Solving Another Mixed-Integer Nonlinear Programming Problem in Sample Surveys

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear integer programming problem from Raghav, Ali, and Bari [65, pp. 32-33]:

Minimize

X(9)+X(10)

subject to


 (.7893 + (X(5) - 1) * .06) / X(1) + (.4411 + (X(6) - 1) * .02) / X(2) + (.3576 + (X(7) - 1) * .02) / X(3) + (.3973 + (X(8) - 1) * .03) / X(4) - X(9) <= .00655


 (.8316 + (X(5) - 1) * .06) / X(1) + (.3358 + (X(6) - 1) * .02) / X(2) + (.1062 + (X(7) - 1) * .01) / X(3) + (.2791 + (X(8) - 1) * .02) / X(4) - X(10) <= .00454


 (2.4 +.9/X(5)* X(1) + (3.4 + .8 / X(6)) * X(2) + (4 + 1.25 / X(7)) * X(3) + (4.6 + 1.68 / X(8)) * X(4)))  <=5000


2<=X(1)<=1214

2<=X(2)<=822

2<=X(3)<=1028

2<=X(4)<=786

where X(1) through X(4) are integer variables, X(4) through X(8) are continuous variables and >1, and X(9) and X(10) are >=0.

One notes line 234, line 239, and  line 233, which are 234 X(9) = (.7893 + (X(5) - 1) * .06) / X(1) + (.4411 + (X(6) - 1) * .02) / X(2) + (.3576 + (X(7) - 1) * .02) / X(3) + (.3973 + (X(8) - 1) * .03) / X(4) - .00655,

239 X(10) = (.8316 + (X(5) - 1) * .06) / X(1) + (.3358 + (X(6) - 1) * .02) / X(2) + (.1062 + (X(7) - 1) * .01) / X(3) + (.2791 + (X(8) - 1) * .02) / X(4) - .00454, and

233 X(5) = .9 * X(1) * (1 / (5000 - 2.4 * X(1) - (3.4 + .8 / X(6)) * X(2) - (4 + 1.25 / X(7)) * X(3) - (4.6 + 1.68 / X(8)) * X(4))), respectively, from the long inequality constraints above, respectively.  That is an attempt to find active constraints.


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
    111 REM FOR J44 = 1 TO 4


    114 A(1) = 2 + FIX(RND * 1213)

    115 A(2) = 2 + FIX(RND * 821)

    116 A(3) = 2 + FIX(RND * 1027)

    117 A(4) = 2 + FIX(RND * 785)


    118 FOR J44 = 5 TO 10


        119 A(J44) = 1 + (RND * 2)


    120 NEXT J44

    128 FOR I = 1 TO 50000

        129 FOR KKQQ = 1 TO 10

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

            153 J = 1 + FIX(RND * 10)
            154 IF J < 4.5 THEN GOTO 162 ELSE GOTO 156

            155 REM 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) - INT(RND * 15) ELSE X(J) = A(J) + INT(RND * 15)

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

        169 NEXT IPP

        172 X(1) = INT(X(1))
        174 X(2) = INT(X(2))
        176 X(3) = INT(X(3))
        178 X(4) = INT(X(4))

        215 FOR J44 = 1 TO 10


            216 IF X(J44) < 0 THEN 1670
        219 NEXT J44
        220 FOR J44 = 1 TO 4

            221 IF X(J44) < 2 THEN 1670

        223 NEXT J44

        225 IF X(1) > 1214 THEN 1670
        226 IF X(2) > 822 THEN 1670


        227 IF X(3) > 1028 THEN 1670

        229 IF X(4) > 786 THEN 1670

        230 FOR J44 = 5 TO 8

            231 IF X(J44) <= 1## THEN 1670

        232 NEXT J44

        233 X(5) = .9 * X(1) * (1 / (5000 - 2.4 * X(1) - (3.4 + .8 / X(6)) * X(2) - (4 + 1.25 / X(7)) * X(3) - (4.6 + 1.68 / X(8)) * X(4)))


        234 X(9) = (.7893 + (X(5) - 1) * .06) / X(1) + (.4411 + (X(6) - 1) * .02) / X(2) + (.3576 + (X(7) - 1) * .02) / X(3) + (.3973 + (X(8) - 1) * .03) / X(4) - .00655


