1. No category

1
The Robust Cold Standby Redundancy Allocation in
Series-Parallel Systems with Budgeted Uncertainty
Mohammad Javad Feizollahi, Student Member, IEEE, Roya Soltani, Hadi Feyzollahi
February 17, 2015
Abstract—This paper studies a redundancy allocation problem
(RAP) with cold standby strategy in non-repairable seriesparallel systems. We assume that the components’ reliabilities
are uncertain values in a budgeted uncertainty set, with unknown probability distributions. Because the system reliability is
a nonlinear function of the components’ reliabilities, classical
robust optimization approaches cannot be directly applied to
construct the robust counterpart of this problem. Therefore,
this paper for the first time proposes linear MIP and binary
equivalent models for the cold standby RAP; and by exploiting
the problem structure, robust counterparts are developed to deal
with budgeted uncertainty in this problem. Then, two exact
solution methods are proposed: one of them solves a mixed
integer programming (MIP) model iteratively in a Benders’
decomposition framework, and the other one solves a single
binary linear model. The validity and the performance of the
proposed approach are tested through a Monte Carlo simulation
and computational results.
Index Terms—Cold standby redundancy allocation, robust
optimization, budgeted uncertainty, mixed integer nonlinear
programming, series-parallel system.
RAP
CSRAP
CROP
MIP
MINLP
CSRAPMIP
CSRAPBL
MIPRB
MIPRC
BR
n
J
I
rj
rej
rˆj
ACRONYMS AND A BBREVIATIONS
Redundancy Allocation Problem
Cold Standby RAP
Constrained Redundancy Optimization
Mixed Integer (Linear) Programming
Mixed Integer Non-Linear Programming
MIP equivalent of CSRAP
Binary Linear equivalent of CSRAP
Basic Benders decomposition
MIPRB with ‘Lazy Cut Callback’
Method of solving binary linear robust
counterpart of uncertain CSRAPBL
N OTATIONS
number of subsystems in series
set of subsystem
set of constraints
reliability of a component in subsystem j
uncertain rj
largest possible value of rej
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M. J. Feizollahi is with School of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA (e-mail:[email protected]).
R. Soltani is with Department of Industrial Engineering, Iran University of
Science & Technology, Tehran, Iran (e-mail: [email protected]).
H. Feyzollahi is with Department of Industrial & Systems Engineering,
University at Buffalo, Buffalo, NY, USA (e-mail: [email protected]).
δj
xj
ξ
UΓ
largest possible difference of rej from rˆj
number of components in subsystem j
normalized perturbation vector
budgeted uncertainty set with protection level
Γ
lj
minimum value of xj
uj
maximum value of xj
S
set of integer solutions x, which satisfy Minimum, and maximum limits lj , and uj resp.
P
set of all feasible solutions x
Rj (.)
reliability function of subsystem j
R(.)
overall system reliability function
gij (.)
an increasing convex function of xj , which
represents the effect of xj in ith constraint
bi
maximum available resource in ith constraint
g¯ij (.)
a piece-wise linear convex function of xj
which coincides with gij (.) at integer xj
aijk , bijk constants representing kth segment of g¯ij (.)
P
set of all feasible solutions x for CSRAP
fj (.)
logarithm of Rj (.)
f (x)
logarithm of R(x)
f¯j (.)
a piece-wise linear concave function of xj
which coincides with fj (.) at integer xj
a0jk , b0jk constants representing kth segment of f¯j (.)
f (x, ξ) f (x) under scenario ξ
f Rob (x) robust value of f (x, ξ) for a solution x
RC(x)
ξx
S
yj
qij
χjk
wjk
gijk
robustness cost of a solution x
worst-case scenario for solution x
set of all worst case scenarios for x ∈ P
a continuous variable ≤ f¯j (xj )
a continuous variable ≥ g¯ij (xj )
binary variable, which is 1 if xj = lj + k,
and 0 otherwise
coefficient of χjk in the log reliability function
coefficient of χjk in the ith constraint
I. I NTRODUCTION
I
N the design stage of systems, some methods are implemented to ensure reliability for a predetermined time period
under specified circumstances. One way to enhance a system’s
reliability is the allocation of redundant components in subsystems [1]. There are two main redundancy strategies: active,
and standby. In the case of active strategy, all components
work simultaneously; while in the standby strategy, one of the
redundant components begins to work only when the operating
2
the above assumptions, the system reliability is
" xj
#
Y
X (− ln(rj ))l−1
,
R(x) =
rj
(l − 1)!
j∈J
(1)
l=1
and the classical CSRAP can be formulated as (2)-(3).
max R(x)
(2)
x∈S
s.t.
