Chapter 7 Dynamic Job Shops

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Chapter 7Dynamic Job Shops

• Advantages/Disadvantages
• Planning, Control and Scheduling
• Open Queuing Network Model

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Definition of a Job Shop

• Several types of machines
• Similar machines grouped, focused on particular
  operations
• Storage for work in process (no blocking)
• Material handling between machine groups
• Different job types have different routings among
  machine groups may revisit the same machine
  group
• Manufacturing process in evolution
• Job is typically a batch of identical pieces
• May have to set up machines between different job
  types

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Advantages Disadvantages

• Advantages (machine groups)
• Ease of supervision/operation by skilled workers
• High utilization of expensive machines
• Flexibility broad scope of products
• Disadvantages (flexibility)
• High WIP and long flow times
• Conflict between resource utilization and
  customer service
• Difficult to meet promise dates
• Cost of variety reduced learning, difficult
  scheduling

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Planning, Control, Scheduling

• Focus on produce-to-order (if produce-to-stock,
  treat PA as a customer order)
• Design and layout
• evolutionary
• based on real or perceived bottlenecks
• Order acceptance and resource planning
• Quoted delivery date need distribution of flow
  time
• Required resource levels complicated by random
  order arrivals

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Planning, Control, Scheduling (cont.)

• Loading and resource commitment
• Schedule a new job on critical machine groups
  first this schedule determines sequencing on
  noncritical groups and timing of release to the
  shop
• Forward load jobs on all machines, looking ahead
  from present
• Backward load jobs on all machines, working back
  from due dates
• Production schedule based on deterministic
  processing and transport times slack time built
  in to handle uncertainty
• Allocation and resource assignment
• Assign jobs at a machine group to worker/machine
• Often from resource perspective, ignoring promise
  dates

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Why Models?

• Estimate flow times for use in quoting delivery
  dates
• Identify policies
• order release
• scheduling
• sequencing
• to reduce unnecessary flow time and work in
  process
• Model a particular policy, then observe numbers
  of jobs in system, at each machine group, and in
  transition between machine groups

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Job Assumptions

1. A job released to the system goes directly to a
   machine center
2. Job characteristics are statistically independent
   of other jobs
3. Each job visits a specified sequence of machine
   centers
4. Finite processing times, identically distributed
   for each job of specific type at a given machine
   center
5. Jobs may wait between machine centers

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Machine Assumptions

• A machine center consists of a number (perhaps
  one) of identical machines
• Each machine operates independently of other
  machines can operate at its own maximum rate
• Each machine is continuously available for
  processing jobs
• (in practice, rate in 2 is adjusted down to
  account for breakdowns, maintenance, etc.)

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Operation Assumptions

1. Each job is indivisible
2. No cancellation or interruption of jobs
3. No preemption
4. Each job is processed on no more than one machine
   at a time
5. Each machine center has adequate space for jobs
   awaiting processing there
6. Each machine center has adequate output space for
   completed jobs to wait

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Aggregation of Job Flow

• Follow Section 7.3.2 in the text to determine

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Example

1 0.15 3 1 2 - - 6 4 1 - -
2 0.10 1 2 3 1 2 8 6 1 2 7
3 0.05 2 3 1 3 - 4 9 4 2 -
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Job Shop Capacity

• Capacity is the maximum tolerable total arrival
  rate, ?
• Job arrival rates to machine centers satisfy
• Let vi be the average number of times an
  aggregate job visits machine center i during its
  stay in the system.

unaffected by variability!
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Jackson Open Queuing Network

• Assume
• A single class (aggregated) of jobs arrives from
  outside according to a Poisson process with rate
  ?
• The service times of jobs at machine center i
  are iid exponential random variables with mean
  1/?i
• If there are n jobs present at machine center i,
  then the total processing rate is ?iri(n) (e.g.,
  if there are ci identical machines then ri(n)
  minn, ci)
• Service protocol (queue discipline) is
  independent of job service time requirements,
  e.g., FCFS or random

