Solving an Instance of Brown’s Almost Linear System of Equations

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Floudas et al. [7, p. 329, Test Problem 5, Brown’s almost linear system]:

2 * X(1) + X(2) + X(3) + X(4) +X(5) = 6

X(1) + 2 * X(2) + X(3) + X(4) + X(5) = 6

X(1) + X(2) + 2 * X(3) + X(4) + X(5) = 6

X(1) + X(2) + X(3) + 2 * X(4) + X(5) = 6

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

-2<=X(i)<=2, i=1, 2, 3. 4, 5,

X(1) through X(5) are continuous 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 31996. STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37

 

33 FOR J44 = 1 TO 5

34 A(J44) = -2 + RND * 4

37 NEXT J44

 

128 FOR I = 1 TO 30000

 

129 FOR KKQQ = 1 TO 5

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

 

181 j = 1 + FIX(RND * 5)

189 r = (1 – RND * 2) * A(j)
190 X(j) = A(j) + (RND ^ (RND * 10)) * r

 

195 NEXT IPP

211 REM FOR J44 = 1 TO 5

214 REM X(J44) = INT(X(J44))

 

218 REM NEXT J44

 

225 FOR J44 = 1 TO 5

226 IF X(J44) < -2## THEN 1670

227 IF X(J44) > 2## THEN 1670
228 NEXT J44
229 GOTO 243

 

243 X(5) = -2 * X(1) – X(2) – X(3) – X(4) + 6

 

244 IF X(5) < -2## THEN 1670

245 IF X(5) > 2## THEN 1670

 

246 LHS2 = X(1) + 2 * X(2) + X(3) + X(4) + X(5) – 6
247 LHS3 = X(1) + X(2) + 2 * X(3) + X(4) + X(5) – 6

248 LHS4 = X(1) + X(2) + X(3) + 2 * X(4) + X(5) – 6
249 LHS5 = X(1) * X(2) * X(3) * X(4) * X(5) – 1

 

455 POBA = -ABS(LHS2) – ABS(LHS3) – ABS(LHS4) – ABS(LHS5)

 

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 < -.00005 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 [55]. The complete output through JJJJ = -31950.94000000785 is shown below:

.9163308616021965          .9163308616021965          .9163308616021977
.916330861602197          1.418345691989016          -1.992059048559714D-05
-31994.34000000091

.9163294344125231          .916329434412523          .9163294344125231
.9163294344125231          1.418352827937385          -2.111942530818542D-05
-31977.69000000357

1.00003885624662          1.00003885624662          1.00003885624662
1.00003885624662          .9998057187669032          -3.887738545503105D-05
-31957.13000000686

1.000014301929638          1.000014301929638          1.00001430192964
1.000014301929638          .9999284903518086          -1.430479334700883D-05
-31950.94000000785

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 [55], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31950.94000000785 was 8 minutes, total, including the time for “Creating .EXE file.” One can compare the computational results above with those on p. 329 of Floudas et al. [7].

Acknowledgment

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

References

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

[6] 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] C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer, C. A. Schweiger (1999). Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers 1999.

[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, Hao-Chun Lu, Chia-Hui Huang, Nian-Ze Hu (2009). A superior representation method for piecewise lineat functions. INFORMS Journal on Computing 21 (2): 314-321 (2009).

[30] 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.
[31] 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.
[32] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[33] 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.
[34] 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.
[35] 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.

[36] 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.
[37] 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/
[38] 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.

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

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.

[47] J. Smith (1985). Chemical Engineering Kinetics. Butterworth, Stoneham, MA.

[48] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[49] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[50] 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
[51] 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.
[52] 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.

[53] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chai-Chien Lee (2010). An efficient global optimization approach for solving mixed-integer nonlinear programming problems. Proceedings of the 40th International Conference on Computers and Industrian Engineering, Japan, pp.1-4, 2010.
Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.

[54] Pei-Chun Wang, Jung-Fa Tsai (2011). Global optimization of mixed-integer nonlinear programming for engineering design problems. Proceedings of 2011 International Conference on System Science and Engineeing, Macau, China – June 2011.

[55] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[56] 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.
[57] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving an Integer Version of Brown’s Almost Linear System of Equations

Jsun Yui Wong

The computer program listed below seeks to solve the immediately following integer nonlinear system of equations:

2 * X(1) + X(2) + X(3) + X(4) +X(5) = 6

X(1) + 2 * X(2) + X(3) + X(4) + X(5) = 6

X(1) + X(2) + 2 * X(3) + X(4) + X(5) = 6

X(1) + X(2) + X(3) + 2 * X(4) + X(5) = 6

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

-32<=X(i)<=32, i=1, 2, 3. 4, 5,

X(1) through X(5) are integer variables.

The problem above is based on Floudas et al.’s Test Problem 5 on p. 329 [7, Brown’s almost linear system], which is as follows

2 * X(1) + X(2) + X(3) + X(4) +X(5) = 6

X(1) + 2 * X(2) + X(3) + X(4) + X(5) = 6

X(1) + X(2) + 2 * X(3) + X(4) + X(5) = 6

X(1) + X(2) + X(3) + 2 * X(4) + X(5) = 6

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

-2<=X(i)<=2, i=1, 2, 3. 4, 5,

X(1) through X(5) are continuous 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 31996. STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37

33 FOR J44 = 1 TO 5

34 A(J44) = -32 + RND * 64

37 NEXT J44

 

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 5

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

 

181 j = 1 + FIX(RND * 5)

189 r = (1 – RND * 2) * A(j)
190 X(j) = A(j) + (RND ^ (RND * 10)) * r

 

195 NEXT IPP

211 FOR J44 = 1 TO 5

214 X(J44) = INT(X(J44))

 

218 NEXT J44

 

225 FOR J44 = 1 TO 5

226 IF X(J44) < -32## THEN 1670

227 IF X(J44) > 32## THEN 1670
228 NEXT J44
229 GOTO 243

 

243 X(5) = -2 * X(1) – X(2) – X(3) – X(4) + 6

 

244 IF X(5) < -32## THEN 1670

245 IF X(5) > 32## THEN 1670

 

246 LHS2 = X(1) + 2 * X(2) + X(3) + X(4) + X(5) – 6
247 LHS3 = X(1) + X(2) + 2 * X(3) + X(4) + X(5) – 6

248 LHS4 = X(1) + X(2) + X(3) + 2 * X(4) + X(5) – 6
249 LHS5 = X(1) * X(2) * X(3) * X(4) * X(5) – 1

 

455 POBA = -ABS(LHS2) – ABS(LHS3) – ABS(LHS4) – ABS(LHS5)

 

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 < 0 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 [55]. The complete output through JJJJ = -31836.4000000262 is shown below:

1       1       1       1       1
0          -31959.75000000644

1       1       1       1       1
0          -31881.09000001903

1       1       1       1       1
0          -31836.4000000262

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 [55], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31836.4000000262 was 6 minutes, total, including the time for “Creating .EXE file.”

