Branch and Bound Algorithm for Constrained Redundancy
Optimization in Series Systems
Narges asgharian
Az_asgharian@yahoo.com
Farshad kiumarci
Mehrab esmaeli
Abstract
This paper deals the problem of constrained redundancy allocation of series
system. For maximizing the overall system reliability under limited resource
constraints, the problem is formulated as a nonlinear integer programming
problem. By exploiting special features of the problem, we derive a new necessary condition
for optimal redundancy assignments. This condition leads to a new fathoming condition in the
branch and bound method that may result in a significant reduction of computational efforts ,
as evidenced in our numerical calculation for linearly constrained redundancy optimization
problems.
Keywords: reliability optimization, series system, Redundancy allocation, branch and
bound algorithm, fathoming condition
1. Introduction
We consider in this paper the constrained redundancy optimization problem in series
systems. The goal of the problem is to determine an optimal redundancy assignment so as to
maximize the overall system reliability under certain limited resource constraints. This kind
of problems is often encountered in the design of various engineering systems. The
components with redundancy in a series setting can be independent subsystems or basic
elements in an overall system. The components in figure 1, for example, can represent
electronic parts in a section of circuits, coolers and filters in a lubrication system, valves in a
pipeline (see, e.g., Birolini, 1999; Sundararajan, 1991) or subsystems of a complicated
communication networks. Typically, the adding of redundant components is constrained to
cost, volume and weight limitations.
Mathematically, the constrained redundancy optimization problem for such a series sys- tem
can be formulated as follows:
max
𝑛
R(x ) = ∏𝑖=1( 1 − (1 − ri )xi
𝑛
s.t. Ci (x ) = ∑𝑖=1( cij (x j ) ≤ bi) , i = 1, . . . , m,
(1)
(2)
∈ X ⊂ Zn X ⊂ Zn ,
where ri ∈ (0, 1) is the component reliability in the i th subsystem in series, 1−(1−ri )xi is the
reliability of the i th subsystem when having xi identical components, x = (x 1 , x 2 , . . . , xn )
represents a redundancy assignment, R(x ) is the overall systems reliability when adopting
redundancy assignment x , cij (x j ) is a strictly increasing function of x j that represents the i
th resource consumed in j th subsystem, bi is the i th total available resource, and X is a
subset of Zn , the positive integer vector set in Rn . Denote by S the feasible region of the
problem, i.e., the set of x satisfying (2).
Chern (1992) showed that problem (1) and (2) is NP-hard. The solution schemes for this
problem proposed in the early days include dynamic programming (see Bellman and Dreyfus,
1958) and greedy heuristic algorithm (see Fox, 1966). When there are multiple resource
constraints, the curse of dimensionality prevents an efficient adoption of dynamic programming.
Variant marginal allocation scheme based on Fox (1966) are abundant in the literature (see, e.g.,
Tillman et al., 1977; Ohtagaki et al., 1995). These heuristic algorithms do not guarantee the
optimality of the solution. Ghare and Taylor (1969) formulated the optimal redundancy problem
in series systems as a zero-one integer programming and solved it by a branch and bound
procedure. Branch and bound methods for general reliability systems were also investigated by
Nakagawa et al. (1978) and Tzafestas (1980). Using the increasing property of the constraint
functions, Misra and Sharma (1991) proposed a search algorithm to scan the entire feasible
region by successively increasing the number of each redundancy component by one and
skipping all the feasible points before reaching the last feasible point that does not violate the
constraints. By doing so, the method in Misra and Sharma (1991) enumerates the feasible
redundancy assignments close to the boundary of the feasible region and determines the best
redundancy assignment by computing and comparing the corresponding systems reliability. A
new necessary optimality condition for problem (1) and (2) is obtained in this paper by
exploiting special features of the problem.
2. A necessary optimality condition
The following is always true in constrained redundancy optimization. Increasing the number of
parallel paths in one subsystem, while keeping all other subsystems unchanged, will increase
both the overall systems reliability and the resource consumed. This point can be enhanced
by observing that all R(x ) and Ci (x ), i = 1, . . . , m, are strictly increasing functions of x .
