Substance

Solving Another Integer Nonlinear Program

Jsun Yui Wong

The computer program listed below seeks to solve the following problem based on the first problem on p. 2462 of Varshney and Mradula [85, p. 2462]:

Minimize

- ( -.4388203 / X(1) - 2.663113 / X(2) - 49.60277 / X(3) - 2.66616 / X(4) - 9.938173 / X(5) - .002437891 / X(6) - .08937845 / X(7) - 1.206554 / X(8) - .044436 / X(9) - .2094497 / X(10) )

subject to

4 * X(1) + 4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4) <= 10000

2<= X(1) <= 395
2<= X(2) <= 382
2<=X(3) <= 439
2<=X(4) <= 368
2<=X(5) <= 416

2<=X(6) <= 30
2<=X(7) <= 30
2<=X(8) <= 30
2<=X(9) <= 30
2<=X(10) <= 30

X(1) through X(10) are general integer variables.

The following computer program was used to solve the problem above.

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 = -4E+250
121 FOR J44 = 1 TO 5

122 A(J44) = RND * 50
123 NEXT J44
124 FOR J44 = 6 TO 10

125 A(J44) = RND * 3
126 NEXT J44

128 FOR I = 0 TO FIX(RND * 20000)

129 FOR KKQQ = 1 TO 10

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

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

143 j = 1 + FIX(RND * 10)

154 IF RND < .5 THEN GOTO 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 * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)

164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP

171 FOR J44 = 1 TO 10
173 X(J44) = INT(X(J44))
174 IF X(J44) < 2 THEN 1670
175 NEXT J44

188 IF X(1) > 395 THEN 1670
189 IF X(2) > 382 THEN 1670
190 IF X(3) > 439 THEN 1670
191 IF X(4) > 368 THEN 1670
192 IF X(5) > 416 THEN 1670

193 IF X(6) > 30 THEN 1670
194 IF X(7) > 30 THEN 1670
195 IF X(8) > 30 THEN 1670
196 IF X(9) > 30 THEN 1670
197 IF X(10) > 30 THEN 1670

227 IF 4 * X(1) + 4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4) > 10000 THEN 1670

499 P = -.4388203 / X(1) - 2.663113 / X(2) - 49.60277 / X(3) - 2.66616 / X(4) - 9.938173 / X(5) - .002437891 / X(6) - .08937845 / X(7) - 1.206554 / X(8) - .044436 / X(9) - .2094497 / X(10)

1111 IF P <= M THEN 1670

1420 M = P

1444 FOR KLX = 1 TO 10

1455 A(KLX) = X(KLX)
1456 NEXT KLX
1488 REM
1489 LHS = 4 * X(1) + 4.9 * X(2) + 5.9 * X(3) + 7.75 * X(4) + 8.92 * X(5) + 6 * X(6) + 7 * X(7) + 9 * X(8) + 11 * X(9) + 12 * X(10) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4) + 100 * (X(6) / 4 + X(7) / 4 + X(8) / 4 + X(9) / 4 + X(10) / 4)

1557 GOTO 128
1670 NEXT I

1889 IF M < -.8441043 THEN 1999
1926 PRINT A(1), A(2), A(3), A(4), 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 [91]. The complete output of a single run through JJJJ= -31962 is shown below:

13       32       136       31       59
2       6       21       4       9
-.8441042055052964       -31993

13 32 136 31 59
2 6 21 4 9
-.8441042055052964 -31987

13 32 136 31 59
2 6 21 4 9
-.8441042055052964 -31972

13 32 136 31 59
2 6 21 4 9
-.8441042055052964 -31962
.
.
.

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 [91], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31962 was 2 seconds, not including the time for “Creating .EXE file" (18 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Varshney and Mradula [85, p. 2462].

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

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

[12] 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.

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

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

[15] 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.

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

[17] 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.

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

[19] 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

[20] 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.)

[21] 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.

[22] 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.

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

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

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[33] 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.

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

[35] 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

[36] 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).

[37] 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

[38] 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

[39] 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.

[40] 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.

[41] 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.

[42] 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.

[43] 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.

[44] 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.

[45] 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.

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

[47] 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.

[48] 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.

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

[50] 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.

[51] 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.

[52] 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

[53] 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.

[54] 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.

[55] 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.

[56] 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.

[57] 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.

[58] 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.)

[59] 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.

[60] 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.

[61] 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.

[62] 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.

[63] 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.

[64] 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/

[65] 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.

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

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

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

[69] 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.

[70] 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.

[71] 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.

[72] 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.

[73] 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).

[74] 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.

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

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

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

[78] 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.

[79] Har3i 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

[80] 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.

[81] 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.

[82] 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

[83] 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.

[84] 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.

[85] 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.

[86] 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.

[87] 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.

[88] 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

[89] 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.

[90] 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.

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

[92] 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 Multivariate Stratified Sampling Design Problem

Jsun Yui Wong

The two computer programs listed below seek to solve a multi-objective nonlinear multivariate sampling design problem
involving general integer variables and one continuous variable.

1. Stage 1

The first computer program listed below seeks to solve the following Example 1-Stage 1 problem in Varshney and Mradula [85, p. 2459]:

Minimize

- (-443.0974013 / X(1) - 270.715638 / X(2) - 204.9419482 / X(3) - 240.6056486 / X(4) - 10.7780449 / X(5) - 2.849638294 / X(6) - 3.415699137 / X(7) - 5.070828722 / X(8))

subject to

2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4) + 100 * (X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) <= 5000


2<= X(1) <= 962
2<=X(2) <= 962
2<=X(3) <= 962
2<=X(4) <= 962

2<=X(5) <= 23
2<=X(6) <= 28
2<=X(7) <= 23
2<=X(8) <=29

X(1) through X(8) are general integer variables.

