Substance

Solving a Mixed Integer Nonlinear Programming Problem in Process Synthesis, Second Edition

Jsun Yui Wong

 

The computer program listed below seeks to solve the following problem in Li and Tsai [25, p. 12, Example 4]:

Minimize (X(4) - 1) ^ 2 + (X(5) - 2) ^ 2 + (X(6) - 1) ^ 2 - LOG(X(7) + 1) + (X(1) - 1) ^ 2 + (X(2) - 2) ^ 2 + (X(3) - 3) ^ 2

subject to

X(4) + X(5) + X(6) + X(1) + X(2) + X(3)<=5
X(6) ^ 2 + X(1) ^ 2 + X(2) ^ 2 + X(3) ^ 2<=5.5

X(4) + X(1)<=1.2
X(5) + X(2)<=1.8
X(6) + X(3)<=2.5
X(7) + X(1)<=1.2
X(5) ^ 2 + X(2) ^ 2<=1.64
X(6) ^ 2 + X(3) ^ 2<=4.25
X(5) ^ 2 + X(3) ^ 2<=4.64

0<=X(1) <=1.2

0<=X(2) <=1.8

0<=X(3) <=2.5

X(4) through X(7) are binary integer variables.

X(8) through X(16) below are slack variables added.

For the purpose of getting some domino effect, the computer program here tries to find an optimal value for X(10) of line 321, which is 321 X(10) = 1.2 - X(4) - X(1), earlier than the computer program of the first edtion. When one compares X(10) to X(8), to find an optimal X(10) is relatively easy because only optimal X(1) and optimal X(4) are needed.

 

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 -31999.96 STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37

 

77 A(1) = RND * 1.2
78 A(2) = RND * 1.8
79 A(3) = RND * 2.5
111 A(4) = FIX(RND * 2)

112 A(5) = FIX(RND * 2)

113 A(6) = FIX(RND * 2)
114 A(7) = FIX(RND * 2)

128 FOR I = 1 TO 1500

 

129 FOR KKQQ = 1 TO 7

 

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

181 j = 1 + FIX(RND * 7)

183 IF j < 4 THEN GOTO 184 ELSE GOTO 197

 

184 r = (1 - RND * 2) * A(j)
185 X(j) = A(j) + (RND ^ (RND * 10)) * r

 

191 GOTO 222

 

197 IF A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0

 

222 NEXT IPP

268 IF X(1) < 0## THEN 1670

 

269 IF X(1) > 1.2## THEN 1670
278 IF X(2) < 0## THEN 1670
289 IF X(2) > 1.8## THEN 1670
291 IF X(3) < 0## THEN 1670
293 IF X(3) > 2.5## THEN 1670

 

311 REM X(8) = 5 - X(4) - X(5) - X(6) - X(1) - X(2) - X(3)
313 REM X(9) = 5.5 - X(6) ^ 2 - X(1) ^ 2 - X(2) ^ 2 - X(3) ^ 2

 

321 X(10) = 1.2 - X(4) - X(1)

322 X(13) = 1.2 - X(7) - X(1)

323 X(11) = 1.8 - X(5) - X(2)
325 X(12) = 2.5 - X(6) - X(3)
327 REM X(13) = 1.2 - X(7) - X(1)
329 X(14) = 1.64 - X(5) ^ 2 - X(2) ^ 2
331 X(15) = 4.25 - X(6) ^ 2 - X(3) ^ 2
333 X(16) = 4.64 - X(5) ^ 2 - X(3) ^ 2

 

353 X(9) = 5.5 - X(6) ^ 2 - X(1) ^ 2 - X(2) ^ 2 - X(3) ^ 2
355 X(8) = 5 - X(4) - X(5) - X(6) - X(1) - X(2) - X(3)

 

401 FOR J47 = 8 TO 16

404 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

409 NEXT J47

 

443 POBA = -(X(4) - 1) ^ 2 - (X(5) - 2) ^ 2 - (X(6) - 1) ^ 2 + LOG(X(7) + 1) - (X(1) - 1) ^ 2 - (X(2) - 2) ^ 2 - (X(3) - 3) ^ 2 + 1000000 * (X(8) + X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15) + X(16))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 16

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

1670 NEXT I
1889 IF M < -5 THEN 1999

 

1907 PRINT A(1), A(2), A(3)

1908 PRINT A(4), A(5), A(6), A(7), A(8)
1909 PRINT A(9), A(10), A(11), A(12)
1910 PRINT A(13), A(14), A(15), A(16), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [51]. The complete output through JJJJ = -31999.9600000001 is shown below:

.1999999999978652          .7999999999534856          1.907878401288323
1           1          0          1          0
0          0          0          0
0          0          0          0          -4.579582405927654
-32000

.1999999999661386          .7999999978912403          1.907878399774086
1          1          0          1          0
0         0          0          0
0         0          0          0          -4.579582414235267
-31999.99

.199999999924534          .799999999914449          1.907878401599777
1           1          0          1          0
0          0          0          0
0          0          0          0          -4.579582405458381
-31999.98

.1999999992970735          .7999999996643608          1.907878402594483
1           1          0          1          0
0          0          0          0
0          0          0          0         -4.57958240488985
-31999.97000000001

.1999999998539791          .7999999999443966          1.907878395902781
1          1           0          1          0
0         0           0          0
0         0           0          0          -4.57958241794302
-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 [51], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =-31999.96000000001 was 2 or 3 seconds, not including the time for “Creating .EXE file” (10 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Table 2 of Li and Tsai [25, p. 12].

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] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] 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.
[11] 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.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] E. L. Hannan (1981), Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235-246.

[14] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[15] James P. Ignizio, Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ, 1982.

[16] James P. Ignizio, Stephen C. Daniels (1983). Fuzzy multicriteria integer programming via fuzzy generalized networks. Fuzzy Sets and Systems 10 (1983) 261-270.

[17] James P. Ignizio, Tom M. Cavalier, Linear Programming. Prentice-Hall, Englewood Cliffs, NJ, 1994.

[18] 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.

[19] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

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

[21] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

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

[23] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[24] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[25] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization, 33: 1-13 (2005).
[26] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[27] 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.

[28] Han-Lin Li, Hao-Chun Lu (2009). Global optimization for generalized geometric progams with mixed free-sign variables. Operations Research 57 (3): 701-713 (2009).

[29] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[30] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[31] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[32] 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.
[33] 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.
[34] 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.

[35] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[36] 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/
[37] 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.

[38] Katta Murty, Operations Research: Deterministic Optimization Models. Prentice-Hall, 1995.

[39] 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…/
[40] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[41] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[42] 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.
[43] 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.

[44] Vikas Sharma (2012), Multobjective integer nonlinear fractional programming problem: A cutting plane approach. OPSEARCH of the Operational Research Society of India (April-June 2012) 49 (2) 133-153.

