class 3 notes

209
Mean and Range Charts When standard deviation is not known, must estimate using R. Now, we check if variation between samples (i.e., the average) is stable with a Mean (xbar) chart: Calculate the average for each sample: Then, determine the average range: Finally, determine control limits (LCL, UCL): th th datapointsin sam ple # ofdata pointsin sam ple i i x i 1 2 # ofsam ples x x x R A x R A x 2 2 LCL UCL R A z x 2 (“x-bar- bar”)

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Page 1: Class 3 Notes

Mean and Range Charts• When standard deviation is not known, must estimate using R.

• Now, we check if variation between samples (i.e., the average) is stable with a Mean (xbar) chart:• Calculate the average for each sample:

• Then, determine the average range:

• Finally, determine control limits (LCL, UCL):

th

th

data points in sample

# of data points in samplei

ix

i

1 2

# of samples

x xx

RAx

RAx

2

2

LCL

UCL

RAz x 2

(“x-bar-bar”)

Page 2: Class 3 Notes

Example: Mean chart using Step 1: Calculate sample means

10.124

08.1211.1210.1211.121

x

12.124

11.1210.1211.1215.122

x

11.124

15.1211.1209.1209.123

x

10.124

10.1208.1210.1212.124

x

12.124

12.1213.1214.1209.125

x

Page 3: Class 3 Notes

Example: Mean chart using RStep 2: Calculate the overall mean:

Step 2: Choose the appropriate factor:

Step 3: Calculate the control limits:

12.10 12.12 12.11 12.10 12.1212.11

5x

144.12046.073.011.12UCL 2 RAx

076.12046.073.011.12LCL 2 RAx

24 0.73n A

Page 4: Class 3 Notes

Example: Mean chart using R

12.144

12.11

12.076

x

LCLx

UCLx

Variation between samples is in control.

Page 5: Class 3 Notes

Mean and Range Charts

• Establishing Mean and Range charts (initial method):

1) Obtain a large (typically 20-25) sample

2) Determine control limits for Mean and Range charts

3) Check if data is inside limits on both

• Inside limits on Range chart, then variation within samples is stable

• Inside limits on Mean chart, then variation between samples is stable

4) If not, investigate process, correct and return to (1), otherwise assume process is stable and use control charts.

Page 6: Class 3 Notes

Mean and Range Charts

• Using Mean and Range charts:Periodically,

Collect Sample Data

Plot Sample Data on R-

Chart

Is Data Inside Limits?

Investigate Source of Variation

No

Plot Sample Data on Mean

Chart

Is Data Inside Limits?

Investigate Source of Variation

No

Within Samples

Between Samples

Yes

Yes

Page 7: Class 3 Notes

Example: Mean and Range ChartThe quality inspector has taken samples over the last 5 weeks.

Using the control charts found previously, determine if the process needs to be investigated.

Week

Obs. 6 7 8 9 10

1 12.10 12.11 12.13 12.15 12.18

2 12.14 12.08 12.07 12.11 12.16

3 12.13 12.09 12.12 12.09 12.15

4 12.07 12.12 12.10 12.04 12.12

Page 8: Class 3 Notes

Example: Mean and Range ChartStep 1: Calculate sample ranges

Min value in week 6: 12.07

Max value in week 6: 12.14

6 12.14 12.07 0.07R

Min value in week 7: 12.08

Max value in week 7: 12.12

7 12.12 12.08 0.04R

Min value in week 8: 12.07

Max value in week 8: 12.13

8 12.13 12.07 0.06R

Min value in week 9: 12.04

Max value in week 9: 12.15

9 12.15 12.04 0.11R

Min value in week 10: 12.12

Max value in week 10: 12.18

10 12.18 12.12 0.06R

Page 9: Class 3 Notes

Example: Mean and Range Chart

0.105

0 LCLR

UCLR

Step 2: Plot on Range chart

0.046 R

6 7 8 9 10

Variation within sample is out of control

STOP! Investigate source of variation.

Page 10: Class 3 Notes

Example: Mean and Range ChartA review of the data in week 9 revealed that the final entry was, in

fact, a typo. It should have be 12.14, not 12.04. Is the process in control.

Week

Obs. 6 7 8 9 10

1 12.10 12.11 12.13 12.15 12.18

2 12.14 12.08 12.07 12.11 12.16

3 12.13 12.09 12.12 12.10 12.15

4 12.07 12.12 12.10 12.14 12.12

Page 11: Class 3 Notes

Example: Mean and Range ChartStep 1: Calculate sample ranges

Min value in week 6: 12.07

Max value in week 6: 12.14

6 12.14 12.07 0.07R

Min value in week 7: 12.08

Max value in week 7: 12.12

7 12.12 12.08 0.04R

Min value in week 8: 12.07

Max value in week 8: 12.13

8 12.13 12.07 0.06R

Min value in week 9: 12.10

Max value in week 9: 12.15

9 12.15 12.10 0.05R

Min value in week 10: 12.12

Max value in week 10: 12.18

10 12.18 12.12 0.06R

Page 12: Class 3 Notes

Example: Mean and Range Chart

0.105

0 LCLR

UCLR

Step 2: Plot on Range chart

0.046 R

6 7 8 9 10

Variation within samples is stable.

Page 13: Class 3 Notes

Example: Mean and Range ChartStep 3: Calculate sample means

6

12.10 12.14 12.13 12.0712.110

4x

7

12.11 12.08 12.09 12.1212.100

4x

8

12.13 12.07 12.12 12.1012.105

4x

9

12.15 12.11 12.10 12.1412.125

4x

10

12.18 12.16 12.15 12.1212.153

4x

Page 14: Class 3 Notes

Example: Mean chart using R

12.144

12.11

12.076

x

LCLx

UCLx

6 7 8 9 10

Variation between sample is out of control

STOP! Investigate source of variation.

Page 15: Class 3 Notes

In-Class Exercises: Mean/Range ChartThe weight (in grams) of toner bottles is measured in the following 5 samples (assume that it is a continuous random variable). Each sample composed of 4 observations, one from each of four machines. Using 3 control limits determine if the process is in control.

Observation Samples

1 6330 6328 6325 6328 6337

2 6326 6327 6333 6331 6333

3 6334 6333 6330 6327 6338

4 6331 6332 6329 6333 6340

Page 16: Class 3 Notes

Mean and Range Charts

• Table S6.1 (pg. 227):

# observations per sample

Mean Factor (for x chart)

Upper Range (for R chart)

Lower Range (for R chart)

n A2 D4 D3

2 1.88 3.27 0

3 1.02 2.57 0

4 0.73 2.28 0

5 0.58 2.11 0

6 0.48 2.00 0

7 0.42 1.92 0.08

8 0.37 1.86 0.14

9 0.34 1.82 0.18

Page 17: Class 3 Notes

In-Class Exercises: Mean/Range Charts

Page 18: Class 3 Notes

In-Class Exercises: Mean/Range Charts

LCLR=

UCLR=

R=

Page 19: Class 3 Notes

In-Class Exercises: Mean/Range Chart

Page 20: Class 3 Notes

In-Class Exercises: Mean/Range Charts

LCLxbar=

UCLxbar=

x =

Page 21: Class 3 Notes

In-Class Exercises: Mean/Range Charts

LCLxbar=

UCLxbar=

x =

Page 22: Class 3 Notes

Updating Control Charts

Page 23: Class 3 Notes

Updating Control Charts

• Typically, an initial sample of 15-20 data points is collected and used to establish charts.

• If process remains “in control” for next 15-20, charts can be updated.

• Centerline of “old” (original) is tested against centerline of “new”

• If significantly (statistically) different, use “new” data to make new chart(s).

• If not significantly different, use both “old” and “new” data to make chart(s).

