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Reliability optimization of multi-state weighted k-out-of-n systems by fuzzy mathematical programming and genetic algorithm

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Abstract

This paper proposes two fuzzy mathematical models for reliability optimization of multi-state weighted k-out-of-n systems. The optimizing problems are treated as finding the minimal cost and maximal reliability system structures subjected to the fuzzy reliability requirement and the fuzzy cost available. In this approach, we consider fuzzy constraint for reliability that enables us to apply different weights for the fuzzy constraint and the objective function with their importance factors. Universal generating function technique is employed for computing the system reliability based on the fuzzy mathematical model. And then, genetic algorithm is used as the efficient optimization tool to facilitate the approach in finding optimum solutions for reliability-cost objective function at combinatorial and NP-hard applications. Finally, a numerical example is illustrated for demonstrating the flexibility and effectiveness of the proposed fuzzy mathematical programming approach.

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Abbreviations

n :

The number of components of a system

M :

The highest possible state of each component and system

k j :

The minimum total weight required to ensure that the system is in state j or above

φ :

The structure function of the system representing the state of the system

U total :

The total utility of all components

P :

The component reliability distribution matrix

U :

The utility distribution matrix

p ij :

The probability of component i being in state j

u ij :

The utility of component i in state j

c i :

The cost of component i

f i :

The feasibility of increasing component reliability

\( Z_{j}^{e} \) :

The worth quantity of the objective function

\( \mu_{j}^{z} \) :

Memberships function for the objective function

\( w_{j} \) :

The weight of fuzzy constraint

g i :

The feasibility of increasing component utility

P i max :

The maximum component reliability

P i min :

The minimum component reliability

U i min :

Minimum component weighted-average utility

\( C_{s}^{j} \) :

The expected total cost when the system is in state j or above

\( R_{j} (k_{j} ,n) \) :

The probability of the system being in state j or above, given k j , j and n

\( C_{j} \) :

The cost of the system state being below state j

\( \dot{C}_{s}^{j} \) :

The upper limit for the total cost when the system attains a state of j or above

\( R_{j}^{*} \) :

The minimum required probability for the system to attain a state of j or above

\( t_{j} \) :

Amount of allowable rise for \( \dot{C}_{s}^{j} \) in fuzzy constraint in model 1

\( t_{j}^{*} \) :

Amount of allowable reduction for \( R_{j}^{*} \)in fuzzy constraint in model 2

\( Z_{j}^{u} \) :

The best quantity of objective function

\( \mu_{j} \) :

Memberships function for the fuzzy constraint

\( w_{j}^{z} \) :

The weight of objective function

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Acknowledgements

This study was supported by a Grant from University of Tehran (Grant no. 27775/1/04). The authors are grateful for the support provided by the College of Engineering, University of Tehran, Iran.

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Authors and Affiliations

  1. Department of Industrial Engineering, University College of Engineering, University of Tehran, P. O. Box 1439955961, Tehran, Iran

    V. Ebrahimipur, S. F. Qurayshi, A. Shabani & B. Maleki-Shoja

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  1. V. Ebrahimipur
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  2. S. F. Qurayshi
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  3. A. Shabani
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  4. B. Maleki-Shoja
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Correspondence to V. Ebrahimipur.

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Ebrahimipur, V., Qurayshi, S.F., Shabani, A. et al. Reliability optimization of multi-state weighted k-out-of-n systems by fuzzy mathematical programming and genetic algorithm. Int J Syst Assur Eng Manag 2, 312–318 (2011). https://doi.org/10.1007/s13198-012-0084-y

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