[45] 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.
[46] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[47] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[48] 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
[49] 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.
[50] 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.
[51] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[52] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[53] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Using Discrete Variables To Approximate Continuous Variables, Second Edition

Jsun Yui Wong

 

The computer program listed below seeks to solve the following problem in Li and Tsai [25, p. 11]:

Minimize X(1) ^ -2 * X(2) ^ -.5 * X(3) ^ -1 + 8 * X(1) ^ -1 * X(4) ^ 2 - 8 * X(4)

subject to

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

2 * X(1) + X(2) - X(3) + X(4)<=6

1<= X(1) <=5

3<= X(2) <=7

1<= X(3) <=10

1<= X(4) <=5

X(5) and X(6) below are slack variables added.

Whereas line 128 of the earlier edition is 128 FOR I = 1 TO 3000, here line 128 is 128 FOR I = 1 TO 1000.

 

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO 32000 STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37

 

71 unitsize(1) = .001

73 unitsize(2) = .001

75 unitsize(3) = .001

77 unitsize(4) = .001

 

81 A(1) = 1 + FIX(RND * 1000) * unitsize(1)

83 A(2) = 3 + FIX(RND * 1000) * unitsize(2)

85 A(3) = 1 + FIX(RND * 1000) * unitsize(3)
87 A(4) = 1 + FIX(RND * 1000) * unitsize(4)

 

128 FOR I = 1 TO 1000

 

129 FOR KKQQ = 1 TO 4

 

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

 

181 j = 1 + FIX(RND * 4)

 

187 IF RND < .5 THEN X(j) = A(j) - FIX(RND * 10) * unitsize(j) ELSE X(j) = A(j) + FIX(RND * 10) * unitsize(j)
222 NEXT IPP

 

268 IF X(1) < 1## THEN 1670
269 IF X(1) > 5## THEN 1670
278 IF X(2) < 3## THEN 1670
289 IF X(2) > 7## THEN 1670
291 IF X(3) < 1## THEN 1670
293 IF X(3) > 10## THEN 1670
295 IF X(4) < 1## THEN 1670
297 IF X(4) > 5## THEN 1670

 

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

400 X(6) = 6 - 2 * X(1) - X(2) + X(3) - X(4)
401 FOR J47 = 5 TO 6

402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

 

403 NEXT J47

443 POBA = -X(1) ^ -2 * X(2) ^ -.5 * X(3) ^ -1 - 8 * X(1) ^ -1 * X(4) ^ 2 + 8 * X(4) + 1000000 * (X(5) + X(6))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 6

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

1670 NEXT I
1889 IF M < 9.997 THEN 1999

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [51]. The complete output through JJJJ = -31999.98 is shown below:

4.999999999999981          3.500000000000002          9.999999999999957
2.499999999999951          0          0          9.997861910064664
-32000

 4.999999999999974          3.501000000000004          9.999999999999972
2.499999999999954          0          0          9.997860615440633
-31999.99

 4.999999999999973          3.50000000000002          9.999999999999943
2.499999999999957          0          0          9.997861910064646
-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 [51], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31999.98 was 2 seconds, not including the time for “Creating .EXE file” (9 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Table 1 of Li and Tsai [25, p. 12].

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] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] 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.
[11] 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.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] E. L. Hannan (1981), Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235-246.

[14] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[15] James P. Ignizio, Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ, 1982.

[16] James P. Ignizio, Stephen C. Daniels (1983). Fuzzy multicriteria integer programming via fuzzy generalized networks. Fuzzy Sets and Systems 10 (1983) 261-270.

[17] James P. Ignizio, Tom M. Cavalier, Linear Programming. Prentice-Hall, Englewood Cliffs, NJ, 1994.

[18] 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.

[19] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

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

[21] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

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

[23] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[24] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[25] Han-Lin Li, Jung-Fa Tsai (2005). Treating free variables in generalized geometric global optimization programs. Journal of Global Optimization, 33: 1-13 (2005).

[26] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[27] 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.

[28] Han-Lin Li, Hao-Chun Lu (2009). Global optimization for generalized geometric progams with mixed free-sign variables. Operations Research 57 (3): 701-713 (2009).

[29] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[30] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[31] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[32] 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.
[33] 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.
[34] 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.

[35] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[36] 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/
[37] 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.

[38] Katta Murty, Operations Research: Deterministic Optimization Models. Prentice-Hall, 1995.

[39] 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…/
[40] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[41] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[42] 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.
[43] 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.

[44] Vikas Sharma (2012), Multobjective integer nonlinear fractional programming problem: A cutting plane approach. OPSEARCH of the Operational Research Society of India (April-June 2012) 49 (2) 133-153.

[45] 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.
[46] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[47] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[48] 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
[49] 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.
[50] 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.
[51] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[52] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[53] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving Another Instance of the Same Generalized Geometric Programming Problem

Jsun Yui Wong

 

The computer program listed below seeks to solve the following instance of Li and Lu's Example 3 [27, p. 710]:

Minimize X(1) ^ (-2) * X(2) ^ .5 * X(3) ^ 1.2 - X(1) ^ (-2) * X(2) ^ .5 - X(2) ^ (.5) * X(3) ^ 1.2 - X(1) ^ (-2) * X(3) ^ 1.2

subject to

X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2>=3

X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2<=6

1<=X(i)<=4, i=1, 2, 3

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

See Li and Lu [27, p. 710] for a better description of the present problem.

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

 

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
40 FOR J40 = 1 TO 3

 

41 A(J40) = 1 + FIX(RND * 3)
42 NEXT J40

 

128 FOR I = 1 TO 3000

 

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)

201 r = (1 - RND * 2) * A(j)
205 X(j) = A(j) + (RND ^ (RND * 10)) * r

222 NEXT IPP

255 FOR J41 = 1 TO 3

258 X(J41) = INT(X(J41))

262 NEXT J41

 

265 FOR J41 = 1 TO 3

268 IF X(J41) < 1## THEN 1670
269 IF X(J41) > 4## THEN 1670
272 NEXT J41

308 X(4) = -3 + X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2

 

310 X(5) = 6 - X(1) ^ (-2) * X(2) ^ .5 - X(2) ^ (.5) * X(3) ^ 1.2 - X(1) ^ (-2) * X(3) ^ 1.2

 

401 FOR J47 = 4 TO 5

402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

 

403 NEXT J47

443 POBA = -X(1) ^ (-2) * X(2) ^ .5 * X(3) ^ 1.2 + X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2 + 1000000 * (X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 5

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

1670 NEXT I
1889 IF M < 5 THEN 1999

 

1907 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 [50]. The complete output through JJJJ = -31999.9100000001 is shown below:

3           1          4          0          0
5.389142754202688          -32000

 3          1          4          0          0
5.389142754202688          -31999.94000000001

 3          1          4          0          0
5.389142754202688          -31999.91000000001

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 [50], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31999.91000000001 was 1 or 2 seconds, not including the time for “Creating .EXE file” (9 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Table 3 of Li and Lu [27, p. 711, Experiment 1].

