Working on an Integer Nonlinear Programming Formulation of a Multivariate Stratified Sampling Design

Jsun Yui Wong

The computer program listed below seeks to solve the following integer nonlinear programming problem based on the formulation in Varshney and Mradulah [88, p. 2462]:

Minimize

-( -.4388203 / X(3) – 2.663113 / X(4) – 49.60277 / X(5) – 2.66616 / X(6) – 9.938173 / X(7) – .002437891 / X(8) – .08937845 / X(9) – 1.206554 / X(10) – .044436 / X(11) – .2094497 / X(12) )

subject to

4 * X(3) + 4.9 * X(4) + 5.9 * X(5) + 7.75 * X(6) + 8.92 * X(7) + 6 * X(8) + 7 * X(9) + 9 * X(10) + 11 * X(11) + 12 * X(12) + 100 * (X(3) / 4 + X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4) + 100 * (X(8) / 4 + X(9) / 4 + X(10) / 4 + X(11) / 4 + X(12) / 4) <= 10000

2<=X(1) <= 395
2<= X(2) <= 382
2<=X(3) <= 439
2<=X(4) <= 368
2<=X(4) <= 416
2<=X(8) <= 20
2<= X(9) <= 20
2<=X(10) <= 20
2<=X(11) <= 20
2<=X(12) <= 20
X(3) through X(12) are integer variables.
One notes that the following computer program tries to take advantage of searching through a possibly active constraint (see Bernau [11]), line 297, which is
X(3) = (-(4.9 * X(4) + 5.9 * X(5) + 7.75 * X(6) + 8.92 * X(7) + 6 * X(8) + 7 * X(9) + 9 * X(10) + 11 * X(11) + 12 * X(12) + 100 * (X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4) + 100 * (X(8) / 4 + X(9) / 4 + X(10) / 4 + X(11) / 4 + X(12) / 4)) + 10000) / 29.
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 = 3 TO 12
123 A(J44) = FIX(RND * 80)
124 NEXT J44
126 A(1) = RND ^ (RND * 8)

127 A(2) = RND ^ (RND * 8)

128 FOR I = 0 TO FIX(RND * 80000)
129 FOR KKQQ = 1 TO 12

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

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

143 j = 1 + FIX(RND * 12)

145 REM GOTO 162

154 IF j < 2.5 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 = 3 TO 12

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

175 NEXT J44

179 IF X(3) > 395 THEN 1670

180 IF X(4) > 382 THEN 1670
181 IF X(5) > 439 THEN 1670

182 IF X(6) > 368 THEN 1670
183 IF X(7) > 416 THEN 1670

279 IF X(8) > 20 THEN 1670

280 IF X(9) > 20 THEN 1670
281 IF X(10) > 20 THEN 1670

282 IF X(11) > 20 THEN 1670
283 IF X(12) > 20 THEN 1670
297 X(3) = (-(4.9 * X(4) + 5.9 * X(5) + 7.75 * X(6) + 8.92 * X(7) + 6 * X(8) + 7 * X(9) + 9 * X(10) + 11 * X(11) + 12 * X(12) + 100 * (X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4) + 100 * (X(8) / 4 + X(9) / 4 + X(10) / 4 + X(11) / 4 + X(12) / 4)) + 10000) / 29

298 IF X(3) < 2 THEN 1670

1103 P = -.4388203 / X(3) – 2.663113 / X(4) – 49.60277 / X(5) – 2.66616 / X(6) – 9.938173 / X(7) – .002437891 / X(8) – .08937845 / X(9) – 1.206554 / X(10) – .044436 / X(11) – .2094497 / X(12)

1111 IF P <= M THEN 1670

1420 M = P

1444 FOR KLX = 1 TO 12
1455 A(KLX) = X(KLX)
1456 NEXT KLX

1557 GOTO 128
1670 NEXT I

1677 IF M < -9999999.844 THEN 1999

1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6)
1933 PRINT A(7), A(8), A(9), A(10), A(11), A(12), M, JJJJ
1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [93]. The complete output of a single run through JJJJ= -31549 is shown below:
6.767673214250133D-02 1.556824283641913D-05 13.39551724137935
32 138 31
59 2 6 20 4
8 -.8436034238646266 -31674

6.767257399927002D-03 5182917296402567D-02 13.39551724137935
32 138 31
59 2 6 20 4
8 -.8436034238646266 -31549

.
.
.

