Solving Multi-Objective Mixed-Integer Nonlinear Programs by Example

Jsun Yui Wong

The computer program listed below seeks to solve the following mixed-integer fuzzy programming formulation from page 121 of Ali and Hasan [1]: 
   
Minimize                  X(8)

subject to 

         (1 - .35 ^ (4 + X(1))) * (1 - .45 ^ (2 + X(2))) * (1 - .3 ^ (3 + X(3))) - .0940632 * X(8) >= .8967782

         (1 - .3 ^ (3 + X(4))) * (1 - .45 ^ (2 + X(5))) * (1 - .4 ^ (2 + X(6))) * (1 - .35 ^ (3 + X(7))) - .1678001 * X(8) >= .8311538

         140 * (X(1) + EXP(.25 * X(1))) + 110 * (X(2) + EXP(.25 * X(2))) + 150 * (X(3) + EXP(.25 * X(3))) + 70 * (X(4) + EXP(.25 * X(4))) + 30 * (X(5) + EXP(.25 * X(5))) + 45 * (X(6) + EXP(.25 * X(6))) + 65 * (X(7) + EXP(.25 * X(7))) <= 3000

        1<= X(1) <=4
        1<= X(2) <= 4
        1<= X(3) <= 7
        1<= X(4) <= 5
        1<= X(5) <= 8
        1<= X(6) <= 10
        1<= X(7) <= 7
        X(8) >=0

X(1) through X(7) are integer variables; X(8) is continuous and >=0.



0 REM 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

    111 REM EPSILON = RND * .5

    113 FOR J44 = 1 TO 7

        120 A(J44) = 1 + FIX((RND * 7))
     
    125 NEXT J44
    127 A(8) = RND

    128 FOR I = 1 TO FIX(6000 + RND * 60000)


        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 IF RND < .5 THEN 156 ELSE 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) - FIX(1 + RND * 2) ELSE X(J) = A(J) + FIX(1 + RND * 2)


        169 NEXT IPP


        170 FOR J44 = 1 TO 7

            171 X(J44) = INT(X(J44))
            172 IF X(J44) < 1 THEN 1670
        173 NEXT J44

        174 IF X(1) > 4 THEN 1670
        175 IF X(2) > 4 THEN 1670
        176 IF X(3) > 7 THEN 1670
        177 IF X(4) > 5 THEN 1670
        178 IF X(5) > 8 THEN 1670
        179 IF X(6) > 10 THEN 1670
        180 IF X(7) > 7 THEN 1670
        181 IF X(8) > 1 THEN 1670
       

        411 IF 140 * (X(1) + EXP(.25 * X(1))) + 110 * (X(2) + EXP(.25 * X(2))) + 150 * (X(3) + EXP(.25 * X(3))) + 70 * (X(4) + EXP(.25 * X(4))) + 30 * (X(5) + EXP(.25 * X(5))) + 45 * (X(6) + EXP(.25 * X(6))) + 65 * (X(7) + EXP(.25 * X(7))) > 3000 THEN 1670



        418 IF (1 - .35 ^ (4 + X(1))) * (1 - .45 ^ (2 + X(2))) * (1 - .3 ^ (3 + X(3))) - .0940632 * X(8) < .8967782 THEN 1670



        419 IF (1 - .3 ^ (3 + X(4))) * (1 - .45 ^ (2 + X(5))) * (1 - .4 ^ (2 + X(6))) * (1 - .35 ^ (3 + X(7))) - .1678001 * X(8) < .8311538 THEN 1670


        452 PDU = X(8)


        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

        1557 GOTO 128
    1670 NEXT I

    1889 IF M < .93 THEN 1999


    1923 PRINT A(1), A(2), A(3), A(4), A(5)
    1924 PRINT A(6), A(7), A(8), M, JJJJ
1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [79].  The complete output of a single run through JJJJ= -31979 is showm below:

2      4      2      2      6
5      2      .9409117      .9409117      -31989

2      4      2      2      6
5      2      .9409117      .9409117      -31986

2      4      2      2      6
5      2      .9409117      .9409117      -31985

2      4      2      2      7
4      2      .9318493      .9318493      -31983

2      4      2      2      5
4      3      .9344354      .9344354      -31979

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 [79], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31979 was 3 seconds, not including the time for “Creating .EXE file" (20 seconds, total, including the time for “Creating .EXE file").  One can compare the computational results above with those on page 121 of  Ali and Hasan [1].   