Page 24: Class 3 Notes

Nelson’s Rules

Page 25: Class 3 Notes

Control Charts

• Nelson Rules* – set of 8 control chart rules for determining when a process is out of control• Developed from Western Electric (later AT&T) rules

• Look for statistically significant patterns in charts

• Several use an A-B-C classification:UCL

LCL

Center Line

A

B

C

C

B

A

+ + 2+ 3

– 3– 2–

* Nelson, Lloyd S. "The Shewhart Control Chart—Tests for Special Causes," Journal of Quality Technology, vol. 16(4), pp. 237 – 239, October 1984

Note: Assumes Normal Distribution (e.g., works for Xbar, but not for R, c or p.

Page 26: Class 3 Notes

Nelson Rules

• Rule 1: – single point falls on/outside the 3 sigma limit

UCL

LCL

Center Line

A

B

C

C

B

A

Conclusion: one observation significantly out of control

Page 27: Class 3 Notes

Nelson Rules

• Rule 2: two out of three consecutive points, all on the same side of the center line, fall in zone A (or beyond)

UCL

LCL

Center Line

A

B

C

C

B

A

Conclusion: medium tendency to be mediumly out of control

Page 28: Class 3 Notes

Nelson Rules

• Rule 3: four out of five consecutive points, all on the same side of the center line, fall in zone B (or beyond)

UCL

LCL

Center Line

A

B

C

C

B

A

Conclusion: strong tendency to be slightly out of control

Page 29: Class 3 Notes

Nelson Rules

• Rule 4: eight consecutive points, all on the same side of the center line

UCL

LCL

Center Line

A

B

C

C

B

A

Conclusion: bias exists

Page 30: Class 3 Notes

Nelson Rules

• Rule 5: long series of points (about 14) are high, low, high, low, … without any interruption in the sequence

UCL

LCL

Center Line

A

B

C

C

B

A

Conclusion: oscillation; not likely random noise

Page 31: Class 3 Notes

Nelson Rules

• Rule 6: six (or more) consecutive points all increasing or all decreasing

UCL

LCL

Center Line

A

B

C

C

B

A

Conclusion: trend exists

First point to have increased

Page 32: Class 3 Notes

Nelson Rules

• Rule 7: fifteen (or more) consecutive points fall in zone C either above or below the centerline

UCL

LCL

Center Line

A

B

C

C

B

A

Conclusion: stratification, more variability expected

Page 33: Class 3 Notes

Nelson Rules

• Rule 8: eight (or more) consecutive points fall on both sides of the centerline, with none falling in zone C

UCL

LCL

Center Line

A

B

C

C

B

A

Conclusion: mixture (i.e., two different processes; one high, one low)

Page 34: Class 3 Notes

Control Charts

• As noted, these rules apply to (approx.) normal data

• Exist similar rules for charts based on non-normal data

• Tend to be less reactive

• E.g., for c-charts and p-charts:

1) 1 point outside limits

2) 9 points on one side of centerline

3) 6 points in a row increasing or decreasing

4) 14 points alternating high-low

Page 35: Class 3 Notes

Run Tests

Page 36: Class 3 Notes

Run Tests

• Run – uninterrupted sequence of observations with a certain observable characteristic• Typically, there are two characteristics studied:

• Above or below the median (A/B)

• Increase or decrease from previous observation (Up/Down)

• Run test – test of existence and patterns in runs to detect abnormalities and non-randomness in a process • Compares actual count normal range of counts.

• Analysts often supplement control tests with a run test.

• Dr. Deming highly recommended their use

Page 37: Class 3 Notes

Run Tests

Counting runs:Counting Above/Below Median Runs (7 runs)

U U D U D U D U U D

B A A B A B B B A A B

Counting Up/Down Runs (8 runs)

Page 38: Class 3 Notes

Run Tests

Too many runs:

Too few runs:

Above/Below Median (10 runs)

B A B A B B A B A B A

Above/Below Median (2 runs)

B B B B B B A A A A A

Page 39: Class 3 Notes

Six Sigma

Page 40: Class 3 Notes

Six Sigma

• Why 99% isn’t good enough

20,000 lost articles of mail per hour Unsafe drinking water almost 15

minutes out of each day 5,000 incorrect surgical operations

per week. 2 short or long landings at most major

airports each day No electricity for almost 7 hours each

month

2 308,7333 66,8034 6,2005 2336 3.4

DefectsDefects % Good% Good

69.1%93.3%

99.38%99.976%

99.9996%

“99% Good”

Source: The Nature of Six Sigma Quality. M. Harry, Ph.d

Per million

(Note: #’s allow for a shift in the mean.)

Page 41: Class 3 Notes

Six Sigma

• Precise definition: 3.4 defects per million opportunities• Nearest customer specification limit is 6 standard deviations

from the mean output of the process

• General definition: a measurement/data-driven method for removing defects in any process

• Part of most TQM efforts, but has taken on a life of its own (e.g., Lean – Six Sigma)

• Two main processes: DMAIC and DMADV

• Employees classified (trained/tested) as Six Sigma Green Belts, Black Belts and Master Black Belts

Page 42: Class 3 Notes

Process Capability

Page 43: Class 3 Notes

Process Capability

• Control vs. Capability:

Process Control

Measure of sample statistics variation relative

to process variation

Internal

Process Capability

Measure of process variation

relative to customer

specifications

External

Page 44: Class 3 Notes

Process Capability

• Process variability – natural variability in a process, measured by • Known or estimated by

• Specifications (or tolerances) – range of acceptable values established by customer requirements (or engineering design)

• Process capability – measure of process variability relative to specification• Measures how well the process satisfies the specifications• Process must be in control before capability can be measured

2 3xn A n R

Page 45: Class 3 Notes

Process Capability

• Let’s say we know what “normal” variation is:

UCL

3

LCL

Can we simply build units until a point falls outside the limits?

We establish control charts.

Page 46: Class 3 Notes

Process Capability

• What if the specs look like this:

UCL

3

LCL

USL LSL

Process variation exceeds specifications

USL = Upper Specification Limit UCL =Upper Control Limit

LSL = Lower Specification Limit LCL = Lower Control Limit

We would produce parts outside of specs and not know there is a problem.

Page 47: Class 3 Notes

Process Capability

• What if the specs look like this:

UCL

3

LCLUSL/ LSL/

Process variation equals specifications

USL = Upper Specification Limit UCL =Upper Control Limit

LSL = Lower Specification Limit LCL = Lower Control Limit

Better, but do not know there is a problem until it is too late (also, still 3/1000 out of spec, if normal.)

Page 48: Class 3 Notes

Process Capability

• What if the specs look like this:

UCL

3

LCL

USL LSL

Process variation less than specifications

USL = Upper Specification Limit UCL =Upper Control Limit

LSL = Lower Specification Limit LCL = Lower Control Limit

Now, we have a warning before parts are out of spec (also, very few out of spec, if normal.)

Page 49: Class 3 Notes

Process Capability

• Numerically:

• Process capability ratio:

• Traditionally, process was capable if Cp ≥ 1.33 (spec limits are +4 )

• Six Sigma requires Cp ≥ 2.00 (spec limits are +6 , hence the name):

6

LSLUSL pC

UCL

3

LCL

USL LSL

6USL-LSL

Page 50: Class 3 Notes

Process variation less than specifications

Process variation matches specifications

Process Capability

• Visually:

UCL

3

LCLUSL LSL

Process variation exceeds specifications

3

USL/UCL LSL/LCL UCL

3

LCLUSL LSL

Cp ≤ 1 Cp = 1 Cp ≥ 1

Page 51: Class 3 Notes

• Traditional vs. Six-Sigma:

Traditional Requirements

Process Capability

USL

Cp ≥ 1.33

UCL

3

LCLLSL

4Six-Sigma

Requirements

Cp ≥ 2.00

USL UCL

3

LCLLSL

6

Notice that the spec limits do not change (they are what they are), but what is a “capable” process does (require less normal variation.)

Page 52: Class 3 Notes

Example: Process CapabilityThree machines can perform a specific job. The tolerance range

for the job is 0.8 mm (with of specs; i.e., +/- 0.4 mm.) If the machines have the correct mean and the following standard deviations, which are capable of performing the job?