Fig. 1. Series-parallel configuration of components with cold standby strategy
in a reliability system.
X
gij (xj ) ≤ bi , ∀ i ∈ I,
(3)
j∈J
where
one fails (see Fig. 1). For more study on redundancy allocation
problem (RAP) with an active strategy, readers are referred to
the works [2]–[4], to name a few. Research articles related to
the standby strategy in RAP are reviewed in [5].
The standby strategy in turn is classified into hot, warm,
and cold [5]. Cold standby redundancy is an effective system
design strategy in which the redundant component has no
probability of failure prior to being put into operation. Many
systems, such as space exploration and satellite systems [6],
textile manufacturing systems [7], and carbon recovery systems in fertilizer plants [8], use the cold standby redundancy
allocation problem (CSRAP) as an effective strategy to achieve
high reliability.
For standby systems, different time to failure distributions
such as normal, lognormal, exponential, Erlang, and Weibull
can be considered. In case of the cold standby strategy for
nonrepairable systems, [9] studied systems designed with
components that have phase-type time-to-failure distributions.
In [10], the author studied the problem of determining optimal
design configurations for CSRAP with an Erlang time to
failure distribution. The choice of redundancy strategy and
imperfect switching of cold standby redundant components
with an Erlang distribution in CSRAP are studied in [11].
Binary programming formulations to deal with these problems
are proposed in [10], and [11]. In the recent literature, [12]
and [13] proposed a multi-objective model, and an entropy
based redundancy allocation with choices of the redundancy
strategy where the times to failures of the cold standby part
follow Erlang distributions.
In this paper, we consider a series-parallel CSRAP presented
in Fig. 1 with the following assumptions.
•
•
•
•
•
•
•
All the components in one subsystem have same type.
Components’ times to failure follow exponential distributions, which is most suitable for electronic components
(e.g., transistors, capacitors, resistors, and integrated circuits) used in control systems, power generators, etc.
The standby strategy is the cold type.
The switch reliability to a cold standby component is
perfect.
The components are in one of two states: functioning, or
nonfunctioning (i.e. binary state).
There is no repair or preventive maintenance.
The replacement time is negligible.
According to [10] and [14], for a CSRAP in Fig. 1 under
S = x ∈ Zn+ : lj ≤ xj ≤ uj , ∀j ∈ J .
(4)
Let P = {x ∈ S : (3) is satisfied} be the set of all
feasible solutions. Without loss of generality, we can assume
that R(x) > 0 for all x ∈ P . Then, by taking the logarithm
of R(x), we obtain
X
ln R(x) = f (x) =
fj (xj ),
(5)
j∈J
where,
"
fj (xj ) = ln rj
xj
X
(− ln(rj ))l−1
l=1
(l − 1)!
#
, ∀j ∈ J.
(6)
In the remainder of this paper, we consider the below
reformulation of CSRAP.
" xj
#
X
X (− ln(rj ))l−1
CSRAP: max f (x) =
ln rj
x∈S
(l − 1)!
j∈J
l=1
(7)
X
s.t.
gij (xj ) ≤ bi , ∀ i ∈ I.
j∈J
Linear MIP, and binary equivalent models of CSRAP, which
we denote by CSRAPMIP , and CSRAPBL , respectively, are
presented in the Appendix.
One of the important factors that needs to be taken into
consideration in system designs is the intrinsic uncertainty that
exists in components’ reliabilities [15]. The uncertainty can
stem from human errors, measuring errors, production technology, improper storage, and the like. Different approaches
such as stochastic programming, fuzzy programming, interval
programming, and robust optimization have been proposed by
researchers in the literature to deal with uncertainty. [16] carried out a comprehensive review about reliability optimization
problems, especially redundancy allocation problems under
uncertainty, and discussed about the pros and cons of different
approaches. Here, we mention some of the prominent works;
and for more study, the readers are referred to the survey
provided in [16].
In stochastic programming approaches, it is assumed that
the probability distribution or at least mean and variance
of the uncertain parameters are precisely known. Regarding
random lifetimes, some stochastic programming models for
active and standby RAP have been provided in [17]. Also,
[18] did stochastic comparisons for systems with active and
cold standby strategies considering random lifetimes. In the
case of random reliabilities with known means and standard
3
deviations, multi-criteria approaches have been proposed (e.g.