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] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] 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.
[11] 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.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] E. L. Hannan (1981), Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235-246.

[14] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[15] James P. Ignizio, Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ, 1982.

[16] James P. Ignizio, Stephen C. Daniels (1983). Fuzzy multicriteria integer programming via fuzzy generalized networks. Fuzzy Sets and Systems 10 (1983) 261-270.

[17] James P. Ignizio, Tom M. Cavalier, Linear Programming. Prentice-Hall, Englewood Cliffs, NJ, 1994.

[18] 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.

[19] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

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

[21] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

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

[23] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[24] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[25] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] 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.

[27] Han-Lin Li, Hao-Chun Lu (2009). Global optimization for generalized geometric progams with mixed free-sign variables. Operations Research 57 (3): 701-713 (2009).

[28] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[29] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[30] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[31] 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.
[32] 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.
[33] 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.

[34] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[35] 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/
[36] 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.

[37] Katta Murty, Operations Research: Deterministic Optimization Models. Prentice-Hall, 1995.

[38] 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…/
[39] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[40] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[41] 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.
[42] 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.

[43] Vikas Sharma (2012), Multobjective integer nonlinear fractional programming problem: A cutting plane approach. OPSEARCH of the Operational Research Society of India (April-June 2012) 49 (2) 133-153.

[44] 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.
[45] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[46] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[47] 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
[48] 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.
[49] 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.
[50] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[51] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[52] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving Another Generalized Geometric Programming Problem

Jsun Yui Wong

 

The computer program listed below seeks to solve the following instance of Li and Lu's Example 3 [27, p. 710]:

Minimize X(1) ^ (-2) * X(2) ^ .5 * X(3) ^ 1.2 - X(1) ^ (-2) * X(2) ^ .5 - X(2) ^ (.5) * X(3) ^ 1.2 - X(1) ^ (-2) * X(3) ^ 1.2

subject to

X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2>=4

X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2<=8

1<=X(i)<=8, i=1, 2, 3

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

See Li and Lu [27, p. 710] for a better description of the present problem.

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

 

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
40 FOR J40 = 1 TO 3

 

41 A(J40) = 1 + FIX(RND * 3)
42 NEXT J40

 

128 FOR I = 1 TO 3000

 

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)

201 r = (1 - RND * 2) * A(j)
205 X(j) = A(j) + (RND ^ (RND * 10)) * r

 

222 NEXT IPP

255 FOR J41 = 1 TO 3

258 X(J41) = INT(X(J41))

262 NEXT J41

 

265 FOR J41 = 1 TO 3

268 IF X(J41) < 1## THEN 1670
269 IF X(J41) > 8## THEN 1670

272 NEXT J41

308 X(4) = -4 + X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2

 

310 X(5) = 8 - X(1) ^ (-2) * X(2) ^ .5 - X(2) ^ (.5) * X(3) ^ 1.2 - X(1) ^ (-2) * X(3) ^ 1.2

401 FOR J47 = 4 TO 5

402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

 

403 NEXT J47

443 POBA = -X(1) ^ (-2) * X(2) ^ .5 * X(3) ^ 1.2 + X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2 + 1000000 * (X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 5

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

1670 NEXT I
1889 IF M < 7.4 THEN 1999

1907 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 [50]. The complete output through JJJJ = -31999.4300000001 is shown below:

8          2          4          0          0
7.452201139687744          -31999.95000000001

8          2          4          0          0
7.452201139687744          -31999.4600000001

8          2         4          0          0
7.452201139687744          -31999.4300000001

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 [50], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31999.4300000001 was 3 seconds, not including the time for “Creating .EXE file” (10 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Table 3 of Li and Lu [27, p. 711, Experiment 2].

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.

[6] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] 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.
[11] 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.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] E. L. Hannan (1981), Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235-246.

[14] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[15] James P. Ignizio, Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ, 1982.

[16] James P. Ignizio, Stephen C. Daniels (1983). Fuzzy multicriteria integer programming via fuzzy generalized networks. Fuzzy Sets and Systems 10 (1983) 261-270.

[17] James P. Ignizio, Tom M. Cavalier, Linear Programming. Prentice-Hall, Englewood Cliffs, NJ, 1994.

[18] 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.

[19] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

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

[21] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

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

[23] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[24] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[25] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] 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.

[27] Han-Lin Li, Hao-Chun Lu (2009). Global optimization for generalized geometric progams with mixed free-sign variables. Operations Research 57 (3): 701-713 (2009).

[28] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[29] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[30] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[31] 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.
[32] 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.
[33] 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.

[34] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[35] 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/
[36] 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.

[37] Katta Murty, Operations Research: Deterministic Optimization Models. Prentice-Hall, 1995.

[38] 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…/
[39] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[40] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[41] 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.
[42] 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.

[43] Vikas Sharma (2012), Multobjective integer nonlinear fractional programming problem: A cutting plane approach. OPSEARCH of the Operational Research Society of India (April-June 2012) 49 (2) 133-153.

[44] 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.
[45] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[46] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[47] 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
[48] 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.
[49] 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.
[50] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[51] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[52] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving a Generalized Geometric Programming Problem Involving Mixed Free-Sign Variables, Second Edition

Jsun Yui Wong

 

The computer program listed below seeks to solve Lin and Lu's Example 2 [27, p. 710], which is shown immediately below:

Minimize X(0) ^ 3 * X(1) * X(2) ^ 3 * X(3) + X(0) ^ 3 * X(1) * X(2) * X(3) ^ 2

subject to

X(0) ^ 3 * X(1) * X(2) ^ 2 + X(2) * X(3)<=-500

- X(0) ^ 3 * X(1) * X(2) + X(2) ^ 2 * X(3)<=500

-5<=X(0) <=5

-5<=X(1) <=5

X(2) epsilon {-1, 0, 1, 4, 5, 6, 7.5, 8, 9, 10}

X(3) epsilon { -27, -18, -9, -7, -4, -1, 1, 3, 4, 5}.

