Substance

Solving Another Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following fractional programming problem in Wang and Chu [30, Example 6]:

Maximize      ((3 * X(1) + 4 * X(2) + 50) / (3 * X(1) + 5 * X(2) + 4 * X(3) + 50)) - ((3 * X(1) + 5 * X(2) + 3 * X(3) + 50) / (5 * X(1) + 5 * X(2) + 4 * X(3) + 50)) - ((X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(2) + 4 * X(3) + 50)) - ((4 * X(1) + 3 * X(2) + 3 * X(3) + 50) / (3 * X(2) + 3 * X(3) + 50))

subject to

 6 * X(1)+ 3 * X(2) + 3 * X(3)< =10,

 10 * X(1) + 3 * X(2) + 8 * X(3)< =10,

X(1) >= 0, X(2)>=0, X(3) >= 0.

X(4) through X(5) below are 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
70 FOR J44 = 1 TO 3

72 A(J44) = RND * 1000

 

73 NEXT J44

 

128 FOR I = 1 TO 100

 

129 FOR KKQQ = 1 TO 3

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

 

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

196 FOR J99 = 1 TO 3

201 IF X(J99) < 0 THEN 1670

 

204 NEXT J99

 

308 X(4) = 10 - 6 * X(1) - 3 * X(2) - 3 * X(3)

 

309 X(5) = 10 - 10 * X(1) - 3 * X(2) - 8 * X(3)

 

325 FOR J99 = 4 TO 5

327 XX(J99) = X(J99)

 

330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

331 NEXT J99

 

359 POBA = ((3 * X(1) + 4 * X(2) + 50) / (3 * X(1) + 5 * X(2) + 4 * X(3) + 50)) - ((3 * X(1) + 5 * X(2) + 3 * X(3) + 50) / (5 * X(1) + 5 * X(2) + 4 * X(3) + 50)) - ((X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(2) + 4 * X(3) + 50)) - ((4 * X(1) + 3 * X(2) + 3 * X(3) + 50) / (3 * X(2) + 3 * X(3) + 50)) + 1000000 * (X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 5

 

1456 XXX(KLX) = XX(KLX)

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

1670 NEXT I

 

1889 IF M < -1000 THEN 1999

 

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

1903 PRINT XXX(4), XXX(5)
1912 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [31]. The complete output through JJJJ = -31999.99 is shown below:

2.79219283830006D-16           3.333333239940138          4.322134112965536D-15
0          0
2.801795710115663D-7          2.801795476245388D-07
-1.900000002101347          -32000

6.023568307138324D-15          3.333333330522641          9.1999714915622D-16
0          0
8.432040303785981.D-09          8.432010834804484D-09
-1.900000000063241          -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 [31], the wall-clock time for obtaining the output through JJJJ= -31999.99 was 2 seconds, not including the time for "Creating the .EXE file." One can compare the computational results above to the results in TABLE II of Wang and Chu [30, Example 6].

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 Sinulation 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] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[13] 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.
[14] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[15] 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.
[16] 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.
[17] 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.
[18] 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/
[19] 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.
[20] 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…/
[21] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[22] 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.
[23] 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.
[24] 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.
[25] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[26] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[27] 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
[28] 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.
[29] 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.
[30] 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.
[31] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[32] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[33] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[34] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[35] 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.
[36] 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 Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following fractional programming problem in Wang and Chu [30, Example 2]:

Maximize ((3 * X(1) + 4 * X(3) + 50) / (4 * X(1) + 4 * X(2) + 5 * X(3) + 50)) + ((4 * X(1) + 3 * X(2) + 3 * X(3) + 50) / (3 * X(2) + 3 * X(3) + 50)) + ((X(1) + 2 * X(2) + 5 * X(3) + 50) / (X(1) + 5 * X(2) + 5 * X(3) + 50)) + ((X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(2) + 4 * X(3) + 50))

subject to

2 * X(1)+ X(2) + 5 * X(3) < =10,

X(1)+ 6 * X(2) + 3 * X(3) <=10,

5 * X(1) + 9 * X(2) + 2 * X(3)< =10,

9 * X(1) + 7 * X(2) + 3 * X(3) <=10,

X(1) >= 0, X(2)> =0, X(3) >= 0.

