Scheduling
THE SCHEDULING ACTIVITY
AT MACHINE SHOP
Objectives (goals) of Scheduling

As it was previously mentioned, there are three
primary objectives (goals) which apply to all
scheduling problems:



Job completions should not be “late” (about due
dates).
The the length of time during which a job stays in
the system should be “minimum” (about flow time)
It is desired to fully utilize the capacities of work
centers (about work center utilization).
2
Kinds of Scheduling Problems
Studies about scheduling activity point out that there
are two fundamental kinds of scheduling problems.
Static Scheduling
Problems
where,
problem consists of a fixed set of
jobs which are to be all completed.
The number of jobs does not change.
Dynamic Scheduling
Problems
deals with
scheduling of a continuous situation,
which means that, new jobs are
continually added to the production
environment.
3
Static Scheduling Approach
 The static scheduling problem contains a fixed set of jobs
which are to be completed.
 The typical basic assumptions in this kind of problem can
be stated as follows;



... entire set of jobs arrive simultaneously, and
all respective work centers are available at that time.
Majority of the static scheduling problems use a criterion
called “minimum make span” which can be defined as the
minimum total time to process all subject jobs. This
criterion is related with the scheduling objective which is
about “flow time”.
4
Static Scheduling Approach
Deterministic
Solutions
Static Scheduling
Problems
use known and nonvarying
process times.
Methods
Producing
Optimum Results
Methods
Utilizing Heuristic
Procedures
Stochastic
Solutions
use process times which are
subject to random variations.
scheduling decisions are made
sequentially rather than at once
called dispatching or sequencing rules.
only applicable to
relatively small problems.
5
Dynamic Scheduling Approach
Dynamic Scheduling
Problems
Deterministic
Solutions
Stochastic
Analytical approaches had been
Solutions
developed which are “based on
queuing models that provide expected
steady state conditions for certain kinds
of situations and time distributions”.
Typically, the criteria used in such queuing cases are; Avarage Flow Time
Average Work-in-process or number of jobs in the system and Work/ machine
center utilization.
To find a solution to dynamic scheduling problems, usually the approach is,
to use different “dispatching rules” at the work centers.
6
Job Shop Dispatching and Common Priority
Sequencing Rules
At any given point in time, there exists a set of “n” jobs to be scheduled on
“m” machines.
This means, at any time there are (n!)m possible ways to schedule the jobs.
7
Job Shop Dispatching and Common Priority
Sequencing Rules
One method to generate schedules in job shops is with “dispatching”
which allows the schedule for a work station to evolve over a period of time.
The decision about which job to process next is made with simple priority rules whenever
the work station becomes available for further processing.
Major advantage of “dispatching” method is that latest information on operating conditions of
the work center can be incorporated into the schedule as it evolves.
8
Job Shop Dispatching and Common Priority
Sequencing Rules
Common Priority
Sequencing
Rules
MULTIPLE
DIMENSION
RULES
SINGLE DIMENSION
RULES
First Come First
Served (FCFS)
(single dimension)
Earliest Due Date
(EDD)
(single dimension)
Shortest Processing
Time (SPT)
(single dimension)
Slack per Remaining
Operations(S/OP)
(multiple dimension)
Critical Ratio
(CR)
(multiple dimension)
9
Job Shop Dispatching and Common Priority
Sequencing Rules

SINGLE DIMENSION
RULES

MULTIPLE
DIMENSION
RULES
They base the priority on single
aspect of the job, such as, arrival
time, the due date, or the processing
time at the current work center.
They incorporate information about
the remaining work centers at which
the job must be processed, in
addition to the processing time at the
present work station or the due date
considered by single dimension rules.
10
Job Shop Dispatching and Common Priority
Sequencing Rules

Critical Ratio
(CR)
(multiple dimension)

The critical ratio is calculated by dividing
“the time remaining until the due date of the
job” by the “remaining total shop time for
the job”.
CR = [Due date of the job-Today’s date] / Σ
remaining shop time
 CR < 1.0 implies that the job is behind
schedule,
 CR = 1.0 implies that the job is on schedule
CR > 1.0 implies that the job is ahead of
schedule.

The job with the lowest CR is scheduled
next.
11
Job Shop Dispatching and Common Priority
Sequencing Rules

There may be different ways to set
the due dates of jobs. For example, a
due date might have been calculated
by computerized methods as in MRP
or might have been simply
determined by the customer.

