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1 Lecture 2 Scheduling 1 Classification - Examples -1-1 r j C max given: n jobs with processing times p 1,..., p n and release dates r 1,..., r n jobs have to be scheduled without preemption on one machine taking into account the earliest starting times of the jobs, such that the makespan is minimized n = 4, p = (2, 4, 2, 3), r = (5, 4, 0, 3)

2 Lecture 2 Scheduling 2 Classification - Examples -1-1 r j C max given: n jobs with processing times p 1,..., p n and release dates r 1,..., r n jobs have to be scheduled without preemption on one machine taking into account the earliest starting times of the jobs, such that the makespan is minimized n = 4, p = (2, 4, 2, 3), r = (5, 4, 0, 3) Feasible Schedule with C max = 12 (schedule is optimal)

3 Lecture 2 Scheduling 3 Classification - Examples -2- F 2 w j T j given n jobs with weights w 1,..., w n and due dates d 1,..., d n operations (i, j) with processing times p ij, i = 1, 2; j = 1,..., n jobs have to be scheduled on two machines such that operation (2, j) is schedules on machine 2 and does not start before operation (1, j), which is scheduled on machine 1, is finished and the total weighted tardiness ( is ) minimized n = 3, p =, w = (3, 1, 5), d = (6, 8, 4) 3 4 1

4 Lecture 2 Scheduling 4 Classification - Examples -2- F 2 w j T j given n jobs with weights w 1,..., w n and due dates d 1,..., d n operations (i, j) with processing times p ij, i = 1, 2; j = 1,..., n jobs have to be scheduled on two machines such that operation (2, j) is schedules on machine 2 and does not start before operation (1, j), which is scheduled on machine 1, is finished and the total weighted tardiness ( is ) minimized n = 3, p =, w = (3, 1, 5), d = (6, 8, 4) M wj T j = 3(8 6) + (12 9) M (4 4) = d 3 d 1 d 2

5 Lecture 2 Scheduling 5 Classes of Schedules Nondelay Schedules: A feasible schedule is called a nondelay schedule if no machine is kept idle while a job/an operation is waiting for processing Example: P 3 prec C max n = 6 p = (1, 1, 2, 2, 3, 3)

6 Lecture 2 Scheduling 6 Classes of Schedules Nondelay Schedules: A feasible schedule is called a nondelay schedule if no machine is kept idle while a job/an operation is waiting for processing Example: P 3 prec C max n = 6 p = (1, 1, 2, 2, 3, 3) Best nondelay: M1 M2 M Optimal M1 M2 M

7 Lecture 2 Scheduling 7 Classes of Schedules Remark: restricted to non preemptive schedules Active Schedules: A feasible schedule is called active if it is not possible to construct another schedule by changing the order of processing on the machines and having at least one job/operation finishing earlier and no job/operation finishing later. Semi-Active Schedules: A feasible schedule is called semi-active if no job/operation can be finishing earlier without changing the order of processing on any one of the machines. -2-

8 Lecture 2 Scheduling 8 Classes of Schedules Examples of (semi)-active schedules: Prec: 1 2; 2 3 not semi-active M1 M

9 Lecture 2 Scheduling 9 Classes of Schedules Examples of (semi)-active schedules: Prec: 1 2; 2 3 not semi-active semi-active M1 M2 M1 M

10 Lecture 2 Scheduling 10 Classes of Schedules Examples of (semi)-active schedules: Prec: 1 2; 2 3 not semi-active semi-active active M1 M2 M1 M2 M1 M

11 Lecture 2 Scheduling 11 Classes of Schedules Properties: every nonpreemptive nondelay schedule is active every active schedule is semiactive if the objective criterion is regular, the set of active schedules contains an optimal schedule (regular = non decreasing with respect to the completion times) Summary: All schedules -4- x non- delay active optimal schedule Semi active

12 Lecture 2 Scheduling 12 Research topics for Scheduling determine boarder line between polynomially solvable and NP-hard models for polynomially solvable models find the most efficient solution method (low complexity) for NP-hard models develop enumerative methods (DP, branch and bound, branch and cut,...) develop heuristic approached (priority based, local search,...) consider approximation methods (with quality guarantee)

13 Lecture 2 Scheduling 13 Intermezzo: Complexity Theory -1- mathematical framework to study the difficulty of algorithmic problems Notations/Definitions problem: generic description of a problem (e.g. 1 C j ) instance of a problem: given set of numerical data (e.g. n, p 1,..., p n ) size of an instance I: length of the string necessary to specify the data (Notation: I ) binary encoding: I = n + log(p 1 ) log(p n ) unary encoding: I = n + p p n

14 Lecture 2 Scheduling 14 Intermezzo: Complexity Theory -2- Notations/Definitions efficiency of an algorithm: upper bound on number of steps depending on the size of the instance (worst case consideration) big O-notation: for an O(f(n)) algorithm a constant c > 0 and an integer n 0 exist, such that for an instance I with size n = I and n n 0 the number of steps is bounded by cf(n) Example: 7n n + 10 log(n) is O(n 3 ) (pseud)polynomial algorithm: O(p( I )) algorithm, where p is a polynomial and I is coded binary (unary) Example: an O(n log( p j )) algorithm is a polynomial algorithm and an O(n p j ) algorithm is a pseudopolynomial algorithm

15 Lecture 2 Scheduling 15 Intermezzo: Complexity Theory -3- Classes P and N P a problem is (pseudo)polynomial solvable if a (pseudo)polynomial algorithm exists which solves the problem Class P: contains all decision problems which are polynomial solvable Class N P: contains all decision problems for which - given an yes instance - the correct answer, given a proper clue, can be verified by a polynomial algorithm Remark: each optimization problem has a corresponding decision problem by introducing a threshold for the objective value (does a schedule exist with objective smaller k?)