 X(4) through X(7) below are 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
70 FOR J44 = 1 TO 3

72 A(J44) = RND * 1000

 

73 NEXT J44

 

128 FOR I = 1 TO 100

 

129 FOR KKQQ = 1 TO 3

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

 

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

196 FOR J99 = 1 TO 3

199 REM X(J99) = INT(X(J99))

201 IF X(J99) < 0 THEN 1670

 

204 NEXT J99

 

308 X(4) = 10 - 2 * X(1) - X(2) - 5 * X(3)

 

309 X(5) = 10 - X(1) - 6 * X(2) - 3 * X(3)

 

311 X(6) = 10 - 5 * X(1) - 9 * X(2) - 2 * X(3)

 

313 X(7) = 10 - 9 * X(1) - 7 * X(2) - 3 * X(3)

325 FOR J99 = 4 TO 7

327 XX(J99) = X(J99)

 

330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

331 NEXT J99

 

359 POBA = ((3 * X(1) + 4 * X(3) + 50) / (4 * X(1) + 4 * X(2) + 5 * X(3) + 50)) + ((4 * X(1) + 3 * X(2) + 3 * X(3) + 50) / (3 * X(2) + 3 * X(3) + 50)) + ((X(1) + 2 * X(2) + 5 * X(3) + 50) / (X(1) + 5 * X(2) + 5 * X(3) + 50)) + ((X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(2) + 4 * X(3) + 50)) + 1000000 * (X(4) + X(5) + X(6) + X(7))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 7

 

1456 XXX(KLX) = XX(KLX)

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

1670 NEXT I

 

1889 IF M < -10000 THEN 1999

 

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

1902 PRINT A(6), A(7)

 

1903 PRINT XXX(4), XXX(5), XXX(6), XXX(7)
1912 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [31]. The complete output through JJJJ = -31999.99 is shown below:

1.11111034571914        7.445754727588355D-16       9.523536033130662D-15
0       0
0       0
7.777779308561671       8.888889654280826        4.444448271404274
6.888527705996357D-06
4.09070288421722       -32000

1.111111035544948       4.072399707656941D-15       9.859579221084657D-15
0       0
0       0
7.77777792891005       8.888888964454997        4.444444822275204
6.800954094123853D-07
4.090702941563836       -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 [31], the wall-clock time for obtaining the output through JJJJ= -31999.99 was 2 seconds, not including the time for "Creating the .EXE file." One can compare the computational results above to the results in TABLE II of Wang and Chu [30, Example 2].

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 Sinulation 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] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[13] 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.
[14] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[15] 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.
[16] 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.
[17] 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.
[18] 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/
[19] 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.
[20] 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…/
[21] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[22] 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.
[23] 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.
[24] 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.
[25] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[26] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[27] 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
[28] 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.
[29] 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.
[30] 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.
[31] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[32] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[33] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[34] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[35] 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.
[36] 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 Integer Fractional Programming Problem

Jsun Yui Wong

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

Minimize (63 * X(1) - 18 * X(2) + 39) / (13 * X(1) + 26 * X(2) + 13) + (37 * X(1) + 73 * X(2) + 13) / (13 * X(1) + 13 * X(2) + 13)

subject to

5 * X(1) - 3 * X(2) = 3,

1.5<= X(1)<=3,

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

The problem above is based on Example 3 in Wang and Shen [31, page 287], which is

Minimize (63 * X(1) - 18 * X(2) + 39) / (13 * X(1) + 26 * X(2) + 13) + (37 * X(1) + 73 * X(2) + 13) / (13 * X(1) + 13 * X(2) + 13)

subject to

5 * X(1) - 3 * X(2) = 3,

1.5<= X(1)<=3.

 

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 2

 

74 A(J40) = (RND * 10)

 

77 NEXT J40

 

128 FOR I = 1 TO 500

 

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
222 NEXT IPP
223 X(2) = (-3 + 5 * X(1)) / 3

224 FOR j41 = 1 TO 2

 

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

 

235 NEXT j41
237 IF X(2) = (-3 + 5 * X(1)) / 3 THEN 256 ELSE GOTO 1670

 

256 FOR J47 = 1 TO 1

257 IF X(J47) < 1.5 THEN 1670

259 NEXT J47

266 IF X(1) > 3 THEN 1670

 

348 POBA = -(63 * X(1) - 18 * X(2) + 39) / (13 * X(1) + 26 * X(2) + 13) - (37 * X(1) + 73 * X(2) + 13) / (13 * X(1) + 13 * X(2) + 13)

 

466 P = POBA

 

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 2

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

1670 NEXT I
1889 IF M < -1000 THEN 1999

1900 PRINT A(1), A(2)

 

1922 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [32]. The complete output through JJJJ = -31999.98 is shown below:

3          4
-5          -32000

3          4
-5          -31999.99

3          4
-5          -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 [32], the wall-clock time for obtaining the output through JJJJ= -31999.98 was 2 seconds, not including the creating .EXE file time. One can compare the computational results here with those in Wang and Shen [31, page 286, Table 2].

