Jsun Yui Wong
The computer program listed below seeks to solve the following 3-objective integer program from Hwang, Lee, Tillman, and Lie [38, p. 434, Example 1] and Lee [44, p. 61, Example 3-1]:
Minimize {(X(8)+X(9)), (X(7))}, which have priority 1 and priority two, respectively
1.2 * X(1) + 2.3 * X(2) + 3.4 * X(3) + 4.5 * X(4) + X(5) -X(8) =35.6
X(1) + X(2) + X(3) + X(4) + X(6) -X(9) =16
(1 - .2 ^ (X(1) + 1)) * (1 - .3 ^ (X(2) + 1)) * (1 - .25 ^ (X(3) + 1)) * (1 - .15 ^ (X(4) + 1)) + X(7) -X(10) =1
where X(1) through X(4) are non-negative general integer variables and X(5) through X(10) are >=0.
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)
81 FOR JJJJ = -32000 TO 32000
85 RANDOMIZE JJJJ
86 M = -3E+50
92 A(1) = 1 + FIX(RND * 5)
93 A(2) = 1 + FIX(RND * 5)
94 A(3) = 1 + FIX(RND * 5)
96 A(4) = 1 + FIX(RND * 5)
111 FOR J44 = 5 TO 10
114 A(J44) = (RND * 20)
117 NEXT J44
128 FOR I = 1 TO 9000
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 4)
153 J = 1 + FIX(RND * 10)
154 IF J > 4 THEN GOTO 156 ELSE GOTO 162
155 REM GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
162 IF RND < .5 THEN X(J) = A(J) - INT(RND * 3) ELSE X(J) = A(J) + INT(RND * 3)
164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
172 X(1) = INT(X(1))
174 X(2) = INT(X(2))
176 X(3) = INT(X(3))
178 X(4) = INT(X(4))
188 IF X(1) < 1 THEN 1670
189 IF X(2) < 1 THEN 1670
190 IF X(3) < 1 THEN 1670
192 IF X(4) < 1 THEN 1670
208 IF X(1) > 5 THEN 1670
209 IF X(2) > 5 THEN 1670
210 IF X(3) > 5 THEN 1670
211 IF X(4) > 5 THEN 1670
215 FOR J44 = 5 TO 10
216 IF X(J44) < 0 THEN 1670
218 IF X(J44) > 50 THEN 1670
219 NEXT J44
311 X(8) = -35.6 + 1.2 * X(1) + 2.3 * X(2) + 3.4 * X(3) + 4.5 * X(4) + X(5)
315 X(9) = -16 + X(1) + X(2) + X(3) + X(4) + X(6)
322 X(10) = -1 + (1 - .2 ^ (X(1) + 1)) * (1 - .3 ^ (X(2) + 1)) * (1 - .25 ^ (X(3) + 1)) * (1 - .15 ^ (X(4) + 1)) + X(7)
326 FOR J44 = 8 TO 10
328 IF X(J44) < 0 THEN 1670
329 IF X(J44) > 50 THEN 1670
330 NEXT J44
437 PDU = -X(7) - 70000 * X(8) - 70000 * X(9) - 99999999 * X(5) * X(8) - 99999999 * X(6) * X(9) - 99999999 * X(7) * X(10)
466 P = PDU
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 10
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -.0099 THEN 1999
1936 PRINT A(5), A(6), A(7), A(8), A(9), A(10)
1956 PRINT A(1), A(2), A(3), A(4), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [77]. The complete output of a single run through JJJJ= -31761 is shown below:
1.300000000000001 3 9.578981157062501D-03
0 0 6.505213034913927D-19
3 5 3 2 -9.578981157685634D-03
-31973
.1000000000000016 2 8.309210620084375D-03
1.52655665885959D-16 0 8.131516293641283D-19
4 5 3 2 -8.30921215800258D-03
-31761
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 [77], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31761 was 48 seconds, not including the time for “Creating .EXE file" (55 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Hwang, Lee, Tillman, and Lie [38, p. 436, Example 1].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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Friday, January 25, 2019
Tuesday, January 8, 2019
Solving a Three-Objective Binary Programming Problem
Jsun Yui Wong
The computer program listed below seeks to solve the following 3-objective integer (binary) program from Boland, Charkhgard, and Savelsbergh [9, p. 863, Example 2]:
Minimize ( 4 * X(1) + 2 * X(2) + X(3))
minimize (X(1) + 2 * X(2) + 4 * X(3))
minimize (2 * X(1) + 3 * X(2) + 6 * X(3))
subject to
X(1) + X(2) + X(3) >= 2
X(i) = 0 or 1, for i=1, 2, 3.
