SYSTEM RELIABILITY OPTIMIZATION
CONSIDERING UNCERTAINTY: MINIMIZATION OF
THE COEFFICIENT OF VARIATION FOR SERIESPARALLEL SYSTEMS
Hatice Tekiner-Mogulkoc,
David W. Coit
Presented by Chia-Ling Lee
Model

defender要防止attacker進行破壞與竊取，其採取的方法可
能也是使用切塊和備份的動作。切割的大小可以不同，每
塊備份的數量可以不同。再將切成的資料塊以及備份檔案
放置不同的location，每個location的容量不一樣，可容納
的資料量不同。
Model
Defender side
 Give parameter：
capacity of each node, defense budget
 Objective：
最大化被攻擊者最小化的data survivability , data security。
 Decision variables：
 每份資料切割的塊數
 每塊資料的大小
 每塊資料備份數量
 資料塊放置位置
 Constraint：
 總防禦資源
 節點容量
Model
Attacker side
 Give parameter：
attack budget
 Objective：
最小化data survivability , data security
 Decision variables：
 破壞data survivability or data security
 使用的攻擊策略
 攻擊目標
 攻擊目標時的勝率 (透過contest success function反算)
 Constraint：
攻擊者預算
Agenda
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Introduction
Redundancy allocation problem for series-parallel system
Mathematical formulation
Developed approaches to minimize the coefficient of
variation
Problems without component mixing
 Problems with component mixing
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Neighborhood generation
Numerical examples
conclusion
Introduction
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Most system reliability optimization studies assume that
the reliability values of the components are
deterministic, i.e., known with certainty.
In practice, the reliability of a component is generally
estimated from field or test data, and therefore there is
some uncertainty associated with the estimates.
Decision-makers clearly prefer solutions with lower
estimation uncertainty.
New design optimization methods are needed that
explicitly consider uncertainty
Introduction
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In this paper, the variance of the component and
system reliability estimate is used as a metric to
characterize estimation uncertainty, and a new
formulation proposed to minimize the coefficient of
variation(CV).
There are several studies where estimation
uncertainty is considered as a part of the system
reliability optimization model.
Introduction
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Coit and Smith , and Coit define an ∝∗ 100%percentile of
the system reliability distribution and/or system time-tofailure distribution, then use these lower-bound limits as
objective functions to maximize.
Their algorithm uses a genetic algorithm to search the
solution space, and an adaptive penalty function to
encourage evaluation of feasible solutions near the
feasible region.
For these models, it is necessary to specify an ∝ value.
Often, it is unclear what value of to select, although the
particular value chosen impacts the optimal solution.
Introduction
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Another approach is to consider the problem using multiobjective optimization models when reliabilities are
uncertain.
Marseguerra et al. formulate the problem as a multiobjective optimization problem with two objectives:
maximizing the reliability estimate, and minimizing its
associated variance.
In their proposed algorithm, genetic algorithms and Monte
Carlo simulation are combined to identify recommended
solutions, and to obtain a set of Pareto optimal solutions.
Introduction
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Coit et al. use the weighted objective method with
iteratively changing weights to obtain a Pareto
optimal set.
Difficulties with these approaches are that it is
necessary to select objective function weights to
combine the two objective functions, or to select one
preferred solution from a larger Pareto set. Often,
it is not clear what weights to use, or which solution
to select from the Pareto set.
Introduction
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Taguchi et al. formulated the system reliability
optimization with interval coefficients.
In their formulation, problem coefficients were not
known specifically, and only an interval with a
minimum and maximum was available.
They used a genetic algorithm to find solutions.
There have also been several research efforts [6]–
[8] that use fuzzy sets to consider uncertainty in
system reliability optimization problems.
Introduction
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Another approach is to consider the coefficient of
variation(CV) of the system reliability estimate.
Advantages of this approach are that it is not
necessary to select an value or objective function
weights.
The recommended solution is one with a low
standard deviation, but a high estimate of the
system reliability.
Introduction
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Tekiner and Coit present a solution for the problems
where component mixing is not allowed, and the
objective function is to minimize the CV of the system
reliability estimate with respect to minimize system
reliability constraint, and some other system level
constraints.
They convert the problem into a linear integer
programming problem.
The optimal solution of this formulation has a high
reliability estimate with low estimation variability.
Introduction
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In this paper, we extend this work to present a heuristic for the
problems where component mixing is allowed, and the
objective is to find a minimum CV for the system reliability
estimates.
This formulation has certain benefits compared to other
reliability optimization models that considered uncertainty.
The main advantage of the proposed heuristic is that it
provides a convenient method to obtain a solution, with high
estimated reliability, and low variance.
On the other hand, when a lower percentile of system
reliability is used as an objective function [2], the resulting
model is a difficult nonlinear integer programming problem
with potentially different solutions for different percentile
levels.
Redundancy allocation problem for series-parallel system
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fig. 1 presents a typical series-parallel system.
For each subsystem, there are multiple component choices
available, and the problem is the determination of the
component choice and redundancy level for each
subsystem.
Redundancy allocation problem for series-parallel system
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The redundancy allocation problem has been formulated
with many different objective functions, constraints,
problem applications, etc.
Solutions have been determined using mathematical
programming (e.g., dynamic programming, integer
programming), simple heuristics, and heuristic search
(e.g.,genetic algorithms, Tabu search).
However, in almost all cases, it has been assumed that
reliability estimates are known exactly, i.e., they are
deterministic.
Redundancy allocation problem for series-parallel system
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
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The redundancy allocation problem has most often been
formulated such that mixing of components is not allowed.
This means that, if there are functionally equivalent
component choices available, once a component type is
selected, only the same type can be used to provide
redundancy.
Coit and Smith demonstrated that mixing components can
provide better solutions, with higher system reliability.
Redundancy allocation problem for series-parallel system
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The CV is a well-known statistical metric to consider the
uncertainty of a random variable.
The objective function for our new formulation can be
𝑉
stated as C𝑉 = .
𝑅
The resulting problem is often a highly constrained
nonlinear integer programming problem that is difficult to
solve.
Tekiner and Coit show that this problem can be converted
to an equivalent linear integer programming problem by
using the square of the CV for those problems where
mixing of components is not allowed.
In this paper, we are also using the same objective of
minimizing (𝐶𝑉)2 = 𝑉/𝑅2 for those problems where
mixing components is allowed.
Redundancy allocation problem for series-parallel system
Redundancy allocation problem for series-parallel system
Mathematical formulation