[19]–[21]). The interval programming approaches assume that
the uncertain data are within intervals with known lower and
upper bounds (e.g. see [22]–[27]).
The fuzzy programming approaches assume fuzzy lifetimes
(e.g. [28] and [29] ), fuzzy reliability (e.g. [30] and [31]), and
fuzzy goals and constraints (e.g. [32]–[34]). Assuming random
fuzzy lifetimes for components, various fuzzy methods for
CSRAP have been developed in [35], [36], and [28]. In [15],
the reliability of cold standby systems under partial information about probabilities of times to failure of components was
analyzed, and imprecise reliability models were proposed.
In the case of robust optimization, two main approaches
are considered for problems with discrete variables. Some
robustness measures are based on regret (e.g. see [14], [37]–
[40]), and some are based on the actual performance of the
realized values for uncertain parameters (e.g. see [41]–[46])
and try to make a trade off between the optimality robustness
and the feasibility robustness. To the best of our knowledge,
just a few works exist in the literature of robust reliability
optimization problems. The former approach in the form of
minmax regret method and with respect to interval uncertainty,
was implemented to the RAP with active, cold standby and
mix of strategies in [38], [14], and [39], respectively. Concerning the latter approach, [43] developed robust models for RAP
with an active strategy under fuzzy uncertainty. In another
work, [42] considered an uncertainty set, and developed robust
counterparts for the RAP with an active strategy. In that paper,
authors described motivations behind using a robust optimization approach with budgeted uncertainty to deal with uncertain
RAPs, and compared solutions of their robust method with
conventional stochastic approaches.
Because in most cases a cold standby strategy is desirable,
and it yields higher system reliability, in this paper, we propose
a robust optimization for uncertain CSRAP, and develop exact
solution methods to solve the resulting models. Similar to
[42], in this paper we consider robust optimization under a
budgeted uncertainty approach. We refer the readers to [42]
for a detailed discussion about the motivation and advantages
of using these approaches in system reliability problems.
The rest of this paper is organized as follows. In Section
II, we introduce mathematical formulation of the uncertain
and robust CSRAP. Then, various formulations and solution
methods for robust counterparts of uncertain CSRAP are
proposed in Section III. In Section IV, we present experimental
results of running the proposed robust methods on different test
cases. In Section V, we summarize with the main conclusions,
and plans for future research.
II. U NCERTAIN AND ROBUST CSRAP
In reality, the reliabilities of components are not exactly
known, and are estimated within an uncertainty set. Next, we
introduce budgeted uncertainty for components’ reliabilities,
and describe uncertain CSRAP.
A. CSRAP with budgeted uncertainty
In this paper, it is assumed that rej , the uncertain component
reliability in subsystem j, is not known exactly, and it takes a
random value in [ˆ
rj − δj , rˆj ]. To normalize the uncertainty set,
suppose that rej = rˆj − δj ξj , where ξ is a perturbation vector
including n s-independent random variables between 0 and 1.
For a given protection level Γ ∈ [0, n], the set




X
UΓ = ξ ∈ R n :
ξj ≤ Γ, 0 ≤ ξj ≤ 1, ∀j ∈ J
(8)


j∈J
is called a budgeted uncertainty set. This type of uncertainty
which has adjustable conservativeness was proposed by Bertsimas and Sim [41]. While Γ = 0 gives the most optimistic
solution (deterministic problem with nominal data), Γ = n
generates the most pessimistic one.
In the presence of uncertain component reliabilities, it is
assumed that f (.) and fj (.) are functions of both xj and ξj
as
#
"
xj
X
(− ln(ˆ
rj − δj ξj ))l−1
, (9)
fj (xj , ξj ) = ln (ˆ
rj − δj ξj )
(l − 1)!
l=1
and
f (x, ξ) =
X
fj (xj , ξj ).
(10)
j∈J
For a given scenario ξ ∈ UΓ , the classic models CSRAP (7)
can be represented as
Problem CSRAP(ξ) : max {f (x, ξ) : x ∈ P } .
(11)
Note that uncertainty in component reliabilities affects the optimality of the solutions for the problem CSRAP(ξ). Moreover,
any x ∈ P is a feasible solution of CSRAP(ξ), for all possible
realizations of ξ ∈ UΓ .
B. Robust CSRAP under budgeted uncertainty
Similar to [42], for any given solution x ∈ P , its robust
reliability, which is denoted by RRob (x) (or its substitute as
f Rob (x)), is defined as the worst possible reliability of x over
all possible scenarios ξ ∈ UΓ , i.e.
f Rob (x) = min f (x, ξ).
