Sunday, April 29, 2018

Solving a Posynomial Programming Problem

Jsun Yui Wong

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

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

subject to

           1<= X(1) through X(5) <=100.

The problem is based on Example 1 in Lu, Li, Gounaris, and Floudas [32, p. 152]:


0 DEFDBL A-Z

2 DEFINT K

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

12 FOR JJJJ = -32000 TO -31999.9588888 STEP .01


    14 RANDOMIZE JJJJ

    16 M = -1D+37


    22 FOR j44 = 1 TO 5

        25 A(j44) = 1 + FIX(RND * 100)

    28 NEXT j44
 
    128 FOR i = 1 TO 3000


        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)

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


            191 GOTO 222


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


        222 NEXT IPP


        266 FOR j44 = 1 TO 5


            268 IF X(j44) < 1## THEN 1670


            269 IF X(j44) > 100## THEN 1670


        272 NEXT j44
        445 POBA = -X(1) * X(2) * X(3) * X(4) * X(5) + X(2) ^ .5 * X(4) ^ .5 + 3 * X(1) + X(5)

        466 P = POBA

        1111 IF P <= M THEN 1670


        1450 M = P
        1454 FOR KLX = 1 TO 5

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

    1670 NEXT i
    1889 REM IF M < -99999999 THEN 1999

    1907 PRINT A(1), A(2), A(3)

    1908 PRINT A(4), A(5), M, JJJJ
   
1999 NEXT JJJJ

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

99.99999999999722         1.000000000000556         1.000000000000017
1.000000000000067         1.000000000000007         201.99999999993
-32000

99.9999999999999           1.000000000005671         1.000000000000205
1.000000000000124         1                                    201.9999999994027
-31999.99

99.99999999999101         1.000000000000556         1.0000000000000327
1.000000000000781         1.000000000000007         201.99999999997141
-31999.98

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [51], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =-31999.98 was 3 seconds, not including the time for “Creating .EXE file”  (10 seconds, total, including the time for “Creating .EXE file”).  One can compare the computational results above with those in Table 1 of Lu, Li, Gounaris, and Floudas [32, p. 153]. 
       

Acknowledgment

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


References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[26] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[27] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.

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

[29] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[30] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[31] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[32] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[33] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[34] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.

[35] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[36] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[37] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

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

[39] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[40] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[41] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[42] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[43] C. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.

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

[45] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[46] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[47] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[48] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[49] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[50] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[51] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[52] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[53] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

Friday, April 27, 2018

Using Discrete Variables To Approximate the Continuous Variables of an Insulated Steel Tank Design

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Li and Tsai [25, p. 11, Example 2]:

Minimize         400 * X(1) ^ .9 + 1000 - 22 * (X(2) + 14.7) ^ 1.2 + X(4)

subject to

         X(2) = EXP(-3950 / (X(3) + 460) + 11.86)

       144*( 80 - X(3)   =X(1)*X(4)

         0<=X(1) <=15.1
     
         14.7<=X(2) <=94.2
       
        -459.67<=X(3) <=80
       
         0<=X(4).



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 -31980 STEP .01

    14 RANDOMIZE JJJJ

    16 M = -1D+37


    66 REM GOTO 111

    71 unitsize(1) = .001

    73 unitsize(2) = .001

    75 unitsize(3) = .001

    77 unitsize(4) = .001





    81 A(1) = 0 + FIX(RND * 1000) * unitsize(1)


    83 A(2) = 14.7 + FIX(RND * 1000) * unitsize(2)


    85 A(3) = -459.67 + FIX(RND * 1000) * unitsize(3)

    86 A(4) = 0 + FIX(RND * 1000) * unitsize(4)

    87 REM A(4) = 0 + FIX(RND * 15.1)
    89 GOTO 128

    111 A(1) = 0 + (RND * 15)


    112 A(2) = 14.7 + (RND * 79.5)

    113 A(3) = -459.67 + (RND * 539.67)
    114 A(4) = 0 + (RND * 10)

    128 FOR I = 1 TO 3000


        129 FOR KKQQ = 1 TO 4


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



            181 j = 1 + FIX(RND * 4)

