Production Planning

1
Production Planning Control
Operations Scheduling
Professor JIANG Zhibin Department of Industrial
Engineering Logistics Management Shanghai Jiao
Tong University
2
Operations Scheduling

• Contents
• Introduction
• Job Shop Scheduling Terminology
• Sequencing Rules
• Sequencing Theory for a Single Machine
• Sequencing Theory for Multiple Machines
• Assembly Line Balancing
• Advanced Topics for Operations scheduling

3
Introduction-What is Operations Scheduling ?

• Implement the production orders generated in MRP
  under given objectives
• Allocate production resources (machine, workers
  et al.) to production orders (jobs or tasks and
  their due dates) in an optimized manners
• The results are time allocations of production
  resources to different jobs (job sequences on
  each production resources)
• All the orders can be completed while all
  production resources are utilized with their
  loads being balanced.

4
Introduction-Objectives of Job Shop Scheduling

• Objectives of operations scheduling
• Meet due date
• Minimize WIP inventory
• Minimize the average flow time through the
  systems
• Provide for high machine/worker (time)
  utilization (minimize idle time)
• Reduce setup cost
• Minimize production and worker costs

• Discussion
• 1) and 3) aim at providing a high level of
  costumer service
• 2), 4), 5) and 6) are to provide a high level of
  workshop efficiency
• Impossible to optimize all above objectives
  simultaneously
• Proper trade off between cost and quality is
  one of the most challenging strategic issues
  facing a firm today

5
Introduction-Objectives of operations Scheduling

• Discussion (Cont.)
• Some of these objectives conflicts, e.g.
• Reduce WIP inventory ? Worker idle time may
  increase or machine utilization may decrease
• Reasons differences in the throughput rate from
  one part of the system to another may force the
  faster operations to wait.

As an example, if there is no buffer for WIP
between 1 and 2, what happens?
6
Introduction-Functions of Scheduling and Control

• The following functions must be performed in
  scheduling and controlling a shop floor
• Allocating orders, equipments, and personnel to
  work centers or other specified location-Short
  term capacity planning
• Determining the sequence of orders (i. e. job
  priorities)
• Initializing performance of the scheduled work,
  commonly termed the dispatching of jobs
• Shop-floor control, involving
• Reviewing the status and controlling the progress
  of orders as they are being worked on
• Expediting the late and critical orders
• Revising the schedules in light of changes in
  order status.

7
Introduction-Elements of the Shop Floor
Scheduling Problems

• The classic approaches to shop floor scheduling
  focuses on the following six elements
• Job arrival patterns static or dynamic
• Static jobs arrive in batch
• Dynamic jobs arrive over time interval according
  to some statistical distribution.
• Numbers and variety of machines in the shop floor
• If there is only one machine or if a group of
  machines can be treated as one machine, the
  scheduling problem is much more simplified
• As number of variety of machines increase, the
  more complex the scheduling problems is likely to
  become.

8
Introduction-Elements of the Shop Floor
Scheduling Problems

• (Continued)
• Ratio of workers to machines
• Machine limited system more workers than machine
  or equal number workers and machines
• Labor-limited system more machines than worker.
• Flow pattern of jobs flow shop or job shop
• Flow shop all jobs follow the same paths from
  one machine to the next
• Job shop no similar pattern of movement of jobs
  from one machine to the next.

9
Introduction-Elements of the Shop Floor
Scheduling Problems

• (Continued)
• Job sequencing
• Sequencing or priority sequencing the process of
  determining which job is started first on some
  machines or work center by priority rule
• Priority rule the rule used for obtaining a job
  sequencing
• Priority rule evaluation criteria
• To meet corresponding objectives of scheduling
• Common standard measures
• Meeting due date of customers or downstream
  operations
• Minimizing flow time (the time a job spends in
  the shop flow)
• Minimizing WIP
• Minimizing idle time of machines and workers
  (Maximizing utilization).

10
Introduction-Elements of the Job Shop Scheduling
Problems

• Flow shop
• Each of the n jobs must be processed through the
  m machines in the same order.
• Each job is processed exactly once on each
  machine.

• An assembly line is a classic example of flow
  shop
• Every cars go through all the stations one by one
  in the same sequences
• Same tasks are performed on each car in each
  station
• Its operations scheduling is simplified as
  assembly line balancing
• An assembly balancing problem is to determine the
  number of stations and to allocate tasks to each
  station.

