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Practical Set Partitioning and Column Generation

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PRACTICAL SET PARTITIONING AND COLUMN GENERATION Andrew Mason [email protected], www.esc.auckland.ac.nz/Mason Linkoping 1999, Auckland 2000, Auckland 2001 Set Partitioning Problems Given a set of objects (with index set I), find a minimal cost partition of I into mutually disjoint subsets. Example: Copying 2 CD’s onto C60 tapes. ENIGMA MCMXC Mins ENIGMA the CROSS of Mins 1 The Voice of Enigma 2.13 changes 2 Sadeness 4.25 1 Second Chapter 2.27 3 Find Love 4.82 2 The Eyes of Truth 7.22 4 Sadeness (Reprise) 2.80 3 Return to Innocence 4.28 5 Callas Went Away 4.48 4 I Love You… I'll Kill You 8.85 6 Mea Culpa 4.87 5 Silent Warrior 6.17 7 The Voice & The Snake 1.75 6 The Dream of the Dolphin 2.78 8 Knocking on Forbidden 4.45 7 Age of Loneliness 5.37 Doors 8 Out from the Deep 4.88 9 Way to Eternity 2.30 9 The CROSS of changes 2.38 10 Hallelujah 4.25 Total: 44.20 11 The Rivers of Belief 3.52 12 Sadeness II 2.72 13 Mea Culpa II 6.07 14 Principles of Lust 4.83 15 The Rivers of Belief II 7.07 Total: 60.30 Set of objects: Possible Subsets: Cost of Subsets (assuming minimisation objective): This particular problem is known as: (c) A. Mason www.esc.auckland.ac.nz/Mason/ 1+ This material is not to be distrubted; contact the author for the latest version and permission to distribute. Formal Set Partitioning Definition: Given: 1/ I={1…m} 2/ a collection of subsets P={P1, P2, …, Pn}, where each PjÍP 3/ a cost function c(Pj) then JÌ{1, …, n} defines a partition of I if and only if: 1/ (all elements in a subset) 2/ We seek a minimum cost partition: min sum_j in J c(Pj) st J partitioning I Integer Programming (IP) Formulation of Set Partitioning: · Rows correspond to elements of I · Columns are elements of P · aij=1 if element I is in Pj, aij=0 otherwise · cj=cost of Pj · xj=1 if Pj is in the partition · all items must be included in soln Variables: Matrix Coefficients: Right hand side: Constraints: LP Dual Variables: Note: X integer => will be Binary to satisfy constraints Set Partitioning Example 1: Airline Planning (Pairings, Tours of Duty) Problem Partition ……………..….………………..… into …….. (These tours will later be allocated to people.) (c) A. Mason www.esc.auckland.ac.nz/Mason/ 2+ This material is not to be distrubted; contact the author for the latest version and permission to distribute. Mon Tue Wed Mon 00 06 00 00 0000 0600 1200 1800 0000 0600 1200 1800 0000 0600 1200 1800 AKL AKL CHC CHC WLG WLG HNL HNL LAX LAX MEL MEL SYD SYD SIN SIN LHR LHR BNE BNE CNS CNS DPS DPS FKF FKF Example and Picture from Air New Zealand Costs include: Rules for building columns include: Integer Programming Formulation: min x st x Set Partitioning Example 2: Political Districting (Images stolen from http://www.elections.org.nz/elections/general/electorates/index.html) (c) A. Mason www.esc.auckland.ac.nz/Mason/ 3+ This material is not to be distrubted; contact the author for the latest version and permission to distribute. Partition ……….…census area units…………………….into ……………electorates…… Costs: Shape of electorate, natural unit (eg not split by rivers), deviation from desired populationsize! Possible Additional Constraint: Must have 61 electorates Other Set Partitioning Examples: Vehicle Routing - columns are routes, rows are deliveries to make Bin Packing Set Packing Problems Variables: Integer (0/1) Matrix Coefficients: Binary Right hand side: Binary Constraints: LP Dual Variables: Set Packing Example: Cutting of Boards 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 6 7 8 9 10 6 7 8 9 10 6 7 8 9 10 11 12 13 14 15 11 12 13 14 15 11 12 13 14 15 11 12 13 14 15 max x st 1 2 3 4 5 x 6 7 8 9 10 11 12 13 14 15 (c) A. Mason www.esc.auckland.ac.nz/Mason/ 4+ This material is not to be distrubted; contact the author for the latest version and permission to distribute. Set Covering Problems Variables: Integer (0/1) Matrix Coefficients: Binary Right hand side: Binary Constraints: LP Dual Variables: Set Covering Example: Mail Deliveries · Must walk along each street in town to deliver mail for that street. · Each person