Operations Scheduling

1
Operations Scheduling Chapter 8
2
The Hierarchy of Production Decisions

• The logical sequence of operations in factory
  planning corresponds to the sequencing of
  chapters in a production management text book.
• All planning starts with the demand forecast.
• Demand forecasts are the basis for the top level
  (aggregate) planning.
• The Master Production Schedule (MPS) is the
  result of disaggregating aggregate plans down to
  the individual item level.
• Based on the MPS, MRP is used to determine the
  size and timing of component and subassembly
  production.
• Detailed shop floor schedules are required to
  meet production plans resulting from the MRP.

3
Hierarchy of Production Decisions
4
Scheduling

• Scheduling Establishes the timing of the use of
  equipment, facilities and human activities in an
  organization
• Effective scheduling can yield
• Cost savings
• Increases in productivity
• Improved customer satisfaction

5
Scheduling Techniques

• Scheduling techniques are designed to
  disaggregate the master production schedule into
  time-phased daily or hourly activities.
• A detailed production schedule must include when
  and where each activity must take place in order
  to meet the master schedule.

6
Scheduling Activities

• Scheduling involves the following major
  activities
• Routing (determining where the work is going to
  be done).
• Short-run capacity planning.
• Short-run machine, manpower and production
  scheduling.
• Determining the sequence of activities
  (determining when the work is to be done).
• Dispatching (issuing the order to begin work).
• Controlling the progress of orders and monitoring
  the process to determine that operations are
  running according to plan.
• Revising the schedule based on changes in order
  status of jobs, material and/or capacity
  availability and various other reasons.
• Expediting (speeding the progress of the work
  order) late, critical jobs.

7
Elements of Scheduling

• Elements of Scheduling Problems
• 1. Job arrival patterns (static vs. dynamic).
  Dynamic arrival pattern means that more jobs will
  arrive in the system during the time those
  currently in the system are being processed. In a
  static system, all jobs that will ever enter the
  system are known. Most job shops are dynamic.
• 2. Ratio of workers to machines (machine limited
  vs. labor limited environment).
• 3. Priority rules for scheduling.
• 4. Flow patterns of jobs through the plant.
• a. Flow shop All jobs follow the same pattern
  of flow through the system. In a flow shop,
  routing is not typically a problem.
• b. Job shop Each job follows its own
  specified pattern. Job shop is more difficult to
  analyze than the flow shop.

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Goals of Production Scheduling

• High Customer Service on-time delivery
• Low Inventory Levels WIP and FGI
• High Utilization of machines and labor

9
Meeting Due Dates Measures

• Service Level
• Used typically in make to order systems.
• Fraction of orders which are filled on before
  their due dates.
• Fill Rate
• Used typically in make to stock systems.
• Fraction of demands met from stock.

• Lateness
• Used in shop floor control.
• Difference between order due date and completion
  date.
• Average lateness has little meaning.
• Better measure is lateness variance.
• Tardiness
• Used in shop floor control.
• Is equal to the lateness of a job if it is late
  and zero, otherwise.
• Average tardiness is meaningful but unintuitive.

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Basic Definitions

• Throughput (TH) for a line, throughput is the
  average quantity of good (non-defective) parts
  produced per unit time.
• Work in Process (WIP) inventory between the
  start and endpoints of a product routing.
• Raw Material Inventory (RMI) material stocked at
  beginning of routing.
• Finished Goods Inventory (FGI) material held in
  inventory prior to shipping to the customer.
• Cycle Time (CT) time between release of the job
  at the beginning of the routing until it reaches
  an inventory point at the end of the routing.
• Makespan The total amount of time to process a
  fixed number of jobs.
• Littles Law TH WIP/CT ? WIPTHCT ?
  (L?w)
• where ?TH and wCT

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Reducing WIP and Cycle Time

• Less WIP Equals Shorter Cycle Times (Littles
  Law)
• Shorter cycle time means
• Less WIP
• Better responsiveness
• All of which reduce costs and improve sales
  revenue

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Classic Scheduling Assumptions

• MRP/ERP
• Benefits Simple paradigm, hierarchical
  approach.
• Problems
• MRP assumes that lead times are an attribute of
  the part, independent of the status of the shop.
• MRP uses pessimistic lead time estimates.

