Short Term Scheduling

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Short Term Scheduling
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Characteristics

• Planning horizon is short
• Multiple unique jobs (tasks) with varying
  processing times and due dates
• Multiple unique jobs sharing the same set of
  resources (machines)
• Time is treated as continuous (not discretized
  into periods)
• Varying objective functions

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Characteristics (Continued)

• Common in make-to-order environments with high
  product variety
• Common as a support tool for MRP in generating
  detailed schedules once orders have been released

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Example

• Two jobs, A and B
• Two machines M1 and M2
• Jobs are processed on M1 and then on M2
• Job A 9 minutes on M1 and 2 minutes on M2
• Job B 4 minutes on M1 and 9 minutes on M2

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Example
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Example (Continued)
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Challenge
As the number of jobs increases, complete
enumeration becomes difficult 3! 6, 4! 24,
5! 120, 6! 720, 10! 3,628,800,
while 13! 6,227,020,800 25!
15,511,210,043,330,985,984,000,000
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Classification of Scheduling Problems

• Number of jobs
• Number of machines
• Type of production facility
• Single machine
• Flow shop
• parallel machines
• job shop
• Job arrivals
• Static
• Dynamic
• Performance measures

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A Single Machine Example

Jobs 1 2 3 4 5 6 Processing time, pj 12
8 3 10 4 18 Release time, rj -20 -15
12 -10 3 2 Due date, dj 10 2 72 -8
8 60
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The Single Machine Problem

• Single machine scheduling problems seek an
  optimal sequence (for a given criterion) in which
  to complete a given collection of jobs on a
  single machine that can accommodate only one job
  at a time.

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Decision Variables

• xj time job j is started (relative to time 0
  now), xj ? max(0, rj) for all values of j.

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Sequencing constraints

• - (start time of j) (processing time of j) lt
  start time of j
• 
  or
• - (start time of j) (processing time of j) lt
  start time of j

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Sequencing constraints

• - (start time of j) (processing time of j) ?
  start time of j
• 
  or
• - (start time of j) (processing time of j) ?
  start time of j
• xj pj ? xj or xj pj ? xj

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Disjunctive variables

• - Introduce disjunctive variables yjj, yjj
  1 if job j is scheduled before job j and yjj
  0 otherwise.
• xj pj ? xj M(1 - yjj),
• xj pj ? xj Myjj,
• for all pairs of j and j (for every j and every
  j gt j), M is a large positive constant

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Example
Formulate sequencing constraints for a problem
with three jobs, j 1, , 3 with processing
times 14, 3, and 7.
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Due date constraints
xj pj ? dj, for all values of j
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Examples of Performance measures
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Example
Jobs 1 2 3 Processing time 15 6 9 Release
time 5 10 0 Due date 20 25 36 Start time
9 24 0
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Objective functions
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Objective functions
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Formulation Minimizing Makespan (Maximum
Completion Time)
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A Formulation with a Linear Objective Function
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• Similar formulations can be constructed with
  other min-max objective functions, such as
  minimizing maximum lateness or maximum tardiness.
  
• Other objective functions involving minimizing
  means (other than mean tardiness) are already
  linear.

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The Job Shop Scheduling Problem

• N jobs
• M Machines
• A job j visits in a specified sequence a subset
  of the machines

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Notation

• pjm processing time of job j on machine m,
• xjm start time of job j on machine m,
• yj,j,m 1 if job j is scheduled before job j
  on machine m,
• M(j) The subset of the machines visited by job
  j,
• SS(m, j) the set of machines that job j visits
  after visiting machine m

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Formulation
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Solution Methods

• Small to medium problems can be solved exactly
  (to optimality) using techniques such as branch
  and bound and dynamic programming
• Structural results and polynomial (fast)
  algorithms for certain special cases
• Large problems in general may not solve within a
  reasonable amount of time (the problem belongs to
  a class of combinatorial optimization problems
  called NP-hard)
• Large problems can be solved approximately using
  heuristic approaches

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Single Machine Results

• Makespan
• Not affected by sequence
• Average Flow Time
• Minimized by performing jobs according to the
  shortest processing time (SPT) order
• Average Lateness
• Minimized by performing in shortest processing
  time (SPT) order
• Maximum Lateness (or Tardiness)
• Minimized by performing in earliest due date
  (EDD) order.
• If there exists a sequence with no tardy jobs,
  EDD will do it

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Single Machine Results (Continued)

• Average Weighted Flow Time
• Minimized by performing according to the
  smallest processing time ratio (processing
  time/weight) order
• Average Tardiness
• No simple sequencing rule will work

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Two Machine Results

• Given a set of jobs that must go through a
  sequence of two machines, what sequence will
  yield the minimum makespan?

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Johnsons Algorithm

• A Simple algorithm (Johnson 1954)
• 1. Sort the processing times of the jobs on the
  two machines in two lists.
• 2. Find the shortest processing time in either
  list and remove the corresponding job from both
  lists.
• If the job came from the first list, place it in
  the first available position in the sequence.
• If the job came from the second list, place it in
  the last available position in sequence.
• 3. Repeat until are all jobs have been sequenced.
• The resulting sequence minimizes makespan.

