Friday, April 27, 2018

Using Discrete Variables To Approximate the Continuous Variables of an Insulated Steel Tank Design

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Li and Tsai [25, p. 11, Example 2]:

Minimize         400 * X(1) ^ .9 + 1000 - 22 * (X(2) + 14.7) ^ 1.2 + X(4)

subject to

         X(2) = EXP(-3950 / (X(3) + 460) + 11.86)

       144*( 80 - X(3)   =X(1)*X(4)

         0<=X(1) <=15.1
     
         14.7<=X(2) <=94.2
       
        -459.67<=X(3) <=80
       
         0<=X(4).



0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO -31980 STEP .01

    14 RANDOMIZE JJJJ

    16 M = -1D+37


    66 REM GOTO 111

    71 unitsize(1) = .001

    73 unitsize(2) = .001

    75 unitsize(3) = .001

    77 unitsize(4) = .001





    81 A(1) = 0 + FIX(RND * 1000) * unitsize(1)


    83 A(2) = 14.7 + FIX(RND * 1000) * unitsize(2)


    85 A(3) = -459.67 + FIX(RND * 1000) * unitsize(3)

    86 A(4) = 0 + FIX(RND * 1000) * unitsize(4)

    87 REM A(4) = 0 + FIX(RND * 15.1)
    89 GOTO 128

    111 A(1) = 0 + (RND * 15)


    112 A(2) = 14.7 + (RND * 79.5)

    113 A(3) = -459.67 + (RND * 539.67)
    114 A(4) = 0 + (RND * 10)

    128 FOR I = 1 TO 3000


        129 FOR KKQQ = 1 TO 4


            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        133 FOR IPP = 1 TO (1 + FIX(RND * 3))



            181 j = 1 + FIX(RND * 4)

            201 REM r = (1 - RND * 2) * A(j)
            205 REM X(j) = A(j) + (RND ^ (RND * 10)) * r


            187 IF RND < .5 THEN X(j) = A(j) - FIX(RND * 10) * unitsize(j) ELSE X(j) = A(j) + FIX(RND * 10) * unitsize(j)


        222 NEXT IPP

        225 X(3) = (144 * 80 - X(1) * X(4)) / 144

        227 IF (-3950 / (X(3) + 460) + 11.86) > 85 THEN GOTO 1670


        233 X(2) = EXP(-3950 / (X(3) + 460) + 11.86)


        268 IF X(1) < 0## THEN 1670


        269 IF X(1) > 15## THEN 1670
        278 IF X(2) < 14.7## THEN 1670
        289 IF X(2) > 94.2## THEN 1670
        291 IF X(3) < -459.67## THEN 1670
        293 IF X(3) > 80## THEN 1670
        295 IF X(4) < 0## THEN 1670
     
     
        443 POBA = -400 * X(1) ^ .9 - 1000 - 22 * (X(2) - 14.7) ^ 1.2 - X(4)


        466 P = POBA

        1111 IF P <= M THEN 1670


        1450 M = P
        1454 FOR KLX = 1 TO 4

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX
        1557 GOTO 128

    1670 NEXT I
    1889 IF M < -5194.867## THEN 1999


    1907 PRINT A(1), A(2), A(3), A(4), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [51]. The complete output through JJJJ = -31980.59000000311 is shown below:

3.469446951953614D-18      94.1778659402041      80
0      -5194.866244203784      -31990.85000000147

1.734723475976807D-18      94.1778659402041      80
8.673617379884036D-18      -5194.866244203783      -31985.74000000228

8.673617379884036D-19      94.1778659402041      80
0      -5194.866244203783      -31980.77000000308

1.734723475976807D-18      94.1778659402041      80
0      -5194.866244203783      -31980.59000000311

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 [51], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =  -31980.59000000311 was 11 seconds, not including the time for “Creating .EXE file”  (19 seconds, total, including the time for “Creating .EXE file”).  One can compare the computational results above with those on p. 11 of Li and Tsai [25, Example 2]. 
       

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

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