The present paper treats (4 * X(1) + 2 * X(2) + X(3))
as the principal objective--see Sharma, Dahiya, and Verma [67, p. 1914] and Haimes, Lasdon, Wismer [28]--and treats (X(1) + 2 * X(2) + 4 * X(3)) and (2 * X(1) + 3 * X(2) + 6 * X(3)) as two additional constraints, Sharma, Dahiya, and Verma [67, p. 1914] and Haimes, Lasdon, and Wismer [28].
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)
81 FOR JJJJ = -32000 TO 32000
85 RANDOMIZE JJJJ
86 M = -3E+50
89 epsi1 = -15 + RND * 30
90 epsi2 = -15 + RND * 30
92 A(1) = INT(RND * 1)
93 A(2) = INT(RND * 1)
94 A(3) = INT(RND * 1)
128 FOR I = 1 TO 100
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 2)
153 J = 1 + FIX(RND * 3)
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
162 REM IF RND < .5 THEN X(J) = A(J) - INT(RND * 4) ELSE X(J) = A(J) + INT(RND * 4)
164 IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
172 X(1) = INT(X(1))
174 X(2) = INT(X(2))
176 X(3) = INT(X(3))
188 IF X(1) < 0## THEN 1670
189 IF X(2) < 0## THEN 1670
190 IF X(3) < 0## THEN 1670
208 IF X(1) > 1 THEN 1670
209 IF X(2) > 1 THEN 1670
200 IF X(3) > 1 THEN 1670
226 IF X(1) + X(2) + X(3) < 2 THEN 1670
235 IF (X(1) + 2 * X(2) + 4 * X(3)) > epsi1 THEN 1670
237 IF (2 * X(1) + 3 * X(2) + 6 * X(3)) > epsi2 THEN 1670
435 PDU = (-4 * X(1) - 2 * X(2) - X(3))
466 P = PDU
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 3
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1527 gg01star = -(-4 * X(1) - 2 * X(2) - X(3))
1529 gg02star = (X(1) + 2 * X(2) + 4 * X(3))
1533 gg03star = (2 * X(1) + 3 * X(2) + 6 * X(3))
1557 GOTO 128
1670 NEXT I
1889 IF M < -99999999999 THEN 1999
1924 PRINT gg01star, gg02star, gg03star
1956 PRINT A(1), A(2), A(3), -M, JJJJ
1996 REM
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [75]. The complete output of a single run through JJJJ= -31938 is shown below:
3 6 9
0 1 1 3 -31995
3 6 9
0 1 1 3 -31985
3 6 9
0 1 1 3 -31984
6 3 5
1 1 0 6 -31970
3 6 9
0 1 1 3 -31946
5 5 8
1 0 1 5 -31938
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 [75], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31938 was 1 second, not including the time for “Creating .EXE file" (8 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Boland, Charkhgard, and Savelsbergh [9, p. 863, Example 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
[8] Ritu Arora, S. R. Arora (2014). A cutting plane approach for multi-objective integer indefinite quadratic programming problem: OPSEARCH of the Operational Research Society of India (April-June 2015), 52(2):367-381.
[9] Natahia Boland, Hadi Charkhgard, Martin Savelsbergh (2019). Preprocessing and cut generation techniques for multi-objective binary programming. European Journal of Operational Rearch 274 (2019) 858-875.
