Substance

An Algorithm for Solving Nonlinear Programming Problems and Integer Nonlinear Programming Problems: An 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 4].

Maximize

((-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))

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 -31999.90 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 100

 

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

 

368 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))
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:

1.000002253320367 1.00000282499549 0
0 0 5.516639267545698 -32000

1.000000069357661          1.000000052190488          0
0          0          5.5166660265338          -31999.99

1.000000736913784 1.000011426716021 0
0 0 5.51659630942257 -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

The continuous case above can be transformed to the integer case 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 -31999.90 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 100

 

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

 

368 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))
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 = -31999.90000000002 is shown below:

1          1          0          0          0
5.516666666666667          -32000

1 1 0 0 0
5.516666666666667 -31999.99

1 1 0 0 0
5.516666666666667 -31999.98

1 1 0 0 0
5.516666666666667 -31999.97000000001

1 1 0 0 0
5.516666666666667 -31999.96000000001

1 1 0 0 0
5.516666666666667 -31999.95000000001

1 1 0 0 0
5.516666666666667 -31999.94000000001

1 1 0 0 0
5.516666666666667 -31999.93000000001

1 1 0 0 0
5.516666666666667 -31999.92000000001

1 1 0 0 0
5.516666666666667 -31999.91000000001

1 1 0 0 0
5.516666666666667 -31999.90000000002

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= -31999.90000000002 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] 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.

 

Solving Another Nonlinear Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Jong [12, p. 17, Problem 8]:

Minimize (X(1) ^ 2 - 4 * X(1) + 2 * X(2) ^ 2 - 8 * X(2) + 3 * X(3) ^ 2 - 12 * X(3) - 56) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 2 * X(2) + X(3) + 20) + ((2 * X(1) ^ 2 - 16 * X(1) + X(2) ^ 2 - 8 * X(2) - 2) / (2 * X(1) + 4 * X(2) + 6 * X(3)))

subject to

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

- X(1) - X(2) + X(3)<=4,

X(1)>=1,

X(2)>=1,

X(3)>=1.

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

Whereas line 199 of the preceding paper is 199 X(J99) = INT(X(J99)), here line 199 is 199 REM X(J99) = INT(X(J99)).

 

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 * 50

73 NEXT J44

 

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

191 NEXT IPP

 

196 FOR J99 = 1 TO 3

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

201 IF X(J99) < 1 THEN 1670

204 NEXT J99

205 REM

 

311 X(4) = 10 - X(1) - X(2) - X(3)

 

315 X(5) = 4 + X(1) + X(2) - X(3)
333 FOR J44 = 4 TO 5
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

 

366 POBA = -(X(1) ^ 2 - 4 * X(1) + 2 * X(2) ^ 2 - 8 * X(2) + 3 * X(3) ^ 2 - 12 * X(3) - 56) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 2 * X(2) + X(3) + 20) - ((2 * X(1) ^ 2 - 16 * X(1) + X(2) ^ 2 - 8 * X(2) - 2) / (2 * X(1) + 4 * X(2) + 6 * X(3))) + 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 < -99999 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5)
1912 PRINT M, JJJJ

1999 NEXT JJJJ

 This BASIC computer program was run with qb64v1000-win [33]. The complete output through JJJJ = -31999.98 is shown below:

1.821631971584992          1.0000000000000462          1.0000000000045943
0          0
6.11983426932261          -32000

1.82163192140177             1.0000000000003429           1.0000000000000018
0          0
6.119834269363134          -31999.99

1.821631985088472          1.00000000000000011          1.000000000000444
0          0
6.119834269360302          -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 [33], the wall-clock time for obtaining the output through
JJJJ= -31999.98 was two seconds, not including the time for "Creating .EXE file." One can compare the computational results above to the results in Jong [12, p. 18].

