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Current trends in deterministic scheduling by Chung-Yee Lee, Lei Lei , Michael Pinedo

Current trends in deterministic scheduling by Chung-Yee Lee, Lei Lei , Michael Pinedo. Emrah Zarifoğlu 97021730. Deterministic Scheduling. Set of jobs Set of machines Certain performance measures. Notation. α │ β │ γ α machine configuration

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Current trends in deterministic scheduling by Chung-Yee Lee, Lei Lei , Michael Pinedo

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  1. Current trends in deterministic scheduling by Chung-Yee Lee, Lei Lei, Michael Pinedo Emrah Zarifoğlu 97021730

  2. Deterministic Scheduling • Set of jobs • Set of machines • Certain performance measures

  3. Notation • α│β│γ • αmachine configuration • β processing restrictions and constraints • γ performance mesure to be optimized

  4. Notation • Jj: job j, j = 1,…, n • Mj: machine i, i = 1,…, m • Cj: the completion time for J j • wj: the weight for Jj • dj: the due date of Jj • Lj: the lateness of Jj= Cj– dj • L max: maximum lateness = max{Lj, j = 1,…, n} • C max: makespan = max{Cj: j = 1,…, n}

  5. Complexity • Polynomial time algorithmA known algorithm that is guaranteed to terminate within a number of steps which is a polynomial function of the size of the problem • NPnon-deterministic polynomial time, A set or property of computational decision problems solvable by a non-deterministic Turing Machine in a number of steps that is a polynomial function of the size of the input • NP-hard if solving a problem in polynomial time would make it possible to solve all problems in class NP in polynomial time

  6. Recent Developments in Scheduling Theory • New trend: extending classical algorithms to models related to real problems • Two popular areas • Scheduling with a 1-job-on-r-machine pattern • Jobs processed simultaneously on several machines (r positive integer) • Several jobs processed by a single processor (0<r≤1) • Machine scheduling with availability constraints • Possibility of machine unavailability due to maintenance • Machine availability in giventime windows

  7. Scheduling with 1-job-on-r-machine pattern • r positive integer • Diagnosable microprocessor systems • semiconductor circuit design team workforce planning • Berth allocation (one vessel for several berths) • 0<r≤1 • Berth allocation (several vessels share one berth)

  8. 1-job-on-r-machine (r positive integer) • multiprocessor task system • Nonfix-fixed number of machines working simultaneously but machines required are not specified • Example: Pm│nonfix│C maxan m-parallel-machine scheduling problem where each job can be processed simultaneouslyby a fixed number of machines with the objective minimizing the makespan. • Fix-fixation of the set of machines for particular jobs • P2│fix│Σwj Cjdenotes a 2-parallel-machine scheduling problem where each job can be processed simultaneously by a specific set of machines, and the objective is to minimize the total weighted completion time.

  9. The machine set not fixed • Pm│nonfix│C max • Pm│prmp,nonfix│C max (r=1,2) polynomial algorithms (Blazewicz et al. 1984) • Pm│nonfix,pj=1│C max (m≥r≥2) linear integer programming or dynamic programming polynomial in n (Blazewicz et al. 1986) • Pm│nonfix│C max (m=2,3,4; pj is a function of nonincreasing function of number of mehines used and determined before) NP-hard (Du and Leung 1989) • Pm│nonfix│C max (r=1,2; same speed processors) O(nm + n logn) (Blazewicz et al. 1990) • Pm│nonfix│C max (r=1,k; same speed processors) O(nm + n logn) (Blazewicz et al. 1990)

  10. The machine set not fixed (cont’d) • Pm│nonfix│ Σwj Cj • (m=2) NP-hard (Lee and Cai 1996) • P2│nonfix│ ΣCj  dynamic programming NP-hard O(nP3s+1) (Lee and Cai 1996) • P2│nonfix│ ΣCj (pj=p) O(nlogn) (Lee and Cai 1996) • Pm│nonfix│L max • Pm│nonfix,rj,dj,prmp│L max -> linear programmşng to check feasibility (Plehn 1990) • P2│nonfix│L max (EDD rule) dynamic programming NP-hard O(nP3s+1logP) (Plehn 1990) • P2│nonfix, pj=1 │L max  O(nlogn) (Plehn 1990)

