Jsun Yui Wong
Hannan’s 3-objective problem [13, Example 1, pp. 243-245] has been transformed by Yang, Ignizio, and Kim [47, p. 50] to the following problem:
Maximize lambda
subject to
lambda<= 1 – (6 – (3 * X(1) + X(2) + X(3))) / (2),
lambda<= 1 – (20 / 3 – (3 * X(1) + X(2) + X(3))) / (10 / 3),
lambda <= 1 – (7 – (3 * X(1) + X(2) + X(3))) / (5),
lambda < = 1 – (7 – (X(1) – X(2) + 2 * X(3))) / (5),
lambda <= 1 – (8 – (X(1) – X(2) + 2 * X(3))) / (20 / 3),
lambda <= 1 – (9 / 2 – (X(1) + 2 * X(2))) / (5 / 2),
lambda <= 1 – (5 – (X(1) + 2 * X(2))) / (5),
4 * X(1)+ 2 * X(2) + 3 * X(3)<=10,
X(1) + 3 * X(2) + 2 * X(3)<=8,
X(3) <=5,
X(1), X(2), X(3>=0.
The computer program listed below seeks to solve the formulation above–X(5) through X(13) 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 -31999. STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
19 FOR J44 = 1 TO 4
22 A(J44) = RND * 2
31 NEXT J44
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))
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
223 FOR j41 = 1 TO 3
225 IF X(j41) < 0 THEN 1670
235 NEXT j41
239 IF X(3) > 5 THEN 1670
309 X(5) = -X(4) + 1 – (6 – (3 * X(1) + X(2) + X(3))) / (2)
311 X(6) = -X(4) + 1 – (20 / 3 – (3 * X(1) + X(2) + X(3))) / (10 / 3)
313 X(7) = -X(4) + 1 – (7 – (3 * X(1) + X(2) + X(3))) / (5)
315 X(8) = -X(4) + 1 – (7 – (X(1) – X(2) + 2 * X(3))) / (5)
317 X(9) = -X(4) + 1 – (8 – (X(1) – X(2) + 2 * X(3))) / (20 / 3)
319 X(10) = -X(4) + 1 – (9 / 2 – (X(1) + 2 * X(2))) / (5 / 2)
321 X(11) = -X(4) + 1 – (5 – (X(1) + 2 * X(2))) / (5)
323 X(12) = 10 – 4 * X(1) – 2 * X(2) – 3 * X(3)
325 X(13) = 8 – X(1) – 3 * X(2) – 2 * X(3)
337 FOR J44 = 5 TO 13
338 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
339 NEXT J44
443 POBA = X(4) + 1000000 * (X(10) + X(11) + X(12) + X(13) + X(5) + X(6) + X(7) + X(8) + X(9))
466 P = POBA
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 13
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < .2631 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1901 PRINT A(6), A(7), A(8), A(9), A(10)
1905 PRINT A(11), A(12), A(13), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [46]. The complete output through JJJJ = -31999.82000000003 is shown below:
.5000473359910486 1.078919963472388 1.947323576334272
.263154905037441 0
0 0 0 0 0
0 0 0 .263154905037441
-31999.82000000003
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 [46], the wall-clock time for obtaining the output through
JJJJ= -31999.82000000003 was 70 seconds, total, including the time for “Creating .EXE file.” One can compare the computational results above to those on page 51 of Yang, Ignizio, and Kim [47].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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