Substance

Optimization of Parameters of a Machining/Grinding Process Using This Blog's Mixed-Integer Nonlinear Programming (MINLP) Algorithm

                                                                            Jsun Yui Wong

The computer program listed below seeks to solve the following problem of maximizing material removal rate (MRR) from Mellal and Williams [36, p. 930, Experiment 9 with SR MAX=.4, ND MAX=11 in Table 1 on p. 933]:

Maximize

X(1) * X(2)

subject to

.4 - .145 * (X(1) ^ .1939) * (X(2) ^ .7071) * (X(3) ^ (-.2343))<= .4

29.67 * (X(1) ^ .4167) * (X(2) ^ (-.8333))<=11

5<= X(1) <= 30

8.6<= X(2) <= 13.4

120<= X(3) <= 500.

X(4) and X(5) below are slack variables added.

 

0 DEFDBL A-Z

1 DEFINT K

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

 

81 FOR JJJJ = -32000 TO 32000

 

89 RANDOMIZE JJJJ
90 M = -3E+30

96 A(1) = 5 + (RND * 25)

98 A(2) = 8.6 + (RND * 4.8)

99 A(3) = 120 + (RND * 380)

128 FOR I = 1 TO 20000

 

129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 2)

 

153 J = 1 + FIX(RND * 3)

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

 

169 NEXT IPP

 

238 IF X(1) < 5 THEN 1670

239 IF X(1) > 30 THEN 1670

 

248 IF X(2) < 8.6 THEN 1670

249 IF X(2) > 13.4 THEN 1670

258 IF X(3) < 120 THEN 1670

259 IF X(3) > 500 THEN 1670

351 X(4) = 11 - 29.67 * (X(1) ^ .4167) * (X(2) ^ (-.8333))

354 X(5) = .4 - .145 * (X(1) ^ .1939) * (X(2) ^ .7071) * (X(3) ^ (-.2343))

 

356 FOR J44 = 4 TO 5

 

357 IF X(J44) < 0## THEN X(J44) = X(J44) ELSE X(J44) = 0##

359 NEXT J44

443 PDU = X(1) * X(2) + 1000000 * (X(4) + X(5))

 

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 5

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

1557 GOTO 128
1670 NEXT I
1889 IF M < -5555555 THEN 1999

1904 PRINT A(1), A(2), A(3), A(4), A(5), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [50]. The complete output of a single run through JJJJ= -31998 is shown below:

16.58831155724903       13.40000009536743       444.280898770321
0       0       222.2833764491217       -32000

16.58831155724903       13.40000009536743       378.4072604067686
0       0       222.2833764491217       -31999

16.58831155724903       13.40000009536743      352.0148967133522
0       0       222.2833764491217       -31998

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [50], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 2 or 3 seconds, not including the time for “Creating .EXE file" (8 seconds, including the time for “Creating .EXE file"). One can compare the computational results above with those in Mellal and Williams [36, Experiment 9 in Table 1 on p. 933].

Acknowledgment

 

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[24] 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

[25] 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

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] 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.

[33] 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.

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

[35] 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.

[36] 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.

[37] 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.

[38] 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.

[39] 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.

[40] 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.

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

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

[43] 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.

[44] 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.

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

[46] 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.

[47] 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.

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

[49] 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.

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

[51] 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.

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

 

Parameter Optimization of Abrasive Jet Machining Process in Ductile Materials Using This Blog's Mixed-Integer Nonlinear Programming (MINLP) Algorithm

                                                                   Jsun Yui Wong

The computer program listed below seeks to solve the following problem of maximizing material removing rate (MRR) in Mellal and Williams [36, p. 937]:

Maximize

1.0436D-06 * 1.6 * (2.7D-06 / (1.5 ^ 2 * 1150 ^ 1.5 * (2.48D-06) ^ .5)) * X(1) * X(3) ^ 3

subject to

1 - (25.82 / 2) * (2.48D-06 / 1150) ^ .5 * X(2) * X(3)>=0

.0000167<= X(1) <= .0005

.005<= X(2) <= .075

150000 <= X(3) <= 400000.

X(4) below is a slack variable added.

 

0 DEFDBL A-Z

1 DEFINT K

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

 

81 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ
90 M = -3E+30
95 REM FOR J44 = 1 TO 3

96 A(1) = .0000167 + (RND * .000483)

98 A(2) = .005 + (RND * .07)

99 A(3) = 150000 + (RND * 250000)

128 FOR I = 1 TO 20000

129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 2)

 

153 J = 1 + FIX(RND * 3)

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

 

169 NEXT IPP

 

238 IF X(1) < .0000167 THEN 1670

239 IF X(1) > .0005 THEN 1670

 

248 IF X(2) < .005 THEN 1670

249 IF X(2) > .075 THEN 1670

258 IF X(3) < 150000 THEN 1670

259 IF X(3) > 400000 THEN 1670

351 X(4) = 1 - (25.82 / 2) * (2.48D-06 / 1150) ^ .5 * X(2) * X(3)

356 FOR J44 = 4 TO 4

 

357 IF X(J44) < 0## THEN X(J44) = X(J44) ELSE X(J44) = 0##

359 NEXT J44

443 PDU = 1.0436D-06 * 1.6 * (2.7D-06 / (1.5 ^ 2 * 1150 ^ 1.5 * (2.48D-06) ^ .5)) * X(1) * X(3) ^ 3 + 1000000 * (X(4))

 

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 4

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < -5555555 THEN 1999

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

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [50]. The complete output of a single run through JJJJ= -31998 is shown below:

5.0000052852556D-04       4.99999967123504D-03       333600.6296866373
0       .6056383025744083       -32000

5.0000052852556D-04       4.999999680738293D-03       333600.629052579
0       .605638299121087       -31999

5.0000052852556D-04       4.99999966298563D-03       333600.6302370388
0       .6056383055721026       -31998

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [50], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 2 or 3 seconds, not including the time for “Creating .EXE file" (8 seconds, including the time for “Creating .EXE file"). One can compare the computational results above with those in Mellal and Williams [36, p. 937, Table 9].

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[24] 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

[25] 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

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] 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.

[33] 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.

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

[35] 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.

[36] 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.

[37] 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.

[38] 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.

[39] 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.

[40] 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.

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

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

[43] 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.

[44] 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.

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

[46] 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.

[47] 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.

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

[49] 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.

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

[51] 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.

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

Parameter Optimization of Ultrasonic Machining Process Using This Blog's Mixed-Integer Nonlinear Programming (MINLP) Algorithm, Second Edition

                                                                   Jsun Yui Wong

The computer program listed below seeks to solve the following problem of maximizing material removing rate (MRR) in Mellal and Williams [36, p. 939]:

Maximize

((4.963 * 20 ^ .25 * .1 ^ .75) / (6900 * (1 + .246)) ^ .75) * X(4) ^ .75 * X(1) ^ .75 * X(2) ^ .25 * X(3) * X(5)

subject to

1 - ((1154.7) / ((20 * 6900 * (1 + .246)) ^ .5 * .8)) * (X(4) * X(1) * X(3) / X(2)) ^ .5>=0

.005 <= X(1) <= .1

.05<= X(2) <= .5

.007 <= X(3) <= .15

4.5 <= X(4) <= 45

10000 <= X(5) <= 40000.

X(6) below is a slack variable added.

One notes that following Rao, Pawar, and Davim [43, p. 1124], the present paper uses lambda=.246, as shown above.

Whereas line 128 of the earlier edition is 128 FOR I = 1 TO 20000, here line 128 is 128 FOR I = 1 TO 10000.

 

0 DEFDBL A-Z

1 DEFINT K

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

 

81 FOR JJJJ = -32000 TO 32000

 

89 RANDOMIZE JJJJ
90 M = -3E+30
95 REM FOR J44 = 1 TO 3

96 A(1) = .005 + (RND * .095)

97 REM NEXT J44

98 A(2) = .05 + (RND * .45)

99 A(3) = .007 + (RND * .143)

 

111 A(4) = 4.5 + (RND * 40.5)

113 A(5) = 10000 + (RND * 30000)

 

128 FOR I = 1 TO 10000

129 FOR KKQQ = 1 TO 5
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)

 

153 J = 1 + FIX(RND * 5)

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

 

169 NEXT IPP

 

238 IF X(1) < .005 THEN 1670

239 IF X(1) > .1 THEN 1670

 

248 IF X(2) < .05 THEN 1670

249 IF X(2) > .5 THEN 1670

258 IF X(3) < .007 THEN 1670

259 IF X(3) > .15 THEN 1670

 

268 IF X(4) < 4.5 THEN 1670

269 IF X(4) > 45 THEN 1670

278 IF X(5) < 10000 THEN 1670

279 IF X(5) > 40000 THEN 1670

 

351 X(6) = 1 - ((1154.7) / ((20 * 6900 * (1 + .246)) ^ .5 * .8)) * (X(4) * X(1) * X(3) / X(2)) ^ .5

 

356 FOR J44 = 6 TO 6

 

357 IF X(J44) < 0## THEN X(J44) = X(J44) ELSE X(J44) = 0##

359 NEXT J44

443 PDU = ((4.963 * 20 ^ .25 * .1 ^ .75) / (6900 * (1 + .246)) ^ .75) * X(4) ^ .75 * X(1) ^ .75 * X(2) ^ .25 * X(3) * X(5) + 1000000 * X(6)

 

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 6

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < -5555555 THEN 1999

1901 REM GOTO 1950

1904 PRINT A(1), A(2), A(3), A(4), A(5), A(6), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [50]. The complete output of a single run through JJJJ= -31998 is shown below:

3.084754945653472D-02       .5000000298023206      .1499958910733272
8.91884814961422       40000       0       4.006409756142836
-32000

3.831797974129483D-02       .5000000298023223       .1499889469497716
7.180540606138552       40000       0       4.00636338572702
-31999

2.263357415902976D-02       .500000029802322       .1499988721329657
12.1553555644334       40000       0       4.006429662117143
-31998

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [50], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 3 seconds, not including the time for “Creating .EXE file" (8 seconds, including the time for “Creating .EXE file"). One can compare the computational results above with those in Mellal and Williams [36, p. 940, Table 12].

