IEEE TRANSACTIONS ON RELIABILITY, VOL. 61, NO. 4, DECEMBER 2012
957
The Robust Deviation Redundancy Allocation Problem With Interval Component Reliabilities Mohammad Javad Feizollahi and Mohammad Modarres
Abstract—We propose a robust deviation framework to deal with uncertain component reliabilities in the constrained redundancy optimization problem (CROP) in series-parallel reliability systems. The proposed model is based on a linearized binary version of standard nonlinear integer programming formulations of this problem. We extend the linearized model to address uncertainty by assuming that the component reliabilities belong to an interval uncertainty set, where only upper and lower bounds are known for each component reliability, and develop a Min-Max regret model to handle data uncertainty. A key challenge is that, because the deterministic model involves nonlinear functions of the uncertain data, classical robust deviation approaches cannot be applied directly to find robust solutions. We exploit problem structures to develop four exact solution methods, and present computational results demonstrating their performance.
smallest possible value of largest possible value of set of all possible scenarios for number of components in subsystem minimum value of maximum value of overall system reliability function an increasing convex function of , which represents the effect of in th constraint maximum available resource in th constraint
Index Terms—Interval data, nonlinear integer programming, redundancy allocation, robust deviation.
set of all feasible solutions logarithm of
ACRONYMS
under scenario
CROP
Constrained Redundancy Optimization Problem
RCROP
Robust CROP
MIP
Mixed Integer Programming
B
Classical Benders Decomposition
BC
Benders Decomposition with Cut Callback
worst-case alternative for solution
AB
Accelerated Benders Decomposition
ABC
Accelerated Benders Decomposition with Cut Callback
binary variable, which is 1 if otherwise
optimal solution of CROP under scenario regret of a solution
under scenario
robustness cost of a solution worst-case scenario for solution
NOTATION
, and 0
coefficient of
in the objective function
coefficient of
in the th constraint
lower bound of the RCROP
number of subsystems in series
upper bound of the RCROP
set of subsystem set of constraints
I. INTRODUCTION
reliability of a component in subsystem
G
uncertain
Manuscript received October 23, 2011; revised April 04, 2012 and June 08, 2012; accepted June 10, 2012. Date of publication October 09, 2012; date of current version November 27, 2012. This research was partially supported by INSF. Associate Editor: G. Levitin. M. J. Feizollahi is with the Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran (e-mail:
[email protected]). M. Modarres is with the Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran, and is also with the Center of Excellence in Power System Management and Control (e-mail:
[email protected]). Digital Object Identifier 10.1109/TR.2012.2221032
ENERALLY, reliability is a system performance measure. By increasing the complexity of systems, the consequences of their unreliable behavior have become severe in terms of cost, effort, lives, etc., hence the need for improving the products and systems reliability have become very important [1]. Series configuration for reliability systems is often encountered in industrial and engineering applications. In this configuration, the reliability of the system can be improved by allowing redundancy incorporation [1]. Fig. 1 illustrates a series system with parallel redundant components, where the system functions if there is a path formed by functioning parts which
0018-9529/$31.00 © 2012 IEEE
958
IEEE TRANSACTIONS ON RELIABILITY, VOL. 61, NO. 4, DECEMBER 2012
Due to the monotonicity of the logarithm operator, we can replace the objective in CROP to obtain the following nonlinear integer programming equivalent formulations involving separable functions. In the remainder of this paper, we consider the below reformulation of CROP.
Fig. 1. Series-parallel configuration of components in a reliability system.
