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The Robust Redundancy Allocation Problem in Series-Parallel Systems With Budgeted Uncertainty Mohammad Javad Feizollahi, Student Member, IEEE, Shabbir Ahmed, and Mohammad Modarres

Abstract—We propose a robust optimization framework to deal with uncertain component reliabilities in redundancy allocation problems in series-parallel systems. The proposed models are based on linearized versions of standard mixed integer nonlinear programming (MINLP) formulations of these problems. We extend the linearized models to address uncertainty by assuming that the component reliabilities belong to a budgeted uncertainty set, and develop robust counterpart models. A key challenge is that, because the models involve nonlinear functions of the uncertain data, classical robust optimization approaches cannot apply directly to construct their robust optimization counterparts. We exploit problem structure to develop robust counterparts and exact solution methods, and present computational results demonstrating their performance. Index Terms—Budgeted uncertainty, mixed integer nonlinear programming, redundancy allocation, robust optimization, seriesparallel system.

ACRONYMS

AND

ABBREVIATIONS

MIPR_C

Benders Decomposition with Cut Callback

BR

Robust Counterpart of uncertain BL_CROP or BL_COST NOTATIONS number of subsystems in series set of subsystem set of constraints reliability of a component in subsystem uncertain largest possible value of largest possible difference of

from

number of components in subsystem budgeted uncertainty set with protection level for perturbation vector

CROP

Constrained Redundancy Optimization Problem

COST

Cost minimization reliability problem

MIP

Mixed Integer (Linear) Programming

MINLP

Mixed Integer Non-Linear Programming

MIP_CROP

MIP equivalent of CROP

MIP_COST

MIP equivalent of COST

an increasing convex function of , which represents the effect of in th constraint

BL_CROP

Binary Linear equivalent of CROP

maximum available resource in th constraint

BL_COST

Binary Linear equivalent of COST

RCROP

Robust CROP

a piece-wise linear convex function of coincides with at integer s

RCOST

Robust COST

constants representing th segment of

MIPR_B

Classical Benders Decomposition

an increasing convex function of the components cost of subsystem

Manuscript received January 2013; revised July 25, 2013; accepted September 08, 2013. Date of publication January 17, 2014; date of current version February 27, 2014. The work of M. J. Feizollahi and M. Modarres was supported in part by the INSF. The work of S. Ahmed was supported by the Air Force of Scientific Research under Grant FA9550-08-1-0117 and the National Science Foundation under Grant CMMI-0758234. Associate Editor: G. Levitin. M. J. Feizollahi and S. Ahmed are with H. Milton Stewart School of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA (e-mail: [email protected]; [email protected]). M. Modarres with the Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran. He is also with the Center of Excellence in Power System Management and Control (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TR.2014.2299191

minimum value of maximum value of reliability function of subsystem overall system reliability function

which

representing

overall system cost function minimum required system reliability in COST logarithm of set of all feasible solutions

for CROP

set of all feasible solutions scenario

for COST under

set of all robust feasible solutions to COST logarithm of logarithm of

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As described in [1]–[4], for a given reliability system with subsystems in series (like Fig. 1), known component reliability , and the to-be-determined number of components in subsystem noted as , the overall system reliability is given by (1) Fig. 1. Series-parallel configuration of components in a reliability system.

Then, CROP can be formulated as a piece-wise linear concave function of coincides with at integer s

which

(2) (3)

constants representing th segment of under scenario and

robust reliability of a solution robustness cost of a solution

(4)

worst-case scenario for solution a continuous variable a continuous variable binary variable, which is 1 if otherwise

, and 0

coefficient of

in the log reliability function

coefficient of

in the th constraint

The set denotes the specified bounds on the number of components in each subsystem. Note that CROP assumes that each constraint (3) is specified by a separable convex (may be linear) increasing function of the decision variables . Analogously, COST can be formulated as (5) (6)

lower bound of the RCROP upper bound of the RCROP expected value of a random variable variance of a random variable I. INTRODUCTION

S

YSTEM designers desire to achieve architectures and configurations with the highest possible reliability. In emerging and complex systems, to prevent severe consequences such as losing money, effort, lives, and other resources as a result of system failure, improving products and systems reliability is vital [1]. Series configurations for reliability systems are often encountered in industrial and engineering applications. In this configuration, the reliability of the system can be improved by incorporating redundancy [1], [2]. We consider the redundancy allocation problem in a series-parallel system with subsystems in series, and each subsystem consisting of statistically identical parallel components with a specific component reliability parameter (see Fig. 1). In this configuration, the system functions if there is a path through the system formed by functioning parts [2]. The goal of the problem is to determine the number of components in the subsystems to achieve the desired overall system reliability, with resource constraints. The problem of maximizing the overall system reliability, subject to some limited resource constraints, is known as the Constrained Redundancy Optimization Problem (CROP) [1]–[4]. A closely related problem of cost minimization subject to a minimum reliability requirement is known as the COST model [1], [3], [4].