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] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[3] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[4] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[5] Yuelin Gao, Siqiao Jin (2013). A global optimization algorithm for sum of linear ratios problem. Journal of Applied Mathematics, Volume 2013. Hindawi Publishing Company.
[6] 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.

[7] 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.

[8] 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.

[9] Hongwei Jiao, Zhankui Wang, Yongqiang Chen (2013). Global Optimization Algorithm for Sum of Generalized Polynomial Ratios Problems. Applied Mathematical Modelling 37 (2013) 187-197.

[10] Hongwei Jiao, Yongqiang Chen (2014). A New Linearizing Method for Sum of Ratios Problem with Coefficients. Journal of Applied Mathematics, Volume 2014, Article ID 490297, 10 pages. http://dx.doi.org/10.1155/2014/490297, Hindawi Publishing Corporation.

[11] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[12] 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.
[13] 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 (1416) pp. 922-934.
[15] 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.
[16] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[17] 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.
[18] 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.
[19] 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.
[20] 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.
[21] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[22] 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.
[23] 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.
[24] Pei-Ping Shen, Chun-Feng Wang (2006). Global optimization for sum of linear ratios problem with coefficients. Appliedd Mathematics and Computation, Vol. 176 (2006 ) 219-229.
[25] 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.
[26] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[27] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[28] 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
[29] 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.
[30] 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.
[31] Wang Chun-Feng, Shen Pei-Ping (2008). A global optimization algorithm for linear fractional programming. Applied Mathematics and Computation, Vol. 204 (2008 ) 281-287.
[32] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[33] 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 Integer Fractional Programming Problem

Jsun Yui Wong

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

Maximize -(63 * X(1) - 18 * X(2) + 39) / (13 * X(1) + 26 * X(2) + 13) - (13 * X(1) + 26 * X(2) + 13) / (37 * X(1) + 73 * X(2) + 13) + (37 * X(1) + 73 * X(2) + 13) / (13 * X(1) + 13 * X(2) + 13) + (13 * X(1) + 13 * X(2) + 13) / (63 * X(1) - 18 * X(2) + 39)

subject to

5 * X(1) - 3 * X(2) = 3,

1.5<= X(1)<=3,

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

The problem above is based on Example 4 in Shen and Wang [24, page 228], which is

maximize -(63 * X(1) - 18 * X(2) + 39) / (13 * X(1) + 26 * X(2) + 13) - (13 * X(1) + 26 * X(2) + 13) / (37 * X(1) + 73 * X(2) + 13) + (37 * X(1) + 73 * X(2) + 13) / (13 * X(1) + 13 * X(2) + 13) + (13 * X(1) + 13 * X(2) + 13) / (63 * X(1) - 18 * X(2) + 39)

subject to

5 * X(1) - 3 * X(2) = 3,

1.5<= X(1)<=3.

 

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 2

 

74 A(J40) = (RND * 10)

77 NEXT J40

 

128 FOR I = 1 TO 500

 

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 * 1)

 

183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP
223 X(2) = (-3 + 5 * X(1)) / 3

224 FOR j41 = 1 TO 2

 

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

 

235 NEXT j41
237 IF X(2) = (-3 + 5 * X(1)) / 3 THEN 256 ELSE GOTO 1670

 

256 FOR J47 = 1 TO 1

257 IF X(J47) < 1.5 THEN 1670

259 NEXT J47

266 IF X(1) > 3 THEN 1670

 

348 POBA = -(63 * X(1) - 18 * X(2) + 39) / (13 * X(1) + 26 * X(2) + 13) - (13 * X(1) + 26 * X(2) + 13) / (37 * X(1) + 73 * X(2) + 13) + (37 * X(1) + 73 * X(2) + 13) / (13 * X(1) + 13 * X(2) + 13) + (13 * X(1) + 13 * X(2) + 13) / (63 * X(1) - 18 * X(2) + 39)

