Solving a Multi-Objective Geometric Programming Problem in Ota and Ojha [47] Using the Mixed-Integer Nonlinear Programming (MINLP) Algorithm of the Present Blog

Jsun Yui Wong

The computer program listed below seeks to solve the following multi-objective geometric programming problem from Ota and Ojha [47,  p. 1081]: 
       
Minimize    10 * X(1) ^ -1 + 12 * X(2) ^ -1

minimize      8 * X(1) ^ -2 + 6 * X(2) ^ -1

subject to 

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

 where  X(1),  X(2) > 0. 

One notes line 89, which is 89 IF RND < 1 / 6 THEN w1 = 0 ELSE IF RND < 1 / 5 THEN w1 = .1 ELSE IF RND < 1 / 4 THEN w1 = .2 ELSE IF RND < 1 / 3 THEN w1 = .5 ELSE IF RND < 1 / 2 THEN w1 = .9 ELSE w1 = 1.


0 DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000

    85 RANDOMIZE JJJJ

    87 M = -3E+50

    89 IF RND < 1 / 6 THEN w1 = 0 ELSE IF RND < 1 / 5 THEN w1 = .1 ELSE IF RND < 1 / 4 THEN w1 = .2 ELSE IF RND < 1 / 3 THEN w1 = .5 ELSE IF RND < 1 / 2 THEN w1 = .9 ELSE w1 = 1
    90 w2 = 1 - w1


    92 A(1) = .0000001 + (RND * 4)
    93 A(2) = .0000001 + (RND * 4)


    128 FOR I = 1 TO 30000


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


            153 J = 1 + FIX(RND * 2)

            154 REM GOTO 162

            156 r = (1 - RND * 2) * A(J)

            158 X(J) = A(J) + (RND ^ (RND * 10)) * r

            161 REM GOTO 169

            162 REM IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1

        169 NEXT IPP


        177 IF X(1) < .0000001## THEN 1670
        187 IF X(2) < .0000001## THEN 1670
     

        222 IF .25 * X(1) * X(2) ^ -1 + .75 * X(2) > 1 THEN 1670

        411 PDU = w1 * (-10 * X(1) ^ -1 - 12 * X(2) ^ -1) + (1 - w1) * (-8 * X(1) ^ -2 - 6 * X(2) ^ -1)



        466 P = PDU

        1111 IF P <= M THEN 1670

        1450 M = P


        1454 FOR KLX = 1 TO 2

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX


        1557 GOTO 128
    1670 NEXT I

    1889 IF M < -99999 THEN 1999


    1924 PRINT M, A(1), A(2)

    1933 PRINT w1, w2, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [59].  The output of a single run through JJJJ= -31961 is summarized below:

-17.1911792420965      1.183297411830931      .8903002371209567
.5      .5      -32000
.
.
.

-20.88759170290532      1.124079270701997      .9307714833786028
 .9            9.999999999999998D-02      -31993
.
.
.

-14.30142068934547      1.222727273320694      .8586788711508603
.2      .8      -31991
.
.
.

-21.78119633074406    1.105822576725803      .9420517041490557
1      0         -31983
.
.
.

-13.3158562388016         1.232373343855897        .8501151058145975
.1     .9      -31966              2
.
.
.

-12.31888654377727       1.244991556294527      .838268703405557
0     1      -31961 

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 [59], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31961 was 4 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 Ota and Ojha [47, p. 1082, Table 1].


Acknowledgment

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

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem.  European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem.  Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem.  Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem.  Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes.  INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[6] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database.  Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[7] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study.  Princeton and Oxford: Princeton University Press, 2006.

[8] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming.  New York: John Wiley and Sons, Inc., 1968.

[9] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost.  Operations Research, Vol. 14, No. 1 (Jan. - Feb., 1966), pp. 52-58.

[10]  Ta-Cheng Chen (2006).  IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.

[11]  Leandro dos Santos Coelho (2009),  Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.

[12]  William Conley (1981).  Optimization:  A Simplified Approach.  Published 1981 by Petrocelli Books in New York.

[13]  Lino Costa, Pedro (2001).  Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems.

[16]  Anoop K. Dhingra (1992).  Optimal apportionment of reliability and redundancy in series systems under multiple objections. IEEE Transactions on Reliability, Vol. 41, No. 4, 1992 December, pp. 576-582.

[17] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming.  John Wiley, New York (1967).

[18]  C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[19] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[20] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[21] Diptesh Ghosh, Ravi Kothari, Population Heuristics for the Corridor Allocation Problem, W.P. No. 2012-09-02, September 2012.  Retrieved on September 14 2012 from Google search.

[22]  Ignacio E. Grossmann.  Overview of Mixed-integer Nonlinear Programming.  https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[23]  R. Gupta, R. Malhotra (1995).  Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.

[24]  M. Hashish, M. P. duPlessis (1979).  Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses.  Transactions of ASME:  Journal of Engineering for Industry 101 (1979) 311-318.

[25] David M. Himmelblau, Applied Nonlinear Programming.  New York: McGraw-Hill Book Company, 1972.

[26] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Berlin: Springer-Verlag, 1981.

[27] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout.  Les Cahiers du GERAD.  Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[28] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout.  Retrieved from www.optimization-online.org./DB_HTML/2012/04/3432.html

[29]  N. K. Jain, V. K. Jain, K. Deb (2007).  Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms.  International Journal of Machine Tools and Manufacture 47 (2007), 900-919.

[30] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[31]  Adhe Kania, Kuntjoro Adji Sidarto (2016).  Solving mixed integer  nonlinear programming problems using spiral dynamics optimization algorithm.  AIP Conference Proceedings 1716, 020004 (2016).
https://doi.org/10.1063/1.4942987.  Published by the American Institute of Physics.

[32]  A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[33] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[34]  Gia-Shi Liu (2006),  A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.

[35]  Yubao Liu, Guihe Qin (2014),  A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[36]  Rein Luus (1975).  Optimization of System Reliability by a New Nonlinear Integer Programming Procedure. IEEE Transactions on Reliability, Vol. R-24, No. 1, April 1975, pp. 14-16.

[37] Milos  Madic, Miroslav  Radovanovic (2014).  Optimization of machining processes using pattern search algorithm.
International Journal of Industrial Engineering Computations 5 (2014) 223-234.  Homepage: www.GrowingScience.com/ijiec

[38]  F. Masedu, M Angelozzi (2008).  Modelling optimum fraction assignment in the 4X100 m relay race by integer linear programming.   Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.