  11. The machine set fixed • Pm│fix│C max • Branch and bound algorithm (Bozoki and Richard 1970) • P│fix, pj=1 │C max  Np-hard (Krawczyk and Kubale 1985) • Pm│fix│C max (only 2-machine jobs) NP-hard (Kubale 1987) • P3│fix│C max  NP-hard (Blazewicz et al. 1992) • P3│fix│C max (block-constraints) pseudopolynomial O(nP) (Hoogeveen et al. 1994) • Pm│fix, pj=1 │C max  polynomial (Hoogeveen et al. 1994) • P2│fix, rj│C max  NP-hard (Hoogeveen et al. 1994) • Pm│fix│C max (precedence constraints) branch and bound algorithms (Krämer 1995)

  12. The machine set fixed (cont’d) • Pm│fix│ Σwj Cj • Pm│fix│ Σwj Cj  integer programming (Dobson and KArmarkar 1989) • P2│fix│ ΣCj  NP-hard (Hoogeveen et al. 1994) • P│fix, pj=1 │ ΣCj  NP-hard (Hoogeveen et al. 1994) • P2│prmp, fix│ ΣCj  O(nlogn) (Cai et al. 1996) • Pm│fix, pj=1 │ ΣCj  polynomial (Brucker 1995

  13. Machine Scheduling with Availability Constraints • Mostly assumed available machines but may not be true (e.g., machine breakdown-stochastic, preventive maintenance-deterministic) • Assume machine i unavailable from sik until tik (0≤ sik ≤ tik, 0≤k≤ni) • “unavailability constraints”↔”machines are available in time windows” • Resumable (r-a)If a job cannot be finished before the next down period of a machine and the job can continue after the machine has become available again • Nonresumable (nr-a) if the job has to restart rather than continue

  14. Machine Scheduling with Availability Constraints (Cont’d) • Pm│prmp│ feasibility (different availability intervals)  O(nlogm) (Schmidt 1984) • 1│nr-a│ΣCj (one unavailability period) NP-hard (Adiri et al. 1989) • P││Cmax (at most one unavailability period that is at the beginning)  classical LPT by tight error bound ½, modified LPT by bound 1/3 (Lee at al. 1991) • P││ΣCj (at most one unavailability period that is at the beginning) SPT algorithm (Kaspi and Montreuil 1988, Liman 1991) • P2││ΣCj (one machine always available, other available from time zero to a fixed oint) NP-hard dynamic programming (Lee and Liman 1993)

  15. Machine Scheduling with Availability Constraints (Cont’d) • Pm││ΣCj (machine i available in a time window) SPT (Mosheiov 1994) • F2│r-a│Cmax (at least one machine always available) NP-hard pseudopolynomial dynamic programming algorithm (Lee 1996b) • F2│nr-a│Cmax (at least one machine always available) NP-hard pseudopolynomial dynamic programming algorithm (Lee 1996b) • 1│r-a│ΣCj  SPT (Lee 1996a) • 1│r-a│Lmax EDD (Lee 1996a) • 1│nr-a│Cmax NP-hard (Lee 1996a) • Pm│prmp, r-a│Cmax O(n + m logm) (Schmidt 1984)

  16. Recent developments in search algorithms • Complexity not easy to formulate as mathematical programming • Classical techniquesoften not solvable in real time • By improvement of computing technology • Neighbourhood search technique • Local improvement with minor changes • Simulated annealing, tabu search, genetic algorithms • Constraint-guided heuristic search technique • Not to find optimal but to find good feasible schedules • Formulationbased on list of rules and consraints • Focus on partial solutions and extending them to a complete feasible solution • Based on measurements of flexibility and constraining factors • Expert systems-scheduling systems based on this search technique

  17. General concepts in neighboorhood searchtechniques • The mapping of the data in a format suitable for the algorithm • the description of a schedule has to be both conciseand unambiguous • The neighbourhood design • The knowledge required centers mainly on those aspects of the schedule thathave the greatest impact on the objective • The search process within the neighbourhood • Given all the schedules in the neighbourhood, a search has to be conducted thatleads to the next schedule in the search process • The acceptance-rejection criterion • Whenever a schedule within the neighbourhood is selected, a decisionhas to be made whether or not to accept the schedule