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


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


        222 NEXT IPP

        225 X(3) = (144 * 80 - X(1) * X(4)) / 144

        227 IF (-3950 / (X(3) + 460) + 11.86) > 85 THEN GOTO 1670


        233 X(2) = EXP(-3950 / (X(3) + 460) + 11.86)


        268 IF X(1) < 0## THEN 1670


        269 IF X(1) > 15## THEN 1670
        278 IF X(2) < 14.7## THEN 1670
        289 IF X(2) > 94.2## THEN 1670
        291 IF X(3) < -459.67## THEN 1670
        293 IF X(3) > 80## THEN 1670
        295 IF X(4) < 0## THEN 1670
     
     
        443 POBA = -400 * X(1) ^ .9 - 1000 - 22 * (X(2) - 14.7) ^ 1.2 - X(4)


        466 P = POBA

        1111 IF P <= M THEN 1670


        1450 M = P
        1454 FOR KLX = 1 TO 4

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

    1670 NEXT I
    1889 IF M < -5194.867## THEN 1999


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

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [51]. The complete output through JJJJ = -31980.59000000311 is shown below:

3.469446951953614D-18      94.1778659402041      80
0      -5194.866244203784      -31990.85000000147

1.734723475976807D-18      94.1778659402041      80
8.673617379884036D-18      -5194.866244203783      -31985.74000000228

8.673617379884036D-19      94.1778659402041      80
0      -5194.866244203783      -31980.77000000308

1.734723475976807D-18      94.1778659402041      80
0      -5194.866244203783      -31980.59000000311

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [51], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =  -31980.59000000311 was 11 seconds, not including the time for “Creating .EXE file”  (19 seconds, total, including the time for “Creating .EXE file”).  One can compare the computational results above with those on p. 11 of Li and Tsai [25, Example 2]. 
       

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[26] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[27] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.

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

[29] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[30] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[31] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[32] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[33] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[34] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.

[35] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[36] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[37] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

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

[39] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[40] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[41] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[42] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[43] C. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.

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

[45] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[46] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[47] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[48] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[49] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[50] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[51] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[52] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[53] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

Thursday, April 26, 2018

Using Discrete Variables To Approximate Continuous Variables

Jsun Yui Wong

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

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

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

         2 * X(1) + X(2) - X(3) + X(4)<=6
     
        1<= X(1) <=5
       
        3<= X(2) <=7
       
        1<= X(3) <=10
       
        1<= X(4) <=5

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


0 DEFDBL A-Z

2 DEFINT K

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

12 FOR JJJJ = -32000 TO 32000 STEP .01

    14 RANDOMIZE JJJJ

    16 M = -1D+37


    71 unitsize(1) = .001

    73 unitsize(2) = .001

    75 unitsize(3) = .001

    77 unitsize(4) = .001


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

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

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


    128 FOR I = 1 TO 3000


        129 FOR KKQQ = 1 TO 4


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



            181 j = 1 + FIX(RND * 4)



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

        222 NEXT IPP



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


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

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

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


        403 NEXT J47

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


        466 P = POBA

        1111 IF P <= M THEN 1670


        1450 M = P
        1454 FOR KLX = 1 TO 6

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

    1670 NEXT I
    1889 IF M < 9.997 THEN 1999

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

1999 NEXT JJJJ


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

4.999999999999981      3.500000000000002      9.999999999999957
2.499999999999951      0      0                         9.997861910064664
-32000

4.999999999999986      3.49999999999997       9.99999999999996
2.499999999999954      0      0                         9.997861910064673
-31999.99

4.999999999999971      3.50000000000001       9.999999999999952
2.499999999999962      0      0                         9.997861910064643
-31999.98

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [51], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =  -31999.98 was 2 seconds, not including the time for “Creating .EXE file”  (10 seconds, total, including the time for “Creating .EXE file”).  One can compare the computational results above with those in Table 1 of Li and Tsai [25, p. 12]. 
       

