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Selection of Job Shop Scheduling
Problem Using Fuzzy Linguistic Variables
Vikas S. Jadhav^{1} and V. H. Bajaj^{2}
{^{1}Research Student, ^{2}Professor} Department of
Statistics, Dr.B.A.M.University, Aurangabad431004(M.S) INDIA.
Corresponding Addresses:
^{1}[email protected],
^{2}[email protected]
Abstract:
Jobshop scheduling (JSS) is a
difficult problem, both theoretically and practically. The theoretical
problems stem from the search for optimal schedules to a
“Minimum /limited number of resources (Machines)” to
complete the “Maximum works (Tasks)” with customers
satisfaction. This paper,
concentrates on JSS Problem under fuzzy approach to solve a real life
tailor JSSP formulated. The scheduling problem is a very common
problem of a tailor shop, such that we have to satisfy the multiple
conflicting objectives, which are to “Minimize the
job lateness or tardiness” and “Maximize the customer
satisfaction” in the best possible manner. Here we try
“to find out an optimal scheduling sequence to perform
the jobs that arrive at the shop”.
In this paper, the
customer priority is based on the “fuzzy linguistic variables
“It is expressed as Bad, Low, Medium, High, Very High etc. Fuzzy sets
are used for modeling uncertainty due to vagueness. Fuzzy membership
functions are used to define how well a value “fits” into a fuzzy set.A
new technique is proposed based on the concept of fuzzy linguistic
membership functions. This JSS model is more practical and realistic
in manufacturing areas.
Numerical example is given to demonstrate the effectiveness of the new
developed model.
Key Words:
Job Shop Scheduling, Fuzzy
Linguistic Variables, Optimal Scheduling, Fuzzy Sets, Vagueness.
1. Introduction
Scheduling
consists of planning and arranging jobs in an orderly sequence of
operations in order to “Meet the customer's requirements”
[1]. The schedule of jobs and the control of their flows through a
production process are the most significant role in any modern
manufacturing systems. In a single machine scheduling, there is only
one machine to process all jobs to optimize the objective function,
say minimizing the sum of the maximum earliness and tardiness [2]. It
is well known that the optimal solution of single objective models can
be quite different to those models consist of multi objectives. In
fact, the decision maker often wants to minimize the
earliness/tardiness penalty or total flow time. Each of these
objectives is valid from a general point of view. Since these
objectives conflict with each other, a solution may perform well for
one objective or it gives inferior results for others. For this
reason, scheduling problems have a multiobjective nature. In decision
making situations, the high degree of fuzziness and uncertainties are
included in the data set.
The fuzzy set theory
provides a framework for handling the uncertainties of this type
[3].Bellman and Zadeh [4] presented some applications of fuzzy theory
to the various decisionmaking processes in a fuzzy environment.
Zimmerman [5,6] presented a fuzzy optimization technique to a linear
programming (LP) problem with single and multiobjectives. The fuzzy
set theory has been applied to formulate and solve problems in various
areas such as artificial intelligence, image processing, robotics,
pattern recognition, (Hannan, [7] and Yager, [8]). Different
approaches to multiobjective single machine problems with fuzzy
parameters have been presented in the literature during the last
decade.
2. Job Shop Scheduling Problem (JSSP)
The jobshop
scheduling problem, also known as the sequencing problem [5], is
concerned with allocating limited resources to operations over time
[9]. Although scheduling has an important role in the field of
production and operations management, it is a difficult problem both
theoretically and practically. Theoretical scheduling problems are
related to the search for optimal schedules, subject to a limited
number of constraints in a huge search space. These problems suffer
from excessive combinatorial complexity. Practical scheduling problems
are complex due to the number and variety of the constraints
themselves, many of which are soft rather than hard physical
constraints [10]. Informally, the jobshop scheduling problem can be
described as follows.
We are given
n jobs (Tasks) and mmachines (Processors). Each job
consists of a sequence of operations that must be processed on m
machines in a given order. Each operation must be executed
uninterrupted on a given machine for a given period of time and each
machine can only handle at most one operation at a time. The problem
is to find a schedule, an allocation of the operations of n jobs
to certain time intervals on m machines, with minimum
overall time [9,10].
