The computer program listed below seeks to solve the following nonlinear integer fractional programming problem:
Minimize (-X(1) ^ 2 * X(2) ^ .5 + 2 * X(1) * X(2) ^ (-1) - X(2) ^ 2 + (2.8 * X(1) ^ (-1)) * X(2) + 7.5) / (X(1) * X(2) ^ 1.5 + 1) + (X(2) + .1) / (X(1) ^ 2 * X(2) ^ (-1) - 3 * X(1) ^ (-1) + 2 * X(1) * X(2) ^ 2 - 9 * X(2) ^ (-1) + 12)
subject to
2 * X(1) ^ (-1)+ X(1) * X(2)<=4,
X(1) + 3 * X(1) ^ (-1) * X(2)<=5
X(1) ^ 2 - 3 * X(2) ^ 3<=2,
1<= X(1), X(2) <=3,
X(1) and X(2) are integer variables.
The problem above is based on Example 5 in Hou, Shen, and Wang [12], which is
Minimize (-X(1) ^ 2 * X(2) ^ .5 + 2 * X(1) * X(2) ^ (-1) - X(2) ^ 2 + (2.8 * X(1) ^ (-1)) * X(2) + 7.5) / (X(1) * X(2) ^ 1.5 + 1) + (X(2) + .1) / (X(1) ^ 2 * X(2) ^ (-1) - 3 * X(1) ^ (-1) + 2 * X(1) * X(2) ^ 2 - 9 * X(2) ^ (-1) + 12)
subject to
2 * X(1) ^ (-1)+ X(1) * X(2)<=4,
X(1) + 3 * X(1) ^ (-1) * X(2)<=5
X(1) ^ 2 - 3 * X(2) ^ 3<=2,
1<= X(1), X(2) <=3.
X(3) through X(5) 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
71 FOR J40 = 1 TO 2
74 A(J40) = (1 + RND * 2)
77 NEXT J40
128 FOR I = 1 TO 100
129 FOR KKQQ = 1 TO 2
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 1))
181 J = 1 + FIX(RND * 2)
183 R = (1 - RND * 2) * A(J)
187 X(J) = A(J) + (RND ^ (RND * 10)) * R
222 NEXT IPP
223 FOR J41 = 1 TO 2
225 X(J41) = INT(X(J41))
235 NEXT J41
256 FOR J47 = 1 TO 2
257 IF X(J47) < 1 THEN 1670
258 IF X(J47) > 3 THEN 1670
259 NEXT J47
311 X(3) = 4 - 2 * X(1) ^ (-1) - X(1) * X(2)
313 X(4) = 5 - X(1) - 3 * X(1) ^ (-1) * X(2)
315 X(5) = 2 - X(1) ^ 2 + 3 * X(2) ^ 3
322 FOR J44 = 3 TO 5
325 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
327 NEXT J44
333 REM
337 REM
339 REM
341 POBA = -(-X(1) ^ 2 * X(2) ^ .5 + 2 * X(1) * X(2) ^ (-1) - X(2) ^ 2 + (2.8 * X(1) ^ (-1)) * X(2) + 7.5) / (X(1) * X(2) ^ 1.5 + 1) - (X(2) + .1) / (X(1) ^ 2 * X(2) ^ (-1) - 3 * X(1) ^ (-1) + 2 * X(1) * X(2) ^ 2 - 9 * X(2) ^ (-1) + 12) + 1000000 * (X(3) + X(4) + X(5))
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 5
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 REM GOTO 128
1670 NEXT I
1889 REM IF M < -.000000000003 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1902 PRINT M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [36]. The complete output through JJJJ = -31999.97000000001 is shown below:
1 1 0 0 0
-5.516666666666667 -32000
1 1 0 0 0
-5.516666666666667 -31999.99
1 1 0 0 0
-5.516666666666667 -31999.98
2 1 0 0 0
-2.749122807017544 -31999.97000000001
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 [36], the wall-clock time for obtaining the output through JJJJ= -31999.97000000001 was 2 seconds, not including the creating .EXE file time. One can compare the computational results here with those in Table 2 of Hou, Shen, and Wang [12].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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