1. Supply Management & Model Theory Certified Professional Logistician
This topic is Supply Management and model theory. Inventory management concentrated on the wholesale level of supply. Supply management will concentrate on supply at retail levels of support. Retail supply levels are located closer to the customers, who typically place an order when they need the item. Retail levels of support are characterized with an order size of 1 each for the item when needed. It is typically the wholesale supply level that only waits to order an Economic Order Quantity (EOQ) of the item.
Bernard Price
Certified Professional Logistician

2. Operational Availability
The Probability that the Equipment is Operating or in a Committable Condition to Operate at any Random Point in Time
Quantitative Expression of User Need
Operational Availability (Ao) is the probability that the equipment is operating or in a committable condition when not operating at any random point in calendar time. A specified equipment Ao is a quantitative expression of User need for the system.

3. Derivation of Operational Availability
UP TIME
UP TIME / SYSTEM FAILURE
Ao = =
UP TIME + DOWN TIME
(UP TIME + DOWN TIME) / SYSTEM FAILURE
UP TIME / SYSTEM FAILURE =
MCTBF
SYSTEM
DOWN TIME / SYSTEM FAILURE =
MSRT
SYSTEM
Operational Availability (Ao) represents the percentage of Up Time divided by Total Time. Total time is Up Time plus Down Time.
For computing Ao, the percentage of Up Time divided by Total Time is mathematically the same as the Up Time per system failure divided by the Up Time plus Down Time per system failure. The Up Time per system failure is equal to the system’s Mean Calendar Time Between Failure (MCTBF). MCTBF is a function of the reliability of system and the system’s operating usage time. The Down Time per system failure is equal to the Mean System Restoral Time (MSRT).
MCTBF
- Mean Calendar Time Between System Failure
SYSTEM
MSRT
- Mean System Restoral Time per System Failure
SYSTEM

4. Mean System Restoral Time Breakdown in Models
MSRTSYSTEM = MTRSYSTEM + CWTSYSTEM
MTR - Mean Time to Restore with 100% Stock Availability Forward
CWT- Mean Customer Wait Time at Forward Level per System Failure
MTRSYSTEM = MTTRSYSTEM + MRDTSYSTEM
MTTR - Mean Time to Repair when all Resources with Equipment
MRDT - Mean Restoral Delay Time with Spares Available Forward
The Mean System Restoral Time (MSRT) is further broken down in models. The MSRT is equal to the system’s Mean Time to Restore (MTR) when spares are available at the most forward level of supply support plus the Customer Wait Time (CWT) per system failure when spares are not available forward in stock.
The Mean Time to Restore with 100% Stock Availability forward includes the system’s Mean Time to Repair (MTTR) maintainability when spares, people and appropriate test equipment and tools are with the system plus support efficiency factors that contribute to a restoral delay time when spares are located at the forward supply level.

5. Restoral Delay Time Contributors
Spares are Not Collocated with Equipment
Spares are Delivered Forward to Restore
Contact Maintenance Team Restores Equipment
Equipment is Evacuated to Restore
Some ILS Elements May Not Be Satisfactory
Personnel Lacking Appropriate Skills
Personnel Not Available
Non-Functioning TMDE Forward
Forward Repair Documentation Insufficient
Non-design contributors to the restoral delay time are mentioned on this slide. When spares stocked forward are not collocated with the system, additional time is needed to get spares delivered forward to restore the system, or have a Contact Maintenance Team come out to restore the system, or have the system evacuated so that it can be restored and returned to service.
Other MSRT contributors may be related to some ILS elements not being performed satisfactorily. Lack of appropriate training leads to possibly the personnel lacking appropriate skills. Possibly the personnel doing repair are not available immediately. Another delay time factor may be non-functioning Test, Measurement and Diagnostic Equipment (TMDE) placed forward. If the forward repair manual documentation is insufficient, it also may contribute to the restoral delay time.

