1. Contents College 4
§4.1, §4.2, §4.4, §4.6
Extra literature on resource constrained project scheduling (will be handed out)

2. Project Scheduling Project definition:
A complex and large scale one-of-a-kind product or service, made up by a number of component activities (jobs), that entails a considerable financial effort and must be time-phased, i.e. scheduled, according to specified precedence and resource requirements (Hax and Candea, 1984)

3. Project properties
Project goals: quality, time, costs, customer satisfaction
Network of activities/jobs
Limited resource capacity
Project life-cycle:
Order acceptance
Engineering and process planning
Material and resource scheduling
Project execution
Evaluation & service

4. Project examples Construction Production Management Research
Maintenance
Installation, implementation

5. Hierarchical planning
Strategic resource planning
Strategic
Rough-cut process planning
Rough-cut capacity planning
Tactical
Engineering & process planning
Project scheduling
Tactical/ operational
RCCP: Order acceptance
Detailed scheduling: van de individuele mensen
Detailed scheduling
Operational

6. Project breakdown structure
Main Activity
Main Activity
Main Activity
RCCP
Activity
Activity
Activity
Activity
Activity
Activity
Project
Scheduling
Main Activity = werkpakket: RCCP
Activity: Project scheduling
RCCP: grove werkvoorbereiding
Project scheduling: gedetailleerde werkvoorbereiding

7. Project scheduling topics
Project representation (precedence graph)
Critical Path Method (CPM)
Resource-constrained project scheduling
standard problem
methods
extensions

8. Project representation / precedence graph
“job on node”-representation: 1 2 4 6 3 5 7
Rules for “job on node” networks:
network contains no directed cycles
event numbering
network contains no redundant arcs

9. Project representation example 1
“job on node”-representation: 1 2 4 6 3 5 7 1 2 4 6
JON:
knoop = job
pijl = relatie
JOA:
knoop = begin of einde van een job
pijl = job
kenmerken JOA:
geen overbodige pijlen
geen cykels
lastig
niet unieke netwerken
(alleen interpretatie, niet afleiding??)
LINEAIRE vs. NETWERK precedentie relaties
“job on arc”-representation: 3 7 5

10. Project representation example 2
“job on node”-representation: 1 2 4 6 3 5
“job on arc”-representation: 6 1 4 2 5 3

11. Project scheduling Without resource constraints relatively easy
With resource constraints very complex:
when jobs share resources with limited availability, these jobs cannot be processed simultaneously  draw disjunctive arcs
Example 4.6.1: 1 4 2 5 3

12. Project scheduling without resource constraints: critical path method (CPM)
Critical job: job without slack
Critical path = chain of critical jobs
Forward procedure
Backward procedure
Slack = speling

13. Project scheduling without resource constraints: critical path method (CPM)
Critical path method initialization:
determine earliest starting time for all jobs
determine latest completion time for all jobs
determine which jobs have no slack
2 CPM solution methods:
Forward procedure
Backward procedure

14. Critical path method Forward CPM procedure Backward CPM procedure
STEP1: For each job that has no predecessors:
STEP2: compute for each job j:
STEP3:
STEP1: For each job that has no successors:
STEP2: compute for each job j:
STEP3: Verify that:
Bij AON netwerk kun je de jobs op volgorde afwerken

15. Critical path method example
Sink
(project start) 2 1 3 4 1
Earliest completion time = earliest starting time + p(j) !!! S 2 4 6 T
Source
(project start) 1 2 3 5

16. Critical path method example (cont.)
Critical job:
C’’ = C’ = S’+p 2 1 3 3 4 1
Legend:
Earliest completion time = earliest starting time + p(j) !!! S 2 4 6 T
duration 3 3 7 7 8 8 8 j 1 2 S’
C’’ 3 5 6 3 8

