June 2010
Probabilistic Margin Management
Prepared by:
ARES Corporation
1331 Gemini Street, Suite 120
Houston, TX 77058
Table of Contents
Executive Summary ................................................................................................... 3
Problem Statement ..................................................................................................... 3
ARES Solution ........................................................................................................... 4
Merits of ARES Solution ........................................................................................... 4
Technical Description ................................................................................................ 4
Detailed PMM Steps .................................................................................................. 5
PMM Implementation ................................................................................................ 8
Conclusion ................................................................................................................. 9
References .................................................................................................................. 9
ARES Probability Margins Management Process
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Executive Summary
System can be viewed as a combination of functions
or subsystems. Interactions between functions are an
essential part of the overall design. Through function
interdependencies, function margins are affected by
changes in margins of other interrelated functions.
To optimize a system with respect to performance
and margins, it is necessary to model the effects of
these margin interdependencies, so both baseline and
proposed margin optimizations can be properly
assessed.
An effective PMM process requires the collaboration
with system and subsystem leads, experts and subsystem managers, to determine the appropriate
Technical Performance Parameters (TPPs), margin
interdependencies and the associated margin
sensitivities. This includes regular reports and
recommendations made to the margins manager in
order to aid in responsible margin reallocation and
optimization of the system.
The Probabilistic Margin Management (PMM)
process provides a number of benefits compared to
traditional margin management that are designed to
aid program management throughout the entire
system development process.





Margins are uniquely and precisely defined by a
quantitative metric eliminating subjectivity
This process allows for complete margin
management by treating each margin as a
“bankable resource” that can be used in a tradeoff process to optimize cost, schedule and
performance goals
PMM quantifies the major technical requirement
drivers,
Optimizes each requirements margin to improve
the success of meeting project goals and
objectives
Enhances the program management decision
process clarifying the interactions between cost
schedule and technical risks.
Problem Statement
The design and development of any complex
multidisciplinary system is inherently difficult and
ARES Probability Margins Management Process
uncertain because no single person has all of the
detailed knowledge required for all disciplines. It is
therefore necessary to establish and maintain a set of
margins for all system requirements in order to
properly account for the changes with a known
value, changes with an unknown value and changes
that you do not even know about (the unknown
unknowns). Today most margins are established
using discrete values derived from historical data,
expert judgment, or taken directly from industry
standards. There are two primary faults with this
approach. The first is that margins themselves are
uncertain and are not universal across various
systems. As a result, incorrect margins are often
utilized when important programmatic decisions are
being made during all parts of the program life cycle.
Secondly,
margins
cannot
be
established
independently. The existence of interactions and
uncertainty within complex systems must be
accounted for otherwise the total impact will be
underestimated. The process depicted in Figure 1
illustrates how the margin interactions are developed
and the continuously managed through the design
process.
Define Technical
Performance Parameters
Develop Margin
Interactions Matrix
Develop/Revise
Transfer Function
Distributions
Update TPP &
Margin Estimates
Assess Design
Changes &
Maturation
Figure 1. ARES PMM Process Flowchart
For example, the program manager maintains a set of
margins for each subsystem in case of unexpected
problems. As system development advances the
program manager can allocate additional or
redistribute the available margin based on current
design information. The PMM process can help
program managers make better decisions about the
effects of allocating margin and develop an
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optimized distribution that maximizes the likelihood
of meeting all requirements.
The ARES’ PMM process can help to develop
greater confidence in the initial margin allocation,
provide a greater understanding of the impacts of
interactions on the system and help the manager
make better decisions about the reassessment of TPP
margins.
ARES Solution
The ARES’ PMM process provides a unified system
level approach to TPP margin control. The PMM
process is a comprehensive and practical method for
capturing the effects of margin interactions as well
as the probabilistic and unique nature of margin
assessment. ARES’ PMM process supports the
assessment of baseline margins and optimization of
the major sensitivity effects by allowing the margin
estimates to be expressed in terms of a statistical
confidence similar to the way risk is managed.
For accurate margin management, it is also
necessary to capture the probabilistic nature of
margin estimates and the corresponding interactions.
System margins estimates can be suitably
represented as probability distributions. When
margin estimates are developed, the estimate can be
characterized as having a quantified amount of
confidence for achieving a specific value.
Merits of ARES Solution
The benefits of ARES PMM process can be
summarized as follows:




Probabilistic instead of deterministic providing
an alternative method when adequate information
is not available.
Account for complex interactions generally not
readily available
Provide program management the ability to make
informed decisions regarding the system’s ability
to meet TPP requirements
Use sensitivity analysis to show the TPPs
affected most by a margin change

Determine optimal margin allocation which
results in the greatest confidence of achieving all
requirements.
