Jsun Yui Wong
The following computer program seeks to solve the problem considered by Sandgren [44], Lee and Geem [30], Mahdavi, Fesanghary, and Damangir [32], Jaberipour and Khorram [19], and Gholizadeh and Barzegar [15], among others. This problem has discrete variables and continuous variables.
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
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
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<-31995 THEN 162 ELSE 145
145 IF B=3 OR B=4 GOTO 164
147 IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625 ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
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
194 X(5)=X(1) -.0193*X(3)
195 X(6)=X(2) -.00954*X(3)
199 X(7)=-1296000! + 1.3333333333333#*3.141592654#*X(3)^3 + 3.141592654#*X(3) ^2 *X(4)
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569 P= -.6224*X(1)*X(3)*X(4) -1.7781*X(2)*X(3) ^2 -3.1611*X(1) ^2 *X(4) -19.84*X(1) ^2 *X(3) +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),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 top-two candidate solutions through JJJJ=-30062 are shown below. What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.
1.125 .625 58.28628654289647
43.71393346695399 7.462646913716786D-05 6.894882643287964D-02
1.736438644002192D-04 -7197.965676984811 -31656
1.125 .625 58.28628935954246
43.71391795959724 7.457210786745994D-05 6.894879956207689D-02
4.156150680501014D-06 -7197.965503888046 -30062
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 ten minutes.
The following computer program tries to improve on the solution shown above at JJJJ=-30062. The A(7)= 4.156150680501014D-06 shown above suggests that at optimality it is likely that A(7)=0; in other words, the corresponding constraint is active. That is tried with line 200 of the following computer program.
While line 199 of the computer program above is
199 X(7)=-1296000! + 1.3333333333333#*3.141592654#*X(3)^3 + 3.141592654#*X(3) ^2 *X(4),
line 199 below is
199 REM X(7)=-1296000! + 1.3333333333333#*3.141592654#*X(3)^3 + 3.141592654#*X(3) ^2 *X(4).
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
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
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<-31995 THEN 162 ELSE 145
145 IF B=3 OR B=4 GOTO 164
147 IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625 ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
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
194 X(5)=X(1) -.0193*X(3)
195 X(6)=X(2) -.00954*X(3)
199 REM X(7)=-1296000! + 1.3333333333333#*3.141592654#*X(3)^3 + 3.141592654#*X(3) ^2 *X(4)
200 X(4)= ( + 1296000! - 1.3333333333333#*3.141592654#*X(3)^3 ) / (+ 3.141592654#*X(3) ^2 )
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569 P= -.6224*X(1)*X(3)*X(4) -1.7781*X(2)*X(3) ^2 -3.1611*X(1) ^2 *X(4) -19.84*X(1) ^2 *X(3) +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),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=-31988 is shown below. What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.
1.125 .625 58.29015090845521
43.69268114487037 4.421098581675054D-08 6.891196038545272D-02
0 -7197.729199130497 -31995
1.125 .625 58.29015308692593
43.69266916514552 2.166499452060933D-09 6.891193960284212D-02
0 -7197.729065837913 -31994
1.125 .625 58.29013414146686
43.69277334899321 3.678138735530023D-07 6.891212034252163D-02
0 -7197.730225041295 -31991
1.125 .625 58.2901531892509
43.69266860244574 1.916273528745904D-10 6.891193862666185D-02
0 -7197.729059577026 -31988
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 four seconds.
Similarly the residual A(5)= 7.462646913716786D-05 at JJJJ=31656 of the first computer program listed above suggests that at optimality it is likely that A(5)=0; in other words, the corresponding constraint is active. That is tried with line 192 of the following computer program; the following line 194 is just 194 REM X(5)=X(1) -.0193*X(3).