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[26] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[27] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.

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

[29] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[30] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[31] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[32] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[33] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[34] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.

[35] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[36] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[37] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

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

[39] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[40] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[41] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[42] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[43] C. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.

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

[45] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[46] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[47] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[48] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[49] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[50] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[51] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[52] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[53] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

Wednesday, April 25, 2018

Solving a Signomial Discrete Programming Problem


Jsun Yui Wong

The computer program listed below seeks to solve the following signomial discrete optimization problem from Tsai, Li, and Hu [46, p. 619, Example 2]: 

Minimize              .6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2 + 3.1661 * X(1) ^ 2 * X(4) + 19.84 * X(1) ^ 2 * X(3)

subject to

        - X(1) + .0193 * X(3)<=0

        - X(2) + .00954 * X(3)<=0

        - 3.141592654 * X(3) ^ (2) * X(4) - (4 / 3) * 3.141592654 * X(3) ^ 3 + 750 * 1728<=0

         -240+X(4)<=0 

         1<=X(1)<=1.375

         0.625<=X(2)<=1

         48<=X(3)<=52

         90<=X(4)<=112

where X(1) and X(2) are discrete variables with discreteness of .0625, and X(3) and X(4) are integer variables.

X(5) through X(7) 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
    31 unitsize(1) = .0625

    33 unitsize(2) = .0625

    35 unitsize(3) = 1


    37 unitsize(4) = 1


    61 A(1) = 1 + FIX(RND * 7) * unitsize(1)

    63 A(2) = .625 + FIX(RND * 7) * unitsize(2)

    65 A(3) = 48 + FIX(RND * 5)


    67 A(4) = 90 + FIX(RND * 23)


    128 FOR I = 1 TO 3000


        129 FOR KKQQ = 1 TO 4


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


            181 j = 1 + FIX(RND * 4)


            183 REM r = (1 - RND * 2) * A(j)
            187 REM X(j) = A(j) + (RND ^ (RND * 10)) * r
            197 IF RND < .5 THEN X(j) = A(j) - FIX(RND * 3) * unitsize(j) ELSE X(j) = A(j) + FIX(RND * 3) * unitsize(j)

        222 NEXT IPP


        268 IF X(1) < 1## THEN 1670
        269 IF X(1) > 1.375## THEN 1670


        272 IF X(2) < .625## THEN 1670
        273 IF X(2) > 1## THEN 1670


        274 IF X(3) < 48## THEN 1670
        275 IF X(3) > 52## THEN 1670
        284 IF X(4) < 90## THEN 1670
        285 IF X(4) > 112## THEN 1670


        308 X(5) = X(1) - .0193 * X(3)


        309 X(6) = X(2) - .00954 * X(3)
        319 X(7) = 3.141592654 * X(3) ^ (2) * X(4) + (4 / 3) * 3.141592654 * X(3) ^ 3 - 750 * 1728

        401 FOR J47 = 5 TO 7

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


        403 NEXT J47

        443 POBA = -.6224 * X(1) * X(3) * X(4) - 1.7781 * X(2) * X(3) ^ 2 - 3.1661 * X(1) ^ 2 * X(4) - 19.84 * X(1) ^ 2 * X(3) + 1000000 * (X(5) + X(6) + X(7))


        466 P = POBA

        1111 IF P <= M THEN 1670


        1450 M = P
        1454 FOR KLX = 1 TO 7

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

    1670 NEXT I
    1889 IF M < -7080 THEN 1999


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

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [51]. The complete output through JJJJ =  -31999.96000000001 is shown below:

1     .625       51     91       0
0     0       -7079.0373125      -32000

1     .625       51     91       0
0     0       -7079.0373125      -31999.98

1     .625       51     91       0
0     0       -7079.0373125      -31999.97000000001

1     .625       51     91       0
0     0       -7079.0373125      -31999.96000000001

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [51], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =  -31999.96000000001 was 2 seconds, not including the time for “Creating .EXE file”  (10 seconds, total, including the time for “Creating .EXE file”).  One can compare the computational results above with those in Table 1 of Tsai, Li, and Hu [46, p. 620]. 
       

