Mean and Range Charts

Figure S6.5

Steps In Creating Control Charts

Take samples from the population and compute the appropriate sample statistic

Use the sample statistic to calculate control limits and draw the control chart

Plot sample results on the control chart and determine the state of the process (in or out of control)

Investigate possible assignable causes and take any indicated actions

Continue sampling from the process and reset the control limits when necessary

Control Charts for Attributes
• For variables that are categorical
• Measurement is typically counting defectives
• Charts may measure
• Percent defective (p-chart)
• Number of defects (c-chart)

p(1 - p)

n

sp =

UCLp = p + zsp

^

^

LCLp = p - zsp

^

where p = mean fraction defective in the sample

z = number of standard deviations

sp = standard deviation of the sampling distribution

n = sample size

^

Control Limits for p-Charts

Population will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics

Sample Number Fraction Sample Number Fraction

Number of Errors Defective Number of Errors Defective

1 6 .06 11 6 .06

2 5 .05 12 1 .01

3 0 .00 13 8 .08

4 1 .01 14 7 .07

5 4 .04 15 5 .05

6 2 .02 16 4 .04

7 5 .05 17 11 .11

8 3 .03 18 3 .03

9 3 .03 19 0 .00

10 2 .02 20 4 .04

Total = 80

(.04)(1 - .04)

100

80

(100)(20)

sp = = .02

p = = .04

^

p-Chart for Data Entry

.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

UCLp= 0.10

UCLp = p + zsp= .04 + 3(.02) = .10

^

Fraction defective

p = 0.04

LCLp = p - zsp = .04 - 3(.02) = 0

^

LCLp= 0.00

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2 4 6 8 10 12 14 16 18 20

Sample number

p-Chart for Data Entry

.11 –

.10 –

.09 –

.08 –

.07 –

.06 –

.05 –

.04 –

.03 –

.02 –

.01 –

.00 –

UCLp= 0.10

UCLp = p + zsp= .04 + 3(.02) = .10

^

Fraction defective

p = 0.04

LCLp = p - zsp = .04 - 3(.02) = 0

^

LCLp= 0.00

| | | | | | | | | |

2 4 6 8 10 12 14 16 18 20

Sample number

p-Chart for Data Entry

Possible assignable causes present

UCLc = c + 3 c

LCLc = c -3 c

where c = mean number defective in the sample

Control Limits for c-Charts

Population will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics

c = 54 complaints/9 days = 6 complaints/day

UCLc = c + 3 c

= 6 + 3 6

= 13.35

UCLc= 13.35

14 –

12 –

10 –

8 –

6 –

4 –

2 –

0 –

Number defective

LCLc = c - 3 c

= 6 - 3 6

= 0

c = 6

LCLc= 0

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9

Day

c-Chart for Cab Company
Managerial Issues andControl Charts
• Select points in the processes that need SPC
• Determine the appropriate charting technique
• Set clear policies and procedures

Three major management decisions:

Using an x-chart and R-chart:

• Observations are variables
• Collect 20 - 25 samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R-chart
• Track samples of n observations each
Which Control Chart to Use

Variables Data

Which Control Chart to Use

Attribute Data

• Using the p-chart:
• Observations are attributes that can be categorized in two states
• We deal with fraction, proportion, or percent defectives
• Have several samples, each with many observations
Which Control Chart to Use

Attribute Data

• Using a c-Chart:
• Observations are attributes whose defects per unit of output can be counted
• The number counted is a small part of the possible occurrences
• Defects such as number of blemishes on a desk, number of typos in a page of text, flaws in a bolt of cloth

Upper control limit

Target

Lower control limit

Patterns in Control Charts

Normal behavior. Process is “in control.”

Figure S6.7

Upper control limit

Target

Lower control limit

Patterns in Control Charts

One plot out above (or below). Investigate for cause. Process is “out of control.”

Figure S6.7

Upper control limit

Target

Lower control limit

Patterns in Control Charts

Trends in either direction, 5 plots. Investigate for cause of progressive change.

Figure S6.7

Upper control limit

Target

Lower control limit

Patterns in Control Charts

Two plots very near lower (or upper) control. Investigate for cause.

Figure S6.7

Upper control limit

Target

Lower control limit

Patterns in Control Charts

Run of 5 above (or below) central line. Investigate for cause.

Figure S6.7

Upper control limit

Target

Lower control limit

Patterns in Control Charts

Erratic behavior. Investigate.

Figure S6.7