11
Introduction-Job Shop
A job shop is organized by machines which
are grouped according to their functions.
12
Introduction-Job Shop
Job A
Job B

• Not all jobs are assumed to require exactly the
  same number of operations, and some jobs may
  require multiple operations on a single machine
  (Reentrant system, Job B twice in work center 3
  ).
• Each job may have a different required sequencing
  of operations.
• No all-purpose solution algorithms for solving
  general job shop problems
• Operations scheduling of shop floor usually means
  job shop scheduling

13
Job Shop Scheduling Terminology

• Parallel processing versus sequential processing
• Sequencing Processing the m machines are
  distinguishable, and different operations are
  performed by different machines.
• Parallel processing The machines are identical,
  and any job can be processed on any machine.

• M1, M2, M3, and M4 are different
• Job A has 2 operations which should be processed
  on different Machines M1and M2
• Job B has 3 operations which should be processed
  on different Machines M3, M2 and M4

• M1, M2, M3, and M4 are identical
• Jobs A and B can be processed on any one of the 4
  machines

14
Job Shop Scheduling Terminology

• 2 Flow time
• The flow time of job i is the time that elapses
  from the initiation of that job on the first
  machine to the completion of job i.
• The mean flow time, which is a common measure of
  system performance, is the arithmetic average of
  the flow times for all n jobs

Mean Flow Time(F1F2F3)/3
Machines
Job 1
Job 2
Job 3
M1
Job 1
Job 2
Job 3
M2
F1 FT of Job 1
Time
F2 FT of Job 2
F3 FT of Job 3
15
Job Shop Scheduling Terminology

• 3. Make-span
• The make-span is the time required to complete a
  group of jobs (all n jobs).
• Minimizing the make-span is a common objective in
  multiple-machine sequencing problems.

Make-span of the 3 jobs
16
Job Shop Scheduling Terminology

• 4. Tardiness and lateness
• Tardiness is the positive difference between the
  completion time and the due date of a job.
• Lateness refers to the difference between the job
  completion time and its due date and differs from
  tardiness in that lateness can be either positive
  or negative.
• If lateness is positive, it is tardiness when it
  is negative, it is earliness

Latenessgt0---Tardiness
Latenesslt0---Earliness
When the completion of Job is earlier than due
date, the tardiness is 0
17
Sequencing Rules

• FCFS (first come-first served)
• Jobs are processed in the sequence in which they
  entered the shop
• The simplest and nature way of sequencing as in
  queuing of a bank

• SPT (shortest processing time)
• Jobs are sequenced in increasing order of their
  processing time
• The job with shortest processing time is first,
  the one with the next shortest processing time is
  second, and so on

• EDD (earliest due date)
• Jobs are sequenced in increasing order of their
  due dates
• The job with earliest due date is first, the one
  with the next earliest due date is second, and so
  on

18
Sequencing Rules

• CR (Critical ratio)
• Critical ratio is the remaining time until due
  date divided by processing time
• Scheduling the job with the smallest CR next

CRiRemaining time of Job i/Processing time of
Job i (Due date of Job i-current
time)/Processing time of Job i

• CR provides the balance between SPT and EDD,
  such that the task with shorter remaining time
  and longer processing time takes higher priority
• CR will become smaller as the current time
  approaches due date, and more priority will given
  to one with longer processing time
• For a job, if the numerator of its CR is
  negative ( the job has been already later), it is
  naturally scheduled next
• If more than one jobs are later, higher
  priority is given to one that has shorter
  processing time (SPT).

19
Sequencing Rules

• Example 5.1
• A machine center in a job shop for a local
  fabrication company has five unprocessed jobs
  remaining at a particular point in time. The jobs
  are labeled 1, 2, 3, 4, and 5 in the order that
  they entered the shop. The respective processing
  times and due dates are given in the table below.
• Sequence the 5 jobs by above 4 rules and compare
  results based on mean flow time, average
  tardiness, and number of tardy jobs

20
Sequencing RulesFCFS
Mean Flow time268/553.6 Average
tardiness121/524.2 No. of tardy jobs3.
1 11
61 0
2 40
45 0
3 71
31 40
4 72
33 39
5 74
32 42
Totals 268

121
21
Sequencing RulesSPT
Mean Flow time135/527.0 Average
tardiness43/58.6 No. of tardy jobs1.
4 1 1
33
0
5 2 3
32
0
1 11 14
61
0
2 29 43
45
0
3 31 74
31
43
Totals 135

43
22
Sequencing RulesEDD
Mean Flow time235/547.0 Average
tardiness33/56.6 No. of tardy jobs4.
3 31 31
31
0
5 2 33
32
1
4 1 34
33
1
2 29 63
45
18
1 11 74
61
13
Totals 235

33
23
Sequencing RulesCR
Current time should be reset after scheduling one
job
24
Sequencing RulesCR
Mean Flow time289/557.8 Average
tardiness87/517.4 No. of tardy jobs4.
Both Jobs 4 and 5 are later, however Job 4 has
shorter processing time and thus is scheduled
first Finally, job 1 is scheduled last.
25
Sequencing RulesSummary

• Discussions
• SPT results in smallest mean flow time
• EDD yields the minimum maximum tardiness (42,
  43, 18, and 31 for the 4 different rules)
• Always true? Yes!