Acknowledgment

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

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Solving Multi-Objective Mixed-Integer Nonlinear Programs by Example

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from page 193 of Ali, Raghav, and Bari [1; the first problem on p. 193]: 
   
Minimize                 X(6) + X(7) + X(8) + X(9) + X(10)

subject to 

  (1 - (1 - .9) ^ (1 + X(1))) +X(6)= .9999

        (1 - (1 - .85) ^ (2 + X(2)))+X(7) = .9994

        (1 - (1 - .85) ^ (2 + X(3))) +X(8)= .9994

        (1 - (1 - .8) ^ (2 + X(4)))+X(9) = .9984

        (1 - (1 - .85) ^ (1 + X(5)))+X(10) = .9994
     
         (3 * (X(1) + EXP(.25 * X(1)))) + (4 * (X(2) + EXP(.25 * X(2)))) + (3 * (X(3) + EXP(.25 * X(3)))) + (5 * (X(4) + EXP(.25 * X(4)))) + (4 * (X(5) + EXP(.25 * X(5)))) <= 60

         (8 * (X(1) + EXP(.25 * X(1)))) + (7 * (X(2) + EXP(.25 * X(2)))) + (8 * (X(3) + EXP(.25 * X(3)))) + (4 * (X(4) + EXP(.25 * X(4)))) + (6 * (X(5) + EXP(.25 * X(5)))) <= 90

        0<= X(1) <= 3
        0<= X(2) <= 2
        0<= X(3) <= 2
        0<= X(4) <= 2
        0<= X(5) <= 3

X(1) through X(5) are integer variables; X(6) through X(10) are continuous and >=0.


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

    120 A(1) = FIX((RND * 4))
    121 A(2) = FIX((RND * 4))
    122 A(3) = FIX((RND * 4))
    123 A(4) = FIX((RND * 4))

    124 A(5) = FIX((RND * 4))

    128 FOR I = 1 TO FIX(2000 + RND * 2000)


        129 FOR KKQQ = 1 TO 5

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        135 FOR IPP = 1 TO FIX(1 + RND * 4)


            143 J = 1 + FIX(RND * 5)
            147 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) - FIX(1 + RND * 0) ELSE X(J) = A(J) + FIX(1 + RND * 0)


        169 NEXT IPP


        170 FOR J44 = 1 TO 5

            171 X(J44) = INT(X(J44))
            172 IF X(J44) < 0 THEN 1670
        174 NEXT J44


        176 IF X(1) > 3 THEN 1670
        177 IF X(2) > 2 THEN 1670
        178 IF X(3) > 2 THEN 1670
        179 IF X(4) > 2 THEN 1670
        180 IF X(5) > 3 THEN 1670

        248 X(6) = -(1 - (1 - .9) ^ (1 + X(1))) + .9999
        250 X(7) = -(1 - (1 - .85) ^ (2 + X(2))) + .9994

        251 X(8) = -(1 - (1 - .85) ^ (2 + X(3))) + .9994


        252 X(9) = -(1 - (1 - .8) ^ (2 + X(4))) + .9984
        254 X(10) = -(1 - (1 - .85) ^ (1 + X(5))) + .9994

        409 IF (3 * (X(1) + EXP(.25 * X(1)))) + (4 * (X(2) + EXP(.25 * X(2)))) + (3 * (X(3) + EXP(.25 * X(3)))) + (5 * (X(4) + EXP(.25 * X(4)))) + (4 * (X(5) + EXP(.25 * X(5)))) > 60 THEN 1670
        411 IF (8 * (X(1) + EXP(.25 * X(1)))) + (7 * (X(2) + EXP(.25 * X(2)))) + (8 * (X(3) + EXP(.25 * X(3)))) + (4 * (X(4) + EXP(.25 * X(4)))) + (6 * (X(5) + EXP(.25 * X(5)))) > 90 THEN 1670
        454 PDU = -X(6) - X(7) - X(8) - X(9) - X(10)