  18. Simulated Annealing and Tabu Search • Very similar • Difference in acceptance-rejection criteria • Simulated annealing-probabilistic process • Tabu search-deterministic process • Simulated annealing job shop scheduling problems with the makespan objective • Tabu search single machine, parallel machine, flow shop, flexible flow shop and job shop problems with objectives that include the makespan, the total weighted completion time, as well as the total weighted tardiness

  19. Simulated Annealing and Tabu Search (Cont’d) • Jm││Cmax  simulated annealing (Matsuo et al. 1987) • 1││ Σwj Cj  tabu search (Potts and Van Wassenhove 1997) • 1│ sjk│ Σwj Cj  tabu search (Laguna et al. 1991, 1993) • 1│ sjk│ ΣTj  tabu search (Laguna et al. 1991, 1993) • Pm││ Σwj Cj  tabu searcH (Barnes and Laguna 1992, Barnes et al. 1995) • Flow shop problem  tabu search (Adenso-Dias 1992, Nowicki and Smutnicki 1994) • Jm││Cmax  tabu search (Dell’Amico and Trubian 1993, Nowicki and Smutnicki 1993, Taillard 1994, Dauzère-Pérèsand Paulli 1997) • Job shop problem with multiple-purpose machines  tabu search (Hurink, Jurisch and Thole 1994)

  20. Genetic Algorithms • Mimics natural evolutionary process • Population, generation, mutation, crossover, fitness, reproduction, chromosome • Jm││Cmax genetic algorithms (Lawton 1992, Della Croce et al. 1992,Bean1994,Bierwirth1995), Herrmann et al. 1995) • Job shop with machine learning genetic algorithms (Lee et al. 1995) • Real life applications  genetic algorithms (Bean 1994, Kettani and Jobin 1995, Herrmann et al. 1995)

  21. Constraint-guided Heuristic Search • popularization of artificial intelligence techniquesand languages (e.g., PROLOG) • Focus on finding feasible schedules rather than optimal ones • The most severe constraints in the beginning, the least severe constraints for the final part • Sometimes necessary to break some constraints • Soft constraints (constraint relaxation) • Hard constraints • List implied constraints as soon as possible (constraint propagation) • Consistency checking • Dealing with inconsistencies conflict resolution

  22. Recent Developments in Scheduling Practice • Flexible-resource scheduling (Daniels and Mazzola 1993, 1994, Ozdamar and Ulusoy 1995, Daniels et al. 1996, 1997, Alidaee and Ahmadian 1997, Alidaee and Kochenberger 1997, Armstrong et al. 1997a, 1997b) • Scheduling variable-speed machines (Trick 1994) • Scheduling with finite capacity input and output buffers (Hall et al. 1993, 1994,1997, Nawijn and Bass 1994) • Scheduling of machine and material handling operations (Egbelu 1987, Matsuo, Shang and Sullivan (1991), Hall et al. 1993, Lei et al. 1993, 1995, Blazewicz et al. 1994, Hall et al. 1994, and Crama 1995) • İntegrating scheduling with batching and lot sizing (Potts and Van Wassenhove 1992)

  23. Machine scheduling with material handling operations • Resourcesmachines and materialhandling transporters • Cost of material handling 80% • To reduce cost deal with the issues: • Sequencingthat specifies the order in which jobs are processed at machiningcenters • Schedulingthat makes time-phased routing and dispatching of transporters forjob pick-up and delivery • Facility layout and flowpath designthat makes efficient operations possible.

  24. Machine scheduling with material handling operations (Cont’d) • Knumber of transporters in a system • Jtotal number of job types • ntotal number of jobs to be processed • nmpsthe number of jobs in a minimal part set • Πmintheobjective of minimizing the production cycle time of an MPS in a repetitive process • twa manufacturing environment where the starting time of each material handlingoperation must be confined within a time window • nwtthe constraint that jobs arenot allowed to wait in process

  25. Machine scheduling with material handling operations (Cont’d) • The problem is to find a simultaneous feasible schedulefor job sequencing and time-phased dispatching and routing of transporters so that agiven objective is optimized • Work divided into: • Robotic cell scheduling • has the fewest constraints, and is also theone for which most analytical results are available • identify the optimal job inputsequence and the robot operation sequence with respect to certain objective functions • Scheduling of Automated Guided Vehicles (AGVs) • deals with an automated job shop withnon-zero buffers at machining centers and multiple AGVs traveling on a sharednetwork. • Cyclic scheduling of hoists subject to time-window constraints • deals withthe scheduling of multiple hoists in a flexible flowshop • tw, nwt and collision-free constraints