[10] Borndorfer, Prof. Dr. Ralf. Solving Multi-Objective Integer Programs. www.zib.de/projects/solving-multi-objective-integer-programs
[11] Jerome Bracken, Garth P. McCormick, Selected Applications of Nonlinear Programming. New York: John Wiley and Sons, Inc., 1968.
[12] R. C. Carlson and G. L. Nemhauser, Scheduling To Minimize Interaction Cost. Operations Research, Vol. 14, No. 1 (Jan. - Feb., 1966), pp. 52-58.
[13] Ta-Cheng Chen (2006). IAs based approach for reliability redundany allocation problems. Applied Mathematics and Computation 182 (2006) 1556-1567.
[14] 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.
[15] William Conley (1981). Optimization: A Simplified Approach. Published 1981 by Petrocelli Books in New York.
[16] 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.
[17] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[18] Pintu Das, tapan kumar Roy (2014). Multi-objective geometric programming and its application in gravel box problem. Journal of global research in computer science volume 5. no.7, july 2014. www.jgrcs.info
[19] 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.
[20] 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.
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[22] 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.
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[26] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf
[27] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
[28] Y. Y. Haimes, L. S. Lasdon, D. A. Wismer (1971). On a bicriterion formulation of the problems of of integrated system identification and system optimization. Ieee Transactions on Systems, Man, and Cybernetics. 1971; 1(3); 296-297.
[29] M. Hashish, M. P. duPlessis (1979). Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses. Transactions of ASME: Journal of Engineering for Industry 101 (1979) 311-318.
[30] David M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[31] R. Gupta, R. Malhotra (1995). Multi-criteria integer linear fractional programming problem, Optimization, 35:4, 373-389.
[321] M. Hashish, M. P. duPlessis (1979). Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses. Transactions of ASME: Journal of Engineering for Industry
[33] M. Hashish, M. P. duPlessis (1979). Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses. Transactions of ASME: Journal of Engineering for Industry 101 (1979) 311-318.
[34] David M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[35] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.
[36] 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
[37] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout. Retrieved from www.optimization-online.org./DB_HTML/2012/04/3432.html
[38] N. K. Jain, V. K. Jain, K. Deb (2007). Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms. International Journal of Machine Tools and Manufacture 47 (2007), 900-919.
[39] 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.
[40] 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.
[41] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[42] 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.
[43] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[44] 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.
[45] 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.
[46] Milos Madic, Miroslav Radovanovic (2014). Optimization of machining processes using pattern search algorithm.
International Journal of Industrial Engineering Computations 5 (2014) 223-234. Homepage: www.GrowingScience.com/ijiec
[47] 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.
[48] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html
[49] Kaisa Miettinen, Petrie Eskelinen, Francisco Ruiz, Mariano Luque (2010). NAUTILUS Journmethod: An interactive technique in multiobjective optimization based on the nadir point. European Journal of Operational Research 206 (2010) 426-434.
[50] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.
[51] Mohamed Arezki Mellal, Edward J. Williams (2016). Parameter optimization of advanced machining processes using cuckoo optimization algorithm and hoopla heuristic. Journal of Intelligent Manufacturing (2016) 27 (5): 927-942.
[52] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[53] 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.
[54] A. Moradi, A. M. Nafchi, A. Ghanbarzadeh (2015). Multi-objective optimization of truss structures using the bee algorithm. (One can read this via Goodle search.)
[55] 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.
[56] Subhash C. Narula, H. Roland Weistroffer (1989). A flexible method for nonlinear criteria decisionmaking problems. Ieee Transactions on Systems, Man and Cybernetics, vol. 19 , no. 4, July/August 1989, pp. 883-887.
[57] 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.
