Jsun Yui Wong
The computer program listed below seeks to solve the following problem from Mellal and Zio [34, p. 218, Table 5]:
Maximize PROD * (1 - (1 - X(31)) ^ X(11)) * (1 - (1 - X(32)) ^ X(12)) * (1 - (1 - X(33)) ^ X(13)) * (1 - (1 - X(34)) ^ X(14)) * (1 - (1 - X(35)) ^ X(15)) * (1 - (1 - X(36)) ^ X(16)) * (1 - (1 - X(37)) ^ X(17)) * (1 - (1 - X(38)) ^ X(18)) * (1 - (1 - X(39)) ^ X(19)) * (1 - (1 - X(40)) ^ X(20))
where PROD = (1 - (1 - X(21)) ^ X(1)) * (1 - (1 - X(22)) ^ X(2)) * (1 - (1 - X(23)) ^ X(3)) * (1 - (1 - X(24)) ^ X(4)) * (1 - (1 - X(25)) ^ X(5)) * (1 - (1 - X(26)) ^ X(6)) * (1 - (1 - X(27)) ^ X(7)) * (1 - (1 - X(28)) ^ X(8)) * (1 - (1 - X(29)) ^ X(9)) * (1 - (1 - X(30)) ^ X(10))
subject to
2 * X(1) ^ 2 + 5 * X(2) ^ 2 + 5 * X(3) ^ 2 + 4 * X(4) ^ 2 + 4 * X(5) ^ 2 + 1 * X(6) ^ 2 + 1 * X(7) ^ 2 + 4 * X(8) ^ 2 + 4 * X(9) ^ 2 + 3 * X(10) ^ 2 + 3 * X(11) ^ 2 + 1 * X(12) ^ 2 + 1 * X(13) ^ 2 + 3 * X(14) ^ 2 + 4 * X(15) ^ 2 + 5 * X(16) ^ 2 + 1 * X(17) ^ 2 + 4 * X(18) ^ 2 + 2 * X(19) ^ 2 + 1 * X(20) ^ 2<=600
8 * X(1) * EXP(X(1) / 4) + (9) * X(2) * EXP(X(2) / 4) + (6) * X(3) * EXP(X(3) / 4) + (10) * X(4) * EXP(X(4) / 4) + (8) * X(5) * EXP(X(5) / 4) + 9 * X(6) * EXP(X(6) / 4) + (9) * X(7) * EXP(X(7) / 4) + (7) * X(8) * EXP(X(8) / 4) + (9) * X(9) * EXP(X(9) / 4) + (8) * X(10) * EXP(X(10) / 4) + 9 * X(11) * EXP(X(11) / 4) + (8) * X(12) * EXP(X(12) / 4) + (7) * X(13) * EXP(X(13) / 4) + (10) * X(14) * EXP(X(14) / 4) + (6) * X(15) * EXP(X(15) / 4) + 7 * X(16) * EXP(X(16) / 4) + (7) * X(17) * EXP(X(17) / 4) + (8) * X(18) * EXP(X(18) / 4) + (9) * X(19) * EXP(X(19) / 4) + (9) * X(20) * EXP(X(20) / 4)<=900
-XADDC - XADDB - XADD + (.6 / 10 ^ 5) * ((-1000 / LOG(X(21))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) + (.1 / 10 ^ 5) * ((-1000 / LOG(X(22))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) + (1.2 / 10 ^ 5) * ((-1000 / LOG(X(23))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) + (.3 / 10 ^ 5) * ((-1000 / LOG(X(24))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) + (2.9 / 10 ^ 5) * ((-1000 / LOG(X(25))) ^ 1.5) * (X(5) + EXP(X(5) / 4))<=700
where
XADDC = -(1.3 / 10 ^ 5) * ((-1000 / LOG(X(36))) ^ 1.5) * (X(16) + EXP(X(16) / 4)) - (1.9 / 10 ^ 5) * ((-1000 / LOG(X(37))) ^ 1.5) * (X(17) + EXP(X(17) / 4)) - (2.7 / 10 ^ 5) * ((-1000 / LOG(X(38))) ^ 1.5) * (X(18) + EXP(X(18) / 4)) - (2.8 / 10 ^ 5) * ((-1000 / LOG(X(39))) ^ 1.5) * (X(19) + EXP(X(19) / 4)) - (1.5 / 10 ^ 5) * ((-1000 / LOG(X(40))) ^ 1.5) * (X(20) + EXP(X(20) / 4))