  26. The robotic cell scheduling problem • The no-buffer case • F2(1)│J>1│Πmin polynomial (Sethi et al. 1992) • F2(1)│J>1│Πmin O(n4mps) to optimize robot moves and job sequence (Hall et al. 1996a) • F2(1)│J>1│Cmax Gilmore and Gomory algorithm O(n3) (Kise et al. 1991) • F2(1)│J>1│Cmax(transportation between machines job dependent) NP-hard (Ganesharajah et al. 1995) • The finite buffer case • F2(1)│J>1│Cmax(fixed job inputsequence) Np-hard branch-and-bound algorithm to determine the sequence of robot moves (King et al. 1993)

  27. Scheduling of automated guided vehicles • process of flexible manufacturing • circulate on a network of guidepathsconnecting machine centers, and transport tools and jobs among the centers • AGV flowpaths • Unidirectional(→) • Bi-directional (↔) • Netwok configurations • Single-loop • Multi-loop

  28. Analytical approaches to AGV scheduling • Unidirected flowpath case • Deadlines are fixed single-loop network determined in O(nlogn) (Blazewicz et al. 1991) • Deadlines are fixed, collision-free routing two-loop network determined with dynamic programming (Blazewicz et al. 1994) • Bi-directional flowpath case • Send an AGV from a source location to a machine center Dijkstra’s algorithm polynomial O(K4m2) (Kim and Tanchoco 1991) • Column generation based heuristic approach (Krishnamurthy et al. 1993) • Two-AGV scheduling problem to minimize makespan dynamic programming (LAngevin et al. 1994)

  29. Heuristic rules for AGV and machine scheduling • AGV dispatching rules • Work center-initiated • Vehicle-initiated • Pull-based-vehicle selects a work center with the highest need for job replenishment • Push based-vehicle first selects a job to move and then a work center towhich the job should be sent. • Plan conflict-free vehicle routes(Taghaboni and Tanchoco 1988) • Testing various machine and AGVscheduling rules against different scheduling criteria via simulation experiments(Sabuncuoglu and Hommertzheim 1989, 1992b) • Hierarchical approach forreal-time on-line AGV scheduling problems (Sabuncuoglu and Hommertzheim 1992a) • Artificial intelligence and expert systems techniques  review (Kusiak 1989) • Approach based on a Hopfield neural network with simulated annealing (Chung and Fischer 1995)

  30. The hoist scheduling problem • Considered as a special class of Jm(K)│J>1│Πminproblems with tw and nwt constraints. • Interval processing time a decision variable selected from a given range as job processing time • A common objective of hoist scheduling inpractice to minimize the cycle time of a repetitive process for producing a givenMPS

  31. The hoist scheduling problem (Cont’d) • Fm(K)│J=1, nwt, tw│Πmin • Fm(1)│J=1, nwt, tw│Πmin NP-hard (Lei and Wang 1989) • Mixed integer program (Philips and Unger 1976) • Branch-and-bound procedure that solves a large number of LP subproblems (Shapiro and Nuttle 1988) • Branch-and-bound procedure that solves relaxations of LPs (Armstrong et al. 1991, 1994) • Fm(K)│J>1, nwt, tw│Πmin • Heuristic dispatching rules and expert systems • Expert systems (Yih 1990, Yih and Thesen 1991)

  32. Some Conclusions • Changing r from 1 to a positive integerincreases the complexity of problem • no relationship between the complexity of the nonfix and the fixmodels. • Most nonpreemptive problems are NP-hard • For preemptive problems, most polynomialalgorithms based on linear integer programming techniques

  33. Future Research Directions • Branch-and-bound techniques, dynamic programming, heuristicalgorithms with an error bound analysis • The nonpre-emptivecase with different job release times and different machine available timewindows • semi-resumable case where someextra setup time may be required when a job restarts. • Extension of the existing modelsto more complicated job shop and open shop problems • combining machine availability constraints with human resourceconstraints • comparing neighbourhood search techniqueswith constraint-guided heuristic search techniques • where the optimal home positionof a transporter after a delivery is • How to coordinate the machine and material handlingoperations to minimize the machine, transporter, and job waiting time • How to con-structcollision-free schedules when jobs arrive dynamically

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