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 [93], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31549 was 40 seconds, not including the time for “Creating .EXE file” (60 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Varshney and Mradulah [88, p. 2462], where one can see ( 13 32 138 31 59 2 6 20 4 8 ).

Now 299 X(3) = INT(X(3)) is added to the computer program above as follows:
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 = 3 TO 12
123 A(J44) = FIX(RND * 80)

124 NEXT J44
126 A(1) = RND ^ (RND * 8)

127 A(2) = RND ^ (RND * 8)
128 FOR I = 0 TO FIX(RND * 80000)
129 FOR KKQQ = 1 TO 12

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

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

143 j = 1 + FIX(RND * 12)

145 REM GOTO 162

154 IF j < 2.5 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 = 3 TO 12

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

175 NEXT J44

179 IF X(3) > 395 THEN 1670

180 IF X(4) > 382 THEN 1670
181 IF X(5) > 439 THEN 1670

182 IF X(6) > 368 THEN 1670
183 IF X(7) > 416 THEN 1670
279 IF X(8) > 20 THEN 1670

280 IF X(9) > 20 THEN 1670
281 IF X(10) > 20 THEN 1670

282 IF X(11) > 20 THEN 1670
283 IF X(12) > 20 THEN 1670
297 X(3) = (-(4.9 * X(4) + 5.9 * X(5) + 7.75 * X(6) + 8.92 * X(7) + 6 * X(8) + 7 * X(9) + 9 * X(10) + 11 * X(11) + 12 * X(12) + 100 * (X(4) / 4 + X(5) / 4 + X(6) / 4 + X(7) / 4) + 100 * (X(8) / 4 + X(9) / 4 + X(10) / 4 + X(11) / 4 + X(12) / 4)) + 10000) / 29

298 IF X(3) < 2 THEN 1670

299 X(3) = INT(X(3))
1103 P = -.4388203 / X(3) – 2.663113 / X(4) – 49.60277 / X(5) – 2.66616 / X(6) – 9.938173 / X(7) – .002437891 / X(8) – .08937845 / X(9) – 1.206554 / X(10) – .044436 / X(11) – .2094497 / X(12)
1111 IF P <= M THEN 1670

1420 M = P

1444 FOR KLX = 1 TO 12
1455 A(KLX) = X(KLX)
1456 NEXT KLX

1557 GOTO 128
1670 NEXT I

1677 IF M < -9999999.844 THEN 1999

1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6)
1933 PRINT A(7), A(8), A(9), A(10), A(11), A(12), M, JJJJ
1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [93]. The complete output of a single run through JJJJ= -31355 is shown below:

.0563297356712307 1.747293307767763D-05 13
32 136 31
60 2 6 20 4
9 -.8441695596215193 -31674

.0012965640604338 .4224148405903696 13
32 136 31
60 2 6 20 4
9 -.8441695596215193 -31355

.
.
.
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 [93], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31355 was 55 seconds, not including the time for “Creating .EXE file” (70 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Varshney and Mradulah [88, p. 2462], where one can see ( 13 32 138 31 59 2 6 20 4 8 ).

Acknowledgment

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

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[85] Mohamed Tawhid, Vimal Savsani (2018). Epsilon-constraint heat transfer search (epsilon-HTS) algorithm for solvinfg multi-objective engineering design problems. journal of computational design and engineering 5 (2018) 104-119.

[86] Frank A. Tillman, Ching-Lai Hwang, Way Kuo (1977). Determining Component Reliability and Redundancy for Optimun System Reliability. IEEE Transactions on Reliability, Vol. R-26, No. 3, Augusr 1977, pp. 162-165.

[87] Rahul Varshney, Srikant Gupta, Irfan Ali (2017). An optimum multivariate-multiobjective stratified samplinr design: fuzzy programming approach. Pakistan Journal of Statistics and Operations Research, pp. 829-855, December 2017

[88 ] Rahul Varshney, Mradula (2019 May 25). Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function, Journal of Statistical Computation and Simulation (2019) 89:13, pp. 2454-2467.

