[Help-glpk] Linear programming with endogenous constraints

[Dan bole]

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 + X1_y2*OP1 _y2    +  X2*(Cap2 + OP2)

 

Subject to:

1. P2 = X1_y2 + X2    (Total production in year 2 is production of widgets using capacity installed in year )
2.  0 <= P2 <= D2 (Total widgets produced in year 2 must equal demand in year 2, where demand is exogenously determined)
3. 0 <= X1_y2 + X2 <= C2 (Total production must be less than factor capacity)
4. 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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