Solving Multi-Objective Geometric Programming Problems with Weighted Mean Method by Example

                                                            Jsun Yui Wong

The computer program listed below seeks to solve the following multi-objective geometric programming problem on pages 83-84 of Ojha and Biswal [58, Example 1]:

Minimize                  w1 * (4 * X(1) + 10 * X(2) + 4 * X(3) + 2 * X(4)) + w2 * (1 / (X(1) * X(2) * X(3)))

subject to 

         X(1) ^ 2 / X(4) ^ 2 + X(2) ^ 2 / X(4) ^ 2 <= 1

         100 / (X(1) * X(2) * X(3)) <= 1

         X(1), X(2), X(3), X(4)>0

         w1+ w2 =1, w1, w2>0.


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), AL(50), SW(50), SV(50), C2(22), C3(22), C4(22), C5(22)

81 FOR JJJJ = -32000 TO 32000

    85 RANDOMIZE JJJJ
   
    91 M = -3E+50
    92 IF RND < .2 THEN w1 = .1 ELSE IF RND < .4 THEN w1 = .2 ELSE IF RND < .6 THEN w1 = .3 ELSE IF RND < .8 THEN w1 = .4 ELSE w1 = .5

    93 w2 = 1 - w1

    104 REM w2 = .2
    120 A(1) = RND * 8
    121 A(2) = RND * 8
    122 A(3) = ((RND * 8))
    123 A(4) = ((RND * 8))

    128 FOR I = 1 TO FIX(31500 + RND * 1000)

        129 FOR KKQQ = 1 TO 4
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        135 FOR IPP = 1 TO FIX(1 + RND * 3)


            143 J = 1 + FIX(RND * 4)

            149 REM IF J < 5 THEN GOTO 162 ELSE GOTO 156

            156 r = (1 - RND * 2) * A(J)

            158 X(J) = A(J) + (RND ^ (RND * 15)) * r

            161 GOTO 169

            162 IF RND < .5 THEN X(J) = A(J) - FIX(1 + RND * 2) ELSE X(J) = A(J) + FIX(1 + RND * 2)


        169 NEXT IPP


        180 IF X(1) < 0## THEN 1670
        183 IF X(2) < 0## THEN 1670

        185 IF X(3) < 0## THEN 1670
        186 IF X(4) < 0## THEN 1670

        404 IF 100 / (X(1) * X(2) * X(3)) > 1 THEN 1670
        405 IF X(1) ^ 2 / X(4) ^ 2 + X(2) ^ 2 / X(4) ^ 2 > 1 THEN 1670

        462 PDU = -w1 * (4 * X(1) + 10 * X(2) + 4 * X(3) + 2 * X(4)) - w2 * (1 / (X(1) * X(2) * X(3)))

        466 P = PDU

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 4

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX

        1462 ggg10 = (4 * X(1) + 10 * X(2) + 4 * X(3) + 2 * X(4))

        1465 ggg20 = (1 / (X(1) * X(2) * X(3)))


        1557 GOTO 128
    1670 NEXT I
    1890 IF M < -888888888 THEN GOTO 1999
    1892 REM IF ggg10 > 80 THEN GOTO 1999

    1899 PRINT ggg10, ggg20, JJJJ

    1921 PRINT w1, w2, A(1), A(2)
   
    1924 PRINT A(3), A(4), M

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [77].  The output of a single run through JJJJ= -31380 is summarized below:

88.15562469901354      9.999999999999997D-03      -32000
.4      .6      4.719555136150587      2.868805245195353
7.385805551278306      5.523064748672223      -35.26824987960541

88.11130563143098      .01      -31987
.1      .9      4.827953883258128      2.868362464746813
7.221091635824544      5.61574945381609        -8.8201305631431

88.50150329803067      .01      -31982
.4      .6      4.53943321573826       3.05910344029511
7.201188950556504      5.463990114950264       -35.40660131921227

89.08585157360382      .010   -31979
.2      .8      6.242948946227302      2.550393104955556.
6.280628289719224      6.74380579013108      -17.82517031472077
.
.
.
.
.
.
87.98776799193816      .01      -31380
.4      .6      5.083801172404946      2.68363355020535
7.32973427366338      5.747645352805643      -35.20110719677526

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 [77], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31380 was 85 seconds, not including the time for “Creating .EXE file" (100 seconds, total, including the time for “Creating .EXE file").  One can compare the computational results above with those in Table 2 of Ojha and Biswal [58, p. 84, Table 2 and below].

Acknowledgment

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

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