[89] Rahul Varshney, Najmussehar, M. J. Ahsan (2012). An optimum multivariate stratified double sampling design in presence of non-response, Optimization Letters (2012) 6: pp. 993-1008.

[90] Rahul Varshney, M. G. M. Khan, Ummatul Fatima, M. J. Ahsan (2015). Integer compromise allocation in multivariate stratified surveys, Annals of Operations Research (2015) 226:659-668.

[91] V. Verma, H. C. Bakhshi, M. C. Puri (1990) Ranking in integer linear fractional programming problems, ZOR – Methods and Models of Operations Research (1990)
34:325-334.

[92] Tawan Wasanapradit, Nalinee Mukdasanit, Nachol Chaiyaratana, Thongchai Srinophakun (2011). Solving mixed-integer nonlinear programming problems using improved genetic algorithms. Korean Journal of Chemical Engineering 28 (1):32-40 January 2011.

[93] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64.

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Solving a Multiobjective Mixed-Integer Nonlinear Programming Formulation of a Multivariate Stratified Survey with Stochastic Quadratic Cost Function

 

Jsun Yui Wong

The computer program listed below seeks to solve the following multi-objective mixed integer nonlinear programming problem from Ali, Raghav, and Bari [1, p. 974]:

Minimize

X(5) + X(6)

subject to

552.640 / X(1) + 136.277 / X(2) + 274.114 / X(3) + 2588.343 / X(4) – X(5) <= 21.29
14926.197 / X(1) + 165.39749 / X(2) + 130.202 / X(3) + 3084.324 / X(4) -X(6) <= 74.04
(X(1) + X(2) + 1.5 * X(3) + 2 * X(4) + .5 * X(1) ^ .5 + .5 * X(2) ^ .5 + X(3) ^ .5 + 1.5 * X(4) ^ .5) + 2.33 * (.25 * X(1) ^ 2 + .25 * X(2) ^ 2 + .35 * X(3) ^ 2 + .45 * X(4) ^ 2 + .125 * X(1) + .125 * X(2) + .175 * X(3) + .225 * X(4)) ^ .5 <= 1200

2<=X(1) <= 1419
2<= X(2) <= 619
2<=X(3) <= 1253
2<=X(4) <= 899
X(1) through X(4) are integer variables
X(5) >= 0
X(6) >= 0.
One notes that the following computer program takes advantage of searching through an active constraint, line 407, which is
407 X(6) = 14926.197 / X(1) + 165.39749 / X(2) + 130.202 / X(3) + 3084.324 / X(4) – 74.04; see Bernau [11].

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 = 1 TO 4
123 A(J44) = RND * 200
124 NEXT J44
125 FOR J44 = 5 TO 6
126 A(J44) = RND * 10
127 NEXT J44
128 FOR I = 0 TO FIX(RND * 50000)
129 FOR KKQQ = 1 TO 6
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
135 FOR IPP = 1 TO FIX(1 + RND * 3.3)
143 j = 1 + FIX(RND * 6)

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 = 1 TO 4

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

175 NEXT J44

179 IF X(1) > 1419 THEN 1670

180 IF X(2) > 619 THEN 1670

182 IF X(3) > 1253 THEN 1670
183 IF X(4) > 899 THEN 1670
290 IF X(5) < 0 THEN 1670
291 IF X(6) < 0 THEN 1670

407 X(6) = 14926.197 / X(1) + 165.39749 / X(2) + 130.202 / X(3) + 3084.324 / X(4) – 74.04

409 IF X(6) < 0 THEN 1670

411 IF 552.640 / X(1) + 136.277 / X(2) + 274.114 / X(3) + 2588.343 / X(4) – X(5) > 21.29 THEN 1670

511 IF (X(1) + X(2) + 1.5 * X(3) + 2 * X(4) + .5 * X(1) ^ .5 + .5 * X(2) ^ .5 + X(3) ^ .5 + 1.5 * X(4) ^ .5) + 2.33 * (.25 * X(1) ^ 2 + .25 * X(2) ^ 2 + .35 * X(3) ^ 2 + .45 * X(4) ^ 2 + .125 * X(1) + .125 * X(2) + .175 * X(3) + .225 * X(4)) ^ .5 > 1200 THEN 1670