Technical Description
ARES continuous margins management process is
outlined in Figure 1. To perform an interdependency
assessment, it is first necessary to define the TPPs to
be modeled. A Margin Interaction Display is then
created to define which margins interact with one
another. Transfer functions for each interaction are
developed through subject matter expert interviews
and historical data analysis. TPP estimates are
gathered to complete the model. This baseline
model is used to assess design modifications and
maturation and is continuously updated according to
the latest TPP estimates.
PMM is used to evaluate a system through a
mathematical model which captures the effect of
probabilistic function margin interactions. Normal
distributions are used to represent either margin or
Current Best Estimate (CBE) distributions. PMM
can be used at either the system-level or componentlevel, because the margin interaction calculations can
be performed on any integrated system, with the
resulting margins rolling up into the overall margin
calculation of the system. This process is similar to
that used to perform a Probability Risk Assessment
(PRA).
Matrix algebra is employed to solve for cascading
margin effects due to interactions. The core of the
mathematical solution is provided in the following
equation. Using matrix algebra to build the vector
dependencies and incorporating them into a single
equation, the resulting margin effects to the N order
can be determined.
{Σ(∆𝑀)} = ([𝐶] + [𝐶][𝐶] + ⋯ + [𝐶]𝑁 ){∆𝑀0 }
Where:
{∆𝑀0 } = vector of the initial margin change
{Σ(∆𝑀)} = the total margin change after N orders of
effect are calculated
[C] = a square matrix of margin transfer functions
ARES Probability Margins Management Process
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The interaction effects continue regardless of how
many orders of effect are considered. But for
practical cases with realistic transfer functions, the
additional margin change effect decreases as the
number of order effects considered increases. Since
there is a point of diminishing returns to the
calculation at higher orders, a maximum order of
effect is chosen (typically 5) as providing a
reasonable estimate of the effect, beyond which
further calculation provides minimal added value.
Detailed PMM Steps
Baseline Assessment
Define Functions
Capture Margin Estimates for each Function
Develop Margin Interaction Display
Capture Transfer Function Distributions
Perform Interaction Assessment for Baseline Case
Perform sensitivity assessments to
determine margin drivers
Refine Optimization
Optimization Assessment
Determine Optimization Criteria
margin rolling up into the margin calculation of the
overall system functions.
Step 2. Capture Margin Distributions
The PMM process is also designed to use
distributions on either known margins or the CBE of
design parameters. If CBE distributions are utilized,
they are transformed into margin distributions by
using the appropriate margin definition equation
relevant to each design parameter as shown below.
After the PMM assessment is performed, the
resulting CBE design parameters can be determined
by using the inverse of the margin definition
equations:
𝐶𝐵𝐸 𝑀𝑎𝑟𝑔𝑖𝑛 =
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐵𝑒𝑠𝑡 𝐸𝑠𝑡𝑖𝑚𝑎𝑡𝑒 − 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡
𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑚𝑒𝑛𝑡
Margin-related input parameters are represented as
Normal distributions to account for the level of
uncertainty in the analysis, as shown in Figure 3.
Normal distributions are used to represent margin
and CBE distributions, consistent with industryaccepted design development procedures such as
Design for Six Sigma [1].
Modify resources or requirements
Perform Interaction Assessment for Optimization Case
No
Assess Optimization Results:
Adequate?
Yes
Optimization Complete
Figure 2. ARES PMM Flowchart
Step 1. Define Functions
PMM can be used to analyze any level of system
detail. As with any analytical modeling effort, this
analysis provides the most benefit if it is performed
at the level of detail which content specific decisions
are made.. The methodology can also be performed
at varying levels of detail based on the fidelity of
information available for each subsystem. PMM
allows use of different levels of detail, because the
margin interaction calculations can be performed at
subsystem levels, with the resulting subsystem
ARES Probability Margins Management Process
Figure 3. Sample Input Distribution
Step 3. Margin Interaction Display
It is necessary to have a method of graphically
capturing the interactions of the margins of
functions. Figure 4 demonstrates how PMM captures
the interactions of margins for any system. The red
nodes on the matrix indicate interactions among the
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system-level functions. The interactions flow in a
clockwise direction. For instance, a change in the
Thermal Management margin has an effect on the
Power, Propulsion and Gross Vehicle Weight
(GVW) margins and the GVW margin effects the
Tank Structure margin. This Margin Interaction
Display (MID) is tiered such that the margin
interactions within individual functions can also be
described with a MID.
Tank Structure
terms of the relative effects of margins. For the
specific case of the same input element as the output
element, the value of the transfer is zero, since the
margin change of an element will have no direct
additional change on the input element. The
interactions are captured in a square matrix in order
to facilitate the mathematical calculations.
