Limit set - Revision history

165.91.114.83: /* Definition for flows */

2014-06-01T15:35:04Z

Definition for flows

← Older revision		Revision as of 17:35, 1 June 2014
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	~~{{Expert-subject\|Mathematics\|date=November 2008}}~~		Alyson Meagher is the name [http://medialab.zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- authentic psychic readings] her parents gave her but she doesn't like when people use her complete title. To perform lacross is one of the things she loves most. Her family members lives in Ohio but her spouse desires them to move. Office supervising is exactly where her primary income comes from but she's currently utilized for an additional 1.<br><br>Here is my page: free tarot [http://www.publicpledge.com/blogs/post/7034 love psychic readings] ([https://www.machlitim.org.il/subdomain/megila/end/node/12300 cool training])
	A convenient '''notation for theoretic scheduling problems''' was introduced by [[Ronald Graham]], [[Eugene Lawler]], [[Jan Karel Lenstra]] and [[Alexander Rinnooy Kan]] in.<ref name="initial specification" /> It consists of three fields: [[Alpha (letter)\|α]], [[Beta (letter)\|β]] and [[Gamma\|γ]].

	~~Each field may be a comma separated list of words. The α field describes the machine environment, β the job characteristics, and γ the objective function.~~

	~~== Machine environment ==~~

	~~=== Single stage problems ===~~
	~~Each job comes with a given processing time.~~

	~~; 1~~
	~~: there is a single machine~~
	~~; P~~
	~~: there are <math>m</math> parallel identical machines~~
	~~; Q~~
	~~: there are <math>m</math> parallel machines with different given speeds, length of job <math>i</math> on machine <math>j</math>~~ is the ~~processing time <math>p_{i}</math> divided by speed <math>s_j</math>~~
	~~; R~~
	~~: there are <math>m</math> parallel unrelated machines, there are given processing times <math>p_{ij}</math> for job <math>i</math> on machine <math>j</math>~~

	~~The last two letters might be followed by the number of machines which is then fixed, here <math>m</math> stands then for a fixed number.~~

	~~=== Multi-stage problem ===~~
	~~; O :~~ [~~[Open Shop Scheduling\|open shop problem]]~~
	~~; F~~ : ~~[[Flow Shop Scheduling Problem\|flow shop problem]]~~
	~~; J : [[Job Shop Scheduling\|job shop problem]]~~

	~~== Job characteristics ==~~
	~~The processing time may be equal for all jobs (<math>p_i=p<~~/~~math>, or <math>p_{ij}=p<~~/~~math>) or even of unit length (<math>p_i=1</math>, or <math>p_{ij}=1</math>)~~. ~~This makes a difference because all release times, deadlines are assumed to be integer~~.

	~~; <math>r_i<~~/~~math>~~
	~~: for each job a release time is given before which it cannot be scheduled, default is 0.~~
	~~; <math>d_i<~~/~~math>~~
	~~: for each job a deadline is given after which it cannot be scheduled~~. ~~If the objective~~ is ~~<math>\sum U_i</math> for example, then this field is implicitly assumed.~~
	~~; pmtn~~
	~~: the jobs may be preempted and execution resumed later, possibly on a different machine~~
	~~; <math>size_i</math>~~
	~~: Each job comes with a number~~ of ~~machines on which it must be scheduled at~~ the ~~same time, default is 1~~.

	~~Precedence relations might be given for the jobs,~~ in ~~form of a partial order, meaning that if i~~ is ~~a predecessor of i~~' ~~in that order, i' can start only when i is completed.~~

	~~; prec~~
	: an ~~arbitrary precedence relation is given~~
	~~; sp-tree, tree, intree, outtree, chain~~
	~~: specific partial orders~~

	~~== Objective functions ==~~
	Most objective functions depend on the deadline <math>d_i</math> and the completion time <math>C_i</math> of job <math>i</math>. We define lateness <math>L_i=C_i-d_i</math>, earliness <math>E_i = \max\{0, d_i-C_i\}</math>, tardiness <math>T_i = \max\{0, C_i-d_i\}</math>, unit penalty <math>U_i = 0</math> if <math>C_i\le d_i</math> and <math>U_i=1~~</math> otherwise~~. ~~The common objective functions are~~ <~~math~~>~~C_\max, L_\max, E_\max, T_\max, \sum C_i, \sum L_i, \sum E_i, \sum T_i~~<~~/math~~> ~~or weighted version of these sums, where every job comes with a priority <math>w_i</math>.~~

