Solving a Multi-Objective Nonlinear Preemptive Goal Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following multi-objective nonlinear preemptive (lexicographic) goal program from Baykasoglu [11, p. 60, Example 5]: 

Lexmin             

 {z1= X(5), z2= X(6) }

subject to

        - X(5) + (2 * X(2) * X(4) + X(3) * (X(1) - 2 * X(4)))  =127.46

        - X(6)  + ((6000) / (X(3) * (X(1) - 2 * X(4)) ^ 3 + 2 * X(2) * X(4) * (4 * X(4) ^ 2 + 3 * X(1) * (X(1) - 2 * X(4)))))  =.0059

16 - ((180000 * X(1)) / (X(3) * (X(1) - 2 * X(4)) ^ 3 + 2 * X(2) * X(4) * (4 * X(4) ^ 2 + 3 * X(1) * (X(1) - 2 * X(4))))) - ((15000 * X(2)) / ((X(1) - 2 * X(4)) * X(3) ^ 3 + 2 * X(4) * X(2) ^ 3))    >= 0

        10<= X(1) >= 80

        10<= X(2) > 50

        .9  <=  X(3) <= 5

        .9   <=  X(4) <= 5

        X(5),  X(6) >=0.

One notes that line 1287, which is 1287 P = -100000000 * X(5) - X(6), is a use of Hillier and Lieberman's streamlined procedure [40, pp. 289-291] for preemptive goal programming.


0 DEFDBL A-Z
1 REM        DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000


    85 RANDOMIZE JJJJ
    87 M = -4E+250
    121 FOR J44 = 1 TO 6

        122 A(J44) = RND * 50


    123 NEXT J44


    128 FOR I = 0 TO FIX(RND * 300000)


        129 FOR KKQQ = 1 TO 6

            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

            143 j = 1 + FIX(RND * 6)

            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 * 2.3) ELSE X(j) = A(j) + FIX(1 + RND * 2.3)

        169 NEXT IPP


        291 X(5) = -127.46 + (2 * X(2) * X(4) + X(3) * (X(1) - 2 * X(4)))

        293 X(6) = -.0059 + ((6000) / (X(3) * (X(1) - 2 * X(4)) ^ 3 + 2 * X(2) * X(4) * (4 * X(4) ^ 2 + 3 * X(1) * (X(1) - 2 * X(4)))))

        311 IF 16 - ((180000 * X(1)) / (X(3) * (X(1) - 2 * X(4)) ^ 3 + 2 * X(2) * X(4) * (4 * X(4) ^ 2 + 3 * X(1) * (X(1) - 2 * X(4))))) - ((15000 * X(2)) / ((X(1) - 2 * X(4)) * X(3) ^ 3 + 2 * X(4) * X(2) ^ 3)) < 0## THEN 1670

        312 IF X(1) > 80 THEN 1670

        313 IF X(2) > 50 THEN 1670

        314 IF X(3) > 5 THEN 1670

        315 IF X(4) > 5 THEN 1670

        316 IF X(1) < 10 THEN 1670

        317 IF X(2) < 10 THEN 1670

        318 IF X(3) < .9 THEN 1670

        319 IF X(4) < .9 THEN 1670
        331 FOR J44 = 1 TO 6


            332 IF X(J44) < 0 THEN 1670

        334 NEXT J44


        1287 P = -100000000 * X(5) - (X(6))

        1311 IF P <= M THEN 1670
        1420 M = P

        1442 FOR KLX = 1 TO 6

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

        1557 GOTO 128
    1670 NEXT I
    1676 REM IF A(1) = 295 THEN 1677 ELSE GOTO 1999

    1677 IF M < -.0000375 GOTO 1999

    1931 PRINT A(1), A(2), A(3), A(4), A(5)
    1933 PRINT A(6), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [97]. The complete output of a single run through JJJJ= -31703 is shown below:

61.75321130232686                40.79664138028355               .899999946652803
.9008541889653118                4.649058915617843D-16     
3.740344944116402D-05         -3.74499400303202D-05         -31963

61.9285422063957                  40.73971507539157               .9000105827495516
.900152969925244                  4.024558464266193D-16      
1.009776776223317D-05         -1.013801334687583D-05       -31703
.
.
.

