Jsun Yui Wong
The complete computer program below tries to solve Problem 9 in Lawler and Bell [4 and 5]. Line 181 through line 1111 of the following computer program partly describe the problem.
The following computer program has been modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [8]. Line 501 through line 711 are illustrative. The intent behind line 501 through line 518 is to produce domino effect; X(9) through X(26) are slack variables. Looking at line 501, one sees that it is not hard to knock down X(9), domino one.
0 REM DEFDBL A-Z
2 DEFINT I,J,K,X
3 DIM B(519),N(519),A(2002),H(519),L(519),U(519),X(2002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
20 FOR J44=1 TO 8
21 A(J44)=FIX(RND*8)
22 NEXT J44
128 FOR I=1 TO 100
129 FOR KKQQ=1 TO 8
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO FIX(RND*8)
181 J=1+FIX(RND*8)
183 REM R=(1-RND*2)*A(J)
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 REM GOTO 222
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
396 NEXT IPP
443 FOR J44=1 TO 8
446 IF X(J44)<0 THEN 1670
449 NEXT J44
501 X(9)= +7 -X(1)
502 X(10)= +7 -X(3)
503 X(11)= +7 -X(4)
504 X(12)= +7 -X(6)
505 X(13)= +7 -X(8)
506 X(14)= +15 - X(2)
507 X(15)= +15 - X(5)
508 X(16)= +15 - X(7)
509 X(17)= -41 + 11* X(1) +7* X(4) + 13* X(6)
510 X(18)= -12 + 2* X(1) + 2* X(4) +8*X(8)
511 X(19)= -42 +3* X(2) + 5* X(5) + 7* X(8)
512 X(20)= -13 + 4* X(3)*X(7) + X(5)
513 X(21)= 31 - X(3) -4* X(5) -2* X(6) - 9* X(8)
514 X(22)= 73 - 12* X(2) - 8* X( 2 )*X(8) -2* X(3) * X(6)
515 X(23)= -60 + 6* X(2) +9* X(4) *X(6) + 5* X(7)
516 X(24)= -53 +6* X(2)*X(7) +9* X(3)+5*X(5)
517 X(25)= 47 -9* X(1)* X(8) - 6* X(3) *X(5) - 4* X(3)*X(7)
518 X(26)= 69 -2* X(1) -4*X(2) - 7* X(4) -3*X(5) - X(7)
519 REM X(27)= -4 + X(1) +2* X(2) + X(4)
554 FOR J44=9 TO 26
555 IF X(J44)<0 THEN PS(J44)=ABS(X(J44)) ELSE PS(J44)=0
556 NEXT J44
703 POBAT= - X(1) -X(2) -X(3) -X(4) -X(5) -X(6)-X(7)-X(8)
711 POBA= POBAT -999999999#*( PS(9)+PS(10)+PS(11)+PS(12)+PS(13)+PS(14)+PS(15)+PS(16)+ PS(17)+PS(18)+PS(19)+PS(20)+PS(21)+PS(22) +PS(23)+PS(24)+PS(25)+PS(26) )
866 P=POBA
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBAT=POBAT
1454 FOR KLX=1 TO 26
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1899 IF M<-20 THEN 2222
2001 PRINT A(1),A(2),A(3),A(4),A(5)
2002 PRINT A(6),A(7),A(8),A(9),A(10)
2003 REM PRINT A(11),A(12),A(13),A(14),A(15)
2004 REM PRINT A(16),A(17),A(18),A(19),A(20)
2005 REM PRINT A(21),A(22),A(23),A(24),A(25)
2022 PRINT A(26),M,JJJJ,PPOBAT
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31956 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.
3 4 1 3 6
1 2 0 4 6
6 -20 -31979 -20
3 4 1 3 6
1 2 0 4 6
6 -20 -31977 -20
2 4 1 4 6
1 2 0 5 6
1 -20 -31975 -20
3 4 1 3 6
1 2 0 4 6
6 -20 -31966 -20
2 4 1 4 6
1 2 0 5 6
1 -20 -31956 -20
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31956 was three seconds.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[3] 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.
[4] 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.
[5] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May-June, 1967), p. 578.
[6] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[7] 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.
[8] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
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April 2012
04/24/2012
The Domino Method of General Integer Nonlinear Programming Applied to Problem 9 of Lawler and Bell
Posted at 10:32 AM | Permalink | Comments (0)
The Domino Method of General Integer Nonlinear Programming Applied to Problem 7 of Lawler and Bell
Jsun Yui Wong
The complete computer program below tries to solve Problem 7 in Lawler and Bell [4]. Line 181 through line 1111 of the following computer program partly describe the problem.
The following computer program has been modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [8]. Line 501 through line 711 are illustrative. The intent behind line 501 through line 518 is to produce domino effect; X(9) through X(26) are slack variables. Looking at line 501, one sees that it is not hard to knock down X(9), domino one.
