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An Application of Cuckoo Search Algorithm for Series System with Cost and Multiple Choices Constraints
Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 86 (2016) 453 – 456
2016 International Electrical Engineering Congress, iEECON2016, 2-4 March 2016, Chiang Mai, Thailand
An Application of Cuckoo Search Algorithm for Series System with Cost and Multiple Choices Constraints Mana Sopaa, Niwat Angkawisittpanb* a Faculty of Engineering, Mahasarakham University, Maha Sarakham, Thailand, 44150. Research Unit for Computational Electromagnetics and Optical Systems, Faculty of Engineering, Mahasarakham University, Maha Sarakham, Thailand, 44150.
b
Abstract This paper presents an application of Cuckoo Search Algorithm (CSA) for solving reliability optimization problems for a series system. The aim of research is to select hardware from multiple choices for each subsystem to maximize the overall of system reliability and under cost constraints. CSA is a new metaheuristic algorithm inspired from behavior of Cuckoos laid their eggs in the nests of other species with the Lévy flight distribution. The effectiveness of the proposed algorithm has been evaluated with two groups of examples from the existing literature. The computational results show that the CSA has effectively been able to find the global optimal solution. 2016The TheAuthors. Authors. Published Elsevier © 2016 Published by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of iEECON2016. Peer-review under responsibility of the Organizing Committee of iEECON2016 Keywords: Cuckoo Search Algorithm, Reliability Optimization Problems, A Series System, Metaheuristic;
1. Introduction The reliability optimization problems that are presented by different needs of engineering and industry such as electrical systems, hardware design, etc. The design of reliability needs to consider various factors in many aspects such as cost, performance and other characteristics for maximizing the overall of system. For example, to maximize the system reliability subjected to cost and performance constraints [1] and subjected to cost with multiple choices [2].
* Corresponding author. Tel.: +66-042-341615; fax: +66-042-341614. E-mail address: [email protected]
1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of iEECON2016 doi:10.1016/j.procs.2016.05.079
454
Mana Sopa and Niwat Angkawisittpan / Procedia Computer Science 86 (2016) 453 – 456
For the purposes of this study is to deal with the reliability optimization problems for a series system with multiple choices and cost constraints. The problem is formulated as binary integer nonlinear programming. During the past there were theories and methods to solve problems with different techniques which are as follows: AS algorithm [3] and in some cases have been applied this problem to the yield optimization by using the Bee Algorithm (BA) [4]. 2. Problem Formulation A series system of n subsystem, there are different technologies of hardware available with reliabilities and different costs. The objective is to select hardware for each subsystem in order to the overall of system reliability and subject to the cost constraints. The notations are introduced below: Notations n Ni Cij Rij Rsys
number of subsystems number of hardware available for subsystems i cost of a subsystem i using hardware j reliability of a subsystem i using hardware j the overall of a series system reliability
Define the objective function to solve this problems with the equation. n
i 1
n
Subject to
§ ¨ ¨ ©
·
Ni
¦ X R ¸¸
Maximize Rsys
Ni
¦¦ X C ij
ij
ij
j 1
ij
¹
dC
(1)
i 1 j 1
Ni
¦X
ij
1, i 1, 2, ...,n
(2)
j 1
X ij
^0,1`, i
1, 2, ...,n and j 1, 2, ..., Ni
(3)
The terms of the Eq. 1: represents the cost constraint and C is integer, Eq. 2: represents the multiple choices constraints, Eq. 3: the variables for decision. When the result of a solution conformable all constraints, it is called the set of feasible solution and otherwise, it is called the set of infeasible solution. 3. Description of the Proposed Approach The CSA is a new metaheuristic optimization algorithm inspired by the behavior of some Cuckoo bird species with the Lévy flight. It has been presented by Yang and Deb [5] and it was applied to solve engineering design optimization problems. The CSA is based on three concepts of rules [6]: 1) Each Cuckoo bird lays one egg (a solution) at a time, and continues to lay their egg in the available host nests by randomly, 2) A Cuckoo egg with high quality (the better solution) in the best nests will be inherited to the next generation, and 3) An egg laid by a Cuckoo (called alien egg) is discovered by probability. In this case, the host bird species can abandon the nest or either throws the alien egg away and builds a new nest. Its simplicity, this last assumption can be fined by the fraction of the nests are replaced by new nests (called new random solutions). The details of CSA can be described in the following steps:
455
Mana Sopa and Niwat Angkawisittpan / Procedia Computer Science 86 (2016) 453 – 456
Step 1: The first step, initialize the CSA parameters. The parameters consist of nests (n), discovery probability (prob), step size (α), and the maximum of iterations and stop criterion. Step 2: Generate initial Cuckoo’s eggs in the nests of host birds. The first position of the nests is assigned by randomly with decision variables using Eq. 4.