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 [97], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31703 was 90 seconds, not including the time for “Creating .EXE file.”  One can compare the computational results above with those in Baykasoglu [11, p. 61].       
 
Acknowledgment

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

References

[1]  Irfan Ali, Yashpal Singh Raghav, Abdul Bari (2013).  Compromise allocation in multivariate stratified surveys with stochastic quadratic cost function, Journal of Statistical Computation and Simulation 2013,   Vol. 83,  No. 5, pp. 962-976.

[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] Adil Baykasoglu (2005), Preemptive goal programming using simulated annealing, Engineering Optimization, 37:1, 49-63.

[12] Adil Baykasoglu, S. Owen, N. Gindy (1999), Solution of goal programming models using a basic taboo search algorithm, Journal of the Operational Research Society (1999) 50, 960-963.

[13]  H. Bernau (1990 ).  Active constraint strategies in optimization. Geographical data inversion methods and applications.  pp. 15-31.

[14]  Victor Blanco,  Justo Puerto (2011).  Some algebraic methods for solving multiobjective polynomial integer programs, Journal of Symbolic Computation, 46 (2011), pp. 511-533.

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

[16] Regina S. Burachik, C. Yalcin Kaya, M. Mustafa Rizvi (March 19, 2019),  Algorithms for Generating Pareto Fronts of Multi-objective Integer and Mixed-Integer Programming Problems, arXiv: 1903.07041v1 [math.OC] 17 Mar 2019.

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

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

[19] 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.

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

[21] 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.

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

[23]  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

[24] 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.)

[25] 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.

[26] 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.

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

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

[29] 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.

[30] 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.

[31] 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.

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

[33] 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.

[34]  Shazia Ghufran, Saman Khowaja, M. J. Ahsan (2014).  Compromise allocation in multiobjective stratified sample surveys under two stage randomized response model, Optim Lett (2014) 8:343-357.

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

[36]  Neha Gupta, Irfan Ali, Abdul Bari (2013).  An optimal chance constraint multivariate stratified sampling design using auxiliary infotmation.  Journal of Mathematical Modelling and Algorithms, January 2013.

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

[38] 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.

[39]  Ibrahim M. Hezam, Osama Abdel Raouf, Mohey M. Hadhoud (September 2013).  A new compound swarm intelligence algorithm for for solving global optimization problems.  International Journal of Computers and Technology, Vol. 10, No. 9, 2013.


[40] Frederick S. Hillier, Gerald J. Liebermann (1990).  Introduction to mathematical programming.  McGraw-Hill Publishing Company, New York.


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

[42] 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

[43]  Sana  Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015).  An optimum multivariate stratified sampling design.  Research Journal of Mayhematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[44] 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

[45] Golam Reza Jahanshahloo, Farhad Hosseinzadeh, Nagi Shoja, Ghasem Tohidi (2003)  A method for solvong 0-1 multiple objective liner programming problem using DEA. Journal of the Operations Research Society of Japan (2003), 46 (2): 189-202.  www.orsj.or.jp/~archive/pdf/e_mag/Vol.46_02_189.pdf

[46] 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.

[47] 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.

[48] 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.

[49] 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.

[50] 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.

[51]  M. G. M. Khan, E. A. Khan, M. J. Ahsan (2003).  An optimal multivariate stratified sampling design using dynamic programming.  Australian and New Zealand Journal of  Statistics, vol. 45, no. 1, 2003, pp. 107-113.

[52]  M. G. M. Khan, T. Maiti, M. J. Ahsan (2010).  An optimal multivariate stratified sampling design using auxiliary information:  an integer solution using goal programming approach.  Journal of Official Statistics, vol. 26, no. 4, 2010, pp. 695-708.

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

[54] 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.

[55]  F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs.  European Journal of Operational Research 126 (2000) 51-68.

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

[57] 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.

[58] 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.

[59] 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

[60] 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.

[61] 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.

[62] 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.

[63] 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.

[64] 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.

[65]  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.)

[66] 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.

[67]  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.

[68] 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.

[69]  A. K. Ojha, K. K. Biswal (2010).  Multi-objective geometric programming problem with weighted mean method.  (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp.  82-86, 2010.

[70] Rashmi Ranjan Ota, jayanta kumar dASH, A. K. Ojha (2014). A hybrid optimization techniques for solving non-linear multi-objective optimization problem.  amo-advanced modelling and optimization, volume 16,number 1, 2014.

