Substance

Here Is Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems

                                                          Jsun Yui Wong

The computer program listed below seeks to solve the following mixed-integer nonlinear programming problem from Dhingra [13, pp. 577-578], Deep et al. [12, p. 517, Problem 18], and Liu and Qin [27, p. 2054, Problem 3]:

Maximize (1 - (1 - X(5)) ^ X(1)) * (1 - (1 - X(6)) ^ X(2)) * (1 - (1 - X(7)) ^ X(3)) * (1 - (1 - X(8)) ^ X(4))

subject to

1 * X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 2 * X(4) ^ 2<=250

(1 / 10 ^ 5) * (-1000 / LOG(X(5))) ^ 1.5 * (X(1) + EXP(X(1) / 4)) + (2.3 / 10 ^ 5) * (-1000 / LOG(X(6))) ^ 1.5 * (X(2) + EXP(X(2) / 4)) + (.3 / 10 ^ 5) * (-1000 / LOG(X(7))) ^ 1.5 * (X(3) + EXP(X(3) / 4)) + (2.3 / 10 ^ 5) * (-1000 / LOG(X(8))) ^ 1.5 * (X(4) + EXP(X(4) / 4))<=400

6 * X(1) * EXP(X(1) / 4) + (6) * X(2) * EXP(X(2) / 4) + (8) * X(3) * EXP(X(3) / 4) + (7) * X(4) * EXP(X(4) / 4)<=500

1<= X(i) <= 10, i=1, 2, 3, 4

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

.5<= X(5), X(6), X(7), X(8)<=.999999.

The formulation above has the operating time = 1000 hours (see above and see line 346 below), which comes from Dhingra [13, p. 578, Table 1], whereas the operating time (T) of the preceding paper is 100 hours; this 100 hours comes from Deep et al. [12, p. 517].

X(9) through X(11) below are slack variables added.

Here line 128 is 128 FOR I = 1 TO 10000 . The sequence of line 341 through line 346 aims to get domino effect.

 

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 4

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

99 NEXT J44

115 FOR J44 = 5 TO 8

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

119 NEXT J44

128 FOR I = 1 TO 10000

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

153 j = 1 + FIX(RND * 8)
154 REM GOTO 164
155 REM IF j > 4.5 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 4

327 X(J44) = INT(X(J44))

328 IF X(J44) < 1 THEN 1670

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

336 FOR J44 = 5 TO 8

 

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

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

 

341 X(9) = 250 - 1 * X(1) ^ 2 - 2 * X(2) ^ 2 - 3 * X(3) ^ 2 - 2 * X(4) ^ 2

 

343 X(10) = 500 - 6 * X(1) * EXP(X(1) / 4) - (6) * X(2) * EXP(X(2) / 4) - (8) * X(3) * EXP(X(3) / 4) - (7) * X(4) * EXP(X(4) / 4)

 

346 X(11) = 400 - (1 / 10 ^ 5) * (-1000 / LOG(X(5))) ^ 1.5 * (X(1) + EXP(X(1) / 4)) - (2.3 / 10 ^ 5) * (-1000 / LOG(X(6))) ^ 1.5 * (X(2) + EXP(X(2) / 4)) - (.3 / 10 ^ 5) * (-1000 / LOG(X(7))) ^ 1.5 * (X(3) + EXP(X(3) / 4)) - (2.3 / 10 ^ 5) * (-1000 / LOG(X(8))) ^ 1.5 * (X(4) + EXP(X(4) / 4))

 

355 FOR J44 = 9 TO 11

 

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

359 NEXT J44

 

393 PDU = (1 - (1 - X(5)) ^ X(1)) * (1 - (1 - X(6)) ^ X(2)) * (1 - (1 - X(7)) ^ X(3)) * (1 - (1 - X(8)) ^ X(4)) + 1000000 * (X(9) + X(10) + X(11))

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 11

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < .9999545 THEN 1999

 

1904 PRINT A(1), A(2), A(3)
1905 PRINT A(4), A(5), A(6)
1906 PRINT A(7), A(8), A(9)
1907 PRINT A(10), A(11)

1909 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [39]. The complete output through JJJJ= -31780 is shown below:

5          5          4
6          .9016273076268694          .8890706419149593
.9474732745979748          .8492447889666785          0
0         0
.999954639844179          -31927

5          6          4
5         .900666728589303            .8514942826029884
949588769897003          88675597250079          0
0          0
.9999545208401468          -31780

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 [39], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31780 was 124 seconds, not including the time for “Creating .EXE file” (132 seconds, total, including the time for “Creating .EXE file” ). One can compare the computational results above with those in Dhingra [13, p. 578, Table 2], in Deep et al. [12, p. 517, Problem-18], and in Liu and Qin [27, p. 2054, Table IX]:

 

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] 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.

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

[12] 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.

[13] 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.

[14] 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.

[15] 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.

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

[17] 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.

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

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

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

[21] 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

[22] 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

[23] 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.

[24] 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.

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

[26] 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.

[27] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problems, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[28] 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.

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

[30] 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.

[31] 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.

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

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

[34] 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.

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

[36] 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.

[37] 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

[38] 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.

