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Hybrid Genetic Algorithm for
Jobshop Production Problem
CHAPTER No. 01
INTRODUCTION
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Hybrid Genetic Algorithm for
Jobshop Production Problem
INTRODUCTION
“Scheduling is broadly defined as the process of assigning a set of tasks to
resources over a period of time” (Pinedo, 2001). It can also be termed as an allocation of
the operations to time intervals on the machines.
“Scheduling is the allocation of resources over time to perform a collection of
tasks… Scheduling is a decision making function: it is the process of determining a
schedule…Scheduling is a body of theory: it is a collection of principles, models,
techniques and logical conclusions that provide insight to the scheduling function. ”
(Baker, 1974)
Manufacturing industries are the backbone in the economic structure of a nation,
as they contribute to both increasing GDP/GNP and providing employment. Productivity,
which directly affects the growth of GDP, and benefits from a manufacturing system, can
be maximized if the available resources are utilized in an optimized manner. Optimized
utilization of resources can only be possible if there is proper scheduling system in place.
This makes scheduling a highly important aspect of a manufacturing system.
Effective scheduling plays a very important role in today’s competitive
manufacturing world. Performance criteria such as machine utilization, manufacturing
lead times, inventory costs, meeting due dates, customer satisfaction, and quality of
products are all dependent on how efficiently the jobs are scheduled in the system.
Hence, it becomes increasingly important to develop effective scheduling approaches that
help in achieving the desired objectives.
Several types of manufacturing shop configurations exist in real world. Based on
the method of meeting customer’s requirements they are classified as either open or
closed shops. In an open shop the products are built to order where as in a closed shop the
demand is met with existing inventory. Based on the complexity of the process, the shops
are classified as single-stage, single-machine, parallel machine, multi-stage flow shop
and multi-stage job shop. The single-stage shop configurations require only one operation
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Hybrid Genetic Algorithm for
Jobshop Production Problem
to be performed on the machines. In multi-stage flow shops, several tasks are performed
for each job and there exists a common route for every job. In multi-stage job shops, an
option of selecting alternative resource sets and routes for the given jobs is provided.
Hence the job shop allows flexibility in producing a variety of parts. The
processing complexity increases as we move from single stage shops to job shops.
Various methods have been developed to solve the different types of scheduling problems
in these different shop configurations for the different objectives. These range from
conventional methods such as mathematical programming & priority rules to meta-
heuristic and artificial intelligence-based methods.
Job shop scheduling is one of the widely studied and most complex combinatorial
optimization problems. The JSSP is not only very hard, but it is one of the worst
members in the class.
An indication of this is given by the fact that one 10 X 10 problem formulated by
Muth and Thompson remained unsolved for over 20 years.
A vast amount of research has been performed in this particular area to effectively
schedule jobs for various objectives. A large number of small to medium companies still
operate as job shops. Despite the extensive research carried out it appeared that many
such companies continue to experience difficulties with their specific JSSP. Therefore
developing effective scheduling methods that can provide good schedules with less
computational time is still a requirement. Most of the real world manufacturing
companies aim at successfully meeting the customer needs while improving the
performance efficiency.
Informally, the problem can be described as follow, that we are given a set of jobs
and a set of machines. Each job consists of a chain of operations, each of which needs to
be processed during an uninterrupted time period of a given length on a given machine.
Each machine can process at most one operation at a time.
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Hybrid Genetic Algorithm for
Jobshop Production Problem
The objective is to find a schedule of minimum makespan, determining a
sequence of jobs that optimize designed performance measures such as makespan, mean
flow time and mean utilization. However, the most widely used measure is makespan.
A JSSP consists of ‘m’ machines and ‘n’ jobs. The possible number of solutions
to JSSP can be calculated by the formula (n!)m
. Definitely, each and every solution is not
feasible, and more than one optimal solution may exist. So the number of alternative
solutions grows at a much faster rate than the number of jobs and the number of
machines, thus, it is infeasible to evaluate all solutions (i.e., complete enumeration) even
for a reasonable sized practical JSSP. In the earlier days of solving JSSP, Akers et al
(1955) and Friedman, and Giffler et al (1960) and Thompson explored only a subset of
the alternative solutions in order to suggest acceptable schedules. Although such an
approach was computationally expensive, it could solve the problems much quicker than
a human could do at that time. After that, the branch-and-bound (B&B) algorithm was
widely popular for solving JSSPs, using the concept of omitting a subset of solutions
comprising those that were out of bounds. Among them, Carlier and Pinson solved a
10×10 JSSP optimally for the first time, as mentioned above, a problem that was
proposed in 1963 by Muth and Thompson. They considered the n×m JSSP as ‘m’ one-
machine problems and evaluated the best preemptive solution for each machine. Their
algorithm relaxed the constraints in all other machines except the one under
consideration. The concept of converting an ‘m’ machines problem to a one-machine
problem was also used by Emmons and Carlier. As the complexity of this algorithm is
directly dependent on the number of machines that’s why it is not computationally
cheaper for large scale problems.
Although the above algorithms can achieve optimum or near optimum makespan,
they are computationally expensive, remaining out of reach for large problems, even with
current computational powers. For this reason, numerous heuristic and meta-heuristic
approaches have been proposed in the last few decades. These approaches do not
guarantee optimality, but provide a good quality solution within a reasonable period of
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Hybrid Genetic Algorithm for
Jobshop Production Problem
time. Examples of such approaches applied to JSSPs are genetic algorithms (GA), Tabu
search (TS), shifting bottleneck (SB), greedy randomized adaptive search procedure
(GRASP) and simulated annealing (SA). Of all these we chose Genetic Algorithms
(GA).
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Hybrid Genetic Algorithm for
Jobshop Production Problem
CHAPTER No. 02
LITERATURE REVIEW
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Hybrid Genetic Algorithm for
Jobshop Production Problem
LITERATURE REVIEW
INTRODUCTION:
In this chapter we present the work that has been done in the past for solving
scheduling problems.
Scheduling is one of the most widely researched areas of operational research,
which is largely due to the rich variety of different problem types within the field. A
search on the Web of Science for publications with “scheduling” as topic yields over 200
publications for every year since 1996, and 300 publications in 2005, 2006 and 2007.
Arguably, the field of scheduling traces back to the early twentieth century with Gantt
(1916), thus explicitly discusses a scheduling problem. However, it was about forty years
later that a sustained collection of publications on scheduling started to appear.
Nevertheless, scheduling has a long history relative to the lifetime of the main operational
research journals, with several landmark publications appearing in the mid 1950s.
To provide some insight into the development of the field over the time, the
decade- wise view was taken showing development work in this field in each decade.
1950-1959:
Johnson (1954) provided the starting point to scheduling as an independent area
within operational research. He considered the production model now called the flow
shop. Smith (1956) addressed the single machine problem of minimizing the sum of the
completion times, thus introducing a rule known as the Shortest Processing Time rule
(SPT rule). Moreover, in 1956 Smith also introduced a rule for scheduling often referred
to as Smith’s rule or the SWPT rule. McNaughton (1959) studied problems of scheduling
jobs on m identical parallel machines.
McNaughton gave a simple algorithm that finds an optimal preemptive schedule.
In late 1950s Land independently developed the concept of Branch and Bound for
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Hybrid Genetic Algorithm for
Jobshop Production Problem
solving scheduling problems. In late 1958 Eastman also used Branch and Bound
technique for solving Travelling salesman problem (TSP).