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[24] 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

[25] 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

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] 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.

[33] 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.

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

[35] 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.

[36] 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.

[37] 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.

[38] 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.

[39] 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.

[40] 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.

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

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

[43] 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.

[44] 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.

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

[46] 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.

[47] 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.

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

[49] 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.

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

[51] 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.

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

 

This Mixed-Integer Nonlinear Programming (MINLP) Solver Applied to Another Instance of a Large-Scale Reliability-Redundancy Allocation Problem, Third Edition

                                                                      Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Mellal and Williams [36, pp. 187-188, Table 7.1]:

Maximize PRODU

where PRODU is described below by line 411 through line 420

subject to

SUMV<=500

where SUMV is described below by line 338 through line 341

SUMW<=800

where SUMW is described below by line 344 through line 347

SUMM<=600

where SUMM is described below by line 351 through line 354

1<= X(i) <=10, for i=1 through 20 and these Xs are integers;

.5<= X(i) <=1, for i=21, 22, 23, ..., 40.

X(41) through X(43) below are slack variables added.

Whereas line 124 of the preceding paper is 124 NEWN = INT(18000 + RND * 2000), here line 124 is 124 NEWN = INT(18000 + RND * 4000).

 

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)

11 SV(1) = 2: SV(2) = 5: SV(3) = 5: SV(4) = 4: SV(5) = 4: SV(6) = 1: SV(7) = 1: SV(8) = 4: SV(9) = 4: SV(10) = 3: SV(11) = 3: SV(12) = 1: SV(13) = 1: SV(14) = 3: SV(15) = 4: SV(16) = 5: SV(17) = 1: SV(18) = 4: SV(19) = 2: SV(20) = 1

12 SW(1) = 8: SW(2) = 9: SW(3) = 6: SW(4) = 10: SW(5) = 8: SW(6) = 9: SW(7) = 9: SW(8) = 7: SW(9) = 9: SW(10) = 8: SW(11) = 9: SW(12) = 8: SW(13) = 7: SW(14) = 10: SW(15) = 6: SW(16) = 7: SW(17) = 7: SW(18) = 8: SW(19) = 9: SW(20) = 9

 

13 AL(1) = .6 / 10 ^ 5: AL(2) = .1 / 10 ^ 5: AL(3) = 1.2 / 10 ^ 5: AL(4) = .3 / 10 ^ 5: AL(5) = 2.9 / 10 ^ 5: AL(6) = 1.7 / 10 ^ 5: AL(7) = 2.6 / 10 ^ 5: AL(8) = 2.5 / 10 ^ 5: AL(9) = 1.3 / 10 ^ 5: AL(10) = 1.8 / 10 ^ 5: AL(11) = 2.4 / 10 ^ 5: AL(12) = 1.3 / 10 ^ 5: AL(13) = 1.2 / 10 ^ 5: AL(14) = 2.1 / 10 ^ 5: AL(15) = .9 / 10 ^ 5: AL(16) = 1.3 / 10 ^ 5: AL(17) = 1.9 / 10 ^ 5: AL(18) = 2.7 / 10 ^ 5: AL(19) = 2.8 / 10 ^ 5: AL(20) = 1.5 / 10 ^ 5

81 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ
90 M = -3E+30

95 FOR J44 = 1 TO 20

97 A(J44) = FIX(1 + RND * 10)

99 NEXT J44

115 FOR J44 = 21 TO 40

117 A(J44) = .5 + RND * .499999##

119 NEXT J44

124 NEWN = INT(18000 + RND * 4000)

 

128 FOR I = 1 TO NEWN

 

129 FOR KKQQ = 1 TO 40
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 5)

 

153 J = 1 + FIX(RND * 40)

 

155 IF J > 20.5 THEN GOTO 156 ELSE GOTO 164

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

 

161 GOTO 169

 

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

 

169 NEXT IPP
326 FOR J44 = 1 TO 20

327 X(J44) = CINT(X(J44))

 

328 IF X(J44) < 1 THEN 1670

329 IF X(J44) > 10 THEN 1670
331 NEXT J44

332 FOR J44 = 21 TO 40

 

333 IF X(J44) < .5## THEN 1670

334 IF X(J44) > .999999## THEN 1670
335 NEXT J44

 

338 SUMV = 0
339 FOR J44 = 1 TO 20

 

340 SUMV = SUMV + SV(J44) * X(J44) ^ 2

 

341 NEXT J44
343 X(41) = 500 - SUMV

 

344 SUMW = 0
345 FOR J44 = 1 TO 20

 

346 SUMW = SUMW + SW(J44) * X(J44) * EXP(X(J44) / 4)

347 NEXT J44

349 X(42) = 800 - SUMW

 

351 SUMM = 0
352 FOR J44 = 1 TO 20

 

353 SUMM = SUMM + AL(J44) * ((-1000 / LOG(X(J44 + 20))) ^ 1.5) * (X(J44) + EXP(X(J44) / 4))

 

354 NEXT J44
355 X(43) = 600 - SUMM

 

356 FOR J44 = 41 TO 43

 

357 IF X(J44) < 0## THEN X(J44) = X(J44) ELSE X(J44) = 0##

359 NEXT J44

411 PRODU = 1
415 FOR J44 = 1 TO 20

 

418 PRODU = PRODU * (1 - ((1 - X(J44 + 20)) ^ X(J44)))

420 NEXT J44

 

444 PDU = PRODU + 1000000 * (X(41) + X(42) + X(43))

 

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 43

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < .814 THEN 1999

1904 PRINT A(1), A(2), A(3), A(4), A(5)

1907 PRINT A(6), A(7), A(8), A(9), A(10)
1908 PRINT A(11), A(12), A(13), A(14), A(15)

 

1927 PRINT A(16), A(17), A(18), A(19), A(20)

 

1944 PRINT A(21), A(22), A(23), A(24), A(25)

1947 PRINT A(26), A(27), A(28), A(29), A(30)
1948 PRINT A(31), A(32), A(33), A(34), A(35)

1949 PRINT A(36), A(37), A(38), A(39), A(40)

1950 PRINT A(41), A(42), A(43), M, JJJJ, NEWN

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [48]. Each of the two candidate solutions of a single run through JJJJ= -31845 is either partly shown or fully shown below; one notes line 1889 above is 1889 IF M < .814 THEN 1999.

2       2       3       2       3
2       3       3       2       3
3       3       2       2       3 
3       2       3       3       2
.
.
.
.7766005292605046       .8803252932796247  
0       0       0       .8142553564339847
-31932       18102

 

2       2       3       2       3
2       3       3       2       3
3       2       3       2       2 
3       3       3       3       2
.9037502591217865       .9411411241722917       .8174715943209683
.923204685534939      .7762579732456564
.876418617035402       .7837650917670971       .7872546015125201
.8868017843377017       .7998951257646731 
.7843798064754277       .87253799902962       .8163687424814584
.8755144861893616       .8967757851721541
.8144373990327487       .7966283367110538       .7826143058405968
.781738482679067       .8774727904647764
0       0       0        .8171076605401475
-31845       20164

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 [48], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31845 was 43 minutes, total, including the time for “Creating .EXE file." One can compare the computational results above with those in Mellal and Williams [36, p. 193, Table 7.2].

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[24] 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

[25] 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

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] 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.

[33] 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.

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

[35] 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.

[36] 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.

[37] 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.

[38] 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.

[39] 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.

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

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

[42] 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.

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

[44] 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.

[45] 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.

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

[47] 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.

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

[49] 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.

