Substance

Solving a Fractional Programming Problem in 20 Variables

Jsun Yui Wong

Based on the computer program in Wong [39], the computer program listed below seeks to solve the following problem from Jaberipour and Khorram [13, pp. 741-742, Example 4]:

Maximize R1 + R2 + R3, defined in lines 301, 303, and 305 below,

1<=X(i)<=5, i=1,..., 20.

One notes that line 223 through line 235 in Wong [39] are 223 FOR j41 = 1 TO 20, 225 X(j41) = INT(X(j41)), and 235 NEXT j41, respectively, here line 223 through line 235 are 223 REM FOR j41 = 1 TO 20, 225 REM X(j41) = INT(X(j41)), and 235 REM NEXT j41, respectively.

 

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

 

71 FOR J40 = 1 TO 20

 

74 A(J40) = (1 + RND * 4)
75 REM A(2) = (2 + RND * 200)

77 NEXT J40

 

128 FOR I = 1 TO 50000

 

129 FOR KKQQ = 1 TO 20

 

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

 

181 J = 1 + FIX(RND * 20)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP

 

223 REM FOR j41 = 1 TO 20

 

225 REM X(j41) = INT(X(j41))

 

235 REM NEXT j41

 

256 FOR J47 = 1 TO 20

257 IF X(J47) < 1 THEN 1670
258 IF X(J47) > 5 THEN 1670

 

259 NEXT J47

260 SUM1 = 0

 

261 FOR J44 = 1 TO 20
265 SUM1 = SUM1 + J44 * X(J44)

 

269 NEXT J44

270 SUM2 = 0

 

271 FOR J44 = 1 TO 20
275 SUM2 = SUM2 + X(J44)

 

279 NEXT J44

280 SUM3 = 0

 

281 FOR J44 = 1 TO 10
285 SUM3 = SUM3 + J44 * X(2 * J44)

 

289 NEXT J44

290 SUM4 = 0

 

291 FOR J44 = 1 TO 10
295 SUM4 = SUM4 + X(2 * J44 - 1)

 

299 NEXT J44

301 R1 = SUM1 / SUM2

303 R2 = SUM3 / SUM4

305 R3 = SUM4 / SUM3

307 GOTO 347

 

311 X(3) = 6 - 2 * X(1) - X(2)
313 X(4) = 8 - 3 * X(1) - X(2)

 

315 X(5) = 1 - X(1) + X(2)

322 FOR J44 = 3 TO 5

 

325 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

327 NEXT J44

 

343 REM

347 POBA = R1 + R2 + R3

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 20

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 REM GOTO 128

1670 NEXT I
1889 IF M < 38.5 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5)
1901 PRINT A(6), A(7), A(8), A(9), A(10)
1910 PRINT A(11), A(12), A(13), A(14), A(15)
1911 PRINT A(16), A(17), A(18), A(19), A(20)

1922 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [38]. The complete output through JJJJ = -31999.99 is shown below:

1.000000023530792         1.000000024888448          1.000000000215454
4.99999989275312          1.000000013115532
4.999999050387178          1.000000004188031          4.999999988517495
1.000000056133371          4.999999959762943
1.000000000922831          4.999999615140256          1.000000020098937
4.99999984112548          1.000000000652795
4.999999972860543          1.000000034610626          4.999999991891256
1.000000003723272          4.999999999750646
38.60118504510297             -32000

 

1.00000000253903 1.00000007567784 1.0000000016618036
4.999999332843052 1.000000011102316
4.999999060760837 1.000000000755814 4.99999999608387
1.00000001700265 4.999999967689313
1.000000128005934 4.999999915208235 1.00000002506695
4.999999998575465 1.00000033396696
4.999999991851928 1.000000000964161 4.999999899827006
1.00000008758909 4.999999999953226
38.60118400559084 -31999.99

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 [38], the wall-clock time for obtaining the output through JJJJ= -31999.99 was 10 seconds, total, including the time for "Creating .EXE file." One can compare the computational results above with those on page 742 of Jaberipour and Khorram [13].

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] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[18] 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.
[19] 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.
[20] 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.
[21] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[22] 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.
[23] 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.
[24] 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.
[25] 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/
[26] 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.
[27] 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…/
[28] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[29] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[30] 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.
[31] 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.
[32] 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.
[33] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[34] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[35] 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
[36] 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.
[37] 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.
[38] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[39] 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.

Searching with Computers for Integer Solution/Solutions of a Sum of Ratios Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following problem:

Minimize ((X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(3)) / (X(3) ^ 2 + 5 * X(1) * X(2))) + ((X(1) + 1) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20))

subject to

X(1) ^ 2 + X(2) ^ 2 + X(3)<=5,

(X(1) - 2) ^ 2 + X(2) ^ 2 + X(3) ^ 2<=5,

1<=X(1) <=33333,

1<=X(2) <= 33333,

1<=X(3) <= 33333,

X(1) through X(3) are integer variables.

The problem above is based on Jong's [16, page 16/21] Problem 6, which is as follows:

Minimize ((X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(3)) / (X(3) ^ 2 + 5 * X(1) * X(2))) + ((X(1) + 1) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20))

subject to

X(1) ^ 2 + X(2) ^ 2 + X(3)<=5,

(X(1) - 2) ^ 2 + X(2) ^ 2 + X(3) ^ 2<=5,

1<=X(1) <=3,

1<=X(2) <= 3,

1<=X(3) <= 2.

 

X(4) and X(5) below are added slack variables.

One notes the following line 193 through line 195 and line 201 through line 218.

