IE 514 Production
Scheduling
Introduction
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Contact Information
Siggi Olafsson
3018 Black Engineering
294-8908
olafsson@iastate.edu
http://www.public.iastate.edu/~olafsson
OH: MW 10:30-12:00
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Administration (Syllabus)
Text
Prerequisites
Assignments
Homework
Project
Final Exam
35%
40%
35%
Computing
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What is Scheduling About?
Applied operations research
Models
Algorithms
Solution using computers
Implement algorithms
Draw on common databases
Integration with other systems
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Application Areas
Procurement and production
Transportation and distribution
Information processing and
communications
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Manufacturing Scheduling
Short product life-cycles
Quick-response manufacturing
Manufacture-to-order
More complex operations must be
scheduled in shorter amount of time with
less room for errors!
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Scope of Course
Levels of planning and scheduling
Long-range planning (several years),
middle-range planning (1-2 years),
short-range planning (few months),
scheduling (few weeks), and
reactive scheduling (now)
These functions are now often integrated
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Scheduling Systems
Enterprise Resource Planning (ERP)
Common for larger businesses
Materials Requirement Planning (MRP)
Very common for manufacturing companies
Advanced Planning and Scheduling (APS)
Most recent trend
Considered “advanced feature” of ERP
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Scheduling Problem
Allocate scarce resources to tasks
Combinatorial optimization problem
Maximize profit
Subject to constraints
Mathematical techniques and heuristics
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Our Approach
Scheduling Problem
Problem Formulation
Model
Solve with Computer Algorithms
Conclusions
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Scheduling Models
Project scheduling
Job shop scheduling
Flexible assembly systems
Lot sizing and scheduling
Interval scheduling, reservation,
timetabling
Workforce scheduling
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General Solution
Techniques
Mathematical programming
Linear, non-linear, and integer programming
Enumerative methods
Branch-and-bound
Beam search
Local search
Simulated annealing/genetic algorithms/tabu
search/neural networks.
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Scheduling System Design
Order
master file
Databases
Shop floor
data collection
Database Management
Schedule
generation
Automatic Schedule Generator
Schedule Editor
User interfaces
Performance
Evaluation
Graphical Interface
User
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LEKIN
On disk with book
Generic job shop scheduling system
User friendly windows environment
C++ object oriented design
Can add own routines
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Advanced Topics
Uncertainty, robustness, and reactive
scheduling
Multiple objectives
Internet scheduling
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Topic 1
Setting up the Scheduling
Problem
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Modeling
Three components to any model:
Decision variables
This is what we can change to affect the system,
that is, the variables we can decide upon
Objective function
E.g, cost to be minimized, quality measure to be
maximized
Constraints
Which values the decision variables can be set to
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Decision “Variables”
Three basic types of solutions:
A sequence: a permutation of the jobs
A schedule: allocation of the jobs in a more
complicated setting of the environment
A scheduling policy: determines the next
job given the current state of the system
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Model Characteristics
Multiple factors:
Number of machine and resources,
configuration and layout,
level of automation, etc.
Our terminology:
Resource = machine (m)
Entity requiring the resource = job (n)
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Notation
Static data:
Processing time (pij)
Release date (rj)
Due date (dj)
Weight (wj)
Dynamic data:
Completion time (Cij)
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Machine Configuration
Standard machine configurations:
Single machine models
Parallel machine models
Flow shop models
Job shop models
Real world always more complicated.