3. Assumptions of JobShop Scheduling Problems (JSSP):
In a classical JSSP, n jobs
are processed on mmachines, in which the main assumptions are
listed below:
a)
All jobs are available at time zero.
b)
Each job is processed by one machine at a time.
c)
A machine can process only one operation at a time.
d)
The processing times are fixed and sequence independent.
e)
The processing order of
each job is given and fixed.
3.1 Indices and Parameters
The following notations and
definitions are used to describe the jobshop scheduling problem.
n  Number of jobs
m  Number of machines
p_{i } Processing time of
job, i (i=1,2,…n)
d_{i } Fuzzy Due Date of
job, i (i=1,2,…n)
μ_{CP
} Fuzzy Customer Priority
μ_{C
} Fuzzy Membership
function ‘Close’
μ_{D
} Fuzzy Membership
function ‘Distant’
μ_{sp }
Sequencing Priority.
3.2 Decision
Variables
The fuzzy logic variables
may have a membership value of not only o or 1 that is; the degrees of
truth of a statement can range between 0 and 1 and is not constrained
to the two truth values of classic propositional logic.
…... (1)
The fuzzy set theory [3,
11] is based on the extension of the classical definition of the set.
In the classical set theory, each element of a universe X
either belongs to a set A or not, whereas in the fuzzy set
theory, an element belongs to a set A with a certain membership
degree.
Definition:
A fuzzy set
Ain X is defined by: A={( x,μ_{
A}(x))x
∈X } where,
μ_{ A}(x):
X →[0,1] A μ is called the “Membership
function” of A and μ_{
A}(x)is
the “Degree of membership” to which x
belongs to A.
4. Proposed Methodology:
In this paper, the fuzzy methodology
is applied to the scheduling of jobs; the objective is the
determination of an optimal sequence for dynamic job arrivals such
that potentially conflicting priorities are satisfied. The jobshop
scheduling process includes the most important elements in many
manufacturing areas such as:
(A)Due Date (B) Customer Priority (C)
Processing Time.
A. Due Date (d_{i})
Due date is the number
of days at any point, after which customer will arrive at the job shop
expecting his job is completed.
As per the usual method normally, the
due dates is set as 28 days from the date of the order.
The membership functions of ‘Closed’ and ‘Distant’
were assigned depending on the due date allocated.
The fuzzy sets C = Close and D = Distant membership functions.
Hence given the universe
U = [∞, 28 days] € Z.
Membership function can
be defined as:
Membership of ‘Close’
Membership of ‘Distant’
μ_{c}(x)=1.0x ≤
0 μ_{d} (x) = 0 x ≤ 7
μ_{c}(x)=1.0x/10
0< x<10 μ_{d} (x) = x/14 – 0.50 7 < x < 21
μ_{c}(x)=0 x ≥ 10
μ_{d} (x) = 1.0 21 ≤ x ≤
28
The selection of a ‘trapezoidal’
form of membership function for ‘close’ is based on the assumption
that the critically the closeness of an impending due date increases
linearly with time up to the point at which the job is ‘late’. The
‘distant’ function represents to avoid too early completion causing
stock holding problem.
B. Customer Priority
(μ_{CP})
The customer
priority is based on the importance of a customer and based on the
fuzzy values. The customer priority CP contains the five different
fuzzy values. The customer priority fuzzy values are given below.
Customer Priority (C_{P})
= {Bad, Low, Medium, High, Very High}
μ_{cp} 
0.0 
0.25 
0.50 
0.75 
1.0 
Cp 
Bad 
Low 
Medium 
High 
Very High 
C.Processing
Time (p_{i})
“Depending
upon the processing time assigned to one particular job three fuzzy
functions were assigned, for fuzzy set such
as Short, Medium, and long”.
It is the time required
to process job (j) on any machine. The processing time (P_{j}),
will normally include both actual processing time and setup time.
It is assumed that at
the time of scheduling the exact processing times are unknown.
a)
Sequence Priority Rules (μ_{sp})
The sequencing priority
is defined as:
μ_{sp}(c, d) =
Min {μ_{c},
μ_{cp}}
∨ Min{μ_{cp},
μ_{d}},
..(2)
DueDate
Cust. Priority 
*‘CLOSE’ 
‘DISTANT’ 
Bad (B) 
Reject 
Reject 
Low (L) 
Seq.Quite High 
Seq.V.Low 
* Medium (M) 
*Seq.High 
Seq. Low 
High (H) 
Seq.V.High 
Seq.Quite Low 
Very Important
(VI) 
Seq. Extremely
High 
Seq.Medium 
5. Exact Solution Flow Chart Approach:
·
An Illustrative
Example
Let us
consider a single tailor machine jobshop Problem, where data about
six pending jobs given in a single machine.
Table1: Single Machine Six
Jobs waiting to be processed.
No. of Jobs 
Safari Suit(1) 
Repair work (2) 
Three piece Suit (3) 
Two piece Suit (4) 
Shirt Trouser(5) 
Ladies Suit(6) 
Due Date 
0 
10 
6 
28 
26 
14 
Processing Time 
5 
1 
8 
6 
2 
4 
Cust.Priority 
M 
L 
VH 
VH 
H 
L 
FuzzyCP 
0.5 
0.25 
1.0 
1.0 
0.75 
0.25 
Fuzzy C 
1.0 
0.0 
0.4 
0.0 
0.0 
0.0 
Fuzzy D 
0.0 
0.21 
0.0 
1.0 
1.0 
0.5 
MaxMin 
Close 
Distant 
Close 
Distant 
Distant 
Distant 
Sequence
μ_{sp} 
H 
VL 
EH 
M 
QL 
VL 
Step1 Lets Job 1:
Calculate and compare the fuzzy value
of ‘customer priority’
with ‘close’, ‘distant’(As per
mentioned flow chart steps).
Min {μ_{C},
μ_{CP})
∨Min {μ_{CP},
μ_{D}}
*
CP=
0.5 (Customer Priority is Medium)
C=
1.0 (Membership of ‘Close’)
D=
0.0 (Membership of ‘Distant’)
(Min {1.0, 0.5} = 0.5)∨
(Min {0.5, 0.0} = 0.0)
Max {0.5,
0.0} = 0.5 ‘Close’
Thus: Medium
and Close, Sequence:
High.
(According to sequencing priority
rules).
Similarly we can calculate the
priorities values of other jobs as will, which are listed in Table 1
and from that, it is quite clear that Job 3 should be processes
first, as it has the highest sequencing priority, is
processed–duration 8 days. The sequencing
priority comes out to be :3 > 1 > 4 > 5 > 2 >6.
Repeat all the tasks (Jobs) in step 1,
for the remaining five jobs.
No. of Jobs 
Safari
Suit (1) 
Repair work (2) 
Two piece Suit (4) 
Shirt Trouser(5) 
Ladies Suit(6) 
Due Date 
8 
2 
20 
18 
6 
Process Time 
5 
1 
6 
2 
4 
Cust.Priority 
M 
L 
VH 
H 
L 
Fuzzy CP 
0.5 
0.25 
1.0 
0.75 
0.25 
Fuzzy DDC 
1.0 
0.8 
0.0 
0.0 
0.4 
Fuzzy DDD 
0.0 
0.0 
0.93 
0.79 
0.0 
MaxMin 
Close 
Close 
Distant 
Distant 
Close 
Sequence 
H 
QH 
M 
QL 
QL 
The current sequencing priority is
given by: 1 > 2 > 6 > 4 > 5. Thus Job1 is processedduration
5 days.
Similarly we draw tables and calculate
sequence priority every time and after whole process, the final
sequence priority (μ_{sp})
comes to be:
Sequence Priority (μ_{sp})
= 3 >1 > 2 > 6 > 5 > 4.
Three Piece Suit > Safari Suit >
Repair Work > Ladies Suit> Shirt Trouser > Two Piece Suit.