The mathematical formulation of the redundancy
allocation problem to minimize CV (by minimizing (𝐶𝑉)2 )
Developed approaches to minimize the
coefficient of variation


This problem is a highly constrained nonlinear
integer programming problem. In this paper, we
propose a neighborhood search heuristic to solve
this problem.
Our heuristic consists of two phases:
 to
determine an optimal solution for the case when
component mixing is not allowed.
 to search for neighborhood solutions for each subsystem
and use linear integer programming to select the set of
subsystem solutions that collectively minimize CV.
Developed approaches to minimize the
coefficient of variation
Problems without component mixing:
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The optimum solution can be found when component
mixing is not allowed by use of an equivalent problem
formulation, and linear integer programming.
For the problem, each subsystem consists of 𝑛𝑖 units of
component type , 𝑗 ∈ {1,2, … , 𝑠𝑖 }.
To convert the problem into an equivalent linear integer
programming problem, first new parameters are defined.
These parameters are 𝑣𝑖𝑗𝑘 , 𝑟𝑖𝑗𝑘 , ∝𝑖𝑗𝑘 , 𝛽𝑖𝑗𝑘 , 𝑛𝑚𝑎𝑥𝑖𝑗 .
Developed approaches to minimize the
coefficient of variation
Problems without component mixing:
Developed approaches to minimize the
coefficient of variation
Problems without component mixing
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An equivalent parameters are objective function can be
determined by subtracting one from (𝐶𝑉)2 , and taking
the logarithm of the result.
Because 𝑣𝑖𝑗𝑘 , 𝑟𝑖𝑗𝑘 , ∝𝑖𝑗𝑘 and 𝛽𝑖𝑗𝑘 are constants for a
particular problem, the problem can be converted into a
linear 0-1 integer programming problem by defining new
decision variables.
This is the first time the system reliability optimization
problem considering uncertainty has been transformed
into a linear problem.
Developed approaches to minimize the
coefficient of variation
26
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New decision variables, and the equivalent linear integer
programming model, can be presented as follows.
Developed approaches to minimize the
coefficient of variation
Problems without component mixing
 Numerical examples [9] show that the solutions
obtained by maximizing reliability are not the same
as the ones obtained by minimizing the CV.
 The redundancy allocations obtained by minimizing
the CV are shifted toward to the components which
have less uncertain reliability estimates.
Developed approaches to minimize the
coefficient of variation
Problems with component mixing:
 Because component mixing may improve the
reliability, it is desirable to find solutions in which
component mixing is allowed.
 However, the model to minimize the CV cannot be
converted into a linear model.
 Therefore, we provide a heuristic to solve the
problem when component mixing is allowed. This
heuristic includes three phases.
Developed approaches to minimize the
coefficient of variation
Problems with component mixing (1):
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The first phase is to solve the problem assuming that the
component mixing is not allowed.
For this phase, the model presented in (3) is solved to
minimize the CV.
consider that the optimum solution 𝑋 ∗ obtained in Phase 1
is as follows.
where represents the optimum solution found for
subsystem i.
where
represents the number of component type 𝑗 in
subsystem i.
Developed approaches to minimize the
coefficient of variation
Problems with component mixing (1):
 Because component mixing is not allowed, the value
of
is one for only one type of component 𝑗, and
zero for others
 Consider that the optimum solution found for
subsystem i is 𝑦𝑖𝑗𝑘 = 1 . Therefore,,
for
component type l, and
for all other
component type 𝑗 .
Developed approaches to minimize the
coefficient of variation
Problems with component mixing (2):
 In this phase, a neighborhood for the solution obtained
in Phase 1 is generated.
 The neighborhood is a set of subsystem solutions that
are near to or close to the Phase 1 solution using some
defined definition for “neighborhood.”
 The solutions in the neighborhood consider all
component choices.
 The goal is to find a set of promising or likely solutions
that are near to a solution that was already found to
be optimal for a constricted version of the problem.
Developed approaches to minimize the
coefficient of variation
Problems with component mixing (2):

