Hi to all,
I'm contending with difficulties of modelling a problem.

I've the following SET with the following properties:

PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD

where :
QUANTITY = the quantity that I have to rent (the solver will suggest how
many of each product I will rent)
Monthly Cost = the money I've to pay each month to rent this product, this
is a fixed amount for each product and period
RENT_PERIOD = how many months I will use the product

for example I can have:

PRODUCT= PROD1, MONTHLY_COST = 100, RENT_PERIOD = 5 months
PRODUCT= PROD1, MONTHLY_COST = 80, RENT_PERIOD = 6 months
PRODUCT= PROD1, MONTHLY_COST = 75, RENT_PERIOD = 7 months

PRODUCT= PROD2, MONTHLY_COST = 110, RENT_PERIOD = 5 months
PRODUCT= PROD2, MONTHLY_COST = 120, RENT_PERIOD = 6 months
PRODUCT= PROD2, MONTHLY_COST = 130, RENT_PERIOD = 7 months

than, I have various constraint about the QUANTITY of each Product (I omit
them, as they are not important here).

than I've the Objective function that is:

MIN = SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) / SUM(PRODUCT:
QUANTITY * RENT_PERIOD)

i.e. I want to minimize the weighed average of the MONTHLY_COST.

This objective will make my problem NOT LINEAR!!

Any way/chance to transform it in a Linear problem?

thanks a lot for any suggestion.

JPC

From:	Steve
Subject:	Re: Linearization of a Weighed average
Date:	18 Jan 2005 07:55:41 -0800

JPC,

You may have lost me here. But you are optimizing cost. While demand
is a constraint. If you set demand constraints (PC's required) for
each month in your operating profile, the solver will be forced to pick
the best combination of cost-period combinations.

And if you do have month-specific demands it will pay you to create a
month-specific index.

BTW, generally you can not formulate a linear model in which decision
variables are multiplied or divided by each other. (There is an
exception called quadratic programming, but forget about that for now.)

Steve

P.S. I am busy so if you have another question, I may not get to it
till tomorrow.

From:	JPC
Subject:	Re: Linearization of a Weighed average
Date:	Tue, 18 Jan 2005 17:03:29 +0100

Many thanks Steve!

I'm going to think better about your helpful assertions.

Sorry for all my annoying questions.

thanks
JPC

From:	Bob Daniel
Subject:	Re: Linearization of a Weighed average
Date:	Tue, 18 Jan 2005 09:35:50 -0000

I do suggest you buy a book on MP modeling. I would suggest (but then I
would)
Applications of optimization with Xpress-MP
Christelle Guéret, Christian Prins & Marc Sevaux
Translated and revised by Susanne Heipcke
Dash Optimization, 2002, ISBN 0-9543503-0-8

I have put the section on ratio objective functions at
http://www.blisworthhouse.co.uk/OR/Bits_of_Book/ratio_obj_fn.pdf

Regards
Bob Daniel

"JPC" wrote in message
news:csh3t2$gus$1@atlantis.cu.mi.it...
>
>
> Hi to all,
> I'm contending with difficulties of modelling a problem.
>
> I've the following SET with the following properties:
>
> PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD
>
> where :
> QUANTITY = the quantity that I have to rent (the solver will suggest how
> many of each product I will rent)
> Monthly Cost = the money I've to pay each month to rent this product,
this
> is a fixed amount for each product and period
> RENT_PERIOD = how many months I will use the product
>
> for example I can have:
>
> PRODUCT= PROD1, MONTHLY_COST = 100, RENT_PERIOD = 5 months
> PRODUCT= PROD1, MONTHLY_COST = 80, RENT_PERIOD = 6 months
> PRODUCT= PROD1, MONTHLY_COST = 75, RENT_PERIOD = 7 months
>
> PRODUCT= PROD2, MONTHLY_COST = 110, RENT_PERIOD = 5 months
> PRODUCT= PROD2, MONTHLY_COST = 120, RENT_PERIOD = 6 months
> PRODUCT= PROD2, MONTHLY_COST = 130, RENT_PERIOD = 7 months
>
>
> than, I have various constraint about the QUANTITY of each Product (I omit
> them, as they are not important here).
>
>
> than I've the Objective function that is:
>
>
> MIN = SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) / SUM(PRODUCT:
> QUANTITY * RENT_PERIOD)
>
> i.e. I want to minimize the weighed average of the MONTHLY_COST.
>
> This objective will make my problem NOT LINEAR!!
>
> Any way/chance to transform it in a Linear problem?
>
> thanks a lot for any suggestion.
>
> JPC
>
>
>

From:	JPC
Subject:	Re: Linearization of a Weighed average
Date:	Tue, 18 Jan 2005 13:40:45 +0100

Hi Bob,

can you give me a little help with the modelling?
I wrote the following (omitting the not important information):

SETS:
PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD, Y
!where the values of the variables MONTHLY_COST and RENT_PERIOD are loaded
from a file, so they are frozen, while the QUANTITY and Y variables(I
intoduced the Y variable following your help) must be calculated by the
solver;

!Objective;
MIN = SUM(PRODUCT: MONTHLY_COST * RENT_PERIOD * Y ) ;

!Constraints:;
1= SUM(PRODUCT: RENT_PERIOD * Y);

FOR (PRODUCT: Y = QUANTITY * D ); !--> this last condition makes my problem
not linear;

How to write it in a linear way?
thanks again, I promise that after this I will buy a good book!
The problem is that is not easy to find a good book here in my little town
in Italy...

