Seeking the Two (Far Apart) Extreme Points of a Four-Bar Truss Problem Involving Two Objectives

Jsun Yui Wong

The two computer programs listed below seek the two extreme points of the four-bar truss problem involving two objectives in Tawhid and Savsani [62, p. 112] and in Sadollah, Eskandar, and Kim [54, Section 4.2.1]: 
                                             
Minimize            200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4))

minimize             (10*200 / 2E+05) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4))

subject to

        1<= X(1), X(4) <=3     

        2^.5 <= X(2), X(3) <=3.   

Whereas line 437 of the first computer is 437 PDU = -200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4)), line 437 of the second computer program is 437 PDU = -(2000 / (2 * 10 ^ 5)) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4)).
 

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
   
    91 epsi = (RND * .1)


    92 A(1) = 1 + RND * 2

    93 A(2) = 2 ^ .5 + RND * 1.5857
    95 A(3) = 2 ^ .5 + RND * 1.5857

    96 A(4) = 1 + RND * 2

    128 FOR I = 1 TO 6000



        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 * 15)) * r

            161 GOTO 169

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

        169 NEXT IPP
        171 GOTO 188
        172 X(1) = INT(X(1))
        174 X(2) = INT(X(2))
        176 X(3) = INT(X(3))
        177 X(4) = INT(X(4))

        188 IF X(1) < 1## THEN 1670

        189 IF X(2) < 2 ^ .5## THEN 1670
        192 IF X(3) < 2 ^ .5## THEN 1670

        193 IF X(4) < 1## THEN 1670

        195 IF X(1) > 3## THEN 1670


        196 IF X(2) > 3## THEN 1670

        197 IF X(3) > 3## THEN 1670


        198 IF X(4) > 3## THEN 1670


        318 IF (2000 / 2E+05) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4)) > epsi THEN 1670

        437 PDU = -200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4))
        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

        1527 gg01star = 200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4))


        1529 gg02star = (2000 / 2E+05) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4))


        1557 GOTO 128

    1670 NEXT I

    1889 IF M < -99999999999 THEN 1999

   
    1914 PRINT gg01star, gg02star, JJJJ

    1924 PRINT A(1), A(2), A(3), A(4)

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [64].  The complete output of a single run through JJJJ= -31999 is shown below:

1174.199989102033         3.414213562373133D-02      -32000 
1                                    1.414213562373096             1.414213562373125
1.000000000000001

1174.199989102031         3.414213562373095D-02      -31999 
1.000000000000001         1.414213562373095             1414213562373096
1

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 [64], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999 was 2 seconds, not including the time for “Creating .EXE file" (7 seconds, total, including the time for “Creating .EXE file").  One can compare the computational results above with those in Tawhid and Savsani [62, p. 112; Figure 8, p. 113], Sadollah, Eskandar, and Kim [54, Figure 10].


Whereas line 437 above is 437 PDU = -200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4)), line 437  below is 437 PDU = -(2000 / (2 * 10 ^ 5)) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4)).
 

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 = -3D+50
   
    90 epsi = (RND * 2000)

    91 REM epsi = (RND * .1)


    92 A(1) = 1 + RND * 2

    93 A(2) = 2 ^ .5 + RND * 1.5857
    95 A(3) = 2 ^ .5 + RND * 1.5857

    96 A(4) = 1 + RND * 2

    128 FOR I = 1 TO 6000



        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 * 15)) * r

            161 GOTO 169

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

        169 NEXT IPP
     

        188 IF X(1) < 1## THEN 1670

        189 IF X(2) < 2 ^ .5## THEN 1670
        192 IF X(3) < 2 ^ .5## THEN 1670

        193 IF X(4) < 1## THEN 1670

        195 IF X(1) > 3## THEN 1670


        196 IF X(2) > 3## THEN 1670

        197 IF X(3) > 3## THEN 1670


        198 IF X(4) > 3## THEN 1670

        436 IF 200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4)) > epsi THEN 1670

        437 PDU = -(2000 / (2 * 10 ^ 5)) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4))


        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

        1527 gg01star = 200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4))


