ISE480 Sequencing and Scheduling

Transcription

1 ISE480 Sequencing and

2 DETERMINISTIC MODELS ISE480 Sequencing and Spring Term 2

3 Models 3

4 Framework and Notation 4

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10 Machine environment a Single machine and machines in parallel 1 single machine Pm Qm Rm m identical machines in parallel m machines in parallel w/different speeds v i m unrelated machines in parallel 10

11 Machine environment Machines in series: Fm FFc Jm FJc Om Flow shop: all jobs processed in the same order on the machines Flexible Flow shop: same as flow shop but with c stages of parallel machines Job shop: each job has its own routing Flexible Job shop: same as job shop but with c stages of parallel machines Open shop: each job has to processed on all machines but no routing restrictions 11

12 Processing characteristics and constraints b could be empty! r j s jk s ijk prmp prec Release dates sequence dependent setup times sequence and machine dependent setup times preemption precedence constraints 12

13 Processing characteristics and constraints brkdwn M j prmu block nwt recrc breakdowns machine eligibility restrictions permutation blocking no waiting recirculation 13

14 Objectives g Performance measures of individual jobs C j completion time of job j L j lateness = C j d j T j tardiness = max(l j, 0) E j earliness = max(-l j, 0) U j h j (C j ) unit penalty = 1 if C j > d j and 0 otherwise h j is a non-decreasing cost function 14

15 Objective Functions L j T j U j d j C j C j d j Lateness Tardiness Unit Penalty 1 d C j j Fig. Due date related penalty functions 15

16 Objectives g (cont.) Functions to be minimized C max = max C j L max = max L j Σw j C j Σw j (1-e -rc j ) Σw j T j Σw j U j Σw j ' E j + Σw j '' T j makespan maximum lateness total weighted completion time total weighted discounted Cj total weighted tardiness weighted number of tardy jobs total weighted earliness and tardiness 16

17 Regular objective functions Regular objective functions non-decreasing in C 1,,C n most objective functions considered in this class are regular Non-regular objective functions Example: Σw j ' E j + Σw j '' T j Much harder to solve! 17

18 Description of a Problem a b g machine environment constraints, processing characteristics Examples: Paper bag factory FF3 r j w j T j Gate assignment P m r j, M j w j T j Tasks in a CPU 1 r j, prmp w j C j Traveling Salesman 1 s jk C max objective (to be minimized) 18

19 Classes of Schedules A SEQUENCE usually corresponds to a permutation of the n jobs or the order in which the jobs are to be processed on a given machine A SCHEDULE usually refers to an allocation of jobs within a more complicated setting of machines 19

20 Classes of Schedules A POLICY prescribes an appropriate action for any one of the states the system may be in (stochastic settings). 20

21 Classes of Schedules Nondelay (greedy) schedule No machine is kept idle while a task is waiting for processing. An optimal schedule need not be nondelay!! Example: P2 prec C max jobs p j

22 Precedence Constraints Original Schedule (nonpreemptive) jobs p j

23 Precedence Constraints Reduced Processing Time jobs p j The processing time of each job is reduced by 1 unit

24 Precedence Constraints Use of 3 Machines jobs p j machines are used instead of 2 with the original processing times

25 Active Schedule Defined A feasible nonpreemptive schedule is called ACTIVE if it is not possible to construct another schedule by changing the order of processing on the machines and having at least one task finishing earlier without any task finishing later. There is at least one optimal and active schedule for Jm g if the objective function is regular. Example : Consider a job shop with three machines and two jobs. Job 1 needs 1 time unit on machine 1 and 3 time units on machine 2. Job 2 needs 2 time units on machine 3 and 3 time units on machine 2. Both jobs have to be processed last on machine 2. 25

26 Example of an Active Schedule Machine 1 1 Machine Machine t It is clear that this schedule is active as reversing the sequence of the two jobs on machine 2 postpones the processing of job 2. However, the schedule is neither nondelay nor optimal. Machine 2 remains idle until time 2 while there is a job available for processing at time 1. 26

27 Semi active Schedule No task can be completed earlier without changing the order of processing on any one of the machines. Example: Consider again a schedule with three machines and two jobs. The routing of the two jobs is the same as in the previous example. The processing times of job 1 on machines 1 and 2 are both equal to 1. The processing times of job 2 on machines 2 and 3 are both equal to 2. 27

28 Example of a Semi active Schedule Machine 1 1 Machine Machine t Consider the schedule under which job 2 is processed on machine 2 before job 1. This implies that job 2 starts its processing on machine 2 at time 2 and job 1 starts its processing on machine 2 at time 4. This schedule is semi-active. However, it is not active as job 1 can be processed on machine 2 without delaying the processing of job 2 on machine 2. 28

29 Venn Diagram of Classes of Schedules for Job Shops Optimal Schedules Semi-active X X Nondelay Active All Schedules A Venn diagramm of the three classes of nonpreemptive schedules; the nondelay schedules, the active schedules, and the semi-active schedules 29