travels through the system. We consider the redundancy allocation problem in a series-parallel system with subsystems in series, and each subsystem consisting of identical parallel components. Each subsystem component has a specified reliability, and the goal is to determine the number of components in each subsystem to achieve the desired overall system reliability. The problem of maximizing the overall system reliability, subject to some limited resource constraints such as budget or technical specifications, is known as the constrained redundancy optimization problem (CROP) (see [1]–[3]). For a reliability system with subsystems in series, and the given specified reliability of a component in subsystem , , and the to-be-determined number of components in subsystem , the overall system reliability as a function of the number of components is given by
and, CROP can be formulated as (see [1]–[3]) (1) (2) and (3) where is the set of subsystems in series, is the set of all constraints defined by , an increasing convex function of for all , and the set denotes the specified bounds on the number of components in each subsystem. Note that CROP assumes that each constraint (2) is specified by a separable convex (may be linear) function of the decision variables . CROP is known to be NP-hard (see Chern, [4]), and heuristic and exact solution methods for it have been extensively studied (see [5] for a survey). We assume throughout that the feasible region of CROP is such that for all feasible solutions . Then, by taking the logarithm of , we obtain
where,
(4) In much of the existing literature, it is assumed that all problem data are known with certainty. However in reality there is considerable uncertainty and inaccuracy in the estimation of the problem parameters, particularly the component reliabilities. In the presence of data uncertainty, decision makers may not feel confident if they rely on solutions of deterministic optimization models which only use some estimation of uncertain data. In particular, because of the nonlinear relation between component reliabilities and system reliability in CROP, we know that small perturbations in component reliabilities (which are unavoidable because of measurement errors) may cause large deviations in the overall system reliability, which are not desirable. Fuzzy programming [6]–[8], stochastic programming [9]–[12], penalty function technique [13], system-reliability confidence-intervals [14], multi-objective models that maximizes the system reliability estimate while minimizing its associated variance [15], [16], and robust designs with two level optimization [17] are some approaches that have been used in the literature to consider uncertain component reliabilities in system reliability problems. Although these approaches mitigate uncertainty issue in some way, they have their own difficulties. The first concern is finding membership and distribution functions of fuzzy and stochastic parameters, which is not so trivial. The second and more important concern is solving the resulting problems, which are difficult to solve, even for very small instances. Therefore, they are mostly solved approximately by some heuristic methods. To handle uncertainty, in this paper we use the Min-Max regret (also known as robust deviation) approach to find conservative solutions to hedge against variations of the input data. Min-Max regret combinatorial optimization problems have been studied extensively over the past two decades (see, e.g., [18]–[22], and the references therein). The motivation for using the Min-Max regret approach as a modeling tool is thoroughly discussed in the book [22]; This approach is suitable in situations where the decision maker may feel regret if he or she makes a wrong decision. Maximum regret can also serve as an indicator of how much the performance of a decision can be improved if all uncertainties could be resolved [22]. In Min-Max regret models for interval data, it is assumed that, for each objective function coefficient (the reliability of each component in CROP model), an uncertainty interval is specified, and the set of scenarios is the Cartesian product of all uncertainty intervals. In this approach, we aim to find a solution with minimum largest deviation over all possible scenarios for
FEIZOLLAHI AND MODARRES: DEVIATION REDUNDANCY ALLOCATION PROBLEM WITH COMPONENT RELIABILITIES
component reliabilities, between the value of the solution and the optimal value of the corresponding scenario. This paper is organized as follows. In Section II, we present interval reliability CROP, and some basic definitions and exploit its structure to find the worst case scenario for a given feasible solution. Then, in Section III, we propose robust CROP in a Min-Max regret framework, formulate it as an integer programming model, and develop 4 exact algorithms to solve it. Section IV contains experimental results comparing the performance of these exact algorithms. Finally, we present some concluding remarks in Section V. II. INTERVAL RELIABILITY CROP In interval reliability CROP, it is assumed that the reliability of component , , is not known exactly, and it is a random value in . Let be a scenario which is a realization of the component reliabilities, be the set of all possible vectors (set of scenarios), and be the set of all feasible solutions .
959
In the sequel, we propose a binary linear model to find the worst-case scenario and alternative for a given feasible solution. A. Finding Worst-Case Scenario and Alternative for a Given Solution Let us first describe some parameter details, as well as a proposition, which are useful in developing desired model. For any , the worst-case scenario, , maximize over all , and the worst-case alternative is equal to . If there is more than one optimal solution as , take any of them as . For a given pair of solutions and , let denote the reliability vector that maximizes (with obviously defined elements). Then, for any , . Let for any . Proposition 1: For any given pair of solutions , , it holds that
. and,
Let nario ,
be the objective value of solution , and ; that is,
under sce-
(See the Appendix A for the proof of the proposition.) , From Proposition (1), for a given solution depends on and , and cannot be determined before finding . From the definition of in (3), takes an integer value in for all . Therefore, we can write
(5) Problem CROP : For a given scenario , the classic CROP model of (4) can be represented as Maximize
where (7) By substituting the integer variable with the binary variables , we obtain linear binary equivalent formulations of CROP as
Definition: 1) Denote the optimal solution under scenario as . In other words, is the solution which maximizes over for a given scenario . 2) Robust deviation or regret of a solution under scenario , , is defined as
3) Robustness cost of a solution , imum robust deviation as
, which is its max(8) where
Problem RCROP: The Min-Max regret version of the CROP problem, which we call the “robust CROP” (RCROP), is as follows. Minimize
(6)
, (9)
and (10)
960
IEEE TRANSACTIONS ON RELIABILITY, VOL. 61, NO. 4, DECEMBER 2012
The correctness of the above reformulation follows from noting that
Hence,
Therefore, (14) can be reformulated as (15)
(11) In (11), the third identity follows from the fact that is a . Similarly, it can be shown binary variable, and . that By modifying in (9), we can exploit the binary linear equivalent of the CROP model of (8) to find for a given solution . This modification can be done as follows.
