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Supplement 6 – Statistical Process Control

PowerPoint presentation to accompany

Heizer/Render

Principles of Operations Management, 6e

Operations Management, 8e

© 2006 Prentice Hall, Inc.

Statistical Process Control (SPC)

- Control Charts for Variables
- Setting Mean Chart Limits (x-Charts)
- Setting Range Chart Limits (R-Charts)
- Process Capability
- Process Capability Ratio (Cp)
- Process Capability Index (Cpk )

Statistical Process Control (SPC)

- Variability is inherent in every process
- Natural or common causes
- Special or assignable causes
- SPC charts provide statistical signals when assignable causes are present
- SPC approach supports the detection and elimination of assignable causes of variation

Inspection

- Inspection is the activity that is done to ensure that an operation is producing the results expected (post production activity).
- Where to inspect
- At point of product design
- At point of product production (source)
- At point of product assembly
- At point of product dispatch to customer
- At point of product reception by customer

What to Inspect

- Variables of an entity
- Degree of deviation from a target (continuum scale)
- Lifespan of device
- Reliability or accuracy of device
- Attributes of an entity
- Classifies attributes into discrete classes such as good versus bad, or pass or fail
- Maximum weight of bag at airport
- Minimum height for exit seat

Data Used for Quality Judgments

Variables

Attributes

- Characteristics that can take any real value
- May be in whole or in fractional numbers
- Continuous random variables, e.g. weight, length, duration, etc.

- Defect-related characteristics
- Classify products as either good or bad or count defects
- Categorical or discrete random variables

How to Inspect: Methods

- Visual inspection
- Manual inspection (weigh, count )
- Mechanical inspection (machine-based)
- Testing of device

Disadvantages of Inspection

- Most inspections are not done at the source
- Errors are discovered after its too late
- No link between error and the cause of errors
- Errors are often too costly to correct
- As products grow in number more staff are needed for inspection
- As products grow in number, more time is needed for inspection
- Most inspections involve the inspection of good parts as well as bad ones
- CHALLENGE: How could one achieve high product quality without having to inspect all goods produced?

Statistical Process Control

- A statistics-based approach for monitoring and inspecting results of a process, through the gathering, structuring, and analyses of product variables/attributes, as well as the taking of corrective actionat the source, during the production process.

Class Example

- What can we learn from the results of the bodyguards?

Common Cause Variations

- Also called natural causes
- Affects virtually all production processes
- Generally this requires some change at the systemic level of an organization
- Managerial action is often necessary
- The objective is to discover avoidable common causes present in processes
- Eliminate (when possible) the root causes of the common variations, e.g. different arrival times of suppliers, different arrival times of customers, weight of products poured in box by machine

Assignable Variations

- Also called special causes of variation
- Generally this is caused by some change in the local activity or process
- Variations that can be traced to a specific reason at a localized activity
- The objective is to discover special causes that are present
- Eliminate the root causes of the special variations, e.g. machine wear, material quality, fatigued workers, misadjusted equipment
- Incorporate the good process control

Each of these represents one sample of five boxes of cereal

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Frequency

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Weight

SPC and Statistical SamplesTo conduct inspection using the SPC approach, one has to compute averages for several small samples instead of using data from individual items: 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

SPC SamplingTo measure the process, we take samples of same size at different times. We plot the mean of each sample for each point in time

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

Figure S6.1

Variation

Shape

Frequency

Weight

Weight

Weight

Attributes of Distributions(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

Frequency

Time

Weight

Identifying Presence of Common Sources of Variation(d) If only common 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

Identifying Presence of Special Sources of Variation(e) If assignable causes are present, the process output is not stable over time and is not predicable

Figure S6.1

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

(a) In statistical control and capable of producing within control limits

Frequency

Upper Control Limit

Lower Control Limit

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

(c) Out of statistical control and incapable of producing within limits

Size

(weight, length, speed, etc.)

Interpreting SPC ChartsFigure S6.2

Three population distributions

Mean of sample means = x

Beta

Standard deviation of the sample means

Normal

= sx =

Uniform

s

n

| | | | | | |

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

95.45% fall within ± 2sx

99.73% of all x

fall within ± 3sx

Population and Sampling DistributionsDistribution of sample means

Figure S6.3

Sampling distribution of means

Process distribution of means

x = m

(mean)

Sampling DistributionFigure S6.4

Steps In Creating Control Charts

Take representative sample from output of a process over a long period of time, e.g. 10 units every hour for 24 hours.

