# Probabilistic Approach in Mathematical Programming Model to Solve Redundancy Allocation Problems in a Series-Parallel System with All Unit Discount Policy

Document Type: Research Paper

Authors

Department of Industrial Engineering, Amirkabir University of Technology, Garmsar Campus, Iran

Abstract

Nowadays, designing and implementing systems with premier features and higher reliability is deemed to be a basic principle for the engineers and users. Because regarding this point can result in the proper use of a system during its lifetime. Reliability refers to a measure of quality versus time factor that is computed by a probability of working without any failure in a given time and under some specific conditions. Since a system consists of many items, which withstand several stress factors, manufacturers commonly employ different solution approaches to increase the reliability. One of the main strategies in this regard is to consider some additional items in parallel to original ones. This is also called redundancy allocation. Different items have different reliability, cost, weight, and importance level. So, making decision on assigning redundant would be of interest under limited budget and volume or weight in system development.  Some redundant item is not turned on until the main item works correctly. Hence, a switch and sensors would be considered to monitor the status of the main item and to decide when the redundant must start working. This type is called standby item. In today’s competitive world, offering a system with lower total expense, given that its high reliability is maintained, can make the company popular with the customers. Although in recent years, research on optimization have been presented with all unit discount for components of a system, this paper not only addresses the active redundancy strategy, but also it discusses a combination of components with active or cold redundancy strategy. It uses a system that generates the all unit discount to the sum of the components with the two strategies mentioned. Additionally, in order to make the problem more realistic to the real world, failure rate and cost parameters were considered uncertain. To solve two models with the aims of maximizing the reliability and minimizing the cost, chance constrained approach was employed for the constraints of cost and reliability. The proposed model was solved with accurate method using GAMS software. With regard to the model’s proper treatment of changes in effective factors proposed in the model, it is concluded that this model is exploitable for optimizing stability in mass production industries where applying the global discount policy leads to some benefits.