nestij (0) ROUND X j min ( X j max X j min ). rand1
(4)
Where nestij(0) are the initial values for the jth decision variables and for the ith nest; the rounding function used to round up or round down jth values as the integer for the problem. Xjmax and Xjmin are the maximum and the minimum of search space values for the jth decision variables (jth technology) and rand1 is a random number returns from the interval [0,1]. Step 3: Generate new Cuckoo by Lévy flights. All the host nests are replaced with the quality of new Cuckoo’s eggs, which generated by Lévy flights distribution from their positions using Eq. 5
nesti (τ 1)
ROUND nesti (τ ) D.SL.(nesti (τ ) nestbest (τ )).rand 2
(5)
Where nesti (W) is the ith current nest position, α is the step size parameter; SL is a random state transition based on the Lévy flights behavior. The Lévy flight provides a random walk while the random step length is drawn from Lévy flight distribution, the step length SL can be calculated according to Eq. 6 and rand2 is a random number with standard normal distribution and nestbest is the best nest. 1β
u
SL
v
1E
; u ~ N (0,V u2 ) where V u
*(1 β ) sin(Sβ / 2) ½ ® ( β 1) / 2 ¾ ¯ *>(1 β ) / 2@ β 2 ¿
and v ~ N (0, V v2 ) where V v
1
(6)
Where β is number in the interval [1, 2]; u and v are drawn from normal distributions. Step 4: Discovery the alien eggs. For each solution in terms of the probability matrix K, the alien eggs are abandoned using Eq. 7. K ij
1 ® ¯0
if rand prob if rand t prob
(7)
Where prob is the discovering probability and rand are random numbers in the interval [0,1]. Early eggs are replaced considering by the newly quality eggs generated ones from their current position by random walks with step size using Eq. 8. nestt 1
ROUND nestt S. * K ; S
nests(randpm1(n),:) nests(randpm2(n),:) .rand
(8)
Where randpm1 and randpm2 are random numbers with a permutation function applied for different rows, nests and discovering probability matrix K. Step 5: Stop criteria. The processes of CSA are alternatively performed until the stopping criterion is true. 4. Computation results In order to evaluate the CSA, the data sets are presented in Tables 5 and 6 by Nahas & Nourelfath [3] in references. The system tried to adjust the number of host nests n = 5, 10, 15, 20, 40 and the probability prob = 0.1, 0.25, 0.5, 0.70. The results of the best solution are from the parameters: Host nests = 20, prob = 0.70, Iterations = 2000, α = 0.01 and β = 1.5. The results are illustrated in Fig. 1 (a). and Fig. 1 (b).