1960-1969:
Roy and Sussman (1964) represented the job shop problem through disjunctive
graph formulation. Lomnicki (1965) introduced the concept of flow shop scheduling with
the help of branch and bound method. Further the work was developed by Ignall and
Scharge (1965), providing an algorithm for minimizing the sum of completion times of
the jobs in a two machine flow shop problem. Brooks and White (1965) proposed active
schedule generation branching. McMahon and Burton (1967) introduced a job-based
bound for 3 jobs to be used in combination with the machine-based bound. Nabeshima
(1967) improved the machine-based bound by including any idle time resulting from
processing the operations on the preceding machine. Conway et al (1967) classified the
scheduling environments according to the types of information.
1970-1979:
Held and Karp (1971) used Lagrangean relaxation for TSP. Bruno et al (1974)
showed that the problem with two identical parallel machines is NP hard. Held et al
(1974) used iterative technique known as sub gradient optimization. Lenstra et al (1977)
systematically studied complexity issues for scheduling problems and gave their
classification. Lageweg et al (1978) independently discovered two machine bound.
Miliotis (1976, 1978) used polyhedral approach for solving TSP. Graham et al (1979)
summarized the development in scheduling.
1980-1989:
Crowder and Padberg (1980), Gr¨otschel (1980) and Padberg and Hong (1980)
obtained optimal solutions to instances with up to 100 or more cities in TSP. Hariri and
Potts (1983) applied multiplier adjustment method to TSP. Potts and Van Wassenhove
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Hybrid Genetic Algorithm for
Jobshop Production Problem
(1985) applied mathematical techniques to the problems of scheduling a single machine.
Kirkpatrick et al (1983) and Cerny (1985) proposed simulated annealing as an
optimization technique. Glover (1986) introduced Tabu search. Adams, Balas and
Zawack (1988) proposed Shifting Bottleneck technique. Goldberg (1989) used genetic
algorithms for optimization. Matsuo et al (1989) used transpose neighbourhood within
simulated annealing.
1990-1999:
Falkenauer and Bouffouix (1991) proposed implementation of GA for the JSP
with release time and due dates. Storer et al (1992) used data perturbation and heuristic
set representations in genetic algorithms, together with a hybrid representation.
Falkenauer and Bouffouix (1991), Yamada and Nakano (1992) and Della Croce et al
(1995) designed genetic algorithms based on priority representations. Yamada et al
(1994) used backtracking in simulated annealing algorithm. Smith (1992) and Dorndorf
and Pesch (1995) used heuristic set representation. Dorndorf and Pesch (1995) proposed
two different implementations of GA.
2000-2009:
Potts and Kovalyov (2000), gave a detailed account of the models and results in
scheduling. Zhou and Feng et al (2001) proposed a hybrid heuristic GA for JSSP.
Congram et al (2002) introduced dynasearch as a local method. Schuurman and
Vredeveld (2001) provided worst-case bounds for problems of minimizing the makespan
on parallel machines. Anderson and Potts (2004) showed the competitive ratio of the so-
called delayed Smith’s rule (SWPT). Zhang proposed et al (2005) a genetic simulated
algorithm to solve the JSSP by combining the GA and simulated annealing. Atkin et al
(2007) considered the take-off problem at Heathrow. Chen and Hall (2007) considered
two-stage assembly system where manufacturing is assumed to be a non-bottleneck
operation.
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Hybrid Genetic Algorithm for
Jobshop Production Problem
CONCLUSION:
A major research during the past decades has involved defining the boundary
between polynomially solvable problems and those that are NP-hard. For classical
scheduling problems, this activity is almost complete, with very few problems still having
an open complexity status.
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Hybrid Genetic Algorithm for
Jobshop Production Problem
CHAPTER No. 03
METHODOLOGY
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Hybrid Genetic Algorithm for
Jobshop Production Problem
METHODOLOGY
3.1 INTRODUCTION:
In this chapter, we consider the minimization of makespan as the objective of
JSSP. The Job-Shop Scheduling problem (JSSP) considers a set of jobs to be processed
on a set of machines. Each job is defined by an ordered set of operations and each
operation is assigned to a machine with a predefined constant processing time. The order
of the operation within the jobs and its correspondent machines are fixed and independent
from job to job. To solve the problem we need to find a sequence of operations on each
respecting some constraints and optimizing some objective function. it is assumed that
two consecutive operations of the same job are assigned to different machines, each
machine can only process one operation at a time and different machines cannot process
the same job simultaneously. We will adopt the maximum of the completion time of all
jobs “ MAKESPAN” as the objective function.
The JSSP is a well-known difficult combinatorial optimization problem. Many
algorithms have been proposed for solving JSSP in the last few decades, including
algorithms based on evolutionary techniques.
However, there is room for improvement in solving medium to large scale
problems effectively. We present a HGA that includes a heuristic job ordering with a
Genetic Algorithm. We apply HGA to a number of benchmark problems. It is found that
the algorithm is able to improve the solution obtained by traditional genetic algorithm.
3.2 MATHEMATICAL COMPLEXITY OF FINDING THE
OPTIMAL SEQUENCE:
In Job shop scheduling problem we consider the well-known n×m static problem, in
which n jobs must be processed exactly once on each of m machines. Each job is routed
through each of the m machines in some pre-defined order. The processing of a job on a
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Hybrid Genetic Algorithm for
Jobshop Production Problem
machine is called an operation. An operation must be processed on machine for an
integral duration. Once processing is initiated, an operation cannot be pre-empted, and
concurrency is not allowed.
In Job shop scheduling problems there are “n” number of jobs and “m” number of
machines. Number of possible sequences can be found out by (n!)m
. As it is shown by the
number of sequences is very large and this is not an easy task to calculate the makespan
each of them.
In any job shop, a job passes through a sequence of work centers as specified in its
routing and it may wait for the required resources at those work centers. The total waiting
time of the job in the entire process usually constitutes a major part of production lead
time. This undesirable time is usually large, particularly for job shops with high-mix,
low-volume production. It is not easy to measure the total job waiting time in such shops
because:
1) Jobs with diverse routings are processed simultaneously.
2) The process time of an operation of a job may vary with both job and work
center.
3) Product mix keeps changing frequently.
4) Resources have limited capacity. This complexity makes it difficult to accurately
predict job progress on shop floor.
3.3 PROBLEM DEFINITION:
1. Every job has a unique sequence on m machines. There are no alternate routings.
2. There is only one machine of each type in the shop.
3. Processing times for all jobs are known and constant.
4. All jobs are available for processing at time zero.
5. Transportation time between machines is zero.
6. Each machine can perform only one operation at a time on any job.
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Hybrid Genetic Algorithm for
Jobshop Production Problem
7. An operation of a job can be performed by only one machine.
8. Operation cannot be interrupted.
9. A job does not visit the same machine twice.
10. An operation of a job cannot be performed until its preceding operations are
completed.
11. Each machine is continuously available for production.
12. There is no restriction on queue length for any machine.
13. There are no limiting resources other than machines/workstations.
14. The machines are not identical and perform different operations.
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Hybrid Genetic Algorithm for
Jobshop Production Problem
Solution Approaches for job shop scheduling
problems
Approximate
Constraint
Neural networks
Artificial
intelligence
Priority
dispatch rules
Genetic
Algorithms
Threshold
Algorithms
Tabu Search
Problem space
based methods
Bottleneck based
heuristics
Local search and
meta heuristic
methods
Exact Methods
Branch and bound
techniques
Mathematical
Formulations
Efficient Methods
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Hybrid Genetic Algorithm for
Jobshop Production Problem
3.4 TYPES OF ALGORITHMS FOR SOLVING JOB SHOP
SCHEDULING PROBLEM:
The types of algorithms for Job shop scheduling problems are as follow:
1) Fluid synchronization Algorithm(FSA)
2) Asymptotically optimal Algorithm
3) Rollout Algorithm
4) Genetic Algorithm
3.4.1 FLUID SYNCHRONIZATION ALGORITHM (FSA):
In fluid synchronization algorithm, rounding an optimal solution to a fluid
relaxation in which discrete jobs are replaced with the flow of a continuous fluid, and use
ideas from fair queuing in the area of communication networks in order to ensure that the
discrete schedule is close to the one implied by the fluid relaxation. FSA produces a
schedule with makespan at most Cmax+(I+2)Pmax Jmax.
