Direct Finding Multiple Optimal Solutions in One Run of a General Integer
Linear Program: an Illustration
Jsun Yui Wong
Similar to the computer program of the preceding paper, the computer
program listed below aims to solve directly the following problem from Tsai,
Lin, and Hu [89, pp. 806-807, Example 2]:
Maximize
10 * (X(1) + X(2) + X(3)) + 8 *
(X(4) + X(5) + X(6)) - 4 * (X(1) + X(4)) - 5 * (X(2) + X(5)) - 6 * (X(3) +
X(6))
subject to
X(1) + X(4) <= 400
X(2) + X(5) <= 500
X(3) + X(6) <= 300
.2 * (X(1) + X(2) + X(3))
>= X(1)
.1 * (X(1) + X(2) + X(3))
>= X(2)
.2 * (X(1) + X(2) + X(3))
<= X(3)
.4 * (X(4) + X(5) + X(6)) >=
X(4)
.5 * (X(4) + X(5) + X(6)) >=
X(6)
X(1), X(2), X(3), X(4), X(5),
X(6) >=0
X(1), X(2), X(3), X(4), X(5),
X(6) are integers.
0 REM DEFDBL A-Z
1 REM DEFINT I, J, K
2 DIM B(99), N(99), A(200), H(99), L(99), U(99), X(200), D(111), P(111),
PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22),
PX(44), J44(44), PN(22), NN(22)
9 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3D+30
102 A(1) = FIX(RND * 501)
104 A(2) = FIX(RND * 501)
106 A(3) = FIX(RND * 501)
107 A(4) = FIX(RND * 501)
108 A(5) = FIX(RND * 501)
109 A(6) = FIX(RND * 501)
128 FOR I = 1 TO 50000
129 FOR KKQQ = 1 TO 6
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
134 FOR IPP = 1 TO FIX(1 +
RND * 4)
136 B = 1 + FIX(RND * 6)
144 IF RND < 1 / 2
THEN 160 ELSE GOTO 167
160 R = (1 - RND * 2) *
A(B)
164 X(B) = A(B) + (RND ^
(RND * 15)) * R
165 GOTO 188
167 IF RND < .5 THEN
X(B) = A(B) - FIX(RND * 4) ELSE X(B) = A(B) + FIX(RND * 4)
169 GOTO 188
188 NEXT IPP
211 FOR J44 = 1 TO 6
212 X(J44) = INT(X(J44))
213 IF X(J44) < 0 THEN
1670
214 IF X(J44) > 500
THEN 1670
215 NEXT J44
241 IF X(1) + X(4) > 400
THEN 1670
242 IF X(2) + X(5) > 500
THEN 1670
243 IF X(3) + X(6) > 300
THEN 1670
244 IF .2 * (X(1) + X(2) +
X(3)) < X(1) THEN 1670
245 IF .1 * (X(1) + X(2) +
X(3)) < X(2) THEN 1670
246 IF .2 * (X(1) + X(2) +
X(3)) > X(3) THEN 1670
247 IF .4 * (X(4) + X(5) +
X(6)) < X(4) THEN 1670
248 IF .5 * (X(4) + X(5) +
X(6)) < X(6) THEN 1670
278 PD1 = 10 * (X(1) + X(2) +
X(3)) + 8 * (X(4) + X(5) + X(6)) - 4 * (X(1) + X(4)) - 5 * (X(2) + X(5)) - 6 *
(X(3) + X(6))
479 P = PD1
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1670 NEXT I
1677 IF M < 4514 THEN 1999
1904 PRINT A(1), A(2), A(3),
A(4), A(5), A(6), M, JJJJ
1999 NEXT JJJJ
This computer program was run with qb64v1000-win [101]. Its complete output of one run through JJJJ=
-22324 is shown below:
85 42 300
305 458
0 4514 -27021
85 42 299
306 458
1 4516 -26988
85 42 299
306 458
1 4516 -26308
85 41 300
306 459
0 4516 -26173
85 42 299
306 458
1 4516 -25725
85 42 300
305 458
0 4514 -22324
Two distinct optimal solutions are shown above.
Above there is no rounding by hand; it is just straight copying by hand
from the monitor screen. On a personal computer with Processor Intel Pentium
CPU G620@ 2.60GHz, 4.00 GB of RAM, 64-bit Operating System, and QB64v1000-win
[101], the wall-clock time (not CPU time) for obtaining the output through JJJJ
= -22324 was 15 minutes, counting from "Starting program...". One can compare the computational results above
with those in Tsai, Lin, and Hu
[89, p. 807, Table 2, Example 2].
Acknowledgement
I would like to acknowledge the encouragement of Roberta Clark and Tom
Clark.
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