456
Mana Sopa and Niwat Angkawisittpan / Procedia Computer Science 86 (2016) 453 – 456 1
1
0.965134
0.9
0.9
0.86544
0.8 0.8
Reliability
Reliability
0.7
0.7 0.6 0.5 0.4
0.6 0.5 0.4 0.3
0
200
400
600
800
1000
Iterations
1200
1400
1600
1800
2000
0.2
0
200
400
600
800
1000
Iterations
1200
1400
1600
1800
2000
Fig. 1. (a) ; (b) Evolution of the best solutions using the CSA
Fig. 1 (a), the search space is larger than 1.572¯1012. The result is obtained the global optimal equal to 0.965134. The selected technologies are: 3-3-4-4-3-3-2-2-3-2-2-4-4-4-2. Fig. 1 (b), the search space is larger than 1.932¯1020. The result is obtained the global optimal equal to 0.86544. The selected technologies are: 3-3-3-4-2-3-2-2-3-1-2-3-44-1-2-3-3-5-2-3-2-2-3-1. 5. Conclusions In this paper, the optimal reliability in this problem with the application of a new approach called Cuckoo Search Algorithm (CSA) was studied. It is seen that the proposed CSA has the ability to find the global optimal solution in computational time and the accurate results. Acknowledgements The authors would like to thank Research Unit for Computational Electromagnetics and Optical Systems, Faculty of Engineering, Mahasarakham University. References 1. R. Meziane et al. Reliability optimization using ant colony algorithm under performance and cost constraints. Electric Power Systems Research; 2005, p. 1-8. 2. N. Ruan and X. Sun. An exact algorithm for cost minimization in series reliability systems with multiple component choices. Applied Mathematics and Computation; 2006, p. 732-741. 3. N. Nahas and M. Nourelfath. Ant system for reliability optimization of a series system with multiple-choice and budget constraints. Reliability Engineering and System Safety; 2005, p. 1-12. 4. W. Wongthatsanekorn and N. Matheekrieangkrai. Bee Algorithm for Solving Yield Optimization Problem for Hard Disk Drive Component under Budget and Supplier’s Rating Constraints and Hueristic Performance Comparison. Lecture Notes in Electrical Engineering; 2011, p. 203-216. 5. X.-S. Yang, and S. Deb. Cuckoo search via Lévy Flights. Proceedings of world congress on nature & biologically inspired computing (NaBIC2009India), USA, IEEE Publications; 2009, p. 210-214. 6. X.-S. Yang, and S. Deb. Engineering Optimisation by Cuckoo Search, Int. J. Mathematical Modelling and Numerical Optimisation; 2010, p. 330–343.
ScienceDirect Procedia Computer Science 86 (2016) 453 – 456
2016 International Electrical Engineering Congress, iEECON2016, 2-4 March 2016, Chiang Mai, Thailand
An Application of Cuckoo Search Algorithm for Series System with Cost and Multiple Choices Constraints Mana Sopaa, Niwat Angkawisittpanb* a Faculty of Engineering, Mahasarakham University, Maha Sarakham, Thailand, 44150. Research Unit for Computational Electromagnetics and Optical Systems, Faculty of Engineering, Mahasarakham University, Maha Sarakham, Thailand, 44150.
b
Abstract This paper presents an application of Cuckoo Search Algorithm (CSA) for solving reliability optimization problems for a series system. The aim of research is to select hardware from multiple choices for each subsystem to maximize the overall of system reliability and under cost constraints. CSA is a new metaheuristic algorithm inspired from behavior of Cuckoos laid their eggs in the nests of other species with the Lévy flight distribution. The effectiveness of the proposed algorithm has been evaluated with two groups of examples from the existing literature. The computational results show that the CSA has effectively been able to find the global optimal solution. 2016The TheAuthors. Authors. Published Elsevier © 2016 Published by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of iEECON2016. Peer-review under responsibility of the Organizing Committee of iEECON2016 Keywords: Cuckoo Search Algorithm, Reliability Optimization Problems, A Series System, Metaheuristic;
1. Introduction The reliability optimization problems that are presented by different needs of engineering and industry such as electrical systems, hardware design, etc. The design of reliability needs to consider various factors in many aspects such as cost, performance and other characteristics for maximizing the overall of system. For example, to maximize the system reliability subjected to cost and performance constraints [1] and subjected to cost with multiple choices [2].