Where Cmax is the lower bound provided by the fluid relaxation, I is the number
of distinct job types, Jmax is the maximum number of stages of any job-type, and Pmax
is the maximum processing time over all tasks. Computational results based on all
benchmark instances chosen from the OR library when N jobs from each job-type are
present. The results suggest that FSA has a relative error of about 10% for N = 10, 1% for
N = 100, 0.01% for N = 1000. In comparison to eight different dispatch rules that have
similar running times as FSA, FSA clearly dominates them. In comparison to the shifting
bottleneck heuristic whose running time and memory requirements are several orders of
magnitude larger than FSA, the shifting bottleneck heuristic produces better schedules for
small N (up to 10), but fails to provide a solution for larger values of N.
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Hybrid Genetic Algorithm for
Jobshop Production Problem
There are following restrictions on the schedule.
1. The schedule must be non-preemptive. That is, once a machine begins processing
a stage of a job, it must complete that stage before doing anything else.
2. Each machine may work on at most one task at any given time.
3. The stages of each job must be completed in order.
3.4.2 ASYMPTOTICALLY OPTIMAL ALGORITHM:
In computer science, an algorithm is said to be asymptotically optimal if, roughly
speaking, for large inputs it performs at worst a constant factor (independent of the input
size) worse than the best possible algorithm. If the input data have some a
priori properties which can be exploited in construction of algorithms, in addition to
comparisons, then asymptotically faster algorithms may be possible. A consequence of an
algorithm being asymptotically optimal is that, for large enough inputs, no algorithm can
outperform it by more than a constant factor. For this reason, asymptotically optimal
algorithms are often seen as the "end of the line" in research, the attaining of a result that
cannot be dramatically improved upon. Conversely, if an algorithm is not asymptotically
optimal, this implies that as the input grows in size, the algorithm performs increasingly
worse than the best possible algorithm.
3.4.3 ROLLOUT ALGORITHM:
Rollout algorithms for combinatorial optimization developed by Bertsekas et al.
(1997), or the equivalent pilot method developed by Duin and Vob (1999), are
metaheuristic methods aimed at improving solutions of known heuristics. Rollout
algorithms improve the performance of heuristics by sequential application of the
heuristic. Rollout algorithms can be very useful when exact methods are too slow and
solutions obtained by existing heuristics are not good enough.
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Hybrid Genetic Algorithm for
Jobshop Production Problem
3.4.4 GENETIC ALGORITHM (GA):
The GA was first introduced by John Holland (1975). It is a stochastic heuristics,
which encompass semi-random search method whose mechanism is based on the
simplifications of evolutionary process observed in nature. As opposed to many other
optimization methods, GA works with a population of solutions instead of just a single
solution. GA assigns a value to each individual in the population according to a problem-
specific objective function. A survival-of-the-fittest step selects individuals from the old
population. A reproduction step applies operators such as crossover or mutation to those
individuals to produce a new population that is fitter than the previous one. GA is an
optimization method of searching based on evolutionary process. In applying GA, we
have to analyze specific properties of problems and decide on a proper representation, an
objective function, and a construction method of initial population, a genetic operator and
a genetic parameter. The following sub-sections describe in detail how the GA is
developed to solve the JSSP problem.
As genetic algorithm deals with a lot of individuals, it gives different solutions of
a problem. The same case happens for the job-shop scheduling problem. Most of the
time, it improves the quality of solution if the appropriate genetic operators is applied
with appropriate problems like job-shop scheduling within the reasonable period of time
where other methods may take longer.
The term makespan refers to the cumulative time to complete all the operations on
all machines. It is a measure of the time period from the starting time of the first
operation to the ending time of the last operation. The objective of the problem is to find
out a valid schedule that yields the minimum makespan. Sometimes there may be
multiple solutions that have the minimum makespan, but the goal is to find out any one of
them it is not necessary to find all possible optimum solutions.
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Hybrid Genetic Algorithm for
Jobshop Production Problem
3.5 REPRESENTATION:
The first step in constructing the GA is to define an appropriate genetic
representation (coding).We use integer base representation scheme, in which each
chromosomes represent a solution to a problem having length equal to the product of total
number of machines and a total number jobs.
If a problem consists of 3 jobs 3 machines then each number would exist three
times in each solution.
Jobs/machines M1 M2 M3
J1 1 2 3
J2 3 2 1
J3 1 3 2
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Hybrid Genetic Algorithm for
Jobshop Production Problem
SOLUTIONS:
1 2 2 3 2 3 1 1 3
3 3 1 2 1 2 1 3 2
3 2 1 2 1 1 3 3 2
1 2 3 1 3 3 2 2 1
3 1 3 2 2 3 1 2 1
3 1 1 2 3 2 3 2 1
Here, 1 implies operation of job J1, and 2 implies operation of job J2 , and 3
implies operation of job J3. Because there are three operations in each job, it appears the
three times in a chromosome. Such as number 2 being repeated the three times in a
chromosome, it implies three operations of job J2. The first number 2 represents the first
operation of job J2 which processes on the machine 3. The second number 2 represents
the second operation of job J2 which processes on the machine 2, and so on. The
representation for such problem is based on two-row structure, as following:
1 2 2 3 2 3 1 1 3
1 -1 2-1 2-2 3-1 2-3 3-2 1-2 1-3 3-3
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Hybrid Genetic Algorithm for
Jobshop Production Problem
MACHINE SEQUENCE:
1 2 2 1 1 3 2 3 2
3.6 GENETIC OPERATORS:
Crossover
Mutation
Selection
3.6.1 POPULATION INITIALIZTION:
A population is initiated of legal solutions, selected by choosing random input
values. There are no fixed rules for how large the population should be. The answer is
dependent upon the type of problem. For a simple problem with a regular search space
a small population of 40 to 100 will probably be sufficient. For larger more complex
problems and especially those with irregular search space larger populations of 400 or
more are recommended. The clue is diversity – a diverse population, i.e. a large one
will tend to search out niches – in engineering terms that means finding elusive,
difficult to find solutions to problems.
3.6.2 CROSSOVER:
Crossover selects genes from parent chromosomes and creates a new offspring.
Various crossover operators can be used such as single-point crossover, two-point
crossover, partial-mapped crossover (PMX), order-crossover, cycle-crossover and job
based order crossover. We use single point crossover technique. That is to choose
randomly single crossover point and every integer after this point copy from a first parent
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Hybrid Genetic Algorithm for
Jobshop Production Problem
and then every integer before a crossover point copy from a second parent. Keeping in
view that, no number repeats in child chromosome more than the number of machines. In
our algorithm the crossover rate is varied according to the problem. After the crossover
is done, fitness values for the child chromosomes are calculated and the result is
compared with the parent’s fitness values. If the fitness value is better than the parent’s
fitness value, then it replaces the parent otherwise remains same.
Parent 1 3 1 1 2 3 2 3 2 1
Parent 2 1 2 2 3 2 3 1 1 3
Child1 3 1 1 2 3 2 3 1 2
child 2 2 1 1 2 2 3 3 1 3
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Hybrid Genetic Algorithm for
Jobshop Production Problem
3.6.3 MUTATION:
After crossover operation, the string is subjected to mutation operation. The
mutation operation is critical to the success of the GA since it diversifies the search
directions and avoids convergence to local optima. We select a parent, and an operation is
get randomly. It is analogous to biological mutation. Once the children are created during
crossover, the mutation operator is applied to each child. Each gene has a user-specified
mutation probability Mutation operator alters a chromosome locally to create a better
string. We adopted swap mutation procedure, where in each column of the solution two
genes are randomly picked and their values are swapped. Bit wise mutation is performed.