The redundancy allocation problems CROP and COST are known to be NP-hard [5], and many heuristic and exact solution methods for them have been extensively studied (see [6] for a survey). Similar to [2] and [4], we assume throughout that the feasible region of CROP and COST is such that for all feasible solutions . Then, by taking the logarithm of , we obtain (7) where (8) Due to the monotonicity of the logarithm operator, we can replace the non-separable function with separable function in the objective (2) of CROP, and the constraint (6) of COST to obtain the following MINLP equivalent formulations involving separable functions [2]. In the remainder of this paper, we consider the below reformulations of CROP and COST.

(9)

(10)

FEIZOLLAHI et al.: THE ROBUST REDUNDANCY ALLOCATION PROBLEM IN SERIES-PARALLEL SYSTEMS WITH BUDGETED UNCERTAINTY

In many previously published works, it is assumed that all problem data are known with certainty. However, in reality there is considerable uncertainty, even inaccuracy, regarding the estimation of the problem parameters, particularly the component reliabilities [2]. In particular, because of the nonlinear relation between component reliabilities and the system reliability in CROP and COST, 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 [2]. In [2], the authors implemented robust deviation or Min-Max regret to address data uncertainty in component reliabilities in CROP. As it was reviewed in [2], fuzzy programming [7]–[9], stochastic programming [10]–[17], penalty function technique [18], system-reliability confidence-intervals [19], multi-objective models that maximizes the system reliability estimate while minimizing its associated variance [20]–[22], robust designs with two level optimization [23], and minimization of the coefficient of variation [24], [25] are some approaches that have been used in the literature to consider uncertain component reliabilities in system reliability problems. Different methods for optimizing system reliability under uncertainty were reviewed and critiqued in [24]. The effects of uncertainty in both component reliability and load demand on system reliability for systems with series–parallel structures was investigated in [26]. Uncertainty in the distribution parameters of components’ reliabilities is explored in [27]. In [28], stochastic comparisons are used for redundancy allocations in series and parallel systems. Although these approaches mitigate uncertainty issues in some way, they each have their own limitations. In stochastic programming approaches for reliability optimization problems, different techniques such as maximization of a lower bound on system reliability [29], maximization of expected reliability and minimization of variance [20], [22], Pareto optimality [20], minimization of a coefficient of variance [24], [25], and optimization with chance constrained [16], [17] each were exploited. In these approaches, probability distributions which can be obtained from historical data and statistical methods were assumed for component reliabilities. In most of these approaches, the resulting formulations are non-convex hard optimization problems. Different heuristics were developed to find local optimal solutions for these models. In this paper, we propose a robust optimization framework for dealing with uncertainty in the component reliabilities. Our robust models are based on linearized versions of deterministic MINLP formulations (9) and (10) of CROP and COST. We address the uncertainty by assuming that the component reliabilities belong to a budgeted uncertainty set (see [30], [31]), and develop robust counterpart models. A robust optimization approach is proposed in [32] to consider uncertainty in component costs and weights in CROP. In contrast, our model considers uncertainty in component reliabilities. A key challenge is that, because the objective function of CROP (9) and the constraint of COST (10) are nonlinear functions of the reliabilities, unlike [32], classical robust optimization approaches cannot apply directly to construct the robust optimization counterparts. We exploit problem structure to develop robust coun-

