Jsun Yui Wong
The computer program listed below seeks to solve the following problem based on Chen [10, p. 1562, P3, Complex (Bridge) System] and on Liu and Qin [30, p. 2053, Problem 2, Complex (Bridge) System]:
Maximize O(1) * O(2) + O(3) * O(4) + O(1) * O(4) * O(5) + O(2) * O(3) * O(5) - O(1) * O(2) * O(3) * O(4) - O(1) * O(2) * O(3) * O(5) - O(1) * O(2) * O(4) * O(5) - O(1) * O(3) * O(4) * O(5) - O(2) * O(3) * O(4) * O(5) + 2 * O(1) * O(2) * O(3) * O(4) * O(5)
where O(J44) = 1 - ((1 - (X(J44 + 5)))) ^ X(J44)--see line 391 through line 394 below
subject to
1 * X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 4 * X(4) ^ 2 + 2 * X(5) ^ 2<=110
(2.33 / 10 ^ 5) * ((-1000 / LOG(X(6))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) + (1.45 / 10 ^ 5) * ((-1000 / LOG(X(7))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) + (.541 / 10 ^ 5) * ((-1000 / LOG(X(8))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) + (8.05 / 10 ^ 5) * ((-1000 / LOG(X(9))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) + (1.95 / 10 ^ 5) * ((-1000 / LOG(X(10))) ^ 1.5) * (X(5) + EXP(X(5) / 4))<=175
7 * X(1) * EXP(X(1) / 4) + (8) * X(2) * EXP(X(2) / 4) + (8) * X(3) * EXP(X(3) / 4) + (6) * X(4) * EXP(X(4) / 4) + (9) * X(5) * EXP(X(5) / 4)<=200
.5<= X(i) <= 1, 6<=i<=10
where X(1) through X(5) are positive general integer variables with X(i)=1, 2, 3, ..., 10.
X(11) through X(13) 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)
81 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3E+30
95 FOR J44 = 1 TO 5
97 A(J44) = FIX(1 + RND * 10)
99 NEXT J44
115 FOR J44 = 6 TO 10
117 A(J44) = .5 + RND * .49999
119 NEXT J44
128 FOR I = 1 TO 100000
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 j = 1 + FIX(RND * 10)
155 IF j > 5.5 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 5
327 X(J44) = CINT(X(J44))
328 IF X(J44) < 1 THEN 1670
329 IF X(J44) > 10 THEN 1670
331 NEXT J44
336 FOR J44 = 6 TO 10
338 IF X(J44) < .5## THEN 1670
339 IF X(J44) > .999999## THEN 1670
340 NEXT J44
341 X(11) = 110 - 1 * X(1) ^ 2 - 2 * X(2) ^ 2 - 3 * X(3) ^ 2 - 4 * X(4) ^ 2 - 2 * X(5) ^ 2
343 X(12) = 200 - 7 * X(1) * EXP(X(1) / 4) - (8) * X(2) * EXP(X(2) / 4) - (8) * X(3) * EXP(X(3) / 4) - (6) * X(4) * EXP(X(4) / 4) - (9) * X(5) * EXP(X(5) / 4)
346 X(13) = 175 - (2.33 / 10 ^ 5) * ((-1000 / LOG(X(6))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) - (1.45 / 10 ^ 5) * ((-1000 / LOG(X(7))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) - (.541 / 10 ^ 5) * ((-1000 / LOG(X(8))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) - (8.05 / 10 ^ 5) * ((-1000 / LOG(X(9))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) - (1.95 / 10 ^ 5) * ((-1000 / LOG(X(10))) ^ 1.5) * (X(5) + EXP(X(5) / 4))
355 FOR J44 = 11 TO 13
357 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
359 NEXT J44
391 FOR J44 = 1 TO 5
393 O(J44) = 1 - ((1 - (X(J44 + 5)))) ^ X(J44)
394 NEXT J44
401 PDU = O(1) * O(2) + O(3) * O(4) + O(1) * O(4) * O(5) + O(2) * O(3) * O(5) - O(1) * O(2) * O(3) * O(4) - O(1) * O(2) * O(3) * O(5) - O(1) * O(2) * O(4) * O(5) - O(1) * O(3) * O(4) * O(5) - O(2) * O(3) * O(4) * O(5) + 2 * O(1) * O(2) * O(3) * O(4) * O(5) + 1000000 * (X(11) + X(12) + X(13))
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 < .99988961 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), A(12), A(13)
1909 PRINT M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [45]. The complete output of a single run through JJJJ= -31917 is shown below:
3 3 2
4 1 .8271828129627745
.8584470687531135 .9146456241665405 .6473915423816696
.7083633509542693
.9998896175115296 -31982
3 3 2
4 1 .8275566506233765
.8585263665874964 .914785253366289 .6470983072525923
.7044017880815086
.9998896208547377 -31917
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 [45], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31917 was 5 minutes, total, including the time for “Creating .EXE file." One can compare the computational results above with those in Chen [10, p. 1565, Table 6] and in Liu and Qin [30, p. 2054, Table VIII].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.
