A case study on Machine scheduling and sequencing using Meta heuristics

Sarfaraz ahmad Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 4) January 2016, pp.75-78
www.ijera.com 75 | P a g e
A case study on Machine scheduling and sequencing using
Meta heuristics
Sarfaraz ahmad*, Ashok kumar jha**
*(Department of Manufacturing engineering, NIT Jamshedpur, Jamshedpur-14)
** (Department of Manufacturing engineering, NIT Jamshedpur , India)
ABSTRACT
Modern manufacturing systems are constantly increasing in complexity and become more agile in nature such
system has become more crucial to check feasibility of machine scheduling and sequencing because effective
scheduling and sequencing can yield increase in productivity due to maximum utilization of available resources
but when number of machine increases traditional scheduling methods e.g. Johnson‟s ,rule is becomes in
effective Due to the limitations involved in exhaustive enumeration, for such problems meta-heuristics has
become greater choice for solving NP hard problems because of their multi solution and strong neighbourhood
search capability in a reasonable time.
Keywords - Manufacturing systems, Meta-heuristics, productivity
I. INTRODUCTION
Production scheduling is a key for organizational
productivity, which prepares a calendar for producing
components/products. The scheduling problems are
classified into single machine scheduling, flow shop
scheduling job shop scheduling, open shop
scheduling, and hybrid scheduling. In this paper,
the open shop scheduling problem is considered. The
open shop scheduling problem is alternatively called
as moderated job shop scheduling problem
(Panneerselvam [1]), since all the machines will not
have 100% utilization all the time hence the
machines which have similar processing capabilities
will be grouped and a batch of single operation jobs
will be scheduled on these machines while doing so
the organization may be keen in optimizing any one
of the following measures of performances
 Minimizing the sum of the completion times of
all the jobs.
 Minimizing total tardiness.
 Minimizing total lateness.
 Minimizing the total number of tardy jobs.
 Minimizing makespan.
Classification of Scheduling Problems
According to French (1982), the general scheduling
problem is to find a sequence, in which the jobs (e.g.,
a basic task) pass between the resources (e.g.,
machines), which is a feasible schedule, and optimal
with respect to some performance criterion. Graves
(1981) introduced a functional classification scheme
for scheduling problems. This scheme categorizes
problems using the following dimensions:
a) Requirement generation
b) Processing complexity
c) Scheduling criteria
d) Parameter variability
e) Scheduling environment
(a) Based on requirements generation, it can be
classified as an open shop or a closed shop. An open
shop is "build to order" and no inventory is stocked
and when orders are filled from existing inventory it
is called closed shop. Closed shop can further
classified into job shop and flow shop and detailed
classification of scheduling problem is shown in
1.1Classification of scheduling problems based on
requirement generations
(b) Processing complexity refers to the number of
processing steps and workstations associated with the
production process. This dimension can be
decomposed further as follows:
RESEARCH ARTICLE OPEN ACCESS