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

 

This Mixed-Integer Nonlinear Programming (MINLP) Solver Applied to Another Instance of a Large-Scale Reliability-Redundancy Allocation Problem, Second Edition

                                                                      Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Mellal and Williams [36, pp. 187-188, Table 7.1]:

Maximize PRODU

where PRODU is described below by line 411 through line 420

subject to

SUMV<=500

where SUMV is described below by line 338 through line 341

SUMW<=800

where SUMW IS DESCRIBED below by line 344 through line 347

SUMM<=600

where SUMM is described below by line 351 through line 354

1<= X(i) <=10, for i=1 through 20 and these Xs are integers;

.5<= X(i) <=1, for i=21, 22, 23, ..., 40.

X(41) through X(43) below are slack variables added.

Whereas line 124 of the earlier edition is 124 NEWN = INT(15000 + RND * 25000), here line 124 is 124 NEWN = INT(18000 + RND * 2000).

 

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)

11 SV(1) = 2: SV(2) = 5: SV(3) = 5: SV(4) = 4: SV(5) = 4: SV(6) = 1: SV(7) = 1: SV(8) = 4: SV(9) = 4: SV(10) = 3: SV(11) = 3: SV(12) = 1: SV(13) = 1: SV(14) = 3: SV(15) = 4: SV(16) = 5: SV(17) = 1: SV(18) = 4: SV(19) = 2: SV(20) = 1

12 SW(1) = 8: SW(2) = 9: SW(3) = 6: SW(4) = 10: SW(5) = 8: SW(6) = 9: SW(7) = 9: SW(8) = 7: SW(9) = 9: SW(10) = 8: SW(11) = 9: SW(12) = 8: SW(13) = 7: SW(14) = 10: SW(15) = 6: SW(16) = 7: SW(17) = 7: SW(18) = 8: SW(19) = 9: SW(20) = 9

 

13 AL(1) = .6 / 10 ^ 5: AL(2) = .1 / 10 ^ 5: AL(3) = 1.2 / 10 ^ 5: AL(4) = .3 / 10 ^ 5: AL(5) = 2.9 / 10 ^ 5: AL(6) = 1.7 / 10 ^ 5: AL(7) = 2.6 / 10 ^ 5: AL(8) = 2.5 / 10 ^ 5: AL(9) = 1.3 / 10 ^ 5: AL(10) = 1.8 / 10 ^ 5: AL(11) = 2.4 / 10 ^ 5: AL(12) = 1.3 / 10 ^ 5: AL(13) = 1.2 / 10 ^ 5: AL(14) = 2.1 / 10 ^ 5: AL(15) = .9 / 10 ^ 5: AL(16) = 1.3 / 10 ^ 5: AL(17) = 1.9 / 10 ^ 5: AL(18) = 2.7 / 10 ^ 5: AL(19) = 2.8 / 10 ^ 5: AL(20) = 1.5 / 10 ^ 5

81 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ
90 M = -3E+30

95 FOR J44 = 1 TO 20

97 A(J44) = FIX(1 + RND * 10)

99 NEXT J44

115 FOR J44 = 21 TO 40

117 A(J44) = .5 + RND * .499999##

119 NEXT J44

124 NEWN = INT(18000 + RND * 2000)

 

128 FOR I = 1 TO NEWN

 

129 FOR KKQQ = 1 TO 40
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 5)

 

153 J = 1 + FIX(RND * 40)

 

155 IF J > 20.5 THEN GOTO 156 ELSE GOTO 164

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

 

161 GOTO 169

 

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

 

169 NEXT IPP
326 FOR J44 = 1 TO 20

327 X(J44) = CINT(X(J44))

 

328 IF X(J44) < 1 THEN 1670

329 IF X(J44) > 10 THEN 1670
331 NEXT J44

332 FOR J44 = 21 TO 40

 

333 IF X(J44) < .5## THEN 1670

334 IF X(J44) > .999999## THEN 1670
335 NEXT J44

 

336 REM X(41) = 600 - 2 * X(1) ^ 2 - 5 * X(2) ^ 2 - 5 * X(3) ^ 2 - 4 * X(4) ^ 2 - 4 * X(5) ^ 2 - 1 * X(6) ^ 2 - 1 * X(7) ^ 2 - 4 * X(8) ^ 2 - 4 * X(9) ^ 2 - 3 * X(10) ^ 2 - 3 * X(11) ^ 2 - 1 * X(12) ^ 2 - 1 * X(13) ^ 2 - 3 * X(14) ^ 2 - 4 * X(15) ^ 2 - 5 * X(16) ^ 2 - 1 * X(17) ^ 2 - 4 * X(18) ^ 2 - 2 * X(19) ^ 2 - 1 * X(20) ^ 2

 

337 REM X(42) = 900 - 8 * X(1) * EXP(X(1) / 4) - (9) * X(2) * EXP(X(2) / 4) - (6) * X(3) * EXP(X(3) / 4) - (10) * X(4) * EXP(X(4) / 4) - (8) * X(5) * EXP(X(5) / 4) - 9 * X(6) * EXP(X(6) / 4) - (9) * X(7) * EXP(X(7) / 4) - (7) * X(8) * EXP(X(8) / 4) - (9) * X(9) * EXP(X(9) / 4) - (8) * X(10) * EXP(X(10) / 4) - 9 * X(11) * EXP(X(11) / 4) - (8) * X(12) * EXP(X(12) / 4) - (7) * X(13) * EXP(X(13) / 4) - (10) * X(14) * EXP(X(14) / 4) - (6) * X(15) * EXP(X(15) / 4) - 7 * X(16) * EXP(X(16) / 4) - (7) * X(17) * EXP(X(17) / 4) - (8) * X(18) * EXP(X(18) / 4) - (9) * X(19) * EXP(X(19) / 4) - (9) * X(20) * EXP(X(20) / 4)

 

 

338 SUMV = 0
339 FOR J44 = 1 TO 20

 

340 SUMV = SUMV + SV(J44) * X(J44) ^ 2

 

341 NEXT J44
343 X(41) = 500 - SUMV

 

 

344 SUMW = 0
345 FOR J44 = 1 TO 20

 

346 SUMW = SUMW + SW(J44) * X(J44) * EXP(X(J44) / 4)

347 NEXT J44

349 X(42) = 800 - SUMW

 

351 SUMM = 0
352 FOR J44 = 1 TO 20

 

353 SUMM = SUMM + AL(J44) * ((-1000 / LOG(X(J44 + 20))) ^ 1.5) * (X(J44) + EXP(X(J44) / 4))

 

354 NEXT J44
355 X(43) = 600 - SUMM

 

356 FOR J44 = 41 TO 43

 

357 IF X(J44) < 0## THEN X(J44) = X(J44) ELSE X(J44) = 0##

359 NEXT J44

411 PRODU = 1
415 FOR J44 = 1 TO 20

 

418 PRODU = PRODU * (1 - ((1 - X(J44 + 20)) ^ X(J44)))

420 NEXT J44

 

444 PDU = PRODU + 1000000 * (X(41) + X(42) + X(43))

 

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 43

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < .814 THEN 1999

1904 PRINT A(1), A(2), A(3), A(4), A(5)

1907 PRINT A(6), A(7), A(8), A(9), A(10)
1908 PRINT A(11), A(12), A(13), A(14), A(15)

 

1927 PRINT A(16), A(17), A(18), A(19), A(20)

 

1944 PRINT A(21), A(22), A(23), A(24), A(25)

1947 PRINT A(26), A(27), A(28), A(29), A(30)
1948 PRINT A(31), A(32), A(33), A(34), A(35)

1949 PRINT A(36), A(37), A(38), A(39), A(40)

1950 PRINT A(41), A(42), A(43), M, JJJJ, NEWN

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [48]. Each of the two candidate solutions of a single run through JJJJ= -31290 is either partly shown or fully shown below; one notes line 1889 above is 1889 IF M < .814 THEN 1999.

.
.
.
.7907428079479922          .8775501954862495
0          0          0          .814570385760469
-31298          19576

 

2          2          3          2          3
2          3          3          2          3
3          2          3          2          2
3          3          3          3          2
.9069685105991184          .94214699219746          .8195226343476363
.919218435986863          .7794731767229812
.8776110482788236          .786168638383807          .789569513855273
.8715705414774302          .7998417501586546
.7771629261754043          .8839798515775967          .8181484385415423
.8681280396148318          .9019711940360983
.8140657710685832          .7901343022350663          .7908939254855466
.7854312372618298          .8767042592557286
0          0          0          .8168415910058556
-31290          19224

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 [48], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31290 was 3 hours and 3 minutes, total, including the time for “Creating .EXE file." One can compare the computational results above with those in Mellal and Williams [36, p. 193, Table 7.2].

Acknowledgment

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

References

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[24] 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

[25] 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

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] 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.

[33] 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.

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

[35] 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.

[36] 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.

[37] 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.

[38] 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.

[39] 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.

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

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

[42] 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.

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

[44] 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.

[45] 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.

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

[47] 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.

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

[49] 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.