bye
JPC

--

"Bob Daniel" wrote in message
news:csilbe$1g8$1$8300dec7@news.demon.co.uk...
> I do suggest you buy a book on MP modeling. I would suggest (but then I
> would)
> Applications of optimization with Xpress-MP
> Christelle Guéret, Christian Prins & Marc Sevaux
> Translated and revised by Susanne Heipcke
> Dash Optimization, 2002, ISBN 0-9543503-0-8
>
> I have put the section on ratio objective functions at
> http://www.blisworthhouse.co.uk/OR/Bits_of_Book/ratio_obj_fn.pdf
>
> Regards
> Bob Daniel
>
> "JPC" wrote in message
> news:csh3t2$gus$1@atlantis.cu.mi.it...
> >
> >
> > Hi to all,
> > I'm contending with difficulties of modelling a problem.
> >
> > I've the following SET with the following properties:
> >
> > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD
> >
> > where :
> > QUANTITY = the quantity that I have to rent (the solver will suggest
how
> > many of each product I will rent)
> > Monthly Cost = the money I've to pay each month to rent this product,
> this
> > is a fixed amount for each product and period
> > RENT_PERIOD = how many months I will use the product
> >
> > for example I can have:
> >
> > PRODUCT= PROD1, MONTHLY_COST = 100, RENT_PERIOD = 5 months
> > PRODUCT= PROD1, MONTHLY_COST = 80, RENT_PERIOD = 6 months
> > PRODUCT= PROD1, MONTHLY_COST = 75, RENT_PERIOD = 7 months
> >
> > PRODUCT= PROD2, MONTHLY_COST = 110, RENT_PERIOD = 5 months
> > PRODUCT= PROD2, MONTHLY_COST = 120, RENT_PERIOD = 6 months
> > PRODUCT= PROD2, MONTHLY_COST = 130, RENT_PERIOD = 7 months
> >
> >
> > than, I have various constraint about the QUANTITY of each Product (I
omit
> > them, as they are not important here).
> >
> >
> > than I've the Objective function that is:
> >
> >
> > MIN = SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) / SUM(PRODUCT:
> > QUANTITY * RENT_PERIOD)
> >
> > i.e. I want to minimize the weighed average of the MONTHLY_COST.
> >
> > This objective will make my problem NOT LINEAR!!
> >
> > Any way/chance to transform it in a Linear problem?
> >
> > thanks a lot for any suggestion.
> >
> > JPC
> >
> >
> >
>
>

From:	Bob Daniel
Subject:	Re: Linearization of a Weighed average
Date:	Wed, 19 Jan 2005 10:34:15 -0000

From the incomplete info you have given I guess your problem is to minimise
the cost of covering demand for products p in month m. Your decision
variables are say h(p,m',t), the number of product p to hire in month m' for
a duration of t months.
I don't understand why these isn't a separate problem for each p. If the
rental cost depends on the total hired in month m' over all p, then of
course you just have 1 problem.
All (all!) you have to do is to get the right summations to find the total
of each p you have in month m (clue: not those for which m'>m, not those
whose hire period has finished). We leave getting the summation limits right
as an exercise for you, AFTER you have used the internet to order a book,
and got the italian post office to deliver it to you.
The ratio obj fn looks a (wrong) side issue.
Regards
Bob

"JPC" wrote in message
news:csj04d$jdk$1@atlantis.cu.mi.it...
> Hi Bob,
>
> can you give me a little help with the modelling?
> I wrote the following (omitting the not important information):
>
> SETS:
> PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD, Y
> !where the values of the variables MONTHLY_COST and RENT_PERIOD are loaded
> from a file, so they are frozen, while the QUANTITY and Y variables(I
> intoduced the Y variable following your help) must be calculated by the
> solver;
>
> !Objective;
> MIN = SUM(PRODUCT: MONTHLY_COST * RENT_PERIOD * Y ) ;
>
> !Constraints:;
> 1= SUM(PRODUCT: RENT_PERIOD * Y);
>
> FOR (PRODUCT: Y = QUANTITY * D ); !--> this last condition makes my
problem
> not linear;
>
>
> How to write it in a linear way?
> thanks again, I promise that after this I will buy a good book!
> The problem is that is not easy to find a good book here in my little town
> in Italy...
>
> bye
> JPC
>
> --
>
> "Bob Daniel" wrote in message
> news:csilbe$1g8$1$8300dec7@news.demon.co.uk...
> > I do suggest you buy a book on MP modeling. I would suggest (but then I
> > would)
> > Applications of optimization with Xpress-MP
> > Christelle Guéret, Christian Prins & Marc Sevaux
> > Translated and revised by Susanne Heipcke
> > Dash Optimization, 2002, ISBN 0-9543503-0-8
> >
> > I have put the section on ratio objective functions at
> > http://www.blisworthhouse.co.uk/OR/Bits_of_Book/ratio_obj_fn.pdf
> >
> > Regards
> > Bob Daniel
> >
> > "JPC" wrote in message
> > news:csh3t2$gus$1@atlantis.cu.mi.it...
> > >
> > >
> > > Hi to all,
> > > I'm contending with difficulties of modelling a problem.
> > >
> > > I've the following SET with the following properties:
> > >
> > > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD
> > >
> > > where :
> > > QUANTITY = the quantity that I have to rent (the solver will suggest
> how
> > > many of each product I will rent)
> > > Monthly Cost = the money I've to pay each month to rent this product,
> > this
> > > is a fixed amount for each product and period
> > > RENT_PERIOD = how many months I will use the product
> > >
> > > for example I can have:
> > >
> > > PRODUCT= PROD1, MONTHLY_COST = 100, RENT_PERIOD = 5 months
> > > PRODUCT= PROD1, MONTHLY_COST = 80, RENT_PERIOD = 6 months
> > > PRODUCT= PROD1, MONTHLY_COST = 75, RENT_PERIOD = 7 months
> > >
> > > PRODUCT= PROD2, MONTHLY_COST = 110, RENT_PERIOD = 5 months
> > > PRODUCT= PROD2, MONTHLY_COST = 120, RENT_PERIOD = 6 months
> > > PRODUCT= PROD2, MONTHLY_COST = 130, RENT_PERIOD = 7 months
> > >
> > >
> > > than, I have various constraint about the QUANTITY of each Product (I
> omit
> > > them, as they are not important here).
> > >
> > >
> > > than I've the Objective function that is:
> > >
> > >
> > > MIN = SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) /
SUM(PRODUCT:
> > > QUANTITY * RENT_PERIOD)
> > >
> > > i.e. I want to minimize the weighed average of the MONTHLY_COST.
> > >
> > > This objective will make my problem NOT LINEAR!!
> > >
> > > Any way/chance to transform it in a Linear problem?
> > >
> > > thanks a lot for any suggestion.
> > >
> > > JPC
> > >
> > >
> > >
> >
> >
>
>

From:	JPC
Subject:	Re: Linearization of a Weighed average
Date:	Wed, 19 Jan 2005 13:43:18 +0100

Bob,
thanks for your suggestions.
I want just let you know that I've already modelled the problem, and the
primary goal, i.e. the demand covering for each month, has been altready
reached.
The problem of the demand covering is separated for each Prodoct, as you
have observed.
But the cost, I think, to be minimised, is the weighed average cost,
considering all the Products.
Am I wrong in this assumption?

