J. Smith (1985) Problem 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 * (-50000)), bbb = 1.344D+09, and 100<=X(1)<=1000.

The above is Floudas et al.'s case with delta H=-50000.
 
     
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 * (-50000))

        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 =  -31999.60000000006 is summarized below:


347.3178415262124          -1.230173977224622D-09          -31999.97000000001
.
.
.

445.495508123506            -2.307041812087216D-06          -31999.64000000006
.
.
.

300.4328148991304          -1.424801611329268D-10          -31999.60000000006

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 =  -31999.60000000006 was 1 second, not including the time for “Creating .EXE file” (7 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 GlobalOptimization.  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.

Avriel and Williams (1971) Heat Exchanger Network Problem in Floudas et al. [7]

Jsun Yui Wong

The computer program listed below seeks to solve the following problem of eight continuous variables from Floudas et al. [7, pp. 51-52]:

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

subject to

        X(4) + X(6) - 100 - 300<=0

        - X(4) + X(5) + X(7) - 300<=0

        X(8) -X(5 -600+ 500<=0

        X(1) - X(1) * X(6) + ((1D+05) / 120) * X(4) - ((1D+05) / 120) * 100<=0

        X(2) * X(4) - X(2) * X(7) - ((1D+05) / 80) * X(4) + ((1D+05) / 80) * X(5)<=0


        X(3) * X(5) - X(3) * X(8) - ((1D+05) / 40) * X(5) + ((1D+05) / 40) * 500<=0 

        100<= X(1) <= 10000       
       1000<= X(2) <= 10000
     
       1000<= X(3) <= 10000

 10<= X(i) <= 1000, i=4, 5, 6, 7, 8.

 X(9) through X(14) below are slack variables.


0 DEFDBL A-Z

2 DEFINT K

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

12 FOR JJJJ = -32000 TO 32000 STEP .01

    14 RANDOMIZE JJJJ

    16 M = -1D+37

    92 A(1) = 100 + FIX(RND * 9901)
    93 A(2) = 1000 + FIX(RND * 9001)
    94 A(3) = 1000 + FIX(RND * 9001)


    95 A(4) = 10 + FIX(RND * 991)
    96 A(5) = 10 + FIX(RND * 991)

    97 A(6) = 10 + FIX(RND * 991)

    98 A(7) = 10 + FIX(RND * 991)
    99 A(8) = 10 + FIX(RND * 991)


    128 FOR I = 1 TO 20000


        129 FOR KKQQ = 1 TO 8

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

            181 j = 1 + FIX(RND * 8)

            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) > 10000## THEN 1670

        232 IF X(2) < 1000## THEN 1670
        233 IF X(2) > 10000## THEN 1670
        234 IF X(3) < 1000## THEN 1670


        235 IF X(3) > 10000## THEN 1670


        236 FOR J44 = 4 TO 8
            237 IF X(J44) < 10 THEN GOTO 1670
            238 IF X(J44) > 1000 THEN GOTO 1670


        244 NEXT J44


        246 X(9) = -X(4) - X(6) + 100 + 300


        247 X(10) = X(4) - X(5) - X(7) + 300



        248 X(11) = -X(8) + X(5) + 600 - 500

        251 X(12) = -X(1) + X(1) * X(6) - ((1D+05) / 120) * X(4) + ((1D+05) / 120) * 100

        253 X(13) = -X(2) * X(4) + X(2) * X(7) + ((1D+05) / 80) * X(4) - ((1D+05) / 80) * X(5)


        255 X(14) = -X(3) * X(5) + X(3) * X(8) + ((1D+05) / 40) * X(5) - ((1D+05) / 40) * 500
        260 IF X(1) < 100 THEN 1670

        261 IF X(1) > 10000## THEN 1670
        262 IF X(2) < 1000## THEN 1670
        263 IF X(2) > 10000## THEN 1670
        264 IF X(3) < 1000## THEN 1670


        265 IF X(3) > 10000## THEN 1670


        266 FOR J44 = 4 TO 8
            267 IF X(J44) < 10 THEN GOTO 1670
            268 IF X(J44) > 1000 THEN GOTO 1670


        269 NEXT J44


        450 FOR J47 = 9 TO 14


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



        452 NEXT J47


        453 POBA = -X(1) - X(2) - X(3) + 1000000 * (X(13) + X(14) + 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 14

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

    1670 NEXT I
    1889 IF M < -7512.27 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), A(13), A(14), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [54]. The complete output through JJJJ =  -31990.270000000156 is shown below:

1018.685845387        1000                        5493.58146949264
264.4621419608414   280.256741275952   135.5378318428512
284.2054002855197   380.2567412506235
0   0   0   0   0
0      -7512.267314879637                        -31999.24000000012

1022.80236632016      1000                        5489.437573718315
264.7605209044411   280.4224971073827   135.2394568180514
284.3380143800206   380.4224970818268
0   0   0   0   0
0      -7512.239940038475                          -31995.11000000078

1021.27990286284      1000                        5490.96997836527
264.6502811873032   280.3612014084006   135.3496207763103
284.2890692001287   380.3612011611714
0   0   0   0   0
0      -7512.249881228108                         -31990.93000000145

1024.91756653315         1000                                        5487.316568929566
264.913320300725                   280.5073372429737        135.0866538000017
284.4058796825021                 380.5073372429025
0      0      0      0      0
0         -7512.234135462716                                            -31990.270000000156

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 =  -31990.270000000156 was 47 minutes, total, including the time for “Creating .EXE file.”  One can compare the computational results above with those on p. 52 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] 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 3 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) - 13 * 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) - 13 * 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 < 746 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 =   -31999.62000000006 is shown below:

50      150      0      0      0
200      1.5      0
200      0      0      750       -31999.92000000001

50      150      0      0      0
200      1.5      0
200      0      0      750       -31999.89000000002

50      150      0      0      0
200      1.5      0
200      0      0      750       -31999.62000000006

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 =   -31999.62000000006
was 8 seconds, not including the time for “Creating .EXE file”  (15 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 3]. 
       
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 Design Problem 2 in Visweswaran and Floudas [52]

Jsun Yui Wong

The computer program listed below seeks to solve the following problem in Visweswaran and Floudas [52, pp. 52-54, Test Problem 2], which is as follows:

Minimize           1300 * (1000 / down1) ^ .6 + 1300 * (600 / down2) ^ .6

where down1 = .0333333333333333333333333 * (X(1) * X(2)) + (X(1) + X(2)) / 6,

down2 = .0333333333333333333333333 * (X(3) * X(4)) + (X(3) + X(4)) / 6

subject to

         X(9) + X(13)=10

        X(9)+ X(15) - X(10)    =0


        X(13) +x(11)- X(14)=0
        X(12)  + X(11) - X(10)=0


         X(16) +x(15)- X(14) =0


         150 * X(9) + X(8) * X(15) - X(5) * X(10)=0


          150 * X(13) + X(7) * X(11) - X(6) * X(14)=0

         X(10) * (X(7) - X(5))=1000

         X(14) * (X(8) - X(6))=600


         X(1) + X(7)=500

         X(2) + X(5)=250


         X(3) + X(8)=350
         X(4) + X(6)=200   

 
        10<= X(1) <= 350
        10<= X(2) <= 350

        10<= X(3) <= 200
        10<= X(4) <= 200

           150<= X(i) <=310, i=5, 6, 7, 8
     
 
          0<=X(i) <=10, i=9, 10, 11,..., 16.
           

The purpose of the sequence of line 247 through line 268 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


    95 FOR J44 = 1 TO 2
        96 A(J44) = 10 + RND * 340
    98 NEXT J44



    101 FOR J44 = 3 TO 4
        104 A(J44) = 10 + RND * 190
    105 NEXT J44


    111 FOR J44 = 5 TO 8
        114 A(J44) = 150 + RND * 160
    115 NEXT J44


    121 FOR J44 = 9 TO 16
        124 A(J44) = 0 + RND * 10
    125 NEXT J44


    128 FOR I = 1 TO 12000



        129 FOR KKQQ = 1 TO 16

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

            181 j = 1 + FIX(RND * 16)
            185 REM IF RND < .5 THEN X(j) = A(j) - FIX(RND * 6) ELSE X(j) = A(j) + FIX(RND * 6)
            189 r = (1 - RND * 2) * A(j)
            190 X(j) = A(j) + (RND ^ (RND * 10)) * r
        222 NEXT IPP



