Scheduling Single Machine Scheduling. Tim Nieberg


 Bertina Curtis
 5 years ago
 Views:
Transcription
1 Scheduling Single Machine Scheduling Tim Nieberg
2 Single machine models Observation: for nonpreemptive 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
3 Single machine models: 1 w j C j Given: n jobs with processing times p 1,...,p n and weights w 1,...,w n Consider case: w 1 =... = w n (= 1):
4 Single machine models: 1 w j C j Given: n jobs with processing times p 1,...,p n and weights w 1,...,w n Consider special case: w 1 =... = w n (= 1): SPTrule: shortest processing time first Theorem: SPT is optimal for 1 C j Proof: by an exchange argument (on board) Complexity: O(n log(n))
5 Single machine models: 1 w j C j General case WSPTrule: 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 NPhard in the strong sense (see [Brucker 2004])
6 Single machine models: 1 prec f max Given: n jobs with processing times p 1,...,p n precedence constraints between the jobs 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
7 Single machine models: 1 prec f max Given: n jobs with processing times p 1,...,p n precedence constraints between the jobs regular functions f 1,...,f n objective criterion f max = max{f 1 (C 1 ),...,f n (C n )} Observation: Method completion time of last job = p j plan backwards from p j to 0 from all available jobs (jobs from which all successors have already been scheduled), schedule job which is cheapest on that position
8 Single machine models: 1 prec f max S set of already scheduled jobs (initial: S = ) J set of all jobs, which successors have been scheduled (initial: all jobs 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 f 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 =.
9 Single machine models: 1 prec f max Theorem: Algorithm 1 prec f max is optimal for 1 prec f max Proof: on the board Complexity: O(n 2 )
10 Single machine models: Maximum Lateness Problem 1 L max : Earliest due date first (EDD) is optimal for 1 L max (Jackson s EDD rule) Proof: special case of Lawler s algorithm
11 Single machine models: Maximum Lateness Problem 1 L max : Earliest due date first (EDD) is optimal for 1 L max (Jackson s EDD rule) Proof: special case of Lawler s algorithm Problem 1 r j C max : 1 r j C max 1 L max define d j := K r j, with constant K > maxr j reversing the optimal schedule of this 1 L max problem gives an optimal schedule for the 1 r j C max problem
12 Single machine models: Maximum Lateness Problem 1 prec L max : if d j < d k whenever j k, any EDD schedule respects the precedence constraints, i.e. in this case EDD is optimal defining d j := min{d j,d k p k } if j k does not increase L max in any feasible schedule
13 Single machine models: Maximum Lateness Problem 1 prec L max : if d j < d k whenever j k, any EDD schedule respects the precedence constraints, i.e. in this case EDD is optimal defining d j := min{d j,d k p k } if j k does not increase L max in any feasible schedule Algorithm 1 prec L max 1 make due dates consistent: set d j = min{d j,min k j k d k p k } 2 apply EDD rule with modified due dates
14 Single machine models: Maximum Lateness Remarks on Algorithm 1 prec L max leads to an optimal solution Step 1 can be realized in O(n 2 ) problem 1 prec L max can be solved without knowledge of the processing times, whereas Lawler s Algorithm (which also solves this problem) in general needs this knowledge (Exercise), Problem 1 r j,prec C max 1 prec L max
15 Single machine models: Maximum Lateness Problem 1 r j L max : problem 1 r j L max is NPhard Proof: by reduction from 3PARTITION (on the board)
16 Single machine models: Maximum Lateness Problem 1 pmtn,r j L max : preemptive EDDrule: at each point in time, schedule an available job (job, which release date has passed) with earliest due date. preemptive EDDrule leads to at most k preemptions (k = number of distinct release dates)
17 Single machine models: Maximum Lateness Problem 1 pmtn,r j L max : preemptive EDDrule: at each point in time, schedule an available job (job, which release date has passed) with earliest due date. preemptive EDDrule leads to at most k preemptions (k = number of distinct release dates) preemptive EDD solves problem 1 pmtn,r j L max Proof (on the board) uses following results: L max r(s) + p(s) d(s) for any S {1,..., n}, where r(s) = min j S r j, p(s) = j S p j, d(s) = max j S d j preemptive EDD leads to a schedule with L max = max S {1,...,n} r(s) + p(s) d(s)
18 Single machine models: Maximum Lateness Remarks on preemptive EDDrule for 1 pmtn,r j L max : can be implemented in O(n log(n)) is an online algorithm after modification of release and duedates, preemptive EDD solves also 1 prec,pmtn,r j L max
19 Single machine models: Maximum Lateness Approximation algorithms for problem 1 r j L max : a polynomial algorithm A is called an αapproximation for problem P if for every instance I of P algorithm A yields an objective value f A (I) which is bounded by a factor α of the optimal value f (I); i.e. f A (I) αf (I)
20 Single machine models: Maximum Lateness Approximation algorithms for problem 1 r j L max : a polynomial algorithm A is called an αapproximation for problem P if for every instance I of P algorithm A yields an objective value f A (I) which is bounded by a factor α of the optimal value f (I); i.e. f A (I) αf (I) for the objective L max, αapproximation does not make sense since L max may get negative for the objective T max, an αapproximation with a constant α implies P = N P (if T max = 0 an αapproximation is optimal)