XADDB = -(2.4 / 10 ^ 5) * ((-1000 / LOG(X(31))) ^ 1.5) * (X(11) + EXP(X(11) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(32))) ^ 1.5) * (X(12) + EXP(X(12) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(33))) ^ 1.5) * (X(13) + EXP(X(13) / 4)) - (2.1 / 10 ^ 5) * ((-1000 / LOG(X(34))) ^ 1.5) * (X(14) + EXP(X(14) / 4)) - (.9 / 10 ^ 5) * ((-1000 / LOG(X(35))) ^ 1.5) * (X(15) + EXP(X(15) / 4))
XADD = -(1.7 / 10 ^ 5) * ((-1000 / LOG(X(26))) ^ 1.5) * (X(6) + EXP(X(6) / 4)) - (2.6 / 10 ^ 5) * ((-1000 / LOG(X(27))) ^ 1.5) * (X(7) + EXP(X(7) / 4)) - (2.5 / 10 ^ 5) * ((-1000 / LOG(X(28))) ^ 1.5) * (X(8) + EXP(X(8) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(29))) ^ 1.5) * (X(9) + EXP(X(9) / 4)) - (1.8 / 10 ^ 5) * ((-1000 / LOG(X(30))) ^ 1.5) * (X(10) + EXP(X(10) / 4))
1<= X(i) <=10, for i=1 through 20, and these Xs are integers;
.5<= X(i) <=1, for i=21, 22, 23, ..., 40.
X(41) through X(43) below are slack variables added.
0 DEFDBL A-Z
1 DEFINT K
2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(22), CC(20), RR(20), WW(20)
81 FOR JJJJ = -32000 TO 32000
89 RANDOMIZE JJJJ
90 M = -3E+30
95 FOR J44 = 1 TO 20
97 A(J44) = FIX(1 + RND * 10)
99 NEXT J44
115 FOR J44 = 21 TO 40
117 A(J44) = .5 + RND * .499999##
119 NEXT J44
128 FOR I = 1 TO 40000
129 FOR KKQQ = 1 TO 40
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
151 FOR IPP = 1 TO FIX(1 + RND * 5)
153 j = 1 + FIX(RND * 40)
155 IF j > 20.5 THEN GOTO 156 ELSE GOTO 164
156 r = (1 - RND * 2) * A(j)
158 X(j) = A(j) + (RND ^ (RND * 10)) * r
161 GOTO 169
164 IF RND < .5 THEN X(j) = A(j) - 1 ELSE X(j) = A(j) + 1
169 NEXT IPP
326 FOR J44 = 1 TO 20
327 X(J44) = CINT(X(J44))
328 IF X(J44) < 1 THEN 1670
329 IF X(J44) > 10 THEN 1670
331 NEXT J44
336 FOR J44 = 21 TO 40
338 IF X(J44) < .5## THEN 1670
339 IF X(J44) > .999999## THEN 1670
340 NEXT J44
342 X(41) = 600 - 2 * X(1) ^ 2 - 5 * X(2) ^ 2 - 5 * X(3) ^ 2 - 4 * X(4) ^ 2 - 4 * X(5) ^ 2 - 1 * X(6) ^ 2 - 1 * X(7) ^ 2 - 4 * X(8) ^ 2 - 4 * X(9) ^ 2 - 3 * X(10) ^ 2 - 3 * X(11) ^ 2 - 1 * X(12) ^ 2 - 1 * X(13) ^ 2 - 3 * X(14) ^ 2 - 4 * X(15) ^ 2 - 5 * X(16) ^ 2 - 1 * X(17) ^ 2 - 4 * X(18) ^ 2 - 2 * X(19) ^ 2 - 1 * X(20) ^ 2