	~~== Examples ==~~
	~~Adapted from <ref name="initial specification">{{cite conference~~
	~~\| last1 = Graham \| first1 = R. L.~~
	~~\| last2 = Lawler \| first2 = E. L.~~
	~~\| last3 = Lenstra \| first3 = J.K.~~
	~~\| last4 = Rinnooy Kan \| first4 = A.H.G.~~
	~~\| title = Optimization and Approximation in Deterministic Suquencing and Scheduling~~: ~~a Survey~~
	~~\| booktitle = Proceedings of the Advanced Research Institute on Discrete Optimization and Systems Applications of the Systems Science Panel of NATO and of the Discrete Optimization Symposium~~
	~~\| publisher = Elsevier~~
	~~\| year = 1979~~
	~~\| pages = (5) 287–326~~
	~~}}<~~/~~ref>~~

	~~; 1\|prec\|<math>L_\max<~~/~~math>: a single machine, general precedence constraint, minimizing maximum lateness~~.

	~~; R\|pnmt\|<math>\sum C_i<~~/~~math>: variable number of unrelated parallel machines, allowing preemption, minimizing total completion time.~~

	~~; J3\|<math>p_{ij}<~~/~~math>\|<math>C_\max<~~/~~math>: 3-machines job shop with unit processing times, minimizing maximum completion time.~~

	~~== References ==~~
	* B. Chen, C.N. Potts and G.J. Woeginger. "A review of machine scheduling: Complexity, algorithms and approximability". ''Handbook of Combinatorial Optimization'' (Volume 3) (Editors: D.-Z. Du and P. Pardalos), 1998, Kluwer Academic Publishers. 21-169. ISBN 0-7923-5285-8 (~~HB) 0-7923-5019-7 (Set)~~
	*[~~http~~://www.~~informatik~~.~~uni-osnabrueck~~.de/~~knust~~/~~class~~/ ~~Peter Brucker, Sigrid Knust. Complexity results for scheduling problems~~]
	~~{{Reflist}}~~

	~~[[Category:Theoretical computer science]]~~
	~~[[Category:Scheduling (computing~~)]]

en>BD2412: /* Examples */Fixing links to disambiguation pages using AWB

2013-12-16T20:06:52Z

Examples: Fixing links to disambiguation pages using AWB

← Older revision		Revision as of 22:06, 16 December 2013
Line 1:		Line 1:
	The ~~writer~~ is ~~called Irwin Wunder but it's not~~ the ~~most masucline name out~~ there. ~~Doing ceramics~~ is ~~what her family~~ and ~~her appreciate~~. ~~Years ago we moved to North Dakota. She~~ is a ~~librarian but she~~'~~s always needed her own business~~.<br><br>~~my web blog~~ ... ~~[http~~://~~www~~.~~articlestunner~~.~~com~~/~~cures~~-~~for~~-~~the~~-~~yeast~~-~~infection~~-~~suggestions~~-to-~~use~~-~~now~~/ ~~std testing at home~~]		{{Expert-subject\|Mathematics\|date=November 2008}}
			A convenient '''notation for theoretic scheduling problems''' was introduced by [[Ronald Graham]], [[Eugene Lawler]], [[Jan Karel Lenstra]] and [[Alexander Rinnooy Kan]] in.<ref name="initial specification" /> It consists of three fields: [[Alpha (letter)\|α]], [[Beta (letter)\|β]] and [[Gamma\|γ]].

			Each field may be a comma separated list of words. The α field describes the machine environment, β the job characteristics, and γ the objective function.

			== Machine environment ==

			=== Single stage problems ===
			Each job comes with a given processing time.

			; 1
			: there is a single machine
			; P
			: there are <math>m</math> parallel identical machines
			; Q
			: there are <math>m</math> parallel machines with different given speeds, length of job <math>i</math> on machine <math>j</math> is the processing time <math>p_{i}</math> divided by speed <math>s_j</math>
			; R
			: there are <math>m</math> parallel unrelated machines, there are given processing times <math>p_{ij}</math> for job <math>i</math> on machine <math>j</math>

			The last two letters might be followed by the number of machines which is then fixed, here <math>m</math> stands then for a fixed number.

