6/16/2019
• Topics
– Ch. 12: Safety inventory
Today
• Exclude pages 330331,333336, 345
• Handouts
– TECH492592_2019Su0723_Figure 122
– TECH492592_2019Su0723_Figure 123
• Assignments / Announcements (Due)
–
–
–
–
Managing Uncertainty in a Supply
Chain: Safety Inventory (Ch. 12)
Read Ch. 12
Note “NORMSDIST” in Figure 123 should be “NORMDIST”
No late submission for Assignments 7, 8, 9, and 10
Assignment 7 (Due 7/30, 11:59 PM)
• Show calculations or attach Excel files to receive full credit  START EARLY
• Question in the following slides
• Similar to Examples 121, 122, 123
TECH 492/592 – Manufacturing
Distribution Applications
– Assignment 8 (Due 7/30, 11:59 PM)
• Show calculations or attach Excel files to receive full credit  START EARLY
• Question in the following slides
• Similar to Examples 124, 125
– Assignment 9 (Due 7/30, 11:59 PM)
July 23, 2019
• Show calculations or attach Excel files to receive full credit  START EARLY
• Question in the following slides
• Similar to Examples 127, 1213 (123, 125)
– Assignment 10 (Due 7/30, 11:59 PM)
• Show calculations or attach Excel files to receive full credit  START EARLY
• Question in the following slides
• Similar to Example 129, 1211, 1212
Assignment 7
• Suppose Walmart,
Assignment 8
• Suppose Walmart,
– Sells Samsung Galaxy cell phones
– Uses continuous review policy
• Suppose for Galaxy cell phones at Walmart,
– Sells Samsung Galaxy cell phones
– Uses continuous review policy
• Suppose for Galaxy cell phones at Walmart,
– Average weekly demand is 1000 phones
– Standard deviation of weekly demand is 400 phones
– Lead time for the Samsung to replenish cell phones is 4 weeks
– Average weekly demand is 1000 phones
– Standard deviation of weekly demand is 400 phones
– Lead time for the Samsung to replenish cell phones is 4 weeks
• Samsung guarantees to deliver cell phones 4 weeks after cell phones are ordered
• Questions
• Samsung guarantees to deliver cell phones 4 weeks after cell phones are ordered
• Questions
Suppose that the reorder point (ROP) is 5000 phones and the lot size (Q) is 3000
phones
– (71) What is safety inventory?
– (72) What is cycle inventory?
– (73) What is average inventory carried by Walmart?
– (74) What is Cycle Service Level (CSL)?
Now suppose that Walmart would like to achieve CSL of 0.99
– (75) What should the safety inventory (ss) be if Walmart wishes to achieve
CSL of 0.99?
– (76) What should ROP be if Walmart wishes to achieve CSL of 0.99?
Suppose that the reorder point (ROP) is 5000 phones and the lot size (Q) is 3000
phones
– (81) What is fill rate (fr)?
Now suppose that Walmart would like to achieve fr of 0.99 when the lot size (Q)
is 3000 phones
– (82) What should the safety inventory (ss) be if Walmart wishes to achieve fr
of 0.99?
– (83) What should ROP be if Walmart wishes to achieve fr of 0.99?
Assignment 9
• Suppose Walmart,
– Sells Samsung Galaxy cell phones
– Uses continuous review policy
Assignment 10
•
– Operates 400 stores
– Sells Garmin GPS at $100 at each store
• Suppose for Galaxy cell phones at Walmart,
– Average weekly demand is 1000 phones
– Standard deviation of weekly demand is 400 phones
– Lot size (Q) is 3000 phones
•
•
• Samsung guarantees to deliver cell phones 2 weeks after cell phones are ordered
– (93) What should the safety inventory (ss) be if Walmart wishes to achieve CSL of
0.99?
– (94) What should OUL be if Walmart wishes to achieve CSL of 0.99?
Suppose at each Sam’s Club store, for Garmin GPS,
– Average weekly demand is 50 units
•
Demand at each store is independent
– Standard deviation of weekly demand is 10 units
– Lead time for Garmin to replenish cell phones is 5 weeks
Now suppose that the lead time is reduced to the average lead time of 2 weeks but
the standard deviation of lead time is 1 week
• Samsung targets to deliver cell phones 2 weeks after cell phones are ordered but cannot
guarantee
Assume cost of Garmin GPS is $100
– Uses continuous review policy
– Incurs inventory holding cost per year of 20% of product price
• Questions
– (91) What should the safety inventory be if Walmart wishes to achieve CSL of
0.99?
– (92) What should the safety inventory be if Walmart wishes to achieve fr of 0.99?
Now suppose Walmart, uses periodic review policy with review interval of 2 weeks
when lead time for the Samsung to replenish cell phones is 2 weeks
Suppose Sam’s Club,
•
•
Garmin guarantees to deliver GPS 5 weeks after GPS are ordered
Questions
– (101) What should the safety inventory (ss) at each store be if Sam’s Club wishes to achieve CSL of
0.95 at each store?
– (102) What is total annual holding cost of safety inventory of GPS (total of all Sam’s Club stores)?
Suppose now Sam’s Club decides to sell Garmin GPS only online and ships all GPS from a central
warehouse but wishes to increase CSL to 0.99 at the central warehouse
– (103) What is the aggregated demand at the central warehouse?
– (104) What is the standard deviation of the aggregate demand at the central warehouse?
– (105) What should the safety inventory (ss) at the central warehouse be if Sam’s Club wishes to
achieve CSL of 0.99 at the central warehouse?
– (106) What is the total annual holding cost of safety inventory of GPS at the central warehouse?
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6/16/2019
Topics: Managing Uncertainty in a
Supply Chain: Safety Inventory
•
•
•
•
•
•
The role of safety inventory in a supply chain
Factors affecting the level of safety inventory
Determining appropriate level of safety inventory
Impact of supply uncertainty on safety inventory
Impact of aggregation on safety inventory
Impact of replenishment policies on safety
inventory
• Managing safety inventory in a multiechelon
supply chain
• Managerial levers to reduce safety inventory
Important Questions
• How is cycle service e level (CSL) calculated?
• Need to be able to solve problem similar to:
– Example 122 (Calculate CSL for continuous review
policy)
– Example 123 (Calculate safety inventory ss and
ROP from CSL for continuous review policy)
•
•
•
Important Questions
What is safety inventory?
What is average inventory when we include safety inventory?
How three measures of product availability differ?
– Product fill rate (fr)
– Order fill rate
– Cycle service level (CSL)
•
•
What is lead time?
What is the formula to calculate expected demand over k periods of time when:
•
Assuming independent, what is the formula to calculate standard deviation of
demand over k periods of time when:
•
•
What is coefficient of variation (cv)?
What is standard deviation of demand during lead time when:
•
How two inventory replenishment policies differ?
•
•
What is reorder point (ROP)?
Need to be able to solve problem similar to Example 121
– Demands of each period are different?
– Demands of each period are the same?
– Demands of each period are different?
– Demands of each period are the same?
– Lead time is certain (only standard deviation of demand)?
– Lead time is uncertain (standard deviation of demand and standard deviation of lead time)?
– Continuous review
– Periodic review
Important Questions
• How is fill rate(fr) calculated?
• What is expected shortage per replenishment
cycle (ESC)?
• How is ESC calculated?
• How do we use GOALSEEK to find safety
inventory ss for given fr (or ESC calculated from fr
and Q)?
• Need to be able to solve problem similar to:
– Example 124 (Calculate fr for continuous
replenishment policy)
– Example 125 (Calculate safety inventory ss and ROP
from fr for continuous replenishment policy)
Important Questions
• What is standard deviation of demand during lead
time when:
– Lead time is certain (only standard deviation of demand)?
– Lead time is uncertain (standard deviation of demand and
standard deviation of lead time)?
• What is periodic replenishment policy?
• Need to be able to solve problem similar to:
– Example 127 (Calculate safety inventory ss and ROP from
fr for continuous replenishment policy when there are
uncertainties about both demand and lead time)
– Example 1213 (Calculate safety inventory ss and OUL
from CSL for periodic replenishment policy)
Important Questions
• What is standard deviation of demand during
lead time when:
– Demand is decentralized (nonaggregate)?
– Demand is centralized (aggregate)?
• Need to be able to solve problem similar to:
– Example 129
• Calculate safety inventory ss of aggregate and nonaggregate
demand
• Calculate annual inventory, transportation, facility costs
– Inventory may also include cycle inventory
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6/16/2019
Important Questions
• What is component commonality?
• What is postponement?
• Need to be able to solve problem similar to:
– Example 1211 (component commonality)
• Identify the number of products that use same component
• Apply demand aggregation for the common component
• For each common component, calculate safety inventory ss for the
common components based on the aggregate demand
• Calculate total safety inventory for all components
The Role of Safety Inventory in a
Supply Chain
– Example 1212 (postponement)
• Identify component that need to be shared (commonality) to
enable postponement
• Apply demand aggregation for the common component
• For each common component, calculate safety inventory ss for the
common components based on the aggregate demand
• Calculate total safety inventory for all components
The Role of Safety Inventory in a
Supply Chain
• Safety inventory is:
– Inventory for satisfying the actual demand that exceeds the forecasted
demand for a given period
• Safety inventory is needed because:
– Demand is uncertain and
– Product shortage may occur if the actual demand exceeds the forecasted
demand
• Inventory profile with and without safety inventory is illustrated below
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 = 𝑆𝑎𝑓𝑒𝑡𝑦 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 + 𝐶𝑦𝑐𝑙𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦
Without safety inventory (Ch. 11)
With safety inventory (Ch. 12)
TradeOff in Safety Inventory Decisions
• Large amount of safety inventory
– Increases product availability; thus increases profit
– Increases inventory holding costs; thus reduces profit
• Making good decisions on safety inventory is critically
important when:
– Product lifecycles are short
– Demand is volatile
because a large amount of safety inventory
– Can help account for demand volatility, but
– Can hurt company if new products introduced to the
market make the products in the inventory obsolete and
worthless
•
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Factors that Affect Safety Inventory
Decisions
• Increased ease of searching products
– Customers buy products from competitors if the product is not
available
• For example, if one does not find a book in Amazon.com, he/she can search
BarnesandNoble.com
– In this case, companies increase safety inventory to increase product
availability
• Increased product variety
– Market becomes heterogeneous
• Demand for individual product is very unstable and difficult to forecast
– In this case, companies increase safety inventory
• Reduced product lifecycle
– Products in the inventory can easily become obsolete and worthless
– In this case, companies try not to carry too much safety inventory
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Key Questions in Safety Inventory
Decisions
• Key questions in safety inventory decisions
1. What is the appropriate level of product
availability?
2. How much safety inventory is needed for the
desired level of product availability?