        239 X(10) = (.8316 + (X(5) - 1) * .06) / X(1) + (.3358 + (X(6) - 1) * .02) / X(2) + (.1062 + (X(7) - 1) * .01) / X(3) + (.2791 + (X(8) - 1) * .02) / X(4) - .00454


        326 FOR J44 = 1 TO 10

            328 IF X(J44) < 0 THEN 1670

        330 NEXT J44

        333 FOR J44 = 1 TO 4

            334 IF X(J44) < 2 THEN 1670

        336 NEXT J44


        471 P = -X(9) - X(10)

        1111 IF P <= M THEN 1670

        1450 M = P

        1454 FOR KLX = 1 TO 10

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

        1557 GOTO 128
    1670 NEXT I

    1889 IF M < -.00011997 THEN 1999

    1933 PRINT A(1), A(2), A(3), A(4)

    1936 PRINT A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ 

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [78]. The complete output of a single run through JJJJ= -31965 is shown below:

527  311  220  247
2.161809919487455          2.08329930063022         2.127126519609546
2.144582945096825         4.343164689580447D-05              7.626989558445075D-05
-1.197015424802552D-04   -31978

529        311        220        246
2.170670355426797               2.08392912799284                 2.127769788581231
2.136546310641485               4.449669608975741D-05         7.51952696422561D-05 
-1.196919657320135D-04   -31971

529  311  220  246
2.170670355426797               2.08392912799284                  2.12776979823337
2.136546310641485               4.449669657008732D-05         7.51952691619263D-05
-1.196919657320136D-04   -31965

.
.
.


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 [78], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31965 was 8 seconds, not including the time for “Creating .EXE file.”  One can compare the computational results above with those in Raghav, Ali, and Bari [65, p. 33], where one can see the following numbers:  529, 311, 220, 246, 0.0001196, 0.000044, 0.000075.


Acknowledgment

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

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Solving Another Mixed-Integer Nonlinear Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear integer programming problem from Raghav, Ali, and Bari [65, p. 35]:

Minimize

 (((1 / 611.57) * ((493.33 + (X(5) - 1) * 37) / X(1) + (275.68 + (X(6) - 1) * 13.78) / X(2) + (223.52 + (X(7) - 1) * 13.97) / X(3) + (248.29 + (X(8) - 1) * 17.38) / X(4))) - .006701) ^ 2 / .006701 ^ 2 + ((1 / 994.14) * ((831.61 + (X(5) - 1) * 62.37) / X(1) + (335.76 + (X(6) - 1) * 16.79) / X(2) + (106.17 + (X(7) - 1) * 6.64) / X(3) + (279.11 + (X(8) - 1) * 19.54) / X(4)) - .004537) ^ 2 / .004537 ^ 2

subject to

(2.4 +.9/X(5)* X(1) + (3.4 + .8 / X(6)) * X(2) + (4 + 1.25 / X(7)) * X(3) + (4.6 + 1.68 / X(8)) * X(4)))  <=5000

2<=X(1)<=1214

2<=X(2)<=822

2<=X(3)<=1028

2<=X(4)<=786

where X(1) through X(4) are integer variables and X(4) through X(8) are continuous variables and are >1.

One notes line 233, which is 233 X(5) = .9 * X(1) * (1 / (5000 - 2.4 * X(1) - (3.4 + .8 / X(6)) * X(2) - (4 + 1.25 / X(7)) * X(3) - (4.6 + 1.68 / X(8)) * X(4))) from the long inequality constraint above.  That is an attempt to find an active constraint.