Acknowledgment

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

References

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

[6] 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] C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer, C. A. Schweiger (1999). Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers 1999.

[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, Hao-Chun Lu, Chia-Hui Huang, Nian-Ze Hu (2009). A superior representation method for piecewise lineat functions. INFORMS Journal on Computing 21 (2): 314-321 (2009).

[30] 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.
[31] 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.
[32] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[33] 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.
[34] 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.
[35] 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.

[36] 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.
[37] 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/
[38] 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.

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

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.

[47] J. Smith (1985). Chemical Engineering Kinetics. Butterworth, Stoneham, MA.

[48] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[49] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[50] 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
[51] 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.
[52] 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.

[53] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chai-Chien Lee (2010). An efficient global optimization approach for solving mixed-integer nonlinear programming problems. Proceedings of the 40th International Conference on Computers and Industrian Engineering, Japan, pp.1-4, 2010.
Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.

[54] Pei-Chun Wang, Jung-Fa Tsai (2011). Global optimization of mixed-integer nonlinear programming for engineering design problems. Proceedings of 2011 International Conference on System Science and Engineeing, Macau, China – June 2011.

[55] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[56] 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.
[57] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

J. Smith (1985) Problem with Delta H=-35958 in Floudas et al. [7]

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Floudas et al. [7, p. 333, Test Problem 9], “where the goal is to find all steady state temperatures of a nonisothermal CSTR,” Floudas et al. [7, p. 333].

((bbb / 298) * X(1) * EXP(-7548.1193 / X(1))) – (((bbb * (1 + aaa * 298) / (aaa * 298))) * (EXP(-7548.1193 / X(1)))) + X(1) / 298 – 1 = 0

where aaa = -1000 / (3 * (-35958)), bbb = 1.344D+09, and 100<=X(1)<=1000.

The above is Floudas et al.’s case with delta H=-35958.

 

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

92 A(1) = 100 + (RND * 900)

128 FOR I = 1 TO 50

129 FOR KKQQ = 1 TO 1

130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + 0)

 

181 j = 1 + FIX(RND * 0)

189 r = (1 – RND * 2) * A(j)
190 X(j) = A(j) + (RND ^ (RND * 10)) * r

 

222 NEXT IPP
230 IF X(1) < 100 THEN 1670

231 IF X(1) > 1000## THEN 1670
236 FOR J44 = 1 TO 1
237 IF X(J44) < 100 THEN GOTO 1670
238 IF X(J44) > 1000 THEN GOTO 1670

 

239 NEXT J44

241 aaa = -1000 / (3 * (-35958))

 

243 bbb = 1.344D+09

245 LHS = ((bbb / 298) * X(1) * EXP(-7548.1193 / X(1))) – (((bbb * (1 + aaa * 298) / (aaa * 298))) * (EXP(-7548.1193 / X(1)))) + X(1) / 298 – 1

 

260 IF X(1) < 100 THEN 1670

261 IF X(1) > 1000## THEN 1670

 

455 POBA = -ABS(LHS)

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 1

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1477 IF M > -.0000001## THEN 1908

1557 GOTO 128

1670 NEXT I
1889 REM IF M < -7512.24 THEN 1999

1908 PRINT A(1), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [55]. The output through JJJJ = -31999.79000000003 is summarized below:

 

382.5467783448055          -2.087069864403334D-08          -32000
.
.
.

379.0874970073848          -4.237852460273195D-08          -31999.90000000002
.
.
.

299.6385833262814          -2.165712769525412D-08          -31999.79000000003

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 [55], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.79000000003 was 1 second, not including the time for “Creating .EXE file” (6 seconds, total, including the time for “Creating .EXE file.”) One can compare the computational results above with those on p. 334 of Floudas et al. [7].

Acknowledgment

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

References

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

[6] 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] C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer, C. A. Schweiger (1999). Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers 1999.

[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, Hao-Chun Lu, Chia-Hui Huang, Nian-Ze Hu (2009). A superior representation method for piecewise lineat functions. INFORMS Journal on Computing 21 (2): 314-321 (2009).

[30] 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.
[31] 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.
[32] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[33] 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.
[34] 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.
[35] 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.

[36] 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.
[37] 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/
[38] 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.

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

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.

[47] J. Smith (1985). Chemical Engineering Kinetics. Butterworth, Stoneham, MA.

[48] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[49] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[50] 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
[51] 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.
[52] 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.

[53] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chai-Chien Lee (2010). An efficient global optimization approach for solving mixed-integer nonlinear programming problems. Proceedings of the 40th International Conference on Computers and Industrian Engineering, Japan, pp.1-4, 2010.
Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.

[54] Pei-Chun Wang, Jung-Fa Tsai (2011). Global optimization of mixed-integer nonlinear programming for engineering design problems. Proceedings of 2011 International Conference on System Science and Engineeing, Macau, China – June 2011.

[55] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

[56] 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.
[57] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Ben-Tal et al. (1994) Problem 1 in Floudas et al. [7]

Jsun Yui Wong

The computer program listed below seeks to solve the following problem of seven continuous variables from Floudas et al. [7, pp. 38-39]:

Maximize (9 – 6 * X(1) – 16 * X(2) – 15 * X(3)) * X(4) + (15 – 6 * X(1) – 16 * X(2) – 15 * X(3)) * X(5) – X(6) + 5 * X(7)

subject to

X(3) * X(4) + X(3) * X(5)<=50

X(4) +- X(6)<=100

X(5) + X(7)<=200

(3 * X(1) + X(2) + X(3) – 2.5) * X(4) – .5 * X(6)<=0

(3 * X(1) + X(2) + X(3) – 1.5) * X(5) + .5 * X(7)<=0

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

0<= X(1) <= 1

0<= X(2) <= 1

0<= X(3) <= 1

0<= X(4) <= 100

0<= X(5) <= 200

0<= X(6) <= 100

0<= X(7) <= 200.