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

 

This Mixed-Integer Nonlinear Programming (MINLP) Solver Applied to a Large-Scale Reliability-Redundancy Allocation Problem, Fourth Edition

                                                                  Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Zhang, Hu, Shao, Li, and Wang [48, p. 2860, Table 3, P5] and from Mellal and Zio [34, p. 218, Table 5]:

Maximize PROD * (1 - (1 - X(31)) ^ X(11)) * (1 - (1 - X(32)) ^ X(12)) * (1 - (1 - X(33)) ^ X(13)) * (1 - (1 - X(34)) ^ X(14)) * (1 - (1 - X(35)) ^ X(15)) * (1 - (1 - X(36)) ^ X(16)) * (1 - (1 - X(37)) ^ X(17)) * (1 - (1 - X(38)) ^ X(18)) * (1 - (1 - X(39)) ^ X(19)) * (1 - (1 - X(40)) ^ X(20))

where PROD = (1 - (1 - X(21)) ^ X(1)) * (1 - (1 - X(22)) ^ X(2)) * (1 - (1 - X(23)) ^ X(3)) * (1 - (1 - X(24)) ^ X(4)) * (1 - (1 - X(25)) ^ X(5)) * (1 - (1 - X(26)) ^ X(6)) * (1 - (1 - X(27)) ^ X(7)) * (1 - (1 - X(28)) ^ X(8)) * (1 - (1 - X(29)) ^ X(9)) * (1 - (1 - X(30)) ^ X(10))

subject to

2 * X(1) ^ 2 + 5 * X(2) ^ 2 + 5 * X(3) ^ 2 + 4 * X(4) ^ 2 + 4 * X(5) ^ 2 + 1 * X(6) ^ 2 + 1 * X(7) ^ 2 + 4 * X(8) ^ 2 + 4 * X(9) ^ 2 + 3 * X(10) ^ 2 + 3 * X(11) ^ 2 + 1 * X(12) ^ 2 + 1 * X(13) ^ 2 + 3 * X(14) ^ 2 + 4 * X(15) ^ 2 + 5 * X(16) ^ 2 + 1 * X(17) ^ 2 + 4 * X(18) ^ 2 + 2 * X(19) ^ 2 + 1 * X(20) ^ 2<=600

8 * X(1) * EXP(X(1) / 4) + (9) * X(2) * EXP(X(2) / 4) + (6) * X(3) * EXP(X(3) / 4) + (10) * X(4) * EXP(X(4) / 4) + (8) * X(5) * EXP(X(5) / 4) + 9 * X(6) * EXP(X(6) / 4) + (9) * X(7) * EXP(X(7) / 4) + (7) * X(8) * EXP(X(8) / 4) + (9) * X(9) * EXP(X(9) / 4) + (8) * X(10) * EXP(X(10) / 4) + 9 * X(11) * EXP(X(11) / 4) + (8) * X(12) * EXP(X(12) / 4) + (7) * X(13) * EXP(X(13) / 4) + (10) * X(14) * EXP(X(14) / 4) + (6) * X(15) * EXP(X(15) / 4) + 7 * X(16) * EXP(X(16) / 4) + (7) * X(17) * EXP(X(17) / 4) + (8) * X(18) * EXP(X(18) / 4) + (9) * X(19) * EXP(X(19) / 4) + (9) * X(20) * EXP(X(20) / 4)<=900

-XADDC - XADDB - XADD + (.6 / 10 ^ 5) * ((-1000 / LOG(X(21))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) + (.1 / 10 ^ 5) * ((-1000 / LOG(X(22))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) + (1.2 / 10 ^ 5) * ((-1000 / LOG(X(23))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) + (.3 / 10 ^ 5) * ((-1000 / LOG(X(24))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) + (2.9 / 10 ^ 5) * ((-1000 / LOG(X(25))) ^ 1.5) * (X(5) + EXP(X(5) / 4))<=700

where XADDC = -(1.3 / 10 ^ 5) * ((-1000 / LOG(X(36))) ^ 1.5) * (X(16) + EXP(X(16) / 4)) - (1.9 / 10 ^ 5) * ((-1000 / LOG(X(37))) ^ 1.5) * (X(17) + EXP(X(17) / 4)) - (2.7 / 10 ^ 5) * ((-1000 / LOG(X(38))) ^ 1.5) * (X(18) + EXP(X(18) / 4)) - (2.8 / 10 ^ 5) * ((-1000 / LOG(X(39))) ^ 1.5) * (X(19) + EXP(X(19) / 4)) - (1.5 / 10 ^ 5) * ((-1000 / LOG(X(40))) ^ 1.5) * (X(20) + EXP(X(20) / 4))

XADDB = -(2.4 / 10 ^ 5) * ((-1000 / LOG(X(31))) ^ 1.5) * (X(11) + EXP(X(11) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(32))) ^ 1.5) * (X(12) + EXP(X(12) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(33))) ^ 1.5) * (X(13) + EXP(X(13) / 4)) - (2.1 / 10 ^ 5) * ((-1000 / LOG(X(34))) ^ 1.5) * (X(14) + EXP(X(14) / 4)) - (.9 / 10 ^ 5) * ((-1000 / LOG(X(35))) ^ 1.5) * (X(15) + EXP(X(15) / 4))

XADD = -(1.7 / 10 ^ 5) * ((-1000 / LOG(X(26))) ^ 1.5) * (X(6) + EXP(X(6) / 4)) - (2.6 / 10 ^ 5) * ((-1000 / LOG(X(27))) ^ 1.5) * (X(7) + EXP(X(7) / 4)) - (2.5 / 10 ^ 5) * ((-1000 / LOG(X(28))) ^ 1.5) * (X(8) + EXP(X(8) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(29))) ^ 1.5) * (X(9) + EXP(X(9) / 4)) - (1.8 / 10 ^ 5) * ((-1000 / LOG(X(30))) ^ 1.5) * (X(10) + EXP(X(10) / 4))

1<= X(i) <=10, for i=1 through 20 and these Xs are integers;

.5<= X(i) <=1, for i=21, 22, 23, ..., 40.

X(41) through X(43) below are slack variables added.

Whereas line 128 of the preceding paper is 128 FOR I = 1 TO 20000, here line 128 is 128 FOR I = 1 TO 30000.

 

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)

81 FOR JJJJ = -32000 TO 32000

 

89 RANDOMIZE JJJJ
90 M = -3E+30

95 FOR J44 = 1 TO 20

97 A(J44) = FIX(1 + RND * 10)

99 NEXT J44

115 FOR J44 = 21 TO 40

117 A(J44) = .5 + RND * .499999##

119 NEXT J44

128 FOR I = 1 TO 30000

 

129 FOR KKQQ = 1 TO 40
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 5)

 

153 j = 1 + FIX(RND * 40)

 

155 IF j > 20.5 THEN GOTO 156 ELSE GOTO 164

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

 

161 GOTO 169

 

164 IF RND < .5 THEN X(j) = A(j) - 1 ELSE X(j) = A(j) + 1

 

169 NEXT IPP
326 FOR J44 = 1 TO 20

327 X(J44) = CINT(X(J44))

 

328 IF X(J44) < 1 THEN 1670

329 IF X(J44) > 10 THEN 1670
331 NEXT J44

336 FOR J44 = 21 TO 40

 

338 IF X(J44) < .5## THEN 1670

339 IF X(J44) > .999999## THEN 1670
340 NEXT J44

 

342 X(41) = 600 - 2 * X(1) ^ 2 - 5 * X(2) ^ 2 - 5 * X(3) ^ 2 - 4 * X(4) ^ 2 - 4 * X(5) ^ 2 - 1 * X(6) ^ 2 - 1 * X(7) ^ 2 - 4 * X(8) ^ 2 - 4 * X(9) ^ 2 - 3 * X(10) ^ 2 - 3 * X(11) ^ 2 - 1 * X(12) ^ 2 - 1 * X(13) ^ 2 - 3 * X(14) ^ 2 - 4 * X(15) ^ 2 - 5 * X(16) ^ 2 - 1 * X(17) ^ 2 - 4 * X(18) ^ 2 - 2 * X(19) ^ 2 - 1 * X(20) ^ 2

 

343 X(42) = 900 - 8 * X(1) * EXP(X(1) / 4) - (9) * X(2) * EXP(X(2) / 4) - (6) * X(3) * EXP(X(3) / 4) - (10) * X(4) * EXP(X(4) / 4) - (8) * X(5) * EXP(X(5) / 4) - 9 * X(6) * EXP(X(6) / 4) - (9) * X(7) * EXP(X(7) / 4) - (7) * X(8) * EXP(X(8) / 4) - (9) * X(9) * EXP(X(9) / 4) - (8) * X(10) * EXP(X(10) / 4) - 9 * X(11) * EXP(X(11) / 4) - (8) * X(12) * EXP(X(12) / 4) - (7) * X(13) * EXP(X(13) / 4) - (10) * X(14) * EXP(X(14) / 4) - (6) * X(15) * EXP(X(15) / 4) - 7 * X(16) * EXP(X(16) / 4) - (7) * X(17) * EXP(X(17) / 4) - (8) * X(18) * EXP(X(18) / 4) - (9) * X(19) * EXP(X(19) / 4) - (9) * X(20) * EXP(X(20) / 4)