This formulation has constraints (15) (other than the constraints ). In Benders decomposition approach (see [23]), we relax these constraints, starting with a small subset of them, and progressively taking new constraints (Benders cuts) into account. The corresponding relaxed formulation is called the master problem; it is defined by using a subset instead of in (15). After solving the master problem, a sub problem is solved that seeks the constraint (cut) that is most violated by the obtained solution. After incorporating this constraint, the master problem is re-solved, and so on. When the optimal solution to the master problem does not violate any constraints of the original formulation, it must be an optimal solution to the original problem. Solving a master problem provides a lower bound, and solving a sub problem provides an upper bound for the optimal value of the RCROP. Hence, the master problem is
(12)
Recall that in (12) is a fixed, known integer value. In the following proposition, we will prove the validity of (12). Proposition 2: The optimal value of the model (8) with (10) and (12) is equal to , and , where is its optimal solution. (See the Appendix B for the proof of the proposition.)
where is a subset of . By exploiting (7), we propose the binary linear equivalent of the master problem as
for all III. ROBUST CONSTRAINED REDUNDANCY OPTIMIZATION PROBLEM RCROP in (6) can be rewritten as
(16)
or alternatively, (13)
where
,
(14)
(17)
The latter formulation has an infinite number of constraints (14). Here, we provide a tractable version of (13) and (14). By taking advantage of , which was defined before as equal to , it is easy to see that
and are calculated from (10). Let , and be the lower, and upper bounds of the Problem RCROP, respectively. In Benders decomposition approach, at each iteration after finding a solution and for the master problem, we update . Then, solve a sub problem to find the worst-case scenario and alternative of , and update . If , add the appropriate cut to the master problem, and so on. Next, we present four algorithms to solve RCROP in Benders decomposition framework.
for all
and
FEIZOLLAHI AND MODARRES: DEVIATION REDUNDANCY ALLOCATION PROBLEM WITH COMPONENT RELIABILITIES
A. Exact Solution Algorithms Here, we present four algorithms to solve RCROP in Benders decomposition framework. In the next section, we will evaluate the performance of these algorithms. Algorithm 1. (B) Classical Benders Decomposition: Step 0. Initial Setting. Set , , and , where is an arbitrary feasible solution. from (17), and from (10). Calculate Step 1. Master Problem. Solve the master problem (16), with a general Mixed Integer Programming (MIP) solver like CPLEX (see [24]). Let be its optimal solution. Because , is a lower bound for RCROP. Obviously, does not decrease by replacing with , where . Hence, at each iteration, we can update the lower bound as . , the Step 2. Sub Problem. The sub problem finds worst-case alternative for , by solving a regular instance of the CROP model (8) with (10) and (12). Recall that is equal to the optimal value of this model. After finding the worst-case alternative for , it is straightforward to find by Proposition (1), and consequently to find . Because and is feasible with respect to constraints (14), is an upper bound for RCROP. In fact, we should update the upper bound if it is greater than . Therefore, at each iteration, if , we update the upper bound as . , the optimal solution Step 3. Stopping Criteria If for the Problem RCROP is found, and stop; otherwise, go to Step 4. Step 4. Add Benders Cut. The most violated constraint to be included in the master problem (16) is
961