17. Resource constraints Suppose jobs require a resource:
resource requirements 6 1 2 3 4 5 6 5 4 3 2 1 1 2 3 4 5 6 7 8

18. Resource constraints (cont.)
Suppose :
Cmax increases by 2 6 5 4 2 3 1 5 6 2 3 4 1 1 2 3 4 5 6 7 8 9 10

19. Resource-Constrained Project Scheduling Problem (RCPSP)
n jobs j=1,…,n
N resources i=1,…,N
Rk: availability of resource k
pj: duration of job j
Rkj:requirement of resource k for job j
Pj: (immediate) predecessors of job j

20. RCPSP Goal: minimize makespan: Restrictions:
no job may start before T=0
precedence relations
finite resource capacity

21. RCPSP example 4 1 2 6 5 2 3 4
Op ieder moment voor alle resources genoeg beschikbaar 2 4 6 8 10 12 2 1 3 5 2 4 6 2 4 6 8 10 12

22. Disjunctive arcs Suppose R1=4. These jobs cannot be performed
simultaneously:
job 1 & 3
job 3 & 6
job 4 & 5
job 5 & 6 1
CPM werkt nu niet meer! 2 4 6
disjunctive arcs 3 5

23. Priority-rule-based scheduling
Generation scheme
serial
parallel
Priority rule
latest finish time
minimum slack
Sampling procedure
Generation scheme: hoe bouw ik een schedule op
Priority: welke job ga ik aan het schedule toevoegen

24. Serial scheduling method
Each stage represents a job  n stages
completed set of jobs: scheduled jobs
decision set: jobs of which all predecessors have been scheduled
remaining set: other jobs
procedure:
1. Start with an empty schedule
2. Select job from decision set with highest priority, and schedule it as early as possible
3. Repeat step 2 if the decision set is not empty

25. Serial scheduling method example (1)
Decision set 2 1 3
Prioriteiten zijn gegeven (niet volgens bekende prioriteitsregel) 2 2 4 6 8

26. Serial scheduling method example (2)
Decision set 2 1 3 1 2 2 4 6 8

27. Serial scheduling method example (3)
Decision set 2 1 3 3 1 2 2 4 6 8

28. Serial scheduling method example (4)2 1 3 3 1 2 2 2 4 6 8

29. Parallel scheduling method
At most n stages
Each stage n represents:
1. partial schedule
2. schedule time tn
3. four disjoint sets of jobs:
completed set: scheduled jobs, completed at tn
active set: scheduled jobs, not completed yet
decision set: all unscheduled jobs, that could be scheduled
remaining set: all unscheduled jobs, that cannot be scheduled

30. Parallel scheduling method
procedure:
1. Start with an empty schedule.
2. Let T be the first time in which an unscheduled job may start. Let D be the collection of jobs that may be started on T, and of which all predecessors are scheduled
3. Select the job from D with the highest priority, and schedule it from time T
4. Repeat step 2 if there remain jobs to be scheduled

31. Parallel scheduling method example (1)
Decision set 2 1 3 2 2 4 6 8

32. Parallel scheduling method example (2)
Decision set 2 1 3 1 2 2 4 6 8

33. Parallel scheduling method example (3)
Decision set 2 1 3 2 1 2 2 4 6 8

34. Parallel scheduling method example (4)2 1 3 2 3 1 2 2 4 6 8

35. Priority-rule-based scheduling
Generation scheme
serial
parallel
Priority rule
latest finish time
minimum slack
Sampling procedure
Generation scheme: hoe bouw ik een schedule op
Priority: welke job ga ik aan het schedule toevoegen

36. Priority-rule-based scheduling: priority rules
Latest finish time (LFT) priority rule:
Minimum slack (MS) priority rule
current earliest starting time

37. MS priority rule example (1)
serial scheduling scheme 2 1 3
Serial scheduling scheme 1 2 2 4 6 8

38. MS priority rule example (2)1 3 3 1 2 2 4 6 8

39. MS priority rule example (3)2 1 3 3 1 2 2 2 4 6 8

40. Priority-rule-based scheduling
Generation scheme
serial
parallel
Priority rule
latest finish time
minimum slack
Sampling procedure
Generation scheme: hoe bouw ik een schedule op
Priority: welke job ga ik aan het schedule toevoegen

41. Priority-rule-based scheduling
Multi-pass priority-rule-based heuristics
multi-priority rule procedures
1 scheduling scheme, x priority rules
 generate x schedules, keep the best found
sampling procedures
1 scheduling scheme, 1 priority rule
jobs are randomly selected from the decision set

42. Sampling procedure Random sampling Biased random sampling
all jobs have the same probability:
Biased random sampling
job with highest priority has highest probability, however not proportionally
Regret-based random sampling
job with highest priority proportionally has the highest probability
Dit is extra stof uitgedeeld bij 2e college. Ook op webpage!!