 

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] 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.
[13] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[14] 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.
[15] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[16] 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.
[17] 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.
[18] 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.
[19] 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/
[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] 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…/
[22] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[23] 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.
[24] 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.
[25] 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.
[26] 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.
[27] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[28] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[29] 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
[30] 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.
[31] 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.
[32] 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.
[33] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[34] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[35] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. ht5p://myblogsubstance.typepad.com/substance/2012/04/12/
[36] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[37] 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.
[38] 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.
[39] 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, Nonlinear, and Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve Jong's [12, p. 17] Problem 8 plus the restriction that the three variables are integer variables. Thus the present problem is as follows:

Minimize  (X(1) ^ 2 - 4 * X(1) + 2 * X(2) ^ 2 - 8 * X(2) + 3 * X(3) ^ 2 - 12 * X(3) - 56) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 2 * X(2) + X(3) + 20) + ((2 * X(1) ^ 2 - 16 * X(1) + X(2) ^ 2 - 8 * X(2) - 2) / (2 * X(1) + 4 * X(2) + 6 * X(3)))

subject to

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

- X(1) - X(2) + X(3)<=4,

X(1) >=1,

X(2) >=1,

X(3) >=1,

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

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

One notes line 199, which is 199 X(J99) = INT(X(J99)).

 

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 * 50

73 NEXT J44

 

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

191 NEXT IPP

 

196 FOR J99 = 1 TO 3

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

201 IF X(J99) < 1 THEN 1670

204 NEXT J99

205 REM

 

311 X(4) = 10 - X(1) - X(2) - X(3)

 

315 X(5) = 4 + X(1) + X(2) - X(3)
333 FOR J44 = 4 TO 5
336 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

339 NEXT J44

366 POBA = -(X(1) ^ 2 - 4 * X(1) + 2 * X(2) ^ 2 - 8 * X(2) + 3 * X(3) ^ 2 - 12 * X(3) - 56) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 2 * X(2) + X(3) + 20) - ((2 * X(1) ^ 2 - 16 * X(1) + X(2) ^ 2 - 8 * X(2) - 2) / (2 * X(1) + 4 * X(2) + 6 * X(3))) + 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 < -99999 THEN 1999

1900 PRINT A(1), A(2), A(3), A(4), A(5)
1912 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [34]. The complete output through JJJJ = -31999.98 is shown below:

2       1       1       0       0
6.107142857142857          -32000

2       1       1       0       0
6.107142857142857          -31999.99

1       1       1       0       0
5.81140350877193          -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 [34], the wall-clock time for obtaining the output through JJJJ= -31999.98 was two 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] 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] 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.
[13] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[14] 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.
[15] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[16] 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.
[17] 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.
[18] 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.
[19] 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/
[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] 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…/
[22] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[23] 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.
[24] 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.
[25] 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.
[26] 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.
[27] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[28] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[29] 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
[30] 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.
[31] 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.
[32] 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.
[33] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[34] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[35] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. ht5p://myblogsubstance.typepad.com/substance/2012/04/12/
[36] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[37] 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.
[38] 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.
[39] 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

Similar to the computer in Wong [35], the computer program listed below seeks to solve the following problem from Ishii, Ibaraki, and Mine [12, p. 235].

Maximize (7 * X(1) + 9 * X(2) + 3) / (3 * X(1) + 4 * X(2) + 2)

subject to

X(3) = 6 - 2 * X(1) - 3 * X(2),

X(4) = 5 - 3 * X(1) - 2 * X(2),

X(1), X(2), X(3), X(4) >= 0, integer.

One notes line 199, which is 199 X(J99) = INT(X(J99)).

 

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 2

72 A(J44) = RND * 1000

 

73 NEXT J44

 

128 FOR I = 1 TO 100

 

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 * 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 2

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

201 IF X(J99) < 0 THEN 1670

203 IF X(J99) > 1000 THEN 1670

204 NEXT J99

207 X(3) = 6 - 2 * X(1) - 3 * X(2)
208 IF X(3) < 0 THEN 1670

209 X(4) = 5 - 3 * X(1) - 2 * X(2)
210 IF X(4) < 0 THEN 1670

360 POBA = (7 * X(1) + 9 * X(2) + 3) / (3 * X(1) + 4 * X(2) + 2)

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1452 M = P
1454 FOR KLX = 1 TO 4

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

1670 NEXT I

 

1889 IF M < -99999 THEN 1999

 

1900 PRINT A(1), A(2), A(3), A(4)
1912 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [33]. The complete output through JJJJ =-29794.04000035311 is shown below:

1          1          1          0
2.111111111111111          -31850.64000002391

1          0          4          2
2          -31201.87000012776

1          0          4          2
2          -30692.26000020933

0          2          0          1
2.1          -30652.44000021571

1          1          1          0
2.111111111111111          -29794.04000035311

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 [33], the wall-clock time for obtaining the output through
JJJJ= -29794.04000035311 was 30 seconds, total.

