The basics of scheduling OUTLINE Questions? Exam results Go over Exam The basics of scheduling

Exam Results

Exam Results

Basics of scheduling Definitions Creating a schedule Difference between optimum and good Do optima exist? Performance objectives Basic rules SPT EDD Moore Lawler Johnson Heuristics

Introduction to Scheduling The general job shop problem All problems are subsets or relaxations of basic assumptions Organized research and study of this area followed W.W.II n jobs {J1, J2, ….Jn} a job is a task or lot or batch that is to processed as a unit. If it is broken into several jobs, we restart the problem with each part a new job m machines {M1, M2, ….Mm} a machine is a processor or resource that performs a specific function required by a job

Introduction to Scheduling (continued) Processing of a job on a machine is called an operation nm operations oij is the ith job on the jth machine Each job passes through each machine once and only once (Real jobs may repeat or skip) Each job has a prescribed order in which it is processed by the machines - This is called its Technological Constraint (TC). Other names are processing order, routing, process plan. General Job Shop - each job has its own processing order Flow Shop - all jobs have the same processing order Permutation Shop - all machines see the jobs in the same order

Introduction to Scheduling (continued) A production line with all machines tied together is an example of a permutation shop If all jobs have the same processing order, but the machines are not tied together, we have a Flow shop, because some machines may not see all jobs in the same order. (They may pass each other between machines – think of cars and successive toll booths)

Examples – Gantt charts J2 J1 J3 Flow Permutation J2 J1 J3 M1 M2 J2 J1 J3 Flow non-Permutation J1 J3 J2 M1 M2 J1 J2 J2 J1 General J2 J1 J2 J1

Assumptions 1. No two operations of a job may be processed simultaneously 2. Once an operation is started, it must be finished (no preemption) 3. Jobs may not be cancelled 4. Each job has m operations, one on each machine 5. Processing time pij is independent of the schedule (no set - up dependency) 6. In process inventory is allowed 7. There is only one of each machine

Assumptions (continued) 8. Machines may be idle 9. No machine may process more than one operation at a time 10. Machines are always available (no breakdowns) 11. All parameters are known and are deterministic - no randomness - no changes in the technological constraints We relax these as required by our situation and solve a different problem

Notations and definitions di - due date of Job i ri - ready time of Job i ai - allowance = di - ri si - slack = di - remaining operations pij - the time required to process oij Wik - the waiting time of Job i preceding its kth operation (not the work on Mk): J1’s time = W11+p11+W12 +p13+W13 +p12+W14+p1 17, if the TC is M1 to M3 to M2 to M17 We designate the kth operation as oij(k)

Notations and definitions (continued) Ci is the completion time of Job i Fi is the flow time of Job i = Ci - ri Even though the English words have an identical meaning, we distinguish between Lateness and Tardiness Lateness Li =Ci - di, therefore maybe positive or negative depending on whether we complete a job before or after its due date

Notations and definitions (continued) Tardiness is non- zero only if the job is completed after its due date: Ti = max{Li, 0} We also define Earliness as Ei = Max{-Li, 0} The weight or importance of a job is indicated either by wi or Some of our definitions refer to instants in time Completion, readiness Others refer to elapsed time Processing, Waiting, Flow

Notations and definitions (continued) Scheduling - the ordering of operations subject to restrictions and providing start and finishing times for each operation

Notations and definitions (continued)- Schematic Processing Order - Mm-1, Mj, Mm …..M1, M2 M1 M2 Oi1(m-1) Oi2(m) Wi m Mj Mm-1 Oi j(2) Wi 2 Oi m-1(1) pi m-1(1) Oi m(3) Mm ri Ci

Measures Optimality or goodness of schedules only makes sense if we define the measure under which we are considering optimality or goodness. There are three broad categories of measures: Completion time Due dates Inventory or utilization We also define a general class of measures called regular measures

Measures (continued) Based on completion Time Maximum Flow time Maximum Completion time Average Flow time Average Completion time Weighted Average Completion time

Measures (continued) Based on due dates Average Lateness Maximum Lateness Average Tardiness Maximum Tardiness Number of tardy jobs

Measures (continued) Regular measures Always are minimized Are non decreasing in completion times Examples Average and maximum completion time Average and maximum flow time Average and maximum lateness Average and maximum tardiness Number of tardy jobs

Measures (continued) Given two sets of completions times obtained under two schedules generated for the same problem: C and C’ if Ci <= Ci’ implies that R(C)<=R(C’) then R is regular

Inventory Inventory can exist in many places Transit Raw material Work in Process (WIP) Finished Goods (FG) Distribution Retail Location Spares

Classification notation All problems can be classified as n/m/A/B where n - number of jobs m - number of machines A - pattern F - Flow Shop P - Permutation G - General Job Shop B - Measure Cmax, Fmax etc.

Classification notation (continued) for example n/3/F/Fbar means any number of jobs on three machines in a flow shop being measured on the basis of average flow time

Some further definitions Jobs and ready times fixed = Static Parameters known and fixed = Deterministic Random arrival of jobs = Dynamic Uncertain processing times = stochastic

Kinds of scheduling Taking sequences and placing them in a schedule is called time tabling (creating a Gantt chart) Semiactive - process each job as soon as it can be (slide to the left on the chart) Active - No operation can be started earlier without delaying some other operation Non-delay - no machine is kept idle Non-feasible - does not meet Technological Constraints Number of possible schedules (including non-feasible ones) = (n!)m

Optimality Since for any given problem there are a countable number of possible schedules (as long as we do not allow preemption or unnecessary delays) there must be an optimum (or optima) because we can (theoretically) compare all possible schedules and select the best one If we look at the space that contains our schedules and attempt to locate the optimum we find that:

Optimality (continued) all possible Optimal feasible Active semi active non - delay

Equivalent measures Two measures are equivalent if a schedule optimal with respect to one is also optimal with respect to the other and vice versa. Cbar, Fbar, Wbar, Lbar are equivalent Note that a schedule optimal to Lmax is optimal with Tmax, but not vice versa Cmax, Nbarp and I bar are also equivalent Cbar, Fbar, Lbar, Nbaru, Nbarw are equivalent for one machine The choice of measure depends on the circumstances