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
    111 REM FOR J44 = 1 TO 4


    114 A(1) = 2 + FIX(RND * 1213)

    115 A(2) = 2 + FIX(RND * 821)

    116 A(3) = 2 + FIX(RND * 1027)

    117 A(4) = 2 + FIX(RND * 785)


    118 FOR J44 = 5 TO 8



        119 A(J44) = 1 + (RND * 2)



    120 NEXT J44



    128 FOR I = 1 TO 50000


        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)

            154 IF J < 4.5 THEN GOTO 162 ELSE GOTO 156

            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 * 15) ELSE X(J) = A(J) + INT(RND * 15)


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

        169 NEXT IPP

        172 X(1) = INT(X(1))
        174 X(2) = INT(X(2))
        176 X(3) = INT(X(3))
        178 X(4) = INT(X(4))

        215 FOR J44 = 1 TO 8


            216 IF X(J44) < 0 THEN 1670
        219 NEXT J44
        220 FOR J44 = 1 TO 4

            221 IF X(J44) < 2 THEN 1670

        223 NEXT J44

        225 IF X(1) > 1214 THEN 1670
        226 IF X(2) > 822 THEN 1670


        227 IF X(3) > 1028 THEN 1670

        229 IF X(4) > 786 THEN 1670

        230 FOR J44 = 5 TO 8

            231 IF X(J44) <= 1## THEN 1670

        232 NEXT J44

        233 X(5) = .9 * X(1) * (1 / (5000 - 2.4 * X(1) - (3.4 + .8 / X(6)) * X(2) - (4 + 1.25 / X(7)) * X(3) - (4.6 + 1.68 / X(8)) * X(4)))


        326 FOR J44 = 1 TO 8

            328 IF X(J44) < 0 THEN 1670

        330 NEXT J44

        333 FOR J44 = 1 TO 4

            334 IF X(J44) < 2 THEN 1670

        336 NEXT J44



        467 P = -(((1 / 611.57) * ((493.33 + (X(5) - 1) * 37) / X(1) + (275.68 + (X(6) - 1) * 13.78) / X(2) + (223.52 + (X(7) - 1) * 13.97) / X(3) + (248.29 + (X(8) - 1) * 17.38) / X(4))) - .006701) ^ 2 / .006701 ^ 2 - ((1 / 994.14) * ((831.61 + (X(5) - 1) * 62.37) / X(1) + (335.76 + (X(6) - 1) * 16.79) / X(2) + (106.17 + (X(7) - 1) * 6.64) / X(3) + (279.11 + (X(8) - 1) * 19.54) / X(4)) - .004537) ^ 2 / .004537 ^ 2


        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 < -.0007871 THEN 1999

    1933 PRINT A(1), A(2), A(3), A(4)

    1936 PRINT A(5), A(6), A(7), A(8), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [78]. The output of a single run through JJJJ= -31960 is summarized below:


540  313  211  248
2.147905655365354        2.114890065618677          2.166363222778617
2.20927598670702         -2.787029044276058D-04    -31978
.
.
.

541  313  211  247
2.146059528132376      2.108092383599432            2.156239922890985  
2.19301124463398       -2.786887217092242D-04    -31960         
.
.
.

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 [78], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31960 was 8 seconds, not including the time for “Creating .EXE file.”  One can compare the computational results above with those in Raghav, Ali, and Bari [65, p. 35].

Acknowledgment

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

References

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Solving Another Multi-Objective Nonlinear Mixed-Integer Preemptive Goal Programming Problem

Jsun Yui Wong

The computer program listed below seeks to solve the following nonlinear integer goal programming problem from Lee [44, p. 70 and p. 77, Example 3-3]:

Minimize

[  (X(10)+X(11)+X(12)), (X(9)) ] 

subject to

X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 4 * X(4) ^ 2 + 2 * X(5) ^ 2 + X(6)-X(10)=110

7 * (X(1) + EXP(X(1) / 4)) + 7 * (X(2) + EXP(X(2) / 4)) + 5 * (X(3) + EXP(X(3) / 4)) + 9 * (X(4) + EXP(X(4) / 4)) + 4 * (X(5) + EXP(X(5) / 4)) + X(7)-X(11)=175

7 * (X(1) * EXP(X(1) / 4)) + 8 * (X(2) * EXP(X(2) / 4)) + 8 * (X(3) * EXP(X(3) / 4)) + 6 * (X(4) * EXP(X(4) / 4)) + 9 * (X(5) * EXP(X(5) / 4)) + X(8)-X(12)=200

(1 – (1 – .80) ^ X(1)) * (1 – (1 – .85) ^ X(2)) * (1 – (1 – .90) ^ X(3)) * (1 – (1 – .65) ^ X(4)) * (1 – (1 – .75) ^ X(5)) + X(9)-X(13)=1 

where X(1) through X(5) are positive general integer variables and X(6) through X(13) are >=0.