The usefulness of this approach is that it does not produce illegal solution.
We pick chromosomes from population randomly and swap any 2 genes randomly
taking in account that the gene values are not same.
3 2 1 2 1 1 3 3 2
3 2 1 3 1 1 2 3 2
3.6.4 FITNESS FUNCTION:
Fitness function is defined of each chromosome so as to determine which with
reproduce and survive into the next generation. It is relevant to the objective function to
be optimized. The greater the fitness of a chromosome is greater the probability to
survive. In this report, the fitness function is defined as:
Fitness=1/Cmax
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Hybrid Genetic Algorithm for
Jobshop Production Problem
3.6.5 SELECTION:
The selection operator involves randomly choosing members of the population to
enter a mating pool. The operator is carefully formulated to ensure that better members of
the population (with higher fitness) have a greater probability of being selected for
mating, but that worse members of the population still have a small probability of being
selected. Having some probability of choosing worse members is important to ensure that
the search process is global and does not simply converge to the nearest local optimum.
Selection is one of the important aspects of the GA process, and there are several ways
for the selection: some of these are Tournament selection, ranking selection, and
Proportional selection. In the proportional selection a string is selected for the mating
with a probability proportional to its fitness. There are many ways of proportional
selection: the most popular are Roulette Wheel Selection (RWS), Stochastic Reminder
Roulette Wheel Selection (SRRWS), and Stochastic Universal Sampling (SUS). We used
Roulette Wheel Selection (RWS).
ROULETTE WHEEL SELECTION:
Roulette wheel probabilistically selects individuals based on their fitness values
Fi. A real-valued interval, S, is determined as either the sum of the individuals expected
selection.
Probabilities S =∑Pi, where ∑ Pi =
or the sum of the fitness values S=∑Fi over all
the individuals in the current population. Individuals are then mapped one-to-one into
contiguous intervals in the range. The size of each individual interval corresponds to the
fitness value of the associated individual. The circumference of the roulette wheel is the
sum of all fitness values of the individuals. The fittest individual occupies the largest
interval, whereas the least fit have correspondingly smaller intervals within the roulette
wheel. To select an individual, a random number is generated in the interval and the
individual whose segment spans the random number is selected. This process is repeated
until the desired number of individuals has been selected.
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Hybrid Genetic Algorithm for
Jobshop Production Problem
Roulette wheel Selection
Fig No. 1
26
Hybrid Genetic Algorithm for
Jobshop Production Problem
Fig No. 2
Start
J = 0
(Column)
Cmax = 0
I = 0
(Row)
Select
Chrom [ I,j
]
Identify job (x)
Operation (0)
Machine (k)
EST0 = MATK = JAT = 0
CT0 = EST0 + PT0
EST0 = MATK
EST0 = JATK
Cmax = CT0
MATk = 0
JATk = 0
I=jobs
I = i+1
J = j+1
Makespan = Cmax END J<Machs
Is it the first
operation of
job x and also
on Machine k
?
MATK>JATK
CT0 > Cmax
NO
YES
YES
NO
YES NO
NO
NO
YES
YES
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Hybrid Genetic Algorithm for
Jobshop Production Problem
3.7 LOCAL SEARCH HEURISTIC (LSH):
Local search techniques have been proven useful in solving combinatorial
problems. Local search methods are applied to a neighborhood of a current solution. In
each generation, the solution with the minimum makespan value (best of the lot) is
further improved by subjecting it to the LSH. The process of local improvement is started
with the first two genes in the first row of the solution provided by GA, as the best of the
population .These two genes are swapped and after that the solution is decoded and the
corresponding makespan value is determined .if this makespan value is smaller than the
original makespan value of the solution then the change is stored otherwise genes are
reverted back to their original position. Now the same procedure is repeated with the first
and third gene of the same solution in the same row. This process is repeated
continuously until processing for the first gene against all the other genes in the solution
is completed. On completion, the next gene is considered and same process is repeated
.This repetition is kept continued until at least half of the genes are tested against all the
other genes. The reason to keep it down to half of the total number of genes is that by the
time first 50% of the operations are scheduled a trend has developed and the last 50%
follow the same trend and therefore do not affect the makespan value.
Though LSH is very effective but it is helped a great deal by evolution of GA. It
is GA that is responsible to search out a comparatively better solution which after being
subjected to local improvement is converted into an even better one. The possibility
getting trapped in local optimum is remote. It is to be noted that this locally improved
procedure, in each generation, does not replace any solution in the main population and
therefore plays no role in the evolution of GA. In other words the local improvement
procedure and the evolution of GA are kept separate so that the natural evolution of GA
is not affected by local improvement. This prevents GA from getting trapped in local
minimum.
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Hybrid Genetic Algorithm for
Jobshop Production Problem
Fig No. 3
Selection by using
stochastic Universal
Sampling
In case the
resulting solution
is illegal then
repair
Evaluation and
placing back in
population
Mutation Evaluation and
placing back in
population
Decoding &
Calculating
fitness values
Randomly generate
initial pop Gen 0 START
Selection the
chromosomes with the
most minimum
Identify the best
chromosome STOP
LSH
Crossover
Gen 0 + 1
Gen 0
LSH
been
applied
it
prevousl
y
Gen < Gen
Max
YES
YES
NO
YES
NO
NO
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Hybrid Genetic Algorithm for
Jobshop Production Problem
CHAPTER No. 04
RESULTS
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Hybrid Genetic Algorithm for
Jobshop Production Problem
RESULTS
The HGA was implemented in MATLAB version 7.8 (R2009) on a computer
with a 2.4 GHz Intel Core i3 processor, manufactured by Acer. Following table shows the
experimental results in which the population size varies according to the size of problem
and the crossover rate is 0.90, mutation rate 0.80, the maximum generation is 100 and the
maximum number of generations is selected as the stopping criterion. In this process
from one generation to the next generation, the crossover and mutation is repeated until
the maximum number of generations is satisfied.
31
Hybrid Genetic Algorithm for
Jobshop Production Problem
S.
No.
