Solving Another 3-Objective Integer Nonlinear Programming Problem with the Epsilon-Constraint Method

Jsun Yui Wong

The computer program listed below seeks to solve the following 3-objective integer nonlinear programming problem from Arora and Arora [8,  p. 377]:   

Maximize               (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)
       

maximize                  (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)
   

maximize                  (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)
     

subject to


         3 * X(1) - 2 * X(2) + X(3) + X(4) <= 9

         X(2) + 3 * X(3) + X(4) <= 7


         X(1) + X(2) + X(3) + X(4) <= 12


         2 * X(2) + X(3) <= 10


         0 <=X(1) <=5

        0<= X(2)<=3

         0 <=X(3) <=3

        0<= X(4)<=1

 X(1) through X(4) are general integer variables.


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
   
    91 w1 = 10# + FIX(RND * 201)
    92 w2 = 10# + FIX(RND * 201)

    95 A(1) = FIX(RND * 6)
    97 A(2) = FIX(RND * 4)

    98 A(3) = FIX(RND * 4)
    99 A(4) = FIX(RND * 2)


    128 FOR I = 1 TO 100000


        129 FOR KKQQ = 1 TO 4
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        151 FOR IPP = 1 TO FIX(1 + RND * 4)


            153 J = 1 + FIX(RND * 4)
            154 GOTO 162

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

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

            161 REM GOTO 169

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

        169 NEXT IPP
        171 X(1) = INT(X(1))
        173 X(2) = INT(X(2))

        174 X(3) = INT(X(3))
        175 X(4) = INT(X(4))



        177 IF X(1) < 0## THEN 1670
        187 IF X(2) < 0## THEN 1670

        188 IF X(3) < 0## THEN 1670
        189 IF X(4) < 0## THEN 1670


        192 IF X(1) > 5 THEN 1670

        194 IF X(2) > 3 THEN 1670

        196 IF X(3) > 3 THEN 1670
        198 IF X(4) > 1 THEN 1670


        226 IF 3 * X(1) - 2 * X(2) + X(3) + X(4) > 9 THEN 1670



        228 IF X(2) + 3 * X(3) + X(4) > 7 THEN 1670

        230 IF X(1) + X(2) + X(3) + X(4) > 12 THEN 1670

        232 IF 2 * X(2) + X(3) > 10 THEN 1670
        244 IF (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11) < w1 THEN 1670

        255 IF (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9) < w2 THEN 1670

        417 PDU = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)


        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


        1522 g01star = (X(1) + X(2) - 2 * X(3) - 3 * X(4) + 15) * (-X(2) + X(3) + X(4) + 10)


        1524 g02star = (X(1) + X(2) - X(3) - 2 * X(4) + 12) * (-X(2) + 2 * X(3) + 2 * X(4) + 11)


        1526 g03star = (X(1) - 3 * X(2) - X(3) + X(4) + 9) * (2 * X(2) + 2 * X(3) + 2 * X(4) + 9)


        1557 GOTO 128
    1670 NEXT I

    1889 IF M < -99999999999 THEN 1999
    1922 PRINT g01star, g02star, g03star, JJJJ
    1928 PRINT A(1), A(2), A(3), A(4)

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [60].  The output of a single run through JJJJ= -31478 is summarized below:

171      187      90      -31995
4      2      1      0     

180      165      108      -31994
3      0      0      0
.
.
.

171      187      90      -31670
4      2      1      0     

180      165      108      -31667
3      0      0      0

165      196      105      -31666
3      1      2      0
.
.
.

130      170      150      -31478
2      0      2      1     

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 [60], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31478 was 50 seconds, not including the time for “Creating .EXE file" (59 seconds, total, including the time for “Creating .EXE file"). One can compare the computational results above with those in Arora and Arora  [8, pp. 377-380].


 Acknowledgment

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

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