(i) One stage, one processor
(ii) One stage multiple processors
(iii) Multistage flow shop
(iv) Multistage job shop
The one stage, one processor and one stage,
multiple processors problems require one processing
step that must be performed on a single resource or
multiple resources respectively. In the multistage,
flow shop problem each job consists of several tasks,
which require processing by distinct resources; but
there is a common route for all jobs. Finally, in the
multistage, job shop situation, alternative resource
sets and routes can be chosen, possibly for the same
job, allowing the production of different part types.
(c) Scheduling criteria, states the desired objectives
to be met. “They are numerous, complex, and often
conflicting”. Some commonly used scheduling
criteria include the following: Minimize total
tardiness, Minimize the number of late jobs,
Maximize system/resource utilization, Minimize in-
process inventory, Balance resource usage, Maximize
production rate etc.
(d) The dimension „parameters variability‟ indicates
the degree of uncertainty of the various parameters of
the scheduling problem. If the degree of uncertainty
is insignificant (i.e. the uncertainty in the various
quantities is several orders of magnitude less than the
quantities themselves), the scheduling problem could
be called deterministic. For example, the expected
processing time is six hours, and the variance is one
minute. Otherwise, the scheduling problem could be
called stochastic.
(e) The dimension, scheduling environment, defined
the scheduling problem as static or dynamic.
Scheduling problems in which the number of jobs to
be considered and their ready times are available are
called static. On the other hand, scheduling problems
in which the number of jobs and related
characteristics change over time are called dynamic.
II. Mathematical model and problem
descriptions
Assumption
 Once an operation is started on the machine it
must be completed before another operation can
begin on that machine
 All processing time on the machine are known,
deterministic, finite and independent of sequence
of the jobs to be processed.
 The first machine is assumed to be ready
whichever and whatever job is to be processed
on it first.
 Each job is processed through each of the m
machines once and only once.
Problem descriptions
A general job shop problem suppose having n jobs
{J1, J2, J3 --------- Jn} to be processed through m
machine {M1, M2, M3 ----------- Mm}. Technological
constraints demand that each job should be processed
through the machine in a particular order and gives
an important special case named as flow shop. Thus
in case of flow shop jobs pass between the machine
in the same order i.e. if J1 must be processed on M1
before machine M2 then the same the true for all jobs.
This technological constraint therefore gives the form
like :
Job processing order
J1 M1 M2 M3 ……………… M m
J2 M1 M2 M3 ……………… M m
…………………………………………………
Jn M1 M2 M3 ……………… M m
______________________________________
For a general job shop problem defined above the
number of possible sequences are (n!) m
, where n is
number of jobs and m is the number of machines.
With the above technological constraints in case of
flow shop number of different sequence reduces to
(n!). This reduced number is quite large for even
moderate size problems and recognized to be NP
hard (Garey et al., 1976; Gonzalez and Sahni, 1978;
Pinedo, 2005 and several others)[3,4,5]. There has
been an attempt to solve this problem with typical
objective function being the minimization of average
flow time, minimizing the time required to complete
all the jobs or makespan, minimizing average
lateness values or tardiness, minimizing maximum
tardiness, and minimizing the number of tardy jobs.
III. Case study
In a manufacturing shop generally a number of
job passes through a number of machine .let in a
manufacturing shop there are three shop namely M1,
M2, M3..And each job must passes through it only
once. The processing time of each job is given below
JOBS Duration (Hours)
Machine M1 Machine M2 Machine M3
J1 17 19 13
J2 15 11 12
J3 14 21 16
J4 20 16 20
J5 16 17 17
Palmer heuristics approach
A heuristic developed by Palmer (1965)[6], in an
effort to use Johnson‟s rule, is built around the notion
of a slope index. The slope index gives a large value
to jobs that have a tendency of progressing from
small to large operating times as they move through
the stages. The sequence of operations is given by
priority to jobs having the strongest tendency to
progress from short times to long times. This means
that the job sequence can be generated based upon an
non-increasing order of the slope indices.

Let S (j) be the slope index for job j and Ot
j be the
operating time of job j at stage t. Palmer‟s slope
index is calculated as follows:
Let assign +2 to M3 , 0 to M2 and -2 to M1 then we
have to calculate slope S(J) of each job
S(J1) ={17×(-2)+(19×0)+(13×2)}=-8
S(J2) ={15×(-2)+(11×0)+(12×2)}=-6
S(J3) ={14×(-2)+(21×0)+(16×2)}=4
S(J4) ={20×(-2)+(16×0)+(20×2)}=0
S(J5) ={16×(-2)+(17×0)+(17×2)}=2
Then we have to arrange the job according to
decreasing order of their slope
J3-J5-J4-J2-J1
J3 starts at 0 finishes at 14 at machine M1 then goes to
M2 now goes to M2 takes 21 finishes at 35,goes to M3
takes another 16 finishes at 51.
J5 starts at M1 at 14 takes another 16 finishes at 30
then goes to M3 wait till 35 takes another 17 finishes
at 52. M3 is already available so takes another 17
finesse at 69.
J4 starts at M1 at 30 takes another 20finishes at 50
then goes to M2 wait till 52 takes another16 finishes
at 68 . then goes to M3 wait till 69 takes another 20
finishes at 89.
then goes to M2 wait till 68 takes another11 finishes
at 79 . then goes to M3 wait till 89 takes another 12
finishes at 101.
M2is already available so takes another 19fineses at
101 goes to M3 takes another 13 finishes at 114.
So makespan associated with this sequence is 114.
Since this heuristics solution need not be optimal so
we have to find the goodness factor in order to know
how good it is.
Goodness factor is defines as a ratio of difference
between heuristic solution and optimum solution to
optimum solution. Since we don‟t know the optimum
solution so we can replace optimum solution with
lower bound.
Lower bound of M1=total processing time at m1
+minimum (sum of processing time of m2
+m3)=82+23=105
Lower bound of M2= total processing time at m2
+minimum processing time of m1 + minimum
processing time of m3=84+14+12=110
Lower bound of M3=total processing time at m
+minimum (sum of processing time of m1 +m2)
=26+78=104
Maximum bound is best one so we choose lower
bound 110. So goodness factor {(114-
110)÷110}=0.0364 e.g. 3.64%.
If we willing to accept the solution about 3.64% of
optimum then we should go for palmer heuristics.
Campbell, Dudek, and Smith(CDS)
Campbell, Dudek, and Smith (1970) develop one of
the most significant heuristic methods for flow shop
problems with makespan criterion, in the following
denoted by CDS. Its strength lies in two properties:
(1) it uses Johnson‟s rule in a heuristic fashion, and
(2) it generally creates several schedules from which
a “best” schedule can be chosen. In so doing, k – 1
sub-problem are created and Johnson‟s rule is applied
to each of the sub-problems. Thus, k – 1 job
sequences are generated. Since Johnson‟s algorithm
is a two-stage algorithm, a k stage problem must be
collapsed into a two-stage problem. Let g be a
counter for the k – 1 sub-problems, the operating
times for the “first” stage are denoted as a(j, g),
where j denotes the job and g denotes the g-th sub
problem. Similarly, b(j, g) denotes the “second” stage
operating times of job j and sub-problem g. Given
these notations, the operating times are calculated by
the following two formulas:
For each of the sub-problems, Johnson‟s
algorithm provides a job sequence using the values
a(j, g) and b(j, g). Once Johnson‟s sequence is
created, the problem is then returned to the
consideration of all k stages. Since in the given
problem there are three machine (k=3) so number of
possible job sequences is (k-1) is 2. Since Johnson‟s
algorithm is a two-stage algorithm, a k stage problem
must be collapsed into a two-stage problem.
M1+M2 M2+M3
J1 36 32
J2 26 23
J3 35 37
J4 36 36
J5 33 34
For this sub problem job sequence according to
Johnson‟s algorithms [8]
The smallest one in the above mention table is 23 so
J2 goes to extreme right row then next smallest
number is 32 so J1 goes to 2nd
extreme right row in
this manner we fill the remaining vacant row
according to Johnson‟s algorithms.
J5 J3 J4 J1 J2
b.