        466 P = PDU

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 10

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

        1557 GOTO 128
    1670 NEXT I
    1890 IF M < -99999 THEN 1999


    1923 PRINT A(1), A(2), A(3), A(4)

    1924 PRINT A(5), M, JJJJ
    1927 PRINT A(6), A(7), A(8), A(9), A(10)

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [78].  The complete output of a single run through JJJJ= -31997 is showm below:

1     1     1     2
2      -1.822499999999983D-02      -32000
9.900000000000006D-03      2.774999999999956D-03      2.774999999999956D-03
-4.7271214720368D-17        2.774999999999956D-03

1     1     1     2
2      -1.822499999999983D-02      -31999
9.900000000000006D-03      2.774999999999956D-03      2.774999999999956D-03
-4.7271214720368D-17        2.774999999999956D-03

1     1     1     2
2      -1.822499999999983D-02      -31998
9.900000000000006D-03      2.774999999999956D-03      2.774999999999956D-03
-4.7271214720368D-17        2.774999999999956D-03

1     1     1     2
2      -1.822499999999983D-02      -31997
9.900000000000006D-03      2.774999999999956D-03      2.774999999999956D-03
-4.7271214720368D-17        2.774999999999956D-03

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 [78], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31997 was 2 seconds, not including the time for “Creating .EXE file" (16 seconds, total, including the time for “Creating .EXE file").  One can compare the computational results above with those on page 193 of  Ali, Raghav, and Bari [1].   


Acknowledgment

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

References

[1]  Irfan  Ali, Yashpal Singh Raghav, Abdul Bari (2011).  Integer goal programming approach for finding a compromise allocation of repairable components.  International Journal of Engineering, Science and Technology, Vol. 3, No. 6, 2011, pp. 184-195.

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

[3] 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.

[4] 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.

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

[6] Oscar Augusto, Bennis Fouad,  Stephane Caro (2012).  A new method for decision making in multi-objective optimization problems.  Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.

[7] 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.

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

[9] 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.

[10] 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.

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

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

[13] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
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[16] 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.

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

[18]  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

[19] Kalyanmoy Deb, Amrit Pratap, Subrajyoti Moitra (2000).  Mechanical component design for  multi objectives using elitist non-dominated sorting GA. Proceedings of the Parallel Probl.Solv. Nat. PPSN VI Conference, Paris, France, pp. 859-868 (2000).  (Please see pp. 1-10 in Technical Report No. 200002 via Google search.)

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[35] 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

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[40] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem:  OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.

[41]  Mohammed Faisal Khan, Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2012)  Allocation in multivariate stratified surveys with non-linear random cost function.     American Journal of operations research, 2012, 2, 100-105. 

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

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[44] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
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[46] 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.

[47] Milos Madic, Miroslav Radovanovic (2014). Optimization of machining processes using pattern search algorithm.
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[49] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html

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[51] 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.

[52] 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.

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[54] 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.

[55]  A. Moradi, A. M. Nafchi, A. Ghanbarzadeh (2015).  Multi-objective optimization of truss structures using the bee algorithm.  (One can read this via Goodle search.)

[56] 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.

[57]  Subhash C. Narula, H. Roland Weistroffer (1989).  A flexible method for nonlinear criteria decisionmaking problems. Ieee Transactions on Systems, Man and Cybernetics, vol. 19 , no. 4, July/August 1989, pp. 883-887.

[58] 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.

