Jsun Yui Wong
The following computer program seeks to solve the case II formulation of the welded beam design problem in Kashan [27, p. 1790]. Table 7 in Kashan [27, p. 1784] is very informative.
One notes that while line 245 of the computer program of the preceding paper is
245 JNEW= 2*2^.5*X(1)*X(2) * ( X(2)^2/12 + ( (X(1)+X(3) )/2 )^2 ), here line 246 is
246 JNEW= 2*(X(1)*X(2)/2^.5 ) * ( X(2)^2/12 + ( (X(1)+X(3) )/2 )^2 )). For details see page 1790 of Kashan {27}.
0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM DIM PE(35,35)
5 REM DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
105 A(1)=FIX(RND*10)
106 A(2)=FIX(RND*10)
107 A(3)=FIX(RND*10) *.5
109 A(4)=FIX(RND*10) *.5
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144 IF JJJJ<-32222 THEN 162 ELSE 164
145 IF B=3 OR B=4 GOTO 150 ELSE GOTO 162
150 IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2)*.5 ELSE X(B)=A(B)+FIX(1+RND*2)*.5
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625 ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162 IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 4
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B) -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
181 REM IF RND<.5 THEN 182 ELSE 191
182 REM X(5)= -.125 +X(1)
183 REM GOTO 201
191 REM X(5)=0
193 REM X(1)=.125
201 IF RND<.5 THEN 203 ELSE 207
203 X(6)=-X(1)+X(4)
204 GOTO 210
207 X(6)=0
209 X(4)=X(1)
210 IF ABS (( X(4)*X(3)^2 ) )<.0000001 THEN 1670
211 X(7)=30000-504000!/( X(4)*X(3)^2 )
213 IF ABS (( (30000000# )*X(4) * X(3)^3 ) ) <.0000001 THEN 1670
217 X(8)=.25-( 65856000#/( (30000000# )*X(4) * X(3)^3 ) )
221 PCBXB= ( ( 4.013*30000000#*( X(3)^2*X(4)^6 /36 )^.5 ) /196) * ( 1- ( (X(3)/28)* .790569415#) )
222 X(9)=-6000+PCBXB
229 IF ABS( (2^.5 *X(1)*X(2) ) )<.0000001 THEN 1670
231 TAUP=6000/ (2^.5 *X(1)*X(2) )
235 MNEW=6000*(14+X(2)/2 )
239 RNEW= ( X(2)^2/4 + ( (X(1)+X(3) )/2 )^2 )^.5
243 IF 2*(X(1)*X(2)/2^.5 ) * ( X(2)^2/12 + ( (X(1)+X(3) )/2 )^2 ) > 1000000000# THEN 1670
245 REM JNEW= 2*2^.5*X(1)*X(2) * ( X(2)^2/12 + ( (X(1)+X(3) )/2 )^2 )
246 JNEW= 2*(X(1)*X(2)/2^.5 ) * ( X(2)^2/12 + ( (X(1)+X(3) )/2 )^2 )
249 IF ABS (JNEW) <.0000001 THEN 1670
251 TAUPP=(MNEW*RNEW)/JNEW
253 IF ABS (2*RNEW ) <.0000001 THEN 1670
255 IF ( TAUP^2+2*TAUP*TAUPP*X(2)/2*RNEW +TAUPP^2 )<1E-09 THEN 1670
259 IF ABS (2*RNEW ) <.0000001 THEN 1670
261 TAUBXB= ( TAUP^2+2*TAUP*TAUPP*X(2)/ (2*RNEW ) +TAUPP^2 )^.5
281 X(10)= 13600-TAUBXB
301 REM X(11)= 5 -.10471*X(1)^2 -.04811 * X(3)*X(4) * (14+ X(2) )
321 IF X(1)<0 THEN 1670
322 IF X(2)<0 THEN 1670
323 IF X(3)<0 THEN 1670
324 IF X(4)<0 THEN 1670
325 IF X(1)>50 THEN 1670
326 IF X(2)>50 THEN 1670
327 IF X(3)>50 THEN 1670
328 IF X(4)>50 THEN 1670
421 FOR J44=5 TO 11
422 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
423 NEXT J44
425 PTOTAL=P(5)+P(6)+P(7) +P(8)+P(9)+P(10) +P(11)
569 P= - 1.10471*X(2)*X(1)^2-.04811*X(3)*X(4)*(14+X(2) ) +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 11
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-2.73 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1911 PRINT A(7),A(8),A(9)
1912 PRINT A(10),A(11)
1989 PRINT M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31997 is shown below. What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.