The problem, IMHO, is due to the fact that when I run the solver, the demand
is defined ONLY for the Current Year (2005).
A lot of products bought during the 2005, will have the End of the hire
period in 2006.
But in 2006 I've no demand to satisfy (or better, we will have the 2006
demand only in early autumn 2005), so that if I ask to the solver the best
total cost (i.e. sum of Monthly Cost * Prod.qty * hire duration months),
in my mind the solver, for those products ending in 2006, will try to reduce
the "hire duration months", rather then chosing a best Monthly Cost.
But, now, while I'm writing, I think that the solution can be the following:
I've to calculate the Best Total Cost only for the whole 2005, not taking
into account the hire period after the 2005.
Total Cost = SUM(Monthly Cost * Product Qty * Hire Period); but the Hire
period to be considered must be limited to Dec 2005.
What do you think about that? Is it correct?

Regarding the book, does it contain also a sort of Tricks library (where to
find for example how to linearise some conditions)?
For me is very important to know all the possible Tricks, because really,
since I'm a computer programmer, I did not spent a lot of time on the
modelling (defining sets, variables, sum operations, an so on, are quite
simple tasks), but I really spent a lot of time on the linearization of sum
constraints, on the transformation of some constraints into goal objectives
, and so on.
So I really need a book that contains, besides a lto of example, also a sort
of reference about all the possible Tricks.
For example, I needed to linearise the Absolute Value of a variable. I
searched the Internet and I spent one day to find the simple (all is simple
when you know it) solution. If I had a good book with a a sort of "Tricks
Reference", I think I would have saved a lot of time.

thanks again

--

"Bob Daniel" wrote in message
news:csld4v$p5$1$8302bc10@news.demon.co.uk...
> From the incomplete info you have given I guess your problem is to
minimise
> the cost of covering demand for products p in month m. Your decision
> variables are say h(p,m',t), the number of product p to hire in month m'
for
> a duration of t months.
> I don't understand why these isn't a separate problem for each p. If the
> rental cost depends on the total hired in month m' over all p, then of
> course you just have 1 problem.
> All (all!) you have to do is to get the right summations to find the total
> of each p you have in month m (clue: not those for which m'>m, not those
> whose hire period has finished). We leave getting the summation limits
right
> as an exercise for you, AFTER you have used the internet to order a book,
> and got the italian post office to deliver it to you.
> The ratio obj fn looks a (wrong) side issue.
> Regards
> Bob
>
> "JPC" wrote in message
> news:csj04d$jdk$1@atlantis.cu.mi.it...
> > Hi Bob,
> >
> > can you give me a little help with the modelling?
> > I wrote the following (omitting the not important information):
> >
> > SETS:
> > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD, Y
> > !where the values of the variables MONTHLY_COST and RENT_PERIOD are
loaded
> > from a file, so they are frozen, while the QUANTITY and Y variables(I
> > intoduced the Y variable following your help) must be calculated by the
> > solver;
> >
> > !Objective;
> > MIN = SUM(PRODUCT: MONTHLY_COST * RENT_PERIOD * Y ) ;
> >
> > !Constraints:;
> > 1= SUM(PRODUCT: RENT_PERIOD * Y);
> >
> > FOR (PRODUCT: Y = QUANTITY * D ); !--> this last condition makes my
> problem
> > not linear;
> >
> >
> > How to write it in a linear way?
> > thanks again, I promise that after this I will buy a good book!
> > The problem is that is not easy to find a good book here in my little
town
> > in Italy...
> >
> > bye
> > JPC
> >
> > --
> >
> > "Bob Daniel" wrote in message
> > news:csilbe$1g8$1$8300dec7@news.demon.co.uk...
> > > I do suggest you buy a book on MP modeling. I would suggest (but then
I
> > > would)
> > > Applications of optimization with Xpress-MP
> > > Christelle Guéret, Christian Prins & Marc Sevaux
> > > Translated and revised by Susanne Heipcke
> > > Dash Optimization, 2002, ISBN 0-9543503-0-8
> > >
> > > I have put the section on ratio objective functions at
> > > http://www.blisworthhouse.co.uk/OR/Bits_of_Book/ratio_obj_fn.pdf
> > >
> > > Regards
> > > Bob Daniel
> > >
> > > "JPC" wrote in message
> > > news:csh3t2$gus$1@atlantis.cu.mi.it...
> > > >
> > > >
> > > > Hi to all,
> > > > I'm contending with difficulties of modelling a problem.
> > > >
> > > > I've the following SET with the following properties:
> > > >
> > > > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD
> > > >
> > > > where :
> > > > QUANTITY = the quantity that I have to rent (the solver will
suggest
> > how
> > > > many of each product I will rent)
> > > > Monthly Cost = the money I've to pay each month to rent this
product,
> > > this
> > > > is a fixed amount for each product and period
> > > > RENT_PERIOD = how many months I will use the product
> > > >
> > > > for example I can have:
> > > >
> > > > PRODUCT= PROD1, MONTHLY_COST = 100, RENT_PERIOD = 5 months
> > > > PRODUCT= PROD1, MONTHLY_COST = 80, RENT_PERIOD = 6 months
> > > > PRODUCT= PROD1, MONTHLY_COST = 75, RENT_PERIOD = 7 months
> > > >
> > > > PRODUCT= PROD2, MONTHLY_COST = 110, RENT_PERIOD = 5 months
> > > > PRODUCT= PROD2, MONTHLY_COST = 120, RENT_PERIOD = 6 months
> > > > PRODUCT= PROD2, MONTHLY_COST = 130, RENT_PERIOD = 7 months
> > > >
> > > >
> > > > than, I have various constraint about the QUANTITY of each Product
(I
> > omit
> > > > them, as they are not important here).
> > > >
> > > >
> > > > than I've the Objective function that is:
> > > >
> > > >
> > > > MIN = SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) /
> SUM(PRODUCT:
> > > > QUANTITY * RENT_PERIOD)
> > > >
> > > > i.e. I want to minimize the weighed average of the MONTHLY_COST.
> > > >
> > > > This objective will make my problem NOT LINEAR!!
> > > >
> > > > Any way/chance to transform it in a Linear problem?
> > > >
> > > > thanks a lot for any suggestion.
> > > >
> > > > JPC
> > > >
> > > >
> > > >
> > >
> > >
> >
> >
>
>

From:	Bob Daniel
Subject:	Re: Linearization of a Weighed average
Date:	Wed, 19 Jan 2005 15:06:32 -0000

"JPC" wrote in message
news:cslkl7$8dh$1@atlantis.cu.mi.it...
> Bob,
> thanks for your suggestions.
> I want just let you know that I've already modelled the problem, and the
> primary goal, i.e. the demand covering for each month, has been altready
> reached.
> The problem of the demand covering is separated for each Prodoct, as you
> have observed.
> But the cost, I think, to be minimised, is the weighed average cost,
> considering all the Products.
> Am I wrong in this assumption?

