Document Type : Research Paper



In this paper, a mathematical model is proposed for project portfolio
selection and resource availability cost problem to scheduling activities in
order to maximize net present value of the selected projects preserving
precedence and resource constraints. Since the developed model belongs to
NP-hard problems list, so a genetic based meta-heuristic algorithm is
proposed to tackle the developed model. In the proposed algorithm beside
common operators of genetic algorithms such as crossover & mutation, some
intelligent operators are utilized for local search in computed resources and
shifting the activities with negative cash flows. The key parameters of the
algorithm are calibrated using Taguchi method to accelerate convergence of
the proposed algorithm. Then, the algorithm is used to solve 90 test
problems consisting 30 small-scale, 30 middle-scale and 30 large scale
problems to examine the algorithm’s performance. It is observed that, in
small problems, the obtained solutions from the proposed genetic algorithm
have been comparably better than the local optimum solutions stemmed
from Lingo software. On the other hand, for the middle and large size
problems which there is no local optimum available within the limited CPU
time, robustness of the proposed algorithm is appropriate


C.L. Hwang, F.A. Tillman, W.K. Wei and C.H. Lie. Optimal
schedule maintenance policy based on multiple-criteria decision-making.
IEEE transaction on reliability, Vol. R-28, No 5, 1979; 394-397.
M. Sakawa. Multi-objective optimization by the surrogate worth
trade-off method. IEEE Transaction on reliability, Vol. 27, No 5, 1978;
T. Inagaki, K. Inoue and H. Akashi. Interactive optimization of
system reliability under multiple objectives. IEEE Transaction on
reliability,Vol.R-27, NO 4 1978;264-267.
M.Sakawa. Multi-objective reliability and redundancy optimization of
a seies-parallel system by the surrogate worth trade-off method.
Microelectronic and Reliability, Vol.17,No 4, 1978;465-467.
Y.Nakagawa. Studies on optimal design of high reliability system
single and multiple objective nonlinear integer programming.
Ph.D.thesis Kyoto university. 1978.
M.Sakawa. Optimal reliability-design of a series-parallel system by a
large-scale multi-objective optimization method. IEEE Transaction on
Reliability, Vol.R-30 1981; 173-174.
M.Sakawa. Interactive multi-objective optimization by Sequential
Proxy Optimization Technique (SPOT). IEEE Transaction on
Reliability, Vol.R-31, 1982;461-464.
K.B.Misra and U.Sharma. An efficient approach for multiple criteria
redundancy optimization problems. Microelectronic and Reliability,
Vol.31,No 2-3, 1991; 285-294.
K.B.Misra and U.Sharma. Multi-criteria optimization for combined
reliability and redundancy allocation in systems employing mixed
redundancies. Microelectronic and Reliability, Vol.31, 1991;323-335.
A. K. Dhingra. Optimal apportionment of reliability and redundancy
in series systems under multiple objectives. IEEE Transaction on
Reliability, Vol.41,No 4, 1992;576-582.
M.Sasaki, M.Gen and M.Ida. A method for solving reliability
optimization problem by fuzzy multi-objective 0-1 linear programming.
 … 119
Electronics and Communications in Japan Part II: Fundamental
Electronic Science 1992; 106-116.
G.S.Mahaparta, T.K.Roy. Fuzzy multi-objective mathematical
programming on reliability optimization model. Applied Mathematics
and Computing, Vol. 174, No 1, 2006;643-659
Daniel Salazar, Claudio M.Rocco, Blas j.Galvan. Optimization of
constrained multiple-objective reliability problems using evolutionary
algorithms. Reliability engineering and system safety, Vol.91, No 9,
Zhigang tian, Ming j zuo. Redundancy allocation for multi-state
systems using physical programming and genetic algorithms. Reliability
engineering and system safety, Vol.91, No 9, 2006;1049-1056.
David.W.Coit and Abdullah konak. Multiple weighted objectives
heuristic for the redundancy allocation problem. IEEE transaction on
reliability, Vol.55,No 3, 2006;551-558
Jian-jua zhao, Zhaoheng liu, My-thein dao. Reliability optimization
using multi-objective ant colony system approaches. Reliability
engineering and system safety, Vol.92, No 1, 2007;109-120.
Tavakkoli-Moghaddam, R., Safari J. and Sassani, F., Reliability
optimization of series-parallel systems with a choice of redundancy
strategies using a genetic algorithm, Reliability Engineering and System
Safety, Vol. 93, No 4, 2008; 550-556.
Sadjadi, S.J and Soltani, R. An efficient heuristic versus a robust
hybrid meta-heuristic for general framework of serial-parallel
redundancy problem. Reliability engineering and system safety, Vol.94,
No 11, 2009;1703-1710.
Amir azaron, Cahit perkgoz, Hideki Katagiri, Kouske Kato,
Masatoshi Sakawa. Multi-objective reliability optimization for
dissimilar-unitcold-standby system using a genetic algorithm.
Computers & operation research Vol.36,No 5, 2009;1562-1571.
Zai Wang, Tianshi Chen. Ke Tang, Xin Yao.A multi-objective
Approach to redundancy allocation problem in parallel-series
systems. IEEE transaction 2009;582-589.