[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.
[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.
[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.
[5] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.
[6] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html
[7] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.
[8] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming. New York: John Wiley and Sons, Inc., 1968.
[9] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. - Feb., 1966), pp. 52-58.
[10] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[11] Leandro dos Santos Coelho (2009), Self-Organizing Migrating Strategies Applied to Reliability-Redundany Optimization of Systems. IEEE Transactions on Reliability, Vol. 58, No. 3, 2009 September, pp. 501-519.
[12] 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.
[13] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[14] 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.
[15] 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.
[16] 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.
[17] 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.
[18] C. A. Floudas, P. M. Pardalos, A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer-Verlag, 1990.
[19] 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.
[20] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf
[21] David M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[22] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.
[23] 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
[24] 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
[25] 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.
[26] 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.
[27] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[28] 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.
[29] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[30] Yubao Liu, Guihe Qin (2014), A hybrid TS-DE algorithm for reliability redundancy optimization problem, Journal of Computers, 9, No. 9, September 2014, pp. 2050-2057.
[31] 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.
[32] F. Masedu, M Angelozzi (2008). Modelling optimum fraction assignment in the 4X100 m relay race by integer linear programming. Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.
[33] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html
[34] 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.
[35] Yuji Nakagawa, Mitsunori Hikita, Hiroshi Kamada (1984). Surrogate Constraints for Reliability Optimization Problems with Multiple Constraints. IEEE Transactions on Reliability, Vol. R-33, No. 4, October 1984, pp. 301-305.
[36] 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.
[37] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[38] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[39] 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.
[40] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.
[41] 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.
[42] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[43] 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
[44] 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.
[45] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[46] Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990). Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.
Saturday, July 28, 2018
Saturday, July 21, 2018
This Mixed-Integer Nonlinear Programming Solver Applied to a 4x100 Meter Relay Race
Jsun Yui Wong
The computer program listed below seeks to solve the following problem from Masedu and Angelozzi [30, p. 75] and Kania and Sidarto [24, p. 020004-7, Problem 4]:
Minimize T(1)*X(1)+T(2)*X(2)+T(3)*X(3)+...+T(24)*X(24)
where the Ts are given times
subject to
X(1) +X(5) + X(9) + X(13) + X(17) + X(21) = 1
X(2) +X(6) + X(10) + X(14) + X(18) + X(22) = 1
X(3) +X(7) + X(11) + X(15) + X(19) + X(23) = 1
X(4) +X(8) + X(12) + X(16) + X(20) + X(24) = 1
X(1) + X(2) + X(3) + X(4)<=1
X(5) + X(6) + X(7) + X(8)<=1
X(9) + X(10) + X(11) + X(12)<=1
X(13) + X(14) + X(15) + X(16)<=1
X(17) + X(18) + X(19) + X(20)<=1
X(21) + X(22) + X(23) + X(24)<=1
X(1) through X(24) are zero-one integer variables.