I don't see why you want to do this. Is it linked to what follows?

>
> The problem, IMHO, is due to the fact that when I run the solver, the
demand
> is defined ONLY for the Current Year (2005).
> A lot of products bought during the 2005, will have the End of the hire
> period in 2006.
> But in 2006 I've no demand to satisfy (or better, we will have the 2006
> demand only in early autumn 2005), so that if I ask to the solver the best
> total cost (i.e. sum of Monthly Cost * Prod.qty * hire duration months),
> in my mind the solver, for those products ending in 2006, will try to
reduce
> the "hire duration months", rather then chosing a best Monthly Cost.
> But, now, while I'm writing, I think that the solution can be the
following:
> I've to calculate the Best Total Cost only for the whole 2005, not taking
> into account the hire period after the 2005.
> Total Cost = SUM(Monthly Cost * Product Qty * Hire Period); but the Hire
> period to be considered must be limited to Dec 2005.
> What do you think about that? Is it correct?

You are going through the "how to model" learning process at an accelerated
pace. Now you have come to the classic "what do I do about end effects?"
problem. It's the same in production planning, when the "best" thing to do
is to leave the system with no stock, as this minimizes costs. There is no
correct answer, but ignoring 2006 is probably a bad idea. Your suggestion
would lead to long hires going well into 2006, as they have the lowest cost
per period in 2005.
A conventional, easy idea is to duplicate the 2005 demand into 2006, on the
hypothesis that the best forecast of tomorrow is today.

> Regarding the book, does it contain also a sort of Tricks library (where
to
> find for example how to linearise some conditions)?

Yes.
But it doesn't have "all possible tricks". That would be too big a claim. A
lot, though.

> For me is very important to know all the possible Tricks, because really,
> since I'm a computer programmer, I did not spent a lot of time on the
> modelling (defining sets, variables, sum operations, an so on, are quite
> simple tasks), but I really spent a lot of time on the linearization of
sum
> constraints, on the transformation of some constraints into goal
objectives
> , and so on.
> So I really need a book that contains, besides a lto of example, also a
sort
> of reference about all the possible Tricks.
> For example, I needed to linearise the Absolute Value of a variable. I
> searched the Internet and I spent one day to find the simple (all is
simple
> when you know it) solution. If I had a good book with a a sort of "Tricks
> Reference", I think I would have saved a lot of time.
>
> thanks again
>
> --
>
> "Bob Daniel" wrote in message
> news:csld4v$p5$1$8302bc10@news.demon.co.uk...
> > From the incomplete info you have given I guess your problem is to
> minimise
> > the cost of covering demand for products p in month m. Your decision
> > variables are say h(p,m',t), the number of product p to hire in month m'
> for
> > a duration of t months.
> > I don't understand why these isn't a separate problem for each p. If the
> > rental cost depends on the total hired in month m' over all p, then of
> > course you just have 1 problem.
> > All (all!) you have to do is to get the right summations to find the
total
> > of each p you have in month m (clue: not those for which m'>m, not those
> > whose hire period has finished). We leave getting the summation limits
> right
> > as an exercise for you, AFTER you have used the internet to order a
book,
> > and got the italian post office to deliver it to you.
> > The ratio obj fn looks a (wrong) side issue.
> > Regards
> > Bob
> >
> > "JPC" wrote in message
> > news:csj04d$jdk$1@atlantis.cu.mi.it...
> > > Hi Bob,
> > >
> > > can you give me a little help with the modelling?
> > > I wrote the following (omitting the not important information):
> > >
> > > SETS:
> > > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD, Y
> > > !where the values of the variables MONTHLY_COST and RENT_PERIOD are
> loaded
> > > from a file, so they are frozen, while the QUANTITY and Y variables(I
> > > intoduced the Y variable following your help) must be calculated by
the
> > > solver;
> > >
> > > !Objective;
> > > MIN = SUM(PRODUCT: MONTHLY_COST * RENT_PERIOD * Y ) ;
> > >
> > > !Constraints:;
> > > 1= SUM(PRODUCT: RENT_PERIOD * Y);
> > >
> > > FOR (PRODUCT: Y = QUANTITY * D ); !--> this last condition makes my
> > problem
> > > not linear;
> > >
> > >
> > > How to write it in a linear way?
> > > thanks again, I promise that after this I will buy a good book!
> > > The problem is that is not easy to find a good book here in my little
> town
> > > in Italy...
> > >
> > > bye
> > > JPC
> > >
> > > --
> > >
> > > "Bob Daniel" wrote in message
> > > news:csilbe$1g8$1$8300dec7@news.demon.co.uk...
> > > > I do suggest you buy a book on MP modeling. I would suggest (but
then
> I
> > > > would)
> > > > Applications of optimization with Xpress-MP
> > > > Christelle Guéret, Christian Prins & Marc Sevaux
> > > > Translated and revised by Susanne Heipcke
> > > > Dash Optimization, 2002, ISBN 0-9543503-0-8
> > > >
> > > > I have put the section on ratio objective functions at
> > > > http://www.blisworthhouse.co.uk/OR/Bits_of_Book/ratio_obj_fn.pdf
> > > >
> > > > Regards
> > > > Bob Daniel
> > > >
> > > > "JPC" wrote in message
> > > > news:csh3t2$gus$1@atlantis.cu.mi.it...
> > > > >
> > > > >
> > > > > Hi to all,
> > > > > I'm contending with difficulties of modelling a problem.
> > > > >
> > > > > I've the following SET with the following properties:
> > > > >
> > > > > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD
> > > > >
> > > > > where :
> > > > > QUANTITY = the quantity that I have to rent (the solver will
> suggest
> > > how
> > > > > many of each product I will rent)
> > > > > Monthly Cost = the money I've to pay each month to rent this
> product,
> > > > this
> > > > > is a fixed amount for each product and period
> > > > > RENT_PERIOD = how many months I will use the product
> > > > >
> > > > > for example I can have:
> > > > >
> > > > > PRODUCT= PROD1, MONTHLY_COST = 100, RENT_PERIOD = 5 months
> > > > > PRODUCT= PROD1, MONTHLY_COST = 80, RENT_PERIOD = 6 months
> > > > > PRODUCT= PROD1, MONTHLY_COST = 75, RENT_PERIOD = 7 months
> > > > >
> > > > > PRODUCT= PROD2, MONTHLY_COST = 110, RENT_PERIOD = 5 months
> > > > > PRODUCT= PROD2, MONTHLY_COST = 120, RENT_PERIOD = 6 months
> > > > > PRODUCT= PROD2, MONTHLY_COST = 130, RENT_PERIOD = 7 months
> > > > >
> > > > >
> > > > > than, I have various constraint about the QUANTITY of each Product
> (I
> > > omit
> > > > > them, as they are not important here).
> > > > >
> > > > >
> > > > > than I've the Objective function that is:
> > > > >
> > > > >
> > > > > MIN = SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) /
> > SUM(PRODUCT:
> > > > > QUANTITY * RENT_PERIOD)
> > > > >
> > > > > i.e. I want to minimize the weighed average of the MONTHLY_COST.
> > > > >
> > > > > This objective will make my problem NOT LINEAR!!
> > > > >
> > > > > Any way/chance to transform it in a Linear problem?
> > > > >
> > > > > thanks a lot for any suggestion.
> > > > >
> > > > > JPC
> > > > >
> > > > >
> > > > >
> > > >
> > > >
> > >
> > >
> >
> >
>
>