.2080101623961738 6.938305143773716 9.005127341555024
.2080101623961738 0 0
120.9856908663273 .2355482658009589 187.5135967737112
3.356329366020418 0
-2.218553454852496 -32000
.2088698247814008 6.950872136033321 8.968635053415744
.2088698247814008 0 0
1.321100559933257 .2354313491672269 247.7800307485399
27.20272961996238 0
-2.223158690383047 -31999
.2638885266630925 5.746340576392275 8.110526277757888
.2638885266630925 0 0
965.6601089670321 .2344078329711995 5763.846807381042
133.6346463987993 0
-2.475312941692234 -31998
.2013035201344999 6.386625002912507 10.00181038796852
.2013732745782423 0 6.975444374234321D-05
4980.912263959436 .2391047698010023 2.908515966737468D-06
59.35303075265756 0
-2.261333188873962 -31997
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output shown above was forty-five seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[41] Keith Meintjes and Alexander P. Morgan, Chemical Equilibrium Systems as Numerical Test Problems. ACM Transactions on Mathematical Software, Vol. 16, No. 2, June 1990, Pages 143-151.
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[43] Yuanbin Mo, Hetong Liu, Qin Wang,Conjugate Direction Particle Swarm Optimization Solving Systems of Nonlinear Equations. Computers and Mathematics with Applications 57 (2009) 1877-1882.
[44] Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud, An Introduction to Interval Analysis. Cambridge University Press, 2009.
[45] Panos Y. Papalambros, Douglass J. Wilde, Principles of Optimal Design: Modeling and Computation, Second Edition. Cambridge University Press, 2000.
[46] W. L. Price, Global Optimization by Controlled Random Search. Journal of Optimization Theory and Applications, July 1983, Volume 40, Issue 3, pp. 333-348.
[47] Singiresu S. Rao, Ying Xiong (2005), A Hybrid Genetic Algorithm for Mixed-Discrete Design Optimization. ASME Journal of Mechanical Design, Vol. 127, November 2005, pp. 1100-1112.
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[49] 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.
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[52] Jung-Fa Tsai (2010), Global Optimization for Signomial Discrete Programming Problems in Engineering Design. Engineering Optimization, 42:9, pp. 833-843.
[53] L.-W. Tsai, A. P. Morgan, Solving the Kinematics of the Most General Six- and Five-Degree-of Freedom Manipulators by Continuation Methods. Journal of Mechanisms, Transmissions, and Automation in Design, June 1985, Vol. 107, pp. 189-200.