[39]  MathWorks, Mixed Integer Optimization.  https://www.mathworks.com/help/gads;mixed-integer-optimization.html

[40]  Mohamed Arezki Mellal, Enrico Zio (2016).  A Guided Stochastic Fractal Search Approach for System Reliability Optimization.  Reliability Engineering 
and System Safety 152 (2016) 213-227.

[41]  Mohamed Arezki Mellal, Edward J. Williams (2016).  Parameter optimization of advanced machining processes using cuckoo optimization algorithm and hoopla heuristic.  Journal of Intelligent Manufacturing (2016) 27 (5): 927-942. 

[42]  Mohamed Arezki Mellal, Edward J. Williams (2018).  Large-scale reliability-redundancy allocation optimization problem using three soft computing methods.  In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering.  Published July 2018, CRC Press.     

[43] 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.

[44] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[45] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[46]  A. K. Ojha, K. K. Biswal (2010).   Multi-objective geometric programming problem with weighted mean method  (IJCSIS) International Journal of Computer Science and Information Security Vol. 7, No. 2, 2010, pp.82-86.  http://sites.google.com/site/ijcsis/

[47]  Rashmi Ranjan Ota,  A. K. Ojha (2015).  A comparative study on optimization techniques for solving multi-objective geometric programming problems.  Applied Mathematical Sciences Vol. 9 2015, no. 22, 1077-1085.  http://sites.google.com/site/ijcsis/

[48] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP).  https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

[49] Panos Y. Papalambros,  Douglass J. Wilde, Principles of Optimal Design, Second Edition.  Cambridge University Press, 2000.

[50]  R. V. Rao, P. J. Pawar, J. P. Davim (2010).  Parameter optimization of ultrasonic machining process using nontraditional optimization algorihms.  Materials and Manufacturing Processes, 25 (10),1120-1130.

[51] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

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

[53]  Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017).  A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[54] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm.    Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.

[55] G. Stephanopoulos, A. W. Westerberg, The Use of Hestenes' Method of Multipliers to Resolve
Dual Gaps in Engineering System Optimization.  Journal of Optimization Theory and Applications,  Vol.15, No. 3, pp. 285-309, 1975.

[56]  Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977).  Determining Component Reliability and Redundancy for  Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[57]  Hardi Tambunan, Herman Mawengkang (2016).  Solving Mixed Integer Non-Linear Programming Using Active Constraint.  Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281.  http://www.ripublication.com/gjpam.htm

[58] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011).  Solving mixed-integer nonlinear programming problems using improved genetic algorithms.  Korean  Joutnal of  Chemical Engineering  28 (1):32-40 January 2011.

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

[60]  Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990).  Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.

[61]  ZHANG HongQi, HU ZiangTao, SHAO XiaoDong, LI ZiCheng, WANG YuHui  (November 2013).  IPSO-based hybrid approaches for reliability-redundany allocation problems.  SCIENCE CHINA Technologial Sciences, November 2013, Vol. 56, No. 11:  2854-2864

Solving a Geometric Programming Problem Using the Mixed-Integer Nonlinear Programming (MINLP) Algorithm of the Present Blog

Jsun Yui Wong

The computer program listed below seeks to solve the following geometric programming problem from Duffin, Peterson, and Zener [17,  p. 9, Problem 3]:.
       
Minimize  

40 / (X(1) * X(2) * X(3)) - 40 * X(2) * X(3)

subject to     

 X(1) * X(3) / 2 + X(1) * X(2) / 4 <= 1

X(1), X(2), and X(3) are positive.


0 DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000

    89 RANDOMIZE JJJJ

    90 M = -3E+50

    92 A(1) = .05 + (RND * 10)
    93 A(2) = .05 + (RND * 10)

    96 A(3) = .05 + (RND * 10)

    128 FOR I = 1 TO 30000

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


            153 J = 1 + FIX(RND * 3)

            154 REM GOTO 162

            156 r = (1 - RND * 2) * A(J)

            158 X(J) = A(J) + (RND ^ (RND * 10)) * r

            161 REM GOTO 169

            162 REM IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1

        169 NEXT IPP


        177 IF X(1) < .01## THEN 1670
        187 IF X(2) < .01## THEN 1670
        197 IF X(3) < .01## THEN 1670


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



        388 PDU = -40 / (X(1) * X(2) * X(3)) - 40 * X(2) * X(3)



        466 P = PDU

        1111 IF P <= M THEN 1670

        1450 M = P


        1454 FOR KLX = 1 TO 3

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX


        1557 GOTO 128
    1670 NEXT I

    1889 IF M < -99999 THEN 1999

    1904 PRINT M, A(1), A(2), A(3), JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [58].  The complete output of a single run through JJJJ= -31996 is shown below:

-60.0035965507729     1.98746175106339      1.01426394174926
.4991767034582993   -32000

-60.00615195678768   1.997919203893426    1.018536286959685
.4917733381288832   -31998

-60.00185643195058   2.011165860632116   .9945176109173011
.4971892602873506   -31997

-60.0003103495981     1.9974493491549      .9980041209737496
.5022748934675165   -31996
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 [58], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31996 was 1 or 2 seconds, not including the time for “Creating .EXE file" (8 seconds, including the time for “Creating .EXE file").  One can compare the computational results above with those in Duffin, Peterson, and Zener [17, p. 9].


Acknowledgment

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

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem.  European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem.  Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem.  Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem.  Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes.  INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[6] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database.  Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[7] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study.  Princeton and Oxford: Princeton University Press, 2006.

[8] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming.  New York: John Wiley and Sons, Inc., 1968.

[9] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost.  Operations Research, Vol. 14, No. 1 (Jan. - Feb., 1966), pp. 52-58.

[10]  Ta-Cheng Chen (2006).  IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.

[11]  Leandro dos Santos Coelho (2009),  Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.

[12]  William Conley (1981).  Optimization:  A Simplified Approach.  Published 1981 by Petrocelli Books in New York.

[13]  Lino Costa, Pedro (2001).  Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[14] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[15]  Kusum Deep, Krishna Pratap Singh, M. L.  Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems.  Applied Mathematics and Computation 212 (2009) 505-518.