[71] 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/

[72] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system.  OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.
 
[73] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

[74] S. Pant, D. Anand, A. Kishor, S. B. Singh (2015).  A particle swarm algorithm for optimization of complex system reliability, International Journal of Performability Engineering, Volume 11, Number 1, Jan.  2015, Pp. 33-42.
 
[75] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

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

[77]  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 Statitiscal Computation ans Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370. 

[78] 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.

[79] 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.

[80] 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.

[81]  Shafiullah, Irfan Ali, Abdul Bari (2015).  Fuzzy  geometric programming approach in multi-objective multivariate stratified sample surveys in presence of non-respnse, International Journal of Operations Research, Vol. 12, No. 2, pp.  021-035 (2015).

[82] 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.

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

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

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

[86] 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.

[87] Sonja Surjanovic, Derek Bingham (August 2017).  Virtual Library of Simulation Experiments: Test Functions and Datasets--Optimization Test Problems (Sum of Different Powers Function).  sfu.ca/~ssurjano/index.html.  Or just google sum of different powers function.

[88] 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

[89] 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.

[90] 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.

[91]  Rahul Varshney, Srikant Gupta,  Irfan  Ali (2017).  An optimum multivariate-multiobjective stratified samplinr design:  fuzzy programming approach.  Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

[92]  Rahul Varshney,  Mradula (2019 May 25).  Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13,  pp.2454-2467.

[93]  Rahul Varshney, Najmussehar, M. J. Ahsan (2012).  An optimum multivariate stratified double sampling design in presence of non-response, Optimization Letters (2012) 6: pp. 993-1008.

[94]  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.

[95]  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.

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

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

[98] 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 Three-Objective Preemptive Goal Programming Problem from the Literature

Jsun Yui Wong

The computer program listed below seeks to solve the following multi-objective integer preemptive (lexicographic) goal program from Baykasoglu, Owen, and Gindy [12, p. 971, Test problem A-3]: 

Lexmin             

{  z1=(X(5) + X(7) + X(10)), z2=( X(12)), z3=( X(13) )  }

subject to

         X(5) - X(6) + X(1) + X(2)=30

         X(7) - X(8) + X(3) + X(4)=30

         X(9) - X(10) + 3 * X(1) + 2 * X(3)=120

         X(11) - X(12) + 3 * X(2) + 2 * X(4)=20

        X(13) - X(14) + 10 * X(1) + 9 * X(2) + 8 * X(3) + 7 * X(4)=800
   
        X(1),  X(2),  X(3), . . .,  X(14) >= 0 and are integers.


One notes that line 1300, which is 1300 P = -100000000000000 * (X(5) + X(7) + X(10)) - 10000000000 * X(12) - X(13), is a use of Hillier and Lieberman's streamlined procedure [38, pp. 289-291] for preemptive goal programming.


0 DEFDBL A-Z
1 REM        DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

    85 RANDOMIZE JJJJ
    87 M = -4E+250
    121 FOR J44 = 1 TO 14


        122 A(J44) = FIX(RND * 10)


    123 NEXT J44
 

    128 FOR I = 0 TO FIX(RND * 60000)


        129 FOR KKQQ = 1 TO 14

            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

            143 j = 1 + FIX(RND * 14)

            144 REM     IF RND < .5 THEN GOTO 156 ELSE GOTO 162

            145 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 * 5.3) ELSE X(j) = A(j) + FIX(1 + RND * 5.3)

        169 NEXT IPP
     
        281 X(5) = 30 + X(6) - X(1) - X(2)

        283 X(7) = 30 + X(8) - X(3) - X(4)

        285 X(9) = 120 + X(10) - 3 * X(1) - 2 * X(3)

        287 X(11) = 20 + X(12) - 3 * X(2) - 2 * X(4)


        293 X(13) = 800 + X(14) - 10 * X(1) - 9 * X(2) - 8 * X(3) - 7 * X(4)
       
        331 FOR J44 = 1 TO 14


            332 IF X(J44) < 0 THEN 1670

            333 X(J44) = INT(X(J44))

        334 NEXT J44



        1300 P = -100000000000000 * (X(5) + X(7) + X(10)) - 10000000000 * X(12) - X(13)

   
        1311 IF P <= M THEN 1670
        1420 M = P

        1442 FOR KLX = 1 TO 14


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

        1557 GOTO 128
    1670 NEXT I
 

    1931 PRINT A(1), A(2), A(3), A(4), A(5)
    1932 PRINT A(6), A(7), A(8), A(9), A(10)
    1933 PRINT A(11), A(12), A(13), A(14)