[59] 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/

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

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

[62]  Ciara Pike-Burke.  Multi-Objective Optimization.  https://www.lancaster.ac.uk/pg/pikeburc/report1.pdf.

[63]  Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014)  Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the the case of the non-response, Journal of Statistical Computation and Simulation 84:1, 22-36. 

[64] 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.

[65] 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.

[66] 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.

[67] 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.

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

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

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

[71] 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.

[72] 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

[73] 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.

[74] 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.

[75]  Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys. Annals of Operations Research (2015) 226:659-668.
 
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34:325-334.

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Solving Another Integer Nonlinear Programming Problem in Multivariate Stratified Sampling

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from page 32 of Raghav, Ali, and Bari [61; the first problem on p. 32]: 
   
Minimize                  ((.7893 + (X(5) - 1) * .06) / X(1) + (.4411 + (X(6) - 1) * .02) / X(2) + (.3576 + (X(7) - 1) * .02) / X(3) + (.3973 + (X(8) - 1) * .03) / X(4))
   
subject to 

         ((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) <= 5000

        2<= X(1) <= 1214
        2<= X(2) <= 822
        2<= X(3) <= 1028
        2<= X(4) <= 786.

X(1) through X(4) are integer variables, and X(5) through X(8) are continuous and >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

    120 A(1) = 2 + ((RND * 500))
    121 A(2) = 2 + ((RND * 500))
    122 A(3) = 2 + ((RND * 500))
    123 A(4) = 2 + ((RND * 500))

    124 A(5) = 1 + ((RND * 3))
    125 A(6) = 1 + ((RND * 3))

    126 A(7) = 1 + ((RND * 3))
    127 A(8) = 1 + ((RND * 3))

    128 FOR I = 1 TO FIX(120000 + RND * 120000)

        129 FOR KKQQ = 1 TO 8
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        151 FOR IPP = 1 TO FIX(1 + RND * 4)


            153 J = 1 + FIX(RND * 8)

            154 IF RND < .5 THEN GOTO 162 ELSE GOTO 156
            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) - FIX(1 + RND * 2) ELSE X(J) = A(J) + FIX(1 + RND * 2)


        169 NEXT IPP


        170 FOR J44 = 1 TO 4

            171 X(J44) = INT(X(J44))
            172 IF X(J44) < 2 THEN 1670
        173 NEXT J44

        174 IF X(1) > 1214 THEN 1670
        175 IF X(2) > 822 THEN 1670
        176 IF X(3) > 1028 THEN 1670
        177 IF X(4) > 786 THEN 1670


        190 FOR J44 = 5 TO 8

            195 IF X(J44) < 1 THEN 1670

        199 NEXT J44


        423 IF ((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) > 5000 THEN 1670


        441 PDU = -((.7893 + (X(5) - 1) * .06) / X(1) + (.4411 + (X(6) - 1) * .02) / X(2) + (.3576 + (X(7) - 1) * .02) / X(3) + (.3973 + (X(8) - 1) * .03) / X(4))

        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

        1557 GOTO 128
    1670 NEXT I

    1889 IF M < -.007 THEN 1999

    1923 PRINT A(1), A(2), A(3), A(4)

    1924 PRINT A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ


.This BASIC computer program was run with QB64v1000-win [76].  The output of a single run through JJJJ= -31994 is summarized below:

.
.
.
480        307        255        247
2.134167209270545     2.20709981832645     2.316044120635944
2.10987713445238       -6.550469146403757D-03      -31997
.
.
.
482        307        254        247
2.126968989359377     2.24545298061206     2.304703072891653
2.11596151613346       -6.55043373300427D-03      -31994

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 [76], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31994 was 17 seconds, not including the time for “Creating .EXE file" (30 seconds, total, including the time for “Creating .EXE file").  One can compare the computational results above with those on page 32 of Raghav, Ali, and Bari [61].   


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] Oscar Augusto, Bennis Fouad,  Stephane Caro (2012).  A new method for decision making in multi-objective optimization problems.  Pesquisa Operacional, Sociedade Brasileira de Pesquisa Operacional, 2012 32 (3), pp.331-369.