[16]  Anoop K. Dhingra (1992).  Optimal apportionment of reliability and redundancy in series systems under multiple objections. IEEE Transactions on Reliability, Vol. 41, No. 4, 1992 December, pp. 576-582.

[17] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming.  John Wiley, New York (1967).

[18]  C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[19] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[20] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[21] Diptesh Ghosh, Ravi Kothari, Population Heuristics for the Corridor Allocation Problem, W.P. No. 2012-09-02, September 2012.  Retrieved on September 14 2012 from Google search.

[22]  Ignacio E. Grossmann.  Overview of Mixed-integer Nonlinear Programming.  https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[23]  R. Gupta, R. Malhotra (1995).  Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[24]  M. Hashish, M. P. duPlessis (1979).  Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses.  Transactions of ASME:  Journal of Engineering for Industry 101 (1979) 311-318.

[25] David M. Himmelblau, Applied Nonlinear Programming.  New York: McGraw-Hill Book Company, 1972.

[26] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Berlin: Springer-Verlag, 1981.

[27] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout.  Les Cahiers du GERAD.  Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[28] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout.  Retrieved from www.optimization-online.org./DB_HTML/2012/04/3432.html

[29]  N. K. Jain, V. K. Jain, K. Deb (2007).  Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms.  International Journal of Machine Tools and Manufacture 47 (2007), 900-919.

[30] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[31]  Adhe Kania, Kuntjoro Adji Sidarto (2016).  Solving mixed integer  nonlinear programming problems using spiral dynamics optimization algorithm.  AIP Conference Proceedings 1716, 020004 (2016).
https://doi.org/10.1063/1.4942987.  Published by the American Institute of Physics.

[32]  A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[33] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[34]  Gia-Shi Liu (2006),  A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.

[35]  Yubao Liu, Guihe Qin (2014),  A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[36]  Rein Luus (1975).  Optimization of System Reliability by a New Nonlinear Integer Programming Procedure. IEEE Transactions on Reliability, Vol. R-24, No. 1, April 1975, pp. 14-16.

[37] Milos  Madic, Miroslav  Radovanovic (2014).  Optimization of machining processes using pattern search algorithm.
International Journal of Industrial Engineering Computations 5 (2014) 223-234.  Homepage: www.GrowingScience.com/ijiec

[38]  F. Masedu, M Angelozzi (2008).  Modelling optimum fraction assignment in the 4X100 m relay race by integer linear programming.   Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.

[39]  MathWorks, Mixed Integer Optimization.  https://www.mathworks.com/help/gads;mixed-integer-optimization.html

[40]  Mohamed Arezki Mellal, Enrico Zio (2016).  A Guided Stochastic Fractal Search Approach for System Reliability Optimization.  Reliability Engineering
and System Safety 152 (2016) 213-227.

[41]  Mohamed Arezki Mellal, Edward J. Williams (2016).  Parameter optimization of advanced machining processes using cuckoo optimization algorithm and hoopla heuristic.  Journal of Intelligent Manufacturing (2016) 27 (5): 927-942.  

[42]  Mohamed Arezki Mellal, Edward J. Williams (2018).  Large-scale reliability-redundancy allocation optimization problem using three soft computing methods.  In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering.  Published July 2018, CRC Press.      

[43] 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.

[44] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[45] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[46]  A. K. Ojha, K. K. Biswal (2010).   Multi-objective geometric programming problem with weighted mean method  (IJCSIS) International Journal of Computer Science and Information Security Vol. 7, No. 2, 2010, pp.82-86.  http://sites.google.com/site/ijcsis/

[47] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP).  https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

[48] Panos Y. Papalambros,  Douglass J. Wilde, Principles of Optimal Design, Second Edition.  Cambridge University Press, 2000.

[49]  R. V. Rao, P. J. Pawar, J. P. Davim (2010).  Parameter optimization of ultrasonic machining process using nontraditional optimization algorihms.  Materials and Manufacturing Processes, 25 (10),1120-1130.

[50] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

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

[52]  Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017).  A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[53] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm.     Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.

[54] G. Stephanopoulos, A. W. Westerberg, The Use of Hestenes' Method of Multipliers to Resolve
Dual Gaps in Engineering System Optimization.  Journal of Optimization Theory and Applications,  Vol.15, No. 3, pp. 285-309, 1975.

[55]  Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977).  Determining Component Reliability and Redundancy for  Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[56]  Hardi Tambunan, Herman Mawengkang (2016).  Solving Mixed Integer Non-Linear Programming Using Active Constraint.  Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281.  http://www.ripublication.com/gjpam.htm

[57] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011).  Solving mixed-integer nonlinear programming problems using improved genetic algorithms.  Korean  Joutnal of  Chemical Engineering  28 (1):32-40 January 2011.

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

[59]  Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990).  Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.

[60]  ZHANG HongQi, HU ZiangTao, SHAO XiaoDong, LI ZiCheng, WANG YuHui  (November 2013).  IPSO-based hybrid approaches for reliability-redundany allocation problems  SCIENCE CHINA Technologial Sciences, November 2013, Vol. 56, No. 11:  2854-2864

Solving a Multi-Objective Geometric Programming Problem with the Mixed-Integer Nonlinear Programming (MINLP) Algorithm of the Present Blog, Second Edition

Jsun Yui Wong

The computer program listed below seeks to solve the following 2-criteria geometric programming problem:
        
Maximize                 (((-4 * X(1) - 10 * X(2) - 4 * X(3) - 2 * X(4)) / (1)))

maximize                     X(1) ^ -1 * X(2) ^ -1 * X(3) ^ -1

with given weights of .1 and .9, respectively,

subject to     
   
         X(1) ^ 2 * X(4) ^ -2 + X(2) ^ 2 * X(4) ^ -2<=1

         100 * X(1) ^ -1 * X(2) ^ -1 * X(3) ^ -1<=1

X(1) through X(4) > 0.

The problem above is based on Example 1 of Ojha and Biswal [45,  pp. 83-84].
   
Whereas line 158 of the earlier edition is 158 X(J) = A(J) + (RND ^ (RND * 10)) * r, here line 158 is 158 X(J) = A(J) + .01## + (RND ^ (RND * 10)) * r.       