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terparts and exact solution methods, and present computational results demonstrating their performance. This paper is organized as follows. In Section II, we provide Mixed Integer Programming (MIP) and binary linear equivalent models of CROP and COST. Then, in Section III, we introduce uncertainty in component reliabilities, and explore some structural properties of uncertain redundancy allocation problems. Section IV contains various formulations which we have proposed as robust counterparts of uncertain CROP and COST, and different solution methods to solve these problems. In Section V, we present experimental results comparing the performance of proposed methods, and the quality of robust solutions. Finally, we present some concluding remarks in Section VI. II. LINEAR EQUIVALENT MODELS OF CROP AND COST In this section, we consider the separable MINLP reformulations (9) and (10) of CROP and COST, and present two linear MIP and binary equivalent models of them. A. MIP Equivalent Models Because is a univariate concave function of the bounded integer variable , we can substitute by a piece-wise linear function that consists of line segments connecting consequent points for integer . Fig. 2 depicts and versus for a subsystem with , , and . Note that, in one hand, the piece-wise linear function is equal to at integer values of , for all . On the other hand, in all feasible solutions , the need to take integer values. Then, for all feasible values of . Therefore, by substituting with , we can develop MIP equivalents of CROP and COST, which we call MIP_CROP and MIP_COST, respectively. MIP_CROP is formulated as follows. (11) (12) (13) (14) where

and are continuous variables. Moreover, for all , , and , and are constant parameters given by (15)

and for all

, (16)

MIP_CROP has continuous variables, ables, and constraints, where . MIP_COST can be formulated analogously.

integer vari-

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remainder of the paper, we deal with uncertain component reliabilities. A. Uncertainty Sets

Fig. 2. Diagram of and .

and

for a subsystem with

,

B. Binary Linear Equivalent Models A binary linear equivalent model to CROP is proposed in [2], which will be presented in the sequel. From the definition of in (4), takes an integer value in for all . Therefore, we can write

(17) By substituting the integer variable with the binary variables , we obtain linear binary equivalent formulations of CROP, and COST, called BL_CROP, and BL_COST, respectively. BL_CROP is formulated as (18)

(19) (20) (21) ,

where

(22) and

, (23)

The correctness of the above reformulation is shown in [2]. Note that BL_CROP has binary variables, and linear constraints. The BL_COST can be formulated similarly. III. UNCERTAIN CROP AND COST The CROP and COST models, and their linear equivalents, are valid as long as the exact values of component reliabilities, , are known precisely. However, those reliabilities are typically only estimated within most likely intervals. In the

To address data uncertainty, in the robust optimization approach, it is assumed that uncertain parameters of the model are (affine) functions of some perturbation factors. Contrary to stochastic programming, in robust optimization, no probability distribution is assigned to uncertain data. Depending on the set that perturbation factors belong in, various robust optimization approaches have been developed. For example, Soyster used a box uncertainty set or interval data [33], which is the easiest but also the most pessimistic approach. In contrast, other authors (see e.g., [34] and [35]) used ellipsoidal uncertainty sets to reduce the conservativeness of Soyster’s Model. Using ellipsoidal uncertainty sets results in conic quadratic programming models, when the original model is a linear programming model. Thus, this approach cannot be directly applied on MIP or MINLP models. Later, Bertsimas and Sim ([30], and [31]) introduced a new robust optimization approach that uses a budgeted uncertainty set, which has the same complexity as the original model, and has adjustable conservativeness. Because CROP and COST are MINLP models (and their linear equivalents are MIP models), we consider a budgeted uncertainty set, and develop robust counterparts of CROP and COST models considering this type of uncertainty set. Note that the basic motivation of using a robust optimization approach is the lack of statistical knowledge about uncertain parameters in the model, which can be the case for some CROP and COST problems. Even in the case of the availability of these probability distributions, solving robust counterparts of the model is almost always easier than solving its non-convex stochastic or chance constrained equivalent. In other words, stochastic and chance constrained models can be approximated by robust models. Therefore, at the expense of over-conservatism of the solutions, the modeler can obtain safe solutions without requiring precise probability distribution functions and solving challenging non-convex stochastic models. However, conservatism of the robust counterpart can be adjusted in the budgeted uncertainty approach. Next, we describe the box (interval) and budgeted uncertainty sets for component reliabilities. 1) Box Uncertainty Set: Suppose that , the uncertain component reliability in subsystem , is not known exactly, and it takes a random value in . To normalize the uncertainty set, suppose that , where are independent random variables belonging to the box (24) We call a perturbation vector. Box uncertainty is the most conservative and pessimistic approach in the robust optimization literature, and uses the worst-case optimization approach (see [30] and [31]). Because decision variables are non-negative in CROP and COST, to obtain the robust counterpart in the presence of box uncertainty, it is enough to replace with its worst case value i.e., , and then solve the classic CROP and COST models with known methods.

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2) Budgeted Uncertainty Set: Bertsimas and Sim [30] introduced this type of uncertainty set to construct robust counterparts for LP and MIP models. Budgeted uncertainty is less conservative than box uncertainty, and its conservativeness is adjustable. They used the fact that it is unlikely all uncertain data take on their worst case values. In this type of uncertainty, in addition to box constraints (24), there is a budget to the sum of the perturbation elements as follows.