starts at 5am, must be finished by 7am. · If two people walk a street, only 1 does the deliveries (hence ‘covering’) Columns: Cost: Set Covering vs Set Packing If changing any 1 in a column into a 0 gives a valid no more expensive column, then the set covering and set packing solutions are the same. Set Partitioning/Covering/Packing Generalisations: Different Possibilities: Right Hand Side: Binary or Integer (or Real) Variables: Binary or Integer A-Matrix coefficients: Binary or Integer (or Real) Constraints: Mix of <, =, > (c) A. Mason www.esc.auckland.ac.nz/Mason/ 5+ This material is not to be distrubted; contact the author for the latest version and permission to distribute. Generalised Set Covering Example: Single-Day Shift Generation Columns are: Costs: 1 1 1 1 b1 1 1 1 1 1 1 1 1 b2 1 1 1 1 1 1 1 1 1 1 1 1 b3 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … b4 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … b5 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … b6 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … b7 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … b8 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … b9 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … b10 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … b11 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … b12 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … b13 1 … 1 1 1 … 1 1 1 … 1 1 1 … b14 1 1 1 … 1 1 1 … 1 1 1 … b15 1 1 1 … 1 1 1 … 1 1 1 … 1 1 … 1 1 1 … 1 1 1 … 1 … 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … 1 1 1 … 1 1 … 1 1 1 … A= 1 … 1 1 1 … 1 1 1 … b= : : : 1 1 1 … 1 1 1 … : 1 1 1 … 1 1 1 … 1 1 1 … 1 1 … 1 … . bm-6 . bm-5 bm-4 bm-3 1 : : 1 : : 1 : : 1 : : bm-2 1 1 1 1 1 1 1 1 bm-1 3 hour shifts 1 4 hour shifts 1 5 hour shifts 1 8 hour shifts 1 bm Variables: Matrix Coefficients: Right hand side: Constraints: NS+Mon-Min Part time 25 Full time Smoothed Workload 20 Arrivals Departures 15 10 Officers on Duty 5 0 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 Generalised Set Covering: Example from NZ Customs (c) A. Mason www.esc.auckland.ac.nz/Mason/ 6+ This material is not to be distrubted; contact the author for the latest version and permission to distribute. Generalised Set Covering Example: Personalised Single-Day Shift Generation Note: The Gx=e constraints are known as…GUB (generalised upper bnd) OR convexity min x st x x Variables: Matrix Coefficients: Right hand side: Constraints: (c) A. Mason www.esc.auckland.ac.nz/Mason/ 7+ This material is not to be distrubted; contact the author for the latest version and permission to distribute. Set Partitioning Example: ToD Allocation to Crew (“Rostering”) The problem here is take the optimal Tours of Duty (ToDs) produced earlier, and allocate them to staff, eg cabin crew. We assume that each ToD requires 4 cabin crew. ToDs Week 1 Week 2 Week 3 Week 4 ToDs Week 1 Week 2 1 1 W 2 2 k 3 3 W 4 4 k 5 5 6 6 W 7 7 k 8 8 9 9 10 10 W k 11 11 12 12 W T F S S M W T F S S M W T F S S M W T F S S M W T F S S M W T F S S M T T T T min x st x 1 2 3 4 5 x 6 7 8 9 10 11 12 Variables: Matrix Coefficients: Right hand side: Constraints: (c) A. Mason www.esc.auckland.ac.nz/Mason/ 8+ This material is not to be distrubted; contact the author for the latest version and permission to distribute. Generalised Set Covering Example: Group Single-Day Shift Generation Building shifts with couples who prefer to work together. min x st x x Variables: Matrix Coefficients: Right hand side: Constraints: Fractional matrix coefficients can arise, eg… (c) A. Mason www.esc.auckland.ac.nz/Mason/ 9+ This material is not to be distrubted; contact the author for the latest version and permission to distribute. Elastic Constraints Eg for generalised set partitioning: Introduce costed slack and surplus variables Min (c | cslack | csurplus) (x xslack xsurplus ) St. (A | I | -I) (x xslack xsurplus ) = b min c_slack c_surplus min st 1 -1 st 1 -1 1 -1 1 -1 1 -1 Notes: don’t normally enforce integrality of slack/surplus variables (happens naturally) can put bounds on u and s, and/or piecewise linear costs This problem is more stable in the sense that the LP Dual Variables are now….