The immediately following computer program is used to solve the problem above.

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 = -4E+250

121 FOR J44 = 1 TO 4

122 A(J44) = RND * 50
123 NEXT J44
124 FOR J44 = 5 TO 8

125 A(J44) = RND * 10
126 NEXT J44

128 FOR I = 0 TO FIX(RND * 20000)

129 FOR KKQQ = 1 TO 8

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

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 j = 1 + FIX(RND * 8)

154 IF RND < .5 THEN GOTO 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 * 2.3) ELSE X(j) = A(j) + FIX(1 + RND * 2.3)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP

171 FOR J44 = 1 TO 8
173 X(J44) = INT(X(J44))
174 IF X(J44) < 2 THEN 1670
175 NEXT J44

188 IF X(1) > 962 THEN 1670
189 IF X(2) > 962 THEN 1670
190 IF X(3) > 962 THEN 1670
191 IF X(4) > 962 THEN 1670

193 IF X(5) > 23 THEN 1670
194 IF X(6) > 28 THEN 1670
195 IF X(7) > 23 THEN 1670
196 IF X(8) > 29 THEN 1670

228 IF 2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4) + 100 * (X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) > 5000 THEN 1670

444 P = (-443.0974013 / X(1) - 270.715638 / X(2) - 204.9419482 / X(3) - 240.6056486 / X(4) - 10.7780449 / X(5) - 2.849638294 / X(6) - 3.415699137 / X(7) - 5.070828722 / X(8))

1111 IF P <= M THEN 1670

1420 M = P

1444 FOR KLX = 1 TO 8

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

1557 GOTO 128
1670 NEXT I

1889 IF M < -3333333333 THEN 1999

1926 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), M, JJJJ
1999 NEXT

This BASIC computer program was run with QB64v1000-win [91]. The output of a single run through JJJJ= -31995 is shown below:

45 32 45 27 10
3 5 7 -35.20728629122804
-32000

49 38 32 36 8
4 4 4 -33.43612311589425
-31999

49 383 33 35 8
4 4 4 -33.43300616527918
-31998

49 383 33 35 8
4 4 4 -33.43300616527918
-31997

50       37       32       35       8
4       4       5       - 33.38522558962201
-31996

50 37 32 35 8
4 4 5 -33.38522558962201
-31995
.
.
.

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 [91], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31995 was 2 seconds, not including the time for “Creating .EXE file" (18 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Varshney and Mradula [85, p. 2459].

2. Stage 2

The following computer program seeks to solve the following Example 1-Stage 2 problem in Varshney and Mradula [85, p. 2460]:

Minimize

- ( -746.921876 / X(1) - 329.7093584 / X(2) - 97.34568757 / X(3) - 270.4706345 / X(4) - 18.16836996 / X(5) - 3.470624826 / X(6) - 1.62242812657002 / X(7) - 5.700241329 / X(8) - X(9) )

subject to

2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4) + 100 * (X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) <= 5000

-(-443.0974013 / X(1) - 270.715638 / X(2) - 204.9419482 / X(3) - 240.6056486 / X(4) - 10.7780449 / X(5) - 2.849638294 / X(6) - 3.415699137 / X(7) - 5.070828722 / X(8)) - X(9) <= 33.3852

2<= X(1) <= 962
2<=X(2) <= 962
2<=X(3) <= 962
2<=X(4) <= 962

2<=X(5) <= 23
2<=X(6) <= 28
2<=X(7) <= 23
2<=X(8) <=29
X(9)>=0

X(1) through X(8) are general integer variables.

The immediately following computer program was used to solve the problem above.

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 = -4E+250
100 A(9) = RND

121 FOR J44 = 1 TO 4

122 A(J44) = RND * 60
123 NEXT J44
124 FOR J44 = 5 TO 8

125 A(J44) = RND * 10
126 NEXT J44

128 FOR I = 0 TO FIX(RND * 200000)

129 FOR KKQQ = 1 TO 9

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

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 j = 1 + FIX(RND * 9)

154 IF RND < .5 THEN GOTO 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 * 2.3) ELSE X(j) = A(j) + FIX(1 + RND * 2.3)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP

171 FOR J44 = 1 TO 8
173 X(J44) = INT(X(J44))
174 IF X(J44) < 2 THEN 1670
175 NEXT J44

177 IF X(9) < 0## THEN 1670

188 IF X(1) > 962 THEN 1670
189 IF X(2) > 962 THEN 1670
190 IF X(3) > 962 THEN 1670
191 IF X(4) > 962 THEN 1670

193 IF X(5) > 23 THEN 1670
194 IF X(6) > 28 THEN 1670
195 IF X(7) > 23 THEN 1670
196 IF X(8) > 29 THEN 1670

228 IF 2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4) + 100 * (X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) > 5000 THEN 1670

444 IF -(-443.0974013 / X(1) - 270.715638 / X(2) - 204.9419482 / X(3) - 240.6056486 / X(4) - 10.7780449 / X(5) - 2.849638294 / X(6) - 3.415699137 / X(7) - 5.070828722 / X(8)) - X(9) > 33.3852 THEN 1670

499 P = -746.921876 / X(1) - 329.7093584 / X(2) - 97.34568757 / X(3) - 270.4706345 / X(4) - 18.16836996 / X(5) - 3.470624826 / X(6) - 1.62242812657002 / X(7) - 5.700241329 / X(8) - X(9)

1111 IF P <= M THEN 1670

1420 M = P

1444 FOR KLX = 1 TO 9

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

1557 GOTO 128
1670 NEXT I

1889 IF M < -38.87 THEN 1999

1926 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), M, JJJJ, A(9)
1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [91]. The output of a single run through JJJJ= -31254 is shown below:

54       38      27       35       8
4       4       5       -38.86298075883173
-31773       .3370461039270914

54 38 27 35 8
4 4 5 -38.86298075882438
-31254 .3370461039197465
.
.
.