In the following computer program, D of line 81 stands for discreteness, and X(8) through X(12) 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

81 D(1) = .0001

 

85 D(2) = .0001

86 D(3) = .0001

87 D(4) = .001

88 D(5) = .001

89 D(6) = .001

90 D(7) = .001

 

91 A(1) = FIX(RND * 1.01)
93 A(2) = FIX(RND * 1.01)
94 A(3) = FIX(RND * 1.01)

95 A(4) = FIX(RND * 101)
96 A(5) = FIX(RND * 201)

 

98 A(6) = FIX(RND * 101)
99 A(7) = FIX(RND * 201)

 

128 FOR I = 1 TO 10000

 

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)

189 REM r = (1 – RND * 2) * A(j)
190 REM X(j) = A(j) + (RND ^ (RND * 10)) * r
195 IF RND < .5 THEN X(j) = A(j) – INT(RND * 10) * D(j) ELSE X(j) = A(j) + INT(RND * 10) * D(j)

222 NEXT IPP
223 REM FOR J44 = 4 TO 7
224 REM X(J44) = INT(X(J44))

225 REM NEXT J44

 

226 FOR J44 = 1 TO 7
227 IF X(J44) < 0## THEN 1670

229 NEXT J44
235 IF X(1) > 1## THEN 1670

 

236 IF X(2) > 1## THEN 1670

 

237 IF X(3) > 1## THEN 1670

 

238 IF X(4) > 100## THEN 1670

 

239 IF X(5) > 200## THEN 1670
240 IF X(6) > 100## THEN 1670

 

241 IF X(7) > 200## THEN 1670

 

246 X(1) = 1 – X(2) – X(3)

247 X(8) = 100 – X(4) – X(6)

249 X(9) = 200 – X(5) – X(7)
251 X(10) = 50 – X(3) * X(4) – X(3) * X(5)

 

277 X(11) = -(3 * X(1) + X(2) + X(3) – 2.5) * X(4) + .5 * X(6)
279 X(12) = -(3 * X(1) + X(2) + X(3) – 1.5) * X(5) – .5 * X(7)

 

326 FOR J44 = 1 TO 7
327 IF X(J44) < 0## THEN 1670

329 NEXT J44
335 IF X(1) > 1## THEN 1670

 

336 IF X(2) > 1## THEN 1670

 

337 IF X(3) > 1## THEN 1670

 

338 IF X(4) > 100## THEN 1670

 

339 IF X(5) > 200## THEN 1670
340 IF X(6) > 100## THEN 1670

 

341 IF X(7) > 200## THEN 1670

 

450 FOR J47 = 8 TO 12

 

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

452 NEXT J47

 

453 POBA = (9 – 6 * X(1) – 16 * X(2) – 15 * X(3)) * X(4) + (15 – 6 * X(1) – 16 * X(2) – 15 * X(3)) * X(5) – X(6) + 5 * X(7) + 1000000 * (X(8) + X(9) + X(10) + X(11) + X(12))

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 12

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

1670 NEXT I
1889 IF M < 449.95 THEN 1999

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8)
1908 PRINT A(9), A(10), A(11), A(12), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [54]. The complete output through JJJJ = -31999.97000000001 is shown below:

1.27675647831893D-15          .4999999999999991          .4999999999999996
1.413799632921098D-15          99.99999999999945          9.999999970941873D-04
100.00000000001          0
-9.407585821463727D-12          0          0          -5.506706202140776D-12
449.9989850857623          -32000

2.109423746787797D-15          .4999999999999997          .5000000000000001
9.99999998975422D-04          99.998999999999983          9.9999999968006617D-04
99.999000000000885          0
0          0          0          -4.931166586175095D-12
449.98799506889          -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 [54], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.97000000001 was 4 seconds, not including the time for “Creating .EXE file” (12 seconds, total, including the time for “Creating .EXE file.”) One can compare the computational results above with those on pp. 39-40 of Floudas et al. [7, pp. 39-40].

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] C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer, C. A. Schweiger (1999). Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers 1999.

[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, Hao-Chun Lu, Chia-Hui Huang, Nian-Ze Hu (2009). A superior representation method for piecewise lineat functions. INFORMS Journal on Computing 21 (2): 314-321 (2009).

[30] 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.
[31] 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.
[32] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[33] 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.
[34] 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.
[35] 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.

[36] 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.
[37] 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/
[38] 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.

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

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.
[47] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[48] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[49] 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
[50] 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.
[51] 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.

[52] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chai-Chien Lee (2010). An efficient global optimization approach for solving mixed-integer nonlinear programming problems. Proceedings of the 40th International Conference on Computers and Industrian Engineering, Japan, pp.1-4, 2010.
Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.

[53] Pei-Chun Wang, Jung-Fa Tsai (2011). Global optimization of mixed-integer nonlinear programming for engineering design problems. Proceedings of 2011 International Conference on System Science and Engineeing, Macau, China – June 2011.

[54] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[55] 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.
[56] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving the Haverly Pooling Problem Case 2 in Floudas et al. [7]

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Floudas et al. [7, pp. 34-36]:

Maximize 9 * X(8) + 15 * X(9) – 6 * X(1) – 16 * X(2) – 10 * (X(3) + X(4))

subject to

X(5) + X(6) – X(1) – X(2)=0

X(8) – X(5) – X(3)=0

X(9) – X(6) – X(4)=0

X(7) * X(5) + 2 * X(3) -2.5 * X(8) <=0

X(7) * X(6) + 2 * X(4) -1.5 * X(9) <=0

X(7)*X(5) +X(7)*X(6)) -3 * X(1) – X(2)=0

0<= X(8) <=600

0<= X(9) <=200

0<= X(i) <= 500, i=1, 2, 3, …, 7.

X(10) and X(11) below are slack variables.

One notes line 224, which is 224 X(J44) = INT(X(J44)); using discrete/integer variables to approximate continuous variables can be fruitful sometimes.