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Constraints
Precedence constraints
Routing constraints
Material-handling constraints
Storage/waiting constraints
Machine eligibility
Tooling/resource constraints
Personnel scheduling constraints
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Other Characteristics
Sequence dependent setup
Preemptions
preemptive resume
preemptive repeat
Make-to-stock versus make-to-order
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Objectives and
Performance Measures
Throughput (TP) and makespan (Cmax)
Due date related objectives
Work-in-process (WIP), lead time
(response time), finished inventory
Others
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Throughput and Makespan
Throughput
Defined by bottleneck machines
Makespan
Cmax  maxC1 , C2 ,...,Cn 
Ci  maxCi1 , Ci 2 ,...,Cim , i  1,...,n
Minimizing makespan tends to maximize
throughput and balance load
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Due Date Related
Objectives
Lateness L j  C j  d j
Minimize maximum lateness (Lmax)
Tardiness Tj  maxC j  d j ,0
Minimize the weighted tardiness
n
w T
j 1
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Due Date Penalties
Tardiness
Lateness
Lj
Tj
Cj
Cj
dj
dj
Late or Not
In practice
Uj
1
Cj
Cj
dj
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WIP and Lead Time
Work-in-Process (WIP) inventory cost
Minimizing WIP also minimizes average
lead time (throughput time)
Minimizing lead time tends to minimize
the average number of jobs in system
Equivalently, we can minimize sum of the
completion times:
n
n
C j
 w jC j
i 1
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Other Costs
Setup cost
Personnel cost
Robustness
Finished goods inventory cost
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Topic 2
Solving Scheduling
Problems
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Classic Scheduling Theory
Look at a specific machine environment
with a specific objective
Analyze to prove an optimal policy or to
show that no simple optimal policy exists
Thousands of problems have been studied
in detail with mathematical proofs!
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Example: single machine
Lets say we have
Single machine (1), where
the total weighted completion time should be
minimized (SwjCj)
We denote this problem as
1||  wj C j
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Optimal Solution
Theorem: Weighted Shortest Processing
time first - called the WSPT rule - is
optimal for 1 ||  wj C j
 Note: The SPT rule starts with the job that has
the shortest processing time, moves on the job
with the second shortest processing time, etc.
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Proof (by contradiction)
Suppose it is not true and schedule S is optimal
Then there are two adjacent jobs, say job j
followed by job k such that
wj
wk

p j pk
Do a pairwise interchange to get schedule S ’
j
k
t  p j  pk
t
k
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t  p j  pk
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Proof (continued)
The weighted completion time of the two jobs under S is
(t  p j )wj  (t  p j  pk )wk
The weighted completion time of the two jobs under S ‘ is
(t  pk )wk  (t  p j  pk )wj
Now:
(t  p j ) w j  (t  p j  pk ) wk  (t  p j ) w j  p j wk  (t  pk ) wk
 (t  p j ) w j  pk w j  (t  pk ) wk
 (t  pk ) wk  (t  pk  p j ) w j
Contradicting that S is optimal.
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Complexity Theory
Classic scheduling theory draws heavily on
complexity theory
The complexity of an algorithm is its
running time in terms of the input
parameters (e.g., number of jobs and
number of machines)
Big-Oh notation, e.g., O(n2m)
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Polynomial versus NP-Hard
Time
O(n2 )
n
O(2 )
O(n)
O(logn)
Problemsize (n)
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Scheduling in Practice
Practical scheduling problems cannot be
solved this easily!
Need:
Heuristic algorithms
Knowledge-based systems
Integration with other enterprise functions
However, classic scheduling results are
useful as a building block
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General Purpose
Scheduling Procedures
Some scheduling problems are easy
Simple priority rules
Complexity: polynomial time
However, most scheduling problems are
hard
Complexity: NP-hard, strongly NP-hard
Finding an optimal solution is infeasible in
practice  heuristic methods
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Types of Heuristics
Simple Dispatching Rules
Composite Dispatching Rules
Branch and Bound
Beam Search
Simulated Annealing
Tabu Search
Genetic Algorithms
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Methods
Improvement
Methods
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Topic 3
Dispatching Rules
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Dispatching Rules
Prioritize all waiting jobs
job attributes
machine attributes
current time
Whenever a machine becomes free: select
the job with the highest priority
Static or dynamic
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Release/Due Date Related
Earliest release date first (ERD) rule
variance in throughput times
Earliest due date first (EDD) rule
maximum lateness
Minimum slack first (MS) rule
Current
Time
maxd j  p j  t,0
maximum lateness
Processing
Time
Deadline
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Processing Time Related
Longest Processing Time first (LPT) rule
balance load on parallel machines
makespan
Shortest Processing Time first (SPT) rule
sum of completion times
WIP
Weighted Shortest Processing Time first
(WSPT) rule
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Processing Time Related
Critical Path (CP) rule
precedence constraints
makespan
Largest Number of Successors (LNS) rule
precedence constraints
makespan
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Other Dispatching Rules
Service in Random Order (SIRO) rule
Shortest Setup Time first (SST) rule
makespan and throughput
Least Flexible Job first (LFJ) rule
makespan and throughput
Shortest Queue at the Next Operation
(SQNO) rule
machine idleness
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Discussion
Very simple to implement
Optimal for special cases
Only focus on one objective
Limited use in practice
Combine several dispatching rules
Composite Dispatching Rules
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Example
Single Machine with
Weighted Total Tardiness
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Setup
Problem: 1 ||  w jTj
No efficient algorithm (NP-Hard)
Branch and bound can only solve very
small problems (<30 jobs)
Are there any special cases we can solve?