Both machine A and B have Cp ≥ 1.33, so they are capable by traditional requirements.

Machine σ Capability 6σ Cp

A 0.10

B 0.06

C 0.16

0.10(6) = 0.60

0.06(6) = 0.36

0.16(6) = 0.96

0.80/0.60=1.

33

0.80/0.36=2.

22

0.80/0.96=0.

83However, only machine B is capable under Six Sigma.

Page 53: Class 3 Notes

In-Class Exercise: Process CapabilityAs part of an insurance company’s training program, participants

learn how to conduct an analysis of client’s insurability. The goal is to have participants achieve a time in the range of 30 to 45 minutes. For now, assume that all participants averaged 37.5 minutes. Test results for three participants were: Armand, a had a standard deviation of 3 minutes; Jerry, a standard deviation of 2.5 minutes; and Melissa, a standard deviation of 1.8 minutes.

a) Which of the participants would you judge to be capable, using the traditional rules? Six Sigma rules?

Page 54: Class 3 Notes

In-Class Exercise: Process Capability

Page 55: Class 3 Notes

Process Capability

• Cp can be misleading:

UCL

3

LCLUSL LSL

Process is “capable” (Cp ≥ 1.33), but mean (sometimes called X) is not centered.

Mean

Page 56: Class 3 Notes

Process Capability

• For uncentered processes:

• Process capability index:

• Looks at process variation relative to the closest spec limit:

• Used in addition to Cp

• Traditionally, process is capable if Cpk ≥ 1.0 (closest SL +3 )

• Six Sigma requires Cpk ≥ 1.5 (closest SL +4.5), though newer versions require Cpk ≥ 2.0.

USL Mean Mean LSLmin ,

3 3pkC

UCL

3

LCL

USL LSL

3Mean-LSL

UCL

3

LCL

USL LSL

3USL-Mean

Page 57: Class 3 Notes

Process Capability

Page 58: Class 3 Notes

Example: Process CapabilityThe output for the three machines in the previous example were

found to not be centered a 1.45. Analysis of the process reveals the means and standard deviations below. Spec limits are 1.0 and 1.8. Are the processes capable to six sigma levels?

Machine B meets traditional and (original) Six Sigma requirements.

Machine σ X Cpk

A 0.10 1.29

B 0.06 1.35

C 0.16 1.33

Min{(1.8-1.29)/(3*0.10),(1.29-1.0)/

(3*0.10)=0.97Min{(1.8-1.35)/(3*0.06),

(1.35-1.0)/(3*0.06)=1.94Min{(1.8-1.33)/(3*0.16),

(1.33-1.0)/(3*0.16)=0.69

Page 59: Class 3 Notes

In-Class Exercise: Process CapabilityAs part of an insurance company’s training program, participants

learn how to conduct an analysis of client’s insurability. The goal is to have participants achieve a time in the range of 30 to 45 minutes. Now, the means are also different. Test results for three participants were: Armand, a mean of 38 minutes and a standard deviation of 3 minutes; Jerry, a mean of 37 minutes and a standard deviation of 2.5 minutes; and Melissa, a mean of 37.5 minutes and a standard deviation of 1.8 minutes.

a) Which of the participants would you judge to be capable, using the traditional rules? Original Six-Sigma rules?

Page 60: Class 3 Notes

In-Class Exercise: Process Capability

Page 61: Class 3 Notes

Putting It Together

Page 62: Class 3 Notes

Putting It Together

• Stability and Capability:Collect Initial Sample Data

(20+)

Create/Plot Control Charts

Investigate Source of Variation

No

Calculate Cp and Cpk

Is Process Capable?

Take Steps to Reduce

Variability

No

VerifyStability

VerifyCapability

Yes

Yes

Monitor

Is Process Stable?

Page 63: Class 3 Notes

Putting It Together

• Spec limits are what they are, so we focus on changing the process mean or reducing the process variability:

UCL

3

LCLUSL LSL

UCL

3

LCLUSL LSL

UCL

3New

LCLUSL LSL

Page 64: Class 3 Notes

Putting It Together

• Improving process capability: requires changing the process target value (move X) and/or reducing the process variability (lower )• Simplify: eliminate steps, reduce the number of parts, use

modular design

• Standardize: use standard parts, procedures

• Mistake-proof: design parts that can only be assembled the correct way

• Upgrade equipment: replace worn out equipment, take advantage of technological improvements

• Automate: substitute automated for manual processing

Page 65: Class 3 Notes

Supply Chain Management

Page 66: Class 3 Notes

SCM Job Outlook• U.S. News & World Report describes supply chain

management as one of the 20 hottest job tracks for this century.

• Logistics--the second largest employment sector in the United States (Council of Logistics Management).

• Bureau of Labor Statistics projects, for 2001-2012:• “Employment growth will be concentrated in the service-

providing sector of the economy … transportation and warehousing are other service -providing industries that are projected to grow faster than average”.

Page 67: Class 3 Notes

SCM Job Outlook

Source: www.odinjobs.com

Page 68: Class 3 Notes

SCM Job Outlook

Source: www.careersinsupplychain.org

Page 69: Class 3 Notes

Supply Chain

Page 70: Class 3 Notes

Supply Chain• Supply chain – sequence of organizations (facilities,

functions, and activities) that are involved in producing and delivering a product or service

• Begins with raw materials

• Ends with delivery to the end consumer

• A.k.a. Value Chains

Page 71: Class 3 Notes

Manufacturing Supply Chain

Page 72: Class 3 Notes

Example: LNG Supply Chain

Page 73: Class 3 Notes

Service vs. Manufacturing

Supplier

Supplier

Supplier

Supplier

Supplier

StorageMfg

StorageDist.

StorageRet.

StorageServ.

Cust

Cust

Page 74: Class 3 Notes

Hybrid Supply Chain

Supplier

SupplierStorage

Serv.

Supplier

Supplier

Supplier

StorageMfg

StorageDist.

StorageRet. C

ustomer

Page 75: Class 3 Notes

Supply Chain

Purchasing Production Distribution

Value-AddingSupply Demand

• In addition to traditional focus on internal production, firms now focus on two additional external areas:

Supply and Demand• Supply starts with most basic raw materials, ends with

the beginning of internal operations

• Demand starts with product delivery to organization’s immediate customer, ends with final customer in chain

Page 76: Class 3 Notes

Supply Chain

Supplier

Supplier

Supplier

StorageMfg

StorageDist.

StorageRet.

Cust

• Closer organization is to final customer, shorter its demand component and longer its supply component

Demand

Supply

Demand

Supply

Page 77: Class 3 Notes

• Three types of flows in SCs:

Supply Chain

Supplier

Supplier

Supplier

Mfg Dist. Ret.

End C

ustomer

Physical materials – value added along the way

Money – member take their “piece of the pie”

Information – (dis)aggregated, updated

◄◄◄ UPSTREAM DOWNSTREAM ►►►

Page 78: Class 3 Notes

Supply Chain Management

Page 79: Class 3 Notes

Supply Chain Management• Supply Chain Management – efficient integration of

suppliers, factories, warehouses and stores so that merchandise is produced and distributed:

In the right quantities

To the right locations

At the right time

Goal: maximize total profits• Share a bigger “pie”

Page 80: Class 3 Notes

Supply Chain Management• SCM involves all three flows:

Physical Material•Order timing

•Quantities

•Holding of surplus

•Event (interruption) management

•Returns/repairs

•Replacement parts

Money•Extending credit

•Contract terms

•Buybacks

•Payment timing

•Repair/service costs

Information•POS data

•Inventory levels

•Forecasts

•Shipment status

•Capabilities

•Limitations

Page 81: Class 3 Notes

Supply Chain Management• Organizations within supply chains have different,

conflicting objectives:

• Manufacturers: long run production, high quality, high productivity, low production cost

• Distributors: low inventory, reduced transportation costs, quick replenishment capability

• Customers: shorter order lead time, high in-stock inventory, large variety of products, low prices

