Hybridized Gravitational Search Algorithms with Real Coded Genetic Algorithms for Integer and Mixed Integer Optimization Problems

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.

Appendix A: Benchmark Functions

Problem 1:

$$ {\text{Min}}\;f(x,y) = 2x + y $$

$$ \begin{aligned} {\text{Subject to}}:\, & 1.25 - x^{2} - y \le 0 \\ & x + y \le 1.6 \\ & 0 \le x \le 1.6 \\ & y \in \left\{ {0,1} \right\} \\ \end{aligned} $$

The global optima is $(x,y;f) = (0.5,1;2)$.

Problem 2:

$$ {\text{Min}}\;f(x,y) = - y + 2x - \ln (x/2) $$

$$ \begin{aligned} {\text{Subject to}}:\, & - x - \ln ({x \mathord{\left/ {\vphantom {x 2}} \right. \kern-0pt} 2}) + y \le 0, \\ & 0.5 \le x \le 1.5, \\ & y \in \left\{ {0,1} \right\}. \\ \end{aligned} $$

The global optima is $(x,y;f) = (1.375,1;2.124)$.

Problem 3:

$$ {\text{Min}}\;f(x,y) = - 0.7y + 5(x_{1} - 0.5)^{2} + 0.8 $$

$$ \begin{aligned} {\text{Subject to}}:\, & - \exp (x_{1} - 0.2) - x_{2} \le 0, \\ & x_{2} + 1.1y \le - 1.0, \\ & x_{1} - 1.2y \le 0.2, \\ & 0.2 \le x_{1} \le 1.0, \\ & - 2.22554 \le x_{2} \le - 1.0, \\ & y \in \left\{ {0,1} \right\}. \\ \end{aligned} $$

The global optima is $(x_{1} ,x_{2} ,y;f) = (0.94194, - 2.1,1;1.07654)$.

Problem 4:

$$ {\text{Min}}\;f(x) = (x_{1} - 10)^{3} + (x_{2} - 20)^{3} $$

$$ \begin{aligned} {\text{Subject to}}:\, & (x_{1} - 5)^{2} + (x_{2} - 5)^{2} - 100 \ge 0.0, \\ & - (x_{1} - 6)^{2} - (x_{2} - 5)^{2} + 82.81 \ge 0.0, \\ & 13 \le x_{1} \le 100, \\ & 0 \le x_{2} \le 100. \\ \end{aligned} $$

The global optima is $(x_{1} ,x_{2} ;f) = (14.095,0.84296; - 6961.81381)$.

Problem 5:

$$ {\text{Min}}\;f(x) = x_{1}^{2} + x_{1} x_{2} + 2x_{2}^{2} - 6x_{1} - 2x_{2} - 12x_{3} , $$

$$ \begin{aligned} {\text{Subject to}}:\, & 2x_{1}^{2} + x_{2}^{2} \le 15.0, \\ & - x_{1} + 2x_{2} + x_{3} \le 3.0, \\ & 0 \le x_{i} \le 10,\,{\text{integer}}\,\,i = 1, \ldots ,3. \\ \end{aligned} $$

The global optima is $(x_{1} ,x_{2} ,x_{3} ;f) = (2,0,5; - 68)$.

Problem 6:

$$ {\text{Min}}\;f(x) = (x_{1} + 2x_{2} + 3x_{3} - x_{4} )(2x_{1} + 5x_{2} + 3x_{3} - 6x_{4} ), $$

$$ \begin{aligned} {\text{Subject to}}:\, & x_{1} + 2x_{2} + x_{3} + 3x_{4} \ge 4.0, \\ & x \in \left\{ {0,1} \right\}^{4} \\ \end{aligned} $$

The global optima is $(x_{1} ,x_{2} ,x_{3} ,x_{4} ;f) = (0,0,1,1; - 6)$.