0 REM DEFDBL A-Z
2 DEFINT I,J,K,X
3 DIM B(519),N(519),A(2002),H(519),L(519),U(519),X(2002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
20 FOR J44=1 TO 8
21 A(J44)=FIX(RND*8)
22 NEXT J44
128 FOR I=1 TO 100
129 FOR KKQQ=1 TO 8
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO FIX(RND*8)
181 J=1+FIX(RND*8)
183 REM R=(1-RND*2)*A(J)
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 REM GOTO 222
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
396 NEXT IPP
443 FOR J44=1 TO 8
446 IF X(J44)<0 THEN 1670
449 NEXT J44
501 X(9)= +7 -X(1)
502 X(10)= +7 -X(3)
503 X(11)= +7 -X(4)
504 X(12)= +7 -X(6)
505 X(13)= +7 -X(8)
506 X(14)= +15 - X(2)
507 X(15)= +15 - X(5)
508 X(16)= +15 - X(7)
509 X(17)= -41 + 11* X(1) +7* X(4) + 13* X(6)
510 X(18)= -12 + 2* X(1) + 2* X(4) +8*X(8)
511 X(19)= -42 +3* X(2) + 5* X(5) + 7* X(8)
512 X(20)= -13 + 4* X(3)*X(7) + X(5)
513 X(21)= 31 - X(3) -4* X(5) -2* X(6) - 9* X(8)
514 X(22)= 73 - 12* X(2) - 8* X( 2 )*X(8) -2* X(3) * X(6)
515 X(23)= -60 + 6* X(2) +9* X(4) *X(6) + 5* X(7)
516 X(24)= -53 +6* X(2)*X(7) +9* X(3)+5*X(5)
517 X(25)= 47 -9* X(1)* X(8) - 6* X(3) *X(5) - 4* X(3)*X(7)
518 X(26)= 69 -2* X(1) -4*X(2) - 7* X(4) -3*X(5) - X(7)
519 REM X(27)= -4 + X(1) +2* X(2) + X(4)
554 FOR J44=9 TO 26
555 IF X(J44)<0 THEN PS(J44)=ABS(X(J44)) ELSE PS(J44)=0
556 NEXT J44
703 POBAT= -3* X(1)*X(4) -X(2) -X(2)*X(5) -X(2)*X(7) -6*X(3)*X(8) -X(5)*X(7) -11*X(6)
711 POBA= POBAT -999999999#*( PS(9)+PS(10)+PS(11)+PS(12)+PS(13)+PS(14)+PS(15)+PS(16)+ PS(17)+PS(18)+PS(19)+PS(20)+PS(21)+PS(22) +PS(23)+PS(24)+PS(25)+PS(26) )
866 P=POBA
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBAT=POBAT
1454 FOR KLX=1 TO 26
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1899 IF M<-83 THEN 2222
2001 PRINT A(1),A(2),A(3),A(4),A(5)
2002 PRINT A(6),A(7),A(8),A(9),A(10)
2003 PRINT A(11),A(12),A(13),A(14),A(15)
2004 PRINT A(16),A(17),A(18),A(19),A(20)
2005 PRINT A(21),A(22),A(23),A(24),A(25)
2022 PRINT A(26),M,JJJJ,PPOBAT
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31912 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.
2 4 1 4 6
1 2 0 5 6
3 6 7 11 9
13 22 0 0 1
4 23 10 34 3
1 -83 -31975 -83
2 4 1 4 6
1 2 0 5 6
3 6 7 11 9
13 22 0 0 1
4 23 10 34 3
1 -83 -31943 -83
2 4 1 4 6
1 2 0 5 6
3 6 7 11 9
13 22 0 0 1
4 23 10 34 3
1 -83 -31912 -83
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31912 was five seconds.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[3] 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.
[4] 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.
[5] E. L. Lawler, M. D. Bell, Errata: A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 15, No. 3 (May-June, 1967), p. 578.
[6] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[7] 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.
[8] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
Posted at 05:35 AM | Permalink | Comments (0)
04/23/2012
The Domino Method of General Integer Nonlinear Programming Applied to Problem 5 of Lawler and Bell
Jsun Yui Wong
The computer program below tries to solve Problem 5 in Lawler and Bell [8]. Line 181 through line 1111 of the following computer program partly describe the problem.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [20]. Line 501 through line 711 are illustrative. The intent behind line 501 through line 513 is to produce domino effect; X(8) through X(20) are slack variables. Looking at line 501, one sees that it is not hard to knock down X(8), domino one.
0 REM DEFDBL A-Z
2 DEFINT I,J,K,X
3 DIM B(519),N(519),A(2002),H(519),L(519),U(519),X(2002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
20 FOR J44=1 TO 7
21 A(J44)=FIX(RND*8)
22 NEXT J44
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 100
129 FOR KKQQ=1 TO 7
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO FIX(RND*7)
181 J=1+FIX(RND*7)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 REM X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 REM GOTO 222
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
396 NEXT IPP
443 FOR J44=1 TO 7
446 IF X(J44)<0 THEN 1670
449 NEXT J44
501 X(8)= +7 -X(1)
502 X(9)= +7 -X(2)
503 X(10)= +7 -X(3)
504 X(11)= +15 -X(4)
505 X(12)= +15 -X(5)
506 X(13)= +7 - X(6)
507 X(14)= -6 + X(1) + X(2) +X(3)
508 X(15)= +7 - 3* X(1) -2* X(3) - X(5)
509 X(16)= -7 + X(1) *X(6)+ X(2) +3* X(5)
510 X(17)= -8 + X(4) + X(5) + 6*X(6)
511 X(18)= +20 -3* X(1)*X(3) -6* X(4) - 4* X(5)
512 X(19)= 15 -4* X(1) - 2* X(3) - X(6) * X(7)
513 X(20)= -25 +4* X(2)* X(7) + 3* X(4)*X(5)
554 FOR J44=8 TO 20
555 IF X(J44)<0 THEN PS(J44)=ABS(X(J44)) ELSE PS(J44)=0
556 NEXT J44
703 POBA2= - X(1)^2 - X(2)^2 -X(3)^2 -X(4)^2 - X(5)^2 -X(6)^2- X(7)^2
711 POBA= POBA2 -999999999#*( PS(8)+PS(9)+PS(10)+PS(11)+PS(12)+PS(13)+PS(14)+PS(15)+PS(16)+ PS( 17)+PS(18 )+PS(19 )+PS(20 ) )
759 POB1=POBA
863 P1NEWMAY=POB1
866 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 20
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1899 IF M<-26 THEN 2222
2001 PRINT A(1),A(2),A(3),A(4),A(5)
2002 PRINT A(6),A(7),A(8),A(9),A(10)
2003 PRINT A(11),A(12),A(13),A(14),A(15)
2004 PRINT A(16),A(17),A(18),A(19),A(20)
2011 PRINT M,JJJJ,PPOBA2
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31983 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.
1 4 1 1 1
1 2 6 3 6
14 14 6 0 1
1 0 7 7 10
-25 -31996 -25
1 4 1 1 1
1 2 6 3 6
14 14 6 0 1
1 0 7 7 10
-25 -31987 -25
1 4 1 1 1
1 2 6 3 6
14 14 6 0 1
1 0 7 7 10
-25 -31983 -25
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31983 was two seconds.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed-Discrete Engineering Optimization Problems, Engineering Optimization, 43:5, 507-523.
[6] 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.
[7] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[8] 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.
[9] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[10] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[11] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[12] 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.
[13] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[14] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[15] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[16] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[17] P. B. Thanedar, G. N. Vanderplaats (1995), Survey of Discrete Variable Optimization for Structural Design. Journal of Structural Engineering, ASCE, Volume 121 (2), pp. 301-306.
[18] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[19] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[20] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[21] Ji-Hui Zhang, Xin-He Xu, An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26 (1999) 645-663.