The job with the earliest due date is
the job which is to be scheduled next.
Earliest Due Date
(EDD)
(single dimension)
12
Job Shop Dispatching and Common Priority
Sequencing Rules

First Come First
Served (FCFS)
(single dimension)
The job which arrives the work
station earlier than the others has the
highest priority and is to be
scheduled next.
13
Job Shop Dispatching and Common Priority
Sequencing Rules

Slack per Remaining
Operations(S/OP)
(multiple dimension)


Slack is the difference between “the time
remaining until the due date of the job” and
“remaining total shop time for the job”
including that of the operation being
scheduled.
The job’s priority is determined by dividing
the slack by the number of the operations
that remain including the operation which is
being scheduled.
S/RO = [(Due date – Today’s date)-Total
shop time remaining]/ Number of remaining
operations
14
Job Shop Dispatching and Common Priority
Sequencing Rules
Shortest Processing
Time (SPT)
(single dimension)

King of the priority sequencing rules!

The job which has the shortest
processing time is scheduled next.
15
Job Shop Dispatching and Common Priority
Sequencing Rules

There are nalso some other rules most of which can be
considered as variants of the already mentioned ones ;
Random (R)
(single dimension)

The principle is picking any job in
the queue with equal probability.

This rule is often used as a
benchmark for other rules. In other
words, it is used for evaluation of
other rules with respect to each other
by using reference values obtained
from using R rule.
16
Job Shop Dispatching and Common Priority
Sequencing Rules

There are nalso some other rules most of which can be
considered as variants of the already mentioned ones ;

Least Work
Remaining (LWR)
(multiple dimension)

It is an extension of SPT rule in the
sense that it considers all processing
time remaining until the job is
completed.
This means that, the next job to be
processed is the one which has least
work remaining along its routing
including the work in current station.
17
Job Shop Dispatching and Common Priority
Sequencing Rules

There are nalso some other rules most of which can be
considered as variants of the already mentioned ones ;
Fewest Operations
Remaining (FOR)
(multiple dimension)

This rule is again another variant of
SPT rule. It considers the number of
successive operations.

The next job to be picked is the one
which has smallest number of
successive operations including the
operation being scheduled.
18
Job Shop Dispatching and Common Priority
Sequencing Rules

There are nalso some other rules most of which can be
considered as variants of the already mentioned ones ;

Slack Time (ST)
(multiple dimension)
It is a variant of EDD rule. It subtracts
“remaining total shop time for the job”
including that of the operation being
scheduled from the “time remaining until the
due date”. The resulting value is called
“slack”.


Slack time =(due date - today’s date) (remaining processing time)
Jobs are run in the order of smallest amount
of slack.
19
Job Shop Dispatching and Common Priority
Sequencing Rules

There are nalso some other rules most of which can be
considered as variants of the already mentioned ones ;


Next Queue (NQ)
(multiple dimension)

This is a different kind of rule which is based
on machine utilization.
It considers the “next queues” at each of the
succeeding work centers to which the jobs
will go.
It selects the job which is going to the
smallest queue. Here, the measure of the
queue might be either in hours or perhaps in
number of jobs.
20
Job Shop Dispatching and Common Priority
Sequencing Rules

There are nalso some other rules most of which can be
considered as variants of the already mentioned ones ;
Least Setup (LSU)
(single dimension)

This is another rule for which the objective is
to maximize machine (capacity) utilization.

It selects the job which has minimum setup
time on the machine.

It should be noted that, this rule clearly
considers dependencies between setup times
and job sequence (setup time for a specific
job may change according to sequence).
21
Job Shop Dispatching and Common Priority
Sequencing Rules

There are nalso some other rules most of which can be
considered as variants of the already mentioned ones ;