X(25) through X(30) 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(33), CC(20), RR(20), WW(20), T(33)
11 T(1) = 12.27
12 T(2) = 11.57
13 T(3) = 11.54
14 T(4) = 12.07
15 T(5) = 11.34
16 T(6) = 11.45
17 T(7) = 12.45
18 T(8) = 12.34
19 T(9) = 11.29
20 T(10) = 11.50
21 T(11) = 11.45
22 T(12) = 11.52
23 T(13) = 12.54
24 T(14) = 12.34
25 T(15) = 12.32
26 T(16) = 11.57
27 T(17) = 12.20
28 T(18) = 11.22
29 T(19) = 12.07
30 T(20) = 12.03
31 T(21) = 11.54
32 T(22) = 11.48
33 T(23) = 11.56
34 T(24) = 12.30
81 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3E+30
95 FOR J44 = 1 TO 24
97 A(J44) = FIX(RND * 2)
99 NEXT J44
128 FOR I = 1 TO 500
129 FOR KKQQ = 1 TO 24
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 j = 1 + FIX(RND * 24)
154 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 A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0
169 NEXT IPP
251 X(1) = -X(5) - X(9) - X(13) - X(17) - X(21) + 1
252 X(2) = -X(6) - X(10) - X(14) - X(18) - X(22) + 1
253 X(3) = -X(7) - X(11) - X(15) - X(19) - X(23) + 1
254 X(4) = -X(8) - X(12) - X(16) - X(20) - X(24) + 1
256 FOR J44 = 1 TO 4
257 IF X(J44) < 0 THEN GOTO 1670
259 NEXT J44
261 X(25) = 1 - X(1) - X(2) - X(3) - X(4)
262 X(26) = 1 - X(5) - X(6) - X(7) - X(8)
263 X(27) = 1 - X(9) - X(10) - X(11) - X(12)
264 X(28) = 1 - X(13) - X(14) - X(15) - X(16)
265 X(29) = 1 - X(17) - X(18) - X(19) - X(20)
266 X(30) = 1 - X(21) - X(22) - X(23) - X(24)
301 SUMM = 0
303 FOR J44 = 1 TO 24
305 SUMM = SUMM + T(J44) * X(J44)
333 NEXT J44
355 FOR J44 = 25 TO 30
357 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
359 NEXT J44
395 PDU = -SUMM + 1000000 * (X(25) + X(26) + X(27) + X(28) + X(29) + X(30))
466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 30
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -45.70 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)
1908 PRINT A(17), A(18), A(19), A(20)
1909 PRINT A(21), A(22), A(23), A(24)
1910 PRINT A(25), A(26), A(27), A(28)
1911 PRINT A(29), A(30), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [42]. The complete output of a single run through JJJJ= -31954 is shown below:
0 0 0 0
1 0 0 0
0 0 1 0
0 0 0 1
0 1 0 0
0 0 0 0
0 0 0 0
0 0 -45.58 -31998
0 0 0 0
1 0 0 0
0 0 1 0
0 0 0 1
0 1 0 0
0 0 0 0
0 0 0 0
0 0 -45.58 -31995
0 0 0 0
1 0 0 0
0 0 1 0
0 0 0 1
0 1 0 0
0 0 0 0
0 0 0 0
0 0 -45.58 -31961
0 0 1 0
1 0 0 0
0 0 0 1
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
0 0 -45.62 -31954
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 [42], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31954 was 3 seconds, not including the time for “Creating .EXE file” (12 seconds, total, including the time for “Creating .EXE file” ). One can compare the computational results above with those in Masedu and Angelozzi [30, p. 76, Figure 1, Table 3] and in Kania and Sidarto [24, p. 020004-8, 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] 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] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[28] 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.
[29] 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.
[30] F. Masedu, M. Angelozzi (2008). Modelling Optimum Fraction Assignment in the 4X100 M Relay Race by Integer Linear Programming. Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.
[31] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html
[32] 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.
[33] 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.
[34] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[35] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[36] 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.
[37] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.
[38] 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.
