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IE 514 Production

Scheduling

Introduction

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Contact Information

Siggi Olafsson

3018 Black Engineering

294-8908

[email protected]

http://www.public.iastate.edu/~olafsson

OH: MW 10:30-12:00

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Administration (Syllabus)

Text

Prerequisites

Assignments

Homework 35%

Project 40%

Final Exam 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 andcommunications

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Manufacturing Scheduling

Short product life-cycles

Quick-response manufacturing

Manufacture-to-order

More complex operations must bescheduled in shorter amount of time withless room for errors!

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Scope ofCourse

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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OurApproach

Scheduling Problem

Model

Conclusions

Problem Formulation

Solve with Computer Algorithms

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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

Databases

Schedulegeneration

User interfaces

Database Management

Automatic Schedule Generator

Schedule EditorPerformance

Evaluation

Graphical Interface

User

Order

master file

Shop floor

data collection

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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 reactivescheduling

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 jobsA 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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OtherCharacteristics

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

Minimizing makespan tends to maximizethroughput and balance load

_ a_ a niiiii

n

,...,1,,...,,max

,...,,max

21

21max

!!

!

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Due Date Related

Objectives

Lateness

Minimize maximum lateness (Lmax)

Tardiness

Minimize the weighted tardiness

jjj dCL

_ a0,max jjj dCT !

!

n

j

jjTw1

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Due Date Penalties

jdjC

jL

jdjC

j

jdjC

jU1

jdjC

In practice

Tardiness

Late or Not

Lateness

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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 thecompletion times:

n

i

jC1

n

j

jjCw1

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OtherCosts

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 environmentwith a specific objective

Analyze to prove an optimal policy or toshow that no simple optimal policy exists

Thousands of problems have been studiedin 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 beminimized (7wjCj)

We denote this problem as

jjCw||1

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Optimal Solution

Theorem: Weighted Shortest Processingtime first - called the WSPT rule - is

optimal for

Note: The SPT rule starts withthe job thathasthe shortestprocessingtime, moves onthe jobwiththe secondshortestprocessingtime, etc.

jjCw||1

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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 jfollowed by job k such that

Do a pairwise interchange to get schedule S

k

k

j

j

p

w

p

w

j k

k j

kj ppt

kj ppt

t

t

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Proof (continued)

The weighted completion time of the two jobs under S is

The weighted completion time of the two jobs under S is

Now:

Contradicting that S is optimal.

jkjkk wpptwpt )()(

kkjjj wpptwpt )()(

jjkkk

kkjkjj

kkkjjjkkjjj

wpptwpt

wptwpwpt

wptwpwptwpptwpt

)()(

)()(

)()()()(

!

"

!

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Complexity Theory

Classic scheduling theory draws heavily oncomplexity theory

The complexity of an algorithm is itsrunning time in terms of the inputparameters (e.g., number of jobs and

number of machines) Big-Oh notation, e.g., O(n2m)

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Polynomial versus NP-Hard

)(log nO

)(nO

)(2

nO)2(n

O

)(sizeroblem n

Time

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Scheduling in Practice

Practical scheduling problems cannot besolved 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 arehard

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

Construction

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

maximum lateness_ a0,max tpd jj

Current

Time

Processing

TimeDeadline

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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:

No efficient algorithm (NP-Hard)

Branch and bound can only solve verysmall problems (

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Case 1: Tight Deadlines

Assume dj=0

Then

We know that WSPT is optimal for thisproblem!

_ a_ a

!!!

!

jjjj

jj

jjj

CwTw

CC

dCT

,0max

,0max

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Conclusion

The WSPT is optimal in the extreme caseand 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 sufficientlyspread out then the MS rule

is optimal (proof a bit harder)

Conclusion: The MS rule should be a goodheuristic whenever deadlines are widelyspread out

_ a0,maxarg tpdj jjselect

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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 rulesinto one composite dispatching rule

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Apparent Tardiness Cost

(ATC) Dispatching Rule

New ranking index

When machine becomes free: Compute index for all remaining jobs

Select job with highest value

Scaling constant

_ a

)(

0,maxexp)(

tpK

tpd

p

wtI

jj

j

j

j

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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

and the due date range factormax

1CdX

max

minmax

C

ddR

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Choosing K

Usually 1.5 e Ke 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 jobshops

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 bestschedule 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

If jobs have different release dates, which

we denote

then the problem is NP-Hard

What makes it so much more difficult?

max||1 L

max||1 Lrj

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max||1 Lrj

Jobs 1 2 3

jp 4 2 5

jr 0 3 5

jd 8 14 10

1 2 3

1r 2r 3r

0 5 10 15

1d 3d 2d

_ a _ a_ a _ a 55,4,4max1015,1410,84max

,,max,,max 332211321max

!!!