343 X(42) = 900 - 8 * X(1) * EXP(X(1) / 4) - (9) * X(2) * EXP(X(2) / 4) - (6) * X(3) * EXP(X(3) / 4) - (10) * X(4) * EXP(X(4) / 4) - (8) * X(5) * EXP(X(5) / 4) - 9 * X(6) * EXP(X(6) / 4) - (9) * X(7) * EXP(X(7) / 4) - (7) * X(8) * EXP(X(8) / 4) - (9) * X(9) * EXP(X(9) / 4) - (8) * X(10) * EXP(X(10) / 4) - 9 * X(11) * EXP(X(11) / 4) - (8) * X(12) * EXP(X(12) / 4) - (7) * X(13) * EXP(X(13) / 4) - (10) * X(14) * EXP(X(14) / 4) - (6) * X(15) * EXP(X(15) / 4) - 7 * X(16) * EXP(X(16) / 4) - (7) * X(17) * EXP(X(17) / 4) - (8) * X(18) * EXP(X(18) / 4) - (9) * X(19) * EXP(X(19) / 4) - (9) * X(20) * EXP(X(20) / 4)
344 XADDC = -(1.3 / 10 ^ 5) * ((-1000 / LOG(X(36))) ^ 1.5) * (X(16) + EXP(X(16) / 4)) - (1.9 / 10 ^ 5) * ((-1000 / LOG(X(37))) ^ 1.5) * (X(17) + EXP(X(17) / 4)) - (2.7 / 10 ^ 5) * ((-1000 / LOG(X(38))) ^ 1.5) * (X(18) + EXP(X(18) / 4)) - (2.8 / 10 ^ 5) * ((-1000 / LOG(X(39))) ^ 1.5) * (X(19) + EXP(X(19) / 4)) - (1.5 / 10 ^ 5) * ((-1000 / LOG(X(40))) ^ 1.5) * (X(20) + EXP(X(20) / 4))
345 XADDB = -(2.4 / 10 ^ 5) * ((-1000 / LOG(X(31))) ^ 1.5) * (X(11) + EXP(X(11) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(32))) ^ 1.5) * (X(12) + EXP(X(12) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(33))) ^ 1.5) * (X(13) + EXP(X(13) / 4)) - (2.1 / 10 ^ 5) * ((-1000 / LOG(X(34))) ^ 1.5) * (X(14) + EXP(X(14) / 4)) - (.9 / 10 ^ 5) * ((-1000 / LOG(X(35))) ^ 1.5) * (X(15) + EXP(X(15) / 4))
348 XADD = -(1.7 / 10 ^ 5) * ((-1000 / LOG(X(26))) ^ 1.5) * (X(6) + EXP(X(6) / 4)) - (2.6 / 10 ^ 5) * ((-1000 / LOG(X(27))) ^ 1.5) * (X(7) + EXP(X(7) / 4)) - (2.5 / 10 ^ 5) * ((-1000 / LOG(X(28))) ^ 1.5) * (X(8) + EXP(X(8) / 4)) - (1.3 / 10 ^ 5) * ((-1000 / LOG(X(29))) ^ 1.5) * (X(9) + EXP(X(9) / 4)) - (1.8 / 10 ^ 5) * ((-1000 / LOG(X(30))) ^ 1.5) * (X(10) + EXP(X(10) / 4))
350 X(43) = XADDC + XADDB + XADD + 700 - (.6 / 10 ^ 5) * ((-1000 / LOG(X(21))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) - (.1 / 10 ^ 5) * ((-1000 / LOG(X(22))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) - (1.2 / 10 ^ 5) * ((-1000 / LOG(X(23))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) - (.3 / 10 ^ 5) * ((-1000 / LOG(X(24))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) - (2.9 / 10 ^ 5) * ((-1000 / LOG(X(25))) ^ 1.5) * (X(5) + EXP(X(5) / 4))
355 FOR J44 = 41 TO 43
357 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0
359 NEXT J44
422 PROD = (1 - (1 - X(21)) ^ X(1)) * (1 - (1 - X(22)) ^ X(2)) * (1 - (1 - X(23)) ^ X(3)) * (1 - (1 - X(24)) ^ X(4)) * (1 - (1 - X(25)) ^ X(5)) * (1 - (1 - X(26)) ^ X(6)) * (1 - (1 - X(27)) ^ X(7)) * (1 - (1 - X(28)) ^ X(8)) * (1 - (1 - X(29)) ^ X(9)) * (1 - (1 - X(30)) ^ X(10))
