Hillier and Lieberman's Streamlined Procedure for Solving Multi-Objective Mixed-Integer Nonlinear/Linear Preemptive Goal Programming Problems

Jsun Yui Wong

1.  A Problem from Markland [62]   

The first computer program listed below seeks to solve the immediately following problem from Markland [62, p. 283, Problem 4] with the streamlined procedure of Hillier and Lieberman [42, pp. 289-291]:
   
Minimize     

 P1* X(10) +P2 * X(6) +P3*X(7) +P4*X(9) +P5* X(8)

where P1>>P2>>P3>>P4>>P5

subject to

         2.00 * X(1) + .1 * X(2) + 1.1 * X(3) + .08 * X(4) + .75 * X(5) <= 25

         225 * X(1) + 200 * X(2) + 175 * X(3) + 150 * X(4) + 400 * X(5) <= 10000

        .15 * X(1) + .25 * X(2) + .15 * X(3) + .05 * X(4) + .08 * X(5) >= 9.50

   X(1) +    X(6) = 100
   X(2)  +      X(7) = 100

   X(3)     +    X(8) = 100

   X(4)    +  X(9) = 100

    X(5)   +  X(10) = 100

All variables are nonnegative.

The formulation above is from Markland [62, pp. 809-810].

0 DEFDBL A-Z
1 REM        DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

    85 RANDOMIZE JJJJ
    87 M = -4E+250
    120 FOR J44 = 1 TO 10

        121 A(J44) = FIX(RND * 40)

    123 NEXT J44

    128 FOR I = 0 TO FIX(RND * 90000)


        129 FOR KKQQ = 1 TO 10

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

        135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

            143 j = 1 + FIX(RND * 10)

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

            145 REM GOTO 162
            154 REM     IF j > 3.5 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 * 5.3) ELSE X(j) = A(j) + FIX(1 + RND * 5.3)

        169 NEXT IPP
       

        221 FOR J44 = 1 TO 12

            231 REM     X(J44) = INT(X(J44))


            232 IF X(J44) < 0 THEN 1670

        234 NEXT J44


        250 X(6) = 100 - X(1)
        252 X(7) = 100 - X(2)

        253 X(8) = 100 - X(3)

        254 X(9) = 100 - X(4)

        255 X(10) = 100 - X(5)

        257 FOR J44 = 1 TO 10
            258 REM      X(J44) = INT(X(J44))

            259 IF X(J44) < 0 THEN 1670
        260 NEXT J44


        263 IF 2.00 * X(1) + .1 * X(2) + 1.1 * X(3) + .08 * X(4) + .75 * X(5) > 25 THEN 1670

        265 IF 225 * X(1) + 200 * X(2) + 175 * X(3) + 150 * X(4) + 400 * X(5) > 10000 THEN 1670
        266 IF .15 * X(1) + .25 * X(2) + .15 * X(3) + .05 * X(4) + .08 * X(5) < 9.50 THEN 1670

        1415 P = -10 ^ 80 * (X(10)) - 10 ^ 60 * X(6) - 10 ^ 40 * (X(7)) - 10 ^ 20 * X(9) - X(8)
     

        1417 IF P <= M THEN 1670
        1420 M = P


        1442 FOR KLX = 1 TO 10


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

        1557 GOTO 128
    1670 NEXT I
    1672 IF M < -4D+200 THEN 1999

    1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [102]. The complete output of a single run through JJJJ=-28987 is shown below:

.
.
.
.0059093962162189         35.71146723582667       5.46674869668203D-03
0                                    7.138293643361265       99.99409060378378
64.28853276417333         99.99453325130332       100
92.86170635663873        -9.286170635663874D+81      -28987                               
.
.
.


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 [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -28987 was 5 minutes, not including the time for “Creating .EXE file.”   One can compare the computational results above with those in Markland [62, pp. 809-810].         

Remark 3 and remark 4 on page 198 of Winston and Venkataramanan are noteworthy [103].