        1529 gg02star = (2000 / (2 * 10 ^ 5)) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4))


        1557 GOTO 128

    1670 NEXT I

    1889 IF M < -99999999999 THEN 1999
   

    1914 PRINT gg01star, gg02star, JJJJ

    1924 PRINT A(1), A(2), A(3), A(4)

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [64].  The complete output of a single run through JJJJ= -31991 is shown below:


1733.610508480852            2.761423749153971D-03              -31994
1.01467784230918      3      1.414213562373095
3

1727.854823855953            2.761423749153968D-03            -31991
1.000288630946933     3     1.414213562373095
3

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 [64], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31991 was 2 seconds, not including the time for “Creating .EXE file" (7 seconds, total, including the time for “Creating .EXE file").  One can compare the computational results above with those in Tawhid and Savsani [62, p. 112; Figure 8, p. 113], Sadollah, Eskandar, and Kim [54, Figure 10].


Acknowledgment

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

References

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Euclidean-Distance Approach for Solving Multi-Objective Nonlinear Programming Problems

Jsun Yui Wong

The computer program listed below seeks to solve the following two-objective nonlinear programming problem in Siwale [58, pp. 3-7, Example 1]: 
 
                                               n
Minimize                   1 - EXP( - sigma   ( X(i) -  1/(n^.5)        )^2      where n=8
                                               i=1
   
                                               n
Minimize                   1 - EXP( - sigma   ( X(i) +  1/(n^.5)        )^2      where n=8
                                               i=1


subject to

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



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 FOR J44 = 1 TO 8
        92 A(J44) = -2 + (RND * 4)
    93 NEXT J44

    128 FOR I = 1 TO 1000


        129 FOR KKQQ = 1 TO 8
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        151 FOR IPP = 1 TO FIX(1 + RND * 3)


            153 J = 1 + FIX(RND * 8)
            154 REM GOTO 162

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

            158 X(J) = A(J) + (RND ^ (RND * 15)) * 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


        331 sumone = 0

        333 FOR J44 = 1 TO 8

            335 sumone = sumone + (X(J44) - 1 / (8 ^ .5##)) ^ 2

        339 NEXT J44
        351 sumtwo = 0

        353 FOR J49 = 1 TO 8

            355 sumtwo = sumtwo + (X(J49) + 1 / (8 ^ .5##)) ^ 2


        359 NEXT J49

        411 REM

        444 PDU = -((1 - EXP(-sumone)) ^ 2 + (1 - EXP(-sumtwo)) ^ 2) ^ .5##
        466 P = PDU

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 8

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

        1488 gg01star = ((1 - EXP(-sumone)))


        1489 gg02star = ((1 - EXP(-sumtwo)))


        1557 GOTO 128

    1670 NEXT I

    1889 IF M < -9999995 THEN 1999


    1914 PRINT gg01star, gg02star

    1924 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [64].  The complete output of a single run through JJJJ= -31999 is shown below:

 .6321205515416997                 .632120566115416
-5.798466420806267D-12         -5.133093831204474D-09         7.256627994216514D-09
-8.676635200914182D-09          1.919631719002274D-08         7.599427192875048D-09
 2.228175579711506D-09          5.547345525396628D-09         -.8939534673502063
-32000

 .632120571426406                  .6321205462307094
 4.448388921193392D-10        -5.571622696701695D-09         7.200104234886022D-11
-1.236779307884272D-08        -8.895145818296203D-09         5.597197540865425D-10
-1.035884249561153D-09        -2.163515332683085D-08         -.8939534673502063
-31999

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 [64], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999 was 2 seconds, not including the time for “Creating .EXE file" (6 seconds, total, including the time for “Creating .EXE file").  One can compare the computational results above with those in the table on p. 4 of Siwale [58, p. 4, Example 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] Ritu Arora, S. R. Arora (2014). A cutting plane approach for multi-objective integer indefinite quadratic programming problem:  OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.