Problem 7:

$$ \begin{aligned} {\text{Min}}\;f(y_{1} ,v_{1} ,v_{2} ) & = 7.5y_{1} + 5.5(1 - y_{1} ) + 7v_{1} + 6v_{2} \\ & \quad + 50\frac{{y_{1} }}{{0.9\left[ {1 - \exp ( - 0.5v_{1} )} \right]}} \\ & \quad + 50\frac{{1 - y_{1} }}{{0.8\left[ {1 - \exp ( - 0.4v_{2} )} \right]}} \\ \end{aligned} $$

$$ \begin{aligned} {\text{Subject to}}:\, & 0.9\left[ {1 - \exp ( - 0.5v_{1} )} \right] - 2y_{1} \le 0, \\ & 0.8\left[ {1 - \exp ( - 0.4v_{2} )} \right] - 2(1 - y_{1} ) \le 0, \\ & v_{1} \le 10y_{1} , \\ & v_{2} \le 10(1 - y_{1} ), \\ & v_{1} ,v_{2} \ge 0, \\ & y_{1} \in \left\{ {0,1} \right\}. \\ \end{aligned} $$

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:

$$ \begin{aligned} {\text{Min}}\;f(x,y) & = (y_{1} - 1)^{2} + (y_{2} - 1)^{2} + (y_{3} - 1)^{2} - \ln (y_{4} + 1) \\ & \quad + (x_{1} - 1)^{2} + (x_{2} - 2)^{2} + (x_{3} - 3)^{2} \\ \end{aligned} $$

Subject to:

$$ \begin{aligned} & y_{1} + y_{2} + y_{3} + x_{1} + x_{2} + x_{3} \le 5.0, \\ & y_{3}^{2} + x_{1}^{2} + x_{2}^{2} + x_{3}^{2} \le 5.5, \\ & y_{1} + x_{1} \le 1.2, \\ & y_{2} + x_{2} \le 1.8, \\ & y_{3} + x_{3} \le 2.5, \\ & y_{4} + x_{1} \le 1.2, \\ & y_{2}^{2} + x_{2}^{2} \le 1.64, \\ & y_{3}^{2} + x_{3}^{2} \le 4.25, \\ & y_{2}^{2} + x_{3}^{2} \le 4.64, \\ & x_{1} ,x_{2} ,x_{3} \ge 0, \\ & y_{1} ,y_{2} ,y_{3} ,y_{4} \in \left\{ {0,1} \right\}. \\ \end{aligned} $$

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:

$$ {\text{Min}}\;f(x) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + x_{5}^{2} , $$

$$ \begin{aligned} {\text{Subject to}}:\, & x_{1} + 2x_{2} + x_{4} \ge 4.0, \\ & x_{2} + 2x_{3} \ge 3.0, \\ & x_{1} + 2x_{5} \ge 5.0, \\ & x_{1} + 2x_{2} + 2x_{3} \le 6.0, \\ & 2x_{1} + x_{3} \le 4.0, \\ & x_{1} + 4x_{5} \le 13.0, \\ & 0 \le x_{i} \le 3,\,\,i = 1,2, \ldots ,5;\,{\text{integer}} .\\ \end{aligned} $$

The global optimal solution is $(x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ;f) = (1,1,1,1,2;8)$.

Problem 10:

$$ {\text{Min}}\;f(x) = x_{1} x_{7} + 3x_{2} x_{6} + x_{3} x_{5} + 7x_{4} , $$

Subject to:

$$ \begin{aligned} & x_{1} + x_{2} + x_{3} \ge 6.0, \\ & x_{4} + x_{5} + 6x_{6} \ge 8.0, \\ & x_{1} x_{6} + x_{2} + 3x_{5} \ge 7.0, \\ & 4x_{2} x_{7} + 3x_{4} x_{5} \ge 25.0, \\ & 3x_{1} + 2x_{3} + x_{5} \ge 7.0, \\ & 3x_{1} x_{3} + 6x_{4} + 4x_{5} \le 20.0, \\ & 4x_{1} + 2x_{3} + x_{6} x_{7} \le 15.0, \\ & 0 \le x_{1} ,x_{2} ,x_{3} \le 4, \\ & 0 \le x_{4} ,x_{5} ,x_{6} \le 2,\, \\ & 0 \le x_{7} \le 6,\, \\ & x_{i} ;i = 1,2, \ldots ,7;\,{\text{integers}} .\\ \end{aligned} $$

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:

$$ {\text{Min}}\;f(x) = \exp ( - x_{1} ) + x_{1}^{2} - x_{1} x_{2} - 3x_{2}^{2} - 6x_{2} + 4x_{1} , $$

$$ \begin{aligned} {\text{Subject to}}:\, & 2x_{1} + x_{2} \le 8.0, \\ & - x_{1} + x_{2} \le 2.0, \\ & 0 \le x_{1} ,x_{2} \le 3, \\ & x_{1} ,x_{2} \,{\text{integers}} .\\ \end{aligned} $$

The global optimal solution is $(x_{1} ,x_{2} ;f) = (1,3; - 42.632)$.