M1 M2
J1 17 13
J2 15 12
J3 14 16
J4 20 20
J5 16 17
For this sub problem job sequence according to
Johnson‟s algorithms.
J3 J5 J4 J1 J2
From above discussed two sequences best one is J2-
J5-J4-J1-J2 because of their minimum completion time
i.e. 114
Since we already defined goodness factor and we
have lower bound is 110.
So goodness factor {(114-110)÷110}=0.0364 e.g.
3.64%.
If we willing to accept the solution about 3.64% of
optimum then we should go for palmer heuristics.
IV. Conclusion
Meta heuristics approach tends to move
relatively quickly towards very good solutions, so it
provides a very efficient way of dealing with large
complicated problems. It is very useful in cases
where traditional methods get stuck at local minima
and common area of application is combinatorial
optimization problems. However is does not
guarantee for optimum solution but if we willing to
accept 0 to 4% of optimum solution then this types of
methods is very useful.
Acknowledgements
I would like to express my deep sense of
gratitude and indebtedness to my supervisors Mr.
Ashok kumar jha, Associate Professor and Head of
Department (HOD) of Manufacturing Engineering
Dr. Amresh kumar, National Institute of Technology
,Jamshedpur, for their esteemed guidance, invaluable
encouragement, moral support, free to act on my
ideas and scholarly inputs from early stage of
research work that instill confidence in me during the
research work
Journal Papers:
[1] Panneerselvam, R. (1999) Heuristic for
Moderated Job Shop Scheduling Problem to
Minimize Makespan. Industrial Engineering
Journal, 28, 26-29.Note that the journal title,
volume number and issue number are set in
italics.
[2] French, S., 1982. Sequencing and
Scheduling: An Introduction to the
Mathematics of the Job-Shop. Ellis
Horwood Limited.
[3] Garey, M.R., Johnson, D.S. and Sethi, R.R.,
1976. The complexity of flow shop and Job
shop scheduling. Operations Research
1,117–129.
[4] Gonzalez, T. and Sen, T., 1978. Flow shop
and job shop schedules: complexity and
approximations. Operations Research 26,
36–52.
[5] Pinedo, M., 2005. Scheduling theory,
algorithms, and system, (Prentice-Hall).
[6] Palmer, D.S., 1965. Sequencing jobs
through a multi-stage process in the
minimum total time a quick method of
obtaining a near optimum. Operations
Research Quarterly 16(1), 101–107.
[7] Johnson, D.S., Aragon, C.R., McGeoeh,
L.A. and Schevon, C., 1989. Optimization
by simulated annealing: an experimental
evaluation; Part I. Graph partitioning.
Operations Research 37(6), 865-892.
[8] Johnson, S.M., 1954. Optimal two-and
three-stage production schedules with setup
times included. Naval Research Logistics
Quarterly 1, 61-68.
Theses:
[9] A.k dhingra , Multi objective flow shop
scheduling using meta heuristics, doctoral
diss., nit kurukshetra, Haryana, IN, 2006.

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