CROP( ), and COST( ), respectively. Before developing the robust counterpart models, let us define some important concepts. Definitions: Robust Reliability of a Solution : For any given solution , its robust reliability, which is denoted by (or its substitute as ), is defined as the worst possible reliability of over all possible scenarios ,

(25)

(33)

where is a given protection level, and the modeller can adjust the conservativeness of the model by changing it. While gives the most optimistic solution (deterministic problem with nominal data), generates the most pessimistic one, just like in the box uncertainty set. In fact, a budgeted uncertainty set with protection level equals the box uncertainty set.

(or for simplicity represented as ) be the scenario Let which minimizes . We call it the worst case scenario for the solution . Robustness Cost of a Solution : We name the difference between and (i.e., the deterministic cost) as the robustness cost of solution , and represent it by , (34)

B. Uncertain Crop and Cost Models In the presence of uncertain component reliabilities, it is assumed that and are functions of both and as (26) (27)

be the set of all possible values (scenarios) for , when Let belongs to a budgeted uncertainty set with protection level . Let be the set of all feasible solutions of CROP. Then, (28) and, (29) , let be the set of feasible solutions For a given of COST, corresponding to the scenario . Thus, (30) For a given scenario , the classic models CROP (9) and COST (10) can be represented as

Note that the uncertainty set is an approximation of the true uncertainty in the data. Thus, for a given feasible solution , it is possible that the realized reliability is less than its robust reliability with respect to the approximated uncertainty set. Violation Probability of Solution : Depending on the approximated uncertainty set, the actual uncertainty set, and the probability distribution of uncertain data with in it, there is a probability that the realized reliability is less than its robust reliability. We name this probability the violation probability of solution , which is equal to

(35) For example, suppose that the real uncertainty set is a box uncertainty set, and we have modeled it by a budgeted uncertainty set. It is clear that, if up to of the change within their bounds, and up to one changes by , then the realized reliability of solution will be greater or equal to its robust reliability. Otherwise, realized reliability can be less than the robust reliability of solution . Bertsimas and Sim [31] discussed details on the violation probability, and presented some analytical estimates of it. In Section V, we calculate this probability empirically by Monte Carlo simulation. Robust Feasible Solution: In the problem CROP, the feasibility of a solution depends on the realization of . The solutions which are always feasible for all are called robust feasible solutions. Denote the set of all robust feasible solutions of COST by . Then,

(31) and (32) Note that uncertainty in component reliabilities affects the optimality, and feasibility of a solution for the Problems

(36)

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C. Calculating Worst-Case Scenario Solution

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for a Feasible

IV. ROBUST CROP AND COST

An important step in developing the robust counterpart models is the calculation of the worst case realization from . To find the worst-case scenario, let us first consider some of its properties, and then present and prove some propositions. is a decreasing concave function of Proposition 1: , for any . (See the Appendix for the proof of this proposition.) Proposition 2: For any given feasible solution , . for a given solution Proof: The worst-case scenario minimizes over . From Proposition (1), is a decreasing function of for all . Thus the budget constraint (25) has to be tight at the minimizer. Proposition 3: In the worst case scenario, , at most one can be in , and the others should be equal either to 0 or 1. (See the Appendix for the proof of this proposition.) , then in the worst Proposition 4: If is an integer in case scenario the of are equal to 1, and the others are equal to 0. . Then Proof: Suppose there exists because is an integer, by Proposition (2), there exists that , while by Proposition (3) it is a contradiction. Now, to develop a model to find a worst case scenario, let us define as

In the robust optimization approach, it is assumed that uncertain parameters take their worst possible cases. Therefore, to construct the general form of robust CROP and COST, must be substituted by in the CROP and COST models, or their linear equivalents. Then, we formulate the robust CROP model, which we represent by RCROP, as (41) Similarly, RCOST, the robust COST model, can be formulated as

(42) The general robust models (41) and (42) are not tractable, and cannot be solved directly by common optimization software. Thus, we exploit our linear MIP and binary equivalent models in Section II to develop tractable robust counterparts. A. Robust Counterpart of MIP Equivalent Models If there were just a finite number of scenarios for values of perturbation vector , by using the model (41), we can develop a robust counterpart of uncertain MIP_CROP as

(37) In the case of nonintegral , where as define

, we can also (43) where,

and

,

(38) When is an integer, by Proposition 4 and Proposition 2, in a worst-case scenario, the of are each equal to 1, and the others are equal to 0. Thus, it is enough to find the of the with greatest values for , and set for them and for the others, or solve the model (39) When is not an integer, by Proposition 4 and Proposition , 2, in a worst-case scenario, exactly one is equal to of are equal to 1 while the others are equal to and the 0. Therefore, we can solve the following model to obtain the worst-case scenario. (40) In both models (39) and (40), the optimal objective value , the robustness cost of . equals In the next section, we develop robust counterparts for uncertain CROP and COST models in the presence of budgeted uncertainty.

