A Hybrid Evolutionary Optimization Model for Solving Job Shop Scheduling Problem using GA and SA

IOSR Journal of Computer Engineering (IOSR-JCE)
e-ISSN: 2278-0661,p-ISSN: 2278-8727, Volume 17, Issue 6, Ver. I (Nov – Dec. 2015), PP 16-24
www.iosrjournals.org
DOI: 10.9790/0661-17611624 www.iosrjournals.org 16 | Page
A Hybrid Evolutionary Optimization Model for Solving Job
Shop Scheduling Problem using GA and SA
Dr. S. Jayasankari
Assistant Professor, Department of Computer Science,
P.K.R. Arts College for Women, Gobichettipalayam, Tamilnadu, India
Email id: jai.nivi.ravi@gmail.com
Abstract: The heuristic optimization techniques were commonly used in solving several optimization
problems. The present work aims to develop a hybrid algorithm to solve the scheduling optimization problem of
JSSP. There are different variants of these algorithms that were addressed in several previous works. The
impacts of these two kinds (Genetic Algorithm (GA) and Simulated Annealing (SA) based optimization model)
of initial condition on the performance of these two algorithms were studied using the convergence curve and
the achieved makespan. Even though genetic algorithm performed better than other evolutionary algorithms, it
has some weakness. During running GA, sometimes, it will produce same result without any improvement. SA
has a mechanism to overcome from that situation. During SA, if same result will be repeated, then it is rapidly
changing the change in temperature variable and re-initiates another random search. By using this feature of
SA, it has been implemented a hybrid based evolutionary model for solving JSSP by improving GA.
Comparison has been made with the performance of the proposed SA-GA-Hybrid model with GA as well as SA.
Keywords: Job Shop Scheduling Problem, Optimization, Genetic Algorithm, Simulated Annealing.
I. Introduction
Scheduling is an optimization method to make the best possible way to use the limited resources by
making a suitable allocation of the constraint resources over a period of time. Those problems are combinatorial
optimization problems, where a set of possible solutions is distinct or can be reduced to a discrete one, and the
goal is to find the best possible solution. Scheduling falls under the NP-hard class of combinatorial optimization
problems category. Developing of job processing schedules in a scheduling environment is a very difficult task.
The number of possible schedules increases with the increase in the number of jobs and related operations. This
growth makes it practically impossible to use mathematical programming or thorough search-based approaches
to find the global optimum schedules.
In the recent competitive atmosphere in manufacturing and service industries, the operational
sequencing and scheduling has become a crucial for survival in the marketplace. Companies have to produce
their product advance as opposed to due date. Otherwise, it will affect upon reputation of a business. At the same
time, the activities and operations need to be scheduled with the intention that the available resources will be
used in an efficient manner. As a result, there is a great good scheduling algorithm and heuristics are invented.
Most of the principal practical scheduling problems exist in stochastic and dynamic environment.
Stochastic is a problem in that some of the variables are undefined while dynamic problem is when jobs
arrive randomly. In another way, the problems with ready time is known and fixed are known problems such as
static and for problem where all the parameter such as processing times are known and fixed is called
deterministic problems. Regardless of this, it is impossible to predict exactly when jobs will become available for
processing.
The core objective in solving the job shop scheduling problem is to find the sequence for each operation
on each machine that optimizes the objective function. The objective function that has been used in scheduling
the job shop problem is minimization of makespan value or the time to complete all jobs.
1.1 Problem Definition
The Job Shop Scheduling Problem (JSSP) may be described as follows: Given „n‟ jobs, each composed
of several operations that must be processed on „m‟ machines. Each operation uses one of the „m‟ machines for a
fixed duration. Each machine can process at most one operation at a time and once an operation initiates
processing on a given machine it must complete processing on that machine without interruption. The
operations of a given job have to be processed in a given order. The problem consists in finding a
schedule of the operations on the machines, taking into account the precedence constraints, that
minimizes the makespan (Cmax), that is, the finish time of the last operation completed in the schedule.
The focus on job-shop scheduling problems composed of the following elements:

A Hybrid Evolutionary Optimization model for Solving Job Shop Scheduling Problem using GA…
 Jobs: J = {J1 , · · · , Jn } is a set of n jobs to be scheduled. Each job Ji consists of a predetermined sequence
of operations. Oi,j is the operation j of Ji. All jobs are released at time 0.
 Machines: M = {M1 , · · · , Mm } is a set of m machines. Each machine can process only one operation at a
time. And each operation can be processed without interruption during its performance on one of the set of
machines. All machines are available at time 0.
 Constraints: The constraints are rules that limit the possible assignments of the operations. They can be
divided mainly into following situations:
- Each operation can be processed by only one machine at a time (disjunctive constraint).
- Each operation, which has started, runs to completion (non-preemption condition).
- Each machine performs operations one after another (capacity constraint).
- Although there are no precedence constraints among operations of different jobs, the predetermined
sequence of operation for each job forces each operation to be scheduled after all predecessor operations
(precedence/conjunctive constraint).
- The machine constraints emphasize the operations can be processed only by the machine from the given
set (resource constraint).
 Objective: The objective is to find a schedule that has minimum time required to complete all operations
(minimum makespan).
1.2 Different Approaches for Solving Scheduling Problems
Job shop scheduling problem is a NP-hard problem with no easy solution. Branch-and-bound, Tabu
search, and biologically inspired approaches such as GA, Swarm Intelligence and other stochastic model such as
Simulated Annealing algorithm were proposed for achieving possible solutions to complex problems such as job
shop scheduling.
SA is a stochastic heuristic algorithm which is used to resolve combinatorial optimization problems.
Simulated Annealing optimization is analogous to the annealing of metals. This Simulated Annealing is different
from other algorithms, such that SA uses a probability mechanism to have power over the process of jumping out
of the local minimum. In the process of searching, SA is not only accepting better solutions, but also accepting
worse solutions randomly with a certain probability. At high temperatures, the probability of accepting better
solutions is comparatively big. With the decrease of the temperature, the chance of accepting worse solutions also
move downs, and when the temperature closes in upon zero, SA no longer accepts any worse solution. These
make SA have more chance to avoid getting trapped in a local minimum and avoid the limitation of other local
search algorithms and the gradient algorithms. Because of its qualities above, SA has become an enormously
accepted method for solving large-sized and practical problems like job shop scheduling, timetabling, traveling
salesman and packing problem. However, like many other search algorithms, SA may get trapped in a local
minimum or take a long time to find a reasonable solution. For these reasons, SA is often used as a part of a
hybrid method.
II. The JSSP And The Hybrid Evolutionary Model For Solving JSSP
Let J = {0, 1, …, n, n+1} denote the set of operations to be scheduled and M = {1,..., m} the set of
machines. The operations 0 and n+1 are dummy, have no duration and represent the initial and final
operations. The operations are interrelated by two kinds of constraints. First, the precedence constraints, which
force each operation j to be scheduled after all predecessor operations, Pj, are completed. Second,
operation j can only be scheduled if the machine it requires is idle. Further, let dj denote the (fixed)
duration (processing time) of operation j. Let Fj represent the finish time of operation j. A schedule can be
represented by a vector of finish times (F1, , Fm, ... , Fn+1). Let A(t) be the set of operations being processed at
time t, and let rj,m = 1 if operation j requires machine m to be processed and rj,m = 0 otherwise.
The conceptual model of the JSP can be described the following way (Goncalvesetal 2005):
Minimize Fn+1 (Cmax) ……….(1)
Subject to:
Fk ≤ Fj – dj j-1,…,n+2 ; k ϵ Pj ……….(2)
 ( ) ,jA t j m
r ≤ 1 m ϵ M ; t ≥ 0 ……….(3)