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

 

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

 

40 A(0) = -5 + RND * 10

 

41 A(1) = -5 + RND * 10

 

43 IF RND < .5 THEN A(2) = 0 ELSE A(2) = 9

 

 

46 IF RND < .5 THEN A(3) = -9 ELSE A(3) = 3

 

 

128 FOR I = 1 TO 3000

 

129 FOR KKQQ = 0 TO 3

 

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

 

181 j = FIX(RND * 4)

200 IF j = 0 THEN GOTO 201 ELSE IF j = 1 THEN GOTO 204 ELSE IF j = 2 THEN GOTO 208 ELSE GOTO 210
201 r = (1 - RND * 2) * A(0)
202 X(0) = A(0) + (RND ^ (RND * 10)) * r
203 GOTO 222

204 r = (1 - RND * 2) * A(1)
205 X(1) = A(1) + (RND ^ (RND * 10)) * r

207 GOTO 222

208 IF RND < .1 THEN X(j) = -1 ELSE IF RND < 1 / 9 THEN X(j) = 0 ELSE IF RND < 1 / 8 THEN X(j) = 1 ELSE IF RND < 1 / 7 THEN X(j) = 4 ELSE IF RND < 1 / 6 THEN X(j) = 5 ELSE IF RND < 1 / 5 THEN X(j) = 6 ELSE IF RND < 1 / 4 THEN X(j) = 7.5 ELSE IF RND < 1 / 3 THEN X(j) = 8 ELSE IF RND < 1 / 2 THEN X(j) = 9 ELSE X(j) = 10

209 GOTO 222

 

210 IF RND < .1 THEN X(j) = -27 ELSE IF RND < 1 / 9 THEN X(j) = -18 ELSE IF RND < 1 / 8 THEN X(j) = -9 ELSE IF RND < 1 / 7 THEN X(j) = -7 ELSE IF RND < 1 / 6 THEN X(j) = -4 ELSE IF RND < 1 / 5 THEN X(j) = -1 ELSE IF RND < 1 / 4 THEN X(j) = 1 ELSE IF RND < 1 / 3 THEN X(j) = 3 ELSE IF RND < 1 / 2 THEN X(j) = 4 ELSE X(j) = 5

 

222 NEXT IPP

258 IF X(0) < -5## THEN 1670
259 IF X(0) > 5## THEN 1670

 

268 IF X(1) < -5## THEN 1670
269 IF X(1) > 5## THEN 1670

 

308 X(4) = -500 - X(0) ^ 3 * X(1) * X(2) ^ 2 - X(2) * X(3)

309 X(5) = 500 + X(0) ^ 3 * X(1) * X(2) - X(2) ^ 2 * X(3)

 

401 FOR J47 = 4 TO 5

402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

 

403 NEXT J47

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

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 0 TO 5

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

1670 NEXT I
1889 IF M < 360000 THEN 1999

1907 PRINT A(0), A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [50]. The complete output through JJJJ = -31999.76000000004 is shown below:

-4.879113209656626          4.537200067061007          1
-27          0          0          369953.9999999876
-31999.89000000002

4.997259945889812          -4.222938844609257          1
-27          0          0          369953.9999998124
-31999.76000000004

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 [50], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =-31999.76000000004 was 3 seconds, not including the time for “Creating .EXE file” (11 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on page 710 of Li and Lu [27].

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] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] 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.
[11] 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.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] E. L. Hannan (1981), Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235-246.

[14] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[15] James P. Ignizio, Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ, 1982.

[16] James P. Ignizio, Stephen C. Daniels (1983). Fuzzy multicriteria integer programming via fuzzy generalized networks. Fuzzy Sets and Systems 10 (1983) 261-270.

[17] James P. Ignizio, Tom M. Cavalier, Linear Programming. Prentice-Hall, Englewood Cliffs, NJ, 1994.

[18] 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.

[19] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

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

[21] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

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

[23] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[24] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[25] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] 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.

[27] Han-Lin Li, Hao-Chun Lu (2009). Global optimization for generalized geometric progams with mixed free-sign variables. Operations Research 57 (3): 701-713 (2009).

[28] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[29] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[30] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[31] 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.
[32] 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.
[33] 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.

[34] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[35] 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/
[36] 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.

[37] Katta Murty, Operations Research: Deterministic Optimization Models. Prentice-Hall, 1995.

[38] 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…/
[39] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[40] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[41] 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.
[42] 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.

[43] Vikas Sharma (2012), Multobjective integer nonlinear fractional programming problem: A cutting plane approach. OPSEARCH of the Operational Research Society of India (April-June 2012) 49 (2) 133-153.

[44] 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.
[45] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[46] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[47] 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
[48] 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.
[49] 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.
[50] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[51] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[52] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving a Generalized Geometric Programming Problem Involving Mixed Free-Sign Variables

Jsun Yui Wong

 

The computer program listed below seeks to solve Li and Lu's Example 1 [27, p. 709], which is shown immediately below:

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

subject to

X(1) * X(2) ^ 2 + X(2) * X(3)<=-500

- X(1) * X(2) + X(2) ^ 2 * X(3)<=500

-5<=X(1)<=5

X(2) epsilon {-1, 0, 1, 4, 5, 6, 7.5, 8, 9, 10}

X(3) epsilon { -27, -18, -9, -7, -4, -1, 1, 3, 4, 5}.

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

 

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

 

41 A(1) = -5 + RND * 10

 

43 IF RND < .1 THEN A(2) = -1 ELSE IF RND < 1 / 9 THEN A(2) = 0 ELSE IF RND < 1 / 8 THEN A(2) = 1 ELSE IF RND < 1 / 7 THEN A(2) = 4 ELSE IF RND < 1 / 6 THEN A(2) = 5 ELSE IF RND < 1 / 5 THEN A(2) = 6 ELSE IF RND < 1 / 4 THEN A(2) = 7.5 ELSE IF RND < 1 / 3 THEN A(2) = 8 ELSE IF RND < 1 / 2 THEN A(2) = 9 ELSE A(2) = 10

 

46 IF RND < .1 THEN A(3) = -27 ELSE IF RND < 1 / 9 THEN A(3) = -18 ELSE IF RND < 1 / 8 THEN A(3) = -9 ELSE IF RND < 1 / 7 THEN A(3) = -7 ELSE IF RND < 1 / 6 THEN A(3) = -4 ELSE IF RND < 1 / 5 THEN A(3) = -1 ELSE IF RND < 1 / 4 THEN A(3) = 1 ELSE IF RND < 1 / 3 THEN A(3) = 3 ELSE IF RND < 1 / 2 THEN A(3) = 4 ELSE A(3) = 5

 

128 FOR I = 1 TO 3000

 

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)

200 IF j = 1 THEN GOTO 201 ELSE IF j = 2 THEN GOTO 208 ELSE GOTO 210

201 r = (1 - RND * 2) * A(1)
205 X(1) = A(1) + (RND ^ (RND * 10)) * r

207 GOTO 222

208 IF RND < .1 THEN X(j) = -1 ELSE IF RND < 1 / 9 THEN X(j) = 0 ELSE IF RND < 1 / 8 THEN X(j) = 1 ELSE IF RND < 1 / 7 THEN X(j) = 4 ELSE IF RND < 1 / 6 THEN X(j) = 5 ELSE IF RND < 1 / 5 THEN X(j) = 6 ELSE IF RND < 1 / 4 THEN X(j) = 7.5 ELSE IF RND < 1 / 3 THEN X(j) = 8 ELSE IF RND < 1 / 2 THEN X(j) = 9 ELSE X(j) = 10