The following computer program uses the streamlined procedure of Hillier and Lieberman [34, pp. 289-291] for multi-objective preemptive goal programming problems.  See line  458, which is  458 P = -10 ^ 15 * (X(10) + X(11) + X(12)) - X(9).


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

    92 A(1) = FIX(RND * 9)

    93 A(2) = FIX(RND * 9)
    94 A(3) = FIX(RND * 9)

    96 A(4) = FIX(RND * 9)
    97 A(5) = FIX(RND * 9)

    111 FOR J44 = 6 TO 13

        114 A(J44) = (RND * 50)

    117 NEXT J44

    128 FOR I = 1 TO 500000

        129 FOR KKQQ = 1 TO 13
            130 X(KKQQ) = A(KKQQ)
        131 NEXT KKQQ
        151 FOR IPP = 1 TO FIX(1 + RND * 5)

            153 J = 1 + FIX(RND * 13)
            154 IF J > 5 THEN GOTO 156 ELSE GOTO 162
            155 REM 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) - INT(RND * 4) ELSE X(J) = A(J) + INT(RND * 4)
            164 REM IF A(J) = 0 THEN X(J) = 1 ELSE X(J) = 0

        169 NEXT IPP

        172 X(1) = INT(X(1))
        174 X(2) = INT(X(2))
        176 X(3) = INT(X(3))
        178 X(4) = INT(X(4))
        179 X(5) = INT(X(5))
        188 IF X(1) < 1 THEN 1670

        189 IF X(2) < 1 THEN 1670
        190 IF X(3) < 1 THEN 1670
        192 IF X(4) < 1 THEN 1670
        194 IF X(5) < 1 THEN 1670

        215 FOR J44 = 6 TO 13

            216 IF X(J44) < 0 THEN 1670
        219 NEXT J44

        311 X(10) = -110 + X(1) ^ 2 + 2 * X(2) ^ 2 + 3 * X(3) ^ 2 + 4 * X(4) ^ 2 + 2 * X(5) ^ 2 + X(6)
        315 X(11) = -175 + 7 * (X(1) + EXP(X(1) / 4)) + 7 * (X(2) + EXP(X(2) / 4)) + 5 * (X(3) + EXP(X(3) / 4)) + 9 * (X(4) + EXP(X(4) / 4)) + 4 * (X(5) + EXP(X(5) / 4)) + X(7)

        319 X(12) = -200 + 7 * (X(1) * EXP(X(1) / 4)) + 8 * (X(2) * EXP(X(2) / 4)) + 8 * (X(3) * EXP(X(3) / 4)) + 6 * (X(4) * EXP(X(4) / 4)) + 9 * (X(5) * EXP(X(5) / 4)) + X(8)

        322 X(13) = -1 + (1 - (1 - .80) ^ X(1)) * (1 - (1 - .85) ^ X(2)) * (1 - (1 - .90) ^ X(3)) * (1 - (1 - .65) ^ X(4)) * (1 - (1 - .75) ^ X(5)) + X(9)
        326 FOR J44 = 1 TO 13

            328 IF X(J44) < 0 THEN 1670

        330 NEXT J44


        458 P = -10 ^ 15 * (X(10) + X(11) + X(12)) - X(9)


        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 REM IF M < -109900 THEN 1999
   
    1933 PRINT A(1), A(2), A(3), A(4), A(5)

    1936 PRINT A(6), A(7), A(8), A(9)

    1946 PRINT A(10), A(11), A(12), A(13), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [77]. The best candidate solution through JJJJ= -31949 is shown below:

.
.
.

3        2        2        3        3
27           28.87534441934497        7.518918241159362
.0955327034546875       
0        0        0        3.089976191583688D-18
-.0955327034546875      -31949
.
.
.


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 [77], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31949 was 2 minutes, not including the time for “Creating .EXE file.”  One can compare the computational results above with those in Lee [44, p. 77, Example 3-3].

Acknowledgment

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

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.