TARDINESS
(single dimension)
is the difference between a
late job’s due date and its
completion time
22
Job Shop Dispatching and Common Priority
Sequencing Rules
Altough priority sequencing rules
seem to be simple, the actual task of
scheduling hundreds of jobs through
hundreds of work centers requires
intensive data gathering and
manipulation.
Computers are needed to track the data
and to maintain valid priorities.
Any priority sequencing rule can be used
to schedule any number of work centers
with the dispatching procedure.
The scheduler needs information on each
job’s processing requirements;
“the job’s due date”,
“its routing”,
“the standard setup at each operation”,
“the processing time at each op.”,
“the expected waiting time at each op”;
“whether alternative work centers could be
used at each operation”,
“the components and raw materials required
at each operation”,
and “the current status of the job”.
23
Single
(One)-Machine Case
“single-machine” environment is the
simple and special case of all other or more complex environments.
Single (One) Machine Case
Job 1
Job 2
Job n
Priority sequencing
rule
Machine
A
The results of single-machine problems not only provide insight to the single-machine environment,
but also, provide a basis for heuristics of more complex environments.
In most of the practical applications, scheduling problems in complex environments
are broken down into smaller sub problems each of which becomes a single-machine case.
24
Single

(One)-Machine Case
Problem Definition for Single-Machine Case

Majority of the studies on single-machine scheduling is based
on the static problem of how to best schedule a fix set of jobs
through a single machine with the assumptions that;
Job 1
Job 2
Job n
Fixed set of jobs
are available at
the start of the
scheduling
period
Priority sequencing
rule
Machine A is available
when the fixed set of
jobs arrive.
Machine
A
setup times are
independent of
the sequence.
25
Single

(One)-Machine Case
What is meant by saying “setup times are
independent of sequence”?

Simply, it means, setup times does not alter the total make-span
of the entire set of jobs no matter in which sequence the jobs are
processed.

In this case, the total make-span equals the sum of all setup and
run times for any sequence of jobs
26
Single

(One)-Machine Case
What is meant by saying “setup times are
independent of sequence”?
A DRILL
MACHINE
CASE
27
Single

(One)-Machine Case
What is meant by saying “setup times are
independent of sequence”?

Where setup times are sequence dependent, the total make-span
is altered with a change in sequence of jobs.
A FURNACE CASE
28
Single

(One)-Machine Case
What is meant by saying “setup times are
independent of sequence”?

As a result one may come up with the conclusion that
development of analytical models for the case of
single-machine problems where setup is sequence
dependent is rather complex.

In this respect, it is fortunate that, in most of the
single-machine environments make-span does not
depend on the sequence (sequence independent).
29
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)


If the objective is to “minimize the make-span”
(minimize the total time to run the entire set of jobs), it
is clear that make-span will be equal to sum of all
“setup+run” times and will be same for all sequencing
options.
However, if the objective is to “minimize the average
time each job spends at the machine”, it can be shown
that this can be achieved by sequencing jobs in
ascending order according to their total processing time
(setup+run time).
30
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

We can show this fact by the following example :
Average Flow Time = Sum of flow times / Number of jobs
where, Sum of Flow Times = Σ [waiting time+(setup time+processing time)] values
for each job in the system
31
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

In this example :
The Criterion was ...
“average time in the system”
The Objective was ...
The Rule used was ...
“minimize the average time each
job spends at the machine”,
SPT
The Measure used was ...
Average Flow Time = Sum of flow
times / Number of jobs
The average time in the system will always be minimized by processing
the next job which has the shortest processing time at the current operation.
32
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

SPT also proves to be very usefull when the objective is
“minimizing the average number of jobs in the system”
in a single-machine case.

This time, the measure which we are going to use is
“average work-in-process (wip) inventory” for which the
formulation is;
Avarage wip inventory = Sum of flow times / Make-span
33
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Example :
34
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)


Here, at this point, we should also note that “average
wip inventory” and “average flow time” measures are
directly related with each other. That is, when the value
of one decreases, the value of the other also decreases
and vice versa.
Compare the results in previous example and following
example.
35
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Example :
36
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)



When the objective is to “minimize the average job
lateness”, again SPT seems to yield better overall results
in sequencing jobs for the single-machine case.
Ofcourse, to introduce the lateness criterion, we must
first establish due dates for the jobs.
We can again demonstrate what is meant by “better
overall results” by comparing the following examples
37
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Example : (we have added 2 new columns and 3 new measures)
where; Average Minutes Early = (Sum of Minutes Early) / Number of Jobs
Average Minutes Past Due = (Sum of Minutes Past Due) / Number of Jobs
Average Total Inventory=(Sum of Time in System with respect to actual pickup time)/Makespan
38
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Example (continued):
“Average Total Inventory” will be equal to “Average Wip Inventory” of the system when sum of time
in system is equal to sum of flow times. That is, when individual actual pickup times are equal to
individual flow times.
39
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Example (continued):
 SPT rule is overweighing values with respect to average flow time,
average minutes early, average wip inventory and average total inventory.
 On the other hand, EDD rule provides lower average minutes past due and
lower maximum minutes past due (EDD: 25, SPT: 45).
40
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Summary of research studies for EDD rule:


EDD rule performs well with respect to
 the percentage of jobs past due and
 the variance of hours past due.
 easy adjustment of schedules when due dates change.
It does not perform well with
 flow time,
 wip inventory or
 work center utilization.
41
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Summary of research studies for SPT rule:

SPT rule tends to
 minimize the average flow time
 minimize the wip inventory
 minimize the percentage of jobs past due
 maximize work center (and hence shop) utilization
42
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Summary of research studies for SPT rule:

It has disadvanteges of
 tendency to produce a large variance in past due hours
because the larger jobs might have to wait a long time
for processing.
 providing no opportunity to adjust schedules when due
dates changed.
 possibility of increasing total inventory since it tends to
push all work to the finished state.
43
Application of Single-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

FCFS rule performs poorly with respect to all performance
measures. However, this result is something expected since it
does not consider any of the job characteristics.
44
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Priority sequencing rules such as CR and S/RO, also consider
information about the remaining work centers at which the job
must be processed in addition to the processing time at the
current work center or the due date which is considered by
single-dimension rules.

Thus, since they apply more than one aspect of the job, they
are multiple-dimension rules.

In order to see how multiple dimension rules are used and to
compare their results against the results of SPT, FCFS and
EDD rules, one can review the following case study.
45
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Case Study: NC Milling Machine at Taylor Machine Shop

Taylor Machine Shop processes engine blocks for
automotive industry. These blocks are manufactured from
blank casting material and the initial metal removal
process is performed on the NC Milling Machine.

After the nc milling operation, according to its type and
size, each engine block ondergoes a series of operations
successively at different work centers.

Customers request to pickup their engine blocks on
prescheduled due dates.
46
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Case Study: NC Milling Machine at Taylor Machine Shop

Currently there are six different blank castings at the NC
Milling Machine and the machine is available for
processing respective jobs.


Setup times for each engine block is identical, so they are
considered to be independent of the sequence and included
in the respective processing time.
At Taylor Machine Shop, the current M-day is 1400 and
the net daily work is 7,5 hrs.
47
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Case Study: NC Milling Machine at Taylor Machine Shop

What we want to do is, to obtain the results of CR, S/OP,
SPT, FCFS and EDD sequencing rules for these six jobs
and compare them to make a decision on which sequence
to use.
48
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The first step is to calculate the CR and S/OP values for each job by
using the given data as in below:
CR = [Time remaining until due date]/[Shop time remaining] = 15/6 = 2.33 and
S/OP = [Time remaining until the due date - Shop time remaining] / # of
operations remaining = [14-6]/10 = 8/10 = 0.80 for Job 1.
49
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The second step is, to squence six jobs at the NC Milling Machine
according to five specified priority sequencing rules
OR
50
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The second step is, to squence six jobs at the NC Milling Machine
according to five specified priority sequencing rules
51
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

When sequencing the jobs with respect to EDD rule, we
might expect to see the results of two separate sequencing
cases and compare them with each other.


In the first case, we can use the “customer due date”s
of the jobs (which are usually completion dates of the
respective shop orders).
In the second case, we can estimate pickup (due) times
for each job (which are usually the completion times of
the nc milling operation as calculated by Shop Floor
Control Module).
52
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The approach which we may use in estimating the subject
pickup times from NC Milling Machine is given as in
below:
53
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The third step is to calculate “job flow time”, “hours early” and “hours
past due” values and to make an assumption of “actual pickup hours”
of each job with respect to each priority sequencing rule. The results
of this step are given as follows:
54
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The third step ...
55
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The third step ...
56
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The third step ...
57
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The third step ...
58
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The third step ...
59
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)

Solution

The fourth step is to calculate the values of the measures for each
sequencing rule.
The average time spent (avg flow time), the average lateness (avg past due hours), the average number
of jobs in the system (avg wip inventory) and the average total number of jobs in the system (avg total
inventory) will be all minimized if we use the SPT rule.
On the other hand, both multiple-dimension rules (CR and S/OP) yield better results with respect to average
early time when compared to EDD rule by maximizing it. Also, variance in average past due hours
(average lateness) for both of these rules are lower than the other rules and S/OP rule minimized this
variance.
60
Application of Multiple-Dimension Rules (Analysis of SingleMachine Case where setup times are independent of sequence)
END OF SINGLE MACHINE CASE
61
The Two-Machine (flow shop) Case