21 Single machine models: Maximum Lateness The headbodytail problem (1 r j,d j < 0 L max ) n jobs job j: release date r j (head), processing time p j (body), delivery time q j (tail) starting time S j r j ; completion time C j = S j + p j delivered at C j + q j goal: minimize max n j=1 C j + q j
22 Single machine models: Maximum Lateness The headbodytail problem (1 r j,d j < 0 L max ), (cont.) define d j = q j, i.e. the due dates get negative! result: min n j=1 C j + q j = min n j=1 C j d j = min n j=1 L j = L max headbodytail problem equivalent with 1 r j L max problem with negative due dates Notation: 1 r j,d j < 0 L max an instance of the headbodytail problem defined by n triples (r j,p j,q j ) is equivalent to an inverse instance defined by n triples (q j,p j,r j ) for the headbodytail problem considering approximation algorithms makes sense
23 Single machine models: Maximum Lateness The headbodytail problem (1 r j,d j < 0 L max ), (cont.) L max r(s) + p(s) + q(s) for any S {1,...,n}, where r(s) = min r j, p(s) = p j, q(s) = min q j j S j S j S (follows from L max r(s) + p(s) d(s))
24 Single machine models: Maximum Lateness Approximation ratio for EDD for problem 1 r j,d j < 0 L max structure of a schedule S t = r(q) C c Q c critical job c of a schedule: job with L c = maxl j critical sequence Q: jobs processed in the interval [t,c c ], where t is the earliest time that the machine is not idle in [t,c c ] if q c = min j Q q j the schedule is optimal since then L max (S) = L c = C c d c = r(q) + p(q) + q(q) L max Notation: L max denotes the optimal value
25 Single machine models: Maximum Lateness Approximation ratio for EDD for problem 1 r j,d j < 0 L max structure of a schedule Q b c... 0 r(q) S b C c Q interference job b: last scheduled job from Q with q b < q c Lemma: For the objective value L max (EDD) of an EDD schedule we have: (Proofs on the board) 1 L max (EDD) L max < q c 2 L max (EDD) L max < p b Theorem: EDD is 2approximation algorithm for 1 r j,d j < 0 L max
26 Single machine models: Maximum Lateness Approximation ratio for EDD for problem 1 r j,d j < 0 L max Remarks: EDD is also a 2approximation for 1 prec, r j, d j < 0 L max (uses modified release and due dates) by an iteration technique the approximation factor can be reduced to 3/2
27 Single machine models: Maximum Lateness Enumerative methods for problem 1 r j L max we again will use headbodytail notation Simple branch and bound method: branch on level i of the search tree by selecting a job to be scheduled on position i if in a node of the search tree on level i the set of already scheduled jobs is denoted by S and the finishing time of the jobs from S by t, for position i we only have to consider jobs k with r k < min j / S (max{t, r j} + p j ) lower bound: solve for remaining jobs 1 r j, pmtn L max search strategy: depth first search + selecting next job via lower bound
28 Single machine models: Maximum Lateness Advanced b&bmethods for problem 1 r j L max node of search tree = restricted instance restrictions = set of precedence constraints branching = adding precedence constraints between certain pairs of jobs after adding precedence constraints, modify release and due dates apply EDD to instance given in a node critical sequence has no interference job: EDD solves instance optimal backtrack critical sequence has an interference job: branch
29 Single machine models: Maximum Lateness Advanced b&bmethods for problem 1 r j L max (cont.) branching, given set Q, critical job c, interference job b, and set Q of jobs from Q following b Q b c... 0 r(q) S b C c Q L max = S b +p b +p(q )+q(q ) < r(q )+p b +p(q )+q(q ) if b is scheduled between jobs of Q the value is at least r(q ) + p b + p(q ) + q(q ); i.e. worse than the current schedule
30 Single machine models: Maximum Lateness Advanced b&bmethods for problem 1 r j L max (cont.) branching, given sequence Q, critical job c, interference job b, and set Q of jobs from Q following b Q b c... 0 r(q) S b C c Q L max = S b +p b +p(q )+q(q ) < r(q )+p b +p(q )+q(q ) if b is scheduled between jobs of Q the value is at least r(q ) + p b + p(q ) + q(q ); i.e. worse than the current schedule branch by adding either b Q or Q b
31 Single machine models: Maximum Lateness Advanced b&bmethods for problem 1 r j L max (cont.) lower bounds in a node: maximum of lower bound of parent node r(q ) + p(q ) + q(q ) r(q {b}) + p(q {b}) + q(q {b}) using the modified release and due dates upper bound UB: best value of the EDD schedules discard a node if lower bound UB search strategy: select node with minimum lower bound
32 Single machine models: Maximum Lateness Advanced b&bmethods for problem 1 r j L max (cont.) speed up possibility: let k / Q {b} with r(q ) + p k + p(q ) + q(q ) UB if r(q ) + p(q ) + p k + q k UB then add k Q if r k + p k + p(q ) + q(q ) UB then add Q k
33 Single machine models: Number of Tardy Jobs Problem 1 U j : Structure of an optimal schedule: set S 1 of jobs meeting their due dates set S 2 of jobs being late jobs of S 1 are scheduled before jobs from S 2 jobs from S 1 are scheduled in EDD order jobs from S 2 are scheduled in an arbitrary order Result: a partition of the set of jobs into sets S 1 and S 2 is sufficient to describe a solution