Posted at 06:16 AM | Permalink | Comments (0)
04/22/2012
The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design , Fourth Edition
Jsun Yui Wong
The computer program below tries to solve the stepped cantilever beam problem considered by Thanedar and Vanderplaats [17], Chen and Chen [2], and Jaberipour and Khorram [5], among others. Line 181 through line 1111 of the following computer program partly describe the problem.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [20]. Line 505 through line 711 are illustrative. Domino one through domino eleven are slack variables X(11) through X(21) at line 505 through line 577, respectively. One notes that line 505 and line 521 indicate that when domino one, X(11), is knocked down, domino two, X(12), is down.
This paper features line 133, which is 133 FOR IPP=1 TO FIX(RND*10).
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(2002),H(519),L(519),U(519),X(2002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
21 A(1)=1+RND*4
22 A(2)=30+RND*35
23 A(3)=1+RND*4
24 A(4)=30+RND*35
25 A(5)=1+FIX(RND*5 )
26 A(6)=30+FIX(RND*36 )
27 IF RND<.25 THEN A(7)=2.4 ELSE IF RND<.33 THEN A(7)=2.6 ELSE IF RND<.5 THEN A(7)=2.8 ELSE A(7)=3.1
28 IF RND<.25 THEN A(8)=45 ELSE IF RND<.33 THEN A(7)=50 ELSE IF RND<.5 THEN A(7)=55 ELSE A(7)=60
29 IF RND<.25 THEN A(9)=2.4 ELSE IF RND<.33 THEN A(9)=2.6 ELSE IF RND<.5 THEN A(9)=2.8 ELSE A(9)=3.1
30 IF RND<.25 THEN A(10)=45 ELSE IF RND<.33 THEN A(10)=50 ELSE IF RND<.5 THEN A(10)=55 ELSE A(10)=60
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 5000
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO FIX(RND*10)
181 J=1+FIX(RND*10)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 REM X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05
185 IF J=1 THEN 301 ELSE IF J=2 THEN 311 ELSE IF J=3 THEN 321 ELSE IF J=4 THEN 331 ELSE IF J=5 THEN 341 ELSE IF J=6 THEN 351 ELSE IF J=7 THEN 361 ELSE IF J=8 THEN 371 ELSE IF J=9 THEN 381 ELSE 391
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 REM GOTO 222
189 REM X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 REM NEXT IPP
301 REM
302 R=(1-RND*2)*A(J)
303 X(J)=A(J)+ (RND^(RND*10))*R
305 GOTO 396
311 REM
312 R=(1-RND*2)*A(J)
313 X(J)=A(J)+ (RND^(RND*10))*R
315 GOTO 396
321 REM
322 R=(1-RND*2)*A(J)
323 X(J)=A(J)+ (RND^(RND*10))*R
325 GOTO 396
331 REM
332 R=(1-RND*2)*A(J)
333 X(J)=A(J)+ (RND^(RND*10))*R
335 GOTO 396
341 X(J)=1+FIX(RND*5 )
345 GOTO 396
351 X(J)=30+FIX(RND*36 )
355 GOTO 396
361 IF RND<.25 THEN X(J)=2.4 ELSE IF RND<.33 THEN X(J)=2.6 ELSE IF RND<.5 THEN X(J)=2.8 ELSE X(J)=3.1
365 GOTO 396
371 IF RND<.25 THEN X(J)=45 ELSE IF RND<.33 THEN X(J)=50 ELSE IF RND<.5 THEN X(J)=55 ELSE X(J)=60
375 GOTO 396
381 IF RND<.25 THEN X(J)=2.4 ELSE IF RND<.33 THEN X(J)=2.6 ELSE IF RND<.5 THEN X(J)=2.8 ELSE X(J)=3.1
385 GOTO 396
391 IF RND<.25 THEN X(J)=45 ELSE IF RND<.33 THEN X(J)=50 ELSE IF RND<.5 THEN X(J)=55 ELSE X(J)=60
396 NEXT IPP
443 FOR J44=1 TO 10
446 IF X(J44)<1 THEN 1670
449 NEXT J44
505 X(11)=-X(2)+20*X(1)
521 X(12)= -4.2857+(X(1)*X(2)^2) / 1000
536 X(13)=-X(4)+20*X(3)
537 X(14)= -2.1429+(X(3)*X(4)^2) / 1000
547 X(15)=-X(6)+20*X(5)
548 X(16)= -10.7143+(X(5)*X(6)^2) / 1000
558 X(17)=-X(8)+20*X(7)
559 X(18)= -8.5714+(X(7)*X(8)^2) / 1000
569 X(19)=-X(10)+20*X(9)
570 X(20)= -6.4268+(X(9)*X(10)^2) / 1000
577 X(21)= - 10000*( 244/(X(5)*X(6)^3 ) +148/ ( X(7)*X(8)^3 ) +76/(X(9)*X(10)^3 ) +28/ ( X(1)*X(2)^3 ) +4/ (X(3)*X(4)^3) ) +1086
601 FOR J44=11 TO 21
603 IF X(J44)<0 THEN PS(J44)=ABS(X(J44)) ELSE PS(J44)=0
605 NEXT J44
703 POBA2=-100*( X(5)*X(6)+X(7)*X(8)+X(9)*X(10) +X(1)*X(2) +X(3)*X(4) )
711 POBA= POBA2 -999999999#*( PS(11)+PS(12)+ PS(13)+PS(14)+PS(15)+PS(16)+PS(17)+PS(18)+PS(19)+PS(20)+PS(21) )
759 POB1=POBA
863 P1NEWMAY=POB1
866 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 21
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1899 IF M<-63911! THEN 2222
1999 PRINT A(1),A(2),A(3),A(4),A(5)
2000 PRINT A(6),A(7),A(8),A(9),A(10)
2001 PRINT A(11),A(12),A(13),A(14),A(15)
2002 PRINT A(16),A(17),A(18),A(19),A(20)
2003 PRINT A(21),M,JJJJ,PPOBA2
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-30663 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.
2.204837 44.08823 1.750975 34.98332 3
60 3.1 55 2.6 50
8.525848E-03 0 3.617859E-02
0 0
8.570004E-02 7 .8060999 2
7.320023E-02
1075.011 -63896.23 -31181 -63896.23
2.207199 44.06465 1.752492 34.96821 3
60 3.1 55 2.6 50
7.932663E-02 2.384186E-06 8.163834E-02
4.768372E-06 0
8.570004E-02 7 .8060999 2
7.320023E-02
1075.01 -63904.09 -30663 -63904.09
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-30663 was five hours.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed-Discrete Engineering Optimization Problems, Engineering Optimization, 43:5, 507-523.