 

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

84 A(1) = 0 + RND * 500

85 A(2) = 0 + RND * 500

86 A(3) = 0 + RND * 500

87 A(4) = 0 + RND * 500

88 A(5) = 0 + RND * 500

89 A(6) = 0 + RND * 500

90 A(7) = 0 + RND * 500

91 A(8) = 0 + RND * 600

 

92 A(9) = 0 + RND * 200

128 FOR I = 1 TO 20000

 

 

129 FOR KKQQ = 1 TO 9

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

181 j = 1 + FIX(RND * 9)

 

189 r = (1 – RND * 2) * A(j)

190 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP
223 FOR J44 = 1 TO 9
224 X(J44) = INT(X(J44))

225 NEXT J44

 

226 FOR J44 = 1 TO 9
227 IF X(J44) < 0## THEN 1670

229 NEXT J44

235 IF X(1) > 500## THEN 1670

 

236 IF X(2) > 500## THEN 1670

 

237 IF X(3) > 500## THEN 1670

 

238 IF X(4) > 500## THEN 1670

 

239 IF X(5) > 500## THEN 1670

 

240 IF X(6) > 500## THEN 1670

 

241 IF X(7) > 500## THEN 1670

 

242 IF X(8) > 600## THEN 1670

 

243 IF X(9) > 200## THEN 1670

 

247 X(8) = X(3) + X(5)

249 X(9) = X(4) + X(6)
251 X(1) = X(5) + X(6) – X(2)

264 X(7) = ((3 * X(1) + X(2)) / ((X(5) + X(6))))
326 FOR J44 = 1 TO 9
327 IF X(J44) < 0## THEN 1670

329 NEXT J44

335 IF X(1) > 500## THEN 1670

 

336 IF X(2) > 500## THEN 1670

 

337 IF X(3) > 500## THEN 1670

 

338 IF X(4) > 500## THEN 1670

 

339 IF X(5) > 500## THEN 1670

 

340 IF X(6) > 500## THEN 1670

 

341 IF X(7) > 500## THEN 1670

 

342 IF X(8) > 600## THEN 1670

 

343 IF X(9) > 200## THEN 1670

 

447 X(10) = 2.5 * X(8) – X(7) * X(5) – 2 * X(3)
448 X(11) = 1.5 * X(9) – X(7) * X(6) – 2 * X(4)

 

450 FOR J47 = 10 TO 11

 

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

 

452 NEXT J47

 

456 POBA = 9 * X(8) + 15 * X(9) – 6 * X(1) – 16 * X(2) – 10 * (X(3) + X(4)) + 1000000 * (X(10) + X(11))
466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 11

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

1670 NEXT I
1889 IF M < 598 THEN 1999

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8)
1908 PRINT A(9), A(10), A(11), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [54]. The complete output through JJJJ = -31992.39000000122 is shown below:

300       0       300       0      300
0       3       600
0       0       0       600       -31997.01000000048

299       0       299       0       299
0       3       598
0       0       0       598       -31996.8900000005

299       0       299       0       299
0       3       598
0       0       0       598       -31995.27000000076

299       0       299       0       299
0       3       598
0       0       0       598       -31992.44000000121

300       0       300       0       300
0      3       600
0       0       0       600       -31992.39000000122

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 [54], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31992.39000000122
was 72 seconds, not including the time for “Creating .EXE file” (80 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on p. 36 of Floudas et al. [7, p. 36, Case 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] C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A Meyer, C. A. Schweiger (1999). Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers 1999.

[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, Hao-Chun Lu, Chia-Hui Huang, Nian-Ze Hu (2009). A superior representation method for piecewise lineat functions. INFORMS Journal on Computing 21 (2): 314-321 (2009).

[30] 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.
[31] 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.
[32] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[33] 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.
[34] 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.
[35] 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.

[36] 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.
[37] 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/
[38] 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.

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

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.
[47] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[48] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[49] 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
[50] 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.
[51] 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.

[52] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chai-Chien Lee (2010). An efficient global optimization approach for solving mixed-integer nonlinear programming problems. Proceedings of the 40th International Conference on Computers and Industrian Engineering, Japan, pp.1-4, 2010.
Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.

[53] Pei-Chun Wang, Jung-Fa Tsai (2011). Global optimization of mixed-integer nonlinear programming for engineering design problems. Proceedings of 2011 International Conference on System Science and Engineeing, Macau, China – June 2011.

[54] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[55] 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.
[56] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving the Heat Exchanger Network Problem in Ryoo and Sahinidis [43] Plus the Requirement That the 12 Variables Have Integer Values

Jsun Yui Wong

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

Minimize 1200 * (800 / down1) ^ .6 + 1200 * (1000 / down2) ^ .6

where down1 = 2.5 * (.666666666666666666666666 * (320 – X(10)) * (300 – X(9)) ^ .5 + ((320 – X(10)) + (300 – X(9))) / 6),

down2 = .2 * (.666666666666666666666666 * (340 – X(12)) * (300 – X(11)) ^ .5 + ((340 – X(12)) + (300 – X(11))) / 6)

subject to

X(1) + X(2)=10

X(1) + X(6)=X(3)

X(2) + X(5)=X(4)
X(5) + X(7)=X(3)
X(6) + X(8)=X(4)

100 * X(1) + X(10) * X(6) =X(9)*X(3)
100 * X(2) + X(10) * X(5) =X(11)*X(4)

X(3)* (X(10) – X(9)=800

X(4)* (X(12) – X(11))=1000

0<= X(i) <=10, i=1, 2, 3,…, 8

100<= X(9) <= 290
100<= X(10) <= 310

100<= X(11) <= 290
100<= X(12) <= 330,

where X(1) through X(12) have integer values.

The purpose of the sequence of line 246 through line 269 is to produce some domino effect.

