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# Topic 11 - PowerPoint PPT Presentation

Topic 11. Basic Project Scheduling. Project Scheduling. Jobs subject to precedence constraints Job on arc format (most common). 2. 5. 6. 0. 1. 4. 7. 2. 3. Overview. Critical Path Method (CPM) Program Evaluation and Review Technique (PERT). No resource Constraints. Project

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### Topic 11

Basic Project Scheduling

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• Jobs subject to precedence constraints

• Job on arc format (most common)

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5

6

0

1

4

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2

3

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Critical Path

Method (CPM)

Program Evaluation

and Review Technique

(PERT)

No resource

Constraints

Project

Scheduling

Heuristic Resource

Leveling

Integer Programming

Formulations

Resource

Constraints

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• Not allowed: no two jobs can have the same starting and ending node!

• Need to introduce a dummy job.

B

A

D

C

B

A

D

C

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A

E

B

D

C

Is this correct?

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A

B

E

C

D

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B

E

D

C

Job on Node Network

• No need for a dummy node

• Less used traditionally

• Recently more popular

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• Think of unlimited machines in parallel

• … and n jobs with precedence constraints

• Processing times pj as before

• Objective to minimize makespan

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• Forward procedure:

• Starting at time zero, calculate the earliest each job can be started

• The completion time of the last job is the makespan

• Backward procedure

• Starting at time equal to the makespan, calculate the latest each job can be started so that this makespan is realized

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Step 1:

Set at time t = 0 for all jobs j with no predecessors,

Sj’=0 and set Cj’ = pj.

Step 2:

Compute for each job j

Cj’ = Sj’ + pj.

Step 3:

The optimal makespan is

STOP

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Step 1:

Set at time t = Cmax for all jobs j with no successors,

Cj’’= Cmax and set Sj’’ = Cmax - pj.

Step 2:

Compute for each job j

Sj’’ = Cj’’ - pj.

Step 3:

Verify that

STOP

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• The forward procedure gives the earliest possible starting time for each job

• The backwards procedures gives the latest possible starting time for each job

• If these are equal the job is a critical job.

• If these are different the job is a slack job, and the difference is the float.

• A critical path is a chain of jobs starting at time 0 and ending at Cmax.

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Example

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7

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11

10

Forward Procedure

5+6=11

11+12=23

23+10=33

33+9=42

5

43+8=51

14+12=26

26+10=36

51+5=56

5+9=14

43+7=50

36+7=43

14+7=21

26+6=32

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9

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8

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12

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10

Backwards Procedure

24-12=12

34-10=24

43-9=34

51-8=43

14-9=5

56-5=51

36-10=26

43-7=36

56

26-12=14

56-5=51

51-8=43

35-10=26

43-7=36

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7

10

13

Critical Path

1

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12

14

3

11

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### Topic 12

Variable Processing Times

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• Assumed the processing times were fixed

• More money  shorter processing time

• Start with linear costs

• Processing time

• Marginal cost

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Resources (money)

Processing

time

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• Objective: minimum cost of project

• Time/Cost Trade-Off Heuristic

• Good schedules

• Works also for non-linear costs

• Linear programming formulation

• Optimal schedules

• Non-linear version not easily solved

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Cut set

Sink node

Source (dummy) node

Minimal cut set

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• Step 1:

• Set all processing times at their maximum

• Determine all critical paths with these processing times

• Construct the graph Gcp of critical paths

• Continue to Step 2

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• Step 2:

• Determine all minimum cut sets in Gcp

• Consider those sets where all processing times are larger than their minimum

• If no such set STOP; otherwise continue to Step 2

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• Step 3:

• For each minimum cut set:

• Compute the cost of reducing all processing times by one time unit.

• Take the minimum cut set with the lowest cost

• If this is less than the overhead per time unit go on to Step 4; otherwise STOP

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• Step 4:

• Reduce all processing times in the minimum cut set by one time units

• Determine the new set of critical paths

• Revise graph Gcp and go back to Step 2

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Maximum Processing Times

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Maximum Processing Times

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14

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Critical Path Subgraph (Gcp)

C1=7

C12=2

C6=3

C9=4

C14=8

C11=2

C3=4

Cut sets: {1},{3},{6},{9},

{11},{12},{14}.

Minimum cut

set with lowest cost

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6

9

14

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12

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Critical Path Subgraph (Gcp)

C1=7

C12=2

C6=3

C9=4

C14=8

C11=2

C13=4

C3=4

Cut sets: {1},{3},{6},{9},

{11},{12,13},{14}.

Minimum cut

set with lowest cost

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3

6

9

14

12

13

11

Critical Path Subgraph (Gcp)

C1=7

C12=2

C6=3

C9=4

C14=8

C11=2

C13=14

C3=4

Reduce processing

time until = 4

Reduce processing time

next on job 6

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3

6

9

7

2

12

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14

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Critical Path Subgraph (Gcp)

C2=2 C4=3 C7=4

C10=5

C1=7

C12=2

C6=3

C9=4

C14=8

C11=2

C13=4

C3=4

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• The heuristic does not guaranteed optimum

• Here total cost is linear

• Want to minimize

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Minimize

subject to

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### Topic 13

PERT

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• Assumed processing times deterministic

• Processing time of j random with mean mj and variance sj2.

• Want to determine the expected makespan

• Assume we have

pja = most optimistic processing time

pjm = most likely processing time (mode)

pjb = most pessimistic processing time

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• Estimate expected processing time

• Apply CPM with expected processing times

• Let Jcp be a critical path

• Estimate expected makespan

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• Estimate the variance of processing times

• and the variance of the makespan

• Assume it is normally distributed

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• Potential problems with PERT:

• Always underestimates project duration

• other paths may delay the project

• Non-critical paths ignored

• critical path probability

• critical activity probability

• Activities are not always independent

• same raw material, weather conditions, etc.

• Estimates by be inaccurate

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• No resource constraints:

• Critical Path Method (CPM)

• Simple deterministic

• Time/cost trade-offs

• Linear cost (heuristic or exact)

• Non-linear cost (heuristic)

• Accounting for randomness (PERT)

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### Topic 14

Adding Resource Constraints

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• Renewable resources

• Very hard problem

• No LP

• Can develop an IP

• Let job n+1 be dummy job (sink) and

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• Let H bound the makespan, e.g.

• Completion time of job j is

and the makespan

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Minimize

Subject to

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• The IP cannot be solved

• (Almost) always resource constraints

• Heuristic:

Resource constraint  Precedence constraint

• Example: pouring foundation and sidewalk

• both require same cement mixer

• delaying foundation delays building

• precedence constraint: pour foundation first

• not a logical constraint

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• Say n jobs need the same resource

• Could otherwise all be done simultaneously

• Add (artificial) precedence constraints

• Have n! possibilities

• Will we select the optimal sequence?

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• Resource leveling

• Solve with no resource constraints

• Plot the resource use as a function of time

• If infeasible suggest precedence constraints

• Longest job

• Minimum slack

• User adds constraints

• Start over

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Uses resource

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Resource Profile

Capacity

Time

10 20 30

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### Topic 15

Job Shop Scheduling

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• Have m machines and n jobs

• Each job visits some or all of the machines

• Customer order of small batches

• Wafer fabrication in semiconductor industry

• Hospital

• Very difficult to solve

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• Constraints

• Job follows a specific route

• One job at a time on each machine

Machine 1

(1,1)

(1,2)

(1,3)

Machine 2

(2,3)

(2,1)

(2,2)

Machine 3

(3,1)

(3,3)

Machine 4

(4,3)

(4,2)

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Each job follows a specific route through the job shop ...

(1,1)

(2,1)

(3,1)

Sink

Source

(1,2)

(2,2)

(4,2)

(2,3)

(1,3)

(4,3)

(3,3)

(Conjuctive arcs)

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(2,1)

(3,1)

Sink

Source

(1,2)

(2,2)

(4,2)

(2,3)

(1,3)

(4,3)

(3,3)

Graph Representation

Machine constraints must also be satisfied ...

(Disjunctive arcs)

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• Select one of each pair of disjunctive arcs

• The longest path in this graph G(D) determines the makespan

(1,1)

(2,1)

(3,1)

Sink

Source

(1,2)

(2,2)

(4,2)

(2,3)

(1,3)

(4,3)

(3,3)

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• Are all selections feasible?

(1,1)

(2,1)

(3,1)

Sink

Source

(1,2)

(2,2)

(4,2)

(2,3)

(1,3)

(4,3)

(3,3)

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Minimize

Subject to

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• Exact solution

• Branch and Bound

• 20 machines and 20 jobs

• Dispatching rules (16+)

• Shifting Bottleneck

• Search heuristics

• Tabu, SA, GA, etc.

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### Topic 16

Types of Schedules

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• A schedule is nondelay if no machine is idled when there is an operation available

• A schedule is called active if no operation can be completed earlier by altering the sequence on machines and not delaying other operations

• For “regular” objectives the optimal schedule is always active but not necessarily nondelay

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Semiactive

Optimum

Nondelay

Active

All Schedules

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Machine 1

(1,1)

(2,1)

Machine 2

(2,3)

(2,2)

(2,1)

Machine 3

(3,2)

0 2 4 6 8

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Machine 1

(1,1)

Machine 2

(2,3)

(2,2)

(2,1)

Machine 3

(3,2)

0 2 4 6 8

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Machine 1

(1,1)

Machine 2

(2,3)

(2,1)

(2,2)

Machine 3

(3,2)

0 2 4 6 8

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### Topic 17

Branch & Bound for Job Shops

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• Operation (i,j) with duration pij

• Minimize makespan

• Branch by generating all active schedules

• Notation

• Let W denote operations whose predecessors have been scheduled

• Let rij be the earliest possible starting time of (i,j) in W.

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Step 1. (Initialize)

Let W contain the first operation of each job

Let rij = 0 for all (i,j)W.

Step 2. (Machine selection)

Compute the current partial schedule

and let i* denote the machine where minimum achieved

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Step 3. (Branching)

Let W’ denote all operations on machine i such that

For each operation in W’ consider a partial schedule with that operation next on i*

For each partial schedule, delete operation from W and include immediate follower in W. Go back to Step 2.

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Selection of (i*,j)

Selection of (i*,l)

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(1,1)

(2,1)

(3,1)

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Source

(2,2)

(1,2)

(4,2)

(3,2)

Sink

0

3

7

(1,3)

4

(2,3)

(4,3)

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• Level 1

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(1,1)

(2,1)

(3,1)

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Source

(2,2)

(1,2)

(4,2)

(3,2)

Sink

0

3

7

(1,3)

4

(2,3)

(4,3)

Disjunctive Arcs

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8

(1,1)

(2,1)

(3,1)

4

0

4

0

8

3

5

6

Source

(2,2)

(1,2)

(4,2)

(3,2)

Sink

0

4

3

7

(1,3)

4

(2,3)

(4,3)

Disjunctive Arcs

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No disjunctive arcs

(1,1) scheduled first

on machine 1

LB = 24

(1,3) scheduled first

on machine 1

LB = 26

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• Level 2:

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(1,1)

(2,1)

(3,1)

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Source

(2,2)

(1,2)

(4,2)

(3,2)

Sink

0

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7

(1,3)

4

(2,3)

(4,3)

Disjunctive Arcs

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No disjunctive arcs

(1,1) scheduled first

on machine 1

LB = 24

(1,3) scheduled first

on machine 1

LB = 26

(1,1) first on M1 and

(2,2) first on M2

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• Lower bounds

• The length of the critical path in G(D’)

• Quick but not very tight

• Linear programming relaxation

• A maximum lateness problem (see book)

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### Topic 18

The Shifting Bottleneck Heuristic

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• Minimize makespan in a job shop

• Let M denote the set of machines

• Let M0  M be machines for which disjunctive arcs have been selected

• Basic idea:

• Select a machine in M - M0 to be included in M0

• Sequence the operations on this machine

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8

(1,1)

(2,1)

(3,1)

4

0

0

8

3

5

6

Source

(2,2)

(1,2)

(4,2)

(3,2)

Sink

0

3

7

(1,3)

4

(2,3)

(4,3)

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Set up a nonpreemptive single machine maximum lateness

problem for Machine 1:

Optimum sequence is 1,2,3 with Lmax(1)=5

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Set up a nonpreemptive single machine maximum lateness

problem for Machine 2:

Optimum sequence is 2,3,1 with Lmax(2)=5

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• Similarly,

 Either Machine 1 or Machine 2 is the bottleneck

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10

8

(1,1)

(2,1)

(3,1)

4

0

10

0

8

3

5

6

Source

(2,2)

(1,2)

(4,2)

(3,2)

Sink

0

3

3

7

(1,3)

4

(2,3)

(4,3)

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Set up a nonpreemptive single machine maximum lateness

problem for Machine 2:

Optimum sequence is 2,1,3 with Lmax(2)=1

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Similarly,

 Either Machine 2 or Machine 3 is the bottleneck

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10

8

(1,1)

(2,1)

(3,1)

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10

0

8

8

0

8

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5

6

Source

(2,2)

(1,2)

(4,2)

(3,2)

Sink

0

3

3

7

(1,3)

4

(2,3)

(4,3)

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• Procedure continues until all the disjunctive arcs have been added

• Extremely effective

• Fast

• Good solutions

• ‘Just a heuristic’

• No guarantee of optimum

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(•,•,•)

(1,•,•)

(2,•,•)

(3,•,•)

(1,2,3)

(1,3,2)

(3,1,2)

(3,2,1)

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• The solution is actually a little bit more complicated than before

• Precedence constraints because of other (already scheduled) machines

• Delay precedence constraints

(see example in book)

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• Shifting bottleneck can be applied generally

• Basic idea

• Solve problem “one variable at a time”

• Determine the “most important” variable

• Find the best value of that variable

• Move on to the “second most important” ….

• Here we treat each machine as a variable

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### Topic 19

Shifted Bottleneck for Total Weighted Tardiness Objective

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• We now apply a shifted bottleneck procedure to a job shop with total weighted tardiness objective

• Need n sinks in disjunctive graph

• Machines scheduled one at a time

• Given current graph calculate completion time of each job

• Some n due dates for each operation

• Piecewise linear penalty function

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• Machine criticality

• Solve a single machine problem

• Piecewise linear cost function

• May have delayed precedence constraints

• Generalizes single-machine with n jobs, precedence constraints, and total weighted tardiness objective

• ATC rule

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Earliest time machine can be used

Ranks jobs

 good schedule

Scaling constant

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• Criticality = subproblem objective function

• Simple

• More effective ways, e.g.

• Add disjunctive arcs for each machine

• Calculate new completion times and

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5

10

4

Sink

(1,1)

(2,1)

(3,1)

5

0

4

5

6

Source

(3,2)

(1,2)

(2,2)

Sink

0

3

7

Sink

(3,3)

5

(2,3)

(1,3)

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• Solve using dispatching rule

• Use K=0.1

• Have t = 4,

• For machine 1 this results in

Schedule

first

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Schedule

first

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10

4

Sink

(1,1)

(2,1)

(3,1)

5

5

0

4

5

6

Source

(3,2)

(1,2)

(2,2)

Sink

5

0

3

7

Sink

(3,3)

5

(2,3)

(1,3)

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Schedule

first

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5

10

4

Sink

(1,1)

(2,1)

(3,1)

5

5

10

0

4

5

6

Source

(3,2)

(1,2)

(2,2)

Sink

5

3

0

3

7

Sink

(3,3)

5

(2,3)

(1,3)

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5

10

4

Sink

(1,1)

(2,1)

(3,1)

5

5

10

4

0

4

5

6

Source

(3,2)

(1,2)

(2,2)

Sink

5

5

3

0

3

7

Sink

(3,3)

5

(2,3)

(1,3)

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Machine 1

1,1

1,2

1,3

Machine 2

2,3

2,1

2,2

Machine 3

3,3

3,2

3,1

0 5 10 15 20 25 30

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• Objective function

• Try finding a better schedule using LEKIN

(optimal value = 18)

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### Topic 20

Random Search for Job Shop Scheduling

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• Popular to use genetic algorithms, simulated annealing, tabu search, etc.

• Do not work very well

• Problems defining the neighborhood

• Do not exploit special structure

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• Approximately nm neighbors!

• Simply too inefficient

5

10

4

Sink

(1,1)

(2,1)

(3,1)

5

5

10

4

0

4

5

6

Source

(3,2)

(1,2)

(2,2)

Sink

5

5

3

0

3

7

Sink

(3,3)

5

(2,3)

(1,3)

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• Random search methods can be applied

• Use ‘critical path’ neighborhood

• Can eliminate many neighbors immediately

• Specialized methods usually better

• Random search = ‘giving up’ !

• Traveling Salesman Problem (TSP)

• Very well studied

• Lin-Kernighan type heuristics (1970)

• Order of 1000 times faster than random search

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Percentage over known optimum:

Winner

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• Partitioning

• by scheduling the bottleneck machine first

• Random sampling

• using randomized dispatching rules

• Calculating the promising index

• incorporating local improvement heuristic

• Can incorporate any special structure!

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### Topic 21

Special Case: Flow Shops

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Stage 1 Stage 2 Stage 3 Stage 4

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• Very common in applications:

• Paper mills

• Steel lines

• Bottling lines

• Food processing lines

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• Exact solutions

• Simple flow shop with makespan criterion

• Two machine case (Johnson’s rule)

• Realistic problems require heuristic approaches

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• Multiple objectives usual

• Meet due dates

• Maximize throughput

• Minimize work-in-process (WIP)

Setting for job

j on Machine i

IE 514

• Identify bottlenecks

• Compute time windows at bottleneck stage

• Compute machine capacity at bottleneck

• Schedule bottleneck stage

• Schedule non-bottlenecks

IE 514

• In practice usually known

• Schedule downstream bottleneck first

• Determining the bottleneck

• loading

• number of shifts

• downtime due to setups

• Bottleneck assumed fixed

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• Due date

• Shipping day

• Multiply remaining processing times with a safety factor

• Release date

• Status sj of job j

• Release date if sj = l

Decreasing

function -

determined

empirically

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• Capacity of each machine at bottleneck

• Speed

• Number of shifts

• Setups

• Two cases:

• Identical machines

• Non-identical machines

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• Jobs selected one at a time

• Setup time

• Due date

• Capacity

• For example ATCS rule

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• Determined by sequence at bottleneck stage

• Minor adjustments

• Adjacent pairwise interchanges to reduce setup

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• Representation: graph w/disjoint arcs

• Solution methods

• Branch-and-bound

• Shifted bottleneck heuristic

• Beam search

• Can incorporate bottleneck idea & dispatching rules

• Random search methods

• Special case: flow shops

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