1. 2-1 Scheduling Constraints

2. 2-2 Outline Activities Temporal constraints Resources Resource constraints (mono-activity) Resource constraints (two activities)

3. 2-3 Activities Interval (block) activities Splittable activities (with interruption cost?) A time A AAA

4. 2-4 Temporal constraints Earliest and latest start and end times Fixed or variable durations Precedence constraints Precedence constraints with minimal delays Precedence constraints with fixed delays Maximal delays

5. 2-5 Resources Unary resources Discrete resources State resources Energetic resources

6. 2-6 Unary resources Main resources in 4/10 problems Example: one individual machine or person AB C time

7. 2-7 Discrete resources Main resources in 5/10 problems Example: group of identical machines A B D time C

8. 2-8 State resources Main resources in 1/20 problems Example: oven with different temperatures D time A B C

9. 2-9 Energetic resources Main resources in 1/20 problems Example: number of man-days per week AB time C

10. 2-10 Resource constraints Mono-activity –Resource requirement –Resource provision –Resource consumption/production –Periods during which a resource is not (or not fully) available –Optional or variable requirements –Transition times

11. 2-11 Resource requirement Unary resource: A requires R Discrete resource: A requires c units of R State resource: A requires R... –in a given state s –in any of a given set of states {s 1... s n } –not in a given state s –not in any of a given set of states {s 1... s n } Energetic resource: A requires w units of R per time period (e.g., 4 man-months per month)

12. 2-12 Resource provision Similar to resource requirement when part of the problem is to make the resources available for the requiring activities The required capacity can never exceed the provided capacity

13. 2-13 Resource consumption Unary resource: A consumes R Discrete resource: A consumes c units of R The consumed capacity is no longer available for any other activity The resource is required from the start time of the activity up to the end of the schedule

14. 2-14 Resource production Unary resource: A produces R Discrete resource: A produces c units of R The produced capacity becomes available when the activity ends The resource is provided from the end time of the activity up to the end of the schedule

15. 2-15 Capacity constraints Periods during which a resource is not fully available –Maintenance periods –Vacations –Forbidden states (at night) Periods during which some minimal amount must be required or provided

16. 2-16 Optional or variable requirements Optional activities –Resource alternatives –Sub-contracts

17. 2-17 Optional or variable requirements Time versus capacity tradeoffs 2 people during 3 days or 3 people during 2 days

18. 2-18 Optional or variable requirements Variable requirement over time Example: 8 person-days with either 2 or 3 people at any time

19. 2-19 Transition times Tool setup between two tasks on the same machine State change (oven temperature, color used in a painting shop) Cleaning

20. 2-20 Resource constraints Two activities –Percentage constraints –Synchronization constraints

21. 2-21 Percentage constraints  % of activity A on resource R A must be done before (or after): – B starts – B ends –  % of activity B on resource R B are done Often complements variable requirements over time

22. 2-22 Synchronization constraints When A executes, B requires (at least, at most) c 1 units of resource R When A does not execute, B requires (at least, at most) c 2 units of resource R