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

[40] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[41] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[42] Jsun Yui Wong (2011, July 23). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem. Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[43] Jsun Yui Wong (2011 July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[44] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[45] Jsun Yui Wong (2012, September 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[46] Jsun Yui Wong (2013 January 10). The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization. http://myblogsubstance.typepad.com/substance/2013/01/

 

Here Is Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems

                                                               Jsun Yui Wong

The computer program listed below seeks to solve the following integer nonlinear programming problem in Deep et al. [12, p. 514, Problem-13]:

Minimize    EXP(-X(1)) + X(1) ^ 2 - X(1) * X(2) - 3 * X(2) ^ 2 - 6 * X(2) + 4 * X(1)

subject to

2 * X(1) + X(2) <=8

- X(1) + X(2) <=2

0<= X(1), X(2) <=3

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

X(3) and X(4) below are slack variables added.

The sequence of line 341 through line 344 aims to obtain domino effect.

 

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
122 A(1) = FIX(RND * 2)

124 A(2) = FIX(RND * 2)

128 FOR I = 1 TO 100

129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 2)
153 j = 1 + FIX(RND * 2)
154 GOTO 164
155 IF j < 2.6 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 2

327 X(J44) = INT(X(J44))

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

329 IF X(J44) > 3## THEN 1670
331 NEXT J44

341 X(3) = 2 + X(1) - X(2)

344 X(4) = 8 - 2 * X(1) - X(2)
355 FOR J44 = 3 TO 4

 

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

359 NEXT J44
389 PDU = -EXP(-X(1)) - X(1) ^ 2 + X(1) * X(2) + 3 * X(2) ^ 2 + 6 * X(2) - 4 * X(1) + 1000000 * (X(3) + 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 REM GOTO 128
1670 NEXT I
1889 IF M < -999 THEN 1999

1904 PRINT A(1), A(2), A(3)
1905 PRINT A(4), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [37]. The complete output through JJJJ= -31996 is shown below:

1          3          0
0          42.63212055882856          -32000

1          3          0
0          42.63212055882856          -31999

1          3          0
0          42.63212055882856          -31998

1          3          0
0          42.63212055882856          -31997

1          3          0
0          42.63212055882856          -31996

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 [37], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31996 was 1 or 2 seconds, not including the time for “Creating .EXE file” (9 seconds, total, including the time for “Creating .EXE file” ). One can compare the computational results above with those in Deep et al. [12, p. 514, Problem-13].

 

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] 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.

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

[12] 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.

[13] 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.

[14] 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.

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

[16] 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.

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

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

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

[20] 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

[21] 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

[22] 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.

[23] 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.

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

[25] 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.

[26] 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.

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

[28] 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.

[29] 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.

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

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

[32] 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.

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

[34] 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.

[35] 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

[36] 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.

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

[38] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[39] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[40] Jsun Yui Wong (2011, July 23). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem. Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[41] Jsun Yui Wong (2011 July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[42] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[43] Jsun Yui Wong (2012, September 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[44] Jsun Yui Wong (2013 January 10). The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization. http://myblogsubstance.typepad.com/substance/2013/01/

 

Here Is Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems, Second Edition

                                                           Jsun Yui Wong

The computer program listed below seeks to solve the following mixed-integer nonlinear programming problem from Deep et al. [12, p. 517, Problem-18] and Liu and Qin[26, p. 2054, Problem 3]:

Maximize     (1 - (1 - X(5)) ^ X(1)) * (1 - (1 - X(6)) ^ X(2)) * (1 - (1 - X(7)) ^ X(3)) * (1 - (1 - X(8)) ^ X(4))

subject to

1 * X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 2 * X(4) ^ 2<=250

(1 / 10 ^ 5) * (-100 / LOG(X(5))) ^ 1.5 * (X(1) + EXP(X(1) / 4)) + (2.3 / 10 ^ 5) * (-100 / LOG(X(6))) ^ 1.5 * (X(2) + EXP(X(2) / 4)) + (.3 / 10 ^ 5) * (-100 / LOG(X(7))) ^ 1.5 * (X(3) + EXP(X(3) / 4)) + (2.3 / 10 ^ 5) * (-100 / LOG(X(8))) ^ 1.5 * (X(4) + EXP(X(4) / 4))<=400

6 * X(1) * EXP(X(1) / 4) + (6) * X(2) * EXP(X(2) / 4) + (8) * X(3) * EXP(X(3) / 4) + (7) * X(4) * EXP(X(4) / 4)<=500

1<= X(i) <= 10, i=1, 2, 3, 4

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

.5<= X(5), X(6), X(7), X(8)<=.999999.

X(9) through X(11) below are slack variables added.

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

The sequence of line 341 through line 346 aims to take advantage of domino effect.