 

466 P = POBA

 

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 2

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

1670 NEXT I
1889 IF M < -1000 THEN 1999

1900 PRINT A(1), A(2)

 

1922 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [31]. The complete output through JJJJ = -31999.98 is shown below:

3 4
3.291666666666667 -32000

3 4
3.291666666666667 -31999.99

3 4
3.291666666666667 -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 [31], the wall-clock time for obtaining the output through
JJJJ= -31999.98 was 2 seconds, not including the creating .EXE file time. One can compare the computational results here with those in Shen and Wang [24, page 227].

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] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[3] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[4] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.
[5] Yuelin Gao, Siqiao Jin (2013). A global optimization algorithm for sum of linear ratios problem. Journal of Applied Mathematics, Volume 2013. Hindawi Publishing Company.
[6] 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.

[7] 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.

[8] 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.

[9] Hongwei Jiao, Zhankui Wang, Yongqiang Chen (2013). Global Optimization Algorithm for Sum of Generalized Polynomial Ratios Problems. Applied Mathematical Modelling 37 (2013) 187-197.

[10] Hongwei Jiao, Yongqiang Chen (2014). A New Linearizing Method for Sum of Ratios Problem with Coefficients. Journal of Applied Mathematics, Volume 2014, Article ID 490297, 10 pages. http://dx.doi.org/10.1155/2014/490297, Hindawi Publishing Corporation.

[11] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[12] 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.
[13] 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 (1416) pp. 922-934.
[15] 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.
[16] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[17] 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.
[18] 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.
[19] 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.
[20] 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.
[21] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[22] 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.
[23] 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.
[24] Pei-Ping Shen, Chun-Feng Wang (2006). Global optimization for sum of linear ratios problem with coefficients. Appliedd Mathematics and Computation, Vol. 176 (2006 )219-229.
[25] 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.
[26] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[27] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[28] 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
[29] 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.
[30] 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.
[31] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[32] 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 Linear Ratios Problem

Jsun Yui Wong

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

Maximize (3 * X(1) + 5 * X(2) + 3 * X(3) + 50) / (3 * X(1) + 4 * X(2) + 5 * X(3) + 50) + (3 * X(1) + 5 * X(2) + 50) / (3 * X(1) + 5 * X(2) + 3 * X(3) + 50) + (4 * X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(1) + 4 * X(2) + 3 * X(3) + 50)

subject to

6 * X(1) + 3 * X(2) + 3 * X(3)<=10,

10 * X(1) + 3 * X(2) + 8 * X(3)<=10,

X(1), X(2), X(3)>=0,

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

The problem above is based on Example 3 in Jiao and Chen [16, page 7], which is

Maximize (3 * X(1) + 5 * X(2) + 3 * X(3) + 50) / (3 * X(1) + 4 * X(2) + 5 * X(3) + 50) + (3 * X(1) + 5 * X(2) + 50) / (3 * X(1) + 5 * X(2) + 3 * X(3) + 50) + (4 * X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(1) + 4 * X(2) + 3 * X(3) + 50)

subject to

6 * X(1) + 3 * X(2) + 3 * X(3)<=10,

10 * X(1) + 3 * X(2) + 8 * X(3)<=10,

X(1), X(2), X(3)>=0,

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

X(4) and X(5) below are slack variables. One notes line 325, which is 325 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0.

 

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 3

 

74 A(J40) = (RND * 1000)

 

75 REM

77 NEXT J40

 

128 FOR I = 1 TO 500

 

129 FOR KKQQ = 1 TO 3

 

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

 

181 J = 1 + FIX(RND * 3)

 

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

 

223 FOR j41 = 1 TO 3

 

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

 

235 NEXT j41

 

256 FOR J47 = 1 TO 3

257 IF X(J47) < 0 THEN 1670

259 NEXT J47

 

311 X(4) = 10 - 6 * X(1) - 3 * X(2) - 3 * X(3)
313 X(5) = 10 - 10 * X(1) - 3 * X(2) - 8 * X(3)

322 FOR J44 = 4 TO 5

 

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

 

327 NEXT J44

 