 

344 XADDC = -(1.3 / 10 ^ 5) * ((-1000 / LOG(X(36))) ^ 1.5) * (X(16) + EXP(X(16) / 4)) - (1.9 / 10 ^ 5) * ((-1000 / LOG(X(37))) ^ 1.5) * (X(17) + EXP(X(17) / 4)) - (2.7 / 10 ^ 5) * ((-1000 / LOG(X(38))) ^ 1.5) * (X(18) + EXP(X(18) / 4)) - (2.8 / 10 ^ 5) * ((-1000 / LOG(X(39))) ^ 1.5) * (X(19) + EXP(X(19) / 4)) - (1.5 / 10 ^ 5) * ((-1000 / LOG(X(40))) ^ 1.5) * (X(20) + EXP(X(20) / 4))

 

345 XADDB = -(2.4 / 10 ^ 5) * ((-1000 / LOG(X(31))) ^ 1.5) * (X(11) + EXP(X(11) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(32))) ^ 1.5) * (X(12) + EXP(X(12) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(33))) ^ 1.5) * (X(13) + EXP(X(13) / 4)) - (2.1 / 10 ^ 5) * ((-1000 / LOG(X(34))) ^ 1.5) * (X(14) + EXP(X(14) / 4)) - (.9 / 10 ^ 5) * ((-1000 / LOG(X(35))) ^ 1.5) * (X(15) + EXP(X(15) / 4))

 

348 XADD = -(1.7 / 10 ^ 5) * ((-1000 / LOG(X(26))) ^ 1.5) * (X(6) + EXP(X(6) / 4)) - (2.6 / 10 ^ 5) * ((-1000 / LOG(X(27))) ^ 1.5) * (X(7) + EXP(X(7) / 4)) - (2.5 / 10 ^ 5) * ((-1000 / LOG(X(28))) ^ 1.5) * (X(8) + EXP(X(8) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(29))) ^ 1.5) * (X(9) + EXP(X(9) / 4)) - (1.8 / 10 ^ 5) * ((-1000 / LOG(X(30))) ^ 1.5) * (X(10) + EXP(X(10) / 4))

 

350 X(43) = XADDC + XADDB + XADD + 700 - (.6 / 10 ^ 5) * ((-1000 / LOG(X(21))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) - (.1 / 10 ^ 5) * ((-1000 / LOG(X(22))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(23))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) - (.3 / 10 ^ 5) * ((-1000 / LOG(X(24))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) - (2.9 / 10 ^ 5) * ((-1000 / LOG(X(25))) ^ 1.5) * (X(5) + EXP(X(5) / 4))

 

355 FOR J44 = 41 TO 43

 

357 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

359 NEXT J44

 

422 PROD = (1 - (1 - X(21)) ^ X(1)) * (1 - (1 - X(22)) ^ X(2)) * (1 - (1 - X(23)) ^ X(3)) * (1 - (1 - X(24)) ^ X(4)) * (1 - (1 - X(25)) ^ X(5)) * (1 - (1 - X(26)) ^ X(6)) * (1 - (1 - X(27)) ^ X(7)) * (1 - (1 - X(28)) ^ X(8)) * (1 - (1 - X(29)) ^ X(9)) * (1 - (1 - X(30)) ^ X(10))

 

433 PDU = PROD * (1 - (1 - X(31)) ^ X(11)) * (1 - (1 - X(32)) ^ X(12)) * (1 - (1 - X(33)) ^ X(13)) * (1 - (1 - X(34)) ^ X(14)) * (1 - (1 - X(35)) ^ X(15)) * (1 - (1 - X(36)) ^ X(16)) * (1 - (1 - X(37)) ^ X(17)) * (1 - (1 - X(38)) ^ X(18)) * (1 - (1 - X(39)) ^ X(19)) * (1 - (1 - X(40)) ^ X(20)) + 1000000 * (X(41) + X(42) + X(43))

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 43

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < .891 THEN 1999

 

1904 PRINT A(1), A(2), A(3), A(4), A(5)

1907 PRINT A(6), A(7), A(8), A(9), A(10)
1908 PRINT A(11), A(12), A(13), A(14), A(15)

 

1927 PRINT A(16), A(17), A(18), A(19), A(20)

1944 PRINT A(21), A(22), A(23), A(24), A(25)

1947 PRINT A(26), A(27), A(28), A(29), A(30)
1948 PRINT A(31), A(32), A(33), A(34), A(35)

1949 PRINT A(36), A(37), A(38), A(39), A(40)

1950 PRINT A(41), A(42), A(43), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [46]. The three candidate solutions of a single run through JJJJ= -30925 are summarized below:

2          2          3          2          3
3          3          3          2          3
3          3          3          3          3
3          3          3          3          2
.
.
.
0          0          0          .8914396704441848
-31633

 

2          2          3          2          3
3          3          3          2          3
3          3          3          3          3
3          3          3          3          2
.
.
.
0          0          0          .8914860141865848
-31289

 

2          2          3          2          3
3          3          3          2          3
3          3          3          3          3
3          3          3          3          2
.923125398817832             .9522575417680427          .8426220151835963
.9347738052604837          .804949115578434
.8318993375898558          .8102148116771514          .8128144273398658
.9001994010573906          .8259286617188014
.8192437813972897          .8414294193692286          .8411257634893566
.8198136653907575          .8497380430733504
.8354409394150434          .8254578815492992          .8122105889522838
.8070668816681016          .8974909671008575
0          0          0          .8915961374457698
-30925

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 [46], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -30925 was 5 hours and a half, total, including the time for “Creating .EXE file." One can compare the computational results above with those in Mellal and Zio [34, p. 224, Table 14].

Acknowledgment

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

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[40] 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.

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

[42] 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.

[43] 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.

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

[45] 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.

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

[47] 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.

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

This Mixed-Integer Nonlinear Programming (MINLP) Solver Applied to a Large Scale Reliability-Redundancy Allocation Problem, Third Edition

                                                                      Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Mellal and Zio [34, p. 218, Table 5]:

Maximize PROD * (1 - (1 - X(31)) ^ X(11)) * (1 - (1 - X(32)) ^ X(12)) * (1 - (1 - X(33)) ^ X(13)) * (1 - (1 - X(34)) ^ X(14)) * (1 - (1 - X(35)) ^ X(15)) * (1 - (1 - X(36)) ^ X(16)) * (1 - (1 - X(37)) ^ X(17)) * (1 - (1 - X(38)) ^ X(18)) * (1 - (1 - X(39)) ^ X(19)) * (1 - (1 - X(40)) ^ X(20))

where PROD = (1 - (1 - X(21)) ^ X(1)) * (1 - (1 - X(22)) ^ X(2)) * (1 - (1 - X(23)) ^ X(3)) * (1 - (1 - X(24)) ^ X(4)) * (1 - (1 - X(25)) ^ X(5)) * (1 - (1 - X(26)) ^ X(6)) * (1 - (1 - X(27)) ^ X(7)) * (1 - (1 - X(28)) ^ X(8)) * (1 - (1 - X(29)) ^ X(9)) * (1 - (1 - X(30)) ^ X(10))

subject to

2 * X(1) ^ 2 + 5 * X(2) ^ 2 + 5 * X(3) ^ 2 + 4 * X(4) ^ 2 + 4 * X(5) ^ 2 + 1 * X(6) ^ 2 + 1 * X(7) ^ 2 + 4 * X(8) ^ 2 + 4 * X(9) ^ 2 + 3 * X(10) ^ 2 + 3 * X(11) ^ 2 + 1 * X(12) ^ 2 + 1 * X(13) ^ 2 + 3 * X(14) ^ 2 + 4 * X(15) ^ 2 + 5 * X(16) ^ 2 + 1 * X(17) ^ 2 + 4 * X(18) ^ 2 + 2 * X(19) ^ 2 + 1 * X(20) ^ 2<=600

8 * X(1) * EXP(X(1) / 4) + (9) * X(2) * EXP(X(2) / 4) + (6) * X(3) * EXP(X(3) / 4) + (10) * X(4) * EXP(X(4) / 4) + (8) * X(5) * EXP(X(5) / 4) + 9 * X(6) * EXP(X(6) / 4) + (9) * X(7) * EXP(X(7) / 4) + (7) * X(8) * EXP(X(8) / 4) + (9) * X(9) * EXP(X(9) / 4) + (8) * X(10) * EXP(X(10) / 4) + 9 * X(11) * EXP(X(11) / 4) + (8) * X(12) * EXP(X(12) / 4) + (7) * X(13) * EXP(X(13) / 4) + (10) * X(14) * EXP(X(14) / 4) + (6) * X(15) * EXP(X(15) / 4) + 7 * X(16) * EXP(X(16) / 4) + (7) * X(17) * EXP(X(17) / 4) + (8) * X(18) * EXP(X(18) / 4) + (9) * X(19) * EXP(X(19) / 4) + (9) * X(20) * EXP(X(20) / 4)<=900