[6] 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.
[7] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[8] 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.
[9] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[10] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[11] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[12] 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.
[13] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[14] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[15] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[16] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[17] P. B. Thanedar, G. N. Vanderplaats (1995), Survey of Discrete Variable Optimization for Structural Design. Journal of Structural Engineering, ASCE, Volume 121 (2), pp. 301-306.
[18] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[19] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[20] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[21] Ji-Hui Zhang, Xin-He Xu, An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26 (1999) 645-663.
Posted at 08:40 AM | Permalink | Comments (0)
04/21/2012
Errata to: The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design , Second Edition
Jsun Yui Wong
The output should read as follows:
2.205983 44.07694 1.751892 34.97447 3
60 3.1 55 2.6 50
4.271698E-02 3.242493E-05 6.336975E-02
3.767014E-05 0
8.570004E-02 7 .8060999 2
7.320023E-02
1075.011 -63900.45 -31900 -63900.45
2.204677 44.08984 1.751359 34.9795 3
60 3.1 55 2.6 50
3.711701E-03 2.384186E-06 4.768372E-02
1.907349E-06 0
8.570004E-02 7 .8060999 2
7.320023E-02
1075.011 -63896.55 -31480 -63896.55
Posted at 02:59 PM | Permalink | Comments (0)
The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design , Third Edition
Jsun Yui Wong
The computer program below tries to solve the stepped cantilever beam problem considered by Thanedar and Vanderplaats [17], Chen and Chen [2], and Jaberipour and Khorram [5], among others. Line 181 through line 1111 of the following computer program partly describe the problem.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [20]. Line 505 through line 711 are illustrative. Domino one through domino eleven are slack variables X(11) through X(21) at line 505 through line 577, respectively. One notes that line 505 and line 521 indicate that when domino one, X(11), is knocked down, domino two, X(12), is down.
In line 401 through line 430 the lower bounds and the upper bounds of the ten variables are shown.
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(2002),H(519),L(519),U(519),X(2002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
21 A(1)=1+RND*4
22 A(2)=30+RND*35
23 A(3)=1+RND*4
24 A(4)=30+RND*35
25 A(5)=1+FIX(RND*5 )
26 A(6)=30+FIX(RND*36 )
27 IF RND<.25 THEN A(7)=2.4 ELSE IF RND<.33 THEN A(7)=2.6 ELSE IF RND<.5 THEN A(7)=2.8 ELSE A(7)=3.1
28 IF RND<.25 THEN A(8)=45 ELSE IF RND<.33 THEN A(7)=50 ELSE IF RND<.5 THEN A(7)=55 ELSE A(7)=60
29 IF RND<.25 THEN A(9)=2.4 ELSE IF RND<.33 THEN A(9)=2.6 ELSE IF RND<.5 THEN A(9)=2.8 ELSE A(9)=3.1
30 IF RND<.25 THEN A(10)=45 ELSE IF RND<.33 THEN A(10)=50 ELSE IF RND<.5 THEN A(10)=55 ELSE A(10)=60
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 5000
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO 5
181 J=1+FIX(RND*10)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 REM X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05
185 IF J=1 THEN 301 ELSE IF J=2 THEN 311 ELSE IF J=3 THEN 321 ELSE IF J=4 THEN 331 ELSE IF J=5 THEN 341 ELSE IF J=6 THEN 351 ELSE IF J=7 THEN 361 ELSE IF J=8 THEN 371 ELSE IF J=9 THEN 381 ELSE 391
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 REM GOTO 222
189 REM X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 REM NEXT IPP
301 REM
302 R=(1-RND*2)*A(J)
303 X(J)=A(J)+ (RND^(RND*10))*R
305 GOTO 396
311 REM
312 R=(1-RND*2)*A(J)
313 X(J)=A(J)+ (RND^(RND*10))*R
315 GOTO 396
321 REM
322 R=(1-RND*2)*A(J)
323 X(J)=A(J)+ (RND^(RND*10))*R
325 GOTO 396
331 REM
332 R=(1-RND*2)*A(J)
333 X(J)=A(J)+ (RND^(RND*10))*R
335 GOTO 396
341 X(J)=1+FIX(RND*5 )
345 GOTO 396
351 X(J)=30+FIX(RND*36 )
355 GOTO 396
361 IF RND<.25 THEN X(J)=2.4 ELSE IF RND<.33 THEN X(J)=2.6 ELSE IF RND<.5 THEN X(J)=2.8 ELSE X(J)=3.1
365 GOTO 396
371 IF RND<.25 THEN X(J)=45 ELSE IF RND<.33 THEN X(J)=50 ELSE IF RND<.5 THEN X(J)=55 ELSE X(J)=60
375 GOTO 396
381 IF RND<.25 THEN X(J)=2.4 ELSE IF RND<.33 THEN X(J)=2.6 ELSE IF RND<.5 THEN X(J)=2.8 ELSE X(J)=3.1
385 GOTO 396
391 IF RND<.25 THEN X(J)=45 ELSE IF RND<.33 THEN X(J)=50 ELSE IF RND<.5 THEN X(J)=55 ELSE X(J)=60
396 NEXT IPP
401 IF X(1)<1 THEN 1670
402 IF X(2)<30 THEN 1670
403 IF X(3)<1 THEN 1670
404 IF X(4)<30 THEN 1670
405 IF X(5)<1 THEN 1670
406 IF X(6)<30 THEN 1670
407 IF X(7)<2.4 THEN 1670
408 IF X(8)<45 THEN 1670
409 IF X(9)<2.4 THEN 1670
410 IF X(10)<45 THEN 1670
421 IF X(1)>5 THEN 1670
422 IF X(2)>65 THEN 1670
423 IF X(3)>5 THEN 1670
424 IF X(4)>65 THEN 1670
425 IF X(5)>5 THEN 1670
426 IF X(6)>65 THEN 1670
427 IF X(7)>3.1 THEN 1670
428 IF X(8)>60 THEN 1670
429 IF X(9)>3.1 THEN 1670
430 IF X(10)>60 THEN 1670
505 X(11)=-X(2)+20*X(1)
521 X(12)= -4.2857+(X(1)*X(2)^2) / 1000
536 X(13)=-X(4)+20*X(3)
537 X(14)= -2.1429+(X(3)*X(4)^2) / 1000
547 X(15)=-X(6)+20*X(5)
548 X(16)= -10.7143+(X(5)*X(6)^2) / 1000
558 X(17)=-X(8)+20*X(7)
559 X(18)= -8.5714+(X(7)*X(8)^2) / 1000
569 X(19)=-X(10)+20*X(9)
570 X(20)= -6.4268+(X(9)*X(10)^2) / 1000
577 X(21)= - 10000*( 244/(X(5)*X(6)^3 ) +148/ ( X(7)*X(8)^3 ) +76/(X(9)*X(10)^3 ) +28/ ( X(1)*X(2)^3 ) +4/ (X(3)*X(4)^3) ) +1086
601 FOR J44=11 TO 21
603 IF X(J44)<0 THEN PS(J44)=ABS(X(J44)) ELSE PS(J44)=0
605 NEXT J44
703 POBA2=-100*( X(5)*X(6)+X(7)*X(8)+X(9)*X(10) +X(1)*X(2) +X(3)*X(4) )
711 POBA= POBA2 -999999999#*( PS(11)+PS(12)+ PS(13)+PS(14)+PS(15)+PS(16)+PS(17)+PS(18)+PS(19)+PS(20)+PS(21) )
759 POB1=POBA
863 P1NEWMAY=POB1
866 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 21
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1899 IF M<-63911! THEN 2222
1999 PRINT A(1),A(2),A(3),A(4),A(5)
2000 PRINT A(6),A(7),A(8),A(9),A(10)
2001 PRINT A(11),A(12),A(13),A(14),A(15)
2002 PRINT A(16),A(17),A(18),A(19),A(20)
2003 PRINT A(21),M,JJJJ,PPOBA2
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The complete output through JJJJ=-31855 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.