The ARES methodology utilizes a transfer function
which is expressed as changes in margin. The
following are advantages of using changes in margin
as the inputs and outputs of the transfer functions.

Thermal
Management

Power
Propulsion

GVW
Volume
Figure 4. Margin Interaction Display
Step 4. Transfer Function Distributions
After identifying which functions and elements have
margin interactions for the system or subsystem of
interest, it is necessary to quantify the amount of
interactions. The margin interactions are quantified
in terms of transfer functions. The transfer function,
a concept used in block diagram system modeling, is
defined as the ratio of an elemental block's output to
its input. To evaluate the output response of an
element, the input is multiplied by the value of the
transfer function ratio. This provides an analytical
tool that is easily integrated into a model based
system engineering paradigm.
Parameters are expressed as dimensionless ratios
to provide consistent measuring
Transfer functions are captured in terms of the
effect of margin change which is more practical
than capturing transfer functions as complex
mathematical relationships
Margin interaction matrices that use margin
change as inputs and outputs are rolled-up into
higher level system margin interaction matrices.
The Transfer Function Relationship Diagram,
provided in Figure 5, is utilized to help visualize the
transfer functions in terms of their relationship to the
input and outputs of the system elements.
Tank Structure
Tank Structure
ARES Probability Margins Management Process
Power
2
Propulsion
3
GVW
4
Volume
5
6
Δ M1
Δ M1
Δ M2
Δ M6
1
Δ M2
Thermal
Management
Δ M1
2
Power
Δ M3
Δ M3
Δ M2
Δ M4
3
Propulsion
Δ M4
Δ M4
Δ M4
Δ M4
Δ M1
Δ M2
Δ M3
Δ M5
Δ M5
Δ M1
Δ M5
Δ M5
Δ M5
Δ M1
Δ M2
Δ M3
Δ M4
Δ M6
4
GVW
ARES margin interaction transfer function ratios are
defined as the direct change in margin of the output
parameter due to a change in the margin of the input
parameter. The value of this transfer function can be
captured practically for an output element as a
function of a different input element, by thinking in
Thermal
Management
1
5
Volume
Δ M6
Δ M6
Δ M6
Δ M1
Δ M3
Δ M5
6
Figure 5. Transfer Function Relationships
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The system and subsystem transfer functions are
both expressed as Triangular distributions, which are
defined by the lowest, the most likely, and the
highest possible Transfer Function ratios and are
shown in Figure 6 [3].
outputs are the distributions of margins for each of
the system functions (including interactions) and a
table providing the change in margin for each
parameter based on the change in margin of the other
parameters. The baseline margin distributions will
vary from the input modified optimum margin
distributions due to the interaction effects between
distributions that are not initially considered.
The remaining steps in the PMM process are
designed to take the margin interaction assessment
provided in Steps 1 to 5 and optimize the allocation
of margin. An optimized solution is defined as the
design which has the maximum probability of
meeting all of the requirements.
Figure 6. Notional Transfer Function Distribution
The transfer function distribution estimates are
developed during interviews with system and
subsystem experts in order to establish the bounds of
each Triangular distribution. These results are then
organized into a Transfer Function Matrix similar to
the one provided in Figure 7.
Input
Thermal
Management
Power
Propulsion
GVW
Volume
Tank
Structure
0
.09 .10 .11
0
.09 .10 .11
.09 .10 .11
.09 .10 .11
Thermal
Management
.09 .10 .11
0
.09 .10 .11
.09 .10 .11
.09 .10 .11
0
Power
0
0
0
.09 .10 .11
.09 .10 .11
.09 .10 .11
Propulsion
0
0
.09 .10 .11
0
.09 .10 .11
0
GVW
0
0
0
.09 .10 .11
0
.09 .10 .11
Output
Tank
Structure
Volume
0
0
0
0
.09 .10 .11
0
Figure 7. Transfer Function Matrix (notional)
Step 5. Monte Carlo Simulation
Monte Carlo simulation for the baseline margins is
performed utilizing a large set of trials. The primary
ARES Probability Margins Management Process
Step 6. Optimization Criteria
It is necessary to define the goals of the optimization
effort. The definitions should be stated in terms of
the resulting margins of the system. Each function
may have different optimization requirements
depending on whether it is characterized as a target
or a constraint. A target function margin is the
desired function margin to optimize. A constraint
function margin is a function margin which must not
exceed a certain threshold, or which must be within a
certain bound.