1011 P = -X(5) – X(6)

1111 IF P <= M THEN 1670

1420 M = P

1444 FOR KLX = 1 TO 6
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1557 GOTO 128
1670 NEXT I
1889 IF M < -7.636 THEN 1999
1931 PRINT A(1), A(2), A(3), A(4)
1937 PRINT A(5), A(6), M, JJJJ
1999 NEXT

This BASIC computer program was run with QB64v1000-win [96]. The complete output of a single run through JJJJ= 7692 is shown below:
299       53       51      151
5.645685280709436       1.98006580996213       -7.625751090671566
-16043
299       53       51       151
5.645685280718897       1.98006580996213       -7.6257510906681027
1979
299       53       51       151
5.645685280709422       1.98006580996213       -7.625751090671552
7692

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 [96], the wall-clock time (not CPU time) for obtaining the output through JJJJ = 7692 was 50 minutes, total. One can compare the computational results above with those in Ali, Raghav, and Bari [1, p. 974], where one can find (299       53       51       151).

Acknowledgment

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

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Solving an Integer Nonlinear programming Formulation of a System Maintenance Problem

 

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Khan, Ali, Raghav, and Shoeb [46, p. 102]:

Maximize

(1 – .4 ^ (2 + X(1))) * (1 – .3 ^ (2 + X(2))) * (1 – .1 ^ (4 + X(3)))

subject to

25 * (X(1) + EXP(.25 * X(1))) + 30 * (X(2) + EXP(.25 * X(2))) + 40 * (X(3) + EXP(.25 * X(3))) <= 500

5 * (X(1) + EXP(.25 * X(1))) + 10 * (X(2) + EXP(.25 * X(2))) + 15 * (X(3) + EXP(.25 * X(3))) <= 135

0<= X(i) <= 5, i=1, 2,

0<= X(3) <= 3,

and X(1) through X(3) are integer variables.
0 DEFDBL A-Z

1 REM DEFINT K

2 DIM N(99), A(35222), X(35012), D(111), P(111), PS(33), J44(35202), J(35211), AA(99), STIO(35303)

81 FOR JJJJ = -32000 TO 32000

85 RANDOMIZE JJJJ

87 M = -4E+250

121 FOR J44 = 1 TO 3
123 A(J44) = FIX(RND * 4)

126 NEXT J44

128 FOR I = 0 TO FIX(RND * 1000)

129 FOR KKQQ = 1 TO 3

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

135 FOR IPP = 1 TO FIX(1 + RND * 2.3)

143 j = 1 + FIX(RND * 3)
145 GOTO 162

154 IF RND < .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 * 1.3) ELSE X(j) = A(j) + FIX(1 + RND * 1.3)
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 = 1 TO 3

172 IF X(J44) < 0 THEN 1670
173 IF X(J44) > 5 THEN 1670

174 NEXT J44
175 IF X(3) > 3 THEN 1670
176 IF 25 * (X(1) + EXP(.25 * X(1))) + 30 * (X(2) + EXP(.25 * X(2))) + 40 * (X(3) + EXP(.25 * X(3))) > 500 THEN 1670
179 IF 5 * (X(1) + EXP(.25 * X(1))) + 10 * (X(2) + EXP(.25 * X(2))) + 15 * (X(3) + EXP(.25 * X(3))) > 135 THEN 1670
999 P = (1 – .4 ^ (2 + X(1))) * (1 – .3 ^ (2 + X(2))) * (1 – .1 ^ (4 + X(3)))

1114 IF P <= M THEN 1670

1420 M = P

1444 FOR KLX = 1 TO 3

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

1557 GOTO 128
1670 NEXT I

1889 REM IF M < -155 THEN 1999

1931 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ
This BASIC computer program was run with QB64v1000-win [94]. The complete output of a single run through JJJJ= -31997 is shown below:

4 4 0 .9950784681854016
-32000

5 1 1 .971396122741632
-31999

5       4       0       .9975340310141606
-31998

5 4 0 .9975340310141606
-31997

.
.
.

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 = -31997 was 2 seconds, not including the time for “Creating .EXE file” (20 seconds, total, including the time for “Creating .EXE file”). One can compare the computational results above with those in Khan, Ali, Raghav, and Shoeb [46, p. 104].

Acknowledgment

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

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