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Fix a quantity that is data in one model and a decision in another#

Write one spec in which a quantity, such as a plant's size, is chosen by the solver in one study and given by the data in another. There are two ways:

  • Pin it in the data where the solver may keep the quantity as a variable. The file and the call stay the same, and equal bounds hold the variable at one value.
  • Fix it with spec.fix where the model must not have the variable: a Benders subproblem, a myopic step or a rolling window. The quantity becomes a parameter, so size * on is linear and size may stand in a bound.

Equal bounds in the data#

  1. Declare the quantity as a variable, with named bounds.
dimensions:
  plant: { dtype: str }
parameters:
  size_min: { dims: [plant] }
  size_max: { dims: [plant] }
variables:
  size:
    dims: [plant]
    bounds: { lower: size_min, upper: size_max }

  1. Write every rule against the variable. rate - relmax * size <= 0 is one equation whether size is chosen or given.

  2. Pin it in the data where it is given. Attach size_min and size_max as the same value for a plant whose size is fixed. Equal bounds pin a variable (variables).

A pinned variable is still a variable: size * on is variable * variable, and size cannot stand in another variable's bounds:. Where a bound has to come from it, fix it with spec.fix.

spec.fix#

  1. Write the spec with the quantity as a variable. Here size exists only for a candidate plant:
version: 0
dimensions:
  plant: { dtype: str }
  snapshot: { dtype: int }
parameters:
  size_min: { dims: [plant] }
  size_max: { dims: [plant] }
  relmax: { dims: [snapshot, plant] }
  demand: { dims: [snapshot] }
  invest: { dims: [plant] }
  fuel: { dims: [plant] }
  candidate: { dims: [plant], dtype: bool }
variables:
  size:
    dims: [plant]
    where: candidate
    bounds: { lower: size_min, upper: size_max }
  rate:
    dims: [snapshot, plant]
    bounds: { lower: 0 }
constraints:
  capacity:
    dims: [snapshot, plant]
    expression: rate - relmax * size <= 0
  supply:
    dims: [snapshot]
    expression: sum(rate, over=plant) >= demand
objective:
  sense: minimize
  expression: sum(invest * size) + sum(fuel * rate)
  1. Call spec.fix with every name to fix, in one call.
import mathspec as ms

subproblem = ms.to_spec('plant.yaml').fix('size')

The spec it returns differs from the file in two places:

parameters:
  size: { dims: [plant], dtype: float, missing: absent }
assumptions:
  size_within_bounds:
    holds: size >= size_min AND size <= size_max
    where: candidate
  • size is a parameter under the same name, so every expression goes on reading it. A binary or integer variable becomes an int parameter.
  • size keeps what it meant outside its mask. The variable did not exist outside candidate, so capacity was not built there. The parameter is missing: absent, so a plant with no row is still not built, and every row stays as written. A variable with no where: becomes a parameter that needs every row.
  • The bounds become an assumption on the numbers you attach.
  • A constraint that named only fixed variables becomes an assumption under its own name, because it now compares numbers.
  1. Attach size as data, one value for each candidate plant, as for any parameter.