[54] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/
[55] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/
[56] Jsun Yui Wong (2011, May 15). The Domino Method Applied to Solving a Nonlinear System of Ten Equations of a Model of Propane-in-Air Combustion, Fourth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/15/
[57] Jsun Yui Wong (2011, May 5). The Domino Method Applied to Solving a Nonlinear System of Five Equations, Third Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/05/05/
[58] Jsun Yui Wong (2011, July 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to an Alkylation-Process Model, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/07/27/
[59] Jsun Yui Wong (2012, April 10). The Domino Method of General Integer Nonlinear Programming Applied to a Coil Compression Spring Design. Retrieved from http://myblogsubstance.typepad.com/substance/2012/04/
[60] Jsun Yui Wong (2012, April 21). The Domino Method of General Integer Nonlinear Programming Applied to a Five-Step Cantilever Beam Design. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/04/21/
[61] Jsun Yui Wong (2012, April 24). The Domino Method of General Integer Nonlinear Programming Applied to Problem 10 of Lawler and Bell. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2012/4/24/
[62] Jsun Yui Wong (2013, January 14). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Integer Variables and Continuous Variables, Revised Edition. Retrieved from http://myblogsubstance.typepad.com/substance/2013/01/
[63] Jsun Yui Wong (2013, January 24). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Programming Problem with Eight 0-1 Variables and Nine Continuous Variables. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2013/01/24/
The following computer program seeks to solve the case II formulation of the welded beam design problem in Kashan [27, p. 1790]. Table 7 in Kashan [27, p. 1784] is very informative.
One notes that while line 245 of the computer program of the preceding paper is
245 JNEW= 2*2^.5*X(1)*X(2) * ( X(2)^2/12 + ( (X(1)+X(3) )/2 )^2 ), here line 246 is
246 JNEW= 2*(X(1)*X(2)/2^.5 ) * ( X(2)^2/12 + ( (X(1)+X(3) )/2 )^2 )). For details see page 1790 of Kashan {27}.
0 DEFDBL A-Z
1 DEFINT I,J,K
2 DIM B(99),N(99),A(2002),H(99),L(99),U(99),X(2002),D(111),P(111),PS(33),J(99),AA(99),HR(32),HHR(32),LHS(44),PLHS(44),LB(22),UB(22),PX(44),J44(44)
4 REM DIM PE(35,35)
5 REM DIM SD(35,35)
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-3D+30
105 A(1)=FIX(RND*10)
106 A(2)=FIX(RND*10)
107 A(3)=FIX(RND*10) *.5
109 A(4)=FIX(RND*10) *.5
126 IMAR=10+FIX(RND*1500)
128 FOR I=1 TO IMAR
129 FOR KKQQ=1 TO 4
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
139 FOR IPP=1 TO FIX(1+RND*3)
140 B=1+FIX(RND*4)
144 IF JJJJ<-32222 THEN 162 ELSE 164
145 IF B=3 OR B=4 GOTO 150 ELSE GOTO 162
150 IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2)*.5 ELSE X(B)=A(B)+FIX(1+RND*2)*.5
152 GOTO 179
154 IF RND<.5 THEN X(2)=A(2)-FIX(1+RND*2)*.0625 ELSE X(2)=A(2)+FIX(1+RND*2)*.0625
156 GOTO 179
162 IF RND<.5 THEN X(B)=A(B)-FIX(1+RND*2) ELSE X(B)=A(B)+FIX(1+RND*2)
163 GOTO 179
164 FOR J44=1 TO 4
165 IF ABS(X(J44))>32000 THEN 1670
166 NEXT J44
176 IF RND<.9 THEN R=(1-RND*2)*A(B) ELSE IF RND<.5 THEN R=(1-RND*2)*(A(B) -.1) ELSE R=(1-RND*2)*(A(B) +.1)