 

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
77 FOR J44 = 1 TO 8
79 A(J44) = INT(RND * 10)
82 NEXT J44

89 A(9) = 100 + INT(RND * 190)

90 A(10) = 100 + INT(RND * 210)

91 A(11) = 100 + INT(RND * 190)

 

92 A(12) = 100 + INT(RND * 230)

 

128 FOR I = 1 TO 20000

 

129 FOR KKQQ = 1 TO 12

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

181 j = 1 + FIX(RND * 12)
185 IF RND < .5 THEN X(j) = A(j) – FIX(RND * 10) ELSE X(j) = A(j) + FIX(RND * 10)
189 REM r = (1 – RND * 2) * A(j)

190 REM X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP
223 FOR J44 = 1 TO 12
224 X(J44) = INT(X(J44))

225 NEXT J44

 

226 FOR J44 = 1 TO 8
227 IF X(J44) < 0## THEN 1670
228 IF X(J44) > 10## THEN 1670

229 NEXT J44

 

230 IF X(9) < 100## THEN 1670

234 IF X(9) > 290# THEN 1670

236 IF X(10) < 100## THEN 1670

238 IF X(10) > 310## THEN 1670

 

240 IF X(11) < 100## THEN 1670

242 IF X(11) > 290## THEN 1670

 

244 IF X(12) < 100## THEN 1670

245 IF X(12) > 330## THEN 1670

 

246 X(1) = 10 – X(2)

247 X(3) = X(1) + X(6)

249 X(4) = X(2) + X(5)
251 X(7) = X(3) – X(5)

 

253 X(8) = X(4) – X(6)

 

263 X(9) = (100 * X(1) + X(10) * X(6)) / X(3)

 

267 X(11) = (100 * X(2) + X(10) * X(5)) / X(4)

269 X(12) = (1000 + X(4) * X(11)) / X(4)

326 FOR J44 = 1 TO 8
327 IF X(J44) < 0## THEN 1670
328 IF X(J44) > 10## THEN 1670

329 NEXT J44

 

330 IF X(9) < 100## THEN 1670

334 IF X(9) > 290# THEN 1670

336 IF X(10) < 100## THEN 1670

338 IF X(10) > 310## THEN 1670

 

340 IF X(11) < 100## THEN 1670

342 IF X(11) > 290## THEN 1670

 

344 IF X(12) < 100## THEN 1670

345 IF X(12) > 330## THEN 1670

 

453 IF (320 – X(10)) * (300 – X(9)) < 0 THEN 1670
454 IF (340 – X(12)) * (300 – X(11)) < 0 THEN 1670

455 down1 = (2.5 * ((.6666666666666666 * ((320 – X(10)) * (300 – X(9)) ^ .5) + ((320 – X(10)) + (300 – X(9))) / 6)))

458 down2 = (.2 * ((.6666666666666666 * ((340 – X(12)) * (300 – X(11)) ^ .5) + ((340 – X(12)) + (300 – X(11))) / 6)))

464 POBA = -1200 * (800 / down1) ^ .6 – 1200 * (1000 / down2) ^ .6 – 1000000 * ABS(X(3) * (X(10) – X(9)) – 800)

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 12

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

1670 NEXT I
1889 IF M < -5674 THEN 1999

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8)
1908 PRINT A(9), A(10), A(11), A(12), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [54]. The complete output through JJJJ = -31999.85000000002 is shown below:

10       0       10       10       10
0       0       10
100       180       180       280       -5469.56415818952
-31999.99

10       0       10       10       10
0       0       10
100       180       180       280       -5469.56415818952
-31999.94000000001

10       0       10       10       10
0       0       10
100       180       180       280       -5469.56415818952
-31999.85000000002

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 [54], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.85000000002
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 p. 565 of Ryoo and Sahinidis [43, p. 565, Example 16].

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, Hao-Chun Lu, Chia-Hui Huang, Nian-Ze Hu (2009). A superior representation method for piecewise lineat functions. INFORMS Journal on Computing 21 (2): 314-321 (2009).

[30] 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.
[31] 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.
[32] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[33] 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.
[34] 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.
[35] 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.

[36] 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.
[37] 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/
[38] 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.

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

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.
[47] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[48] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[49] 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
[50] 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.
[51] 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.

[52] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chai-Chien Lee (2010). An efficient global optimization approach for solving mixed-integer nonlinear programming problems. Proceedings of the 40th International Conference on Computers and Industrian Engineering, Japan, pp.1-4, 2010.
Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.

[53] Pei-Chun Wang, Jung-Fa Tsai (2011). Global optimization of mixed-integer nonlinear programming for engineering design problems. Proceedings of 2011 International Conference on System Science and Engineeing, Macau, China – June 2011.

[54] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[55] 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.
[56] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving the Heat Exchanger Network Problem in Ryoo and Sahinidis [43]

 

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Ryoo and Sahinidis [43, p. 565, Example 16], which is as follows:

Minimize 1200 * (800 / down1) ^ .6 + 1200 * (1000 / down2) ^ .6

where down1 = 2.5 * (.666666666666666666666666 * (320 – X(10)) * (300 – X(9)) ^ .5 + ((320 – X(10)) + (300 – X(9))) / 6),

down2 = .2 * (.666666666666666666666666 * (340 – X(12)) * (300 – X(11)) ^ .5 + ((340 – X(12)) + (300 – X(11))) / 6)

subject to

X(1) + X(2)=10

X(1) + X(6)=X(3)

X(2) + X(5)=X(4)
X(5) + X(7)=X(3)
X(6) + X(8)=X(4)

100 * X(1) + X(10) * X(6) =X(9)*X(3)
100 * X(2) + X(10) * X(5) =X(11)*X(4)

X(3)* (X(10) – X(9)=800

X(4)* (X(12) – X(11))=1000

0<= X(i) <=10, i=1, 2, 3,…, 8

100<= X(9) <= 290
100<= X(10) <= 310

100<= X(11) <= 290
100<= X(12) <= 330.

The purpose of the sequence of line 246 through line 269 is to produce some domino effect.

