Jsun Yui Wong
The computer program listed below seeks to solve the following problem in Lachhwani [17, p. 54, Model II]:
Minimize X(3) + X(4), which are two deviational variables,
subject to
(2 * X(1) + 20 * X(2) + 12) * (X(1) + 10 * X(2) + 17) * X(7)+ .43653*X(3) - .43653 * X(5) - 1.67289=0,
(2 * X(1) + 20 * X(2) + 12) * (3 * X(1) + 30 * X(2) + 51) * X(8)+ .43653* X(4) - .43653 * X(6) - 2.50934=0,
((-2 * X(1) - 5 * X(2) + 15) * (2 * X(1) + 5 * X(2) + 11))*X(7)=1,
((-4 * X(1) - 10 * X(2) + 30) * (2 * X(1) + 5 * X(2) + 11))*X(8)=1,
X(1) + 15 * X(2)<=2,
3 * X(1) + 20 * X(2)<=4,
X(1), X(2), X(3), X(4), X(5), X(6)>=0,
X(7), X(8)>0.
X(9) and X(10) below are added slack variables.
0 DEFDBL A-Z
2 DEFINT K
3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)
12 FOR JJJJ = -32000 TO 32000 STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
51 FOR J40 = 1 TO 8
55 A(J40) = RND
56 NEXT J40
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 8
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 j = 1 + FIX(RND * 8)
183 r = (1 - RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP
224 FOR J47 = 1 TO 6
226 IF X(J47) < 0 THEN 1670
227 NEXT J47
231 X(7) = 1 / ((-2 * X(1) - 5 * X(2) + 15) * (2 * X(1) + 5 * X(2) + 11))
234 X(8) = 1 / ((-4 * X(1) - 10 * X(2) + 30) * (2 * X(1) + 5 * X(2) + 11))
238 X(3) = (-(2 * X(1) + 20 * X(2) + 12) * (X(1) + 10 * X(2) + 17) * X(7) + .43653 * X(5) + 1.67289) / .43653
240 X(4) = (-(2 * X(1) + 20 * X(2) + 12) * (3 * X(1) + 30 * X(2) + 51) * X(8) + .43653 * X(6) + 2.50934) / .43653
244 FOR J47 = 1 TO 6
246 IF X(J47) < 0 THEN 1670
247 NEXT J47
267 X(9) = 2 - X(1) - 15 * X(2)
268 X(10) = 4 - 3 * X(1) - 20 * X(2)
337 FOR J44 = 9 TO 10
338 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
339 NEXT J44
451 POBA = -X(3) - X(4) + 1000000 * (X(9) + X(10))
466 P = POBA
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1457 M = P
1559 GOTO 128
1670 NEXT I
1889 IF M < -.0001 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5), A(6)
1902 PRINT A(7), A(8), A(9), A(10), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [39]. The complete output through JJJJ =-31999.97000000001 is shown below:
.7998398353134477 8.001067764577013D-02 1.6366542727326D-05
3.600378059966596D-05 2.544311441144984D-22 7.590194109784202D-23
5.917159765808536D-03 2.958579882904268D-03 0
0 -5.237032332699196D-05 -31999.99
.8000359848619804 7.999336715433525D-02 0
1.1453966507151D-05 2.878576451779904D-22 9.636898226792295D-23
5.917159763366334D-03 2.958579881683167D-03 0
0 -1.1453966507151D-05 -31999.97000000001
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [39], the wall-clock time for obtaining the output through JJJJ= -31999.97000000001 was 10 seconds, total, including the time for "Creating .EXE file." One can compare the computational results above with those on pp. 54-55 of Lachhwani [17].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[6] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[7] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[8] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[9] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[11] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.
[12] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.
[13] Majid Jaberipour, Esmaile Khorram (2010). Solving the sum of ratios problems by a harmony search algorithm. Journal of Computational and Applied Mathematics 234 (2010) 733-742.
[14] Hongwei Jiao, Zhankui Wang, Yongqiang Chen (2013). Global Optimization Algorithm for Sum of Generalized Polynomial Ratios Problems. Applied Mathematical Modelling 37 (2013) 187-197.
[15] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.
[16] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.
[17] K. Lachhwani (2014). FGP approach to multi objectve quadratic fraction programming problem. Indian Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57.
Available online at http://ijim.srbiau.ac.ir/
[18] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[19] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[20] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[21] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[22] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[23] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[24] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[25] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.
[26] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[27] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.
[28] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[29] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[30] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[31] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[32] c. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.
[33] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[34] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[35] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[36] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[37] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[38] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[39] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[40] Jsun Yui Wong (2018, January 1). Solving a Nonlinear Integer Fractional Programming Problem with 20 General Integer Variables. http://nonlinearintegerprogrammingsolver.blogspot.ca/2018/01/solving-nonlinear-integer-fractional.html.
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