2. An Integer Version of the Same Problem
 
One notes the following line 144, which is 144 REM IF RND < .5 THEN GOTO 156 ELSE GOTO 162.

0 DEFDBL A-Z
1 REM        DEFINT X, A
2 DIM N(99), A(45222), X(45012), D(111), P(111), PS(33), J44(45202), J(45211), AA(99), STIO(45303)
81 FOR JJJJ = -32000 TO 32000

    85 RANDOMIZE JJJJ
    87 M = -4E+250
    120 FOR J44 = 1 TO 10

        121 A(J44) = FIX(RND * 40)


    123 NEXT J44

    128 FOR I = 0 TO FIX(RND * 90000)


        129 FOR KKQQ = 1 TO 10

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

        135 FOR IPP = 1 TO FIX(1 + RND * 3.3)

            143 j = 1 + FIX(RND * 10)

            144 REM IF RND < .5 THEN GOTO 156 ELSE GOTO 162


            145 GOTO 162
            154 REM     IF j > 3.5 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 * 5.3) ELSE X(j) = A(j) + FIX(1 + RND * 5.3)


        169 NEXT IPP
        170 REM   GOTO 221

        171 REM X(1) = 100
        173 REM
        177 REM
        178 REM  X(2) = 140

        179 REM  IF X(3) > 20 THEN 1670

        221 FOR J44 = 1 TO 12

            231 REM     X(J44) = INT(X(J44))


            232 IF X(J44) < 0 THEN 1670

        234 NEXT J44


        250 X(6) = 100 - X(1)
        252 X(7) = 100 - X(2)

        253 X(8) = 100 - X(3)

        254 X(9) = 100 - X(4)

        255 X(10) = 100 - X(5)

        257 FOR J44 = 1 TO 10
            258 REM      X(J44) = INT(X(J44))

            259 IF X(J44) < 0 THEN 1670
        260 NEXT J44


        263 IF 2.00 * X(1) + .1 * X(2) + 1.1 * X(3) + .08 * X(4) + .75 * X(5) > 25 THEN 1670

        265 IF 225 * X(1) + 200 * X(2) + 175 * X(3) + 150 * X(4) + 400 * X(5) > 10000 THEN 1670
        266 IF .15 * X(1) + .25 * X(2) + .15 * X(3) + .05 * X(4) + .08 * X(5) < 9.50 THEN 1670

        272 REM  IF 100 * X(1) + 60 * X(2) > 600 THEN 1670

        1411 REM   P = -10 ^ 30 * X(3) - 10 ^ 30 * X(6) - 10 ^ 15 * X(7) - 10 ^ 15 * X(9) - X(11)
        1413 REM     P = -10 ^ 30 * X(4) - 10 ^ 20 * X(8) - 10 ^ 10 * X(5) - X(9)

        1415 P = -10 ^ 80 * (X(10)) - 10 ^ 60 * X(6) - 10 ^ 40 * (X(7)) - 10 ^ 20 * X(9) - X(8)
        1416 REM P = -10 ^ 30 * X(4) - 10 ^ 20 * (X(6)) - 10 ^ 10 * (X(9)) - X(11)


        1417 IF P <= M THEN 1670
        1420 M = P


        1442 FOR KLX = 1 TO 10


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

        1557 GOTO 128
    1670 NEXT I
    1672 IF M < -4D+200 THEN 1999

    1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [102]. The complete output of a single run through JJJJ=-29408 is shown below:

.
.
.
0  27  16  1  4
100  73  84  99  96
-9.6D+81    -29556

0  36  0  0  7
100  64  100  100  93
-9.3D+81    -29408
.
.
.


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 [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -29408 was 5 minutes, not including the time for “Creating .EXE file.”   One can compare the computational results above with those in Markland [62, pp. 809-810].         
     
Remark 3 and remark 4 on page 198 of Winston and Venkataramanan are noteworthy [103].

Acknowledgement

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

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