 

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 4

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

99 NEXT J44

115 FOR J44 = 5 TO 8

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

 

119 NEXT J44

128 FOR I = 1 TO 10000

 

129 FOR KKQQ = 1 TO 8
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 j = 1 + FIX(RND * 8)
154 GOTO 164
155 IF j > 4.5 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 4

327 X(J44) = INT(X(J44))

328 IF X(J44) < 1 THEN 1670

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

336 FOR J44 = 5 TO 8

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

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

 

341 X(9) = 250 - 1 * X(1) ^ 2 - 2 * X(2) ^ 2 - 3 * X(3) ^ 2 - 2 * X(4) ^ 2

 

343 X(10) = 500 - 6 * X(1) * EXP(X(1) / 4) - (6) * X(2) * EXP(X(2) / 4) - (8) * X(3) * EXP(X(3) / 4) - (7) * X(4) * EXP(X(4) / 4)

 

346 X(11) = 400 - (1 / 10 ^ 5) * (-100 / LOG(X(5))) ^ 1.5 * (X(1) + EXP(X(1) / 4)) - (2.3 / 10 ^ 5) * (-100 / LOG(X(6))) ^ 1.5 * (X(2) + EXP(X(2) / 4)) - (.3 / 10 ^ 5) * (-100 / LOG(X(7))) ^ 1.5 * (X(3) + EXP(X(3) / 4)) - (2.3 / 10 ^ 5) * (-100 / LOG(X(8))) ^ 1.5 * (X(4) + EXP(X(4) / 4))

 

355 FOR J44 = 9 TO 11

 

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

359 NEXT J44

393 PDU = (1 - (1 - X(5)) ^ X(1)) * (1 - (1 - X(6)) ^ X(2)) * (1 - (1 - X(7)) ^ X(3)) * (1 - (1 - X(8)) ^ X(4)) + 1000000 * (X(9) + X(10) + X(11))

 

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 11

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < .99997 THEN 1999

 

1904 PRINT A(1), A(2), A(3)
1905 PRINT A(4), A(5), A(6)
1906 PRINT A(7), A(8), A(9)
1907 PRINT A(10), A(11)

1909 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [38]. The complete output through JJJJ= -31566 is shown below:

3          5          6
5          .9875437701173425          .901094528524518
.8955196874460578          .9257634707522393          0
0          0
.999985047310636          -31908

5          3          6
5          .9533781878424883          .9881799292187095
.8992470862545967          .9633787841532827         0
0          0
.99999701640704          -31566 .

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 [38], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31566 was 10 seconds, not including the time for “Creating .EXE file” (20 seconds, total, including the time for “Creating .EXE file” ). One can compare the computational results above with those in Deep et al. [12, p. 517, Problem-18] and in Liu and Qin[26, p. 2054, Table IX].

 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] 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.

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

[12] 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.

[13] 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.

[14] 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.

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

[16] 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.

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

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

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

[20] 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

[21] 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

[22] 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.

[23] 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.

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

[25] 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.

[26] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problems, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.

[27] 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.

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

[29] 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.

[30] 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.

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

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

[33] 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.

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

[35] 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.

[36] 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

[37] 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.

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

[39] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[40] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[41] Jsun Yui Wong (2011, July 23). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem. Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[42] Jsun Yui Wong (2011 July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[43] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[44] Jsun Yui Wong (2012, September 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[45] Jsun Yui Wong (2013 January 10). The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization. http://myblogsubstance.typepad.com/substance/2013/01/

 

Here Is Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems

                                                       Jsun Yui Wong

The computer program listed below seeks to solve the following mixed-integer nonlinear programming problem in Costa and Oliveira [10, p. 263, Problem 4'] and in Deep et al. [12, p. 513, Problem 7]:

Minimize 7.5 * X(3) + 5.5 * (1 - X(3)) + 7 * X(1) + 6 * X(2) + (50 * (1 - X(3))) / (.8 * (1 - EXP(-.4 * X(2)))) + (50 * X(3)) / (.9 * (1 - EXP(-.5 * X(1))))

subject to

.9 * (1 - EXP(-.5 * X(1))) - 2 * X(3) <=0

.8 * (1 - EXP(-.4 * X(2))) - 2 * (1 - X(3)) <=0

X(1)<= 10 * X(3)

X(2) <= 10 * (1 - X(3))

X(1), X(2) >=0

X(3) = 0 or 1.

X(4) through X(7) below are slack variables added.

One notes line 1557, which is 1557 REM GOTO 128.

 

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
107 A(1) = (RND * 10)

109 A(2) = (RND * 10)

 

124 A(3) = FIX(RND * 2)

 

128 FOR I = 1 TO 2000

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)

155 IF j < 2.6 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 A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0

169 NEXT IPP
311 IF X(1) < 0 THEN 1670
318 IF X(1) > 10 THEN 1670
321 IF X(2) < 0 THEN 1670
325 IF X(2) > 10 THEN 1670

 

327 X(3) = INT(X(3))

328 IF X(3) < 0 THEN 1670

329 IF X(3) > 1 THEN 1670

341 X(4) = 10 * X(3) - X(1)

344 X(5) = 2 * X(3) - .9 * (1 - EXP(-.5 * X(1)))

348 X(6) = 10 * (1 - X(3)) - X(2)

349 X(7) = 2 * (1 - X(3)) - .8 * (1 - EXP(-.4 * X(2)))

355 FOR J44 = 4 TO 7

 

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

359 NEXT J44

388 PDU = -7.5 * X(3) - 5.5 * (1 - X(3)) - 7 * X(1) - 6 * X(2) - (50 * X(3)) / (.9 * (1 - EXP(-.5 * X(1)))) - (50 * (1 - X(3))) / (.8 * (1 - EXP(-.4 * X(2)))) + 1000000 * (X(4) + X(5) + X(6) + X(7))

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 7

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

 

1557 REM GOTO 128
1670 NEXT I
1889 IF M < -999 THEN 1999

1904 PRINT A(1), A(2), A(3)
1905 PRINT A(4), A(5), A(6), A(7), M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [37]. Selected candidate solutions through JJJJ= -31865--including the top two at JJJJ= -31982 and at JJJJ= -31865--are shown below:

2.989677255015183D-03          1.292600229866133D-21          1
0          0          -1.292600229866133D-21          0
-1.#IND          -32000

2.95365408829768D-03          9.666616845124944D-20          1
0          0          -9.666616845124944D-20         0
-1.#IND          -31999

3.51424595545633          3.848600950808451D-16          1
0          0          -3.848600950808451D-16          -8.881784197001253D-17
-99.23963505432462          -31982

1.857074168246347D-16          3.733553514788621          0
-1.857074168246347D-16          -9.99200722162641D-17          0
0          -108.5051801099387          -31980

3.516896435023995          1.639094979724525D-16          1
0          0          -1.639094979724525D-16          -8.881784197001253D-17
-99.23965258038535          -31865

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 [37], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31865 was 2 seconds, not including the time for “Creating .EXE file” (10 seconds, total, including the time for “Creating .EXE file” ). One can compare the computational results above with those in Costa and Oliveira [10, p. 263, Problem 4].

 

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] 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.

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

[12] 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.

[13] 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.

[14] 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.

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

[16] 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.

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

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

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

[20] 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

[21] 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

[22] 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.

[23] 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.

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

[25] 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.

[26] 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.

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

[28] 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.

[29] 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.

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

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

[32] 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.

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

[34] 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.

[35] 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

[36] 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.

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

[38] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[39] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[40] Jsun Yui Wong (2011, July 23). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem. Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[41] Jsun Yui Wong (2011 July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[42] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[43] Jsun Yui Wong (2012, September 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[44] Jsun Yui Wong (2013 January 10). The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization. http://myblogsubstance.typepad.com/substance/2013/01/

 

Here Is Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems, Second Edition

                                                       Jsun Yui Wong

The computer program listed below seeks to solve the following mixed-integer nonlinear programming problem in Costa and Oliveira [10, p. 263, Problem 4]:

Minimize 7.5 * X(6) + 5.5 * X(7) + 7 * X(1) + 6 * X(2) + 5 * (X(3))

subject to

X(6) +X(7)= 1

X(8) = (.9 * (1 - EXP(-.5 * X(1))))*X(4)

X(9) = (.8 * (1 - EXP(-.4 * X(2))))*X(5)

X(8) + X(9) = 10

X(4) + X(5) = X(3)

X(8) * X(6) + X(9) * X(7) = 10

X(1) <= 10 * X(6)
X(2) <= 10 * X(7)

X(4) <= 20 * X(6)

X(5) <= 20 * X(7)

X(1), X(2), X(3), X(4), X(5), X(8), X(9) >=0

X(6), X(7) = 0 or 1.

X(10) through X(13) below are slack variables added.

One notes that line 128 here is 128 FOR I = 1 TO 500.

 

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

101 FOR J44 = 1 TO 9

108 A(J44) = RND * 10

110 NEXT J44

113 FOR J44 = 6 TO 7

115 A(J44) = FIX(RND * 2)

117 NEXT J44

 

128 FOR I = 1 TO 500

 

129 FOR KKQQ = 1 TO 9
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 6)
153 j = 1 + FIX(RND * 9)

155 IF j = 6 THEN GOTO 164 ELSE IF j = 7 THEN GOTO 164 ELSE GOTO 156

 

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

161 GOTO 169

 

164 IF A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0

169 NEXT IPP

326 FOR J44 = 1 TO 9

 

328 IF X(J44) < 0 THEN 1670

330 NEXT J44

 

332 FOR J44 = 6 TO 7

333 X(J44) = CINT(X(J44))

334 IF X(J44) < 0 THEN 1670

335 IF X(J44) > 1 THEN 1670

 

336 NEXT J44

 

341 X(6) = 1 - X(7)
343 X(8) = -X(9) + 10

 

348 X(4) = X(8) / (.9 * (1 - EXP(-.5 * X(1))))

 

349 X(5) = X(9) / (.8 * (1 - EXP(-.4 * X(2))))

351 X(3) = X(4) + X(5)

 

353 X(10) = 10 * X(6) - X(1)
354 X(11) = 10 * X(7) - X(2)

355 X(12) = 20 * X(6) - X(4)

356 X(13) = 20 * X(7) - X(5)

357 lhs = X(8) * X(6) + X(9) * X(7) - 10

358 FOR J44 = 10 TO 13

 

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

360 NEXT J44

392 PDU = -7.5 * X(6) - 5.5 * X(7) - 7 * X(1) - 6 * X(2) - 5 * (X(3)) + 1000000 * (X(10) + X(11) + X(12) + X(13)) - 1000000 * ABS(lhs)

 

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 13

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

 

1557 GOTO 128
1670 NEXT I

1889 IF M < -100 THEN 1999

1904 PRINT A(1), A(2), A(3)
1905 PRINT A(4), A(5), A(6)
1906 PRINT A(7), A(8), A(9)
1907 PRINT M, JJJJ
1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [37]. The complete output through JJJJ =-31998 is shown below:

3.514236950337757          1.388541455072557D-16          13.42799528025653
13.42799528025653          4.83482994862401D-21          1
0          10          4.294191623287711D-37
-99.23963505378582          -32000

3.514236978955354         1.388466982025888D-16          13.4279952401919
13.4279952401919          3.919591590247976D-22          1
0          10          3.48129666449635D-38
-99.23963505378582          -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 [37], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 1 second, not including the time for “Creating .EXE file” (9 seconds, total, including the time for “Creating .EXE file” ). One can compare the computational results above with those in Costa and Oliveira [10, p. 263, Problem 4].