 

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

72 A(1) = 1 + RND * 33333

75 A(2) = 1 + RND * 33333

77 A(3) = 1 + RND * 33333
128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

 

181 J = 1 + FIX(RND * 3)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

193 X(1) = INT(X(1))
194 X(2) = INT(X(2))
195 X(3) = INT(X(3))

 

201 IF X(1) < 1 THEN 1670
203 IF X(1) > 33333 THEN 1670

211 IF X(2) < 1 THEN 1670
213 IF X(2) > 33333 THEN 1670
217 IF X(3) < 1 THEN 1670
218 IF X(3) > 33333 THEN 1670

311 X(4) = 5 - X(1) ^ 2 - X(2) ^ 2 - X(3)

314 X(5) = 5 - (X(1) - 2) ^ 2 - X(2) ^ 2 - X(3) ^ 2
333 FOR J44 = 4 TO 5

 

336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44
388 REM POBA = -((X(1) - 1) ^ 2 + (X(2) - 1) ^ 2 + 1) * ((X(1) - 2) ^ 2 + (X(2) - 3) ^ 2 + 1) * ((X(1) - 4) ^ 2 + (X(2) - 2) ^ 2 + 1) + 1000000 * (X(3))
389 POBA = -((X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(3)) / (X(3) ^ 2 + 5 * X(1) * X(2))) - ((X(1) + 1) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20)) + 1000000 * (X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 5

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -111111 THEN 1999

 

1900 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [40]. The complete output through JJJJ = -31999.95000000001 is shown below:

1    1    1    0    0
-.8333333333333334       -32000

1    1    1    0    0
-.8333333333333334       -31999.99

1    1    1    0    0
-.8333333333333334       -31999.98

1    1    1    0    0
-.8333333333333334       -31999.97000000001

1    1    1    0    0
-.8333333333333334       -31999.96000000001

1    1    1    0    0
-.8333333333333334       -31999.95000000001

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 [40], the wall-clock time for obtaining the output through
JJJJ= -31999.95000000001 was about 2 seconds, not including the time for "Creating .EXE file."

 

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Mohamed Abdel-Baset, Ibrahim M. Hezam (2015). An Improved flower pollination algorithm for ratios optimization problems. Applied Mathematics and Information Sciences Letters: An International Journal, 3, No. 2, 83-91 (2015). http://dx.doi.org/10.12785/amisl/030206.
[2] 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.
[3] Harold P. Benson (2002). Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22: 343-364 (2002)
[4] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[5] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[6] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[7] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[8] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[9] Mirjam Dur, Charoenchai Khompatraporn, Zelda B. Zabinsky (2007). Solving fractional problems with dynamic multistart improving hit-and-run. Annals of Operations Research (2007) 156:25-44.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] Yun-Chol Jong (2012). An efficient global optimization algorithm for nonlinear sum-of-ratios problem. www.optimization-online.org/DB_FILE/2012/08/3586.pdf.
[17] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[18] 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.
[19] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[20] 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.
[21] 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.
[22] 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.
[23] 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/
[24] 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.
[25] 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…/
[26] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[27] 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.
[28] 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.
[29] Lizhen Shao, Matthias Ehrgott (2014). An objective space cut and bound algorithm for convex multiplicative programmes. Journal of Global Optimization (2014) 58:711-728.
[30] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei (2009). A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[31] Pei Ping Shen, Yuan Ma, Yongqiang Chen (2011). Global optimization for the generalized polynomial for the sum of ratios problem. Journal of Global Optimization (2011) 50:439-455.
[32] Peiping Shen, Tongli Zhang, Chunfeng Wang (2017). Solving a class of generalized fractional programming problems using the feasibility of linear programs.
Journal of Inequalities and Applications (2017) 207:147.
[33] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[34] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[35] 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
[36] 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.
[37] 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.
[38] Chun-Feng Wang, Xin-Yue Chu (2017). A new branch and bound method for solving sum of linear ratios problem. IAENG International Journal of Applied Mathematics 47:3, IJAM_47_3_06.
[39] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[42] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

 

Solving a Sum of Ratios Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Jong [16, page 16/21, Problem 6]:

Minimize ((X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(3)) / (X(3) ^ 2 + 5 * X(1) * X(2))) + ((X(1) + 1) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20))

subject to

X(1) ^ 2 + X(2) ^ 2 + X(3)<=5,

(X(1) - 2) ^ 2 + X(2) ^ 2 + X(3) ^ 2<=5,

1<=X(1) <=3,

1<=X(2) <= 3,

1<=X(3) <= 2.

 X(4) and X(5) 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

72 A(1) = 1 + RND * 2

75 A(2) = 1 + RND * 2

77 A(3) = 1 + RND * 1
128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 3

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))

 

181 J = 1 + FIX(RND * 3)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

 

201 IF X(1) < 1 THEN 1670
203 IF X(1) > 3 THEN 1670

211 IF X(2) < 1 THEN 1670
213 IF X(2) > 3 THEN 1670
217 IF X(3) < 1 THEN 1670
218 IF X(3) > 2 THEN 1670

311 X(4) = 5 - X(1) ^ 2 - X(2) ^ 2 - X(3)

314 X(5) = 5 - (X(1) - 2) ^ 2 - X(2) ^ 2 - X(3) ^ 2
333 FOR J44 = 4 TO 5

 

336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44
388 REM POBA = -((X(1) - 1) ^ 2 + (X(2) - 1) ^ 2 + 1) * ((X(1) - 2) ^ 2 + (X(2) - 3) ^ 2 + 1) * ((X(1) - 4) ^ 2 + (X(2) - 2) ^ 2 + 1) + 1000000 * (X(3))
389 POBA = -((X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(3)) / (X(3) ^ 2 + 5 * X(1) * X(2))) - ((X(1) + 1) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20)) + 1000000 * (X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 5

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -111111 THEN 1999

 

1900 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [40]. The complete output through JJJJ = -31999.98 is shown below:

1.000000000000163        1.230116180585231       1.000000000000347
0       0       -.8185653156048536       -32000

1.000000000000002       1.23011617184374       1.000000000000016
0       0       -.8185653156048015       -31999.99

 1.000000000000041       1.230116173223621       1.000000000000097
0       0       -.8185653156048143      -31999.98

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 [40], the wall-clock time for obtaining the output through
JJJJ= -31999.98 was about 2 seconds, not including the time for "Creating .EXE file." One can compare the computational results above to the results in Jong [16, page 17/21].

 

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Mohamed Abdel-Baset, Ibrahim M. Hezam (2015). An Improved flower pollination algorithm for ratios optimization problems. Applied Mathematics and Information Sciences Letters: An International Journal, 3, No. 2, 83-91 (2015). http://dx.doi.org/10.12785/amisl/030206.
[2] 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.
[3] Harold P. Benson (2002). Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22: 343-364 (2002)
[4] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[5] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[6] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[7] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[8] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[9] Mirjam Dur, Charoenchai Khompatraporn, Zelda B. Zabinsky (2007). Solving fractional problems with dynamic multistart improving hit-and-run. Annals of Operations Research (2007) 156:25-44.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] Yun-Chol Jong (2012). An efficient global optimization algorithm for nonlinear sum-of-ratios problem. www.optimization-online.org/DB_FILE/2012/08/3586.pdf.
[17] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[18] 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.
[19] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[20] 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.
[21] 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.
[22] 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.
[23] 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/
[24] 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.
[25] 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…/
[26] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[27] 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.
[28] 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.
[29] Lizhen Shao, Matthias Ehrgott (2014). An objective space cut and bound algorithm for convex multiplicative programmes. Journal of Global Optimization (2014) 58:711-728.
[30] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei (2009). A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[31] Pei Ping Shen, Yuan Ma, Yongqiang Chen (2011). Global optimization for the generalized polynomial for the sum of ratios problem. Journal of Global Optimization (2011) 50:439-455.
[32] Peiping Shen, Tongli Zhang, Chunfeng Wang (2017). Solving a class of generalized fractional programming problems using the feasibility of linear programs.
Journal of Inequalities and Applications (2017) 207:147.
[33] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[34] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[35] 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
[36] 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.
[37] 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.
[38] Chun-Feng Wang, Xin-Yue Chu (2017). A new branch and bound method for solving sum of linear ratios problem. IAENG International Journal of Applied Mathematics 47:3, IJAM_47_3_06.
[39] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[42] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

 

Solving for Integer Solutions of an Integer Nonlinear Programming Problem

Jsun Yui Wong

 The computer program listed below seeks to solve the following problem:

Minimize ((X(1) - 1) ^ 2 + (X(2) - 1) ^ 2 + 1) * ((X(1) - 2) ^ 2 + (X(2) - 3) ^ 2 + 1) * ((X(1) - 4) ^ 2 + (X(2) - 2) ^ 2 + 1)

 subject to

X(1) + 2*X(2) <=10,

0<=X(1) <= 33333,

0<=X(2) <= 33333,

X(1) and X(2) are integer variables.

The problem above is based on Shen, Zhang, and Wang's [32, p. 12 of 16] Example 4, which is as follows:

Minimize ((X(1) - 1) ^ 2 + (X(2) - 1) ^ 2 + 1) * ((X(1) - 2) ^ 2 + (X(2) - 3) ^ 2 + 1) * ((X(1) - 4) ^ 2 + (X(2) - 2) ^ 2 + 1)

subject to

X(1) + 2*X(2) <=10,

0<=X(1) <=10,

0<=X(2) <= 4.

X(3) below is an added slack variable.

One notes line 72 and line 75, which are 72 A(1) = RND * 33333 and 75 A(2) = RND * 33333, respectively.

One also notes line 201 through 213, which are as follows:
201 IF X(1) < 0 THEN 1670
203 IF X(1) > 33333 THEN 1670
211 IF X(2) < 0 THEN 1670
213 IF X(2) > 33333 THEN 1670.

 

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

72 A(1) = RND * 33333

75 A(2) = RND * 33333

128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

 

181 J = 1 + FIX(RND * 2)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP
193 X(1) = INT(X(1))

194 X(2) = INT(X(2))

 

201 IF X(1) < 0 THEN 1670
203 IF X(1) > 33333 THEN 1670

211 IF X(2) < 0 THEN 1670
213 IF X(2) > 33333 THEN 1670

311 X(3) = 10 - X(1) - 2 * X(2)

 

333 FOR J44 = 3 TO 3

 

336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44
388 POBA = -((X(1) - 1) ^ 2 + (X(2) - 1) ^ 2 + 1) * ((X(1) - 2) ^ 2 + (X(2) - 3) ^ 2 + 1) * ((X(1) - 4) ^ 2 + (X(2) - 2) ^ 2 + 1) + 1000000 * (X(3))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 3

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -111111 THEN 1999

 

1900 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [40]. The complete output through JJJJ = -31999.92000000001 is shown below:

2    0    0    -270     -32000

2    2    0    -30    -31999.99

2    0    0    -270    -31999.98

0    2    0    -306    -31999.97000000001

0    2    0    -306    -31999.96000000001

2    0    0    -270    -31999.95000000001

0    1    0    -324    -31999.94000000001

2    0    0    -270    -31999.93000000001

2    2    0    -30    -31999.92000000001

The best integer solution shown above is at JJJJ=-31999.99, for example.

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 [40], the wall-clock time for obtaining the output through
JJJJ= -31999.92000000001 was about 2 seconds, not including the time for "Creating .EXE file."

 Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Mohamed Abdel-Baset, Ibrahim M. Hezam (2015). An Improved flower pollination algorithm for ratios optimization problems. Applied Mathematics and Information Sciences Letters: An International Journal, 3, No. 2, 83-91 (2015). http://dx.doi.org/10.12785/amisl/030206.
[2] 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.
[3] Harold P. Benson (2002). Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22: 343-364 (2002)
[4] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[5] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[6] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[7] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[8] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[9] Mirjam Dur, Charoenchai Khompatraporn, Zelda B. Zabinsky (2007). Solving fractional problems with dynamic multistart improving hit-and-run. Annals of Operations Research (2007) 156:25-44.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] Yun-Chol Jong (2012). An efficient global optimization algorithm for nonlinear sum-of-ratios problem. www.optimization-online.org/DB_FILE/2012/08/3586.pdf.
[17] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[18] 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.
[19] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[20] 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.
[21] 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.
[22] 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.
[23] 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/
[24] 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.
[25] 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…/
[26] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[27] 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.
[28] 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.
[29] Lizhen Shao, Matthias Ehrgott (2014). An objective space cut and bound algorithm for convex multiplicative programmes. Journal of Global Optimization (2014) 58:711-728.
[30] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei (2009). A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[31] Pei Ping Shen, Yuan Ma, Yongqiang Chen (2011). Global optimization for the generalized polynomial for the sum of ratios problem. Journal of Global Optimization (2011) 50:439-455.
[32] Peiping Shen, Tongli Zhang, Chunfeng Wang (2017). Solving a class of generalized fractional programming problems using the feasibility of linear programs.
Journal of Inequalities and Applications (2017) 207:147.
[33] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[34] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[35] 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
[36] 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.
[37] 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.
[38] Chun-Feng Wang, Xin-Yue Chu (2017). A new branch and bound method for solving sum of linear ratios problem. IAENG International Journal of Applied Mathematics 47:3, IJAM_47_3_06.
[39] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[42] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

 

Integer Solutions of the Two Preceding Problems

Jsun Yui Wong

I.  (X(1) - 2) ^ 2 + (X(2) - 3) ^ 2<=1

The computer program listed below seeks to solve the following problem:

Minimize (5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4)

subject to

(X(1) - 2) ^ 2 + (X(2) - 3) ^ 2<=1,

0<=X(1) <= 3,

0<=X(2) <= 3,

X(1) and X(2) are integer variables.

The problem above is Example 5 in Shen, Zhang, and Wang [32, p. 12 of 16] plus the integer restriction.

X(3) below is an added slack variable.

One notes line 193 and line 194, which are 193 X(1)=INT(X(1)) and 194 X(2)=INT(X(2)), respectively.

 

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

72 A(1) = RND * 3

75 A(2) = RND * 3

128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

 

181 J = 1 + FIX(RND * 2)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

193 X(1)=INT(X(1))

 

194 X(2)=INT(X(2))

 

201 IF X(1) < 0 THEN 1670
203 IF X(1) > 3 THEN 1670

211 IF X(2) < 0 THEN 1670
213 IF X(2) > 3 THEN 1670

311 X(3) = 1 - (X(1) - 2) ^ 2 - (X(2) - 3) ^ 2

 

333 FOR J44 = 3 TO 3

 

336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

339 NEXT J44

378 POBA = -(5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4) + 1000000 * (X(3))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 3

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -111111 THEN 1999

 

1900 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [40]. The complete output through

JJJJ = -31999.4600000001 is shown below:

2          3          0          -78246          -31999.80000000003

2          3          0          -78246          -31999.73000000004

2          3          0          -78246          -31999.58000000007

2          3          0          -78246          -31999.53000000008

2          3          0          -78246          -31999.51000000008

2          3          0          -78246          -31999.4600000001

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 [40], the wall-clock time for obtaining the output through
JJJJ= -31999.4600000001 was about 2 seconds, not including the time for "Creating .EXE file."

 

II.   (X(1) - 2) ^ 2 + (X(2) - 2) ^ 2<=1

The computer program listed below seeks to solve the following problem:

Minimize (5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4)

subject to

(X(1) - 2) ^ 2 + (X(2) - 2) ^ 2<=1,

0<=X(1) <= 3,

0<=X(2) <= 3,

X(1) and X(2) are integer variables.

The problem above is Example 5.2 in Shao and Ehrgott [29, p. 723] plus the integer restriction.

X(3) below is an added slack variable.

One notes line 193 and line 194, which are 193 X(1)=INT(X(1)) and 194 X(2)=INT(X(2)), respectively.

 

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

72 A(1) = RND * 3

75 A(2) = RND * 3

128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

 

181 J = 1 + FIX(RND * 2)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

193 X(1)=INT(X(1))

 

194 X(2)=INT(X(2))

 

201 IF X(1) < 0 THEN 1670
203 IF X(1) > 3 THEN 1670

211 IF X(2) < 0 THEN 1670
213 IF X(2) > 3 THEN 1670

311 X(3) = 1 - (X(1) - 2) ^ 2 - (X(2) - 2) ^ 2

 

333 FOR J44 = 3 TO 3

 

336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

339 NEXT J44

378 POBA = -(5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4) + 1000000 * (X(3))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 3

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -111111 THEN 1999

 

1900 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [40]. The complete output through JJJJ = -31999.33000000011 is shown below:

2          3          0          -78246          -31999.80000000003

2          3          0          -78246          -31999.33000000011

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 [40], the wall-clock time for obtaining the output through
JJJJ= -31999.33000000011 was about 2 seconds, not including the time for "Creating .EXE file."