34 Single machine models: Number of Tardy Jobs Algorithm 1 U j 1 enumerate jobs such that d 1... d n ; 2 S 1 := ; t := 0; 3 FOR j:=1 TO n DO 4 S 1 := S 1 {j}; t := t + p j ; 5 IF t > d j THEN 6 Find job k with largest p k value in S 1 ; 7 S 1 := S 1 \ {k}; t := t p k ; 8 END 9 END
35 Single machine models: Number of Tardy Jobs Remarks Algorithm 1 U j Principle: schedule jobs in order of increasing due dates and always when a job gets late, remove the job with largest processing time; all removed jobs are late complexity: O(n log(n)) Example: n = 5; p = (7,8,4,6,6); d = (9,17,18,19,21)
36 Single machine models: Number of Tardy Jobs Remarks Algorithm 1 U j Principle: schedule jobs in order of increasing due dates and always when a job gets late, remove the job with largest processing time; all removed jobs are late complexity: O(n log(n)) Example: n = 5; p = (7,8,4,6,6); d = (9,17,18,19,21) d 3 20
37 Single machine models: Number of Tardy Jobs Remarks Algorithm 1 U j Principle: schedule jobs in order of increasing due dates and always when a job gets late, remove the job with largest processing time; all removed jobs are late complexity: O(n log(n)) Example: n = 5; p = (7,8,4,6,6); d = (9,17,18,19,21) d 5
38 Single machine models: Number of Tardy Jobs Remarks Algorithm 1 U j Principle: schedule jobs in order of increasing due dates and always when a job gets late, remove the job with largest processing time; all removed jobs are late complexity: O(n log(n)) Example: n = 5; p = (7,8,4,6,6); d = (9,17,18,19,21) Algorithm 1 U j computes an optimal solution Proof on the board
39 Single machine models: Weighted Number of Tardy Jobs Problem 1 w j U j problem 1 w j U j is NPhard even if all due dates are the same; i.e. 1 d j = d w j U j is NPhard Proof on the board (reduction from PARTITION) priority based heuristic (WSPTrule): schedule jobs in increasing p j /w j order
40 Single machine models: Weighted Number of Tardy Jobs Problem 1 w j U j problem 1 w j U j is NPhard even if all due dates are the same; i.e. 1 d j = d w j U j is NPhard Proof on the board (reduction from PARTITION) priority based heuristic (WSPTrule): schedule jobs in increasing p j /w j order WSPT may perform arbitrary bad for 1 w j U j :
41 Single machine models: Weighted Number of Tardy Jobs Problem 1 w j U j problem 1 w j U j is NPhard even if all due dates are the same; i.e. 1 d j = d w j U j is NPhard Proof on the board (reduction from PARTITION) priority based heuristic (WSPTrule): schedule jobs in increasing p j /w j order WSPT may perform arbitrary bad for 1 w j U j : n = 3;p = (1,1,M); w = (1 + ǫ,1,m ǫ);d = (1 + M,1 + M,1 + M) wj U j (WSPT)/ w j U j (opt) = (M ǫ)/(1 + ǫ)
42 Single machine models: Weighted Number of Tardy Jobs Dynamic Programming for 1 w j U j assume d 1... d n as for 1 U j a solution is given by a partition of the set of jobs into sets S 1 and S 2 and jobs in S 1 are in EDD order Definition: F j (t) := minimum criterion value for scheduling the first j jobs such that the processing time of the ontime jobs is at most t F n (T) with T = n j=1 p j is optimal value for problem 1 w j U j Initial conditions: F j (t) = { for t < 0; j = 1,...,n 0 for t 0; j = 0 (1)
43 Single machine models: Weighted Number of Tardy Jobs Dynamic Programming for 1 w j U j (cont.) if 0 t d j and j is late in the schedule corresponding to F j (t), we have F j (t) = F j 1 (t) + w j if 0 t d j and j is on time in the schedule corresponding to F j (t), we have F j (t) = F j 1 (t p j )
44 Single machine models: Weighted Number of Tardy Jobs Dynamic Programming for 1 w j U j (cont.) if 0 t d j and j is late in the schedule corresponding to F j (t), we have F j (t) = F j 1 (t) + w j if 0 t d j and j is on time in the schedule corresponding to F j (t), we have F j (t) = F j 1 (t p j ) summarizing, we get for j = 1,...,n: { min{f j 1 (t p j ),F j 1 (t) + w j } F j (t) = F j (d j ) for 0 t d j for d j < t T (2)
45 Single machine models: Weighted Number of Tardy Jobs DPalgorithm for 1 w j U j 1 initialize F j (t) according to (1) 2 FOR j := 1 TO n DO 3 FOR t := 0 TO T DO 4 update F j (t) according to (2) 5 wj U j (OPT) = F n (d n )
46 Single machine models: Weighted Number of Tardy Jobs DPalgorithm for 1 w j U j 1 initialize F j (t) according to (1) 2 FOR j := 1 TO n DO 3 FOR t := 0 TO T DO 4 update F j (t) according to (2) 5 wj U j (OPT) = F n (d n ) complexity is O(n n j=1 p j) thus, algorithm is pseudopolynomial
47 Single machine models: Total Tardiness Basic results: 1 T j is NPhard preemption does not improve the criterion value 1 pmtn T j is NPhard idle times do not improve the criterion value Lemma 1: If p j p k and d j d k, then an optimal schedule exist in which job j is scheduled before job k. Proof: exercise this lemma gives a dominance rule
48 Single machine models: Total Tardiness Structural property for 1 T j let k be a fixed job and Ĉk be latest possible completion time of job k in an optimal schedule define ˆd j = { d j max{d k,ĉ k } for j k for j = k Lemma 2: Any optimal sequence w.r.t. ˆd 1,..., ˆd n is also optimal w.r.t. d 1,...,d n. Proof on the board
49 Single machine models: Total Tardiness Structural property for 1 T j (cont.) let d 1... d n let k be the job with p k = max{p 1,...,p n } Lemma 1 implies that an optimal schedule exists where {1,...,k 1} k Lemma 3: There exists an integer δ, 0 δ n k for which an optimal schedule exist in which {1,...,k 1,k+1,...,k+δ} k and k {k+δ+1,...,n}. Proof on the board
