Jsun Yui Wong
The computer program listed below seeks to solve the following problem in Nigdeli, Bekdas, and Yang [9, pp. 33-34].
Minimize (5000) / (((X(4) * (X(2) - 2 * X(3)) ^ 3 / 12) + (X(1) * X(3) ^ 3 / 6) + (2 * X(1) * X(3) * ((X(2) - X(3)) / 2) ^ 2)))
subject to
2 * X(1) * X(4) + X(4) * (X(2) - 2 * X(3))<=300
(180000 * X(2)) / (((X(4) * (X(2) - 2 * X(3)) ^ 3) + 2 * X(1) * X(4) * (4 * X(3) ^ 2 + 3 * X(2) * (X(2) - 2 * X(3)))) + (15000 * X(1)) / ((X(4) ^ 3 * (X(2) - 2 * X(3))) + (2 * X(4) * X(1) ^ 3)))<=6
10<=X(1) <= 50
10<=X(2) <= 80
.9<=X(3) <= 5
.9<=X(4) <= 5.
The X(5) and X(6) 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
22 REM FOR J44 = 1 TO 4
68 A(1) = 20 + RND * 20
70 A(2) = 30 + RND * 30
72 A(3) = 1.9 + RND * 2.1
74 A(4) = 1.9 + RND * 2.1
71 REM NEXT J44
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 * 3))
181 J = 1 + FIX(RND * 4)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP
249 X(5) = 300 - 2 * X(1) * X(4) - X(4) * (X(2) - 2 * X(3))
251 X(6) = 6 - (180000 * X(2)) / (((X(4) * (X(2) - 2 * X(3)) ^ 3) + 2 * X(1) * X(4) * (4 * X(3) ^ 2 + 3 * X(2) * (X(2) - 2 * X(3)))) - (15000 * X(1)) / ((X(4) ^ 3 * (X(2) - 2 * X(3))) + (2 * X(4) * X(1) ^ 3)))
253 IF X(1) < 10 THEN GOTO 1670
255 IF X(2) < 10 THEN GOTO 1670
257 IF X(3) < .9 THEN GOTO 1670
259 IF X(4) < .9 THEN GOTO 1670
261 IF X(1) > 50 THEN GOTO 1670
263 IF X(2) > 80 THEN GOTO 1670
265 IF X(3) > 5 THEN GOTO 1670
267 IF X(4) > 5 THEN GOTO 1670
268 FOR J99 = 5 TO 6
269 IF X(J99) < 0 THEN X(J99) = X(J99) ELSE X(J99) = 0
270 NEXT J99
318 POBA = -(5000) / (((X(4) * (X(2) - 2 * X(3)) ^ 3 / 12) + (X(1) * X(3) ^ 3 / 6) + (2 * X(1) * X(3) * ((X(2) - X(3)) / 2) ^ 2))) + 1000000 * X(6) + 1000000 * X(5)
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 REM IF M < -1.339964 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4)
1901 PRINT A(5), A(6), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [15]. The complete output through JJJJ=-31999.98 is shown below:
49.99999989957234 79.99999999999893 4.999999999999993
1.764705884437955
0 0 -6.625958177414923D-03 -32000
49.9999884666348 79.99999999890551 4.999999999999994
1.764706121811363
0 0 -6.625959531848595D-03 -31999.99
50 79.99999998885822 5 1.7647058824686
0 0 -6.625958167535851D-03 -31999.98
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.
The solution above at JJJJ=-31999.98 can be compared to the four solutions presented in Table 5 of Nigdeli, Bekdas, and Yang [9, p. 34]. .
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [15], the wall-clock time for obtaining the output through JJJJ= -31999.98 was 8 seconds.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
References
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[15] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.
[16] Jsun Yui Wong (2012, April 12). The Domino Method of General Integer Nonlinear Programming Applied to a Nonlinear Fractional Programming Problem from the Literature. http://myblogsubstance.typepad.com/substance/2012/04/12/
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