[39] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[40] 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
[41] 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.
[42] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[43] Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990). Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.
The computer program listed below seeks to solve the following problem from Masedu and Angelozzi [30, p. 75] and Kania and Sidarto [24, p. 020004-7, Problem 4]:
Minimize T(1)*X(1)+T(2)*X(2)+T(3)*X(3)+...+T(24)*X(24)
where the Ts are given times
subject to
X(1) +X(5) + X(9) + X(13) + X(17) + X(21) = 1
X(2) +X(6) + X(10) + X(14) + X(18) + X(22) = 1
X(3) +X(7) + X(11) + X(15) + X(19) + X(23) = 1
X(4) +X(8) + X(12) + X(16) + X(20) + X(24) = 1
X(1) + X(2) + X(3) + X(4)<=1
X(5) + X(6) + X(7) + X(8)<=1
X(9) + X(10) + X(11) + X(12)<=1
X(13) + X(14) + X(15) + X(16)<=1
X(17) + X(18) + X(19) + X(20)<=1
X(21) + X(22) + X(23) + X(24)<=1
X(1) through X(24) are zero-one integer variables.
X(25) through X(30) 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(33), CC(20), RR(20), WW(20), T(33)
11 T(1) = 12.27
12 T(2) = 11.57
13 T(3) = 11.54
14 T(4) = 12.07
15 T(5) = 11.34
16 T(6) = 11.45
17 T(7) = 12.45
18 T(8) = 12.34
19 T(9) = 11.29
20 T(10) = 11.50
21 T(11) = 11.45
22 T(12) = 11.52
23 T(13) = 12.54
24 T(14) = 12.34
25 T(15) = 12.32
26 T(16) = 11.57
27 T(17) = 12.20
28 T(18) = 11.22
29 T(19) = 12.07
30 T(20) = 12.03
31 T(21) = 11.54
32 T(22) = 11.48
33 T(23) = 11.56
34 T(24) = 12.30
81 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3E+30
95 FOR J44 = 1 TO 24
97 A(J44) = FIX(RND * 2)
99 NEXT J44
128 FOR I = 1 TO 500
129 FOR KKQQ = 1 TO 24
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 j = 1 + FIX(RND * 24)
154 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 A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0
169 NEXT IPP
251 X(1) = -X(5) - X(9) - X(13) - X(17) - X(21) + 1
252 X(2) = -X(6) - X(10) - X(14) - X(18) - X(22) + 1
253 X(3) = -X(7) - X(11) - X(15) - X(19) - X(23) + 1
254 X(4) = -X(8) - X(12) - X(16) - X(20) - X(24) + 1
256 FOR J44 = 1 TO 4
257 IF X(J44) < 0 THEN GOTO 1670
259 NEXT J44
261 X(25) = 1 - X(1) - X(2) - X(3) - X(4)
262 X(26) = 1 - X(5) - X(6) - X(7) - X(8)
263 X(27) = 1 - X(9) - X(10) - X(11) - X(12)
264 X(28) = 1 - X(13) - X(14) - X(15) - X(16)
265 X(29) = 1 - X(17) - X(18) - X(19) - X(20)
266 X(30) = 1 - X(21) - X(22) - X(23) - X(24)
301 SUMM = 0
303 FOR J44 = 1 TO 24
305 SUMM = SUMM + T(J44) * X(J44)
333 NEXT J44
355 FOR J44 = 25 TO 30
357 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
359 NEXT J44
395 PDU = -SUMM + 1000000 * (X(25) + X(26) + X(27) + X(28) + X(29) + X(30))
466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 30
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -45.70 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)
1908 PRINT A(17), A(18), A(19), A(20)
1909 PRINT A(21), A(22), A(23), A(24)
1910 PRINT A(25), A(26), A(27), A(28)
1911 PRINT A(29), A(30), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [42]. The complete output of a single run through JJJJ= -31954 is shown below:
0 0 0 0
1 0 0 0
0 0 1 0
0 0 0 1
0 1 0 0
0 0 0 0
0 0 0 0
0 0 -45.58 -31998
0 0 0 0
1 0 0 0
0 0 1 0
0 0 0 1
0 1 0 0
0 0 0 0
0 0 0 0
0 0 -45.58 -31995
0 0 0 0
1 0 0 0
0 0 1 0
0 0 0 1
0 1 0 0
0 0 0 0
0 0 0 0
0 0 -45.58 -31961
0 0 1 0
1 0 0 0
0 0 0 1
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
0 0 -45.62 -31954
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 [42], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31954 was 3 seconds, not including the time for “Creating .EXE file” (12 seconds, total, including the time for “Creating .EXE file” ). One can compare the computational results above with those in Masedu and Angelozzi [30, p. 76, Figure 1, Table 3] and in Kania and Sidarto [24, p. 020004-8, 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] 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] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[28] 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.
