Course Introduction Scheduling Outline Complexity Hierarchy DMP204 SCHEDULING, TIMETABLING AND ROUTING 1. Course Introduction Lecture 1 2. Scheduling Introduction to Scheduling: Terminology, Problem Classification Classification 3. Complexity Hierarchy 2 Course Introduction Course Introduction Scheduling Scheduling Outline Course Presentation Complexity Hierarchy Complexity Hierarchy Communication media Black Board (BB). What we use: Mail 1. Course Introduction Announcements Course Documents (for Photocopies) 2. Scheduling Blog – Lecture Diary Problem Classification Electronic hand in of the exam project Web-site http://www.imada.sdu.dk/~marco/DM204/ 3. Complexity Hierarchy Lecture plan and slides Literature and Links Exam documents 3 4
Course Introduction Course Introduction Scheduling Scheduling Evaluation Complexity Hierarchy Complexity Hierarchy Final Assessment (10 ECTS) Schedule Oral exam: 30 minutes + 5 minutes defense project meant to assess the base knowledge Third quarter 2008 Fourth quarter 2008 Group project: Tuesday 10:15-12:00 Wednesday 12:15-14:00 free choice of a case study among few proposed ones Friday 8:15-10:00 Friday 10:15-12:00 Deliverables: program + report meant to assess the ability to apply ∼ 27 lectures Schedule: Project hand in deadline + oral exam in June 5 6 Course Introduction Course Introduction Scheduling Scheduling Course Content Course Material Complexity Hierarchy Complexity Hierarchy General Optimization Methods Mathematical Programming, Constraint Programming, Heuristics Literature Problem Specific Algorithms (Dynamic Programming, Branch and Bound) B1 Pinedo, M. Planning and Scheduling in Manufacturing and Services Springer Verlag, 2005 Scheduling B2 Pinedo, M. Scheduling: Theory, Algorithms, and Systems Springer Single and Parallel Machine Models New York, 2008 Flow Shops and Flexible Flow Shops B3 Toth, P. & Vigo, D. (ed.) The Vehicle Routing Problem SIAM Job Shops Monographs on Discrete Mathematics and Applications, 2002 Resource-Constrained Project Scheduling Slides Timetabling Interval Scheduling, Reservations Class exercises (participatory) Educational Timetabling Workforce and Employee Timetabling Transportation Timetabling Vehicle Routing Capacited Vehicle Routing Vehicle Routing with Time Windows 7 8
Course Introduction Course Introduction Scheduling Scheduling Course Goals and Project Plan The problem Solving Cycle Complexity Hierarchy Complexity Hierarchy How to Tackle Real-life Optimization Problems: The real Formulate (mathematically) the problem problem Model the problem and recognize possible similar problems Search in the literature (or in the Internet) for: Modelling Experimental complexity results (is the problem NP -hard?) Analysis solution algorithms for original problem Mathematical solution algorithms for simplified problem Model Quick Solution: Design solution algorithms Heuristics Test experimentally with the goals of: Algorithm configuring Implementation tuning parameters Mathematical Design of Theory comparing good Solution studying the behavior (prediction of scaling and deviation from Algorithms optimum) 9 10 Course Introduction Course Introduction Scheduling Problem Classification Scheduling Problem Classification Outline Scheduling Complexity Hierarchy Complexity Hierarchy Manufacturing 1. Course Introduction Project planning Single, parallel machine and job shop systems Flexible assembly systems 2. Scheduling Automated material handling (conveyor system) Problem Classification Lot sizing Supply chain planning Services 3. Complexity Hierarchy ⇒ different algorithms 11 12
Course Introduction Course Introduction Scheduling Problem Classification Scheduling Problem Classification Problem Definition Visualization Complexity Hierarchy Complexity Hierarchy Constraints Scheduling are represented by Gantt charts Activities Resources machine-oriented Objectives M 1 J 1 J 2 J 3 J 4 J 5 Problem Definition Given: a set of jobs J = { J 1 , . . . , J n } that have to be processed M 2 J 1 J 2 J 3 J 4 J 5 by a set of machines M = { M 1 , . . . , M m } Find: a schedule , M 3 J 1 J 2 J 3 J 4 J 5 i.e. , a mapping of jobs to machines and processing times time subject to feasibility and optimization constraints. 0 5 10 15 20 or job-oriented Notation: ... n, j, k jobs m, i, h machines 14 15 Course Introduction Scheduling Problem Classification Data Associated to Jobs Complexity Hierarchy Processing time p ij Release date r j Due date d j (called deadline, if strict) Weight w j A job J j may also consist of a number n j of operations O j 1 , O j 2 , . . . , O jn j and data for each operation. Associated to each operation a set of machines µ jl ⊆ M Data that depend on the schedule (dynamic) Starting times S ij Completion time C ij , C j 17