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/~archive/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] Hiroaki Ishii, Toshihide Ibaraki, Hisashi Mine (1976). A primal cutting plane algorithm for Integer Fractional Programming problems. Journal of the Operations Research Society of Japan, Volume 19, No. 3, September 1976, pp. 228-244. WWW.orsj.or.jp/~archive/pdf/e_mag/Vol.19_03_228.pdf.
[13] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[14] 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 (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.
[25] 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.
[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] 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.
[32] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[33] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[34] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. ht5p://myblogsubstance.typepad.com/substance/2012/04/12/
[35] Jsun Yui Wong (2017, September 25). Solving an integer fractional programming problem using the method of this blog. https://computerprogramsandresults.wordpress.com/2017/09/25/solving...
[36] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[37] 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.
[38] 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.
[39] 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, Nonlinear, and Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve Shen, Ma, and Chen's [25, p. 452] Example 3 plus the restriction that the two variables are integer variables. Thus the present problem is as follows:

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

X(3) through X(5) below are slack variables.

One notes line 199, which is 199 X(J99) = INT(X(J99)).

 

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 2

72 A(J44) = 1 + RND * 2

 

73 NEXT J44

 

128 FOR I = 1 TO 100

 

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

 

196 FOR J99 = 1 TO 2

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

201 IF X(J99) < 1 THEN 1670

203 IF X(J99) > 3 THEN 1670
204 NEXT J99

 

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

 

 

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

 

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

 

325 FOR J99 = 3 TO 5

327 XX(J99) = X(J99)

 

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

 

331 NEXT J99

 

364 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 < -1111 THEN 1999

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

1912 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [33]. The complete output through JJJJ = -31999.96000000001 is shown below:

1          1          0          0          0
3.333333333333334          -32000

1 1 0 0 0
3.333333333333334 -31999.99

1 1 0 0 0
3.333333333333334 -31999.98

1 1 0 0 0
3.333333333333334 -31999.97000000001

1 1 0 0 0
3.333333333333334 -31999.96000000001

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 [33], the wall-clock time for obtaining the output through JJJJ= -31999.96000000001 was one or two seconds, not including the time for "Creating the .EXE file."

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 (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.
[25] 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.
[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] 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.
[32] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[33] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[34] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. ht5p://myblogsubstance.typepad.com/substance/2012/04/12/
[35] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[36] 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.
[37] 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.
[38] 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 an Integer Nonlinear Fractional Programming Problem

 Jsun Yui Wong

The computer program listed below seeks to solve Wang and Zhang's [31, p. 329] Example 2 plus the restriction that the two variables are integer variables. Thus the present problem is as follows:

Maximize (-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)

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

X(3) through X(5) below are slack variables.

One notes line 199, which is 199 X(J99) = INT(X(J99)).

 

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 2

72 A(J44) = 1 + RND * 2

 

73 NEXT J44

 

128 FOR I = 1 TO 100

 

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

 

196 FOR J99 = 1 TO 2

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

201 IF X(J99) < 1 THEN 1670

203 IF X(J99) > 3 THEN 1670
204 NEXT J99

 

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

 

 

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

 

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

 

325 FOR J99 = 3 TO 5

327 XX(J99) = X(J99)

 

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

 

331 NEXT J99

 

364 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))

 

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 < 5 THEN 1999

 

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

1912 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with qb64v1000-win [32]. The complete output through JJJJ = -31999.96000000001 is shown below:

1          1          0          0          0
5.516666666666667          -32000

1 1 0 0 0
5.516666666666667 -31999.99

1 1 0 0 0
5.516666666666667 -31999.98

1 1 0 0 0
5.516666666666667 -31999.97000000001

1 1 0 0 0
5.516666666666667 -31999.96000000001

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= -31999.96000000001 was 2 seconds, not including the time for "Creating the .EXE file."

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] Yan-Jun Wang, Ke-Cun Zhang (2004). Global optimization of nonlinear sum of ratios problem. Applied Mathematics and Computation 158 (2004) 319 330.
[32] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[33] 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/
[34] Helen Wu (2015). Geometric Programming. https://optimization.mccormick.northwstern.edu/index.php/Geometric_Programming.
[35] 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.
[36] 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.
[37] 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 Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve Wang and Chu's [30] Example 4 plus the restriction that the three variables are integer variables. Thus the present problem is as follows:

Maximize ((3 * X(1) + 4 * X(2) + 50) / (4 * X(1) + 3 * X(2) + 2 * X(3) + 50)) + ((3 * X(1) + 5 * X(2) + 3 * X(3) + 50) / (3 * X(1) + 4 * X(2) + 5 * 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) >= 0,  X(2)> =0,  X(3) >= 0,

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

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

One notes line 199, which is 199 X(J99) = INT(X(J99)).

