Re: [Help-glpk] Linear programming with endogenous constraints

[glpk xypron]

Hello Dan,

when using GLPK it is not necessary to start programming in 
Python or any other programming language. GLPK comes with the
standalone solver glpsol. You can formulate your linear problem
in the math programming language GMPL described in doc/gmpl.pdf.

With GMPL you can write your constraints very much like you formulate
them in a mathematical equation.

See the examples provided with the GLPK distribution.

Resorting to programming is necessary for implementation
of heuristics like column generation or for tight integration
into other software.

GLPK can be directly used together with C. Language bindings
for several other languages exist. For Python have a look at
http://www.dcc.fc.up.pt/~jpp/code/python-glpk/

> how to develop variable bounds that are
> endogenous
A constraint may contain multiple variables, e.g.

production - capacity <= 0

Best regards

Xypron

-------- Original-Nachricht --------
> Datum: Thu, 13 May 2010 17:46:10 -0700
> Betreff: [Help-glpk] Linear programming with endogenous constraints

> 
> I have been learning about linear programming
> using python and GLPK and was hoping that someone might be able to point
> me
> towards a reference to learn how to develop variable bounds that are
> endogenous
> to other structural variables (preferably using these programs, not only
> am I new to linear programming, but I am relatively new to computer
> programming)?  I am initially interested in cases where the endogenous 
> relationship
> is between structural variables.  I am able to write the code for
> constraints when
> they are exogenous, but am having trouble writing the code for a
> constraint
> when a structural variable is limited by another structural variable.  If
> it is helpful, I have included a simplified
> example below (not code, just the concept).  
> 
>  
> 
> The example I am working with is to minimize
> the costs associated with production of widgets over two years, assuming
> that
> you can choose to produce from 2 factories with costs (capital and
> operating) that
> decrease through time.  Both capital and
> operating costs increase linearly with production (cost per unit * number
> of
> units = total cost).  At the start,
> neither factor exists, so you need to add capacity to meet widget demand. 
> Each factory has a total production capacity
> limit and you can add capacity at different times.  For instance, you can
> build 40% of total
> capacity in year 1, and then the remaining 60% in year 2.  Since
> capital/operating costs decline in year
> 2, you build the remaining 60% at lower cost (operating costs are related
> to
> the year of construction, so the operating costs in year 2 will be
> different
> for the 40% and 60%). In year 2, you can also decide to back-off
> production at
> the 40% of the factory built in year 1 and produce only from the 60% built
> in
> year 2 (and thus not incur the operating costs from the 40%).  However,
> the capital costs are amortized, so
> if you decide to build anything in the first year, you will incur capital
> costs
> in the second year.  
> 
>  
> 
> How you allocate the operating costs in year 2
> therefore depends on your decisions in year 1. 
> The maximum production from capacity added in the first year is limited
> by the production capacity installed in year 1. 
> So my minimization function looks something like (for brevity, I wrote
> the equation for only 1 factory and only for year 2):
> 
>  
> 
> Minimize:
> 
> TC2 = X1*Cap1 + X1y2*OP1 y2    +  X2*(Cap2
> + OP2)
> 
>  
> 
> Subject
> to:
> 
> P2 = X1y2 + X2    (Total production in year 2 is production of widgets
> using capacity installed in year ) 0 <= P2 <= D2 (Total widgets produced
> in year 2 must equal demand in year 2, where demand is exogenously
> determined)0 <= X1y2 + X2 <= C2 (Total production
> must be less than factor capacity)0<= X1 y2 < = X1Production in year 2
> from capacity added in
> year 1 is limited by capacity added in year 1 (where the capacity added in
> year
> is equal to the number of widgets produced)
> Variable Descriptions:
> 
> P2
>             = Total widget production
> 
> D2
>            = Widget demand in year 2
> 
> C2
>            = Production capacity in year 2
> 
> X1
>             =
> # of widgets produced using capacity installed in year 1
> 
> Cap1
>              =
> Capital costs associated with capacity installed in year 1 
> 
> OP1 y2        = Operating costs associated with capacity
> installed in year 1
> 
> X2
>             = # of widgets produced using
> capacity installed in year 2
> 
> Cap1
>              =
> Capital costs associated with capacity installed in year 2 
> 
> OP2           = Operating costs associated with capacity
> installed in year 2                                     
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