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

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

[11] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
.
[12] 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.

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

[14] 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.

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

[16]  Pintu Das, tapan kumar Roy (2014).  Multi-objective geometric programming and its application in gravel box problem.  Journal of global research in computer science volume 5. no.7, july 2014.  www.jgrcs.info

[17] Kalyanmoy Deb, Amrit Pratap, Subrajyoti Moitra (2000).  Mechanical component design for  multi objectives using elitist non-dominated sorting GA. Parallel Probl.Solv. Nat. PPSN VI 1917 (2000) 859-868.       

[18] 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.

[19] 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.

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

[21] 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.

[22] 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.

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

[24] 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.

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

[26] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[27] 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.

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

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

[30] 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

[31] 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

[32] 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.

[33] 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.

[34] 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.

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

[36] 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.

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

[38] 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.

[39] 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.

[40] 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

[41] 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.

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

[43] 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.

[44] 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.

[45] 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.

[46] 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.

[47] 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.

[48] 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.

[49] 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/

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

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

[52] 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.

[53] 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.

[54] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015).  Water cycle algorithm for solvinfg constrained multi-objective optimization problems.  Applied Soft Computing 27 (2015) 279-298.     

[55] 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.

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

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

[58] Isaac Siwale.  A Note on Multi-Objective Mathematical Problems. https://www.aptech.com/wp-content/uploads/2012/09/MultObjMath.pdf

[59] 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.

[60] 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.

[61] 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

[62] Mohamed Tawhid, Vimal  Savsani (2018).    Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems.   Journal of computational design and engineering 5 (2018) 104-119.     

[63] 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.

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

[65] 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.

Solving a 2-Objective Integer Nonlinear Programming Problem with the Epsilon-Constraint Method

Jsun Yui Wong

The computer program listed below seeks to solve the following 2-objective integer nonlinear programming problem from Sharma, Dahiya, and Verma [54,  p. 1926, Example 2]:   

Miniimize                 - (-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 RANDOMIZE JJJJ

    87 M = -3E+50
    88 epsi = RND * 10


    92 A(1) = (RND * 14)

    93 A(2) = (RND * 14)


    128 FOR I = 1 TO 3000

        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 * 15)) * 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

        172 X(1) = INT(X(1))
        174 X(2) = INT(X(2))


        188 IF X(1) < 0## THEN 1670

        189 IF X(2) < 0## THEN 1670


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

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




        228 IF X(1) ^ 2 + X(1) * X(2) + X(2) = 0## THEN 1670

        229 IF (X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(2)) / (X(1) ^ 2 + X(1) * X(2) + X(2)) > epsi THEN 1670

        431 PDU = (-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)

        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

        1527 gg01star = (-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)

        1529 gg02star = (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 < -99999999999 THEN 1999

    1924 PRINT -gg01star, gg02star, epsi


    1956 PRINT A(1), A(2), -M, JJJJ


1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [60].  The output of a single run through JJJJ= -31946 is summarized below:


.
.
.

-1      3      9.123768210411072
0      3      -1      -31959 

-1      1.8      7.35616147518158
1      2       -1      -31957

.2293577981651376      1      8.78101944923401
5      0       .2293577981651376          -31953

.2293577981651376      1      1.881211400032044
5      0       .2293577981651376          -31951

.2293577981651376      1      7.111677527427673
5      0       .2293577981651376          -31950

.2293577981651376      1      6.083215475082398
5      0       .2293577981651376          -31947

.2105263157894737      1.6      1.755475401878357
2      2        .2105263157894737        -31946

.
.
.

