ΣΧΕΔΙΑΣΜΟΣ ΚΑΙ ΠΡΟΓΡΑΜ- ΜΑΤΙΣΜΟΣ ΠΑΡΑΓΩΓΗΣ Βραχυχρόνιος Προγραμματισμός Παραγωγής

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1 ΣΧΕΔΙΑΣΜΟΣ ΚΑΙ ΠΡΟΓΡΑΜ- ΜΑΤΙΣΜΟΣ ΠΑΡΑΓΩΓΗΣ Βραχυχρόνιος Προγραμματισμός Παραγωγής Γιώργος Λυμπερόπουλος Πανεπιστήμιο Θεσσαλίας Τμήμα Μηχανολόγων Μηχανικών 1

2 PRODUCTION PLANNING AND SCHEDULING Short-Term Production Scheduling George Liberopoulos University of Thessaly Department of Mechanical Engineering 2

3 Hierarchy of Production Decisions Forecasts of future demands Aggregate planning Master Production Schedule MRP (schedule of production quantities by product and time period) Detailed job shop scheduling 3

4 Characteristics of job shop scheduling problems Job arrival pattern (static/dynamic problems) Number and variety of machines in the shop Number of workers in the shop (ability, speed of workers) Particular flow patterns (serial/mixed/rework) Evaluation of alternative rules 4

5 Objectives of job shop scheduling 1. Meet due dates 2. Minimize average flow time through the system 3. Minimize work-in-process (WIP) inventory 4. Minimize machine/worker idle time (maximize utilization) 5. Reduce setup times 6. Minimize production and worker costs 7. Provide for accurate job status information Impossible to optimize all objectives simultaneously 1-2 high customer service 3-6 high level of plant efficiency (minimize cost) Conflicting objectives: e.g. 3 & 4 5

6 Job shop terminology Flow shop n jobs must be processed through m machines in the same order Job shop Jobs may not all require m operations Some jobs may require multiple operations on the same machine Each job may have different sequencing of operations Parallel vs. sequential processing Flow time of job i Time from the initiation of the first job until the completion of job ii Makespan Flow time of job completed last Tardiness & Lateness Lateness = Completion time due date Tardiness = (Completion time due date) + 6

7 Myopic sequencing rules First come first served (FCFS) Shortest Processing Time (SPT) Earliest Due Date (EDD) Smallest Critical Ratio (CR) CR = (Due date Current time) / Processing time 7

8 tt ii = Processing time for job ii dd ii = Due date for job ii WW ii = Waiting time for job ii FF ii = Flow time for job ii FF ii = WW ii + tt ii LL ii = Lateness of job ii LL ii = FF ii dd ii TT ii = Tardiness of job ii TT ii = (LL ii ) + = max(ll ii, 0) EE ii = Earliness of job ii EE ii = ( LL ii ) + = max( LL ii, 0) Performance Measures Examples Maximum Tardiness TT max = max(tt 1, TT 2,, TT nn ) Mean flow time FF = 1 nn ii=1 nn FF ii 8

9 Some results Rule that minimizes mean flow time FF : SPT The following measures are equivalent 1. Mean flow time FF = 1 nn ii=1 nn 2. Mean waiting time WW = 1 nn ii=1 nn 3. Mean lateness LL = 1 nn ii=1 nn Rule that minimizes maximum lateness LL max : EDD LL ii ( LL max = max(ll 1, LL 2,, LL nn )) FF ii WW ii 9

10 Minimize number of tardy jobs: No simple rule Moore s algorithm 1. Sequence jobs according to EDD to get initial solution. 2. Find first tardy job, say [ii]. If none exists, GOTO Consider jobs 1, 2,, [ii]. Reject job with largest processing time. GOTO Optimal sequence: Current sequence + rejected jobs in any order 10

11 Moore s algorithm: Example ii dd ii tt ii ii dd ii tt ii Completion time Jobs to consider Largest processing time First tardy job Reject job 1 11

12 Moore s algorithm: Example (cont d) ii dd ii tt ii Completion time Jobs to consider Largest processing time Reject job 5 First tardy job 12

13 Moore s algorithm: Example (cont d) ii dd ii tt ii Completion time No tardy jobs! Optimal sequence: 2, 3, 4, 6 {1, 5} Number of tardy jobs: 2 13