2.209339 44.04339 1.750053 34.99278 3
60 3.1 55 2.6 50
.143383 1.859665E-05 8.274078E-03
2.932549E-05 0
8.570004E-02 7 .8060999 2
7.320023E-02
1075.01 -63904.6 -31945 -63904.6
2.207571 44.06111 1.750792 34.98518 3
60 3.1 55 2.6 50
9.032059E-02 3.671646E-05 3.065872E-02
3.576279E-06 0
8.570004E-02 7 .8060999 2
7.320023E-02
1075.01 -63901.98 -31855 -63901.98
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31855 was forty minutes.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed-Discrete Engineering Optimization Problems, Engineering Optimization, 43:5, 507-523.
[6] 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.
[7] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[8] 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.
[9] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[10] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[11] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[12] 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.
[13] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[14] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[15] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[16] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[17] P. B. Thanedar, G. N. Vanderplaats (1995), Survey of Discrete Variable Optimization for Structural Design. Journal of Structural Engineering, ASCE, Volume 121 (2), pp. 301-306.
[18] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[19] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[20] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[21] Ji-Hui Zhang, Xin-He Xu, An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26 (1999) 645-663.
Posted at 02:51 PM | Permalink | Comments (0)
The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design , Second Edition
Jsun Yui Wong
The computer program below tries to solve the stepped cantilever beam problem considered by Thanedar and Vanderplaats [17], Chen and Chen [2], and Jaberipour and Khorram [5], among others. Line 181 through line 1111 of the following computer program partly describe the problem.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [20]. Line 505 through line 711 are illustrative. Domino one through domino eleven are slack variables X(11) through X(21) at line 505 through line 577, respectively. One notes that line 505 and line 521 indicate that when domino one, X(11), is knocked down, domino two, X(12), is down.
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(2002),H(519),L(519),U(519),X(2002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
21 A(1)=1+RND*4
22 A(2)=30+RND*35
23 A(3)=1+RND*4
24 A(4)=30+RND*35
25 A(5)=1+FIX(RND*5 )
26 A(6)=30+FIX(RND*36 )
27 IF RND<.25 THEN A(7)=2.4 ELSE IF RND<.33 THEN A(7)=2.6 ELSE IF RND<.5 THEN A(7)=2.8 ELSE A(7)=3.1
28 IF RND<.25 THEN A(8)=45 ELSE IF RND<.33 THEN A(7)=50 ELSE IF RND<.5 THEN A(7)=55 ELSE A(7)=60
29 IF RND<.25 THEN A(9)=2.4 ELSE IF RND<.33 THEN A(9)=2.6 ELSE IF RND<.5 THEN A(9)=2.8 ELSE A(9)=3.1
30 IF RND<.25 THEN A(10)=45 ELSE IF RND<.33 THEN A(10)=50 ELSE IF RND<.5 THEN A(10)=55 ELSE A(10)=60
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 5000
129 FOR KKQQ=1 TO 10
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO 5
181 J=1+FIX(RND*10)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 REM X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05
185 IF J=1 THEN 301 ELSE IF J=2 THEN 311 ELSE IF J=3 THEN 321 ELSE IF J=4 THEN 331 ELSE IF J=5 THEN 341 ELSE IF J=6 THEN 351 ELSE IF J=7 THEN 361 ELSE IF J=8 THEN 371 ELSE IF J=9 THEN 381 ELSE 391
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 REM GOTO 222
189 REM X(J)=A(J)+PA
190 REM X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 REM NEXT IPP
301 REM
302 R=(1-RND*2)*A(J)
303 X(J)=A(J)+ (RND^(RND*10))*R
305 GOTO 396
311 REM
312 R=(1-RND*2)*A(J)
313 X(J)=A(J)+ (RND^(RND*10))*R
315 GOTO 396
321 REM
322 R=(1-RND*2)*A(J)
323 X(J)=A(J)+ (RND^(RND*10))*R
325 GOTO 396
331 REM
332 R=(1-RND*2)*A(J)
333 X(J)=A(J)+ (RND^(RND*10))*R
335 GOTO 396
341 X(J)=1+FIX(RND*5 )
345 GOTO 396
351 X(J)=30+FIX(RND*36 )
355 GOTO 396
361 IF RND<.25 THEN X(J)=2.4 ELSE IF RND<.33 THEN X(J)=2.6 ELSE IF RND<.5 THEN X(J)=2.8 ELSE X(J)=3.1
365 GOTO 396
371 IF RND<.25 THEN X(J)=45 ELSE IF RND<.33 THEN X(J)=50 ELSE IF RND<.5 THEN X(J)=55 ELSE X(J)=60
375 GOTO 396
381 IF RND<.25 THEN X(J)=2.4 ELSE IF RND<.33 THEN X(J)=2.6 ELSE IF RND<.5 THEN X(J)=2.8 ELSE X(J)=3.1
385 GOTO 396
391 IF RND<.25 THEN X(J)=45 ELSE IF RND<.33 THEN X(J)=50 ELSE IF RND<.5 THEN X(J)=55 ELSE X(J)=60
396 NEXT IPP
443 FOR J44=1 TO 10
446 IF X(J44)<1 THEN 1670
449 NEXT J44
505 X(11)=-X(2)+20*X(1)
521 X(12)= -4.2857+(X(1)*X(2)^2) / 1000
536 X(13)=-X(4)+20*X(3)
537 X(14)= -2.1429+(X(3)*X(4)^2) / 1000
547 X(15)=-X(6)+20*X(5)
548 X(16)= -10.7143+(X(5)*X(6)^2) / 1000
558 X(17)=-X(8)+20*X(7)
559 X(18)= -8.5714+(X(7)*X(8)^2) / 1000
569 X(19)=-X(10)+20*X(9)
570 X(20)= -6.4268+(X(9)*X(10)^2) / 1000
577 X(21)= - 10000*( 244/(X(5)*X(6)^3 ) +148/ ( X(7)*X(8)^3 ) +76/(X(9)*X(10)^3 ) +28/ ( X(1)*X(2)^3 ) +4/ (X(3)*X(4)^3) ) +1086
601 FOR J44=11 TO 21
603 IF X(J44)<0 THEN PS(J44)=ABS(X(J44)) ELSE PS(J44)=0
605 NEXT J44