-XADDC - XADDB - XADD + (.6 / 10 ^ 5) * ((-1000 / LOG(X(21))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) + (.1 / 10 ^ 5) * ((-1000 / LOG(X(22))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) + (1.2 / 10 ^ 5) * ((-1000 / LOG(X(23))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) + (.3 / 10 ^ 5) * ((-1000 / LOG(X(24))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) + (2.9 / 10 ^ 5) * ((-1000 / LOG(X(25))) ^ 1.5) * (X(5) + EXP(X(5) / 4))<=700

 

where XADDC = -(1.3 / 10 ^ 5) * ((-1000 / LOG(X(36))) ^ 1.5) * (X(16) + EXP(X(16) / 4)) - (1.9 / 10 ^ 5) * ((-1000 / LOG(X(37))) ^ 1.5) * (X(17) + EXP(X(17) / 4)) - (2.7 / 10 ^ 5) * ((-1000 / LOG(X(38))) ^ 1.5) * (X(18) + EXP(X(18) / 4)) - (2.8 / 10 ^ 5) * ((-1000 / LOG(X(39))) ^ 1.5) * (X(19) + EXP(X(19) / 4)) - (1.5 / 10 ^ 5) * ((-1000 / LOG(X(40))) ^ 1.5) * (X(20) + EXP(X(20) / 4))

XADDB = -(2.4 / 10 ^ 5) * ((-1000 / LOG(X(31))) ^ 1.5) * (X(11) + EXP(X(11) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(32))) ^ 1.5) * (X(12) + EXP(X(12) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(33))) ^ 1.5) * (X(13) + EXP(X(13) / 4)) - (2.1 / 10 ^ 5) * ((-1000 / LOG(X(34))) ^ 1.5) * (X(14) + EXP(X(14) / 4)) - (.9 / 10 ^ 5) * ((-1000 / LOG(X(35))) ^ 1.5) * (X(15) + EXP(X(15) / 4))

XADD = -(1.7 / 10 ^ 5) * ((-1000 / LOG(X(26))) ^ 1.5) * (X(6) + EXP(X(6) / 4)) - (2.6 / 10 ^ 5) * ((-1000 / LOG(X(27))) ^ 1.5) * (X(7) + EXP(X(7) / 4)) - (2.5 / 10 ^ 5) * ((-1000 / LOG(X(28))) ^ 1.5) * (X(8) + EXP(X(8) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(29))) ^ 1.5) * (X(9) + EXP(X(9) / 4)) - (1.8 / 10 ^ 5) * ((-1000 / LOG(X(30))) ^ 1.5) * (X(10) + EXP(X(10) / 4))

1<= X(i) <=10, for i=1 through 20 and these Xs are integers;

.5<= X(i) <=1, for i=21, 22, 23, ..., 40.

X(41) through X(43) below are slack variables added.

Whereas line 128 of the preceding paper is 128 FOR I = 1 TO 35000, here line 128 is 128 FOR I = 1 TO 20000.

 

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)

81 FOR JJJJ = -32000 TO 32000

 

89 RANDOMIZE JJJJ
90 M = -3E+30

95 FOR J44 = 1 TO 20

97 A(J44) = FIX(1 + RND * 10)

99 NEXT J44

115 FOR J44 = 21 TO 40

117 A(J44) = .5 + RND * .499999##

119 NEXT J44

128 FOR I = 1 TO 20000

 

129 FOR KKQQ = 1 TO 40
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 5)

 

153 j = 1 + FIX(RND * 40)

 

155 IF j > 20.5 THEN GOTO 156 ELSE GOTO 164

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

 

161 GOTO 169

 

164 IF RND < .5 THEN X(j) = A(j) - 1 ELSE X(j) = A(j) + 1

 

169 NEXT IPP
326 FOR J44 = 1 TO 20

327 X(J44) = CINT(X(J44))

 

328 IF X(J44) < 1 THEN 1670

329 IF X(J44) > 10 THEN 1670
331 NEXT J44

336 FOR J44 = 21 TO 40

 

338 IF X(J44) < .5## THEN 1670

339 IF X(J44) > .999999## THEN 1670
340 NEXT J44

 

342 X(41) = 600 - 2 * X(1) ^ 2 - 5 * X(2) ^ 2 - 5 * X(3) ^ 2 - 4 * X(4) ^ 2 - 4 * X(5) ^ 2 - 1 * X(6) ^ 2 - 1 * X(7) ^ 2 - 4 * X(8) ^ 2 - 4 * X(9) ^ 2 - 3 * X(10) ^ 2 - 3 * X(11) ^ 2 - 1 * X(12) ^ 2 - 1 * X(13) ^ 2 - 3 * X(14) ^ 2 - 4 * X(15) ^ 2 - 5 * X(16) ^ 2 - 1 * X(17) ^ 2 - 4 * X(18) ^ 2 - 2 * X(19) ^ 2 - 1 * X(20) ^ 2

 

343 X(42) = 900 - 8 * X(1) * EXP(X(1) / 4) - (9) * X(2) * EXP(X(2) / 4) - (6) * X(3) * EXP(X(3) / 4) - (10) * X(4) * EXP(X(4) / 4) - (8) * X(5) * EXP(X(5) / 4) - 9 * X(6) * EXP(X(6) / 4) - (9) * X(7) * EXP(X(7) / 4) - (7) * X(8) * EXP(X(8) / 4) - (9) * X(9) * EXP(X(9) / 4) - (8) * X(10) * EXP(X(10) / 4) - 9 * X(11) * EXP(X(11) / 4) - (8) * X(12) * EXP(X(12) / 4) - (7) * X(13) * EXP(X(13) / 4) - (10) * X(14) * EXP(X(14) / 4) - (6) * X(15) * EXP(X(15) / 4) - 7 * X(16) * EXP(X(16) / 4) - (7) * X(17) * EXP(X(17) / 4) - (8) * X(18) * EXP(X(18) / 4) - (9) * X(19) * EXP(X(19) / 4) - (9) * X(20) * EXP(X(20) / 4)

 

344 XADDC = -(1.3 / 10 ^ 5) * ((-1000 / LOG(X(36))) ^ 1.5) * (X(16) + EXP(X(16) / 4)) - (1.9 / 10 ^ 5) * ((-1000 / LOG(X(37))) ^ 1.5) * (X(17) + EXP(X(17) / 4)) - (2.7 / 10 ^ 5) * ((-1000 / LOG(X(38))) ^ 1.5) * (X(18) + EXP(X(18) / 4)) - (2.8 / 10 ^ 5) * ((-1000 / LOG(X(39))) ^ 1.5) * (X(19) + EXP(X(19) / 4)) - (1.5 / 10 ^ 5) * ((-1000 / LOG(X(40))) ^ 1.5) * (X(20) + EXP(X(20) / 4))

 

345 XADDB = -(2.4 / 10 ^ 5) * ((-1000 / LOG(X(31))) ^ 1.5) * (X(11) + EXP(X(11) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(32))) ^ 1.5) * (X(12) + EXP(X(12) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(33))) ^ 1.5) * (X(13) + EXP(X(13) / 4)) - (2.1 / 10 ^ 5) * ((-1000 / LOG(X(34))) ^ 1.5) * (X(14) + EXP(X(14) / 4)) - (.9 / 10 ^ 5) * ((-1000 / LOG(X(35))) ^ 1.5) * (X(15) + EXP(X(15) / 4))

 

348 XADD = -(1.7 / 10 ^ 5) * ((-1000 / LOG(X(26))) ^ 1.5) * (X(6) + EXP(X(6) / 4)) - (2.6 / 10 ^ 5) * ((-1000 / LOG(X(27))) ^ 1.5) * (X(7) + EXP(X(7) / 4)) - (2.5 / 10 ^ 5) * ((-1000 / LOG(X(28))) ^ 1.5) * (X(8) + EXP(X(8) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(29))) ^ 1.5) * (X(9) + EXP(X(9) / 4)) - (1.8 / 10 ^ 5) * ((-1000 / LOG(X(30))) ^ 1.5) * (X(10) + EXP(X(10) / 4))

 

350 X(43) = XADDC + XADDB + XADD + 700 - (.6 / 10 ^ 5) * ((-1000 / LOG(X(21))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) - (.1 / 10 ^ 5) * ((-1000 / LOG(X(22))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(23))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) - (.3 / 10 ^ 5) * ((-1000 / LOG(X(24))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) - (2.9 / 10 ^ 5) * ((-1000 / LOG(X(25))) ^ 1.5) * (X(5) + EXP(X(5) / 4))

 