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Mohamed Abdel-Baset, Ibrahim M. Hezam (2015). An Improved flower pollination algorithm for ratios optimization problems. Applied Mathematics and Information Sciences Letters: An International Journal, 3, No. 2, 83-91 (2015). http://dx.doi.org/10.12785/amisl/030206.
[2] 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.
[3] Harold P. Benson (2002). Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22: 343-364 (2002)
[4] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[5] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[6] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[7] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[8] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[9] Mirjam Dur, Charoenchai Khompatraporn, Zelda B. Zabinsky (2007). Solving fractional problems with dynamic multistart improving hit-and-run. Annals of Operations Research (2007) 156:25-44.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] Yun-Chol Jong (2012). An efficient global optimization algorithm for nonlinear sum-of-ratios problem. www.optimization-online.org/DB_FILE/2012/08/3586.pdf.
[17] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[18] 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.
[19] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[20] 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.
[21] 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.
[22] 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.
[23] 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/
[24] 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.
[25] 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…/
[26] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[27] 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.
[28] 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.
[29] Lizhen Shao, Matthias Ehrgott (2014). An objective space cut and bound algorithm for convex multiplicative programmes. Journal of Global Optimization (2014) 58:711-728.
[30] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei (2009). A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[31] Pei Ping Shen, Yuan Ma, Yongqiang Chen (2011). Global optimization for the generalized polynomial for the sum of ratios problem. Journal of Global Optimization (2011) 50:439-455.
[32] Peiping Shen, Tongli Zhang, Chunfeng Wang (2017). Solving a class of generalized fractional programming problems using the feasibility of linear programs.
Journal of Inequalities and Applications (2017) 207:147.
[33] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[34] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[35] 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
[36] 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.
[37] 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.
[38] Chun-Feng Wang, Xin-Yue Chu (2017). A new branch and bound method for solving sum of linear ratios problem. IAENG International Journal of Applied Mathematics 47:3, IJAM_47_3_06.
[39] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[42] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

 

Another Version of the Preceding Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Shao and Ehrgott [29, p. 723, Example 5.2]:

Minimize (5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4)

subject to

(X(1) - 2) ^ 2 + (X(2) - 2) ^ 2<=1,

0<=X(1) <= 3,

0<=X(2) <= 3.

One notes the big 2 above; in the preceding paper, it is 3.

X(3) below is an added slack variable.

 

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

72 A(1) = RND * 3

75 A(2) = RND * 3

128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

 

181 J = 1 + FIX(RND * 2)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP
201 IF X(1) < 0 THEN 1670
203 IF X(1) > 3 THEN 1670

211 IF X(2) < 0 THEN 1670
213 IF X(2) > 3 THEN 1670

311 X(3) = 1 - (X(1) - 2) ^ 2 - (X(2) - 2) ^ 2

 

333 FOR J44 = 3 TO 3

 

336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

339 NEXT J44

378 POBA = -(5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4) + 1000000 * (X(3))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 3

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -111111 THEN 1999

 

1900 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [40]. The complete output through JJJJ = -31999.98 is shown below:

2.109949904817913          2.993937130019061        0
-76987.16781058605          -32000

2.104782367447188          2.994495176193511        0
-76969.52690960531          -31999.99

2.097450896297014          2.995240334196171        0
-76958.16722934964          -31999.98  

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 [40], the wall-clock time for obtaining the output through
JJJJ= -31999.98 was 1 or 2 seconds, not including the time for "Creating .EXE file."

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Mohamed Abdel-Baset, Ibrahim M. Hezam (2015). An Improved flower pollination algorithm for ratios optimization problems. Applied Mathematics and Information Sciences Letters: An International Journal, 3, No. 2, 83-91 (2015). http://dx.doi.org/10.12785/amisl/030206.
[2] 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.
[3] Harold P. Benson (2002). Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22: 343-364 (2002)
[4] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[5] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[6] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[7] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[8] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[9] Mirjam Dur, Charoenchai Khompatraporn, Zelda B. Zabinsky (2007). Solving fractional problems with dynamic multistart improving hit-and-run. Annals of Operations Research (2007) 156:25-44.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[11] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] Yun-Chol Jong (2012). An efficient global optimization algorithm for nonlinear sum-of-ratios problem. www.optimization-online.org/DB_FILE/2012/08/3586.pdf.
[17] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[18] 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.
[19] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[20] 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.
[21] 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.
[22] 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.
[23] 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/
[24] 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.
[25] 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…/
[26] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[27] 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.
[28] 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.

[29] Lizhen Shao, Matthias Ehrgott (2014). An objective space cut and bound algorithm for convex multiplicative programmes. Journal of Global Optimization (2014) 58:711-728.

[30] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei (2009). A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.

[31] Pei Ping Shen, Yuan Ma, Yongqiang Chen (2011). Global optimization for the generalized polynomial for the sum of ratios problem. Journal of Global Optimization (2011) 50:439-455.

[32] Peiping Shen, Tongli Zhang, Chunfeng Wang (2017). Solving a class of generalized fractional programming problems using the feasibility of linear programs.
Journal of Inequalities and Applications (2017) 207:147.

[33] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[34] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[35] 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
[36] 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.
[37] 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.
[38] Chun-Feng Wang, Xin-Yue Chu (2017). A new branch and bound method for solving sum of linear ratios problem. IAENG International Journal of Applied Mathematics 47:3, IJAM_47_3_06.
[39] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[42] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

 

Solving Another Nonlinear Programming Problem Involving a Product of 3 Nonlinear Functions

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Shen, Zhang, and Wang [31, p. 12 of 16, Example 5]:

Minimize (5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4)

subject to

(X(1) - 2) ^ 2 + (X(2) - 3) ^ 2<=1,

0<=X(1) <= 3,

0<=X(2) <= 3.

X(3) below is an added slack variable.

