Sunday, December 9, 2018

Seeking the Two (Far Apart) Extreme Points of a Four-Bar Truss Problem Involving Two Objectives


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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