System reliability optimization considering uncertainty: Minimization of the coefficient of variation for series-parallel systems HaticeTekiner-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 • Giveparameter： 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 • Giveparameter： 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 Coitdeﬁne 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 – 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 Coitpresent 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 , 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.
Redundancy allocation problem for series-parallel system • 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.,geneticalgorithms, 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 • 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 • 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 variation Problems 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 subsystemconsists 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 variation Problems 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.
Developed approaches to minimize the coefficient of variation • 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  show that the solutions obtained by maximizing reliability are not the same as the ones obtained byminimizing 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): • 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.
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 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 .
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): • Denoteas 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
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 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 • 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 add all possible two-component combinations. In this step, redundancy allocations with (1+2)=3 components are generated.
Neighborhood generation • 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 • 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 • For the second example, we solve the same problem with
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.