703 POBA2=-100*( X(5)*X(6)+X(7)*X(8)+X(9)*X(10) +X(1)*X(2) +X(3)*X(4) )
711 POBA= POBA2 -999999999#*( PS(11)+PS(12)+ PS(13)+PS(14)+PS(15)+PS(16)+PS(17)+PS(18)+PS(19)+PS(20)+PS(21) )
759 POB1=POBA
863 P1NEWMAY=POB1
866 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 21
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1899 IF M<-63911! THEN 2222
1999 PRINT A(1),A(2),A(3),A(4),A(5)
2000 PRINT A(6),A(7),A(8),A(9),A(10)
2001 PRINT A(11),A(12),A(13),A(14),A(15)
2002 PRINT A(16),A(17),A(18),A(19),A(20)
2003 PRINT A(21),M,JJJJ,PPOBA2
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The top-two candidate solutions through JJJJ=-31480 are shown below. What follows is a hand copy; immediately below there is no rounding by hand.
2.205983 44.07694 1.751892 34.97447 3
60 3.1 55 2.6 50
4.271698E-02 6.336975E-02 0
7 2
3.242493E-05 3.767014E-05 8.570004E-02
.8060999 7.320023E-02
1075.011 -63900.45 -31900 -63900.45
2.204677 44.08984 1.751359 34.9795 3
60 3.1 55 2.6 50
3.711701E-03 4.768372E-02 0
7 2
2.384186E-06 1.907349E-06 8.570004E-02
.8060999 7.320023E-02
1075.011 -63896.55 -31480 -63896.55
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31480 was two hours and a half.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] M. Jaberipour, E. Khorram (2011), A New Harmony Search Algorithm for Solving Mixed-Discrete Engineering Optimization Problems, Engineering Optimization, 43:5, 507-523.
[6] 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.
[7] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[8] 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.
[9] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[10] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[11] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[12] 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.
[13] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[14] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[15] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[16] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[17] P. B. Thanedar, G. N. Vanderplaats (1995), Survey of Discrete Variable Optimization for Structural Design. Journal of Structural Engineering, ASCE, Volume 121 (2), pp. 301-306.
[18] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[19] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[20] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[21] Ji-Hui Zhang, Xin-He Xu, An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26 (1999) 645-663.
Posted at 06:37 AM | Permalink | Comments (0)
04/19/2012
The Domino Method of General Integer Nonlinear Programming Applied to a Test Function of Fifty Continuous Variables and Fifty General Integer Variables, Second Edition
Jsun Yui Wong
The following computer program tries to solve Problem 4 of Chen and Chen [2, pp. 50-53]. Line 181 through line 1111 and line 1772 through line 1868 of the following computer program partly describe the problem. Noteworthy is line 184, which is 184 X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [18].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(1002),H(519),L(519),U(519),X(1002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
71 FOR J44=1 TO 100
74 A(J44)=-RND*5
77 NEXT J44
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 200
129 FOR KKQQ=1 TO 100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO 2
181 J=1+FIX(RND*100)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
243 FOR J44=1 TO 100
246 IF X(J44)<-10 THEN 1670
248 IF X(J44)>10 THEN 1670
249 NEXT J44
355 PSUM=0
357 FOR J44=1 TO 100
359 PSUM=PSUM+(1/100)*( X(J44)^4-16*X(J44)^2+5*X(J44) )
366 NEXT J44
403 POBA2=-PSUM
411 POBA= POBA2
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 100
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1772 FOR J44=51 TO 100
1777 AA(J44) = CINT (A(J44) )
1788 NEXT J44
1791 PSUM1=0
1792 FOR J44=1 TO 50
1793 PSUM1=PSUM1+( A(J44)^4-16*A(J44)^2+5*A(J44) )
1794 NEXT J44
1799 PSUM2=0
1857 FOR J44=51 TO 100
1859 PSUM2=PSUM2+( AA(J44)^4-16*AA(J44)^2+5*AA(J44) )
1866 NEXT J44
1868 PFUN=(1/100)*(PSUM1+PSUM2)
1871 PRINT A(1),A(2),A(3),A(4),A(5)
1872 PRINT A(6),A(7),A(8),A(9),A(10)
1873 PRINT A(11),A(12),A(13),A(14),A(15)
1874 PRINT A(16),A(17),A(18),A(19),A(20)
1875 PRINT A(21),A(22),A(23),A(24),A(25)
1876 PRINT A(26),A(27),A(28),A(29),A(30)
1877 PRINT A(31),A(32),A(33),A(34),A(35)
1878 PRINT A(36),A(37),A(38),A(39),A(40)
1879 PRINT A(41),A(42),A(43),A(44),A(45)
1880 PRINT A(46),A(47),A(48),A(49),A(50)
1881 PRINT AA(51),AA(52),AA(53),AA(54),AA(55)
1882 PRINT AA(56),AA(57),AA(58),AA(59),AA(60)
1883 PRINT AA(61),AA(62),AA(63),AA(64),AA(65)
1884 PRINT AA(66),AA(67),AA(68),AA(69),AA(70)
1885 PRINT AA(71),AA(72),AA(73),AA(74),AA(75)
1886 PRINT AA(76),AA(77),AA(78),AA(79),AA(80)
1887 PRINT AA(81),AA(82),AA(83),AA(84),AA(85)
1888 PRINT AA(86),AA(87),AA(88),AA(89),AA(90)
1889 PRINT AA(91),AA(92),AA(93),AA(94),AA(95)
1899 PRINT AA(96),AA(97),AA(98),AA(99),AA(100),M,JJJJ,PFUN,PSUM1,PSUM2
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The output through JJJJ=-31997 is briefly shown below. What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.
