Monday, July 16, 2018

Mixed-Integer Nonlinear Programming Solver Applied to a Reliability-Redundancy Problem with Five Stages


Jsun Yui Wong

The computer program listed below seeks to solve the following mixed-integer nonlinear programming problem, which is based on Table 1 of Tillman, Hwang, and Kuo [38, p. 165]:     
         
Maximize            (1 - (1 - X(6)) ^ X(1)) * (1 - (1 - X(7)) ^ X(2)) * (1 - (1 - X(8)) ^ X(3)) * (1 - (1 - X(9)) ^ X(4)) * (1 - (1 - X(10)) ^ X(5))

subject to

         1 * X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 4 * X(4) ^ 2 + 2 * X(5) ^ 2<=110


                 (2.33 / 10 ^ 5) * ((-1000 / LOG(X(6))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) + (1.45 / 10 ^ 5) * ((-1000 / LOG(X(7))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) + (.541 / 10 ^ 5) * ((-1000 / LOG(X(8))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) + (8.05 / 10 ^ 5) * ((-1000 / LOG(X(9))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) + (1.95 / 10 ^ 5) * ((-1000 / LOG(X(10))) ^ 1.5) * (X(5) + EXP(X(5) / 4))<=175

 7 * X(1) * EXP(X(1) / 4) + (8) * X(2) * EXP(X(2) / 4) + (8) * X(3) * EXP(X(3) / 4) + (6) * X(4) * EXP(X(4) / 4) + (9) * X(5) * EXP(X(5) / 4)<=200


            1<= X(i) <= 10, i=1, 2, 3, 4; 5, X(1) through X(5) are integer variables

.5<= X(6), X(7), X(8), X(9), X(10)<=.999999.

X(11), X(12), and X(13) 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 5

        97 A(J44) = FIX(1 + RND * 10)

    99 NEXT J44

    115 FOR J44 = 6 TO 10

        117 A(J44) = .5 + RND * .499999

    119 NEXT J44

    128 FOR I = 1 TO 40000


        129 FOR KKQQ = 1 TO 10
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        151 FOR IPP = 1 TO FIX(1 + RND * 3)


            153 j = 1 + FIX(RND * 10)


            155 IF j > 5.5 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 5

            327 X(J44) = INT(X(J44))


            328 IF X(J44) < 1 THEN 1670

            329 IF X(J44) > 10 THEN 1670
        331 NEXT J44

        336 FOR J44 = 6 TO 10


            338 IF X(J44) < .5## THEN 1670

            339 IF X(J44) > .999999## THEN 1670
        340 NEXT J44


        341 X(11) = 110 - 1 * X(1) ^ 2 - 2 * X(2) ^ 2 - 3 * X(3) ^ 2 - 4 * X(4) ^ 2 - 2 * X(5) ^ 2


        343 X(12) = 200 - 7 * X(1) * EXP(X(1) / 4) - (8) * X(2) * EXP(X(2) / 4) - (8) * X(3) * EXP(X(3) / 4) - (6) * X(4) * EXP(X(4) / 4) - (9) * X(5) * EXP(X(5) / 4)


        346 X(13) = 175 - (2.33 / 10 ^ 5) * ((-1000 / LOG(X(6))) ^ 1.5) * (X(1) + EXP(X(1) / 4)) - (1.45 / 10 ^ 5) * ((-1000 / LOG(X(7))) ^ 1.5) * (X(2) + EXP(X(2) / 4)) - (.541 / 10 ^ 5) * ((-1000 / LOG(X(8))) ^ 1.5) * (X(3) + EXP(X(3) / 4)) - (8.05 / 10 ^ 5) * ((-1000 / LOG(X(9))) ^ 1.5) * (X(4) + EXP(X(4) / 4)) - (1.95 / 10 ^ 5) * ((-1000 / LOG(X(10))) ^ 1.5) * (X(5) + EXP(X(5) / 4))


        355 FOR J44 = 11 TO 13


            357 IF X(J44) < 0 THEN X(J44) = X(J44) ELSE X(J44) = 0

        359 NEXT J44


        411 PDU = (1 - (1 - X(6)) ^ X(1)) * (1 - (1 - X(7)) ^ X(2)) * (1 - (1 - X(8)) ^ X(3)) * (1 - (1 - X(9)) ^ X(4)) * (1 - (1 - X(10)) ^ X(5)) + 1000000 * (X(11) + X(12) + X(13))


        466 P = PDU
        1111 IF P <= M THEN 1670
        1452 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 < .931677 THEN 1999

    1904 PRINT A(1), A(2), A(3)
    1905 PRINT A(4), A(5), A(6)
    1906 PRINT A(7), A(8), A(9)
    1907 PRINT A(10), A(11), A(12), A(13)

    1909 PRINT M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [41].  The complete output through JJJJ= -31266 is shown below:

3      2      2
3      3     .7795519252219796
.8727194718321425      .9022987392717032      .7114015479912691
.786765140967794      0      0      0
.931677103177272        -31376

3      2      2
3      3     .7789622643589318
.871993590446706       .9031635093885894       .7118740906368367
.7866751378019727      0      0      0
.931679616709972         -31266

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 [41], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31266 was 23 minutes, total, including the time for “Creating .EXE file."  One can compare the computational results above with those in
Tillman, Hwang, and Kuo [38, p. 165, Table 2] and with those in Xu, Kuo, and Lin [42, p. 52, Table 2].

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

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