[58] Rashmi Ranjan Ota, A. K. Ojha (2015). A comparative study on optimization techniques for solving multi-objective geometric programming problems. Applied mathematical sciences, vol. 9, 2015, no. 22, 1077-1085. http://sites.google.com/site/ijcsis/
[59] Rashmi Ranjan Ota, A. K. Ojha (2015). A comparative study on optimization techniques for solving multi-objective geometric programming problems. Applied mathematical sciences, vol. 9, 2015, no. 22, 1077-1085. http://sites.google.com/site/ijcsis/
[60] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[61] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[62] Ciara Pike-Burke. Multi-Objective Optimization. https://www.lancaster.ac.uk/pg/pikeburc/report1.pdf.
[63] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining process using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.
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The computer program listed below seeks to solve the following 3-objective integer (binary) program from Boland, Charkhgard, and Savelsbergh [9, p. 863, Example 2]:
Minimize ( 4 * X(1) + 2 * X(2) + X(3))
minimize (X(1) + 2 * X(2) + 4 * X(3))
minimize (2 * X(1) + 3 * X(2) + 6 * X(3))
subject to
X(1) + X(2) + X(3) >= 2
X(i) = 0 or 1, for i=1, 2, 3.
The present paper treats (4 * X(1) + 2 * X(2) + X(3))
as the principal objective--see Sharma, Dahiya, and Verma [67, p. 1914] and Haimes, Lasdon, Wismer [28]--and treats (X(1) + 2 * X(2) + 4 * X(3)) and (2 * X(1) + 3 * X(2) + 6 * X(3)) as two additional constraints, Sharma, Dahiya, and Verma [67, p. 1914] and Haimes, Lasdon, and Wismer [28].
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)
81 FOR JJJJ = -32000 TO 32000
85 RANDOMIZE JJJJ
86 M = -3E+50
89 epsi1 = -15 + RND * 30
90 epsi2 = -15 + RND * 30
92 A(1) = INT(RND * 1)
93 A(2) = INT(RND * 1)
94 A(3) = INT(RND * 1)
128 FOR I = 1 TO 100
129 FOR KKQQ = 1 TO 3
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 2)
153 J = 1 + FIX(RND * 3)
154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
162 REM IF RND < .5 THEN X(J) = A(J) - INT(RND * 4) ELSE X(J) = A(J) + INT(RND * 4)
164 IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0
169 NEXT IPP
172 X(1) = INT(X(1))
174 X(2) = INT(X(2))
176 X(3) = INT(X(3))
188 IF X(1) < 0## THEN 1670
189 IF X(2) < 0## THEN 1670
190 IF X(3) < 0## THEN 1670
208 IF X(1) > 1 THEN 1670
209 IF X(2) > 1 THEN 1670
200 IF X(3) > 1 THEN 1670
226 IF X(1) + X(2) + X(3) < 2 THEN 1670
235 IF (X(1) + 2 * X(2) + 4 * X(3)) > epsi1 THEN 1670
237 IF (2 * X(1) + 3 * X(2) + 6 * X(3)) > epsi2 THEN 1670
435 PDU = (-4 * X(1) - 2 * X(2) - X(3))
466 P = PDU
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 3
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1527 gg01star = -(-4 * X(1) - 2 * X(2) - X(3))
1529 gg02star = (X(1) + 2 * X(2) + 4 * X(3))
1533 gg03star = (2 * X(1) + 3 * X(2) + 6 * X(3))
1557 GOTO 128
1670 NEXT I
1889 IF M < -99999999999 THEN 1999
1924 PRINT gg01star, gg02star, gg03star
1956 PRINT A(1), A(2), A(3), -M, JJJJ
1996 REM
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [75]. The complete output of a single run through JJJJ= -31938 is shown below:
3 6 9
0 1 1 3 -31995
3 6 9
0 1 1 3 -31985
3 6 9
0 1 1 3 -31984
6 3 5
1 1 0 6 -31970
3 6 9
0 1 1 3 -31946
5 5 8
1 0 1 5 -31938
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 [75], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31938 was 1 second, not including the time for “Creating .EXE file" (8 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Boland, Charkhgard, and Savelsbergh [9, p. 863, Example 2].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[17] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[18] Pintu Das, tapan kumar Roy (2014). Multi-objective geometric programming and its application in gravel box problem. Journal of global research in computer science volume 5. no.7, july 2014. www.jgrcs.info
[19] 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.