Fj ≥ 0 j=1,….,n+1 ……….(4)
The objective function (1) minimizes the finish time of operation n+1 (the last operation), and
therefore minimizes the makespan. Constraints (2) impose the precedence relations between operations
and constraints (3) state that one machine can only process one operation at a time. Finally (4) forces the finish
times to be non- negative. The JSP is amongst the hardest combinatorial optimization problems. The JSP is NP-
hard (Lenstra and Rinnooy Kan, 1979), and has also proven to be computationally challenging.
2.1 Simulated Annealing
Simulated Annealing (SA) is motivated by an analogy to annealing in solids. The idea of SA comes
from a paper published by Metropolis etc al in 1953 [Metropolis, 1953]. The algorithm in this paper simulated
the cooling of material in a heat bath. This is a process known as annealing. If a solid is heated up to the melting
point of that metal and then cool it, the structural properties of the solid depend on the rate of cooling. If the liquid
metal is cooled very slowly, large crystals will be formed. However, if the liquid is cooled quickly the crystals
will contain imperfections. Metropolis's algorithm simulated the material as a system of particles. This algorithm
simulates the cooling process by slowly lowering the temperature of the system until it converges to a steady,
frozen state. In 1982, Kirkpatrick et al (Kirkpatrick, 1983) took the idea of the Metropolis algorithm and applied
it to optimization problems. The idea is to use simulated annealing to search for feasible solutions and converge
to an optimal solution.
Simulated annealing refers to the annealing process done on a computer by simulation. In this model, a
parameter T, equivalent to temperature in annealing, is reduced slowly. The law of thermodynamics state that at
temperature, t, the probability of an increase in energy of magnitude, δE, is given by (Kirkpatrick et al 1983):
P(δE) = exp(-δE /kt)
Where k is a constant known Boltzmann's constant.
The simulation in the Metropolis algorithm calculates the new energy of the system. If the energy has
decreased then the systems move to this state. If the energy has increased then the new state is established using
the probability returned by the above formula. A certain number of iterations are carried out at each temperature
and then the temperature is decreased. This is repeated until the system freezes into a steady state.
The following equation is directly used in simulated annealing, although it is usual to drop the Boltzmann
constant as this was only introduced into the equation to cope with different materials. Therefore, the probability
of accepting a worse state is given by the equation (Kirkpatrick et al 1983):
P = exp(-c/t) > r
Where
c = the change in the evaluation function
t = the current temperature
r = a random number between 0 and 1
The probability of accepting a worse move is a function of both the temperature of the system and of the
change in the cost function. It can be valued that as the temperature of the system decreases the probability of
accepting a worse move is decreased. This is the same as slowly moving to a frozen state in physical annealing.
Also note, that if the temperature is zero then only better moves will be accepted which effectively makes
simulated annealing act like hill climbing. The following algorithm is mentioned by Russell, 1995.
Function SIMULATED-ANNEALING(Problem, Schedule) returns a solution state
Inputs :Problem, a problem
Schedule, a mapping from time to temperature
Local Variables :Current, a node
Next, a node
T, a “temperature” controlling the probability of downward steps
Current = MAKE-NODE(INITIAL-STATE[Problem])
For t = 1 to  do
T = Schedule[t]
If Termination Condition then
return Current
Next = a randomly selected successor of Current
E = fittness[Next] – fittness[Current]
if E > 0 then
Current = Next

else if (exp(-E/T) > probability then
Current = Next
2.2 Genetic Algorithm (GA)
Genetic algorithms are adaptive methods, which may be used to solve search and optimization
problems (Beasley et al. (1993)). They are based on the genetic process of biological organisms. Over many
generations, natural populations evolve according to the principles of natural selection, i.e. Survival of the fittest,
first clearly stated by Charles Darwin in The Origin of Species. By simulating this process, genetic algorithms
are able to evolve solutions to real world problems.
Before a genetic algorithm can be run, an appropriate encoding (or representation) for the problem must
be developed. A fitness function is also required, which assigns a figure of merit to each encoded solution.
During the run, parents must be selected for reproduction, and recombined to generate offspring. It is
assumed that a solution to a problem may be represented as a set of parameters. These parameters
(known as genes) are coupled together to form a string of values (chromosome). In genetic terminology, the set
of parameters represented by a particular chromosome is referred to as an individual. The fitness of an
individual depends on its chromosome and is evaluated by the fitness function.
The individuals, during the reproductive phase, are selected from the population and recombined,
producing offspring, which involve the next generation. Parents are randomly selected from the population
using a scheme, which favors fitter individuals. Having selected two parents, their chromosomes are
recombined, typically using mechanisms of crossover and mutation. Mutation is usually applied to some
individuals, to guarantee population diversity. The basic genetic pseudo code is as follows.
Genetic Algorithm
{
Generate initial population Pt
Evaluate population Pt
While stopping criteria not satisfied Repeat
{
Select elements from Pt to copy into Pt+l
Crossover elements of Pt and put into Pt+l
Mutation elements of Pt and put into Pt+l
Evaluate new population Pt+l
Pt = Pt+l
}
}
2.3 Proposed Hybrid Evolutionary Optimization Model
The following steps explains the Proposed SA-GA-Hybrid Model for solving JSSP
1. Initialize the solution X of size [p x (rxc)] (where r x c is the size of the JSSP and p is the population size)
2. For Each Generation Repeat 3 to 6
3. Xmin  X1 ; Fmin  Fittness(X1) ; Current  X1
4. Do GA based perturbation on X and produce Xnew
T = Schedule[t]
If SA_Termination_Condition then
Go to 8
Xnew= GeneticOperationsOn(X)
5. Calculate Fitness of Xnew
Fnew= Fittness(Xnew)
Where
Fnew be the p x 1 matrix which will contain the
individual fitness values.
6. Main SA based Check
[FSorted, Index]  sort(Fnew) ;
Next  FSorted (top)
E = fittness[Next] – fittness[Current]
if E > 0 then
Current Next
else if (exp(-E/T) > probability then

