Jsun Yui Wong
The two computer programs listed below seek the two extreme points of the four-bar truss problem involving two objectives in Tawhid and Savsani [62, p. 112] and in Sadollah, Eskandar, and Kim [54, Section 4.2.1]:
Minimize 200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4))
minimize (10*200 / 2E+05) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4))
subject to
1<= X(1), X(4) <=3
2^.5 <= X(2), X(3) <=3.
Whereas line 437 of the first computer is 437 PDU = -200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4)), line 437 of the second computer program is 437 PDU = -(2000 / (2 * 10 ^ 5)) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4)).
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)
81 FOR JJJJ = -32000 TO 32000
85 RANDOMIZE JJJJ
87 M = -3E+50
91 epsi = (RND * .1)
92 A(1) = 1 + RND * 2
93 A(2) = 2 ^ .5 + RND * 1.5857
95 A(3) = 2 ^ .5 + RND * 1.5857
96 A(4) = 1 + RND * 2
128 FOR I = 1 TO 6000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 J = 1 + FIX(RND * 4)
154 REM GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
162 IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1
169 NEXT IPP
171 GOTO 188
172 X(1) = INT(X(1))
174 X(2) = INT(X(2))
176 X(3) = INT(X(3))
177 X(4) = INT(X(4))
188 IF X(1) < 1## THEN 1670
189 IF X(2) < 2 ^ .5## THEN 1670
192 IF X(3) < 2 ^ .5## THEN 1670
193 IF X(4) < 1## THEN 1670
195 IF X(1) > 3## THEN 1670
196 IF X(2) > 3## THEN 1670
197 IF X(3) > 3## THEN 1670
198 IF X(4) > 3## THEN 1670
318 IF (2000 / 2E+05) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4)) > epsi THEN 1670
437 PDU = -200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4))
466 P = PDU
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1527 gg01star = 200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4))
1529 gg02star = (2000 / 2E+05) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4))
1557 GOTO 128
1670 NEXT I
1889 IF M < -99999999999 THEN 1999
1914 PRINT gg01star, gg02star, JJJJ
1924 PRINT A(1), A(2), A(3), A(4)
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [64]. The complete output of a single run through JJJJ= -31999 is shown below:
1174.199989102033 3.414213562373133D-02 -32000
1 1.414213562373096 1.414213562373125
1.000000000000001
1174.199989102031 3.414213562373095D-02 -31999
1.000000000000001 1.414213562373095 1414213562373096
1
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [64], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31999 was 2 seconds, not including the time for “Creating .EXE file" (7 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Tawhid and Savsani [62, p. 112; Figure 8, p. 113], Sadollah, Eskandar, and Kim [54, Figure 10].
Whereas line 437 above is 437 PDU = -200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4)), line 437 below is 437 PDU = -(2000 / (2 * 10 ^ 5)) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4)).
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)
81 FOR JJJJ = -32000 TO 32000
85 RANDOMIZE JJJJ
87 M = -3D+50
90 epsi = (RND * 2000)
91 REM epsi = (RND * .1)
92 A(1) = 1 + RND * 2
93 A(2) = 2 ^ .5 + RND * 1.5857
95 A(3) = 2 ^ .5 + RND * 1.5857
96 A(4) = 1 + RND * 2
128 FOR I = 1 TO 6000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 3)
153 J = 1 + FIX(RND * 4)
154 REM GOTO 162
156 r = (1 - RND * 2) * A(J)
158 X(J) = A(J) + (RND ^ (RND * 15)) * r
161 GOTO 169
162 IF RND < .5 THEN X(J) = A(J) - 1 ELSE X(J) = A(J) + 1
169 NEXT IPP
188 IF X(1) < 1## THEN 1670
189 IF X(2) < 2 ^ .5## THEN 1670
192 IF X(3) < 2 ^ .5## THEN 1670
193 IF X(4) < 1## THEN 1670
195 IF X(1) > 3## THEN 1670
196 IF X(2) > 3## THEN 1670
197 IF X(3) > 3## THEN 1670
198 IF X(4) > 3## THEN 1670
436 IF 200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4)) > epsi THEN 1670
437 PDU = -(2000 / (2 * 10 ^ 5)) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4))
466 P = PDU
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 4
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1527 gg01star = 200 * (2 * X(1) + (2 * X(2)) ^ .5 + X(3) ^ .5 + X(4))
1529 gg02star = (2000 / (2 * 10 ^ 5)) * (2 / X(2) + 2 * 2 ^ .5 / X(2) - 2 * 2 ^ .5 / X(3) + 2 / X(4))
1557 GOTO 128
1670 NEXT I
1889 IF M < -99999999999 THEN 1999
1914 PRINT gg01star, gg02star, JJJJ
1924 PRINT A(1), A(2), A(3), A(4)
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [64]. The complete output of a single run through JJJJ= -31991 is shown below:
1733.610508480852 2.761423749153971D-03 -31994
1.01467784230918 3 1.414213562373095
3
1727.854823855953 2.761423749153968D-03 -31991
1.000288630946933 3 1.414213562373095
3
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and QB64v1000-win [64], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31991 was 2 seconds, not including the time for “Creating .EXE file" (7 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Tawhid and Savsani [62, p. 112; Figure 8, p. 113], Sadollah, Eskandar, and Kim [54, Figure 10].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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