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 [91], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31254 was 2.5 minutes, total, including the time for “Creating .EXE file." One can compare the computational results above with those in Varshney and Mradula [85, p. 2460].

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

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

[12] 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.

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

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

[15] 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.

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

[17] 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.

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

[19] 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

[20] 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.)

[21] 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.

[22] 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.

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

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

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[33] 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.

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

[35] 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

[36] 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).

[37] 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

[38] 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

[39] 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.

[40] 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.

[41] 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.

[42] 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.

[43] 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.

[44] 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.

[45] 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.

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

[47] 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.

[48] 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.

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

[50] 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.

[51] 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.

[52] 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

[53] 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.

[54] 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.

[55] 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.

[56] 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.

[57] 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.

[58] 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.)

[59] 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.

[60] 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.

[61] 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.

[62] 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.

[63] 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.

[64] 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/

[65] 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.

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

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

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

[69] 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.

[70] 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.

[71] 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.

[72] 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.

[73] 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).

[74] 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.

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

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

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

[78] 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.

[79] Har3i 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

[80] 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.

[81] 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.

[82] 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

[83] 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.

[84] 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.

[85] 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.

 [86] 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.

[87] 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.

[88] 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

[89] 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.

[90] 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.

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

[92] 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.

A Multiobjective Integer Nonlinear Programming Approach to Another Multivariate Stratified Sampling Design Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following integer nonlinear program from Ghufran, Khowaga, and Ahsan [30, p. 605]:

Minimize

.5 * (1 / 225600205681.21) * ((1 - X(1) / 8) * 190177129.163709 / X(1) + (1 - X(2) / 34) * 3075358778.397211 / X(2) + (1 - X(3) / 45) * 8750902957.838071 / X(3) + (1 - X(4) / 12) * 451381009.970472 / X(4))
+ .5 * (1 / 2484564.0625) * ((1 - X(1) / 8) * 5073.889916 / X(1) + (1 - X(2) / 34) * 588180.653800 / X(2) + (1 - X(3) / 45) * 221961.923300 / X(3) + (1 - X(4) / 12) * 5705.062673 / X(4))

subject to

15 * X(1) + 7 * X(2) + 5 * X(3) + 9 * X(4) + 10 * X(1) ^ .5 + 5 * X(2) ^ .5 + 2 * X(3) ^ .5 + 6 * X(4) ^ .5 <= 200

2 <=X(1) <=8

2<= X(2)<=34

2 <=X(3) <=45

2<= X(4)<=12

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

0 DEFDBL A-Z

1 DEFINT K

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

81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ

87 M = -4E+250

111 FOR J44 = 1 TO 4
113 A(J44) = FIX(RND * 3)

121 NEXT J44

128 FOR I = 0 TO FIX(RND * 1000)

129 FOR KKQQ = 1 TO 4

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

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 J = 1 + FIX(RND * 4)

154 IF RND < .5 THEN GOTO 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 * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
171 X(1) = INT(X(1))
173 X(2) = INT(X(2))
175 X(3) = INT(X(3))
176 X(4) = INT(X(4))

177 IF X(1) < 2## THEN 1670
187 IF X(2) < 2## THEN 1670
188 IF X(3) < 2## THEN 1670
189 IF X(4) < 2## THEN 1670

191 IF X(1) > 8## THEN 1670
192 IF X(2) > 34## THEN 1670
193 IF X(3) > 45## THEN 1670
194 IF X(4) > 12## THEN 1670

226 IF 15 * X(1) + 7 * X(2) + 5 * X(3) + 9 * X(4) + 10 * X(1) ^ .5 + 5 * X(2) ^ .5 + 2 * X(3) ^ .5 + 6 * X(4) ^ .5 > 200 THEN 1670

419 PDU1 = -.5 * (1 / 225600205681.21) * ((1 - X(1) / 8) * 190177129.163709 / X(1) + (1 - X(2) / 34) * 3075358778.397211 / X(2) + (1 - X(3) / 45) * 8750902957.838071 / X(3) + (1 - X(4) / 12) * 451381009.970472 / X(4))

421 PDU2 = -.5 * (1 / 2484564.0625) * ((1 - X(1) / 8) * 5073.889916 / X(1) + (1 - X(2) / 34) * 588180.653800 / X(2) + (1 - X(3) / 45) * 221961.923300 / X(3) + (1 - X(4) / 12) * 5705.062673 / X(4))

422 PDU = PDU1 + PDU2

466 P = PDU

1111 IF P <= M THEN 1670

1420 M = P

1444 FOR KLX = 1 TO 4

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

1557 GOTO 128
1670 NEXT I

1889 IF M < -.018 THEN 1999
1926 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [87]. The complete output of a single run through JJJJ= -31988 is shown below:

2    9    9    2    -1.735801682977117D-02
-31995

2 9 9 2 -1.735801682977117D-02
-31991

2 9 9 2 -1.735801682977117D-02
-31988

.
.
.