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[25] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.

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

[28] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[29] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[30] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[31] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[32] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[33] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.

[34] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[35] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[36] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

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

[38] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[39] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[40] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[41] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[42] C. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.

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

[44] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[45] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).

[46] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002).  Global optimization for signomial discrete programming problems in engineering design. Engineering Optimization 34:6, 613-622 (2002).

[47] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[48] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[49] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[50] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[51] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[52] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[53] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

Monday, April 23, 2018

Solving a Generalized Geometric Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve Li and Lu's Example 3 [27, p. 710], which is shown immediately below: 

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

subject to

         X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2>=31

        X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2<=62

         1<=X(i)<=32,    i=1, 2, 3

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

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

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


0 DEFDBL A-Z

2 DEFINT K

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

12 FOR JJJJ = -32000 TO 31999.900 STEP .01


    14 RANDOMIZE JJJJ

    16 M = -1D+37
    40 FOR J40 = 1 TO 3


        41 A(J40) = 1 + FIX(RND * 32)


    42 NEXT J40

    128 FOR I = 1 TO 3000



        129 FOR KKQQ = 1 TO 3


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


            181 j = 1 + FIX(RND * 3)

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

         
        222 NEXT IPP

        255 FOR J41 = 1 TO 3

            258 X(J41) = INT(X(J41))
           
        262 NEXT J41



        265 FOR J41 = 1 TO 3

            268 IF X(J41) < 1## THEN 1670
            269 IF X(J41) > 32## THEN 1670


        272 NEXT J41

        308 X(4) = -31 + X(1) ^ (-2) * X(2) ^ .5 + X(2) ^ (.5) * X(3) ^ 1.2 + X(1) ^ (-2) * X(3) ^ 1.2


        310 X(5) = 62 - X(1) ^ (-2) * X(2) ^ .5 - X(2) ^ (.5) * X(3) ^ 1.2 - X(1) ^ (-2) * X(3) ^ 1.2

     

        401 FOR J47 = 4 TO 5

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


        403 NEXT J47

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


        466 P = POBA

        1111 IF P <= M THEN 1670


        1450 M = P
        1454 FOR KLX = 1 TO 5

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

    1670 NEXT I
    1889 IF M < 61.78 THEN 1999

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

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [50]. The complete output through JJJJ = -31999.91000000001 is shown below:

32     26     8      0    0
61.78578762313648      -31999.96000000001

32     26     8      0    0
61.78578762313648      -31999.93000000001

32     26     8      0    0
61.78578762313648      -31999.91000000001

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [50], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =  -31999.91000000001 was 3 seconds, not including the time for “Creating .EXE file”  (10 seconds, total, including the time for “Creating .EXE file”).  One can compare the computational results above with those in Table 3 of Li and Lu [27, p. 711, Experiment 4]. 
       

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[25] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.

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

[28] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[29] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[30] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[31] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[32] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[33] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.

[34] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[35] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[36] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

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

[38] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[39] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[40] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[41] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[42] C. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.

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

[44] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[45] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[46] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[47] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[48] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[49] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[50] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[51] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[52] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.

Thursday, April 12, 2018

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

Jsun Yui Wong

The computer program listed below seeks to solve the last instance in Table 5 of Lu, [32, p. 114]: 

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

subject to

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

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

         -6<=X(1) <= 6.75

        -6<= X(2) <= 6.75

        -1<= X(3) <= 9.2

        -9<= X(4) <= 6.3

where X(1) through X(4) are free-sign discrete variables; here the number of discrete values is 512. 