 

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

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

[12] 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.

[13] 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.

[14] 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.

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

[16] 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.

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

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

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

[20] 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

[21] 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

[22] 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.

[23] 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.

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

[25] 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.

[26] 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.

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

[28] 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.

[29] 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.

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

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

[32] 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.

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

[34] 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.

[35] 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

[36] 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.

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

[38] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[39] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[40] Jsun Yui Wong (2011, July 23). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem. Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[41] Jsun Yui Wong (2011 July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[42] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[43] Jsun Yui Wong (2012, September 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[44] Jsun Yui Wong (2013 January 10). The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization. http://myblogsubstance.typepad.com/substance/2013/01/

 

Here Is Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems

                                                    Jsun Yui Wong

Problems

The computer program listed below seeks to solve the following mixed-integer nonlinear programming problem in Wasanapradit et al. [35, p. 39, Test Problem 5]:

Maximize       X(3) - 2 * X(1) - X(2)

subject to

X(1) - 2 * EXP(-X(2))=0

- X(1) + X(2) + X(3)<=0

.5<= X(1) <= 1.4

X(3) = 0 or 1.

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)
81 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ
90 M = -3E+30

102 A(1) = .5 + (RND * .9)

 

107 A(2) = -10 + (RND * 20)
108 FOR J44 = 3 TO 3

 

124 A(J44) = FIX(RND * 2)

126 NEXT J44

 

128 FOR I = 1 TO 500

 

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)

155 IF j < 2.5 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 A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0

169 NEXT IPP
311 IF X(1) < .5## THEN 1670
318 IF X(1) > 1.4## THEN 1670

 

326 FOR J44 = 3 TO 3

327 X(J44) = INT(X(J44))

328 IF X(J44) < 0 THEN 1670

329 IF X(J44) > 1 THEN 1670
333 NEXT J44

336 X(1) = 2 * EXP(-X(2))
338 IF X(1) < .5## THEN 1670
340 IF X(1) > 1.4## THEN 1670

 

343 X(4) = X(1) - X(2) - X(3)

355 FOR J44 = 4 TO 4

 

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

359 NEXT J44

399 PDU = X(3) - 2 * X(1) - X(2) + 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 < -9 THEN 1999

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

1907 PRINT A(4), M, JJJJ
1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [36]. The complete output through JJJJ =-31991 is shown below:

.8526055031677633          .852605500660183           0
0          -2.55781650699571         -31996

1.374822528188588          .3748225281800124          1
0          -2.12446758455719          -31994

1.374822528525458          .3748225279349844          1
0         -2.1244675849859          -31991

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 [36], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31991 was 2 seconds, not including the time for “Creating .EXE file” (11 seconds, total, including the time for “Creating .EXE file” ). One can compare the computational results above with those in Wasanapradit et al. [35, p. 39, Test Problem 5].

 

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

[11] 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.

[12] 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.

[13] 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.

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

[15] 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.

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

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

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

[19] 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

[20] 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

[21] 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.

[22] 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.

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

[24] 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.

[25] 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.

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

[27] 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.

[28] 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.

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

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

[31] 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.

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

[33] 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.

[34] 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

[35] 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.

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

[37] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[38] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[39] Jsun Yui Wong (2011, July 23). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem. Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[40] Jsun Yui Wong (2011 July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[41] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[42] Jsun Yui Wong (2012, September 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[43] Jsun Yui Wong (2013 January 10). The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization. http://myblogsubstance.typepad.com/substance/2013/01/

 

Here Is Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems

                                                                 Jsun Yui Wong

The computer program listed below seeks to solve the following mixed integer optimization problem from Deep et al. [11, p. 512, Problem 2]:

Minimize - X(2) + 2 * X(1) - LOG(X(1) / 2)

subject to

- X(1) - LOG(X(1) / 2) + X(2)<=0

.5<= X(1) <= 1.5

X(2) = 0 or 1.

X(3) 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)
81 FOR JJJJ = -32000 TO 32000

 

89 RANDOMIZE JJJJ
90 M = -3E+30
107 A(1) = .5 + (RND * 1)

 

124 A(2) = FIX(RND * 2)

 

128 FOR I = 1 TO 1000

 

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

155 IF j < 1.6 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 A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0

169 NEXT IPP
311 IF X(1) < .5 THEN 1670
318 IF X(1) > 1.5 THEN 1670

 

327 X(2) = INT(X(2))

328 IF X(2) < 0 THEN 1670

329 IF X(2) > 1 THEN 1670

 

341 X(3) = X(1) + LOG(X(1) / 2) - X(2)

355 FOR J44 = 3 TO 3

 

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

359 NEXT J44
396 PDU = X(2) - 2 * X(1) + LOG(X(1) / 2) + 1000000 * (X(3))

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 3

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

 

1557 GOTO 128
1670 NEXT I
1889 REM IF M < .94561335 THEN 1999

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

1907 PRINT M, JJJJ
1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [35]. The complete output through JJJJ =-31998 is shown below:

1.374822528203618          1          0
-2.124467584576316          -32000

1.374822528207108          1          0
-2.124467584580757          -31999

1.374822528187324          1          0
-2.12446758455558          -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 [35], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 1 second, not including the time for “Creating .EXE file” (9 seconds, total, including the time for “Creating .EXE file” ). One can compare the computational results above with those in Deep et al. [11, p. 512, Problem 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] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[11] 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.