Some researchers, e.g., Misra and Sharma (1991), have noticed that an optimal redundancy
assignment of (1) and (2) always locates close to the boundary of the feasible region due to
the monotonicity of R(x ) and Ci (x ), i = 1, 2, . . . , m. A feasible
redundancy assignment x is said to be noninferior if there exists no other feasible y ∈ S such
that yi ≥ xi , (i = 1, . . . , n), with at least one strict inequality. It is easy to see that x is
noninferior iff there does not exist a j such that x + e j ∈ S, where e j denotes the j th unit
vector in Rn .
Proposition 1. Any optimal redundancy assignment of (1) and (2) must be noninferior.
To prove Proposition 2, we need the following lemma.
Lemma 1.
(i) If 0 < a < b < 1 and 0 < p < q , then
(1 − bq )(1 − a p ) > (1 − b p )(1 − a q ).
(3)
(ii) If 0 < a < b < 1 and 1 < p + 1 ≤ q , then
(1 − b p+1 )(1 − a q −1 ) > (1 − b p ) (1 − a q ).
Proposition 2.
(4)
R(u (i, j ) (x )) > R(x ) if i < j, ri < r j , and x j ≥ xi + 1.
Proof:
j ) b(xp ),)(1R(x
p+1the
(1 − b By
)(1definition
− a q −1 of
) >u (i,
(1 −
− a) qand
). R(u (i, j ) (x )) are different only in their i th
and j th factors. Thus R (u (i, j ) (x )) > R(x ) is equivalent to
[1 − (1 − ri )xi +1 ][1 − (1 − r j )x j −1 ] > [1 − (1 − ri )xi ][1 − (1 − r j )x j ].
Since ri < r j and xi + 1 ≤ x j , we imply (9) by applying Lemma 1(ii) with a = 1 − r j
b = 1 − ri , p = xi , q = x j . Thus, the conclusion of Proposition 2 is true.
The significance of Proposition 2 is that for a feasible redundancy assignment x with i < j , ri
< r j , and xi < x j (xi + 1 ≤ x j by the integrality of xi and x j ), if the unit decreasing
transformation is feasible, i.e., u (i, j ) (x ) ∈ S, then the overall systems reliability can get
higher via replacing x by u (i, j ) (x ).
We therefore obtain, from Propositions 1 and 2, the following theorem.
Theorem 1 (Necessary optimality condition). An optimal redundancy assignment of (1)
and (2) must be both noninferior and maximal decreasing.
3. Branch and bound algorithm
The experiments of branch and bound algorithms for convex integer programming are reported in Gupta and Ravindran (1985). General branch and bound algorithms have been investigated for solving (1) and (2) by several authors (see Ghare and Taylor, 1969; Nakagawa et
al., 1978; Tzafestas, 1980).
We outline here a general branch and bound algorithm for solving (1) and (2). The algorithm starts by finding a global optimal solution x ∗ of the continuous relaxation problem of (1)
and (2). If the solution is integral, then it is an optimal solution to (1) and (2). Otherwise
a fractional variable, x ∗ say,j is chosen and two continuous relaxation sub problems are generated by adding an additional bound constraint x j ≤ xj ∗ or x j ≥ xj ∗ + 1, respectively,
where xj ∗ denotes the integral part of xj ∗ . One of the generated sub problems is chosen to
be solved next. If its optimal solution is integral and its objective value is better than that of
the incumbent (current integer solution with the best objective function value), then it
becomes the new incumbent. Otherwise the subproblem is divided again, and the process is
repeated until no subproblem remains to be solved. The process of forming the continuous
relaxation subproblems is called branching. A node stores the information necessary for describing and solving the corresponding subproblem. In the conventional branch and bound
algorithms, a node is eliminated or fathomed from the node tree for further consideration if
one of the following three conditions holds: (a) the corresponding continuous relaxation
subproblem generates an integral optimal solution, (b) its global optimal value is less than or
equal to the lower bound associated with the current incumbent, or (c) it is infeasible.
The maximal decreasing property derived in this paper provides a basis for adding an
additional fathoming condition in the branch and bound method for solving (1) and (2).
Using this fathoming condition may significantly speed up the convergence of the algorithm by
further eliminating certain nodes that do not generate an optimal solution of (1) and (2).
Let N = (cz, vlb, vub) denote a node in a branch and bound algorithm for solving (1) and
(2), where cz is the optimal objective function value of the continuous relaxation at the
father node, and vectors vlb, vub ∈ Rn are the lower bound and upper bound on the decision
variables, respectively.
