Sequencing Problem. Introduction The selection of an appropriate order for a series of jobs to be...
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Transcript of Sequencing Problem. Introduction The selection of an appropriate order for a series of jobs to be...
Introduction
• The selection of an appropriate order for a series of
jobs to be done on a finite number of service
facilities, in some pre-assigned order, is called
sequencing.
• A practical situation may corresponds to an industry
producing a number of products, each of which is to
be processed through different machines, of course,
finite in number.
• The general sequencing problem may be defined as:
Let there be n jobs to be performed one at a time on
Introduction…
on each of m machines. The sequence (order) of the
machines in which each job should be performed is
given. The actual or expected time required by the
jobs on each of the machines is also given. The
general sequencing problem, therefore, is to find
the sequence out of (n!)m possible sequences
which minimize the total elapsed time between
the start of the job in the first machine and the
completion of the job on the last machine.
Assumptions
• Each job once started on a machine, is to be
performed up to completion on that machine.
• The processing time on each machine is known.
Such a time is independent of the order of the jobs
in which they are to be processed.
• The time taken by each job in changing over from
one machine to another is negligible.
Assumptions…
• A job starts on the machine as soon as the job and
the machine both are idle and job is next to the
machine and the machine is also next to the job.
• No machine may process more than one job
simultaneously.
• The order of completion of job has no significance,
i.e. no job is to be given priority. The order of
completion of jobs is independent of sequence of
jobs.
Basic Terms
• Number of Machines: It refers to the no. of service
facilities through which a job must pass before it is
assumed to be completed.
• Processing Order: It refers to the order (sequence)
in which given machines are required for
completing the job.
• Processing Time: It indicates the time required by a
job on each machine.
Basic Terms…
• Total Elapsed Time: It is the time interval between
starting the 1st job and completing the last job
including the idle time in a particular order by the
given set of machines.
• Idle time on a machine: It is the time for which a
machine does not have a job to process, i.e. idle
time from the end of job (i-1) to the start of job i.
Basic Terms…
• Processing Time: It indicates the time required by a
job on each machine.
• No passing rule: It refers to the rule of maintaining
the order in which jobs are to be processed on given
machines.
Processing n jobs through 2 machines
• Let there be n jobs, each of which is to be processed
through 2 machines, say M1 & M2 in the order M1M2.
That is, each job has to pass through the same
sequence of operations.
• Or, a job is assigned on M1first and after it has been
completely machine M1, it is assigned to machine M2.
• If the machine M2 is not free at the moment for
processing the same job, then the job has to wait in a
waiting line for its turn on machine M2 (no passing).
Processing n jobs through 2 machines…
• Let tij (i=1,2 and j=1,2,…,n) be the time required for
processing jth job on the ith machine.
• Since passing is not allowed, therefore machine
M1will remain busy in processing all the n jobs one
by one while machine M2 may remain idle after
completion of one job and before starting of another
job.
• Thus the objective is to minimize the idle time od
the 2nd machine.
Processing n jobs through 2 machines…
• Let X2j be the time for which machine M2 remains
idle after finishing (j-1)th job and before starting
processing jth job (j=1,2,…,n).
• Total Elapsed Time (T) is
• The problem is to minimize T. However, since
is the total time for which machine M2has to work
and is thus fixed, it does not form a part of the
optimization problem.
n
1j2j
n
1j2j Xt
n
1j2jt
Processing n jobs through 2 machines…
• Thus the problem reduces to that of minimize
• A convenient procedure for obtaining a sequence of
performing jobs to minimize is well illustrated
by the following Gantt Chart:
n
1j2jX
n
1j2jX
11
1
1j2j
2
1j1j
3
1j
2
1j2j1j
2
1j2j
3
1j
2
1j2j1j232221
2221222113121123
1121112112112221
2121121122
22
2121121121211211
22
11 21
t,tt,ttMax.
X,ttMax.XXX
0 ,XXtttttMax. X Similarly,
tX since , t,tttMax.XX
Thus
0 ,tXttMax. X
:as written bemay Xfor expression Thus
otherwise 0
tXt tif tXtt
tX
clear that isit chart, theFrom
X
Optimum sequence Algorithm
• List the jobs along with their processing times in a
table.