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


0 DEFDBL A-Z

2 DEFINT K

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

12 FOR JJJJ = -32000 TO 32000 STEP .01

    14 RANDOMIZE JJJJ

    16 M = -1D+37
    31 unitsize(1) = 12.75 / 511

    32 unitsize(2) = 12.75 / 511

    33 unitsize(3) = 10.2 / 511
    34 unitsize(4) = 15.3 / 511


    61 A(1) = -6 + FIX(RND * 512) * unitsize(1)
    63 A(2) = -6 + FIX(RND * 512) * unitsize(2)

    65 A(3) = -1 + FIX(RND * 512) * unitsize(3)



    67 A(4) = -9 + FIX(RND * 512) * unitsize(4)

    128 FOR I = 1 TO 6000


        129 FOR KKQQ = 1 TO 4


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


            181 j = 1 + FIX(RND * 4)


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


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

        222 NEXT IPP


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


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


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


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


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


        401 FOR J47 = 5 TO 6

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


        403 NEXT J47

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


        466 P = POBA

        1111 IF P <= M THEN 1670


        1450 M = P
        1454 FOR KLX = 1 TO 6

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

    1670 NEXT I
    1889 IF M < 72813 THEN 1999

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

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [49]. The complete output through JJJJ =-31995.78000000068 is shown below:

2.084148727984344      -4.927103718199606      6.046183953033268
6.3      0      0                72813.98768768994
-31999.77000000004

2.084148727984344      -4.92710371819961       6.04618395303327
6.299999999999998      0      0                72813.98768768997
-31995.78000000068

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [49], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =-31995.78000000068 was 13 seconds, not including the time for “Creating .EXE file,” (was 18 seconds, total, including the time for “Creating .EXE file.”)  One can compare the computational results above with those on page 114 of Lu [32, Table 5].           
                                               
                                               
Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[25] Han-Lin Li, Jung-Fa Tsai, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[27] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[28] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[29] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[30] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[31] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[32] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.

[33] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[34] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[35] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

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

[37] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[38] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[39] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[40] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[41] C. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.

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

[43] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[44] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[45] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[46] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[47] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[48] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[49] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[50] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[51] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.


Monday, April 9, 2018

Solving Another Free-Sign Pure Discrete Signomial Programming Problem--Lu's Example 4

Jsun Yui Wong

The computer program listed below seeks to solve the immediately following formulation from Lu [32, pp. 116-118 and Table 7]:

Minimize            .25 * 3.141592654 ^ 2 * X(1) ^ (2) * X(2) * X(3)+ .5 * 3.141592654 ^ 2 * X(1) ^ (2) * X(2)

subject to

        (3.141592654) ^ (-1) * 1000 * (8 * X(1) ^ (-3) * X(2) ^ (2) + 2.92 * X(1) ^ (-2) * X(2) - 4.92 * X(1) ^ (-1)) - 189000 * (X(2) - X(1))<=0

        8 * (11.5 * 10 ^ 6) ^ (-1) * 1000 * X(1) ^ (-4) * X(2) ^ 3 * X(3) + 1.05 * X(1) * X(3) + 2.1 * X(1) - 14<=0

        .009 - X(1)<=0

         X(2) - 4<=0
     
        3 * X(1) - X(2)<=0

        8 * (11.5 * 10 ^ 6) ^ (-1) * 300 * X(1) ^ (-4) * X(2) ^ 3 * X(3) - 6<=0

        1.25 - 8 * (11.5 * 10 ^ 6) ^ (-1) * (1000 - 300) * X(1) ^ (-4) * X(2) ^ 3 * X(3)<=0

         .009<=X(1) <=.5

         .6<=X(2) <=4

         1<=X(3) <=120   

 where X(1) and X(2) are discrete variables; here the number of discrete values for X(1) is 500; the number of discrete values for X(2) is 250. 

 X(3) is an integer variable.

