{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# PeterBrucker - On the Complexity of Scheduling Peter...

This preview shows pages 1–13. Sign up to view the full content.

On the Complexity of Scheduling Peter Brucker University of Osnabrueck Germany

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
1.Scheduling Problems In a scheduling problem one has to find time slots in which activities (or jobs) should be processed under given constraints. The main constraints are resource constraints and precedence constraints between activities. A quite general scheduling problem is the Resource Constrained Project Scheduling Problem (RCPSP) which can be formulated as follows:
The RCPSP We have • Activities j = 1, . .. , n with processing times p j . Resources k = 1, . .. , r. A constant amount of R k units of resource k is available at any time. During processing, activity j occupies r jk units of resource k for k = 1, . .. , r. Precedence constrains i j between some activities i, j with the meaning that activity j cannot start before i is finished. .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The RCPSP The objective is to determine starting times S j for all activities j in such a way that at each time t the total demand for resource k is not greater than the availability R k for k = 1, . .. , r, the given precedence constraints are fulfilled, i. e. S i + p i S j if i j ,
The RCPSP • some objective function f( C 1 , . .. , C n ) is minimized where C j = S j + p j is the completion time of activity j. The fact that activities j start at time S j and finish at time S j + p j implies that the activities j are not preempted. We may relax this condition by allowing preemption (activity splitting).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
The RCPSP If preemption is not allowed the vector S = (S j ) defines a schedule. S is called feasible if all resource and precedence constraints are fulfilled. One has to find a feasible schedule which minimizes the objective function f( C 1 , . .. , C n ). In project planning f( C 1 , . .. , C n ) is often replaced by the makespan C max which is the maximum of all C j - values.
An Example Consider a project with n = 4 activities, r = 2 resources with capacities R 1 = 5 and R 2 = 7, a precedence relation 2 3 and the following data:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
An Example A corresponding schedule with minimal makespan 2 3
The RCPSP • The constraints S i + p i S j may be replaced by S i + d ij S j (positive and negative time- lags). With time-lags we may model release times r j or deadlines d j . We may have more than one objective function (multi-criteria optimization). There are other generalizations.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
RCPSP with Multiple Modes • Associated with each activity j is a set M j of modes (processing alternatives). • The processing time p jm and per period usage r jkm of resource k for activity j depends on mode m. One has to assign a mode to each activity and to schedule the activities in the assigned modes.
Applications Production scheduling Robotic cell scheduling Computer processor scheduling Timetabling Personnel scheduling Railway scheduling Air traffic control etc.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Assumptions All data are assumed to be integers. We consider only off-line scheduling
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 61

PeterBrucker - On the Complexity of Scheduling Peter...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online