Keywords

Main Subjects

### References

1. Islami Baladeh, A.A., Seyed Esfahani, M.M. and Farsi, M.A. (2014). "A Scenario-Based Model for Redundancy Allocation with Choice of Redundancy Strategies", Journal of Industrial Engineering, Vol. 48, No. 1, PP. 91-98.
2. Kuo, W. and Wan, R. (2007). "Recent advances in optimal reliability allocation", Chapter, Computational Intelligence in Reliability Engineering, Vol. 37, No. 2 of the series Studies in Computational Intelligence, PP. 1-36.
3. Abouei Ardakan, M. and Hamadani Z. (2014). “Reliability optimization of series-parallel systems with mixed redundancy strategy in subsystems”,Reliability Engineering and System Safety, Vol. 130, No. 1, PP. 132–139.
4. Shaghaghi Nazarloo, F., Amiri, M. and Azimi, P. (2014). "Development of a simulation-based method to solve the problem of efficient allocation of surplus repairable systems", the tenth International Conference on Industrial Engineering,Iran Industrial Engineering Society, Amir Kabir University of Technology, Tehran, Iran, PP. 27-28.
5. Yahyatabar Arabi, A.A and Eshraghniaye Jahromi, A. (2013). “Availability optimization of a series system with multiple repairable load sharing subsystems considering redundancy and repair facility allocation”, International Journal of System Assurance Engineering Management, Vol.4, No 3, PP. 262–274.
6. Soltani, R., Sadjadi S.J. and Tavakkoli-Moghaddam, R. (2014). “Interval programming for the redundancy allocation with choices of redundancy strategy and component type under uncertainty: Erlang time to failure distribution”, Applied Mathematics and Computation, Vol. 244, No. 1, PP. 413-421.
7. Ghazi Mir Saeed, M., Najafi, A.A. and Shahryari, H. (2014). "Providing exact solution of k of n on the issue of allocation strategy excess surplus", Industrial Management, Vol. 6, No. 1, PP. 97-110.
8. Feizollahi, M.J., Soltani, R. and Feyzollahi, H. (2015). “The Robust Cold Standby Redundancy Allocation in Series-Parallel Systems With Budgeted Uncertainty",IEEE Transactions on Reliability., Vol. 64, No 2, PP. 1-9.
9. .Zhang, E. and Chen, Q. (2015) “Multi-objective reliability redundancy allocation in an interval environment using particle swarm optimization”, Reliability Engineering and System Safety, Vol. 145, No. 1, PP. 83–92.
10. Kong, X et al. (2015). “Solving the redundancy allocation problem with multiple strategy choices using a new simplified particle swarm optimization”, Reliability Engineering and System Safety, Vol. 144, No. 1, PP. 147–158.
11. Latif Shabgahi, G. R., Aslansefat, K. and Bahar Gogani, M. (2015). "Reliability and Safety Modelling in Reliable Systems Supported with Cold Standby Spares by a Markov Model", Journal of Industrial Engineering, Vol. 49, No. 2, PP. 273-285.
12. Pourkarim Guilani, P. et al. (2016). “Redundancy allocation problem of a system with increasing failure rates of components based on Weibull distribution: a simulation-based optimization approach”, Reliability Engineering and System Safety, Vol. 152, No. 1, PP. 187–196.
13. Chatwattanasiri, N., Coit, D.W. and Wattanapongsakorn, N. (2016). “System redundancy optimization with uncertain stress-based component reliability: Minimization of regret”, Reliability Engineering and System Safety., Vol. 154, No. 1, PP. 73-83.
14. Jahromi, A.E. and Feizabadi, M. (2017). “Optimization of multi-objective redundancy allocation problem with non-homogeneous components”, Computers & Industrial Engineering., Vol. 108, No. 1, PP. 111–123.
15. Gholinezhad., H. and Zeinal Hamadani, A. (2017). “A new model for the redundancy allocation problem with component mixing and mixed redundancy strategy”, Reliability Engineering and System Safety., Vol. 164, No. 1, PP.66–73..
16. Xiang, Q et al. (2017). “Reliability-redundancy-location allocation with maximum reliability and minimum cost using search techniques”, Information and Software Technology, Vol. 82, No. 1, PP. 36–54.
17. Huang, C.L. (2015). “A particle-based simplified swarm optimization algorithm for reliability redundancy allocation problems”, Reliability Engineering and System Safety., Vol. 142, No. 1, PP. 221-230.
18. Faghih-Roohi, Sh. et al. (2015). “Dynamic availability assessment and optimal component design of multi-state weighted k-out-of-n systems”, Reliability Engineering and System Safety., Vol. 123, No. 1, PP. 57-62.
19. Salmasnia، A., Ameri، E. and Akhavan Niaki, T. (2015). “A Robust Loss Function Approach for a Multi-Objective Redundancy Allocation Problem”, Applied Mathematical Modelling, Vol.40, No 1, PP. 635–645.
20. Azadeh,A. et al. (2015). “A multi-objective optimization problem for multi-state series-parallel systems: A two-stage flow-shop manufacturing system”, Reliability Engineering and System Safety., Vol. 136, No. 1, PP. 62-74.
21. Mogulkoc, T. and W.Coit, D. (2011). “System Reliability Optimization Considering Uncertainty: Minimization of the Coefficient of Variation for Series-Parallel Systems”, IEEE Transactions ON Reliability, Vol. 60, No. 3, PP. 667 – 674.
22. Smith, C.O. (1976). “Introduction to Reliability in Design” 1th. Ed”, Chapter 4&5, McGraw-Hill Pub. Co., New York.
23. Ekhtiari, M. (2010). "multi-objective Contingency planning for optimization problem to determine the number of manpower in production systems workshop”, Journal of Industrial Management Studies, Vol. 19, No. 1, PP. 189 to 216.
24. Soltani, R., Sadjadi, J. and Tofigh, A.A. (2013). “A model to enhance the reliability of the serial parallel systems with component mixing”, Applied Mathematical Modelling, Vol. 38, No. 3, PP. 1064–1076.
25. Sadjadi, J. and Soltani, R (2014). “Minimum-Maximum regret redundancy allocation with the choice of redundancy strategy and multiple choice of component type under uncertainty”, Computers & Industrial Engineering, Vol. 79, No. 1, PP. 204–213.
26. Ghelich, I. and Ghelich, F. (2015). “The chance of solving approach and two-stage constraints in the allocation of multi-period model Mkanyaby- blood facilities with uncertainties in demand”, Eighth International Conference of Iranian Operations Research Society, Ferdowsi University of Mashhad, Mashhad, Iran, pp. 63-66.
27. Amiri, M and Khajeh, M, (2015). “Developing a bi-objective optimization model for solving the availability allocation problem in repairable series–parallel systems by NSGA II”, Journal of Industrial Engineering International, March 2016, Vol. 12, No. 1, PP. 61–69.