Note that is the value of the perturbation vector in the sceis a continuous nario ‘ ’. Because in budgeted uncertainty set, and contains an infinite number of scenarios, (43) has an infinite number of constraints. We can relax these constraints, and within a just use a finite subset of worst case scenarios Benders decomposition [36] framework. Below, we present a Benders method to find the optimal solution for a robust counterpart of uncertain MIP_CROP. 1) Procedure 1: (MIPR_B) Benders method to find the optimal solution of robust counterpart of uncertain MIP_CROP. Step 1. Step 0. Choose an arbitrary possible scenario . Set . Step 2. Step 1. Solve the relaxed model (43), and let be its optimal solution. using Step 3. Step 2. Find the worst case scenario model (39) for integer , and model (40) for non integer . Step 4. Step 3. Calculate . If it is less than , and go to Step 1; otherthen set wise, go to step 4. Step 5. Step 4. Stop, as the current solution is robust-optimal.

FEIZOLLAHI et al.: THE ROBUST REDUNDANCY ALLOCATION PROBLEM IN SERIES-PARALLEL SYSTEMS WITH BUDGETED UNCERTAINTY

Proposition 5: For any budgeted uncertainty set , MIPR_B in Procedure 1 can find the robust solution of models (41) and (42) in a finite number of iterations. Proof: Obviously, Procedure 1 generates a new worst-case scenario , in each iteration. (Otherwise, in Step 3 of the pro, and it terminates.) Because is cedure, finite, there are a finite number of worst-case scenarios. Therefore, Procedure 1 terminates after a finite number of iterations. In addition to the MIPR_B procedure above where the relaxed MIP (43) is solved to optimality in every iteration, we can add the violated cuts within a branch-and-bound method for solving (43) whenever an incumbent solution is found. We name the latter procedure MIPR_C.

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Similarly, the dual of (47) can be written as

(50) Because the optimal objective values of (45) and (49) are equal (from strong duality), then the robust counterpart of BL_CROP for integer is

B. Robust Counterpart of Binary Equivalent Models Considering the binary linear equivalent models of CROP and COST models developed in Section II-B, and by definition of robustness cost, for a given feasible binary solution ,

If is an integer, from Propositions 2 and 4, written as

(44) can be

(45)

(51) In this model, there are binary variables, continlinear constraints. Similarly, because uous variables, and the optimal objective values of (47) and (50) are equal, then the robust counterpart of BL_CROP for a non integer is

, and . The constant parameter where can be calculated in advance as (46) and are variables, and fixed In the model (45), the constants, respectively. In the case of a non integer , (52)

(47) and . The constant parameter where can be calculated in advance as

binary variables, continuous This model has linear constraints. We refer to robust variables, and counterparts (51) and (52) as BR. Robust counterparts of uncertain BL_COST can be developed analogously. Next, we investigate an illustrative example to compare solutions of different stochastic and robust approaches in solving uncertain CROP.

(48)

C. Illustrative Example

(49)

In this subsection, we compare the solutions of the proposed robust CROP model and conventional stochastic approaches. To perform this comparison, we explored a small case of CROP with 6 subsystems in series. Input data of this system including , , , and for each subsystem , and coefficient matrix of five linear constraints, are presented in Table I. The right hand side vector of linear constraints was assumed to be .