Problem
Size =
Job x
machines
Source
Optimal
Makespan
(OM)
Makespa
n found
(M)
solution
gap =
(M-OM)
Num
of
genes
CPU
time
(sec)
1 FT 6 6 x 6 Fisher and
Thompson, 1963
55 55 0
2 LA 1 10 x 5 S. Lawrence, 1984 666 635 -31
3 LA 2 10 x 5 S. Lawrence, 1984 655 664 9
4 LA 3 10 x 5 S. Lawrence, 1984 597 629 32
5 LA 4 10 x 5 S. Lawrence, 1984 590 590 0
6 LA 5 10 x 5 S. Lawrence, 1984 593 593 0
7 LA 6 15 x 5 S. Lawrence, 1984 926 926 0
8 LA 7 15 x 5 S. Lawrence, 1984 890 890 0
9 LA 8 15 x 5 S. Lawrence, 1984 863 863 0
10 LA 9 15 x 5 S. Lawrence, 1984 951 951 0
11 LA 10 15 x 5 S. Lawrence, 1984 958 958 0
12 LA 11 20 x 5 S. Lawrence, 1984 1222 1222 0
13 LA 12 20 x 5 S. Lawrence, 1984 1039 1039 0
14 LA 13 20 x 5 S. Lawrence, 1984 1222 1150
15 LA 14 20 x 5 S. Lawrence, 1984 1292 1292 0
16 LA 15 20 x 5 S. Lawrence, 1984 1207 1207 0
17 LA 16 10 x 10 S. Lawrence, 1984 945 987
18 LA 17 10 x 10 S. Lawrence, 1984 784 819
19 LA 18 10 x 10 S. Lawrence, 1984 848 898
20 LA 19 10 x 10 S. Lawrence, 1984 842 881
21 LA 20 10 x 10 S. Lawrence, 1984 902 939
22 FT 10 10 x 10 Fisher 1963 930 976 46
Table No. 1
32
Hybrid Genetic Algorithm for
Jobshop Production Problem
FT6 No. of jobs: 6
No. of machines: 6 Optimum Makespan: 55
Makespan found: 55
Chart No. 1
Sequence: 3 2 3 6 6 1 3 1 2 4 2 5 6 2 4 5 3 4 5
4 1 6 3 5 1 2 4 2 3 6 6 1 1 4 5 5
Machine Order:
3 1 2 4 6 5
2 3 5 6 1 4
3 4 6 1 2 5
2 1 3 4 5 6
3 2 5 6 1 4
2 4 6 1 5 3
Process time:
1 3 6 7 3 6 8 5 10 10 10 4 5 4 8 9 1 7 5 5 5 3 8 9 9 3 5 4 3 1 3 3 9 10 4 1
33
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 1 No. of Jobs: 10
No. of machines: 5
Optimum Makespan: 666
Makespan found: 444
Chart No. 2
Sequence: 3 3 4 5 1 3 4 2 4 5 5 1 4 2 5 1 1 2 3 5 2 3 1 2 4
Process time:
21 53 95 55 34
21 52 16 26 71
39 98 42 31 12
83 34 64 19 37
54 43 79 92 62
69 77 87 87 93
38 60 41 24 83
17 49 25 44 98
77 79 43 75 96
Machine Order:
2 1 5 4 3
1 4 5 3 2
4 5 2 3 1
2 1 5 3 4
1 4 3 2 5
2 3 5 1 4
4 5 2 3 1
3 1 2 4 5
4 2 5 1 3
5 4 3 2 1
34
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 2 No. of Jobs: 10
No. of machines: 5
Optimum Makespan: 655
Makespan found: 664
Chart No. 3
Sequence: 2 1 2 6 5 7 5 2 4 3 10 2 6 3 8 7 9 1 10 3 3 5 8 9
10 5 1 2 7 4 6 6 8 4 10 9 4 1 5 7 9 10 6 3 8 1 8 7
4 9
Machine Order: 1 4 2 5 3
2 3 5 1 4
3 2 5 1 4
5 1 4 3 2
2 1 5 4 3
5 2 4 1 3
2 1 3 4 5
5 1 3 2 4
5 3 2 4 1
Process time: 20 87 31 76 17
25 32 24 18 81
72 23 28 58 99
86 76 97 45 90
27 42 48 17 46
67 98 48 27 62
28 12 19 80 50
63 94 98 50 80
14 75 50 41 55
72 18 37 79 61
35
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 3 No. of Jobs: 10
No. of machines: 5
Optimum Makespan: 597
Makespan found: 629
Chart No. 4
Sequence:
2 2 8 5 3 2 5 7 1 8 6 5 6 10 7 1 4 5 4 3 9 9 10 2 3
1 4 9 2 5 3 1 6 7 4 9 6 8 8 10 9 10 3 4 1 6 7 7 8
10
Machine Order: 2 3 1 5 4
3 2 1 5 4
3 4 5 1 2
5 1 3 2 4
5 1 2 4 3
5 1 2 3 4
4 3 1 5 2
5 2 1 3 4
5 1 4 3 2
5 2 1 3 4
Process time: 23 45 82 84 38
21 29 18 41 50
38 54 16 52 52
37 54 74 62 57
57 81 61 68 30
81 79 89 89 11
33 20 91 20 66
24 84 32 55 8
56 7 54 64 39
40 83 19 8 7
36
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 4 No. of Jobs: 10
No. of machines: 5
Optimum makespan: 590
Makespan found: 590
Chart No. 5
Sequence:
2 3 9 10 2 6 6 3 4 3 5 2 9 9 1 10 3 8 5 7 1 9 5 8
10 6 4 8 10 9 4 5 10 6 1 2 7 3 8 4 5 6 1 4 7 8 1 2 7
7
Machine Order: 1 3 4 5 2
2 4 5 3 1
2 1 4 5 3
3 5 1 4 2
2 4 5 1 3
4 3 1 5 2
3 2 1 4 5
2 4 1 5 3
3 5 1 2 4
3 5 4 2 1
Process time: 12 94 92 91 7
19 11 66 21 87
14 75 13 16 20
95 66 7 7 77
45 6 89 15 34
77 20 76 88 53
74 88 52 27 9
88 69 62 98 52
61 9 62 52 90
54 5 59 15 88
37
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 5
No. of Jobs: 10
No. of machines: 5
Optimum Makespan: 593
Makespan found:593
Chart No. 6
Sequence:
8 5 3 2 3 8 6 7 1 4 8 8 6 2 6 9 3 9 9 7 5 1 1 2 4
8 10 6 2 1 7 5 4 3 10 7 1 6 5 4 10 2 9 5 3 10 9 10 7
4
Process time:
72 87 95 66 60
5 35 48 39 54
46 20 21 97 55
59 19 46 34 37
23 73 25 24 28
28 45 5 78 83
53 71 37 29 12
12 87 33 55 38
49 83 40 48 07
65 17 90 27 23
Machine Order:
2 1 5 3 4
5 4 1 3 2
2 4 3 1 5
1 4 5 2 3
5 3 4 2 1
4 1 5 2 3
1 4 2 5 3
5 3 4 2 1
3 4 2 1 5
3 4 1 5 2
38
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 6
No. of Jobs: 15
No. of machines: 5
Optimum Makespan: 926
Makespan found: 926
Machine Order:
2 3 5 1 4
4 5 2 3 1
3 1 2 4 5
4 2 5 1 3
5 4 3 2 1
3 2 1 4 5
1 4 2 5 3
1 2 3 5 4
3 4 5 1 2
1 5 4 2 3
5 3 1 4 2
1 5 3 2 4
5 4 2 3 1
5 2 1 3 4
1 2 3 5 4
Process time:
21 34 95 53 55
52 16 71 26 21
31 12 42 39 98
77 77 79 55 66
37 34 64 19 83
43 54 92 62 79
93 69 87 77 87
60 41 38 83 24
98 17 25 44 49