[6] 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.

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

[8] 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.

[9] 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.

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

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

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

[13] 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.

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

[15] 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.

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

[17]  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

[18] Kalyanmoy Deb, Amrit Pratap, Subrajyoti Moitra (2000).  Mechanical component design for  multi objectives using elitist non-dominated sorting GA. Proceedings of the Parallel Probl.Solv. Nat. PPSN VI Conference, Paris, France, pp. 859-868 (2000).  (Please see pp. 1-10 in Technical Report No. 200002 via Google search.)

[19] 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.

[20] 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.

[21]  Wassila Drici, Mustapha Moulai (2019):  An exact method for solving multi-objective integer indefinite quadratic programs, Optimization Methods and Software.

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

[23] 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.

[24] 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.

[24] 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.

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

[26] 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.

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

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

[29] 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.

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

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

[32] 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

[33] 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

[34] Ekta  Jain, Kalpana Dahiya, Vanita  Verma (2018):  Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.

[35] 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.

[36] 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.

[37] 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.

[38] Reena Kapoor, S. R. Arora (2006). Linearization of a 0-1 quadratic fractional programming problem:  OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.

[39]  Mohammed Faisal Khan, Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2012)  Allocation in multivariate stratified surveys with non-linear random cost function.     American Journal of operations research, 2012, 2, 100-105. 

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

[41] 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.

[42] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[43] 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.

[44] 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.

[45] 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

[46] 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.

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

[48]  Kaisa Miettinen, Petrie Eskelinen, Francisco Ruiz, Mariano Luque (2010).  NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point.  European Journal of Operational Research 206 (2010) 426-434.

[49] 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.

[50] 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.

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[53]  A. Moradi, A. M. Nafchi, A. Ghanbarzadeh (2015).  Multi-objective optimization of truss structures using the bee algorithm.  (One can read this via Goodle search.)

[54] 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.

[55]  Subhash C. Narula, H. Roland Weistroffer (1989).  A flexible method for nonlinear criteria decisionmaking problems. Ieee Transactions on Systems, Man and Cybernetics, vol. 19 , no. 4, July/August 1989, pp. 883-887.

[56] 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.

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

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

[60]  Ciara Pike-Burke.  Multi-Objective Optimization.  https://www.lancaster.ac.uk/pg/pikeburc/report1.pdf.

[61]  Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014)  Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the the case of the non-response, Journal of Statistical Computation and Simulation 84:1, 22-36. 

[62] 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.

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Solving Another Multi-Objective Mixed-Integer Nonlinear Programming Problem in Multivariate Stratified Sampling with the Goal Programming Technique

Jsun Yui Wong

The computer program listed below seeks to solve the following problem on pp. 32-34 of Raghav, Ali, and Bari [60]: 

Minimize             X(9) + X(10)
     
subject to 

  ((.7893 + (X(5) - 1) * .06) / X(1) + (.4411 + (X(6) - 1) * .02) / X(2) + (.3576 + (X(7) - 1) * .02) / X(3) + (.3973 + (X(8) - 1) * .03) / X(4)) - X(9) <= .00655   
  ((.8316 + (X(5) - 1) * .06) / X(1) + (.3358 + (X(6) - 1) * .02) / X(2) + (.1062 + (X(7) - 1) * .01) / X(3) + (.2791 + (X(8) - 1) * .02) / X(4)) - X(10) <= .00454
 ((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) <= 5000

        2<= X(1) <= 1214
        2<= X(2) <= 822
        2<= X(3) <= 1028
        2< X(4) <= 786

 X(1) through X(4) are integers and  X(5) through X(8) are continuous and >1.

One of the three inequality constraints above is expected to be binding.  That makes the arrival/testing of line 402, which is 402 X(5) = -.9 * X(1) / ((2.4 * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) - 5000).