6. Relationship of Customer Wait Time (CWT) to Logistics Response Time
CWT = SA1 x 0 + (1 - SA1) x MTTO1
CWT = (1 - SA1 ) x MTTO1
SA1 - Stock Availability at (or Probability of Filling an Order from) Forward Level Stock
MTTO1 - Mean Time to Obtain a Line Replaceable Unit (LRU) Spare at Forward Level Support
The relationship of Customer Wait Time (CWT) per system failure to the Logistics Response Time is shown on this slide. If a failure occurs and the appropriate spare is stocked forward to restore the system, there is no CWT associated with that failure.
However, when a failure occurs and the appropriate spare is not stocked forward, the spare has to be obtained through the logistics chain. Therefore, the forward support customer wait time per system failure is equal to 1 minus the Stock Availability or probability of filling an order at the forward level support multiplied by the Mean Time to Obtain (MTTO) a spare at the forward level support. Spares used to repair systems are call Line Replaceable Units or LRUs.

7. Traditional Supply Flow
DECENTRALIZED
LOCATION
DECENTRALIZED
LOCATION
DECENTRALIZED
LOCATION
DECENTRALIZED
LOCATION
INTERMEDIATE
SUPPORT
INTERMEDIATE
SUPPORT
The Logistics chain influences the Mean Time To Obtain (MTTO) a spare. One part of the logistics chain is the supply chain.
In a 3 level traditional supply chain, the decentralized, forward location will obtain its spare from an intermediate support location. The intermediate support location in turn is re-supplied by the centralized supply location.
In a traditional 2 level supply chain, the centralized stockage location will supply the decentralized, forward location directly. A non-traditional supply chain relies on managed Total Asset Visibility (TAV). With TAV, it is more likely to have one decentralized location supply another decentralized location when no spares are available at the support’s more centrally located higher echelon.
CENTRALIZED
LOCATION

8. Inventory Distribution
STOCKAGE LOCATION
COMMERCIAL
GOVERNMENT
CENTRALIZED LOCATION
INTERMEDIATE SUPPORTS
DECENTRALIZED LOCATION
MANUFACTURER OR
PLANT WAREHOUSE
DISTRIBUTION CENTER OR
REGIONAL WAREHOUSE
RETAIL STORE OR CUSTOMER
DEPOT SUPPORT OR
WHOLESALE LEVEL
GENERAL SUPPORT
DIRECT SUPPORT
(AUTHORIZED STOCKAGE LIST)
ORGANIZATIONAL SUPPORT OR SITE
(PRESCRIBED LOAD LIST)
Traditional supply distribution in Government or commercial industry follow the same supply flow. For a 3 level supply chain in commercial industry, the Original Equipment Manufacturer or its plant warehouse supplies the distribution center or regional warehouse. This intermediate supplier ships to the retail store or customer.
For a 3 level supply chain in Government, the depot or wholesale level of supply supplies the General Support (GS) or Direct Support (DS) level. The intermediate supplier ships to the Organization’s support level or site operating the system. In Government, the Prescribed Load List or PLL contains all the spares initially provisioned at the forward, decentralized location below Brigade Level. The Authorized Stockage List or ASL contains the spares initially provisioned at the Brigade level, which is often modeled as the DS level. Theater level support is often modeled as the GS level.

9. Influence of Maintenance on Supply Support
MTTO, or the logistic pipeline time for an item, is dependent on the item’s maintenance policy as well as its supply time
A removal and replacement maintenance action of an item causes a demand for a spare to occur within the supply system
A successful item repair causes that item to become operable again and placed either into stock or sent back to the customer
An unsuccessful repair action or throwaway of an item requires reprocurement of that item if the stock levels are to be replenished
The Unserviceable Return Rate of reparable items impacts reprocurement or repair because an item not retrograde shipped back for repair leads to an unsuccessful repair action
This slide discusses the influence of maintenance on supply support. The Mean Time To Obtain (MTTO) a spare is essentially the logistic pipeline time for an item. The MTTO this spare is dependent on the item’s maintenance policy as well as its supply time.
A removal and replacement maintenance action to an item causes a demand for a spare to occur within the supply system. Any successful item repair causes that item to become operable again and placed backed into stock or be sent back to the customer. When a customer desires a repair and return action, the repaired item may be placed into the equipment instead of putting it into supply. An unsuccessful repair action or throwaway of an item requires re-procurement of that item if the stock levels are to be replenished. The Unserviceable Return Rate of reparable items impacts both re-procurement and repair because an item not retrograde shipped back for repair leads to an unsuccessful repair action.