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Denote
as the neighborhood(is., set of solutions)
generated for subsystem i around the optimum solution
obtained in Phase 1.
Consider that there are 𝑔𝑖 number of solutions in the set
i.e.,
. . Then we can define an index 𝑙 for
neighborhood solutions such that 𝑙 = 1,2, … , 𝑔𝑖 .
,
Developed approaches to minimize the
coefficient of variation
Problems with component mixing (3):

Given neighborhood solutions for each subsystem i , it is
possible to convert the problem into a linear 0-1 integer
programming problem by defining new parameters, and
new decision variables.
Developed approaches to minimize the
coefficient of variation

Note that
are constants for a particular
problem. Therefore, the problem can be defined as a linear 01 integer programming problem, as follows.
Neighborhood generation
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The heuristic depends on the determination of the
neighborhood,
, for each subsystem.
In this paper, we also propose an approach to generate
the neighborhood around the optimum solution founded
by solving the problem where mixing component is not
allowed(Phase 1).
Neighborhood generation
Neighborhood generation
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Example:
Consider that there are four component types available for
subsystem 𝑖.
Consider that the optimum solution found in the case of no
mixing is (0 0 3 0). That is, three of component type 3 is used
for subsystem 𝑖.
Neighborhood generation
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In this case, case 3 should be applied.
Case 3.1: Keep (3-3)=0 of component type 3, and add all
possible two-component combinations. In this step, redundancy
allocations with (0+2)=2 components are generated.
Neighborhood generation
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Case 3.2: Keep (3-2)=1 of component type 3, and add all
possible two-component combinations. In this step, redundancy
allocations with (1+2)=3 components are generated.
Neighborhood generation
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Case 3.3: Keep (3-1)=2 of component type 3, and add all
possible two-component combinations. In this step, redundancy
allocations with (2+2)=4 components are generated.
The neighborhood has 30 solutions which have 2,3, or 4
components.
Numerical examples
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Table VI presents the component data for the examples. We
use C++ to generate the neighbors, and GAMS with CPLEX to
solve mathematical formulations (3) and (4).
Numerical examples
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As a first example, we solve the problem with 𝑐0 = 182 and
𝑤0 = 159. For comparison purposes, we first use the objective
function of maximizing the system reliability estimate.
Numerical examples
Numerical examples
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For the second example, we solve the same problem with 𝑅0
= 0.95.
Numerical examples
Numerical examples
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The proposed heuristic provides a better solution than the
one where component mixing is not allowed. The solution
that minimizes CV provides a compromise solution offering
both high reliability and low estimation uncertainty.
Therefore, the system reliability is not improved in some
instances where component mixing is allowed.
Because it is also possible that the components with
smaller variance have larger weights, when the weight
limitation increased, these components are selected which
results in lower CV. The components selected have
smaller variances, and it is possible for them to have
relatively smaller reliability.
Conclusion
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Consideration of uncertainty is important for many decision
maker. They want both higher reliability, and lower estimation
variance.
The solution for maximizing reliability and minimizing CV are
not the same.
The redundancy allocations in those having CV the objective
function are shifted toward the components which have smaller
variances.
However, the objective function for this problem also depends
on the minimum system reliability requirement.
A higher minimum system reliability requirement results in a
smaller improvement in the system variance.
Conclusion
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The results show that trying to optimize CV can be
very beneficial for reliability design problems
where a lower system reliability variance is very
important.
THANK YOU