Class 3 Notes
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Transcript of 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”)
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
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
Example: Mean chart using R
12.144
12.11
12.076
x
LCLx
UCLx
Variation between samples is in control.
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.
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
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
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
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.
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
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
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.
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
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.
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
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
In-Class Exercises: Mean/Range Charts
In-Class Exercises: Mean/Range Charts
LCLR=
UCLR=
R=
In-Class Exercises: Mean/Range Chart
In-Class Exercises: Mean/Range Charts
LCLxbar=
UCLxbar=
x =
In-Class Exercises: Mean/Range Charts
LCLxbar=
UCLxbar=
x =
Updating Control Charts
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).
Nelson’s Rules
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.
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
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
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
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
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
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
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
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)
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
Run Tests
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
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)
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
Six Sigma
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.)
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
Process Capability
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
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
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.
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.
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.)
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.)
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
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
• 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.)
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.
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?
In-Class Exercise: Process Capability
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
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
Process Capability
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
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?
In-Class Exercise: Process Capability
Putting It Together
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?
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
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
Supply Chain Management
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”.
SCM Job Outlook
Source: www.odinjobs.com
SCM Job Outlook
Source: www.careersinsupplychain.org
Supply Chain
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
Manufacturing Supply Chain
Example: LNG Supply Chain
Service vs. Manufacturing
Supplier
Supplier
Supplier
Supplier
Supplier
StorageMfg
StorageDist.
StorageRet.
StorageServ.
Cust
Cust
Hybrid Supply Chain
Supplier
SupplierStorage
Serv.
Supplier
Supplier
Supplier
StorageMfg
StorageDist.
StorageRet. C
ustomer
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
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
• 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 ►►►
Supply Chain Management
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”
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
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
Supply Chain Management
Source: Supply Chain Council (www.supply-chain.org)
• Supply Chain Operational Reference (SCOR) Metrics
Bullwhip Effect
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
Bullwhip Effect
Retailer Distributor
Manufacturer
Supplier
Stock
Increased Variability
Orders
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
Bullwhip Effect• Sales promotions:
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)
Purchasing
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
Purchasing
Legal
Acc’tingOp.’s
DataProcess.
Design/ Eng.
Receiving
Suppliers
Purchasing
Q/A
• Interfaces:
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
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
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
Strategic Partnering
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
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)
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
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
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)
E-business
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
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
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
Logistics
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
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
LogisticsUnlike inventory, transportation has remained stable …
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
Logistics• Within a facility:
Rec
eivi
ng
Production Process
Shipping
Storage
Workcenter
Work center
Storage
Work center Workcenter
Storage
Logistics• Within a facility:
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.
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
3PL – Freight Matching
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
Inventory Management
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!
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
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
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
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
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
Counting systems• Of periodic and continuous systems, which do think
are used more and is that changing? Why?
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
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
How much to order:EOQ (Basic)
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)
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.
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
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
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
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
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.
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)
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.
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)
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?
• 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
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
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
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)
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
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
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
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
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?
In-Class: Economic Order QuantityWhat is the total cost to the pet shop for cages?
Ordering “Close to EOQ”
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?
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
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)
Economic Order Quantity• Total costs are “flat” around EOQ:
Q
TC
Range of TC
Range of Q
Q0
Estimation Error with EOQ
Estimation Error with EOQ• EOQ is often based on estimated costs
• What happens when those costs are off?
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
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
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%)
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
Changes in Inputs
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
Economic Order Quantity• EOQ has economies of scale as demand rises:
Demand, D
Holding & Ordering Costs
Price Breaks:EOQ (advanced)
(Not on Exam)
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?
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?
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.
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
Second Step• Find the price whose “best” Q has the lowest TC:
Q
TC
Skip Problems
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
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
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
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
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?
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?
Economic Order Quantity• Additional considerations:
• Multiple products
• Transportation pooling
• Economic production quantity (we will omit)
• When to order (ROP)
• Next …
Normal Distribution(Refresher)
• 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
• 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–
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)
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.
When to order?(ROP)
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.
Reorder Point• Lead time (LT) – time interval between placing an
order and receiving it
• May be constant or variable
LT
• 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
???
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
• 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.
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.
?
Reorder Point• Overall, the ordering process will look like:
Q0
SSROP
Q0
Q0
Q0
Q0
Inventory Position
Inventory On-Hand
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
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
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)
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
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
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
• 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
• Math:
• ROP =
= d*LT + z*dLT
Reorder Point
Average DemandOver LT
Safety Stock+
2LT
22ddLT dLT
Often, one of these is zero
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)
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)
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)
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 =
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.
In-Class Exercise: ROP
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%.
In-Class Exercise: ROP & EOQa) How much and at what inventory level should the place an
order?
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?
In-Class Exercise: ROP & EOQc) Loosely sketch the two situations.
See Solution
Safety Stock• Methods for reducing safety stock:
• Reduce demand variability
• Reduce lead time
• Reduce variability of length of lead time
• Economies of scale (pooling)
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
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)
In-Class Exercise: ROP & EOQc) Loosely sketch the two situations.
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