50 Single machine models: Total Tardiness DPalgorithm for 1 T j Definition: F j (t) := minimum criterion value for scheduling the first j jobs starting their processing at time t by Lemma 3 we get: there exists some δ {1,...,j} such that F j (t) is achieved by scheduling 1 first jobs 1,...,k 1, k + 1,...,k + δ in some order 2 followed by job k starting at t + k+δ l=1 p l p k 3 followed by jobs k + δ + 1,...,j in some order where p k = max j l=1 p l
51 Single machine models: Total Tardiness DPalgorithm for 1 T j (cont.) {1,...,j} {1,...,k 1, k + 1,...,k + δ} k {k + δ + 1,...,j} Definition: J(j, l, k) := {i i {j, j + 1,...,l}; p i p k ; i k}
52 Single machine models: Total Tardiness DPalgorithm for 1 T j (cont.) {1,...,j} {1,...,k 1, k + 1,...,k + δ} k {k + δ + 1,...,j} Definition: J(j, l, k) := {i i {j, j + 1,...,l}; p i p k ; i k} {1,...,j} J(1, k + δ, k) k J(k + δ + 1, j, k)
53 Single machine models: Total Tardiness DPalgorithm for 1 T j (cont.) {1,...,j} J(1, k + δ, k) k J(k + δ + 1, j, k) Definition: J(j, l, k) := {i i {j, j + 1,...,l}; p i p k ; i k} V(J(j, l, k), t) := minimum criterion value for scheduling the jobs from J(j, l, k) starting their processing at time t
54 Single machine models: Total Tardiness DPalgorithm for 1 T j (cont.) t t J(j, l, k) J(j, k + δ, k ) k J(k + δ + 1, l, k ) C k (δ) we get: V (J(j,l,k),t) = min δ where p k = max{p j j J(j,l,k)} and C k (δ) = t + p k + j V(J(j,k +δ,k ) p j {V (J(j,k + δ,k ),t) + max{0,c k (δ) d k } +V (J(k + δ + 1,l,k ),C k (δ)))} V (,t) = 0, V ({j},t) = max{0,t + p j d j }
55 Single machine models: Total Tardiness DPalgorithm for 1 T j (cont.) optimal value of 1 T j is given by V ({1,...,n},0) complexity: at most O(n 3 ) subsets J(j, l, k) at most p j values for t each recursion (evaluation V(J(j, l, k), t)) costs O(n) (at most n values for δ) total complexity: O(n 4 p j ) (pseudopolynomial)
Single machine models: Maximum Lateness 12 Approximation ratio for EDD for problem 1 r j,d j < 0 L max. structure of a schedule Q...
Lecture 4 Scheduling 1 Single machine models: Maximum Lateness 12 Approximation ratio for EDD for problem 1 r j,d j < 0 L max structure of a schedule 0 Q 1100 11 00 11 000 111 0 0 1 1 00 11 00 11 00
More informationClassification  Examples
Lecture 2 Scheduling 1 Classification  Examples 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
More informationScheduling Shop Scheduling. Tim Nieberg
Scheduling Shop Scheduling Tim Nieberg Shop models: General Introduction Remark: Consider non preemptive problems with regular objectives Notation Shop Problems: m machines, n jobs 1,..., n operations
More informationA binary search algorithm for a special case of minimizing the lateness on a single machine
Issue 3, Volume 3, 2009 45 A binary search algorithm for a special case of minimizing the lateness on a single machine Nodari Vakhania Abstract We study the problem of scheduling jobs with release times
More informationSingle machine parallel batch scheduling with unbounded capacity
Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University
More informationAn improved online algorithm for scheduling on two unrestrictive parallel batch processing machines
This is the PrePublished Version. An improved online algorithm for scheduling on two unrestrictive parallel batch processing machines Q.Q. Nong, T.C.E. Cheng, C.T. Ng Department of Mathematics, Ocean
More informationCompletion Time Scheduling and the WSRPT Algorithm
Completion Time Scheduling and the WSRPT Algorithm Bo Xiong, Christine Chung Department of Computer Science, Connecticut College, New London, CT {bxiong,cchung}@conncoll.edu Abstract. We consider the online
More informationReal Time Scheduling Basic Concepts. Radek Pelánek
Real Time Scheduling Basic Concepts Radek Pelánek Basic Elements Model of RT System abstraction focus only on timing constraints idealization (e.g., zero switching time) Basic Elements Basic Notions task
More informationList Scheduling in Order of αpoints on a Single Machine
List Scheduling in Order of αpoints on a Single Machine Martin Skutella Fachbereich Mathematik, Universität Dortmund, D 4422 Dortmund, Germany martin.skutella@unidortmund.de http://www.mathematik.unidortmund.de/
More informationApproximability of TwoMachine NoWait Flowshop Scheduling with Availability Constraints
Approximability of TwoMachine NoWait Flowshop Scheduling with Availability Constraints T.C. Edwin Cheng 1, and Zhaohui Liu 1,2 1 Department of Management, The Hong Kong Polytechnic University Kowloon,
More informationIntroduction to Scheduling Theory
Introduction to Scheduling Theory Arnaud Legrand Laboratoire Informatique et Distribution IMAG CNRS, France arnaud.legrand@imag.fr November 8, 2004 1/ 26 Outline 1 Task graphs from outer space 2 Scheduling
More informationStiffie's On Line Scheduling Algorithm
A class of online scheduling algorithms to minimize total completion time X. Lu R.A. Sitters L. Stougie Abstract We consider the problem of scheduling jobs online on a single machine and on identical
More informationAperiodic Task Scheduling
Aperiodic Task Scheduling JianJia Chen (slides are based on Peter Marwedel) TU Dortmund, Informatik 12 Germany Springer, 2010 2014 年 11 月 19 日 These slides use Microsoft clip arts. Microsoft copyright