Problem 12:

$$ \begin{aligned} {\text{Min}}\;f(x) & = \sum\limits_{i = 1}^{9} {\left[ {\exp \left( { - \frac{{\left( {u_{i} - x_{2} } \right)^{{x_{3} }} }}{{x_{1} }}} \right) - 0.01i} \right]}^{2} , \\ where,u_{i} & = 25 + \left( { - 50\ln (0.01i)} \right)^{2/3} , \\ \end{aligned} $$

$$ \begin{aligned} {\text{Subject to}}:\, & 0.1 \le x_{1} \le 100.0, \\ & 0.0 \le x_{2} \le 25.6, \\ & 0.0 \le x_{3} \le 5.0, \\ & x_{1} ,x_{2} \,{\text{integers}} .\\ \end{aligned} $$

The global optimal solution is $(x_{1} ,x_{2} ,x_{3} ;f) = (50,25,1.5;0.0)$.

Problem 13:

$$ \begin{aligned} {\text{Min}}\;f(x) & = x_{1}^{2} + x_{2}^{2} + 3x_{3}^{2} + 4x_{4}^{2} + 2x_{5}^{2} - 8x_{1} - 2x_{2} \\ & - 3x_{3} - x_{4} - 2x_{5} , \\ \end{aligned} $$

$$ \begin{aligned} {\text{Subject to}}:\, & x_{1} + x_{2} + x_{3} + x_{4} + x_{5} \le 400, \\ & x_{1} + 2x_{2} + 2x_{3} + x_{4} + 6x_{5} \le 800, \\ & 2x_{1} + x_{2} + 6x_{3} \le 200, \\ & x_{3} + x_{4} + 5x_{5} \le 200, \\ & x_{1} + x_{2} + x_{3} + x_{4} + x_{5} \ge 55, \\ & x_{1} + x_{2} + x_{3} + x_{4} \ge 48, \\ & x_{2} + x_{4} + x_{5} \ge 34, \\ & 6x_{1} + 7x_{5} \ge 104, \\ & 0 \le x_{i} \le 99;\,{\text{integers }}\,i = 1,2, \ldots ,5. \\ \end{aligned} $$

the global optimal solution is $(x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ;f) = (16,22,5,5,7;807)$.

Problem 14:

$$ \begin{aligned} {\text{Max}}\;f(x,y) & = - 5.357854x_{1}^{2} - 0.835689y_{1} x_{3} \\ & - 37.29329y_{1} + 40792.141, \\ \end{aligned} $$

$$ \begin{aligned} {\text{Subject to}}:\, & a_{1} + a_{2} y_{2} x_{3} + a_{3} y_{1} x_{2} - a_{4} x_{1} x_{3} \le 92.0, \\ & a_{5} + a_{6} y_{2} x_{3} + a_{7} y_{1} y_{2} + a_{5} x_{1}^{2} \le 110.0, \\ & a_{9} + a_{10} x_{1} x_{3} + a_{11} y_{1} x_{1} + a_{12} x_{1} x_{2} \le 25.0, \\ & 27 \le x_{1} ,x_{2} ,x_{3} \le 45, \\ & y_{1} \in \left\{ {78,79, \ldots ,102} \right\}, \\ & y_{2} \in \left\{ {33,34, \ldots ,45} \right\}. \\ \end{aligned} $$

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:

$$ \begin{aligned} & {\text{Max}}\;f(y) = r_{1} r_{2} r_{3} , \\ & r_{1} = 1 - 0.1^{{y_{1} }} 0.2^{{y_{2} }} 0.15^{{y_{3} }} , \\ & r_{2} = 1 - 0.05^{{y_{4} }} 0.2^{{y_{5} }} 0.15^{{y_{6} }} , \\ & r_{3} = 1 - 0.02^{{y_{7} }} 0.06^{{y_{8} }} , \\ \end{aligned} $$

Subject to:

$$ \begin{aligned} & y_{1} + y_{2} + y_{3} \ge 1, \\ & y_{4} + y_{5} + y_{6} \ge 1, \\ & y_{7} + y_{8} \ge 1, \\ & 3y_{1} + y_{2} + 2y_{3} + 3y_{4} + 2y_{5} + y_{6} + 3y_{7} + 2y_{8} \le 10, \\ & y \in \left\{ {0,1} \right\}^{8} . \\ \end{aligned} $$

The global optimal solution is $(y;f) = (0,1,1,1,0,1,1,0;0.94347)$.