Page 82: Class 3 Notes

Supply Chain Management

Source: Supply Chain Council (www.supply-chain.org)

• Supply Chain Operational Reference (SCOR) Metrics

Page 83: Class 3 Notes

Bullwhip Effect

Page 84: Class 3 Notes

Supply Chain Management• Bullwhip effect – distortion of demand information of

a product while it passes back through the SC• Common pattern: variability of orders to an upstream site

are always greater than those of the downstream site

• Creates problems:• Excess inventory

• Capacity shortages

• Forecasting inaccuracies

• Lost revenues

• Obsolescence

Page 85: Class 3 Notes

Bullwhip Effect

Retailer Distributor

Manufacturer

Supplier

Stock

Increased Variability

Orders

Page 86: Class 3 Notes

Bullwhip Effect• Some (but not all) causes:

• Demand forecast updating• Orders are used as signals of product demand

• Forecasting techniques rely heavily on recent demand observations

• Order batching• Fixed ordering costs and manufacturing setups lead to batching

• Periodic planning/ordering as part inventory management systems

• Economies of scale in transportation (full truckload rates)

• Rationing and shortage gaming• Rationing of orders when demand exceeds manufacturing capacity

• Customers exaggerate their real needs (“gaming”)

• Sales promotions/discounts• Artificially inflates orders without demand

Page 87: Class 3 Notes

Bullwhip Effect• Sales promotions:

Page 88: Class 3 Notes

Bullwhip Effect• Ways to counter:

• Generally, sales information must be shared up SC

• Suppliers give point-of-sale (POS) data

• Electronic Data Interchange (EDI)• Continuous Replenishment Program (CRP)

• Advanced Continuous Replenishment (ACR)

• Vendor Maintained Inventory (VMI)

• Every Day Low Prices (EDLP)• Instead of promotions/sales

• Strategic buffering• E.g., bulk of retail inventory held at distribution center

• Collaborative Forecasting and Replenishment (CFAR)

Page 89: Class 3 Notes

Purchasing

Page 90: Class 3 Notes

Supply Chain Management• Purchasing – obtaining materials, parts, and supplies

needed to produce a product or provide a service

• For manufacturing companies approximately 60% of total product cost comes from purchased parts and materials

• Interfaces with other functional areas and outside suppliers

• Responsible for:

• Vendor selection and monitoring

• Quality of incoming parts, materials or subassemblies

• Communication and timing of deliveries of goods or services

Page 91: Class 3 Notes

Purchasing

Legal

Acc’tingOp.’s

DataProcess.

Design/ Eng.

Receiving

Suppliers

Purchasing

Q/A

• Interfaces:

Page 92: Class 3 Notes

Purchasing• Vendor Selection

1) Vendor evaluation – developing criteria (w/ weights), finding potential suppliers and measuring their potential• Criteria: price, quality, reputation, services, location, transportation

costs/lead times, inventory policy, flexibility, financial stability

2) Vendor development – readying supplier to potentially become an integral part of operations• Provide information, training/technical knowledge, establishing

communications framework

3) Negotiations – establishing terms (cost, quantity, etc.)• Cost-based pricing, market-based pricing, competitive bids

Page 93: Class 3 Notes

Purchasing• Global sourcing

• Why is there increased usage?• Global trade agreements:

• General Agreement on Tariffs and Trade (GATT)• North American Free Trade Agreement (NAFTA)

• Improved technology• Increased speed and reduced cost of distribution• Improved worker skill levels/infrastructure in 2nd/3rd world countries• Growing/emerging markets over seas

• What are some added concerns?• Cultural/legal/language differences• Currency exchange• Reduced oversight/control (QC, safety)• Increased lead time/transportation costs

Page 94: Class 3 Notes

Purchasing• Outsourcing – buying goods or services from outside

sources rather than providing them in-house• What must we consider in making decision?

• Make-vs.-buy analysis (cost to produce, investment)

• Stability of demand and possible seasonality

• Quality available versus quality in-house

• Desire to maintain close control over operations

• Idle capacity, lead times, expertise, stability of technology, compatibility with other in-house operations (skill levels, equipment)

• Control of proprietary information/designs

Page 95: Class 3 Notes

Strategic Partnering

Page 96: Class 3 Notes

Strategic Partnering• Strategic partnering (SP) - two or more firms with

complementary products/services (horizontal) or along supply chain (vertical) join for strategic benefit

• Advantages of SP include:• Fully utilize system knowledge

• Decrease required inventory levels

• Improve service levels

• Decrease work duplication

• Improve forecasts

Page 97: Class 3 Notes

Strategic Partnering• Types of strategic partnering in SCM include:

• Simple POS

• Reverse purchase orders (RPO)

• Quick response

• Vendor managed inventory (VMI)

• Continuous replenishment program (CRP)

• Advanced continuous replenishment (ACR)

Page 98: Class 3 Notes

Strategic Partnering• Simple POS: retailer determines order sizes/timing but

also passes point of sales (POS) data to the supplier• Improves supplier’s forecasts

• Reverse purchase order (RPO): supplier order size/ timing recommendations used by retailer to determine actual order sizes/timing• Example: Panasonic (supplier) and BestBuy (retailer)

• Upside: supplier seems to be providing high service

• Downside: supplier does not see the real demand

Page 99: Class 3 Notes

Supply Chain Management• Continuous Replenishment Program (CRP): vendors

receive POS data, use to prepare shipments at agreed upon intervals to maintain agreed levels of inventory

• Example: Walmart

• Advanced Continuous Replenishment (ACR): suppliers may gradually decrease inventory levels at retailer or distribution center as long as service levels are met. • Inventory levels continuously improved in a structured way

• Example: K-Mart

Page 100: Class 3 Notes

Supply Chain Management• Quick response – vendors receive POS data from

retailers; use to synchronize production and inventory activities at supplier (JIT)• Retailer still prepares individual orders

• POS data used by supplier to improve forecasting/scheduling

• Example: Milliken & Company

• Vendor managed inventory (VMI) – inventory levels monitored and controlled by vendor not retailer• Vendor’s inventory decisions based on actual POS data

• Empirical results: sales increases of 20 to 25 percent, and 30 percent inventory turnover improvements (e.g., Walmart)

Page 101: Class 3 Notes

E-business

Page 102: Class 3 Notes

Supply Chain Management• E-business – use of electronic technology to facilitate

business transactions

• Advantages:

• Access to customers/suppliers outside of local area

• Have a global presence

• Collect detailed information on customer interests

• Explore new opportunities such as new vendors and new customers (an open system)

• Have common standards

• Lower set up costs, especially in many-to-many environment

• Level the playing field for small companies

Page 103: Class 3 Notes

Supply Chain Management• Supply Chain of B2C (e-Retailing)

• With or without inventory?

• Third-party fulfillers (drop shipping)• e.g. Amazon

• Internet-based B2B Applications• Web sites for ordering and order tracking (MRO catalogs)

• One-to-one connections: collaborative requirements planning, vendor managed inventory (VMI), electronic data interchange (EDI)

• Many-to-many connections: industry specific hubs, e-procurement

Page 104: Class 3 Notes

E-Business• Electronic Data Interchange (EDI) - direct computer-

to-computer transmission of inter-organizational transactions.

• Includes: purch. orders, shipping notices, debit/credit memos

• Reduces paperwork, lead time, inventory and variability

• Increases accuracy, timeliness

• Facilitates Just-In-Time (JIT) systems

Page 105: Class 3 Notes

Logistics

Page 106: Class 3 Notes

Supply Chain ManagementLogistics – management of the movement and storage of

materials within the supply chain.• Within a facility

• Between facilities (incoming and/or out-going shipments)

Techniques in logistics:• Distribution requirement planning (DRP)

• Third party logistics (3-PL)

• Reverse logistics

Page 107: Class 3 Notes

LogisticsCosts in logistics:

Inventory: cycle stock, pipeline stock, buffer stock.