			=== Multi-stage problem ===
			; O : [[Open Shop Scheduling\|open shop problem]]
			; F : [[Flow Shop Scheduling Problem\|flow shop problem]]
			; J : [[Job Shop Scheduling\|job shop problem]]

			== Job characteristics ==
			The processing time may be equal for all jobs (<math>p_i=p</math>, or <math>p_{ij}=p</math>) or even of unit length (<math>p_i=1</math>, or <math>p_{ij}=1</math>). This makes a difference because all release times, deadlines are assumed to be integer.

			; <math>r_i</math>
			: for each job a release time is given before which it cannot be scheduled, default is 0.
			; <math>d_i</math>
			: for each job a deadline is given after which it cannot be scheduled. If the objective is <math>\sum U_i</math> for example, then this field is implicitly assumed.
			; pmtn
			: the jobs may be preempted and execution resumed later, possibly on a different machine
			; <math>size_i</math>
			: Each job comes with a number of machines on which it must be scheduled at the same time, default is 1.

			Precedence relations might be given for the jobs, in form of a partial order, meaning that if i is a predecessor of i' in that order, i' can start only when i is completed.

			; prec
			: an arbitrary precedence relation is given
			; sp-tree, tree, intree, outtree, chain
			: specific partial orders

			== Objective functions ==
			Most objective functions depend on the deadline <math>d_i</math> and the completion time <math>C_i</math> of job <math>i</math>. We define lateness <math>L_i=C_i-d_i</math>, earliness <math>E_i = \max\{0, d_i-C_i\}</math>, tardiness <math>T_i = \max\{0, C_i-d_i\}</math>, unit penalty <math>U_i = 0</math> if <math>C_i\le d_i</math> and <math>U_i=1</math> otherwise. The common objective functions are <math>C_\max, L_\max, E_\max, T_\max, \sum C_i, \sum L_i, \sum E_i, \sum T_i</math> or weighted version of these sums, where every job comes with a priority <math>w_i</math>.

			== Examples ==
			Adapted from <ref name="initial specification">{{cite conference
			\| last1 = Graham \| first1 = R. L.
			\| last2 = Lawler \| first2 = E. L.
			\| last3 = Lenstra \| first3 = J.K.
			\| last4 = Rinnooy Kan \| first4 = A.H.G.
			\| title = Optimization and Approximation in Deterministic Suquencing and Scheduling: a Survey
			\| booktitle = Proceedings of the Advanced Research Institute on Discrete Optimization and Systems Applications of the Systems Science Panel of NATO and of the Discrete Optimization Symposium
			\| publisher = Elsevier
			\| year = 1979
			\| pages = (5) 287–326
			}}</ref>

			; 1\|prec\|<math>L_\max</math>: a single machine, general precedence constraint, minimizing maximum lateness.

			; R\|pnmt\|<math>\sum C_i</math>: variable number of unrelated parallel machines, allowing preemption, minimizing total completion time.

			; J3\|<math>p_{ij}</math>\|<math>C_\max</math>: 3-machines job shop with unit processing times, minimizing maximum completion time.

			== References ==
			* B. Chen, C.N. Potts and G.J. Woeginger. "A review of machine scheduling: Complexity, algorithms and approximability". ''Handbook of Combinatorial Optimization'' (Volume 3) (Editors: D.-Z. Du and P. Pardalos), 1998, Kluwer Academic Publishers. 21-169. ISBN 0-7923-5285-8 (HB) 0-7923-5019-7 (Set)
			*[http://www.informatik.uni-osnabrueck.de/knust/class/ Peter Brucker, Sigrid Knust. Complexity results for scheduling problems]
			{{Reflist}}

			[[Category:Theoretical computer science]]
			[[Category:Scheduling (computing)]]

en>Braincricket: Typos and general fixes, typos fixed: , → , (2) using AWB (8062)

2012-07-05T12:23:12Z

Typos and general fixes, typos fixed: , → , (2) using AWB (8062)

New page

The writer is called Irwin Wunder but it's not the most masucline name out there. Doing ceramics is what her family and her appreciate. Years ago we moved to North Dakota. She is a librarian but she's always needed her own business.<br><br>my web blog ... [http://www.articlestunner.com/cures-for-the-yeast-infection-suggestions-to-use-now/ std testing at home]