Jsun Yui Wong
The computer program listed below seeks to solve the following pressure vessel minimization problem in Li et al. [16, pp. 931-932]:
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,
1296000 – 3.141592654 * X(3) ^ 2 * X(4) – (4 / 3) * 3.141592654 * X(3) ^ 3<=0,
– X(2) + .00954 * X(3)<=0,
X(1), X(2) Epsilon { (2.919/(400-1) )* ( l-1 ) : l=1, 2, 3,…, 400 } ,
X(3) Epsilon { (500/(400-1) )* ( l-1 ) : l=1, 2, 3,…, 400 } ,
10<= X(4) <=240,
where 400 is the number of discrete values for X(1), X(2), and X(3).
X(5) through X(7) below are slack variables.
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
67 A(1) = 0 + FIX(RND * 401) * .0072975
69 A(2) = 0 + FIX(RND * 401) * .0072975
70 A(3) = 0 + FIX(RND * 401) * 1.25
77 A(4) = 10 + (RND * 230)
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 4
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
151 J = 1 + FIX(RND * 3)
155 IF J = 1 GOTO 167 ELSE IF J = 2 GOTO 169 ELSE GOTO 173
167 X(1) = 0 + FIX(RND * 401) * .0072975
168 GOTO 191
169 X(2) = 0 + FIX(RND * 401) * .0072975
171 GOTO 191
173 X(3) = 0 + FIX(RND * 401) * 1.25
183 REM r = (1 – RND * 2) * A(J)
187 REM X(J) = A(J) + (RND ^ (RND * 10)) * r
189 REM X(J) = .5 + FIX(RND * 505) * .01
191 NEXT IPP
195 r = (1 – RND * 2) * A(4)
197 X(4) = A(4) + (RND ^ (RND * 10)) * r
201 IF X(1) < 0 THEN 1670
203 IF X(1) > 2.919 THEN 1670
211 IF X(2) < 0 THEN 1670
213 IF X(2) > 2.919 THEN 1670
221 IF X(3) < 0 THEN 1670
223 IF X(3) > 500 THEN 1670
231 IF X(4) < 10 THEN 1670
233 IF X(4) > 240 THEN 1670
305 X(5) = X(1) – .0193 * X(3)
306 X(6) = -1296000 + 3.141592654 * X(3) ^ 2 * X(4) + (4 / 3) * 3.141592654 * X(3) ^ 3
307 X(7) = X(2) – .00954 * X(3)
325 FOR J99 = 5 TO 7
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))
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 7
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 REM GOTO 128
1670 NEXT I
1889 IF M < -5880 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1903 PRINT A(6), A(7), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [33]. The complete output through JJJJ = -31990.7000000015 is shown below:
.7516425 .3721725 38.75 223.1234495163882
0
0 0 -5871.937275375835 -31998.70000000021
.7516425 .3721725 38.75 223.4033250661287
0
0 0 -5877.51151586208 -31997.32000000043
.7516425 .3721725 38.75 223.2190193364292
0
0 0 -5873.840725536562 -31991.60000000135
.7516425 .3721725 38.75 223.384842141064
0
0 0 -5877.143394140504 -31990.7000000015
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 [33], the wall-clock time for obtaining the output through JJJJ= -31990.7000000015 was 100 seconds, including the time for creating the .EXE file. One can compare the computational results here with those in Table 5 of Li et al. [16, p. 932].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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