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 [87], the wall-clock time (not CPU time) for obtaining the feasible solutions through JJJJ = -31988 was 1 or 2 seconds, not including the time for “Creating .EXE file" (17 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Ghufran, Khowaga, and Ahsan [30, p. 606].

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

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

[12] 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.

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

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

[15] 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.

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

[17] 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.

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

[19] 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

[20] 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.)

[21] 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.

[22] 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.

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

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

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

[30] Shazia Ghufran, Saman Khowaga, M. J. Ahsan (2012 ). Optimum multivariate stratified sampling designs with treavel cost: a multiobjective integer nonlinear programming approach. Comunications in Statistics - Simulation and Computation, 41:5, pp. 598-610.

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

[32] 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.

[33] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[33] 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.

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

[35] 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

[36] 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).

[37] 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

[38] 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

[39] 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.

[40] 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.

[41] 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.

[42] 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.

[43] 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.

[44] M. G. M. Khan, M. Rashid, S. Sharma (2019). An optimal multivariate cluster sampling design. Communications in Statistics--Theory and Methods 2019, pp.1-9.

[45] 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.

[46] 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.

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

[48] 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.

[49] 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.

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

[51] 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.

[52] 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.

[53] 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

[54] 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.

[55] 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.

[56] 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.

[57] 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.

[58] 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.

[59] 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.)

[60] mustapha moulai, wassila drici (april 02 2018). an indefinite quadratic optimization over an integer efficient set. optimization 2018 april 2.

[61] 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.

[62] 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.

[63] 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.

[64] 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.

[65] 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.

[66] 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/

[67] 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.

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

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

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

[71] 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.

[72] 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.

[73] 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.

[74] 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.

[75] 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).

[76] 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.

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

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

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

[80] 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.

[81] Har3i 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

[82] 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.

[83] 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.

[84] 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

[85] 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.

[86] 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.

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

[88] 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.

Can the Algorithm Here Handle Twelve Objectives?

Jsun Yui Wong

The computer program listed below tries to help answer that question:

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

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

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

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

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

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

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

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

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

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

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

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

subject to

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

X(2) + 3 * X(3) + X(4) <= 7

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

2 * X(2) + X(3) <= 10

0 <=X(1) <=5

0<= X(2)<=3

0 <=X(3) <=3

0<= X(4)<=1

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

The formulation above comes mostly from Arora and Arora's [9, p. 377] three objective integer indefinite quadratic programmig problem. See also Drici and Moulai [23].

One notes that, for example, objective twelve is the same as objective three; a better illustration would have 12 different objectives.

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 = -4E+250

93 W1 = (1 / 5.5) * RND: W2 = (1 / 5.5) * RND: W3 = (1 / 5.5) * RND

97 W4 = (1 / 5.5) * RND: W5 = (1 / 5.5) * RND: W6 = (1 / 5.5) * RND
98 W7 = (1 / 5.5) * RND: W8 = (1 / 5.5) * RND: W9 = (1 / 5.5) * RND
99 W10 = (1 / 5.5) * RND: W11 = (1 / 5.5) * RND: W12 = (1 / 5.5) * RND

100 IF W1 + W2 + W3 + W4 + W5 + W6 + W7 + W8 + W9 + W10 + W11 > 1 THEN 1999

103 W12 = 1 - W1 - W2 - W3 - W4 - W5 - W6 - W7 - W8 - W9 - W10 - W11

111 FOR J44 = 1 TO 4
113 A(J44) = FIX(RND * 3)

121 NEXT J44

128 FOR I = 0 TO FIX(RND * 1000)

129 FOR KKQQ = 1 TO 4

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

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 J = 1 + FIX(RND * 4)

154 IF RND < .5 THEN GOTO 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 * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
171 X(1) = INT(X(1))
173 X(2) = INT(X(2))

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

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

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

192 IF X(1) > 5 THEN 1670

194 IF X(2) > 3 THEN 1670

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

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

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

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

232 IF 2 * X(2) + X(3) > 10 THEN 1670

417 PDU1 = W1 * (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10) + W2 * (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11) + W3 * (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9) + W4 * (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10) + W5 * (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11) + W6 * (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)

419 PDU2 = W7 * (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10) + W8 * (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11) + W9 * (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9) + W10 * (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10) + W11 * (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11) + W12 * (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)

422 PDU = PDU1 + PDU2

466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 4

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

1522 OBJ1 = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)

1524 OBJ2 = (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)

1526 OBJ3 = (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)

1527 OBJ4 = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)

1528 OBJ5 = (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)

1529 OBJ6 = (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)
1532 OBJ7 = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)

1534 OBJ8 = (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)

1536 OBJ9 = (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)

1537 OBJ10 = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)

1538 OBJ11 = (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)

1539 OBJ12 = (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)

1557 GOTO 128
1670 NEXT I

1889 IF M < -99999999999 THEN 1999
1905 REM PRINT W1, W2, W3
1906 REM PRINT W4, W5, W6

1907 REM PRINT W7, W8, W9

1908 REM PRINT W10, W11, W12

1911 PRINT A(1), A(2), A(3), A(4)

1922 PRINT OBJ1, OBJ2, OBJ3
1923 PRINT OBJ4, OBJ5, OBJ6
1924 PRINT OBJ7, OBJ8, OBJ9
1926 PRINT OBJ10, OBJ11, OBJ12, M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [85]. The output of a single run through JJJJ= -31595 is summarized below:

3 1 1 1
154 182 135
154 182 135
154 182 135
154 182 135 158.0766 -31996

3 1 1 1
154 182 135
154 182 135
154 182 135
154 182 135 157.8094 -31993

3 1 1 1
154 182 135
154 182 135
154 182 135
154 182 135 151.8857 -31992

2 0 2 1
130 170 150
130 170 150
130 170 150
130 170 150 150.7934 -31991

3 1 1 1
154 182 135
154 182 135
154 182 135
154 182 135 153.7238 -31988

3 1 2 0
165 196 105
165 196 105
165 196 105
165 196 105 151.3481 -31987

3 1 1 1
154 182 135
154 182 135
154 182 135
154 182 135 154.8197 -31986

3 1 1 1
154 182 135
154 182 135 147.7507 -31985

3 1 1 1
154 182 135
154 182 135
154 182 135
154 182 135 158.9577 -31984

3 1 2 0
165 196 105
165 196 105
165 196 105
165 196 105 157.5636 -31982

3 1 1 1
154 182 135
154 182 135
154 182 135
154 182 135 153.6401 -31981

2 0 2 1
130 170 150
130 170 150
130 170 150
130 170 150 154.9689 -31979

3 1 2 0
165 196 105
165 196 105
165 196 105
165 196 105 165.163 -31978

3 1 1 1
154 182 135
154 182 135
154 182 135
154 182 135 152.42 -31977

3 1 1 1
154 182 135
154 182 135
154 182 135
154 182 135 156.1118 -31973

3 1 1 1
154 182 135
154 182 135
154 182 135
154 182 135 156.1026 -31971

2 0 1 1
144 165 143
144 165 143
144 165 143
144 165 143 149.4057 -31970

2 0 1 1
144 165 143
144 165 143
144 165 143
144 165 143 148.1087 -31967

3 1 2 0
165 196 105
165 196 105
165 196 105
165 196 105 157.9869 -31964

2 0 2 1
130 170 150
130 170 150
130 170 150
130 170 150 154.0427 -31962

3 1 2 0
165 196 105
165 196 105
165 196 105
165 196 105 161.9647 -31959

3 0 0 0
180 165 108
180 165 108
180 165 108
180 165 108 152.2126 -31956

.
.
.

4 2 1 0
171 187 90
171 187 90
171 187 90
171 187 90 162.0784 -31595

.
.
.

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 [85], the wall-clock time (not CPU time) for obtaining the feasible solutions through JJJJ = -31595 was 2 seconds, not including the time for “Creating .EXE file" (18 seconds, total, including the time for “Creating .EXE file").

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

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

[12] 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.

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

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

[15] 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.

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

[17] 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.

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

[19] 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

[20] 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.)

[21] 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.

[22] 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.

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

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

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[33] 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.

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

[35] 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

[36] 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).

[37] 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

[38] 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

[39] 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.

[40] 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.

[41] 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.

[42] 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.

[43] 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.

[44] 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.

[45] 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.

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

[47] 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.

[48] 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.

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

[50] 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.

[51] 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.

[52] 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

[53] 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.

[54] 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.

[55] 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.

[56] 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.

[57] 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.

[58] 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.)

[59] 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.

[60] 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.

[61] 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.

[62] 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.

[63] 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.

[64] 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/

[65] 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.

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

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

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

[69] 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.

[70] 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.

[71] 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.

[72] 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.

[73] 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).

[74] 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.

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

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

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

[78] 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.

[79] Har3i 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

[80] 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.

[81] 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.

[82] 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

[83] 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.

[84] 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.

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

[86] 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.

Can the Present Algorithm Handle Six Objectives?

Jsun Yui Wong

The computer program listed below tries to help answer that question:

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

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

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

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

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

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

subject to

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

X(2) + 3 * X(3) + X(4) <= 7

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

2 * X(2) + X(3) <= 10

0 <=X(1) <=5

0<= X(2)<=3

0 <=X(3) <=3

0<= X(4)<=1

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

The formulation above comes mostly from Arora and Arora's [9, p. 377] three objective integer indefinite quadratic programmig problem. See also Drici and Moulai [23].

One notes that, for example, objective four is the same as objective one.

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 = -4E+250

93 W1 = RND
95 W2 = RND
96 W3 = RND
97 W4 = RND
98 W5 = RND

100 IF W1 + W2 + W3 + W4 + W5 > 1 THEN 1999

103 W6 = 1 - W1 - W2 - W3 - W4 - W5

111 FOR J44 = 1 TO 4
113 A(J44) = FIX(RND * 3)

121 NEXT J44

128 FOR I = 0 TO FIX(RND * 1000)

129 FOR KKQQ = 1 TO 4

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

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 J = 1 + FIX(RND * 4)

154 IF RND < .5 THEN GOTO 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 * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
171 X(1) = INT(X(1))
173 X(2) = INT(X(2))

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

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

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

192 IF X(1) > 5 THEN 1670

194 IF X(2) > 3 THEN 1670

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

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

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

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

232 IF 2 * X(2) + X(3) > 10 THEN 1670

417 PDU = W1 * (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10) + W2 * (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11) + W3 * (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9) + W4 * (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10) + W5 * (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11) + W6 * (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)