26
Sequencing Theory for A Single Machines
Assuming that n jobs are to be processed through
one machine. For each job i, define the following
quantities

• tiProcessing time for job i, constant for job
  i
• diDue date for job i, constant for job i
• WiWaiting time for job i, the amount of time
  that the job must wait before its processing can
  begin.
• When all the jobs are processed continuously, Wi
  is the sum of the processing times for all of the
  preceding jobs

• FiFlow time for job i, the waiting time plus
  the processing time Fi Wi ti
• LiLateness of job i , Li Fi- di, either
  positive or negative
  
• TiTardiness of job i, the positive part of Li,
  TimaxLi,0
• EiEarliness of job i, the negative part of Li,
  Ei max- Li,0

27
Sequencing Theory for A Single Machines

• Suppose that 4 jobs J1, J2, J3, J4 need to be
  scheduled

• For example a schedule is J3-J2-J1-J4

• Considered as a permutation of integers 1, 2, 3,
  4 3, 2, 1, 4.

• For only a single machine, every schedule can be
  represented by a permutation (ordering) of the
  integers 1, 2, 3, , n.
• There are totally n! (the factorial of n)
  different permutations.

• A permutation of integers 1, 2, ?, n is expressed
  by 1, 2, ?, n, which represents a schedule
• i denotes the integer that put in the ith place
  in the permutation
• In case of a schedule 3, 2, 1, 4, 13, 22,
  31, and 44

28
Sequencing Theory for A Single Machines

• Shortest-Processing-Time Scheduling
• Theorem 8.1 The scheduling rule that minimizes
  the mean flow time F is SPT.

• The mean flow time of all jobs on the schedule is
  given by

29
Sequencing Theory for A Single Machines

• Shortest-Processing-Time Scheduling
• Theorem 8.1 The scheduling rule that minimizes
  the mean flow time F is SPT

• The mean flow time is given by

• The double summation term may be written in a
  different form. Expanding the double summation,
  we obtain
• k1t1
• k2t1 t2
• knt1 t2 tn

• By summing down the column rather than across the
  row, we may rewrite F in the form
• nt1(n-1)t2tn

SPT sequencing rule the job with shortest
processing time t is set first
30
Sequencing Theory for A Single Machines

• Shortest-Processing-Time Scheduling (Cont.)
• Corollary 8.1 The following measures are
  equivalent
• Mean flow time
• Mean waiting time
• Mean lateness
• SPT minimizes mean flow time, mean waiting
  time, and mean lateness for single machine
  sequencing.
• Earliest-Due-Date Scheduling If the objective is
  to minimize the maximum lateness, then the jobs
  should be sequenced according to their due dates.
  That is, d1? d2? dn.

31
Sequencing Theory for A Single Machine

• 3. Minimizing the number of Tardy Jobs An
  algorithm from Moore(1968) that minimizes the
  number of tardy jobs for the single machine
  problem.
• Step1. Sequence the jobs according to the
  earliest due date to obtain the initial solution.
  That is d1? d2?,, ? dn
• Step2. Find the first tardy job in the current
  sequence, say job i. If none exists go to step
  4.
• Step3. Consider jobs 1, 2, , i. Reject the
  job with the largest processing time. Return to
  step2. (Why ?)
• Reason It has the largest effect on the
  tardiness of the Jobi.
• Step4. Form an optimal sequence by taking the
  current sequence and appending to it the rejected
  jobs. (Can be appended in any order?)
• Yes, because we only consider the number of
  tardiness jobs rather than tardiness.