Zhaojun Li,Haitao Liao,David W.Coit. A two-stage approach for
solving multi-objective decision making with applications to system
reliability optimization. Reliability engineering and system safety
Vol.94, No 10, 2009;1585-1592.
Yun-chia liamg and Min-Hua Lo. Multi-objective redundancy
allocation optimization using a variable neighborhood search algorithm.
J heuristic Vol.16, No 3, 2010;511-535.
E.Zio and R.Bazzo. Level diagrams analysis of Pareto front for multiobjective
system redundancy allocation. Reliability engineering and
system safety, Vol.96,No 5, 2011;569-580.
Banu soylu,selda kapan ulusoy. A preference ordered classification
for a multi-objective max-min redundancy allocation problem.
Computers & Operation Research Vol.38 , No 12, 2011; 1855-1866.
Isis Didier lins and Enrique Lopez Droguett.Redundancy allocation
problems considering systems with imperfect repairs using multiobjective
genetic algorithms and discrete event simulation. Simulation
modeling practice and theory,Vol.19,No 1, 2011;362-381.
Laxminarayan Sahoo, Asoke Kumar Bhunia, Parmad Kumar Kapur.
Genetic algorithm based multi-objective reliability optimization in
interval environment. Computers & Industrial Engineering, Vol. 62, No
1, 2012;152–160.
Sajadi, S.J and Soltani, R. Alternative design redundancy allocation
using an efficient heuristic and a honey bee mating algorithm. Expert
Systems with Applications, Vol. 39, No 1 ,2012;990-999.
Chambari, A., Rahmati, S.H,. Najafi, A.A and Karimi, A,. A biobjective
model to optimize reliability and cost of system with a choice
of redundancy strategies, Computers & Industrial Engineering, Vol. 63,
No 1, 2012; 109-119
Safaei, N., Tavakkoli-Moghaddam, R. and Corey Kiassat.,
Annealing-based particle swarm optimization to solve the redundant
reliability, Applied Soft Computing. Vol. 12, No 11,2012; 3462-3471.
Chern MS. On the computational complexity of reliability
redundancy allocation in a series system. Oper Res Lett Vol.11, No
5,1992; 309–315.

E. Atashpas-Gargari, C. Lucas, Colonial competitive algorithm. E
Congress on Evolutionary Computation, 2007. 4661- 4667.
Deb, K., Pratap, A.,Agrawal, S., and Meyarivan, T. A fast and elitist
multi-objective genetic algorithm:NSGA-II.IEEE Transaction on
Evolutionary Computation, Vol.6, No 2 ,2002- 182-197.
Knowels,J.D., and corne, D.W.(1999). The Pareto archived
evolution strategy: A new baseline algorithm for Pareto multi-objective
optimization. In congress on Evolutionary Computation(CEC99),
Volume 1,Piscataway,NJ, 98-105.IEEE Press.
Corne, D.W., Knowles, J.D., and Oates, M.J.(2000). The Pareto
envelope-based selection algorithm for multi objective optimization. In :
M.S. et al.(Ed.), Parallel problem solving from Nature-PPSN VI, Berlin,
839-848. Springer.
Tavakkoli-Moghaddam, R., Azarkish, M. and Sadeghnejad A.,
Solving a multi-objective job shop scheduling problem with sequencedependent
setup times by a Pareto archive PSO combined with genetic
operators and VNS, Int. J. of Advanced Manufacturing Technology, Vol.
53,No 5-8, 2011; 733–750