(12)
ξ∈UΓ
Let ξ x be the scenario which minimizes f (x, ξ) in (12). We
call it the worst case scenario for the solution x.
In the robust optimization approach, it is assumed that uncertain parameters take their worst possible cases. Therefore,
the robust CSRAP model can be formulated as
Robust CSRAP : max f Rob (x) = max min f (x, ξ). (13)
x∈P
x∈P
ξ∈UΓ
Model (13) cannot be solved directly by commercial optimization packages such as CPLEX [47]. Therefore, some
tractable counterparts need to be developed. Prior to developing robust counterparts of the proposed model (13), for a
given x ∈ P , it is required to find a worst case scenario in the
defined uncertainty set. For this purpose, first let us present
the following propositions.
Proposition 1. f (x, ξ) is a decreasing concave function of ξ.
Proof. Because f (x, ξ) is the summation of fj (xj , ξj ) for all
j ∈ J, it is enough to show that fj (xj , ξj ) is a decreasing
4
concave function of ξj , for each j ∈ J. For the sake of
simplicity, let
0 > ηj : = ln r˜j = ln(ˆ
rj − δj ξj );
(−ηj )xj −1
if xj ≥ 1, 0 < αj =
, else αj = 0;
(xj − 1)!
(−ηj )xj −2
, else βj = 0;
if xj ≥ 2, 0 < βj =
(xj − 2)!
xj
X
(−ηj )xj −3
if xj ≥ 3, 0 < γj =
, else γj = 0.
(xj − 3)!
x=3
e
e
b0e
jk = fj (k + 1, ξj ) − fj (k, ξj )


(− ln(ˆ
rj −δj ξje ))k
l=1


,
(− ln(ˆ
rj −δj ξje ))l−1 
k!
(17)
(l−1)!
and
e
0e
a0e
jk = fj (k, ξj ) − bjk k
"
#
k
X
(− ln(ˆ
rj − δj ξje ))l−1
e
= ln (ˆ
rj − δj ξj )
− b0e
jk k.
(l − 1)!
l=1
(18)
∂(γj + βj )
∂αj
= −βj , and
= −γj .
∂ηj
∂ηj
By using this notation, for any j ∈ J, and xj ≥ 1, it holds
that fj (xj , ξj ) = ηj + ln (γj + βj + αj ). Then,
(15)
Then, we can develop a robust counterpart of uncertain
CSRAPMIP according to (19).
max R
x∈S
y,q,R
s.t. R ≤
and
−δj2 (αj2 + βj2 + αj βj + βj γj )
∂ 2 fj (xj , ξj )
=
< 0,
∂ξj2
[(ˆ
rj − δj ξj )(γj + βj + αj )]2


= ln 1 + k
P

(14)
Note that
∂fj (xj , ξj )
−δj αj
< 0,
=
∂ξj
(ˆ
rj − δj ξj )(γj + βj + αj )
1) Robust Counterpart of uncertain CSRAPMIP : Let ξ e ∈
UΓ be the realization of the perturbation vector for the scenario
e. For each j ∈ J and lj ≤ k < uj , and scenario e, let
(16)
which prove fj (xj , ξj ) is a decreasing concave function of
ξj .
Proposition 2. For any given solution x ∈ P , the corresponding worst case scenario ξ x ∈ UΓ has the following properties.
P
x
1)
j∈J ξj = Γ.
2) At most one ξjx can be in (0, 1), and the others should
be equal either to 0 or 1.
3) If Γ is an integer in [0, n], then Γ of ξjx are equal to 1,
and the others are equal to 0.
4) If Γ is not an integer, exactly one ξjx is equal to Γ−bΓc,
and the bΓc of ξjx are equal to 1 while the others are
equal to 0.
Proof. For a decreasing concave function f (x, ξ) over ξ, [42]
showed that the worst case scenario ξ x has properties 1-4. By
Proposition 1, f (x, ξ) is a decreasing concave function of ξ,
which completes the proof.
III. ROBUST C OUNTERPARTS AND S OLUTION M ETHODS
In this section, the robust counterparts and the corresponding solution methods are developed for both cases of linear
MIP and Binary equivalent models (see the Appendix) for
uncertain CSRAP.