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Case 1: Tight Deadlines
Assume dj=0
Then
T j  max0, C j  d j 
 max0, C j   C j
w T  w C
j
j
j
j
We know that WSPT is optimal for this
problem!
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Conclusion
The WSPT is optimal in the extreme case
and should be a good heuristic whenever
due dates are tight
Now lets look at the opposite
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Case 2: “Easy” Deadlines
Theorem: If the deadlines are sufficiently
spread out then the MS rule
jselect  argmaxd j  p j  t,0
is optimal (proof a bit harder)
Conclusion: The MS rule should be a good
heuristic whenever deadlines are widely
spread out
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Composite Rule
Two good heuristics
Weighted Shorted Processing Time (WSPT )
optimal with due dates zero
Minimum Slack (MS)
Optimal when due dates are “spread out”
Any real problem is somewhere in between
Combine the characteristics of these rules
into one composite dispatching rule
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Apparent Tardiness Cost
(ATC) Dispatching Rule
New ranking index
 maxd j  p j  t ,0

I j (t ) 
exp 
pj
K  p (t )


wj
Scaling constant
When machine becomes free:
Compute index for all remaining jobs
Select job with highest value
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Special Cases (Check)
If K is very large:
ATC reduces to WSPT
If K is very small and no overdue jobs:
ATC reduces to MS
If K is very small and overdue jobs:
ATC reduces to WSPT applied to overdue
jobs
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Choosing K
Value of K determined empirically
Related to the due date tightness factor
d
  1
Cmax
and the due date range factor
d max  d min
R
Cmax
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Choosing K
Usually 1.5  K  4.5
Rules of thumb:
Fix K=2 for single machine or flow shop.
Fix K=3 for dynamic job shops.
Adjusted to reduce weighted tardiness
cost in extremely slack or congested job
shops
Statistical analysis/empirical experience
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Topic 4
Branch-and-Bound
& Beam Search
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Branch and Bound
Enumerative method
Guarantees finding the best schedule
Basic idea:
Look at a set of schedules
Develop a bound on the performance
Discard (fathom) if bound worse than best
schedule found before
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Example
Single Machine with
Maximum Lateness
Objective
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Classic Results
The EDD rule is optimal for 1 || Lmax
If jobs have different release dates, which
we denote
1 | rj | Lmax
then the problem is NP-Hard
What makes it so much more difficult?
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1 | rj | Lmax
Jobs
EDD
1
2
0
r2
r3
2
2
3
5
rj
0
3
5
dj
8
14
10
3
5
r1
pj
1
4
10
d1
15
d3
d2
Lmax  maxL1 , L2 , L3   maxC1  d1 , C2  d 2 , C3  d3 
 max4  8,10  14,15  10  max 4,4,5  5
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Can we
improve?
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Delay Schedule
Add a delay
1
3
0
r1
5
r2
r3
2
10
d1
15
d3
d2
Lmax  maxL1 , L2 , L3   maxC1  d1 , C2  d 2 , C3  d3 
 max4  8,16  14,10  10  max 4,2,0  3
What makes this problem hard is that
the optimal schedule is not necessarily
a non-delay schedule
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Final Classic Result
The preemptive EDD rule is optimal for
the preemptive (prmp) version of the
problem
1 | rj , prmp| Lmax
 Note that in the previous example, the preemptive EDD
rule gives us the optimal schedule
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Branch and Bound
The problem
1 | rj | Lmax
cannot be solved using a simple dispatching rule
so we will try to solve it using branch and bound
To develop a branch and bound procedure:
Determine how to branch
Determine how to bound
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Data
Jobs
1
2
3
4
pj
4
2
6
5
rj
0
1
3
5
dj
8
12
11
10
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Branching
(•,•,•,•)
(1,•,•,•)
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(2,•,•,•)
(3,•,•,•)
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(4,•,•,•)
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Branching
(•,•,•,•)
(1,•,•,•)
(2,•,•,•)
(3,•,•,•)
(4,•,•,•)
Discard immediately because
r3  3
r4  5
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Branching
(•,•,•,•)
(1,•,•,•)
(2,•,•,•)
(3,•,•,•)
(4,•,•,•)
Need to develop lower bounds on
these nodes and do further branching.
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Bounding (in general)
Typical way to develop bounds is to relax the
original problem to an easily solvable problem
Three cases:
If there is no solution to the relaxed problem there is
no solution to the original problem
If the optimal solution to the relaxed problem is
feasible for the original problem then it is also
optimal for the original problem
If the optimal solution to the relaxed problem is not
feasible for the original problem it provides a bound
on its performance
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Relaxing the Problem
The problem
1 | rj , prmp| Lmax
is a relaxation to the problem 1 | rj | Lmax
Not allowing preemption is a constraint in the
original problem but not the relaxed problem
We know how to solve the relaxed problem
(preemptive EDD rule)
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Bounding
Preemptive EDD rule optimal for the
preemptive version of the problem
Thus, solution obtained is a lower bound
on the maximum delay
If preemptive EDD results in a nonpreemptive schedule all nodes with higher
lower bounds can be discarded.
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Lower Bounds
Start with (1,•,•,•):
r4  5
Job with EDD is Job 4 but
Second earliest due date is for Job 3
Lmax  5
Job 1
Job 2
Job 3
Job 4
0
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Branching
(•,•,•,•)
Lmax  5
(1,•,•,•)
(2,•,•,•)
(1,2,•,•)
(1,3,•,•)
Lmax  6
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(1,3,4,2)
(3,•,•,•)
(4,•,•,•)
Lmax  7
Lmax  5
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Beam Search
Branch and Bound:
Considers every node
Guarantees optimum
Usually too slow
Beam search
Considers only most promising nodes
 (beam width)
Does not guarantee convergence
Much faster
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Single Machine Example
(Total Weighted Tardiness)
Jobs
1
2
3
4
pj
10
10
13
4
dj
4
2
1
12
wj
14
12
1
12
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Branching
(Beam width = 2)
(•,•,•,•)
(1,•,•,•)
(2,•,•,•)
(3,•,•,•)
(4,•,•,•)
(1,4,2,3)
(2,4,1,3)
(3,4,1,2)
(4,1,2,3)
408
436
814
440
ATC Rule
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Beam Search
(•,•,•,•)
(1,•,•,•)
(1,2,•,•)
(2,•,•,•)
(1,3,•,•)
(1,4,2,3)
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(3,•,•,•)
(1,4,•,•)
(1,4,3,2)
(2,1,•,•)
(4,•,•,•)
(2,3,•,•)
(2,4,1,3)
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(2,4,•,•)
(2,4,3,1)
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Discussion
Implementation tradeoff:
Careful evaluation of nodes (accuracy)
Crude evaluation of nodes (speed)
Two-stage procedure
Filtering
crude evaluation
filter width > beam width
Careful evaluation of remaining nodes
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Topic 5
Random Search
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Construction versus
Improvement Heuristics
Construction Heuristics
Start without a schedule
Add one job at a time
Dispatching rules and beam search
Improvement Heuristics
Start with a schedule
Try to find a better ‘similar’ schedule
Can be combined
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Local Search
0.
1.
Start with a schedule s0 and set k = 0.
Select a candidate schedule
2.
from the neighborhood of sk
If acceptance criterion is met, let
3.
Otherwise let sk 1  sk .
Let k = k+1 and go back to Step 1.
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s  N sk 
sk 1  s.
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Defining a Local Search
Procedure
Determine how to represent a schedule.
Define a neighborhood structure.
Determine a candidate selection
procedure.
Determine an acceptance/rejection
criterion.
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Neighborhood Structure
Single machine
A pairwise interchange of adjacent jobs
(n-1 jobs in a neighborhood)
Inserting an arbitrary job in an arbitrary
position
(  n(n-1) jobs in a neighborhood)
Allow more than one interchange
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Neighborhood Structure
Job shops with makespan objective
Neighborhood based on critical paths
Set of operations start at t=0 and end at
t=Cmax
Machine 1
Machine 2
Machine 3
Machine 4
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Neighborhood Structure
The critical path(s) define the
neighborhood
Interchange jobs on a critical path
Too few better schedules in the
neighborhood
Too simple to be useful
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One-Step Look-Back
Interchange
Jobs on critical path
Machine h
Machine i
Interchange jobs
on critical path
Machine h
Machine i
One-step look-back
interchange
Machine h
Machine i
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Neighborhood Search
Find a candidate schedule from the
neighborhood:
Random
Appear most promising
(most improvement in objective)
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Acceptance/Rejection
Major difference between most methods
Always accept a better schedule?
Sometimes accept an inferior schedule?
Two popular methods:
Simulated annealing
Tabu search
Same except the acceptance criterion!
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Topic 6
Simulated Annealing
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Simulated Annealing (SA)
Notation
S0 is the best schedule found so far
Sk is the current schedule
G(Sk) it the performance of a schedule
Note that
G(Sk )  G(S0 )
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Aspiration Criterion
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Acceptance Criterion
Let Sc be the candidate schedule
If G(Sc) < G(Sk) accept Sc and let
Sk 1  Sc
If G(Sc)  G(Sk) move to Sc with
probability
Always  0
G ( Sk )G ( Sc )
P(Sk 1, Sc )  e
k
Temperature > 0
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Cooling Schedule
The temperature should satisfy
1   2  3  ...  0
Normally we let
k  0
but we sometimes stop when some
predetermined final temperature is
reached
If temperature is decreased slowly
convergence is guaranteed
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SA Algorithm
Step 1:
Set k = 1 and select the initial temperature 1.
Select an initial schedule S1 and set S0 = S1.
Step 2:
Select a candidate schedule Sc from N(Sk).
If G(S0) < G(Sc) < G(Sk), set Sk+1 = Sc and go to Step 3.
If G(Sc) < G(S0), set S0 = Sk+1 = Sc and go to Step 3.
If G(Sc) < G(S0), generate Uk Uniform(0,1);
If Uk  P(Sk, Sc), set Sk+1 = Sc; otherwise set Sk+1 = Sk; go to
Step 3.
Step 3:
Select k+1  k.
Let k=k+1. If k=N STOP; otherwise go to Step 2.
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Discussion
SA has been applied successfully to many
industry problems
Allows us to escape local optima
Performance depends on
Construction of neighborhood
Cooling schedule
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Topic 7
Tabu Search
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Tabu Search
Similar to SA but uses a deterministic
acceptance/rejection criterion
Maintain a tabu list of schedule changes
A move made entered at top of tabu list
Fixed length (5-9)
Neighbors restricted to schedules not
requiring a tabu move
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Tabu List
Rational
Avoid returning to a local optimum
Disadvantage
A tabu move could lead to a better schedule
Length of list
Too short  cycling (“stuck”)
Too long  search too constrained
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Tabu Search Algorithm
Step 1:
Set k = 1. Select an initial schedule S1 and set S0 = S1.
Step 2:
Select a candidate schedule Sc from N(Sk).
If Sk  Sc on tabu list set Sk+1 = Sk and go to Step 3.
Enter Sc  Sk on tabu list.
Push all the other entries down (and delete the last one).
If G(Sc) < G(S0), set S0 = Sc.
Go to Step 3.
Step 3:
Select k+1  k.
Let k=k+1. If k=N STOP; otherwise go to Step 2.
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Single Machine Total
Weighted Tardiness Example
Jobs
1
2
3
4
pj
10
10
13
4
dj
4
2
1
12
wj
14
12
1
12
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Initialization
Tabu list empty L={}
Initial schedule S1 = (2,1,4,3)
20 - 4 = 16
10 - 2 = 8
Deadlines
37 - 1 = 36
24 - 12 = 12
0
10
w T
j
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j
20
30
40
 1416 12 8 1 36 1212  500
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Neighborhood
Define the neighborhood as all schedules
obtain by adjacent pairwise interchanges
The neighbors of S1 = (2,1,4,3) are
(1,2,4,3)
Select the best
(2,4,1,3)
non-tabu schedule
(2,1,3,4)
Objective values are: 480, 436, 652
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Second Iteration
Let S0 =S2 = (2,4,1,3). G(S0)=436.
Let L = {(1,4)}
New neighborhood
Sequence
(4,2,1,3)
(2,1,4,3)
(2,4,3,1)
w T
460
500
608
j
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Third Iteration
Let S3 = (4,2,1,3), S0 = (2,4,1,3).
Let L = {(2,4),(1,4)}
New neighborhood
Sequence
(2,4,1,3)
(4,1,2,3)
(4,2,3,1)
w T
436
440
632
j
Tabu!
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Fourth Iteration
Let S4 = (4,1,2,3), S0 = (2,4,1,3).
Let L = {(1,2),(2,4)}
New neighborhood
Sequence
(1,4,2,3)
(4,2,1,3)
(4,1,3,2)
w T
402
460
586
j
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Fifth Iteration
Let S5 = (1,4,2,3), S0 = (1,4,2,3).
Let L = {(1,4),(1,2)}
This turns out to be the optimal
schedule
Tabu search still continues
Notes:
Optimal schedule only found via tabu list
We never know if we have found the optimum!
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Topic 8
Genetic Algorithm
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Genetic Algorithms
Schedules are individuals that form
populations
Each individual has associated fitness
Fit individuals reproduce and have
children in the next generation
Very fit individuals survive into the next
generation
Some individuals mutate
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Total Weighted Tardiness
Single Machine Example
First Generation
Individual
(sequence)
Fitness
 w T 
j
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Initial population
selected randomly
(2,1,3,4)
(3,4,1,2)
(4,1,3,2)
652
814
586
j
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Mutation Operator
Fittest
Individual
Individual
(2,1,3,4)
(3,4,1,2)
(4,1,3,2)
Fitness
652
814
586
Reproduction
via mutation
(4,3,1,2)
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Cross-Over Operator
Fittest
Individuals
Individual
(2,1,3,4)
(4,3,1,2)
(4,1,3,2)
Fitness
652
758
586
(4,1,3,2)
(2,1,3,2)
Reproduction
via cross-over
(2,1,3,4)
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Infeasible!
(4,1,3,4)
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Representing Schedules
When using GA we often represent
individuals using binary strings
(0 1 1 0 1 0 1 0 1 0 1 0 0 )
(1 0 0 1 1 0 1 0 1 0 1 1 1 )
(0 1 1 0 1 0 1 0 1 0 1 1 1 )
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Discussion
Genetic algorithms have been used very
successfully
Advantages:
Very generic
Easily programmed
Disadvantages:
A method that exploits special structure is
usually faster (if it exists)
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Topic 9
Nested Partitions Method
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The Nested Partitions Method
(Shameless self-promotion!)
