Chapter 2: Model of scheduling problem Components of any model: Decision variables –What we can change to optimize the system, i.e., model output Parameters –model input values, i.e., values that cannot be changed Objective function –e.g. profit, quality to be maximized –e.g. costs, time to be minimized Constraints –determine which decision variable values are allowed

Modeling: Decision variables Three basic types of decision variables: Sequence –permutation of the jobs Schedule –allocation of the jobs in a more complex environment Scheduling policy –determines the next job, given the current state of the system Other variables are regarded as dynamic data, or as auxiliary variables: –Completion time (C ij ) –Lateness (L j ) –Tardiness (T j ) –Makespan (M j )

Modeling: Parameters Typical scheduling parameters: Number of resources (m machines, operators) Configuration and layout Resource capabilities Number of jobs (n) Job processing times (p ij ) Job release and due dates (resp. r ij and d ij ) Job weight (w ij ) or priority Setup times

Modeling: Objective function Objectives and performance measures: Throughput, makespan (C max, weighted sum) Due date related objectives (L max, T max, Σw j T j ) Work-in-process (WIP), lead time (response time), finished inventory Total setup time Penalties due to lateness (Σw j L j ) Idle time Yield Multiple objectives may be used with weights

Modeling: Constraints Precedence constraints (linear vs. network) Routing constraints Material handling constraints (Sequence dependent) Setup times Transport times Preemption Machine eligibility Tooling/resource constraints Personnel (capability) scheduling constraints Storage/waiting constraints Resource capacity constraints prec s jk prmp M j

Machine configurations: Single-machine vs. parallel-machine Flow shop vs. job shop Processing characteristics: Sequence dependent setup times and costs –length of setup depends on jobs –s ijk : setup time for processing job j after k on machine i –costs: waste of material, labor Preemptions –interrupt the processing of one job to process another with a higher priority

Generic notation of scheduling problem MachineJobObjective characteristicscharacteristicsfunction for example: Pm | r j, prmp | Σw j C j (parallel machines) 1 | s jk | C max (sequence dependent setup / traveling salesman) Q2 | prec | Σw j T j (2 machines w. different speed, precedence rel., weighted tardiness)

Scheduling models Deterministic models –input matches realization vs. Stochastic models –distributions of processing times, release and due dates, etc. known in advance –outcome/realization of distribution known at completion

Chapter 3: general-purpose scheduling procedures Dispatching rules Composite dispatching rules Adaptive search (Handouts) Mathematical programming (Appendices A3, A4) Branch-and-bound Beam-Search

Dispatching rules Static rules –not time-dependent –are function of job and/or machine data vs. Dynamic rules –time-dependent –e.g.: “minimum slack first”-priority rule

Dispatching rules Local rules –e.g. only consider current job, machine vs. Global rules –“look ahead”-strategy

Dispatching rules Schedule generation scheme: mechanism by which schedule is constructed serial scheme schedule job-for-job (n stages) vs. parallel scheme (at most T stages) –decision set (all candidates to be scheduled at stage t) –completed set of jobs (jobs that are completed at stage t) –remaining set of jobs (not allowed to be scheduled at t) –active set of jobs (scheduled, but not completed at stage t)

Dispatching rules Service in random order (SIRO) Earliest release date (ERD) Earliest due date (EDD) Minimum slack (MS) Weighted shortest processing time (WSPT) Note: WSPT is optimal in single-machine setting (Smith’s rule) Longest processing time (LPT)

Dispatching rules, cont. Shortest setup time (SST) Least flexible job (LFJ) Greatest resource demand (GRD) Critical path rule (CP) Largest number of successors (LNS) Shortest queue at the next operation (SQNO)

Dispatching rules, overview Dependent on release and due dates: ERD, EDD, MS Dependent on processing times: LPT, SPT, WSPT, CP, LNS Miscellaneous: SST, LFJ, SQNO

Dispatching in more complicated situations Dispatching+routing over non-identical machines Batch and dispatch Dispatching with preemption Dispatching with setups

More advanced dispatching rules Look-ahead with multiple job planning (e.g. including jobs with future release dates, dispatching with a waiting option, worst case slack rule) Adaptive search techniques Composite rules

Composite priority rule that is mixture of 3 basic priority rules: ATC ( apparent tardiness rule ) is comb. of: 1. Weighted Shortest Processing Time First 2. Earliest Due Date First 3. Minimal slack ATCS ( ATC with setups ) 4. Shortest Setup Time First

Composite dispatching: Apparent Tardiness Cost (ATC) ATC combines MS rule and WSPT rule k 1 =due date scaling par. ( look-ahead parameter) k 1 function of Due Date Range factor: average p j of the remaining jobs

Composite dispatching: Apparent Tardiness Cost with Setups (ATCS) ATCS combines MS rule, WSPT rule and SST rule: k 1 =due date scaling par. k 2 =setup time scaling par. k 1 and k 2 functions of: Due Date tightness Due Date Range Setup Time Severity

Dispatching rules: multiple passes drawback of priority rules: may yield bad solutions SOLUTION: Use multiple passes Multi-pass priority rule based methods: 1. Multi-priority rule procedures (repeat dispatching procedure with different disp. rules) 2. Sampling procedures (each job has a probability to be dispatched)

Dispatching rules: sampling Random sampling (each job has the same probability): P j =1/n Biased random sampling (based on priority of job): first sort jobs on non-increasing priority v j [j] is the position of job j in the list  Probability that job is selected: Parameterized regret-based random sampling: accounts for relative difference in priorities of jobs random sampling deterministic

Dispatching rules: sampling (cont.) Parameterized regret-based random sampling: regret factor (v j =priority): probability:  =0: random sampling  =  : deterministic activity selection Kolisch and Drexl: the more passes, the higher  should be taken this method is called the “Adaptive Search”-method normalization