3. What actions can be taken to reduce safety
inventory without hurting product availability?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
3
6/16/2019
Factors Affecting the Level of Safety
Inventory
Factors Affecting the Level of
Safety Inventory
• The appropriate level of safety inventory is
determined by:
– Desired level of product availability (more detail in Ch. 13)
– Uncertainties of both demand and supply
– Inventory replenishment policies
• If desired level of product availability increases:
– Safety inventory increases
• If uncertainty of demand or supply increases:
– Safety inventory increases
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Measuring Product Availability
• Product availability reflects:
Measuring Product Availability
• Product fill rate (fr)
– Fraction of demand that is satisfied from available inventory
– Measured over specified amounts of demand rather than time
– Equivalent to probability that product demanded is satisfied from available
inventory
– A company’s ability to satisfy customer order from
inventory
• A stockout occurs when:
• Order fill rate
– Fraction of orders that is satisfied from available inventory
– Measured over specified number of orders rather than time
– If an order consists of multiple products, order is filled only if all products in
the order is satisfied from the available inventory
– Customer order is not satisfied because product is not
available
• Three measures of product availability are:
• Cycle service level (CSL)
– Fraction of replenishment cycles that end with all the customer demand being
met
– Product fill rate (fr)
– Order fill rate
– Cycle service level (CSL)
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Measuring Demand Uncertainty
• As discussed in Ch. 7, demand has:
– Systematic component and
– Random component
• Random component is estimated by the standard
deviation of forecast errors; thus, demand is modeled
by:
• Replenishment cycle is the interval between two successive replenishment deliveries
– Measured over specified number of replenishment cycles
– Equivalent to probability of not having a stockout in a replenishment cycle
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Demand Distribution over L
Periods
•
Distribution of demand over k periods is modeled using:
•
Distribution of demand for each period i (i=1, …, k) is:
– Distribution of demand during each period
•
𝑘
• Lead time is the time between when an order is placed
and when it is received; it is denoted by:
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Suppose ρij is the correlation coefficient of demand between periods i and j, total
demand during L periods is:
– Normally distributed with
– Mean DL and
– Standard deviation σL
– D: average demand per period
– σD: standard deviation of demand per period (forecast
error)
– L: lead time
– Normally distributed with
– Mean Di and
– Standard deviation σi
𝐷𝐿 =
𝑘
𝐷𝑖
𝜎𝑖2 + 2
𝜎𝐿 =
𝑖=1
𝑖=1
𝜌𝑖𝑗 𝜎𝑖 𝜎𝑗
(12.1)
𝑖>𝑗
𝑘
•
𝜎𝐿 =
Demand in two periods is:
– Perfectly positively correlated if ρij = 1
– Perfectly negatively correlated if ρij = –1
– Independent if ρij = 0
𝑖=1
In this class, we assume
demands between periods are
independent, thus ρij = 0
𝑘
𝜎𝑖2 =
𝜎𝑖
2
𝑖=1
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
4
6/16/2019
Evaluating Demand Distribution over L
Periods
Evaluating Demand Distribution over L
Periods
• If demand during each of L periods is:
• If demand during each of L periods is:
–
–
–
–
–
–
–
–
Independent and
Normally distributed with
Mean D and
Standard deviation σD
• Then, the total demand during the L periods is:
k=L
𝑘=𝐿
𝐷𝐿 =
𝑘=𝐿
𝐷𝑖 = 𝐷1 + 𝐷2 + ⋯ + 𝐷𝑘
𝜎𝑖2 =
𝜎𝐿 =
𝑖=1
𝜎1
2
+ 𝜎2
2
+ ⋯ + 𝜎𝑘
2
𝑖=1
L terms
L terms
𝐼𝑓 𝐷1 = 𝐷2 =…=𝐷𝑘
𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑎𝑙𝑙 𝐷
•
𝐷𝐿 = 𝐷 + 𝐷 + ⋯ + 𝐷 = 𝐿𝐷
– Normally distributed with
– Mean DL and
– Standard deviation σL
in which
𝐷𝐿 = 𝐷 × 𝐿
𝐼𝑓 𝜎1 = 𝜎2 =…=𝜎𝑘
𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑎𝑙𝑙 𝜎𝐷
𝜎𝐿 =
𝜎𝐷
2
+ 𝜎𝐷
2+
⋯ + 𝜎𝐷
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
2
= 𝐿 𝜎𝐷 2
= 𝐿 × 𝜎𝐷
• Another important measure of uncertainty is coefficient of
variation (cv)
– cv is the ratio of standard deviation σ to the mean μ
(12.2)
Evaluating Supply Uncertainty
• Supply uncertainty can be described by:
– Uncertainty of lead time
• Distribution of lead time is:
𝑐𝑣 = 𝜎/𝜇
• Coefficient of variation measures the size of uncertainty
relative to demand
– Product with mean demand of 100 and standard deviation of
100 (cv=1) has a larger demand uncertainty than
– Product with mean demand of 1000 and standard deviation of
100 (cv=0.1)
– Standard deviation alone cannot capture the above difference
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
– Normally distributed
– L: average lead time for replenishment
– sL: standard deviation of lead time
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Demand Distribution with
Uncertain Lead Times
• Assumption
Replenishment Policies
• Replenishment policy consists of decisions
– When to reorder
– How much to reorder
D: average demand per period
σD: standard deviation of demand per period
L: average lead time for replenishment
sL: standard deviation of lead time
• Examples of replenishment policies are
– Continuous review
– Periodic review
• Continuous review
• Demand during the lead time is:
– Normally distributed with
– Mean DL and
– Standard deviation σL
in which
𝐷𝐿 = 𝐷 × 𝐿
𝜎𝐿 =
𝜎𝐿 = 𝐿 × 𝜎𝐷
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Demand Distribution over L
Periods
–
–
–
–
Independent and
Normally distributed with
Mean D and
Standard deviation σD
– Inventory is continuously tracked
– Order of lot size Q is placed when the inventory declines to the reorder point
(ROP)
– Size of order does not change
– Time between order changes
• Periodic review
𝐿𝜎𝐷2 + 𝐷 2𝑠𝐿2
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(12.3)
–
–
–
–
Inventory status is checked at regular periodic intervals
Order is placed to raise the inventory level to a specific threshold
Size of order changes
Time between order does not change
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
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6/16/2019
Evaluating CSL and Fill Rate Given a
Replenishment Policy
• We will study three combinations of:
– Replenishment policy
– Product availability
Fill rate (fr)
Cycle service level (CSL)
Yes
Yes

Yes
Continuous review
Periodic review
Determining Appropriate Level of
Safety Inventory
• We start with continuous review policy
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Determining Appropriate Level of
Safety Inventory
• Replenishment policy is restricted to continuous review
policy
Evaluating Safety Inventory Given a
Reorder Point
• From Equation (12.2)
– Q: reorder lot size
– L: replenishment lead time (number of weeks)
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑑𝑒𝑚𝑎𝑛𝑑 𝑑𝑢𝑟𝑖𝑛𝑔 𝑙𝑒𝑎𝑑 𝑡𝑖𝑚𝑒 = 𝐷 × 𝐿
• Weekly demand is:
– Normally distributed with:
– Mean D and
– Standard deviation σD
• Three cases are discussed next
– Safety inventory given a replenishment policy
– CSL given a replenishment policy
– Fill rate given a replenishment policy
Replenishment
policy is
continuous
review
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
• When the amount of inventory reaches reorder point
(ROP), a new order is placed
• Because, on average, DL products are sold between
when the inventory reaches ROP and when the new
order arrives, safety inventory is:
𝑆𝑎𝑓𝑒𝑡𝑦 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦, 𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷 × 𝐿
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Safety Inventory Given a
Reorder Point
• Work on Example 121 in pages 317318 of
the textbook and check your solution with the
solution in the textbook
(12.4)
Evaluating Safety Inventory Given a
Reorder Point
•
Work on Example 121 in pages 317318 of the textbook and check your solution
with the solution in the textbook
121
121
•
Safety inventory
•
Average inventory
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿
What are: Safety inventory?
Average inventory?
Average flow time (average time spent by a Palm at B&M)?
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 = 𝑐𝑦𝑐𝑙𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 + 𝑠𝑎𝑓𝑒𝑡𝑦 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦
=
•
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑓𝑙𝑜𝑤 𝑡𝑖𝑚𝑒 =
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
𝑄
2
+ 𝑠𝑠
Average flow time
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦
𝐷𝑒𝑚𝑎𝑛𝑑
=
𝑄
+𝑠𝑠
2
𝐷
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
6
6/16/2019
Evaluating Safety Inventory Given a
Reorder Point
•
Work on Example 121 in pages 317318 of the textbook and check your solution
with the solution in the textbook
121
•
Safety inventory
•
Average inventory
•
Average flow time
Determining Appropriate Level of
Safety Inventory
6000−−2500
2500××22==1000
1000
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 ==6000
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 = 𝑐𝑦𝑐𝑙𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 + 𝑠𝑎𝑓𝑒𝑡𝑦 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦
𝑄
10000
10000 1000 = 6000
= + 𝑠𝑠 =
= 2 ++
1000 = 6000
2
2
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑓𝑙𝑜𝑤 𝑡𝑖𝑚𝑒 =
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦
𝐷𝑒𝑚𝑎𝑛𝑑
=
𝑄
+𝑠𝑠
2
𝐷
6000
6000
==
𝑤𝑒𝑒𝑘𝑠
==
2.42.4
𝑤𝑒𝑒𝑘𝑠
2500
2500
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Cycle Service Level (CSL)
given a Reorder Point
•
•
Recall that CSL is equivalent to the probability of not having a stockout in a
replenishment cycle
Stockout does not happen as far as demand during replenishment lead time L is
smaller than ROP; Thus
Evaluating Cycle Service Level (CSL)
given a Reorder Point
• Notation
These are the parameters of a distribution. In our case, we
are using a normal distribution. The parameters required
to define “location” and “shape” of a normal distribution
are mean and standard deviation.
𝐶𝑆𝐿 = 𝑃𝑟𝑜𝑏 𝑑𝑒𝑚𝑎𝑛𝑑 𝑑𝑢𝑟𝑖𝑛𝑔 𝑙𝑒𝑎𝑑 𝑡𝑖𝑚𝑒 𝑜𝑓 𝐿 𝑤𝑒𝑒𝑘𝑠 ≤ 𝑅𝑂𝑃
•
𝐶𝑆𝐿 = 𝐹 𝑅𝑂𝑃, 𝐷𝐿 , 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑅𝑂𝑃, 𝐷𝐿 , 𝜎𝐿 , 1
If the total demand during the L periods is
– Normally distributed with mean DL and standard deviation σL
CSL can be written, using the normal distribution notation in Appendix 12A, as
𝐶𝑆𝐿 = 𝐹 𝑅𝑂𝑃, 𝐷𝐿 , 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑅𝑂𝑃, 𝐷𝐿 , 𝜎𝐿 , 1
•
(12.5)
Excel 2010 uses “NORM.DIST” instead of “NORMDIST”
If weekly demand is independent, then from Equation (12.2)
𝐷𝐿 = 𝐷 × 𝐿
𝜎𝐿 = 𝐿𝜎𝐷
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
“F()” is a general notation for a
cumulative distribution function, which
calculates a probability that an
uncertain variable is less than a specific
value. F() is used for any probability
distribution. In our case we are using it
– distribution. Uncertain
for a normal
variable is the total demand during L
weeks and the specific value is ROP.
