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Reliability Allocation and
Optimization using BlockSim
In
last month's Reliability Basics article, we discussed some
popular reliability allocation methods, including the equal, AGREE, ARINC,
feasibility of objectives and repairable systems apportionment
techniques. We also presented some practical applications using
ReliaSoft's Lambda
Predict software. All of the methods discussed
are based on the assumptions that:
- The components' lifetime
characteristics are described by the exponential
distribution; in other words, the components all exhibit a
constant failure rate behavior.
- The components are all
connected in a series configuration reliability-wise.
These methodologies are widely
used in many reliability prediction standards and are useful
when limited reliability data exists, usually during the initial
design phase of a new product. When there are specific details
concerning the failure rate behavior of the product and there is
a more complex configuration instead of just components in
series, we can go one step further and incorporate a more
detailed analysis of reliability allocation that is free of the
above assumptions. In this article we present a more advanced
method for reliability allocation that is free of these
assumptions and can be performed with ReliaSoft's BlockSim
7 software.
In BlockSim, we can
create complex reliability block diagrams that model advanced
systems. A reliability block diagram represents the reliability
relationships of the components in a system and can comprise
blocks in any combination of series, parallel, stand-by and load
sharing configurations. After constructing the reliability model
of a complex system, we can optimize the allocation of
reliability to different components in order to meet a
reliability specification while considering the cost of the
allocation.
Reliability Improvement
Methodology in BlockSim
There are two approaches when trying to improve the reliability
of a system: fault avoidance and fault tolerance. Fault
avoidance is achieved by using highly reliable components, and
fault tolerance is achieved by adding redundancy. Note that
although adding redundancy can sometimes be the easiest way to
achieve a goal, it is not always the preferred solution, since
it can result in greater system complexity and additional
product costs due to the increased weight or size.
The optimum method of
allocating reliability should take into account the cost or
relative difficulty of improving the reliability of different
subsystems or components. That way we are not simply allocating
reliability to components, but are solving the optimization
problem of the most cost-effective and efficient way of
allocating or improving reliability in order to meet the
required system goal. Note that reliability
requirements are always associated with a specific mission
duration, since reliability is defined as a function of a
specified lifetime variable, such as time, cycles, miles etc.
BlockSim uses the following
default cost function that acts like a penalty function for
increasing a component's reliability. It is assumed that the
cost increases exponentially since it is more difficult to
progressively increase reliability:
where:
- Ci(Ri)
is the penalty (or cost) equation as a function of component
reliability.
- f is the
feasibility (or cost index) of improving a component's
reliability relative to the other components in the system.
- Rmin,i
is the current reliability at the time at which the
optimization is to be performed.
- Rmax,i
is the maximum achievable reliability at the time at which
the optimization is to be performed.
The feasibility term is a
constant that represents the difficulty in increasing a
component's reliability relative to the rest of the components
in the system. Depending on the design complexity, technological
limitations, etc., certain components can be very hard to
improve. Clearly, the more difficult it is to improve the
reliability of the component, the greater the cost. The lower
the feasibility value, the more rapidly the cost function
approaches infinity.
The expression for the overall
system cost at the system reliability goal is the sum of the
costs of its components:
Once the overall cost function
for the system has been defined, the problem becomes one of
minimizing the cost function while remaining within the
constraints defined by the target system reliability and the
reliability ranges for the components.
BlockSim employs a
nonlinear programming technique to minimize the system cost
function. The structural properties of the system are accounted
for in the determination of the optimum solution. For example,
the optimization for a system of three units in series will be
different from the optimization for a system consisting of the
same three units in parallel, in spite of the fact that the cost
functions for the individual components remain the same. The
optimization occurs by varying the reliability values of the
components within their respective constraints of maximum and
minimum reliability in such a way that the overall system goal
is achieved at the minimum overall system cost.
Example
Let's examine the allocation and optimization method in BlockSim
7 through an example. Suppose that we have the reliability block
diagram that is shown in Figure 1.
Figure 1: A Reliability Block Diagram in BlockSim
This block diagram is composed
of 10 blocks that are in a mix of series and parallel
configurations. None of the blocks has a constant failure rate.
Based on these two facts, the reliability allocation methods
described in last month's HotWire article cannot be applied
to this system.
The lifetime characteristics of
each block are entered by double-clicking the block. For
example, Figure 2 shows that Block 1 follows a Weibull
distribution with a shape parameter beta equal to 2.5 and a
characteristic life eta equal to 2000.
Figure 2: Failure Distribution for Block 1
BlockSim can also link
to Weibull++
or ALTA Folios so
that test results for specific components can be applied
directly to the block. For example, by clicking the Weibull++
icon on the Reliability tab of the Block Properties window for
Block 6, we can apply lifetime characteristics based on
the distribution and its associated parameters previously
identified from component level life data analysis.