0 DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000


    89 RANDOMIZE JJJJ

    90 M = -3E+50

    94 A(1) = .0000000001 + (RND * 10)
    95 A(2) = .0000000001 + (RND * 10)
    96 A(3) = .0000000001 + (RND * 10)
    97 A(4) = .0000000001 + (RND * 10)

    128 FOR I = 1 TO 30000


        129 FOR KKQQ = 1 TO 4
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        151 FOR IPP = 1 TO FIX(1 + RND * 3)


            153 J = 1 + FIX(RND * 4)

            154 REM GOTO 162


            156 r = (1 - RND * 2) * A(J)
            158 X(J) = A(J) + .01## + (RND ^ (RND * 10)) * r

            159 REM GOTO 169

            162 REM IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1

        169 NEXT IPP
        177 IF X(1) < .0000000000001## THEN 1670
        178 IF X(2) < .0000000000001## THEN 1670
        179 IF X(3) < .0000000000001## THEN 1670
        180 IF X(4) < .0000000000001## THEN 1670



        263 IF X(1) ^ 2 * X(4) ^ -2 + X(2) ^ 2 * X(4) ^ -2 > 1## THEN 1670


        287 IF 100 * X(1) ^ -1 * X(2) ^ -1 * X(3) ^ -1 > 1## THEN 1670 



        375 PDU = .1 * (((-4 * X(1) - 10 * X(2) - 4 * X(3) - 2 * X(4)) / (1))) + .9 * X(1) ^ -1 * X(2) ^ -1 * X(3) ^ -1


        466 P = PDU

        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 < -999999 THEN 1999

    1904 PRINT M, A(1), A(2), A(3), A(4), JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [57].  The complete output of a single run through JJJJ= -31996 is shown below:

-8.831462777999191      4.487480223481182      2.731716837571951
8.15760991189058         5.25354931107736      -31998
   
-8.796040548514386      5.344602829872222      2.602952480900544
7.188225964043273       5.944782398163001      -31997
      
-8.793029011463105      4.907298442740978      2.747602212847382
7.416676793495234       5.624182931867983      -31996

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 [57], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31996 was 1 or 2 seconds, not including the time for “Creating .EXE file" (10 seconds, including the time for “Creating .EXE file").  One can compare the computational results above with those in Ojha and Biswal [45, p. 84, Table 2].


Acknowledgment

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

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem.  European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem.  Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem.  Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem.  Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes.  INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[6] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database.  Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[7] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study.  Princeton and Oxford: Princeton University Press, 2006.

[8] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming.  New York: John Wiley and Sons, Inc., 1968.

[9] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost.  Operations Research, Vol. 14, No. 1 (Jan. - Feb., 1966), pp. 52-58.

[10]  Ta-Cheng Chen (2006).  IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.

[11]  Leandro dos Santos Coelho (2009),  Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.

[12]  William Conley (1981).  Optimization:  A Simplified Approach.  Published 1981 by Petrocelli Books in New York.

[13]  Lino Costa, Pedro (2001).  Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[14] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[15]  Kusum Deep, Krishna Pratap Singh, M. L.  Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems.  Applied Mathematics and Computation 212 (2009) 505-518.

[16]  Anoop K. Dhingra (1992).  Optimal apportionment of reliability and redundancy in series systems under multiple objections. IEEE Transactions on Reliability, Vol. 41, No. 4, 1992 December, pp. 576-582.

[17]  C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[18] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[19] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[20] Diptesh Ghosh, Ravi Kothari, Population Heuristics for the Corridor Allocation Problem, W.P. No. 2012-09-02, September 2012.  Retrieved on September 14 2012 from Google search.

[21]  Ignacio E. Grossmann.  Overview of Mixed-integer Nonlinear Programming.  https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[22]  R. Gupta, R. Malhotra (1995).  Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[23]  M. Hashish, M. P. duPlessis (1979).  Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses.  Transactions of ASME:  Journal of Engineering for Industry 101 (1979) 311-318.

[24] David M. Himmelblau, Applied Nonlinear Programming.  New York: McGraw-Hill Book Company, 1972.

[25] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Berlin: Springer-Verlag, 1981.

[26] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout.  Les Cahiers du GERAD.  Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[27] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout.  Retrieved from www.optimization-online.org./DB_HTML/2012/04/3432.html

[28]  N. K. Jain, V. K. Jain, K. Deb (2007).  Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms.  International Journal of Machine Tools and Manufacture 47 (2007), 900-919.

[29] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[30]  Adhe Kania, Kuntjoro Adji Sidarto (2016).  Solving mixed integer  nonlinear programming problems using spiral dynamics optimization algorithm.  AIP Conference Proceedings 1716, 020004 (2016).
https://doi.org/10.1063/1.4942987.  Published by the American Institute of Physics.

[31]  A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[32] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[33]  Gia-Shi Liu (2006),  A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.

[34]  Yubao Liu, Guihe Qin (2014),  A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[35]  Rein Luus (1975).  Optimization of System Reliability by a New Nonlinear Integer Programming Procedure. IEEE Transactions on Reliability, Vol. R-24, No. 1, April 1975, pp. 14-16.

[36] Milos  Madic, Miroslav  Radovanovic (2014).  Optimization of machining processes using pattern search algorithm.
International Journal of Industrial Engineering Computations 5 (2014) 223-234.  Homepage: www.GrowingScience.com/ijiec

[37]  F. Masedu, M Angelozzi (2008).  Modelling optimum fraction assignment in the 4X100 m relay race by integer linear programming.   Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.

[38]  MathWorks, Mixed Integer Optimization.  https://www.mathworks.com/help/gads;mixed-integer-optimization.html

[39]  Mohamed Arezki Mellal, Enrico Zio (2016).  A Guided Stochastic Fractal Search Approach for System Reliability Optimization.  Reliability Engineering
and System Safety 152 (2016) 213-227.

[40]  Mohamed Arezki Mellal, Edward J. Williams (2016).  Parameter optimization of advanced machining processes using cuckoo optimization algorithm and hoopla heuristic.  Journal of Intelligent Manufacturing (2016) 27 (5): 927-942.  

[41]  Mohamed Arezki Mellal, Edward J. Williams (2018).  Large-scale reliability-redundancy allocation optimization problem using three soft computing methods.  In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering.  Published July 2018, CRC Press.      

[42] 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.