Course Introduction Course Introduction Scheduling Problem Classification Scheduling Problem Classification Problem Classification α | β | γ Classification Scheme Complexity Hierarchy Complexity Hierarchy Machine Environment α 1 α 2 α 1 α 2 α 1 α 2 | β 1 . . . β 13 | γ single machine/multi-machine ( α 1 = α 2 = 1 or α 2 = m ) A scheduling problem is described by a triplet α | β | γ . parallel machines: identical ( α 1 = P ), uniform p j /v i ( α 1 = Q ), α machine environment (one or two entries) unrelated p j /v ij ( α 1 = R ) β job characteristics (none or multiple entry) multi operations models: Flow Shop ( α 1 = F ), Open Shop γ objective to be minimized (one entry) ( α 1 = O ), Job Shop ( α 1 = J ), Mixed (or Group) Shop ( α 1 = X ) Single Machine Flexible Flow Shop Open, Job, Mixed Shop [R.L. Graham, E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan (1979): ( α = FFc ) Optimization and approximation in deterministic sequencing and scheduling: a survey, Ann. Discrete Math. 4, 287-326.] 18 19 Course Introduction Course Introduction Scheduling Problem Classification Scheduling Problem Classification α | β | γ Classification Scheme α | β | γ Classification Scheme Complexity Hierarchy Complexity Hierarchy Job Characteristics α 1 α 2 | β 1 . . . β 13 β 1 . . . β 13 | γ β 1 . . . β 13 β 1 = prmp presence of preemption (resume or repeat) β 1 . . . β 13 Job Characteristics (2) α 1 α 2 | β 1 . . . β 13 β 1 . . . β 13 | γ β 2 precedence constraints between jobs (with α = P, F ) β 8 = brkdwn machines breakdowns acyclic digraph G = ( V, A ) β 9 = M j machine eligibility restrictions (if α = Pm ) β 2 = prec if G is arbitrary β 10 = prmu permutation flow shop β 2 = { chains, intree, outtree, tree, sp - graph } β 11 = block presence of blocking in flow shop (limited buffer) β 3 = r j presence of release dates β 12 = nwt no-wait in flow shop (limited buffer) β 4 = p j = p preprocessing times are equal β 13 = recrc Recirculation in job shop ( β 5 = d j presence of deadlines) β 6 = { s - batch, p - batch } batching problem β 7 = { s jk , s jik } sequence dependent setup times 20 21
Course Introduction Course Introduction Scheduling Problem Classification Scheduling Problem Classification α | β | γ Classification Scheme α | β | γ Classification Scheme Complexity Hierarchy Complexity Hierarchy γ Objective α 1 α 2 | β 1 β 2 β 3 β 4 | γ γ Makespan: Maximum completion C max = max { C 1 , . . . , C n } γ Objective (always f ( C j ) ) α 1 α 2 | β 1 β 2 β 3 β 4 | γ γ tends to max the use of machines Maximum lateness L max = max { L 1 , . . . , L n } Lateness L j = C j − d j Total completion time � C j (flow time) Tardiness T j = max { C j − d j , 0 } = max { L j , 0 } Total weighted completion time � w j · C j Earliness E j = max { d j − C j , 0 } tends to min the av. num. of jobs in the system, ie, work in � 1 if C j > d j progress, or also the throughput time Unit penalty U j = Discounted total weighted completion time � w j (1 − e − rC j ) 0 otherwise Total weighted tardiness � w j · T j Weighted number of tardy jobs � w j U j All regular functions (nondecreasing in C 1 , . . . , C n ) except E i 22 23 Course Introduction Course Introduction Scheduling Problem Classification Scheduling Problem Classification α | β | γ Classification Scheme Exercises Complexity Hierarchy Complexity Hierarchy Gate Assignment at an Airport Airline terminal at a airport with dozes of gates and hundreds of arrivals each day. Other Objectives α 1 α 2 | β 1 β 2 β 3 β 4 | γ γ γ Non regular objectives Gates and Airplanes have different characteristics Min � w ′ j E j + � w ” j T j (just in time) Airplanes follow a certain schedule Min waiting times Min set up times/costs During the time the plane occupies a gate, it must go through a Min transportation costs series of operations There is a scheduled departure time (due date) Performance measured in terms of on time departures. 24 25
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