“NORMDIST()” is an Excel function that calculates
either probability density or cumulative distribution
at a specific value of an uncertain variable. Again, in
our case uncertain variable is the total demand
during L weeks and the specific value is ROP. Note
that the function is “NORM.DIST()” in Excel 2010.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Cycle Service Level (CSL)
given a Reorder Point
• Graphically,
𝐹 𝑅𝑂𝑃, 𝐷𝐿 , 𝜎𝐿
“1” or “TRUE” calculates
cumulative distribution.
“0” or “FALSE” calculates
probability density.
Evaluating Cycle Service Level (CSL)
given a Reorder Point
• Work on Example 122 in pages 318319 of the textbook and check your
solution with the solution in the textbook
𝑜𝑟
𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑅𝑂𝑃, 𝐷𝐿 , 𝜎𝐿 , 1
122
Reorder Point
can be explained as below
F(ROP, DL, σL) or
NORMDIST(ROP, DL, σL,1)
is the area under the
curve and left of ROP. This
area is the probability that
the total demand is less
than ROP, i.e., probability
that there is no stockout
–
in a replenishment cycle.
Normal distribution of the total demand with
mean DL=DL and the standard deviation σL=σD√L
Total demand during
replenishment cycle
of L weeks
Mean DL=DL
ROP
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
•
•
•
•
•
•
•
•
What is D?
What is σD?
What is L?
What is Q?
What is ROP?
What is DL?
What is σL?
What is CSL?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
7
6/16/2019
Evaluating Cycle Service Level (CSL)
given a Reorder Point
• Work on Example 122 in pages 318319 of the textbook and check your
solution with the solution in the textbook
122
•
•
•
•
•
•
•
•
Evaluating Cycle Service Level (CSL)
given a Reorder Point
• Work on Example 122 in pages 318319 of the textbook and check your
solution with the solution in the textbook
122
Reorder Point
What is D?
What is σD?
What is L?
What is Q?
What is ROP?
What is DL?
What is σL?
What is CSL?
•
•
•
•
•
•
•
•
𝐷=
𝜎𝐷 =
𝐿=
𝑄=
𝑅𝑂𝑃 =
𝐷𝐿 = 𝐷 × 𝐿
𝜎𝐿 = 𝐿 × 𝜎𝐷
𝐶𝑆𝐿 = 𝐹 𝑅𝑂𝑃, 𝐷𝐿 , 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑅𝑂𝑃, 𝐷𝐿 , 𝜎𝐿 , 1
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
What is D?
What is σD?
What is L?
What is Q?
What is ROP?
What is DL?
What is σL?
What is CSL?
Reorder Point
𝐷 = 2500
𝜎𝐷 = 500
𝐿=2
𝑄 = 10000
𝑅𝑂𝑃 = 6000
𝐷𝐿 = 𝐷 × 𝐿 = 2500 × 2 = 5000
𝜎𝐿 = 𝐿 × 𝜎𝐷 = 2 × 500 = 707
𝐶𝑆𝐿 = 𝐹 𝑅𝑂𝑃, 𝐷𝐿 , 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑅𝑂𝑃, 𝐷𝐿 , 𝜎𝐿 , 1
= 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 6000,5000,707,1 = 0.921
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Safety Inventory Given Desired
Cycle Service Level (or Fill Rate)
• In many practical settings, companies design
replenishment policies that achieve a desired
level of product availability
• Thus, appropriate level of safety inventory needs
to be determined to achieve desired level of
product availability (in terms of CSL or fill rate)
• Two cases are discussed:
Evaluating Safety Inventory Given
Desired Cycle Service Level (CSL)
• Goal
– For continuous review replenishment policy,
– Identify appropriate ROP and safety inventory that
achieves the desired cycle service level (CSL)
• Assumption
– L: lead time
– CSL: desired cycle service level
– DL: mean demand during lead time
– σL: standard deviation of demand during lead time
– Safety inventory given desired CSL
– Safety inventory given desired fill rate
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Safety Inventory Given
Desired Cycle Service Level (CSL)
•
•
Recall ROP=DL+ss from Equation (12.4)
Thus, safety inventory ss needs to satisfy the following condition
•
ROP
Given that demand is normally distributed, the above condition can be written as
below using cumulative distribution function F()
Evaluating Safety Inventory Given
Desired Cycle Service Level (CSL)
•
Work on Example 123 in page 320 of the textbook and check your solution with
the solution in the textbook.
123
𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑 𝑑𝑢𝑟𝑖𝑛𝑔 𝑙𝑒𝑎𝑑 𝑡𝑖𝑚𝑒 ≤ 𝐷𝐿 + 𝑠𝑠 = 𝐶𝑆𝐿
𝐹 𝐷𝐿 + 𝑠𝑠, 𝐷𝐿 , 𝜎𝐿 = 𝐶𝑆𝐿
•
Using inverse normal distribution F 1() (check Appendix 12A for the definition of
inverse normal),
𝐷𝐿 + 𝑠𝑠 = 𝐹 −1 𝐶𝑆𝐿, 𝐷𝐿 , 𝜎𝐿
•
→
𝑠𝑠 = 𝐹 −1 𝐶𝑆𝐿, 𝐷𝐿 , 𝜎𝐿 − 𝐷𝐿
Using inverse standard normal distribution FS1(),
𝑠𝑠 = 𝐹𝑆−1 𝐶𝑆𝐿 × 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿
Excel 2010 uses “NORM.S.INV” instead of “NORMSINV”
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(12.6)
• What is D?
• What is σD?
• What is L?
• What is DL?
• What is σL?
• What is CSL?
• What is ss?
• What is ROP?
• Note that we do not need to know Q to calculate ss and ROP
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
8
6/16/2019
Evaluating Safety Inventory Given
Desired Cycle Service Level (CSL)
•
Work on Example 123 in page 320 of the textbook and check your solution with
the solution in the textbook.
123
Evaluating Safety Inventory Given
Desired Cycle Service Level (CSL)
•
Work on Example 123 in page 320 of the textbook and check your solution with
the solution in the textbook.
123
• What is D?
𝐷=
𝜎𝐷 =
• What is σD?
𝐿=
• What is L?
• What is DL?
𝐷𝐿 = 𝐷 × 𝐿
• What is σL?
𝜎𝐿 = 𝐿 × 𝜎𝐷
• What is CSL?
𝐶𝑆𝐿 =
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿
• What is ss?
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 𝑅𝑂𝑃 = 𝑠𝑠 + 𝐷𝐿
• What is ROP?
• Note that we do not need to know Q to calculate ss and ROP
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
• What is D?
𝐷 = 2500
𝜎𝐷 = 500
• What is σD?
𝐿=2
• What is L?
• What is DL?
𝐷𝐿 = 𝐷 × 𝐿 = 2500 × 2 = 5000
• What is σL?
𝜎𝐿 = 𝐿 × 𝜎𝐷 = 2 × 500 = 707
• What is CSL?
𝐶𝑆𝐿 = 0.90
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 0.90 × 707 = 1.2816 × 707 = 906
• What is ss?
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 𝑅𝑂𝑃 = 𝑠𝑠 + 𝐷𝐿 = 906 + 5000 = 5906
• What is ROP?
• Note that we do not need to know Q to calculate ss and ROP
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Fill Rate Given a Reorder
Point
• Recall that product fill rate (fr) is
Determining Appropriate Level of
Safety Inventory
– Fraction of demand that is satisfied from available inventory
– Equivalent to the probability that product demanded is satisfied from available
inventory
• Suppose we define expected shortage per replenishment cycle (ESC) as
the average units of demand per replenishment cycle that are not satisfied
from inventory
• Then, product fill rate is expressed as below using order lot size Q
𝑓𝑟 = 1 −
𝐸𝑆𝐶 𝑄 − 𝐸𝑆𝐶
=
𝑄
𝑄
(12.7)
• Let f(x) be the density function of the demand distribution, then
𝐸𝑆𝐶 =
∞
𝑥=𝑅𝑂𝑃
𝑥 − 𝑅𝑂𝑃 𝑓 𝑥 𝑑𝑥
(12.8)
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Fill Rate Given a Reorder
Point
• Graphically,
Evaluating Fill Rate Given a Reorder
Point
• If the total demand during the L periods is
– Normally distributed with mean DL and standard deviation σL
∞
• Then, ESC can be written as below
𝑥 − 𝑅𝑂𝑃 𝑓 𝑥 𝑑𝑥
𝑥=𝑅𝑂𝑃
f(x)dx is the probability that xROP occurs
can be explained as below Entire expression is the expected value (average units) of
xROP, which is the demand that are not satisfied from
inventory
Normal distribution of the total
demand with mean DL=DL and
the standard deviation σL=σD√L
This height is probability
density f(x) at x
–
ROP
x
Total demand during
replenishment cycle
of L weeks
𝐸𝑆𝐶 = −𝑠𝑠 1 − 𝐹𝑆
𝐸𝑆𝐶 = −𝑠𝑠 1 − 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇
𝑠𝑠
𝜎𝐿
+ 𝜎𝐿 𝑓𝑆
𝑠𝑠
, 0, 1, 1
𝜎𝐿
𝑠𝑠
𝜎𝐿
+ 𝜎𝐿 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇
(12.9)
𝑠𝑠
, 0, 1, 0
𝜎𝐿
(12.10)
where FS() is the standard normal cumulative distribution function and fS() is the
standard normal density function
• If weekly demand is independent, then from Equation (12.2)
𝐷𝐿 = 𝐷 × 𝐿
𝜎𝐿 = 𝐿𝜎𝐷
xROP is the demand that cannot be satisfied
during replenishment cycle of L weeks
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
9
6/16/2019
Evaluating Fill Rate Given a Reorder
Point
•
•
Work on Example 124 in pages 321322 of the textbook
124
Evaluating Fill Rate Given a Reorder
Point
Work on Example 124 in pages 321322 of the textbook
124
Reorder Point
122,
•
•
•
•
•
•
•
•
•
Reorder Point
122,
•
•
•
•
•
•
•
•
•
What is D?
What is σD?
What is L?
What is Q?
What is ROP?
What is DL?
What is σL?
What is safety inventory ss?
What is ESC?
• What is fr?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
𝐷=
What is D?
𝜎𝐷 =
What is σD?
𝐿
=
What is L?
𝑄=
What is Q?
𝑅𝑂𝑃
=
What is ROP?
𝐷𝐿 = 𝐷 × 𝐿
What is DL?
𝜎𝐿 = 𝐿 × 𝜎𝐷
What is σL?
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 = 𝑅𝑂𝑃 − 𝐷 × 𝐿
What is safety inventory ss?
𝐸𝑆𝐶 = −𝑠𝑠 1 − 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/𝜎𝐿 , 0, 1, 1 + 𝜎𝐿 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇
What is ESC?