Problem 16:

$$ \begin{aligned} {\text{Max}}\;f(x) & = 215x_{1} + 116x_{2} + 670x_{3} + 924x_{4} + 510x_{5} \\ & \quad + 600x_{6} + 424x_{7} + 942x_{8} + 43x_{9} + 369x_{10} \\ & \quad + 408x_{11} + 52x_{12} + 319x_{13} + 214x_{14} + 851x_{15} \\ & \quad + 394x_{16} + 88x_{17} + 124x_{18} + 17x_{19} + 779x_{20} \\ & \quad + 278x_{21} + 258x_{22} + 271x_{23} + 281x_{24} + 326x_{25} \\ & \quad + 819x_{26} + 485x_{27} + 454x_{28} + 297x_{29} + 53x_{30} \\ & \quad + 136x_{31} + 796x_{32} + 114x_{33} + 43x_{34} + 80x_{35} \\ & \quad + 268x_{36} + 179x_{37} + 78x_{38} + 105x_{39} + 281x_{40} \\ \end{aligned} $$

Subject to:

$$ \begin{aligned} & 8x_{1} + 11x_{2} + 6x_{3} + x_{4} + 7x_{5} + 9x_{6} + 10x_{7} + 3x_{8} + 11x_{9} \\ & + 11x_{10} + 2x_{11} + x_{12} + 16x_{13} + 18x_{14} + 2x_{15} + x_{16} + x_{17} \\ & + 2x_{18} + 3x_{19} + 4x_{20} + 7x_{21} + 6x_{22} + 2x_{23} + 2x_{24} + x_{25} \\ & + 2x_{26} + x_{27} + 8x_{28} + 10x_{29} + 2x_{30} + x_{31} + 9x_{32} + x_{33} \\ & + 9x_{34} + 2x_{35} + 4x_{36} + 10x_{37} + 8x_{38} + 6x_{39} \\ & + x_{40} \le 25,000, \\ & 5x_{1} + 3x_{2} + 2x_{3} + 7x_{4} + 7x_{5} + 3x_{6} + 6x_{7} + 2x_{8} + 15x_{9} \\ & + 8x_{10} + 16x_{11} + x_{12} + 2x_{13} + 2x_{14} + 7x_{15} + 7x_{16} + 2x_{17} \\ & + 2x_{18} + 4x_{19} + 3x_{20} + 2x_{21} + 13x_{22} + 8x_{23} + 2x_{24} + 3x_{25} \\ & + 4x_{26} + 3x_{27} + 2x_{28} + x_{29} + 10x_{30} + 6x_{31} + 3x_{32} + 4x_{33} \\ & + x_{34} + 8x_{35} + 6x_{36} + 3x_{37} + 4x_{38} + 6x_{39} + 2x_{40} \le 25,000 \\ & 3x_{1} + 4x_{2} + 6x_{3} + 2x_{4} + 2x_{5} + 3x_{6} + 7x_{7} + 10x_{8} + 3x_{9} \\ & + 7x_{10} + 2x_{11} + 16x_{12} + 3x_{13} + 3x_{14} + 9x_{15} + 8x_{16} + 9x_{17} \\ & + 7x_{18} + 6x_{19} + 16x_{20} + 12x_{21} + x_{22} + 3x_{23} + 14x_{24} + 7x_{25} \\ & + 13x_{26} + 6x_{27} + 16x_{28} + 3x_{29} + 2x_{30} + x_{31} + 2x_{32} + 8x_{33} \\ & + 2x_{34} + 2x_{35} + 7x_{36} + x_{37} + 2x_{38} + 6x_{39} + 5x_{40} \le 25,000 \\ & 10 \le x_{i} \le 99;\,\,i = 1,2, \ldots ,20, \\ & 20 \le x_{i} \le 99;\,\,i = 21,22, \ldots ,40. \\ \end{aligned} $$