[29] 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.
[30] F. Masedu, M. Angelozzi (2008). Modelling Optimum Fraction Assignment in the 4X100 M Relay Race by Integer Linear Programming. Italian Journal of Sports Sciences, Anno 13, No. 1, 2008, pp. 74-77.
[31] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html
[32] 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.
[33] 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.
[34] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[35] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[36] 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.
[37] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.
[38] 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.
[39] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[40] 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
[41] 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.
[42] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[43] Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990). Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.
Monday, July 16, 2018
Mixed-Integer Nonlinear Programming Solver Applied to a Reliability-Redundancy Problem with Five Stages
Jsun Yui Wong
The computer program listed below seeks to solve the following mixed-integer nonlinear programming problem, which is based on Table 1 of Tillman, Hwang, and Kuo [38, p. 165]:
Maximize (1 - (1 - X(6)) ^ X(1)) * (1 - (1 - X(7)) ^ X(2)) * (1 - (1 - X(8)) ^ X(3)) * (1 - (1 - X(9)) ^ X(4)) * (1 - (1 - X(10)) ^ X(5))
subject to
1 * X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 4 * X(4) ^ 2 + 2 * X(5) ^ 2<=110
(2.33 / 10 ^ 5) * ((-1000 / LOG(X(6))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) + (1.45 / 10 ^ 5) * ((-1000 / LOG(X(7))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) + (.541 / 10 ^ 5) * ((-1000 / LOG(X(8))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) + (8.05 / 10 ^ 5) * ((-1000 / LOG(X(9))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) + (1.95 / 10 ^ 5) * ((-1000 / LOG(X(10))) ^ 1.5) * (X(5) + EXP(X(5) / 4))<=175
7 * X(1) * EXP(X(1) / 4) + (8) * X(2) * EXP(X(2) / 4) + (8) * X(3) * EXP(X(3) / 4) + (6) * X(4) * EXP(X(4) / 4) + (9) * X(5) * EXP(X(5) / 4)<=200
1<= X(i) <= 10, i=1, 2, 3, 4; 5, X(1) through X(5) are integer variables
.5<= X(6), X(7), X(8), X(9), X(10)<=.999999.