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
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
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<-31995 THEN 162 ELSE 145
145 IF B=3 OR B=4 GOTO 164
147 IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625 ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
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
192 X(3)=X(1) / .0193
194 REM X(5)=X(1) -.0193*X(3)
195 X(6)=X(2) -.00954*X(3)
199 X(7)=-1296000! + 1.3333333333333#*3.141592654#*X(3)^3 + 3.141592654#*X(3) ^2 *X(4)
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569 P= -.6224*X(1)*X(3)*X(4) -1.7781*X(2)*X(3) ^2 -3.1611*X(1) ^2 *X(4) -19.84*X(1) ^2 *X(3) +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),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=-31979 is shown below. What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.
1.125 .625 58.29015319917978
43.69814366322346 0 6.891193853194036D-02
58.44314005206252 -7197.974429174728 -31992
1.125 .625 58.29015319917978
43.69266979995663 0 6.891193853194036D-02
1.336543742218055D-02 -7197.729115083545 -31984
1.125 .625 58.29015319917978
43.69266942413557 0 6.891193853194036D-02
9,353802692203317D-03 -7197.729098240924 -31983
1.125 .625 58.29015319917978
43.69266883596002 0 6.891193853194036D-02
3.075427965086419D-03 -7197.729071881524 -31979
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 five seconds.
In conclusion, guessing correctly before or after a computer run which slack variable equals zero at optimality can shorten computational time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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[48] 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/
[49] 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/
[50] 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/
[51] 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/
[52] 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/
[53] 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/
[54] 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/
[55] 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/
[56] 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/
[57] 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 problem considered by Sandgren [44], Lee and Geem [30], Mahdavi, Fesanghary, and Damangir [32], Jaberipour and Khorram [19], and Gholizadeh and Barzegar [15], among others. This problem has discrete variables and continuous variables.
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
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
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<-31995 THEN 162 ELSE 145
145 IF B=3 OR B=4 GOTO 164
147 IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625 ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
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
194 X(5)=X(1) -.0193*X(3)
195 X(6)=X(2) -.00954*X(3)
199 X(7)=-1296000! + 1.3333333333333#*3.141592654#*X(3)^3 + 3.141592654#*X(3) ^2 *X(4)
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569 P= -.6224*X(1)*X(3)*X(4) -1.7781*X(2)*X(3) ^2 -3.1611*X(1) ^2 *X(4) -19.84*X(1) ^2 *X(3) +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),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 top-two candidate solutions through JJJJ=-30062 are shown below. What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.
1.125 .625 58.28628654289647
43.71393346695399 7.462646913716786D-05 6.894882643287964D-02
1.736438644002192D-04 -7197.965676984811 -31656
1.125 .625 58.28628935954246
43.71391795959724 7.457210786745994D-05 6.894879956207689D-02
4.156150680501014D-06 -7197.965503888046 -30062
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 ten minutes.
The following computer program tries to improve on the solution shown above at JJJJ=-30062. The A(7)= 4.156150680501014D-06 shown above suggests that at optimality it is likely that A(7)=0; in other words, the corresponding constraint is active. That is tried with line 200 of the following computer program.
While line 199 of the computer program above is
199 X(7)=-1296000! + 1.3333333333333#*3.141592654#*X(3)^3 + 3.141592654#*X(3) ^2 *X(4),
line 199 below is
199 REM X(7)=-1296000! + 1.3333333333333#*3.141592654#*X(3)^3 + 3.141592654#*X(3) ^2 *X(4).
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
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
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<-31995 THEN 162 ELSE 145
145 IF B=3 OR B=4 GOTO 164
147 IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625 ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
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
194 X(5)=X(1) -.0193*X(3)
195 X(6)=X(2) -.00954*X(3)
199 REM X(7)=-1296000! + 1.3333333333333#*3.141592654#*X(3)^3 + 3.141592654#*X(3) ^2 *X(4)
200 X(4)= ( + 1296000! - 1.3333333333333#*3.141592654#*X(3)^3 ) / (+ 3.141592654#*X(3) ^2 )
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569 P= -.6224*X(1)*X(3)*X(4) -1.7781*X(2)*X(3) ^2 -3.1611*X(1) ^2 *X(4) -19.84*X(1) ^2 *X(3) +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),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=-31988 is shown below. What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.