Initially, this problem is solved by Monte Carlo technique on a random sample of 2000 points [33] and best solution is obtained at

$$ \left( {\begin{array}{*{20}l} {48} \hfill & {73} \hfill & {16} \hfill & {86} \hfill & {49} \hfill & {99} \hfill & {94} \hfill & {79} \hfill & {98} \hfill & {86} \hfill \\ {94} \hfill & {33} \hfill & {95} \hfill & {80} \hfill & {53} \hfill & {86} \hfill & {87} \hfill & {50} \hfill & {39} \hfill & {78} \hfill \\ {47} \hfill & {72} \hfill & {97} \hfill & {98} \hfill & {73} \hfill & {86} \hfill & {99} \hfill & {81} \hfill & {77} \hfill & {95} \hfill \\ {28} \hfill & {95} \hfill & {58} \hfill & {23} \hfill & {55} \hfill & {70} \hfill & {35} \hfill & {82} \hfill & {32} \hfill & {94} \hfill \\ \end{array} } \right) $$

with $f_{\hbox{max} } = 1030361$. But proposed algorithm found the optimal solution at

$$ \left( {\begin{array}{*{20}l} {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill \\ {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill \\ {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill \\ {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill \\ \end{array} } \right) $$

with $f_{\hbox{max} } = 1352439.$

Problem 17:

$$ \begin{aligned} {\text{Max}}\;f(x) & = 50x_{1} + 150x_{2} + 100x_{3} + 92x_{4} + 55x_{5} + 12x_{6} \\ & \quad + 11x_{7} + 10x_{8} + 8x_{9} + 3x_{10} + 114x_{11} + 90x_{12} \\ & \quad + 87x_{13} + 91x_{14} + 58x_{15} + 16x_{16} + 19x_{17} + 22x_{18} \\ & \quad + 21x_{19} + 32x_{20} + 53x_{21} + 56x_{22} + 118x_{23} + 192x_{24} \\ & \quad + 52x_{25} + 204x_{26} + 250x_{27} + 295x_{28} + 82x_{29} \\ & \quad + 30x_{30} + 29x_{31}^{2} - 2x_{32}^{2} + 9x_{33}^{2} + 94x_{34} + 15x_{35}^{3} \\ & \quad + 17x_{36}^{2} - 15x_{37} - 2x_{38} + x_{39}^{2} + 3x_{40}^{4} + 52x_{41} + 57x_{42}^{2} \\ & \quad - x_{43}^{2} + 12x_{44} + 21x_{45} + 6x_{46} + 7x_{47} - x_{48} + x_{49} + x_{50} \\ & \quad + 119x_{51} + 82x_{52} + 75x_{53} + 18x_{54} + 16x_{55} + 12x_{56} \\ & \quad + 6x_{57} + 7x_{58} + 3x_{59} + 6x_{60} + 12x_{61} + 13x_{62} + 18x_{63} \\ & \quad + 7x_{64} + 3x_{65} + 19x_{66} + 22x_{67} + 3x_{68} + 12x_{69} + 9x_{70} \\ & \quad + 18x_{71} + 19x_{72} + 12x_{73} + 8x_{74} + 5x_{75} + 2x_{76} + 16x_{77} \\ & \quad + 17x_{78} + 11x_{79} + 12x_{80} + 9x_{81} + 12x_{82} + 11x_{83} \\ & \quad + 14x_{84} + 16x_{85} + 3x_{86} + 9x_{87} + 10x_{88} + 3x_{89} + x_{90} \\ & \quad + 12x_{91} + 3x_{92} + 12x_{93} - 2x_{94}^{2} - x_{95} + 6x_{96} + 7x_{97} \\ & \quad + 4x_{98} + x_{99} + 2x_{100} \\ \end{aligned} $$

$$ \begin{aligned} {\text{Subject to}}:\, & \sum\limits_{i = 1}^{100} {x_{i} \le 7500,} \\ & \sum\limits_{i = 1}^{50} {10x_{i} + \sum\limits_{i = 51}^{100} {x_{i} } \le 42,000,} \\ & 0 \le x_{i} \le 99;\,i = 1,2, \ldots ,100. \\ \end{aligned} $$