10. Maintenance Distribution
ECHELON (J)
LOCATION
DECENTRALIZED
LOCATION
Repair at
Organizational
or Unit Level 1
THROW AWAY
P(1)
Ship Out
for Repair
Ship Out
for Repair
INTERMEDIATE
SUPPORT
Repair at
Direct or Regional
Support(s) 2
THROW AWAY
P(2)
Ship Out
for Repair
Another part of the logistics chain that may influence the Mean Time To Obtain spares is the maintenance chain. In maintenance support, the failed system is typically repaired forward at or below the decentralized location.
When a failed LRU is removed, it will either be repaired and returned to bring the equipment back up or a spare LRU will be used to bring the system back up. If the item is repaired and returned, the Customer Wait Time becomes more dependent on the maintenance chain. If a spare is used to restore the system, the removed LRU will either be thrown away or sent back to the intermediate support level or centralized support level to be repaired and placed back into stockage there.
CENTRALIZED
LOCATION
Repair at
Depot or
Contractor 3
THROW AWAY
P(3)
WASHOUT
RATE 3
NOTE: P(J) is percentage of repairs made at echelon J P(J) + Washout Rate = 1 M
J=1

11. Logistics Response Time Terms Driving MTTO
Order & Ship Time (OSTj):
Total time for a lower echelon stock point j to order and receive a shipped spare or component from a higher echelon (more centralized) stock point
Repair Cycle Time (RCTi,j):
Total time from failure occurrence to repair of Item i at maintenance support echelon j until Item i is properly placed
For Removal & Replacement Action, RCT includes all wait times, shipment time to echelon j & repair turnaround time at echelon j & placement of repaired item into stock at echelon j.
For Repair & Return Action, RCT includes all wait times, shipment times to & from echelon j & repair turnaround time at echelon j until placement of repaired item back into equipment
The most used Logistics Response Time terms driving the Mean Time To Obtain spares are defined on this slide.
The Order and Ship Time is the total time for a lower echelon stock point j to order and receive a shipped spare or component from a higher echelon or more centralized stock point when the item is available.
The Repair Cycle Time (RCT) is the total time from failure occurrence to repair of the item at maintenance support echelon j until the item is properly placed. For a removal and replacement action, RCT includes all wait times the shipment time to echelon j and the repair turnaround time at echelon j until the repaired item is placed into stock. For a Repair and Return Action, RCT includes all wait times, shipment times to and from echelon j and the repair turnaround time at echelon j until the repaired item is placed back into equipment.

12. Other Terms Driving the MTTO Spares
Maintenance Task Distribution (MTDi,j): 3 M
Pi,j + RRi = 100%
J=1
Pi,j:
Probability or Percentage of time Item i is repaired at maintenance support echelon j.
Replacement Rate (RRi):
Probability Item i is replaced
Mean Time to Obtain a Back Order (MTTOBOi,3):
Average time to obtain Item i at the wholesale/depot support level when a back order has occurred. (accounts for previous orders or repairs of Item i due in )
The Maintenance Task Distribution of an item shows the probability or percentage of repairs made at each maintenance support level and the percentage thrown away. The sum of the repair percentages at each maintenance support echelon plus the replacement rate covering the percentage not repaired totals to 100%.
Another important Logistics Response Time is the Mean Time To Obtain a Back Order (MTTOBO). This is the average time to obtain an Item at the wholesale or depot support level when a back order has occurred. Since MTTOBO accounts for previous orders or repairs due in, this time is usually shorter than the Procurement Lead Time when replacing the item or the Repair Cycle Time when repairing the item.
The Back Order Rate, which is 1 minus the wholesale level Stock Availability (SA), is the probability or percentage of time an order for the item is not in stock at the wholesale or depot support level.
Back Order Rate (1 - SAi,3):
Probability or Percentage of time an order for Item i is not in stock at the wholesale/depot support level