Current  Next
7. X(Index (top))  Current
go to step 4
8. Finally return the results
Makespan  fittness[Current]
BestSolution  Current
Stop.
The following is an example for a 3 x 3 JSSP.
Table 1 : A 3x3 JSSP
Job Operations routing (processing time)
1 1(3) 2(3) 3(3)
2 1(2) 3(3) 2(4)
3 2(3) 1(2) 3(1)
The one of the optimum schedule of the above problem is [ J1,1, J3,1, J2,1, J1,2, J3,2, J2,2, J2,3, J1,3, J3,3 ]. Here, for
example, J1,2 represents the operation 2 of the Job 1. Figure1shows one of such a optimum solution for the
problem represented by “Gantt-Chart".
Figure 1 : The Gantt-Chart Representation of the Solution of the above 3x3 Problem
If we denote the operations of the job as follows,
Job1: Op1, Op2, Op3
Job2: Op4, Op5, Op6
Job3: Op7, Op8, Op9
Or simply
1 2 3
4 5 6
7 8 9
then, the schedule [ J1,1, J1,2, J1,3, J2,1, J2,2, J2,3, J3,1, J3,2, J3,3] or simply [1, 2, 3, 4, 5, 6, 7, 8, 9] will be
the one of the known worst case schedule which will satisfy all the conditions of the JSSP. But in this case, the
makespan will not be optimum. The schedule [ J1,1, J3,1, J2,1, J1,2, J3,2, J2,2, J2,3, J1,3, J3,3 ] or simply [1, 7, 4, 2, 8, 5,
6, 3, 9] will be the one of the best know optimum solution. Here the strings “1, 2, 3, 4, 5, 6, 7, 8, 9” and “1, 7, 4,
2, 8, 5, 6, 3, 9” represents solutions and known as valid strings.
In GA, a legal string or a illegal string (of numbers) which represent the order of the schedule can be
represented by a chromosome. For example, the known worst case solution can be represented as a chromosome
of GA by a string “1, 2, 3, 4, 5, 6, 7, 8, 9”. Similarly, the chromosome of GA “1, 7, 4, 2, 8, 5, 6, 3, 9” will
represent a legal string which is an optimal solution of JSSP. And for example, the chromosome “3, 9, 4, 2, 1, 5,
6, 7, 8” will be a invalid string which correspond to a illegal operation or schedule since this schedule will not
satisfy the conditions of JSSP.
So, if the initial selection of chromosomes of GA with random values, then there will be lot of invalid
strings in the initial guess. The scope of the evolutionary algorithm is to permute the most optimal string to better
most optimal string which will hopefully make that string as a legal string in proceeding generations/steps and
finally it will end up with a string belongs to a better solution or optimal schedule with minimum makespan.
This assumption will be good and can produce meaningful solutions for lower order scheduling
problems such as 3x3 JSSP or 4x4 JSSP. But, it may produce illegal solutions even after very long runs in the
case of higher order scheduling problems like 15x15 JSSP. Because, if we randomly chose initial population then
there will be much chance for getting all illegal strings in the initial set which belongs to no nearby solution. So
the fitness calculation methods will lead to meaningless fitness values and the selection method will also be
incapable of selecting a better solution in each generation or step. So in each step of the evolutionary process
there will not be any guaranty of getting progressive solution.