* Corresponding author. Tel.: +66-042-341615; fax: +66-042-341614. E-mail address: [email protected]
1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of iEECON2016 doi:10.1016/j.procs.2016.05.079
454
Mana Sopa and Niwat Angkawisittpan / Procedia Computer Science 86 (2016) 453 – 456
For the purposes of this study is to deal with the reliability optimization problems for a series system with multiple choices and cost constraints. The problem is formulated as binary integer nonlinear programming. During the past there were theories and methods to solve problems with different techniques which are as follows: AS algorithm [3] and in some cases have been applied this problem to the yield optimization by using the Bee Algorithm (BA) [4]. 2. Problem Formulation A series system of n subsystem, there are different technologies of hardware available with reliabilities and different costs. The objective is to select hardware for each subsystem in order to the overall of system reliability and subject to the cost constraints. The notations are introduced below: Notations n Ni Cij Rij Rsys
number of subsystems number of hardware available for subsystems i cost of a subsystem i using hardware j reliability of a subsystem i using hardware j the overall of a series system reliability
Define the objective function to solve this problems with the equation. n
i 1
n
Subject to
§ ¨ ¨ ©
·
Ni
¦ X R ¸¸
Maximize Rsys
Ni
¦¦ X C ij
ij
ij
j 1
ij
¹
dC
(1)
i 1 j 1
Ni
¦X
ij
1, i 1, 2, ...,n
(2)
j 1
X ij
^0,1`, i
1, 2, ...,n and j 1, 2, ..., Ni
(3)
The terms of the Eq. 1: represents the cost constraint and C is integer, Eq. 2: represents the multiple choices constraints, Eq. 3: the variables for decision. When the result of a solution conformable all constraints, it is called the set of feasible solution and otherwise, it is called the set of infeasible solution. 3. Description of the Proposed Approach The CSA is a new metaheuristic optimization algorithm inspired by the behavior of some Cuckoo bird species with the Lévy flight. It has been presented by Yang and Deb [5] and it was applied to solve engineering design optimization problems. The CSA is based on three concepts of rules [6]: 1) Each Cuckoo bird lays one egg (a solution) at a time, and continues to lay their egg in the available host nests by randomly, 2) A Cuckoo egg with high quality (the better solution) in the best nests will be inherited to the next generation, and 3) An egg laid by a Cuckoo (called alien egg) is discovered by probability. In this case, the host bird species can abandon the nest or either throws the alien egg away and builds a new nest. Its simplicity, this last assumption can be fined by the fraction of the nests are replaced by new nests (called new random solutions). The details of CSA can be described in the following steps:
455
Mana Sopa and Niwat Angkawisittpan / Procedia Computer Science 86 (2016) 453 – 456
Step 1: The first step, initialize the CSA parameters. The parameters consist of nests (n), discovery probability (prob), step size (α), and the maximum of iterations and stop criterion. Step 2: Generate initial Cuckoo’s eggs in the nests of host birds. The first position of the nests is assigned by randomly with decision variables using Eq. 4.
nestij (0) ROUND X j min ( X j max X j min ). rand1
(4)
Where nestij(0) are the initial values for the jth decision variables and for the ith nest; the rounding function used to round up or round down jth values as the integer for the problem. Xjmax and Xjmin are the maximum and the minimum of search space values for the jth decision variables (jth technology) and rand1 is a random number returns from the interval [0,1]. Step 3: Generate new Cuckoo by Lévy flights. All the host nests are replaced with the quality of new Cuckoo’s eggs, which generated by Lévy flights distribution from their positions using Eq. 5
nesti (τ 1)
ROUND nesti (τ ) D.SL.(nesti (τ ) nestbest (τ )).rand 2
(5)