[43] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[44] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[45]  A. K. Ojha, K. K. Biswal (2010).   Multi-objective geometric programming problem with weighted mean method  (IJCSIS) International Journal of Computer Science and Information Security Vol. 7, No. 2, 2010, pp.82-86.  http://sites.google.com/site/ijcsis/

[46] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP).  https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

[47] Panos Y. Papalambros,  Douglass J. Wilde, Principles of Optimal Design, Second Edition.  Cambridge University Press, 2000.

[48]  R. V. Rao, P. J. Pawar, J. P. Davim (2010).  Parameter optimization of ultrasonic machining process using nontraditional optimization algorihms.  Materials and Manufacturing Processes, 25 (10),1120-1130.

[49] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

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

[51]  Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017).  A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[52] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm.     Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.

[53] G. Stephanopoulos, A. W. Westerberg, The Use of Hestenes' Method of Multipliers to Resolve
Dual Gaps in Engineering System Optimization.  Journal of Optimization Theory and Applications,  Vol.15, No. 3, pp. 285-309, 1975.

[54]  Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977).  Determining Component Reliability and Redundancy for  Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[55]  Hardi Tambunan, Herman Mawengkang (2016).  Solving Mixed Integer Non-Linear Programming Using Active Constraint.  Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281.  http://www.ripublication.com/gjpam.htm

[56] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011).  Solving mixed-integer nonlinear programming problems using improved genetic algorithms.  Korean  Joutnal of  Chemical Engineering  28 (1):32-40 January 2011.

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

[58]  Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990).  Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.

[59]  ZHANG HongQi, HU ZiangTao, SHAO XiaoDong, LI ZiCheng, WANG YuHui  (November 2013).  IPSO-based hybrid approaches for reliability-redundany allocation problems  SCIENCE CHINA Technologial Sciences, November 2013, Vol. 56, No. 11:  2854-2864

Solving a Multi-Objective Geometric Programming Problem with the Mixed-Integer Nonlinear Programming (MINLP) Algorithm of the Present Blog

Jsun Yui Wong

The computer program listed below seeks to solve the following 2-criteria geometric programming problem:
        
Maximize                 (((-4 * X(1) - 10 * X(2) - 4 * X(3) - 2 * X(4)) / (1)))

maximize                     X(1) ^ -1 * X(2) ^ -1 * X(3) ^ -1

with given weights of .1 and .9, respectively,

subject to     
   
         X(1) ^ 2 * X(4) ^ -2 + X(2) ^ 2 * X(4) ^ -2<=1

         100 * X(1) ^ -1 * X(2) ^ -1 * X(3) ^ -1<=1

X(1) through X(4) > 0, and X(5) and  X(6) are slack variables added.

The problem above is based on Example 1 of Ojha and Biswal [45,  pp. 83-84].
        

0 DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000


    89 RANDOMIZE JJJJ

    90 M = -3E+50

    94 A(1) = .0000000001 + (RND * 10)
    95 A(2) = .0000000001 + (RND * 10)
    96 A(3) = .0000000001 + (RND * 10)
    97 A(4) = .0000000001 + (RND * 10)

    128 FOR I = 1 TO 30000


        129 FOR KKQQ = 1 TO 4
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        151 FOR IPP = 1 TO FIX(1 + RND * 3)


            153 J = 1 + FIX(RND * 4)

            154 REM GOTO 162


            156 r = (1 - RND * 2) * A(J)
            158 X(J) = A(J) + (RND ^ (RND * 10)) * r

            159 REM GOTO 169



            162 REM IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1

        169 NEXT IPP
        177 IF X(1) < .0000000000001## THEN 1670
        178 IF X(2) < .0000000000001## THEN 1670
        179 IF X(3) < .0000000000001## THEN 1670
        180 IF X(4) < .0000000000001## THEN 1670



        259 X(5) = 1 - X(1) ^ 2 * X(4) ^ -2 - X(2) ^ 2 * X(4) ^ -2
        260 IF X(5) < 0## THEN X(5) = X(5) ELSE X(5) = 0##


        277 X(6) = 1 - 100 * X(1) ^ -1 * X(2) ^ -1 * X(3) ^ -1
        279 IF X(6) < 0## THEN X(6) = X(6) ELSE X(6) = 0##


        355 PDU = .1 * (((-4 * X(1) - 10 * X(2) - 4 * X(3) - 2 * X(4)) / (1))) + .9 * X(1) ^ -1 * X(2) ^ -1 * X(3) ^ -1 + 1000000 * (X(5) + X(6))


        466 P = PDU

        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 < -999999 THEN 1999

    1904 PRINT M, A(1), A(2), A(3), A(4), JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [57].  The complete output of a single run through JJJJ= -31998 is shown below:

-8.79485518542623       5.290007374394635      2.689553082615962
7.028515426058126        5.934464913145813      -32000

-8.904266516841343      4.441394973630241      2.459477025054927
9.154753053790234       5.076831404091132      -31999

-8.808192539496364      5.063723901777905      2.48616842208646
7.943272034370094       5.64112871475353      -31998

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 [57], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 2 seconds, not including the time for “Creating .EXE file" (10 seconds, including the time for “Creating .EXE file").  One can compare the computational results above with those in Ojha and Biswal [45, p. 184, Table 2].


Acknowledgment

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

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem.  European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem.  Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem.  Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem.  Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes.  INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[6] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database.  Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[7] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study.  Princeton and Oxford: Princeton University Press, 2006.

[8] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming.  New York: John Wiley and Sons, Inc., 1968.

[9] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost.  Operations Research, Vol. 14, No. 1 (Jan. - Feb., 1966), pp. 52-58.

[10]  Ta-Cheng Chen (2006).  IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
.
[11]  Leandro dos Santos Coelho (2009),  Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.

[12]  William Conley (1981).  Optimization:  A Simplified Approach.  Published 1981 by Petrocelli Books in New York.

[13]  Lino Costa, Pedro (2001).  Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[14] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[15]  Kusum Deep, Krishna Pratap Singh, M. L.  Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems.  Applied Mathematics and Computation 212 (2009) 505-518.

[16]  Anoop K. Dhingra (1992).  Optimal apportionment of reliability and redundancy in series systems under multiple objections. IEEE Transactions on Reliability, Vol. 41, No. 4, 1992 December, pp. 576-582.

[17]  C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[18] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[19] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[20] Diptesh Ghosh, Ravi Kothari, Population Heuristics for the Corridor Allocation Problem, W.P. No. 2012-09-02, September 2012.  Retrieved on September 14 2012 from Google search.