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 31994

    85 RANDOMIZE JJJJ

    87 M = -3E+50

    111 REM  EPSILON = RND * .5
    116 A(9) = ((RND * 2))
    117 A(10) = ((RND * 2))

    120 A(1) = 2 + ((RND * 500))
    121 A(2) = 2 + ((RND * 500))
    122 A(3) = 2 + ((RND * 500))
    123 A(4) = 2 + ((RND * 500))

    124 A(5) = 1 + ((RND * 3))
    125 A(6) = 1 + ((RND * 3))

    126 A(7) = 1 + ((RND * 3))
    127 A(8) = 1 + ((RND * 3))

    128 FOR I = 1 TO FIX(120000 + RND * 120000)

        129 FOR KKQQ = 1 TO 10
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        151 FOR IPP = 1 TO FIX(1 + RND * 4)


            153 J = 1 + FIX(RND * 10)

            154 IF J < 4.5 THEN GOTO 162 ELSE GOTO 156
            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) - FIX(1 + RND * 2) ELSE X(J) = A(J) + FIX(1 + RND * 2)


        169 NEXT IPP


        170 FOR J44 = 1 TO 4

            171 X(J44) = INT(X(J44))
            172 IF X(J44) < 2 THEN 1670
        173 NEXT J44

        174 IF X(1) > 1214 THEN 1670
        175 IF X(2) > 822 THEN 1670
        176 IF X(3) > 1028 THEN 1670
        177 IF X(4) > 786 THEN 1670


        190 FOR J44 = 5 TO 8

            192 IF X(J44) <= 1 THEN 1670
        193 NEXT J44

        197 IF X(9) < 0 THEN 1670

        198 IF X(10) < 0 THEN 1670


        402 X(5) = -.9 * X(1) / ((2.4 * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) - 5000)

        403 IF X(5) < 1 THEN 1670


        411 IF ((.7893 + (X(5) - 1) * .06) / X(1) + (.4411 + (X(6) - 1) * .02) / X(2) + (.3576 + (X(7) - 1) * .02) / X(3) + (.3973 + (X(8) - 1) * .03) / X(4)) - X(9) > .00655 THEN 1670

        419 IF ((.8316 + (X(5) - 1) * .06) / X(1) + (.3358 + (X(6) - 1) * .02) / X(2) + (.1062 + (X(7) - 1) * .01) / X(3) + (.2791 + (X(8) - 1) * .02) / X(4)) - X(10) > .00454 THEN 1670


        438 PDU = -X(9) - X(10)


        466 P = PDU

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 10

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

        1557 GOTO 128
    1670 NEXT I

    1889 IF M < -.00011978 THEN 1999


    1923 PRINT A(1), A(2), A(3), A(4)

    1924 PRINT A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [74].  The complete output of a single run through JJJJ= -26354 is shown below:

529   311   220   246
2.170065665950915      2.083348773720887            2.118396262984767
2.143930766863212      4.443891220056763D-05      7.526331226032686D-05     
-1.197022244608945D-04      -26354

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 [74], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -26354 was 2 hours and 5 minutes.  One can compare the computational results above with those on page 33 of Raghav, Ali, and Bari [60].   


Acknowledgment

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

References

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[17]  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

[18] Kalyanmoy Deb, Amrit Pratap, Subrajyoti Moitra (2000).  Mechanical component design for  multi objectives using elitist non-dominated sorting GA. Proceedings of the Parallel Probl.Solv. Nat. PPSN VI Conference, Paris, France, pp. 859-868 (2000).  (Please see pp. 1-10 in Technical Report No. 200002 via Google search.)

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[20] 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.

[21]  Wassila Drici, Mustapha Moulai (2019):  An exact method for solving multi-objective integer indefinite quadratic programs, Optimization Methods and Software.

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[23] 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.

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

[26] 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.

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

[28] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[29] 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.

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

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

[32] 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

[33] 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

[34] Ekta  Jain, Kalpana Dahiya, Vanita  Verma (2018):  Integer quadratic fractional programming problems with bounded variables, Annals of Operations Research (October 2018) 269: pp. 269-295.