A. The case of MIP equivalent for CSRAP
A MIP equivalent of the CSRAP is presented in the Appendix as model (30)-(33). In this subsection, we propose a
robust counterpart for this MIP model under uncertainty. Then,
two solution approaches will be developed.
X
j∈J
e
yj ≤ a0e
jk
yje , ∀e ∈ S,
(19)
+ b0e
jk xj , ∀j ∈ J, lj ≤ k < uj , e ∈ S,
(31) and (33),
where S ⊂ UΓ is the set of all worst case scenarios. Model
(19) has (|S| + m) × n + 1 continuous
variables, ninteger
P
variables, and 2n + (m + |S|) × 1 + j∈J (uj − lj ) linear
constraints. Note that |S| ≤ |P | < ∞, because there is a
corresponding worst-case scenario for any x ∈ P .
2) Solution method for the proposed robust counterpart
of CSRAPMIP : Model (19) has o(|P |) robustness constraints,
where |P | represents the cardinality of the finite set P .
Moreover, the set of worst case scenarios is not known in
advance. Therefore, it is difficult to deal with the problem. Taking this problem’s characteristics into account, we
inevitably resort to a well-known decomposition technique
called Benders decomposition [48]. In this technique, we relax
the complicated (robustness) constraints. After solving the
relaxed model, we add the violated constraints in an iterative
manner. In the basic version of the Benders technique, the
relaxed problem is solved to optimality in each iteration. An
alternative approach is using the lazy constraint callback in
CPLEX [47]. In the latter approach, the violated constraints
are added within a branch-and-bound method whenever an
incumbent solution is found. Two variants of the Benders
decomposition method with, and without considering lazy
constraint callback, respectively called MIPRC , and MIPRB ,
are implemented to solve the robust counterpart of CSRAPMIP .
Procedure 1 shows the steps of the proposed MIPRB .
Procedure 1. (MIPRB ) Basic Benders decomposition.
Step 0 (Initial setting). Pick an arbitrary scenario ξ 0 ∈ UΓ .
Set S = {ξ 0 }.
Step 1 (Master problem). Solve the relaxed model (19) with
S, and let (x∗ , R∗ ) be its optimal solution.
∗
Step 2 (Sub-problem). Find the worst case scenario ξ x using
5
model (25) for integer Γ, and model (24) for non integer Γ.
∗
Step 3 (Stopping criterion). If f (x∗ , ξ x ) = R∗ , STOP; x∗ is
robust-optimal. Otherwise go to Step 4. S
∗
Step 4 (Add Benders cut). Set S ← S ξ x , and add the
corresponding new constraint (Benders cut) to the master
problem. Go to Step 1.
In the Benders decomposition approach presented in Procedure 1, the complicating constraints are relaxed, the algorithm
starts with a small subset of constraints, and progressively new
constraints (Benders cuts) are taken into account. The corresponding relaxed formulation is called the master problem; it
is defined by using a subset S ⊂ S in (19). 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 resolved, 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 robust reliability of the system.
In procedure MIPRB , the master problem (relaxed model
(19) with S) is solved to optimality at each iteration. Then, the
violated cuts are added at the end of the iteration. In contrast,
in procedure MIPRC , the violated cuts are added within a
branch-and-bound method whenever an incumbent solution is
found for solving (19). The idea behind using this variant is
to prevent the algorithm from solving the master problem to
optimality at each iteration. As presented, at each
P iteration of
Procedure 1, n continuous variables and L = j∈J (uj − lj )
linear inequalities are added to the master problem.
B. Binary linear equivalent for CSRAP
For a given solution x ∈ P , the difference between its
nominal and robust reliabilities is called the robustness cost
of x, RC(x), that is
RC(x) = f (x, 0) − f Rob (x) = f (x, 0) − f (x, ξ x ).
(20)
Note that dj > 0 and d0j ≥ 0 are constants for any given x.
Moreover, for all integer values of Γ, we have d0j = 0, ∀j ∈ J.
Therefore, in the case of integer Γ,
RC(x) = max
ξ∈UΓ
dj ξj .
(25)
j∈j
In the Appendix, the binary linear equivalent model (36)(39) which is denoted by CSRAPBL is presented for deterministic CSRAP. Next, we consider two cases of having integer
and non-integer values for the protection level Γ, and develop
a robust counterpart of uncertain CSRAPBL for each case.
1) Robust Counterpart of uncertain CSRAPBL in the case
of integer Γ : In this case, RC(x) can be computed through
(25).