Development of a scheduling procedure for the twomachine case is more complex than for single-machine
systems.
In the two machine case, the schedule which involves
both machines must satisfy the chosen objective. The
following basic assumptions are valid for this case;



Each job always goes from a particular machine to another
machine (i.e. we consider routings),
All jobs are available at the start of the schedule, and
Setup times are independent.
62
The Two-Machine (flow shop) Case

A set of rules has been developed to minimize the makespan in the two-machine case.

Note that, while the minimum make-span does not
depend on job sequencing in the one-machine case, this
is not true for the two-machine case.

It should be also noted that, if total time to run the entire
batch of jobs is to be minimized, this does not ensure
either the average time each job spends in the system or
the average number of jobs in the system will also be
minimized.
63
The Two-Machine (flow shop) Case

Minimizing the make-span has two basic advantages;

The group of jobs is completed in the minimum time

The utilization of the two-machine flow shop is maximized.
Utilizing the first machine continuously until it processes the
last job, minimizes the idle time on the second machine.
64
The Two-Machine (flow shop) Case
JOHNSON’s RULE

“Johnson’s Rule” is a procedure that minimizes makespan in scheduling a group of jobs on two machines.

The sequence of jobs on the two machines should be
identical and that the priority assigned to a job should
therefore be the same at both.

Is based on the assumption of a known set of jobs, each
with a known processing time and available to begin
processing on the first machine.
65
The Two-Machine (flow shop) Case
JOHNSON’s RULE (Procedure)

Step 1: Select the job with the minimum processing time
on either machine 1 or machine 2. If this time is
associated with machine 1, schedule this job first. If it is
for machine 2, schedule this job last in the series of jobs
to be run. Remove this job from further consideration.
 Step 2: Select the job with next smallest processing time
and proceed as in step 1 (if for machine 1, schedule it
next, if for machine 2, schedule it as near to last as
possible). Any ties can be broken randomly.
 Step 3: Continue this process until all of the jobs have
been scheduled.
66
The Two-Machine (flow shop) Case
EXAMPLE (Problem)
Five motors will be repaired at two work centers in the
following manner;
Work Center 1: Dismantle the motor and clean the parts
Work Center 2: Replace the parts as necessary, and test the
motor.
The customer’s shop can not operate until all the motors have
been repaired. For this reason the plant manager of Morris
Machine Company wants to develop a schedule which
minimizes the make-span. The estimated times to repair each
motor is given as in following Figure 47a.
67
The Two-Machine (flow shop) Case
EXAMPLE (Problem)
The estimated times to repair each motor is given as in
following Figure.
68
The Two-Machine (flow shop) Case
EXAMPLE (Solution)
69
The Two-Machine (flow shop) Case
EXAMPLE (Solution)
No other sequence of respective jobs will produce a lower
make-span.
This schedule minimizes the idle time of work center 2 and
gives the fastest repair time for all five motors.
Note that the schedule recognizes that a job can not begin at
work center 2 until it has been completed at work center 1.
70
The m-Machine (flow shop) Case
 The rules applied for two-machine case can be used in
larger flow shop scheduling problems.
 Campbell, Dudek and Smith (CD&S) have developed an
efficient heuristic procedure for this purpose.
 This procedure uses the Johnson’s algorithm to solve a
series of two-machine approximations for the actual
problem of having m-machines and is called “CD&S
Algorithm for m-machines”.
71
The m-Machine (flow shop) Case
CD&S ALGORITHM FOR m-machines
 Step 1: Solve the first problem considering only
machine 1 and machine m ignoring the intervening m-2
machines.
 Step 2: Solve the second problem by pooling the first
two machines (1 and 2) and the last two machines (m-1
and m) to form two dummy machines. Processing time
at the first dummy machine is the sum of the processing
time on machines 1 and 2 for each order. Processing
time at the second dummy machine is the sum of the
processing time at machines m-1 and m for each order.
72
The m-Machine (flow shop) Case
CD&S ALGORITHM FOR m-machines
 Step 3: Continue in this manner until m-1 problems are
solved. In the final problem, the first dummy machine
contains machines 1 through m-1, and the second
dummy machine contains machines 2 through m.
 Step 4: Compute the make-span for each problem
solved and select the best sequence.
73
The m-Machine (flow shop) Case
Example for m-Machines (Problem)
 Let’s assume that we have four jobs which has identical
routings according to which jobs should flow through
five work centers in the same order.
 Again let’s assume that all five jobs are available at the
start of the schedule and their setup times are
independent.
 The respective processing time of each job for each
work center is given as follows;
74
The m-Machine (flow shop) Case
Example for m-Machines (Problem)
 We want to find the schedule which minimizes the
make-span required to process all these jobs.
75
The m-Machine (flow shop) Case
Example for m-Machines (Solution)
Step 1 :
76
The m-Machine (flow shop) Case
Example for m-Machines (Solution)
Step 2 :
77
The m-Machine (flow shop) Case
Example for m-Machines (Solution)
Step 3 :
78
The m-Machine (flow shop) Case
Example for m-Machines (Solution)
Step 4:
79
The m-Machine (flow shop) Case
Example for m-Machines (Solution)
Step 5 :
 Solution of step 1 : Job 2 - Job 1 - Job 4 - Job 3
 Solution of step 2 : Job 1 - Job 4 - Job 2 - Job 3
 Solution of step 3 : Job 1 - Job 4 - Job 3 - Job 2
 Solution of step 4 : Job 2 - Job 1 - Job 4 - Job 3
80
The m-Machine (flow shop) Case
Example for m-Machines (Solution)
81
The m-Machine (flow shop) Case
Some notes for multiple machine flow shop scheduling
problems