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 [60], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31946 was 1 second, not including the time for “Creating .EXE file" (8 seconds, total, including the time for “Creating .EXE file"). The (-1   3) shown above is a  dominated point. One can compare the computational results above with those in Sharma, Dahiya, and Verma [54, 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] Ritu Arora, S. R. Arora (2014). A cutting plane approach for multi-objective integer indefinite quadratic programming problem:  OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.

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

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

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

[12] 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.

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

[14] 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.

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

[16]  Pintu Das, tapan kumar Roy (2014).  Multi-objective geometric programming and its application in gravel box problem.  Journal of global research in computer science volume 5. no.7, july 2014.  www.jgrcs.info

[17] 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.

[18] 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.

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

[20] 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.

[21] 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.

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

[23] 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.

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

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

[26] 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.

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

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

[29] 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

[30] 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

[31] 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.

[32] 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.

[33] 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.

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

[35] 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.

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

[37] 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.

[38] 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.

[39] 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

[40] 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.

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

[42] 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.

[43] 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.

[44] 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.

[45] 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.

[46] 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.

[47] 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.

[48] 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/

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

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

[51] 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.

[52] 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.

[53] 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.

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

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

[56] 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.

[57] 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.

[58] 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

[59] 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.

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

[61] 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.

Solving Another 3-Objective Integer Nonlinear Programming Problem with the Epsilon-Constraint Method

Jsun Yui Wong

The computer program listed below seeks to solve the following 3-objective integer nonlinear programming problem from Arora and Arora [8,  p. 377]:   

Maximize               (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)
       

maximize                  (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)
   

maximize                  (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)
     

subject to


         3 * X(1) - 2 * X(2) + X(3) + X(4) <= 9

         X(2) + 3 * X(3) + X(4) <= 7


         X(1) + X(2) + X(3) + X(4) <= 12


         2 * X(2) + X(3) <= 10


         0 <=X(1) <=5

        0<= X(2)<=3

         0 <=X(3) <=3

        0<= X(4)<=1

 X(1) through X(4) are general integer variables.


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
   
    91 w1 = 10# + FIX(RND * 201)
    92 w2 = 10# + FIX(RND * 201)

    95 A(1) = FIX(RND * 6)
    97 A(2) = FIX(RND * 4)

    98 A(3) = FIX(RND * 4)
    99 A(4) = FIX(RND * 2)


    128 FOR I = 1 TO 100000


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


            153 J = 1 + FIX(RND * 4)
            154 GOTO 162

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

            158 X(J) = A(J) + (RND ^ (RND * 15)) * r

            161 REM GOTO 169

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

        169 NEXT IPP
        171 X(1) = INT(X(1))
        173 X(2) = INT(X(2))

        174 X(3) = INT(X(3))
        175 X(4) = INT(X(4))



        177 IF X(1) < 0## THEN 1670
        187 IF X(2) < 0## THEN 1670

        188 IF X(3) < 0## THEN 1670
        189 IF X(4) < 0## THEN 1670


        192 IF X(1) > 5 THEN 1670

        194 IF X(2) > 3 THEN 1670

        196 IF X(3) > 3 THEN 1670
        198 IF X(4) > 1 THEN 1670


        226 IF 3 * X(1) - 2 * X(2) + X(3) + X(4) > 9 THEN 1670



        228 IF X(2) + 3 * X(3) + X(4) > 7 THEN 1670

        230 IF X(1) + X(2) + X(3) + X(4) > 12 THEN 1670

        232 IF 2 * X(2) + X(3) > 10 THEN 1670
        244 IF (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11) < w1 THEN 1670

        255 IF (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9) < w2 THEN 1670

        417 PDU = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)


        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


        1522 g01star = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)


        1524 g02star = (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)


        1526 g03star = (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)


        1557 GOTO 128
    1670 NEXT I

    1889 IF M < -99999999999 THEN 1999
    1922 PRINT g01star, g02star, g03star, JJJJ
    1928 PRINT A(1), A(2), A(3), A(4)

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [60].  The output of a single run through JJJJ= -31478 is summarized below:

171      187      90      -31995
4      2      1      0     

180      165      108      -31994
3      0      0      0
.
.
.

