SEQUENCING
PROBLEMS
Sequencing
A problem in which it is necessary
to determine the orders or
sequences of jobs in which they
should be performed so as to
minimize the total effectiveness
on the sum of the pertinent costs
is known as sequencing problems.
A Sequencing problem could
involve:
 Jobs in a manufacturing plant
 Aircraft waiting for landing and
clearance
 Maintenance scheduling in a factory
 Programs to be run on a computer
 Customers in a bank.
TERMS USED:
 JOBS – Jobs/items/customers are the primary
stimulus for sequencing.
 Number of Machines –
a machine is
characterized by a certain processing
capability or facilities through which a job
must pass before it is completed in job.
 Processing Time – It means the time each
job requires at each machine.
 Total Elapsed Time – It is the time between
starting the first job and completing the last
one.
 Idle Time – Processing It is the time the
machine remains idle during the total elapsed
time.
 Technological Order – It refers to the order in
which various machines are required for
completing the jobs. It is also known as
Processing order.
 No Passing Rule – It implies that passing is
not allowed i.e. the same order of the jobs is
maintained over each machine. If each of njobs are to be processed on 2 machines A & B
in the order AB then this rule will mean that
each job will go to A first and then to B.
JOB ARRIVAL PATTERN
Job Arrival Pattern can be defined as the
pattern in which a job arrives and joins the
system. The arrival pattern may be fixed and
known, variable but known and variable &
unknown.
Categories of Job Arrival Pattern: Static Arrival Pattern – If all jobs arrive
simultaneously.
 Dynamic Arrival Pattern – Where the jobs
arrive continuously.
ASSUMPTIONS
 No machine can process more than one job at a time.
 The processing times on different machines are





independent of the order in which they are processed.
The time involved in moving a job from one machine
to another is negligibly small.
Each job once started on a machine is to be performed
upto completion on that machine.
All machines are of different types.
All jobs are completely known and are ready for
processing.
A job is processed as soon as possible but only in the
order specified.
Types of Sequencing Problems
1. Problems with ‘n’ jobs through one machine.
2. Problems
3.
4.
5.
6.
with
‘n’
jobs
through
two
machines.
Problems with ‘n’ jobs through three
machines.
Problems with ‘n’ jobs through ‘m’ machines.
Problems with ‘n’ jobs through one machine
(dependent set up time) (as Assignment ).
Problems with two jobs through ‘m’
machines.
Problems with ‘n’ jobs through
one machine
Methods
Shortest
Processing Time
(SPT) Rule
Weighted
Scheduling
Processing Time
(WSPT) Rule
Problems with ‘n’ jobs through
2 machines
Methods
Heuristic
Method
Combinatorial
Method
Heuristic Method
Suggested by S.M.Johnson & Bellman. Procedure is
as follows: 1. Select the smallest processing time occurring in the
list Ai or Bj, if there is a tie select either of the
smallest processing time.
2. If the smallest time is on machine A, then place it
at first place if it is for the B machine place the
corresponding job at the last place. Cross off that
job.
3. If there is a tie for minimum time on both the
machines then select machine A first and machine
B last and if there is tie for minimum on machine A
(same machine) then select any one of these jobs
first and if there is tie for minimum on machine B
then select any of these jobs in the last.
4. Repeat step 2 & 3 to the reduced set of processing
5.
6.
i.
ii.
iii.
times obtained by deleting the processing time for
both the machines corresponding to the jobs
already assigned.
Continue the process placing the job next to the
first or next to the last and so on till all jobs have
been places and it is called optimum sequence.
After finding the optimum sequence we can find
the following:
Total Elapsed Time = Total time between starting
the first job of the optimum sequence on machine
A and completing the last job on machine B.
Idle time on machine A.
Idle time on machine B.
Combinatorial Method
This method has three stages:
1. Compare operation time on machine A (ta) with
machine B (tb). Give the code (Aa) 1 if ta > tb and code 1 if ta < tb.
2. Determine the minimum of the two operating time Ba,
where Ba = min(ta, tb).
3. Give comparatic weightage of different jobs f(a), where
f(a) = Aa/Ba.
the optimal sequence for the job is determined by
arranging the jobs in ascending order of their weights
f(a).
Assumption – the jobs have no priority i.e. no job is
preferred to other job and storage space is available for
all the jobs.
Problems with ‘n’ jobs on 3
machines
Job
1
2
3
4
.
.
i
.
.
n
Machine A Machine B
A1
A2
A3
A4
.
.
Ai
.
.
An
B1
B2
B3
B4
.
.
Bi
.
.
Bn
Machine C
C1
C2
C3
C4
.
.
Ci
.
.
Cn
There is no general solution at present however
previous method given by Johnson can be
applied if the following two conditions are
satisfied:
Condition 1 – Min Ai ≥ Max Bi
and/ or
Condition 2 –
Min Ci ≥ Max Bi
Method: Replace given problem into 2
fictitious machines G & H
where, Gi=Ai+Bi,
Hi=Bi+Ci, and so on
Now apply same procedure and find out
optimal sequence.
Problems with ‘n’ jobs through
‘m’ machines
Job/Machines A
B
C….
M
1
2
3
4
5
A1
A2
A3
A4
A5
B1
B2
B3
B4
B5
C1….
C2….
C3….
C4…..
C5….
M1
M2
M3
M4
M5
n
An
Bn
Cn….
Mn
There is no solution available at present, however if
any of the following conditions are met , we can
proceed further:
Condition 1 –
Min A ≥ Max B,C,M-1
and/ or
Condition 2 –
Min M ≥ Max B,C,M-1
Method: Replace given problem into 2 fictitious
machines G & H
where,
Gi=Ai+Bi+Ci+(M-1),
Hi=Bi+Ci+(M-1)+M, and so on
Now apply same procedure and find out optimal
sequence.
Alternatively convert the given problem into a no.
of 2 machine sub problems.
Problems with ‘n’ jobs through
one machine (dependent set up
time)
These problems can be solved as
the Assignment Problems using
special
case
of
Travelling
Salesman.
Problems with 2 jobs through
‘m’ machines
Graphical method is used for such type of problems.
The steps are given below:
1. Construct a two – dimensional graph where the X –
axis represent the processing time and sequence of
job 1 and the M1 machine; whereas, Y – axis
represents the processing tome and sequence of
job 2 on M2 machine. It may be noted that scale
used must be same for both the machines.
2. Shade the area where machine would be occupied
by the two jobs at the same time.
3. The processing of both jobs can be represented by
a continuous path which consists of horizontal,
vertical and 45 degree diagonal segment.
4. The path starts at the lower left corner and
stops at the upper right corner, while avoiding
the shaded area in the graph. In other words,
the path is not allowed to pass through shaded
areas which correspond to operating both jobs
concurrently on the same machine.
5. Any horizontal movement will indicate the
progress of job 1 whereas job 2 is idle.
6. Any vertical movement will reveal that the job
2 is in progress while job 1 is idle.
7. Minimum elapsed time for any job is the
processing time of the job + Idle time of the
same job.