Saturday, March 24, 2018

Solving a Multiobjective Integer Nonlinear Fractional Programming Problem Involving 10 General Integer Variables and Nonlinear Numerators and Denominators

Jsun Yui Wong

The computer program listed below uses a technique in Murty, [36, pp. 277-278], to solve the immediately following problem:

Maximize 

((X(1) + 2 * X(2) ^ 3 + 3 * X(3) + X(4) + 2 * X(5) + 7 * X(6) + 3 * X(7) + 3 * X(8) + 11 * X(9) + 7 * X(10)) / (X(2) ^ 3 + 5 * X(3) + X(6) + 2 * X(8) + 9 * X(10) + 3)) ^ 3,

((X(1) + 7 * X(3) + 4 * X(4) ^ 3 + 3 * X(5) + 5 * X(7) + 13 * X(9) + 11 * X(10)) / (X(2) + 3 * X(4) ^ 3 + X(6) + 4 * X(8) + 6 * X(10) + 1)) ^ 2,

((X(1) + 3 * X(3) + 5 * X(4) + 6 * X(5) + 7 * X(7) + 4 * X(9) ^ 5 + 7 * X(10) + 2) / (2 * X(1) + X(2) + 8 * X(3) + 6 * X(4) + 2 * X(5) + X(6) + 4 * X(7) + 3 * X(8) + 3 * X(9) ^ 5 + 7 * X(10) + 2)) ^ 3,

which are three objective functions

subject to

         X(1) + X(2) + 7 * X(3) + X(4) + 3 * X(5) + 6 * X(6) + 3 * X(7) + 5 * X(8) + 14 * X(9) + 3 * X(10)<=700
       
         4 * X(1) + X(2) + 2 * X(3) + 9 * X(4) + 7 * X(5) + 9 * X(6) + 4 * X(7) + 7 * X(8) + 7 * X(9) + 11 * X(10)<=900

        .3 * X(1) ^ 3 + X(2) + .1 * X(3) + .1 * X(4) ^ 3 + .1 * X(5) + .1 * X(6) + .5 * X(7) ^ 5 + .1 * X(8) + .2 * X(9) ^ 2 + .1 * X(10)<=5000

         0<=X(i) <= 200, i=1, 2, 3, 4, 5, 6, 7, 8, 9, 10       
       
X(1) through X(10) are general integer variables.

The formulation above is loosely based on the formulation in Sharma [42, p.149].
                                     
The decision makers have determined that the weights are .5, .3, and .2, respectively.

X(11) through X(13) below are slack variables. 

One notes the following line 441, which includes the nonlinear numerators and denominators.   

One also notes the nonlinear constraint .3 * X(1) ^ 3 + X(2) + .1 * X(3) + .1 * X(4) ^ 3 + .1 * X(5) + .1 * X(6) + .5 * X(7) ^ 5 + .1 * X(8) + .2 * X(9) ^ 2 + .1 * X(10)<=5000.
   

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
    51 FOR J40 = 1 TO 10
        55 A(J40) = RND * 200


    56 NEXT J40
    128 FOR I = 1 TO 6000


        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)


            183 r = (1 - RND * 2) * A(j)
            187 X(j) = A(j) + (RND ^ (RND * 10)) * r
        222 NEXT IPP
        246 FOR J47 = 1 TO 10
            247 X(J47) = INT(X(J47))
            248 IF X(J47) < 0 THEN 1670
            249 IF X(J47) > 200 THEN 1670
        251 NEXT J47
        255 REM


        301 X(11) = 700 - X(1) - X(2) - 7 * X(3) - X(4) - 3 * X(5) - 6 * X(6) - 3 * X(7) - 5 * X(8) - 14 * X(9) - 3 * X(10)
        303 X(12) = 900 - 4 * X(1) - X(2) - 2 * X(3) - 9 * X(4) - 7 * X(5) - 9 * X(6) - 4 * X(7) - 7 * X(8) - 7 * X(9) - 11 * X(10)
        305 X(13) = 5000 - .3 * X(1) ^ 3 - X(2) - .1 * X(3) - .1 * X(4) ^ 3 - .1 * X(5) - .1 * X(6) - .5 * X(7) ^ 5 - .1 * X(8) - .2 * X(9) ^ 2 - .1 * X(10)


        401 FOR J47 = 11 TO 13

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


        403 NEXT J47


        441 POBA = .5 * ((X(1) + 2 * X(2) ^ 3 + 3 * X(3) + X(4) + 2 * X(5) + 7 * X(6) + 3 * X(7) + 3 * X(8) + 11 * X(9) + 7 * X(10)) / (X(2) ^ 3 + 5 * X(3) + X(6) + 2 * X(8) + 9 * X(10) + 3)) ^ 3 + .3 * ((X(1) + 7 * X(3) + 4 * X(4) ^ 3 + 3 * X(5) + 5 * X(7) + 13 * X(9) + 11 * X(10)) / (X(2) + 3 * X(4) ^ 3 + X(6) + 4 * X(8) + 6 * X(10) + 1)) ^ 2 + .2 * ((X(1) + 3 * X(3) + 5 * X(4) + 6 * X(5) + 7 * X(7) + 4 * X(9) ^ 5 + 7 * X(10) + 2) / (2 * X(1) + X(2) + 8 * X(3) + 6 * X(4) + 2 * X(5) + X(6) + 4 * X(7) + 3 * X(8) + 3 * X(9) ^ 5 + 7 * X(10) + 2)) ^ 3 + 1000000 * (X(11) + X(12) + X(13))


        466 P = POBA

        1111 IF P <= M THEN 1670


        1450 M = P
        1454 FOR KLX = 1 TO 13

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

    1670 NEXT I
    1889 IF M < 3330000 THEN 1999


    1900 PRINT A(1), A(2), A(3), A(4), A(5)

    1907 PRINT A(6), A(7), A(8), A(9), A(10), A(11), A(12), A(13), M, JJJJ
   
1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [49]. The complete output through JJJJ =   -31893.840000017 is shown below:

21      0      0      0      2 
0      5      0       47       0
0      0     0      3332032.155555523
-31971.12000000462

21      0      0      0      2
0      5      0       47       0
0      0     0      3332032.155555523
-31914.98000001361

21      0      0      0      2
0      5      0       47       0
0      0     0      3332032.155555523
-31893.840000017

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 [49], the wall-clock time for obtaining the output through
JJJJ=  -31893.840000017 was 7 minutes, total, including the time for “Creating .EXE file.”             
                                                
Acknowledgment

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

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