 

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 -31997.7 STEP .01

 

14 RANDOMIZE JJJJ

16 M = -1D+37
77 FOR J44 = 1 TO 8
79 A(J44) = INT(RND * 10)
82 NEXT J44

89 A(9) = 100 + INT(RND * 190)

90 A(10) = 100 + INT(RND * 210)

91 A(11) = 100 + INT(RND * 190)

 

92 A(12) = 100 + INT(RND * 230)

 

128 FOR I = 1 TO 20000

 

129 FOR KKQQ = 1 TO 12

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

181 j = 1 + FIX(RND * 12)
185 REM IF RND < .5 THEN X(j) = A(j) – FIX(RND * 10) ELSE X(j) = A(j) + FIX(RND * 10)
189 r = (1 – RND * 2) * A(j)

190 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP
223 REM FOR J44 = 1 TO 12
224 REM X(J44) = INT(X(J44))

225 REM NEXT J44

 

226 FOR J44 = 1 TO 8
227 IF X(J44) < 0## THEN 1670
228 IF X(J44) > 10## THEN 1670

229 NEXT J44

 

230 IF X(9) < 100## THEN 1670

234 IF X(9) > 290# THEN 1670

236 IF X(10) < 100## THEN 1670

238 IF X(10) > 310## THEN 1670

 

240 IF X(11) < 100## THEN 1670

242 IF X(11) > 290## THEN 1670

 

244 IF X(12) < 100## THEN 1670

245 IF X(12) > 330## THEN 1670

 

246 X(1) = 10 – X(2)

247 X(3) = X(1) + X(6)

249 X(4) = X(2) + X(5)
251 X(7) = X(3) – X(5)

 

253 X(8) = X(4) – X(6)

 

263 X(9) = (100 * X(1) + X(10) * X(6)) / X(3)

 

267 X(11) = (100 * X(2) + X(10) * X(5)) / X(4)

269 X(12) = (1000 + X(4) * X(11)) / X(4)

326 FOR J44 = 1 TO 8
327 IF X(J44) < 0## THEN 1670
328 IF X(J44) > 10## THEN 1670

329 NEXT J44

 

330 IF X(9) < 100## THEN 1670

334 IF X(9) > 290# THEN 1670

336 IF X(10) < 100## THEN 1670

338 IF X(10) > 310## THEN 1670

 

340 IF X(11) < 100## THEN 1670

342 IF X(11) > 290## THEN 1670

 

344 IF X(12) < 100## THEN 1670

345 IF X(12) > 330## THEN 1670

 

453 IF (320 – X(10)) * (300 – X(9)) < 0 THEN 1670
454 IF (340 – X(12)) * (300 – X(11)) < 0 THEN 1670

455 down1 = (2.5 * ((.6666666666666666 * ((320 – X(10)) * (300 – X(9)) ^ .5) + ((320 – X(10)) + (300 – X(9))) / 6)))

458 down2 = (.2 * ((.6666666666666666 * ((340 – X(12)) * (300 – X(11)) ^ .5) + ((340 – X(12)) + (300 – X(11))) / 6)))

464 POBA = -1200 * (800 / down1) ^ .6 – 1200 * (1000 / down2) ^ .6 – 1000000 * ABS(X(3) * (X(10) – X(9)) – 800)

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 12

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

1670 NEXT I
1889 IF M < -5175 THEN 1999

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8)
1908 PRINT A(9), A(10), A(11), A(12), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [54]. The complete output through JJJJ = -31999.90000000002 is shown below:

4.20006340987803         5.79993659012197          4.200063409879566
5.79993659012197          6.222452646496241D-17          1.535673128140334D-12
4.200063409879566          5.799936590120434
100.0000000000696          290.4733147881765          100
272.4156780788134          -5173.82562202685          -31999.90000000002

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [54], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999.90000000002
was 4 seconds, not including the time for “Creating .EXE file” (12 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on p. 565 of Ryoo and Sahinidis [43, p. 565, Example 16].

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, Hao-Chun Lu, Chia-Hui Huang, Nian-Ze Hu (2009). A superior representation method for piecewise lineat functions. INFORMS Journal on Computing 21 (2): 314-321 (2009).

[30] 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.
[31] 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.
[32] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[33] 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.
[34] 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.
[35] 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.

[36] 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.
[37] 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/
[38] 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.

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

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.
[47] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[48] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[49] 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
[50] 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.
[51] 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.

[52] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chai-Chien Lee (2010). An efficient global optimization approach for solving mixed-integer nonlinear programming problems. Proceedings of the 40th International Conference on Computers and Industrian Engineering, Japan, pp.1-4, 2010.
Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.

[53] Pei-Chun Wang, Jung-Fa Tsai (2011). Global optimization of mixed-integer nonlinear programming for engineering design problems. Proceedings of 2011 International Conference on System Science and Engineeing, Macau, China – June 2011.

[54] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[55] 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.
[56] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Using Discrete/Integer Variables To Approximate the 10 Continuous Variables of the Pooling Problem in Ryoo and Sahinidis [43]

 

Jsun Yui Wong

The following computer program adds the restriction that X(1) through X(10) are general integer variables–see line 224–to the following problem in Ryoo and Sahinidis [43, p. 564, Example 7], which is as follows:

Minimize – 9 * X(5) – 15 * X(9) + 6 * X(1) + 16 * X(2) + 10 * X(6)

subject to

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

X(3) + X(7)=X(5)

X(4) +X(8)= X(9)

X(7) + X(8)=X(6)

X(10) * X(3) + 2 * X(7)<=2.5*X(5)

X(10) * X(4) + 2 * X(8)<=1.5*X(9)

3 * X(1) + X(2) =X(10)* (X(3) + X(4))

0<= X(1) <= 300

0<= X(2) <= 300

0<= X(3) <= 100

0<= X(4) <= 200

0<= X(5) <= 100

0<= X(6) <= 300

0<= X(7) <= 100

0<= X(8) <= 200

0<= X(9) <= 200

1<= X(10) <= 3.

X(11) through X(12) below are slack variables added.

One notes line 224, which is 224 X(J44) = INT(X(J44)).

The reasoning behind the sequence of line 246 through line 266 is to produce some domino effect.