171      187      90      -31670
4      2      1      0     

180      165      108      -31667
3      0      0      0

165      196      105      -31666
3      1      2      0
.
.
.

130      170      150      -31478
2      0      2      1     

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 [60], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31478 was 50 seconds, not including the time for “Creating .EXE file" (59 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Arora and Arora  [8, pp. 377-380].


 Acknowledgment

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

References

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[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] Ritu Arora, S. R. Arora (2015). A cutting plane approach for multi-objective integer indefinite quadratic programming problem:  OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.

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

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

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

[14] 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.

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[16]  Pintu Das, tapan kumar Roy (2014).  Multi-objective geometric programming and its application in gravel box problem.  Journal of global research in computer science volume 5. no.7, july 2014.  www.jgrcs.info

[17] 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.

[18] 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.

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

[20] 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.

[21] 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.

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

[23] 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.

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

[25] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
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[26] 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.

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

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

[29] 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

[30] 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

[31] 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.

[32] 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.

[33] 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.

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

[35] 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.

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

[37] 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.

[38] 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.

[39] 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

[40] 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.

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

[42] 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.

[43] 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.

[44] 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.

[45] 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.

[46] 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.

[47] 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.

[48] 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/

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

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

[51] 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.

[52] 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.

[53] 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.

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

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

[56] 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.

[57] 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.

[58] 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

[59] 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.

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

[61] 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.

Solving a Multi-Objective Integer Nonlinear Programming Problem with the Epsilon-Constraint Method

Jsun Yui Wong

The computer program listed below seeks to solve the following 3-objective integer nonlinear programming problem from Sharma [52,  p. 149]:   

Maximize               ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2

maximize               (X(1) / (X(2) + 1)) ^ 2

maximize               ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2

subject to

         X(1) + X(2) <= 5

         4 * X(1) + X(2) <= 8

         0 <=X(1) <=4

        0<= X(2)<=2

 X(1) and X(2) are general integer variables.


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
 

    92 A(1) = FIX(RND * 5)
    93 A(2) = FIX(RND * 3)
    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 * 15)) * 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
        171 X(1) = INT(X(1))
        173 X(2) = INT(X(2))
        177 IF X(1) < 0## THEN 1670
        187 IF X(2) < 0## THEN 1670

        188 IF X(1) > 4## THEN 1670


        222 IF X(2) > 2## THEN 1670

        226 IF X(1) + X(2) > 5 THEN 1670
        227 IF 4 * X(1) + X(2) > 8 THEN 1670

        229 IF (X(1) / (X(2) + 1)) ^ 2 < .3 THEN 1670

        235 IF ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2 < .3 THEN 1670

     
        415 PDU = ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2

        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


        1522 g01star = ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2

        1524 g02star = (X(1) / (X(2) + 1)) ^ 2


        1526 g03star = ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2


        1557 GOTO 128
    1670 NEXT I

    1889 IF M < -99999999999 THEN 1999
    1922 PRINT g01star, g02star, g03star
    1924 PRINT A(1), A(2), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [59].  The complete output of a single run through JJJJ= -31992 is shown below:

4   4   .4444444444444444
2   0   4   -32000

4   4   .4444444444444444
2   0   4   -31999

4   4   .4444444444444444
2   0   4   -31998

1   1   .5625
1   0   1   -31997

4   4   .4444444444444444
2   0   4   -31996

4   4   .4444444444444444
2   0   4   -31995

4   4   .4444444444444444
2   0   4   -31994

1   1   .5625
1   0   1   -31993

1   1   .5625
1   0   1   -31992

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 = -31992 was 2 seconds, not including the time for “Creating .EXE file" (11 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Sharma [52, p. 151].


 Acknowledgment

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

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