466 P = PDU

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 4

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

1522 OBJ1 = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)

1524 OBJ2 = (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)

1526 OBJ3 = (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)

1527 OBJ4 = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)

1528 OBJ5 = (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)

1529 OBJ6 = (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)

1557 GOTO 128
1670 NEXT I

1889 IF M < -99999999999 THEN 1999
1905 PRINT W1, W2, W3
1908 PRINT W4, W5, W6

1911 PRINT A(1), A(2), A(3), A(4)

1922 PRINT OBJ1, OBJ2, OBJ3
1923 PRINT OBJ4, OBJ5, OBJ6, M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [85]. The output of a single run through JJJJ= -30672 is summarized below:

.1064085 4.586798E-02 8.460617E-02
.0845415 .6737858 4.790068E-03
3 1 2 0
165 196 105
165 196 105 181.9455 -31911

4.939985E-02 .1623284 2.392113E-02
2.550775E-02 .6991537 3.968918E-02
3 1 2 0
165 196 105
165 196 105 187.8893 -31888

6.488311E-02 .5156589 .1395888
2.652711-E-02 .0270282 .2263138
3 1 1 1
154 182 135
154 182 135 162.2431 -31825

1.020992E-02 .1162019 2.838945E-02
,2888362 .4539682 .1023943
3 1 2 0
165 196 105
165 196 105 174.8282 -31792

.1374902 .1055158 9.483397E-03
.3109474 5.395526E-02 .3826079
3 1 1 1
154 182 135
154 182 135 151.0155 -31691

217438 .2146174 1.001972E-02
,2003297 8.783489E-02 .2697063
3 1 2 0
165 196 105
165 196 105 157.5941 -31494

.0879547 1.803839E-02 .2301677
2.413869E-02 3.864902E-02 .6010515
2 0 2 1
130 170 150
130 170 150 148.8919 -31361

.1771699 .3938255 4.750842E-02
,11301 2.435428E-02 .2441319
3 1 2 0
165 196 105
165 196 105 157.5941 -31302

.1236852 8.625269E-03 .0900721
.5744303 .1227594 8.042771E-02
3 0 0 0
180 165 108
180 165 108 165.7532 -31299

.
.
.

.2348937 .1120725 3.101206E-02
.3102652 .3017904 9.966135E-03
4 2 1 0
171 187 90
171 187 90 174.3026 -30672

.
.
.

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 [85], the wall-clock time (not CPU time) for obtaining the feasible solutions through JJJJ = -30672 was 2 seconds, not including the time for “Creating .EXE file" (18 seconds, total, including the time for “Creating .EXE file").

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

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

[12] 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.

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

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

[15] 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.

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

[17] 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.

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

[19] 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

[20] 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.)

[21] 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.

[22] 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.

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

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

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[33] 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.

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

[35] 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

[36] 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).

[37] 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

[38] 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

[39] 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.

[40] 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.

[41] 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.

[42] 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.

[43] 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.

[44] 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.

[45] 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.

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

[47] 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.

[48] 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.

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

[50] 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.

[51] 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.

[52] 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

[53] 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.

[54] 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.

[55] 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.

[56] 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.

[57] 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.

[58] 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.)

[59] 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.

[60] 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.

[61] 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.

[62] 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.

[63] 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.

[64] 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/

[65] 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.

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

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

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

[69] 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.

[70] 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.

[71] 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.

[72] 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.

[73] 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).

[74] 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.

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

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

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

[78] 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.

[79] Har3i 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

[80] 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.

[81] 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.

[82] 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

[83] 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.

[84] 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.

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

[86] 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.

Addendum: A Unified Approach for Solving Multi-Objective Integer Nonlinear Programs

Addendum: A Unified Approach for Solving Multi-Objective Integer Nonlinear Programs

Jsun Yui Wong

The first sentence should read as follows:

The four very similar computer programs listed below are about the four examples, respectively, in Blanco and Puerto [10].  

A Unified Approach for Solving Multi-Objective Integer Nonlinear Programs

Jsun Yui Wong

The four computer programs listed below are about the four examples, respectively, in Blanco and Puerto [10].

Each line 128 below is 128 FOR I = 0 TO FIX(RND * 50).

1. Example 3.1 of Blanco and Puerto [10]

The computer program listed below seeks to generate all or some of the feasible solutions of the following problem from Blanco and Puerto [10, p. 516, Example 3.1]:

Minimize ( X(1) ^ 2 - X(2), X(1) - X(2) ^ 2 )

subject to

X(2) - X(1) ^ 4 + 10 * X(1) ^ 3 - 30 * X(1) ^ 2 + 25 * X(1) - 7 >= 0

X(2) - X(1) ^ 3 + 9 * X(1) ^ 2 - 25 * X(1) + 12 <= 0

X(1), X(2) >=0 and are general integer variables.

