Models for Global Constraint Applications - Cork Constraint ...

The first task on machine k is given by variable msucck, and the last task
by mpred k.
Machine dependent duration and machine preference are given extensionally
by functions Di : M ↦→ N and Pi : M ↦→ N. The function Si : I ↦→ N gives the
setup time of task i as a function of the successor variable.
The release date reli is a constant which imposes a hard constraint, since it
is imposed by the availability of raw materials. The due date value duei does
not create a hard constraint, but becomes part of the objective function. The
variable deli is used to indicate the delay of a task.
We first define the domains of the overall schedule ends:
maxm ∈ M (10)
maxe ∈ 0..end (11)
We then have a number of primitive constraints which link the variables of
a task:
∀i∈I : [si, ei, li] ∈ 0..end (12)
∀i∈I : ei = si + di
∀i∈I : li = si + wi
∀i∈I : wi = di + setup i
(13)
(14)
(15)
∀i∈I : element(mi, Di, di) (16)
∀i∈I : element(mi, Pi, pi) (17)
∀i∈I : element(succi, Si,setup i) (18)
∀i∈I : pi ≥ pref min
(19)
Equations 19 are used to impose some global preference rule for all tasks. If the
constant pref min is set to a high value, then tasks will be only considered on their
preferred machines, if it is set low, then tasks can move to more machines. This
gives a simple mechanism to switch from normal operations to “fire fighting”
mode, when optimisation criteria are relaxed in order to deliver products on
time.
The links between start, end, release and due dates are given in the next set
of equations.
∀i∈I : si ≥ reli
∀i∈I : del i ≥ ei − duei
∀i∈I : deli ≤ delay max
(20)
(21)
(22)
Equations 22 allows to control the maximal delay allowed on any task. In
particular, it allows to set total delay to zero, forcing a solution which delivers
all products on time.
3.2.2 Global Constraints
We now introduce the main constraint which deals with the machine assignment,
and which ensures that tasks are not overlapping on machines. This again uses
the diffn [13] constraint already presented in section 2.2.3.
diffn({〈si, mi, wi, 1〉|i ∈ I}) (23)
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