X(4) through X(10) 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


    31 unitsize(1) = .491 / 499
    33 unitsize(2) = 3.4 / 249


    61 A(1) = .009 + FIX(RND * 500) * unitsize(1)

    64 A(2) = .6 + FIX(RND * 250) * unitsize(2)

    67 REM NEXT J42

    97 A(3) = 1 + FIX(RND * 120)
    128 FOR I = 1 TO 3000


        129 FOR KKQQ = 1 TO 3


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


            181 j = 1 + FIX(RND * 2)


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


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

            199 IF RND < .5 THEN GOTO 204 ELSE GOTO 222

            204 IF RND < .5 THEN X(3) = A(3) - FIX(RND * 5) ELSE X(3) = A(3) + FIX(RND * 5)


        222 NEXT IPP

        265 IF X(1) < .009 THEN 1670

        266 IF X(1) > .5 THEN 1670


        268 IF X(2) < .6 THEN 1670

        270 IF X(2) > 4 THEN 1670

        271 IF X(3) < 1 THEN 1670

        273 IF X(3) > 120 THEN 1670


        277 REM  NEXT J43

        307 X(4) = -.009 + X(1)


        309 X(5) = -X(2) + 4

        310 X(6) = -3 * X(1) + X(2)

        313 X(7) = -(3.141592654) ^ (-1) * 1000 * (8 * X(1) ^ (-3) * X(2) ^ (2) + 2.92 * X(1) ^ (-2) * X(2) - 4.92 * X(1) ^ (-1)) + 189000 * (X(2) - X(1))


        321 X(8) = -8 * (11.5 * 10 ^ 6) ^ (-1) * 300 * X(1) ^ (-4) * X(2) ^ 3 * X(3) + 6

        323 X(9) = -1.25 + 8 * (11.5 * 10 ^ 6) ^ (-1) * (1000 - 300) * X(1) ^ (-4) * X(2) ^ 3 * X(3)

        325 X(10) = -8 * (11.5 * 10 ^ 6) ^ (-1) * 1000 * X(1) ^ (-4) * X(2) ^ 3 * X(3) - 1.05 * X(1) * X(3) - 2.1 * X(1) + 14


        401 FOR J47 = 4 TO 10

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


        433 NEXT J47

        447 POBA = -.25 * 3.141592654 ^ 2 * X(1) ^ (2) * X(2) * X(3) - .5 * 3.141592654 ^ 2 * X(1) ^ (2) * X(2) + 1000000 * (X(4) + X(5) + X(6) + X(7) + X(8) + X(9) + X(10))

        466 P = POBA

        1111 IF P <= M THEN 1670

        1450 M = P
        1454 FOR KLX = 1 TO 10

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

    1670 NEXT I
    1889 IF M < -2.64251 THEN 1999


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

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

    1909 PRINT M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [49]. The complete output through JJJJ =-31999.79000000003 is shown below 

.2923827655310621        1.391967871485945      7
0   0   0
0   0   0   0
-2.642500110242702      -31999.96000000001

.2923827655310621        1.391967871485947      7
0   0   0
0   0   0   0
-2.642500110242706      -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 [49], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =-31999.79000000003 was 8 seconds, total, including the time for “Creating .EXE file.”  One can compare the computational results above with those on page 120 of Lu [32, Table 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, Christodoulos A. Floudas (2008). Convex underestimating for posynomial functions of postive variables. Optimization Letters 2, 333-340 (2008).
[26] Han-Lin Li, Jung-Fa Tsai (2008). A distributed computational algorithm for solving portfolio problems with integer variables. European Journal of Operational Research 186 (2008) pp. 882-891.
[27] Han-Lin Li, Shu-Cherng Fang, Yao-Huei Huang, Tiantian Nie (2016). An enhanced logarithmic method for signomial programming with discrete variables. European Journal of Operational Research 255 (2016) pp. 922-934.
[28] Ming-Hua Lin, Jung-Fa Tsai (2011). Finding multiple optimal solutions of signomial discrete programming problems with free variables, Optimization and Engineering (2011) 12:425-443.
[29] Ming-Hua Lin, Jung-Fa Tsai (2014). A deterministic global approach for mixed-discrete structural optimization, Engineering Optimization (2014) 46:7, pp. 863-879.
[30] Hao-Chun Lu, Han-Lin Li, Chrysanthos E. Gounaris, Christodoulos A. Floudas (2010). Convex relaxation for solving posynomial problems. Journal of Global Optimization (2010) 46, pp. 147-154.
[31] Hao-Chun Lu (2012). An efficient convexification method for solving generalized geometric problems. Journal of Industrial and Management Optimization, Volume 8, Number 2, May 2012, pp. 429-455.
[32] Hao-Chun Lu (2017). Improved logarithnic linearizing method for optimization problems with free-sign pure discrete signomial terms. Journal of Global Optimization (2017) 68, pp. 95-123.