-2.894538 -2.916732 -2.897156 -2.90236 -2.884825
-2.88068 -2.900869 -2.904549 -2.888273 -2.890312
-2.912056 -2.90072 -2.91055 -2.891103 -2.906806
-2.925113 -2.911441 -2.916215 -2.898119 -2.901229
-2.882945 -2.902141 -2.889028 -2.882448 -2.927424
-2.904305 -2.925745 -2.892661 -2.885703 -2.912717
-2.900761 -2.916887 -2.915223 -2.917567 -2.905979
-2.891907 -2.918138 -2.924882 -2.885847 -2.896884
-2.919468 -2.922116 -2.890473 -2.893988 -2.895194
-2.893813 -2.878756 -2.919025 -2.90041 -2.887194
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
78.32513 -32000 -78.16301 -3916.301 -3900
In summary, the PFUN values at JJJJ=-32000, JJJJ=-31999, JJJJ=-31998, and JJJJ=-31997 are
-78.16301, -77.95125, -78.16237, and -78.15838, respectively.
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31997 was 90 seconds.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] 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.
[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[7] 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.
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[9] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] 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.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[16] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[19] Ji-Hui Zhang, Xin-He Xu (1999), An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26, 645-663.
Posted at 11:02 AM | Permalink | Comments (0)
The Domino Method of General Integer Nonlinear Programming Applied to a Problem with 100 General Integer Variables
Jsun Yui Wong
The following computer program tries to solve a modified version of Problem 4 of Chen and Chen [2, pp. 50-53]. Here the 100 variables are general integer variables; there 50 variables are continuous variables and 50 are general integer variables. Line 181 through line 1111 and line 1773 through line line 1868 of the following computer program partly describe the problem. Noteworthy is line 184, which is 184 X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [18].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(1002),H(519),L(519),U(519),X(1002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
71 FOR J44=1 TO 100
74 A(J44)=-RND*5
77 NEXT J44
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 200
129 FOR KKQQ=1 TO 100
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO 2
181 J=1+FIX(RND*100)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
243 FOR J44=1 TO 100
246 IF X(J44)<-10 THEN 1670
248 IF X(J44)>10 THEN 1670
249 NEXT J44
355 PSUM=0
357 FOR J44=1 TO 100
359 PSUM=PSUM+(1/100)*( X(J44)^4-16*X(J44)^2+5*X(J44) )
366 NEXT J44
403 POBA2=-PSUM
411 POBA= POBA2
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 100
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1772 REM FOR J44=51 TO 100
1773 FOR J44=1 TO 100
1777 AA(J44) = CINT (A(J44) )
1788 NEXT J44
1789 GOTO 1799
1791 PSUM1=0
1792 FOR J44=1 TO 100
1793 PSUM1=PSUM1+( A(J44)^4-16*A(J44)^2+5*A(J44) )
1794 NEXT J44
1799 PSUM2=0
1857 FOR J44=1 TO 100
1859 PSUM2=PSUM2+( AA(J44)^4-16*AA(J44)^2+5*AA(J44) )
1866 NEXT J44
1868 PFUN=(1/100)*(PSUM1+PSUM2)
1871 PRINT AA(1),AA(2),AA(3),AA(4),AA(5)
1872 PRINT AA(6),AA(7),AA(8),AA(9),AA(10)
1873 PRINT AA(11),AA(12),AA(13),AA(14),AA(15)
1874 PRINT AA(16),AA(17),AA(18),AA(19),AA(20)
1875 PRINT AA(21),AA(22),AA(23),AA(24),AA(25)
1876 PRINT AA(26),AA(27),AA(28),AA(29),AA(30)
1877 PRINT AA(31),AA(32),AA(33),AA(34),AA(35)
1878 PRINT AA(36),AA(37),AA(38),AA(39),AA(40)
1879 PRINT AA(41),AA(42),AA(43),AA(44),AA(45)
1880 PRINT AA(46),AA(47),AA(48),AA(49),AA(50)
1881 PRINT AA(51),AA(52),AA(53),AA(54),AA(55)
1882 PRINT AA(56),AA(57),AA(58),AA(59),AA(60)
1883 PRINT AA(61),AA(62),AA(63),AA(64),AA(65)
1884 PRINT AA(66),AA(67),AA(68),AA(69),AA(70)
1885 PRINT AA(71),AA(72),AA(73),AA(74),AA(75)
1886 PRINT AA(76),AA(77),AA(78),AA(79),AA(80)
1887 PRINT AA(81),AA(82),AA(83),AA(84),AA(85)
1888 PRINT AA(86),AA(87),AA(88),AA(89),AA(90)
1889 PRINT AA(91),AA(92),AA(93),AA(94),AA(95)
1899 PRINT AA(96),AA(97),AA(98),AA(99),AA(100),M,JJJJ,PFUN,PSUM1,PSUM2
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The output through JJJJ=-31997 is shown below. What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
78.32513 -32000 -78 0 -7800
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
78.11271 -31999 -78 0 -7800
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
78.3242 -31998 -78 0 -7800
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
-3 -3 -3 -3 -3
78.31976 -31997 -78 0 -7800
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31997 was 90 seconds.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] 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.
[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[7] 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.
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[9] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] 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.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[16] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[19] Ji-Hui Zhang, Xin-He Xu (1999), An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26, 645-663.