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• Data

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Johnsons Algorithm Example

• Data
• Iteration 1 min time is 4 (job 1 on M1) place
  this job first and remove from both lists

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• Data

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Johnsons Algorithm Example (Continued)

• Iteration 2 min time is 5 (job 3 on M2) place
  this job last and remove from lists
• Iteration 3 only job left is job 2 place in
  remaining position (middle).
• Final Sequence 1-2-3
• Makespan 28

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Gantt Chart for Johnsons Algorithm Example
Short task on M2 to clear out quickly.
Short task on M1 to load up quickly.
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Three Machine Results

• Johnsons algorithm can be extended to three
  machines by creating two composite machines (M1
  M1 M2) and (M2 M2 M3) and then applying
  Johnsons algorithm to these two machines
• Optimality is guaranteed only when certain
  conditions are met
• smallest processing time on M1 is greater or
  equal than largest processing on machine 2, or
• smallest processing time on M3 is greater or
  equal than largest processing on machine 2

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Multi-Machine Results

• Generate M-1 pairs of dummy machines
• Example with 4 machines, we have the following
  three pairs (M1, M4), (M1M2, M3M4), (M1M2M3,
  M2M3M4)
• Apply Johnsons algorithm to each pair and select
  the best resulting schedule out of the M-1
  schedules generated
• Optimality is not guaranteed.

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Dispatching Rules

• In general, simple sequencing rules
  (dispatching rules) do not lead to optimal
  schedules. However, they are often used to solve
  approximately (heuristically) complex scheduling
  problems.
• Basic Approach
• Decompose a multi-machine problem (e.g., a job
  shop scheduling problem) into sub-problems each
  involving a single machine.
• Use a simple dispatching rule to sequence jobs on
  each of these machines.

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Example Dispatching Rules

• FIFO simplest, seems fair.
• SPT Actually works quite well with tight due
  dates.
• EDD Works well when jobs are mostly the same
  size.
• Critical ratio (time until due date/work
  remaining) - Works well for tardiness measures
• Many (100s) others.

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Heuristics Algorithms

• Construction heuristics
• Use a procedure (a set of rules) to construct
  from scratch a good (but not necessarily optimal)
  schedule
• Improvement heuristics
• Starting from a feasible schedule (possibly
  obtained using a construction heuristic), use a
  procedure to further improve the schedule

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Example A Single Machine with Setups

• N jobs to be scheduled on a single machine with a
  sequence dependent setup preceding the processing
  of each job.
• The objective is to identify a sequence that
  minimizes makespan.
• The problem is an instance of the Traveling
  Salesman Problem (TSP).
• The problem is NP-hard (the number of
  computational steps required to solve the problem
  grow exponentially with the number of jobs).

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A Heuristic Algorithm
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A Heuristic Algorithm

• Greedy heuristic Start with an arbitrary job
  from the set of N jobs. Schedule jobs
  subsequently based on next shortest setup time.
  

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A Heuristic Algorithm

• Greedy heuristic Start with an arbitrary job
  from the set of N jobs. Schedule jobs
  subsequently based on next shortest setup time.
  
• Improved greedy heuristic Evaluate sequences
  with all possible starting jobs (N different
  schedules). Choose schedule with the shortest
  makespan.

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A Heuristic Algorithm

• Greedy heuristic Start with an arbitrary job
  from the set of N jobs. Schedule jobs
  subsequently based on next shortest setup time.
  
• Improved greedy heuristic Evaluate sequences
  with all possible starting jobs (N different
  schedules). Choose schedule with the shortest
  makespan.
• Improved heuristic Starting from the improved
  greedy heuristic solution carry out a series of
  pair-wise interchanges in the job sequence. Stop
  when solution stops improving.

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A Problem Instance

• 16 jobs
• Each job takes 1 hour on single machine (the
  bottleneck resource)
• 4 hours of setup to change over from one job
  family to another
• Fixed due dates
• Find a solution that minimizes tardiness

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EDD Sequence

• Average Tardiness 10.375

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A Greedy Search

• Consider all pair-wise interchanges
• Choose one the one that reduces average tardiness
  the most
• Continue until no further improvement is possible

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First Interchange Exchange Jobs 4 and 5.

• Average Tardiness 5.0 (reduction of 5.375!)

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Greedy Search Final Sequence
Average Tardiness 0.5 (9.875 lower than EDD)
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A Better (Due-Date Feasible) Sequence

• Average Tardiness 0

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Computational Times

• Current situation computers can examine
  1,000,000 sequences per second and we wish to
  build a scheduling system that has a response
  time of no longer than one minute. How many jobs
  can we sequence optimally (using a brute force
  approach)?

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Effect of Faster Computers

• Future Situation New computers will be 1,000
  times faster, i.e. it can do 1 billion
  comparisons per second). How many jobs can we
  sequence optimally now?

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Implications for Real Problems

• Computation NP (non-polynomial) algorithms are
  slow to use
• No Technology Fix Faster computers do not help
  on NP algorithm.
• Exact Algorithms Need for specialized algorithms
  that take advantage of the structure of the
  problem to reduce the search space.
• Heuristics Likely to continue to be the dominant
  approach to solving large problems in practice
  (e.g., multi-step exchange algorithms, Genetic
  Algorithms, Simulated Annealing, Tabu Search,
  among others)

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Implications for Real Problems (Continued)

• Robustness NP hard problems have many solutions,
  and presumably many good ones.
• Example 25 job sequence problem. Suppose that
  only one in a trillion of the possible solutions
  is good. This still leaves 15 trillion good
  solutions. Our task is to find one of these.
• Focus on Bottleneck We can often concentrate on
  scheduling the bottleneck process, which
  simplifies the problem closer to a single machine
  case.