209 GOTO 222

 

210 IF RND < .1 THEN X(j) = -27 ELSE IF RND < 1 / 9 THEN X(j) = -18 ELSE IF RND < 1 / 8 THEN X(j) = -9 ELSE IF RND < 1 / 7 THEN X(j) = -7 ELSE IF RND < 1 / 6 THEN X(j) = -4 ELSE IF RND < 1 / 5 THEN X(j) = -1 ELSE IF RND < 1 / 4 THEN X(j) = 1 ELSE IF RND < 1 / 3 THEN X(j) = 3 ELSE IF RND < 1 / 2 THEN X(j) = 4 ELSE X(j) = 5

 

222 NEXT IPP

 

268 IF X(1) < -5## THEN 1670
269 IF X(1) > 5## THEN 1670

 

308 X(4) = -500 - X(1) * X(2) ^ 2 - X(2) * X(3)

309 X(5) = 500 + X(1) * X(2) - X(2) ^ 2 * X(3)

 

401 FOR J47 = 4 TO 5

402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

 

403 NEXT J47

443 POBA = -X(1) * X(2) ^ 3 * X(3) - X(1) * X(2) * X(3) ^ 2 + 1000000 * (X(4) + X(5))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 5

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

1670 NEXT I
1889 IF M < -69999988 THEN 1999

 1907 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 [50]. The complete output through JJJJ = -31999.97000000001 is shown below:

-4.90000000000009          10          -1          0
0          -4851.00000000009          -32000

-4.900000000000031          10          -1          0
0          -4851.000000000031          -31999.98

-4.900000000000367          10          -1          0
0          -4851.000000000364          -31999.97000000001

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [50], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31999.97000000001 was 3 seconds, not including the time for “Creating .EXE file” (10 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on page 710 of Li and Lu [27].

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] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] 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.
[11] 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.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] E. L. Hannan (1981), Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235-246.

[14] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[15] James P. Ignizio, Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ, 1982.

[16] James P. Ignizio, Stephen C. Daniels (1983). Fuzzy multicriteria integer programming via fuzzy generalized networks. Fuzzy Sets and Systems 10 (1983) 261-270.

[17] James P. Ignizio, Tom M. Cavalier, Linear Programming. Prentice-Hall, Englewood Cliffs, NJ, 1994.

[18] 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.

[19] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

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

[21] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

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

[23] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[24] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[25] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] 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.

[27] Han-Lin Li, Hao-Chun Lu (2009). Global optimization for generalized geometric progams with mixed free-sign variables. Operations Research 57 (3): 701-713 (2009).

[28] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[29] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[30] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[31] 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.
[32] 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.
[33] 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.

[34] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[35] 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/
[36] 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.

[37] Katta Murty, Operations Research: Deterministic Optimization Models. Prentice-Hall, 1995.

[38] 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…/
[39] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[40] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[41] 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.
[42] 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.

[43] Vikas Sharma (2012), Multobjective integer nonlinear fractional programming problem: A cutting plane approach. OPSEARCH of the Operational Research Society of India (April-June 2012) 49 (2) 133-153.

[44] 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.
[45] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[46] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[47] 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
[48] 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.
[49] 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.
[50] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[51] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[52] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving a Free-Sign Pure Discrete Signomial Programming Problem--Lu's Example 3 with r= 3000

Jsun Yui Wong

The computer program listed below is about extending and solving Lu's Example 3 for r =3000 discrete values, Lu [32, p. 113 and Table 6]. This problem is shown immediately below:

Minimize 2 * X(1) ^ (.9) * X(2) ^ (-1.5) * X(3) ^ (-3) + 5 * X(4) ^ (-.3) * X(5) ^ 2.6 + 4.7 * X(6) ^ (-1.8) * X(7) ^ (-.5) * X(8)

subject to

7.2 * X(1) ^ (-3.8) * X(2) ^ 2.2 * X(3) ^ (4.3) + .5 * X(4) ^ (-.7) * X(5) ^ (-1.6) + .2 * X(6) ^ (4.3) * X(7) ^ (-1.9) * X(8) ^ (3.5)<=695479

10 * X(1) ^ (2.3) * X(2) ^ 1.7 * X(3) ^ (4.5)<=1963735

.6 * X(4) ^ 2.1 * X(5) ^ (-.4) <=2.84

6.2 * X(6) ^ (4.5) * X(7) ^ (-2.7) * X(8) ^ (-.6) <=13105

3.1 * X(1) ^ (1.6) * X(2) ^ .4 * X(3) ^ (-3.8) <=131

3.7 * X(4) ^ 5.4 * X(5) ^ (-1.3) <=9068

.3 * X(6) ^ (-1.1) * X(7) ^ 3.2 * X(8) ^ (-5.6) <=730

.5<= X(i) <=5.5, i=1 through 8

where X(4) and X(5) are continuous variables, and the other six variables are discrete variables; here the number of discrete values is 3000.

X(9) through X(15) below are slack variables added. One notes line 268, which is 268 IF X(J43) < .5## THEN 1670.

 

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO 32009 STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37

 

25 FOR J41 = 1 TO 8

 

31 unitsize(J41) = 5 / 2999

 

 

32 NEXT J41
55 FOR J42 = 1 TO 8

61 A(J42) = .5 + FIX(RND * 3000) * unitsize(J42)

62 NEXT J42

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 8

 

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

 

181 j = 1 + FIX(RND * 8)

 

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

 

197 IF RND < .5 THEN X(j) = A(j) - FIX(RND * 11) * unitsize(j) ELSE X(j) = A(j) + FIX(RND * 11) * unitsize(j)

199 IF RND < .5 THEN GOTO 201 ELSE GOTO 211

201 r = (1 - RND * 2) * A(4)
203 X(4) = A(4) + (RND ^ (RND * 10)) * r

205 GOTO 222
211 r = (1 - RND * 2) * A(5)
215 X(5) = A(5) + (RND ^ (RND * 10)) * r

 

222 NEXT IPP

 

264 FOR J43 = 1 TO 8
268 IF X(J43) < .5## THEN 1670
269 IF X(J43) > 5.5## THEN 1670

270 NEXT J43

307 X(9) = 2.84 - .6 * X(4) ^ 2.1 * X(5) ^ (-.4)

 

309 X(10) = 9068 - 3.7 * X(4) ^ 5.4 * X(5) ^ (-1.3)

310 X(11) = 730 - .3 * X(6) ^ (-1.1) * X(7) ^ 3.2 * X(8) ^ (-5.6)

 

