1. 6-1 Temporal Constraint Propagation (Preemptive Case)

2. 6-2 Outline New variables –Definition –Implementation Relations between the variables Temporal constraints

3. 6-3 New variables (definition) set(A)  {t such that A executes at time t} W A (t)  1 when t  set(A), 0 otherwise start(A)  min t  set(A) (t) end(A)  max t  set(A) (t + 1) duration(A)  |set(A)| span(A)  end(A)  start(A)

4. 6-4 New variables (implementation) Three possible implementations for set(A) –Explicit set variable set(A) –Explicit Boolean variables W A (t) –Dynamic list of intervals I i (A)  [s i (A), e i (A)) with W(I i (A))  1 if  t  [s i (A), e i (A)), t  set(A) W(I i (A))  0 if  t  [s i (A), e i (A)), t  set(A) W(I i (A))  unknown otherwise Explicit or implicit integer variables for start(A), end(A), duration(A), and span(A)

5. 6-5 Relations between the variables end(A)  start(A)  span(A) duration(A)  span(A) duration(A)  |set(A)| cardinality constraint specific implementation for a list of intervals

6. 6-6 Relations between the variables start(A)  min t  set(A) (t) start(A)  set(A) [t  start min (A)  start max (A)] implies [t  set(A)] [t  start min (A)  set(A)] implies [t  start(A)] [t  start max (A)  set(A)] implies [start(A)  t]  t  set(A), start(A)  t [t  set(A)] implies [start(A)  t] [t  start min (A)] implies [t  set(A)]

7. 6-7 Relations between the variables end(A)  max t  set(A) (t  1) (end(A)  1)  set(A) [t  end min (A)  end max (A)] implies [(t  1)  set(A)] [t  (end min (A)  1)  set(A)] implies [(t  1)  end(A)] [t  end max (A)  1)  set(A)] implies [end(A)  (t  1)]  t  set(A), t  end(A) [t  set(A)] implies [t  end(A)] [end max (A)  t] implies [t  set(A)]

8. 6-8 Relation between the variables pos(A) = {t such that W A (t) can be 1} |{t'  pos(A) such that t'  t}|  duration min (A) implies [t  end(A)] |{t'  pos(A) such that t  t'}|  duration min (A) implies [start(A)  t]

9. 6-9 Temporal constraints Constraints between start and end variables Similar to the non-preemptive case when start(A) and end(A) are explicit Other constraints  t  set(A), t  set(B) (inclusion)  t  set(A), t  set(B) (exclusion)  t  set(A), t  set(B) (coverage)