Transportation: trucking, railroads, airfreight, shipping

Warehousing: packaging, sorting, storage, conveyor belts, robotics, forklifts, etc.

Facilities: distribution centers, warehouses, centralized sorting centers.

Information: order fulfillment, coordination, scheduling

Page 108: Class 3 Notes

LogisticsUnlike inventory, transportation has remained stable …

Page 109: Class 3 Notes

Logistics• Within a supply network:

Demand Location B

Demand Location C

Demand Location A

Demand Location D

Supply Location 1

Supply Location 2

Supply Location 3

Page 110: Class 3 Notes

Logistics• Within a facility:

Rec

eivi

ng

Production Process

Shipping

Storage

Workcenter

Work center

Storage

Work center Workcenter

Storage

Page 111: Class 3 Notes

Logistics• Within a facility:

Page 112: Class 3 Notes

Logistics• Distribution Requirements Planning (DRP) - an

extension of the concepts of MRP* to finished goods inventory management and distribution planning

1) Starts with demand at the end of the channel

2) Works backward through the warehouse system

3) Obtains a time phased replenishment schedule for moving inventories through the warehouse network

• Extremely useful when multiple warehouses are present (e.g. Wal-Mart)

• Many MRP vendors also have a DRP module

* Material Requirements Planning – we will discuss later.

Page 113: Class 3 Notes

Logistics• Third Party Logistics (3PL) - use of an outside

company to perform part or all of a firm’s materials management and product distribution function• Have knowledge/expertise that is expensive or impractical to

gain internally

• Give up control; requires sufficient oversight

• Examples 3PL providers: Ryder Dedicated Logistics and J.B. Hunt

• Examples 3PL users: 3M, Dow Chemical, Kodak and Sears

• New trend: online freight matching

Page 114: Class 3 Notes

3PL – Freight Matching

Page 115: Class 3 Notes

Logistics• Reverse Logistics – backward flow of goods returned

to the supply chain from their final destination

• Reasons for return: defective, unsold, change of mind

• Involves sorting, testing, repairing/reconditioning, restocking, recycling or disposing

• Goal: create/recapture value in returned product

• Methods for managing returns:

• Gatekeeping – screening returns to prevent incorrect acceptance

• Avoidance – managing internal processes (design, quality control, forecasting) to minimize number of items returned

Page 116: Class 3 Notes

Inventory Management

Page 117: Class 3 Notes

Inventory Management• Inventory – stock or store of goods

• Examples:

• Manufacturing: firms carry supplies of raw materials, purchased parts, finished items, spare parts, tools,....

• Department stores: carry clothing, furniture, stationery, tools, appliances,...

• Supermarkets: stock fresh and canned foods, packaged and frozen foods, household supplies, ...

• Hospitals: stock drugs, surgical supplies, life-monitoring equipment, sheets, pillow cases, ... patients!

Page 118: Class 3 Notes

Inventory• Types:

• Raw materials & purchased parts

• Pipeline stock – goods-in-transit to warehouses or customers

• Cycle stock – inventory held due to ordering in batches

• Safety stock – inventory held to buffer against variability

• Work-in-progress (WIP) – partially completed goods

• Finished-goods – completed products (manufacturing firms)

• Merchandise (retail stores)

• Replacement parts, tools, & supplies

Page 119: Class 3 Notes

Inventory• In processes:

TransformationInput OutputRaw materials

• Materials received• Customers waiting

in a bank• Paperwork in in-box

Work-in-Process (WIP)• Semi-finished

products• Customers at the

counter• Paperwork on desk

Finished goods• Products waiting to

be shipped• Customers leaving

the bank• Paperwork in out-box

Page 120: Class 3 Notes

Inventory Management• Functions of Inventory:

• To meet anticipated demand: anticipation stock (avg.)

• To smooth production requirements: seasonal inventories

• To deal with uncertainty (variability):

• To decouple operations: buffer inventories

• To protect against stock-outs: safety stock

• To take advantage of order cycles: cycle stock (batches)

• To permit operations: WIP, pipeline stock

• To take advantage of quantity discounts

• To help hedge against price increases

Page 121: Class 3 Notes

Inventory Management• Requirements to for effective IM:

1) Reliable forecasting system (demand)

2) Accurate supply information:

• Lead times, including variability

• Costs (holding, ordering, shortage)

• Garbage In → Garbage Out

3) System for counting/tracking inventory on hand and on order

4) Classification/priority system of inventory

Page 122: Class 3 Notes

Counting systems• Periodic system: physical count of items made at

periodic intervals (weekly, monthly)• Order placed up to predetermined amount

• Good: economies of scale in processing many items

• Bad: lack control between reviews → carry extra stock

• Continuous (perpetual) system: tracks changes to inventory continuously, monitoring current levels• Fixed quantity is ordered when a certain level is reached

• Good: keep constant count of inventory, fixed order quantity

• Bad: higher tracking costs, periodic counting still required, suppliers may work on periodic system

Page 123: Class 3 Notes

Counting systems• Of periodic and continuous systems, which do think

are used more and is that changing? Why?

Page 124: Class 3 Notes

Example: Classification System• ABC Classification System: classifying inventory

according to some measure of importance and allocating control efforts accordingly

A - very important

B - mod. important

C - least important Annual $ value

of items

AA

BB

CC

High

LowFew

ManyNumber of Items

Page 125: Class 3 Notes

ABC Approach• Class A typically contains about 10-20% of the items

and 60-70% of the annual dollar value

• Receives close attention: frequent reviews make sure customer service levels

• Class C contains about 50-60% of the items, but only 10-15% of the dollar value

• Receives only loose control, usually order in large quantities

• Class B is between the two extremes

Page 126: Class 3 Notes

How much to order:EOQ (Basic)

Page 127: Class 3 Notes

Economic Order Quantity• Economic Order Quantity (EOQ): optimal batch size

for ordering, minimizing holding and ordering costs

• Uses:

• In periodic system: determine (approximate) review interval• Interval = avg. demand rate / EOQ

• Used if an order is placed every review

• In continuous system: determine (approximate) amount to order when inventory falls to predetermined level (ROP)

Page 128: Class 3 Notes

Economic Order Quantity• Assumptions:

1. Ordering in batch from supplier

2. Only one product is involved

3. Relatively constant demand rate • Noise, but no trend/cycles

4. Constant lead time

5. Single delivery for each order

6. A single flat unit price from the supplier • No quantity discounts

7. Unit cost is independent of quantity

Most of these assumptions can be relaxed with minor adjustments. We will see one such case later.

Page 129: Class 3 Notes

Economic Order Quantity• Inputs to EOQ:

• (Fixed) ordering/set-up cost (S): cost of ordering and getting an order into inventory regardless of quantity ordered

• Per-order supplier fee

• Labor to place order – determining how much is needed, typing up invoices, calling in order

• Shipping costs (fixed)

• Moving to storage (whole order)

Typically, S is expressed in $/order

Page 130: Class 3 Notes

Economic Order Quantity• Inputs to EOQ (cont’d):

• Unit (variable) price (P): cost of obtaining and getting it into inventory one unit

• Unit price of the item

• Sales tax paid on unit price

• Shipping costs (variable)

• Inspecting goods for quality and quantity (per unit)

• Moving to storage (single item)

Typically, P is expressed in $/unit

Page 131: Class 3 Notes

Economic Order Quantity• Inputs to EOQ (cont’d):

• Holding (carrying) cost (H): cost of holding a single item in storage for a period of time (e.g., one year)

• Opportunity cost of capital (H = rP)

• Insurance

• Taxes

• Depreciation

• Obsolescence

• Warehouse costs per unit (physical holding cost)

Typically, H is expressed in $/unit/year

Page 132: Class 3 Notes

Economic Order Quantity• Inputs to EOQ (cont’d):

• Average demand rate (D): expected demand during a period of time (e.g., one year)

Typically, D is expressed in units/year

Page 133: Class 3 Notes

Example: EOQ InputsBuildex’s most popular item is a standard sheet of fiberboard. Demand has been constant

at 170,000 per year and is expected to remain so for the next few years.