[21]  Ignacio E. Grossmann.  Overview of Mixed-integer Nonlinear Programming.  https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[22]  R. Gupta, R. Malhotra (1995).  Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[23]  M. Hashish, M. P. duPlessis (1979).  Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses.  Transactions of ASME:  Journal of Engineering for Industry 101 (1979) 311-318.

[24] David M. Himmelblau, Applied Nonlinear Programming.  New York: McGraw-Hill Book Company, 1972.

[25] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Berlin: Springer-Verlag, 1981.

[26] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout.  Les Cahiers du GERAD.  Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[27] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout.  Retrieved from www.optimization-online.org./DB_HTML/2012/04/3432.html

[28]  N. K. Jain, V. K. Jain, K. Deb (2007).  Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms.  International Journal of Machine Tools and Manufacture 47 (2007), 900-919.

[29] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[30]  Adhe Kania, Kuntjoro Adji Sidarto (2016).  Solving mixed integer  nonlinear programming problems using spiral dynamics optimization algorithm.  AIP Conference Proceedings 1716, 020004 (2016).
https://doi.org/10.1063/1.4942987.  Published by the American Institute of Physics.

[31]  A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[32] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[33]  Gia-Shi Liu (2006),  A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.

[34]  Yubao Liu, Guihe Qin (2014),  A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[35]  Rein Luus (1975).  Optimization of System Reliability by a New Nonlinear Integer Programming Procedure. IEEE Transactions on Reliability, Vol. R-24, No. 1, April 1975, pp. 14-16.

[36] Milos  Madic, Miroslav  Radovanovic (2014).  Optimization of machining processes using pattern search algorithm.
International Journal of Industrial Engineering Computations 5 (2014) 223-234.  Homepage: www.GrowingScience.com/ijiec

[37]  F. Masedu, M Angelozzi (2008).  Modelling optimum fraction assignment in the 4X100 m relay race by integer linear programming.   Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.

[38]  MathWorks, Mixed Integer Optimization.  https://www.mathworks.com/help/gads;mixed-integer-optimization.html

[39]  Mohamed Arezki Mellal, Enrico Zio (2016).  A Guided Stochastic Fractal Search Approach for System Reliability Optimization.  Reliability Engineering
and System Safety 152 (2016) 213-227.

[40]  Mohamed Arezki Mellal, Edward J. Williams (2016).  Parameter optimization of advanced machining processes using cuckoo optimization algorithm and hoopla heuristic.  Journal of Intelligent Manufacturing (2016) 27 (5): 927-942.  

[41]  Mohamed Arezki Mellal, Edward J. Williams (2018).  Large-scale reliability-redundancy allocation optimization problem using three soft computing methods.  In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering.  Published July 2018, CRC Press.      

[42] 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.

[43] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[44] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[45]  A. K. Ojha, K. K. Biswal (2010).   Multi-objective geometric programming problem with weighted mean method  (IJCSIS) International Journal of Computer Science and Information Security, Vol. 7, No. 2, 2010, pp. 82-86.  http://sites.google.com/site/ijcsis/

[46] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP).  https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

[47] Panos Y. Papalambros,  Douglass J. Wilde, Principles of Optimal Design, Second Edition.  Cambridge University Press, 2000.

[48]  R. V. Rao, P. J. Pawar, J. P. Davim (2010).  Parameter optimization of ultrasonic machining process using nontraditional optimization algorihms.  Materials and Manufacturing Processes, 25 (10),1120-1130.

[49] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

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

[51]  Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017).  A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[52] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm.     Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.

[53] G. Stephanopoulos, A. W. Westerberg, The Use of Hestenes' Method of Multipliers to Resolve
Dual Gaps in Engineering System Optimization.  Journal of Optimization Theory and Applications,  Vol.15, No. 3, pp. 285-309, 1975.

[54]  Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977).  Determining Component Reliability and Redundancy for  Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[55]  Hardi Tambunan, Herman Mawengkang (2016).  Solving Mixed Integer Non-Linear Programming Using Active Constraint.  Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281.  http://www.ripublication.com/gjpam.htm

[56] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011).  Solving mixed-integer nonlinear programming problems using improved genetic algorithms.  Korean  Journal of  Chemical Engineering  28 (1):32-40 January 2011.

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

[58]  Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990).  Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.

[59]  ZHANG HongQi, HU ZiangTao, SHAO XiaoDong, LI ZiCheng, WANG YuHui  (November 2013).  IPSO-based hybrid approaches for reliability-redundany allocation problems. SCIENCE CHINA Technologial Sciences, November 2013, Vol. 56, No. 11:  2854-2864.

Solving a Bi-Objective Quadratic Fractional Integer Programming Problem with the Mixed-Integer Nonlinear Programming (MINLP) Algorithm of the Present Blog, Second Edition

Jsun Yui Wong

The alternative computer program listed below seeks to solve the following problem from Sharma, Dahiya, and Verma [47,  p. 1926, Example 2]:
        
Minimize       - ((-X(1) ^ 2 + 2 * X(2) ^ 2 - 2 * X(1) * X(2)) / (5 * X(1) ^ 2 + 4 * X(2) ^ 2 + X(1) + X(2) - 21)),

minimize            -((-X(1) ^ 2 - X(2) ^ 2 - 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2)))

subject to     

        X(1) >= 0 and integer


       X(2) >= 0 and integer


         3 * X(1) + 2 * X(2) >= 6


         4 * X(1) + 5 * X(2) <= 20.