Partitioning of all schedules
Similar to branch-and-bound & beam search
Random sampling & local search used to
find which node is most promising
Similar to beam search
Retains only one node (beam width = 1)
Allows for possible backtracking
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Single Machine Total
Weighted Tardiness Example
Jobs
1
2
3
4
pj
10
10
13
4
dj
4
2
1
12
wj
14
12
1
12
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First Iteration
(•,•,•,•)
(1,•,•,•)
(2,•,•,•)
Most promising region
(3,•,•,•)
(4,•,•,•)
Random Samples (uniform sampling, ATC rule, genetic algorithm)
(1,4,2,3)
(1,2,4,3)
(1,4,3,2)
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402
480
554
Promising Index
IE 514
I(1,•,•,•) = 402
117
Notation
We let
s(k) be the most promising region in the k-th iteration
sj(k) be the subregions, j=1,2,…,M
sM+1(k) be the surrounding region
S denote all possible regions (fixed set)
S0 all regions that contain a single schedule
I(s) be the promising index of sS
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Second Iteration
Most promising region s(2)
ss(2))
(•,•,•,•)
s4(2)
(1,•,•,•)
s1(2)
(1,2,•,•)
{(2,•,•,•),(3,•,•,•), (4,•,•,•)}
s2(2)
(1,3,•,•)
s3(2)
(1,4,•,•)
Best Promising Index
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Random Samples
(4,2,1,3)
(2,4,1,3)
(4,1,3,2)
460
436
586
I(s4(2)) = 436
119
Third Iteration
(•,•,•,•)
s3(3)
ss(3))
{(1,2,•,•), (1,4,•,•),(2,•,•,•),(3,•,•,•), (4,•,•,•)}
(1,•,•,•)
Random Samples
s(3)
(1,3,•,•)
s1(3)
(1,3,2,4)
s2(3)
(1,3,4,2)
(4,2,3,1)
(2,4,1,3)
(1,4,2,3)
632
436
402
I(s3(3)) = 402
Best Promising Index
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Fourth Iteration
ss(4))
(•,•,•,•)
s(4)
s4(4)
(1,•,•,•)
s1(4)
(1,2,•,•)
{(2,•,•,•),(3,•,•,•), (4,•,•,•)}
s2(4)
(1,3,•,•)
s3(4)
(1,4,•,•)
Backtracking!
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Finding the Optimal Schedule
The sequence
s (k )

k 1
is a Markov chain
Eventually the singleton region soptS0
containing the best schedule is visited
Absorbing state (never leave sopt)
Finite state space
 visited after finitely many iterations.
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Discussion
Finds best schedule in finite iterations
Only branch-and-bound can also guarantee
If we can calculate/estimate/guarantee the
probability of moving in right direction:
Expected number of iterations
Probability of having found optimal schedule
Stopping rules that guarantee performance
(unique)
Works best if incorporates dispatching rules
and local search (genetic algorithms, etc)
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Topic 10
Mathematical Programming
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Review of Mathematical
Programming
Many scheduling problems can be
formulated as mathematical programs:
Linear Programming (IE 534)
Nonlinear Programming (IE 631)
Integer Programming (IE 632)
See Appendix A in book.
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Linear Programs
Minimize
subject to
c1 x1  c2 x2  ... cn xn
a11 x1  a12 x2  ...  a1n xn  b1
a21 x1  a22 x2  ...  a2 n xn  b1

am1 x1  am 2 x2  ...  amn xn  b1
xj  0
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for j  1,...,n
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Solving LPs
LPs can be solved efficiently
Simplex method (1950s)
Interior point methods (1970s)
Polynomial time
Has been used in practice to solve huge
problems
IE 534
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Nonlinear Programs
Minimize
subject to
f ( x1 , x2 ,..., xn )
g1 ( x1, x2 ,..., xn )  0
g 2 ( x1, x2 ,..., xn )  0

g m ( x1, x2 ,..., xn )  0
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Solving Nonlinear
Programs
Optimality conditions
Karush-Kuhn-Tucker
Convex objective, convex constraints
Solution methods
Gradient methods (steepest descent,
Newton’s method)
Penalty and barrier function methods
Lagrangian relaxation
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Integer Programming
LP where all variables must be integer
Mixed-integer programming (MIP)
Much more difficult than LP
Most useful for scheduling
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Example: Single Machine
One machine and n jobs
Minimize n w j C j
j 1
Define the decision variables
1 if job j startsat timet
x jt  
0 otherwise.
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IP Formulation
Minimize
n Cmax 1
  w (t  p ) x
j 1
subject to
j
t 0
j
C max 1
x
t 0
n

jt
t 1
x
js
j 1 s  max{t  p j , 0}
jt
 1 j
 1 t
x jt  0,1 j , t
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Solving IPs
Solution Methods
Cutting plane methods
Linear programming relaxation
Additional constraints to ensure integers
Branch-and-bound methods
Branch on the decision variables
Linear programming relaxation provides bounds
Generally difficult
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Disjunctive Programming
Conjuctive
All constraints must be satisfied
Disjunctive
At least one of the constraints must be
satisfied
Zero-one integer programs
Use for job-shop scheduling
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