433 PDU = PROD * (1 - (1 - X(31)) ^ X(11)) * (1 - (1 - X(32)) ^ X(12)) * (1 - (1 - X(33)) ^ X(13)) * (1 - (1 - X(34)) ^ X(14)) * (1 - (1 - X(35)) ^ X(15)) * (1 - (1 - X(36)) ^ X(16)) * (1 - (1 - X(37)) ^ X(17)) * (1 - (1 - X(38)) ^ X(18)) * (1 - (1 - X(39)) ^ X(19)) * (1 - (1 - X(40)) ^ X(20)) + 1000000 * (X(41) + X(42) + X(43))
466 P = PDU
1111 IF P <= M THEN 1670
1452 M = P
1454 FOR KLX = 1 TO 43
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < .890 THEN 1999
1904 PRINT A(1), A(2), A(3), A(4), A(5)
1907 PRINT A(6), A(7), A(8), A(9), A(10)
1908 PRINT A(11), A(12), A(13), A(14), A(15)
1927 PRINT A(16), A(17), A(18), A(19), A(20)
1944 PRINT A(21), A(22), A(23), A(24), A(25)
1947 PRINT A(26), A(27), A(28), A(29), A(30)
1948 PRINT A(31), A(32), A(33), A(34), A(35)
1949 PRINT A(36), A(37), A(38), A(39), A(40)
1950 PRINT A(41), A(41), A(43), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [46]. The top-two candidate solutions of a single run through JJJJ= -31706 are shown below:
2 2 3 2 3
2 3 3 2 3
3 3 3 3 3
3 3 3 3 3
.9176676059378108 .9529493608779073 .8414426000843975
.9350445768679673 .8020502969111866
.8959271422787656 .8105255951980664 .810916654841777
.904252400517493 .8243589561627621 .
.8175257801246864 .8432949169984238 .8474666184722103
.819240905662896 .8484380456464606
.8395842574160096 .823304323945264 .807493413855936
.8069133497713611 .8367136339197568
0 0 0 .8909194266571648
-31900
2 2 3 2 3
3 3 3 2 3
3 3 3 3 3
3 3 3 3 2
.9207435079905487 .9498790771031488 .840974035964459
.9357658299762895 .807662549706277
.825183271601671 .8143538750318156 .8120130770240397
.9068879444864856 .819553712721343
.8118850433518717 .8407556540903034 .8480743971425285
.8206862478868184 .848582388518259
.8438357935349793 .8257649447020704 .813061425750956
.807172657789285 .8930113078512505
0 0 0 .8913278111036525
-31706
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 (not CPU time) for obtaining the output through JJJJ = -31706 was 103 minutes, total, including the time for “Creating .EXE file." One can compare the computational results above with those in Mellal and Zio [34, p. 224, Table 14].
Acknowledgment
I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.
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