 

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 X(J99) = INT(X(J99))

201 IF X(J99) < 0 THEN 1670
203 REM IF X(J99) > 10 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) / (4 * X(1) + 3 * X(2) + 2 * X(3) + 50)) + ((3 * X(1) + 5 * X(2) + 3 * X(3) + 50) / (3 * X(1) + 4 * X(2) + 5 * 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))

 

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 = -3999.96000000001 is shown below:

0 0 0 0 0
10 10
3 -32000

0 0 0 0 0
10 10
3 -31999.99

0 0 0 0 0
10 10
3 -31999.98

0 1 0 0 0
7 7
3.000349406009783 -3999.97000000001

0          3          0          0          0
1          1
3.002460360852925          -3999.96000000001

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= -3999.96000000001 was 2 seconds, not including the time for "Creating the .EXE file."

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 Wang and Chu's [30] Example 6 plus the restriction that the three variables are integer variables. Thus the present problem is as follows:

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

X(4) through X(5) below are slack variables.

One notes line 199, which is 199 X(J99)=INT(X(J99)).
.

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 X(J99)=INT(X(J99))

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 = -31999.90000000002 is shown below:

0 0 0 0 0
10 10
-2 -32000

0 0 0 0 0
10 10
-2 -31999.99

0 0 0 0 0
10 10
-2 -31999.98

0 1 0 0 0
7 7
-1.963636363636364 -31999.97000000001

0          3          0          0          0
1          1
-1.907692307692308          -31999.96000000001

0 0 0 0 0
10 10
-2 -31999.95000000001

0 0 0 0 0
10 10
-2 -331999.94000000001

0 0 0 0 0
10 10
-2 -31999.93000000001

0 0 0 0 0
10 10
-2 -31999.92000000001

0 0 0 0 0
10 10
-2 -31999.91000000001

0 3 0 0 0
1 1
-1.907692307692308 -31999.90000000002

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= -31999.90000000002 was 2 seconds, not including the time for "Creating the .EXE file."

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 an Integer Fractional Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve Wang and Chu's [30] Example 2 plus the restriction that the three variables are integer variables. Thus the present problem is as follows:

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

X(4) through X(7) below are slack variables.

One notes line 199, which is 199 X(J99) = INT(X(J99)).

 

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 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 = -31999.96000000001 is shown below:

1          0          0          0          0
0          0
8          9          5          1
4.081481481481482          -32000

0          0          0          0          0
0          0
10         10          10          10
4          -31999.99

1 0 0 0 0
0 0
8 9 5 1
4.081481481481482 -31999.98

1 0 0 0 0
0 0
8 9 5 1
4.081481481481482 -31999.97000000001

0 0 0 0 0
0 0
10 10 10 10
4 -31999.96000000001

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= -31999.96000000001 was 2 seconds, not including the time for "Creating the .EXE file."

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 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 4]:

Maximize ((3 * X(1) + 4 * X(2) + 50) / (4 * X(1) + 3 * X(2) + 2 * X(3) + 50)) + ((3 * X(1) + 5 * X(2) + 3 * X(3) + 50) / (3 * X(1) + 4 * X(2) + 5 * 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) >= 0, X(2)> =0, X(3) >= 0.

X(4) and 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 -31999 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
203 REM IF X(J99) > 10 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) / (4 * X(1) + 3 * X(2) + 2 * X(3) + 50)) + ((3 * X(1) + 5 * X(2) + 3 * X(3) + 50) / (3 * X(1) + 4 * X(2) + 5 * 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))

 

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:

5.599804986187998D-15          3.333332349810712          3.742777017698164D-15
0          0
2.950567817945802D-06          2.950567776139278D-06
3.002923975208128          -32000

1.01350324964244D-14          3.333330847652206          3.39851779191871D-15
0          0
7.457043310417812D-06          7.457043252569016D-06
3.002923973069783          -3999.99

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 4].

Acknowledgment

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

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