1.125 .625 58.29015090845521
43.69268114487037 4.421098581675054D-08 6.891196038545272D-02
0 -7197.729199130497 -31995
1.125 .625 58.29015308692593
43.69266916514552 2.166499452060933D-09 6.891193960284212D-02
0 -7197.729065837913 -31994
1.125 .625 58.29013414146686
43.69277334899321 3.678138735530023D-07 6.891212034252163D-02
0 -7197.730225041295 -31991
1.125 .625 58.2901531892509
43.69266860244574 1.916273528745904D-10 6.891193862666185D-02
0 -7197.729059577026 -31988
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 four seconds.
Similarly the residual A(5)= 7.462646913716786D-05 at JJJJ=31656 of the first computer program listed above suggests that at optimality it is likely that A(5)=0; in other words, the corresponding constraint is active. That is tried with line 192 of the following computer program; the following line 194 is just 194 REM X(5)=X(1) -.0193*X(3).
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
107 A(1)=FIX(RND*30)*.0625
109 A(2)=FIX(RND*30)*.0625
111 A(3)=40+FIX(RND*41)
113 A(4)=20+FIX(RND*41)
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<-31995 THEN 162 ELSE 145
145 IF B=3 OR B=4 GOTO 164
147 IF B=1 GOTO 150 ELSE IF B=2 GOTO 154
150 IF RND<.5 THEN X(1)=A(1)-FIX(1+RND*2)*.0625 ELSE X(1)=A(1)+FIX(1+RND*2)*.0625
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
192 X(3)=X(1) / .0193
194 REM X(5)=X(1) -.0193*X(3)
195 X(6)=X(2) -.00954*X(3)
199 X(7)=-1296000! + 1.3333333333333#*3.141592654#*X(3)^3 + 3.141592654#*X(3) ^2 *X(4)
212 IF X(1)<1.1 THEN 1670
215 IF X(2)<.6 THEN 1670
217 IF X(3)<40 THEN 1670
218 IF X(3)>80 THEN 1670
219 IF X(4)<20 THEN 1670
220 IF X(4)>60 THEN 1670
221 FOR J44=5 TO 7
222 IF X(J44)<0 THEN P(J44)=9000000!*X(J44) ELSE P(J44)=0
223 NEXT J44
225 PTOTAL=P(5)+P(6)+P(7)
569 P= -.6224*X(1)*X(3)*X(4) -1.7781*X(2)*X(3) ^2 -3.1611*X(1) ^2 *X(4) -19.84*X(1) ^2 *X(3) +PTOTAL
1111 IF P<=M THEN 1670
1452 M=P
1454 FOR KLX=1 TO 7
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M<-7200 THEN 1999
1904 PRINT A(1),A(2),A(3)
1907 PRINT A(4),A(5),A(6)
1989 PRINT A(7),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=-31979 is shown below. What follows is a hand copy from the computer-monitor screen; below there is no rounding by hand.
1.125 .625 58.29015319917978
43.69814366322346 0 6.891193853194036D-02
58.44314005206252 -7197.974429174728 -31992
1.125 .625 58.29015319917978
43.69266979995663 0 6.891193853194036D-02
1.336543742218055D-02 -7197.729115083545 -31984
1.125 .625 58.29015319917978
43.69266942413557 0 6.891193853194036D-02
9,353802692203317D-03 -7197.729098240924 -31983
1.125 .625 58.29015319917978
43.69266883596002 0 6.891193853194036D-02
3.075427965086419D-03 -7197.729071881524 -31979
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 five seconds.
In conclusion, guessing correctly before or after a computer run which slack variable equals zero at optimality can shorten computational time.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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[48] 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/
[49] 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/
[50] 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/
[51] 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/
[52] 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/
[53] 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/
[54] 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/
[55] 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/
[56] 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/
[57] 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/
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