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Classic Scheduling Assumptions (cont.)

• Classic Scheduling (only classic in academia)
• Benefits Optimal schedules
• Problems Bad assumptions.
• All jobs available at the start of the problem.
• Deterministic processing times.
• No setups.
• No machine breakdowns.
• No preemption.
• No cancellation.

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Objectives in Job Shop Scheduling

• Meet due dates
• Minimize work-in-process (WIP) inventory
• Minimize average flow time
• Maximize machine/worker utilization
• Reduce set-up times for changeovers
• Minimize direct production and labor costs
• (note that these objectives can be conflicting)

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Measures to Evaluate Performance of a Scheduling
Method

• Service Level Fraction of orders filled on
  before their due dates (used in make-to-order
  systems)
• Fill Rate Fraction of demand that are met from
  inventory without backorder (used in
  make-to-stock systems)
• Job Flow Time Time elapsed from the release of a
  job until it is completed.
• Lateness Difference between completion time and
  due date of a job (may be negative).
• Tardiness The positive difference between the
  completion time and the due date of a job.
• Makespan Flow time of the job completed last.

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Measures to Evaluate Performance of a Scheduling
Method

• Production Rate
• Utilization
• Keep in mind that high utilization means high
  return on investment. This is good provided that
  the equipment is utilized to increase revenue.
  Otherwise, high utilization only helps to
  increase inventory, not profits.

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Terminology

• Flow shop n jobs processed through m machines in
  the same sequence
• Job shop the sequencing of jobs through machines
  may be different, and there may be multiple
  operations on some machines.
• Parallel processing vs. sequential processing
  parallel processing means that the machines are
  identical, any job can be processed on any
  machine.

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Common Sequencing Rules

• FCFS. First Come First Served. Jobs processed in
  the order they come to the shop.
• SPT. Shortest Processing Time. Jobs with the
  shortest processing time are scheduled first.
• EDD. Earliest Due Date. Jobs are sequenced
  according to their due dates.
• CR. Critical Ratio. Compute the ratio of
  processing time of the job and remaining time
  until the due date. Schedule the job with the
  largest CR value next.

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Scheduling Service Operations Vs Manufacturing
Operations

• Scheduling service systems presents certain
  problems not generally encountered in
  manufacturing systems. This is primarily due to
• The inability to store services
• The random nature of customer requests
• To avoid problems such as long delays,
  unsatisfied customers, service systems rely on
  appoinment systems and reservation systems.

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High-Volume Systems

• Flow system High-volume system with Standardized
  equipment and activities
• Flow-shop scheduling Scheduling for high-volume
  flow system

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High-Volume Systems

• Examples of high-volume products include autos,
  personal computers, televisons.
• In process industries, examples include petroleum
  refining, sugar refining.
• A major issue in design of high-volume (flow)
  systems is line balancing.

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Success Factors in High-Volume Systems

• Process and product design
• Preventive maintenance
• Rapid repair when breakdown occurs
• Minimization of quality problems
• Reliability and timing of supplies

23
Intermediate-Volume Systems

• Outputs are between standardized high-volume
  systems and made-to-order job shops
• The volume of output is not large enough to
  justify continuous production.
• Examples include canned foods, paint and
  cosmetics.

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Intermediate-Volume Systems

• The three basic issues in these systems are
• 1.Run size, 2.Timing, and 3.Sequence of jobs
• Economic run size

h is defined as h h(1- ?/P)
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Scheduling Low-Volume Systems

• Loading - assignment of jobs to process centers
• Sequencing - determining the order in which jobs
  will be processed
• Job-shop scheduling
• Scheduling for low-volume systems with many
  variations in requirements

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Gantt Load Chart
Figure 15.2

• Gantt chart - used as a visual aid for loading
  and scheduling

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Loading

• Infinite loading Assigning jobs to work centers
  without considering the capacity of work center.
• Finite loading Takes into acccount the capacity
  of work center.
• Forward scheduling Scheduling ahead from some
  point in time.
• Backward scheduling. Scheduling backwards from
  due dates.