More informationCommon Approaches to RealTime Scheduling
Common Approaches to RealTime Scheduling Clockdriven timedriven schedulers Prioritydriven schedulers Examples of priority driven schedulers Effective timing constraints The EarliestDeadlineFirst
More informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
More informationComplexity Theory. IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar
Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline Goals Computation of Problems Concepts and Definitions Complexity Classes and Problems Polynomial Time Reductions Examples
More informationOnline Scheduling for Cloud Computing and Different Service Levels
2012 IEEE 201226th IEEE International 26th International Parallel Parallel and Distributed and Distributed Processing Processing Symposium Symposium Workshops Workshops & PhD Forum Online Scheduling for
More informationOffline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com
More informationScheduling Realtime Tasks: Algorithms and Complexity
Scheduling Realtime Tasks: Algorithms and Complexity Sanjoy Baruah The University of North Carolina at Chapel Hill Email: baruah@cs.unc.edu Joël Goossens Université Libre de Bruxelles Email: joel.goossens@ulb.ac.be
More informationDuplicating and its Applications in Batch Scheduling
Duplicating and its Applications in Batch Scheduling Yuzhong Zhang 1 Chunsong Bai 1 Shouyang Wang 2 1 College of Operations Research and Management Sciences Qufu Normal University, Shandong 276826, China
More informationThe Trip Scheduling Problem
The Trip Scheduling Problem Claudia Archetti Department of Quantitative Methods, University of Brescia Contrada Santa Chiara 50, 25122 Brescia, Italy Martin Savelsbergh School of Industrial and Systems
More informationDynamic programming. Doctoral course Optimization on graphs  Lecture 4.1. Giovanni Righini. January 17 th, 2013
Dynamic programming Doctoral course Optimization on graphs  Lecture.1 Giovanni Righini January 1 th, 201 Implicit enumeration Combinatorial optimization problems are in general NPhard and we usually
More informationIntroducción a Calendarización en Sistemas Paralelas, Grids y Nubes
CICESE Research Center Ensenada, Baja California Mexico Introducción a Calendarización en Sistemas Paralelas, Grids y Nubes Dr. Andrei Tchernykh CICESE Centro de Investigación Científica y de Educación
More informationFairness in Routing and Load Balancing
Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More information8.1 Min Degree Spanning Tree
CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree
More informationA Beam Search Heuristic for MultiMode Single Resource Constrained Project Scheduling
A Beam Search Heuristic for MultiMode Single Resource Constrained Project Scheduling Chuda Basnet Department of Management Systems The University of Waikato Private Bag 3105 Hamilton chuda@waikato.ac.nz
More informationMinimizing the Number of Machines in a UnitTime Scheduling Problem
Minimizing the Number of Machines in a UnitTime Scheduling Problem Svetlana A. Kravchenko 1 United Institute of Informatics Problems, Surganova St. 6, 220012 Minsk, Belarus kravch@newman.basnet.by Frank
More information3. Scheduling issues. Common approaches /1. Common approaches /2. Common approaches /3. 2012/13 UniPD / T. Vardanega 23/01/2013. RealTime Systems 1
Common approaches /1 3. Scheduling issues Clockdriven (timedriven) scheduling Scheduling decisions are made beforehand (off line) and carried out at predefined time instants The time instants normally
More information5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1
5 INTEGER LINEAR PROGRAMMING (ILP) E. Amaldi Fondamenti di R.O. Politecnico di Milano 1 General Integer Linear Program: (ILP) min c T x Ax b x 0 integer Assumption: A, b integer The integrality condition
More informationScheduling a sequence of tasks with general completion costs
Scheduling a sequence of tasks with general completion costs Francis Sourd CNRSLIP6 4, place Jussieu 75252 Paris Cedex 05, France Francis.Sourd@lip6.fr Abstract Scheduling a sequence of tasks in the acceptation
More informationDynamic TCP Acknowledgement: Penalizing Long Delays
Dynamic TCP Acknowledgement: Penalizing Long Delays Karousatou Christina Network Algorithms June 8, 2010 Karousatou Christina (Network Algorithms) Dynamic TCP Acknowledgement June 8, 2010 1 / 63 Layout
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. #approximation algorithm.
Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of three
More informationDimensioning an inbound call center using constraint programming
Dimensioning an inbound call center using constraint programming Cyril Canon 1,2, JeanCharles Billaut 2, and JeanLouis Bouquard 2 1 Vitalicom, 643 avenue du grain d or, 41350 Vineuil, France ccanon@fr.snt.com
More informationA Linear Programming Based Method for Job Shop Scheduling
A Linear Programming Based Method for Job Shop Scheduling Kerem Bülbül Sabancı University, Manufacturing Systems and Industrial Engineering, OrhanlıTuzla, 34956 Istanbul, Turkey bulbul@sabanciuniv.edu
More informationBatch Scheduling of Deteriorating Products
Decision Making in Manufacturing and Services Vol. 1 2007 No. 1 2 pp. 25 34 Batch Scheduling of Deteriorating Products Maksim S. Barketau, T.C. Edwin Cheng, Mikhail Y. Kovalyov, C.T. Daniel Ng Abstract.