 

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

72 A(1) = RND * 3

75 A(2) = RND * 3

128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

 

181 J = 1 + FIX(RND * 2)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP
201 IF X(1) < 0 THEN 1670
203 IF X(1) > 3 THEN 1670

211 IF X(2) < 0 THEN 1670
213 IF X(2) > 3 THEN 1670

311 X(3) = 1 - (X(1) - 2) ^ 2 - (X(2) - 3) ^ 2

 

333 FOR J44 = 3 TO 3

 

336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

339 NEXT J44

378 POBA = -(5 * (X(1) - 0) ^ 4 + (X(2) - 0) ^ 4) * (3 * (X(1) - 5) ^ 4 + 10 * (X(2) - 3) ^ 4) * (7 * (X(1) - 2) ^ 4 + 2 * (X(2) - 4) ^ 4) + 1000000 * (X(3))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 3

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -111111 THEN 1999

 

1900 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [39]. The complete output through JJJJ = -31999.98 is shown below:

2.303818821880575          2.99999999999985          0
-72439.08209361823          -32000

2.303818824389833          2.999999999999993          0
-72439.08209358303          -31999.99

2.303818825410235          2.999999999999985          0
-72439.0820935849          -31999.98

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= -31999.98 was 1 or 2 seconds, not including the time for "Creating .EXE file." One can compare the computational results above to the results in Shen, Zhang, and Wang [31, p. 10 of 16, Table 1, Example 5].

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Mohamed Abdel-Baset, Ibrahim M. Hezam (2015). An Improved flower pollination algorithm for ratios optimization problems. Applied Mathematics and Information Sciences Letters: An International Journal, 3, No. 2, 83-91 (2015). http://dx.doi.org/10.12785/amisl/030206.
[2] 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.
[3] Harold P. Benson (2002). Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22: 343-364 (2002)
[4] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[5] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[6] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[7] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[8] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[9] Mirjam Dur, Charoenchai Khompatraporn, Zelda B. Zabinsky (2007). Solving fractional problems with dynamic multistart improving hit-and-run. Annals of Operations Research (2007) 156:25-44.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[11] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] Yun-Chol Jong (2012). An efficient global optimization algorithm for nonlinear sum-of-ratios problem. www.optimization-online.org/DB_FILE/2012/08/3586.pdf.
[17] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[18] 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.
[19] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[20] 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.
[21] 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.
[22] 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.
[23] 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/
[24] 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.
[25] 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…/
[26] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[27] 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.
[28] 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.
[290] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei (2009). A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[30] Pei Ping Shen, Yuan Ma, Yongqiang Chen (2011). Global optimization for the generalized polynomial for the sum of ratios problem. Journal of Global Optimization (2011) 50:439-455.

[31] Peiping Shen, Tongli Zhang, Chunfeng Wang (2017). Solving a class of generalized fractional programming problems using the feasibility of linear programs.
Journal of Inequalities and Applications (2017) 207:147.

[32] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[33] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[34] 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
[35] 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.
[36] 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.
[37] Chun-Feng Wang, Xin-Yue Chu (2017). A new branch and bound method for solving sum of linear ratios problem. IAENG International Journal of Applied Mathematics 47:3, IJAM_47_3_06.
[38] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[39] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[40] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[41] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

 

Solving a Nonlinear Fractional Programming Problem from the Literature

 Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Shen, Ma, and Chen [30, p. 452, Example 3]:

Maximize ((-X(1) ^ 2 + 3 * X(1) - 2 * X(2) ^ 2 + 3 * X(2) + 3.5) / (X(1) + 1)) + ((X(2)) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20))

subject to

3 * X(1) + X(2)<=8,

X(1) -X(1) ^ -1 * X(2)<=1,

2 * X(1) * X(2) ^ -1 + X(2)<=6,

1<=X(1) <= 3,

.1<=X(2) <= 3.

X(3) through X(5) 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

72 A(1) = 1 + RND * 2

75 A(2) = 1 + RND * 2

128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

 

181 J = 1 + FIX(RND * 2)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

 

201 IF X(1) < 1 THEN 1670
203 IF X(1) > 3 THEN 1670

211 IF X(2) < 1 THEN 1670
213 IF X(2) > 3 THEN 1670

311 X(3) = 8 - 3 * X(1) - X(2)

313 X(4) = 1 - X(1) + X(1) ^ -1 * X(2)

 

315 X(5) = 6 - 2 * X(1) * X(2) ^ -1 - X(2)

 

333 FOR J44 = 3 TO 5
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

 

339 NEXT J44

378 POBA = ((-X(1) ^ 2 + 3 * X(1) - 2 * X(2) ^ 2 + 3 * X(2) + 3.5) / (X(1) + 1)) + ((X(2)) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20)) + 1000000 * (X(3) + X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 5

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -11111 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [38]. The complete output through JJJJ = -31999.98 is shown below:

1.000000000000004          1          0          0
0          3.333333333333333          -32000

1.000000000000276          1.000000000000125          0
0          0          3.333333333332976          -31999.99

1.00000000000098          1.000000000000064          0
0          0          3.333333333332207          -31999.98

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 [38], the wall-clock time for obtaining the output through
JJJJ= -31999.98 was 1 or 2 seconds, not including the time for "Creating .EXE file." One can compare the computational results above to the results in Shen, Ma, and Chen [30, p. 452, Table 1, Example 3].