Algorithm 3. (AB) Accelerated Benders Decomposition: Rei et al. [25] used local branching throughout the solution process to accelerate the classical form of Benders decomposition method. We used this approach as follows. Consider the Algorithm 1 (B), and make the following adjustments. In Step 1, save all incumbent solutions throughout the solution process of the master problem (16). Experimental results show that the optimal solution of the master problem is often found well before the solution process completes. Therefore, there is no need to solve the master problem exactly, and we can set a time or node limit or both in CPLEX. In Step 2, instead of solving the sub problem for the optimal solution of the master problem, solve the sub problem for a certain number of incumbents which are found in Step 1, and update , if needed. In Step 4, add all violated cuts associated with incumbents described in Step 2. This updated algorithm is implemented in two phases. At the first phase, we should specify the time or node limits or both, and the number of cuts that we want to add at each iteration. We set a time limit for this phase. The first phase terminates if it reaches its time limit, or no new incumbent is found. Then, in the second phase, we relax time and node limits on the master problem, and put a time limit on the second phase. This algorithm can be treated as an exact method to solve RCROP if we do not set a limit on the second phase. The last algorithm is a combination of Algorithms 2 (BC) and 3 (AB). Algorithm 4. (ABC) Accelerated Benders Decomposition With Cut Callback: In this algorithm, we combine Algorithms 2 (BC) and 3 (AB) as follows. At first, we run the first phase of Algorithm 3 (AB), then we switch to Algorithm 2 (BC). In the next section, we report experimental results of running these four algorithms on 3 classes, 12 groups, and 240 randomly generated instances. IV. EXPERIMENTAL RESULTS
or its linear equivalent (18)
To generate the test instances, we adopt the same approach as [3]. Two sets of linear and nonlinear constraints are considered. • Linear constraints: where , , . • Nonlinear constraints:
, calculate from (17), add the most Set violated constraint (18) to the master problem (16), and go to Step 1. Another approach to complement Benders decomposition method is adding cuts to the branch-and-bound tree, while we are solving the master problem. Algorithm 2. (BC) Benders Decomposition With Cut Callback: In addition to Algorithm 1 (B) above, where the relaxed RCROP is solved to optimality in each iteration, we can add the Benders cuts within a branch-and-bound method for solving RCROP whenever an incumbent solution is found. In this approach, just one master problem has to be solved, and Benders cuts are added at incumbent nodes of the branch-and-bound tree. There is another approach in Benders framework which accelerates solution times and the process of adding cuts.
where , . The upper bound of reliability of each component, , and interval length, , belong to [0.80,0.98], and , respectively. All of these coefficients in all instances are randomly generated from a uniform distribution. We assume that , and . We set to assure feasibility, where for linear constraints, and for nonlinear constraints. We coded our proposed algorithms in
962
IEEE TRANSACTIONS ON RELIABILITY, VOL. 61, NO. 4, DECEMBER 2012
TABLE I EXPERIMENTAL RESULTS FOR
EXPERIMENTAL RESULTS FOR
EXPERIMENTAL RESULTS FOR
AND
AND
INSTANCES WITH
AND
TABLE II INSTANCES WITH
, AND
TABLE III INSTANCES WITH
C++, and called CPLEX 12.1 to solve MIP sub problems when needed. All algorithms were conducted on a UNIX machine by applying restrictions of 10 GB of RAM, and a single processor (Thread) 2.27 GHz CPU. Similar to [3], we consider 3 classes of instances. • : instances with linear constraints, and subsystems. We generated 6 groups of this class for . • : instances with non-linear constraints, and subsystems. Similar to the group, we generated 6 groups of this class for . • : instances with linear constraints, and 90 subsystems. This class is considered in 4 groups for . The output reported in Tables I–IV are obtained by running the algorithms for 20 instance of each group. For instances, in and groups with , we ran all 4 algorithms, and set a limit of 7200 seconds of total solution time on each instance; while for the others we just ran Algorithms 2–4 (BC, AB, and ABC), and set a limit of 10,800 seconds total time on the algorithms. In Algorithms 3 (AB), and 4 (ABC), we put a time limit on the first phase equal to 20% of the total time limit. At this phase, we applied a time limit equal to seconds on each master problem. At each iteration of
, AND
#
OF
(SOLVED TO THE OPTIMALITY)
(NOT SOLVED TO THE OPTIMALITY)