More informationCSC2420 Spring 2015: Lecture 3
CSC2420 Spring 2015: Lecture 3 Allan Borodin January 22, 2015 1 / 1 Announcements and todays agenda Assignment 1 due next Thursday. I may add one or two additional questions today or tomorrow. Todays agenda
More informationScheduling Sporadic Tasks with Shared Resources in HardRealTime Systems
Scheduling Sporadic Tasks with Shared Resources in HardReal Systems Kevin Jeffay * University of North Carolina at Chapel Hill Department of Computer Science Chapel Hill, NC 275993175 jeffay@cs.unc.edu
More informationOnline machine scheduling with batch setups
Online machine scheduling with batch setups Lele Zhang, Andrew Wirth Department of Mechanical Engineering The University of Melbourne, VIC 3010, Australia Abstract We study a class of scheduling problems
More informationRealTime Scheduling (Part 1) (Working Draft) RealTime System Example
RealTime Scheduling (Part 1) (Working Draft) Insup Lee Department of Computer and Information Science School of Engineering and Applied Science University of Pennsylvania www.cis.upenn.edu/~lee/ CIS 41,
More informationNear Optimal Solutions
Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NPComplete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More informationPlanning and Scheduling in the Digital Factory
Institute for Computer Science and Control Hungarian Academy of Sciences Berlin, May 7, 2014 1 Why "digital"? 2 Some Planning and Scheduling problems 3 Planning for "oneofakind" products 4 Scheduling
More informationFactors to Describe Job Shop Scheduling Problem
Job Shop Scheduling Job Shop A work location in which a number of general purpose work stations exist and are used to perform a variety of jobs Example: Car repair each operator (mechanic) evaluates plus
More informationLecture Outline Overview of realtime scheduling algorithms Outline relative strengths, weaknesses
Overview of RealTime Scheduling Embedded RealTime Software Lecture 3 Lecture Outline Overview of realtime scheduling algorithms Clockdriven Weighted roundrobin Prioritydriven Dynamic vs. static Deadline
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationTwo Algorithms for the StudentProject Allocation Problem
Two Algorithms for the StudentProject Allocation Problem David J. Abraham 1, Robert W. Irving 2, and David F. Manlove 2 1 Computer Science Department, CarnegieMellon University, 5000 Forbes Ave, Pittsburgh
More informationToday s topic: So far, we have talked about. Question. Task models
Overall Stucture of Real Time Systems So far, we have talked about Task...... Task n Programming Languages to implement the Tasks RunTIme/Operating Systems to run the Tasks RTOS/RunTime System Hardware
More informationA Note on Maximum Independent Sets in Rectangle Intersection Graphs
A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,
More informationA Computer Application for Scheduling in MS Project
Comput. Sci. Appl. Volume 1, Number 5, 2014, pp. 309318 Received: July 18, 2014; Published: November 25, 2014 Computer Science and Applications www.ethanpublishing.com Anabela Tereso, André Guedes and
More informationR u t c o r Research R e p o r t. A Method to Schedule Both Transportation and Production at the Same Time in a Special FMS.
R u t c o r Research R e p o r t A Method to Schedule Both Transportation and Production at the Same Time in a Special FMS Navid Hashemian a Béla Vizvári b RRR 32011, February 21, 2011 RUTCOR Rutgers
More informationA simpler and better derandomization of an approximation algorithm for Single Source RentorBuy
A simpler and better derandomization of an approximation algorithm for Single Source RentorBuy David P. Williamson Anke van Zuylen School of Operations Research and Industrial Engineering, Cornell University,
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationResearch Article Batch Scheduling on TwoMachine Flowshop with MachineDependent Setup Times
Hindawi Publishing Corporation Advances in Operations Research Volume 2009, Article ID 153910, 10 pages doi:10.1155/2009/153910 Research Article Batch Scheduling on TwoMachine Flowshop with MachineDependent
More informationConstraintBased Scheduling: A Tutorial
ConstraintBased Scheduling: A Tutorial Claude Le Pape ILOG S. A. 9, rue de Verdun F94253 Gentilly Cedex Tel: +33 1 49 08 29 67 Fax: +33 1 49 08 35 10 Email: clepape@ilog.fr Given a set of resources with
More informationUniversity Dortmund. Robotics Research Institute Information Technology. Job Scheduling. Uwe Schwiegelshohn. EPIT 2007, June 5 Ordonnancement
University Dortmund Robotics Research Institute Information Technology Job Scheduling Uwe Schwiegelshohn EPIT 2007, June 5 Ordonnancement ontent of the Lecture What is ob scheduling? Single machine problems
More informationA Branch and Bound Algorithm for Solving the Binary Bilevel Linear Programming Problem
A Branch and Bound Algorithm for Solving the Binary Bilevel Linear Programming Problem John Karlof and Peter Hocking Mathematics and Statistics Department University of North Carolina Wilmington Wilmington,
More information! Solve problem to optimality. ! Solve problem in polytime. ! Solve arbitrary instances of the problem. !approximation algorithm.
Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NPhard problem What should I do? A Theory says you're unlikely to find a polytime algorithm Must sacrifice one of
More informationClustering and scheduling maintenance tasks over time
Clustering and scheduling maintenance tasks over time Per Kreuger 20080429 SICS Technical Report T2008:09 Abstract We report results on a maintenance scheduling problem. The problem consists of allocating
More informationPartitioned realtime scheduling on heterogeneous sharedmemory multiprocessors
Partitioned realtime scheduling on heterogeneous sharedmemory multiprocessors Martin Niemeier École Polytechnique Fédérale de Lausanne Discrete Optimization Group Lausanne, Switzerland martin.niemeier@epfl.ch
More informationRUN: Optimal Multiprocessor RealTime Scheduling via Reduction to Uniprocessor
1 RUN: Optimal Multiprocessor RealTime Scheduling via Reduction to Uniprocessor Paul Regnier, George Lima, Ernesto Massa Federal University of Bahia, State University of Bahia Salvador, Bahia, Brasil
More informationMultiRobot Task Scheduling
MultiRobot Task Scheduling Yu ("Tony") Zhang Lynne E. Parker Distributed Intelligence Laboratory Electrical Engineering and Computer Science Department University of Tennessee, Knoxville TN, USA IEEE
More informationHeuristic Algorithms for Open Shop Scheduling to Minimize Mean Flow Time, Part I: Constructive Algorithms
Heuristic Algorithms for Open Shop Scheduling to Minimize Mean Flow Time, Part I: Constructive Algorithms Heidemarie Bräsel, André Herms, Marc Mörig, Thomas Tautenhahn, Jan Tusch, Frank Werner OttovonGuerickeUniversität,
More informationHow To Find An Optimal Search Protocol For An Oblivious Cell
The Conference Call Search Problem in Wireless Networks Leah Epstein 1, and Asaf Levin 2 1 Department of Mathematics, University of Haifa, 31905 Haifa, Israel. lea@math.haifa.ac.il 2 Department of Statistics,
More informationBATCH SCHEDULING IN CUSTOMERCENTRIC SUPPLY CHAINS
Journal of the Operations Research Society of Japan 2006, Vol. 49, No. 3, 174187 BATCH SCHEDULING IN CUSTOMERCENTRIC SUPPLY CHAINS Esaignani Selvarajah George Steiner McMaster University (Received July
More informationIntegrated support system for planning and scheduling... 2003/4/24 page 75 #101. Chapter 5 Sequencing and assignment Strategies
Integrated support system for planning and scheduling... 2003/4/24 page 75 #101 Chapter 5 Sequencing and assignment Strategies 5.1 Overview This chapter is dedicated to the methodologies used in this work
More informationThe StudentProject Allocation Problem
The StudentProject Allocation Problem David J. Abraham, Robert W. Irving, and David F. Manlove Department of Computing Science, University of Glasgow, Glasgow G12 8QQ, UK Email: {dabraham,rwi,davidm}@dcs.gla.ac.uk.
More informationScheduling Parallel Jobs with Linear Speedup
Scheduling Parallel Jobs with Linear Speedup Alexander Grigoriev and Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands. Email: {a.grigoriev,m.uetz}@ke.unimaas.nl
More informationGuessing Game: NPComplete?
Guessing Game: NPComplete? 1. LONGESTPATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTESTPATH: Given a graph G = (V, E), does there exists a simple
More informationA Nearlinear Time Constant Factor Algorithm for Unsplittable Flow Problem on Line with Bag Constraints
A Nearlinear Time Constant Factor Algorithm for Unsplittable Flow Problem on Line with Bag Constraints Venkatesan T. Chakaravarthy, Anamitra R. Choudhury, and Yogish Sabharwal IBM Research  India, New
More informationAlgorithm Design and Analysis
Algorithm Design and Analysis LECTURE 27 Approximation Algorithms Load Balancing Weighted Vertex Cover Reminder: Fill out SRTEs online Don t forget to click submit Sofya Raskhodnikova 12/6/2011 S. Raskhodnikova;
More information1 Approximating Set Cover
CS 05: Algorithms (Grad) Feb 224, 2005 Approximating Set Cover. Definition An Instance (X, F ) of the setcovering problem consists of a finite set X and a family F of subset of X, such that every elemennt
More informationCSE 4351/5351 Notes 7: Task Scheduling & Load Balancing
CSE / Notes : Task Scheduling & Load Balancing Task Scheduling A task is a (sequential) activity that uses a set of inputs to produce a set of outputs. A task (precedence) graph is an acyclic, directed
More information6.852: Distributed Algorithms Fall, 2009. Class 2
.8: Distributed Algorithms Fall, 009 Class Today s plan Leader election in a synchronous ring: Lower bound for comparisonbased algorithms. Basic computation in general synchronous networks: Leader election
More informationProduction Planning Solution Techniques Part 1 MRP, MRPII
Production Planning Solution Techniques Part 1 MRP, MRPII Mads Kehlet Jepsen Production Planning Solution Techniques Part 1 MRP, MRPII p.1/31 Overview Production Planning Solution Techniques Part 1 MRP,
More informationSteiner Tree Approximation via IRR. Randomized Rounding
Steiner Tree Approximation via Iterative Randomized Rounding Graduate Program in Logic, Algorithms and Computation μπλ Network Algorithms and Complexity June 18, 2013 Overview 1 Introduction Scope Related
More informationThe power of points in preemptive single machine scheduling
JOURNAL OF SCHEDULING J. Sched. 22; 5:121 133 (DOI: 1.12/jos.93) The power of points in preemptive single machine scheduling Andreas S. Schulz 1; 2; ; and Martin Skutella 1 Massachusetts Institute of
More informationAbstract Title: Planned Preemption for Flexible Resource Constrained Project Scheduling
Abstract number: 0150551 Abstract Title: Planned Preemption for Flexible Resource Constrained Project Scheduling Karuna Jain and Kanchan Joshi Shailesh J. Mehta School of Management, Indian Institute