!! dCdCdCLLLL

EDD

Can we

improve?

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Delay Schedule

1 23

1r 2r 3r

0 5 10 15

1d 3d 2d

_ a _ a_ a _ a 30,2,4max1010,1416,84max

,,max,,max 332211321max

!!!

!! dCdCdCLLLL

Add a delay

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 forthe preemptive (prmp) version of the

problem

Note that in the previous example, the preemptive EDDrule gives us the optimal schedule

max|,|1 L pr prj

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Branch and Bound

The problem

cannot be solved using a simple dispatching ruleso 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

max||1 Lrj

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Data

Jobs 1 2 3 4

jp 4 2 6 5

jr 0 1 3 5

jd 8 12 11 10

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Branching

(,,,)

(1,,,) (2,,,) (3,,,) (4,,,)

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Branching

(,,,)

(1,,,) (2,,,) (3,,,) (4,,,)

Discard immediately because

5

3

4

3

!

!

r

r

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Branching

(,,,)

(1,,,) (2,,,) (3,,,) (4,,,)

Need to developlower bounds on

these nodes and do further branching.

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Bounding (in general)

Typical way to develop bounds is to relax theoriginal problem to an easily solvable problem

Three cases:

If there is no solution to the relaxed problem there isno solution to the original problem

If the optimal solution to the relaxed problem isfeasible 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 boundon its performance

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Relaxing the Problem

The problem

is a relaxation to the problem

Not allowing preemption is a constraint in theoriginal problem but not the relaxed problem

We know how to solve the relaxed problem(preemptive EDD rule)

max||1 Lrj

max|,|1 L pr prj

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Bounding

Preemptive EDD rule optimal for thepreemptive version of the problem

Thus, solution obtained is a lower boundon the maximum delay

If preemptive EDD results in a non-

preemptive schedule all nodes with higherlower bounds can be discarded.

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Lower Bounds

Start with (1,,,):

Job with EDD is Job 4 but

Second earliest due date is for Job 3

54

!r

Job 1

Job 2

Job 3Job 4

0 10 20

5max uL

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Branching

(,,,)

(1,,,) (2,,,) (3,,,) (4,,,)

(1,2,,) (1,3,,)

5max uL

7max uL

6max uL 5max !L(1,3,4,2)

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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

jp 10 10 13 4

jd 4 2 1 12

jw 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)

ATC Rule

408 436 814 440

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Beam Search

(,,,)

(1,,,) (2,,,) (3,,,) (4,,,)

(1,2,,) (1,3,,) (1,4,,) (2,1,,) (2,3,,) (2,4,,)

(2,4,3,1)(2,4,1,3)(1,4,3,2)(1,4,2,3)

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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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Local Search

0. Start with a schedule s0 and set k =0.

1. Select a candidate schedule

from the neighborhood of sk2. If acceptance criterion is met, let

Otherwise let

3. Let k = k+1 and go back to Step 1.

ksNs

.1 ssk!

.1 kk

ss !

Defining a Local Search

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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

( e n(n-1)jobs ina neighborhood)

Allow more than one interchange

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Job shops with makespan objective

Neighborhood based on critical paths

Set of operations start at t

=0and end att=Cmax

Machine 1

Machine 2

Machine 3

Machine 4

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The critical path(s) define theneighborhood

Interchange jobs on a critical path Too few better schedules in the

neighborhood

Too simple to be useful

One-Step Look-Back

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One StepLook Back

Interchange

Machine h

Machine i

Machine h

Machine i

Jobs on critical path

Interchange jobson critical path

Machine h

Machine i

One-step look-backinterchange

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Neighborhood Search

Find a candidate schedule from theneighborhood:

RandomAppear 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

)()( 0SGSG k u

Aspiration Criterion

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Acceptance Criterion

LetSc be the candidate schedule

If G(Sc)< G(Sk)acceptSc and let

If G(Sc)u G(Sk)move to Sc withprobability

ck SS !1

k

ck SS

ck eSSF

)()(

1),(

!

Temperature > 0

Always e 0

Cooling Schedule

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Cooling Schedule

The temperature should satisfy

Normally we let

but we sometimes stop when somepredetermined final temperature isreached

If temperature is decreased slowlyconvergence is guaranteed

0...321 "uuu

0pk

F

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SAAlgorithm

Step 1:Setk=1 and select the initial temperatureF1.