32
Sequencing Theory for A Single Machine
Example 8.3
Solution
33
Sequencing Theory for A Single Machine
Example 8.3 Solution (Cont.)
The optimal sequence 2, 3, 4, 6, 5, 1 or 2, 3,
4, 6, 1, 5. In each case the number of tardy jobs
is exactly 2.
34
Sequencing Theory for A Single Machines
Precedence constraints Lawlers Algorithm
Minimizing maximum lateness
Minimizing maximum tardiness
gi is any non-decreasing function of the flow
time Fi
The Algorithm

• First schedules the job to be completed last,
  then the job to be completed next to last, and so
  on. At each stage one determines the set of jobs
  not required to precede any other. Call this set
  V. among the set V, choose the job k that
  satisfies

e.g. the job among V that has smallest
tardiness, if arranged on position n.
The processing time of the current sequence
35
Sequencing Theory for A Single Machines
The Algorithm (Cont.)

• Consider the remaining jobs and again determine
  the set of jobs that are not required to precede
  any other remaining job.

• The value of t is then reduced by tk and the job
  scheduled next to last is now determined.

• The process is continued until all jobs are
  scheduled.

Note As jobs are scheduled, some of the
precedence constraints may be relaxed, so the set
V is likely to change at each iteration.
36
Sequencing Theory for A Single Machines
Example 8.4
37
Sequencing Theory for A Single Machines
Example 8.4
Step1 find the job scheduled last(sixth)
t 23432115
Step2 find the job scheduled fifth
t 15-213
38
Sequencing Theory for A Single Machines
Example 8.4
Step3 find the job scheduled fourth
Because job3 is no longer on the list, Job 2 now
because a candidate.
t 13-49
Step4 find the job scheduled third
Because job6 has been scheduled, Job 4 now
because a candidate along with Job 2.
t 9-18
39
Sequencing Theory for A Single Machines
Example 8.4
Step5 find the job scheduled second
The optimal sequence 1-2-4-6-3-5
40
Sequencing Theory for Multiple Machines

• Assume that n jobs are to be processed through m
  machines. The number of possible schedules is
  astonishing, even for moderate values of both n
  and m.
• For each machine, there is n! different ordering
  of the jobs if the jobs may be processed on the
  machines in any order, there are totally (n!)m
  possible schedules. (n5, m5, 25 billion
  possible schedules)
• Even with the availability of inexpensive
  computing today, enumerating all feasible
  schedules for even moderate-sized problems is
  impossible or, at best, impractical.

41
Sequencing Theory for Multiple Machines

• Gantt chart
• Suppose that two jobs, I and J, are to be
  scheduled on two machines, 1 and 2, the
  processing times are

• Assume that both jobs must be processed first
  on machine 1 and then on machine 2. There are
  four possible schedules.

42
Sequencing Theory for Multiple Machines
43
Sequencing Theory for Multiple Machines

• Scheduling n Jobs on Two Machines
• Theorem 8.2 The optimal solution for
  scheduling n jobs on two machines is always a
  permutation schedule.
• A very efficient algorithm for solving the
  two-machine problem was discovered by
  Johnson(1954).
• Denote the machines by A and B
• The jobs must be processed first on machine A and
  then on machine B.
• Define
• AiProcessing time of job i on machine A
• BiProcessing time of job i on machine B
• Rule Job i precedes job i1 if min(Ai,
  Bi-1)ltmin(Ai1,Bi)
• List the values of Ai and Bi in two columns.
• Find the smallest remaining element in the two
  columns. If it appears in column A, then schedule
  that job next. If it appears in column B, then
  schedule that job last.
• Cross off the jobs as they are scheduled. Stop
  when all jobs have been scheduled.

44
Sequencing Theory for Multiple Machines
Example 8.5
Optimal sequence 2 4 3 5
1
45
Sequencing Theory for Multiple Machines

• Extension to Three Machines
• The three-machine problem can be reduced to a
  two-machine problem if
• the following condition is satisfied
• min Ai?max Bi or min Ci?max Bi

• It is only necessary that either one of these
  conditions be satisfied. If that is the case,
  then the problem is reduced to a two-machine
  problem
• Define AiAiBi, BiBiCi
• Solve the problem using the rules described
  above for two-machines, treating Ai and Bi as
  the processing times.
• The resulting permutation schedule will be
  optimal for the three-machine problem.
• If the condition are not satisfied, this method
  will usually give reasonable, but possibly
  sub-optimal results.

46
Sequencing Theory for Multiple Machines

• The Two-Job Flow Shop Problem assume that two
  jobs are to be processed through m machines. Each
  job must be processed by the machines in a
  particular order, but the sequences for the two
  jobs need not be the same.
• Draw a Cartesian coordinate system with the
  processing times corresponding to the first job
  on the horizontal axis and the processing times
  corresponding to the second job on the vertical
  axis.
• Block out areas corresponding to each machine at
  the intersection of the intervals marked for that
  machine on the two axes.
• Determine a path from the origin to the end of
  the final block that does not intersect any of
  the blocks and that minimizes the vertical
  movement. Movement is allowed only in three
  directions horizontal, vertical, and 45-degree
  diagonal. The path with minimum vertical distance
  corresponds to the optimal solution.

