Tuesday, May 15, 2018

Solving the Pooling Problem in Ryoo and Sahinidis [43]

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Ryoo and Sahinidis [43, p. 564, Example 7]:

Minimize            - 9 * X(5) - 15 * X(9) + 6 * X(1) + 16 * X(2) + 10 * X(6)

subject to

          X(1)+X(2) = X(3) + X(4)

         X(3) + X(7)=X(5)

         X(4) +X(8)= X(9)

          X(7) + X(8)=X(6)

         X(10) * X(3) + 2 * X(7)<=2.5*X(5)

         X(10) * X(4) + 2 * X(8)<=1.5*X(9)

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

    0<= X(1) =  300

    0<= X(2) =  300

    0<= X(3) =  100

    0<= X(4) =  200

    0<= X(5) =  100

    0<= X(6) =  300

    0<= X(7) =  100

    0<= X(8) =  200

    0<= X(9) =  200

    1<= X(10) = 3.

X(11) and X(12) below are slack variables added.


0 DEFDBL A-Z

2 DEFINT K

3 DIM B(99), N(99), A(99), H(99), L(99), U(99), X(1111), D(111), P(111), PS(33)

12 FOR JJJJ = -32000 TO 32000 STEP .01



    14 RANDOMIZE JJJJ

    16 M = -1D+37

    84 A(1) = 0 + RND * 300

    85 A(2) = 0 + RND * 300

    86 A(3) = 0 + RND * 100

    87 A(4) = 0 + RND * 200

    88 A(5) = 0 + RND * 100

    89 A(6) = 0 + RND * 300

    90 A(7) = 0 + RND * 100

    91 A(8) = 0 + RND * 200


    92 A(9) = 0 + RND * 200

    93 A(10) = 1 + RND * 2


    128 FOR I = 1 TO 100000


        129 FOR KKQQ = 1 TO 10

            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        133 FOR IPP = 1 TO (1 + FIX(RND * 3))

            181 j = 1 + FIX(RND * 10)

            189 r = (1 - RND * 2) * A(j)

            190 X(j) = A(j) + (RND ^ (RND * 10)) * r
        222 NEXT IPP


        226 FOR J44 = 1 TO 9
            227 IF X(J44) < 0## THEN 1670

        229 NEXT J44
        230 IF X(10) < 1## THEN 1670

        234 IF X(10) > 3## THEN 1670

        235 IF X(1) > 300## THEN 1670


        236 IF X(2) > 300## THEN 1670


        237 IF X(3) > 100## THEN 1670


        238 IF X(4) > 200## THEN 1670


        239 IF X(5) > 100## THEN 1670
        240 IF X(6) > 3000## THEN 1670


        241 IF X(7) > 100## THEN 1670


        242 IF X(8) > 200## THEN 1670


        243 IF X(9) > 200## THEN 1670


        246 X(6) = X(7) + X(8)

        247 X(3) = X(5) - X(7)

        249 X(4) = X(9) - X(8)
        251 X(1) = X(3) + X(4) - X(2)

        261 IF ((3 * X(1) + X(2)) / ((X(3) + X(4)))) < 1 THEN 1670

        264 IF ((3 * X(1) + X(2)) / ((X(3) + X(4)))) > 3 THEN 1670

        266 X(10) = ((3 * X(1) + X(2)) / ((X(3) + X(4))))



        447 X(11) = 2.5 * X(5) - X(10) * X(3) - 2 * X(7)
        448 X(12) = 1.5 * X(9) - X(10) * X(4) - 2 * X(8)


        450 FOR J47 = 11 TO 12


            451 IF X(J47) < 0 THEN X(J47) = X(J47) ELSE X(J47) = 0


        452 NEXT J47


        453 POBA = 9 * X(5) + 15 * X(9) - 6 * X(1) - 16 * X(2) - 10 * X(6) + 1000000 * (X(11) + X(12))

        466 P = POBA

        1111 IF P <= M THEN 1670


        1450 M = P
        1454 FOR KLX = 1 TO 12

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX
        1557 GOTO 128

    1670 NEXT I
    1889 IF M < 395 THEN 1999


    1907 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8)
    1908 PRINT A(9), A(10), A(11), A(12), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [54]. The complete output through JJJJ =  -31997.93000000033 is shown below:

2.941007286239028D-04      100.0002941070896             6.361036753569613D-09
100.0005882014572             6.361036753569613D-09      99.99941179854277
6.838581110170434D-52      99.99941179854277
200                                    1.000005881979974            0      0
399.999411754016             -31998.41000000026

4.358111960129918D-04      100.0004358212102               1.001421964597826D-08
100.000871622392               1.001421964597826D-08        99.99912837760802
5.282118877464382D-74       99.99912837760802
200                                     1.000005881979947            0      0
399.9991283075091              -31997.93000000033

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 [54], the wall-clock time (not CPU time) for obtaining the output through
JJJJ =   -31997.93000000033
was 77 seconds, not including the time for “Creating .EXE file”  (85 seconds, total, including the time for “Creating .EXE file”).  One can compare the computational results above with those on p. 564 of Ryoo and Sahinidis [43, p. 564, Example 7]. 
       
Acknowledgment

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

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