So we believe that the random selection of initial solution/seed in an evolutionary algorithm will not lead
to a better result in higher order scheduling problems such as 15x15 JSSP. So in this work, we have evaluated the
performance of two evolutionary optimization techniques SA and GA with different initial conditions.
III. Results And Analysis
In the experiments, we have given the known worst case solution as initial “seed” for the evolutionary
process. We expect that, the system will be capable of producing at latest one meaningful legal string in every
generation/step/iteration and hence there will be a much good probability of achieving a better solution in the
succeeding generations or steps.
3.1 Analysis of Performance with Different Problem Size
The GA was run for 100 generations with population size of 100. The SA was run for 3000 iterations
since this simple SA will only handle one solution at a time.
The following figure shows a result (makespan=68) found by the GA based method for a 6x6 problem.
Figure 2 : Solution found by GA for a 6x6 Problem.
The following figure show a s result (makespan=77) found by the SA based method for a 6x6 problem.
Figure 3 : Solution found by SA for a 6x6 Problem
The following figure shows a result (makespan=62) found by the SA-GA-Hybrid based method for the same
6x6 problem.
Figure 4 : A Solutions found by SA-GA-Hybrid for a 6x6 Problem .
The following figure shows another result (makespan=61) found by the SA-GA-Hybrid based method for the
same 6x6 problem.

Figure 5: Another Solutions found by SA-GA-Hybrid for a 6x6 Problem
The Following fugures show the average fitness of the two methods over generations. If we see these plots, we
can realize that both the algorithms behaves in a similar fashion.
Figure 6 : The Average Fitness Plot of the Solutions found by GA and SA-GA-Hybrid algorithm for a 6x6
Problem .
The Following fugures show the best fitness of the two methods over generations. If we see these plots, we can
realize that proposed SA-GA-Hybrid algorithm performes little bit better than the normal GA .
Figure 7 : The Best Fitness Plot of the Solutions found by GA and SA-GA-Hybrid algorithm for a 6x6 Problem
3.2 Performance in terms of speed

To measure the performance in terms of speed, the model was run with problems of different sizes and the results
were tabulated as follows:
Table 2 : The time taken for different JSSP size
Sl.No JSSP Size
Time Taken(sec)
GA SA
SA-GA
Hybrid
1 3x3 2.87 8.44 2.86
2 4x4 3.61 9.14 3.47
3 6x6 5.70 10.51 5.61
4 10x10 16.75 12.72 16.85
5 15x15 58.28 19.82 55.73
The following figure shows the performance in terms of time with respect to different JSSP problem
size. If we carefully see the lines of GA and SA-GA Hybrid, then we can say that the performance of SA-GA
Hybrid is little bit better than GA in some cases and almost equal in some cases.
Figure 8 : JSSP Size vs Time Chart
3.3 Convergence Capability of the Algorithms
The convergence was measured in terms of makespan for different size of the problems was tabulated
below. We only considered the convergence up to the first 100 iterations in the case of GA and SA-GA-Hybrid
model during its run.
To measure the performance in terms of convergence, the model was tested with problems of different
sizes and the results were tabulated as follows:
Table 3 : Startup with known worst case solution
Sl.
No
JSSP
Achieved Optimal Solution
(Makespan)
Size
Known
optimum
value
GA SA
SA-GA
Hybrid
1 3x3 12 12 12 12
2 4x4 272 272 286 272
3 6x6 55 68 72 61
4 10x10 902 2214 2899 2011
5 15x15 1268 6771 8241 6637
The following figure shows the performance in terms of Makespan with respect to different JSSP problem size.
J S S P S ize vs C o n ve rg e n c e in te rm s o f M a k e s p a n
0
1 0 0 0
2 0 0 0
3 0 0 0
4 0 0 0
5 0 0 0
6 0 0 0
7 0 0 0
8 0 0 0
9 0 0 0
1 0 0 0 0
3 x3 4 x4 6 x6 1 0 x1 0 1 5 x1 5
J S S P S ize
Makespan.
G A S A S A -G A H yb rid
Figure 9: JSSP Size Vs Makespan Chart.

Even though they arrived solution is far from the optimal solution, the better performance in the case of
SA-GA-Hybrid is very obvious.
IV. Conclusion And Future Work
The algorithms have successfully implemented two basic evolutionary models for solving JSSP using
GA & SA, and also the proposed SA-GA-Hybrid. They arrived results showing that the models produced optimal
or near-optimal solutions medium level job shop scheduling problems in a shot duration. While initializing with
known, worst case solution, the evolutionary process was capable of converging into meaningful and more
optimum solutions. Further, as shown in the convergence curves in the previous section, The SA-GA-Hybrid
performed in a very better way than GA. Future works may address the issued involved in improving the
performance of the SA-GA-Hybrid for high dimensional problems.
Acknowledgement
My sincere thanks to the management and principal of P.K.R. Arts College for Women,
Gobichettipalayam and friends, who have motivated & helped me towards the topic and development of this
research paper.
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