The computer program listed below uses a technique in Murty, [36, pp. 277-279], 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) + 3 * X(11)) / (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(12)) / (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) + X(13) + 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) + X(11) + 7 * X(12) + X(13)<=70000
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) + X(11) + 6 * X(12) + X(13)<=90000
.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) + X(11) + X(12) + 3 * X(13)<=500000
0<=X(i) <= 1000, i=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
X(1) through X(13) are general integer variables.
The formulation above is loosely based on the formulation in Sharma [42, p.149].
The decision makers have agreed that the weights X(14) through X(16) are fuzzy as follows:
.45 + RND * .1
.25 + RND * .1
.15 + RND * .1, respectively.
X(17) through X(19) below are slack variables.
One notes the following line 441, which includes the nonlinear numerators and denominators and the fuzzy weights--X(14), X(15), and X(16).
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) + X(11) + X(12) + 3 * X(13)<=500000.
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 32111 STEP .01
14 RANDOMIZE JJJJ
16 M = -1D+37
51 FOR J40 = 1 TO 13
55 A(J40) = RND * 1000
56 NEXT J40
61 A(14) = .45 + RND * .1
63 A(15) = .25 + RND * .1
65 A(16) = .15 + RND * .1
128 FOR I = 1 TO 6000
129 FOR KKQQ = 1 TO 16
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 FOR IPP = 1 TO (1 + FIX(RND * 3))
181 j = 1 + FIX(RND * 16)
183 r = (1 - RND * 2) * A(j)
187 X(j) = A(j) + (RND ^ (RND * 10)) * r
222 NEXT IPP
233 X(16) = 1 - X(14) - X(15)
246 FOR J47 = 1 TO 13
247 X(J47) = INT(X(J47))
248 IF X(J47) < 0 THEN 1670
249 IF X(J47) > 1000 THEN 1670
251 NEXT J47
268 IF X(14) < .45 THEN 1670
269 IF X(14) > .55 THEN 1670
272 IF X(15) < .25 THEN 1670
273 IF X(15) > .35 THEN 1670
274 IF X(16) < .15 THEN 1670
275 IF X(16) > .25 THEN 1670
301 X(17) = 70000 - 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) - X(11) - 7 * X(12) - X(13)
303 X(18) = 90000 - 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) - X(11) - 6 * X(12) - X(13)
305 X(19) = 500000 - .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) - X(11) - X(12) - 3 * X(13)
401 FOR J47 = 17 TO 19
402 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0
403 NEXT J47
441 POBA = X(14) * ((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) + 3 * X(11)) / (X(2) ^ 3 + 5 * X(3) + X(6) + 2 * X(8) + 9 * X(10) + 3)) ^ 3 + X(15) * ((X(1) + 7 * X(3) + 4 * X(4) ^ 3 + 3 * X(5) + 5 * X(7) + 13 * X(9) + 11 * X(10) + X(12)) / (X(2) + 3 * X(4) ^ 3 + X(6) + 4 * X(8) + 6 * X(10) + 1)) ^ 2 + X(16) * ((X(1) + 3 * X(3) + 5 * X(4) + 6 * X(5) + 7 * X(7) + 4 * X(9) ^ 5 + 7 * X(10) + X(13) + 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 + 90000000000 * (X(17) + X(18) + X(19))
466 P = POBA
1111 IF P <= M THEN 1670
1450 M = P
1454 FOR KLX = 1 TO 19
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < 86811100000 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
1909 PRINT A(14), A(15), A(16)
1910 PRINT A(17), A(18), A(19)
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [49]. The complete output through JJJJ = -31999.37000000001 is shown below:
70 0 0 121 1000
0 8 0 1000 0
1000 0 0 86845999407.83077
-31999.96000000001
.550000041723147 .2500011811805727 .1999987770962803
0 0 0
66 0 0 126 1000
0 7 0 1000 0
1000 997 57 86813867914.30338
-31999.52000000008
.5500000415514993 .2500037636007895 .1999961948477112
0 0 0
70 0 0 121 1000
0 8 0 1000 0
1000 0 0 86845999406.17258
-31999.4300000001
.5500000417126456 .25000150443893 .1999984538484245
0 0 0
68 0 0 123 1000
0 8 0 1000 0
1000 994 0 86845999407.77126
-31999.3700000001
.55000004172277 .2500020029690194 .1999979553082107
0 0 0
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 = -31999.37000000001 was 30 seconds, not 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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