348 POBA = (3 * X(1) + 5 * X(2) + 3 * X(3) + 50) / (3 * X(1) + 4 * X(2) + 5 * X(3) + 50) + (3 * X(1) + 5 * X(2) + 50) / (3 * X(1) + 5 * X(2) + 3 * X(3) + 50) + (4 * X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(1) + 4 * X(2) + 3 * X(3) + 50) + 1000000 * (X(4) + X(5) + X(6))

 

 

466 P = POBA

 

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 5

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

1670 NEXT I
1889 IF M < -1000 THEN 1999

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

1922 PRINT 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:

0   0   0   0   0
3       -32000

0   0   0   0   0
3       -31999.99

0   0   0   0   0
3       -31999.98

0   0   0   0   0
3       -31999.97000000001

0   0   0   0   0
3       -31999.96000000001

0   0   0   0   0
3       -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 2 seconds, not including the creating .EXE file time. One can compare the computational results here with those in Jiao and Chen [16, page 7].

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 Sinulation 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.i publishinny.xxxxxxxxxxxxxxxg cpompa
[11] Yuelin Gao, Siqiao Jin (2013). A global optimization algorithm for sum of linear ratios problem. Journal of Applied Mathematics, Volume 2013. Hindawi Publishing Company.
[12] 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.

[13] 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.

[14] 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.

[15] Hongwei Jiao, Zhankui Wang, Yongqiang Chen (2013). Global Optimization Algorithm for Sum of Generalized Polynomial Ratios Problems. Applied Mathematical Modelling 37 (2013) 187-197.

[16] Hongwei Jiao, Yongqiang Chen (2014). A New Linearizing Method for Sum of Ratios Problem with Coefficients. Journal of Applied Mathematics, Volume 2014, Article ID 490297, 10 pages. http://dx.doi.org/10.1155/2014/490297, Hindawi Publishing Corporation.

[17] 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.

[18] Ali Husseinzadeh Kashan (2015). An effective algorithm for constrained optimization based on optics inspired optimization (OIO). Computer-Aided Design 63 (2015) 52-71.

[19] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[20] 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.
[21] 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.
[22] 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.
[23] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[24] 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.
[25] 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.
[26] 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.
[27] 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/
[28] 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.
[29] 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…/
[30] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[31] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[32] 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.
[33] 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.
[34] 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.
[35] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[36] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[37] 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
[38] 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.
[39] 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.
[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[42] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[43] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A 40 benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[44] 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.
[45] 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 Integer Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear integer fractional programming problem:

Maximize (-X(1) ^ 2 + 3 * X(1) - 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

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

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

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

X(1) >= 1,

X(2) > =2,

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

The problem above is based on Example 2 in Jiao, Wang, and Chen [13, p. 195], which is

Maximize (-X(1) ^ 2 + 3 * X(1) - 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

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

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

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

X(1) >= 1,

X(2) > =2.

X(3) through X(5) below are 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

 

71 REM FOR J40 = 1 TO 2

 

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

77 REM NEXT J40

 

128 FOR I = 1 TO 5000

129 FOR KKQQ = 1 TO 2

 

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

 

181 J = 1 + FIX(RND * 2)

 

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

 

223 FOR J41 = 1 TO 2

 

225 X(J41) = INT(X(J41))

 

235 NEXT J41

 

256 REM FOR J47 = 1 TO 2

257 IF X(1) < 1 THEN 1670
258 IF X(2) > 2 THEN 1670

 

259 REM NEXT J47
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 POBA = (-X(1) ^ 2 + 3 * X(1) - 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

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

1670 NEXT I
1889 IF M < 3 THEN 1999

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

1902 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [37]. The complete output through JJJJ = -31999.06000000015 is shown below:

1      2       0       0       0
4.035714285714286          -31999.10000000014

1       1       0       0       0
3.833333333333334          -31999.06000000015

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 [37], the wall-clock time for obtaining the output through JJJJ= -31999.06000000015 was 2 seconds, not including the creating .EXE file time. One can compare the computational results here with those in Table 1 of Jiao, Wang, and Chen [13, p. 194].

Acknowledgment

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

References

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[37] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[38] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
[39] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[40] Xin-She Yang, Christian Huyck, Mehmet Karamanoglu, Nawaz Khan (2014). True global optimality of the pressure vessel design problem: A 40 benchmark for bio-inspired optimisation algorithms. https://arxiv.org/pdf/1403.7793.pdf.
[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.