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 = -4E+250

98 W1 = RND

99 REM W2 = RND

101 REM IF W1 + W2 > 1## THEN 1670

103 W2 = 1 - W1

111 FOR J44 = 1 TO 2
113 A(J44) = FIX(RND * 11)

115 REM A(J44) = RND

121 NEXT J44

128 FOR I = 0 TO FIX(RND * 50)

129 FOR KKQQ = 1 TO 2

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

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 J = 1 + FIX(RND * 2)

154 IF RND < .5 THEN GOTO 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 * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
164 REM IF A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0
169 NEXT IPP

293 FOR J44 = 1 TO 2

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

296 X(J44) = INT(X(J44))

297 NEXT J44

324 IF X(2) - X(1) ^ 4 + 10 * X(1) ^ 3 - 30 * X(1) ^ 2 + 25 * X(1) - 7 < 0## THEN 1670

334 IF X(2) - X(1) ^ 3 + 9 * X(1) ^ 2 - 25 * X(1) + 12 > 0## THEN 1670

539 REM P = W1 * (3 * X(1) + X(2) + 2 * X(3) + X(4)) + W2 * (X(1) - X(2) + 2 * X(3) + 4 * X(4)) + W3 * (-X(1) + 5 * X(2) + X(3) + 2 * X(4))

541 P = W1 * (-X(1) ^ 2 + X(2)) + W2 * (-X(1) + X(2) ^ 2)

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 2

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

1457 OBJ1 = -(-X(1) ^ 2 + X(2))

1458 OBJ2 = -(-X(1) + X(2) ^ 2)

1459 REM OBJ3 = (-X(1) + 5 * X(2) + X(3) + 2 * X(4))

1557 GOTO 128
1670 NEXT I
1890 IF M <= -10 ^ 40 THEN GOTO 1999

1933 REM PRINT W1, W2

1936 PRINT A(1), A(2), OBJ1, OBJ2, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [85]. The feasible solutions of a single run through JJJJ= -31990 is shown below:

5 11 14 -116 -32000

4 8 8 -60 -31999

1 5 -4 -24 -31998

1 5 -4 -24 -31997

5 13 12 -164 -31996

5 13 12 -164 -31994

4 8 8 -60 -31993

1 5 -4 -24 -31990

.
.
.
.

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 [85], the wall-clock time (not CPU time) for obtaining the feasible solutions through JJJJ = -31990 was 2 seconds, not including the time for “Creating .EXE file" (18 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Blanco and Puerto [10, p. 517].

2. Example 4.1 of Blanco and Puerto [10]

The computer program listed below seeks to generate all or some of the feasible solutions of the following problem from Blanco and Puerto [10, p. 521, Example 4.1]:

Minimize

( -5 * X(1) + 8 * X(2) + 5 * X(1) * X(2), 4 * X(1) + 2 * X(1) * X(2) )

subject to

-X(1) + X(1) ^ 3 + X(2) ^ 4 >= 0

X(1), X(2) epsilon { 0, 1, 2 }.

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 = -4E+250

98 W1 = RND

103 W2 = 1 - W1

111 FOR J44 = 1 TO 2
113 A(J44) = FIX(RND * 3)

115 REM A(J44) = RND

121 NEXT J44

128 FOR I = 0 TO FIX(RND * 50)

129 FOR KKQQ = 1 TO 2

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

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 J = 1 + FIX(RND * 2)

154 IF RND < .5 THEN GOTO 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 * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
164 REM IF A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0
169 NEXT IPP

293 FOR J44 = 1 TO 2

294 IF X(J44) < 0## THEN 1670
295 IF X(J44) > 2 THEN 1670

296 X(J44) = INT(X(J44))

297 NEXT J44

324 IF -X(1) + X(1) ^ 3 + X(2) ^ 4 < 0## THEN 1670

547 P = W1 * (5 * X(1) - 8 * X(2) - 5 * X(1) * X(2)) + W2 * (-4 * X(1) - 2 * X(1) * X(2))

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 2

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

1457 OBJ1 = -(5 * X(1) - 8 * X(2) - 5 * X(1) * X(2))

1458 OBJ2 = -(-4 * X(1) - 2 * X(1) * X(2))

1557 GOTO 128
1670 NEXT I
1890 IF M <= -10 ^ 40 THEN GOTO 1999

1933 REM PRINT W1, W2

1936 PRINT A(1), A(2), OBJ1, OBJ2, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [85]. The feasible solutions of a single run through JJJJ= -31990 is shown below:

2 0 -10 8 -32000

2 0 -10 8 -31999

2 0 -10 8 -31998

2 0 -10 8 -31997

2 0 -10 8 -31996

0 0 0 0 -31995

1 0 -5 4 -31994

0 0 0 0 -31993

2 0 -10 8 -31992

0 0 0 0 -31991

2 0 -10 8 -31990

.
.
.

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 [85], the wall-clock time (not CPU time) for obtaining the feasible solutions through JJJJ = -31990 was 2 seconds, not including the time for “Creating .EXE file" (18 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Blanco and Puerto [10, p. 522].

3. Based on Example 4.2 of Blanco and Puerto [10]

The computer program listed below seeks to generate all or some of the feasible solutions of the immediately following problem:

Minimize ( -10 * X(2) + 3 * X(1) * X(2), 2 * X(1) + 4 * X(2) - 6 * X(1) * X(2) )

subject to

-4 * X(2) + 132 * X(1) ^ 4 - 143 * X(1) ^ 3 + 40 * X(1) >= 0

X(1), X(2) epsilon {0, 1, 2}.