Step 7. Sensitivity Assessments
In order to optimize the margins it is necessary to
understand the system functions and subsystem
elements which have the greatest impact on the
margins. In particular it is necessary to understand
how each of the function and element margins
affects the target margin. It is also necessary to
understand how each of the function and element
margins affects margins which are primary
constraints.
Tornado effect diagrams are useful for establishing
sensitivity relationships by providing a ranking of
the design factors that have the greatest influence on
the variability of a particular variable. Figure 8
shows a notional tornado effect diagram. The use of
this type of diagram allows decision makers to
quickly assess which factors have the greatest impact
on the design parameter of interest.
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final assessment provides the margin confidence
levels for the final optimized configuration.
Step 10. Assess Optimization Results
The results of the optimization are compared to the
optimization objective, to determine if the results
meet the requirements, or if further refinement of the
optimization is required. In order for the
optimization to be accepted as final, the optimization
goals of the target margins must be achieved, and the
constraints on the other margins must not be
violated. Once these requirements are achieved the
optimization is complete.
Figure 8. Sensitivity Analysis (notional)
Step 8. Alter Resources/ Requirements
The results of the tornado diagram for a target
optimization margin show the amount of margin
increase which will result from a change in another
function. If driving TPPs are identified which
provide a beneficial increase in the target margin, but
which have an excess margin themselves, then the
margin of the driving TPPs can be changed to cause
an optimum improvement in the target TPP. The
driving TPP’s margin can be changed through either
a change in the resource or the requirement for the
function. The driving TPP’s change to its resource
or requirement which is required to achieve its
change in margin is calculated through the margin
calculation definition.
Step 9. Interaction Assessment
Monte Carlo simulations for the baseline margins are
performed utilizing a large set of trials. The primary
outputs of the interaction assessment are the
distributions of margins for each of the system
functions (including interactions) as done in Step 5.
The baseline margin distributions vary from the
input modified optimum margin distributions due to
the interaction effects between distributions. This
ARES Probability Margins Management Process
Step 11. Refine Optimization
If the results of the optimization either do not
achieve the margin goals for the target margins or
violate the constraints of other margins, the
optimization must undergo a further iteration. At this
point the process loops back to step 7. In this case
the sensitivity studies are utilized to determine what
parameter(s) will have the largest desired effect on
the optimization variable.
PMM Implementation
The ARES PMM process reduces the error
associated with arbitrarily assigning system and
subsystem margins and allows the project margins
manager to make better decisions about reallocation
of margins. This process can be integrated into the
traditional system engineering process or a model
based process using the existing system engineering
tools and databases.
PMM also probabilistically identifies untenable
requirements or margin allocations earlier in the
design/development cycle which allows the manager
to better control system scope or cost growth. By
addressing the design margins or requirement
problems early in the life of the system the cost of
addressing the design inconsistencies is dramatically
reduced.
As described in the Technical Description section
above, ARES collaborates with the system and
subsystem managers to determine the appropriate
TPPs. ARES interviews system and subsystem leads
and experts to determine and develop suitable
interaction and transfer functions. ARES then
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develops the margins baseline and presents a
standard margins report to the system and (as
appropriate) subsystem managers. Additional
guidance is provided which includes advising and
assisting management in the organization and the
prioritization of individual margins for optimization.
ARES has effectively implemented the PMM
process in support of Lockheed Martin’s System
Engineering group during PDRR by writing the
Margins Management Plan and facilitating the
system-level margins management activities and
coordinating activities across the ground and space
segment. Additionally, ARES utilized the PMM
process to estimate the uncertainty in the initial IPT
mass estimates for Northrop Grumman. Including
establishing confidence levels for the base level
estimates, establish driving factors for all margins,
and determined the margin sensitivity to key
assumptions.
References
1. Yang, K., EI-Haik, B., Design for Six Sigma: A
Roadmap for Product Development, McGraw-Hill,
2003
2. Billinton, R., Allan, R., Reliability Evaluation of
Engineering Systems: Concepts and Techniques,
Second Edition, Plenum Press, 1992
3. Elsayed, E., Reliability Engineering, Addison
Wesley Longman, 1996
Conclusion
The PMM process provides several distinct benefits
that aid program management throughout the entire
system development process. Utilizing the ARES
PMM process, decision makers are able to gain a
better understanding of how the allocation of
margins impacts the system and determine how
margins should best be allocated to provide the
greatest confidence that all requirements will be
achieved. Additionally, uncertainty distributions are
utilized to establish which requirements are at the
greatest risk of being violated and require the
greatest initial margin reserves. As a result of
implementing this process, better decisions are made
up front which, in turn, reduces the number of design
changes required during later design phases, and
reduces the risk associated with cost-overruns and
slipped schedules.
ARES Probability Margins Management Process
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