The dual model of (45) is

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Fig. 3. Probability distributions of system reliability for solutions nent reliabilities.

in simulation results from normal, triangular, and uniform distributions for compo-

TABLE I INPUT DATA FOR CROP EXAMPLE WITH 6 SUBSYSTEMS IN SERIES

TABLE II OPTIMAL SOLUTIONS OF THE EXAMPLE DIFFERENT OBJECTIVE FUNCTIONS

FOR

For stochastic approaches, we assumed that , and , which are the mean, and vari. For a given soluance of a uniform distribution in and its variance tion , the expected system reliability were calculated by substituting , , and , in related formulas in [20], and [21]. By enumerating all possible solutions, there were 7715 feasible solutions in the . Then, the following system objective functions for this example were considered to be optimized among all feasible solutions: , • , • • , , • • , •

,

where in and , in . and Table II represents the optimal solutions of this example with . In respect to different objective functions this table, , and are respectively the theoretical expected values, and variances of system reliabilities.

have the maximum Among all feasible solutions, , and , and minimum , respectively. also maximizes for , and for . , minimizes the coefficient In addition to minimizing ), and maximizes for , of variation ( for . Solution maximizes and for , and for . The maximum for is achieved by . Finally, optimal robust of for , and are obsolutions of tained by , and , respectively. Note that, in this example, robust solutions and have also been found by other objecwas only detected by the robust approach tive functions, but . in We simulated the system reliability for 10,000 iterations by generating each component reliability , from normal, symmetric, triangular, and uniform distributions with mean, and , and , respectively. The results variance equal to of the simulations are presented in Table III, and Fig. 3. In and are empirical estimations Table III, , respectively, based on of the expectation, and variance of , and are the simulation results. In this table, , respectively. is the 0.05, and 0.01 quantiles of in the simulation. According to this minimum observed have the largest , table, as we expected, , and , respectively, among in all and smallest has the best , three simulations. Robust solution , and in all simulations. Although and of solutions and are very close to each other, is significantly outperforming with respect to . In Fig. 3, probability distributions of system reliability for are depicted, which are empirically obsolutions tained from simulation. Based on this figure, and Table III, we is higher conclude that, although the expected reliability of than other solutions, it is not a reliable solution, and has too much variation in its system reliability results. In contrary, has the best variance, but the worst expected reliability. On provide better , , the other hand, and values. Although their expected reliabilities are smaller than , and their variance values of the reliabilities are larger than , they are favourable for risk averse, conservative decision makers. Note that, in this example, we enumerated all feasible solutions, and calculated the expected value and variance of the system reliability for each feasible solution, which is not possible for large scale systems. For a system with 20 subsystems,

FEIZOLLAHI et al.: THE ROBUST REDUNDANCY ALLOCATION PROBLEM IN SERIES-PARALLEL SYSTEMS WITH BUDGETED UNCERTAINTY

TABLE III OPTIMAL SOLUTION OF THE EXAMPLE WITH 6 SUBSYSTEMS SERIES IN DIFFERENT APPROACHES

IN

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TABLE IV NON-DECREASING BEHAVIOUR OF VARIANCE OF SYSTEM RELIABILITY

In the next section, we examine the performance of all proposed robust methods and explore the quality of robust solutions. V. EXPERIMENTAL RESULTS

where the number of components can vary from to , there are possible solutions. Therefore, because of the non-convex, non-monotone, and non-linear nature of the variance functions, researchers develop heuristic methods such as genetic algorithm (e.g., [6], [11], [12], [20], [22]) to solve stochastic reliability problems. Thus, it is possible to be trapped in a local optimal solution in these methods. In contrast, the proposed robust algorithms in this paper are exact methods, and can solve to optimality the equivalent MIP and binary counterparts of uncertain CROP and COST models by applying a general MIP solver. In the next section, we will demonstrate the performance of the proposed methods. Another disadvantage of stochastic reliability models is the , which we next illusnon-monotone behaviour of trate. In this example of uncertain CROP with 6 subsystems, with and consider the feasible solutions in Table IV. Although is larger than (i.e., , , and there exists a such that for all ), . This result also holds for and , and so on. Therefore, based on , is is preferred to , and so on. Similar phepreferred to , and . For example, for nomenon can happen for , has better than , or for all any , is preferred to regarding . But we know that, for any outcome of uncertain component reliabiliis less than , and so on. ties , the system reliability of and are feasible, there is no gain Therefore, because both in choosing . In contrast, it is not possible to prefer a solution to a larger solution like in the proposed robust optilike mization approaches because, for any and , , then

or the worst case reliability of is greater than . In other words, the solutions of robust approaches are always maximal.