Fiberboard is purchased from a supplier down river for $30 per sheet. Buildex hires barges to transport the boards to its central warehouse Each barge costs $1800 per trip plus $4 per sheet. Placing an order requires about five hours of employee time. Once at the warehouse, sheets are unloaded at a rate of 10/hour/employee. The unloading equipment used by each employee is rented at a rate of $25 per hour.

The employees working in the warehouse have several tasks:

1. Security guards and general maintenance require about 7,000 hours per year, and this does not depend upon how much inventory is stored in the building.

2. Placing the sheets into storage, after they are unloaded, can be done at the rate of about 25 per employee per hour. No additional equipment is necessary for this.

3. Removing a sheet from inventory and preparing it for shipment requires about ½ hr

4. Materials used to ship one board to a customer costs $1 and delivery costs are about $2 per sheet

The average cost per hour of labor is approximately $12. Buildex usually hires temporary workers to process large deliveries from the supplier and large orders from customers.

Page 134: Class 3 Notes

Example: EOQ InputsItem S P HPurchasing cost ($30/sheet) X (X)

Barge cost ($1800/order) X

Barge cost ($4/sheet) X (X)

Ordering labor ($12/hour, 5 hr/order) X

Warehouse guards/maintenance($12/hour, 7000 hours/year)

Unloading labor($12/hour, 10 sheets/employee/hr)

X (X)

Unloading equipment($25/hour, 10 sheets/employee/hr)

X (X)

Placing in storage($12/hour, 25 sheets/employee/hr)

X (X)

Removing from storage($12/hour, 2 sheets/employee/hr)

Material/delivery to cust. ($3/sheet)

Page 135: Class 3 Notes

Economic Order Quantity• Inputs summary

• S: ordering cost

• P: unit cost

• H: holding costs

• D: average demand rate

• If Q is the order quantity (batch size), then we want to find the optimal value, Q0, which we call the EOQ.

Page 136: Class 3 Notes

Economic Order Quantity

Q

Time

Small Q (frequent orders)

Big Q (infrequent orders)Q

Incur high ordering costs

(many S’s)

Incur low holding costs (low avg. inv.)

Incur low ordering costs

(few S’s)

Incur high holding costs (high avg. inv.)

Note: total inventory ordered is same in both case and is equal to total demand (in long-run)

Page 137: Class 3 Notes

Economic Order Quantity• Total cost (per period) for order size Q:

Holdingcost perperiod

Total cost =Orderingcost perperiod

+Purchasing

cost perperiod

+

PD+ =

S

# ordersper period

H average

inventory

What are these?

Page 138: Class 3 Notes

• Average inventory:

Economic Order Quantity

Q

Right after an order, we have

Q units, Q/2 above Q/2.

Right before an order, we have 0 units, Q/2 below Q/2.

For each time we have an amount above Q/2, there is a time we have that amount below Q/2.

Q/2

Average inventory = Q/2

Page 139: Class 3 Notes

Economic Order Quantity

Q

Time

• Orders per period:

Total demand per period is D, so the average amount ordered per period must also be D.

(# orders/period)(quantity per order) = D

(# orders/period) = D/Q

Q

Page 140: Class 3 Notes

Economic Order Quantity• Total cost (per period) for order size Q:

Holdingcost perperiod

Total cost =Orderingcost perperiod

+Purchasing

cost perperiod

+

PD+ =

S

# ordersper period

H average

inventory

PD Q

D S

2

Q H = TC

Does not depend on Q

We just need to minimize these

Page 141: Class 3 Notes

Economic Order Quantity• Holding & ordering costs:

Order Quantity (Q)

Co

st p

er p

erio

d

Q0

Holding costs

Ordering costs

Sum of costs (H&O)

Page 142: Class 3 Notes

Economic Order Quantity• Finding the optimal value:

At min (Q0), slope = 0:

PD Q

D S

2

Q H = TC

0 Q

D S

2

H =

dQ

dTC2

H

DS2 =Q0

Page 143: Class 3 Notes

Example: Economic Order QuantityPiddling Manufacturing assembles security monitors. It

purchases 3,600 LCD monitors per year at $65 each. Ordering costs are $31, and the annual carrying costs are 20 percent of the purchasing price. Compute the optimal quantity.

D =P =

S =

H =

3,600 units/yr

$65/unit

$31/order

(.20)($65) = $13/unit/yr

Q0 = √2DS/H = √2(3,600)($31)/$13 = 131 units

r = 0.20

Page 144: Class 3 Notes

Example: Economic Order QuantityWhat is Piddling’s annual cost of holding and ordering?

D =P =

S =H =

3,600 units/yr

$65/unit

$31/order$13/unit/yr

H&O Costs = H(Q0/2) + S(D/Q0)

= $1,704/yr

= (13)(131/2) + (31)(3600/131)

Q0 = 131 units

Page 145: Class 3 Notes

Example: Economic Order QuantityWhat is Piddling’s total annual cost?

D =P =

S =H =

3,600 units/yr

$65/unit

$31/order$13/unit/yr

TC = H(Q0/2) + S(D/Q0) + DP

= $235,704/yr

= $1,704 + (3600)($65)

Q0 = 131 units

Page 146: Class 3 Notes

In-Class: Economic Order QuantityA pet shop sells bird cages at a rate of 18 per week.

Their supplier charges $60 per cage and the cost of placing an order is $45. Due to their limited space, they store cages at a nearby warehouse at a cost of $3 per cage per year. Annual cost of capital is 20%. What is the optimal ordering size?

Page 147: Class 3 Notes

In-Class: Economic Order QuantityWhat is the total cost to the pet shop for cages?

Page 148: Class 3 Notes

Ordering “Close to EOQ”

Page 149: Class 3 Notes

Ordering “Close to EOQ”• EOQ is just an approximate number

• Not always practical to order exactly

• What is impact of order close, but convenient amount?

Page 150: Class 3 Notes

Example 2: Economic Order QuantityDell uses 1,350 Intel Atom processors per week. The price of each unit is $100. Placing an order costs $2,000 and the opp. cost of capital is 20% per year. There is also a physical holding of $10 per unit per year.

D =P =S =H =

(1,350 units/wk)(52 wks/yr)$100/unit$2,000/order(.20)($100) + $10 = $30/unit/yr

Q0 = √2DS/H = √2(70,200)($2,000)/$30 = 3,059 units

= 70,200 units/yr

H&O Costs = H(Q0/2) + S(D/Q0)

= $30(3059/2) + $2000(70200/3059)

= $91,782/yr

Page 151: Class 3 Notes

Example 2: Economic Order QuantityRemember that the # of orders per year is:

Dell would rather place 24 orders per year (twice/month). What would it cost to switch to this order frequency?

D/Q0 = (70,200 units/yr)/(3,059 units/order)= 22.94 orders/yr

H&O Costs == $30(2925/2) + $2000(70200/2925)

= $91,875/yr

Q = D/(# orders/yr) = 70200/24 = 2925

Cost = $91,875 – $91,782 = $93! (0.1%)

H(Q/2) + S(D/Q)

Page 152: Class 3 Notes

Economic Order Quantity• Total costs are “flat” around EOQ:

Q

TC

Range of TC

Range of Q

Q0

Page 153: Class 3 Notes

Estimation Error with EOQ

Page 154: Class 3 Notes

Estimation Error with EOQ• EOQ is often based on estimated costs

• What happens when those costs are off?

Page 155: Class 3 Notes

Example: Estimation ErrorK-Mart stocks a model of cordless phones. Demand is estimated to be 150 units/month. They are purchased from a supplier for $50. They have recently switched to an EDI system with the supplier and the manager estimated that placing an order will now cost $20. Cost of capital is 20% per year. What is the optimal order quantity given these costs?