𝑠𝑠/𝜎𝐿 , 0, 1, 0
• What is fr?
𝑓𝑟 = (𝑄 − 𝐸𝑆𝐶)/𝑄
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Evaluating Fill Rate Given a Reorder
Point
•
Work on Example 124 in pages 321322 of the textbook
124
Reorder Point
122,
•
•
•
•
•
•
•
•
•
𝐷 = 2500
What is D?
𝜎𝐷 = 500
What is σD?
𝐿
=2
What is L?
𝑄 = 10000
What is Q?
𝑅𝑂𝑃
= 6000
What is ROP?
𝐷𝐿 = 𝐷 × 𝐿 = 2500 × 2 = 5000
What is DL?
𝜎𝐿 = 𝐿 × 𝜎𝐷 = 2 × 500 = 707
What is σL?
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 = 𝑅𝑂𝑃 − 𝐷 × 𝐿 = 6000 − 5000 = 1000
What is safety inventory ss?
𝐸𝑆𝐶 = −𝑠𝑠 1 − 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/𝜎𝐿 , 0, 1, 1 + 𝜎𝐿 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/𝜎𝐿 , 0, 1, 0
What is ESC?
1000
1000
•
What is fr?
= −1000 1 − 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇
707
, 0, 1, 1
+ 707 × 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇
707
, 0, 1, 0 = 25
𝑓𝑟 = (𝑄 − 𝐸𝑆𝐶)/𝑄
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Fill Rate Given a Replenishment Policy
•
• Goal
Work on Example 124 in pages 321322 of the textbook
124
Evaluating Safety Inventory Given
Desired Fill Rate
Reorder Point
– For continuous review replenishment policy,
– Identify appropriate ROP and safety inventory that
achieves the desired fill rate (fr)
122,
• Assumption
•
•
•
•
•
•
•
•
•
𝐷 = 2500
What is D?
= 1000 × (10.9214) + 707 × 0.1467
𝜎𝐷 = 500
What is σD?
= 1000 × 0.0786 + 707 × 0.1467
𝐿=2
What is L?
= 78.6 + 103.7
𝑄 = 10000
= 25.1
What is Q?
𝑅𝑂𝑃 = 6000
What is ROP?
𝐷
=
𝐷
×
𝐿
=
2500
×
2
=
5000
𝐿
What is DL?
𝜎𝐿 = 𝐿 × 𝜎𝐷 = 2 × 500 = 707
What is σL?
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 = 𝑅𝑂𝑃 − 𝐷 × 𝐿 = 6000 − 5000 = 1000
What is safety inventory ss?
𝐸𝑆𝐶 = −𝑠𝑠 1 − 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/𝜎𝐿 , 0, 1, 1 + 𝜎𝐿 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/𝜎𝐿 , 0, 1, 0
What is ESC?
1000
1000
•
What is fr?
= −1000 1 − 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇
, 0, 1, 1 + 707 × 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇
707
𝑓𝑟 = (𝑄 − 𝐸𝑆𝐶)/𝑄 = (10000 − 25)/10000 = 0.9975
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
707
– L: lead time
– Q: replenishment lot size
– fr: desired fill rate
– DL: mean demand during lead time
– σL: standard deviation of demand during lead time
, 0, 1, 0 = 25
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
10
6/16/2019
Evaluating Safety Inventory Given
Desired Fill Rate
•
•
First, from definition of fr in Equation (12.7), ESC can be written as
with the solution in the textbook
125
𝐸𝑆𝐶 = 1 − 𝑓𝑟 𝑄
•
•
Evaluating Safety Inventory Given
Desired Fill Rate
Work on Example 125 in pages 322323 of the textbook and check your solution
Then, calculate required ESC for given Q and desired fr
Then, find safety inventory ss that solves:
– Equation (12.9) or
– Excel equivalent Equation (12.10) for the required ESC
𝐸𝑆𝐶 = −𝑠𝑠 1 − 𝐹𝑆
𝑠𝑠
𝜎𝐿
+ 𝜎𝐿 𝑓𝑆
𝑠𝑠
𝜎𝐿
𝑠𝑠
𝐸𝑆𝐶 = −𝑠𝑠 1 − 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇
, 0, 1, 1
𝜎𝐿
•
•
(12.9)
𝑠𝑠
+ 𝜎𝐿 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇
, 0, 1, 0
𝜎𝐿
(12.10)
Unfortunately, there is no closedform solution for finding ss for the given ESC and
σL
Thus, we need to try different values of ss using Excel until we find ss that satisfies
Equation (12.10)
– Use “GOALSEEK” (simplified version of Solver) as illustrated in Example 125
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
•
Evaluating Safety Inventory Given
Desired Fill Rate
Work on Example 125 in pages 322323 of the textbook and check your solution
with the solution in the textbook
•
•
•
•
•
•
•
•
•
•
What is D?
What is σD?
What is L?
What is Q?
What is DL?
What is σL?
What is fr?
What is ESC?
What is ss?
What is ROP?
•
Evaluating Safety Inventory Given
Desired Fill Rate
Work on Example 125 in pages 322323 of the textbook and check your solution
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
125
•
•
•
•
•
•
•
•
•
•
What is D?
What is σD?
What is L?
What is Q?
What is DL?
What is σL?
What is fr?
What is ESC?
What is ss?
What is ROP?
with the solution in the textbook
125
𝐷=
𝜎𝐷 =
𝐿=
𝑄=
𝐷𝐿 = 𝐷 × 𝐿
𝜎𝐿 = 𝐿 × 𝜎𝐷
𝑓𝑟 =
𝐸𝑆𝐶
𝐸𝑆𝐶 = (1 − 𝑓𝑟) × 𝑄
𝑄
𝐸𝑆𝐶 = −𝑠𝑠 1 − 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/𝜎𝐿 , 0, 1, 1 + 𝜎𝐿 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/𝜎𝐿 , 0, 1, 0
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 𝑅𝑂𝑃 = 𝑠𝑠 + 𝐷𝐿
𝑓𝑟 = 1 −
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
•
•
•
•
•
•
•
•
•
•
What is D?
What is σD?
What is L?
What is Q?
What is DL?
What is σL?
What is fr?
What is ESC?
What is ss?
What is ROP?
•
Evaluating Safety Inventory Given
Desired Fill Rate
Work on Example 125 in pages 322323 of the textbook and check your solution
𝐷 = 2500
𝜎𝐷 = 500
𝐿=2
𝑄 = 1000
𝐷𝐿 = 𝐷 × 𝐿 = 2500 × 2 = 5000
𝜎𝐿 = 𝐿 × 𝜎𝐷 = 2 × 500 = 707
𝑓𝑟 = 0.975
𝐸𝑆𝐶
𝐸𝑆𝐶 = (1 − 𝑓𝑟) × 𝑄 = (1 − 0.975) × 10000 = 250
𝑄
𝐸𝑆𝐶 = −𝑠𝑠 1 − 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/𝜎𝐿 , 0, 1, 1 + 𝜎𝐿 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/𝜎𝐿 , 0, 1, 0
250
=
−𝑠𝑠
×
1
−
𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/707, 0, 1, 1 + 707 × 𝑁𝑂𝑅𝑀𝐷𝐼𝑆𝑇 𝑠𝑠/707,0, 1, 0
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
𝑓𝑟 = 1 −
with the solution in the textbook
125
NORMDIST(D3/B3, 0, 1, 1)
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
•
•
•
•
•
•
•
•
•
•
What is D?
What is σD?
What is L?
What is Q?
What is DL?
What is σL?
What is fr?
What is ESC?
What is ss?
What is ROP?
𝐷 = 2500
𝜎𝐷 = 500
𝐿=2
𝑄 = 1000
𝐷𝐿 = 𝐷 × 𝐿 = 2500 × 2 = 5000
𝜎𝐿 = 𝐿 × 𝜎𝐷 = 2 × 500 = 707
𝑓𝑟 = 0.975
𝐸𝑆𝐶
𝑓𝑟 = 1 −
𝐸𝑆𝐶 = (1 − 𝑓𝑟) × 𝑄 = (1 − 0.975) × 10000 = 250
𝑄
𝑠𝑠 = 67
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 𝑅𝑂𝑃 = 𝑠𝑠 + 𝐷𝐿 = 67 + 5000 = 5067
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
11
6/16/2019
Impact of Desired Product Availability
and Uncertainty on Safety Inventory
• Two key factors that affect the required level
of safety inventory are
– Desired level of product availability and
– Uncertainty
Impact of Product Availability on
Safety Inventory
• As the desired product availability increases
– Required safety inventory ss also increases
because
– Supply chain needs to be able to accommodate
unusual high demand or low supply
• Thus, it is important to
– Be aware of the products that require high level of
product availability and
– Hold high safety inventories only for those
products
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Uncertainty on Safety
Inventory
• From Equation (12.6), we observe that safety inventory ss increases
as the standard deviation of demand during lead time σL increases
𝑠𝑠 =
𝐹𝑆−1
𝐶𝑆𝐿 × 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿
(12.6)
• Standard deviation of demand during lead time σL is influenced by
lead time L and standard deviation of periodic demand σD as shown
in Equation (12.2)
𝜎𝐿 = 𝐿𝜎𝐷
(12.2)
Approaches to Reduce Safety
Inventory
• Based on above discussions, two managerial
levers to reduce safety inventory without
adversely affecting product availability are
– Reduce supplier lead time L
– Reduce underlying uncertainty of demand
(represented by σD)
• From both Equations (12.2) and (12.6), we observe that, if demand
is independent over time, safety inventory ss increases
– Linearly with σD and
– Proportional to square root of L, 𝐿
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Approaches to Reduce Safety
Inventory
• Reduce supplier lead time
– If lead time L decreases by a factor of k, safety inventory
reduces by a factor of 𝑘
– Reduction of supplier lead time requires significant effort from
the supplier; thus:
• It is important for the retailer to share some of the resulting benefits
with the suppliers
Impact of Supply Uncertainty on
Safety Inventory
• Reduce underlying uncertainty of demand (represented by
σD)
– If σD decreases by a factor of k, safety inventory reduces by a
factor of k
– Reduction of σD may be achieved by:
• Better market intelligence and
• Use of more sophisticated forecasting methods
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
12
6/16/2019
Impact of Supply Uncertainty on Safety
Inventory
• So far, uncertainty discussed is the demand
uncertainty σD due to forecast error
– Lead time L was fixed, i.e., certain
– However, lead time L can also be uncertain
Impact of Supply Uncertainty on Safety
Inventory
• Given
– Continuous review replenishment policy
• Assumption
–
–
–
–
D: average demand per period
σD: standard deviation of demand per period
L: average lead time for replenishment
sL: standard deviation of lead time
• Demand during the lead time is
– Normally distributed with
– Mean DL and
– Standard deviation σL
in which
• Now, we consider the case in which the lead
time L is uncertain
𝐷𝐿 = 𝐷𝐿
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
𝜎𝐿 =
𝐿𝜎𝐷2 + 𝐷2 𝑠𝐿2
(12.3)
• Given distribution of demand in Equation (12.3) and desired CSL, we
obtain required safety inventory using Equation (12.6)
• Given distribution of demand in Equation (12.3) and desired fill rate, we
obtain required safety inventory using procedure outlined in Example 125
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
GOALSEEK
Impact of Supply Uncertainty on Safety
Inventory
Impact of Supply Uncertainty on Safety
Inventory
•
•
Work on Example 127 in page 327 of the textbook and check your solution with
the solution in the textbook
127
• What is D?