[20] 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.
[21] R. J. Duffin, E. L. Peterson, C. Zener (1967), Geometric Programming. John Wiley, New York (1967).
[22] 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.
[23] 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.
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[25] 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.
[26] Ignacio E. Grossmann. Overview of Mixed-integer Nonlinear Programming. https://egon.cheme.cmu.edu/ewo/docs/EWOMINLPGrossmann.pdf
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[28] Y. Y. Haimes, L. S. Lasdon, D. A. Wismer (1971). On a bicriterion formulation of the problems of of integrated system identification and system optimization. Ieee Transactions on Systems, Man, and Cybernetics. 1971; 1(3); 296-297.
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[321] M. Hashish, M. P. duPlessis (1979). Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses. Transactions of ASME: Journal of Engineering for Industry
[33] M. Hashish, M. P. duPlessis (1979). Prediction equations relating high velocity jet cutting performance to stand-off-distance and multipasses. Transactions of ASME: Journal of Engineering for Industry 101 (1979) 311-318.
[34] David M. Himmelblau, Applied Nonlinear Programming. New York: McGraw-Hill Book Company, 1972.
[35] Willi Hock, Klaus Schittkowski, Test Examples for Nonlinear Programming Codes. Berlin: Springer-Verlag, 1981.
[36] 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
[37] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout. Retrieved from www.optimization-online.org./DB_HTML/2012/04/3432.html
[38] N. K. Jain, V. K. Jain, K. Deb (2007). Optimization of process parameters of mechanical type advanced machining processes using genetic algorithms. International Journal of Machine Tools and Manufacture 47 (2007), 900-919.
[39] 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.
[40] 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.
[41] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[42] 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.
[43] Gia-Shi Liu (2006), A combination method for reliability-redundancy optimization, Engineering Optimization, 38:04, 485-499.
[44] 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.
[45] 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.
[46] Milos Madic, Miroslav Radovanovic (2014). Optimization of machining processes using pattern search algorithm.
International Journal of Industrial Engineering Computations 5 (2014) 223-234. Homepage: www.GrowingScience.com/ijiec
[47] 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.
[48] MathWorks, Mixed Integer Optimization. https://www.mathworks.com/help/gads;mixed-integer-optimization.html
[49] Kaisa Miettinen, Petrie Eskelinen, Francisco Ruiz, Mariano Luque (2010). NAUTILUS Journmethod: An interactive technique in multiobjective optimization based on the nadir point. European Journal of Operational Research 206 (2010) 426-434.
[50] Mohamed Arezki Mellal, Enrico Zio (2016). A Guided Stochastic Fractal Search Approach for System Reliability Optimization. Reliability Engineering
and System Safety 152 (2016) 213-227.
[51] Mohamed Arezki Mellal, Edward J. Williams (2016). Parameter optimization of advanced machining processes using cuckoo optimization algorithm and hoopla heuristic. Journal of Intelligent Manufacturing (2016) 27 (5): 927-942.
[52] Mohamed Arezki Mellal, Edward J. Williams (2018). Large-scale reliability-redundancy allocation optimization problem using three soft computing methods. In Mangey Ram, Editor, in Modeling and simulation based analysis in reliability engineering. Published July 2018, CRC Press.
[53] 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.
[54] A. Moradi, A. M. Nafchi, A. Ghanbarzadeh (2015). Multi-objective optimization of truss structures using the bee algorithm. (One can read this via Goodle search.)
[55] 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.
[56] Subhash C. Narula, H. Roland Weistroffer (1989). A flexible method for nonlinear criteria decisionmaking problems. Ieee Transactions on Systems, Man and Cybernetics, vol. 19 , no. 4, July/August 1989, pp. 883-887.
