Thursday, January 2, 2020

Multivariate Stratified Sampling Design in the Presence of Nonresponse by Example



Jsun Yui Wong

1.  First Stage

The computer program listed below seeks to solve the following nonlinear integer programming problem:

Minimize

443.09704013 / X(1)+ 270.715638 / X(2) + 204.9419482 / X(3) + 240.6056486 / X(4) + 10.7780449 / X(5) + 2.849638294 / X(6) + 3.415699137 / X(7) + 5.070828722 / X(8) 

subject to 
 
 2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) <= 5000

2<=X(1)<=1214

2<=X(2)<=822

2<=X(3)<=1028

2<=X(4)<=786

2<=X(5)<=40

2<=X(6)<=40

2<=X(7)<=40

2<=X(8)<=40

where X(1) through X(8) are integer variables.

The problem above is based on the formulation on p. 2459 in Varshney and Mradula [90, p. 2459, Example 1].

0 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
    86 M = -3E+50
    118 FOR J44 = 1 TO 8
        119 A(J44) = 2 + (RND * 8)
    120 NEXT J44
    128 FOR I = 1 TO 30000
        129 FOR KKQQ = 1 TO 8
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        151 FOR IPP = 1 TO FIX(1 + RND * 5)

            153 J = 1 + FIX(RND * 8)

            155 GOTO 162

            156 r = (1 - RND * 2) * A(J)

            158 X(J) = A(J) + (RND ^ (RND * 15)) * r

            161 GOTO 169

            162 REM  IF RND < .5 THEN X(J) = A(J) - INT(RND * 15) ELSE X(J) = A(J) + INT(RND * 15)
            163 IF RND < .5 THEN X(J) = A(J) - INT(RND * 5) ELSE X(J) = A(J) + INT(RND * 5)

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

        169 NEXT IPP

        215 FOR J44 = 1 TO 8
            216 X(J44) = INT(X(J44))
            217 REM     X(4) = INT(X(4))
            218 IF X(J44) < 2 THEN 1670
        219 NEXT J44

        225 IF X(1) > 1214 THEN 1670
        226 IF X(2) > 822 THEN 1670
        227 IF X(3) > 1028 THEN 1670
        228 IF X(4) > 786 THEN 1670
        229 IF X(5) > 40 THEN 1670

        230 IF X(6) > 40 THEN 1670
        233 IF X(7) > 40 THEN 1670

        234 IF X(8) > 40 THEN 1670
        248 IF 2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) > 5000 THEN 1670

        326 FOR J44 = 1 TO 8
            328 IF X(J44) < 2 THEN 1670

        330 NEXT J44

        478 P = -443.09704013 / X(1) - 270.715638 / X(2) - 204.9419482 / X(3) - 240.6056486 / X(4) - 10.7780449 / X(5) - 2.849638294 / X(6) - 3.415699137 / X(7) - 5.070828722 / X(8)

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 8

            1455 A(KLX) = X(KLX)
        1456 NEXT KLX
        1557 GOTO 128
    1670 NEXT I
    1889 IF M < -999999999 THEN 1999
    1933 PRINT A(1), A(2), A(3), A(4), A(5)
    1936 PRINT A(6), A(7), A(8), M, JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [95].  The complete output of a single run through JJJJ= -31995 is shown below:

50  38  33  34   7
4  4  5  -33.3932561317882
-32000
50  37  32  35  8
4  4  5  -33.38521836622201  
-31999
49  39  32  34   8
4  4  5  -33.39305167216097
-31998
50  37  32  35  8
4  4  5  -33.38521836622201
-31997
50  37  32  35  8
4  4  5  -33.38521836622201
-31996
50  37  32  35  8
4  4  5  -33.38521836622201
-31995
.
.
.


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 [95], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31995 was 4 seconds, not including the time for “Creating .EXE file.”  One can compare the computational results above with those in Varshney and Mradula [90, p. 2459], where one can see the following numbers:  50, 37, 32, 35, 8, 4, 4, 5, 33.3852.


2.  Second Stage

The computer program listed below seeks to solve the following nonlinear mixed-integer programming problem:

Minimize

 746.921876 / X(1) + 329.7093584 / X(2) + 97.34568757 / X(3) + 270.4706345 / X(4) + 18.16836996 / X(5) + 3.470624826 / X(6) + 1.622428126 / X(7) + 5.700241329 / X(8)+ X(9)

 subject to
  
  443.09704013 / X(1) + 270.715638 / X(2) + 204.9419482 / X(3) + 240.6056486 / X(4) + 10.7780449 / X(5) + 2.849638294 / X(6) + 3.415699137 / X(7) + 5.070828722 / X(8)     -X(9)    <= 33.38521836622201, which comes from stage 1

  2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) <= 5000

2<=X(1)<=1214

2<=X(2)<=822

2<=X(3)<=1028

2<=X(4)<=786

2<=X(5)<=40

2<=X(6)<=40

2<=X(7)<=40

2<=X(8)<=40

X(9)>=0

where X(1) through X(8) are integer variables, and X(9) is a continuous variable.