TABLE IV AND INSTANCES CUTS AND OPTIMALITY GAP FOR WITH AND (AVERAGE AMONG ALL 20 INSTANCES FOR EACH GROUP)
accelerated Benders Method, if the number of incumbents was less than 8, we just selected the minimum of that number and 4; otherwise, we chose half of them to add associated cuts to the master problem. After termination of the first phase, we just use the remaining time out of the total time limit as a restriction on the second phase (master problem) in Algorithm 3 (Algorithm 4, respectively). and The results of running 4 algorithms on instances in groups with are presented in Table I, where “# of cuts” is the number of cuts generated during problem solving, ” is “ ” is the total solution time in seconds, and “ the portion of time spent on solving the sub problem. Except for
FEIZOLLAHI AND MODARRES: DEVIATION REDUNDANCY ALLOCATION PROBLEM WITH COMPONENT RELIABILITIES
4 instances out of 20 instances in group , all other instances were solved to optimality within 7200 seconds by all 4 algorithms. The average optimality gap, and the number of cuts for these 4 instances were 1.38%, and 253, respectively. Algorithm 1 (B) could not solve these 4 instances in group , but other algorithms solved them optimally within 7200 seconds. Thus, in Table I, we reported the average results only for 16 instances of under Algorithm 1 (B), which are solved to optimality, but all other results are the average of 20 instances. Regarding Table I, Algorithm 1 B has the least number of cuts, and the most total solution times. As we expected, by increasing , the number of cuts, and the total solution times indecreases in both and groups, crease, and which means that, by increasing the size of problem, most of the solution time is spent on the master problem. Therefore, it seems interesting to improve the performance of methods solving the master problem. Overall, Algorithms BC, ABC, AB, and B are the fastest to slowest methods.For larger instances, we just ran the Algorithms BC, ABC, and AB on 20 instances in each group; and the results are shown in Tables II–IV. In Table II, the results for optimally solved instances within 10,800 seconds are shown. On the other hand, Table III contains the results for instances that could not be solved optimally in that time limit. In Tables III and IV, “gap%” is the average optimality gap for the instance that cannot be solved optimally. Regarding the average optimality gap in Tables IV, we conclude that, in the sense of quality of solutions for large instances, Algorithm BC outperforms Algorithm AB, and Algorithm ABC outperforms both of them. Regarding Tables I–IV, we can find an exact solution of moderate size RCROP instances, within a reasonable time and memory limit. Hence, this approach can be considered as a practical one with appealing, reliable solutions; it is possible to obtain exact solutions of moderate instances and near optimal solutions for relatively large instances with practical sizes. To our knowledge, this is the first time that a Min-Max regret approach is proposed to be used in the CROP model, so there is no direct benchmark formulations or solution algorithms in the literature to compare with ours. Here, we compare the performance of our proposed methods to solve RCROP with the algorithm that Sun et al. [3] developed to solve CROP. They compared the speed of their algorithm with other methods in the literature, and showed that their algorithm outperforms others in the literature. They coded their algorithm in Fortran 90, and ran it on a Pentium IV PC (2 GHz and 256 Mb RAM) to solve the classical CROP on similar instances that we generated, except that they ignored uncertainty in the instances. Their results are briefly presented in Table V, where is the solution time in seconds for classical CROP instances. By comparing the results in Table V with Tables I–IV, we conclude that solving RCROP is much more challenging than CROP. It is natural, and expected, because to solve the RCROP instances, we need to iteratively solve the growing master problems, each one being a CROP instance. Note that the optimal solution of RCROP has less risk, and better quality than CROP, in the sense of the worst-case objective value.