96 77 79 75 43
28 35 95 76 07
61 10 95 09 35
59 16 91 59 46
43 52 28 27 50
87 45 39 9 41
39
Hybrid Genetic Algorithm for
Jobshop Production Problem
Chart No. 7
Sequence:
2 5 12 12 3 8 1 14 10 4 5 9 1 13 14 8 3 7 12 15 4 1 6 4
2 3 11 6 1 7 14 8 13 5 11 3 2 15 10 9 10 15 1 13 4 6 3 2
9 7 5 4 14 15 15 9 11 14 10 13 6 2 7 8 6 11 9 13 8 10 12
12 7 5 11
40
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 7 No. of Jobs: 20
No. of machines: 5
Optimum Makespan: 890
Makespan found: 890
Machine Order:
1 4 2 4 3
1 2 5 4 3
4 1 3 2 5
1 2 5 4 3
4 2 1 3 5
2 3 1 5 4
3 2 1 5 4
3 4 5 1 2
5 1 3 2 4
5 1 2 4 3
5 1 2 3 4
4 3 1 5 2
5 2 1 3 4
5 1 4 3 2
5 2 1 3 4
Process time:
47 57 71 96 14
75 60 22 79 65
32 33 69 31 58
44 34 51 58 47
29 44 62 17 08
15 40 97 38 66
58 39 57 20 50
57 32 87 63 21
56 84 90 85 61
15 20 67 30 70
84 82 23 45 38
50 21 18 41 29
06 52 52 38 54
37 54 57 74 62
57 61 81 30 68
41
Hybrid Genetic Algorithm for
Jobshop Production Problem
Chart No. 8
Sequence:
2 7 10 13 13 6 12 3 2 1 8 4 13 5 4 15 10 15 9 7 13 5 12
8 3 2 14 9 1 11 6 14 4 3 9 15 1 15 11 2 7 13 4 9 14 10 7
2 1 14 11 12 8 11 6 15 10 6 14 3 12 12 5 7 10 8 1 9 8 5
4 5 6 3 11
42
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 8 No. of Jobs: 15
No. of machines: 5
Optimum Makespan: 863
Makespan found: 863
Chart No. 9
Sequence:
6 2 3 11 4 9 2 14 12 15 6 6 11 5 14 8 10 10 8 7 12 2 1
9 15 11 13 2 4 7 4 03 5 8 14 9 1 2 1 12 1 5 4 9 6 8 15
7 5 13 13 13 6 7 15 11 9 12 1 8 3 7 13 14 12 5 3 3 11 10
4 10 14 15 10
Machine Order:
4 3 1 5 2
3 2 1 4 5
2 4 1 5 3
3 5 1 2 4
3 5 4 2 1
5 4 3 2 1
5 4 1 2 3
4 3 1 2 5
4 1 5 3 2
5 3 4 1 2
1 2 5 4 3
1 5 3 4 2
1 4 5 3 2
4 2 1 5 3
3 1 3 2 5
Process time:
92 94 12 91 7
21 19 87 11 66
14 13 75 16 20
95 66 7 77 7
34 89 6 45 15
88 77 20 53 76
9 27 52 88 74
69 52 62 88 98
90 62 9 61 52
5 54 59 88 15
41 50 78 53 23
38 72 91 68 71
45 95 52 25 6
30 66 23 36 17
95 71 76 8 88
43
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 9 No. of Jobs: 15
No. of machines: 5
Optimum Makespan: 951
Makespan found: 951
Machine Order:
2 4 3 1 5
4 2 3 5 1
5 4 2 3 1
1 2 3 4 5
1 5 3 4 2
1 4 2 3 5
4 3 5 2 1
3 2 4 5 1
3 5 1 2 4
3 5 4 2 1
5 4 3 1 2
2 1 4 5 3
5 4 1 2 3
4 2 3 1 5
1 2 3 5 4
Process time:
66 85 84 62 19
59 64 46 13 25
88 80 73 53 41
14 67 57 74 47
84 64 41 84 78
63 28 46 26 52
10 17 73 11 64
67 97 95 38 85
95 46 59 65 93
43 85 32 85 60
49 41 61 66 90
17 23 70 99 49
40 73 73 98 68
57 9 7 13 98
37 85 17 79 41
44
Hybrid Genetic Algorithm for
Jobshop Production Problem
Chart No. 10
Sequence:
15 2 8 3 5 13 10 1 14 6 10 2 14 9 2 7 12 4 11 14 1 15 3
11 5 7 13 8 15 12 9 15 10 3 4 6 1 11 14 7 3 4 14 8 6 8
7 9 6 11 5 10 13 12 2 3 12 9 4 1 5 13 9 11 7 12 15 2 8
10 13 5 1 6 4
45
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 10 No. of Jobs: 20
No. of machines: 5
Optimum Makespan: 958
Makespan found: 958
Chart No. 11
Sequence:
5 13 11 11 6 15 1 14 12 4 2 12 8 6 4 9 12 1 10 3 4 4 9
2 13 1 15 3 4 12 14 5 7 3 10 13 11 6 2 1 11 8 9 2 15 6 7
8 15 5 9 8 7 2 6 7 13 5 13 10 7 11 15 2 14 3 3 1 5 10 9
10 8 14 14
Machine Order:
2 3 4 1 5
2 1 5 4 3
1 2 3 5 4
4 2 3 1 5
3 1 2 4 5
4 5 3 1 2
2 5 1 3 4
3 4 2 5 1
1 4 5 2 3
3 5 4 1 2
1 5 4 3 2
3 1 2 5 4
4 3 2 5 1
2 3 5 1 4
4 3 1 5 2
Process time:
58 44 5 9 58
89 97 96 77 84
77 87 81 39 85
57 21 31 15 73
48 40 49 70 71
34 82 80 10 22
91 75 55 17 7
62 47 72 35 11
64 75 50 90 94
67 20 15 12 71
52 93 68 29 57
70 58 93 7 77
27 82 63 6 95
87 56 36 26 48
76 36 36 15 8
46
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 11 No. of Jobs: 20
No. of machines: 5
Optimum Makespan: 1222
Makespan found: 1222
Chart No. 12
Sequence: 3 12 1 4 7 18 17 16 12 17 10 10 10 11 15 14 18 4 9 8 13 14 7
6 20 8 11 9 5 7 15 5 9 2 18 18 14 17 20 5 1 14 15 10 6 4
5 3 9 19 10 1 2 3 6 4 11 20 2 13 16 8 1 14 3 15 7 19 4
11 8 17 20 6 15 13 11 13 12 12 16 18 8 5 20 19 13 19 2 6 7 2
17 3 16 12 1 19 9 16
Machine Order:
3 2 1 4 5 1 4 2 5 3 1 2 3 5 4 3 4 5 1 2 1 5 4 2 3 5 3 1 4 2 1 5 3 2 4 5 4 2 3 1 5 2 1 3 4 1 2 3 5 4 1 4 2 5 3 5 3 1 2 4 2 3 5 1 4 3 2 5 1 4 5 1 4 3 2 2 1 5 4 3 5 2 4 1 3 2 1 3 4 5 5 1 3 2 4 5 3 2 4 1
Process time:
34 21 53 55 95 21 52 71 16 26 12 42 31 98 39 66 77 79 55 77 83 37 34 19 64 79 43 92 62 54 93 77 87 87 69 83 24 41 38 60 25 49 44 98 17 96 75 43 77 79 95 76 7 28 35 10 95 61 9 35 91 59 59 46 16 27 52 43 28 50 9 87 41 39 45 54 20 43 14 71 33 28 26 78 37 89 33 8 66 42 84 69 94 74 27 81 45 78 69 96
47
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 12 No. of Jobs: 20
No. of machines: 5
Optimum Makespan: 1039
Makespan found: 1039
Chart No. 13
Sequence: 12 14 19 14 8 16 8 11 18 5 1 19 15 9 16 10 11 17 10 20 12 3 18