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Mohamed Abdel-Baset, Ibrahim M. Hezam (2015). An Improved flower pollination algorithm for ratios optimization problems. Applied Mathematics and Information Sciences Letters: An International Journal, 3, No. 2, 83-91 (2015). http://dx.doi.org/10.12785/amisl/030206.
[2] 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.
[3] Harold P. Benson (2002). Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22: 343-364 (2002)
[4] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[5] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[6] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[7] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[8] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[9] Mirjam Dur, Charoenchai Khompatraporn, Zelda B. Zabinsky (2007). Solving fractional problems with dynamic multistart improving hit-and-run. Annals of Operations Research (2007) 156:25-44.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[11] 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.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] Yun-Chol Jong (2012). An efficient global optimization algorithm for nonlinear sum-of-ratios problem. www.optimization-online.org/DB_FILE/2012/08/3586.pdf.
[17] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[18] 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.
[19] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[20] 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.
[21] 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.
[22] 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.
[23] 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/
[24] 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.
[25] 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…/
[26] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[27] 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.
[28] 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.
[290] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei (2009). A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[30] Pei Ping Shen, Yuan Ma, Yongqiang Chen (2011). Global optimization for the generalized polynomial for the sum of ratios problem. Journal of Global Optimization (2011) 50:439-455.
[31] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[32] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[33] 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
[34] 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.
[35] 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.
[36] Chun-Feng Wang, Xin-Yue Chu (2017). A new branch and bound method for solving sum of linear ratios problem. IAENG International Journal of Applied Mathematics 47:3, IJAM_47_3_06.
[37] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics  and Computation 158 (2004) 319 330.Applied Mathematics 
[38] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[39] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.

[40] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

 

Solving Another Nonlinear Fractional Programming Problem

 

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Shen, Ma, and Chen [30, p. 453, Example 5]:

Minimize (.25) * ((-X(1) ^ 2 + 3 * X(1) + 2 * X(2) ^ 2 + 3 * X(2) + 3.5) / (X(1) + 1)) + (-1.75) * ((X(2)) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20)) + 1000000 * (X(3) + X(4) + X(5))

subject to

3 * X(1) + X(2)<=8,

X(1) -X(1) ^ -1 * X(2)<=1,

2 * X(1) * X(2) ^ -1 + X(2)<=6,

1<=X(1) <= 3,

1<=X(2) <= 3.

X(3) through X(5) 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

72 A(1) = 1 + RND * 3

 

75 A(2) = 1 + RND * 3

 

128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

 

181 J = 1 + FIX(RND * 2)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

201 IF X(1) < 1 THEN 1670
203 IF X(1) > 3 THEN 1670

211 IF X(2) < 1 THEN 1670
213 IF X(2) > 3 THEN 1670

 

311 X(3) = 8 - 3 * X(1) - X(2)

313 X(4) = 1 - X(1) + X(1) ^ -1 * X(2)

 

315 X(5) = 6 - 2 * X(1) * X(2) ^ -1 - X(2)

 

333 FOR J44 = 3 TO 5
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

 

339 NEXT J44

 

368 REM POBA = ((-X(1) ^ 2 * X(2) ^ .5 + 2 * X(1) * X(2) ^ -1 - X(2) ^ 2 + 2.8 * X(1) ^ -1 * X(2) + 7.5) / (X(1) * X(2) ^ 1.5 + 1)) + ((X(2) + .1) / (X(1) ^ 2 * X(2) ^ -1 - 3 * X(1) ^ -1 + 2 * X(1) * X(2) ^ 2 - 9 * X(2) ^ -1 + 12)) + 1000000 * (X(3) + X(4) + X(5))

378 POBA = -(.25) * ((-X(1) ^ 2 + 3 * X(1) + 2 * X(2) ^ 2 + 3 * X(2) + 3.5) / (X(1) + 1)) - (-1.75) * ((X(2)) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20)) + 1000000 * (X(3) + X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 5

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -11111 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

 This BASIC computer program was run with qb64v1000-win [38]. The complete output through JJJJ = -31999.98 is shown below:

1.618033988749902          1.000000000000027           0
0          0          -.8838686827047476          -32000

1.618047492817476          1.00003019519546          0
0          0          -.8838771292750162          -31999.99

1.618034137822611          1.00000033333675          0
0          0          -.8838687759462024          -31999.98

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 [38], the wall-clock time for obtaining the output through
JJJJ= -31999.98 was 1 or 2 seconds, not including the time for "Creating .EXE file."

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Mohamed Abdel-Baset, Ibrahim M. Hezam (2015). An Improved flower pollination algorithm for ratios optimization problems. Applied Mathematics and Information Sciences Letters: An International Journal, 3, No. 2, 83-91 (2015). http://dx.doi.org/10.12785/amisl/030206.
[2] 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.
[3] Harold P. Benson (2002). Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22: 343-364 (2002)
[4] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[5] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[6] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[7] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[8] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[9] Mirjam Dur, Charoenchai Khompatraporn, Zelda B. Zabinsky (2007). Solving fractional problems with dynamic multistart improving hit-and-run. Annals of Operations Research (2007) 156:25-44.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[11] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Sinulation 18 (2013) 89-98.
[12] 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.
[13] 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.
[14] 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.
[15] 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.
[16] Yun-Chol Jong (2012). An efficient global optimization algorithm for nonlinear sum-of-ratios problem. www.optimization-online.org/DB_FILE/2012/08/3586.pdf.
[17] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[18] 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.
[19] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[20] 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.
[21] 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.
[22] 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.
[23] 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/
[24] 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.
[25] 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…/
[26] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[27] 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.
[28] 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.
[290] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei (2009). A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[30] Pei Ping Shen, Yuan Ma, Yongqiang Chen (2011). Global optimization for the generalized polynomial for the sum of ratios problem. Journal of Global Optimization (2011) 50:439-455.
[31] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[32] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[33] 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
[34] 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.
[35] 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.
[36] Chun-Feng Wang, Xin-Yue Chu (2017). A new branch and bound method for solving sum of linear ratios problem. IAENG International Journal of Applied Mathematics 47:3, IJAM_47_3_06.
[37] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[38] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[39] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[40] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.

 

 

An Algorithm for Solving Nonlinear Programming Problems and Integer Nonlinear Programming Problems: Another Illustration

Jsun Yui Wong

1. Continuous Case

The computer program listed below seeks to solve the following problem from Shen, Ma, and Chen [30, p. 453, Example 6].

