Solving Sandgren's Pressure Vessel Design Problem

Jsun Yui Wong

The computer program listed below seeks to solve Example 3 in Lu [11, pp. 20-21]:   

Minimize         .6224 * X(1) * X(3) * X(4) + 1.7781 * X(2) * X(3) ^ 2+ 3.1661 * X(1) ^ 2 * X(4) + 19.84 * X(1) ^ 2 * X(3)

subject to

        - X(1) + .0193 * X(3)<=0


        -X(2) + .00954 * X(3)<=0,

        1296000 - 3.141592654 * X(3) ^ 2 * X(4) - (4 / 3) * 3.141592654 * X(3) ^ 3<=0


        -240 + X(4)<=0,

       
        .0625<=X(1) <= 6.1875,


       .0625<= X(2) <= 6.1875 ,

       
         10<=X(3) <= 200,


         10<=X(4) <= 200,

        where X(1) and X(2) are discrete variables with discreteness of .0625, and X(3) and X(4) are integer variables.

One notes line 67 and line 68 below.  X(5) through X(8) below are slack variables, which are added.


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 32000 STEP .01

    14 RANDOMIZE JJJJ
    16 M = -1D+37

    22 REM


    67 A(1) = .0625 + INT(RND * 98) * .0625

    68 A(2) = .0625 + INT(RND * 98) * .0625

    77 A(3) = 10 + FIX(RND * 191)
    78 A(4) = 10 + FIX(RND * 191)

    79 REM


    128 FOR I = 1 TO 10000



        129 FOR KKQQ = 1 TO 4

            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        133 FOR IPP = 1 TO (1 + FIX(RND * 4))

            181 REM J = 1 + FIX(RND * 2)

            183 REM r = (1 - RND * 2) * A(J)
            187 REM X(J) = A(J) + (RND ^ (RND * 10)) * r


            189 REM IF RND < .333 THEN X(1) = .009 + INT(RND * 250) * .002 ELSE IF RND < .5 THEN X(2) = .6 + INT(RND * 170) * .02 ELSE X(3) = 1 + FIX(RND * 120)


            190 IF RND < .5 THEN 192

            191 X(1) = .0625 + INT(RND * 98) * .0625



            192 IF RND < .5 THEN 194

            193 X(2) = .0625 + INT(RND * 98) * .0625


            194 IF RND < .5 THEN 196
            195 X(3) = 10 + FIX(RND * 191)

            196 IF RND < .5 THEN 200

            198 X(4) = 10 + FIX(RND * 191)



        200 NEXT IPP


        201 IF X(1) < .0625 THEN 1670
        203 IF X(1) > 6.1875 THEN 1670
        255 REM

        258 IF X(2) < .0625 THEN 1670
        266 IF X(2) > 6.1875 THEN 1670

        277 IF X(3) < 10 THEN 1670
        288 IF X(3) > 200 THEN 1670


        289 IF X(4) < 10 THEN 1670
        291 IF X(4) > 200 THEN 1670



        304 REM X(4) = 10 - X(1) - X(2) - X(3)



        306 X(5) = X(1) - .0193 * X(3)



        307 X(6) = X(2) - .00954 * X(3)

        317 X(7) = -1296000 + 3.141592654 * X(3) ^ 2 * X(4) + (4 / 3) * 3.141592654 * X(3) ^ 3


        320 X(8) = 240 - X(4)



        321 REM X(9) = -8 * (11.5D+06) ^ -1 * 300 * X(1) ^ -4 * X(2) ^ 3 * X(3) + 6


        323 REM X(10) = -1.25 + 8 * (11.5D+06) ^ -1 * 700 * X(1) ^ -4 * X(2) ^ 3 * X(3)


        325 FOR J99 = 5 TO 8


            330 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0

        331 NEXT J99


        359 POBA = -.6224 * X(1) * X(3) * X(4) - 1.7781 * X(2) * X(3) ^ 2 - 3.1661 * X(1) ^ 2 * X(4) - 19.84 * X(1) ^ 2 * X(3) + 1000000 * (X(5) + X(6) + X(7) + X(8))



        466 P = POBA

        1111 IF P <= M THEN 1670



        1452 M = P
        1454 FOR KLX = 1 TO 8


            1459 A(KLX) = X(KLX)
        1460 NEXT KLX
        1557 REM GOTO 128

    1670 NEXT I

    1889 IF M < -6075 THEN 1999
    1900 PRINT A(1), A(2), A(3), A(4), A(5)
    1903 PRINT A(6), A(7), A(8)
    1907 PRINT M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with qb64v1000-win [24]. The complete output through JJJJ = -31997.69000000037 is shown below:

.8125      .4375      42      178      0
0      0      0
-6074.99836015625      -31999.83000000003

.8125      .4375      42      178      0
0      0      0
-6074.99836015625      -31997.69000000037

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.  One can compare the results above with the results given in Lu [11, Table 5, p. 21; Table 6, p. 21].

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [24], the wall-clock time for obtaining the output through JJJJ=  -31997.69000000037 was 18 seconds, including the seconds for creating the .EXE file.   

Acknowledgment

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

References

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