The problem above is based on the formulation on p. 2460 in Varshney and Mradula [90, p. 2460, Example 1].

One notes line 238, which is  238 X(9) = -(-443.09704013 / X(1) - 270.715638 / X(2) - 204.9419482 / X(3) - 240.6056486 / X(4) - 10.7780449 / X(5) - 2.849638294 / X(6) - 3.415699137 / X(7) - 5.070828722 / X(8)) - 33.38521836622201  from the first long inequality constraint above.  That is a search for an active constraint for the goal of optimization; see Bernau [11].


0 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
    86 M = -3E+50
    118 FOR J44 = 1 TO 8
        119 A(J44) = 2 + (RND * 8)

    120 NEXT J44
    123 A(9) = RND

    128 FOR I = 1 TO 30000

        129 FOR KKQQ = 1 TO 9

            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        151 FOR IPP = 1 TO FIX(1 + RND * 5)

            153 J = 1 + FIX(RND * 9)
            154 IF J > 8.5 THEN GOTO 156 ELSE GOTO 163

            155 REM GOTO 162

            156 r = (1 - RND * 2) * A(J)

            158 X(J) = A(J) + (RND ^ (RND * 15)) * r

            161 GOTO 169

            162 REM  IF RND < .5 THEN X(J) = A(J) - INT(RND * 15) ELSE X(J) = A(J) + INT(RND * 15)
            163 IF RND < .5 THEN X(J) = A(J) - INT(RND * 5) ELSE X(J) = A(J) + INT(RND * 5)

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

        169 NEXT IPP

        215 FOR J44 = 1 TO 8
            216 X(J44) = INT(X(J44))
            217 REM     X(4) = INT(X(4))
            218 IF X(J44) < 2 THEN 1670
        219 NEXT J44
        222 IF X(9) < 0## THEN 1670
        225 IF X(1) > 1214 THEN 1670
        226 IF X(2) > 822 THEN 1670
        227 IF X(3) > 1028 THEN 1670
        228 IF X(4) > 786 THEN 1670
        229 IF X(5) > 40 THEN 1670

        230 IF X(6) > 40 THEN 1670
        233 IF X(7) > 40 THEN 1670

        234 IF X(8) > 40 THEN 1670
        238 X(9) = -(-443.09704013 / X(1) - 270.715638 / X(2) - 204.9419482 / X(3) - 240.6056486 / X(4) - 10.7780449 / X(5) - 2.849638294 / X(6) - 3.415699137 / X(7) - 5.070828722 / X(8)) - 33.38521836622201

        248 IF 2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4) > 5000 THEN 1670

        326 FOR J44 = 1 TO 8
            328 IF X(J44) < 2 THEN 1670

        330 NEXT J44

        478 P = -746.921876 / X(1) - 329.7093584 / X(2) - 97.34568757 / X(3) - 270.4706345 / X(4) - 18.16836996 / X(5) - 3.470624826 / X(6) - 1.622428126 / X(7) - 5.700241329 / X(8) - X(9)

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 9

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

        1533 dol = 2.4 * X(1) + 3.4 * X(2) + 4 * X(3) + 4.6 * X(4) + 3 * X(5) + 4 * X(6) + 5 * X(7) + 6 * X(8) + 100 * (X(1) / 4 + X(2) / 4 + X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4 + X(8) / 4)

        1557 GOTO 128
    1670 NEXT I
    1889 IF M < -999999999 THEN 1999
    1933 PRINT A(1), A(2), A(3), A(4)
    1936 PRINT A(5), A(6), A(7), A(8), A(9), M, JJJJ, dol
1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [95].  The complete output of a single run through JJJJ= -31995 is shown below:

54  38  27  35
8  4  4   5  .3370234572013086
 -38.86295811196344      -32000      4992.8 
54  38  27  35
8  4  4   5  .3370234572013086
-38.86295811196344      -31999      4992.8
54  38  27  35
8  4  4   5  .3370234572013086
-38.86295811196344      -31998      4992.8
54  38  27  35
8  4  4   5  .3370234572013086
-38.86295811196344      -31997      4992.8
54  38  27  35
8  4  4   5  .3370234572013086
-38.86295811196344      -31996      4992.8
54  38  27  35
8  4  4   5  .3370234572013086
-38.86295811196344      -31995      4992.8
.
.
.


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 [95], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31995 was 5 seconds, not including the time for “Creating .EXE file.”  One can compare the computational results above with those in Varshney and Mradula [90, p. 2460; Table 4, p. 2465], where one can see the following numbers:  54, 38, 27, 35, 8, 4, 4, 5, 0.3516, 38.8776, 4992.8.


Acknowledgment

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

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