[12] 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.

[13] 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.

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

[15] 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.

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

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

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

[19] 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

[20] 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

[21] 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.

[22] 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.

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

[24] 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.

[25] 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.

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

[27] 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.

[28] 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.

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

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

[31] 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.

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

[33] 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.

[34] 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

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

[36] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[37] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[38] Jsun Yui Wong (2011, July 23). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem. Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[39] Jsun Yui Wong (2011 July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[40] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[41] Jsun Yui Wong (2012, September 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[42] Jsun Yui Wong (2013 January 10). The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization. http://myblogsubstance.typepad.com/substance/2013/01/

 

Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems, Fourth Edition

                                                            Jsun Yui Wong

 

The computer program listed below seeks to solve the following system reliability problem based on Luus [25, p. 15] and on Tambunan and Mawengkang [32, p. 5276]:

Maximize

15
product (1 - (1 - reliability(i)) ^ X(i))
i=1

subject to

15
sum cost(i) * X(i) <=400
i=1

15
sum weight(i) * X(i) <=414
i=1

X(i) =1, 2, 3, ..., 14, i=1,..., 15.

X(1) through X(15) are general integer variables.

X(16) and X(17) below are slack variables added.

One notes that here each stage (of the 15 stages) can have either 1, 2, 3,..., or 14 units; see line 107 and line 314, which are 107 A(J44) = 1 + FIX(RND * 14) and 314 IF X(J44) > 14 THEN 1670. One also notes line 388, which is 388 SUMOBJ = SUMOBJ * (1 - (1 - RR(J44)) ^ X(J44)), a product of fifteen terms.

 

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)

 

11 RR(1) = .90: RR(2) = .75: RR(3) = .65: RR(4) = .80: RR(5) = .85: RR(6) = .93: RR(7) = .78: RR(8) = .66: RR(9) = .78: RR(10) = .91: RR(11) = .79: RR(12) = .77: RR(13) = .67: RR(14) = .79: RR(15) = .67

 

31 CC(1) = 5: CC(2) = 4: CC(3) = 9: CC(4) = 7: CC(5) = 7: CC(6) = 5: CC(7) = 6: CC(8) = 9: CC(9) = 4: CC(10) = 5: CC(11) = 6: CC(12) = 7: CC(13) = 9: CC(14) = 8: CC(15) = 6

 

51 WW(1) = 8: WW(2) = 9: WW(3) = 6: WW(4) = 7: WW(5) = 8: WW(6) = 8: WW(7) = 9: WW(8) = 6: WW(9) = 7: WW(10) = 8: WW(11) = 9: WW(12) = 7: WW(13) = 6: WW(14) = 5: WW(15) = 7

 

88 FOR JJJJ = -32000 TO 32000

 

89 RANDOMIZE JJJJ
90 M = -3E+30

 

106 FOR J44 = 1 TO 15

 

107 A(J44) = 1 + FIX(RND * 14)

109 NEXT J44

 

128 FOR I = 1 TO 1000

 

 

129 FOR KKQQ = 1 TO 15
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 j = 1 + FIX(RND * 15)
154 REM r = (1 - RND * 2) * A(j)
155 REM X(j) = A(j) + (RND ^ (RND * 10)) * r
157 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 2) ELSE X(j) = A(j) + FIX(1 + RND * 2)

159 NEXT IPP
311 FOR J44 = 1 TO 15
312 X(J44) = INT(X(J44))

 

313 IF X(J44) < 1 THEN 1670

314 IF X(J44) > 14 THEN 1670

317 NEXT J44

 

371 SUMCON = 0
374 FOR J44 = 1 TO 15
376 SUMCON = SUMCON + CC(J44) * X(J44)

 

377 NEXT J44
378 X(16) = 400 - SUMCON
379 IF X(16) < 0 THEN X(16) = X(16) ELSE X(16) = 0
380 SUMWW = 0
381 FOR J44 = 1 TO 15
382 SUMWW = SUMWW + WW(J44) * X(J44)

 

383 NEXT J44
384 X(17) = 414 - SUMWW
385 IF X(17) < 0 THEN X(17) = X(17) ELSE X(17) = 0

 

 

386 SUMOBJ = 1
387 FOR J44 = 1 TO 15
388 SUMOBJ = SUMOBJ * (1 - (1 - RR(J44)) ^ X(J44))

 

389 NEXT J44

438 PDU = SUMOBJ + 1000000 * (X(16) + X(17))

 

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 17

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < .94561335 THEN 1999

 

1904 PRINT A(1), A(2), A(3), A(4)
1905 PRINT A(5), A(6), A(7), A(8)
1906 PRINT A(9), A(10), A(11), A(12)

1907 PRINT A(13), A(14), A(15), A(16), A(17)

1915 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [33]. The complete output through JJJJ = -31927 is shown below:

3          4          6          4
3          2          4          5
4          2          3          4
5          4          5          0          0
.9456133574581372          -31982

3          4          6          4
3          2          4          5
4          2          3          4
5          4          5          0          0
.9456133574581372          -31976

3          4          6          4
3          2          4          5
4          2          3          4
5          4          5          0          0
.9456133574581372          -31951

3          4          6          4
3          2          4          5
4          2          3          4
5          4          5          0          0
.9456133574581372          -31927

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 [33], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31927 was 3 seconds, not including the time for “Creating .EXE file” (11 seconds, total, including the time for “Creating .EXE file” ).