177 IF RND<.1 THEN X(B)=CINT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=CINT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
178 REM IF RND<.1 THEN X(B)=INT(A(B))-FIX(1+RND*3) ELSE IF RND<.1 THEN X(B)=INT(A(B))+FIX(1+RND*3) ELSE X(B)=A(B)+(RND^(RND*10) )*R
179 NEXT IPP
181 REM IF RND<.5 THEN 182 ELSE 191
182 REM X(5)= -.125 +X(1)
183 REM GOTO 201
191 REM X(5)=0
193 REM X(1)=.125
201 IF RND<.5 THEN 203 ELSE 207
203 X(6)=-X(1)+X(4)
204 GOTO 210
207 X(6)=0
209 X(4)=X(1)
210 IF ABS (( X(4)*X(3)^2 ) )<.0000001 THEN 1670
211 X(7)=30000-504000!/( X(4)*X(3)^2 )
213 IF ABS (( (30000000# )*X(4) * X(3)^3 ) ) <.0000001 THEN 1670
217 X(8)=.25-( 65856000#/( (30000000# )*X(4) * X(3)^3 ) )
221 PCBXB= ( ( 4.013*30000000#*( X(3)^2*X(4)^6 /36 )^.5 ) /196) * ( 1- ( (X(3)/28)* .790569415#) )
222 X(9)=-6000+PCBXB
229 IF ABS( (2^.5 *X(1)*X(2) ) )<.0000001 THEN 1670
231 TAUP=6000/ (2^.5 *X(1)*X(2) )
235 MNEW=6000*(14+X(2)/2 )
239 RNEW= ( X(2)^2/4 + ( (X(1)+X(3) )/2 )^2 )^.5
243 IF 2*(X(1)*X(2)/2^.5 ) * ( X(2)^2/12 + ( (X(1)+X(3) )/2 )^2 ) > 1000000000# THEN 1670
245 REM JNEW= 2*2^.5*X(1)*X(2) * ( X(2)^2/12 + ( (X(1)+X(3) )/2 )^2 )
246 JNEW= 2*(X(1)*X(2)/2^.5 ) * ( X(2)^2/12 + ( (X(1)+X(3) )/2 )^2 )
249 IF ABS (JNEW) <.0000001 THEN 1670
251 TAUPP=(MNEW*RNEW)/JNEW
253 IF ABS (2*RNEW ) <.0000001 THEN 1670
255 IF ( TAUP^2+2*TAUP*TAUPP*X(2)/2*RNEW +TAUPP^2 )<1E-09 THEN 1670
259 IF ABS (2*RNEW ) <.0000001 THEN 1670
261 TAUBXB= ( TAUP^2+2*TAUP*TAUPP*X(2)/ (2*RNEW ) +TAUPP^2 )^.5
281 X(10)= 13600-TAUBXB
301 REM X(11)= 5 -.10471*X(1)^2 -.04811 * X(3)*X(4) * (14+ X(2) )
321 IF X(1)<0 THEN 1670
322 IF X(2)<0 THEN 1670
323 IF X(3)<0 THEN 1670
324 IF X(4)<0 THEN 1670
325 IF X(1)>50 THEN 1670
326 IF X(2)>50 THEN 1670
327 IF X(3)>50 THEN 1670
328 IF X(4)>50 THEN 1670
421 FOR J44=5 TO 11
422 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
423 NEXT J44
425 PTOTAL=P(5)+P(6)+P(7) +P(8)+P(9)+P(10) +P(11)
569 P= - 1.10471*X(2)*X(1)^2-.04811*X(3)*X(4)*(14+X(2) ) +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 11
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-2.73 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1911 PRINT A(7),A(8),A(9)
1912 PRINT A(10),A(11)
1989 PRINT M,JJJJ
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter for DOS. The complete output through JJJJ=-31997 is shown below. What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.
.2080101623961738 6.938305143773716 9.005127341555024
.2080101623961738 0 0
120.9856908663273 .2355482658009589 187.5135967737112
3.356329366020418 0
-2.218553454852496 -32000
.2088698247814008 6.950872136033321 8.968635053415744
.2088698247814008 0 0
1.321100559933257 .2354313491672269 247.7800307485399
27.20272961996238 0
-2.223158690383047 -31999
.2638885266630925 5.746340576392275 8.110526277757888
.2638885266630925 0 0
965.6601089670321 .2344078329711995 5763.846807381042
133.6346463987993 0
-2.475312941692234 -31998
.2013035201344999 6.386625002912507 10.00181038796852
.2013732745782423 0 6.975444374234321D-05
4980.912263959436 .2391047698010023 2.908515966737468D-06
59.35303075265756 0
-2.261333188873962 -31997
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time for obtaining the output shown above was forty-five seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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