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Loading
29
LOADING (The Assignment Problem)

• In many business situations, management needs to
  assign - personnel to jobs, - jobs to machines, -
  machines to job locations, or - salespersons to
  territories.
• Consider the situation of assigning n jobs to n
  machines.
• When a job i (1,2,....,n) is assigned to machine
  j (1,2, .....n) that incurs a cost Cij.
• The objective is to assign the jobs to machines
  at the least possible total cost.

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The Assignment Problem

• This situation is a special case of the
  Transportation Model and it is known as the
  assignment problem.
• Here, jobs represent sources and machines
  represent destinations.
• The supply available at each source is 1 unit And
  demand at each destination is 1 unit.

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The Assignment Problem
The assignment model can be expressed
mathematically as follows Xij 0, if the job
j is not assigned to machine i 1, if the job j
is assigned to machine i
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The Assignment Problem
33
The Assignment Problem Example

• Ballston Electronics manufactures small
  electrical devices.
• Products are manufactured on five different
  assembly lines (1,2,3,4,5).
• When manufacturing is finished, products are
  transported from the assembly lines to one of the
  five different inspection areas (A,B,C,D,E).
• Transporting products from five assembly lines to
  five inspection areas requires different times
  (in minutes)

34
The Assignment Problem Example

• Ballston Electronics manufactures small
  electrical devices.
• Products are manufactured on five different
  assembly lines (1,2,3,4,5).
• When manufacturing is finished, products are
  transported from the assembly lines to one of the
  five different inspection areas (A,B,C,D,E).
• Transporting products from five assembly lines to
  five inspection areas requires different times
  (in minutes)

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The Assignment Problem Example
Under current arrangement, assignment of
inspection areas to the assembly lines are 1 to
A, 2 to B, 3 to C, 4 to D, and 5 to E. This
arrangement requires 107121719 65 man
minutes.
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The Assignment Problem Example

• Management would like to determine whether some
  other assignment of production lines to
  inspection areas may result in less cost.
• This is a typical assignment problem. n 5
  And each assembly line is assigned to each
  inspection area.
• It would be easy to solve such a problem when n
  is 5, but when n is large all possible
  alternative solutions are n!, this becomes a
  hard problem.

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The Assignment Problem Example

• Assignment problem can be either formulated as a
  linear programming model, or it can be
  formulated as a transportation model.
• However, An algorithm known as Hungarian Method
  has proven to be a quick and efficient way to
  solve such problems.
• This technique is programmed into many computer
  modules such as the one in WINQSB.

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The Assignment Problem Example
WINQSB solution for this problem is as follows
39
Hungarian Method Example
Step 1 Select the smallest value in each
row. Subtract this value from each value in that
row Step 2 Do the same for the columns that do
not have any zero value.
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Hungarian Method Example
If not finished, continue with other columns.
41
Hungarian Method Example

• Step 3 Assignments are made at zero values.
• Therefore, we assign job 1 to machine 1 job 2 to
  machine 3, and job 3 to machine 2.
• Total cost is 51213 30.
• It is not always possible to obtain a feasible
  assignment as in here.

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Hungarian Method Example 2
43
Hungarian Method Example 2

• A feasible assignment is not possible at this
  moment.
• In such a case, The procedure is to draw a
  minimum number of lines through some of the rows
  and columns, Such that all zero values are
  crossed out.

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Hungarian Method Example 2
The next step is to select the smallest uncrossed
out element. This element is subtracted from
every uncrossed out element and added to every
element at the intersection of two lines.
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Hungarian Method Example 2

• We can now easily assign to the zero values.
  Solution is to assign (1 to 1), (2 to 3), (3 to
  2) and (4 to 4).
• If drawing lines do not provide an easy solution,
  then we should perform the task of drawing lines
  one more time.
• Actually, we should continue drawing lines until
  a feasible assignment is possible.