[35] 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.

[36] 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.

[37] 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.

[38] reena kapoor, S. R. Arora (2006). linearization of a 0-1 quadratic fractional programming problem:  OPSEARCH of the Operational Research Society of India (2006), 43(2):190-207.

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

[40] 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.

[41] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[42] 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.

[43] 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.

[44] 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

[45] 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.

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

[47]  Kaisa Miettinen, Petrie Eskelinen, Francisco Ruiz, Mariano Luque (2010).  NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point.  European Journal of Operational Research 206 (2010) 426-434.

[48] 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.

[49] 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.

[50] 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.

[51] 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.

[52]  A. Moradi, A. M. Nafchi, A. Ghanbarzadeh (2015).  Multi-objective optimization of truss structures using the bee algorithm.  (One can read this via Goodle search.)

[53] 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.

[54]  Subhash C. Narula, H. Roland Weistroffer (1989).  A flexible method for nonlinear criteria decisionmaking problems. Ieee Transactions on Systems, Man and Cybernetics, vol. 19 , no. 4, July/August 1989, pp. 883-887.

[55] 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.

[56] 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/

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

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

[59]  Ciara Pike-Burke.  Multi-Objective Optimization.  https://www.lancaster.ac.uk/pg/pikeburc/report1.pdf.

[60]  Yashpal Singh Raghav, Irfan Ali, Abdul Bari (2014)  Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the the case of the non-response, Journal of Statistical Computation and Simulation 84:1, 22-36. 

[61] 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.

[62] 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.

[63] 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.

[64] 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.

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

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

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

[68] 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.

[69] 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

[70] 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.

[71] 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.

[72]  V. Verma, H. C. Bakhshi, M. C. Puri (1990)  Ranking in integer linear fractional programming problems. ZOR - Methods and Models of Operations Research (1990)
34:325-334.

[73] 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.

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

[75] 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 Multi-Objective Mixed-Integer Nonlinear Programming Problem in Multivariate Stratified Sampling

Jsun Yui Wong

The computer program listed below seeks to solve  the following problem based on and different from the problem on p. 34 of Raghav, Ali, and Bari [60, p. 34, with the epsilon-constraint technique]: 

Minimize       (1 / 611.57) * ((493.33 + (X(5) - 1) * 37) / X(1) + (275.68 + (X(6) - 1) * 13.78) / X(2) + (223.52 + (X(7) - 1) * 13.97) / X(3) + (248.29 + (X(8) - 1) * 17.38) / X(4)), the principal objective
     
minimize       (1 / 994.14) * ((831.61 + (X(5) - 1) * 62.37) / X(1) + (335.76 + (X(6) - 1) * 16.79) / X(2) + (106.17 + (X(7) - 1) * 6.64) / X(3) + (279.11 + (X(8) - 1) * 19.54) / X(4)) 

subject to
 
 ((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) <= 5000

        2<= X(1) <= 1214
        2<= X(2) <= 822
        2<= X(3) <= 1028
        2< X(4) <= 786

X(1) through X(4) are integers and  X(5) through X(8) are continuous and >1.

One notes line 111, which is 111 EPSILON = RND * .5.


0 REM 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

    111 EPSILON = RND * .5


    120 A(1) = 2 + ((RND * 500))
    121 A(2) = 2 + ((RND * 500))
    122 A(3) = 2 + ((RND * 500))
    123 A(4) = 2 + ((RND * 500))

    124 A(5) = 1 + ((RND * 5))
    125 A(6) = 1 + ((RND * 5))

    126 A(7) = 1 + ((RND * 5))
    127 A(8) = 1 + ((RND * 5))


    128 FOR I = 1 TO FIX(50000 + RND * 60000)

        129 FOR KKQQ = 1 TO 8
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        151 FOR IPP = 1 TO FIX(1 + RND * 4)


            153 J = 1 + FIX(RND * 8)

            154 IF J < 4.5 THEN GOTO 162 ELSE GOTO 156
            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) - FIX(1 + RND * 0) ELSE X(J) = A(J) + FIX(1 + RND * 0)