[57] 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.
[58] Rashmi Ranjan Ota, A. K. Ojha (2015). A comparative study on optimization techniques for solving multi-objective geometric programming problems. Applied mathematical sciences, vol. 9, 2015, no. 22, 1077-1085. http://sites.google.com/site/ijcsis/
[59] Rashmi Ranjan Ota, A. K. Ojha (2015). A comparative study on optimization techniques for solving multi-objective geometric programming problems. Applied mathematical sciences, vol. 9, 2015, no. 22, 1077-1085. http://sites.google.com/site/ijcsis/
[60] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[61] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[62] Ciara Pike-Burke. Multi-Objective Optimization. https://www.lancaster.ac.uk/pg/pikeburc/report1.pdf.
[63] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining process using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.
[64] 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.
[65] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.
[66] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[67] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[68] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.
[69] Isaac Siwale. A Note on Multi-Objective Mathematical Problems. https://www.aptech.com/wp-content/uploads/2012/09/MultObjMath.pdf
[70] 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.
[71] 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
[72] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. Journal of computational design and engineering 5 (2018) 104-119.
[73] 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.
[74] 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.
[75] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[76] 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.
Friday, January 4, 2019
Solving a Multi-Objective Integer Nonlinear Programming Problem with the Epsilon-Constraint Method, Second Edition
Jsun Yui Wong
The computer program listed below seeks to solve the following 3-objective integer nonlinear programming problem from Sharma [60, p. 149]:
Maximize ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2
maximize (X(1) / (X(2) + 1)) ^ 2
maximize ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2
subject to
X(1) + X(2) <= 5
4 * X(1) + X(2) <= 8
0 <=X(1) <=4
0<= X(2)<=2
X(1) and X(2) are general integer variables.
This paper tries to alleviate the following issue: "One issue with the approach is that it is necessary to preselect which objective to minimize and the epsilon values," Pike-Burke [56, p. 2] on the epsilon-constraint method. One notes that line 89 and line 90, which are 89 EPSI2 = -5 + RND * 10 and 90 EPSI3 = -5 + RND * 10, respectively, try different epsilon values. Also, one notes that here line 229 and line 235 are 229 IF (X(1) / (X(2) + 1)) ^ 2 < EPSI2 THEN 1670 and 235 IF ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2 < EPSI3 THEN 1670, respectively, while line 229 and line 235 of the earlier edition (of October 17, 2018, and of the present blog) are 229 IF (X(1) / (X(2) + 1)) ^ 2 < .3 THEN 1670 and 235 IF ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2 < .3 THEN 1670, respectively.
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)
81 FOR JJJJ = -32000 TO 32000
85 RANDOMIZE JJJJ
87 M = -3E+50
88 EPSI1 = -5 + RND * 10
89 EPSI2 = -5 + RND * 10
90 EPSI3 = -5 + RND * 10
92 A(1) = FIX(RND * 5)
93 A(2) = FIX(RND * 3)
128 FOR I = 1 TO 3200
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 2)
153 J = 1 + FIX(RND * 2)
154 REM GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 REM GOTO 169
162 REM IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1
169 NEXT IPP
171 X(1) = INT(X(1))
173 X(2) = INT(X(2))
177 IF X(1) < 0## THEN 1670
187 IF X(2) < 0## THEN 1670
188 IF X(1) > 4## THEN 1670
222 IF X(2) > 2## THEN 1670
226 IF X(1) + X(2) > 5 THEN 1670
227 IF 4 * X(1) + X(2) > 8 THEN 1670
228 REM IF ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2 < EPSI1 THEN 1670
229 IF (X(1) / (X(2) + 1)) ^ 2 < EPSI2 THEN 1670
235 IF ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2 < EPSI3 THEN 1670
415 PDU = ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2
417 REM
419 REM
466 P = PDU
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 2