        226 FOR J44 = 1 TO 16
            227 X(J44) = INT(X(J44))
        229 NEXT J44

        231 FOR J44 = 1 TO 2
            232 IF X(J44) < 10## THEN 1670
            233 IF X(J44) > 350## THEN 1670

        234 NEXT J44



        235 FOR J44 = 3 TO 4
            236 IF X(J44) < 10## THEN 1670
            237 IF X(J44) > 200## THEN 1670

        238 NEXT J44


        239 FOR J44 = 5 TO 8
            240 IF X(J44) < 150## THEN 1670
            241 IF X(J44) > 310## THEN 1670

        242 NEXT J44

        243 FOR J44 = 9 TO 16
            244 IF X(J44) < 0## THEN 1670
            245 IF X(J44) > 10## THEN 1670

        246 NEXT J44


        247 X(1) = 500 - X(7)

        248 X(2) = 250 - X(5)

        252 X(10) = 1000 / (X(7) - X(5))

        253 X(3) = 350 - X(8)
        254 X(4) = 200 - X(6)


        255 X(14) = 600 / (X(8) - X(6))


        263 X(9) = 10 - X(13)
        264 X(11) = 0 - X(13) + X(14)
        265 X(12) = 0 + X(10) - X(11)

        267 X(15) = 0 - X(9) + X(10)
        268 X(16) = 0 + X(14) - X(15)


        300 FOR J44 = 1 TO 2
            301 IF X(J44) < 10## THEN 1670
            302 IF X(J44) > 350## THEN 1670

        304 NEXT J44



        310 FOR J44 = 3 TO 4
            311 IF X(J44) < 10## THEN 1670
            312 IF X(J44) > 200## THEN 1670

        314 NEXT J44


        320 FOR J44 = 5 TO 8
            321 IF X(J44) < 150## THEN 1670
            322 IF X(J44) > 310## THEN 1670

        324 NEXT J44


        326 FOR J44 = 9 TO 16
            327 IF X(J44) < 0## THEN 1670
            328 IF X(J44) > 10## THEN 1670

        329 NEXT J44


        455 down1 = .0333333333333333333333333 * (X(1) * X(2)) + (X(1) + X(2)) / 6##

        460 down2 = .0333333333333333333333333 * (X(3) * X(4)) + (X(3) + X(4)) / 6##


        464 POBA = -1300## * (1000 / down1) ^ .6## - 1300## * (600 / down2) ^ .6## - 1000000## * ABS(150 * X(9) + X(8) * X(15) - X(5) * X(10)) - 1000000## * ABS(150 * X(13) + X(7) * X(11) - X(6) * X(14))


        466 P = POBA

        1111 IF P <= M THEN 1670


        1450 M = P
        1454 FOR KLX = 1 TO 16

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

    1670 NEXT I
    1889 IF M < -4945 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), A(13), A(14), A(15), A(16), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [55]. The complete output through JJJJ =  -31997.34000000043 is shown below:

190    40       140      50       210
150    310     210
0       10        0         10       10
10     10        0        -4845.462004958673
-31998.44000000025

190    40       140      50       210
150    310     210
0       10        0         10       10
10     10        0        -4845.462004958673
-31997.34000000043 

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 = -31997.34000000043 
was 16 seconds, not including the time for “Creating .EXE file”  (25 seconds, total, including the time for “Creating .EXE file”).  One can compare the computational results above with those on p. 54 of Visweswaran and Floudas [52, p. 54, Test Problem 2]. 
       
Acknowledgment

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

References

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

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

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
[9] Amir Hossein Gandomi, Xin-She Yang, Siamak Taratahari, Amir Hossein Alavi (2013). Firefly Algorithm with Chaos, Communications in Nonlinear Science and Numerical Simulation 18 (2013) 89-98.
[10] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:17-35.
[11] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2013). Erratum to: Cuckoo search algorithm: a metaheuristicapproach to solve structural optimization problem. Engineering with Computers (2013) 29:245.
[12] Chrysanthos E. Gounaris, Christodoulos A. Floudas. Tight convex underestimators for Csquare-continuous problems: II. multivariate functions. Journal of Global Optimization (2008) 42, pp. 69-89.

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

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

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

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

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

[18] Majid Jaberipour, Esmaile Khorram (2010). Solving the sum of ratios problems by a harmony search algorithm. Journal of Computational and Applied Mathematics 234 (2010) 733-742.