 Maximize -(-1.35) * ((X(1) ^ 2 * X(2) ^ .5 - 2 * X(1) * X(2) ^ -1 + X(2) ^ 2 - 2.8 * X(1) ^ -1 * X(2) + 7.5) / (X(1) * X(2) ^ 1.5 + 1)) - (12.99) * ((X(2) + .1) / (X(1) ^ 2 * X(2) ^ -1 - 3 * X(1) ^ -1 + 2 * X(1) * X(2) ^ 2 - 9 * X(2) ^ -1 + 12))

subject to

2 * X(1) ^ -1 + X(1) * X(2)<=4,

X(1) +3 * X(1) ^ -1 * X(2)<=5,

X(1) ^ 2 - 3 * X(2) ^ 3<=2,

1<=X(1) <= 3,

1<=X(2) <= 3.

X(3) through X(5) 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

72 A(1) = 1 + RND * 2

75 A(2) = 1 + RND * 2

128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

 

181 J = 1 + FIX(RND * 2)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

 

201 IF X(1) < 1 THEN 1670
203 IF X(1) > 3 THEN 1670

211 IF X(2) < 1 THEN 1670
213 IF X(2) > 3 THEN 1670

311 X(3) = 4 - 2 * X(1) ^ -1 - X(1) * X(2)

313 X(4) = 5 - X(1) - 3 * X(1) ^ -1 * X(2)

 

315 X(5) = 2 - X(1) ^ 2 + 3 * X(2) ^ 3

 

333 FOR J44 = 3 TO 5
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

339 NEXT J44

378 POBA = -(-1.35) * ((X(1) ^ 2 * X(2) ^ .5 - 2 * X(1) * X(2) ^ -1 + X(2) ^ 2 - 2.8 * X(1) ^ -1 * X(2) + 7.5) / (X(1) * X(2) ^ 1.5 + 1)) - (12.99) * ((X(2) + .1) / (X(1) ^ 2 * X(2) ^ -1 - 3 * X(1) ^ -1 + 2 * X(1) * X(2) ^ 2 - 9 * X(2) ^ -1 + 12)) + 1000000 * (X(3) + X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 5

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < -11111 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [38]. The complete output through JJJJ = -31999.98 is shown below:

2.698672512963932       1.207578072004017       0
0       0       2.332207556891017       -32000

2.698682666462486       1.20758224763645       0
0       0       2.332213594166706       -31999.99

2.698690059295716       1.20758528793531       0
0       0       2.332217989915511       -31999.98

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 [38], the wall-clock time for obtaining the output through JJJJ= -31999.98 was 1 or 2 seconds, not including the time for "Creating .EXE file." One can compare the computational results above to the results in Shen, Ma, and Chen [30, p.452, Table 1].


2. Integer Case

If interested, one can transform the continuous case above to an integer case, for example, by adding line 193 and line 194 below, which are 193 X(1) = INT(X(1)) and 194 X(2) = INT(X(2)) , respectively.

 

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

72 A(1) = 1 + RND * 2

75 A(2) = 1 + RND * 2

128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 2

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 2))

 

181 J = 1 + FIX(RND * 2)

183 r = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * r

191 NEXT IPP

 

193 X(1) = INT(X(1))
194 X(2) = INT(X(2))

 

201 IF X(1) < 1 THEN 1670
203 IF X(1) > 3 THEN 1670

211 IF X(2) < 1 THEN 1670
213 IF X(2) > 3 THEN 1670

311 X(3) = 4 - 2 * X(1) ^ -1 - X(1) * X(2)

313 X(4) = 5 - X(1) - 3 * X(1) ^ -1 * X(2)

 

315 X(5) = 2 - X(1) ^ 2 + 3 * X(2) ^ 3

 

333 FOR J44 = 3 TO 5
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

339 NEXT J44

378 POBA = -(-1.35) * ((X(1) ^ 2 * X(2) ^ .5 - 2 * X(1) * X(2) ^ -1 + X(2) ^ 2 - 2.8 * X(1) ^ -1 * X(2) + 7.5) / (X(1) * X(2) ^ 1.5 + 1)) - (12.99) * ((X(2) + .1) / (X(1) ^ 2 * X(2) ^ -1 - 3 * X(1) ^ -1 + 2 * X(1) * X(2) ^ 2 - 9 * X(2) ^ -1 + 12)) + 1000000 * (X(3) + X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 5

1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 GOTO 128

1670 NEXT I

1889 IF M < 0 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [38]. The complete output through JJJJ = -31999.82000000002 is shown below:

2           1          0          0          0
1.690894736842106         -31999.99

2          1          0          0          0
1.690894736842106         -31999.88000000002

2          1          0          0          0
1.690894736842106          -31999.84000000002

2          1          0          0          0
1.690894736842106          -31999.83000000002

2          1          0          0          0
1.690894736842106          -31999.82000000002

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 [38], the wall-clock time for obtaining the output through JJJJ= -31999.82000000002 was 1 or 2 seconds, not including the time for "Creating .EXE file."

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

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[34] 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.
[35] 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.
[36] Chun-Feng Wang, Xin-Yue Chu (2017). A new branch and bound method for solving sum of linear ratios problem. IAENG International Journal of Applied Mathematics 47:3, IJAM_47_3_06.
[37] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[38] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[39] Xin-She Yang, Amir Hossein Gandomi (2012). Bat algorithm: a novel approach for global engineering optimization. Engineering Computations: International Journal for Computer-Aided Engineering and Software, Vol. 20,No. 5, 2012, pp. 461-483.
[40] B. D. Youn, K. K. Choi (2004). A new response surface methodology for reliability-based design optimization. Computers and Structures 82 (2004) 241-256.