Generating Feasible Solutions of a Multi-Objective Integer Linear Program

         
Jsun Yui Wong

The computer program listed below seeks to generate all or some of the feasible solutions of the following problem with three objectives in Burachik, Kaya, and Rizvi [11, p. 10, Test problem 3]:   

Minimize                -X(1)

Minimize                -X(2)

Minimize                -X(3)

subject to

         (3 * X(1) + 2 * X(2) + 3 * X(3)) <= 18
         (X(1) + 2 * X(2) + X(3)) <= 10
         (9 * X(1) + 20 * X(2) + 7 * X(3)) <= 96
         (7 * X(1) + 20 * X(2) + 9 * X(3)) <= 96

  X(1) through X(3)>=0 and are integer variables.


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 = -4E+250

    98 W1 = FIX(RND * 11) * .1##
    99 W2 = FIX(RND * 11) * .1##

    101 IF W1 + W2 > 1## THEN 1670


    103 W3 = 1 - W1 - W2

    111 FOR J44 = 1 TO 3


        115 A(J44) = FIX(RND * 9)
    121 NEXT J44

    128 FOR I = 0 TO FIX(RND * 11)


        129 FOR KKQQ = 1 TO 3

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

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

            143 j = 1 + FIX(RND * 3)


            154 IF RND < .5 THEN GOTO 156 ELSE GOTO 162


            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)
            164 REM   IF A(j) = 0 THEN X(j) = 1 ELSE X(j) = 0
        169 NEXT IPP

        173 FOR J44 = 1 TO 3

            174 X(J44) = INT(X(J44))


        177 NEXT J44


        293 FOR J44 = 1 TO 3


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

            296 REM   

        297 NEXT J44


        333 REM       


        334 IF (3 * X(1) + 2 * X(2) + 3 * X(3)) > 18 THEN 1670
        335 IF (X(1) + 2 * X(2) + X(3)) > 10 THEN 1670


        336 IF (9 * X(1) + 20 * X(2) + 7 * X(3)) > 96 THEN 1670
        339 IF (7 * X(1) + 20 * X(2) + 9 * X(3)) > 96 THEN 1670


        535 REM 
        537 REM   


        539 P = W1 * (X(1)) + W2 * (X(2)) + W3 * (X(3))


        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 3


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


        1457 OBJ1 = -X(1)

        1458 OBJ2 = -X(2)
        1459 OBJ3 = -X(3)

        1557 GOTO 128
    1670 NEXT I
    1890 IF M <= -10 ^ 40 THEN GOTO 1999

    1933 PRINT A(1), A(2), A(3)

    1934 PRINT OBJ1, OBJ2, OBJ3, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [84].  The feasible solutions of a single run through JJJJ= 32000--the end of the computer program above--are summarized below:

0  1  0
0  -1  0    -31999

1  4   0
-1   -4   0    -31986

2  2  2
-2    -2     -2    -31972

3  2  1
-3   -2    -1     -31956

4  2  0
-4    -2   0    -31952

1   4   1
-1    -4     -1     -31951

0   0    6
0  0   -6    -31931

.
.
.


1  3  3
-1    -3     -3      31927

0  3  1
0    -3   -1      31968

2  3   2
-2    -3    -2      31970

5  1  0
-5   -1     0    31984

3  2   1
-3    -2    -1     31992

2  1  2
-2    -1      -2      31998

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 [84], the wall-clock time (not CPU time) for obtaining the feasible solutions through JJJJ = 32000 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 Burachik, Kaya, and Rizvi [11, pp. 11-12, Figure 3 (a), (b), (c), and (d) and Figure 4 (a), (b), (c), and (d)]. 

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

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