7 20 2 12 13 11 20 15 8 4 1 7 2 13 12 4 9 1 10 11 4 5 19
6 17 13 6 3 9 16 6 14 2 20 7 9 13 17 2 13 8 18 17 3 16 20
15 5 4 8 7 19 12 3 5 14 14 18 9 5 17 16 10 6 15 1 2 6 11
15 3 4 19 18 10 1 7
Machine Order:
2 1 5 3 4
4 5 2 1 3
5 4 2 3 1
2 4 5 3 1
4 2 3 1 5
2 3 4 1 5
2 1 4 5 3
4 5 3 1 2
1 3 2 5 4
1 5 4 3 2
1 3 4 5 2
2 4 5 3 1
2 1 4 5 3
3 5 1 4 2
2 4 5 1 3
4 3 1 5 2
3 2 1 4 5
2 4 1 5 3
3 5 1 2 4
3 5 4 2 1
Process time: 23 82 84 45 38 50 41 29 18 21 16 54 52 38 52 62 57 37 74 54 68 61 30 81 57 89 89 11 79 81 66 91 33 20 20 8 24 55 32 84 7 64 39 56 54 19 40 7 8 83 63 64 91 40 6 42 61 15 98 74 80 26 75 6 87 39 22 75 24 44 15 79 8 12 20 26 43 80 22 61 62 36 63 96 40 33 18 22 5 10 64 64 89 96 95 18 23 15 38 8
48
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 13 No. of Jobs: 20
No. of machines: 5
Optimum Makespan: 1150
Makespan found: 1222
Chart No. 14
Sequence: 10 12 2 20 16 11 8 15 13 17 14 1 5 8 18 19 4 13 12 15 14 9 15
18 20 7 1 2 6 20 17 3 19 2 19 4 6 17 20 15 4 15 8 12 1 3
17 2 5 5 8 4 7 10 19 6 17 2 10 9 14 11 7 8 1 18 9 13 11
7 16 16 3 12 20 13 5 13 4 19 6 9 18 9 3 6 16 14 1 5 16 3
12 7 10 11 14 18 11 10
Machine Order:
4 1 2 5 3 2 1 3 4 5 4 2 1 3 5 3 1 4 2 5 3 4 2 1 5 2 4 3 1 5 4 2 3 5 1 5 4 2 3 1 1 2 3 4 5 1 5 3 4 2 1 4 2 3 5 4 3 5 2 1 3 2 4 5 1 3 5 1 2 4 3 5 4 2 1 5 4 3 1 2 2 1 4 5 3 5 4 1 2 3 4 2 3 1 5 1 2 3 5 4
Process time:
60 87 72 95 66 54 48 39 35 5 20 46 97 21 55 37 59 19 34 46 73 25 24 28 23 78 28 83 45 5 71 37 12 29 53 12 33 55 87 38 48 40 49 83 7 90 27 65 17 23 62 85 66 84 19 59 46 13 64 25 53 73 80 88 41 57 47 14 67 74 41 64 84 78 84 52 28 26 63 46 11 64 10 73 17 38 95 85 97 67 93 65 95 59 46 60 85 43 85 32
49
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 14 No. of Jobs: 20
No. of machines: 5
Optimum Makespan: 1292
Makespan found: 1292
Chart No. 15
Sequence: 8 2 3 18 8 10 11 9 2 11 12 18 1 20 4 1 2 8 12 5 14 19 5
5 13 6 15 9 8 16 20 14 12 3 13 12 6 7 11 10 7 4 6 15 13 1
18 9 4 11 17 14 9 18 3 14 4 8 20 6 10 7 19 13 18 3 9 16 20
15 6 7 10 2 17 19 10 3 19 17 20 1 15 16 17 14 15 16 13 12 11
1 5 7 19 4 16 2 5 17
Process time:
05 58 44 09 58
89 96 97 84 77
81 85 87 39 77
15 57 73 21 31
48 71 70 40 49
10 82 34 80 22
17 55 91 75 07
47 62 72 35 11
90 94 50 64 75
15 67 12 20 71
93 29 52 57 68
77 93 58 70 07
63 27 95 06 82
36 26 48 56 87 36 8 15 76 36 78 84 41 30 76 78 75 88 13 81 54 40 13 82 29 26 82 52 06 06 54 64 54 32 88
Machine order:
4 5 3 1 2 2 5 1 3 4 3 4 2 5 1 1 4 5 2 3 3 5 4 1 2 1 5 4 3 2 3 1 2 5 4 4 3 2 5 1 2 3 5 1 4 4 3 1 5 2 5 3 1 2 4 4 2 1 3 5 2 4 1 5 3 5 1 4 3 2 3 2 5 4 1 5 2 4 1 3 2 1 5 4 3 1 5 3 2 4 2 5 1 4 3 4 2 1 3 5
50
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 15 No. of Jobs: 20
No. of machines: 5
Optimum Makespan: 1207
Makespan found: 1207
Chart No. 16
Sequence:
16 17 7 9 19 5 14 4 13 7 9 14 12 19 15 18 9 17 3 13 1 19 12
4 2 3 15 13 19 5 10 16 6 1 10 15 12 6 12 16 3 5 8 10 13 18
5 1 8 20 13 16 8 6 10 18 11 15 2 1 12 17 15 11 10 20 4 19 17
11 5 1 18 7 14 8 2 20 11 14 3 7 7 9 16 8 6 6 17 2 18 14
20 2 11 4 4 9 3 20
Machine Order: 1 3 2 4 5 3 4 1 5 2 2 5 3 4 1 3 5 1 4 2 3 1 2 4 5 1 5 2 4 3 5 4 2 3 1 1 3 2 5 4 5 1 4 3 2 2 1 5 3 4 1 2 3 5 4 3 1 4 2 5 1 3 2 4 5 1 4 3 2 5 2 1 5 4 3 2 3 5 1 4 2 5 3 1 4 4 1 3 5 2 1 2 3 4 5 2 3 5 1 4
Process time:
6 40 81 37 19 40 32 55 81 9 46 65 70 55 77 21 65 64 25 15 85 40 44 24 37 89 29 83 31 84 59 38 80 30 8 80 56 77 41 97 56 91 50 71 17 40 88 59 7 80 45 29 8 77 58 36 54 96 9 10 28 73 98 92 87 70 86 27 99 96 95 59 56 85 41 81 92 32 52 39 7 22 12 88 60 45 93 69 49 27 21 84 61 68 26 82 33 71 99 44
51
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 16 No. of Jobs: 10
No. of machines: 10
Optimum Makespan: 945
Makespan found: 987
Chart No. 17
Sequence:
8 2 8 3 7 5 1 10 5 1 8 3 6 6 3 7 9 7 3 5 10 8 10 4
3 4 10 1 1 3 6 2 7 5 9 9 10 6 7 1 7 3 10 2 5 8 9 6 9
4 4 1 9 9 3 7 8 8 2 1 6 10 2 7 1 2 8 10 6 7 5 4 5
1 1 4 10 5 2 8 6 4 10 9 9 5 2 6 4 7 5 3 6 4 2 3 2 4
9 8
Process time:
21 71 16 52 26 34 53 21 55 95 55 31 98 79 12 66 42 77 77 39 34 64 62 19 92 79 43 54 83 37 87 69 87 38 24 83 41 93 77 60 98 44 25 75 43 49 96 77 17 79 35 76 28 10 61 09 95 35 7 95 16 59 46 91 43 50 52 59 28 27 45 87 41 20 54 43 14 9 39 71 33 37 66 33 26 8 28 89 42 78 69 81 94 96 27 69 45 78 74 84
Machine Order: 2 7 10 9 8 3 1 5 4 6 5 3 6 10 1 8 2 9 7 4 4 3 9 2 5 10 8 7 1 6 2 4 3 8 9 10 7 1 6 5 3 1 6 7 8 2 5 10 4 9 3 4 6 10 5 7 1 9 2 8 4 3 1 2 10 9 7 6 5 8 2 1 4 5 7 10 9 6 3 8 5 3 9 6 4 8 2 7 10 1 9 10 3 5 4 1 8 7 2 6
52
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 17 No. of Jobs: 10
No. of machines: 10
Optimum Makespan: 784