    1939 PRINT M, JJJJ
1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [97]. The output of a single run through JJJJ= -31912 is summarized below:

22  8  27  3  0
0  0  0  0  0
0   10  271  0
-100000000271  -32000
.
.
.

20  10  30  0  0
0  0  0  0  0
0   10  270  0
-100000000270  -31912
.
.
.


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 [97], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31912 was 10 seconds, not including the time for “Creating .EXE file.”  One can compare the computational results above with those in Baykasoglu, Owen, and Gindy [12, p. 971].       
 
Acknowledgment

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

References

[1]  Irfan Ali, Yashpal Singh Raghav, Abdul Bari (2013).  Compromise allocation in multivariate stratified surveys with stochastic quadratic cost function, Journal of Statistical Computation and Simulation 2013,   Vol. 83,  No. 5, pp. 962-976.

[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] Adil Baykasoglu (2005), Preemptive goal programming using simulated annealing, Engineering Optimization, 37:1, 49-63.

[12] Adil Baykasoglu, S. Owen, N. Gindy (1999), Solution of goal programming models using a basic taboo search algorithm, Journal of the Operational Research Society (1999) 50, 960-973.

[13]  H. Bernau (1990 ).  Active constraint strategies in optimization. Geographical data inversion methods and applications.  pp. 15-31.

[14]  Victor Blanco,  Justo Puerto (2011).  Some algebraic methods for solving multiobjective polynomial integer programs, Journal of Symbolic Computation, 46 (2011), pp. 511-533.

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

[16] Regina S. Burachik, C. Yalcin Kaya, M. Mustafa Rizvi (March 19, 2019),  Algorithms for Generating Pareto Fronts of Multi-objective Integer and Mixed-Integer Programming Problems, arXiv: 1903.07041v1 [math.OC] 17 Mar 2019.

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

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

[19] 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.

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

[21] 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.

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

[23]  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

[24] 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.)

[25] 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.

[26] 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.

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

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

[29] 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.

[30] 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.

[31] 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.

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

[33] 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.

[34]  Shazia Ghufran, Saman Khowaja, M. J. Ahsan (2014).  Compromise allocation in multiobjective stratified sample surveys under two stage randomized response model, Optim Lett (2014) 8:343-357.

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

[36]  Neha Gupta, Irfan Ali, Abdul Bari (2013).  An optimal chance constraint multivariate stratified sampling design using auxiliary infotmation.  Journal of Mathematical Modelling and Algorithms, January 2013.

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

[38] 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.

[39]  Ibrahim M. Hezam, Osama Abdel Raouf, Mohey M. Hadhoud (September 2013).  A new compound swarm intelligence algorithm for for solving global optimization problems.  International Journal of Computers and Technology, Vol. 10, No. 9, 2013.

[40] Frederick S. Hillier, Gerald J. Liebermann (1990).  Introduction to mathematical programming.  McGraw-Hill Publishing Company, New York.

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

[42] 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

[43]  Sana  Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015).  An optimum multivariate stratified sampling design.  Research Journal of Mayhematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[44] 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

[45] Golam Reza Jahanshahloo, Farhad Hosseinzadeh, Nagi Shoja, Ghasem Tohidi (2003)  A method for solvong 0-1 multiple objective liner programming problem using DEA. Journal of the Operations Research Society of Japan (2003), 46 (2): 189-202.  www.orsj.or.jp/~archive/pdf/e_mag/Vol.46_02_189.pdf

[46] 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.

[47] 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.

[48] 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.

[49] 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.

[50] 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.

[51]  M. G. M. Khan, E. A. Khan, M. J. Ahsan (2003).  An optimal multivariate stratified sampling design using dynamic programming.  Australian and New Zealand Journal of  Statistics, vol. 45, no. 1, 2003, pp. 107-113.