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Jackson Network

• If Ni(t) is the number of jobs at machine center
  i at time t, and
• then
• that is, the network has a product-form solution.
  If each m/c center has a single machine, then if
  
• The network can be decomposed into independent
  M/M/1 queues (M/M/ci if multiple m/cs at each
  center)

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Single Machine at Each Station

• Average number of jobs in m/c center i
• Total number of jobs in the system
• Average flow time of a job
• Variance of the number of jobs in the system

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Assigning Tasks to Machine Centers

• Minimize ET or equivalently EN
• s.t. average number of tasks for an arbitrary job
  K
• Then task allocation determines values of vi with
  
• Assume that if a given task is assigned to m/c
  center i then it will require an exponential (?i)
  amount of time
• Optimal

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Achieving the Optimal Visit Rates

• Total set of tasks for all job types
• Let wi be the average number of times task i
  needs to be performed on an arbitrary job
• Ideally, find a partition
• More practically, for each k1,,m, find l(k) as
  the largest task index that satisfies
• Then if

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Generalized Jackson Networks(Approximate
Decomposition Methods)

• The arrival and departure processes are
  approximated by renewal process and each station
  is analyzed individually as GI/G/m queue.
• The performance measures (EN, EW) are
  approximated by 4 parameters . (Dont need the
  exact distribution)
• References
• Whitt, W. (1983a), The Queuing Network
  Analyzer, Bell System Technical Journal, 62,
  2779-2815.
• Whitt, W. (1983b), Performance of Queuing
  Network Analyzer, Bell System Technical Journal,
  62, 2817-2843.
• Bitran, G. and R. Morabito (1996), Open queuing
  networks optimization and performance evaluation
  models for discrete manufacturing systems,
  Production and Operations Management, 5, 2,
  163-193.

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Single-class GI/G/1 OQNS

• Notation
• n number of internal stations in the network.
• For each j, j 1,,n
• expected external arrival rate at station
  j
• ,
• squared coefficient of variation (scv) or
  variability of external interarrival time at
  station j
• Var(A0j)/E(A0j)2,
• expected service rate at station i mj
  1/E(Sj)
• scv or variability of service time at
  station j
• Var(Sj)/E(Sj)2.

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• Notation (cont)
• For each pair (i,j), i 1,,n, j 1,,n
• probability of a job going to station j
  after completing service at station i.
• We assume no immediate feedback,
• - For n nodes, the input consists of n2 4n
  numbers
• The complete decomposition is essentially
  described in 3 steps
• Step 1. Analysis of interaction between stations
  of the networks,
• Step 2. Evaluation of performance measures at
  each station,
• Step 3. Evaluation of performance measures for
  the whole network.

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Step 1

• Determine two parameters for each station j
• (i) the expected arrival rate lj 1/EAj where
  Aj is the interarrival time at the station j.
• (ii) the squared coefficient of variation (scv)
  or variability of interarrival time,
  Var(Aj)/E(Aj)2.
• Consists of 3 sub-steps
• Superposition of arrival processes
• Overall departure process
• Departure splitting according to destinations

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Step 1

• Superposed arrival rate
• can be obtained from the traffic rate
  equations
• for j 1,,n
• where lij pijli is the expected arrival rate
  at station j from station i. We also get
  lj/mj,
• 0 ? lt 1 (the utilization or offered load)
• The expected (external) departure rate to station
  0 from station j is given by
  .
• Throughput or
• The expected number of visits vj E(Kj) lj/l0.

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Step 1(i) (cont.)

• The s.c.v. of arrival time can be approximated by
  Traffic variability equation (Whitt(1983b))
• (1)
• where and

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Step 1(ii) Departure process

• is interdeparture time variability from
  station j. (2)
• Note that if the arrival time and service
  processes are Poisson (i.e., 1),
  then is exact and leads to
• 1.
• Note also that if rj ? 1, then we obtain ?
• On the other hand if rj ? 0, then we obtain
  ?

