hatice tekiner-mogulkoc , david w. coit presented by chia-ling lee

SYSTEM RELIABILITY OPTIMIZATION CONSIDERING UNCERTAINTY: MINIMIZATION OF THE COEFFICIENT OF VARIATION FOR SERIES-PARALLEL SYSTEMS

Hatice Tekiner-Mogulkoc, David W. CoitPresented 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。

ModelDefender 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

ModelAttacker 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 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

Neighborhood generation Numerical examples conclusion

Introduction 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 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 Coit and Smith , and Coit define an percentile of

the system reliability distribution and/or system time-to-failure 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 Another approach is to consider the problem

using multi-objective optimization models when reliabilities are uncertain.

Marseguerra et al. formulate the problem as a multi-objective 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 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 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 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 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 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

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.


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.


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.


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 stated as .

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 for those problems where mixing components is allowed.

Mathematical formulation The mathematical formulation of the

redundancy allocation problem to minimize CV (by minimizing )

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 variationProblems without component mixing: 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 units of component type , .

To convert the problem into an equivalent linear integer programming problem, first new parameters are defined.

These parameters are

Developed approaches to minimize the coefficient of variationProblems without component mixing:

Developed approaches to minimize the coefficient of variation

Problems without component mixing An equivalent parameters are objective

function can be determined by subtracting one from , and taking the logarithm of the result.

Because and 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.

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New decision variables, and the equivalent linear integer programming model, can be presented as follows.


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.


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.


Problems with component mixing (1): 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 obtained in Phase 1 is as follows.

where represents the optimum solution found for subsystem i.

where represents the number of component type in subsystem i.


Problems with component mixing (1): Because component mixing is not

allowed, the value of is one for only one type of component and zero for others

Consider that the optimum solution found for subsystem i is . Therefore,, for component type l, and for all other component type .


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.


Problems with component mixing (2): Denote as the neighborhood(is., set of

solutions) generated for subsystem i around the optimum solution obtained in Phase 1.

Consider that there are number of solutions in the set , i.e., . . Then we can define an index for neighborhood solutions such that


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 0-1 integer programming problem, as follows.

Neighborhood generation 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 Example: Consider that there are four component types

available for subsystem . 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 .

Neighborhood generation 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 Case 3.2: Keep (3-2)=1 of component type 3, and


Neighborhood generation Case 3.3: Keep (3-1)=2 of component type 3, and


The neighborhood has 30 solutions which have 2,3, or 4 components.

Numerical examples 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 As a first example, we solve the problem with

and . For comparison purposes, we first use the objective function of maximizing the system reliability estimate.

Numerical examples

Numerical examples For the second example, we solve the same

problem with

Numerical examples

Numerical examples 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 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 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

hatice tekiner-mogulkoc , david w. coit presented by chia-ling lee

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