• What is σD?
• What is L?
• What is sL?
• What is DL?
• What is σL?
• What is CSL?
• What is ss?
• What is ROP?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Work on Example 127 in page 327 of the textbook and check your solution with
the solution in the textbook
127
• What is D?
𝐷=
𝜎𝐷 =
• What is σD?
𝐿=
• What is L?
𝑠𝐿 =
• What is sL?
𝐷𝐿 = 𝐷 × 𝐿
• What is DL?
𝜎𝐿 = 𝐿(𝜎𝐷 )2+𝐷2(𝑠𝐿 )2
• What is σL?
𝐶𝑆𝐿 =
• What is CSL?
𝑠𝑠
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿
• What is ss?
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 𝑅𝑂𝑃 = 𝑠𝑠 + 𝐷𝐿
• What is ROP?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Supply Uncertainty on Safety
Inventory
Impact of Supply Uncertainty on Safety
Inventory
•
•
Work on Example 127 in page 327 of the textbook and check your solution with
the solution in the textbook
127
• What is D?
𝐷 = 2500
𝜎𝐷 = 500
• What is σD?
𝐿=7
• What is L?
𝑠𝐿 = 7
• What is sL?
𝐷𝐿 = 𝐷 × 𝐿 = 2500 × 7 = 17500
• What is DL?
𝜎𝐿 = 𝐿(𝜎𝐷 )2+𝐷2(𝑠𝐿 )2 = 7 × (500)2+25002 × (7)2= 308000000 = 17550
• What is σL?
𝐶𝑆𝐿 = 0.90
• What is CSL?
1.28155
𝑠𝑠
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 0.90 × 17550 = 22491
• What is ss?
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 𝑅𝑂𝑃 = 𝑠𝑠 + 𝐷𝐿 = 22491 + 17500 = 39991
•
What
is
ROP?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Work on Example 127 in page 327 of the textbook and check your solution with
the solution in the textbook
127
Now suppose sL = 0
• What is D?
𝐷 = 2500
𝜎𝐷 = 500
• What is σD?
𝐿=7
• What is L?
• What is sL?
𝑠𝐿 = 7 → 0
𝐷𝐿 = 𝐷 × 𝐿 = 2500 × 7 = 17500
• What is DL?
𝜎𝐿 = 𝐿(𝜎𝐷 )2+𝐷2(𝑠𝐿 )2= 7 × (500)2+25002 × (0)2= 1750000 = 1323
• What is σL?
𝐶𝑆𝐿 = 0.90
• What is CSL?
1.28155
𝑠𝑠
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 0.90 × 1323 = 1695
• What is ss?
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿 𝑅𝑂𝑃 = 𝑠𝑠 + 𝐷𝐿 = 1695 + 17500 = 19195
•
What
is
ROP?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
13
6/16/2019
Impact of Replenishment Policies on
Safety Inventory
• In this section, we discuss that
Impact of Replenishment Policies
on Safety Inventory
– Periodicreview replenishment policies require
more safety inventory than continuousreview
replenishment policies
for the same level of product availability
• CSL is used as the measure of product
availability
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Continuous Review Policies
Continuous Review Policies
• Goal
• Continuous review
– Inventory is continuously tracked
– Order of lot size Q is placed when the inventory
declines to the reorder point (ROP)
– Size of order does not change
– Time between order changes
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
– For continuous review replenishment policy,
– Identify appropriate reorder point (ROP) and safety inventory
(ss) that achieves the desired service level
• ROP represents the available inventory to meet demand during the
lead time for replenishment
• Assumption
–
–
–
–
–
–
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Continuous Review Policies
Continuous Review Policies
• Recall that if demand during each of L periods is
–
–
–
–
• Recall from Equation (12.4)
Independent and
Normally distributed with
Mean D and
Standard deviation σD
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷𝐿
or
𝑠𝑠 = 𝑅𝑂𝑃 − 𝐷 × 𝐿
(12.4)
Thus,
• Then, the total demand during the L periods is
𝑅𝑂𝑃 = 𝐷𝐿 + 𝑠𝑠
– Normally distributed with
– Mean DL and
– Standard deviation σL
and from Equation (12.6)
𝑠𝑠 = 𝐹𝑆−1 𝐶𝑆𝐿 × 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿
in which
𝐷𝐿 = 𝐷 × 𝐿
D: average demand per period
σD: standard deviation of demand per period
DL: average demand during lead time
σL: standard deviation of demand during lead time
L: average lead time for replenishment
CSL: desired cycle service level
𝜎𝐿 = 𝐿𝜎𝐷
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(12.2)
(12.6)
• Managers using a continuous review policy have to account only for the
uncertainty of demand during the lead time as illustrated by σL in Equation
(12.6)
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
14
6/16/2019
Periodic Review Policies
Periodic Review Policies
• Goal
• Periodic review
– For periodic review replenishment policy,
– Identify appropriate safety inventory (ss) that achieves the
desired service level
– Inventory status is checked at regular periodic
interval T
– Order is placed to raise the inventory level to a
prespecified level called orderupto level (OUL)
– Size of order changes
– Time between order does not change
• Assumption
–
–
–
–
–
–
–
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Periodic Review Policies
• Inventory profile of the periodic review policy is
illustrated below for the case of
– Lead time L=4 and
– Reorder interval T=7
① Replenishment
lot of size DT is
ordered at review
point 1 (day 7)
•
③ Inventory ④ Inventory
increases by DT continues to decrease
until next review
when
replenishment point after T=7 days
from the previous
lot arrives
review point
Periodic Review Policies
⑤
Replenishment
lot of size D’T is
ordered at
review point 2
(day 14)
• Observe that that stockout does not happen as far as
demand during time interval of T+L is smaller than OUL
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑙𝑜𝑡 𝑠𝑖𝑧𝑒 𝑄 = 𝐷𝑇 = 𝐷 × 𝑇
•
14
7
T+L
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Periodic Review Policies
Periodic Review Policies
•
•
As illustrated in the previous slide, stockout does not happen as far as demand
during time period of T+L is smaller than OUL; Thus
Periodic review policy is illustrated in Example 1213
Work on Example 1213 in page 343 of the textbook and check your solution with
the solution in the textbook
1213
𝑃𝑟𝑜𝑏 𝑑𝑒𝑚𝑎𝑛𝑑 𝑑𝑢𝑟𝑖𝑛𝑔 𝑇 + 𝐿 ≤ 𝑂𝑈𝐿 = 𝐶𝑆𝐿
•
(12.20)
OUL
② Inventory
continues to
decrease until
replenishment lot
arrives after L=4
days from the day
replenishment
order is placed
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
•
D: average demand per period
σD: standard deviation of demand per period
DL: average demand during lead time
σL: standard deviation of demand during lead time
L: average lead time for replenishment
T: review interval
CSL: desired cycle service level
Then, using Equation (12.2), the total demand during the T+L periods is
– Normally distributed with
– Mean DT+L and
– Standard deviation σT+L
in which
𝐷𝑇+𝐿 = (𝑇 + 𝐿)𝐷
•
From Equation (12.4)
•
Given the desired CSL,
If lead time is uncertain
with standard deviation sL
𝜎𝑇+𝐿 = 𝑇 + 𝐿𝜎𝐷
𝜎𝑇+𝐿 =
𝑂𝑈𝐿 = 𝐷𝑇+𝐿 + 𝑠𝑠
𝑠𝑠 = 𝐹𝑆−1 𝐶𝑆𝐿 × 𝜎𝑇+𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝑇+𝐿
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
𝑇 + 𝐿 𝜎𝐷
(12.18)
(12.19)
2
+ 𝐷 2 𝑠𝐿
2
•
•
•
•
•
•
•
•
•
What is D?
What is σD?
What is L?
What is T?
What is DT+L?
What is σT+L?
What is CSL?
What is ss?
What is OUL?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
15
6/16/2019
Periodic Review Policies
•
•
Periodic Review Policies
Periodic review policy is illustrated in Example 1213
Work on Example 1213 in page 343 of the textbook and check your solution with
the solution in the textbook
•
•
Periodic review policy is illustrated in Example 1213
Work on Example 1213 in page 343 of the textbook and check your solution with
the solution in the textbook
1213
•
•
•
•
•
•
•
•
•
What is D?
What is σD?
What is L?
What is T?
What is DT+L?
What is σT+L?
What is CSL?
What is ss?
What is OUL?
1213
•
•
•
•
•
•
•
•
•
𝐷=
𝜎𝐷 =
𝐿=
𝑇=
𝐷𝑇+𝐿 = 𝐷 × (𝑇 + 𝐿)
𝜎𝑇+𝐿 = 𝑇 + 𝐿 × 𝜎𝐷
𝐶𝑆𝐿 =
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝑇+𝐿
𝑂𝑈𝐿 = 𝐷𝑇+𝐿 + 𝑠𝑠
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
What is D?
What is σD?
What is L?
What is T?
What is DT+L?
What is σT+L?
What is CSL?
What is ss?
What is OUL?
𝐷 = 2500
𝜎𝐷 = 500
𝐿=2
𝑇=4
𝐷𝑇+𝐿 = 𝐷 × (𝑇 + 𝐿) = 2500 × 6 = 15000
𝜎𝑇+𝐿 = 𝑇 + 𝐿 × 𝜎𝐷 = 6 × 500 = 1225
𝐶𝑆𝐿 = 0.90
1.28155
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝑇+𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 0.90 × 1225 = 1570
𝑂𝑈𝐿 = 𝐷𝑇+𝐿 + 𝑠𝑠 = 15000 + 1570 = 16570
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Continuous vs. Periodic Review Policies
• With continuous review policy, safety inventory is used to cover
demand uncertainty over lead time L
𝑠𝑠 = 𝐹𝑆−1 𝐶𝑆𝐿 × 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿
(12.6)
• With periodic review policy, safety inventory is used to cover
demand uncertainty over review interval and lead time T+L
𝑠𝑠 = 𝐹𝑆−1 𝐶𝑆𝐿 × 𝜎𝑇+𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝑇+𝐿
Impact of Aggregation on Safety
Inventory
(12.19)
• Comparing above two ss, ss in continuous review policy is smaller
than ss in periodic review policy because
𝜎𝐿 < 𝜎𝑇+𝐿
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Supply chain may have different degrees of
inventory aggregation; for example,
– Barnes & Noble sells books at retail stores with
inventory geographically distributed nationwide
– Amazon ships books from few facilities
• Goal of this section is to understand how
aggregation affects
– Forecast accuracy and
– Safety inventory
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Compare safety inventories in the following two ways to serve
demand in the k regions
1.