Acknowledgment

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

References

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

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

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

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

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

[11] 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.

[12] 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.

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

[14] G. Dantzig, R. Fulkerson, S. Johnson, Solution of a Large-Scale Traveling-Salesman Problem.
Operations Research, Vol. 2, No. 4 (Nov., 1954), pp. 393-410.

[15] 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.

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

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

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

[19] 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

[20] 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

[21] 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.

[22] 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.

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

[24] 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.

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

[32] 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

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

[34] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[35] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[36] Jsun Yui Wong (2011, July 23). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem. Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[37] Jsun Yui Wong (2011 July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[38] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[39] Jsun Yui Wong (2012, September 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[40] Jsun Yui Wong (2013 January 10). The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization. http://myblogsubstance.typepad.com/substance/2013/01/

Here Is Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems

Jsun Yui Wong

 

The computer program listed below seeks to solve the following problem from Wang et al. [33, Example 1]:

 Minimize X(1) ^ .5 * X(2) + 3 * LOG(X(1))

subject to

- X(1) + X(2)<=5

X(1) ^ .5 - X(2)<=6

X(1) epsilon {.1, .5, .7, 1.2}

1<=X(2)<=4.

X(3) and X(4) 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)

88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ
90 M = -3E+30

111 IF RND < 1 / 4 THEN A(1) = .1 ELSE IF RND < 1 / 3 THEN A(1) = .5 ELSE IF RND < 1 / 2 THEN A(1) = .7 ELSE A(1) = 1.2

115 A(2) = 1 + RND * 3

 

128 FOR I = 1 TO 500

 

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

153 j = 1 + FIX(RND * 2)
154 r = (1 - RND * 2) * A(j)
161 IF j = 1 THEN GOTO 171 ELSE GOTO 185

171 IF RND < 1 / 4 THEN X(1) = .1 ELSE IF RND < 1 / 3 THEN X(1) = .5 ELSE IF RND < 1 / 2 THEN X(1) = .7 ELSE X(1) = 1.2
177 GOTO 222

 

185 X(j) = A(j) + (RND ^ (RND * 10)) * r

 

222 NEXT IPP

 

313 IF X(2) < 1 THEN 1670

314 IF X(2) > 4 THEN 1670
378 X(3) = 5 + X(1) - X(2)

384 X(4) = 6 - X(1) ^ .5 + X(2)

 

391 FOR J44 = 3 TO 4

 

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

 

399 NEXT J44

438 PDU = -X(1) ^ .5 * X(2) - 3 * LOG(X(1)) + 1000000 * (X(3) + 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 REM IF M < .9 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 [34]. The complete output through JJJJ = -31998 is shown below:

.1          1.000000000001215          0          0
6.591527512964914          -32000

.1          1.000000000014          0          0
6.591527512960871          -31999

.1          1.000000000027958          0          0
6.591527512956457          -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 [34], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 1 (or 2) seconds, not including the time for “Creating .EXE file” (9 seconds, total, including the time for “Creating .EXE file” ). One can compare the computational results above with those in Wang et al. [33, Example 1].

 Acknowledgment

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

References

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

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

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

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

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

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

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

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

[11] 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.

[12] 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.

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

[14] G. Dantzig, R. Fulkerson, S. Johnson, Solution of a Large-Scale Traveling-Salesman Problem.
Operations Research, Vol. 2, No. 4 (Nov., 1954), pp. 393-410.

[15] 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.

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

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

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

[19] 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

[20] 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

[21] 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.

[22] 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.

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

[24] 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.

 

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

 [32] 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

[33] Pei-Chun Wang, Jung-Fa Tsai, Wei-Nung Ma, Chia-Chien Lee (2010). An Efficient Global Optimization Approach for Solving Mixed-Integer Nonlinear Programming Problems. https://ieeexplore.ieee.org/document/5668338/

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

[35] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[36] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[37] Jsun Yui Wong (2011, July 23). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem. Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[38] Jsun Yui Wong (2011 July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[39] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[40] Jsun Yui Wong (2012, September 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[41] Jsun Yui Wong (2013 January 10). The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization. http://myblogsubstance.typepad.com/substance/2013/01/

Everything You Need To Solve Your Mixed-Integer Nonlinear Programming (MINLP) Problems, Third Edition

Jsun Yui Wong

 

The computer program listed below seeks to solve the following problem based on the system reliability problem in Luus [25] and in Tambunan and Mawengkang [32]:

Maximize

15
product (1 - (1 - reliability(i)) ^ X(i))
i=1

subject to

15
sum cost(i) * X(i) <=400
i=1

15
sum weight(i) * X(i) <=414
i=1

X(i) =1, 2, 3,..., 10, i=1,..., 15.

X(1) through X(15) are general integer variables.