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1522 g01star = ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2
1524 g02star = (X(1) / (X(2) + 1)) ^ 2
1526 g03star = ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2
1557 GOTO 128
1670 NEXT I
1889 IF M < -99999999999 THEN 1999
1922 PRINT g01star, g02star, g03star
1924 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [69]. The complete output of a single run through JJJJ= -31955 is shown below:
1 .25 .36
1 1 1 -31992
.1111111111111111 0 .4444444444444444
0 1 .1111111111111111 -31991
1 .25 .36
1 1 1 -31990
1 1 .5625
1 0 1 -31988
0 0 1
0 0 0 -31984
.1111111111111111 0 .4444444444444444
0 1 .1111111111111111 -31883
0 0 1
0 0 0 -31881
4 4 .4444444444444444
2 0 4 -31979
4 4 .4444444444444444
2 0 4 -31974
1 .1111111111111111 .25
1 2 1 -31973
4 4 .4444444444444444
2 0 4 -31971
1 .25 .36
1 1 1 .31968
.25 0 .25
0 2 .25 -31967
1 .25 .36
1 1 1 -31966
1 .1111111111111111 .25
1 2 1 -31962
4 4 .4444444444444444
2 0 4 -31959
0 0 1
0 0 0 -31958
0 0 1
0 0 0 -31957
1 .1111111111111111 .25
1 2 1 -31956
1 1 .5625
1 0 1 -31955
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 [69], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31955 was 2 seconds, not including the time for “Creating .EXE file" (7 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Sharma [60, pp 149-151].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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and System Safety 152 (2016) 213-227.
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[51] Subhash C. Narula, H. Roland Weistroffer (1989). A flexible method for nonlinear criteria decisionmaking problems. Ieee Transactions on Systems, Man and Cybernetics, vol. 19 , no. 4, July/August 1989, pp. 883-887.
[52] 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.
[53] Rashmi Ranjan Ota, A. K. Ojha (2015). A comparative study on optimization techniques for solving multi-objective geometric programming problems. Applied mathematical sciences, vol. 9, 2015, no. 22, 1077-1085. http://sites.google.com/site/ijcsis/
[54] OPTI Toolbox, Mixed Integer Nonlinear Program (MINLP). https://www.inverseproblem.co.nz/OPTI/index.php/Probs/MINLP
[55] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design, Second Edition. Cambridge University Press, 2000.
[56] Ciara Pike-Burke. Multi-Objective Optimization. https://www.lancaster.ac.uk/pg/pikeburc/report1.pdf.
[57] R. V. Rao, P. J. Pawar, J. P. Davim (2010). Parameter optimization of ultrasonic machining process using nontraditional optimization algorihms. Materials and Manufacturing Processes, 25 (10),1120-1130.
[58] 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.
[59] Ali Sadollah, Hadi Eskandar, Joong Hoon Kim (2015). Water cycle algorithm for solvinfg constrained multi-objective optimization problems. Applied Soft Computing 27 (2015) 279-298.
[60] Vikas Sharma (2012). Multiobjective integer nonlinear fractional programming problem: A cutting plane approach, OPSEARCH of the Operational Research Society of India (April-June 2012), 49(2):133-153.
[61] Vikas Sharma, Kalpana Dahiya, Vanita Verma (2017). A ranking algorithm for bi-objective quadratic fractional integer programming problems, Optimization, 66:11, 1913-1929.
[62] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.
[63] Isaac Siwale. A Note on Multi-Objective Mathematical Problems. https://www.aptech.com/wp-content/uploads/2012/09/MultObjMath.pdf
[64] 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.
[65] 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
[66] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. Journal of computational design and engineering 5 (2018) 104-119.
[67] 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.
[68] 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.
[69] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[70] 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 3-objective integer nonlinear programming problem from Sharma [60, p. 149]:
Maximize ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2
maximize (X(1) / (X(2) + 1)) ^ 2
maximize ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2
subject to
X(1) + X(2) <= 5
4 * X(1) + X(2) <= 8
0 <=X(1) <=4
0<= X(2)<=2
X(1) and X(2) are general integer variables.