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    Arista Records LLC et al v. Lime Wire LLC et al Doc. 45 UNITED STATES DISTRICT COURT SOUTHERN DISTRICT OF NEW YORK ARISTA RECORDS LLC; ATLANTIC RECORDING CORPORATION; BMG MUSIC; CAPITOL RECORDS, INC.; ELEKTRA ENTERTAINMENT GROUP INC.; INTERSCOPE RECORDS; LAFACE RECORDS LLC; ECF CASE MOTOWN RECORD COMPANY, L.P.; PRIORITY RECORDS LLC; SONY BMG MUSIC FIRST AMENDED COMPLAINT FOR ENTERTAINMENT; UMG RECORDINGS, INC.; FEDERAL COPYRIGHT VIRGIN RECORDS AMERICA, INC.; and INFRINGEMENT, COMMON LAW WARNER BROS. RECORDS INC., COPYRIGHT INFRINGEMENT, Plaintiffs, UNFAIR COMPETITION, CONVEYANCE MADE WITH v. INTENT TO DEFRAUD AND UNJUST ENRICHMENT LIME WIRE LLC; LIME GROUP LLC; MARK GORTON; GREG BILDSON, and M.J.G. LIME 06 Civ. 05936 (GEL) WIRE FAMILY LIMITED PARTNERSHIP, Defendants. Plaintiffs hereby allege on personal knowledge as to allegations concerning themselves, and on information and belief as to all other allegations, as follows: NATURE OF THE ACTION 1. Plaintiffs are record companies that produce, manufacture, distribute, sell, and license the vast majority of commercial sound recordings in this country. Defendants' business, operated under the trade name "LimeWire" and variations thereof, is devoted essentially to the Internet piracy of Plaintiffs' sound recordings. Defendants designed, promote, distribute, support and maintain the LimeWire software, system/network, and related services to consumers for the well-known and overarching purpose of making and distributing unlimited copies of Plaintiffs' sound recordings REDACTED VERSION - COMPLETE VERSION FILED UNDER SEAL Dockets.Justia.com without paying Plaintiffs anything. Plaintiffs bring this action to stop Defendants' massive and daily infringement of Plaintiffs' copyrights. 2. The scope of infringement caused by Defendants is staggering. Millions of infringing copies of Plaintiff s sound recordings have been made and distributed through LimeWire — copies that can be and are permanently stored, played, and further distributed by Lime Wire's users.
  • 236 Valentine's Day Songs (2015)

    236 Valentine's Day Songs (2015)

    236 Valentine's Day Songs (2015) Song Name Time Artist Album Theme Lay All Your Love On Me 3:02 ABBA The Definitive Collection passion Take a Chance on Me 4:05 Abba Gold romance Love in an Elevator 5:21 Aerosmith Big Ones fun Sweet emotion 5:10 Aerosmith Armageddon passion The Age of Love (Jam & Spoon Watch Out For Stella Mix) 4:08 Age of Love The Age of Love adventure Akon, Colby O'Donis, Kardinal Beautiful 5:13 Offishall Freedom Love Do You Love Me 2:16 Amanda Jenssen Killing My Darlings Love Sweet Pea 2:09 Amos Lee Supply and Demand passion Let Love Fly (Joe Claussell's Sacred Rhythm LP Version) 5:46 Ananda Project Night Blossom adventure True Love 6:05 Angels and Airwaves I-Empire love Reckless (With Your Love) 5:42 Azari & iii Defected in the House Ibiza passion Romeo 3:26 Basement Jaxx The Singles love Love Can Do You No Harm 3:04 Beangrowers Not In A Million Lovers adventure Able to Love (Sfaction Mix) 3:27 Benny Benassi & The Biz No Matter What You Do / Able to Love adventure Crazy in Love 3:56 Beyonce Dangerously In Love Love Cradle of love 4:38 Billy Idol Billy Idol: Greatest Hits Love Where is the Love 4:33 Black Eyed Peas Elephunk World needs love Heart of Glass 4:35 Blondie The Best of Blondie relationship I Feel Love 3:13 Blue Man Group & Venus Hum The Complex romance One Love/People Get Ready 3:02 Bob Marley & The Wailers Exodus romance Thank You For Loving Me 5:09 Bon Jovi Crush relationship Somebody to Love (UK Club Mix) 131/65bpm 4:02 Boogie Pimps Somebody to Love CDM romance Copyright © 2012 Indoor Cycling Association