Where nesti (W) is the ith current nest position, α is the step size parameter; SL is a random state transition based on the Lévy flights behavior. The Lévy flight provides a random walk while the random step length is drawn from Lévy flight distribution, the step length SL can be calculated according to Eq. 6 and rand2 is a random number with standard normal distribution and nestbest is the best nest. 1β
u
SL
v
1E
; u ~ N (0,V u2 ) where V u
*(1 β ) sin(Sβ / 2) ½ ® ( β 1) / 2 ¾ ¯ *>(1 β ) / 2@ β 2 ¿
and v ~ N (0, V v2 ) where V v
1
(6)
Where β is number in the interval [1, 2]; u and v are drawn from normal distributions. Step 4: Discovery the alien eggs. For each solution in terms of the probability matrix K, the alien eggs are abandoned using Eq. 7. K ij
1 ® ¯0
if rand prob if rand t prob
(7)
Where prob is the discovering probability and rand are random numbers in the interval [0,1]. Early eggs are replaced considering by the newly quality eggs generated ones from their current position by random walks with step size using Eq. 8. nestt 1
ROUND nestt S. * K ; S
nests(randpm1(n),:) nests(randpm2(n),:) .rand
(8)
Where randpm1 and randpm2 are random numbers with a permutation function applied for different rows, nests and discovering probability matrix K. Step 5: Stop criteria. The processes of CSA are alternatively performed until the stopping criterion is true. 4. Computation results In order to evaluate the CSA, the data sets are presented in Tables 5 and 6 by Nahas & Nourelfath [3] in references. The system tried to adjust the number of host nests n = 5, 10, 15, 20, 40 and the probability prob = 0.1, 0.25, 0.5, 0.70. The results of the best solution are from the parameters: Host nests = 20, prob = 0.70, Iterations = 2000, α = 0.01 and β = 1.5. The results are illustrated in Fig. 1 (a). and Fig. 1 (b).
456
Mana Sopa and Niwat Angkawisittpan / Procedia Computer Science 86 (2016) 453 – 456 1
1
0.965134
0.9
0.9
0.86544
0.8 0.8
Reliability
Reliability
0.7
0.7 0.6 0.5 0.4
0.6 0.5 0.4 0.3
0
200
400
600
800
1000
Iterations
1200
1400
1600
1800
2000
0.2
0
200
400
600
800
1000
Iterations
1200
1400
1600
1800
2000
Fig. 1. (a) ; (b) Evolution of the best solutions using the CSA
Fig. 1 (a), the search space is larger than 1.572¯1012. The result is obtained the global optimal equal to 0.965134. The selected technologies are: 3-3-4-4-3-3-2-2-3-2-2-4-4-4-2. Fig. 1 (b), the search space is larger than 1.932¯1020. The result is obtained the global optimal equal to 0.86544. The selected technologies are: 3-3-3-4-2-3-2-2-3-1-2-3-44-1-2-3-3-5-2-3-2-2-3-1. 5. Conclusions In this paper, the optimal reliability in this problem with the application of a new approach called Cuckoo Search Algorithm (CSA) was studied. It is seen that the proposed CSA has the ability to find the global optimal solution in computational time and the accurate results. Acknowledgements The authors would like to thank Research Unit for Computational Electromagnetics and Optical Systems, Faculty of Engineering, Mahasarakham University. References 1. R. Meziane et al. Reliability optimization using ant colony algorithm under performance and cost constraints. Electric Power Systems Research; 2005, p. 1-8. 2. N. Ruan and X. Sun. An exact algorithm for cost minimization in series reliability systems with multiple component choices. Applied Mathematics and Computation; 2006, p. 732-741. 3. N. Nahas and M. Nourelfath. Ant system for reliability optimization of a series system with multiple-choice and budget constraints. Reliability Engineering and System Safety; 2005, p. 1-12. 4. W. Wongthatsanekorn and N. Matheekrieangkrai. Bee Algorithm for Solving Yield Optimization Problem for Hard Disk Drive Component under Budget and Supplier’s Rating Constraints and Hueristic Performance Comparison. Lecture Notes in Electrical Engineering; 2011, p. 203-216. 5. X.-S. Yang, and S. Deb. Cuckoo search via Lévy Flights. Proceedings of world congress on nature & biologically inspired computing (NaBIC2009India), USA, IEEE Publications; 2009, p. 210-214. 6. X.-S. Yang, and S. Deb. Engineering Optimisation by Cuckoo Search, Int. J. Mathematical Modelling and Numerical Optimisation; 2010, p. 330–343.