| Re: builtin simplex? [message #33086] |
Thu, 05 December 2002 03:18 |
Carsten Dominik
Messages: 45
Registered: February 1998
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Member |
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>>>> > "BS" == Bringfried Stecklum <stecklum@tls-tautenburg.de> writes:
BS> I recently realized that a piece of code which compiled fine
BS> with IDL5.4 failed under 5.5a because of
BS> "SIMPLEX: incorrect number of arguments."
BS> The reason for that is that there must be an IDL function of
BS> the same name which expects different arguments. I did not find
BS> any reference to it in the help system and it does not show
BS> up with help,/fun. Does anybody know more about it?
BS> The obvious solution was to rename my simplex.pro to
BS> my_simplex.pro.
Hallo Bringfried,
It seems to be a new builtin function:
- Carsten
SIMPLEX
The SIMPLEX function uses the simplex
method to solve linear programming
problems. Given a set of N independent
variables Xi, where i = 1, ..., N, the simplex
method seeks to maximize the following function,
Z = a1X1 + a2X2 + ...aNXN
with the assumption that Xi 0. The Xi
are further constrained by the following
equations:
bj1X1 + bj2X2 + ...bjNXN cj j = 1, 2 ...
, ,M1
bj1X1 + bj2X2 + ...bjNXN cj j = M1 + 1, M1 + 2 ...
, , M1 + M2
bj1X1 + bj2X2 + ...bjNXN= cj j
= M1 + M2 + 1, M1 + M2 + 2 ...
, , M
where M = M1 + M2 + M3 is the total number
of equations, and the constraint values
cj must all be positive.
To solve the above problem using the SIMPLEX
function, the Z equation is rewritten
as a vector:
Zequation = a1 a2 ...aN
The constraint equations are rewritten
as a matrix with N+1 columns and M rows,
where all of the b coefficients
have had their sign reversed:
c1 �b11 �b12...�b1N
c2 �b21 �b22...�b2N
Constraints = : : :
: : :
cM �bM1 �bM2...�bMN
Note
The constraint matrix must be organized so
that the coefficients for the less-than (<)
equations come first, followed by the
coefficients of the greater-than (>) equations,
and then the coefficients of the equal (=) equations.
The Result is a vector of N+1 elements
containing the maximum Z value and the
values of the N independent X variables
(the optimal feasible vector):
Result = Zmax X1 X2...XN
The SIMPLEX function is based on the
routine simplx described in section 10.8 of
Numerical Recipes in C: The Art of Scientific
Computing (Second Edition), published
by Cambridge University Press, and is used by permission.
Syntax
Result = SIMPLEX( Zequation, Constraints, M1, M2, M3
[, Tableau [, Izrov [, Iposv] ] ] [, /DOUBLE]
[, EPS = value] [, STATUS = variable] )
Arguments
Zequation
A vector containing the N coefficients
of the Zequation to be maximized.
Constraints
An array of N+1 columns by M rows containing
the constraint values and coefficients
for the constraint equations.
M1
An integer giving the number of less-than
constraint equations contained in
Constraints. M1 may be zero, indicating
that there are no less than constraints.
M2
An integer giving the number of greater-than
constraint equations contained in
Constraints. M2 may be zero, indicating
that there are no greater than constraints.
M3
An integer giving the number of equal-to
constraint equations contained in
Constraints. M3 may be zero, indicating
that there are no equal to constraints. The
total of M1 + M2 + M3 should equal M,
the number of constraint equations.
Tableau
Set this optional argument to a named
variable in which to return the output array
from the simplex algorithm. For more
detailed discussion about this argument, see
the write-up in section 10.8 of Numerical Recipes in C.
Izrov
Set this optional argument to a named
variable in which to return the output izrov
variable from the simplex algorithm.
For more detailed discussion about this
argument, see the write-up in section
10.8 of Numerical Recipes in C.
Iposv
Set this optional argument to a named
variable in which to return the output iposv
variable from the simplex algorithm.
For more detailed discussion about this
argument, see the write-up in section
10.8 of Numerical Recipes in C.
Keywords
DOUBLE
Set this keyword to use double-precision
for computations and to return a double-
precision result. Set DOUBLE to 0 to use
single-precision for computations and to
return a single-precision result. The
default is /DOUBLE if any of the inputs are
double-precision, otherwise the default is 0.
EPS
Set this keyword to a number close to
machine accuracy, which is used to test for
convergence at each iteration. The default is 10�6.
STATUS
Set this keyword to a named variable to
receive the status of the operation. Possible
status values are:
Value Description
0 Successful completion.
1 The objective function is unbounded.
2 No solution satisfies the given constraints.
3 The routine did not converge.
Table 6-3: SIMPLEX Function Status Values
Example
The following example is taken from Numerical Recipes in C.
Find the Z value which maximizes the equation
Z = X1 + X2 + 3 X3 - 0.5 X4, with the
following constraints:
X1 + 2X3 740
2X2 � 7X4 0
X2 � X3 + 2X4 0.5
X1 + X2 + X3 + X4 = 9
To find the solution, enter the
following code at the IDL command line:
; Set up the Zequation with the X coefficients.
Zequation = [1,1,3,-0.5]
; Set up the Constraints matrix.
Constraints = [ $
[740, -1, 0, -2, 0], $
[ 0, 0, -2, 0, 7], $
[0.5, 0, -1, 1, -2], $
[ 9, -1, -1, -1, -1] ]
; Number of less-than constraint equations.
m1 = 2
; Number of greater-than constraint equations.
m2 = 1
; Number of equal constraint equations.
m3 = 1
;; Call the function.
result = SIMPLEX(Zequation, Constraints, m1, m2, m3)
;; Print out the results.
PRINT, 'Maximum Z value is: ', result[0]
PRINT, 'X coefficients are: '
PRINT, result[1:*]
IDL prints:
Maximum Z value is: 17.0250
X coefficients are:
0.000000 3.32500 4.72500 0.950000
Therefore, the optimal feasible vector
is X1 = 0.0, X2 = 3.325, X3 = 4.725, and
X4 = 0.95.
See Also
AMOEBA, DFPMIN, POWELL
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