311 X(12) = 13105 - 6.2 * X(6) ^ (4.5) * X(7) ^ (-2.7) * X(8) ^ (-.6)

 

321 X(13) = 131 - 3.1 * X(1) ^ (1.6) * X(2) ^ .4 * X(3) ^ (-3.8)

324 X(14) = 1963735 - 10 * X(1) ^ (2.3) * X(2) ^ 1.7 * X(3) ^ (4.5)

 

326 X(15) = 695479 - 7.2 * X(1) ^ (-3.8) * X(2) ^ 2.2 * X(3) ^ (4.3) - .5 * X(4) ^ (-.7) * X(5) ^ (-1.6) - .2 * X(6) ^ (4.3) * X(7) ^ (-1.9) * X(8) ^ (3.5)

401 FOR J47 = 9 TO 15

402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

 

433 NEXT J47
445 POBA = -2 * X(1) ^ (.9) * X(2) ^ (-1.5) * X(3) ^ (-3) - 5 * X(4) ^ (-.3) * X(5) ^ 2.6 - 4.7 * X(6) ^ (-1.8) * X(7) ^ (-.5) * X(8) + 1000000 * (X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15))

466 P = POBA

1111 IF P <= M THEN 1670

1450 M = P
1454 FOR KLX = 1 TO 15

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

1670 NEXT I
1889 IF M < -.73456661 THEN 1999

 

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6)

1908 PRINT A(7), A(8), A(9), A(10), A(11), A(12)

1909 PRINT A(13), A(14), A(15), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [49]. The complete output through JJJJ = -31991.76000000132 is shown below:

.9018006002000623          5.499999999999997             5.499999999999996
1.837253532804801            .5000000000224908          5.5
5.499999999999998           .5000000000000001          0
0          0          0
0          0          0                  -.7345666053491302
-31996.57000000055

.9018006002000616         5.499999999999996          5.499999999999998
1.837253532789092           .500000000000001         5.499999999999997
5.499999999999996          .5         0          0
0          0
0          0          0                 -.7345666052705347
-31991.76000000132

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 [49], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =-31991.76000000132 was 820 seconds, total, including the time for “Creating .EXE file.” One can compare the computational results above with those on page 117 of Lu [32, Table 6].

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.

[6] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] 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.
[11] 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.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] E. L. Hannan (1981), Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235-246.

[14] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[15] James P. Ignizio, Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ, 1982.

[16] James P. Ignizio, Stephen C. Daniels (1983). Fuzzy multicriteria integer programming via fuzzy generalized networks. Fuzzy Sets and Systems 10 (1983) 261-270.

[17] James P. Ignizio, Tom M. Cavalier, Linear Programming. Prentice-Hall, Englewood Cliffs, NJ, 1994.

[18] 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.

[19] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

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

[21] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

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

[23] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[24] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[25] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] 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.
[27] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[28] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[29] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[30] 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.
[31] 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.
[32] 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.

[33] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[34] 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/
[35] 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.

[36] Katta Murty, Operations Research: Deterministic Optimization Models. Prentice-Hall, 1995.

[37] 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…/
[38] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[39] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[40] 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.
[41] 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.

[42] Vikas Sharma (2012), Multobjective integer nonlinear fractional programming problem: A cutting plane approach. OPSEARCH of the Operational Research Society of India (April-June 2012) 49 (2) 133-153.

[43] 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.
[44] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[45] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[46] 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
[47] 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.
[48] 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.
[49] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[50] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[51] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving a Free-Sign Pure Discrete Signomial Programming Problem--Lu's Example 2 with r= 1384

Jsun Yui Wong

The computer program listed below is about extending and solving Lu's Example 2 for r =1384 discrete values [32, p. 111 and Table 5], which is shown immediately below:

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

subject to

X(1) ^ 3 * X(2) * X(3) ^ 2 + X(3) * X(4)<=-500

-X(1) ^ 3 * X(2) * X(3) + X(3) ^ 2 * X(4)<=500

-6<=X(1) <=6.75

-6<=X(2) <=6.75

-1<= X(3) <=9.2

-9<=X(4) <=6.3

where X(1) through X(4) are free-sign discrete variables; here the number of discrete values is 1384 for each discrete variable, [32, p. 114, Table 5].

X(5) and X(6) below are slack variables added. One notes line 268, which is 268 IF X(1) < -6## THEN 1670.

 

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO 32111 STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37
31 unitsize(1) = 12.75 / 1383

33 unitsize(2) = 12.75 / 1383

35 unitsize(3) = 10.2 / 1383

 

37 unitsize(4) = 15.3 / 1383

 

61 A(1) = -6 + FIX(RND * 1384) * unitsize(1)

63 A(2) = -6 + FIX(RND * 1384) * unitsize(2)

65 A(3) = -1 + FIX(RND * 1384) * unitsize(3)

 

67 A(4) = -9 + FIX(RND * 1384) * unitsize(4)

 

128 FOR I = 1 TO 6000

 

129 FOR KKQQ = 1 TO 4

 

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

 

181 j = 1 + FIX(RND * 4)

 

183 REM r = (1 - RND * 2) * A(j)
187 REM X(j) = A(j) + (RND ^ (RND * 10)) * r
197 IF RND < .5 THEN X(j) = A(j) - FIX(RND * 11) * unitsize(j) ELSE X(j) = A(j) + FIX(RND * 11) * unitsize(j)

222 NEXT IPP

 

268 IF X(1) < -6## THEN 1670
269 IF X(1) > 6.75## THEN 1670

 

272 IF X(2) < -6## THEN 1670
273 IF X(2) > 6.75## THEN 1670

 

274 IF X(3) < -1## THEN 1670
275 IF X(3) > 9.2## THEN 1670
284 IF X(4) < -9## THEN 1670
285 IF X(4) > 6.3## THEN 1670

 

308 X(5) = -500 - X(1) ^ 3 * X(2) * X(3) ^ 2 - X(3) * X(4)

 

309 X(6) = 500 + X(1) ^ 3 * X(2) * X(3) - X(3) ^ 2 * X(4)

 

401 FOR J47 = 5 TO 6

402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

 

403 NEXT J47

443 POBA = -X(1) ^ 3 * X(2) * X(3) ^ 3 * X(4) - X(1) ^ 3 * X(2) * X(3) * X(4) ^ 2 + 1000000 * (X(5) + X(6))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 6

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

1670 NEXT I
1889 IF M < 72815.55 THEN 1999

 

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [49]. The complete output through JJJJ = -31950.61000000791 is shown below:

5.339479392624724          -.293383947939263          6.043383947939264
6.299999999999996          0          0          72815.81934346557
-31958.45000000665

5.339479392624726          -.293383947939263          6.043383947939264
6.299999999999995          0          0          72815.81934346558
-31950.61000000791

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 [49], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31950.61000000791 was 220 seconds, including the time for “Creating .EXE file.” One can compare the computational results above with those on page 114 of Lu [32, Table 5].