Consider
the dual variables zj and t for constraints ξj ≤ 1
P
and j∈J ξj ≤ Γ, respectively, in (8). Then, the dual model
of (25) is as follows.
min Γt +
X
zj
j∈J
s.t. t + zj ≥ dj ,
t ≥ 0,
(26)
∀j ∈ J,
zj ≥ 0
∀j ∈ J
For all j ∈ J and 0 ≤ k ≤ uj −lj , let w
ˆjk = fj (lj +k, 0), and
djk = fj (lj + k, 0) − fj (lj + k, 1). Similar to the CSRAPBL
model (36)-(39), in binary robust models, instead of integer
variable xj , we use binary variables χjk , 0 ≤ k ≤ uj − lj ,
which are 1 iff xj = lj + k. This relation can be imposed by
constraints (38) and (39). By incorporating the P
dual model (26)
uj −lj
into the problem (21), and the fact that dj = k=0
djk χjk ,
the robust counterpart of uncertain CSRAPBL , for integer Γ,
is as follows.
max
χ,t,z
Then, the Robust CSRAP can be rewritten as follows.
X
j −lj
X uX
X
w
ˆjk χjk − Γt −
j∈J k=0
zj
j∈J
uj −lj
max f
Rob
x∈P
(x) = max {f (x, 0) − RC(x)} .
x∈P
(21)
s.t. t + zj ≥
X
t ≥ 0,
zj ≥ 0
djk χjk ,
∀j ∈ J,
(27)
k=0
For any j ∈ J, let
dj = f (xj , 0) − f (xj , 1),
(22)
and
∀j ∈ J,
(37), (38), and (39).
(23)
P
Model (27) has n + j∈J (uj − lj ) binary variables, n + 1
continuous variables, and 2n + m linear constraints.
Some useful properties of the worst case scenario ξ x are
presented in Proposition 2. Based on these properties, and
(22)-(23), RC(x) can be computed as




 X

RC(x) = max d0v + max
dj ξj
. (24)
v∈J 


ξ∈UΓ
2) Robust Counterpart of uncertain CSRAPBL in the case of
non-integer Γ: In this case, because Γ−bΓc > 0, d0j is strictly
greater than zero, for all j ∈ J. Moreover, we should use (24)
instead of (25) to compute RC(x). For all j ∈ J, and 0 ≤
k ≤ uj − lj , let d0jk = fj (k, 0) − fj (k, Γ − bΓc). After writing
the dual model of (24), and incorporating it into the problem
(21), the robust counterpart of the uncertain CSRAPBL for a
d0j = f (xj , 0) − f (xj , Γ − bΓc).
ξv =Γ−bΓc
j∈J\{v}
6
non-integer Γ is obtained according to (28).
max
χ,t,z,θ
s.t.
j −lj
X uX
w
ˆjk χjk − θ
j∈J k=0
θ≥
uX
v −lv
d0vk χvk + bΓctv +
k=0
X
zvj , ∀v ∈ J,
j∈J\{v}
uj −lj
tv + zvj ≥
X
djk χjk ,
∀v ∈ J, j ∈ J\{v},
TABLE I
AVERAGE CPU TIME ( IN SECONDS ) FOR LCn AND N Cn INSTANCES OF
THE ROBUST CSRAP
√
√
Integer Γ = b nc
Non Integer Γ = n
group
MIPRB
MIPRC
BR
MIPRB
MIPRC
BR
LC10
0.11
0.08
0.02
0.27
0.40
0.04
LC20
1.07
0.50
0.03
1.97
1.77
0.13
LC30
3.49
1.30
0.04
5.10
4.15
0.28
N C10
0.09
0.08
0.02
0.22
0.37
0.03
N C20
0.48
0.33
0.03
1.24
1.67
0.13
N C30
2.86
1.85
0.04
5.40
5.48
0.26
k=0
θ ≥ 0, tv ≥ 0, zvj ≥ 0,
∀v ∈ J, j ∈ J\{v},
(37), (38), and (39).
(28)
Model (28) has n + j∈J (uj − lj ) binary variables, n2 +
n + 1 continuous variables, and n2 + n + m linear constraints.