43. Biased random sampling
Probability that job j is selected (Pj):
first sort jobs on non-increasing priority
[j] is the position of job j in the list 
random sampling
C = normaliserende constante
deterministic

44. Biased random sampling (example)
C = normaliserende constante

45. Regret-based random sampling
Regret of job = difference between priority value and lowest overall priority value:
Probability that job is selected:
random sampling; deterministic
Nadeel biased sampling is dat het geen rekening houdt met relatieve verschil in de prioriteiten
P(j) altijd >0!!
C = normaliserende constante

46. Regret-based random sampling (example)

47. Time/costs trade-off (§4.4)
Assumptions:
by allocating money (for additional resources) to jobs their processing time (pj) can be reduced
linear relation between allocated money and pj
minimum and maximum processing time
cj = marginal costs of reducing pj:
Paragraaf 4.4

48. Time/costs trade-off heuristic
Definitions:
source and sink in precedence graph
critical path: longest path from source to sink
Gcp = sub-graph of critical path(s)
cut set: set of nodes in sub-graph Gcp whose removal results in disconnecting the source from the sink in the precedence graph
minimal cut set: if putting back 1 node in the graph connects the source to the sink

49. Time/costs trade-off heuristic
STEP 1: Set
Determine Gcp .
STEP 2: Determine all minimum cut sets mcs in Gcp.
Consider only those minimum cut sets of
which all processing times
If there is no such set: STOP
STEP 3: For each mcs compute the costs of
reducing all pj in mcs with one time unit
Let mcs* be the mcs with the lowest costs
If the lowest costs are <c0  apply the
changes, revise Gcp and go to STEP 1

50. Time/costs trade-off heuristic example 4.4.1
processing time 6 12 10 9 2 4 7 10 8 5 12 5 12 10 S 1 14 T 6 9 9 7 7 3 11 13 7 6 5 8
Slightly different solution path (correct!!)

51. Time/costs trade-off heuristic example 4.4.1
processing time 6 12 10 9 2 4 7 10 7 5 12 5 12 10 S 1 14 T 6 9 9 7 7 3 11 13 7 6 5 8
Here I alter the solution path of the book: the book alters job 6 (costs 3), and I alter job 11

52. Time/costs trade-off heuristic example 4.4.1
processing time 6 12 10 9 2 4 7 10 7 5 12 5 12 10 S 1 14 T 6 9 9 6 7 3 11 13 7 6 5 8

53. Time/costs trade-off heuristic example 4.4.1
processing time
job 2 hits
minimum
5min 12 10 9 2 4 7 10 7 5 12 5 12 10 S 1 14 T 6 9 9 5 7 3 11 13 7 6 5 8

54. Time/costs trade-off heuristic example 4.4.1
processing time
5min 11 10 9 2 4 7 10 7 5 12 5 12 10 S 1 14 T 6 9 9
4min 7 3 11 13 7 6 5 8
Makespan can still be reduced with the same cost: for example {12,13}
OPTIMALITY CONDITION: when no P(j) can be increased without enlarging the makespan

55. Time/costs trade-off LP model
Decision variables:
xj = earliest starting time of job j
pj = processing time of job j
Total cost:
Constraints:
precedence relations:
max/min processing times:
Cmax determination:
all variables :
Cmax = auxiliary variable

56. Time/costs trade-off: nonlinear costs
Discrete time-framework:
decreasing convex cost-function
non-decreasing overhead cost-function c0(t)
use the same heuristic as for the linear costs
Continuous time-framework:
non-linear model with the same constraints as the LP-model, but with the objective: