Production Scheduling Nahmias, Chapter 8 ( D t lj l i ) (sv ...

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Production Scheduling
Nahmias, Chapter 8
( (sv. Detaljplanering)
D t lj l i )
Production and Inventory Control (MPS)
MIOF10
Goals of Production Scheduling?
• High Customer Service:
– on-time delivery
– meeting customer due dates
• Low Inventory Levels:
– Low WIP and FGI
• High Utilization:
– Machines
– Personnel
Production Scheduling needs to strike a profitable
balance between these conflicting objectives
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High Customer Service - Meeting Due dates
• Service Level 1: • Lateness:
– Particularly common in – Difference between order due date
make to order systems. y
and completion p date ( (can be
– Fraction of orders which are positive).
filled on or before their due – Average lateness has little meaning
dates .
by itself. Need also to consider the
lateness variance.
• Tardiness:
• Service level 2:
– Is equal to the lateness of a job if it is
– Common in make to stock late and zero, otherwise.
systems.
– Average tardiness is meaningful but
– Fraction of demands met
from stock.
unintuitive.
Scheduling Approaches & Assumptions
• Scheduling as practice dates back to Scientiffic Management
– Serious analysis contingent on computer power - started in 1950’s
• MRP/ERP:
– Benefits: Simple paradigm, paradigm hierarchical approach. approach
– Problems
• MRP assumes that lead times are a constant attribute of the part,
independent of the status of the shop (assumes infinite capacity).
• Strong incentive to use pessimistic lead time estimates to hedge
• Classic Scheduling: (only classic in academia)
– Benefits: “Optimal” schedules (large integer programming problems)
– Problems: Bad assumptions.
• All jobs available at the start of the problem.
• Deterministic processing times.
• No setups.
• No machine breakdowns.
• No cancellation.
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Results for a Single Machine…
• Minimizing Average Cycle Time per job:
– Sequence the jobs in “shortest process time first” (SPT) order.
– The makespan of the considered batch of jobs is not affected
�� Th The makespan= k th the time ti to t process a defined d fi d set t of f jobs j b from f
start to finish
• Minimizing Maximum Lateness (or Tardiness):
– Minimize by performing in “earliest due date” (EDD) order.
– If there exists a sequence with no tardy jobs, EDD will do it.
• Minimize the number of tardy jobs
– Optimal schedule can easily be obtained using Moore’s algorithm
• Mi Minimizing i i i Average A Tardiness: T di
– No simple sequencing rule will work. Problem is NP Hard.
Moore’s Algorithm
• A method for minimizing the number of tardy jobs, when all
jobs are considered equally important
11. Od Order the h jjobs b according di to the h EDD rule. l
2. Stop if no jobs are tardy – the optimal solution is found!
Go to step 6.
3. Find the first tardy job in the sequence.
4. Assuming that this tardy job is the kth in the sequence. Find and
remove job j (j=1, 2, 3, …, k) with the longest processing time.
5. Revise the completion times and return to step 2
6. Insert the removed jobs at the end of the sequence in the order they
were removed
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Moore’s Example
Start with EDD schedule
Job Time Done Due Tardy
2 2 2 6 0
5 5 7 8 0
3 3 10 9 1
4 4 14 11 3
1 6 20 18 2
Job 3 is first late job.
Job 5 is longest of jobs 2,5,3.
Moore’s Example
Job Time Done Due Tardy
2 2 2 6 0
3 3 5 9 0
4 4 9 11 0
1 6 15 18 0
5 5 20 8 12
Average Flow = 51 / 5 = 10.2
Average tardiness = 12 / 5 = 2.4
Number tardy = 1
Max. Tardy =12
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Results for Two Machines in Sequence
• Johnsons Algorithm (Johnson 1954)
• Given a set of jobs that must go through a sequence of two
machines hi Johnsons J h Algorithm Al ith yields i ld the th sequence with ith
minimum makespan
– Makespan is sequence dependent.
1. Sort the times of the jobs on the two machines in two lists.
2. Find the shortest time in either list and remove job from both lists.
� If time came from first list, place job in first available position.
�� If time came from second list, list place job in last available
position in sequence.
3. Repeat until lists are exhausted.
� The resulting sequence will minimize the makespan.
Data:
Example: Johnsons Algorithm
Job Time on M1 Time on M2
1 4 9
2
3
7
6
10
5 p
Solution Illustrated in a Gantt Chart
Machine 1
1 2 3
Johnsons Algorithm:
� Final Sequence: 1-2-3
� Makespan: 28
Machine 2 1 2
3
Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Short task on M1 to
“load up” quickly.
Short task on M2 to
“clear out” quickly.
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Scheduling Problems in General are Hard
• Dilemma:
– NP hard combinatorial problems
– Too hard to find optimal solutions!
– NNeed dsomething thi anyway.
9E+22
8E+22
7E+22
6E+22
5E+22 5
• Classifying “Hardness”:
3E+22
– As the number of variables (n) grow
– Class P: has a polynomial solution.
– Class NP: has no polynomial solution.
2E+22
1E+22
0
• Sequencing problems are NP hard and complexity
grow as n!.
C /10000 d 10000 10
– Compare en /10000 and 10000n10 .
4E+22
56 57 58 59 60 61 62 63
• At n = 40, e n /10000 = 2.4 × 10 13 , 10000n 10 = 1.0 × 10 20
• At n = 80, e n /10000 = 5.5 × 10 30 , 10000n 10 = 1.1 × 10 23
– 3! = 6, 4! = 24, 5! = 120, 6! = 720, … 10! =3,628,800, while
13! = 6,227,020,800
25!= 15,511,210,043,330,985,984,000,000
Example: Computation Times
• A computer can examine 1,000,000 sequences per second and we wish
to build a scheduling system that has response time of no longer than
one minute. How many jobs can we sequence optimally?
Number of Jobs Computer Time
5 0.12 millisec
6 0.72 millisec
7 5.04 millisec
8 40.32 millisec
9 0.36 sec
10 3.63 sec
11 39.92 sec
12 798 7.98 min i
13 1.73 hr
14 24.22 hr
15 15.14 day
� �
20 77,147 years
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Dispatching Rules and Problems
• Optimal Schedules are impossible to find for most real
world problems.
– In practice simple dispatching rules are widely used to control the
material flow.
• Dispatching (körplanering):
– The process of sequencing jobs as they arrive at a machine or
workstation (as opposed to scheduling all the jobs on all machines)
• Dispatching rules:
– FIFO – simplest, seems “fair”.
– SPT – Actually works quite well with tight due dates.
– EDD – WWorks k well llwhen h jobs j b are mostly l the h same size. i
– … and many more…
• Problems with Dispatching:
– Cannot be optimal (can be bad).
– Tends to be myopic (short sighted)
The Bad News on Scheduling
Problem Difficulty
• Most “real-world” production scheduling problems are
NP-hard
– We cannot hope to find optimal solutions of
realistic sized scheduling problems.
– Polynomial approaches, like dispatching, may not work well.
Violation of Assumptions
• Most “real-world” scheduling problems violate the
assumptions made in the classic literature
– There are always more than two machines.
– Process times are not deterministic.
– All jobs are not available at the beginning of the considered time
frame
– Process times are sequence dependent.
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The Good News on Scheduling (I)
�We can control the problem by controlling the environment
– Do not accept all traditional constraints as fixed
– Design the environment to facilitate easier scheduling
– EExactly tl what h tth the Japanese J did when h they th made d a hard h dscheduling h d li
problem much easier by drastically reducing set up times
Strategies that simplifies scheduling…
• Due Dates: We can set the due dates (within reason).
• Job Splitting: We can get smaller jobs by splitting large ones.
– Single machine SPT results imply small jobs “clear out” more quickly
than larger jobs.
– Mechanics of Johnson’s algorithm implies we should start with a small
job and end with a small job.
– Small jobs make for small “move” batches and can be combined to
form larger “process” batches.
The Good News on Scheduling (II)
Strategies that simplifies scheduling… (cont.)
• Feasible Schedules: We do not need to find an optimal
schedule, hdl only l a good dffeasible iblone.
• Focus on Bottleneck: We can often concentrate on
scheduling the bottleneck process, which simplifies
problem closer to single machine case.
– Assumes relatively stable bottlenecks
• Capacity: Capacity can be adjusted dynamically
(overtime, floating workers, use of vendors, etc.) to adapt
facility (somewhat) to schedule.
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A Sequencing Example (I)
• Problem Description:
– 16 jobs
– Each job takes 1 hour on single machine (bottleneck resource)
– 4 hour setup to change families
– Fixed due dates
– Find feasible solution if possible
A Sequencing Example (II)
• EDD Sequence:
Job Due Completion
Number Family Date Time Lateness
1 1 5.00 5.00 0.00
2 1 6.00 6.00 0.00
3 1 10.00 7.00 -3.00
4 2 13.00 12.00 -1.00
5 1 15.00 17.00 2.00
6 2 15.00 22.00 7.00
7 1 22.00 27.00 5.00
8 2 22.00 32.00 10.00
9 1 23.00 37.00 14.00
10 3 29.00 42.00 13.00
11 2 30 30.00 00 47 47.00 00 17 17.00 00
12 2 31.00 48.00 17.00
13 3 32.00 53.00 21.00
14 3 32.00 54.00 22.00
15 3 33.00 55.00 22.00
16 3 40.00 56.00 16.00
• Average Tardiness: 10.375
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A Sequencing Example (III)
• Use a Greedy Approach:
– Consider all pairwise interchanges
– Ch Choose one that th t reduces d average tardiness t di by b maximum i
amount
– Continue until no further improvement is possible
A Sequencing Example (IV)
• First Interchange: Exchange jobs 4 and 5.
Job Due Completion
Number Family Date Time Lateness
1 1 500 5.00 500 5.00 000 0.00
2 1 6.00 6.00 0.00
3 1 10.00 7.00 -3.00
5 1 15.00 8.00 -7.00
4 2 13.00 13.00 0.00
6 2 15.00 14.00 -1.00
7 1 22.00 19.00 -3.00
8 2 22.00 24.00 2.00
9 1 23.00 29.00 6.00
10 3 29.00 34.00 5.00
11 2 30 30.00 00 39 39.00 00 900 9.00
12 2 31.00 40.00 9.00
13 3 32.00 45.00 13.00
14 3 32.00 46.00 14.00
15 3 33.00 47.00 14.00
16 3 40.00 48.00 8.00
• Average Tardiness: 5.0 (reduction of 5.375!)
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A Sequencing Example (V)
• Configuration After Greedy Search:
Job Due Completion
Number Family Date Time Lateness
1 1 5.00 5.00 0.00
2 1 6.00 6.00 0.00
3 1 10.00 7.00 -3.00
5 1 15.00 8.00 -7.00
4 2 13.00 13.00 0.00
6 2 15.00 14.00 -1.00
8 2 22.00 15.00 -7.00
7 1 22.00 20.00 -2.00
9 1 23.00 21.00 -2.00
11 2 30.00 26.00 -4.00
12 2 31.00 27.00 -4.00 4.00
10 3 29.00 32.00 3.00
13 3 32.00 33.00 1.00
14 3 32.00 34.00 2.00
15 3 33.00 35.00 2.00
16 3 40.00 36.00 -4.00
• Average Tardiness: 0.5 (9.875 lower than EDD)
A Sequencing Example (VI)
• A Better (Feasible) Sequence:
Job Due Completion
Number Family Date Time Lateness
1 1 500 5.00 500 5.00 000 0.00
2 1 6.00 6.00 0.00
3 1 10.00 7.00 -3.00
5 1 15.00 8.00 -7.00
4 2 13.00 13.00 0.00
6 2 15.00 14.00 -1.00
8 2 22.00 15.00 -7.00
11 2 30.00 16.00 -14.00
12 2 31.00 17.00 -14.00
7 1 22.00 22.00 0.00
9 1 23 23.00 00 23 23.00 00 000 0.00
13 3 32.00 28.00 -4.00
10 3 29.00 29.00 0.00
16 3 40.00 30.00 -10.00
14 3 32.00 31.00 -1.00
15 3 33.00 32.00 -1.00
• Average Tardiness: 0
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Conclusions Regarding Scheduling
• Scheduling is hard!
– Even simple toy problems generate complicated mathematics.
• Scheduling can be simplified by adjusting the production
environment:
– due date quoting
– flow simplification ⇒ sequencing in place of scheduling
• Finite capacity scheduling is coming.
– But in what form is unclear (shifting bottleneck, genetic
algorithms, rule based systems, etc.).
• Diagnostics are important in scheduling. scheduling
– pure optimization generally impossible
– need good interface to allow human intervention
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