355 FOR J44 = 41 TO 43

 

357 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

359 NEXT J44

 

422 PROD = (1 - (1 - X(21)) ^ X(1)) * (1 - (1 - X(22)) ^ X(2)) * (1 - (1 - X(23)) ^ X(3)) * (1 - (1 - X(24)) ^ X(4)) * (1 - (1 - X(25)) ^ X(5)) * (1 - (1 - X(26)) ^ X(6)) * (1 - (1 - X(27)) ^ X(7)) * (1 - (1 - X(28)) ^ X(8)) * (1 - (1 - X(29)) ^ X(9)) * (1 - (1 - X(30)) ^ X(10))

 

433 PDU = PROD * (1 - (1 - X(31)) ^ X(11)) * (1 - (1 - X(32)) ^ X(12)) * (1 - (1 - X(33)) ^ X(13)) * (1 - (1 - X(34)) ^ X(14)) * (1 - (1 - X(35)) ^ X(15)) * (1 - (1 - X(36)) ^ X(16)) * (1 - (1 - X(37)) ^ X(17)) * (1 - (1 - X(38)) ^ X(18)) * (1 - (1 - X(39)) ^ X(19)) * (1 - (1 - X(40)) ^ X(20)) + 1000000 * (X(41) + X(42) + X(43))

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 43

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < .890 THEN 1999

 

1904 PRINT A(1), A(2), A(3), A(4), A(5)

1907 PRINT A(6), A(7), A(8), A(9), A(10)
1908 PRINT A(11), A(12), A(13), A(14), A(15)

 

1927 PRINT A(16), A(17), A(18), A(19), A(20)

1944 PRINT A(21), A(22), A(23), A(24), A(25)

1947 PRINT A(26), A(27), A(28), A(29), A(30)
1948 PRINT A(31), A(32), A(33), A(34), A(35)

1949 PRINT A(36), A(37), A(38), A(39), A(40)

1950 PRINT A(41), A(41), A(43), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [46]. The two candidate solutions of a single run through JJJJ= -31902 are summarized below:

2 2 3 2 3
2 3 3 2 3
3 3 3 3 3
3 3 3 3 3
.
.
.
0 0 0 .8907301520350084
-31967

2          2          3          2          3
3          3          3          2          3
3          3          3          3          3
3          3          3          3          2
.9212839994325944          .9522288725214245          .8406647171629444
.9340729027257265          .8075354831307674
.8291185516803236          .811792768247496          .8099482598069661
.903556263195004          .823815416082853 .
.823347366046746          .8394017134699163          .8378604946963304
.8221568269844564          .8483111194863011
.8413178585213724          .8229276836078362          .8048764715550408
.8076481217122752          .8996468707728227
0          0          0          .891497978272561
-31902

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 [46], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31902 was 25 minutes and 20 seconds, total, including the time for “Creating .EXE file." One can compare the computational results above with those in Mellal and Zio [34, p. 224, Table 14].

Acknowledgment

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

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Dual Gaps in Engineering System Optimization. Journal of Optimization Theory and Applications, Vol.15, No. 3, pp. 285-309, 1975.

[43] 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.

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

[45] 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.

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

[47] 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.

 

This Mixed-Integer Nonlinear Programming (MINLP) Solver Applied to a Large Scale Reliability-Redundancy Allocation Problem, Second Edition

                                                                                  Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Mellal and Zio [34, p. 218, Table 5]:

Maximize PROD * (1 - (1 - X(31)) ^ X(11)) * (1 - (1 - X(32)) ^ X(12)) * (1 - (1 - X(33)) ^ X(13)) * (1 - (1 - X(34)) ^ X(14)) * (1 - (1 - X(35)) ^ X(15)) * (1 - (1 - X(36)) ^ X(16)) * (1 - (1 - X(37)) ^ X(17)) * (1 - (1 - X(38)) ^ X(18)) * (1 - (1 - X(39)) ^ X(19)) * (1 - (1 - X(40)) ^ X(20))

where PROD = (1 - (1 - X(21)) ^ X(1)) * (1 - (1 - X(22)) ^ X(2)) * (1 - (1 - X(23)) ^ X(3)) * (1 - (1 - X(24)) ^ X(4)) * (1 - (1 - X(25)) ^ X(5)) * (1 - (1 - X(26)) ^ X(6)) * (1 - (1 - X(27)) ^ X(7)) * (1 - (1 - X(28)) ^ X(8)) * (1 - (1 - X(29)) ^ X(9)) * (1 - (1 - X(30)) ^ X(10))

subject to

2 * X(1) ^ 2 + 5 * X(2) ^ 2 + 5 * X(3) ^ 2 + 4 * X(4) ^ 2 + 4 * X(5) ^ 2 + 1 * X(6) ^ 2 + 1 * X(7) ^ 2 + 4 * X(8) ^ 2 + 4 * X(9) ^ 2 + 3 * X(10) ^ 2 + 3 * X(11) ^ 2 + 1 * X(12) ^ 2 + 1 * X(13) ^ 2 + 3 * X(14) ^ 2 + 4 * X(15) ^ 2 + 5 * X(16) ^ 2 + 1 * X(17) ^ 2 + 4 * X(18) ^ 2 + 2 * X(19) ^ 2 + 1 * X(20) ^ 2<=600

8 * X(1) * EXP(X(1) / 4) + (9) * X(2) * EXP(X(2) / 4) + (6) * X(3) * EXP(X(3) / 4) + (10) * X(4) * EXP(X(4) / 4) + (8) * X(5) * EXP(X(5) / 4) + 9 * X(6) * EXP(X(6) / 4) + (9) * X(7) * EXP(X(7) / 4) + (7) * X(8) * EXP(X(8) / 4) + (9) * X(9) * EXP(X(9) / 4) + (8) * X(10) * EXP(X(10) / 4) + 9 * X(11) * EXP(X(11) / 4) + (8) * X(12) * EXP(X(12) / 4) + (7) * X(13) * EXP(X(13) / 4) + (10) * X(14) * EXP(X(14) / 4) + (6) * X(15) * EXP(X(15) / 4) + 7 * X(16) * EXP(X(16) / 4) + (7) * X(17) * EXP(X(17) / 4) + (8) * X(18) * EXP(X(18) / 4) + (9) * X(19) * EXP(X(19) / 4) + (9) * X(20) * EXP(X(20) / 4)<=900

-XADDC - XADDB - XADD + (.6 / 10 ^ 5) * ((-1000 / LOG(X(21))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) + (.1 / 10 ^ 5) * ((-1000 / LOG(X(22))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) + (1.2 / 10 ^ 5) * ((-1000 / LOG(X(23))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) + (.3 / 10 ^ 5) * ((-1000 / LOG(X(24))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) + (2.9 / 10 ^ 5) * ((-1000 / LOG(X(25))) ^ 1.5) * (X(5) + EXP(X(5) / 4))<=700

 

where XADDC = -(1.3 / 10 ^ 5) * ((-1000 / LOG(X(36))) ^ 1.5) * (X(16) + EXP(X(16) / 4)) - (1.9 / 10 ^ 5) * ((-1000 / LOG(X(37))) ^ 1.5) * (X(17) + EXP(X(17) / 4)) - (2.7 / 10 ^ 5) * ((-1000 / LOG(X(38))) ^ 1.5) * (X(18) + EXP(X(18) / 4)) - (2.8 / 10 ^ 5) * ((-1000 / LOG(X(39))) ^ 1.5) * (X(19) + EXP(X(19) / 4)) - (1.5 / 10 ^ 5) * ((-1000 / LOG(X(40))) ^ 1.5) * (X(20) + EXP(X(20) / 4))

XADDB = -(2.4 / 10 ^ 5) * ((-1000 / LOG(X(31))) ^ 1.5) * (X(11) + EXP(X(11) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(32))) ^ 1.5) * (X(12) + EXP(X(12) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(33))) ^ 1.5) * (X(13) + EXP(X(13) / 4)) - (2.1 / 10 ^ 5) * ((-1000 / LOG(X(34))) ^ 1.5) * (X(14) + EXP(X(14) / 4)) - (.9 / 10 ^ 5) * ((-1000 / LOG(X(35))) ^ 1.5) * (X(15) + EXP(X(15) / 4))

XADD = -(1.7 / 10 ^ 5) * ((-1000 / LOG(X(26))) ^ 1.5) * (X(6) + EXP(X(6) / 4)) - (2.6 / 10 ^ 5) * ((-1000 / LOG(X(27))) ^ 1.5) * (X(7) + EXP(X(7) / 4)) - (2.5 / 10 ^ 5) * ((-1000 / LOG(X(28))) ^ 1.5) * (X(8) + EXP(X(8) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(29))) ^ 1.5) * (X(9) + EXP(X(9) / 4)) - (1.8 / 10 ^ 5) * ((-1000 / LOG(X(30))) ^ 1.5) * (X(10) + EXP(X(10) / 4))

1<= X(i) <=10, for i=1 through 20 and these Xs are integers;

.5<= X(i) <=1, for i=21, 22, 23, ..., 40.