P
TABLE II
AVERAGE CPU TIME ( IN SECONDS ) FOR M Cm INSTANCES OF THE
CSRAP SOLVED BY BR
√
√
group
Integer Γ = b nc
Non Integer Γ = n
M C5
0.41
57.87
M C10
0.71
52.47
M C30
0.97
52.02
M C50
1.09
63.13
IV. E XPERIMENTAL R ESULTS
In [42], the authors discussed the motivation and advantages
of using robust optimization approaches in system reliability
problems. The same approaches have been implemented for
this paper, and the quality of the robust solutions has been
proved. However, for the sake of shortening the paper, we refer
the readers to that work. In this section, some test instances for
CSRAP are generated to test the performance of the proposed
robust methods. Similar to [42] and [38], two classes of linear
and nonlinear constraints are considered as follows.
• Linear constraints: g(x) = Ax,
where A = [aij ]m×n , aij ∈ [0, 50], ∀i ∈ I, ∀j ∈ J.
• Nonlinear constraints:
X
g1 (x) =
a1j x2j ≤ b1 ,
j∈J
g2 (x) =
X
a2j (xj + eµj xj ) ≤ b2 ,
(29)
j∈J
g3 (x) =
X
a3j xj eτj xj ≤ b3 ,
j∈J
where µj , τj ∈ [0, 0.01].
In all instances, rˆj , and δj are chosen from [0.80, 0.98], and
[0, rˆj − 0.80], respectively. These coefficients in all of the
instances follow uniform distributions. We set the values of
lj , and uj equal to 1, and 5, respectively. To assure feasibility,
we set b = g(l) + θ(g(u) − g(l)), where θ = 0.7 for linear
constraints, and θ = 0.3 for nonlinear constraints.
We used a PC with an Intel Core i5-2400S CPU 2.50 GHz
processor with 8 GB of RAM, and CPLEX 12.5.1 ( [47],
default parameters), interfaced with C++ to conduct proposed
methods. The average CPU times in Tables I and II, and the
average number of cuts in Table III, are obtained by running
the methods 20 times.
To carry out the experiments, 3 classes of instances for
CSRAP are considered as follows.
• LCn : instances with m = 5 linear constraints, and n
subsystems. 3 groups of this class were generated for
n ∈ {10, 20, 30}.
N Cn : instances with m = 3 nonlinear constraints, and
n subsystems. 3 groups of this class were generated for
n ∈ {10, 20, 30}.
• M Cm : instances with m linear constraints, and 90
subsystems. This class is considered in 4 groups for
m ∈ {5, 10, 30, 50}.
Table I shows the computational performance of three
proposed methods to obtain the robust solution of LCn and
N Cn instances of CSRAP. These three methods are as follows.
• BR: using CPLEX to solve robust counterparts of uncertain CSRAPBL , which are the models (27), and (28) for
integer, and non-integer values of Γ, respectively.
• MIPRB : classical Benders method to find the robust
counterpart of uncertain CSRAPMIP .
• MIPRC : Modified MIPRB with Lazy Cut Callback in
CPLEX.
From Table I, we see that MIPRC slightly outperforms
MIPRB for cases with integer Γ, while for non-integer Γ their
performance is very similar. BR obviously outperforms both
MIPRC and MIPRB . Therefore, for large instances, we ran
only the BR method, and report the results in Table II. In
addition, CPU times for BR are affected considerably when
it solves a model with non-integer Γ, while for MIPRC and
MIPRB the CPU times increase moderately. Table III shows
the average number of cuts generated in MIPRB and MIPRC
to solve LC and NC instances. As illustrated in this table,
the number of cuts increases by increasing the number of
•
TABLE III
AVERAGE NUMBER OF CUTS FOR LCn AND N Cn INSTANCES OF THE
ROBUST CSRAP IN B ENDERS METHODS
√
√
Integer Γ = b nc
Non Integer Γ = n
group
MIPRB
MIPRC
MIPRB
MIPRC
LC10
5.5
6.1
5.3
6.2
LC20
11.8
14.5
11.7
13.1
LC30
15.9
17.1
14.9
16.9
N C10
3.9
4.7
4.3
5.1
N C20
7.5
8.6
8.1
10.0
N C30
12.1
14.5
13.3
16.3
7
subsystems. Moreover, MIPRC needs more cuts than MIPRB .
According to Tables I and II, the size of the robust CSRAP,
i.e. the number of subsystems, has a significant impact on
the complexity of the problem. More subsystems results in
more CPU time. Moreover, CPU time depends on the value
of Γ, specifically its fractionality or integrality. Another factor
that influences the CPU time is the maximum number of
redundant components that are allowed in each subsystem.