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Hillier and Lieberman's Streamlined Procedure for Solving Multi-Objective Mixed-Integer Nonlinear/Linear Preemptive Goal Programming Problems

Jsun Yui Wong

1.  A Problem from Markland [62]   

The first computer program listed below seeks to solve the immediately following problem from Markland [62, p. 283, Problem 4] with the streamlined procedure of Hillier and Lieberman [42, pp. 289-291]:
   
Minimize     

 P1* X(10) +P2 * X(6) +P3*X(7) +P4*X(9) +P5* X(8)

where P1>>P2>>P3>>P4>>P5

subject to

         2.00 * X(1) + .1 * X(2) + 1.1 * X(3) + .08 * X(4) + .75 * X(5) <= 25

         225 * X(1) + 200 * X(2) + 175 * X(3) + 150 * X(4) + 400 * X(5) <= 10000

        .15 * X(1) + .25 * X(2) + .15 * X(3) + .05 * X(4) + .08 * X(5) >= 9.50

   X(1) +    X(6) = 100
   X(2)  +      X(7) = 100

   X(3)     +    X(8) = 100

   X(4)    +  X(9) = 100

    X(5)   +  X(10) = 100

All variables are nonnegative.

The formulation above is from Markland [62, pp. 809-810].

0 DEFDBL A-Z
1 REM        DEFINT X, A
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
    120 FOR J44 = 1 TO 10

        121 A(J44) = FIX(RND * 40)

    123 NEXT J44

    128 FOR I = 0 TO FIX(RND * 90000)


        129 FOR KKQQ = 1 TO 10

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

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

            143 j = 1 + FIX(RND * 10)

            144 IF RND < .5 THEN GOTO 156 ELSE GOTO 162 

            145 REM GOTO 162
            154 REM     IF j > 3.5 THEN GOTO 162 ELSE GOTO 156

            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 * 5.3) ELSE X(j) = A(j) + FIX(1 + RND * 5.3)

        169 NEXT IPP
       

        221 FOR J44 = 1 TO 12

            231 REM     X(J44) = INT(X(J44))


            232 IF X(J44) < 0 THEN 1670

        234 NEXT J44


        250 X(6) = 100 - X(1)
        252 X(7) = 100 - X(2)

        253 X(8) = 100 - X(3)

        254 X(9) = 100 - X(4)

        255 X(10) = 100 - X(5)

        257 FOR J44 = 1 TO 10
            258 REM      X(J44) = INT(X(J44))

            259 IF X(J44) < 0 THEN 1670
        260 NEXT J44


        263 IF 2.00 * X(1) + .1 * X(2) + 1.1 * X(3) + .08 * X(4) + .75 * X(5) > 25 THEN 1670

        265 IF 225 * X(1) + 200 * X(2) + 175 * X(3) + 150 * X(4) + 400 * X(5) > 10000 THEN 1670
        266 IF .15 * X(1) + .25 * X(2) + .15 * X(3) + .05 * X(4) + .08 * X(5) < 9.50 THEN 1670

        1415 P = -10 ^ 80 * (X(10)) - 10 ^ 60 * X(6) - 10 ^ 40 * (X(7)) - 10 ^ 20 * X(9) - X(8)
     

        1417 IF P <= M THEN 1670
        1420 M = P


        1442 FOR KLX = 1 TO 10


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

        1557 GOTO 128
    1670 NEXT I
    1672 IF M < -4D+200 THEN 1999

    1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ

1999 NEXT JJJJ


This BASIC computer program was run with QB64v1000-win [102]. The complete output of a single run through JJJJ=-28987 is shown below:

.
.
.
.0059093962162189         35.71146723582667       5.46674869668203D-03
0                                    7.138293643361265       99.99409060378378
64.28853276417333         99.99453325130332       100
92.86170635663873        -9.286170635663874D+81      -28987                               
.
.
.


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 [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -28987 was 5 minutes, not including the time for “Creating .EXE file.”   One can compare the computational results above with those in Markland [62, pp. 809-810].         

Remark 3 and remark 4 on page 198 of Winston and Venkataramanan are noteworthy [103].


2. An Integer Version of the Same Problem
 
One notes the following line 144, which is 144 REM IF RND < .5 THEN GOTO 156 ELSE GOTO 162.