X(11), X(12), and X(13) 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)
81 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3E+30
95 FOR J44 = 1 TO 5
97 A(J44) = FIX(1 + RND * 10)
99 NEXT J44
115 FOR J44 = 6 TO 10
117 A(J44) = .5 + RND * .499999
119 NEXT J44
128 FOR I = 1 TO 40000
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 j = 1 + FIX(RND * 10)
155 IF j > 5.5 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 5
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 = 6 TO 10
338 IF X(J44) < .5## THEN 1670
339 IF X(J44) > .999999## THEN 1670
340 NEXT J44
341 X(11) = 110 - 1 * X(1) ^ 2 - 2 * X(2) ^ 2 - 3 * X(3) ^ 2 - 4 * X(4) ^ 2 - 2 * X(5) ^ 2
343 X(12) = 200 - 7 * X(1) * EXP(X(1) / 4) - (8) * X(2) * EXP(X(2) / 4) - (8) * X(3) * EXP(X(3) / 4) - (6) * X(4) * EXP(X(4) / 4) - (9) * X(5) * EXP(X(5) / 4)
346 X(13) = 175 - (2.33 / 10 ^ 5) * ((-1000 / LOG(X(6))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) - (1.45 / 10 ^ 5) * ((-1000 / LOG(X(7))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) - (.541 / 10 ^ 5) * ((-1000 / LOG(X(8))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) - (8.05 / 10 ^ 5) * ((-1000 / LOG(X(9))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) - (1.95 / 10 ^ 5) * ((-1000 / LOG(X(10))) ^ 1.5) * (X(5) + EXP(X(5) / 4))
355 FOR J44 = 11 TO 13
357 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
359 NEXT J44
411 PDU = (1 - (1 - X(6)) ^ X(1)) * (1 - (1 - X(7)) ^ X(2)) * (1 - (1 - X(8)) ^ X(3)) * (1 - (1 - X(9)) ^ X(4)) * (1 - (1 - X(10)) ^ X(5)) + 1000000 * (X(11) + X(12) + X(13))
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 < .931677 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), A(12), A(13)
1909 PRINT M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [41]. The complete output through JJJJ= -31266 is shown below:
3 2 2
3 3 .7795519252219796
.8727194718321425 .9022987392717032 .7114015479912691
.786765140967794 0 0 0
.931677103177272 -31376
3 2 2
3 3 .7789622643589318
.871993590446706 .9031635093885894 .7118740906368367
.7866751378019727 0 0 0
.931679616709972 -31266
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 [41], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31266 was 23 minutes, total, including the time for “Creating .EXE file." One can compare the computational results above with those in
Tillman, Hwang, and Kuo [38, p. 165, Table 2] and with those in Xu, Kuo, and Lin [42, p. 52, Table 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] 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] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[28] 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.
[29] 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.
[30] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html
[31] 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.
[32] 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.
[33] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[34] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[35] 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.
[36] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.
[37] 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.
[38] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.
[39] 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
[40] 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.
[41] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[42] Zhongkai Xu, Way Kuo, Hsin-Hui Lin (1990). Optimization Limits in Improving System Reliability. IEEE Transactions on Reliability, Vol. 39, No. 1, 1990 April, pp. 51-60.
Monday, July 2, 2018
Fractional Mixed-Integer Nonlinear Programming Applied to a Reliability-Redundancy Problem, Corrected Edition
Jsun Yui Wong
The computer program listed below seeks to solve the following fractional mixed-integer nonlinear programming problem, a return-on-investment problem, which is based on the reliability/cost ratio problem on pp. 496-497 of Liu [277]; one notes Table 7:
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))) / ((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)))
subject to
1 * X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 2 * X(4) ^ 2<=250
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 / 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
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), X(10), and X(11) below are slack variables added.
Whereas line 128 of the earlier edition is 155 REM IF j > 4.5 THEN GOTO 156 ELSE GOTO 164, here line 128 is 155 IF j > 4.5 THEN GOTO 156 ELSE GOTO 164. This is the correction.
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 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
398 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))) / ((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))) + 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 REM IF M < -999999 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 [40]. The output through JJJJ= -31998 is summarized below:
.4 3 5
3 .5000000000000607 .5000000000000027
.5000000000000038 .5000000000000134 0
0 0
.0386826290706846 -32000
4 3 5
3 .5000000000000003 .5000000000000012
.5000000000000363 .5000000000000001 0
0 0
3.868262907068577D-02 -31999
4 3 5
3 .5000000000000112 .50000000000044
.5000000000000007 .5000000000000022 0
0 0
.0386826290706738 -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 [40], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 4 seconds, not including the time for “Creating .EXE file” (13 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] 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] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[28] 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.
[29] 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.
[30] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html
[31] 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.
[32] 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.
[33] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[34] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[35] 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.
[36] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.
[37] 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.
[38] 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
[39] 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.