In this section, we describe generating test instances for CROP and COST problems, and investigate the performance of all proposed robust methods to solve these instances and explore the quality of robust solutions. To generate the test instances, we adopt the same approach as [2] and [4]. For CROP, two classes of linear and nonlinear constraints are considered. • Linear constraints: , where . • Nonlinear constraints:

(53) . where For the COST model, it is assumed that , where . The upper bound of the reliability of each component , and interval length , belong to , and , respectively. All of these coefficients in all of the instances are randomly generated from uniform distributions. We assume that , and . For CROP instances, we set to assure feasibility, where for linear constraints, and for nonlinear constraints. For COST instances, we set . The proposed solution methods were conducted on a PC with an Intel Core Duo 2.66 GHz processor with 2.99 GB of RAM, and CPLEX 12.1 ([37], default parameters), interfaced with C++. The average CPU times are obtained by running the methods 20 times. Similar to [2], 3 classes of instances for CROP were considered. • : instances with linear constraints, and subsystems. 3 groups of this class were generated for . • : instances with nonlinear constraints, and subsystems. 3 groups of this class were generated for .

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TABLE V AND AVERAGE CPU TIME (IN SECONDS) FOR ROBUST CROP MODEL

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INSTANCES OF THE

Fig. 5. Change in the objective function (percentage) vs. the protection level .

Fig. 4. Violation probability vs the protection level .

•

: instances with linear constraints, and 90 subsystems. This class is considered in 4 groups for . For the COST model, instances with were investigated, which are denoted by the class. Table V presents the performance of three methods of finding the robust solution of and instances of CROP. These three methods are robust counterparts of binary linear equivalent models (BR), classical Benders method to find the robust counterpart of MIP equivalents (MIPR_B), and Benders method with cut callback (MIPR_C). From this table, see that MIPR_C outperforms MIPR_B, and BR outperforms both MIPR_C and MIPR_B especially for large instances. Therefore, for large instances, we just ran the BR method. In addition, CPU times for BR are affected considerably when it solves a model with non-integer , but for MIPR_C and MIPR_B the CPU times do not increase extra ordinarily. To choose the most appropriate protection level , the modeler can rely on theoretical bounds on robustness cost and violation probability for budgeted uncertainty, which are developed in [31]. Although this approach is straightforward, the provided bounds can be very loose. The other method of choosing is by solving the robust models for different values of , and checking the quality of the solutions through Monte Carlo simulation. To illustrate the process, we considered a CROP instance (with and linear constraints), and a COST instance (with ), and solved their robust counterparts for integer protection levels . After finding robust solutions, we analyzed their quality via Monte Carlo simulation by generating component reliabilities from uniform and normal distributions. 10,000 runs of the simulation were done. Figs. 4 and 5 show the results of this simulation. In Fig. 4, the empirical violation probabilities are depicted versus protection level . In this figure, U, and N after COST

and CROP indicate respectively the uniform, and normal distributions in the simulation. It is clear that, by increasing protection level , the violation probability decreases. This probability becomes zero when goes beyond 11 (or 9) for a uniform (or normal) distribution. Fig. 5 shows the percentage of increase in the objective function of the COST model and the percentage of decrease in the objective function of the CROP model versus the protection level . This change is equal to the ratio of the robustness cost to the nominal objective value. Clearly, by adjusting to around 10, a 17% increase in COST and a 2% decrease in CROP objective values occurs. On the other hand, the violation probability is guaranteed to be zero, or a small acceptable value. With regard to Fig. 4, in our test cases with , the experimental violation probabilities are almost fixed when or . Therefore, one cannot expect a decrease in violation probabilities by sacrificing the values of the objective functions in these ranges of . Thus, it is enough to choose a protection level , trading off between violation probabilities, and changes in the objective functions in Figs. 4 and 5, respectively. We note that, in our test cases, , and , for all . Therefore, much higher relative differences are expected in the costs of the components than their reliabilities among subsystems. Consequently, the percentage of increase in the objective function of the COST model is almost always larger than the percentage of decrease in the objective function of the CROP model as the protection level increases in Fig. 5. In [2], a robust deviation framework was proposed for uncertain CROP with interval component reliabilities, and four exact iterative algorithms were developed to find its solution. These algorithms were coded in C++, and CPLEX 12.1 is called to solve the MIP sub problems when needed. They were conducted on a UNIX machine by applying restrictions of 10 GB of RAM, and a single processor (Thread) 2.27 GHz CPU. In Table VIII, the best total solution time, , among those four algorithms for each group is reported from [2]. The BR method in this paper is much faster and adjustable than the methods in [2]. By comparing the result from Table VIII with Tables V and VI, it is clear that, from a solution time point of view, our proposed BR method outperforms the algorithms in [2], especially for larger instances. Note that the BR method solves just one MIP model, and is not an iterative algorithm, while all of the four robust deviation CROP algorithms in [2] are