One notes that the long constraint in Blanco and Puerto [10, p. 524, Example 4.2] is -4 * X(2) + 132 * X(1) ^ 4 - 143 * X(1) ^ 3 + 40 * X(1) <= 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 = -4E+250

98 W1 = RND

103 W2 = 1 - W1

111 FOR J44 = 1 TO 2
113 A(J44) = FIX(RND * 2.3)

121 NEXT J44

128 FOR I = 0 TO FIX(RND * 50)

129 FOR KKQQ = 1 TO 2

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

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 J = 1 + FIX(RND * 2)

154 IF RND < -.5 THEN GOTO 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 * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
164 REM IF A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0
169 NEXT IPP

293 FOR J44 = 1 TO 2

294 IF X(J44) < 0## THEN 1670
295 IF X(J44) > 2## THEN 1670

296 X(J44) = INT(X(J44))

297 NEXT J44

324 IF -4 * X(2) + 132 * X(1) ^ 4 - 143 * X(1) ^ 3 + 40 * X(1) < 0## THEN 1670

547 P = W1 * (10 * X(2) - 3 * X(1) * X(2)) + W2 * (-2 * X(1) - 4 * X(2) + 6 * X(1) * X(2))

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 2

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

1457 OBJ1 = -(10 * X(2) - 3 * X(1) * X(2) )

1458 OBJ2 = -(-2 * X(1) - 4 * X(2) + 6 * X(1) * X(2))

1557 GOTO 128
1670 NEXT I
1890 IF M <= -10 ^ 40 THEN GOTO 1999

1933 REM PRINT W1, W2

1936 PRINT A(1), A(2), OBJ1, OBJ2, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [85]. The feasible solutions of a single run through JJJJ= -31990 is shown below:

2 2 -8 -12 -32000

1 2 -14 -2 -31999

2 2 -8 -12 -31998

0 0 0 0 -31997

1 2 -14 -2 -31996

1 2 -14 -2 -31995

2 2 -8 -12 -31994

1 2 -14 -2 -31993

2 2 -8 -12 -31992

1 2 -14 -2 -31991

2 1 -4 -4 -31990

.
.
.

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 [85], the wall-clock time (not CPU time) for obtaining the feasible solutions through JJJJ = -31990 was 2 seconds, not including the time for “Creating .EXE file" (18 seconds, total, including the time for “Creating .EXE file").

4. Example 5.1 of Blanco and Puerto [10]

The computer program listed below seeks to generate all or some of the feasible solutions of the following problem from Blanco and Puerto [10, p. 526, Example 5.1]:

Minimize ( X(1) ^ 2 + 10 * X(2), -4 * X(1)- X(2) ^ 3 )

subject to

16 * X(2) - 16 * X(1) ^ 3 + 16 * X(1) ^ 2 - 3 * X(1) - 8 >= 0

X(1), X(2) epsilon { 0, 1 }.

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 = -4E+250

98 W1 = RND

103 W2 = 1 - W1

111 FOR J44 = 1 TO 2
113 A(J44) = FIX(RND * 2)

121 NEXT J44

128 FOR I = 0 TO FIX(RND * 50)

129 FOR KKQQ = 1 TO 2

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

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 J = 1 + FIX(RND * 2)

154 IF RND < .5 THEN GOTO 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 * .3) ELSE X(J) = A(J) + FIX(1 + RND * .3)
164 REM IF A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0
169 NEXT IPP

293 FOR J44 = 1 TO 2

294 IF X(J44) < 0## THEN 1670
295 IF X(J44) > 1## THEN 1670

296 X(J44) = INT(X(J44))

297 NEXT J44

324 IF 16 * X(2) - 16 * X(1) ^ 3 + 16 * X(1) ^ 2 - 3 * X(1) - 8 < 0## THEN 1670

545 P = W1 * (-X(1) ^ 2 - 10 * X(2)) + W2 * (4 * X(1) + X(2) ^ 3)

1111 IF P <= M THEN 1670

1450 M = P

1454 FOR KLX = 1 TO 2

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

1457 OBJ1 = -(-X(1) ^ 2 - 10 * X(2))

1458 OBJ2 = -(4 * X(1) + X(2) ^ 3)

1557 GOTO 128
1670 NEXT I
1890 IF M <= -10 ^ 40 THEN GOTO 1999

1933 REM PRINT W1, W2

1936 PRINT A(1), A(2), OBJ1, OBJ2, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [85]. The feasible solutions of a single run through JJJJ= -31990 is shown below:

1 1 11 -5 -32000

0 1 10 -1 -31999

1 1 11 -5 -31998

1 1 11 -5 -31997

1 1 11 -5 -31996

1 1 11 -5 -31995

1 1 11 -5 -31994

0 1 10 -1 -31993

1 1 11 -5 -31992

0 1 10 -1 -31991

1 1 11 -5 -31990

.
.
.

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 [85], the wall-clock time (not CPU time) for obtaining the feasible solutions through JJJJ = -31990 was 2 seconds, not including the time for “Creating .EXE file" (17 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Blanco an Puerto [10, p. 526].

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

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

[12] 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.

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

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

[15] 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.

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

[17] 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.

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

[19] 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

[20] 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.)

[21] 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.

[22] 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.

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

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

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
.
[33] 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.

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

[35] 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

[36] 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).

[37] 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

[38] 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

[39] 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.

[40] 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.

[41] 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.

[42] 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.

[43] 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.

[44] 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.

[45] 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.

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

[47] 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.

[48] 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.

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

[50] 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.

[51] 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.

[52] 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

[53] 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.

[54] 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.

[55] 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.

[56] 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.

[57] 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.

[58] 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.)

[59] 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.

[60] 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.

[61] 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.

[62] 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.

[63] 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.

[64] 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/

[65] 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.

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

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

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

[69] 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.

[70] 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.

[71] 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.

[72] 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.

[73] 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).

[74] 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.

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

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

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

[78] 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.

[79] Har3i 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

[80] 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.

[81] 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.

[82] 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

[83] 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.

[84] 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.

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

[86] 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.