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Sequencing

• Sequencing Determine the order in which jobs at
  a work center will be processed.
• Workstation An area where one person works,
  usually with special equipment, on a specialized
  job.

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Sequencing n jobs on a Single Machine

• Priority rules
• Simple heuristics such as FCFS, SPT, DD, CR are
  used to select the order in which jobs will be
  processed.
• CR (Due Date Current Time)/ Processing Time
• Job time Time needed for setup and processing
  of a job.

48
Example Sequencing Rules

• Use the FCFS, SPT, and Critical Ratio rules to
  sequence the five jobs below. Evaluate the rules
  on the bases of average flow time, average number
  of jobs in the system, and average job lateness.
• 
  (Due Date)
• Job Processing Time Time to Promised
  Completion
• A 6 hours 10 hours
• B 12 16
• C 9 8
• D 14 14
• E 8 7

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Example Sequencing Rules

• FCFS Rule A gt B gt C gt D gt E
• Processing Due Flow
• Job Time Date Time
  Lateness
• A 6 10 6 0
• B 12 16 18 2
• C 9 8 27 19
• D 14 14 41 27
• E 8 7 49 42
• 49 141 90

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Example Sequencing Rules

• FCFS Rule Performance
• Average flow time
• 141/5 28.2 hours
• Average number of jobs in the system
• 141/49 2.88 jobs
• Average job lateness
• 90/5 18.0 hours

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Example Sequencing Rules

• SPT Rule A gt E gt C gt B gt D
• Processing Due Flow
• Job Time Date Time Lateness
• A 6 10 6 0
• E 8 7 14 7
• C 9 8 23 15
• B 12 16 35 19
• D 14 14 49 35
• 49 127 76

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Example Sequencing Rules

• SPT Rule Performance
• Average flow time
• 127/5 25.4 hours
• Average number of jobs in the system
• 127/49 2.59 jobs
• Average job lateness
• 76/5 15.2 hours

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Example Sequencing Rules

• Critical Ratio Rule E gt C gt D gt B gt A
• Processing Promised Flow
• Job Time Completion Time Lateness
• E (.875) 8 7 8 1
• C (.889) 9 8 17 9
• D (1.00) 14 14 31 17
• B (1.33) 12 16 43 27
• A (1.67) 6 10 49 39
• 49 148 93

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Example Sequencing Rules

• Critical Ratio Rule Performance
• Average flow time
• 148/5 29.6 hours
• Average number of jobs in the system
• 148/49 3.02 jobs
• Average job lateness
• 93/5 18.6 hours

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Example Sequencing Rules

• Comparison of Rule Performance
• Average Average Average
• Flow Number of Jobs Job
• Rule Time in System
  Lateness
• FCFS 28.2 2.88 18.0
• SPT 25.4 2.59
  15.2
• CR 29.6 3.02
  18.6
• SPT rule was superior for all 3 performance
  criteria.

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Sequencing n jobs on two machines

• Johnsons Rule technique for minimizing
  completion time for a group of n jobs to be
  processed on two machines or at two work centers.
• Minimizes total idle time
• Johnsons Rule requires satisfying the following
  conditions

57
Johnsons Rule Conditions

• Job time must be known and constant
• Job times must be independent of sequence
• Jobs must follow same two-step sequence
• Job priorities cannot be used
• All units must be completed at the first work
  center before moving to second

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Johnsons Rule Optimum Sequence

1. List the jobs and their times at each work center
2. Find the smallest processing time. If it belongs
   to the first operation of a job schedule that job
   next, otherwise schedule that job last.
3. Eliminate the job from further consideration
4. Repeat steps 2 and 3 until all jobs have been
   scheduled

59
Johnsons Algorithm Example

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

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Johnsons Algorithm Example (cont.)