D =P =

S =H =

150 phones/months

$50/phone

$20/order(.20)($50) = $10/phone/yr

Q0 = √2DS/H = √2(1,800)($20)/$10 = 85 phones

= 1800 phones/year

Page 156: Class 3 Notes

Example: Estimation ErrorThe system was set up to place orders of size 85. The manager, however, was wrong and it will actually cost $25 to place an order. What is the cost of the mistake?

First, we calculate what the EOQ should have been:

D =P =

S =H =

150 phones/months

$50/phone

$25/order(.20)($50) = $10/phone/yr

Q0 = √2DS/H = √2(1,800)($25)/$10 = 95 phones

= 1800 phones/year

Page 157: Class 3 Notes

Example: Estimation ErrorNext, we compare what the cost should have been …

With the actual cost …

H&O Cost (Q = 95) =

= $10(95/2) + $25(1800/95)

= $948.68/yr

H(Q/2) + S(D/Q)

H&O Cost (Q = 85) =

= $10(85/2) + $25(1800/85)

= $954.51/yr

H(Q/2) + S(D/Q)

Cost of Error = $954.51 – $948.68 = $5.73 (0.6%)

Page 158: Class 3 Notes

Estimation Error• Total costs are “flat” around EOQ:

Q

TC

Change in TC

Change in Q

85 95

TC w/ S = $20TC w/ S = $25

Page 159: Class 3 Notes

Changes in Inputs

Page 160: Class 3 Notes

Changes in InputsLooking at EOQ:

• Q0 grows with square root of items in numerator (D, S)

• E.g., doubling S, increases Q0 by just 41.4% (1.414x)

• Q0 grows with 1/square root of H

• E.g., doubling H, decreases Q0 by just 29.5% (0.705x)

0

2DSQ =

H

Page 161: Class 3 Notes

Economic Order Quantity• EOQ has economies of scale as demand rises:

Demand, D

Holding & Ordering Costs

Page 162: Class 3 Notes

Price Breaks:EOQ (advanced)

(Not on Exam)

Page 163: Class 3 Notes

Starting Simple• Often times max order size is limited by transportation

capacity (e.g., barge size for Buildex).

Order Quantity (Q)

Co

st p

er p

erio

d

Q0 Qmax

How much should weorder if limit > Q0?

Qmax

What if limit < Q0?

Page 164: Class 3 Notes

Starting Simple• Sometimes, supplier place a restriction on the

minimum size order that a manufacturer can place.

Order Quantity (Q)

Co

st p

er p

erio

d

Q0 Qmin

How much should weorder if limit < Q0?

Qmin

What if limit > Q0?

Page 165: Class 3 Notes

Putting Them Together• A common practice is to offer price breaks for larger

order sizes. For example:

Now, each price level has a min and a max quantity.How many should we order?

Order Quantity Prices per Unit

1 to 4,999 $100

5,000 to 9,999 $90

10,000 or more $80

Different C’s mean different H’s, giving different EOQ’s.

Page 166: Class 3 Notes

First Step• For each price, where is its EOQ? What Q is best?

Q

TC TC

Q

Q

TC

Q0

Q0

Q0

Above the range In the range

Below the range

Page 167: Class 3 Notes

Second Step• Find the price whose “best” Q has the lowest TC:

Q

TC

Skip Problems

Page 168: Class 3 Notes

Example: Price BreaksDell needs 130,000 E7200’s per year. Placing an order

costs $9,000 and the opportunity cost of capital is 20% per year. There is a physical holding of $6 per unit per year. Intel offers the following unit price options:

Order Quantity Price per Unit

1 to 4,999 $100

5,000 to 9,999 $90

10,000 or more $80

Page 169: Class 3 Notes

Example: Price BreaksH (< 5k) =

H (5-10k) =

H (≥ 10k) =

Q0 (< 5k) =

Q0 (5-10k) =

Q0 (≥ 10k) =

(.20)($100) + $6 = $26/unit/yr

(.20)($90) + $6 = $24/unit/yr

(.20)($80) + $6 = $22/unit/yr

√2(130,000)($9000)/($26) = 9,487 units

√2(130,000)($9000)/($24) = 9,874 units

√2(130,000)($9000)/($22) = 10,313 units

Page 170: Class 3 Notes

Example: Price Breaks• For each range, where is its Q0?

Q

TC TC

Q

Q

TC

Q0

Q0

Q0

< 5,000 5,000-10,000

≥ 10,000

Q* = 5,000 Q* = 9,874

Q* = 10,313

Page 171: Class 3 Notes

Example: Price BreaksTC (< 5k) =

TC (5-10k) =

TC (≥ 10k) =

(26)(5,000/2) + (9,000)(130,000/5,000) + (130,000)($100)

= $13,299,000/yr

(24)(9,874/2) + (9,000)(130,000/9,874) + (130,000)($90)

= $11,936,981/yr

(22)(10,313/2) + (9,000)(130,000/10,313) + (130,000)($80)

= $10,626,892/yr

The optimal order is 10,313

Page 172: Class 3 Notes

Exercise 2: EOQ Quantity DiscountA service garage uses 200 boxes of industrial strength cleaning cloths a year.

The boxes cost $8 each and the company has an annual cost of capital of 25%. Your vendor charges $10 per order, regardless of the quantity ordered.

a) What order quantity minimizes total cost?

b) How many orders per year will they place if they use this quantity?

Page 173: Class 3 Notes

Exercise 2: EOQ Quantity Discountc) You vendor offers a lower unit price of $7 per box, but only for

orders of 55 boxes or more. What quantity should they order?

Page 174: Class 3 Notes

Economic Order Quantity• Additional considerations:

• Multiple products

• Transportation pooling

• Economic production quantity (we will omit)

• When to order (ROP)

• Next …

Page 175: Class 3 Notes

Normal Distribution(Refresher)

Page 176: Class 3 Notes

• Normal – classic “bell shaped curve”• Uniquely defined by:

• mean ( – average value, measure of location

• standard deviation () – measure of spread in distribution

Probability Theory

σ

x

Page 177: Class 3 Notes

• Normal (con’t)• Easier to use Standard (0, 1) Normal:

Probability Theory

1

zx

xz

z-value = “# std. dev.’s from the mean”

x

x

0 x–

0(x–

Page 178: Class 3 Notes

Demand for your product is normally distributed with mean of 100 units and standard deviation of 20 units.

• What is the z value for 127 units?

Z-Value Example

127

x

100

20

127

20

xz

20

100127

35.1 (127 is 1.35 std. dev.’s above 100)

Page 179: Class 3 Notes

Demand for your product is normally distributed with mean of 100 units and standard deviation of 20 units.

• What inventory level gives a 90% chance (probability) of not stocking out?

Z-Value Example

zx 2028.1100 1266.125

100

20

From table, z for 90% is 1.28.

We want 1.28 std. dev.’s above 100.

Page 180: Class 3 Notes

When to order?(ROP)

Page 181: Class 3 Notes

Reorder Point• Reorder Point (ROP) – level of inventory position, at

which a replenishment order (~ Q0) is placed

• Inventory on-hand – inventory held that is available for use

• Inventory position – inventory on-hand, plus orders already placed, but not yet received (pipeline stock)

• We will assume that at most one order is in the pipeline, so we will just track inventory on-hand• Time between orders > time order spends in pipeline

• In reality, inventory position can trigger an order during the lead time of another.

Page 182: Class 3 Notes

Reorder Point• Lead time (LT) – time interval between placing an

order and receiving it

• May be constant or variable

LT

Page 183: Class 3 Notes

• If demand is 100 units/day and lead time is 3 days, how much should we have left to hit 0 as the order arrives?

Reorder Point

Q0

100

3

???

Page 184: Class 3 Notes

Reorder Point

Q0

d

• In general, given constant lead time (LT) and demand (d), when should we place an order?

Want to just run out when order arrives.

LT

???

→ ROP = d*LT

Page 185: Class 3 Notes

• First problem: what if demand rate is not constant?

Reorder Point

When we order, we do not know how much we will have left after LT

LT

ROP

Sometimes lower than avg.

Sometimes higher than avg.

