Logic constructs

Mosel provides the type logctr for defining and working with logic constraints in MIP models. The implementation of these constraints is based on indicator constraints. Logic constraints are composed with linear constraints using the operations and, or, xor, implies, and not as shown in the following example. Mosel models using logic constraints must include the package advmod instead of the Xpress Optimizer library mmxprs.

 uses "advmod"

! **** 'implies', 'not', and 'and' ****
 declarations
  R = 1..3
  C: array(range) of linctr
  x: array(R) of mpvar
 end-declarations

 C(1):= x(1)>=10
 C(2):= x(2)<=5
 C(3):= x(1)+x(2)>=12

 implies(C(1), C(3) and not C(2))
 forall(j in 1..3) C(j):=0              ! Delete the auxiliary constraints

! Same as:
 implies(x(1)>=10, x(1)+x(2)>=12 and not x(2)<=5)

! **** 'or' and 'xor' ****
 declarations
  p: array(1..6) of mpvar
 end-declarations

 forall(i in 1..6) p(i) is_binary

! Choose at least one of projects 1,2,3 (option A)
! or at least two of projects 2,4,5,6 (option B)
  p(1) + p(2) + x(3) >= 1 or p(2) + p(4) + p(5) + p(6) >= 2

! Choose either option A or option B, but not both
  xor(p(1) + p(2) + p(3) >= 1, x(2) + p(4) + p(5) + p(6) >= 2)

These logic constructs, particularly the logic or, can be used for the formulation of minimum or maximum values of a set of variables and also for absolute values:

• Minimum values: y = min{x₁, x₂, ..., x_n} for continuous variables x₁, ..., x_n
  • Logic formulation:

    y ≤ xi ∀ i=1,...,n

    y ≥ x1 or ... or y ≥ xn

• Maximum values: y = max{x₁, x₂, ..., x_n} for continuous variables x₁, ..., x_n
  • Logic formulation:

    y ≥ xi   ∀ i=1,...,n

    y ≤ x1 or ... or y ≤ xn

• Absolute values: y = | x₁ - x₂| for two variables x₁, x₂
  • Modeling y = | x₁ - x₂| is equivalent to y = max{ x₁ - x₂, x₂ - x₁ }
  • Logic formulation:

    y ≥ x1 - x2

    y ≥ x2 - x1

    y ≤ x1 - x2 or y ≤ x2 - x1

• Example implementation with Mosel:

 declarations
  x: array(1..2) of mpvar
  y, u, v: mpvar
  C1, C2: linctr
  C3: logctr
 end-declarations

! Formulation of y = min{x(1), x(2)}
 C1:= y <= x(1)
 C2:= y <= x(2)
 C3:= y >= x(1) or y >= x(2)

! Formulation of u = max{x(1), x(2)}
 C1:= u >= x(1)
 C2:= u >= x(2)
 C3:= u <= x(1) or u <= x(2)

! Formulation of v = |x(1) - x(2)|
 C1:= v >= x(1) - x(2)
 C2:= v >= x(2) - x(1)
 C3:= v <= x(1) - x(2) or v <= x(2) - x(1)