Solving a Multiobjective Mixed-Integer Nonlinear Programming Formulation of a Multivariate Stratified Sample Survey under 2-Stage Randomized Response Model

 

Jsun Yui Wong

The computer program listed below seeks to solve the following multi-objective (mixed) integer nonlinear programming problem [30, p. 343] from Ghufran, Khowaja, and Ahsan [30, p. 355]:   

Minimize

X(1) + X(2) + X(3) + X(4)

subject to

         -(-.001791658449 / X(5) - .038022161 / X(6) - .06002072 / X(7) - .00262104527 / X(8)) - X(1) <= .001807508
         -(-.001918966929 / X(5) - .037963199 / X(6) - .055784166 / X(7) - .00202612295 / X(8)) - X(2) <= .001711371
         -(-.002042358225 / X(5) - .03541605 / X(6) - .066199888 / X(7) - .003824110406 / X(8)) - X(3) <= .001940426
         -(-.002013632209 / X(5) - .028824123 / X(6) - .045554438 / X(7) - .00426626255 / X(8)) - X(4) <= .001522954
        15 * X(5) + 20 * X(6) + 30 * X(7) + 18 * X(8) + 10 * X(5) ^ .5 + 13 * X(6) ^ .5 + 10 * X(7) ^ .5 + 12 * X(8) ^ .5 <= 4000
 2<=X(5) <= 81
2<= X(6) <= 343
 2<=X(7) <= 455
 2<=X(8) <= 121
X(5) through X(8) are integer variables
X(1) >= 0
X(2) >= 0
 X(3) >= 0
 X(4) >= 0.


0 DEFDBL A-Z
1 REM   DEFINT K
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
    121 FOR J44 = 5 TO 8
        123 A(J44) = RND * 50
    124 NEXT J44
    125 FOR J44 = 1 TO 4
        126 A(J44) = RND
    127 NEXT J44
    128 FOR I = 0 TO FIX(RND * 10000)
        129 FOR KKQQ = 1 TO 8
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ

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

            143 j = 1 + FIX(RND * 8)

            145 REM GOTO 162

            154 IF j < 4.5 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 * 3.3) ELSE X(j) = A(j) + FIX(1 + RND * 3.3)
            163 REM   IF RND < .33 THEN X(j) = -1## ELSE IF RND < .5 THEN X(j) = 0## ELSE X(j) = 1##

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

        171 FOR J44 = 5 TO 8

            173 IF X(J44) < 2 THEN 1670
            174 X(J44) = INT(X(J44))

        175 NEXT J44

        179 IF X(5) > 81 THEN 1670

        180 IF X(6) > 343 THEN 1670

        182 IF X(7) > 455 THEN 1670
        183 IF X(8) > 121 THEN 1670
        288 IF X(1) < 0 THEN 1670
        289 IF X(2) < 0 THEN 1670
        290 IF X(3) < 0 THEN 1670
        291 IF X(4) < 0 THEN 1670
        292 IF -(-.001791658449 / X(5) - .038022161 / X(6) - .06002072 / X(7) - .00262104527 / X(8)) - X(1) > .001807508 THEN 1670
        294 REM                         IF -(-035334545 / X(3)xxxxxxxxxxxxxxxx - .109989669 / X(4))-x(2) >.001300567 THEN 1670
        297 IF -(-.001918966929 / X(5) - .037963199 / X(6) - .055784166 / X(7) - .00202612295 / X(8)) - X(2) > .001711371 THEN 1670

        298 IF -(-.002042358225 / X(5) - .03541605 / X(6) - .066199888 / X(7) - .003824110406 / X(8)) - X(3) > .001940426 THEN 1670

        299 IF -(-.002013632209 / X(5) - .028824123 / X(6) - .045554438 / X(7) - .00426626255 / X(8)) - X(4) > .001522954 THEN 1670
        300 IF 15 * X(5) + 20 * X(6) + 30 * X(7) + 18 * X(8) + 10 * X(5) ^ .5 + 13 * X(6) ^ .5 + 10 * X(7) ^ .5 + 12 * X(8) ^ .5 > 4000 THEN 1670
   
        1011 P = -X(1) - X(2) - X(3) - X(4)

        1111 IF P <= M THEN 1670

        1420 M = P

        1444 FOR KLX = 1 TO 8
            1455 A(KLX) = X(KLX)
        1456 NEXT KLX
        1557 GOTO 128
    1670 NEXT I
    1889 IF M < -.00003 THEN 1999
    1931 PRINT A(1), A(2), A(3), A(4)
    1937 PRINT A(5), A(6), A(7), A(8), M, JJJJ
1999 NEXT


This BASIC computer program was run with QB64v1000-win [94]. The complete output of a single run through JJJJ= -31562 is shown below:


4.126025594744503D-06       1.184133101237785D-05         1.9009600339450951D-06
8.915967328510262D-06
15      61     64      20           -2.678428427508356D-05
-31562

.
.
.


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 [94], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31562 was 20 seconds, not including the time for “Creating .EXE file” (40 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Ghufran, Khowaja, and Ahsan [30, p. 355].       

Acknowledgment

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

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