What if ROP = avg. demand?Safety Stock (ss)

Safety stock reduces chance of stockout during lead time

Average Demand

So, we add a little extra.

Page 186: Class 3 Notes

Reorder Point• Second problem: what if lead time is not constant?

When we order, we do not know how long lead time will be

Avg. lead

time

ROP

Lon

ger than

avg.

What if ROP = avg. demandSafety Stock (ss)

Safety stock reduces chance of stockout during lead time

Sh

orter than

avg.

Average Demand

So, we add a little extra.

?

Page 187: Class 3 Notes

Reorder Point• Overall, the ordering process will look like:

Q0

SSROP

Q0

Q0

Q0

Q0

Inventory Position

Inventory On-Hand

Page 188: Class 3 Notes

Reorder Point• Demand rate (d): demand over period (day, week, etc.)

• We will assume it follows a Normal distribution:

d = average demand per period (note: this is D in EOQ)

d = standard deviation of demand per period

d

σd

Page 189: Class 3 Notes

Reorder Point• Lead time (LT): time from placing to receiving an order

• We will assume it follow a Normal distribution

LT = average lead time

LT = standard deviation of lead time

LT

σLT

Page 190: Class 3 Notes

Reorder Point• Demand during lead time (DL): amount of demand seen

between placing an order and receiving it

• Product of random variables: DL = d*LT

ROP

Safety Stock (ss)

Random # of random demands

Creates distribution of demand over lead time (DL)

Page 191: Class 3 Notes

Reorder Point• Demand during lead time (DL): - cont’d

• Under assumptions, this will be a normal:

d*LT = average demand over lead time

dLT = standard deviation of lead time

dLT

σdLT

2 2 2d LTLT d

Page 192: Class 3 Notes

Reorder Point• Distribution of demand during lead time (DL):

d*LT

ss

Service Level (SL)

Risk (stockout)

ROP

The choice of ROP determines the probability of stocking out over the lead time: Pr[DL ≤ ROP] = SL

Page 193: Class 3 Notes

Reorder Point• Choosing the service level (SL):

• Tradeoff:

• Cost of stocking out: goodwill lost, tracking fees, rush shipments

• Cost of carrying more ss: additional holding cost

• Used to determine appropriate z-value

Page 194: Class 3 Notes

• Method:

1) Determine average and std. dev. for demand and lead time

2) Determine service level (given) and corresponding z

3) Calculate average demand over lead time:

4) Determine std. dev. of demand over lead time:

5) Calculate the Reorder Point (ROP)

Reorder Point

2LT

22ddLT dLT

d*LT

Page 195: Class 3 Notes

• Math:

• ROP =

= d*LT + z*dLT

Reorder Point

Average DemandOver LT

Safety Stock+

2LT

22ddLT dLT

Often, one of these is zero

Page 196: Class 3 Notes

Example: Reorder PointThe housekeeping department of a motel stocks washcloths. Usage

can be approximated by a normal distribution that has a mean of 400 and standard deviation of 9 per day. The lead time from a linen supplier is a constant 3 days. The motel maintains a 98% service level. What ROP should they use?

0.98 2.053 (from z table)

d*LT + z dLT = 400(3) + 2.053(15.58)= 1,233 washcloths

SL = → z =

dLT =

ROP =

d = 400 washcloths/day d = 9 washcloths/day Demand:

LT = 3 days LT = 0 daysLead time:

√LTd2 + d2LT

2 = √32 + 40022 = 15.58

(note: always round up)

Page 197: Class 3 Notes

Example 2: Reorder PointThe housekeeping department also stocks soap. Usage has been

relatively constant at 600 bars per day. The lead time follows a normal distribution with a mean of 6 days and standard deviation of 2 days. A service level of 90% is desired. What ROP should they use?

0.90 1.282 (from z table)

d*LT + z dLT = 600(6) + 1.282(1200) = 5,138 bars

SL = → z =

dLT =

ROP =

d = 600 bars/day d = 0 bars/day Demand:

LT = 6 days LT = 2 daysLead time:

√LTd2 + d2LT

2 = √62 + 60022 = 1200

(note: always round up)

Page 198: Class 3 Notes

Example 3: Reorder PointThe motel replaces broken glasses at a rate of 25 per day. Usage

has been tended to be normally distribution with a standard deviation of 3 glasses. The lead time follows a normal distribution with a mean of 10 days and standard deviation of 2 days. What ROP should they use for a service level of 95%?

0.95 1.645 (from z table)

d*LT + z dLT = 25(10) + 1.645(50.89)= 334 glasses

SL = → z =

dLT =

ROP =

d = 25 glasses/day d = 3 glasses/day Demand:

LT = 10 days LT = 2 daysLead time:

√LTd2 + d2LT

2 = √102 + 2522 = 50.89

(note: always round up)

Page 199: Class 3 Notes

Example 3: Reorder PointHow much safety stock in glasses are the carrying?

d*LT + z dLT = 25(10) + 1.65(51) = 334 glassesROP =Recall:

Avg. demand over lead time

Safety stock

1.65(51) = 84 glassesSS =

Page 200: Class 3 Notes

In-Class Exercise: ROPA supermarket has added a new brand of Ketchup (or Catsup, if you prefer) and needs to determine when to place orders. Daily demand fluctuates highly with a mean of 50 bottles and a standard deviation of 15 bottles. The supplier has not proven very reliable. Lead times are 6 days with a standard deviation of 1.5 days. The supermarket uses a service level of 90% (z = 1.282).

a) Find the level of safety stock and re-order point for the supermarket.

Page 201: Class 3 Notes

In-Class Exercise: ROP

Page 202: Class 3 Notes

In-Class Exercise: ROP & EOQA small supermarket has one cash register. On average, they use up 10 rolls of tape per day, but it is not consistent. The standard deviation of usage is 2 rolls per day. They cannot afford to run out frequently, as it requires manual tracking and later entry of sales, so they use a service level of 99% (z = 2.326).

Their supplier charges $2 per roll, plus $10 per order. They guarantee a 3 day delivery time.

The supermarket’s annual cost of capital is 10%.

Page 203: Class 3 Notes

In-Class Exercise: ROP & EOQa) How much and at what inventory level should the place an

order?

Page 204: Class 3 Notes

Exercise 6: ROP & EOQb) The supplier begins to package orders more securely due to

damage in transit. To cover their additional cost, they now charge $22 an order instead of $10. How do the quantities calculated in (a) change with this different cost?

Page 205: Class 3 Notes

In-Class Exercise: ROP & EOQc) Loosely sketch the two situations.

See Solution

Page 206: Class 3 Notes

Safety Stock• Methods for reducing safety stock:

• Reduce demand variability

• Reduce lead time

• Reduce variability of length of lead time

• Economies of scale (pooling)

Page 207: Class 3 Notes

Pooling Safety Stock ExampleTwo towing companies have individual diesel tanks. Each

company uses 100 gal per day. Usage has been tended to be normally distribution with a standard deviation of 20 gal. The lead time from the supplier (same for both) is 7 days. How much safety stock does each carry for a service level of 95%?

0.95 1.645 (from z table)

z dLT = 1.645(52.92) = 87 gal

SL = → z =

dLT =

SS =

d = 100 gal/day d = 20 gal/day Demand:

LT = 7 days LT = 0 daysLead time:

√LTd2 + d2LT

2 = √72 + 10022 = 52.92

Page 208: Class 3 Notes

Pooling Safety Stock ExampleIf they agree to keep one central tank, how much safety stock would

they have to carry for a service level of 95%?

z dLT = 1.645(74.83) = 124 gal

dLT =

SS =

d = (2)100 = 200 gal/day

d = √ 202 + 202

√LTd2 + d2LT

2 = √72 + 20022 = 74.83

= 28.28 gal/day

Reduction in SS = 2(87) – 124 = 50 gals!

(if independent)

Page 209: Class 3 Notes

In-Class Exercise: ROP & EOQc) Loosely sketch the two situations.

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