Posted at 08:07 AM | Permalink | Comments (0)
The Domino Method of General Integer Nonlinear Programming Applied to a Problem with 250 Continuous Variables and 250 General Integer Variables
Jsun Yui Wong
The following computer program is concerned with extending Problem 4 of Chen and Chen [2, pp. 50-53] to 250 continuous variables and 250 general integer variables from 50 continuous variables and 50 general integer variables. Line 181 through line 1111 and line 1772 through line line 1868 of the following computer program partly describe the problem. Noteworthy is line 184, which is 184 X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05.
The following computer program was originally modeled after the nuclear-chain-reaction picture on page 336 of the World Book Dictionary [1] and after the domino method of solving nonlinear systems of equations [18].
0 REM DEFDBL A-Z
2 DEFINT I,J,K
3 DIM B(519),N(519),A(1002),H(519),L(519),U(519),X(1002),D(511),P(511),PS(33),AA(1111)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+37
71 FOR J44=1 TO 500
74 A(J44)=-RND*5
77 NEXT J44
126 REM IMAR=10+FIX(RND*100)
128 FOR I=1 TO 200
129 FOR KKQQ=1 TO 500
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 FOR IPP=1 TO 5
181 J=1+FIX(RND*500)
182 REM GOTO 190
183 REM R=(1-RND*2)*A(J)
184 X(J)=A(J)+(FIX(RND*2))*.05-(FIX(RND*2))*.05
187 REM X(J)=A(J)+ (RND^(RND*10))*R
188 GOTO 222
189 REM X(J)=A(J)+PA
190 X(J)=A(J)+FIX(RND*2 )-FIX(RND*2)
191 REM X(J)=A(J)+FIX(RND*3)-FIX(RND*3)
222 NEXT IPP
243 FOR J44=1 TO 500
246 IF X(J44)<-10 THEN 1670
248 IF X(J44)>10 THEN 1670
249 NEXT J44
355 PSUM=0
357 FOR J44=1 TO 500
359 PSUM=PSUM+(1/500)*( X(J44)^4-16*X(J44)^2+5*X(J44) )
366 NEXT J44
403 POBA2=-PSUM
411 POBA= POBA2
459 POB1=POBA
463 P1NEWMAY=POB1
466 P=P1NEWMAY
1111 IF P<=M THEN 1670
1452 M=P
1453 PPOBA2=POBA2
1454 FOR KLX=1 TO 500
1455 A(KLX)= ( X(KLX) )
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1772 FOR J44=251 TO 500
1777 AA(J44) = CINT (A(J44) )
1788 NEXT J44
1791 PSUM1=0
1792 FOR J44=1 TO 250
1793 PSUM1=PSUM1+( A(J44)^4-16*A(J44)^2+5*A(J44) )
1794 NEXT J44
1799 PSUM2=0
1857 FOR J44=251 TO 500
1859 PSUM2=PSUM2+( AA(J44)^4-16*AA(J44)^2+5*AA(J44) )
1866 NEXT J44
1868 PFUN=(1/500)*(PSUM1+PSUM2)
1899 PRINT A(1),A(2),AA(498),AA(499),AA(500),M,JJJJ,PPOBA2,PFUN,PSUM1,PSUM2
2222 NEXT JJJJ
This BASIC computer program was run with Microsoft's GW BASIC 3.11 interpreter. The output through JJJJ=-31997 is shown below. What follows is a hand copy of the output on the computer-monitor screen; immediately below there is no rounding by hand.
-2.894538 -2.916732 -3 -3 -3
78.32247 -32000 78.32247 -78.16091 -19580.45
-19500
-2.939624 -2.896687 -3 -3 -3
78.2087 -31999 78.2087 -78.04816 -19524.08
-19500
-2.907243 -2.924778 -3 -3 -3
78.03731 -31998 78.03731 -77.86658 -19523.29
-19410
-2.881425 -2.922344 -3 -3 -3
78.31941 -31997 78.31941 -78.15898 -19579.49
-19500
On a personal computer with an Intel 2.66 chip and the IBM basica/D interpreter, version GW BASIC 3.11, the throughput time from JJJJ=-32000 through JJJJ=-31997 was 21 minutes.
References
[1] Clarence L. Barnhart, Robert K. Barnhart (Editors). The World Book Dictionary, 1993 World Book, Inc., a Scott Fetzer company, Chicago London Sydney Toronto.
[2] T. Y. Chen, H. C. Chen (2009): Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.
[3] William Conley, Computer Optimization Techniques. New York/Princeton: Petrocelli Books, Inc., Copyright 1980.
[4] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.
[5] 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.
[6] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[7] 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.
[8] Duan Li, Xiaoling Sun, Nonlinear Integer Programming. Boston: Springer, 2006.
[9] Han-Lin Li, Jung-Fa Tsai, A Distributed Computation Algorithm for Solving Portfolio Problems with Integer Variables. European Journal of Operational Research 186 (2008) 882-891.
[10] Harry M. Markowitz, Alan S. Manne, On the Solution of Discrete Programming Problems. Econometrica, Vol. 25, No. 1 (Jan., 1957), pp. 84-110.
[11] 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.
[12] Christina Oberlin, Stephen J. Wright, Active Set Identification in Nonlinear Programming. SIAM J. Optim., Vol. 17, No. 2, pp. 577-605.
[13] Singiresu S. Rao (2009), Engineering Optimization: Theory and Practice, Fourth Edition. New York: Wiley.
[14] E. Sandgren (1990): Nonlinear Integer and Discrete Programming in Mechanical Design Optimization. Journal of Mechanical Design, June 1990, Vol. 112, pp. 223-229.
[15] Dong Ku Shin, Z. Gurdal, O. H. Griffin Jr. (1990): A Penalty Approach for Nonlinear Optimization with Discrete Design Variables, Engineering Optimization, 16:1,29-42.
[16] Jung-Fa Tsai, Han-Lin Li, Nian-Ze Hu (2002): Global Optimization for Signomial Discrete Programming Problems in Engineering Design, Engineering Optimization, 34:6, 613-622.
[17] S. Wang, K. L. Teo, H. W. J. Lee (1998): A New Approach to Nonlinear Mixed Discrete Programming Problems, Engineering Optimization, 30:3-4. 249-262.
[18] Jsun Yui Wong (2011, April 17). The Domino Method of Solving Nonlinear Systems of Equations. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/04/17/
[19] Ji-Hui Zhang, Xin-He Xu (1999), An Efficient Evolutionary Programming Algorithm. Computers & Operations Research 26, 645-663.
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