Overall results can be summarized
as:
Job 
ThreePiece Suit (3) 
Safari Suit(1) 
Repair work(2) 
Ladies Suit (6) 
Shirt Trouser(5) 
Two Piece Suit(6) 
Cust. Priority 
VH 
M 
L 
L 
H 
VH 
Due Date 
6 
0 
10 
14 
26 
28 
Process Time 
8 
5 
1 
4 
2 
6 
Start Time 
0 
8 
13 
14 
18 
20 
Completion Time 
8 
13 
14 
18 
20 
26 
Lateness 
2 
13 
4 
4 
6 
2 
The dynamic process can be summarized
by considering the sequence priority each time the machine becomes
available:
3 > 1 > 4 > 5 > 2 >6 –Job 3 be
Processed
1 > 2 > 6 > 4 >5 – Job 1 be
Processed
2 > 6 > 4 >5 – Job 2 be
Processed
6 > 4 >5 – Job 6 be
Processed
5 >4 – Job 5
be processed.
6.
Concluding Remarks:
In this paper, a theoretical model has
been presented which demonstrates how fuzzy decision making can
support the dynamic scheduling process. With the objective to
“Minimize the overall lateness (Tardiness)” and “Maximize the customer
satisfaction” of the overall jobs has been formulated, and the optimal
sequence for jobs with fuzzy due dates (d_{i}), fuzzy customer
priority (Cp) and fuzzy Processing time (p_{i}) has been
determined.
A realworld jobshop scheduling
problem defined in tailor JSSP has been analyzed and solved using the
proposed method and exact solution flow chart of (JSSP) approach. It
is shown that the method is very flexible in different aspects and can
be applied in treating reallife scheduling problems. It can be
applied to a problem with any number of jobs, machines &objectives to
be optimsed.
Finally, the proposed method enables
the DM to express performances regarding multiple objectives using
natural language expressions and linguistic variables.
References:
[1]
D.R. Sule, Industrial Scheduling, PWC Publishing Company, 1997.
[2]
R. TavakkoliMoghaddam, G. Moslehi, M. Vasei and A. Azaron,
“Optimal Scheduling for a Single Machine to Minimize the Sum of
Maximum Earliness and Tardiness Considering Idle Insert”, Applied
Mathematics &Computation, Vol. 167, No. 2, pp. 14301450, 2005.
[3]
L.A Zadeh, Fuzzy sets,“Information and Control”,Vol. 8, pp.
338353, 1965.
[4]
R.E. Bellman and L.A. Zadeh, “Decision Making in a Fuzzy
Environment”, Management Sciences, Vol.17,pp. 141164, 1970.
[5]
H.J. Zimmermann, “Description and Optimization of Fuzzy
Systems”, International Journal of General Systems, Vol. 2, pp.
209215, 1976.
[6]
H.J. Zimmermann, “Fuzzy Programming and Linear Programming with
Several Objective Functions”, Fuzzy Sets and Systems, Vol. 1,pp.
4556, 1978.
[7]
E.L. Hannan, “Linear Programming with Multiple Goals”, Fuzzy
Sets & System, Vol.6,pp.235164, 1981.
[8]
R.R. Yager, “Multiple Objective DecisionMaking Using Fuzzy
Sets”, International Journal of ManMachine Studies, Vol. 9,pp.
375382, 1977.
[9]
K. R. Baker, “Introduction to Sequencing and Scheduling”, New
York: Wiley, 1974.
[10]
R. Bellman, A. Esogbue, and I. Nabeshima, “Mathematical Aspects
of Scheduling and Applications”,Oxford, U.K.: Pergamon, 1982.
Author
Biographies:
Mr. Vikas S. Jadhav
Obtained B.Sc. degree in
the year 2006. He received the Master (M.Sc.) degree in Statistics from
the Dr. B.A.M.University, Aurangabad (M.S.) India in the year 2009. He
is currently doing Ph.D.in Statistics, Under the Guidance of Dr. V. H.
Bajaj. Titled “A Study on Job Scheduling Problem using Fuzzy Approach”.
(O.R). He has Published 4 Research Papers in various International
Journals in the relevant Ph.D. topic.
Dr.V.H.Bajaj
Professor and Head Department of
Statistics, Dr.
B.A.M.University, Aurangabad (M.S.) India.
He has guided more than 23 Ph.D. Students in the subject of Statistics,
Computer Science. He has Published 55 Research Papers in various
National and International Journals with impact factor and ISSN No. His
areas of interests include Fuzzy Sets Theory and its Applications,
Operations Research, Industrial Statistics, Data Mining and Data
Warehousing, Data Envelopment Analysis.