13. Mean Time To Obtain (MTTO) Spares at Retail Support Levels
MTTO1 = RCT1 x P1 + (1 - P1) x (OST (1 - SA2) x MTTO2)
FOR LRU THROWAWAY OR REPAIR AT CENTRALIZED LOCATION
FOR LRU REPAIR AT THE INTERMEDIATE LOCATION
OST - ORDER & SHIP TIME
SA - STOCK AVAILABILITY (ORDER FILL RATE)
P - PERCENTAGE REPAIRED AT ECHELON
RCT - REPAIR CYCLE TIME
MTTO2 = OST (1 - SA3) x MTTOBO3
MTTO2 = RCT2 x P2 + (1 - P2) x (OST (1 - SA3) x MTTOBO3)
The Order and Ship Time (OST) and Stock Availability (SA) associated with supply support, the Repair Cycle Time (RCT) associated with maintenance support, and the MTTOBO are instrumental in determining the Mean Time To Obtain (MTTO) spares at retail support levels. On this slide, the subscript 1 represents the forward support level, 2 is the intermediates support level, and 3 is the centralized support level.
The time for the forward support level to obtain a spare depends on the percentage of LRUs repaired and returned there times its RCT. When a spare is removed and replaced, the MTTO a spare depends on its OST from the intermediate support level times its SA there.
The mean time for the intermediate support level to obtain a spare depends on whether its stock is primarily refilled by the supply chain or by the maintenance chain. For an item thrown away or repaired at the centralized location, the OST from the wholesale stockage location and its SA there is more critical than the maintenance chain. For an item repaired at the intermediate location or repaired and returned to the intermediate level the RCT times the percentage repaired tends to be more critical than the supply chain.

14. System Mission Success
When failures occur randomly yielding a constant failure rate, the System Reliability or the Probability of mission success without a system failure is exponentially distributed
For an Exponential Distribution:
P(0) =
Where:
P(0) is the probability of having no failures over a mission time period
λ is the failure rate (failure/ hour) for one system
t is the length of mission time
Metrics are not typically reported on an individual item’s Stock Availability (SA). Therefore, an item’s SA is often computed mathematically. To eventually compute SA, the concept of estimating a system’s Mission Success rate will be introduced first.
When failures occur randomly yielding a constant failure rate, the system Reliability or the Probability of Mission Success without a system failure is exponentially distributed. For an Exponential Distribution, the probability of a system having no failures over a mission time period is equal to the exponential of minus the failure rate times the mission duration time. λ is the failure rate or failures per hour for one system and t is the length of mission time in hours. Therefore, λ times t is the average number of system failures occurring over the length of mission time.
λt is the average number of system failures occurring over the length of mission time

15. Mission Success With Multiple Items
Given that failures occur randomly and are exponentially distributed, item success predictions are based on the Poisson Distribution
For a Poisson Distribution:
Where:
P(x) is the probability of having x failures over the mission time
n is the number of items operating
λ is the failure rate (failure/ hour) for one item
t is the length of mission time
Taking the concept of mission success further, multiple items may be operating, rather than 1 item to accomplish the mission. The Mission Success with multiple Items is based on the Poisson Distribution when failures occur randomly and are exponentially distributed. The Poisson Distribution is used to describe the probability of having an integer number of failures over the duration of the mission time.
The Poisson Distribution formula is shown on this slide. P(x) is the probability of having x failures over the mission time period. n is the number of items operating. λ is the failure rate or failures per hour for one item and t is the length of mission time in hours.
x factorial (x!) is equal to x times (x – 1) times (x – 2) and so on until multiplied by 1. For instance, if x=3, x! is equal to 3 times 2 times 1, which is 6. If x is 0 or 1, x! is equal to 1. n times λ times t is the expected number of failures occurring over the length of mission time.
nλt is the expected number of failures occurring over the length of mission time