 

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 -31990.0321111 STEP .01

14 RANDOMIZE JJJJ

16 M = -1D+37

84 A(1) = 0 + RND * 300

85 A(2) = 0 + RND * 300

86 A(3) = 0 + RND * 100

87 A(4) = 0 + RND * 200

88 A(5) = 0 + RND * 100

89 A(6) = 0 + RND * 300

90 A(7) = 0 + RND * 100

91 A(8) = 0 + RND * 200

 

92 A(9) = 0 + RND * 200

93 A(10) = 1 + RND * 2

 

128 FOR I = 1 TO 20000

 

129 FOR KKQQ = 1 TO 10

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

181 j = 1 + FIX(RND * 10)

189 r = (1 – RND * 2) * A(j)

190 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP
223 FOR J44 = 1 TO 10
224 X(J44) = INT(X(J44))

225 NEXT J44

 

226 FOR J44 = 1 TO 9
227 IF X(J44) < 0## THEN 1670

229 NEXT J44
230 IF X(10) < 1## THEN 1670

234 IF X(10) > 3## THEN 1670

235 IF X(1) > 300## THEN 1670

 

236 IF X(2) > 300## THEN 1670

 

237 IF X(3) > 100## THEN 1670

 

238 IF X(4) > 200## THEN 1670

 

239 IF X(5) > 100## THEN 1670
240 IF X(6) > 3000## THEN 1670

 

241 IF X(7) > 100## THEN 1670

 

242 IF X(8) > 200## THEN 1670

 

243 IF X(9) > 200## THEN 1670

 

246 X(6) = X(7) + X(8)

247 X(3) = X(5) – X(7)

249 X(4) = X(9) – X(8)
251 X(1) = X(3) + X(4) – X(2)

261 IF ((3 * X(1) + X(2)) / ((X(3) + X(4)))) < 1 THEN 1670

264 IF ((3 * X(1) + X(2)) / ((X(3) + X(4)))) > 3 THEN 1670

266 X(10) = ((3 * X(1) + X(2)) / ((X(3) + X(4))))

 

447 X(11) = 2.5 * X(5) – X(10) * X(3) – 2 * X(7)
448 X(12) = 1.5 * X(9) – X(10) * X(4) – 2 * X(8)

 

450 FOR J47 = 11 TO 12

 

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

 

452 NEXT J47

 

453 POBA = 9 * X(5) + 15 * X(9) – 6 * X(1) – 16 * X(2) – 10 * X(6) + 1000000 * (X(11) + X(12))

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 12

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

1670 NEXT I
1889 IF M < 395 THEN 1999

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8)
1908 PRINT A(9), A(10), A(11), A(12), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [54]. The complete output through JJJJ = -31994.69000000085 is shown below:

0       100       0       100       0
100       0       100
200       1       0       0       400
-31998.74000000002

0       100       0       100       0
100       0       100
200       1       0       0       400
-31995.83000000067

0       100       0       100       0
100       0       100
200       1       0       0       400
-31994.93000000081

0       99       0       99       0
99      0      99
198       1       0       0       396
-31994.69000000085

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 [54], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31994.69000000085
was 37 seconds, not including the time for “Creating .EXE file” (45 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on p. 564 of Ryoo and Sahinidis [43, p. 564, Example 7].

Acknowledgment

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

References

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

[6] 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, Hao-Chun Lu, Chia-Hui Huang, Nian-Ze Hu (2009). A superior representation method for piecewise lineat functions. INFORMS Journal on Computing 21 (2): 314-321 (2009).

[30] 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.
[31] 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.
[32] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[33] 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.
[34] 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.
[35] 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.

[36] 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.
[37] 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/
[38] 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.

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

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.
[47] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[48] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[49] 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
[50] 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.
[51] 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.

[52] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chai-Chien Lee (2010). An efficient global optimization approach for solving mixed-integer nonlinear programming problems. Proceedings of the 40th International Conference on Computers and Industrian Engineering, Japan, pp.1-4, 2010.
Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.

[53] Pei-Chun Wang, Jung-Fa Tsai (2011). Global optimization of mixed-integer nonlinear programming for engineering design problems. Proceedings of 2011 International Conference on System Science and Engineeing, Macau, China – June 2011.

[54] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[55] 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.
[56] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Solving a Test Problem in Ryoo and Sahinidis [43] with Mixed Integer Nonlinear Programming

 

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Ryoo and Sahinidis [43, p. 565, Example 15]:

Minimize 2 * X(1) + 3 * X(2) + 1.5 * X(3) + 2 * X(4) – .5 * X(5)

subject to

X(1)^2+ X(3)=1.25

X(2)^1.5+ 1.5 * X(4)=3

X(1) + X(3)<=1.6

1.333 * X(2) + X(4)<=3

– X(3) – X(4) + X(5)<=0

0<=X(1) < = 1.12

0<=X(2) < = 2.10

0<=X(3) < = 1

0<=X(4) < = 1

0<= X(5) <=1,

where X(1) and X(2) are continuous variables, and X(3) through X(5) are 0-1 integer variables.

X(6) through X(8) 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

84 A(1) = 0 + (RND * 1.12)

85 A(2) = 0 + (RND * 2.10)

86 FOR J44 = 3 TO 5

87 IF RND < .5 THEN A(J44) = 0 ELSE A(J44) = 1

96 NEXT J44

128 FOR I = 1 TO 10000

129 FOR KKQQ = 1 TO 5

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

181 j = 1 + FIX(RND * 5)
184 IF j > 3 THEN GOTO 201

189 r = (1 – RND * 2) * A(j)

190 X(j) = A(j) + (RND ^ (RND * 10)) * r
195 GOTO 222

 

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

222 NEXT IPP

 

224 FOR J44 = 1 TO 5
227 IF X(J44) < 0## THEN 1670

229 NEXT J44

231 FOR J44 = 3 TO 5
232 X(J44) = INT(X(J44))

 

233 NEXT J44

236 IF X(1) > 1.12## THEN 1670

 

237 IF X(2) > 2.10## THEN 1670

 

238 IF X(3) > 1## THEN 1670

 

241 IF X(4) > 1## THEN 1670

 

243 IF X(5) > 1## THEN 1670

 

244 X(1) = (1.25 – X(3)) ^ .5
245 X(2) = (3 – 1.5 * X(4)) ^ .66666666666666

246 GOTO 444

 

247 FOR J44 = 1 TO 5
248 IF X(J44) < 0## THEN 1670

249 NEXT J44

276 IF X(1) > 1.12## THEN 1670

 

286 IF X(2) > 2.10## THEN 1670

 

296 IF X(3) > 1## THEN 1670

 

298 IF X(4) > 1## THEN 1670

 

300 IF X(5) > 1## THEN 1670

444 FOR J44 = 6 TO 8

446 X(6) = 1.6 – X(1) – X(3)
447 X(7) = 3 – 1.333 * X(2) – X(4)
448 X(8) = 0 + X(3) + X(4) – X(5)

 

449 NEXT J44

 

450 FOR J47 = 6 TO 8

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

 

452 NEXT J47

 

453 POBA = -2 * X(1) – 3 * X(2) – 1.5 * X(3) – 2 * X(4) + .5 * X(5) + 1000000 * (X(6) + X(7) + X(8))

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 8

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

1670 NEXT I
1889 REM IF M < 11 THEN 1999

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8)

1908 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [54]. The complete output through JJJJ = -31999.99 is shown below:

1.118033988749895          1.310370697104445          0
1          1          0          0          0
-7.667180068813124          -32000

1.118033988749895          1.310370697104445          0
1          1          0          0          0
-7.667180068813124          -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 [54], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31999.99 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 on p. 565 of Ryoo and Sahinidis [43, p. 565, Example 15].

