Abstract
In this paper, the Gravitational Search Algorithm (GSA) is hybridized with real coded Genetic Algorithm to solve Integer and Mixed Integer programming problems. The idea is based on two earlier papers of the authors. In the first paper, the authors proposed a methodology in which the Laplace Crossover and Power Mutation were embedded in Gravitational Search Algorithm and in the second paper, these algorithms were extended for the case of constrained optimization problems. In order to deal with integer variables, a special method is adopted. For dealing with the constraints the Deb’s technique is implemented. The original GSA and three new variants are tested on a set of benchmark problems available in literature. Based on the extensive numerical and graphical analysis of results it is concluded that one of the proposed variants outperform the original GSA and the other proposed variants.
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Acknowledgment
The first author would like to thank Council for Scientific and Industrial Research (CSIR), New Delhi, India, for providing him the financial support vide grant number 09/143(0824)/2012-EMR-I.
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Appendix A: Benchmark Functions
Appendix A: Benchmark Functions
Problem 1:
The global optima is \( (x,y;f) = (0.5,1;2) \).
Problem 2:
The global optima is \( (x,y;f) = (1.375,1;2.124) \).
Problem 3:
The global optima is \( (x_{1} ,x_{2} ,y;f) = (0.94194, - 2.1,1;1.07654) \).
Problem 4:
The global optima is \( (x_{1} ,x_{2} ;f) = (14.095,0.84296; - 6961.81381) \).
Problem 5:
The global optima is \( (x_{1} ,x_{2} ,x_{3} ;f) = (2,0,5; - 68) \).
Problem 6:
The global optima is \( (x_{1} ,x_{2} ,x_{3} ,x_{4} ;f) = (0,0,1,1; - 6) \).
Problem 7:
The objective is to select one between two candidate reactors in order to minimize the production cost. The global optima is \( (y_{1} ,v_{1} ,v_{2} ;f) = (1,3.514237,0;99.239635) \)
Problem 8:
Subject to:
The global optima is \( (x_{1} ,x_{2} ,x_{3} ,y_{1} ,y_{2} ,y_{3} ,y_{4} ;f) = (0.2,1.280624,1.954483,1,0,0,1;3.557463) \).Our algorithm achieves solution \( (x_{1} ,x_{2} ,x_{3} ,y_{1} ,y_{2} ,y_{3} ,y_{4} ;f) = (0.084607,0.798719,2.116424,1,1,0,1;3.3685783) \).
Problem 9:
The global optimal solution is \( (x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ;f) = (1,1,1,1,2;8) \).
Problem 10:
Subject to:
The global optimal solution is \( (x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} ,x_{7} ;f) = (0,2,4,0,2,1,4;14) \).
Problem 11:
The global optimal solution is \( (x_{1} ,x_{2} ;f) = (1,3; - 42.632) \).
Problem 12:
The global optimal solution is \( (x_{1} ,x_{2} ,x_{3} ;f) = (50,25,1.5;0.0) \).
Problem 13:
the global optimal solution is \( (x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ;f) = (16,22,5,5,7;807) \).
Problem 14:
The global optima is \( (x_{1} ,x_{3} ,y_{1} ,;f) = (27,27,78;32217.4) \) and it is obtained with various different feasible combination of \( (x_{2} ,y_{2} ) \).
Problem 15:
Subject to:
The global optimal solution is \( (y;f) = (0,1,1,1,0,1,1,0;0.94347) \).
Problem 16:
Subject to:
Initially, this problem is solved by Monte Carlo technique on a random sample of 2000 points [33] and best solution is obtained at
with \( f_{\hbox{max} } = 1030361 \). But proposed algorithm found the optimal solution at
with \( f_{\hbox{max} } = 1352439. \)
Problem 17:
This is a nonlinear optimization problem with one hundred decision variables. Initially, it is solved by Monte Carlo technique on a random sample of 10000 points [33] and the global optimal solution of this problem is achieved at
with \( f_{\hbox{max} } = 303062435. \)
The global optima of problem 17 is improved by MI-LXPMGSA and it is found at
with \( f_{\hbox{max} } = 3 0 4 1 6 0 0 7 7. \)
Problem 18:
Subject to:
where, \( v_{j} \) is the product of weight and volume per element at stage \( j, \) \( w_{j} \) is the weight of each component at stage \( j, \) and \( C(r_{j} ) \) is the cost of each component with reliability \( r_{j} \) at stage \( j \) as follows:
where \( \alpha_{j} \) and \( \beta_{j} \) are constants representing the physical characteristic of each component at stage \( j \) and \( T \) is the operating time during which the component must not fail. The known optimal solution is \( R(m,r) = 0.999955, \) \( m = [5,5,4,6] \) and \( r = [0.899845,0.887909,0.948990,0.851017]. \) the design data is given below. \( C_{Q} = 400.0,w_{Q} = 500.0,v_{Q} = 250.0,T = 1000\,h. \)
Subsys. | \( 10^{5} .\alpha_{j} \) | \( \beta_{j} \) | \( v_{j} \) | \( w_{j} \) |
---|---|---|---|---|
1 | 1.0 | 1.5 | 1 | 6 |
2 | 2.3 | 1.5 | 2 | 6 |
3 | 0.3 | 1.5 | 3 | 8 |
4 | 2.3 | 1.5 | 2 | 7 |
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Singh, A., Deep, K. (2017). Hybridized Gravitational Search Algorithms with Real Coded Genetic Algorithms for Integer and Mixed Integer Optimization Problems. In: Deep, K., et al. Proceedings of Sixth International Conference on Soft Computing for Problem Solving. Advances in Intelligent Systems and Computing, vol 546. Springer, Singapore. https://doi.org/10.1007/978-981-10-3322-3_9
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