16 Lecture 2 Scheduling 16 Intermezzo: Complexity Theory -4- Polynomial reduction a decision problem P polynomially reduces to a problem Q, if a polynomial function g exists that transforms instances of P to instances of Q such that I is a yes instance of P if and only is g(i) is a yes instance of Q Notation: P Q NP-complete a decision problem P N P is called NP-complete if all problems from the class N P polynomially reduce to P an optimization problem is called NP-hard if the corresponding decision problem is NP-complete

17 Lecture 2 Scheduling 17 Intermezzo: Complexity Theory -5- Examples of NP-complete problems: SATISFIABILITY: decision problem in Boolean logic, Cook in 1967 showed that all problems from N P polynomially reduce to it PARTITION: given n positive integers s 1,..., s n and b = 1/2 n j=1 s j does there exist a subset J I = {1,..., n} such that s j = b = j J j I\J s j

18 Lecture 2 Scheduling 18 Intermezzo: Complexity Theory -6- Examples of NP-complete problems (cont.): 3-PARTITION: given 3n positive integers s 1,..., s 3n and b with b/4 < s j < b/2, j = 1,..., 3n and b = 1/n 3n j=1 s j do there exist disjoint subsets J i I = {1,..., 3n} such that s j = b; i = 1,..., n j J i

19 Lecture 2 Scheduling 19 Intermezzo: Complexity Theory -7- Proofing NP-completeness If an NP-complete problem P can be polynomially reduced to a problem Q N P, than this proves that Q is NP-complete (transitivity of polynomial reductions) Example: P ART IT ION P 2 C max Proof: on the board Famous open problem: Is P = N P? solving one NP-complete problem polynomially, would imply P = N P

20 Lecture 2 Scheduling 20 Single machine models Observation: for non-preemptive problems and regular objectives, a sequence in which the jobs are processed is sufficient to describe a solution Dispatching (priority) rules static rules - not time dependent e.g. shortest processing time first, earliest due date first dynamic rules - time dependent e.g. minimum slack first (slack= d j p j t; t current time) for some problems dispatching rules lead to optimal solutions

21 Lecture 2 Scheduling 21 Single machine models: 1 w j C j -1- Given: n jobs with processing times p 1,..., p n and weights w 1,..., w n Consider case: w 1 =... = w n (= 1):

22 Lecture 2 Scheduling 22 Single machine models: 1 w j C j -1- Given: n jobs with processing times p 1,..., p n and weights w 1,..., w n Consider special case: w 1 =... = w n (= 1): SPT-rule: shortest processing time first Theorem: SPT is optimal for 1 C j Proof: by an exchange argument (on board) Complexity: O(n log(n))

23 Lecture 2 Scheduling 23 Single machine models: 1 w j C j -2- General case WSPT-rule: weighted shortest processing time first, i.e. sort jobs by increasing p j /w j -values Theorem: WSPT is optimal for 1 w j C j Proof: by an exchange argument (exercise) Complexity: O(n log(n)) Further results: 1 tree w j C j can be solved by in polynomial time (O(n log(n))) (see [Brucker 2004]) 1 prec C j is NP-hard in the strong sense (see [Brucker 2004])

24 Lecture 2 Scheduling 24 Single machine models: 1 prec f max -1- Given: n jobs with processing times p 1,..., p n regular functions f 1,..., f n objective criterion f max = max{f 1 (C 1 ),..., f n (C n )} Observation: completion time of last job = p j

25 Lecture 2 Scheduling 25 Single machine models: 1 prec f max -1- Given: n jobs with processing times p 1,..., p n regular functions f 1,..., f n objective criterion f max = max{f 1 (C 1 ),..., f n (C n )} Observation: completion time of last job = p j Method plan backwards from p j to 0 from all available jobs (jobs from which all successors have already been scheduled), schedule the job which is cheapest on that position

26 Lecture 2 Scheduling 26 Single machine models: 1 prec f max -2- S set of already scheduled jobs (initial: S = ) J set of all jobs, which successors have been scheduled (initial: all jobs without successors) t time where next job will be completed (initial: t = p j ) Algorithm 1 prec f max (Lawler s Algorithm) REPEAT select j J such that h j (t) = min k J f k (t); schedule j such that it completes at t; add j to S, delete j from J and update J; t := t p j ; UNTIL J =.

27 Lecture 2 Scheduling 27 Single machine models: 1 prec f max -3- Theorem: Algorithm 1 prec f max is optimal for 1 prec f max Proof: on the board Complexity: O(n 2 )

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