Select an initial schedule S1 and set S0=S1.

Step 2:

Select a candidate schedule Sc from N(Sk).IfG(S0)< G(Sc)< G(Sk), set Sk+1 =Sc and go to Step 3.

IfG(Sc)< G(S0), set S0=Sk+1 =Sc and go to Step 3.

IfG(Sc)< G(S0), generate UkbUniform(0,1);

IfUk eP(Sk, Sc), set Sk+1 =Sc; otherwise set Sk+1 =Sk; go to

Step 3.Step 3:

SelectFk+1 e Fk.

Letk=k+1. Ifk=NSTOP; otherwise go to Step 2.

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Discussion

SA has been applied successfully to manyindustry 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 deterministicacceptance/rejection criterion

Maintain a tabu list of schedule changesA move made entered at top of tabu list

Fixed length (5-9)

Neighbors restricted to schedules notrequiring a tabu move

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Tabu List

Rational

Avoid returning to a local optimum

DisadvantageA tabu move could lead to a better schedule

Length of list

Too short cycling (stuck) Too long search too constrained

Tabu Search Algorithm

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Tabu Search Algorithm

Step 1:Setk=1. Select an initial schedule S1 and set S0=S1.

Step 2:Select a candidate schedule Sc from N(Sk).

If Skp Sc on tabu list set Sk+1 =Sk and go to Step 3.

Enter Sc p Sk on tabu list.

Push all the other entries down (and delete the last one).

IfG(Sc)< G(S0), set S0=Sc.

Go to Step 3.

Step 3:SelectFk+1 e Fk.

Letk=k+1. Ifk=NSTOP; otherwise go to Step 2.

Single Machine Total

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Weighted Tardiness Example

Jobs 1 2 3 4

jp 10 10 13 4

jd 4 2 1 12

jw 14 12 1 12

Initialization

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Initialization

Tabu list empty L={}

Initial schedule S1 = (2,1,4,3)

0 10 20 30 40

Deadlines

20 - 4 = 16

10 - 2 = 8

37 - 1 = 3624 - 12 = 12

!! 50012123618121614jjw

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Neighborhood

Define the neighborhood as all schedulesobtain by adjacent pairwise interchanges

The neighbors ofS1 = (2,1,4,3) are(1,2,4,3)

(2,4,1,3)

(2,1,3,4) Objective values are: 480, 436, 652

Selectthe best

non-tabu schedule

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Second Iteration

LetS0 =S2 = (2,4,1,3). G(S0)=436.

LetL = {(1,4)}

New neighborhood

Seq e ce (4,2,1,3) (2,1,4,3) (2,4,3,1)

Twj 460 500 608

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Third Iteration

LetS3 = (4,2,1,3), S0 = (2,4,1,3).

LetL = {(2,4),(1,4)}

New neighborhood

Seq e ce (2,4,1,3) (4,1,2,3) (4,2,3,1)

Twj 436 440 632

Tabu!

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Fourth Iteration

LetS4 = (4,1,2,3), S0 = (2,4,1,3).

LetL = {(1,2),(2,4)}

New neighborhood

Seq e ce (1,4,2,3) (4,2,1,3) (4,1,3,2)

Twj 402 460 586

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Fifth Iteration

LetS5 = (1,4,2,3), S0 = (1,4,2,3).

LetL = {(1,4),(1,2)}

This turns out to be the optimalschedule

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

Genetic Algorithms

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Genetic Algorithms

Schedules are individuals that formpopulations

Each individual has associated fitness

Fit individuals reproduce and havechildren in the nextgeneration

Very fit individuals survive into the nextgeneration

Some individuals mutate

Total Weighted Tardiness

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Individual

(sequence)

(2,1,3,4) (3,4,1,2) (4,1,3,2)

Fitness

jjTw 652 814 586

Single Machine Example

First GenerationInitialpopulation

selected randomly

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Mutation Operator

Individual (2,1,3,4) (3,4,1,2) (4,1,3,2)

Fitness 652 814 586

(4,3,1,2)

Reproduction

via mutation

Fittest

Individual

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Cross-OverOperator

Individual (2,1,3,4) (4,3,1,2) (4,1,3,2)

Fitness 652 758 586

(4,1,3,2)

(2,1,3,4)

Reproduction

via cross-over

Fittest

Individuals

(2,1,3,2)

(4,1,3,4)

Infeasible!

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Representing Schedules

When using GA we often representindividuals 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 verysuccessfully

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

The Nested Partitions MethodThe Nested Partitions Method

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The Nested Partitions MethodThe Nested Partitions Method(Shameless self(Shameless self--promotion!)promotion!)