47
Sequencing Theory for Multiple Machines
Example 8.7 A regional manufacturing firm
produces a variety of household products. One is
a wooden desk lamp. Prior to packing, the lamps
must be sanded, lacquered, and polished. Each
operation requires a different machine. There are
currently shipments of two models awaiting
processing. The times required for the three
operations for each of the two shipments are
48
Minimizing the flow time is the same as
maximizing the time that both jobs are being
processed. That is equivalent to finding the
path from the origin to the end of block C that
maximizes the diagonal movement and therefore
minimizes either the horizontal or the vertical
movement.
or 10616
or 10(32)15
49
Assembly Line Balancing

• The problem of balancing an assembly line is a
  classic industrial engineering problem.
• The problem is characterized by a set of n
  distinct tasks that must be completed on each
  item.
• The time required to complete task i is a known
  constant ti.
• The goal is to organize the tasks into groups,
  with each group of tasks being performed at a
  single workstation.
• In most cases, the amount of time allotted to
  each workstation is determined in advance, based
  on the desired rate of production of the assembly
  line.

50
Assembly Line Balancing

• Assembly line balancing is traditionally thought
  of as a facilities design and layout problem.
• There are a variety of factors that contribute to
  the difficulty of the problem.
• Precedence constrains some tasks may have to be
  completed in a particular sequence.
• Zoning restriction Some tasks cannot be
  performed at the same workstation.
• Let t1, t2, , tn be the time required to
  complete the respective tasks.
• The total work content (time) associated with the
  production of an item, say T, is given by

51
Assembly Line Balancing

• For a cycle time of C, the minimum number of
  workstations possible is T/C, where the
  brackets indicate that the value of T/C is to be
  rounded to the next larger integer.
• Ranked positional weight technique
• Places a weight on each task based on the total
  time required by this task and all of the
  succeeding tasks
• Tasks are assigned sequentially to stations based
  on these weights-the bigger the weight is, the
  higher the priority is.

52
Assembly Line Balancing
Example 8.11 The Final assembly of Noname
personal computers, a generic mail-order PC
clone, requires a total of 12 tasks. The assembly
is done at the Lubbock, Texas, plant using
various components imported from the Far East.
The network representation of this particular
problem is given in the following figure.
53
Assembly Line Balancing

• ?ti70, and the production rate is a unit /15
  minutes
• The minimum number of workstations 70/155

Precondition The job times and precedence
relationships for this problem are summarized in
the table below.
54
Assembly Line Balancing
The solution precedence requires determining the
positional weight of each task. The positional
weight of task i is defined as the time required
to perform task i plus the times required to
perform all tasks having task i as a predecessor.
t3t7t8t11t1231
The ranking 1, 2, 3, 6, 4, 7, 5, 8, 9, 10, 11, 12
55
Assembly Line Balancing
Profile 1 C15
The ranking 1, 2, 3, 6, 4, 7, 5, 8, 9, 10, 11, 12
56
Assembly Line Balancing
Profile 1 C15
The ranking 1, 2, 3, 6, 4, 7, 5, 8, 9, 10, 11, 12
Cycle Time15
T26
T112
T26
T36
T42
T52

• Evaluate the balancing results by the efficiency
  ?ti/NC
• The efficiencies for Profiles 1 is 77.7.

T52
T612
T91
T85
T77
T104
T104
T116
T127
T127
57
Assembly Line Balancing
Alternative 1 Change cycle time to ensure 5
station balance
Profile 2 Increasing cycle time from 15 to 16

• Increasing the cycle time from 15 to 16, the
  total idle time
• has been cut down from 20 min/units to 10
  resulting in a substantial improvement in
  balancing rate.
• However, the production rate has to be reduced
  from one unit/15 minutes to one unit/16minute

58
Assembly Line Balancing
Alternative 2 Staying with 6 stations, see if a
six-station balance could be obtained by cycle
time less that 15 minutes

• The efficiencies for profile 1 3 are 77.7,
  87.5, and 89.7. Thus the profile 3 is the best
  one.

Profile 2 C13

• 13 minutes appear to be the minimum cycle time
  with six station balance.
• Increasing the number of stations from 5 to 6
  results in a great improvement in production rate

59

• The End !