[52]  M. G. M. Khan, T. Maiti, M. J. Ahsan (2010).  An optimal multivariate stratified sampling design using auxiliary information:  an integer solution using goal programming approach.  Journal of Official Statistics, vol. 26, no. 4, 2010, pp. 695-708.

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

[54] 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.

[55]  F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs.  European Journal of Operational Research 126 (2000) 51-68.

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

[57] 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.

[58] 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.

[59] 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

[60] 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.

[61] 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.

[62] 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.

[63] 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.

[64] 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.

[65]  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.)

[66] 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.

[67]  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.

[68] 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.

[69]  A. K. Ojha, K. K. Biswal (2010).  Multi-objective geometric programming problem with weighted mean method.  (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp.  82-86, 2010.

[70] Rashmi Ranjan Ota, jayanta kumar dASH, A. K. Ojha (2014). A hybrid optimization techniques for solving non-linear multi-objective optimization problem.  amo-advanced modelling and optimization, volume 16,number 1, 2014.

[71] 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/

[72] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system.  OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.
 
[73] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

[74] S. Pant, D. Anand, A. Kishor, S. B. Singh (2015).  A particle swarm algorithm for optimization of complex system reliability, International Journal of Performability Engineering, Volume 11, Number 1, Jan.  2015, Pp. 33-42.
 
[75] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

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

[77]  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 Statitiscal Computation ans Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370. 

[78] 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.

[79] 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.

[80] 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.

[81]  Shafiullah, Irfan Ali, Abdul Bari (2015).  Fuzzy  geometric programming approach in multi-objective multivariate stratified sample surveys in presence of non-respnse, International Journal of Operations Research, Vol. 12, No. 2, pp.  021-035 (2015).

[82] 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.

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

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

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

[86] 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.

[87] Sonja Surjanovic, Derek Bingham (August 2017).  Virtual Library of Simulation Experiments: Test Functions and Datasets--Optimization Test Problems (Sum of Different Powers Function).  sfu.ca/~ssurjano/index.html.  Or just google sum of different powers function.

[88] 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

[89] 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.

[90] 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.

[91]  Rahul Varshney, Srikant Gupta,  Irfan  Ali (2017).  An optimum multivariate-multiobjective stratified samplinr design:  fuzzy programming approach.  Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

[92]  Rahul Varshney,  Mradula (2019 May 25).  Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13,  pp.2454-2467.

[93]  Rahul Varshney, Najmussehar, M. J. Ahsan (2012).  An optimum multivariate stratified double sampling design in presence of non-response, Optimization Letters (2012) 6: pp. 993-1008.

[94]  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.

[95]  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.

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

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

[98] 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 Three-Objective Problem with a Streamedlined Procedure of Preemptive Goal Programming

 

Jsun Yui Wong

The computer program listed below seeks to solve the immediately following integer multi-objective goal programming problem:

Minimize        (X(5) + 2 * X(7) + X(10)) , priority 1

minimize       X(12) , priority 2

minimize       X(13) , priority 3

priority 1>>priority 2>>priority 3

(Or minimize   100000000 * (X(5) + 2 * X(7) + X(10)) + 100000 * X(12) + X(13); see Hillier and Lieberman [38, p. 291].)

subject to

         X(5) - X(6) + X(1) + X(2)=30
         X(7)  - X(8) + X(3) + X(4)=30
         X(9)  - X(10) + 3 * X(1) + 2 * X(3)=120

         X(11)  - X(12) + 3 * X(2) + 2 * X(4)=20
         X(13)  - X(14) + 10 * X(1) + 9 * X(2) + 8 * X(3) + 7 * X(4)=800

X(1),..., X(14) are integer variables and >=0.

The problem above is based on the problem on pp. 4-5 of Mohammad and Hordofa [62].

The computer program listed below uses the streamlined procedure of preemptive goal programming in Hillier and Liberman [38, p. 291]. 