From:	Bob Daniel
Subject:	Re: Linearization of a Weighed average
Date:	Wed, 19 Jan 2005 11:40:28 -0000

Example 10.6 of the book I suggested (see
http://www.dashoptimization.com/home/services/publications/applications_book_ov.html)
does nearly what you want.
Bob
"JPC" wrote in message
news:csj04d$jdk$1@atlantis.cu.mi.it...
> Hi Bob,
>
> can you give me a little help with the modelling?
> I wrote the following (omitting the not important information):
>
> SETS:
> PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD, Y
> !where the values of the variables MONTHLY_COST and RENT_PERIOD are loaded
> from a file, so they are frozen, while the QUANTITY and Y variables(I
> intoduced the Y variable following your help) must be calculated by the
> solver;
>
> !Objective;
> MIN = SUM(PRODUCT: MONTHLY_COST * RENT_PERIOD * Y ) ;
>
> !Constraints:;
> 1= SUM(PRODUCT: RENT_PERIOD * Y);
>
> FOR (PRODUCT: Y = QUANTITY * D ); !--> this last condition makes my
problem
> not linear;
>
>
> How to write it in a linear way?
> thanks again, I promise that after this I will buy a good book!
> The problem is that is not easy to find a good book here in my little town
> in Italy...
>
> bye
> JPC
>
> --
>
> "Bob Daniel" wrote in message
> news:csilbe$1g8$1$8300dec7@news.demon.co.uk...
> > I do suggest you buy a book on MP modeling. I would suggest (but then I
> > would)
> > Applications of optimization with Xpress-MP
> > Christelle Guéret, Christian Prins & Marc Sevaux
> > Translated and revised by Susanne Heipcke
> > Dash Optimization, 2002, ISBN 0-9543503-0-8
> >
> > I have put the section on ratio objective functions at
> > http://www.blisworthhouse.co.uk/OR/Bits_of_Book/ratio_obj_fn.pdf
> >
> > Regards
> > Bob Daniel
> >
> > "JPC" wrote in message
> > news:csh3t2$gus$1@atlantis.cu.mi.it...
> > >
> > >
> > > Hi to all,
> > > I'm contending with difficulties of modelling a problem.
> > >
> > > I've the following SET with the following properties:
> > >
> > > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD
> > >
> > > where :
> > > QUANTITY = the quantity that I have to rent (the solver will suggest
> how
> > > many of each product I will rent)
> > > Monthly Cost = the money I've to pay each month to rent this product,
> > this
> > > is a fixed amount for each product and period
> > > RENT_PERIOD = how many months I will use the product
> > >
> > > for example I can have:
> > >
> > > PRODUCT= PROD1, MONTHLY_COST = 100, RENT_PERIOD = 5 months
> > > PRODUCT= PROD1, MONTHLY_COST = 80, RENT_PERIOD = 6 months
> > > PRODUCT= PROD1, MONTHLY_COST = 75, RENT_PERIOD = 7 months
> > >
> > > PRODUCT= PROD2, MONTHLY_COST = 110, RENT_PERIOD = 5 months
> > > PRODUCT= PROD2, MONTHLY_COST = 120, RENT_PERIOD = 6 months
> > > PRODUCT= PROD2, MONTHLY_COST = 130, RENT_PERIOD = 7 months
> > >
> > >
> > > than, I have various constraint about the QUANTITY of each Product (I
> omit
> > > them, as they are not important here).
> > >
> > >
> > > than I've the Objective function that is:
> > >
> > >
> > > MIN = SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) /
SUM(PRODUCT:
> > > QUANTITY * RENT_PERIOD)
> > >
> > > i.e. I want to minimize the weighed average of the MONTHLY_COST.
> > >
> > > This objective will make my problem NOT LINEAR!!
> > >
> > > Any way/chance to transform it in a Linear problem?
> > >
> > > thanks a lot for any suggestion.
> > >
> > > JPC
> > >
> > >
> > >
> >
> >
>
>

From:	JPC
Subject:	Re: Linearization of a Weighed average
Date:	Wed, 19 Jan 2005 13:08:01 +0100

I'm ordering it!

thanks!