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

$$ \left( {\begin{array}{*{20}l} {51} \hfill & {10} \hfill & {90} \hfill & {85} \hfill & {35} \hfill & {36} \hfill & {75} \hfill & {98} \hfill & {99} \hfill & {30} \hfill \\ {56} \hfill & {23} \hfill & {10} \hfill & {56} \hfill & {98} \hfill & {94} \hfill & {63} \hfill & 8 \hfill & {27} \hfill & {92} \hfill \\ {10} \hfill & {66} \hfill & {69} \hfill & {10} \hfill & {39} \hfill & {38} \hfill & {49} \hfill & 8 \hfill & {95} \hfill & {96} \hfill \\ {86} \hfill & {14} \hfill & 1 \hfill & {55} \hfill & {98} \hfill & {64} \hfill & 8 \hfill & 1 \hfill & {18} \hfill & {99} \hfill \\ {84} \hfill & {78} \hfill & 4 \hfill & {19} \hfill & {85} \hfill & {33} \hfill & {59} \hfill & {95} \hfill & {57} \hfill & {48} \hfill \\ {37} \hfill & {95} \hfill & {62} \hfill & {82} \hfill & {62} \hfill & {62} \hfill & {87} \hfill & {38} \hfill & {95} \hfill & {14} \hfill \\ {91} \hfill & {21} \hfill & {72} \hfill & {85} \hfill & {68} \hfill & {69} \hfill & {30} \hfill & {30} \hfill & {85} \hfill & {93} \hfill \\ {73} \hfill & {19} \hfill & {26} \hfill & {62} \hfill & {94} \hfill & {59} \hfill & {53} \hfill & {11} \hfill & 0 \hfill & 1 \hfill \\ 2 \hfill & {26} \hfill & {43} \hfill & {50} \hfill & {42} \hfill & {93} \hfill & {27} \hfill & {71} \hfill & {61} \hfill & {93} \hfill \\ {44} \hfill & {94} \hfill & {15} \hfill & {92} \hfill & 8 \hfill & {18} \hfill & {42} \hfill & {27} \hfill & {66} \hfill & {49} \hfill \\ \end{array} } \right) $$

with $f_{\hbox{max} } = 303062435.$

The global optima of problem 17 is improved by MI-LXPMGSA and it is found at

$$ \left( {\begin{array}{*{20}l} {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {71} \hfill & {99} \hfill & 0 \hfill & 0 \hfill \\ {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill \\ {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill \\ {99} \hfill & 0 \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & 0 \hfill & 0 \hfill & {99} \hfill & {99} \hfill \\ {99} \hfill & {99} \hfill & 0 \hfill & {99} \hfill & {99} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & 4 \hfill & {99} \hfill \\ {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & 0 \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill \\ {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & 0 \hfill & 0 \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill \\ {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & {99} \hfill & 0 \hfill & {99} \hfill & {99} \hfill & 0 \hfill & 0 \hfill \\ {99} \hfill & 0 \hfill & {99} \hfill & 0 \hfill & 0 \hfill & {99} \hfill & {99} \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right) $$

with $f_{\hbox{max} } = 3 0 4 1 6 0 0 7 7.$

Problem 18:

$$ {\text{Max}}\;R(m,r) = \prod\limits_{j = 1}^{4} {\left\{ {1 - \left( {1 - r_{j} } \right)^{{m_{j} }} } \right\},} $$

Subject to:

$$ \begin{aligned} & g_{1} (m) = \sum\limits_{j = 1}^{4} {v_{j} .m_{j}^{2} \le v_{Q} } , \\ & g_{2} (m,r) = \sum\limits_{j = 1}^{4} {C(r_{j} ).\left( {m_{j} + \exp \,\left( {m_{j} /4} \right)} \right)} \le C_{Q} , \\ & g_{3} (m) = \sum\limits_{j = 1}^{4} {w_{j} .\left( {m_{j} .\exp \,\left( {m_{j} /4} \right)} \right)} \le w_{Q} , \\ & 1 \le m_{j} \le 10:\,\,{\text{intger}};j = 1,2, \ldots ,4, \\ & 0.5 \le r_{j} \le 1 - 10^{ - 6} ;j = 1,2, \ldots ,4, \\ \end{aligned} $$

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:

$$ C(r_{j} ) = \alpha_{j} .\left( {\frac{ - T}{{\ln (r_{j} )}}} \right)^{{\beta_{j} }} $$

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