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, Hao-Chun Lu, Chia-Hui Huang, Nian-Ze Hu (2009). A superior representation method for piecewise lineat functions. INFORMS Journal on Computing 21 (2): 314-321 (2009).

[30] 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.
[31] 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.
[32] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[33] 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.
[34] 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.
[35] 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.

[36] 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.
[37] 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/
[38] 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.

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

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.
[47] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[48] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[49] 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
[50] 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.
[51] 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.

[52] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chai-Chien Lee (2010). An efficient global optimization approach for solving mixed-integer nonlinear programming problems. Proceedings of the 40th International Conference on Computers and Industrian Engineering, Japan, pp.1-4, 2010.
Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.

[53] Pei-Chun Wang, Jung-Fa Tsai (2011). Global optimization of mixed-integer nonlinear programming for engineering design problems. Proceedings of 2011 International Conference on System Science and Engineeing, Macau, China – June 2011.

[54] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[55] 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.
[56] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

 

Nonlinear Programming for Design of Three-Stage Process System with Recycle

 

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Ryoo and Sahinidis [43, p. 566, Example 21]:

Minimize    X(1) ^ .6 + X(2) ^ .6 + X(3) ^ .4 – 4 * X(3) + 2 * X(4) + 5 * X(5) – X(6)

subject to

– 3 * X(1)+X(2) – 3 * X(4)=0

-2 * X(2) +X(3)- 2 * X(5)=0

4 * X(4)-X(6)=0

X(1) + 2 * X(4)<=4

X(2) + X(5)<=4

X(3) + X(6)<=6

0<=X(1) < = 3

0<=X(2) < = 4

0<=X(3) < = 4

0<=X(4) < = 2

0<= X(5) <= 2

0<= X(6) <= 6.

X(7) through X(9) 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

84 A(1) = 0 + (RND * 3)

85 A(2) = 0 + (RND * 4)

86 A(3) = 0 + (RND * 4)

88 A(4) = 0 + (RND * 2)
90 A(5) = 0 + (RND * 2)

98 A(6) = 0 + (RND * 6)

 

128 FOR I = 1 TO 50000

 

129 FOR KKQQ = 1 TO 6

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

181 j = 1 + FIX(RND * 6)

189 r = (1 – RND * 2) * A(j)

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

222 NEXT IPP
228 FOR J44 = 1 TO 6
230 IF X(J44) < 0## THEN 1670

231 NEXT J44

 

233 X(6) = 4 * X(4)
236 X(2) = 3 * X(1) + 3 * X(4)
239 X(3) = 2 * X(2) + 2 * X(5)

 

244 FOR J44 = 1 TO 6
247 IF X(J44) < 0## THEN 1670

249 NEXT J44

 

276 IF X(1) > 3## THEN 1670

 

286 IF X(2) > 4## THEN 1670

 

296 IF X(3) > 4## THEN 1670

 

298 IF X(4) > 2## THEN 1670

 

300 IF X(5) > 2## THEN 1670

311 IF X(6) > 60## THEN 1670

321 X(7) = 4 – X(1) – 2 * X(4)
322 X(8) = 4 – X(2) – X(5)

326 X(9) = 6 – X(3) – X(6)

401 FOR J47 = 7 TO 9

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

 

409 NEXT J47

 

451 POBA = -X(1) ^ .6 – X(2) ^ .6 – X(3) ^ .4 + 4 * X(3) – 2 * X(4) – 5 * X(5) + X(6) + 1000000 * (X(7) + X(8) + X(9))

 

466 P = POBA

1111 IF P <= M THEN 1670

 

1450 M = P
1454 FOR KLX = 1 TO 9

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

1670 NEXT I
1889 IF M < 11 THEN 1999

1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9)

1908 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [54]. The complete output through JJJJ = -31999.96000000001 is shown below:

.166666680865126          1.99999999892839          4
.4999999854443373          1.071610106530635D-09          1.99999994177735
0          0          0
13.40190350372444          -31999.97000000001

.1666708005128221          1.999999998963028          4
.4999958658081871          1.036972537900687D-09          1.999983463232748
0          0          0
13.40189020312117           -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 [54], the wall-clock time (not CPU time) for obtaining the output through
JJJJ = -31999.96000000001 was 9 seconds, not including the time for “Creating .EXE file” (17 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those on p. 566 of Ryoo and Sahinidis [43, p. 566].

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, Hao-Chun Lu, Chia-Hui Huang, Nian-Ze Hu (2009). A superior representation method for piecewise lineat functions. INFORMS Journal on Computing 21 (2): 314-321 (2009).

[30] 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.
[31] 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.
[32] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[33] 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.
[34] 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.
[35] 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.

[36] 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.
[37] 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/
[38] 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.

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

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.
[47] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[48] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[49] 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
[50] 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.
[51] 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.

[52] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chai-Chien Lee (2010). An efficient global optimization approach for solving mixed-integer nonlinear programming problems. Proceedings of the 40th International Conference on Computers and Industrian Engineering, Japan, pp.1-4, 2010.
Date added to IEEE Xplore: 13 December 2010. Publisher:IEEE.

[53] Pei-Chun Wang, Jung-Fa Tsai (2011). Global optimization of mixed-integer nonlinear programming for engineering design problems. Proceedings of 2011 International Conference on System Science and Engineeing, Macau, China – June 2011.

[54] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[55] 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.
[56] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.