0 DEFDBL A-Z
1 REM        DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

    85 RANDOMIZE JJJJ
    87 M = -4E+250
    121 FOR J44 = 1 TO 14

        124 A(J44) = FIX(RND * 50)


    125 NEXT J44

    128 FOR I = 0 TO FIX(RND * 500000)

        129 FOR KKQQ = 1 TO 14

            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

            143 j = 1 + FIX(RND * 14)
            145 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 * 6.3) ELSE X(j) = A(j) + FIX(1 + RND * 6.3)


        169 NEXT IPP

        177 FOR J44 = 1 TO 14


            178 X(J44) = INT(X(J44))


        179 NEXT J44


        291 X(5) = 30 + X(6) - X(1) - X(2)
        293 X(7) = 30 + X(8) - X(3) - X(4)
        295 X(9) = 120 + X(10) - 3 * X(1) - 2 * X(3)

        297 X(11) = 20 + X(12) - 3 * X(2) - 2 * X(4)
        299 X(13) = 800 + X(14) - 10 * X(1) - 9 * X(2) - 8 * X(3) - 7 * X(4)


        1072 FOR J44 = 1 TO 14


            1078 IF X(J44) < 0## THEN 1670


        1079 NEXT J44

        1299 P = -100000000 * (X(5) + 2 * X(7) + X(10)) - 100000 * X(12) - X(13)
        1311 IF P <= M THEN 1670
        1420 M = P

        1442 FOR KLX = 1 TO 14

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

        1557 GOTO 128
    1670 NEXT I

    1677 IF M < -999999999999 THEN 1999
    1931 PRINT A(1), A(2), A(3), A(4), A(5)
    1932 PRINT A(6), A(7), A(8), A(9), A(10)
    1933 PRINT A(11), A(12), A(13), A(14)


    1937 PRINT M, JJJJ
1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [96]. The output of a single run through JJJJ= -31903 is summarized below:

.
.
.

24     6     24     6     0
0     0     0     0     0
0     10     272      0
-1000272       -31966
.
.
.

20    10     30     0     0
0     0     0     0     0
0      10      270      0
-1000270      -31903
.
.
.


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 [96], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31903 was 90 seconds, not including the time for “Creating .EXE file”.  One can compare the computational results above with those in Mohammed and Hordofa [62, p. 6].


Acknowledgment

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

References

[1]  Irfan Ali, Yashpal Singh Raghav, Abdul Bari (2013).  Compromise allocation in multivariate stratified surveys with stochastic quadratic cost function, Journal of Statistical Computation and Simulation 2013,   Vol. 83,  No. 5, pp. 962-976.

[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]  H. Bernau (1990 ).  Active constraint strategies in optimization. Geographical data inversion methods and applications.  pp. 15-31.

[12]  Victor Blanco,  Justo Puerto (2011).  Some algebraic methods for solving multiobjective polynomial integer programs, Journal of Symbolic Computation, 46 (2011), pp. 511-533.

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

[14] Regina S. Burachik, C. Yalcin Kaya, M. Mustafa Rizvi (March 19, 2019),  Algorithms for Generating Pareto Fronts of Multi-objective Integer and Mixed-Integer Programming Problems, arXiv: 1903.07041v1 [math.OC] 17 Mar 2019.

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

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

[17] 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.

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

[19] 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.

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

[21]  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

[22] 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.)

[23] 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.

[24] 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.

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

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

[27] 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.

[28] 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.

[29] 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.

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

[31] 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.

[32]  Shazia Ghufran, Saman Khowaja, M. J. Ahsan (2014).  Compromise allocation in multiobjective stratified sample surveys under two stage randomized response model, Optim Lett (2014) 8:343-357.

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

[34]  Neha Gupta, Irfan Ali, Abdul Bari (2013).  An optimal chance constraint multivariate stratified sampling design using auxiliary infotmation.  Journal of Mathematical Modelling and Algorithms, January 2013.

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

[36] 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.

[37]  Ibrahim M. Hezam, Osama Abdel Raouf, Mohey M. Hadhoud (September 2013).  A new compound swarm intelligence algorithm for for solving global optimization problems.  International Journal of Computers and Technology, Vol. 10, No. 9, 2013.

[38]  Frederick S. Hillier, Gerald J. Liebermann (1990).  Introduction to Mathematical Programming.  McGraw-Hill Publishing Company, New York.

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

[39] 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

[40]  Sana  Iftekhar, M. J. Ahsan, Qazi Mazhar Ali (2015).  An optimum multivariate stratified sampling design.  Research Journal of Mayhematical and Statistical Sciences, vol. 3(1),10-14, January (2015).

