The computer program listed below seeks to solve the following integer problem:
Maximize n1x / d1x + n2x / d2x + n3x / d3x
where
n1x = X(1) ^ 2 - 16 * X(1) + X(2) ^ 2 - 16 * X(2) + X(3) ^ 2 - 16 * X(3) + X(4) ^ 2 - 16 * X(4) + 214,
n2x = X(1) ^ 2 - 16 * X(1) + 2 * X(2) ^ 2 - 20 * X(2) + 3 * X(3) ^ 2 - 60 * X(3) + 4 * X(4) ^ 2 - 56 * X(4) + 586,
n3x = X(1) ^ 2 - 20 * X(1) + X(2) ^ 2 - 20 * X(2) + X(3) ^ 2 - 20 * X(3) + X(4) ^ 2 - 20 * X(4) + 324,
d1x = 2 * X(1) - X(2) - X(3) + X(4) + 2,
d2x = -X(1) + X(2) + X(3) - X(4) + 10,
d3x = X(1) ^ 2 - 4 * X(4),
subject to
6<= X(1) <= 10,
4<= X(2) <= 6,
8<= X(3) <= 12,
6<=X(4) <= 8,
X(1) + X(2) + X(3) + X(4)<=26,
X(1) through X(4) are integer variables.
The integer problem above is based on Example 4 in Shen, Duan, and Pei [23, p. 155].
X(5) below is a slack variable.
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
75 A(1) = 6 + RND * 4
77 A(2) = 4 + RND * 2
78 A(3) = 8 + RND * 4
79 A(4) = 6 + RND * 2
128 FOR I = 1 TO 2000
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
191 NEXT IPP
193 REM GOTO 209
196 FOR J99 = 1 TO 4
199 X(J99) = INT(X(J99))
204 NEXT J99
209 IF X(1) < 6 THEN 1670
212 IF X(1) > 10 THEN 1670
214 IF X(2) < 4 THEN 1670
216 IF X(2) > 6 THEN 1670
218 IF X(3) < 8 THEN 1670
222 IF X(3) > 12 THEN 1670
229 IF X(4) < 6 THEN 1670
23 IF X(4) > 8 THEN 1670
306 X(5) = 26 - X(1) - X(2) - X(3) - X(4)
327 XX(5) = X(5)
330 IF X(5) < 0 THEN X(5) = X(5) ELSE X(5) = 0
340 n1x = X(1) ^ 2 - 16 * X(1) + X(2) ^ 2 - 16 * X(2) + X(3) ^ 2 - 16 * X(3) + X(4) ^ 2 - 16 * X(4) + 214
342 n2x = X(1) ^ 2 - 16 * X(1) + 2 * X(2) ^ 2 - 20 * X(2) + 3 * X(3) ^ 2 - 60 * X(3) + 4 * X(4) ^ 2 - 56 * X(4) + 586
346 n3x = X(1) ^ 2 - 20 * X(1) + X(2) ^ 2 - 20 * X(2) + X(3) ^ 2 - 20 * X(3) + X(4) ^ 2 - 20 * X(4) + 324
349 d1x = 2 * X(1) - X(2) - X(3) + X(4) + 2
352 d2x = -X(1) + X(2) + X(3) - X(4) + 10
355 d3x = X(1) ^ 2 - 4 * X(4)
357 POBA = n1x / d1x + n2x / d2x + n3x / d3x + 1000000 * (X(5))
466 P = POBA
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 5
1456 XXX(KLX) = XX(KLX)
1459 A(KLX) = X(KLX)
1460 NEXT KLX
1557 REM GOTO 128
1670 NEXT I
1889 IF M < -99999 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5), XXX(5)
1902 PRINT M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with qb64v1000-win [29]. The complete output through JJJJ = -31999.9600000001 is shown below:
6 4 8 6 0
2
-2.783333333333333 -32000
6 4 8 6 0
2
-2.783333333333333 -31999.99
6 4 8 6 0
2
-2.783333333333333 -31999.98
6 4 8 6 0
2
-2.783333333333333 -31999.9700000001
6 4 8 6 0
2
-2.783333333333333 -31999.9600000001
Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. The candidate solution above at JJJJ=-32000, for example, is optimal, Senn, Duan, and Pei [23, p. 156].
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [29], the wall-clock time for obtaining the output through JJJJ= -31999.9600000001 was 2 seconds, not including the time for creating the .EXE file.
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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