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Step 1(iii) Departure splitting

• the interarrival time variability at
  station i from station j.
• the departure time variability from
  station j to i.
• (3)
• Equations (1), (2), and (3) form a linear system
  the traffic variability equations in

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Step 2

• Find the expected waiting time by using Kraemer
  Lagenbach-Belz formula modified by Whitt(1983a)
  
• where if lt 1
• if ? 1

Step 3

• Expected lead time for an arbitrary job (waiting
  times service times)
  

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Single-class GI/G/m OQNS with probabilistic
routing

• More general case where there are m identical
  machines at each station i.
• for j
  1,,n
• where

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Single-class GI/G/m OQNS with probabilistic
routing (cont.)

• Then the approximation of expected waiting time
  is similar to GI/G/1 case

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Symmetric Job Shop

• A job leaving m/c center i is equally likely to
  go to any other m/c next
• Then
• So we can approximate each m/c center as M/G/1.
• Then, can be replaced by mean waiting
  time in M/G/1 queue.

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Symmetric Job Shop
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Uniform Flow Job Shop

• All jobs visit the same sequence of machine
  centers. Then
• In this case, M/G/1 is a bad approximation, but
  GI/G/1 works fine.
• The waiting time of GI/G/1 queue can be
  approximated by using equation 3.142, 3.143 or
  3.144 above and the lower and upper bound is

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Uniform-Flow Shop
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Job Routing Diversity

• Assume high work load
• and machine centers each with same number of
  machines, same service time distribution and same
  utilization. Also,
• Q What job routing minimizes mean flow time?
• A1 If then uniform-flow job routing
  minimizes mean flow time
• A2 If then symmetric job routing minimizes
  mean flow time.

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Variance of Flow Time

• The mean flow time is useful information but not
  sufficient for setting due dates. The two
  distributions below have the same mean, but if
  the due date is set as shown, tardy jobs are a
  lot more likely under distribution B!

A
B
Due
ET
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Flow Time of an Aggregate Job

• Pretend we are following the progress of an
  arbitrary tagged job. If Ki is the number of
  times it visits m/c center i, then its flow time
  is (Ki is the r.v. for which vi is the mean)
• Given the values of each Ki, the expected flow
  time is
• and then, unconditioning,

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Variance

• In a similar way, we can find the variance of T
  conditional upon Ki and then uncondition.
• Assume Tij, i 1,,m j 1, and Ki, i
  1,,m are all independent and that, for each i,
  Tij, j 1, are identically distributed.
• Similar to conditional expectation, there is a
  relation for conditional variance

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Using Conditional Variance

• Since T depends on K1, K2, , Km, we can say
• The first term equals

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Using Conditional Variance

• The second term is

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Variance

• The resulting formula for the variance is
• If arrivals to each m/c center are approximately
  Poisson, we can find VarTi from the M/G/1
  transform equation (3.73), p. 61.
• But we still need Cov(Ki, Kj ).

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Markov Chain Model

• Think of the tagged job as following a Markov
  chain with states 1,,m for the machine centers
  and absorbing state 0 representing having left
  the job shop.
• The transition matrix is

Ki is the number of times this M.C. visits state
i before being absorbed let Kji be the number
of visits given X0 j.
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• with probability
• with probability , l 1,,m
• Where if j i and 0 otherwise.
• with probability
• with probability , l 1,,m

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Expectations

• Take expectation of (7.83), we get
• and take expectation of (7.84),
• and

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Because and , we obtain and where
Therefore
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Job Routing Extremes

• Symmetric job shop
• EKi 1,
• If all m/c centers are identical and m is large,
• and we can use an exponential distn to
  approximate T .
• Uniform-flow job shop
• EKi 1,
• Then

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Setting Due Dates

• Use mean and variance of T to obtain approximate
  distribution FT(t) PT lt t, e.g.,
• if VarT lt ET2, fit an Erlang distribution
• if VarT ET2, use an exponential distn
  with mean ET
• if VarT gt ET2, use a hyperexponential
  distribution.
• If td is the due date, what matters is

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Costs Related to Due Dates

• Setting the due date too far away
• Completing job after its due date
• Completing job before its due date
• Total expected cost
• has derivative

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Optimal Due Date