(Clicking the ALTA icon
would allow us, instead, to apply calculated results from
accelerated testing.). Figure 3 shows the component level
testing results for Block 6.
Figure 3: Estimating Lognormal Distribution Parameters for
Block 6 by Associating a Weibull++ Folio
We associate this
life data analysis Folio with Block 6 by clicking the
Transfer and Close button. The parameters of the associated
lognormal distribution are automatically applied to the failure
distribution properties for the specific block, as shown in
Figure 4.
Figure 4: Applying the Associated Life Data Analysis Results
for Block 6
The failure distributions for
all of the blocks are as follows:
Block 1: Weibull
β=2.5;
η=2000,
γ=0
Block 2: Weibull β=3;
η=2500,
γ=0
Block 3: Weibull β=1.5;
η=3000,
γ=0
Block 4: Weibull β=1.9;
η=5000,
γ=0
Block 5: Exponential μ=5000,
γ=0
Block 6: Lognormal μ=7.7142,
σ=0.9653
Block 7: Lognormal μ=7,
σ=1
Block 8: Weibull β=0.9;
η=5000,
γ=0
Block 9: Weibull β=5;
η=6120,
γ=0
Block 10: Weibull β=4.5;
η=7000,
γ=0
We can define these for each
block using the methods shown above.
BlockSim can calculate
very complex reliability equations based on the different
configurations. The equation below is the system reliability
equation for the specific configuration shown in Figure 1:
So far, we have
defined a complex system configuration and defined the different
failure distributions for each of the 10 blocks. Suppose that we
have a system reliability specification of Rsystem =
90% at mission end time = 1000. Using the Quick Calculation Pad,
we find that the system reliability at t = 1000 is 42.17%, as
shown in Figure 5.
Figure 5: Calculating System Reliability at t = 1000
The question now is which
components to redesign for increased reliability (fault
avoidance) or which components to duplicate in parallel
configurations (fault tolerance) in order to meet the specific
reliability requirement at t = 1000.
To determine this, for each
block to be used in the optimization process (in this case, all
blocks), we select the Block to be used in optimized
allocation option on the Reliability Allocation page of the
Other tab in the Block Properties window. We then define a
maximum achievable reliability for each component, which, as
described in the methodology section, is a limiting reliability
that the component will approach but not reach, usually dictated
by technological or financial constraints. It is important to
note that the time associated with this maximum achievable
reliability is the same as that of the overall system
reliability goal, in this case a mission duration t = 1000. We
also choose a feasibility factor for each of the blocks. For
example, for Block 4, the maximum achievable reliability is
0.999 and the slider is moved to 7 on a scale of 1 to 9, meaning
that it is between moderate and hard to improve reliability for
Block 4, as compared to the rest of the blocks.
Figure 6: Reliability Allocation Properties for Block 4
Note that the feasibility used
in the cost function for a particular block is given by f = 1
slider value/100, so the feasibility associated with Block 4 is
0.3.
Once these properties have been
specified for all 10 blocks, we can use the optimization page of
the Quick Calculation Pad. As shown in Figure 7, we define the
mission end time of interest, 1000, and the
associated reliability goal of 90%.
Figure 7: Optimization in the QCP
Figure 8 shows the results of
the optimization process.
Figure 8: Reliability Optimization Results for t = 1000
Column C shows the calculated
goal for each of the blocks in terms of fault avoidance. In
other words, the values in Column C represent the new optimized
goal for each of the blocks so that the 90% system reliability
requirement at t = 1000 is met. Column D presents the
optimization results in terms of fault tolerance. In other
words, the values in Column D represent the number of equivalent
parallel units that would have to be used to meet the 90% system
reliability goal if the reliability of those units could not be
increased from the values in Column B. The system configuration,
the specific failure distributions for each block, the
feasibility factors and the maximum achievable reliability
values have all been factored in to provide the optimum
reliability improvement combination to meet the predefined goal.
Conclusion
In this article we presented a powerful way of allocating and
optimizing reliability using the methodology in ReliaSoft's
BlockSim 7. This approach can handle any type of system
configuration and it does not require unrealistic assumptions
concerning constant failure rates. It uses a non-linear
optimization method to identify the most cost-effective and
efficient way to achieve a system reliability goal.
References
[1] ReliaSoft Corporation, System Analysis Reference,
Tucson, AZ: ReliaSoft Publishing, 2007, pp. 174-197.
[2] ReliaSoft Corporation, "Reliability
Allocation and Optimization," Reliability HotWire,
Issue 6, August 2001.
[3] ReliaSoft Corporation, "Reliability
Allocation Using Lambda Predict," Reliability HotWire,
Issue 98, April 2009.
[4] Mettas, A., "Reliability Allocation and Optimization for
Complex Systems," Proceedings of the Annual Reliability &
Maintainability Symposium, 2000.
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