2.
Have local inventories in each region
Aggregate all inventories into one centralized facility
• Consider the following conditions
–
–
–
–
–
–
–
–
k: number of regions
Di: mean weekly demand in region i, i = 1, …, k
σi: standard deviation of weekly demand in region i, i = 1, …, k
ρij: correlation coefficient of weekly demand between We assume demands
are independent
regions i and j, 1≤i≠j≤k
ρij = 0
L: replenishment lead time
CSL: cycle service level
DL: mean demand during lead time
σL: standard deviation of demand during lead time
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
16
6/16/2019
Impact of Aggregation on Safety
Inventory
Impact of Aggregation on Safety
Inventory
• Based on the conditions in the previous slide and from Equation
(12.2) and Equation (12.6) below, we know that
𝜎𝐿 = 𝐿𝜎𝐷
𝑠𝑠 = 𝐹𝑆−1 𝐶𝑆𝐿 × 𝜎𝐿
(12.2)
(12.6)
𝑠𝑠 = 𝐹𝑆−1 𝐶𝑆𝐿 × 𝐿𝜎𝐷
(12.6)
• If all inventories are aggregated in a central location, distribution of aggregate
demand needs to be analyzed
• Aggregate demand in each period is normally distributed with mean DC,
standard deviation σCD, and variance var(DC) below
• If demand at each of k regions is:
–
–
–
–
Thus,
• If inventories are decentralized,
𝑘
𝐷𝐶 =
𝑇𝑜𝑡𝑎𝑙 𝑠𝑎𝑓𝑒𝑡𝑦 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑜𝑓 𝑘 𝑟𝑒𝑔𝑖𝑜𝑛𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑑𝑒𝑐𝑒𝑛𝑡𝑟𝑎𝑙𝑖𝑧𝑒𝑑 𝑜𝑝𝑡𝑖𝑜𝑛
𝑘
𝐹𝑆−1 𝐶𝑆𝐿 × 𝐿𝜎𝑖 = 𝐹𝑆−1 𝐶𝑆𝐿 × 𝐿
𝑖=1
(12.12)
•
= 𝑘 × 𝑠𝑠 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑟𝑒𝑔𝑖𝑜𝑛 = 𝑘 × 𝐹𝑆−1 𝐶𝑆𝐿 × 𝐿𝜎𝐷
= 𝑘 × 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝐿𝜎𝐷
𝜎𝑖2 + 2
𝑖=1
𝜌𝑖𝑗 𝜎𝑖 𝜎𝑗
𝜎𝐷𝐶 =
𝑣𝑎𝑟 𝐷𝐶
(12.13)
𝑖>𝑗
If demands in k regions are independent (ρij=0) and identically distributed with mean
Di=D, standard deviation σi=σ, then
If demands at all regions are identical
𝐷𝐶 = 𝑘𝐷
𝜎𝐷𝐶 = 𝑘𝜎𝐷
𝜎2
2
+ ⋯ + 𝜎𝑘
k terms
𝐼𝑓 𝐷1 = 𝐷2 =…=𝐷𝑘
𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑎𝑙𝑙 𝐷
𝐼𝑓 𝜎1 = 𝜎2 =…=𝜎𝑘
𝑤ℎ𝑖𝑐ℎ 𝑎𝑟𝑒 𝑎𝑙𝑙 𝜎𝐷
𝜎𝐿 =
𝜎𝐷
2
+ 𝜎𝐷
2+
⋯ + 𝜎𝐷
2
• If H is the holding cost per unit, using Equation
(12.12) and Equation (12.16), the savings per
units are:
(12.15)
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 𝑠𝑎𝑣𝑖𝑛𝑔𝑠 𝑜𝑛 𝑎𝑔𝑔𝑟𝑒𝑔𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑠𝑜𝑙𝑑
𝐹𝑆−1 𝐶𝑆𝐿 × 𝐿 × 𝐻 × 𝑘𝑖=1 𝜎𝑖 − 𝜎𝐷𝐶
=
(12.17)
𝐷𝐶
𝑇𝑜𝑡𝑎𝑙 𝑠𝑎𝑓𝑒𝑡𝑦 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑖𝑛 𝑎𝑔𝑔𝑟𝑒𝑔𝑎𝑡𝑖𝑜𝑛 𝑜𝑝𝑡𝑖𝑜𝑛 𝑠𝑠 = 𝐹𝑆−1 𝐶𝑆𝐿 × 𝐿 × 𝜎𝐷𝐶 (12.16)
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝐿 × 𝜎𝐷𝐶
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example
129
• Work on Example 129 in page 332 of the textbook and
check your solution with the solution in the textbook
•
= 𝑘 𝜎𝐷 2
= 𝑘 × 𝜎𝐷
– Dividing the savings in holding cost by the total
demand kD
• Safety inventory (take into account lead time L period)
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
2
• Holdingcost savings on aggregation per unit sold
are obtained by
𝑘
𝑣𝑎𝑟 𝐷𝐶 =
2+
Impact of Aggregation on Safety
Inventory
• If all inventories are aggregated in a central location, distribution of aggregate
demand needs to be analyzed
• Aggregate demand in each period is normally distributed with mean DC,
standard deviation σCD, and variance var(DC) below
𝐷𝑖
𝜎1
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
𝑖=1
𝜎𝑖2 =
𝑖=1
𝐷𝐶 = 𝐷 + 𝐷 + ⋯ + 𝐷 = 𝑘𝐷
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
𝐷𝐶 =
𝜎𝐷𝐶 =
k terms
𝜎𝑖
𝑖=1
𝑘
If demands at different
regions are not identical
𝑘
𝐷𝑖 = 𝐷1 + 𝐷2 + ⋯ + 𝐷𝑘
𝑖=1
𝑘
=
Independent and
Normally distributed with
Mean Di and
where i=1, 2, …, k
Standard deviation σi
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example
129
• Work on Example 129 in pages 333334 of the textbook
and check your solution with the solution in the textbook
52 weeks per year
• What costs do we need to we calculate?
•
•
52 weeks per year
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
17
6/16/2019
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example
129
• Work on Example 129 in pages 333334 of the textbook
and check your solution with the solution in the textbook
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
52 weeks per year
• What information/number do we have?
• What costs do we need to we calculate?
– Annual inventory holding cost
– Annual transportation cost
– Annual facility cost
•
•
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
• What information/number do we have?
–
–
–
–
–
–
–
–
–
–
𝐷 = 1000
𝜎𝐷 = 300
𝐿=4
ℎ = 0.2
𝐶 = 1000
𝑡𝑟𝑒𝑔𝑖𝑜𝑛𝑎𝑙 = 10 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡
𝑡𝑛𝑎𝑡𝑖𝑜𝑛𝑎𝑙 = 13 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡
𝑓𝑟𝑒𝑔𝑖𝑜𝑛𝑎𝑙 = 150000
𝐶𝑆𝐿 = 0.95
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
–
–
–
–
–
–
–
–
–
–
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
• What information/number do we have?
–
–
–
–
–
–
–
–
–
–
52 weeks per year
Calculate annual inventory
holding cost for:
Number of region k = 4
 Regional distribution center
Weekly demand per region D = 1000
 National distribution center
Standard deviation of weekly demand σD = 300
Supply lead time L = 4
Holding cost h = 0.2
Cost C = 1000
Transportation cost for regional distribution center = 10 per unit
Transportation cost for national distribution center = 13 per unit
Facility operation cost = 150000
Product availability CSL = 0.95
• What information/number do we have?
𝑘=4
Number of region k
Weekly demand per region D
Standard deviation of weekly demand σD
Supply lead time L
Holding cost h
Cost C
Transportation cost for regional distribution center
Transportation cost for national distribution center
Facility operation cost
Product availability CSL
Calculate annual inventory
holding cost for:
 Regional distribution center
 National distribution center
Number of region k = 4
Weekly demand per region D = 1000
Standard deviation of weekly demand σD = 300
Supply lead time L = 4
Regional distribution center
Holding cost h = 0.2
(1) Calculate safety inventory of each region
Cost C = 1000
𝜎𝐿 = 𝐿 × 𝜎𝐷
Transportation cost for regional𝑠𝑠
distribution
center
= 𝜎10
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉
𝐶𝑆𝐿 ×
𝐿 per unit
Transportation cost for national(2)
distribution
= 13 per unit
Calculate center
safety inventory
cost of all regions
𝑠𝑠
=
Facility operation cost = 150000
Product availability CSL = 0.95 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑐𝑜𝑠𝑡 = 𝑠𝑠 × ℎ × 𝐶
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
Calculate annual inventory
holding cost for:
Number of region k = 4
 Regional distribution center
Weekly demand per region D = 1000
 National distribution center
Standard deviation of weekly demand σD = 300
Supply lead time L = 4
Regional distribution center
Holding cost h = 0.2
(1) Calculate safety inventory of each region
Cost C = 1000
𝜎𝐿 = 𝐿 × 𝜎𝐷 = 4 × 300 = 600
Transportation cost for regional𝑠𝑠
distribution
center
= 𝜎10
unit 0.95 × 600 = 986.9
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉
𝐶𝑆𝐿 ×
𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉
𝐿 =per
Transportation
cost
for national(2)
distribution
= 13 per unit
Calculate center
safety inventory
cost of all regions
𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 0.95
= 1.64485
𝑠𝑠
=
986.9
×
4
=
3947.6
Facility operation cost = 150000
Product availability CSL = 0.95 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑐𝑜𝑠𝑡 = 𝑠𝑠 × ℎ × 𝐶 = 3947.6 × 0.2 × 1000 = 789530
• What information/number do we have?
–
–
–
–
–
–
–
–
–
–
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
18
6/16/2019
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
Calculate annual inventory
holding cost for:
Number of region k = 4
 Regional distribution center
Weekly demand per region D = 1000
 National distribution center
Standard deviation of weekly demand σD = 300
Supply lead time L = 4
National distribution center
Holding cost h = 0.2
(1) Calculate aggregate safety inventory of all regions
Cost C = 1000
𝜎𝐷𝐶 = 𝑘 × 𝜎𝐷
Transportation cost for regional𝑠𝑠
distribution
center
= 𝜎10
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉
𝐶𝑆𝐿 ×
𝐿 per unit
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉center
0.95 × = 𝐿13
× per
𝜎𝐷𝐶 unit
Transportation cost for national distribution
Facility operation cost = 150000(2) Calculate safety inventory cost of all regions
Product availability CSL = 0.95 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑐𝑜𝑠𝑡 = 𝑠𝑠 × ℎ × 𝐶
• What information/number do we have?
