PeterBrucker - On the Complexity of Scheduling Peter...

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On the Complexity of Scheduling Peter Brucker University of Osnabrueck Germany
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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:
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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. .
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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 ,
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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).
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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.
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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:
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An Example A corresponding schedule with minimal makespan 2 3
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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.
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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.
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Applications Production scheduling Robotic cell scheduling Computer processor scheduling Timetabling Personnel scheduling Railway scheduling Air traffic control etc.
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Assumptions All data are assumed to be integers. We consider only off-line scheduling
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PeterBrucker - On the Complexity of Scheduling Peter...

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