Hello,
I've this objective function to minimize the weighted average of d(j):
min obj = sum(j,d(j)*w(j)*x(j)) / sum(j,w(j)*x(j))
d(j) and w(j) are data
x(j) is the unique decision variable.
Any trick to transform the objective into a linear expression?
many thanks
JPC

JPC wrote:
Hello,

I've this objective function to minimize the weighted average of d(j):

min obj = sum(j,d(j)*w(j)*x(j)) / sum(j,w(j)*x(j))

d(j) and w(j) are data
x(j) is the unique decision variable.
--
Any trick to transform the objective into a linear expression?

many thanks
JPC
--

The ratio is invariant under scaling of the x variable. *If* you know
that the denominator has to be positive (you didn't say so, but it's
typical in this type of problem), you could constrain the denominator to
equal 1 and minimize the numerator.

/Paul

Thanks Paul.

The denominator is always positive.

I tried your idea but I get no feasible solution.

Please, consider this example with only 3 elements i.e. j=1 to 3:

D1 = 100 , W1= 1
D2 = 200 , W2 = 2
D3 = 300 , W3 = 3

X1 5;

the original NLP obj that gives me an optimal solution is:

MIN = @SUM(ITEMSDATA: W * D * X ) / @SUM(ITEMSDATA: W * X ) ;

Following your suggestion to linearize the obj, I added one constraint
and changed the function:

@SUM(ITEMSDATA: W * X ) = 1;

MIN = @SUM(ITEMSDATA: W * D * X ) ;

but now I get a "No feasible solution found"

Any further help will be really appreciated!

thanks again
JPC

Paul A. Rubin ha scritto:

JPC wrote:
Hello,

I've this objective function to minimize the weighted average of d(j):

min obj = sum(j,d(j)*w(j)*x(j)) / sum(j,w(j)*x(j))

d(j) and w(j) are data
x(j) is the unique decision variable.
--
Any trick to transform the objective into a linear expression?

many thanks
JPC
>
>
>
The ratio is invariant under scaling of the x variable. *If* you know
that the denominator has to be positive (you didn't say so, but it's
typical in this type of problem), you could constrain the denominator to
equal 1 and minimize the numerator.

/Paul

I forgot to mention that there are also some constraints involving the
x(j) decision variable (used in the weighted average) with other
decision variables used in the MIN obj function:

for example:

x1 5;
x1 + s1 >= 10;

and the obj function is composed in a similar way:

MIN =
@SUM(ITEMSDATA: W * D * x ) / @SUM(ITEMSDATA: W * x )
+ @SUM(ITEMSDATA: s ) ;

is still possible to linearize it?
I tried but with no success

JPC wrote:
Thanks Paul.

The denominator is always positive.

I tried your idea but I get no feasible solution.

Please, consider this example with only 3 elements i.e. j=1 to 3:

D1 = 100 , W1= 1
D2 = 200 , W2 = 2
D3 = 300 , W3 = 3

X1 5;

the original NLP obj that gives me an optimal solution is:

MIN = @SUM(ITEMSDATA: W * D * X ) / @SUM(ITEMSDATA: W * X ) ;
--
Following your suggestion to linearize the obj, I added one constraint
and changed the function:

@SUM(ITEMSDATA: W * X ) = 1;

MIN = @SUM(ITEMSDATA: W * D * X ) ;
--
but now I get a "No feasible solution found"

Any further help will be really appreciated!

thanks again
JPC
>
>
>
>
>
Paul A. Rubin ha scritto:

JPC wrote:
Hello,

I've this objective function to minimize the weighted average of d(j):

min obj = sum(j,d(j)*w(j)*x(j)) / sum(j,w(j)*x(j))

d(j) and w(j) are data
x(j) is the unique decision variable.
--
Any trick to transform the objective into a linear expression?

many thanks
JPC
>
>
>
The ratio is invariant under scaling of the x variable. *If* you know
that the denominator has to be positive (you didn't say so, but it's
typical in this type of problem), you could constrain the denominator to
equal 1 and minimize the numerator.

/Paul

JPC wrote:
I forgot to mention that there are also some constraints involving the
x(j) decision variable (used in the weighted average) with other
decision variables used in the MIN obj function:

for example:

x1 5;
x1 + s1 >= 10;

Well, that certainly makes a difference!
and the obj function is composed in a similar way:

MIN =
@SUM(ITEMSDATA: W * D * x ) / @SUM(ITEMSDATA: W * x )
+ @SUM(ITEMSDATA: s ) ;
--
is still possible to linearize it?
I tried but with no success

I don't think so. There's a trick that would work with the new
objective function and with separate constraints on x and s, or with x
and s jointly constrained but no s terms in the objective, but if s is
in the objective *and* x and s are jointly constrained, the trick fails
(by producing either quadratic constraints or a nonlinear objective,
depending on how you handle it.). The trick is described in section
2.10.2 of the excellent book "Applications of optimization with
Xpress-MP", which is available as a free download from Dash
(www.dashoptimization.com).

/Paul

Thank you Paul,

now I know that it is impossible to linearize my problem

bye
JPC

Paul A. Rubin wrote:
JPC wrote:
I forgot to mention that there are also some constraints involving the
x(j) decision variable (used in the weighted average) with other
decision variables used in the MIN obj function:

for example:

x1 5;
x1 + s1 >= 10;

Well, that certainly makes a difference!
and the obj function is composed in a similar way:

MIN =
@SUM(ITEMSDATA: W * D * x ) / @SUM(ITEMSDATA: W * x )
+ @SUM(ITEMSDATA: s ) ;
--
is still possible to linearize it?
I tried but with no success

I don't think so. There's a trick that would work with the new
objective function and with separate constraints on x and s, or with x
and s jointly constrained but no s terms in the objective, but if s is
in the objective *and* x and s are jointly constrained, the trick fails
(by producing either quadratic constraints or a nonlinear objective,
depending on how you handle it.). The trick is described in section
2.10.2 of the excellent book "Applications of optimization with
Xpress-MP", which is available as a free download from Dash
(www.dashoptimization.com).

/Paul

JPC wrote:
Hello,

I've this objective function to minimize the weighted average of d(j):

min obj = sum(j,d(j)*w(j)*x(j)) / sum(j,w(j)*x(j))

d(j) and w(j) are data
x(j) is the unique decision variable.
--
Any trick to transform the objective into a linear expression?

No, but it is just a standard fractional linear programming problem.
Methods of solution have been around for about 50 years. Do a Google
search on "fractional linear programming", and you will see numerous
papers.

R.G. Vickson

many thanks
JPC