The size of the problems which can be treated with analytical methods is
small and have limited applicability in the “real world”.
Computer time required to solve scheduling problems with analytical
methods grows exponentially as the number of jobs or machines increases.
Minimizing the make-span, is not the same as “minimizing average time in
the system” or “minimizing the average number of jobs in the system”.
The static scheduling assumptions (beginning with all machines idle and all
jobs available, and ending with all jobs processed and all machines idle)
clearly influence the results.
82
The m-Machine (flow shop) Case
Some notes for multiple machine flow shop scheduling
problems


The processing times of machines do not reflect any randomness, which
could reduce the applicability of the techniques used for solution.
It is important to note that, the two-machine scheduling rules utilize the
“shortest processing time” logic.The SPT application in the two-machine
case is not exactly the same as it was in the single-machine case.
83
The m-Parallel Machines Case

From the practical point of view, it is important because in
real world the occurrance of parallel formation of
machines (resources) is common.

In this type of formation, all of the individual machines
are capable of performing the same process on the
available set of jobs.
84
The m-Parallel Machines Case

Also, techniques for parallel machines are often used in
decomposition procedures for multistage systems.

When dealing with parallel machines, generally three
primary objectives are considered. These objectives are;
• minimization of the make-span,
• total completion time, and
• maximum lateness.
85
The m-Parallel Machines Case

When dealing with machines in parallel, the make-span
becomes an objective of significant interest in comparision
to single-machine case.

In practice, the scheduler whose job is to deal with
balancing the work load on machines which operate in
parallel is to minimize the make-span. This will ensure a
good balance.
86
The m-Parallel Machines Case

We may consider the scheduling on parallel machines as a
two step procedure, where;
• the first step is to determine which jobs are to be
allocated to which machine, and
• the second step is the determination of priority
sequence of the jobs allocated to each machine.
87
The m-Parallel Machines Case

On parallel machines, “preemption”s play an important
role.
• Here, the word “preemption” refers to the interruption of the
processing of a job (preempt) at any time and put a different job
on the machine. In such a case, the amount of processing on the
preempted job is not lost. When a preempted job is put back on a
machine it only needs the remaining proceesing time.
• In the single-machine case the role of preemption is important
only when jobs are released at different times. However, in the
case where machines are in parallel preemptions are important
even if all the jobs are released at the same time.
88
“m” Parallel Identical Machines :
The Make-span Without Preemptions

The first priority sequencing problem to be considered in
this group has the following characteristics;
• There are “m” identical machines in parallel which are all available at
the start.
• There are “n” jobs each requiring a single operation which can be
performed on any one of the “m” machines, and they are all available
at the start. Preemptions are not allowed.
• Make-span is equal to the completion (flow) time of the last job to
leave the system and,
• The objective is to minimize the make-span, that is balancing the
work load on “m” parallel identical machines.
89
“m” Parallel Identical Machines :
The Make-span Without Preemptions