14 Precedence constraints: Lawler s Algorithm Minimize max gg ii(ff ii ) 1 ii nn gg ii FF ii : non decreasing function Examples: gg ii FF ii = FF ii (flow time) gg ii FF ii = FF ii dd ii = LL ii (lateness) gg ii FF ii = FF ii dd + ii = TT ii (tardiness) 14

15 Precedence constraints: Lawler s Algorithm (cont d) 1. Determine set of jobs, say VV, not required to precede any other. 2. Among the jobs in VV, choose job kk satisfying gg kk ττ = min gg ii(ττ), where ττ = nn ii VV ii=1 tt ii 3. Schedule job kk last. 4. If there are more jobs to schedule, GOTO 1. 15

16 Lawler s Algorithm: Example ii tt ii dd ii Precedence constraints: Problem: Minimize max 1 ii nn FF ii dd ii + (minimize maximum tardiness) 16

17 Lawler s Algorithm: Example (cont d) ii tt ii dd ii VV = 3, 5, 6 2. ττ = tt 1 + tt 2 + tt 3 + tt 4 + tt 5 + tt 6 =15 min (15 9) + = 6, ii = 3 (15 11) + = 4, ii = 5 (15 7) + = 8, ii = 6 3. Optimal sequence: _ 5 = 4 kk =

18 Lawler s Algorithm: Example ii tt ii dd ii VV = 3, 6 2. ττ = tt 1 + tt 2 + tt 3 + tt 4 + tt 6 =13 min (13 9)+ = 4, ii = 3 (13 7) + = 6, ii = 6 = 4 kk = 3 3. Optimal sequence:

19 Lawler s Algorithm: Example ii tt ii dd ii VV = 2, 6 2. ττ = tt 1 + tt 2 + tt 4 + tt 6 = 9 min (9 6)+ = 3, ii = 2 (9 7) + = 2, ii = 6 = 2 kk = 6 3. Optimal sequence: _

20 Lawler s Algorithm: Example ii tt ii dd ii VV = 2, 4 2. ττ = tt 1 + tt 2 + tt 4 = 8 min (8 6)+ = 2, ii = 2 (8 7) + = 1, ii = 4 = 1 kk = 4 3. Optimal sequence:

21 Lawler s Algorithm: Example ii tt ii dd ii VV = 2 2. kk = 2 3. Optimal sequence: _ VV = 1 2. kk = 1 3. Optimal sequence:

22 Note: Sequencing theory for a single Rule that minimizes maximum lateness or maximum tardiness, if there are no precedence constraints: EDD Rule that minimizes maximum flow time (makespan): Any sequence 22

23 Sequencing theory for multiple machines nn jobs through mm machines in series (flow shop): nn! mm possible schedules Example: nn = 5, mm = (~25 billion) possible schedules Special case: mm = 2 machines A B Result: If the objective is to minimize makespan, the sequence of jobs is the same on both machines (permutation schedule) nn! possible schedules 23

24 Sequencing theory for multiple machines Special case: mm = 2 machines AA ii = Processing time of job ii on machine A BB ii = Processing time of job ii on machine B Johnson s Algorithm: Job ii precedes job jj if min AA ii, BB jj < min AA jj, BB ii 1. List the values of AA ii and BB ii in two columns 2. Find smallest remaining value. If in column A, schedule next. If in column B, schedule last. 3. Cross off scheduled jobs. 24

25 Sequencing theory for multiple machines Johnson s Algorithm: Example Job ii AA ii BB ii Optimal schedule: _ 2 _ 4 _ 3 _ 5 _ 1 25

26 Sequencing theory for multiple machines Extension to 3 machines A B C Considerably more complex If the objective is to minimize makespan, the sequence of jobs is the same on both machines (permutation schedule) 3-machine problem can be reduced to 2-machine problem if: min AA ii max BB ii or min CC ii max BB ii Reduction: A B AA ii = AA ii + BB ii and BB ii = BB ii + CC ii If the condition is not satisfied, the reduction yields good but possibly suboptimal solutions 26

27 Sequencing theory for multiple machines Extension to 3 machines (example) Job ii AA ii BB ii CC ii min AA ii = 4 6 = max BB ii or min CC ii = 6 6 = max BB ii Job ii AA ii BB ii Optimal schedule: _ 1 _ 4 _ 5 _ 2 _ 3 27

Factors to Describe Job Shop Scheduling Problem

Factors to Describe Job Shop Scheduling Problem Job Shop Scheduling Job Shop A work location in which a number of general purpose work stations exist and are used to perform a variety of jobs Example: Car repair each operator (mechanic) evaluates plus