1996 reliability optimization of series-parallel systems using a genetic algorithm

Reliability Optimization of
Series-Parallel Systems
Using a Genetic Algorithm
(1996)
Advisor: Wei-Chang Yeh
Student: Jing-Feng Deng
Date: 10/25/2013

Introduction
• This paper is referenced more than 500 times
• A problem-specific genetic (GA) is developed to determine the
optimal design configuration when there are multiple
component choice available for each of several k-out-of-n:G
subsystems.
• A k-out-of-n: G redundancy problem
• A Series-Parallel system

Alternative Solutions

99.06%,
cost=20,
weight=
30

99.14%,
cost=21,
weight=30

Genetic Algorithm (GA)
• 1. Encode the solutions
• 2. Generate an initial population
• 3. Select parent solutions for breeding
• Crossover breeding operator
• Mutation breeding operator
• Cull inferior solutions (elitism)

• 4. Repeat step 3 unitl termination criteria are met

GA: Initial Population
• For a given p, the initial population was determined by
randomly selecting p solution vectors.
• For each solution, s integers between ki and nmax were
randomly selected to represent ni for a particular subsystem.
• ni parts were randomly and uniformly selected from among
the mi available components
• The chosen components were sequenced by their reliability
• Previous experimentation indicated that a population size of
40 converged quickly and produced good solutions.

GA: Crossover Breeding
Operator
• For this GA, parents were selected based on the ordinal
ranking of their objective function.
• A uniform random number, U, between 1 and sqrt(p) was
selected and the solution with the ranking closet to U2 is
selected as a parent.--following the selection procedure of
Tate and Smith (1995)

GA: Mutation Operator
• A mutated component was changed to an index of mi+1 with
50% probability and to a randomly chosen component, from
among the mi choices, with 50% probability

GA: Evolution
• Elitism: A “survival of the fittest” strategy was used.
• The best solution within the population was never chosen for
mutation– to assure that the optimal solution was never
altered via mutation

Numerical Example 1 (1/2)
• Example 1: GA was implemented on the 33 variations of the Fyffe, Hines,
Lee problem (Fyffe et al., 1968) which were attempted by N&M
(Nakagawa & Miyazaki, 1981).
• C = 130
• W = 191 to 159
• 10 trials were performed for 1200 generations and the best solution
from among the 10 trials was used as the final solution.
• 18 children and 22 mutations were produced each generation, Mutation
rate = 0.05
• The GA performance improved by increasing the severity of the penalty
function.
• The GA produced feasible final solutions for all 33 problems while the
N&M model yielded feasible solutions for only 30 of the 33.

• Example 2: This is a P2 problem and the objective is to
minimize cost, given reliability & weight constraints for a
system with 2 subsystems.
• k1=4, k2=2, nmax=8
• The problem is more
difficult than Example 1
in several respects:
• Both subsystems
are k-out-of-n:G
with k>1
• For each subsystem,
there are 10 distinct
components
choices.

• 20 GA trials were performed for 6 cases;
• 15 children and 25 mutations were produced each generation
and the mutation rate was 0.25. In each case the GA was
terminated after 1,200 generations, even if the optimal
solution had not been reached.

• The results show that the GA consistently converged (i.e.,
89%) to the optimal solution.
• The result yield minimum costs which are between 4.4% and
14.2% better than that which could be obtained from any of
the previously presented methods.
• GA identified the optimal solution in 18 out of 20 trials.

Conclusion
• Integer programming algorithms guarantee convergence to an
optimal solution over a smaller search space
• GA can not guarantee that the optimal solution will be
reached.

References
• Bulfin, R. L., & Liu, C. Y. (1985). Optimal allocation of redundant components for
large systems. Reliability, IEEE Transactions on, 34(3), 241-247.
• Coit, D. W., & Smith, A. E. (1996). Reliability optimization of series-parallel
systems using a genetic algorithm. Reliability, IEEE Transactions on, 45(2), 254260, 266.
• Fyffe, D. E., Hines, W. W., & Lee, N. K. (1968). System reliability allocation and a
computational algorithm. Reliability, IEEE Transactions on, 17(2), 64-69.
• Nakagawa, Y., & Miyazaki, S. (1981). Surrogate constraints algorithm for
reliability optimization problems with two constraints. Reliability, IEEE
Transactions on, 30(2), 175-180.
• Tate, D. M., & Smith, A. E. (1995). A genetic approach to the quadratic
assignment problem. Computers & Operations Research, 22(1), 73-83.

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