0 DEFDBL A-Z
1 REM        DEFINT X, A
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
    120 FOR J44 = 1 TO 10

        121 A(J44) = FIX(RND * 40)


    123 NEXT J44

    128 FOR I = 0 TO FIX(RND * 90000)


        129 FOR KKQQ = 1 TO 10

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

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

            143 j = 1 + FIX(RND * 10)

            144 REM IF RND < .5 THEN GOTO 156 ELSE GOTO 162


            145 GOTO 162
            154 REM     IF j > 3.5 THEN GOTO 162 ELSE GOTO 156

            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 * 5.3) ELSE X(j) = A(j) + FIX(1 + RND * 5.3)


        169 NEXT IPP
        170 REM   GOTO 221

        171 REM X(1) = 100
        173 REM
        177 REM
        178 REM  X(2) = 140

        179 REM  IF X(3) > 20 THEN 1670

        221 FOR J44 = 1 TO 12

            231 REM     X(J44) = INT(X(J44))


            232 IF X(J44) < 0 THEN 1670

        234 NEXT J44


        250 X(6) = 100 - X(1)
        252 X(7) = 100 - X(2)

        253 X(8) = 100 - X(3)

        254 X(9) = 100 - X(4)

        255 X(10) = 100 - X(5)

        257 FOR J44 = 1 TO 10
            258 REM      X(J44) = INT(X(J44))

            259 IF X(J44) < 0 THEN 1670
        260 NEXT J44


        263 IF 2.00 * X(1) + .1 * X(2) + 1.1 * X(3) + .08 * X(4) + .75 * X(5) > 25 THEN 1670

        265 IF 225 * X(1) + 200 * X(2) + 175 * X(3) + 150 * X(4) + 400 * X(5) > 10000 THEN 1670
        266 IF .15 * X(1) + .25 * X(2) + .15 * X(3) + .05 * X(4) + .08 * X(5) < 9.50 THEN 1670

        272 REM  IF 100 * X(1) + 60 * X(2) > 600 THEN 1670

        1411 REM   P = -10 ^ 30 * X(3) - 10 ^ 30 * X(6) - 10 ^ 15 * X(7) - 10 ^ 15 * X(9) - X(11)
        1413 REM     P = -10 ^ 30 * X(4) - 10 ^ 20 * X(8) - 10 ^ 10 * X(5) - X(9)

        1415 P = -10 ^ 80 * (X(10)) - 10 ^ 60 * X(6) - 10 ^ 40 * (X(7)) - 10 ^ 20 * X(9) - X(8)
        1416 REM P = -10 ^ 30 * X(4) - 10 ^ 20 * (X(6)) - 10 ^ 10 * (X(9)) - X(11)


        1417 IF P <= M THEN 1670
        1420 M = P


        1442 FOR KLX = 1 TO 10


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

        1557 GOTO 128
    1670 NEXT I
    1672 IF M < -4D+200 THEN 1999

    1931 PRINT A(1), A(2), A(3), A(4), A(5), A(6), A(7), A(8), A(9), A(10), M, JJJJ

1999 NEXT JJJJ

This BASIC computer program was run with QB64v1000-win [102]. The complete output of a single run through JJJJ=-29408 is shown below:

.
.
.
0  27  16  1  4
100  73  84  99  96
-9.6D+81    -29556

0  36  0  0  7
100  64  100  100  93
-9.3D+81    -29408
.
.
.


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 [102], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -29408 was 5 minutes, not including the time for “Creating .EXE file.”   One can compare the computational results above with those in Markland [62, pp. 809-810].         
     
Remark 3 and remark 4 on page 198 of Winston and Venkataramanan are noteworthy [103].

Acknowledgement

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

References

[1]  Irfan Ali, Yashpal Singh Raghav, Abdul Bari (2013).  Compromise allocation in multivariate stratified surveys with stochastic quadratic cost function, Journal of Statistical Computation and Simulation 2013,   Vol. 83,  No. 5, pp. 962-976.

[2]  Irfan Ali, Yashpal Singh Raghav, Abdul Bari (2011).  Integer goal programming approach for finding a compromise allocation of repairable components, International Journal of Engineering, Science and Technology 2011, Vol. 3, No. 6, pp. 184-195.  https://www.researchgate.net> publication > 260348481

[3] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[4] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[5] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[6] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

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