Proposition 3. For a node N = (cz, vlb, vub), suppose that there exist i and j, ri < r j ,
with i < j and vubi < vlb j such that
[cki (xi + 1) − cki (xi )] + [ckj (x j − 1) − ckj (x j )] ≤ 0
(5)
for k ∈ {1, . . . , m} and x ∈ X N = {x ∈ X | vlb ≤ x ≤ vub}. Then the node N can be
fathomed for further branching in a branch and bound method when solving (1) and (2).
4. Computational results
The revised branch and bound algorithm discussed in the previous section has been coded
using FORTRAN 77 and tested on a SUN SPARC station 5.
Problem 1 (Misra and Sharma, 1991)
Problem 2 (Bellman and Dreyfus, 1958)
Problem 3 (Nakagawa et al., 1978)
Problem 4 (Nakagawa et al., 1978)
The second group of test problems is randomly generated with m = 2 and n = 15, 20,25, 30,
and 35. More specifically, for a given n, each cij > 0 in the 2 × n parameter matrix A = (cij )
is generated from the normal distribution N (7.5, 1). Accordingly, each reliability r j ∈ (0, 1)
is generated from the normal distribution N (0.75, 0.2).
Both the conventional branch and bound algorithm and the newly proposed branch and bound
algorithm with use of the new fathoming condition are run to solve the above two groups of
problems. The computational results for the two group of problems are shown in Table 2,
figure 3 and 4, respectively, where R ∗ denotes the optimal systems reliability, NLP the number
(or average number) of continuous nonlinear subproblem.
Figure 1. Average NLPs for different n in the second group of test problems.
Figure 2. Average CPU time for different n in the second group of test problems.
Table 2. Numerical results for Problems 1–4.
Problem
R∗
1
0.695454
2
3
4
0.930803
0.94565
0.969796
NLP (k = 0) NLP (k = 1) CPU (k = 0) CPU (k = 1)
18
5
0.035
0.027
34
533
550
31
220
2175
0.050
9.03
141.68
0.037
4.458
62.20
k= 0: conventional branch and bound algorithm without use of the fathoming condition in
Proposition 3.
k = 1: the proposed new branch and bound algorithm with a use of the fathoming condition in
Proposition 3.
5. Concluding remarks
This paper extends the literature on constrained redundancy optimization for series systems. By
exploiting special features of the problem, we derived a maximal decreasing property for
optimal redundancy assignments. Interestingly, similar optimality conditions have been found
in other optimal allocation problems, such as in the server allocation problem in mul- tiple
center manufacturing systems (see Shanthikumar and Yao, 1988). Our computational results
for linear constrained cases show that the proposed branch and bound algorithm,
incorporated by the new fathoming condition, may substantially reduce the computational
effort both in terms of the number of subproblems, and the CPU time, when compared to
the conventional branch and bound algorithm. A natural extension of the research work
presented in this paper is to investigate similar necessary conditions for optimal redundancy
assignments in complex reliability networks, if they exist.
References
R. Bellman and S. Dreyfus, “Dynamic programming and the reliability of multicomponent
devices,” Opns. Res.vol. 6, pp. 200–206, 1958.
A. Birolini, Reliability Engineering: Theory and Practice, Springer: New York, 1999.
M. S. Chern, “On the computational complexity of reliability redundancy allocation in a
series system,” Opns. Res. Lett. vol. 11, pp. 309–315, 1992.
B. Fox, “Discrete optimization via marginal analysis,” Mgmt. Sci. vol. 13, p. 210–216, 1966.
P. M. Ghare and R. E. Taylor, “Optimal redundancy for reliability in series systems,” Oper.
Res. vol. 17, pp. 838–847, 1969.
P. M. Ghare, R. E. Taylor, “Optimal redundancy for reliability in series system”,
Operations Research, vol 17, 1969, pp 838-847.
F. A. Tillman, C. L. Hwang, W. Kuo, Optimization of System Reliability, 1980; Marcel
Dekker.
K.B. Misra, U. Sharma, An efficient algorithm to solve integer- programming
problems arising in system reliability design, IEEE Transactions on Reliability 40
(1991) 81–91.