Jsun Yui Wong
The computer program listed below uses a technique in Murty, [36, pp. 277-278], to solve the immediately following problem:
Maximize
((X(1) + 2 * X(2) ^ 3 + 3 * X(3) + X(4) + 2 * X(5) + 7 * X(6) + 3 * X(7) + 3 * X(8) + 11 * X(9) + 7 * X(10)) / (X(2) ^ 3 + 5 * X(3) + X(6) + 2 * X(8) + 9 * X(10) + 3)) ^ 3,
((X(1) + 7 * X(3) + 4 * X(4) ^ 3 + 3 * X(5) + 5 * X(7) + 13 * X(9) + 11 * X(10)) / (X(2) + 3 * X(4) ^ 3 + X(6) + 4 * X(8) + 6 * X(10) + 1)) ^ 2,
((X(1) + 3 * X(3) + 5 * X(4) + 6 * X(5) + 7 * X(7) + 4 * X(9) ^ 5 + 7 * X(10) + 2) / (2 * X(1) + X(2) + 8 * X(3) + 6 * X(4) + 2 * X(5) + X(6) + 4 * X(7) + 3 * X(8) + 3 * X(9) ^ 5 + 7 * X(10) + 2)) ^ 3,
which are three objective functions
subject to
X(1) + X(2) + 7 * X(3) + X(4) + 3 * X(5) + 6 * X(6) + 3 * X(7) + 5 * X(8) + 14 * X(9) + 3 * X(10)<=700
4 * X(1) + X(2) + 2 * X(3) + 9 * X(4) + 7 * X(5) + 9 * X(6) + 4 * X(7) + 7 * X(8) + 7 * X(9) + 11 * X(10)<=900
.3 * X(1) ^ 3 + X(2) + .1 * X(3) + .1 * X(4) ^ 3 + .1 * X(5) + .1 * X(6) + .5 * X(7) ^ 5 + .1 * X(8) + .2 * X(9) ^ 2 + .1 * X(10)<=5000
0<=X(i) <= 200, i=1, 2, 3, 4, 5, 6, 7, 8, 9, 10
X(1) through X(10) are general integer variables.
The formulation above is loosely based on the formulation in Sharma [42, p.149].
The decision makers have determined that the weights are .5, .3, and .2, respectively.
X(11) through X(13) below are slack variables.
One notes the following line 441, which includes the nonlinear numerators and denominators.
One also notes the nonlinear constraint .3 * X(1) ^ 3 + X(2) + .1 * X(3) + .1 * X(4) ^ 3 + .1 * X(5) + .1 * X(6) + .5 * X(7) ^ 5 + .1 * X(8) + .2 * X(9) ^ 2 + .1 * X(10)<=5000.
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
51 FOR J40 = 1 TO 10
55 A(J40) = RND * 200
56 NEXT J40
128 FOR I = 1 TO 6000
129 FOR KKQQ = 1 TO 10
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 j = 1 + FIX(RND * 10)
183 r = (1 - RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP
246 FOR J47 = 1 TO 10
247 X(J47) = INT(X(J47))
248 IF X(J47) < 0 THEN 1670
249 IF X(J47) > 200 THEN 1670
251 NEXT J47
255 REM
301 X(11) = 700 - X(1) - X(2) - 7 * X(3) - X(4) - 3 * X(5) - 6 * X(6) - 3 * X(7) - 5 * X(8) - 14 * X(9) - 3 * X(10)
303 X(12) = 900 - 4 * X(1) - X(2) - 2 * X(3) - 9 * X(4) - 7 * X(5) - 9 * X(6) - 4 * X(7) - 7 * X(8) - 7 * X(9) - 11 * X(10)
305 X(13) = 5000 - .3 * X(1) ^ 3 - X(2) - .1 * X(3) - .1 * X(4) ^ 3 - .1 * X(5) - .1 * X(6) - .5 * X(7) ^ 5 - .1 * X(8) - .2 * X(9) ^ 2 - .1 * X(10)
401 FOR J47 = 11 TO 13
402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0
403 NEXT J47
441 POBA = .5 * ((X(1) + 2 * X(2) ^ 3 + 3 * X(3) + X(4) + 2 * X(5) + 7 * X(6) + 3 * X(7) + 3 * X(8) + 11 * X(9) + 7 * X(10)) / (X(2) ^ 3 + 5 * X(3) + X(6) + 2 * X(8) + 9 * X(10) + 3)) ^ 3 + .3 * ((X(1) + 7 * X(3) + 4 * X(4) ^ 3 + 3 * X(5) + 5 * X(7) + 13 * X(9) + 11 * X(10)) / (X(2) + 3 * X(4) ^ 3 + X(6) + 4 * X(8) + 6 * X(10) + 1)) ^ 2 + .2 * ((X(1) + 3 * X(3) + 5 * X(4) + 6 * X(5) + 7 * X(7) + 4 * X(9) ^ 5 + 7 * X(10) + 2) / (2 * X(1) + X(2) + 8 * X(3) + 6 * X(4) + 2 * X(5) + X(6) + 4 * X(7) + 3 * X(8) + 3 * X(9) ^ 5 + 7 * X(10) + 2)) ^ 3 + 1000000 * (X(11) + X(12) + X(13))
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 < 3330000 THEN 1999
1900 PRINT A(1), A(2), A(3), A(4), A(5)
1907 PRINT A(6), A(7), A(8), A(9), A(10), A(11), A(12), A(13), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [49]. The complete output through JJJJ = -31893.840000017 is shown below:
21 0 0 0 2
0 5 0 47 0
0 0 0 3332032.155555523
-31971.12000000462
21 0 0 0 2
0 5 0 47 0
0 0 0 3332032.155555523
-31914.98000001361
21 0 0 0 2
0 5 0 47 0
0 0 0 3332032.155555523
-31893.840000017
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 [49], the wall-clock time for obtaining the output through
JJJJ= -31893.840000017 was 7 minutes, total, including the time for “Creating .EXE file.”
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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