963
TABLE V SOLUTION TIME FOR CLASSICAL CROP WHICH SUN ET AL. REPORTED IN [3]
V. CONCLUSION We proposed and studied the robust deviation (Min-Max regret) CROP in series-parallel reliability systems with interval data. In the proposed model, uncertainty about component reliabilities, which is one of the important concerns of decision makers in finding robust, reliable solutions, is taken into account. The Min-Max regret approach is suitable when the decision maker may feel regret if he or she makes a wrong decision. We proposed a MIP formulation, and developed four exact solution algorithms, by exploiting the problem structures that we identified, to find the solutions with optimal robust deviation. A key challenge is that, because the deterministic CROP model involves nonlinear functions of the uncertain data, classical robust deviation approaches cannot be applied directly to find robust solutions. Finally, we presented computational results demonstrating the performance of the proposed algorithms and quality of solutions. Based of the analysis of numerical results, we can make several conclusions and recommendations. 1) All four proposed exact algorithms are able to efficiently handle problems of small size ( , , , and ). 2) Algorithm B generates the minimum number of Benders cuts, and performs satisfactorily for small instances from a total solution time point of view, but its performance deteriorates as the size of the problem grows. 3) Except for Algorithm B, all other methods can solve medium size instances ( and ) optimally. They can also solve large instances ( , , and ) with reasonable optimality gaps within the time limit. 4) Algorithm BC, which adds the Benders cuts within a branch-and-bound framework, has slightly more cuts than Algorithm B, but it outperforms other methods for medium size instances, and also for large instances which can be solved optimally within the time limit. 5) Algorithm AB generates fewer cuts than Algorithm BC for large instances. For most of the large instances we tested that cannot be solved optimally within the time limit, it obtains better solutions than Algorithm BC from an optimality gap perspective. 6) Algorithm ABC, which is a combination of algorithms BC and AB, appears to be the best algorithm for large instances.
964
IEEE TRANSACTIONS ON RELIABILITY, VOL. 61, NO. 4, DECEMBER 2012
To our knowledge, this is the first time that a Min-Max regret approach is proposed to be used in the CROP model, so there is no benchmark formulations or solution algorithms in the literature to compare with ours. Comparing the solution time for classical CROP with algorithms in the literature and RCROP with our proposed algorithms demonstrates that solving RCROP is much more challenging than CROP. We note that, in the Robust deviation of CROP, even the evaluation of a solution involves solving a regular instance of CROP, which itself is an NP-hard problem. So the applicability of the results of this paper is limited to problem sizes and cases where the classical CROP can be solved efficiently (within seconds or minutes). Further research can be conducted on developing fast heuristics to solve the robust deviation CROP model approximately. Also, generalizing the proposed model and algorithms to the CROP with multiple choice of component type in each subsystem, or with the allowance of using the same type of components in different subsystems or even more complex reliability systems, can be potential future research topics.
APPENDIX B PROOF OF THE PROPOSITION 2 Here we provide a proof for Proposition 2. Proof: Let us replace with in (7). By definition of the robustness cost of , , it holds that
APPENDIX A PROOF OF THE PROPOSITION 1 Here we provide a proof for Proposition 1. . Therefore, Proof: Let
and the first partial derivative of
with respect to
is
(19) Let
be the numerator in (19). Then,
where is as in (12). Here, the third equality is concluded from Proposition (1), and the fifth identity follows from the fact that is a binary variable and . Then, the proof is complete. REFERENCES
where its sign is the same as the sign of because we , for all , have and , . On the other hand, because , is positive (negative) if ( , re. Furthermore, because the denomispectively) for nator of (19) is always positive, the sign of will be determined by the sign of . Thus, is if a monotone increasing (decreasing) function of ( , respectively). Hence, to maximize over , should be (or ) when (or , , can take any value in respectively). When . Then, the proof is complete.
[1] F. A. Tillman, C. L. Hwuang, and W. Kuo, Optimization of System Reliability. : Marcel Dekker, 1980. [2] S. G. Tzafestas, “Optimization of system reliability: A survey of problems and techniques,” International Journal of Systems Science, vol. 11, no. 4, pp. 455–486, 1980. [3] X. Sun, N. Ruan, and D. Li, “An efficient algorithm for nonlinear integer programming problems arising in series-parallel reliability systems,” Optimization Methods and Software, vol. 21, no. 4, pp. 617–633, 2006. [4] M. Chern, “On the computational complexity of reliability redundancy allocation in a series system,” Operations Research Letters, vol. 11, no. 5, pp. 309–315, 1992. [5] W. Kuo and R. Wan, “Recent advances in optimal reliability allocation,” IEEE Trans. Systems, Man and Cybernetics—Part A: Systems and Humans, vol. 37, no. 2, pp. 143–157, 2007. [6] A. K. Dhingra, “Optimal apportionment of reliability and redundancy in series systems under multiple objectives,” IEEE Trans. Reliability, vol. 41, no. 4, pp. 576–582, 1992.