FEIZOLLAHI et al.: THE ROBUST REDUNDANCY ALLOCATION PROBLEM IN SERIES-PARALLEL SYSTEMS WITH BUDGETED UNCERTAINTY

TABLE VI AVERAGE CPU TIME (IN SECONDS) FOR THE ROBUST CROP MODEL

INSTANCES

OF

TABLE VII AVERAGE CPU TIME (IN SECONDS) FOR THE ROBUST COST MODEL

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simulation, other distributions and historical data for component reliabilities can be used to provide realistic simulations. We note that optimizing system reliability is a dynamic, quickly evolving area. In this research, we developed robust redundancy allocation models and methods for series-parallel systems with budgeted uncertainty. As future topics, heuristic methods for reliability problems, such as genetic algorithms or ant colony methods, can be utilized to solve master problems of each iteration. Moreover, extension of the proposed formulations and methods to complex systems with multi-state and multi-choice components is practically interesting. Furthermore, considering other types of uncertainty, dynamic uncertainty and adjustable robust approaches can be considered for future research. APPENDIX

TABLE VIII AVERAGE CPU TIME (IN SECONDS) FOR THE ROBUST DEVIATION CROP MODEL FROM [2]

Proof of the Proposition 1: Here we provide a proof for Proposition 1. Proof: By taking the first derivative of , (54) because , , and for uncertain component reliability , , it is obvious that is always negative. Hence, is a decreasing function of . By taking the second derivative,

iterative, and slow for larger instances. In addition, the proposed robust models in this paper have adjustable protection levels, and the modeller can adjust them based on his risk preferences, while the model in [2] is not so. VI. CONCLUSIONS, AND FUTURE RESEARCH In this paper, we considered the redundancy allocation in series-parallel reliability optimization problems, and developed linear equivalent mixed integer programming and binary models, solvable by general-purpose mixed integer programming software. Then, we considered budgeted uncertainty in component reliabilities, and developed the robust counterparts of the problems. It is worth mentioning that, because the objective function and constraints of the original uncertain problems are not linear functions of uncertain data, classical robust optimization approaches cannot apply directly to construct their robust counterparts. We developed different linear mixed integer programming and binary models to find robust solutions of these problems. Finally, we validated the performance of the model, and examined the quality of robust solutions by simulation. Experimental results showed that, by our proposed methods, robust solutions can be found for uncertain series-parallel reliability optimization problems in a reasonable time (for instances with practical sizes), and the modeller can make trade-offs between violation probability and robustness cost to get a solution with acceptable risk. Although we used the uniform (which is almost the worst) and truncated normal distributions in Monte Carlo

(55) again it is obvious that Then, is a concave function of Proof of the Proposition 3: Here we Proposition 3. Proof: If the proposition is not worst-case scenario, , there exists where . Let claim that where derivative,

is always negative. . provide a proof for true, then in the

and . We is a concave function of , . By taking the first

(56) and by taking the second derivative,

(57) From Proposition 1, and (55), and

. Then, from (57), . Thus, is a strictly concave function of . Therefore, its minimum occurs at the extreme point or

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, and

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Mohammad Javad Feizollahi (S’13) is a PhD student in the H. Milton Stewart School of Industrial & Systems Engineering at the Georgia Institute of Technology. He received his Masters, and Bachelors degrees in Industrial Engineering in 2007, and 2005, respectively, from Sharif University of Technology, Tehran, Iran. 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.

Shabbir Ahmed is a Professor in the H. Milton Stewart School of Industrial & Systems Engineering at the Georgia Institute of Technology. He received his PhD from the University of Illinois at Urbana-Champaign in 2000. His research interests are in optimization, specifically stochastic and integer programming. Dr. Ahmed served as the Chair of the Community of Stochastic Programming, and as a Vice-chair of the INFORMS Optimization Society. He is an Associate Editor for Mathematical Programming A, Mathematical Programming C, Operations Research, and Operations Research Letters; an Area Editor for Surveys in Operations Research and Management Science; and a Department Editor for IIE Transactions. Dr. Ahmed’s honors include the National Science Foundation CAREER award, two IBM Faculty Awards, the Coca-Cola Junior Professorship from ISyE, and the INFORMS Dantzig Dissertation award.

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 PhD in Systems Engineering and Operations Research from the University of California, Los Angles (UCLA), in 1975. His research interests are in operations research, robust optimization, and stochastic dynamic programming.