--

"Bob Daniel" wrote in message
news:cslh15$6sg$1$830fa795@news.demon.co.uk...
> Example 10.6 of the book I suggested (see
>
http://www.dashoptimization.com/home/services/publications/applications_book
_ov.html)
> does nearly what you want.
> Bob
> "JPC" wrote in message
> news:csj04d$jdk$1@atlantis.cu.mi.it...
> > Hi Bob,
> >
> > can you give me a little help with the modelling?
> > I wrote the following (omitting the not important information):
> >
> > SETS:
> > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD, Y
> > !where the values of the variables MONTHLY_COST and RENT_PERIOD are
loaded
> > from a file, so they are frozen, while the QUANTITY and Y variables(I
> > intoduced the Y variable following your help) must be calculated by the
> > solver;
> >
> > !Objective;
> > MIN = SUM(PRODUCT: MONTHLY_COST * RENT_PERIOD * Y ) ;
> >
> > !Constraints:;
> > 1= SUM(PRODUCT: RENT_PERIOD * Y);
> >
> > FOR (PRODUCT: Y = QUANTITY * D ); !--> this last condition makes my
> problem
> > not linear;
> >
> >
> > How to write it in a linear way?
> > thanks again, I promise that after this I will buy a good book!
> > The problem is that is not easy to find a good book here in my little
town
> > in Italy...
> >
> > bye
> > JPC
> >
> > --
> >
> > "Bob Daniel" wrote in message
> > news:csilbe$1g8$1$8300dec7@news.demon.co.uk...
> > > I do suggest you buy a book on MP modeling. I would suggest (but then
I
> > > would)
> > > Applications of optimization with Xpress-MP
> > > Christelle Guéret, Christian Prins & Marc Sevaux
> > > Translated and revised by Susanne Heipcke
> > > Dash Optimization, 2002, ISBN 0-9543503-0-8
> > >
> > > I have put the section on ratio objective functions at
> > > http://www.blisworthhouse.co.uk/OR/Bits_of_Book/ratio_obj_fn.pdf
> > >
> > > Regards
> > > Bob Daniel
> > >
> > > "JPC" wrote in message
> > > news:csh3t2$gus$1@atlantis.cu.mi.it...
> > > >
> > > >
> > > > Hi to all,
> > > > I'm contending with difficulties of modelling a problem.
> > > >
> > > > I've the following SET with the following properties:
> > > >
> > > > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD
> > > >
> > > > where :
> > > > QUANTITY = the quantity that I have to rent (the solver will
suggest
> > how
> > > > many of each product I will rent)
> > > > Monthly Cost = the money I've to pay each month to rent this
product,
> > > this
> > > > is a fixed amount for each product and period
> > > > RENT_PERIOD = how many months I will use the product
> > > >
> > > > for example I can have:
> > > >
> > > > PRODUCT= PROD1, MONTHLY_COST = 100, RENT_PERIOD = 5 months
> > > > PRODUCT= PROD1, MONTHLY_COST = 80, RENT_PERIOD = 6 months
> > > > PRODUCT= PROD1, MONTHLY_COST = 75, RENT_PERIOD = 7 months
> > > >
> > > > PRODUCT= PROD2, MONTHLY_COST = 110, RENT_PERIOD = 5 months
> > > > PRODUCT= PROD2, MONTHLY_COST = 120, RENT_PERIOD = 6 months
> > > > PRODUCT= PROD2, MONTHLY_COST = 130, RENT_PERIOD = 7 months
> > > >
> > > >
> > > > than, I have various constraint about the QUANTITY of each Product
(I
> > omit
> > > > them, as they are not important here).
> > > >
> > > >
> > > > than I've the Objective function that is:
> > > >
> > > >
> > > > MIN = SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) /
> SUM(PRODUCT:
> > > > QUANTITY * RENT_PERIOD)
> > > >
> > > > i.e. I want to minimize the weighed average of the MONTHLY_COST.
> > > >
> > > > This objective will make my problem NOT LINEAR!!
> > > >
> > > > Any way/chance to transform it in a Linear problem?
> > > >
> > > > thanks a lot for any suggestion.
> > > >
> > > > JPC
> > > >
> > > >
> > > >
> > >
> > >
> >
> >
>
>

From:	JPC
Subject:	Re: Linearization of a Weighed average
Date:	Tue, 18 Jan 2005 12:17:39 +0100

Thanks for the suggestions.

Just to be sure to have understood:

does this
guideline(http://www.blisworthhouse.co.uk/OR/Bits_of_Book/ratio_obj_fn.pdf)
make the solution a LP one? I've tried but the problem remain NLP (I'm using
lingo...).

bye
JPC

--

"Bob Daniel" wrote in message
news:csilbe$1g8$1$8300dec7@news.demon.co.uk...
> I do suggest you buy a book on MP modeling. I would suggest (but then I
> would)
> Applications of optimization with Xpress-MP
> Christelle Guéret, Christian Prins & Marc Sevaux
> Translated and revised by Susanne Heipcke
> Dash Optimization, 2002, ISBN 0-9543503-0-8
>
> I have put the section on ratio objective functions at
> http://www.blisworthhouse.co.uk/OR/Bits_of_Book/ratio_obj_fn.pdf
>
> Regards
> Bob Daniel
>
> "JPC" wrote in message
> news:csh3t2$gus$1@atlantis.cu.mi.it...
> >
> >
> > Hi to all,
> > I'm contending with difficulties of modelling a problem.
> >
> > I've the following SET with the following properties:
> >
> > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD
> >
> > where :
> > QUANTITY = the quantity that I have to rent (the solver will suggest
how
> > many of each product I will rent)
> > Monthly Cost = the money I've to pay each month to rent this product,
> this
> > is a fixed amount for each product and period
> > RENT_PERIOD = how many months I will use the product
> >
> > for example I can have:
> >
> > PRODUCT= PROD1, MONTHLY_COST = 100, RENT_PERIOD = 5 months
> > PRODUCT= PROD1, MONTHLY_COST = 80, RENT_PERIOD = 6 months
> > PRODUCT= PROD1, MONTHLY_COST = 75, RENT_PERIOD = 7 months
> >
> > PRODUCT= PROD2, MONTHLY_COST = 110, RENT_PERIOD = 5 months
> > PRODUCT= PROD2, MONTHLY_COST = 120, RENT_PERIOD = 6 months
> > PRODUCT= PROD2, MONTHLY_COST = 130, RENT_PERIOD = 7 months
> >
> >
> > than, I have various constraint about the QUANTITY of each Product (I
omit
> > them, as they are not important here).
> >
> >
> > than I've the Objective function that is:
> >
> >
> > MIN = SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) / SUM(PRODUCT:
> > QUANTITY * RENT_PERIOD)
> >
> > i.e. I want to minimize the weighed average of the MONTHLY_COST.
> >
> > This objective will make my problem NOT LINEAR!!
> >
> > Any way/chance to transform it in a Linear problem?
> >
> > thanks a lot for any suggestion.
> >
> > JPC
> >
> >
> >
>
>

From:	Steve
Subject:	Re: Linearization of a Weighed average
Date:	19 Jan 2005 04:06:04 -0800

Bob,

I think that's what I said. Even if JPC understands that, he still has
to work through the learning process of the model management system to
properly formulate the decision variables and constraints. And if the
problem ends up going IP and it's large, he'll have to tune the solver
too to ensure an acceptable run time, and not to mention managing data
I/O. Quite a chore for a novice.