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] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] 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.
[11] 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.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] E. L. Hannan (1981), Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235-246.

[14] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[15] James P. Ignizio, Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ, 1982.

[16] James P. Ignizio, Stephen C. Daniels (1983). Fuzzy multicriteria integer programming via fuzzy generalized networks. Fuzzy Sets and Systems 10 (1983) 261-270.

[17] James P. Ignizio, Tom M. Cavalier, Linear Programming. Prentice-Hall, Englewood Cliffs, NJ, 1994.

[18] 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.

[19] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

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

[21] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

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

[23] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[24] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[25] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] 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.
[27] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[28] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[29] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[30] 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.
[31] 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.
[32] 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.

[33] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[34] 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/
[35] 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.

[36] Katta Murty, Operations Research: Deterministic Optimization Models. Prentice-Hall, 1995.

[37] 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…/
[38] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[39] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[40] 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.
[41] 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.

[42] Vikas Sharma (2012), Multobjective integer nonlinear fractional programming problem: A cutting plane approach. OPSEARCH of the Operational Research Society of India (April-June 2012) 49 (2) 133-153.

[43] 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.
[44] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[45] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[46] 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
[47] 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.
[48] 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.
[49] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[50] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[51] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving a Free-Sign Pure Discrete Signomial Programming Problem--Lu's Example 3 with r= 2000

Jsun Yui Wong

The computer program listed below is about extending and solving Lu's Example 3 for r =2000 discrete values, [32, p. 113 and Table 6], which is shown immediately below:

Minimize 2 * X(1) ^ (.9) * X(2) ^ (-1.5) * X(3) ^ (-3) + 5 * X(4) ^ (-.3) * X(5) ^ 2.6 + 4.7 * X(6) ^ (-1.8) * X(7) ^ (-.5) * X(8)

subject to

7.2 * X(1) ^ (-3.8) * X(2) ^ 2.2 * X(3) ^ (4.3) + .5 * X(4) ^ (-.7) * X(5) ^ (-1.6) + .2 * X(6) ^ (4.3) * X(7) ^ (-1.9) * X(8) ^ (3.5)<=695479

10 * X(1) ^ (2.3) * X(2) ^ 1.7 * X(3) ^ (4.5)<=1963735

.6 * X(4) ^ 2.1 * X(5) ^ (-.4) <=2.84

6.2 * X(6) ^ (4.5) * X(7) ^ (-2.7) * X(8) ^ (-.6) <=13105

3.1 * X(1) ^ (1.6) * X(2) ^ .4 * X(3) ^ (-3.8) <=131

3.7 * X(4) ^ 5.4 * X(5) ^ (-1.3) <=9068

.3 * X(6) ^ (-1.1) * X(7) ^ 3.2 * X(8) ^ (-5.6) <=730

.5<= X(i) <=5.5, i=1 through 8

where X(4) and X(5) are continuous variables, and the other six variables are discrete variables; here the number of discrete values is 2000.

X(9) through X(15) below are slack variables added. One notes line 268, which is 268 IF X(J43) < .5## THEN 1670.

 

0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO 32000 STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37

 

25 FOR J41 = 1 TO 8

 

31 unitsize(J41) = 5 / 1999

 

32 NEXT J41
55 FOR J42 = 1 TO 8

61 A(J42) = .5 + FIX(RND * 2000) * unitsize(J42)

62 NEXT J42

 

128 FOR I = 1 TO 6000

 

129 FOR KKQQ = 1 TO 8

 

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

 

181 j = 1 + FIX(RND * 8)

 

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

 

197 IF RND < .5 THEN X(j) = A(j) - FIX(RND * 5) * unitsize(j) ELSE X(j) = A(j) + FIX(RND * 5) * unitsize(j)

199 IF RND < .5 THEN GOTO 201 ELSE GOTO 211

201 r = (1 - RND * 2) * A(4)
203 X(4) = A(4) + (RND ^ (RND * 10)) * r

205 GOTO 222
211 r = (1 - RND * 2) * A(5)
215 X(5) = A(5) + (RND ^ (RND * 10)) * r

 

222 NEXT IPP

 

264 FOR J43 = 1 TO 8
268 IF X(J43) < .5## THEN 1670
269 IF X(J43) > 5.5## THEN 1670

270 NEXT J43

307 X(9) = 2.84 - .6 * X(4) ^ 2.1 * X(5) ^ (-.4)

 

309 X(10) = 9068 - 3.7 * X(4) ^ 5.4 * X(5) ^ (-1.3)

310 X(11) = 730 - .3 * X(6) ^ (-1.1) * X(7) ^ 3.2 * X(8) ^ (-5.6)

 

311 X(12) = 13105 - 6.2 * X(6) ^ (4.5) * X(7) ^ (-2.7) * X(8) ^ (-.6)

 

321 X(13) = 131 - 3.1 * X(1) ^ (1.6) * X(2) ^ .4 * X(3) ^ (-3.8)

324 X(14) = 1963735 - 10 * X(1) ^ (2.3) * X(2) ^ 1.7 * X(3) ^ (4.5)

 

326 X(15) = 695479 - 7.2 * X(1) ^ (-3.8) * X(2) ^ 2.2 * X(3) ^ (4.3) - .5 * X(4) ^ (-.7) * X(5) ^ (-1.6) - .2 * X(6) ^ (4.3) * X(7) ^ (-1.9) * X(8) ^ (3.5)

401 FOR J47 = 9 TO 15

402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

 

433 NEXT J47
445 POBA = -2 * X(1) ^ (.9) * X(2) ^ (-1.5) * X(3) ^ (-3) - 5 * X(4) ^ (-.3) * X(5) ^ 2.6 - 4.7 * X(6) ^ (-1.8) * X(7) ^ (-.5) * X(8) + 1000000 * (X(9) + X(10) + X(11) + X(12) + X(13) + X(14) + X(15))

466 P = POBA

1111 IF P <= M THEN 1670

1450 M = P
1454 FOR KLX = 1 TO 15

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

1670 NEXT I
1889 IF M < -.73456699 THEN 1999

 

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6)

1908 PRINT A(7), A(8), A(9), A(10), A(11), A(12)

1909 PRINT A(13), A(14), A(15), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [49]. The complete output through JJJJ = -31999.59000000007 is shown below:

.9002001000500343          5.4974987493747          5.49749874937469
1.83725353308645          .5000000004249298          5.499999999999993
5.49999999999999          .500000000000007          0
0          0          0
0          0          0          -.7345669874578813
-31999.81000000003

.9002001000500357          5.492496248124061          5.499999999999998
1.837253532789008          .5000000000000012          5.499999999999998
5.499999999999983          .500000000000015          0
0          0          0
0          0           0          -.7345669867645721
-31999.59000000007

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 [49], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31999.59000000007 was 33 seconds, not including the time for “Creating .EXE file” (was 40 seconds, including the time for “Creating .EXE file.” One can compare the computational results above with those on page 117 of Lu [32, Table 6].