[33] Kenneth R. MacCrimmon (1973). An overview of multiple objective decision making. In James L. Cochrane, Milan Zeleny (editors), Multiple Criteria Decisiom Making. University of South Carolina Press, Columbia.
[34] Mathworks. Solving a mixed integer engineering design problem using the genetic algorithm – MATLAB & Simulink Example.
https://www.mathworks.com/help/gads/examples/solving-a-mixed-integer-engineering-design-problem-using-the-genetic-algorithm.html/
[35] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

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

[37] Sinan Melih Nigdeli, Gebrail Bekdas, Xin-She Yang (2016). Application of the Flower Pollination Algoritm in Structural Engineering. Springer International Publishing Switzerland 2016. http://www.springer.com/cda/content/document/cda…/
[38] Gideon Oron (1979) An algorithm for optimizing nonlinear contrained zero-one problems to improve wastewater treatment, Engineering Optimization, 4:2, 109-114.
[39] Osama Abdel Raouf, Ibraham M. Hezam (2014). Solving Fractional Programming Problems Based on Swarm Intelligence. Journal of Industrial Engineering International (2014) 10:56.
[40] H. S. Ryoo, N. V. Sahinidis (1995). Global optimization of nonconvex NLP and MINLP with applications in process design. Computers and Chemical Engineering Vol. 19 (5) (1995) pp. 551-566.
[41] C. R. Seshan, V. G. Tikekar (1980) Algorithms for Fractional Programming. Journal of the Indian Institute of Science 62 (B), Feb. 1980, Pp. 9-16.

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

[43] Pei-Ping Shen, Yun-Peng Duan, Yong-Gang Pei. A simplicial branch and duality boundalgorithm for the sum of convex-convex ratios problem. Journal of Computational and Applied Mathematics 223 (2009) 145-158.
[44] P. B. Thanedar, G. N. Vanderplaats (1995). Survey of discrete variable optimization for structural design, Journal of Structural Engineering, 121 (2), 301-306 (1995).
[45] Jung-Fa Tsai (2005). Global optimization of nonlinear fractional programming problems in engineering design. Engineering Optimization (2005) 37:4, pp. 399-409.
[46] Jung-Fa Tsai, Ming-Hua Lin (2007). Finding all solutions of systems of nonlinear equations with free variables. Engineering Optimization (2007) 39:6, pp. 649-659
[47] Jung-Fa Tsai, Ming-Hua Lin, Yi-Chung Hu (2007). On generalized geometric programming problems with non-positive variables. European Journal of Operational Research 178 (2007) pp. 10-19.
[48] Jung-Fa Tsai, Ming-Hua Lin (2008). Global optimization of signomial mixed-integer nonlinear programming with free variables. Journal of Global Optimization (2008) 42 pp. 39-49.
[49] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[50] Taeyong Yang, James P. Ignizio, and Hyun-Joon Kim, Fuzzy programming with nonlinear membership functions: Piecewise linear approximation. Fuzzy Sets and Systems 41 (1991) 39-53.
[51] H.-J. Zimmermann (1978), Fuzzy programming and linear programming with several objective functions. Fuzzy Sets and Systems 1 (1978) 45-55.