So JPC, yes buy a book (www.amazon.com). If you have a short turn
around for this as well as other job responsibilities, you'll probably
have some trouble unless you get someone to help you (not just this
newsgroup.)

Steve

Bob Daniel wrote:
> From the incomplete info you have given I guess your problem is to
minimise
> the cost of covering demand for products p in month m. Your decision
> variables are say h(p,m',t), the number of product p to hire in month
m' for
> a duration of t months.
> I don't understand why these isn't a separate problem for each p. If
the
> rental cost depends on the total hired in month m' over all p, then
of
> course you just have 1 problem.
> All (all!) you have to do is to get the right summations to find the
total
> of each p you have in month m (clue: not those for which m'>m, not
those
> whose hire period has finished). We leave getting the summation
limits right
> as an exercise for you, AFTER you have used the internet to order a
book,
> and got the italian post office to deliver it to you.
> The ratio obj fn looks a (wrong) side issue.
> Regards
> Bob
>
> "JPC" wrote in message
> news:csj04d$jdk$1@atlantis.cu.mi.it...
> > Hi Bob,
> >
> > can you give me a little help with the modelling?
> > I wrote the following (omitting the not important information):
> >
> > SETS:
> > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD, Y
> > !where the values of the variables MONTHLY_COST and RENT_PERIOD are
loaded
> > from a file, so they are frozen, while the QUANTITY and Y
variables(I
> > intoduced the Y variable following your help) must be calculated by
the
> > solver;
> >
> > !Objective;
> > MIN =3D SUM(PRODUCT: MONTHLY_COST * RENT_PERIOD * Y ) ;
> >
> > !Constraints:;
> > 1=3D SUM(PRODUCT: RENT_PERIOD * Y);
> >
> > FOR (PRODUCT: Y =3D QUANTITY * D ); !--> this last condition makes my
> problem
> > not linear;
> >
> >
> > How to write it in a linear way?
> > thanks again, I promise that after this I will buy a good book!
> > The problem is that is not easy to find a good book here in my
little town
> > in Italy...
> >
> > bye
> > JPC
> >
> > --
> >
> > "Bob Daniel" wrote in message
> > news:csilbe$1g8$1$8300dec7@news.demon.co.uk...
> > > I do suggest you buy a book on MP modeling. I would suggest (but
then I
> > > would)
> > > Applications of optimization with Xpress-MP
> > > Christelle Gu=E9ret, Christian Prins & Marc Sevaux
> > > Translated and revised by Susanne Heipcke
> > > Dash Optimization, 2002, ISBN 0-9543503-0-8
> > >
> > > I have put the section on ratio objective functions at
> > > http://www.blisworthhouse.co.uk/OR/Bits_of_Book/ratio_obj_fn.pdf
> > >
> > > Regards
> > > Bob Daniel
> > >
> > > "JPC" wrote in message
> > > news:csh3t2$gus$1@atlantis.cu.mi.it...
> > > >
> > > >
> > > > Hi to all,
> > > > I'm contending with difficulties of modelling a problem.
> > > >
> > > > I've the following SET with the following properties:
> > > >
> > > > PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD
> > > >
> > > > where :
> > > > QUANTITY =3D the quantity that I have to rent (the solver will
suggest
> > how
> > > > many of each product I will rent)
> > > > Monthly Cost =3D the money I've to pay each month to rent this
product,
> > > this
> > > > is a fixed amount for each product and period
> > > > RENT_PERIOD =3D how many months I will use the product
> > > >
> > > > for example I can have:
> > > >
> > > > PRODUCT=3D PROD1, MONTHLY_COST =3D 100, RENT_PERIOD =3D 5 months
> > > > PRODUCT=3D PROD1, MONTHLY_COST =3D 80, RENT_PERIOD =3D 6 months
> > > > PRODUCT=3D PROD1, MONTHLY_COST =3D 75, RENT_PERIOD =3D 7 months
> > > >
> > > > PRODUCT=3D PROD2, MONTHLY_COST =3D 110, RENT_PERIOD =3D 5 months
> > > > PRODUCT=3D PROD2, MONTHLY_COST =3D 120, RENT_PERIOD =3D 6 months
> > > > PRODUCT=3D PROD2, MONTHLY_COST =3D 130, RENT_PERIOD =3D 7 months
> > > >
> > > >
> > > > than, I have various constraint about the QUANTITY of each
Product (I
> > omit
> > > > them, as they are not important here).
> > > >
> > > >
> > > > than I've the Objective function that is:
> > > >
> > > >
> > > > MIN =3D SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) /
> SUM(PRODUCT:
> > > > QUANTITY * RENT_PERIOD)
> > > >
> > > > i.e. I want to minimize the weighed average of the
MONTHLY_COST.
> > > >
> > > > This objective will make my problem NOT LINEAR!!
> > > >
> > > > Any way/chance to transform it in a Linear problem?
> > > >
> > > > thanks a lot for any suggestion.
> > > >
> > > > JPC
> > > >
> > > >
> > > >
> > >
> > >
> >
> >

From:	Steve
Subject:	Re: Linearization of a Weighed average
Date:	18 Jan 2005 05:57:39 -0800

JPC,

These problems always turn out more complicated than they initially
seem. It appears that you have to create a set of integer variables
that are indexed on the PC type (i), the plan type (Rent Period) (j)
and the month that you the lease starts (k). For example Var P1,2,4
would represent leasing PC of type 1 using Lease Plan 2 (6 months),
starting in month 4 (April).