16. Working with Poisson Distributione
P(0) - =
Note:
If the mission reliability is stated, the expected number of failures occurring over the length of mission time can be computed.
P(0) is the mission reliability for all items not failing during the mission
P(1) is the probability of having 1 failure during the mission time
P(0) + P(1) is the probability of mission success with 1 or less failures during the mission time
In working with the Poisson Distribution, the probability of no failure over the mission duration is the Exponential Distribution. The probability of zero failures is equal to the exponential of minus n times λ times t. By taking the natural logarithm of each side of this equation, n λ t is equal to minus the natural logarithm of no failures over the mission duration.
The Poisson probability of 1 failure over the mission duration is equal to n λ t times the exponential of minus n λ t. This is the same as the probability of zero failures over the mission duration times n λ t. The probability of zero failures, P(0), is the mission reliability for all items not failing during the mission duration. P(1) is the probability of having 1 item fail during the mission time. P(0) + P(1) is the cumulative probability of mission success with 1 or less failures during the mission time.
The other mathematical formulas on this slide help to demonstrate how to compute the Poisson Distribution probability of having x failures when the probability of (x – 1) failures already has been computed. The probability of having x items fail during the mission time is equal to the probability of having (x – 1) items fail during the mission duration times n λ t divided by x.
Poisson Distribution generalization:

17. Example:
Suppose an item has a mission reliability of 60.65%. How many items must be used to have at least a 99% chance of succeeding the mission?
P(0) =
0.5 represents the expected number of failures during the mission duration
The following is an example of using the Poisson Distribution to determine the probability of mission success. Suppose an item has a mission reliability of 60.65%. How many items must be used to have at least a 99% chance of succeeding the mission?
The probability of 0 failures for a single item is equal to or 60.65%. λ times t is computed to be 0.5, which represents the expected number of failures during the mission duration.
Using the Poisson Distribution formula, the probability of 1 failure over the mission duration is computed to be or 30.33%. Having 1 additional item to accomplish the mission increases the probability of mission success to 90.98%
Having 1 additional item to accomplish the mission increases the probability of mission success to = 90.98%

18. Example (cont’):
The Δ improvements to mission success by having a second additional item available to accomplish the mission is 7.58%. The probability of missions success with 3 items is:
Having 3 additional items to accomplish the mission increases the probability of mission success to
Continuing the use of the Poisson Distribution formula, the probability of 2 failures over the mission duration is computed to be 7.58%. The improvement change to mission success by having a second additional item available to accomplish the mission is 7.58%. The probability of mission success with 3 items or 2 additional items is 98.56%, which is still less than 99%.
Using the Poisson Distribution formula again, the probability of having 3 failures over the mission duration is computed to be 1.26%. Having 3 additional items to accomplish the mission increases the probability of mission success to 99.82%, which finally exceeds the goal of having at least a 99% chance of succeeding the mission. Therefore, 4 items must be used to have at least 99% chance of succeeding the mission.
Therefore, 4 items must be used to have at least a 99% chance of succeeding the mission

19. Stock Availability / Order Fill Rate Predictions
The probability of filling orders for an item before replenishment spares are obtained
It is assumed that retail level uses an existing spare parts if available, and orders a replacement spare simultaneously
At retail levels, the order quantity is assumed to be 1 (q=1)
Without out a spare the probability of filling an order before replenishment is 0%
The Stock Availability or Order Fill Rate from retail level sparing can analogously be predicted using the Poisson Distribution. The mission time now becomes the average length of time to replenish stock after the supply activity provides a spare to fill a demand. The pipeline time is the Mean Time To Obtain (MTTO) spares. An Order Fill Rate covers the Probability of filling orders for an item before replenishment spares are obtained. It is assumed that retail level will use an existing spare if it is available and the order to replace a spare occurs around the same time. Also, at retail levels, the order quantity is assumed to be 1.
Without a spare, the probability of filling an order before replenishment is 0%. With 1 spare on hand being initially used to fill a demand the probability of not getting another order prior to replenishment is the same as having a mission success with 0 spares available initially and mission time equal to the MTTO spares.
With 1 spare on hand initially the probability of filling all orders prior to replenishment is the same as having a mission success with 0 spares available initially and a mission time equal to the Mean Time To Obtain (MTTO) a spare. (i.e. Fill Rate = )