–
–
–
–
–
–
–
–
–
–
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
–
–
–
–
–
–
–
–
–
–
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
Calculate annual
transportation cost for:
Number of region k = 4
 Regional distribution center
Weekly demand per region D = 1000
 National distribution center
Standard deviation of weekly demand σD = 300
Supply lead time L = 4
Holding cost h = 0.2
Cost C = 1000
Transportation cost for regional distribution center = 10 per unit
Transportation cost for national distribution center = 13 per unit
Facility operation cost = 150000
Product availability CSL = 0.95
• What information/number do we have?
–
–
–
–
–
–
–
–
–
–
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
• What information/number do we have?
– Number of region k = 4
– Weekly demand per region D = 1000
52 weeks per year
• What information/number do we have?
– Number of region k = 4
– Weekly demand per region D = 1000
Calculate annual
transportation cost for:
 Regional distribution center
 National distribution center
– Transportation cost for regional distribution center = 10 per unit
– Transportation cost for national distribution center = 13 per unit
– Facility operation cost = 150000
Regional distribution center transportation cost
– Product availability CSL = 0.95
𝐴𝑛𝑛𝑢𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 × 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑡 = 4 × 1000 × 52 × 10 = 2080000
National distribution center transportation cost
–
•
𝐴𝑛𝑛𝑢𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 × 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑡 = 4 × 1000 × 52 × 13 = 2704000
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Calculate annual
transportation cost for:
 Regional distribution center
 National distribution center
– Transportation cost for regional distribution center = 10 per unit
– Transportation cost for national distribution center = 13 per unit
– Facility operation cost = 150000
Regional distribution center transportation cost
– Product availability CSL = 0.95
𝐴𝑛𝑛𝑢𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 × 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑡
National distribution center transportation cost
–
𝐴𝑛𝑛𝑢𝑎𝑙 𝑑𝑒𝑚𝑎𝑛𝑑 × 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑡
•
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
Calculate annual inventory
holding cost for:
Number of region k = 4
 Regional distribution center
Weekly demand per region D = 1000
 National distribution center
Standard deviation of weekly demand σD = 300
Supply lead time L = 4
National distribution center
Holding cost h = 0.2
(1) Calculate aggregate safety inventory of all regions
Cost C = 1000
𝜎𝐷𝐶 = 𝑘 × 𝜎𝐷 = 4 × 300 = 600
Transportation cost for regional𝑠𝑠
distribution
center
= 𝜎10
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉
𝐶𝑆𝐿 ×
𝐿 per unit
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉center
0.95 × = 𝐿13
× per
𝜎𝐷𝐶 =unit
1.64485 × 2 × 600 = 1973.8
Transportation
cost
for national distribution
𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 0.95
= 1.64485
Facility operation cost = 150000(2) Calculate safety inventory cost of all regions
Product availability CSL = 0.95 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑐𝑜𝑠𝑡 = 𝑠𝑠 × ℎ × 𝐶 = 1973.8 × 0.2 × 1000 = 394765
• What information/number do we have?
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
Calculate facility operation
cost for:
Number of region k = 4
 Regional distribution center
Weekly demand per region D = 1000
 National distribution center
Standard deviation of weekly demand σD = 300
Supply lead time L = 4
Holding cost h = 0.2
Cost C = 1000
Transportation cost for regional distribution center = 10 per unit
Transportation cost for national distribution center = 13 per unit
Facility operation cost = 150000
Product availability CSL = 0.95
• What information/number do we have?
–
–
–
–
–
–
–
–
–
–
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
19
6/16/2019
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
Calculate facility operation
cost for:
– Number of region k = 4
 Regional distribution center
– Weekly demand per region D = 1000
 National distribution center
– Standard deviation of weekly demand σD = 300
–
Supplydistribution
lead time L center
= 4 facility cost
Regional
– Holding cost h = 0.2
𝐴𝑛𝑛𝑢𝑎𝑙 𝑓𝑎𝑐𝑖𝑙𝑖𝑡𝑦 𝑐𝑜𝑠𝑡
–
Cost Cdistribution
= 1000
National
center facility cost
– Transportation cost for regional distribution center = 10 per unit
𝐴𝑛𝑛𝑢𝑎𝑙 𝑓𝑎𝑐𝑖𝑙𝑖𝑡𝑦 𝑐𝑜𝑠𝑡
– Transportation cost for national distribution center = 13 per unit
– Facility operation cost = 150000
– Product availability CSL = 0.95
• What information/number do we have?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
Calculate facility operation
cost for:
– Number of region k = 4
 Regional distribution center
– Weekly demand per region D = 1000
 National distribution center
– Standard deviation of weekly demand σD = 300
–
Supplydistribution
lead time L center
= 4 facility cost
Regional
– Holding cost h = 0.2
𝐴𝑛𝑛𝑢𝑎𝑙 𝑓𝑎𝑐𝑖𝑙𝑖𝑡𝑦 𝑐𝑜𝑠𝑡 = 150000
–
Cost Cdistribution
= 1000
National
center facility cost
– Transportation cost for regional distribution center = 10 per unit
𝐴𝑛𝑛𝑢𝑎𝑙 𝑓𝑎𝑐𝑖𝑙𝑖𝑡𝑦 𝑐𝑜𝑠𝑡 = 0
– Transportation cost for national distribution center = 13 per unit
– Facility operation cost = 150000
– Product availability CSL = 0.95
• What information/number do we have?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
Calculate total cost for:
 Regional distribution center
– Number of region k = 4
 National distribution center
– Weekly demand per region D = 1000
–
Standard
deviationcenter
of weekly
demand
Regional
distribution
facility
cost σD = 300
– Supply
lead
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡
= time L = 4
–
Holding
cost h = 0.2
National
distribution
center facility cost
– Cost
= 1000
𝑇𝑜𝑡𝑎𝑙C𝑐𝑜𝑠𝑡
=
– Transportation cost for regional distribution center = 10 per unit
– Transportation cost for national distribution center = 13 per unit
What do you recommend?
– Facility operation cost = 150000
– Product availability CSL = 0.95
• What information/number do we have?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Impact of Aggregation on Safety
Inventory
• Benefits of demand aggregation is illustrated in Example 129
• Work on Example 129 in pages 333334 of the textbook and check your
solution with the solution in the textbook
52 weeks per year
Calculate total cost for:
 Regional distribution center
– Number of region k = 4
 National distribution center
– Weekly demand per region D = 1000
–
Standard
deviationcenter
of weekly
demand
Regional
distribution
facility
cost σD = 300
– Supply
lead
L =+ 42080000 + 150000 = 3019530
𝑇𝑜𝑡𝑎𝑙 𝑐𝑜𝑠𝑡
= time
789530
$79,235 cost reduction
–
Holding
cost h = 0.2
National
distribution
center facility cost
( = 3,098,765 – 3,019,530)
– Cost
= 1000
𝑇𝑜𝑡𝑎𝑙C𝑐𝑜𝑠𝑡
= 394765 + 2704000 +
0
= 3098765
– Transportation cost for regional distribution center = 10 per unit
– Transportation cost for national distribution center = 13 per unit
What do you recommend?
– Facility operation cost = 150000
– Product availability CSL = 0.95
• What information/number do we have?
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Clarifications: Standard Deviation and
Safety Inventory of Aggregated Demand
•
Standard deviation of aggregated demand
–
–
We assume that demands of different regions (or stores, products, etc.) are independent
First, we calculate “variance of demand in each period at each region” (period is a week if weekly demand is given)
•
Impact of Aggregation on Safety
Inventory
Variance is square of standard deviation 𝜎𝑖
𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑑𝑒𝑚𝑎𝑛𝑑 𝑖𝑛 𝑒𝑎𝑐ℎ 𝑝𝑒𝑟𝑖𝑜𝑑 𝑎𝑡 𝑒𝑎𝑐ℎ 𝑟𝑒𝑔𝑖𝑜𝑛 = (𝜎𝑖 )2 → (𝜎1 )2 𝑎𝑡 𝑟𝑒𝑔𝑖𝑜𝑛 1
𝜎2 2 𝑎𝑡 𝑟𝑒𝑔𝑖𝑜𝑛 2
⋮
(𝜎𝑘 )2 𝑎𝑡 𝑟𝑒𝑔𝑖𝑜𝑛 𝑘
–
Second, we calculate “variance of aggregated demand”, 𝑣𝑎𝑟 𝐷𝐶 , by adding variances of demands for all regions
(assume that there are k regions)
–
Then, “standard deviation of aggregated demand”, 𝜎𝐷𝐶 , is calculated by taking a square root of “variance of
aggregated demand”
𝑣𝑎𝑟 𝐷𝐶 =
𝜎𝐷𝐶 =
•
𝑘
2
𝑖=1(𝜎𝑖 )
𝑣𝑎𝑟 𝐷𝐶 =
= (𝜎1 )2 +(𝜎2 )2 + ⋯ + (𝜎𝑘 )2
𝑘
2
𝑖=1(𝜎𝑖 )
=
This is the formula when demands are independent
(𝜎1 )2 +(𝜎2 )2 + ⋯ + (𝜎𝑘 )2 This is the third formula in Equation (12.13)
Total safety inventory of aggregated demand is calculated using Equation (12.16) which is same as
Equation (12.6) but use 𝜎𝐷𝐶 instead of 𝜎𝐷
𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑠𝑎𝑓𝑒𝑡𝑦 𝑖𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦 𝑜𝑛 𝑎𝑔𝑔𝑟𝑒𝑔𝑎𝑡𝑖𝑜𝑛 𝑠𝑠 = 𝐹𝑆−1 𝐶𝑆𝐿 × 𝐿 × 𝜎𝐷𝐶
This is Equation (12.16)
= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 (𝐶𝑆𝐿) × 𝐿 × 𝜎𝐷𝐶
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
20
6/16/2019
Component Commonality
Component Commonality
• Component commonality
– Use of same components in multiple products
– Effective supply chain strategy to exploit aggregation
and reduce component inventories
• Without component commonality
• Benefits of component commonality is illustrated in
Example 1211
• Work on Example 1211 in pages 337378 of the textbook
and check your solution with the solution in the textbook
•
– Demand for a component is the same as demand for
the finished product in which the component is used
• With component commonality
– Demand for the common component is an
aggregation of the demand for all the finished
products of which the common component is used
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
•
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Component Commonality
Component Commonality
• Benefits of component commonality is illustrated in Example 1211
• Work on Example 1211 in pages 337378 of the textbook and check your
solution with the solution in the textbook
•
• Benefits of component commonality is illustrated in Example 1211
• Work on Example 1211 in pages 337378 of the textbook and check your
solution with the solution in the textbook
•
• 27 servers
• 27 servers
–
–
–
–
Monthly demand per server D
Standard deviation of monthly demand σD
Supply lead time L
Product availability CSL
–
–
–
–
𝐷 = 5000
𝜎𝐷 = 3000
𝐿=1
𝐶𝑆𝐿 = 0.95
• Each server consists of three components (processor, memory unit, hard drive)
• Without commonality (𝑘 = 1)
With commonality (𝑘 = 9)
Each
– Processors
27
3
component
– Memory units
27
3
is used in
– Hard drives
27
3 .