So you would have many variables in order to cover all of the PC type,
plan type, month start combinations. You would then create constraints
by summing the feasible integer variables for each month such that
demand is met. You also have to establish your internal policy for
returning PC's early. It may be cost effective to rent using a 7
month agreement in July and return the machines in December (if you
have a 12 month planning cycle.) And another policy to consider is
whether you can buy multiple leases of the same PC type with the same
or different plans starting on different months. If that is a policy,
then you have a pretty complicated formulation indeed. But it should
be workable with an algebraic modeling system. But you have to be very
careful when you debug to ensure that the proper decision variables and
constraints are being generated.

So I think I see what you want to do. I still do not think that
weighted averaging is correct. Minimize total cost using the
methodology above and it should work for you.

Steve

From:	JPC
Subject:	Re: Linearization of a Weighed average
Date:	Tue, 18 Jan 2005 15:28:42 +0100

Steve,

yes I think you have understood.
For a moment suppose that that we must start the renting in january
2005(now).
We can omit the Starting Rental Month, ok?

My previous SET:
PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD

describes exactly the situation: we have a PC_TYPE (product), and for each
PC type we have different possible RENTAL_PERIOD(months of rent), and
different MONTHLY_COST.
So I have a (very large) matrix of possible combination and the solver
should set the Qty of each possible combination.

In reality my optimisation function is composed, besides a lot of goal
constraints, of two functions:

1) a function that will optimise the monthly demand
2) a function that will optimise the cost.

This 2nd function is a weighed average, because, otherwise, if I consider
the total cost (i.e. QTY * MONTHLY_COST * RENT_PERIOD), then the RENT_PERIOD
will have an important weight in the formula (i.e. the solver will obtain
better solution both using lower Monthly_Cost and/or lower/shorter
RENT_Period).

If I consider only the QTY * MONTHLY_COST, then again the solver can chose
lower qty, instead of chosing the best COST.
Consider that the first function can be optimised in a very different way,
so that many possible combination of the QTY of each row can lead to the
same best optimal Function1 result.

I'm pretty sure that the weighed average IS the solution.

Any way to make it a linear function?

thanks

"Steve" wrote in message
news:1106056659.910066.247610@z14g2000cwz.googlegroups.com...
> JPC,
>
> These problems always turn out more complicated than they initially
> seem. It appears that you have to create a set of integer variables
> that are indexed on the PC type (i), the plan type (Rent Period) (j)
> and the month that you the lease starts (k). For example Var P1,2,4
> would represent leasing PC of type 1 using Lease Plan 2 (6 months),
> starting in month 4 (April).
>
> So you would have many variables in order to cover all of the PC type,
> plan type, month start combinations. You would then create constraints
> by summing the feasible integer variables for each month such that
> demand is met. You also have to establish your internal policy for
> returning PC's early. It may be cost effective to rent using a 7
> month agreement in July and return the machines in December (if you
> have a 12 month planning cycle.) And another policy to consider is
> whether you can buy multiple leases of the same PC type with the same
> or different plans starting on different months. If that is a policy,
> then you have a pretty complicated formulation indeed. But it should
> be workable with an algebraic modeling system. But you have to be very
> careful when you debug to ensure that the proper decision variables and
> constraints are being generated.
>
> So I think I see what you want to do. I still do not think that
> weighted averaging is correct. Minimize total cost using the
> methodology above and it should work for you.
>
> Steve
>

From:	Steve
Subject:	Re: Linearization of a Weighed average
Date:	18 Jan 2005 04:58:49 -0800

JPC,

I'm not really sure what you are trying to do here. But I think that
you may be missing at least one index (Month) in your model. It looks
like you have a set covering problem. If you define and index your
decision variables correctly, you should not need the denominator in
the objective function. You will simply minimize cost. The demand and
resource constraints would be handled as just that - constraints.

Agree with Bob Daniel. Suggest you find a book or web page that deals
with linear programming applications or else hire someone to assist
you. BTW, LINDO has an applications library on their web site:
http://www.lindo.com/ If you bought LINGO from them, they may provide
modeling support.

Steve

From:	JPC
Subject:	Re: Linearization of a Weighed average
Date:	Tue, 18 Jan 2005 14:27:50 +0100

Hi Steve,
I've omitted a lot of index and set here.
Of course I've also Months and a lot of other indexes.

But what is important for me now, is to know if a weighed average can be
linearized or not.

In any case, if you can kindly help me, with the following little model I
want to resolve this simple problem:
given a set of PRODUCTS(Personal Computers), my company needs to LEASE /
RENT, now in january 2005, some Qty of these products(PC), then we will use
them for a certain Period (i.e. for a certain number of months), then we
will return these PCs to the suppliers.
Depending on the PRODUCT (PC type), and depending on the Usage Period
(RENT_PERIOD), we have to pay a MONTHLY_COST for each PC.
At this point we want to be sure to pay the lowest weighed average
MONTHLY_COST.

I've translated the problem in:

SETS:
PRODUCT: QUANTITY, MONTHLY_COST, RENT_PERIOD
where :

QUANTITY = the quantity that I have to rent (the solver will suggest
how
many of each product I will rent), this is the unique variable that must
be calculated by the solver.

MONTHLY_COST = the money I've to pay each month to use each product type,
this is a fixed amount for each row/combination below

RENT_PERIOD = how many months I will use the product, this is a fixed
number for each row

for example I can have:

PRODUCT= PC_TYPE_1 , MONTHLY_COST = 100, RENT_PERIOD = 5 months
PRODUCT= PC_TYPE_1 , MONTHLY_COST = 80, RENT_PERIOD = 6 months
PRODUCT= PC_TYPE_1 , MONTHLY_COST = 75, RENT_PERIOD = 7 months

PRODUCT= PC_TYPE_2 , MONTHLY_COST = 110, RENT_PERIOD = 5 months
PRODUCT= PC_TYPE_2 , MONTHLY_COST = 120, RENT_PERIOD = 6 months
PRODUCT= PC_TYPE_2 , MONTHLY_COST = 130, RENT_PERIOD = 7 months

than, I have various constraint about the QUANTITY of each Product (I
omit
them, as they are not important here).

than I've the Objective function that is:

MIN = SUM(PRODUCT: QUANTITY * MONTHLY_COST * RENT_PERIOD) / SUM(PRODUCT:
QUANTITY * RENT_PERIOD)

In other words, the solver should always try to give priority to the rows
having the lowest MONTHLY_COST.

Hoping I've been clear, and that you have the patience to help me again.

thanks a lot.
JPC