This paper tries to alleviate the following issue: "One issue with the approach is that it is necessary to preselect which objective to minimize and the epsilon values," Pike-Burke [56, p. 2] on the epsilon-constraint method. One notes that line 89 and line 90, which are 89 EPSI2 = -5 + RND * 10 and 90 EPSI3 = -5 + RND * 10, respectively, try different epsilon values. Also, one notes that here line 229 and line 235 are 229 IF (X(1) / (X(2) + 1)) ^ 2 < EPSI2 THEN 1670 and 235 IF ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2 < EPSI3 THEN 1670, respectively, while line 229 and line 235 of the earlier edition (of October 17, 2018, and of the present blog) are 229 IF (X(1) / (X(2) + 1)) ^ 2 < .3 THEN 1670 and 235 IF ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2 < .3 THEN 1670, respectively.
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)
81 FOR JJJJ = -32000 TO 32000
85 RANDOMIZE JJJJ
87 M = -3E+50
88 EPSI1 = -5 + RND * 10
89 EPSI2 = -5 + RND * 10
90 EPSI3 = -5 + RND * 10
92 A(1) = FIX(RND * 5)
93 A(2) = FIX(RND * 3)
128 FOR I = 1 TO 3200
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 2)
153 J = 1 + FIX(RND * 2)
154 REM GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 REM GOTO 169
162 REM IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1
169 NEXT IPP
171 X(1) = INT(X(1))
173 X(2) = INT(X(2))
177 IF X(1) < 0## THEN 1670
187 IF X(2) < 0## THEN 1670
188 IF X(1) > 4## THEN 1670
222 IF X(2) > 2## THEN 1670
226 IF X(1) + X(2) > 5 THEN 1670
227 IF 4 * X(1) + X(2) > 8 THEN 1670
228 REM IF ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2 < EPSI1 THEN 1670
229 IF (X(1) / (X(2) + 1)) ^ 2 < EPSI2 THEN 1670
235 IF ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2 < EPSI3 THEN 1670
415 PDU = ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2
417 REM
419 REM
466 P = PDU
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 2
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1522 g01star = ((2 * X(1) + X(2)) / (X(2) + 2)) ^ 2
1524 g02star = (X(1) / (X(2) + 1)) ^ 2
1526 g03star = ((X(1) + 2) / (2 * X(1) + X(2) + 2)) ^ 2
1557 GOTO 128
1670 NEXT I
1889 IF M < -99999999999 THEN 1999
1922 PRINT g01star, g02star, g03star
1924 PRINT A(1), A(2), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [69]. The complete output of a single run through JJJJ= -31955 is shown below:
1 .25 .36
1 1 1 -31992
.1111111111111111 0 .4444444444444444
0 1 .1111111111111111 -31991
1 .25 .36
1 1 1 -31990
1 1 .5625
1 0 1 -31988
0 0 1
0 0 0 -31984
.1111111111111111 0 .4444444444444444
0 1 .1111111111111111 -31883
0 0 1
0 0 0 -31881
4 4 .4444444444444444
2 0 4 -31979
4 4 .4444444444444444
2 0 4 -31974
1 .1111111111111111 .25
1 2 1 -31973
4 4 .4444444444444444
2 0 4 -31971
1 .25 .36
1 1 1 .31968
.25 0 .25
0 2 .25 -31967
1 .25 .36
1 1 1 -31966
1 .1111111111111111 .25
1 2 1 -31962
4 4 .4444444444444444
2 0 4 -31959
0 0 1
0 0 0 -31958
0 0 1
0 0 0 -31957
1 .1111111111111111 .25
1 2 1 -31956
1 1 .5625
1 0 1 -31955
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 [69], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31955 was 2 seconds, not including the time for “Creating .EXE file" (7 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Sharma [60, pp 149-151].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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