The more redundant components allowed, the more binary
variables needed, and the more CPU time elapses. Also, the
right hand side values of the constraints may affect the solution
time. In this paper, we do not consider component mixing;
but we expect, when the component mixing is allowed in the
model, the complexity of the problem is also increased, which
in turn affects the CPU time of the solution algorithm.
yj ≤ a0jk + b0jk xj ,
qij ≥ aijk + bijk xj ,
∀j ∈ J,
lj ≤ k < uj ,
∀j ∈ J, i ∈ I
lj ≤ k < uj ,
(32)
(33)
where yj , and qij are continuous variables. Moreover, for all
i ∈ I ∪ {0}, j ∈ J, and lj ≤ k < uj , both bijk , and aijk are
constant parameters given by
b0jk = fj (k + 1) − fj (k),
a0jk = fj (k) − b0jk k;
(34)
aijk = gij (k) − bijk k.
(35)
and for all i ∈ I,
bijk = gij (k + 1) − gij (k),
CSRAPMIP has n + mn continuous
variables, n integer variP
ables, and m+2n+(m+1)× j∈J (uj −lj ) linear constraints.
B. Binary Linear Equivalent of CSRAP
V. C ONCLUSIONS , AND F UTURE R ESEARCH
This paper has investigated a redundancy allocation problem with cold standby strategy whose components’ lifetimes
follow exponential distributions. In reality, the estimation
of components’ reliabilities is accompanied by uncertainty
due to some factors such as human errors, manufacturing
technology, temperature, pressure, etc. Therefore, in this paper,
the uncertainty in components’ reliabilities has been taken
into consideration, and a robust optimization framework with
budgeted uncertainty has been developed to deal with this
problem. By exploiting problem structure, a robust optimization approach and two exact solution methods were proposed
for this problem.
The performance of the robust solutions has been examined
via simulation experiments, and the results have shown that
the robust solutions can be found in a reasonable amount
of time. For future study, the model can be extended to
other system structures such as parallel-series, complex, or
bridge. The proposed model formulation needs to be adapted
for each system configuration. Another potential research
direction is considering switch cost and imperfect switching
in the model. Furthermore, the case of multi-state and multichoice components can be considered for the proposed model.
Moreover, the choice of redundancy strategy from active and
standby strategies can be considered to extend the proposed
model. This research has proposed a cold standby system
with an exponential time to failure distribution. Other time to
failure distributions such as Erlang, Gamma, and Lognormal
can be considered for the components. Using heuristic and
meta-heuristic approaches can also be considered as potential
future work .
A PPENDIX
A. MIP Equivalent of CSRAP
The MIP equivalent of CSRAP, which is called CSRAPMIP ,
is formulated as
X
max
yj
(30)
x∈S, y,q
s.t.
X
j∈J
j∈J
qij ≤ bi ,
∀i ∈ I,
(31)
The binary linear equivalent of CSRAP, which is called
CSRAPBL , can be formulated as follows.
max
j −lj
X uX
wjk χjk
(36)
j∈J k=0
uj −lj
s.t.
X
χjk = 1,
∀j ∈ J,
(37)
k=0
j −lj
X uX
gijk χjk ≤ bi ,
∀i ∈ I,
(38)
j∈J k=0
χjk ∈ {0, 1},
∀j ∈ J, 0 ≤ k ≤ uj − lj ,
(39)
where,
wjk = fj (k), ∀j ∈ J, 0 ≤ k ≤ uj − lj ,
(40)
and
gijk = gij (lj + k), ∀i ∈ I, j ∈ J, 0 ≤ k ≤ uj − lj . (41)
P
CSRAPBL has n + j∈J (uj − lj ) binary variables, and
m + n linear constraints.
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Mohammad Javad Feizollahi is a Ph.D. candidate 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 currently working on decentralizing large-scale scheduling problems in
power systems. He is a member of INFORMS and IEEE.
9
Roya Soltani Recieved her Ph.D. in the field of Industrial Engineering
from Iran University of Science & Technology, Tehran, Iran. She obtained
her B.S., and M.S. degrees in the field of Industrial Engineering in 2006,
and 2008, respectively. Her research interests mainly include operations
research, optimization under uncertainty, reliability engineering, scheduling,
transportation planning, reverse logistic and supply chain management, and
heuristic and meta-heuristic applications.
Hadi Feyzollahi is a Ph.D. student at University of Buffalo, New York. He
received his MSc, and BSc in Industrial Engineering from Koc¸ University,
Istanbul, Turkey, and Sharif University of Technology, Tehran, Iran, in 2014,
and 2012, respectively. His research interest includes operations research,
integer programming, disaster management, and robust optimization.