More informationDynamic programming formulation
1.24 Lecture 14 Dynamic programming: Job scheduling Dynamic programming formulation To formulate a problem as a dynamic program: Sort by a criterion that will allow infeasible combinations to be eli minated
More informationA Practical Scheme for Wireless Network Operation
A Practical Scheme for Wireless Network Operation Radhika Gowaikar, Amir F. Dana, Babak Hassibi, Michelle Effros June 21, 2004 Abstract In many problems in wireline networks, it is known that achieving
More informationPeriodic Task Scheduling
Periodic Task Scheduling Radek Pelánek Motivation and Assumptions Examples of Periodic Tasks sensory data acquisition control loops action planning system monitoring Motivation and Assumptions Simplifying
More informationReinforcement Learning
Reinforcement Learning LU 2  Markov Decision Problems and Dynamic Programming Dr. Joschka Bödecker AG Maschinelles Lernen und Natürlichsprachliche Systeme AlbertLudwigsUniversität Freiburg jboedeck@informatik.unifreiburg.de
More informationAbout the inverse football pool problem for 9 games 1
Seventh International Workshop on Optimal Codes and Related Topics September 61, 013, Albena, Bulgaria pp. 15133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute
More informationAnalysis of Algorithms I: Optimal Binary Search Trees
Analysis of Algorithms I: Optimal Binary Search Trees Xi Chen Columbia University Given a set of n keys K = {k 1,..., k n } in sorted order: k 1 < k 2 < < k n we wish to build an optimal binary search
More informationOn the approximability of average completion time scheduling under precedence constraints
On the approximability of average completion time scheduling under precedence constraints Gerhard J. Woeginger Abstract We consider the scheduling problem of minimizing the average weighted job completion
More informationFIND ALL LONGER AND SHORTER BOUNDARY DURATION VECTORS UNDER PROJECT TIME AND BUDGET CONSTRAINTS
Journal of the Operations Research Society of Japan Vol. 45, No. 3, September 2002 2002 The Operations Research Society of Japan FIND ALL LONGER AND SHORTER BOUNDARY DURATION VECTORS UNDER PROJECT TIME
More information1.204 Lecture 10. Greedy algorithms: Job scheduling. Greedy method
1.204 Lecture 10 Greedy algorithms: Knapsack (capital budgeting) Job scheduling Greedy method Local improvement method Does not look at problem globally Takes best immediate step to find a solution Useful
More information4. FixedPriority Scheduling
Simple workload model 4. FixedPriority Scheduling Credits to A. Burns and A. Wellings The application is assumed to consist of a fixed set of tasks All tasks are periodic with known periods This defines
More informationAN ANALYSIS OF THE IMPORTANCE OF APPROPRIATE TIE BREAKING RULES IN DISPATCH HEURISTICS
versão impressa ISSN 01017438 / versão online ISSN 16785142 AN ANALYSIS OF THE IMPORTANCE OF APPROPRIATE TIE BREAKING RULES IN DISPATCH HEURISTICS Jorge M. S. Valente Faculdade de Economia Universidade
More informationA Note on the Bertsimas & Sim Algorithm for Robust Combinatorial Optimization Problems
myjournal manuscript No. (will be inserted by the editor) A Note on the Bertsimas & Sim Algorithm for Robust Combinatorial Optimization Problems Eduardo ÁlvarezMiranda Ivana Ljubić Paolo Toth Received:
More informationOR topics in MRPII. Mads Jepsens. OR topics in MRPII p.1/25
OR topics in MRPII Mads Jepsens OR topics in MRPII p.1/25 Overview Why bother? OR topics in MRPII p.2/25 Why bother? Push and Pull systems Overview OR topics in MRPII p.2/25 Overview Why bother? Push
More informationDiscrete Optimization
Discrete Optimization [Chen, Batson, Dang: Applied integer Programming] Chapter 3 and 4.14.3 by Johan Högdahl and Victoria Svedberg Seminar 2, 20150331 Todays presentation Chapter 3 Transforms using
More informationHow To Calculate A Quotation For A Product Order With A Time Frame
Quotation for Orders with Availability Intervals and LeadTime Sensitive Revenues Pinar Keskinocak R. Ravi Sridhar Tayur School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta,
More informationAperiodic Task Scheduling
Aperiodic Task Scheduling Gerhard Fohler Mälardalen University, Sweden gerhard.fohler@mdh.se RealTime Systems Gerhard Fohler 2005 Non Periodic Tasks So far periodic events and tasks what about others?
More informationA Dynamic and ReliabilityDriven Scheduling Algorithm for Parallel Realtime Jobs on Heterogeneous Clusters
A Dynamic and ReliabilityDriven Scheduling Algorithm for Parallel Realtime Jobs on Heterogeneous Clusters Xiao Qin Department of Computer Science, New Mexico Institute of Mining and Technology, 801 Leroy
More informationATheoryofComplexity,Conditionand Roundoff
ATheoryofComplexity,Conditionand Roundoff Felipe Cucker Berkeley 2014 Background L. Blum, M. Shub, and S. Smale [1989] On a theory of computation and complexity over the real numbers: NPcompleteness,
More informationReinforcement Learning
Reinforcement Learning LU 2  Markov Decision Problems and Dynamic Programming Dr. Martin Lauer AG Maschinelles Lernen und Natürlichsprachliche Systeme AlbertLudwigsUniversität Freiburg martin.lauer@kit.edu
More informationOptimal energy tradeoff schedules
Optimal energy tradeoff schedules Neal Barcelo, Daniel G. Cole, Dimitrios Letsios, Michael Nugent, Kirk R. Pruhs To cite this version: Neal Barcelo, Daniel G. Cole, Dimitrios Letsios, Michael Nugent,
More informationClosest Pair of Points. Kleinberg and Tardos Section 5.4
Closest Pair of Points Kleinberg and Tardos Section 5.4 Closest Pair of Points Closest pair. Given n points in the plane, find a pair with smallest Euclidean distance between them. Fundamental geometric
More information