0 DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000


    85 REM coef = RND


    89 RANDOMIZE JJJJ

    90 M1 = -3E+30
    92 M2 = -3E+30

    96 A(1) = FIX(RND * 6)

    98 A(2) = FIX(RND * 6)

    128 FOR I = 1 TO 10


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


            153 J = 1 + FIX(RND * 2)

            156 REM r = (1 - RND * 2) * A(J)
            158 REM X(J) = A(J) + (RND ^ (RND * 10)) * r
            162 IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1

        169 NEXT IPP


        238 IF X(1) < 0## THEN 1670



        239 IF X(1) > 5## THEN 1670
        248 IF X(2) < 0## THEN 1670

        249 IF X(2) > 5## THEN 1670

        257 IF 3 * X(1) + 2 * X(2) < 6## THEN 1670


        258 IF 4 * X(1) + 5 * X(2) > 20## THEN 1670

        333 REM                                                                                        X(1) = 5

        453 REM PDU = coef * ((-X(1) ^ 2 + 2 * X(2) ^ 2 - 2 * X(1) * X(2)) / (5 * X(1) ^ 2 + 4 * X(2) ^ 2 + X(1) + X(2) - 21)) + (1 - coef) * ((-X(1) ^ 2 - X(2) ^ 2 - 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2)))

        455 PDU1 = ((-X(1) ^ 2 + 2 * X(2) ^ 2 - 2 * X(1) * X(2)) / (5 * X(1) ^ 2 + 4 * X(2) ^ 2 + X(1) + X(2) - 21))
        457 PDU2 = ((-X(1) ^ 2 - X(2) ^ 2 - 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2)))


        466 P1 = PDU1

        477 P2 = PDU2


        1111 IF P1 <= M1 THEN 1670

        1122 IF P2 <= M2 THEN 1670


        1452 M1 = P1

        1432 M2 = P2

        1454 FOR KLX = 1 TO 2

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX


        1496 REM MMRR = ((-X(1) ^ 2 + 2 * X(2) ^ 2 - 2 * X(1) * X(2)) / (5 * X(1) ^ 2 + 4 * X(2) ^ 2 + X(1) + X(2) - 21))



        1504 REM SSR = ((-X(1) ^ 2 - X(2) ^ 2 - 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2)))

        1557 GOTO 128
    1670 NEXT I
    1889 IF M1 < -999999 THEN 1999

    1892 IF M2 < -999999 THEN 1999

    1904 PRINT A(1), A(2), M1, JJJJ

    1907 PRINT A(1), A(2), M2, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [53].  The output of a single run through JJJJ= -31807 is summarized below:

.
.
.

1   2    1   -31818
1   2   -1.8   -31818

0   3   1   -31816
0   3   -3   -31816

0   3   1   -31815
0   3   -3   -31815

0   3   1   -31814
0   3   -3   -31814

4   0   -.253968253968254    -31813
4   0   -1   -31813

2   2     -.2105263157894737   -31810
2   2     -1.6   -31810

5   0   -.2293577981651376    -31807
5   0   -1   -31807

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 [53], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31807 was 1 second, not including the time for “Creating .EXE file" (9 seconds, including the time for “Creating .EXE file").  One can compare the computational results above with those in Sharma, Dahiya, and Verma [47, pp. 1926-1927, Example 2].


Acknowledgment

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

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem.  European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem.  Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem.  Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem.  Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes.  INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[6] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database.  Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[7] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study.  Princeton and Oxford: Princeton University Press, 2006.

[8] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming.  New York: John Wiley and Sons, Inc., 1968.

[9] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost.  Operations Research, Vol. 14, No. 1 (Jan. - Feb., 1966), pp. 52-58.

[10]  Ta-Cheng Chen (2006).  IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
.
[11]  Leandro dos Santos Coelho (2009),  Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.

[12]  William Conley (1981).  Optimization:  A Simplified Approach.  Published 1981 by Petrocelli Books in New York.

[13]  Lino Costa, Pedro (2001).  Evolutionary algorithms approach to the solution of mixed integer non-linear programming problems. Computers and Chemical Engineering, Vol. 25, pp. 257-266, 2001.

[14] George B. Dantzig, Discrete-Variable Extremum Problems.  Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[15]  Kusum Deep, Krishna Pratap Singh, M. L.  Kansal, C. Mohan (2009), A real coded genetic algorithm for solving integer and mixed integer optimization problems.  Applied Mathematics and Computation 212 (2009) 505-518.

[16]  Anoop K. Dhingra (1992).  Optimal apportionment of reliability and redundancy in series systems under multiple objections. IEEE Transactions on Reliability, Vol. 41, No. 4, 1992 December, pp. 576-582.

[17]  C. A. Floudas, A. R. Ciric (1989), Strategies for Overcoming Uncertainties in Heat Exchanger Network Synthesis.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1133-1152, 1989.

[18] C. A. Floudas, A. Aggarwal, A. R. Ciric (1989), Global Optimum Search for Nonconvex NLP and MINLP Problems.  Computers and Chemical Engineering, Vol 13, No. 10, pp. 1117-1132, 1989.

[19] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms.  Springer-Verlag, 1990.

[20] Diptesh Ghosh, Ravi Kothari, Population Heuristics for the Corridor Allocation Problem, W.P. No. 2012-09-02, September 2012.  Retrieved on September 14 2012 from Google search.

[21]  Ignacio E. Grossmann.  Overview of Mixed-integer Nonlinear Programming.  https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf

[22]  M. Hashish, M. P. duPlessis (1979).  Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses.  Transactions of ASME:  Journal of Engineering for Industry 101 (1979) 311-318.

[23] David M. Himmelblau, Applied Nonlinear Programming.  New York: McGraw-Hill Book Company, 1972.

[24] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes.  Berlin: Springer-Verlag, 1981.

[25] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout.  Les Cahiers du GERAD.  Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[26] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout.  Retrieved from www.optimization-online.org./DB_HTML/2012/04/3432.html

[27]  N. K. Jain, V. K. Jain, K. Deb (2007).  Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms.  International Journal of Machine Tools and Manufacture 47 (2007), 900-919.

[28] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008.  Berlin: Springer, 2010.

[29]  Adhe Kania, Kuntjoro Adji Sidarto (2016).  Solving mixed integer  nonlinear programming problems using spiral dynamics optimization algorithm.  AIP Conference Proceedings 1716, 020004 (2016).
https://doi.org/10.1063/1.4942987.  Published by the American Institute of Physics.

[30]  A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems.  Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[31] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems.  Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[32]  Gia-Shi Liu (2006),  A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.

[33]  Yubao Liu, Guihe Qin (2014),  A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[34]  Rein Luus (1975).  Optimization of System Reliability by a New Nonlinear Integer Programming Procedure. IEEE Transactions on Reliability, Vol. R-24, No. 1, April 1975, pp. 14-16.

[35] Milos  Madic, Miroslav  Radovanovic (2014).  Optimization of machining processes using pattern search algorithm.
International Journal of Industrial Engineering Computations 5 (2014) 223-234.  Homepage: www.GrowingScience.com/ijiec

[36]  F. Masedu, M Angelozzi (2008).  Modelling optimum fraction assignment in the 4X100 m relay race by integer linear programming.   Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.

[37]  MathWorks, Mixed Integer Optimization.  https://www.mathworks.com/help/gads;mixed-integer-optimization.html

[37]  Mohamed Arezki Mellal, Enrico Zio (2016).  A Guided Stochastic Fractal Search Approach for System Reliability Optimization.  Reliability Engineering
and System Safety 152 (2016) 213-227.

[38]  Mohamed Arezki Mellal, Edward J. Williams (2016).  Parameter optimization of advanced machining processes using cuckoo optimization algorithm and hoopla heuristic.  Journal of Intelligent Manufacturing (2016) 27 (5): 927-942.  

[39]  Mohamed Arezki Mellal, Edward J. Williams (2018).  Large-scale reliability-redundancy allocation optimization problem using three soft computing methods.  In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering.  Published July 2018, CRC Press.      

[40] 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.

[41] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.

[42] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations," Operations Research 16 (1968), pp. 150-173.

[43] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP).  https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

[44] Panos Y. Papalambros,  Douglass J. Wilde, Principles of Optimal Design, Second Edition.  Cambridge University Press, 2000.

[45]  R. V. Rao, P. J. Pawar, J. P. Davim (2010).  Parameter optimization of ultrasonic machining process using nontraditional optimization algorihms.  Materials and Manufacturing Processes, 25 (10),1120-1130.

[46] H. S. Ryoo, N. V. Sahinidis (1995), Global Optimization of Nonconvex NLPs and MINLPs with Applications in Process Design.  Computers and Chemical Engineering, Vol. 19, No. 5, pp. 551-566, 1995.

[47]  Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017).  A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.

[48] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm.     Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.

[49] G. Stephanopoulos, A. W. Westerberg, The Use of Hestenes' Method of Multipliers to Resolve
Dual Gaps in Engineering System Optimization.  Journal of Optimization Theory and Applications,  Vol.15, No. 3, pp. 285-309, 1975.

[50]  Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977).  Determining Component Reliability and Redundancy for  Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[51]  Hardi Tambunan, Herman Mawengkang (2016).  Solving Mixed Integer Non-Linear Programming Using Active Constraint.  Global Journal of Pure and Applied Mathematics, Volume 12, Number 6 (2016), pp. 5267-5281.  http://www.ripublication.com/gjpam.htm

[52] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011).  Solving mixed-integer nonlinear programming problems using improved genetic algorithms.  Korean  Joutnal of  Chemical Engineering  28 (1):32-40 January 2011.

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

[54]  Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990).  Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.

[55]  ZHANG HongQi, HU ZiangTao, SHAO XiaoDong, LI ZiCheng, WANG YuHui  (November 2013).  IPSO-based hybrid approaches for reliability-redundany allocation problems  SCIENCE CHINA Technologial Sciences, November 2013, Vol. 56, No. 11:  2854-2864

Solving a Bi-Objective Quadratic Fractional Integer Programming Problem with the Mixed-Integer Nonlinear Programming (MINLP) Algorithm of the Present Blog

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Sharma, Dahiya, and Verma [47,  p. 1926, Example 2]:
       
Minimize       - ((-X(1) ^ 2 + 2 * X(2) ^ 2 - 2 * X(1) * X(2)) / (5 * X(1) ^ 2 + 4 * X(2) ^ 2 + X(1) + X(2) - 21)),

minimize            -((-X(1) ^ 2 - X(2) ^ 2 - 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2)))

subject to     

        X(1) >= 0 and integer


       X(2) >= 0 and integer


         3 * X(1) + 2 * X(2) >= 6


         4 * X(1) + 5 * X(2) <= 20.


0 DEFDBL A-Z

1 DEFINT K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000


    85 coef = RND


    89 RANDOMIZE JJJJ

    90 M = -3E+30

    96 A(1) = FIX(RND * 6)

    98 A(2) = FIX(RND * 6)

    128 FOR I = 1 TO 10


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


            153 J = 1 + FIX(RND * 2)

            156 REM r = (1 - RND * 2) * A(J)
            158 REM X(J) = A(J) + (RND ^ (RND * 10)) * r
            162 IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1

        169 NEXT IPP


        238 IF X(1) < 0## THEN 1670



        239 IF X(1) > 5## THEN 1670
        248 IF X(2) < 0## THEN 1670

        249 IF X(2) > 5## THEN 1670

        257 IF 3 * X(1) + 2 * X(2) < 6## THEN 1670


        258 IF 4 * X(1) + 5 * X(2) > 20## THEN 1670



        453 PDU = coef * ((-X(1) ^ 2 + 2 * X(2) ^ 2 - 2 * X(1) * X(2)) / (5 * X(1) ^ 2 + 4 * X(2) ^ 2 + X(1) + X(2) - 21)) + (1 - coef) * ((-X(1) ^ 2 - X(2) ^ 2 - 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2)))



        466 P = PDU
        1111 IF P <= M THEN 1670
        1452 M = P
        1454 FOR KLX = 1 TO 2

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX


        1496 MMRR = ((-X(1) ^ 2 + 2 * X(2) ^ 2 - 2 * X(1) * X(2)) / (5 * X(1) ^ 2 + 4 * X(2) ^ 2 + X(1) + X(2) - 21))



        1504 SSR = ((-X(1) ^ 2 - X(2) ^ 2 - 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2)))

        1557 GOTO 128
    1670 NEXT I
    1889 IF M < -11 THEN 1999

    1904 PRINT A(1), A(2), MMRR, SSR, JJJJ
1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [53].  The output of a single run through JJJJ= -31571 is summarized below:

.
.
.

1       2       1       -1.8        -31582

1       2       1       -1.8        -31581

5       0      -.2293577981651376      -1
-31579

1       2       1       -1.8        -31576

2      2        -.2105263157894737       -1.6
-31574

5       0      -.2293577981651376      -1
-31573

5       0      -.2293577981651376      -1
-31571

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 [53], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31571 was 1 second, not including the time for “Creating .EXE file" (9 seconds, including the time for “Creating .EXE file").  One can compare the computational results above with those in Sharma, Dahiya, and Verma [47, pp. 1926-1927, Example 2].


Acknowledgment

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

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