Computer program to solve nonlinear/linear fractional programming problems: an illustration

 

Computer program to solve nonlinear/linear fractional programming problems:  an illustration  

Jsun Yui Wong

Similar to the computer program of the preceding paper, the computer program listed below aims to solve directly the following fractional programming problem based on Jaberipour and Khorram [42, Example 1, p. 738] and Raouf and Hezam [73, Problem f10, page 5 of 10]: 

 Maximize        ((13 * X(1) + 13 * X(2) + 13) / (37 * X(1) + 73 * X(2) + 13)) ^ -1.4 * ((64 * X(1) - 18 * X(2) + 39) / (13 * X(1) + 26 * X(2) + 13)) ^ 1.2 - ((X(1) + 2 * X(2) + 5 * X(3) + 50) / (X(1) + 5 * X(2) + 5 * X(3) + 50)) ^ .5 * ((X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(2) + 4 * X(3) + 50)) ^ 1.1

subject to

         2 * X(1) + X(2) + 5 * X(3) <= 10 

         5 * X(1)  - 3* X(2)  = 3 

        1.5  <= X(1) <=  3   

       X(1),  X(2),  X(3) >= 0. 

0 DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

    87 RANDOMIZE JJJJ

    88 M = -3D+30

    117 FOR J44 = 1 TO 3

        121 A(J44) = (1 + (RND * 3))

    126 NEXT J44

    127 FOR I = 1 TO 10000

        129 FOR KKQQ = 1 TO 3

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        139 FOR IPP = 1 TO FIX(1 + RND * 3)

            140 B = 1 + FIX(RND * 3)

            156 IF RND < .5 THEN 160 ELSE GOTO 167

            160 r = (1 - RND * 2) * A(B)

            164 X(B) = A(B) + (RND ^ (RND * 15)) * r

            166 GOTO 168

            167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

        168 NEXT IPP

        186 FOR J44 = 1 TO 3

            188 REM    X(J44) = INT(X(J44))

        189 NEXT J44

        422 FOR J44 = 1 TO 3

            435 REM

            436 REM

        447 NEXT J44

        451 X(1) = (3 + 3 * X(2)) / 5

        452 IF X(1) < 0 THEN 1670

        453 IF X(2) < 0 THEN 1670

        454 IF X(3) < 0 THEN 1670

        456 IF X(1) < 1.5 THEN 1670

        457 IF X(1) > 3 THEN 1670

        464 IF 2 * X(1) + X(2) + 5 * X(3) > 10 THEN 1670

        621 IF (X(1) + 2 * X(2) + 5 * X(3) + 50) / (X(1) + 5 * X(2) + 5 * X(3) + 50) < 0 THEN 1670

        623 P = ((13 * X(1) + 13 * X(2) + 13) / (37 * X(1) + 73 * X(2) + 13)) ^ -1.4 * ((64 * X(1) - 18 * X(2) + 39) / (13 * X(1) + 26 * X(2) + 13)) ^ 1.2 - ((X(1) + 2 * X(2) + 5 * X(3) + 50) / (X(1) + 5 * X(2) + 5 * X(3) + 50)) ^ .5 * ((X(1) + 2 * X(2) + 4 * X(3) + 50) / (5 * X(2) + 4 * X(3) + 50)) ^ 1.1

        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 1 TO 3

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

    1670 NEXT I

    1777 IF M < -999999 THEN 1999

    1904 PRINT A(1), A(2), A(3), M, JJJJ     

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output through 

JJJJ=-32000 is shown below.  GW-BASIC also can handle this computer program.

1.499999940395465      1.499999900659180      3.848985065144648D-15

8.279865665019552      -32000

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -32000 was one second, counting from “Starting program…”.  Please see the computational results in Jaberipour and Khorram [42] and  Raouf and Hezam [73, Problem f10, p. 5 of 10].    

All computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

Acknowledgement

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

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Computer program to solve directly geometric programming problems

Computer program to solve directly geometric programming problems         

Jsun Yui Wong

The computer program listed below aims to solve directly the following geometric programming problem from Ravindrain, Phillips, and Solberg [75, page 577]:  

Maxinize

5 * X(1) ^ 2 - X(2) ^ 2 * X(3) ^ 4

subject to

 -5 * X(1) ^ 2 * X(2) ^ -2 + 3 * X(2) ^ -1 * X(3) >= 2

 X(i) >= 0,  i=1, 2, 3, 4.

One notes that line 619 below is 619 P = 5 * X(1) ^ 2 - X(2) ^ 2 * X(3) ^ 4.

0 REM   DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

    87 RANDOMIZE JJJJ

    88 M = -3D+30

    117 FOR J44 = 1 TO 3

        121 A(J44) = 0 + RND * 5

    126 NEXT J44

    127 FOR I = 1 TO 100000

        129 FOR KKQQ = 1 TO 3

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        139 FOR IPP = 1 TO FIX(1 + RND * 3)

            140 B = 1 + FIX(RND * 3)

            156 IF RND < .5 THEN 160 ELSE GOTO 167

            160 r = (1 - RND * 2) * A(B)

            164 X(B) = A(B) + (RND ^ (RND * 15)) * r

            166 GOTO 168

            167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

        168 NEXT IPP

        186 FOR J44 = 1 TO 3

            188 REM   X(J44) = INT(X(J44))

        189 NEXT J44

        422 FOR J44 = 1 TO 3

            435 IF X(J44) < 0 THEN 1670

            436 REM

        455 NEXT J44

        463 IF -5 * X(1) ^ 2 * X(2) ^ -2 + 3 * X(2) ^ -1 * X(3) < 2 THEN 1670

        619 P = 5 * X(1) ^ 2 - X(2) ^ 2 * X(3) ^ 4

        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 1 TO 3

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

    1670 NEXT I

    1777 IF M < .795 THEN 1999

    1904 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output in one run through JJJJ=-31981 is shown below.  GW-BASIC also can handle this computer program.

 .4858663       .4359888      1.193076         .7951868      -32000

 .4853731       .4460661      1.177617         .7952737      -31999

 .4899952       .4470636      1.193125         .795451        -31981

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31981 was 6 seconds, counting from “Starting program…”.  Please see the computations in Ravindrain, Phillips, and Solberg [75, pp. 577-578]. 

All computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System.

Acknowledgement

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

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Computer program to solve nonlinear fractional programming problems

 

Computer program to solve nonlinear fractional programming problems

Jsun Yui Wong

The computer program listed below seeks to solve the following problem from Jong [16, page 16/21, Problem 6]:  

Minimize ((X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(3)) / (X(3) ^ 2 + 5 * X(1) * X(2))) + ((X(1) + 1) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20))

subject to

X(1) ^ 2 + X(2) ^ 2 + X(3)<=5,

(X(1) - 2) ^ 2 + X(2) ^ 2 + X(3) ^ 2<=5,

1<=X(1) <=3,

1<=X(2) <= 3,

1<=X(3) <= 2.

0 REM  DEFDBL A-Z

1 REM DEFINT I, J, K

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J(99), AA(99), HR(32), HHR(32), LHS(44), PLHS(44), LB(22), UB(22), PX(44), J44(44), PN(22), NN(22)

85 FOR JJJJ = -32000 TO 32000

    87 RANDOMIZE JJJJ

    88 M = -3D+30

    117 FOR J44 = 1 TO 3

        121 A(J44) = 1 + RND * 2

    126 NEXT J44

    127 FOR I = 1 TO 5000

        129 FOR KKQQ = 1 TO 3

            130 X(KKQQ) = A(KKQQ)

        131 NEXT KKQQ

        139 FOR IPP = 1 TO FIX(1 + RND * 3)

            140 B = 1 + FIX(RND * 3)

            156 IF RND < .5 THEN 160 ELSE GOTO 167

            160 r = (1 - RND * 2) * A(B)

            164 X(B) = A(B) + (RND ^ (RND * 15)) * r

            166 GOTO 168

            167 IF RND < .5 THEN X(B) = A(B) - FIX(RND * 2) ELSE X(B) = A(B) + FIX(RND * 2)

        168 NEXT IPP

        186 FOR J44 = 1 TO 3

            188 REM   X(J44) = INT(X(J44))

        189 NEXT J44

        201 IF X(1) < 1 THEN 1670

        203 IF X(1) > 3 THEN 1670

        211 IF X(2) < 1 THEN 1670

        213 IF X(2) > 3 THEN 1670

        217 IF X(3) < 1 THEN 1670

        218 IF X(3) > 2 THEN 1670

        222 IF X(1) ^ 2 + X(2) ^ 2 + X(3) - 5 > 0 THEN 1670

        225 IF (X(1) - 2) ^ 2 + X(2) ^ 2 + X(3) ^ 2 - 5 > 0 THEN 1670

        422 FOR J44 = 1 TO 3

            435 REM

            437 REM

        455 NEXT J44

        616 P = -((X(1) ^ 2 + X(2) ^ 2 + 2 * X(1) * X(3)) / (X(3) ^ 2 + 5 * X(1) * X(2))) - ((X(1) + 1) / (X(1) ^ 2 - 2 * X(1) + X(2) ^ 2 - 8 * X(2) + 20))

        1111 IF P <= M THEN 1670

        1452 M = P

        1454 FOR KLX = 1 TO 3

            1455 A(KLX) = X(KLX)

        1456 NEXT KLX

    1670 NEXT I

    1777 IF M < -999999 THEN 1999

    1904 PRINT A(1), A(2), A(3), M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [104]. Its complete output through 

JJJJ=-31995 in one run is shown below.  GW-BASIC also can handle this computer program.

1.000001      1.229833      1      -.8185654      -31999

1.000006      1.230199      1.000001      -.8185661      -31998

1      1.230354      1.000003      -.8185657      -31996

1      1.23025      1      -.8185653      -31995

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen. On a personal computer with QB64v1000-win [104], the wall-clock time (not CPU time) for obtaining the output through JJJJ = -31995 was 2 seconds, counting from “Starting program…”.  Please see the computational results in Jong [16, page 16/21, Problem 6].      

All computational results presented above were obtained from the following computer system:

Processor: Intel (R) Core (TM) i5 CPU M 430 @2.27 GHz 2.26 GHz

Installed memory (RAM): 4.00GB (3.87 GB usable)

System type: 64-bit Operating System. 

Acknowledgment

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

References

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