One of the heuristic methods uses the “longest processing
time first” (LPT) rule for the solution of this problem.
• LPT rule allocates “m” longest jobs to “m” machines
starting from the job which has the longest processing
time. After this, whenever a machine is freed, the largest
of the remaining (unscheduled) jobs is loaded to that
machine.
• Thus, LPT rule tries to place the shorter jobs toward the
end of the schedule where tey can be used for balancing
the work load distribution on the respective “m”
machines.
90
“m” Parallel Identical Machines :
The Make-span Without Preemptions

The make-span obtained by LPT rule is treated as the “worst
case scenario” for the respective problem. Then the following
argument follows;
• If make-span obtained by LPT schedule is assumed to be
the worst case, then there should exist at least one makespan (better solution) which is possibly unknown and can
be called as “optimal make-span”. Ofcourse, the so called
“optimal make-span” should be smaller than the “LPT
Make-span”.
• If we can not determine a smaller make-span, the LPT
Make-span will be assumed to be the optimal solution to
the problem.
91
“m” Parallel Identical Machines :
The Make-span Without Preemptions

This heuristic method for this specific type of scheduling
problem, has some properties which can be shown by
following inequalities;
• MakeSpan(LPT)/MakeSpan(OPT) ≤ (4/3)-(1/3m) ;
This inequality tells us what should be the maximum value of the ratio
of “worst case make-span” to “optimal make-span” by considering
the number of machines. If the value of this ratio is “1”, we can
conclude that “worst case make-span” is equal to “optimal makespan”. If the value is more than “1”, we can conclude that there is an
“optimal make-span” which is smaller than the “worst case makespan” and the value of the “optimal make-span” is;
MakeSpan(OPT) ≥ [ MakeSpan(LPT) / ((4/3)-(1/3m))]
92
“m” Parallel Identical Machines :
The Make-span Without Preemptions

This heuristic method for this specific type of scheduling
problem, has some properties which can be shown by
following inequalities;
• MakeSpan(OPT) ≥ (Σn pj)/m, where pj is the processing time of
individual job.
This inequality also gives an idea about the probable value of the
“optimal make-span”. Right hand side of the inequality simply
divides the sum of all the processing times of “n” jobs by the number
of available identical parallel machines which is denoted by “m”.
93
“m” Parallel Identical Machines :
The Make-span Without Preemptions
Example (Problem)
Let’s assume that, we have 9 jobs for which the processing
times are given as follows and 4 parallel identical machines.
Also assume that no preemptions are allowed. Our problem is
to find a schedule which will provide us the minimum makespan to process all the 9 jobs by utilizing the 4 parallel and
identical machines.
94
“m” Parallel Identical Machines :
The Make-span Without Preemptions
Example (Solution)


We are going to use “longest processing time first” (LPT)
rule.
According to this rule, as the first step, we will allocate “4”
longest jobs to “4” machines starting from the job which has
the longest processing time...
95
“m” Parallel Identical Machines :
The Make-span Without Preemptions
Example (Solution)

... after that, whenever a machine is freed, the largest of the
remaining (unscheduled) jobs is loaded to that machine as
shown in following Figure. Thus, we obtain the LPT Makespan which is 15 hours.
96
“m” Parallel Identical Machines :
The Make-span Without Preemptions
Example (Solution)

In the third step, we can make use of the previously given
inequalities for the analysis of the constructed LPT schedule;
• In our example case, MakeSpan(LPT)= 15 hrs and the number of
parallel identical machines is m=4.
• Thus, the value of the ratio;
MakeSpan(LPT)/MakeSpan(OPT) ≤1.33 – (1/12)
≤ 1.33-0.083
≤ 1,247
• Therefore;
MakeSpan(OPT) ≥ 15/1.247
≥ 12.03 hrs
97
“m” Parallel Identical Machines :
The Make-span Without Preemptions
Example (Solution)