X(41) through X(43) below are slack variables added.

Whereas line 128 of the earlier edition is 128 FOR I = 1 TO 40000, here line 128 is 128 FOR I = 1 TO 35000 .

 

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)

81 FOR JJJJ = -32000 TO 32000

 

89 RANDOMIZE JJJJ
90 M = -3E+30

95 FOR J44 = 1 TO 20

97 A(J44) = FIX(1 + RND * 10)

99 NEXT J44

115 FOR J44 = 21 TO 40

117 A(J44) = .5 + RND * .499999##

119 NEXT J44

128 FOR I = 1 TO 35000

 

129 FOR KKQQ = 1 TO 40
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 5)

 

153 j = 1 + FIX(RND * 40)

 

155 IF j > 20.5 THEN GOTO 156 ELSE GOTO 164

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

 

161 GOTO 169

 

164 IF RND < .5 THEN X(j) = A(j) - 1 ELSE X(j) = A(j) + 1

 

169 NEXT IPP
326 FOR J44 = 1 TO 20

327 X(J44) = CINT(X(J44))

 

328 IF X(J44) < 1 THEN 1670

329 IF X(J44) > 10 THEN 1670
331 NEXT J44

336 FOR J44 = 21 TO 40

 

338 IF X(J44) < .5## THEN 1670

339 IF X(J44) > .999999## THEN 1670
340 NEXT J44

 

342 X(41) = 600 - 2 * X(1) ^ 2 - 5 * X(2) ^ 2 - 5 * X(3) ^ 2 - 4 * X(4) ^ 2 - 4 * X(5) ^ 2 - 1 * X(6) ^ 2 - 1 * X(7) ^ 2 - 4 * X(8) ^ 2 - 4 * X(9) ^ 2 - 3 * X(10) ^ 2 - 3 * X(11) ^ 2 - 1 * X(12) ^ 2 - 1 * X(13) ^ 2 - 3 * X(14) ^ 2 - 4 * X(15) ^ 2 - 5 * X(16) ^ 2 - 1 * X(17) ^ 2 - 4 * X(18) ^ 2 - 2 * X(19) ^ 2 - 1 * X(20) ^ 2

 

343 X(42) = 900 - 8 * X(1) * EXP(X(1) / 4) - (9) * X(2) * EXP(X(2) / 4) - (6) * X(3) * EXP(X(3) / 4) - (10) * X(4) * EXP(X(4) / 4) - (8) * X(5) * EXP(X(5) / 4) - 9 * X(6) * EXP(X(6) / 4) - (9) * X(7) * EXP(X(7) / 4) - (7) * X(8) * EXP(X(8) / 4) - (9) * X(9) * EXP(X(9) / 4) - (8) * X(10) * EXP(X(10) / 4) - 9 * X(11) * EXP(X(11) / 4) - (8) * X(12) * EXP(X(12) / 4) - (7) * X(13) * EXP(X(13) / 4) - (10) * X(14) * EXP(X(14) / 4) - (6) * X(15) * EXP(X(15) / 4) - 7 * X(16) * EXP(X(16) / 4) - (7) * X(17) * EXP(X(17) / 4) - (8) * X(18) * EXP(X(18) / 4) - (9) * X(19) * EXP(X(19) / 4) - (9) * X(20) * EXP(X(20) / 4)

 

344 XADDC = -(1.3 / 10 ^ 5) * ((-1000 / LOG(X(36))) ^ 1.5) * (X(16) + EXP(X(16) / 4)) - (1.9 / 10 ^ 5) * ((-1000 / LOG(X(37))) ^ 1.5) * (X(17) + EXP(X(17) / 4)) - (2.7 / 10 ^ 5) * ((-1000 / LOG(X(38))) ^ 1.5) * (X(18) + EXP(X(18) / 4)) - (2.8 / 10 ^ 5) * ((-1000 / LOG(X(39))) ^ 1.5) * (X(19) + EXP(X(19) / 4)) - (1.5 / 10 ^ 5) * ((-1000 / LOG(X(40))) ^ 1.5) * (X(20) + EXP(X(20) / 4))

 

345 XADDB = -(2.4 / 10 ^ 5) * ((-1000 / LOG(X(31))) ^ 1.5) * (X(11) + EXP(X(11) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(32))) ^ 1.5) * (X(12) + EXP(X(12) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(33))) ^ 1.5) * (X(13) + EXP(X(13) / 4)) - (2.1 / 10 ^ 5) * ((-1000 / LOG(X(34))) ^ 1.5) * (X(14) + EXP(X(14) / 4)) - (.9 / 10 ^ 5) * ((-1000 / LOG(X(35))) ^ 1.5) * (X(15) + EXP(X(15) / 4))

 

348 XADD = -(1.7 / 10 ^ 5) * ((-1000 / LOG(X(26))) ^ 1.5) * (X(6) + EXP(X(6) / 4)) - (2.6 / 10 ^ 5) * ((-1000 / LOG(X(27))) ^ 1.5) * (X(7) + EXP(X(7) / 4)) - (2.5 / 10 ^ 5) * ((-1000 / LOG(X(28))) ^ 1.5) * (X(8) + EXP(X(8) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(29))) ^ 1.5) * (X(9) + EXP(X(9) / 4)) - (1.8 / 10 ^ 5) * ((-1000 / LOG(X(30))) ^ 1.5) * (X(10) + EXP(X(10) / 4))

 

350 X(43) = XADDC + XADDB + XADD + 700 - (.6 / 10 ^ 5) * ((-1000 / LOG(X(21))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) - (.1 / 10 ^ 5) * ((-1000 / LOG(X(22))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(23))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) - (.3 / 10 ^ 5) * ((-1000 / LOG(X(24))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) - (2.9 / 10 ^ 5) * ((-1000 / LOG(X(25))) ^ 1.5) * (X(5) + EXP(X(5) / 4))

 

355 FOR J44 = 41 TO 43

 

357 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

 

359 NEXT J44

 

422 PROD = (1 - (1 - X(21)) ^ X(1)) * (1 - (1 - X(22)) ^ X(2)) * (1 - (1 - X(23)) ^ X(3)) * (1 - (1 - X(24)) ^ X(4)) * (1 - (1 - X(25)) ^ X(5)) * (1 - (1 - X(26)) ^ X(6)) * (1 - (1 - X(27)) ^ X(7)) * (1 - (1 - X(28)) ^ X(8)) * (1 - (1 - X(29)) ^ X(9)) * (1 - (1 - X(30)) ^ X(10))

 

433 PDU = PROD * (1 - (1 - X(31)) ^ X(11)) * (1 - (1 - X(32)) ^ X(12)) * (1 - (1 - X(33)) ^ X(13)) * (1 - (1 - X(34)) ^ X(14)) * (1 - (1 - X(35)) ^ X(15)) * (1 - (1 - X(36)) ^ X(16)) * (1 - (1 - X(37)) ^ X(17)) * (1 - (1 - X(38)) ^ X(18)) * (1 - (1 - X(39)) ^ X(19)) * (1 - (1 - X(40)) ^ X(20)) + 1000000 * (X(41) + X(42) + X(43))

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 43

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < .890 THEN 1999

 

1904 PRINT A(1), A(2), A(3), A(4), A(5)

1907 PRINT A(6), A(7), A(8), A(9), A(10)
1908 PRINT A(11), A(12), A(13), A(14), A(15)

 

1927 PRINT A(16), A(17), A(18), A(19), A(20)

1944 PRINT A(21), A(22), A(23), A(24), A(25)

1947 PRINT A(26), A(27), A(28), A(29), A(30)
1948 PRINT A(31), A(32), A(33), A(34), A(35)

1949 PRINT A(36), A(37), A(38), A(39), A(40)

1950 PRINT A(41), A(41), A(43), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [46]. The two candidate solutions of a single run through JJJJ= -31845 are summarized below:

2          2          3          2          3
2          3          3          2          3
3          3          3          3          3
3          3          3          3          3
...
...
...
...
...
...
...
.8061256745458184          .8291052553502903
0          0          0          .8908194194839857
-31848

2          2          3          2          3
3          3          3          2          3
3          3          3          3          3
3          3          3          3          2
.9239328279432463          .9510848563541263          .838319316771354
.9364877928306918          .8073834812533437
.8284037512635893          .8099425787407948          .8172118422647646
.9002812774412741          .8259188335780018 
.818899422179778            .8402274885282115            .841507732480954
.8178280635282684          .8483294690073486
.8370996040693713          .8198974675173035          .8084115030140404
.8101975610209331          .8997980213081323
0          0          0          .8915402417162928
-31845

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 [46], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31845 was 53 minutes, total, including the time for “Creating .EXE file." One can compare the computational results above with those in Mellal and Zio [34, p. 224, Table 14].

Acknowledgment

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

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