        169 NEXT IPP


        170 FOR J44 = 1 TO 4

            171 X(J44) = INT(X(J44))
            172 IF X(J44) < 2 THEN 1670
        173 NEXT J44

        174 IF X(1) > 1214 THEN 1670
        175 IF X(2) > 822 THEN 1670
        176 IF X(3) > 1028 THEN 1670
        177 IF X(4) > 786 THEN 1670


        190 FOR J44 = 5 TO 8

            192 IF X(J44) <= 1 THEN 1670
        193 NEXT J44


        427 IF (1 / 994.14) * ((831.61 + (X(5) - 1) * 62.37) / X(1) + (335.76 + (X(6) - 1) * 16.79) / X(2) + (106.17 + (X(7) - 1) * 6.64) / X(3) + (279.11 + (X(8) - 1) * 19.54) / X(4)) < EPSILON THEN 1670

        431 IF ((2.4 + (.9 / X(5))) * X(1) + (3.4 + (.8 / X(6))) * X(2) + (4 + (1.25 / X(7))) * X(3) + (4.6 + (1.68 / X(8))) * X(4)) > 5000 THEN 1670

        436 PDU = -(1 / 611.57) * ((493.33 + (X(5) - 1) * 37) / X(1) + (275.68 + (X(6) - 1) * 13.78) / X(2) + (223.52 + (X(7) - 1) * 13.97) / X(3) + (248.29 + (X(8) - 1) * 17.38) / X(4))
        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

        1557 GOTO 128
    1670 NEXT I

    1889 IF M < -.0067016 THEN 1999


    1923 PRINT A(1), A(2), A(3), A(4)

    1924 PRINT A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [74].  The complete output of a single run through JJJJ= -31774 is shown below:

479    308    253    247
2.122339        2.113072      2.156524      2.199984              -6.701558E-03
-31898

481    306    253    248
2.149144        2.110138      2.181549      2.192915              -6.701485E-03
-31774

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 [74], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31774 was 48 seconds, not including the time for “Creating .EXE file" (60 seconds, total, including the time for “Creating .EXE file"). 

Acknowledgment

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

References

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https://doi.org/10.1063/1.4942987. Published by the American Institute of Physics.

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[43] 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.

[44] 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

[45] 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.

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

[47]  Kaisa Miettinen, Petrie Eskelinen, Francisco Ruiz, Mariano Luque (2010).  NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point.  European Journal of Operational Research 206 (2010) 426-434.

[48] 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.

[49] 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.

[50] 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.

[51] 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.

[52]  A. Moradi, A. M. Nafchi, A. Ghanbarzadeh (2015).  Multi-objective optimization of truss structures using the bee algorithm.  (One can read this via Goodle search.)

[53] 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.

[54]  Subhash C. Narula, H. Roland Weistroffer (1989).  A flexible method for nonlinear criteria decisionmaking problems. Ieee Transactions on Systems, Man and Cybernetics, vol. 19 , no. 4, July/August 1989, pp. 883-887.

[55] 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.

[56] 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/

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

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

[59]  Ciara Pike-Burke.  Multi-Objective Optimization.  https://www.lancaster.ac.uk/pg/pikeburc/report1.pdf.

[60]  Yashpal Singh Raghav,  Irfan Ali, Abdul Bari (2014)  Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in the Case of the Non-Response, Journal of Statistical Computation and Simulation 84:1, 22-36. 

[61] 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.

[62] 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.

[63] 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.

[64] 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.

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

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

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

[68] 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.

[69] 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

[70] 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.

[71] 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.

[72]  V. Verma, H. C. Bakhshi, M. C. Puri (1990)  Ranking in integer linear fractional programming problems, ZOR - Methods and Models of Operations Research (1990)
34:325-334.

[73] 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.

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

[75] 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.