[41] 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

[42] Golam Reza Jahanshahloo, Farhad Hosseinzadeh, Nagi Shoja, Ghasem Tohidi (2003)  A method for solvong 0-1 multiple objective liner programming problem using DEA. Journal of the Operations Research Society of Japan (2003), 46 (2): 189-202.  www.orsj.or.jp/~archive/pdf/e_mag/Vol.46_02_189.pdf

[43] 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.

[44] 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.

[45] 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.

[46] 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.

[47] 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.

[48]  M. G. M. Khan, E. A. Khan, M. J. Ahsan (2003).  An optimal multivariate stratified sampling design using dynamic programming.  Australian and New Zealand Journal of  Statistics, vol. 45, no. 1, 2003, pp. 107-113.

[49]  M. G. M. Khan, T. Maiti, M. J. Ahsan (2010).  An optimal multivariate stratified sampling design using auxiliary information:  an integer solution using goal programming approach.  Journal of Official Statistics, vol. 26, no. 4, 2010, pp. 695-708.

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

[51] 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.

[52]  F. H. F. Liu, C. C. Huang, Y. L. Yen (2000): Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs.  European Journal of Operational Research 126 (2000) 51-68.

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

[54] 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.

[55] 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.

[56] 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

[57] Neelam Malhotra, S. R. Arora (1999).  Nonlinear Integer Goal Programming:  Indian Journal of Pure and Applied Mathematics ( IJPAM), 30(1) : 39-50, January 1999.

[58] 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.

[59] 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.

[60] 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.

[61] 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.

[62]  G. T. Mohammed,  B. G. Hordofa (2016).  The modified sequential linear programming method for solving multiple objectives linear programming problems: Pure and Applied Mathematics Journal, 2016; 5(1): pp. 1-8.  (One can read this via Goodle search.)

[63] 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.

[64]  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.)

[65] 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.

[66]  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.

[67] 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.

[68]  A. K. Ojha, K. K. Biswal (2010).  Multi-objective geometric programming problem with weighted mean method.  (IJCSIS) International Journal of Computer Science and Information Security, vol. 7, no. 2, pp.  82-86, 2010.

[69] Rashmi Ranjan Ota, jayanta kumar dASH, A. K. Ojha (2014). A hybrid optimization techniques for solving non-linear multi-objective optimization problem.  amo-advanced modelling and optimization, volume 16,number 1, 2014.

[70] 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/

[71] R. R. Ota, J. C. Pati, A. K. Ojha (2019). Geometric programming technique to optimize power distribution system.  OPSEARCH of the Operational Research Society of India (2019), 56, pp. 282-299.
 
[72] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP

[73] S. Pant, D. Anand, A. Kishor, S. B. Singh (2015).  A particle swarm algorithm for optimization of complex system reliability, International Journal of Performability Engineering, Volume 11, Number 1, Jan.  2015, Pp. 33-42.
 
[74] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.

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

[76]  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 Statitiscal Computation ans Simulation 84:1, 22-36, doi:10.1080/00949655.2012.692370. 

[77] 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.

[78] 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.

[79] 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.

[80]  Shafiullah, Irfan Ali, Abdul Bari (2015).  Fuzzy  geometric programming approach in multi-objective multivariate stratified sample surveys in presence of non-respnse, International Journal of Operations Research, Vol. 12, No. 2, pp.  021-035 (2015).

[81] 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.

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

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

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

[85] 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.

[86] Sonja Surjanovic, Derek Bingham (August 2017).  Virtual Library of Simulation Experiments: Test Functions and Datasets--Optimization Test Problems (Sum of Different Powers Function).  sfu.ca/~ssurjano/index.html.  Or just google sum of different powers function.

[87] 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

[88] 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.

[89] 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.

[90]  Rahul Varshney, Srikant Gupta,  Irfan  Ali (2017).  An optimum multivariate-multiobjective stratified samplinr design:  fuzzy programming approach.  Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

[91 ]  Rahul Varshney,  Mradula (2019 May 25).  Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13,  pp.2454-2467.

[92]  Rahul Varshney, Najmussehar, M. J. Ahsan (2012).  An optimum multivariate stratified double sampling design in presence of non-response, Optimization Letters (2012) 6: pp. 993-1008.

[93]  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.

[94]  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.

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

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

[97] 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.