[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] 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/
[42] 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/
[43] 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/
[44] 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/
[45] 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/
[46] 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/
[47] 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/
The computer program listed below seeks to solve the following fractional mixed-integer nonlinear programming problem, a return-on-investment problem, which is based on the reliability/cost ratio problem on pp. 496-497 of Liu [277]; one notes Table 7:
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))) / ((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)))
subject to
1 * X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 2 * X(4) ^ 2<=250
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 / 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
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), X(10), and X(11) below are slack variables added.
Whereas line 128 of the earlier edition is 155 REM IF j > 4.5 THEN GOTO 156 ELSE GOTO 164, here line 128 is 155 IF j > 4.5 THEN GOTO 156 ELSE GOTO 164. This is the correction.
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 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
398 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))) / ((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))) + 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 REM IF M < -999999 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 [40]. The output through JJJJ= -31998 is summarized below:
.4 3 5
3 .5000000000000607 .5000000000000027
.5000000000000038 .5000000000000134 0
0 0
.0386826290706846 -32000
4 3 5
3 .5000000000000003 .5000000000000012
.5000000000000363 .5000000000000001 0
0 0
3.868262907068577D-02 -31999
4 3 5
3 .5000000000000112 .50000000000044
.5000000000000007 .5000000000000022 0
0 0
.0386826290706738 -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 [40], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31998 was 4 seconds, not including the time for “Creating .EXE file” (13 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] 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] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[28] 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.
[29] 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.
[30] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html
[31] 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.
[32] 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.
[33] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[34] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[35] 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.
[36] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.
[37] 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.
[38] 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
[39] 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.
[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] 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/
[42] 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/
[43] 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/
[44] 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/
[45] 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/
[46] 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/
[47] 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/
Sunday, July 1, 2018
Fractional Mixed-Integer Nonlinear Programming Applied to a Reliability-Redundancy Problem
Jsun Yui Wong
The computer program listed below seeks to solve the following fractional mixed-integer nonlinear programming problem, which is based on the reliability/cost ratio problem on pp. 496-497 of Liu [27]; one notes Table 7:
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))) / ((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)))
subject to
1 * X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 2 * X(4) ^ 2<=250
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 / 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
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), X(10), and X(11) 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)
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
398 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))) / ((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))) + 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 REM IF M ...... 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 [40]. The output through JJJJ= -31813 is summarized below:
4 1 5
3 .5000000000000018 .6269843939688721
.5000000000000053 .5000000000000002 0
0 0
2.977371924220023D-02 -32000
.
.
.
4 3 5
3 .5 .5000000000000001
.5000000000000037 .5000000000000001 0
0 0
3.868262907068585D-02 -31813
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 [40], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31813 was 92 seconds, not including the time for “Creating .EXE file” (100 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] 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] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[28] 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.
[29] 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.
[30] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html
[31] 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.
[32] 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.
[33] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[34] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[35] 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.
[36] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.
[37] 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.
[38] 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
[39] 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.
[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] 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/
[42] 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/
[43] 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/
[44] 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/
[45] 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/
[46] 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/
[47] 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/
The computer program listed below seeks to solve the following fractional mixed-integer nonlinear programming problem, which is based on the reliability/cost ratio problem on pp. 496-497 of Liu [27]; one notes Table 7:
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))) / ((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)))
subject to
1 * X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 2 * X(4) ^ 2<=250
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 / 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
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), X(10), and X(11) 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)
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
398 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))) / ((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))) + 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 REM IF M ...... 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 [40]. The output through JJJJ= -31813 is summarized below:
4 1 5
3 .5000000000000018 .6269843939688721
.5000000000000053 .5000000000000002 0
0 0
2.977371924220023D-02 -32000
.
.
.
4 3 5
3 .5 .5000000000000001
.5000000000000037 .5000000000000001 0
0 0
3.868262907068585D-02 -31813
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 [40], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31813 was 92 seconds, not including the time for “Creating .EXE file” (100 seconds, total, including the time for “Creating .EXE file” ).
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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[40] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[41] 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/
[42] 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/
[43] 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/
[44] 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/
[45] 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/
[46] 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/
[47] 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/
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