Partitioning of all schedules

Similar to branch-and-bound & beam search

Random sampling & local search used tofind which node is most promising

Similar to beam search

Retains only one node (beam width = 1)Allows for possible backtracking


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Weighted Tardiness Example

Jobs 1 2 3 4

jp 10 10 134

jd 4 2 1 12

jw 14 12 1 12

First Iteration

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First Iteration

(,,,)

(1,,,) (2,,,) (3,,,) (4,,,)

(1,4,2,3) 402

Random Samples (uniform sampling,ATC rule, genetic algorithm)

(1,2,4,3) 480(1,4,3,2) 554

I(1,,,) = 402

Promising Index

Mostpromising region

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Notation

We letW(k) be the most promising region in the k-th iteration

Wj

(k) be the subregions, j=1,2,,M

WM+1(k) be the surrounding region

7 denote all possible regions (fixed set)

70 all regions that contain a single schedule

I(W) be the promising index of

W7

Second Iteration

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Second Iteration

(,,,)

(1,,,) {(2,,,),(3,,,),(4,,,)}

(1,3,,) (1,4,,)(1,2,,) (4,2,1,3) 460

Random Samples

(2,4,1,3) 436

(4,1,3,2) 586

Mostpromising region W(2)

Best Promising IndexI(W

4(2)) = 436

W4(2)

W1(2) W2(2) W3(2)

sW(2))

Third Iteration

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Third Iteration

(,,,)

(1,,,) {(1,2,,),(1,4

,,),(2,,,),(3,,,),(4

,,,)}

(1,3,,) (4,2,3,1) 632

Random Samples

(2,4,1,3) 436

(1,4,2,3) 402

(1,3,2,4) (1,3,4,2) I(W3(3)) = 402

W1(3) W2(3)

W3(3)sW(3))

W(3)

Best Promising Index

Fourth Iteration

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Fourth Iteration

(,,,)

(1,,,) {(2,,,),(3,,,),(4,,,)}

(1,3,,) (1,4,,)(1,2,,)

W4(4)

W1(4) W2(4) W3(4)

sW(4))

W(4)

Backtracking!

Finding the Optimal Schedule

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g p

The sequence

is a Markov chain

Eventually the singleton region Wopt7containing the best schedule is visited

Absorbing state (never leave Wopt)

Finite state space

visited after finitely many iterations.

_ ag!1

)(k

kW

Discussion

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Discussion

Finds best schedule in finite iterations Only branch-and-bound can also guarantee

If we can calculate/estimate/guarantee theprobability 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 rulesand local search (genetic algorithms, etc)

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Topic 10

Mathematical Programming

Review ofMathematical

P i

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Programming

Many scheduling problems can beformulated as mathematical programs:

Linear Programming (IE 534)

Nonlinear Programming (IE 631)

Integer Programming (IE 632)

See Appendix A in book.

Li P

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Linear Programs

Minimize

subject tonnxcxcxc ...2211

njx

bxaxaxa

bxaxaxa

bxaxaxa

j

nn

nn

nn

,...,1for0

...

...

...

12211

12222121

11212111

!u

e

e

/

S l i LP

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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

N li P

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Nonlinear Programs

Minimize

subject to

),...,,( 21 nxxxf

0),...,(

0),...,(

0),...,(

2,1

2,12

2,11

!

!

!

n

n

n

xxxg

xxxg

xxxg

/

Solving Nonlinear

P

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Programs

Optimality conditions

Karush-Kuhn-Tucker

Convex objective, convex constraints Solution methods

Gradient methods (steepest descent,Newtons method)

Penalty and barrier function methods

Lagrangian relaxation

I t P i

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Integer Programming

LP where all variables mustbe integer

Mixed-integer programming (MIP)

Much more difficultthan LP Mostuseful for scheduling

E ample Single Machine

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Example: Single Machine

One machine and n jobs

Minimize

Define the decision variables

!n

j jjCw

1

! otherwise.0

attimestartsjobif1 tjxjt

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Solving IPs

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Solving IPs

Solution Methods

Cutting plane methods

Linear programming relaxationAdditional constraints to ensure integers

Branch-and-bound methods

Branch on the decision variables

Linear programming relaxation provides bounds

Generally difficult

Disjunctive Programming

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Disjunctive Programming

Conjuctive

All constraints must be satisfied

DisjunctiveAt least one of the constraints must be

satisfied

Zero-one integer programs

Use for job-shop scheduling

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