X(16) and X(17) below are slack variables added.

One notes line 107, which is 107 A(J44) = 1 + FIX(RND * 10).

 

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)

 

11 RR(1) = .90: RR(2) = .75: RR(3) = .65: RR(4) = .80: RR(5) = .85: RR(6) = .93: RR(7) = .78: RR(8) = .66: RR(9) = .78: RR(10) = .91: RR(11) = .79: RR(12) = .77: RR(13) = .67: RR(14) = .79: RR(15) = .67

 

31 CC(1) = 5: CC(2) = 4: CC(3) = 9: CC(4) = 7: CC(5) = 7: CC(6) = 5: CC(7) = 6: CC(8) = 9: CC(9) = 4: CC(10) = 5: CC(11) = 6: CC(12) = 7: CC(13) = 9: CC(14) = 8: CC(15) = 6

 

51 WW(1) = 8: WW(2) = 9: WW(3) = 6: WW(4) = 7: WW(5) = 8: WW(6) = 8: WW(7) = 9: WW(8) = 6: WW(9) = 7: WW(10) = 8: WW(11) = 9: WW(12) = 7: WW(13) = 6: WW(14) = 5: WW(15) = 7

 

88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ
90 M = -3E+30

 

106 FOR J44 = 1 TO 15

 

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

109 NEXT J44

 

128 FOR I = 1 TO 40000

 

129 FOR KKQQ = 1 TO 15
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 j = 1 + FIX(RND * 15)
154 REM r = (1 - RND * 2) * A(j)
155 REM X(j) = A(j) + (RND ^ (RND * 10)) * r
157 IF RND < .5 THEN X(j) = A(j) - FIX(1 + RND * 2) ELSE X(j) = A(j) + FIX(1 + RND * 2)

159 NEXT IPP
311 FOR J44 = 1 TO 15
312 X(J44) = INT(X(J44))

 

313 IF X(J44) < 1 THEN 1670

314 IF X(J44) > 10 THEN 1670

317 NEXT J44

 

371 SUMCON = 0
374 FOR J44 = 1 TO 15
376 SUMCON = SUMCON + CC(J44) * X(J44)

 

377 NEXT J44
378 X(16) = 400 - SUMCON
379 IF X(16) < 0 THEN X(16) = X(16) ELSE X(16) = 0
380 SUMWW = 0
381 FOR J44 = 1 TO 15
382 SUMWW = SUMWW + WW(J44) * X(J44)

 

383 NEXT J44
384 X(17) = 414 - SUMWW
385 IF X(17) < 0 THEN X(17) = X(17) ELSE X(17) = 0

 

 

386 SUMOBJ = 1
387 FOR J44 = 1 TO 15
388 SUMOBJ = SUMOBJ * (1 - (1 - RR(J44)) ^ X(J44))

 

389 NEXT J44

438 PDU = SUMOBJ + 1000000 * (X(16) + X(17))

 

466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 17

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

 

1557 GOTO 128
1670 NEXT I
1889 IF M < .94 THEN 1999

 

1904 PRINT A(1), A(2), A(3), A(4)
1905 PRINT A(5), A(6), A(7), A(8)
1906 PRINT A(9), A(10), A(11), A(12)

1907 PRINT A(13), A(14), A(15), A(16), A(17)

1915 PRINT M, JJJJ

1999 NEXT JJJJ

 

This BASIC computer program was run with QB64v1000-win [33]. The complete output through JJJJ = -31996 is shown below:

3          4          6          4
3          2          4          5
4          2          3          4
5          4          5          0          0
.9456133574581372          -32000

3 4 6 4
3 2 4 5
4 2 3 4    
5 4 5 0 0
.9456133574581372          -31999

3          4          5          3
3          2          4          5
4          3          3          4
5          5          5          0          0
.9447484845681542          -31998

3 4 5 3
3 2 4 5
4 3 3 4
5 5 5 0 0
.9447484845681542          -31997

3 4 6 4
3 2 4 5
4 2 3 4
5 4 5 0 0
.9456133574581372          -31996

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 [33], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31996 was 3 seconds, not including the time for “Creating .EXE file” (11 seconds, total, including the time for “Creating .EXE file” ).

Acknowledgment

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

References

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

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

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

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

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

[11] 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.

[12] 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.

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

[14] G. Dantzig, R. Fulkerson, S. Johnson, Solution of a Large-Scale Traveling-Salesman Problem.
Operations Research, Vol. 2, No. 4 (Nov., 1954), pp. 393-410.

[15] 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.

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

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

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

[19] 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

[20] 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

[21] 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.

[22] 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.

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

[24] 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.

 

[25] 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.

[26] 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.

[27] 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.

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

[29] 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.

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

[31] 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.

 

[32] 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

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

[32] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[33] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[34] Jsun Yui Wong (2011, July 23). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to Three Instances of the Haverly Pooling Problem. Retrieved from http://myblogsubstance.typepad.com/substance/2011/07/

[35] Jsun Yui Wong (2011 July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/

[36] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/

[37] Jsun Yui Wong (2012, September 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 22 Facilities. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/9/27/

[38] Jsun Yui Wong (2013 January 10). The Domino Method of General Integer Nonlinear Programming Applied to Alkylation Process Optimization. http://myblogsubstance.typepad.com/substance/2013/01/