• 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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Example

• A group of six jobs is to be processed through a
  two-machine flow shop. The first operation
  involves cleaning and the second involves
  painting. Determine a sequence that will minimize
  the total completion time for this group of jobs.
  Processing times are as follows

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• Select the job with the shortest processing time.
  It is job D, with a time of two hours.
• Since the time is at the first center, schedule
  job D first. Eliminate job D from further
  consideration.
• Job B has the next shortest time. Since it is at
  the second work center, schedule it last and
  eliminate job B from further consideration. We
  now have
• The remaining jobs and their times are

64

• The shortest remaining time is six hours for job
  E at work center 1. Thus, schedule that job
  toward the beginning of the sequence (after job
  D). Thus,
• Job C has the shortest time of the remaining two
  jobs. Since it is for the first work center,
  place it third in the sequence. Finally, assign
  the remaining job (F) to the fourth position and
  the result is

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Sequencing Jobs When Setup Times Are
Sequence-Dependent
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Scheduling Difficulties

• Randomness in job arrival times
• Variability in
• Setup times
• Processing times
• Interruptions
• Changes in the set of jobs
• No method for identifying optimal schedule
• Scheduling is not an exact science
• Ongoing task for a manager

67
Classic Dispatching Results

• Optimal Schedules Impossible to find for most
  real problems.
• Dispatching sorts jobs as they arrive at a
  machine.
• 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.
• Many (100?) others.
• Problems with Dispatching
• Cannot be optimal (can be bad).
• Tends to be myopic.

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The Difficulty of Scheduling Problems

• Dilemma
• Too hard for optimal solutions.
• Need something anyway.
• Classifying Hardness
• Class P has a polynomial solution.
• Class NP has no polynomial solution.
• Example Sequencing problems grow as n!.
• Compare en/10000 and 10000n10.
• At n 40, en/10000 2.4 ? 1013, 10000n10 1.0
  ? 1020
• At n 80, en/10000 5.5 ? 1030, 10000n10 1.1
  ? 1023
• 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

en/10000
10000n10
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The Difficulty of Scheduling Problems

• NP stands for non polynomial, meaning that the
  time required to solve such problems is an
  exponential function of the number of jobs rather
  than a polynomial function.
• The problems for which total enumeration is
  hopeless are known in mathematics as NP hard.

70
Computation Times

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

71
Effect of Faster Computers

• Future Situation New computer is 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 algorithms are slow to use.
• No Technology Fix Faster computers dont help on
  NP algorithm.
• Scheduling is Hard Real scheduling problems tend
  to be NP Hard.
• Scheduling is Big Real scheduling problems also
  tend to be quite large impossible to solve
  optimally.

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Implications for Real Problems (cont.)

• Robustness? NP hard problems have many solutions,
  and presumably many good ones.
• Our task is to find one of these.
• Role of Heuristics Polynomial algorithms can be
  used to obtain good solutions. Example
  heuristics include
• Simulated Annealing
• Tabu Search
• Genetic Algorithms

74
The Bad News

• Violation of Assumptions Most real-world
  scheduling problems violate the assumptions made
  in the classic literature
• There are always more than two machines.
• Process times are not deterministic.
• All jobs are not ready at the beginning of the
  problem.
• Process time are sequence dependent.
• Problem Difficulty Most real-world production
  scheduling problems are NP-hard.
• We cannot hope to find optimal solutions of
  realistic sized scheduling problems.
• Polynomial approaches, like dispatching, may not
  work well.

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The Good News

• Due Dates We can set the due dates.
• Job Splitting We can get smaller jobs by
  splitting larger ones.
• Single machine SPT results imply small jobs
  clear out more quickly than larger jobs.
• Mechanics of Johnsons algorithm implies we
  should start with a small job and end with a
  small job.
• Small jobs make for small move batches and can
  be combined to form larger process batches.

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The Good News (cont.)

• Feasible Schedules We do not need to find an
  optimal schedule, only a good feasible one.
• Focus on Bottleneck We can often concentrate on
  scheduling the bottleneck process, which
  simplifies problem closer to single machine case.
• Capacity Capacity can be adjusted dynamically
  (overtime, floating workers, use of vendors,
  etc.) to adapt facility (somewhat) to schedule.

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Minimizing Scheduling Difficulties

• Set realistic due dates
• Focus on bottleneck operations
• Consider lot splitting of large jobs