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### Topic 15

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Job Shop Scheduling

Job Shop Scheduling

- 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

Job Shop Example

- 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)

Graph Representation

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)

(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)

Solving the Problem

- 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)

Feasibility of the Schedule

- 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)

Solution Methods

- Exact solution
- Branch and Bound
- 20 machines and 20 jobs
- Dispatching rules (16+)
- Shifting Bottleneck
- Search heuristics
- Tabu, SA, GA, etc.

Types of Schedules

Definitions

- 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

Active Schedule, not Nondelay

Machine 1

(1,1)

Delay (Operation (2,1) does not fit)

Machine 2

(2,3)

(2,2)

(2,1)

Machine 3

(3,2)

0 2 4 6 8

Branch and Bound

- 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.

Generating Active Schedules

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

Generating Active Schedules

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.

Partitioning Tree

- Level 1

Level 1: select (1,1)

10

8

(1,1)

(2,1)

(3,1)

4

10

0

10

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)

Disjunctive Arcs

10

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

Branching Tree

No disjunctive arcs

(1,1) scheduled first

on machine 1

LB = 24

(1,3) scheduled first

on machine 1

LB = 26

Branching Tree

- Level 2:

Level 2: Select (2,2)

10

8

(1,1)

(2,1)

(3,1)

4

10

0

10

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)

Disjunctive Arcs

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

Lower Bounds

- 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)

The Shifting Bottleneck Heuristic

Shifting Bottleneck

- 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

Selecting a Machine

Set up a nonpreemptive single machine maximum lateness

problem for Machine 1:

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

Set up a nonpreemptive single machine maximum lateness

problem for Machine 2:

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

Set up a nonpreemptive single machine maximum lateness

problem for Machine 2:

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

Discussion

- Procedure continues until all the disjunctive arcs have been added
- Extremely effective
- Fast
- Good solutions
- ‘Just a heuristic’
- No guarantee of optimum

Discussion

- 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)

Discussion

- 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

Shifted Bottleneck for Total Weighted Tardiness Objective

Total Weighted Tardiness

- 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

Machine Selection

- 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

Criticality of Machines

- Criticality = subproblem objective function
- Simple
- More effective ways, e.g.
- Add disjunctive arcs for each machine
- Calculate new completion times and

Subproblem Solutions

- Solve using dispatching rule
- Use K=0.1
- Have t = 4,
- For machine 1 this results in

Schedule

first

Random Search for Job Shop Scheduling

Random Search Methods

- Popular to use genetic algorithms, simulated annealing, tabu search, etc.
- Do not work too well
- Problems defining the neighborhood
- Do not exploit special structure

Defining the Neighborhood

5

10

4

Sink

(1,1)

(2,1)

(3,1)

- Approximately nm neighbors!
- Simply too inefficient

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)

Job Shop with Makespan

- 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

The Nested Partitions Method

- 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!

A Flexible Flow Shop with Setups

Stage 1 Stage 2 Stage 3 Stage 4

Applications

- Very common in applications:
- Paper mills
- Steel lines
- Bottling lines
- Food processing lines

Classical Literature

- Exact solutions
- Simple flow shop with makespan criterion
- Two machine case (Johnson’s rule)
- Realistic problems require heuristic approaches

Objectives

- Multiple objectives usual
- Meet due dates
- Maximize throughput
- Minimize work-in-process (WIP)

Setting for job

j on Machine i

Generating Schedules

- Identify bottlenecks
- Compute time windows at bottleneck stage
- Compute machine capacity at bottleneck
- Schedule bottleneck stage
- Schedule non-bottlenecks

Identifying Bottlenecks

- In practice usually known
- Schedule downstream bottleneck first
- Determining the bottleneck
- loading
- number of shifts
- downtime due to setups
- Bottleneck assumed fixed

Identifying Time Window

- 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

Computing Capacity

- Capacity of each machine at bottleneck
- Speed
- Number of shifts
- Setups
- Two cases:
- Identical machines
- Non-identical machines

Scheduling Bottleneck

- Jobs selected one at a time
- Setup time
- Due date
- Capacity
- For example ATCS rule

Schedule Remaining Jobs

- Determined by sequence at bottleneck stage
- Minor adjustments
- Adjacent pairwise interchanges to reduce setup

Summary: Job Shops

- 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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