Acknowledgment

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

References

[1] Yuichiro Anzai (1974). On Integer Fractional Programming. Journal of the Operations Research Society of Japan, Volume 17, No. 1, March 1974, pp. 49-66. http://www..orsj.or.jp/~archiv/pdf/e_mag/Vol.17_01_049.pdf.
[2] Sjirk Boon. Solving systems of nonlinear equations. Sci. Math. Num-Analysis,1992, Newsgroup Article 3529.
[3] S. S. Chadha (2002). Fractional programming with absolute-value functions. European Journal of Operational Research 141 (2002) pp. 233-238.
[4] Ching-Ter Chang (2002). On the posynomial fractional programming problems. European Journal of Operational Research 143 (2002) pp. 42-52.
[5] Ching-Ter Chang (2006). Formulating the mixed integer fractional posynomial programming, European Journal of Operational Research 173 (2006) pp. 370-386.

[6] Ching-Ter Chang (2017). Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions. Computers and Industrial Engineering 112 (2017) 437-446.

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] 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.
[11] 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.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

[13] E. L. Hannan (1981), Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235-246.

[14] Xue-Ping Hou, Pei-Ping Shen, Chun-Feng Wang (2014). Global Minimization for Generalized Polynomial Fractional Program. Mathmatical Problems in Engineering, Volume 2014. Hindawi Publishing Company.

[15] James P. Ignizio, Linear Programming in Single- and Multiple-Objective Systems. Prentice-Hall, Englewood Cliffs, NJ, 1982.

[16] James P. Ignizio, Stephen C. Daniels (1983). Fuzzy multicriteria integer programming via fuzzy generalized networks. Fuzzy Sets and Systems 10 (1983) 261-270.

[17] James P. Ignizio, Tom M. Cavalier, Linear Programming. Prentice-Hall, Englewood Cliffs, NJ, 1994.

[18] 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.

[19] Sanjay Jain, Kailash Lachhwani (2010). Linear plus fractional multiobjective programming problem with homogeneous constraints using fuzzy approach. Iranian Journal of Operations Research, Vol. 2, No. 1, 2010, pp. 41-49.

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

[21] Ali Husseinzadeh Kashan (2011). An effective algorithm for constrained global optimization and application to mechanical engineering design: League championship algorithm (LCA). Computer-Aided Design 43 (2011) 1769-1792.

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

[23] K. Lachhwani (2012). Fuzzy Goal Programming Approach to Multi Objectve Quadratic Programming Problem. Proceedings of the National Academy of Sciences, India, Section A Physical Sciences (October-December 2012) 82 (4): 317-322.
https://link.springer.com/journal/40010

[24] K. Lachhwani (2014). FGP approach to multi objectve quadratic fractional programming problem. International Journal of Industrial Mathematics, vol. 6, No.1, pp. 49-57. Available online at http://ijim.srbiau.ac.ir/

[25] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] 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.
[27] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[28] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[29] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[30] 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.
[31] 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.
[32] 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.

[33] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[34] 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/
[35] 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.

[36] Katta Murty, Operations Research: Deterministic Optimization Models. Prentice-Hall, 1995.

[37] 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…/
[38] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[39] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[40] 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.
[41] 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.

[42] Vikas Sharma (2012), Multobjective integer nonlinear fractional programming problem: A cutting plane approach. OPSEARCH of the Operational Research Society of India (April-June 2012) 49 (2) 133-153.

[43] 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.
[44] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[45] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[46] 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
[47] 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.
[48] 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.
[49] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[50] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[51] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving Another Free-Sign Pure Discrete Signomial Programming Problem, Corrected Edition

Jsun Yui Wong

Correcting the 2018 April 3 paper, the computer program listed below seeks to solve Example 1 with r =1024 discrete values in Lu [32, p. 107]:

Maximize X(1) ^ (-4 / 3) * X(2) ^ 3 * X(3) - 18 * X(1) - 28 * X(2) - 5.5 * X(3)

subject to

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

X(1) + X(2) + X(3)>=-4

2 < X(1) <= 6

-4.8<= X(2) <= 4

-4<= X(3) <= 4

where all variables are discrete variables. X(2) and X(3) are free-sign discrete variables; here the number of discrete values is 1024.

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 32000 STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37

31 unitsize(1) = 4 / 1023

32 unitsize(2) = 8.8 / 1023

33 unitsize(3) = 8 / 1023

61 A(1) = 2.0000001 + FIX(RND * 1024) * unitsize(1)

63 A(2) = -4.8 + FIX(RND * 1024) * unitsize(2)

65 A(3) = -4 + FIX(RND * 1024) * unitsize(3)

 

128 FOR I = 1 TO 2000

 

129 FOR KKQQ = 1 TO 3

 

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

 

181 j = 1 + FIX(RND * 3)

 

183 REM r = (1 - RND * 2) * A(j)

187 REM X(j) = A(j) + (RND ^ (RND * 10)) * r

 

197 IF RND < .5 THEN X(j) = A(j) - FIX(RND * 5) * unitsize(j) ELSE X(j) = A(j) + FIX(RND * 5) * unitsize(j)

 

222 NEXT IPP

 

268 IF X(1) < 2.0000001## THEN 1670
269 IF X(1) > 6## THEN 1670

 

272 IF X(2) < -4.8## THEN 1670
273 IF X(2) > 4## THEN 1670

 

274 IF X(3) < -4## THEN 1670
275 IF X(3) > 4## THEN 1670

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

310 X(5) = 4 + X(1) + X(2) + X(3)

401 FOR J47 = 4 TO 5

402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0

 

403 NEXT J47

 

445 POBA = X(1) ^ (-4 / 3) * X(2) ^ 3 * X(3) - 18 * X(1) - 28 * X(2) - 5.5 * X(3) + 1000000 * (X(4) + X(5))
466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 5

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

1670 NEXT I
1889 IF M < 157.53 THEN 1999

1907 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 [49]. The complete output through JJJJ = -31999.97000000001 is shown below:

2.000000100000002          -4.791397849462362          -1.208211143695012
0          0          157.5462491147576          -32000

2.0000001           -4.791397849462368          -1.208211143695011
0          0          157.546249114758            -31999.99

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [49], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31999.97000000001 was about 1 second, not including the time for “Creating .EXE file,”
(was 7 seconds, total, including the time for “Creating .EXE file.” ) One can compare the computational results above with those on page 110 of Lu [32, Table 4].

Acknowledgment

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

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