9 servers
–
Total
81
9
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Demand of each
component is
aggregation of 9
servers 𝑘 = 9
Monthly demand per server D
Standard deviation of monthly demand σD
Supply lead time L
Product availability CSL
𝐷 = 5000
𝜎𝐷 = 3000
𝐿=1
𝐶𝑆𝐿 = 0.95
• Safety inventory without commonality (𝑘 = 1)
– Calculate safety inventory of each component
𝜎𝐿 = 𝐿 × 𝜎𝐷
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿
– Multiply by the total number of unique components
𝑠𝑠𝑊𝑖𝑡ℎ𝑜𝑢𝑡𝐶𝑜𝑚𝑚𝑜𝑛𝑎𝑙𝑖𝑡𝑦 = 𝑠𝑠 × 81
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Component Commonality
Component Commonality
• Benefits of component commonality is illustrated in Example 1211
• Work on Example 1211 in pages 337378 of the textbook and check your
solution with the solution in the textbook
•
• Benefits of component commonality is illustrated in Example 1211
• Work on Example 1211 in pages 337378 of the textbook and check your
solution with the solution in the textbook
•
• 27 servers
• 27 servers
–
–
–
–
Monthly demand per server D
Standard deviation of monthly demand σD
Supply lead time L
Product availability CSL
𝐷 = 5000
𝜎𝐷 = 3000
𝐿=1
𝐶𝑆𝐿 = 0.95
• Safety inventory without commonality (𝑘 = 1)
– Calculate safety inventory of each component
–
–
–
–
Monthly demand per server D
Standard deviation of monthly demand σD
Supply lead time L
Product availability CSL
𝜎𝐷𝐶 = 𝑘 × 𝜎𝐷
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿
𝑠𝑠𝑊𝑖𝑡ℎ𝑜𝑢𝑡𝐶𝑜𝑚𝑚𝑜𝑛𝑎𝑙𝑖𝑡𝑦 = 𝑠𝑠 × 81 = 4934.561 × 81 = 399699.4
𝐶𝑆𝐿 = 0.95
– Calculate safety inventory of each component for the aggregated demand
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 0.95 × 3000 = 1.644854 × 3000 = 4934.561
– Multiply by the total number of unique components
𝐿=1
• Safety inventory with commonality (𝑘 = 9)
𝜎𝐿 = 𝐿 × 𝜎𝐷 = 1 × 3000 = 3000
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
𝐷 = 5000
𝜎𝐷 = 3000
𝜎𝐿 = 𝐿 × 𝜎𝐷𝐶
– Multiply by the total number of unique components
𝑠𝑠𝑊𝑖𝑡ℎ𝐶𝑜𝑚𝑚𝑜𝑛𝑎𝑙𝑖𝑡𝑦 = 𝑠𝑠 × 9
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
21
6/16/2019
Component Commonality
• Benefits of component commonality is illustrated in Example 1211
• Work on Example 1211 in pages 337378 of the textbook and check your
solution with the solution in the textbook
•
• Postponement
– Ability of a supply chain to delay product differentiation or
customization
– Until closer to the time the product is sold
– Component commonality is the basis of postponement
• Goal of postponement
• 27 servers
–
–
–
–
Postponement
Monthly demand per server D
Standard deviation of monthly demand σD
Supply lead time L
Product availability CSL
– Have common components in the supply chain
for most of the push phase
– Move product differentiation
as close to the pull phase of
the supply chain as possible
𝐷 = 5000
𝜎𝐷 = 3000
𝐿=1
𝐶𝑆𝐿 = 0.95
• Safety inventory with commonality (𝑘 = 9)
– Calculate safety inventory of each component for the aggregated demand
𝜎𝐿 = 𝐿 × 𝜎𝐷𝐶 = 1 × 9000 = 9000
𝜎𝐷𝐶 = 𝑘 × 𝜎𝐷 = 9 × 3000 = 3 × 3000 = 9000
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 0.95 × 9000 = 1.644854 × 9000 = 14803.68
•
– Multiply by the total number of unique components
𝑠𝑠𝑊𝑖𝑡ℎ𝐶𝑜𝑚𝑚𝑜𝑛𝑎𝑙𝑖𝑡𝑦 = 𝑠𝑠 × 9 = 14803.68 × 9 = 133233.1
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Postponement
Postponement
• Example: Paint
– Final mixing of paint is performed at retail store after the
customer has selected the color
– Instead of at paint factory
• Benefits of postponement is illustrated in Example 1212
• Work on Example 1212 in page 340 of the textbook and
check your solution with the solution in the textbook
•
• Example: Benetton
– Original process
• Thread is dyed
• Then knitted and assembled into garments
– New process
• Greige thread (undyed thread) is knitted and assembled into
garments before dying
• Then garments are dyed after demand is know with great accuracy
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
•
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Postponement
Postponement
• Benefits of postponement is illustrated in Example 1212
• Work on Example 1212 in page 340 of the textbook and check your
solution with the solution in the textbook
•
• Benefits of postponement is illustrated in Example 1212
• Work on Example 1212 in page 340 of the textbook and check your
solution with the solution in the textbook
•
• 100 colors
• 100 colors
–
–
–
–
–
–
–
–
𝐷 = 30
𝜎𝐷 = 10
Weekly demand of each color D
Standard deviation of weekly demand σD
Supply lead time L
Product availability CSL
𝐿=2
𝐶𝑆𝐿 = 0.95
• Each server consists of a base point and a color
• Without postponement (𝑘 = 1) With postponement (𝑘 = 100)
– Base paints
– Total
•
100
100
1
1
Weekly demand of each color D
Standard deviation of weekly demand σD
Supply lead time L
Product availability CSL
𝐷 = 30
𝜎𝐷 = 10
𝐿=2
𝐶𝑆𝐿 = 0.95
• Safety inventory without postponement (𝑘 = 1)
– Calculate safety inventory of each color
𝜎𝐿 = 𝐿 × 𝜎𝐷
Same base
paint is
used in
100 colors
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Demand of the
base paint is
aggregation of 100
colors 𝑘 = 100
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿
– Multiply by the total number of paint colors
𝑠𝑠𝑊𝑖𝑡ℎ𝑜𝑢𝑡𝑃𝑜𝑠𝑡𝑝𝑜𝑛𝑒𝑚𝑒𝑛𝑡 = 𝑠𝑠 × 100
•
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
22
6/16/2019
Postponement
Postponement
• Benefits of postponement is illustrated in Example 1212
• Work on Example 1212 in page 340 of the textbook and check your
solution with the solution in the textbook
•
• Benefits of postponement is illustrated in Example 1212
• Work on Example 1212 in page 340 of the textbook and check your
solution with the solution in the textbook
•
• 100 colors
• 100 colors
–
–
–
–
–
–
–
–
𝐷 = 30
𝜎𝐷 = 10
Weekly demand of each color D
Standard deviation of weekly demand σD
Supply lead time L
Product availability CSL
𝐿=2
𝐶𝑆𝐿 = 0.95
• Safety inventory without postponement (𝑘 = 1)
𝐷 = 30
𝜎𝐷 = 10
Weekly demand of each color D
Standard deviation of weekly demand σD
Supply lead time L
Product availability CSL
𝐿=2
𝐶𝑆𝐿 = 0.95
• Safety inventory with postponement (𝑘 = 100)
– Calculate safety inventory of each color
– Calculate safety inventory of each color for the aggregated demand
𝜎𝐿 = 𝐿 × 𝜎𝐷 = 2 × 10 = 14.14214
𝜎𝐷𝐶 = 𝑘 × 𝜎𝐷
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿= 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 0.95 × 14.14214 = 1.644854 × 14.14214 = 23.26174
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿
– Multiply by the total number of paint colors
– Multiply by the total number of (base) paint
𝑠𝑠𝑊𝑖𝑡ℎ𝑜𝑢𝑡𝑃𝑜𝑠𝑡𝑝𝑜𝑛𝑒𝑚𝑒𝑛𝑡 = 𝑠𝑠 × 100 = 23.26174 × 100 = 2326.174
•
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
𝜎𝐿 = 𝐿 × 𝜎𝐷𝐶
𝑠𝑠𝑊𝑖𝑡ℎ𝑃𝑜𝑠𝑡𝑝𝑜𝑛𝑒𝑚𝑒𝑛𝑡 = 𝑠𝑠 × 1
•
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
Postponement
• Benefits of postponement is illustrated in Example 1212
• Work on Example 1212 in page 340 of the textbook and check your
solution with the solution in the textbook
•
• 100 colors
–
–
–
–
𝐷 = 30
𝜎𝐷 = 10
Weekly demand of each color D
Standard deviation of weekly demand σD
Supply lead time L
Product availability CSL
𝐿=2
𝐶𝑆𝐿 = 0.95
• Safety inventory with postponement (𝑘 = 100)
– Calculate safety inventory of each color for the aggregated demand
𝜎𝐷𝐶 = 𝑘 × 𝜎𝐷 = 100 × 10 = 10 × 10 = 100
𝜎𝐿 = 𝐿 × 𝜎𝐷𝐶 = 2 × 100 = 141.4214
𝑠𝑠 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 𝐶𝑆𝐿 × 𝜎𝐿 = 𝑁𝑂𝑅𝑀𝑆𝐼𝑁𝑉 0.95 × 141.4214 = 1.644854 × 141.4214 = 232.6174
– Multiply by the total number of (base) paint
𝑠𝑠𝑊𝑖𝑡ℎ𝑃𝑜𝑠𝑡𝑝𝑜𝑛𝑒𝑚𝑒𝑛𝑡 = 𝑠𝑠 × 1 = 232.6174 × 1 = 232.6174
•
(Source) Supply Chain Management: Strategy, Planning, and Operation. Sunil Chopra. (2018) 7th Edition. Boston, MA: Pearson Education.
23
Inputs
σD
Q
D
L
10,000
2,500
500
2
Distribution of demand during lead time
DL
σL
5,000
707
Cycle Service Level and Fill Rate
CSL
ESC
fr
0.92
25.13
0.9975
ss
1,000
Input
fr
0,975
Formula
ESC
282,0522
Variable
σL
707
Q
10000
ss
1. Calculate Target ESC from Q and fr as ESC=(1fr)Q
2. Invoke GoalSeek using Data  WhatIfAnalysis  GoalSeek
3. Set Cell to be $A$6 (Expected Shortage per Cycle); set
To value to be Target ESC which is 250 (we want ESC to be 250), and
By changing cell to be $D$3 (we are changing safety inventory)
Input
fr
0,975
Formula
ESC
250
Variable
σL
707
Q
10000
ss
67
1. Calculate Target ESC from Q and fr as ESC=(1fr)Q
2. Invoke GoalSeek using Data  WhatIfAnalysis  GoalSeek
3. Set Cell to be $A$6 (Expected Shortage per Cycle); set
To value to be Target ESC which is 250 (we want ESC to be 250), and
By changing cell to be $D$3 (we are changing safety inventory)
Purchase answer to see full
attachment