20. Stock Availability / Order Fill Rate
From Previous Example
With t = MTTO:
nλt is the expected number of demands occurring over the average replenishment time to obtain a spare
Example Results:
With 0 spares, SA = 0% as there is no stock to fill orders
With 1 spare, SA = 60.65% order fill rate
With 2 spares, SA = 90.98% order fill rate
With 3 spares, SA = 98.56% order fill rate
With 4 spares, SA = 99.82% order fill rate
Using the previous mission success example as a Stock Availability (SA) or Order Fill Rate example, the mission time t is now equal to the retail support supply activity’s Mean Time To Obtain spares for the item. n λ t is now the expected number of demands occurring over the average replenishment time to obtain a spare.
When there are no spares initially stocked, the SA is 0% because there is no stock to fill orders. With 1 spare, the SA yields a 60.65% order fill rate. With 2 spares, the SA yields a 90.98% order fill rate. With 3 spares, the SA yields a 98.56% order fill rate.
Finally, with 4 spares, the SA yields a 99.82% order fill rate. Therefore, the supply activity must have 4 spares available initially to exceed a 99% SA. When 1 spare is given up to satisfy a demand, 3 additional items must be in stock to yield a mission success rate of 99.82% before receiving the replenishment spare.

21. Optimizing Sparing Costs
Trade-off analysis is necessary which varies Retail Level supply stock availabilities to yield varied mixes of PLL and ASL LRU stockage quantity
Increasing ASL order fill rates produces more ASL LRU spare quantities and less PLL Customer Wait Time
The higher cost of sparing more at the centralized location reduces the stock needed at the decentralized locations
The cost savings per decentralized location is magnified by the number of locations being supported by a centralized location
Decreasing ASL order fill rates produces less ASL LRU spares and more PLL Customer Wait Time. More PLL LRU spares are needed to achieve the same availability goal
For optimizing sparing costs to achieve an Ao goal, trade-off analysis is necessary which varies retail level supply stock availabilities to yield mixes to Prescribed Load List (PLL) and Authorized Stockage List (ASL) Line Replaceable Unit (LRU) stockage quantities.
Increasing the ASL order fill rates produces more ASL LRU spare quantities causing less PLL Customer Wait Time. The higher cost of sparing more at the centralized location reduces the stock needed at the decentralized locations. The cost savings per decentralized location is magnified by the number of locations being supported by a centralized location.
Conversely, decreasing ASL order fill rates produces less ASL LRU spares causing more PLL Customer Wait Time. More PLL LRU spares will then be needed to achieve the same availability goal.

22. Multi-Echelon Sparing Optimization to Same Equipment Availability
Total Stock Cost to
Achieve Availability
Min
Cost
Total Second Echelon
Stockage
Sparing Cost
This slide illustrates how multi-echelon sparing optimization reduces the total sparing cost to achieve the same equipment availability requirement or goal.
If a lot of items are put at the most forward retail level in the PLL, then less items are needed at the more centralized retail level in the ASL to achieve the same availability goal. Conversely, when a lot of items are put in the ASL, less items are needed in each PLL that the ASL supports to achieve the same availability goal.
The Army has a sparing to availability model that accomplishes multi-echelon retail level sparing optimization to determine the lowest or minimum total cost mix of spares that achieves the availability goal input.
Total Forward Level
Stockage
Optimum Sparing Mix
Stock Availability at 2nd Echelon

23. Selected Essential-item Stock to Availability Method (SESAME) Usefulness
Optimizes Multi-Echelon Retail Level Initial Sparing to Achieve End Item Ao Requirement or Forward Level Support Stock Availability Goal
-OR-
Evaluates Ao or SA Based on Sparing Mix, LRU Reliabilities and Logistics Response Times
Optimizes Plus Up Sparing to Achieve Ao or SA Goal Given the Present Retail Level Sparing Mix
Maintenance Concept for each Essential Item is Proposed or Known
The Selected Essential-Item Stock to Availability Method (SESAME) Model is the Army’s standard retail supply management tool. Prior to fielding, SESAME can optimize the multi-echelon retail level mix of initial spares to achieve an end item Operational Availability (Ao) requirement or a forward level support Stock Availability (SA) goal.
SESAME can also be used in an evaluation mode to estimate an equipment’s Ao or SA based on the proposed or known LRU sparing mix, each LRU’s failure rate or reliabilities and the Logistics Response Times due to the OST and RCT.
SESAME can also be used after fielding to optimally plus up sparing to achieve the equipment’s Ao or SA if the present retail level sparing mix is found to be inadequate to achieve the needed level of performance based on actual, experienced item demand rates.
To use SESAME, the proposed or known maintenance concept for each essential item must be input in the model.