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

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

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

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

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

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

[25] Han-Lin Li, Jung-Fa Tsai (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]  V. Visweswaran, C. A. Floudas (1996).  Computational results for an efficient implementation of the GOP algorithm and its variants.  In Grossmann I. E. (ed.), Global Optimization in Engineering Design, Kluwer Book Series in Nonconvex Optimization and Its Applications, Chapter 4.

[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 the Pooling Problem in Ryoo and Sahinidis [43]

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Ryoo and Sahinidis [43, p. 564, Example 7]:

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) and X(12) 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 * 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 100000


        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


        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 =  -31997.93000000033 is shown below:

2.941007286239028D-04      100.0002941070896             6.361036753569613D-09
100.0005882014572             6.361036753569613D-09      99.99941179854277
6.838581110170434D-52      99.99941179854277
200                                    1.000005881979974            0      0
399.999411754016             -31998.41000000026

4.358111960129918D-04      100.0004358212102               1.001421964597826D-08
100.000871622392               1.001421964597826D-08        99.99912837760802
5.282118877464382D-74       99.99912837760802
200                                     1.000005881979947            0      0
399.9991283075091              -31997.93000000033

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 =   -31997.93000000033
was 77 seconds, not including the time for “Creating .EXE file”  (85 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.

Heat Exchanger Network Design with Nonlinear Programming

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Ryoo and Sahinidis [43, p. 564, Example 5]:

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

subject to

        100000 * ( X(4)-100) = (120 *X(1)* (300 - X(4)) 

        100000 * (X(5) - X(4))) = 80 *X(2)* (400 - X(5))

        100000 * (500 - X(5)) = 40 *X(3)* (600-500)     

        0<=X(1) < = 15834 
       
        0<=X(2) < = 36250 

        0<=X(3) < = 10000 

        100<=X(4) < = 300 

        100<= X(5) <=400.     


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 * 15836)

    85 A(2) = 0 + (RND * 46250)

    86 A(3) = 0 + (RND * 10000)

    88 A(4) = 100 + (RND * 200)
    90 A(5) = 100 + (RND * 300)

    128 FOR I = 1 TO 50000


        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)


            189 r = (1 - RND * 2) * A(j)

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

        222 NEXT IPP



        224 FOR J44 = 1 TO 3
            227 IF X(J44) < 0## THEN 1670

        229 NEXT J44

        231 IF X(4) < 100## THEN 1670
        232 IF X(5) < 100## THEN 1670



        236 IF X(1) > 15834## THEN 1670



        237 IF X(2) > 36250## THEN 1670


        238 IF X(3) > 10000## THEN 1670


        241 IF X(4) > 300## THEN 1670


        243 IF X(5) > 400## THEN 1670


        244 X(3) = (100000 * (500 - X(5))) / (40 * (100))

        245 X(1) = (100000 * (-100 + X(4))) / (120 * (300 - X(4)))

        246 X(2) = (100000 * (-X(4) + X(5))) / (80 * (400 - X(5)))


        247 FOR J44 = 1 TO 3
            248 IF X(J44) < 0## THEN 1670

        249 NEXT J44

        257 IF X(4) < 100## THEN 1670
        259 IF X(5) < 100## THEN 1670


        276 IF X(1) > 15834## THEN 1670


        286 IF X(2) > 36250## THEN 1670


        296 IF X(3) > 10000## THEN 1670


        298 IF X(4) > 300## THEN 1670


        300 IF X(5) > 400## THEN 1670
        455 POBA = -X(1) - X(2) - X(3)

        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 REM IF M < 11 THEN 1999

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


    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:

579.3067378950841      1359.971236996797      5109.971297583114
182.0175994854676      295.6011480966754
-7049.249272475995      -32000

579.3067211553962      1359.971273893866      5109.971277426733
182.0175980873007      295.6011489029307
-7049.249272475995      -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 =  -31999.96000000001 was 2 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. 564 of Ryoo and Sahinidis [43, p. 564, Example 5]. 
       
Acknowledgment

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

References

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

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

[7] Piya Chootinan, Anthony Chen (2006). Constraint Handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research 33 (2006) 2263-2281.
[8] Amir Hossein Gandomi, Xin-She Yang, Amir Hossein Alavi (2011). Mixed variable structural optimization using Firefly Algorithm, Computers and Structures 89 (2011) 2325-2336.
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[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.