In the third step, we can make use of the previously given
inequalities for the analysis of the constructed LPT schedule;
• Also for inequality, MakeSpan(OPT) ≥ (Σn pj)/m
the value of (Σn pj)/m=48/4=12 hrs
98
“m” Parallel Identical Machines :
The Make-span Without Preemptions
Example (Solution)
 Thus, the results of the analysis tells us that there is a better
schedule for the problem and the make-span of this schedule
is ≥12hrs. By trial and error this better schedule with optimal
make-span can be constructed as follows;
99
“m” Parallel Identical Machines :
The Total Completion Time Without Preemptions
 The second priority sequencing problem to be considered
in this group has the same characteristics as previously
described in this section but, this time the objective is to
minimize the total completion (flow) time.
 Studies on this type of problem has shown that SPT rule
provides an optimal solution for this type of priority
sequencing problem, but, it is not the only rule which will
minimize the “total completion time”.
100
“m” Parallel Identical Machines :
The Total Completion Time Without Preemptions
 Solution method :
• Consider that we have n jobs and m identical parallel
machines where preemptions are not allowed.
• According to the SPT rule, at the start, the smallest job
has to go on machine 1, the second smallest on machine 2
and so on. The (m+1)th smallest job follows the smallest
job on machine 1, the (m+2)th smallest job follows the
second smallest job on machine 2 and so on.
101
“m” Parallel Identical Machines :
The Total Completion Time Without Preemptions
Example (Problem)
 Assume that we have the previous case of 9 jobs which
are given as follows and the same number of parallel
identical machines which is 4.
 Also assume that this time our objective is to minimize
the total completion (flow) time.
102
“m” Parallel Identical Machines :
The Total Completion Time Without Preemptions
Example (Solution)
 Since our objective is to minimize the total completion time, we are
going to use the SPT rule for priority sequencing of the subject 9 jobs
over 4 parallel identical machines.
 First, since the jobs 7, 8 and 9 have the same smallest processing
time, we assign job 7 to machine 1, job 8 to machine 2 and job 9 to
machine 3. The next smallest job is job 5 and is assigned to machine
4. Remaining smallest jobs are job 6 ≤ job 3 ≤ job 4 ≤ job 1 ≤ job2.
So, we assign job 6 as the second job for machine 1, job 3 as the
second job for machine 2, job 4 as the second job for machine 3, job 1
as the second job for machine 4 and job 2 as the third job for machine
1.
103
“m” Parallel Identical Machines :
The Total Completion Time Without Preemptions
Example (Solution)
 The resultant schedule will be as follows;
As can be easily verified from the above Figure, the make-span of the SPT
schedule for this problem is 16 hours. Also, the total completion time of the
schedule is 74 hours.
104
“m” Parallel Identical Machines :
The Total Completion Time Without Preemptions
Example (Solution)
 The comparision of total completion (flow) times;
105
“m” Parallel Identical Machines :
The Make-span With Preemptions
 The third priority sequencing problem which we will
consider is the minimization of make-span problem where
preemptions are allowed. Studies on about the subject
problem showed that, allowing preemptions usually (but
not always) simplifies the analysis of the problem.
106
“m” Parallel Identical Machines :
The Make-span With Preemptions
 The following three step method was developed to
construct a schedule which minimizes the make-span;
• Step 1: Take the “n” jobs and process them one after
another on a single-machine in any sequence. The
make-span is then equal to the sum of the “n”
processing times.
• Step 2: Take this single-machine schedule and cut it
into “m” intervals (parts).
• Step 3: Take the processing sequence of the first
interval as the schedule for machine 1; the processing
sequence of the second interval as the schedule for
machine 2, and so on.
107
“m” Parallel Identical Machines :
The Make-span With Preemptions
Example (Problem)
 Assume that we have 5 jobs as given below and 3 parallel
identical machines. We can unload any of the respective
jobs from one machine before they are completed and
reload them at a later time to perform the remaining
processing. Our objective is to establish a schedule which
minimizes the make-span.
108
“m” Parallel Identical Machines :
The Make-span With Preemptions
Example (Solution)
 Step 1: We construct a single-machine schedule where
make-span is equal to 26 hrs.
109
“m” Parallel Identical Machines :
The Make-span With Preemptions
Example (Solution)
 Step 2: We divide the single-machine schedule into 3
intervals where 3=m (the number of machines).
110
“m” Parallel Identical Machines :
The Make-span With Preemptions
Example (Solution)
 Step 3: We assign the first interval to machine 1, the
second interval to machine 2, and the third interval to
machine 3.
111