FEIZOLLAHI AND MODARRES: DEVIATION REDUNDANCY ALLOCATION PROBLEM WITH COMPONENT RELIABILITIES
[7] K. S. Park, “Fuzzy apportionment of system reliability,” IEEE Trans. Reliability, vol. R-36, no. 1, pp. 129–132, 1987. [8] S. Wang and J. Watada, “Modelling redundancy allocation for a fuzzy random parallel-series system,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 539–557, 2009. [9] A. K. Bhunia, L. Sahoo, and D. Roy, “Reliability stochastic optimization for a series system with interval component reliability via genetic algorithm,” Applied Mathematics and Computation, vol. 216, no. 3, pp. 929–939, 2010. [10] L. Painton and Campbell, “Genetic algorithms in optimization of system reliability,” lEEE Trans. Reliability, vol. 44, no. 2, pp. 172–178, 1995. [11] R. Rubinstein, G. Levitin, A. Lisniaski, and H. Ben-Haim, “Redundancy optimization of static series-parallel reliability models under uncertainty,” IEEE Trans. Reliability, vol. 46, no. 4, pp. 503–511, 1997. [12] A. A. Taflanidis and J. L. Beck, “Stochastic subset optimization for reliability optimization and sensitivity analysis in system design,” Computers and Structures, vol. 87, no. 5/6, pp. 318–331, 2009. [13] R. K. Gupta, A. K. Bhunia, and D. Roy, “A GA based penalty function technique for solving constrained redundancy allocation problem of series system with interval valued reliability of components,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 275–284, 2009. [14] D. W. Coit, “System-reliability confidence-intervals for complex-systems with estimated component-reliability,” IEEE Trans. Reliability, vol. 46, no. 4, pp. 487–493, 1997. [15] D. W. Coit, T. Jin, and N. Wattanapongsakorn, “System optimization with component reliability estimation uncertainty: A Multi-Criteria Approach,” IEEE Trans. Reliability, vol. 53, no. 3, pp. 369–380, 2004. [16] N. Wattanapongskorn and D. W. Coit, “Fault-tolerant embedded system design and optimization considering reliability estimation uncertainty,” Reliability Engineering and System Safety, vol. 92, no. 4, pp. 395–407, 2007. [17] X. Cheng, “Robust reliability optimization of mechanical components using non-probabilistic interval model,” in 2009 IEEE 10th International Conference on Computer-Aided Industrial Design & Conceptual Design, 2009, pp. 821–825. [18] H. Aissi, C. Bazgan, and D. Vanderpooten, “Min-max and min-max regret versions of combinatorial optimization problems: A survey,” European Journal of Operational Research, vol. 197, no. 2, pp. 427–438, 2009.
965
[19] I. Averbakh, “On the complexity of a class of combinatorial optimization problems with uncertainty,” Mathematical Programming, vol. 90, no. 2, pp. 263–272, 2001. [20] A. Kasperski, Discrete Optimization With Interval Data: Minmax Regret and Fuzzy Approach (Studies in Fuzziness and Soft Computing). : Springer, 2008. [21] A. Kasperski and P. Zielinski, “An approximation algorithm for interval data minmax regret combinatorial optimization problems,” Information Processing Letters, vol. 97, no. 5, pp. 177–180, 2005. [22] P. Kouvelis and G. Yu, Robust Discrete Optimization and its Applications. : Kluwer, 1997. [23] J. F. Benders, “Partitioning procedures for solving mixed-variables programming problems,” Numerische Mathematik, vol. 4, no. 1, pp. 238–252, 1962. [24] “IBM ILOG CPLEX V12.1: User’s Manual for CPLEX,” 2009. [25] W. Rei, J. Cordeau, M. Gendreau, and P. Soriano, “Accelerating benders decomposition by local branching,” INFORMS Journal on Computing, vol. 21, no. 2, pp. 333–345, 2009.
Mohammad Javad Feizollahi is a Ph.D. candidate at Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran. He received his Masters, and Bachelors degrees in Industrial Engineering from the same department in 2007, and 2005, respectively. His research interests include operations research and management science, optimization under uncertainty, robust and stochastic optimization, large scale linear and integer programming, and simulation. He is a member of INFORMS.
Mohammad Modarres is a Professor at the Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran. He is also a member of the Center of Excellence in Power System Management & Control. He received his Ph.D. in Systems Engineering and Operations Research from University of California, Los Angles (UCLA) in 1975. His research interests are in operations research, robust optimization, and stochastic dynamic programming.