Compute means and ranges for the variables and calculate the control limits

Draw control limits on the control chart

Plot a chart for the means and another for the mean of ranges on the control chart

Determine state of process (in or out of control)

Investigate possible reasons for out of control events and take corrective action

Continue sampling of process output and reset the control limits when necessary

In-Class Exercise : Control Charts

6/15

6/16

Calculate X bar and R’s for new data

Calculate X double bar and R bar figures for new data

Draw X bar chartCalculate LCL and UCL for X bar chartDraw lines for LCL and UCL and for X double bar in chart

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

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 LimitsSample 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 LimitsFor 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

| | | | | | | | | | | |

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

Out of control

Sample number

Setting Control LimitsControl 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 LimitsSample 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 FactorsTable S6.1

Process average x = 16.01 ounces

Average range R = .25

Sample size n = 5

UCLx = x + A2R

= 16.01 + (.577)(.25)

= 16.01 + .144

= 16.154 ounces

From Table S6.1

Setting Control LimitsProcess average x = 16.01 ounces

Average range R = .25

Sample size n = 5

UCLx = x + A2R

= 16.01 + (.577)(.25)

= 16.01 + .144

= 16.154 ounces

UCL = 16.154

Mean = 16.01

LCLx = x - A2R

= 16.01 - .144

= 15.866 ounces

LCL = 15.866

Setting Control LimitsR – 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 LimitsFor R-Charts

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 LimitsThese 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 spread)

R-chart

LCL

Mean and Range ChartsFigure S6.5

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 mean)

x-chart

LCL

UCL

(R-chart detects increase in dispersion)

R-chart

LCL

Mean and Range ChartsFigure S6.5

Control Charts for Attributes

- For variables that are categorical
- Good/bad, yes/no, acceptable/unacceptable
- Measurement is typically counting defectives
- Charts may measure
- Percent defective (p-chart)
- Number of defects (c-chart)

Target

Lower control limit

Patterns in Control ChartsNormal behavior. Process is “in control.”

Figure S6.7

Target

Lower control limit

Patterns in Control ChartsOne plot out above (or below). Investigate for cause. Process is “out of control.”

Figure S6.7

Target

Lower control limit

Patterns in Control ChartsTrends in either direction, 5 plots. Investigate for cause of progressive change.

Figure S6.7

Target

Lower control limit

Patterns in Control ChartsRun of 5 above (or below) central line. Investigate for cause.

Figure S6.7

Process Capability

- The natural variation of a process should be small enough to produce products that meet the standards required
- A process in statistical control does not necessarily meet the design specifications
- Process capability is a measure of the relationship between the natural variation of the process and the design specifications

Upper Specification - Lower Specification

6s

Cp =

Process Capability Ratio- A capable process must have a Cp of at least 1.0
- Does not look at how well the process is centered in the specification range
- Often a target value of Cp = 1.33 is used to allow for off-center processes
- Six Sigma quality requires a Cp = 2.0

Process mean x = 210.0 minutes

Process standard deviation s = .516 minutes

Design specification = 210 ± 3 minutes

Upper Specification - Lower Specification

6s

Cp =

Process Capability RatioInsurance claims process

Process mean x = 210.0 minutes

Process standard deviation s = .516 minutes

Design specification = 210 ± 3 minutes

Upper Specification - Lower Specification

6s

Cp =

213 - 207

6(.516)

= = 1.938

Process Capability RatioInsurance claims process

Process mean x = 210.0 minutes

Process standard deviation s = .516 minutes

Design specification = 210 ± 3 minutes

Upper Specification - Lower Specification

6s

Cp =

213 - 207

6(.516)

= = 1.938

Process Capability RatioInsurance claims process

Process is capable

3s

Lowerx - Specification Limit

3s

Cpk = minimum of ,

Process Capability Index- A capable process must have a Cpk of at least 1.0
- A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

Process Capability IndexNew Cutting Machine

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

(.251) - .250

(3).0005

Cpk = minimum of ,

Process Capability IndexNew Cutting Machine

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

(.251) - .250

(3).0005

.250 - (.249)

(3).0005

Cpk = minimum of ,

.001

.0015

Cpk = = 0.67

Process Capability IndexNew Cutting Machine

Both calculations result in

New machine is NOT capable

New process mean x = .250 inches

Process standard deviation s = .0005 inches

Upper Specification Limit = .251 inches

Lower Specification Limit = .249 inches

Process Capability ComparisonNew Cutting Machine

Upper Specification - Lower Specification

6s

Cp =

New machine is NOT capable

.251 - .249

.0030

Cp = = 0.66

Acceptance Sampling

- Form of quality testing used for incoming materials or finished goods
- Take samples at random from a lot (shipment) of items
- Inspect each of the items in the sample
- Decide whether to reject the whole lot based on the inspection results
- Only screens lots; does not drive quality improvement efforts

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