• Examine the rows for processing times on machines
M1 and M2, and find the smallest processing time in
each row, i.e. find out min.(t1j, t2j) for all j.
2n2322212
1n1312111
n321
t.......... t t t: M
t.......... t t t: M machineon Time Processing
J .......... J J J : Number Job
Optimum sequence Algorithm…
• If the smallest processing time is for the first
machine M1, then place the corresponding job in the
1st available position in the sequence, otherwise
place 2nd machine M2.
• If there is a tie in selecting the minimum of all the
processing times, then there may be 3 situations
1. Minimum among all processing times is same for
the machines, i.e., min.(t1j, t2j)= t1k = t2r, then process
the kth job first and the rth job last.
Optimum sequence Algorithm…
2. If the tie for minimum occurs among processing
times t1j on machine M1 only, then select arbitrarily
the job to process first.
3. If the tie for minimum occurs among processing
times t2j on machine M2, then select arbitrarily the
job to process last.
• Cross off the jobs already assigned and repeat steps
1 through 4, placing the remaining jobs next to first
or next to last, until all the jobs have been assigned.
Optimum sequence Algorithm…
• Calculate idle time for machines M1 and M2:
Idle time for M1 = Total elapsed time –(time when
the last job in a sequence finishes on M1.
Idle time for M2= Time at which the 1st job in a
sequence finishes on M1 + (time when the jth job
in a sequence starts on M2) – {(time when the (j-
1)th job in a sequences finishes on M2)}
• The total elapsed time to process all jobs through 2
machines as under:
n
2j
Total elapsed time= Time when the nth job in a sequence finishes on machine M2.
job.jth on work starting before and job
1)th-(j processingafter idle remains M machinefor which Time I
M machineon jobjth processingfor required Time tWhere,
I t
M machineon finishes sequence ain jobnth when theTime timeelapsed Total
22j
22j
n
2j2j
n
2j2j
2
ExampleIn a factory, there are six jobs to perform, each of
which should go through 2 machines A & B, in the
order A, B. The processing timings (hrs) for the jobs
are given here. You are required to determine the
sequence for performing the jobs that would
minimize the total elapsed time, T. What is the value
of T?
10 2 2 3 6 5 : B
3 6 5 8 3 1 :A machineon Time Processing
J J J J J J : Job 654321
Solution• The smallest processing time in the given problem is
1 on machine A. So, perform J1 in the beginning.
• The reduced set of processing times becomes
• The min. processing time in this reduced problem is
2 which corresponds to J4 & J5 both on machine B.
• Since the corresponding processing time of J5 on
machine A is larger than the processing time of J4
on machine A, J5 will be processed in the last and J4
10 2 2 3 6 : B
3 6 5 8 3 :A machineon Time Processing
J J J J J : Job 65432
Solution…
shall be processed next to last. The updated job
sequence is
• The remaining processing times are:
• Now, there is a tie among 3 jobs for the smallest
processing time in this reduced problem.
• These corresponds to J2 and J6 on machine A, and to
J1 J4 J5
10 3 6 : B
3 8 3 :A machineon Time Processing
J J J : Job 632
Solution…
J3 on machine B.
• As the corresponding processing time of J6 on
machine B is larger than the corresponding processing
time of J2 on machine B, J6 will be processed next to
J1 and J2 should be placed next. The updated job
sequence is:
• The sequence is optimum one. The total elapsed time
is calculated below:
J1 J6 J2 J3 J4 J5
Job Machine A
In Out
Machine B
In Out
Idle time on B
J1 0 1 1 6 1
J6 1 4 6 16 --
J2 4 7 16 22 --
J3 7 15 22 25 --
J4 15 20 25 27 --
J5 20 26 27 29 --
T=29 hrs
Solution…
• Idle time of machine A is 29-26 = 3 hrs and
• Idle time of machine B is 1 hr
• The Gantt chart for the above problem is