Erratum: Solving a Dual Geometric Programming Problem Using the Algorithm Here

Jsun Yui Wong

The expression

Maximize                   
((((40 * w1 / (X(7) * X(2)))) ^ X(2))) * ((((40 * w1) / (X(7) * X(3))) ^ X(3))) * ((((800 * w2) / (X(7) * X(4))) ^ X(4))) * X(4) ^ X(4) * (1 / (4 * X(5))) ^ X(5) * (1 / X(6)) ^ X(6) * (X(2) + X(3)) ^ (X(2) + X(3)) * (X(5) + X(6)) ^ (X(5) + X(6)) - 1000000 * ABS(-X(2) - X(4) + X(5) + X(6))

should read

Maximize                   
((((40 * w1 / (X(7) * X(2)))) ^ X(2))) * ((((40 * w1) / (X(7) * X(3))) ^ X(3))) * ((((800 * w2) / (X(7) * X(4))) ^ X(4))) * X(4) ^ X(4) * (1 / (4 * X(5))) ^ X(5) * (1 / X(6)) ^ X(6) * (X(2) + X(3)) ^ (X(2) + X(3)) * (X(5) + X(6)) ^ (X(5) + X(6)).
                       

Solving a Dual Geometric Programming Problem Using the Algorithm Here

Jsun Yui Wong

The computer program listed below seeks to solve the following geometricl programming problem on p. 10 of Das and Roy [17]:

Maximize     
 ((((40 * w1) / (X(7) * X(2))) ^ X(2))) * ((((40 * w1) / (X(7) * X(3))) ^ X(3))) * ((((800 * w2) / (X(7) * X(4))) ^ X(4))) * X(4) ^ X(4) * (1 / (4 * X(5))) ^ X(5) * (1 / X(6)) ^ X(6) * (X(2) + X(3)) ^ (X(2) + X(3)) * (X(5) + X(6)) ^ (X(5) + X(6)) - 1000000 * ABS(-X(2) - X(4) + X(5) + X(6))
           
such that 
    w1+w2=1
                X(1) =1
         X(1)   - X(2)    - X(3) - X(4)=0
-X(2) - X(4) + X(5) + X(6)=0
         -X(2) + X(3) - X(4) + X(5)       =0
                 -X(2) + X(3) - X(4) + X(6)  =0
                X(1) through X(6)>=0.

One notes line 192, which is 192 X(7) = 1 .


0 REM     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

    87 M = -3E+50

    88 PL = RND

    90 IF RND < 1 / 9 THEN w1 = .1 ELSE IF RND < 1 / 8 THEN w1 = .2 ELSE IF RND < 1 / 7 THEN w1 = .3 ELSE IF RND < 1 / 6 THEN w1 = .4 ELSE IF RND < 1 / 5 THEN w1 = .5 ELSE IF RND < 1 / 4 THEN w1 = .6 ELSE IF RND < 1 / 3 THEN w1 = .7 ELSE IF RND < 1 / 2 THEN w1 = .8 ELSE w1 = .9

    92 w2 = 1 - w1


    111 FOR J44 = 1 TO 7

        120 A(J44) = ((RND))


    121 NEXT J44

    128 FOR I = 1 TO 20000


        129 FOR KKQQ = 1 TO 7

            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

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


            143 J = 1 + FIX(RND * 7)

            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 * 3) ELSE X(J) = A(J) + FIX(1 + RND * 3)


        169 NEXT IPP
        181 X(1) = 1
        184 X(2) = X(1) - X(3) - X(4)
        188 X(5) = X(2) - X(3) + X(4)
       

        191 X(6) = X(2) - X(3) + X(4)
        192 X(7) = 1

        193 FOR J44 = 1 TO 7

            194 IF X(J44) <= 0## THEN 1670
        197 NEXT J44

        480 PDU = ((((40 * w1) / (X(7) * X(2))) ^ X(2))) * ((((40 * w1) / (X(7) * X(3))) ^ X(3))) * ((((800 * w2) / (X(7) * X(4))) ^ X(4))) * X(4) ^ X(4) * (1 / (4 * X(5))) ^ X(5) * (1 / X(6)) ^ X(6) * (X(2) + X(3)) ^ (X(2) + X(3)) * (X(5) + X(6)) ^ (X(5) + X(6)) - 1000000 * ABS(-X(2) - X(4) + X(5) + X(6))
     
        499 P = PDU

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 7

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

        1557 GOTO 128
    1670 NEXT I
    1890 IF M < -9999999999999 THEN GOTO 1999
    1927 PRINT w1, w2, A(1), A(2), A(3), A(4), A(5), A(6), A(7), M, JJJJ
1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [81].  The output of a single run through JJJJ= -31954 is summarized below:

.8     .2      1       8.340085E-02
.3333333      .5832658      .3333334      .3333334      1
100.7341      32000

.7      .3      1      4.404032E-02   
.3333333       .6226264      .3333334      .3333334    1
122.1631      -31999

.4     .6      1      1.147467E-02   
.3333333       .655192      .3333334      .3333334    1
156.1743      -31998

.7      .3      1      4.400152E-02   
.3333333       .6226652      .3333334      .3333334    1
122.1631      -31997

.3      .7      1      7.303059E-03   
.3333333       .6593636      .3333334      .3333334    1
156.6113      -31996

.2      .8      1      4.238248E-03   
.3333333       .6224284      .3333334      .3333334    1
149.0953      -31995

.7       .3       1      .0440405      .3333333
.6226262      3333334      .3333334      1      122.1631 
-31994
.
.
.
.6     .4      1      2.706528E-02   
.3333333       .6396014      .3333334      .3333334    1
138.4425      -31989

.5     .5     1     1.752967E-02         
.3333333       .649137      .3333334      .3333334    1
149.8429      -31988
.
.
.
.9       .1       1      .2727594     .3333333
.3939073      .3333334      .3333334    1      74.76423 
-31985
.
.
.
.1      .9       1      1.857281E-03   
.3333333       .6648094      .3333334      .3333334       1
127.6965      -31954
.
.
.

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 [81], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31954 was 3 seconds, not including the time for “Creating .EXE file" (17 seconds, total, including the time for “Creating .EXE file").  One can compare the computational results above with those in Das and Roy  [17, p. 10, Table 3]. 

Acknowledgment

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

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Solving a Multi-Objective Geometric Programming Problem in Dual Form

Jsun Yui Wong

The computer program listed below seeks to solve the following formulation in Ota, Pati, and Ojha [62, p. 297]:

Maximize                  ((202.8397 / X(1)) ^ X(1)) * ((91800 / (X(2) * e1)) ^ X(2)) * ((.009 / (X(3) * e1)) ^ X(3)) * ((X(2) + X(3)) ^ (X(2) + X(3)))   

subject to 
   
         X(1) = 1

        -.5 * X(1) -X(2) + .5 * X(3)   =0

                X(1), X(2), X(3)>=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

    87 M = -3E+50
   
    92 IF RND < 1 / 12 THEN e1 = 3.7 ELSE IF RND < 1 / 11 THEN e1 = 10 ELSE IF RND < 1 / 10 THEN e1 = 50 ELSE IF RND < 1 / 9 THEN e1 = 100 ELSE IF RND < 1 / 8 THEN e1 = 200 ELSE IF RND < 1 / 7 THEN e1 = 300 ELSE IF RND < 1 / 6 THEN e1 = 315 ELSE IF RND < 1 / 5 THEN e1 = 320 ELSE IF RND < 1 / 4 THEN e1 = 322 ELSE IF RND < 1 / 3 THEN e1 = 324 ELSE IF RND < 1 / 2 THEN e1 = 324.6 ELSE e1 = 324.69

    111 FOR J44 = 1 TO 3

        120 A(J44) = ((RND * 15))

    121 NEXT J44

    128 FOR I = 1 TO 20000

        129 FOR KKQQ = 1 TO 3
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

        135 FOR IPP = 1 TO FIX(1 + RND * 2)

            143 J = 1 + FIX(RND * 3)

            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 * 3) ELSE X(J) = A(J) + FIX(1 + RND * 3)


        169 NEXT IPP

        181 X(1) = 1

        188 X(2) = -.5 * X(1) + .5 * X(3)


        193 FOR J44 = 1 TO 3

            196 IF X(J44) < 0## THEN 1670


        197 NEXT J44

        478 PDU = ((202.8397 / X(1)) ^ X(1)) * ((91800 / (X(2) * e1)) ^ X(2)) * ((.009 / (X(3) * e1)) ^ X(3)) * ((X(2) + X(3)) ^ (X(2) + X(3)))
        499 P = PDU

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 3

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

        1458 ee1=e1

        1557 GOTO 128
    1670 NEXT I
 
    1925 PRINT ee1, A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [81].  The output of a single run through JJJJ= -31983 is summarized below:

324.6      1     2.174165422141883D-07      1.000000434833084 
5.624022479658293D-03    -32000

315      1     2.378966926785964D-07      1.000000475793385 
5.795421378736472D-03   -31999

324.69      1    2.172261456268032D-07      1.000000434452291 
5.62246357007339D-03    -31998

50      1      5.950421358946745D-05      1.00011900842718
3.651331830432618D-02      -31997
.
.
.

10      1      7.724295390055258D-03      1.015448590780111
.1839443915620831      -31990
.
.
.

3.7      1      2.745562617505041      6.491125235010083
.7020855329854102      -31983
.
.
.

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 [81], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31983 was 2 seconds, not including the time for “Creating .EXE file" (18 seconds, total, including the time for “Creating .EXE file").  One can compare the computational results above with those in Ota, Pati, and Ojha  [62, p. 296, Table 6]. 

Acknowledgment

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

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