Makespan found: 819
Chart No. 18
Sequence: 1 2 5 9 6 1 3 4 2 9 6 7 2 3 5 3 7 5 4 6 1 10 6 5 8
2 4 9 9 3 7 5 7 10 8 2 8 10 9 6 5 3 4 7 3 5 1 6 2 1
8 3 9 2 7 4 8 9 10 1 8 10 5 9 3 2 9 3 1 4 9 8 10 5
5 8 7 4 7 10 2 6 8 1 10 7 7 2 3 6 1 4 6 4 6 1 10 8
4 10
Machine Order: 5 8 10 3 4 9 6 7 2 1 9 6 2 8 3 4 7 10 5 1 3 5 4 2 9 7 8 1 10 6 1 9 4 8 6 3 5 7 2 10 10 1 5 9 7 3 6 4 8 2 4 3 6 1 8 5 9 2 7 10 2 8 9 4 5 6 7 1 3 10 2 8 3 1 9 7 4 10 6 5 3 4 5 10 1 7 8 9 2 6 2 1 6 4 10 8 9 3 7 5
Process time:
18 21 41 45 38 50 84 29 23 82
57 16 52 74 38 54 62 37 54 52
30 79 68 61 11 89 89 81 81 57
91 8 33 55 20 20 32 84 66 24
40 7 19 7 83 64 56 54 8 39
91 64 40 63 98 74 61 6 42 15
80 39 24 75 75 6 44 26 87 22
15 43 20 12 26 61 79 22 8 80
62 96 22 5 63 33 10 18 36 40
96 89 64 95 23 18 15 64 38 8
53
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 18 No. of Jobs: 10
No. of machines: 10
Optimum Makespan: 848
Makespan found:898
Chart No. 19
Sequence: 5 10 3 8 7 4 9 1 2 5 5 7 1 5 10 9 2 1 6 5 6 4 6 1 8
2 9 10 9 1 8 2 7 3 3 6 3 10 2 6 4 6 9 3 7 1 8 10 5
8 9 3 2 1 7 3 6 2 7 6 5 4 4 4 10 7 3 3 8 10 6 1 4 6
9 2 10 4 3 5 5 7 7 8 2 9 1 4 10 5 8 2 9 1 8 8 9 10 7
4
Machine order:
7 1 5 4 8 9 2 6 3 10 4 10 7 6 1 9 5 3 8 2 5 2 9 1 8 7 6 4 10 3 10 2 5 4 9 3 7 1 8 6 4 3 7 10 8 1 5 6 2 9 2 5 1 3 10 7 8 9 6 4 2 4 1 3 10 8 9 5 7 6 6 4 7 2 1 8 9 10 3 5 2 1 8 5 4 6 10 9 7 3 5 9 3 4 2 7 8 10 6 1
Process time: 54 87 48 60 39 35 72 95 66 5
20 46 34 55 97 19 59 21 37 46
45 24 28 28 83 78 23 25 5 73
12 37 38 71 33 12 55 53 87 29
83 49 23 27 65 48 90 7 40 17
66 25 62 84 13 64 46 59 19 85
73 80 41 53 47 57 74 14 67 88
64 84 46 78 84 26 28 52 41 63
11 64 67 85 10 73 38 95 97 17
60 32 95 93 65 85 43 85 46 59
54
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 19 No. of Jobs: 10
No. of machines: 10
Optimum Makespan: 848
Makespan found: 881
Chart No. 20
Sequence: 4 9 8 10 2 1 3 5 9 1 8 6 7 8 10 4 3 5 2 9 4 1 1 9 5
3 8 10 6 4 2 5 2 1 4 8 3 6 8 2 9 9 7 9 9 7 1 4 5 10
4 1 8 3 10 7 7 6 10 2 3 9 3 2 7 6 4 7 2 10 7 4 5 5
6 1 1 6 8 9 3 4 8 10 7 6 10 2 1 3 5 6 7 10 3 5 2 5
6 8
Process time:
44 5 58 97 9 84 77 96 58 89
15 31 87 57 77 85 81 39 73 21
82 22 10 70 49 40 34 48 80 71
91 17 62 75 47 11 7 72 35 55
71 90 75 64 94 15 12 67 20 50
70 93 77 29 58 93 68 57 7 52
87 63 26 6 82 27 56 48 36 95
36 15 41 78 76 84 30 76 36 8
88 81 13 82 54 13 29 40 78 75
88 54 64 32 52 6 54 82 6 26
Machine order:
3 4 6 5 1 8 9 10 2 7
5 8 2 9 1 4 3 6 10 7
10 7 5 4 2 1 9 3 8 6
2 3 8 6 9 5 4 7 10 1
7 2 4 1 3 9 5 8 10 6
8 6 9 3 5 7 4 2 10 1
7 2 5 6 3 4 8 9 10 1
1 6 9 10 4 7 5 8 3 2
6 3 4 7 5 8 9 10 2 1
10 5 7 8 1 3 9 6 4 2
55
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 20 No. of Jobs: 10
No. of machines: 10
Optimum Makespan: 902
Makespan found: 939
Chart No. 21
Sequence: 1 7 4 9 3 8 1 10 7 9 5 6 3 2 8 10 2 7 2 10 5 5 3 1
10 6 9 9 7 5 1 4 2 10 7 4 3 4 8 6 9 5 5 1 9 1 1 10
7 2 2 8 4 10 2 6 7 1 9 3 8 2 6 10 3 5 3 9 8 9 6 8 3
7 10 6 7 3 4 5 8 5 1 4 2 1 4 5 4 6 4 10 8 9 3 8 6 2
7 6
Process time:
9 81 55 40 32 37 6 19 81 40 21 70 65 64 46 65 25 77 55 15 85 37 40 24 44 83 89 31 84 29 80 77 56 8 30 59 38 80 41 97 91 40 88 17 71 50 59 80 56 7 8 9 58 77 29 96 45 10 54 36 70 92 98 87 99 27 86 96 28 73 95 92 85 52 81 32 39 59 41 56 60 45 88 12 7 22 93 49 69 27 21 61 68 26 82 71 44 99 33 84
Machine order: 7 2 5 3 9 4 1 6 10 8
8 3 10 5 2 6 9 1 4 7
3 6 1 4 2 7 5 9 8 10
5 7 8 1 3 6 4 2 10 9
1 7 5 2 3 4 10 9 6 8
3 7 4 6 2 9 1 10 5 8
5 4 2 6 7 8 9 10 1 3
2 8 4 5 7 10 9 1 3 6
4 9 1 3 2 6 5 10 8 7
1 3 4 6 7 10 9 5 8 2
56
Hybrid Genetic Algorithm for
Jobshop Production Problem
FT 10 No. of Jobs: 10
No. of machines: 10
Optimum Makespan: 930
Makespan found: 976
Process time:
29 78 9 36 49 11 62 56 44 21
43 90 75 11 69 28 46 46 72 30
91 85 39 74 90 10 12 89 45 33
81 95 71 99 9 52 85 98 22 43
14 6 22 61 26 69 21 49 72 53
84 2 52 95 48 72 47 65 6 25
46 37 61 13 32 21 32 89 30 55
31 86 46 74 32 88 19 48 36 79
76 69 76 51 85 11 40 89 26 74
85 13 61 7 64 76 47 52 90 45
Machine order:
1 2 3 4 5 6 7 8 9 10
1 3 5 10 4 2 7 6 8 9
2 1 4 3 9 6 8 7 10 5
2 3 1 5 7 9 8 4 10 6
3 1 2 6 4 5 9 8 10 7
3 2 6 4 9 10 1 7 5 8
2 1 4 3 7 6 10 9 8 5
3 1 2 6 5 7 9 10 8 4
1 2 4 6 3 10 7 8 5 9
2 1 3 7 9 10 6 4 5 8
57
Hybrid Genetic Algorithm for
Jobshop Production Problem
Chart No. 22
Sequence: 5 6 4 5 9 1 5 5 9 2 8 4 5 10 8 6 6 2 10 9 7 3 9 8 4
10 9 5 4 5 2 7 10 9 10 6 4 1 7 10 2 3 6 7 9 3 1 8 1
2 4 2 3 7 7 8 3 6 5 9 1 6 10 8 8 7 1 1 10 8 4 3 4 1
2 7 1 4 6 10 2 3 9 5 8 3 9 6 7 2 6 3 7 8 3 2 4 10 5
1
58
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 21 No. of Jobs: 15
No. of machines: 10
Optimum Makespan: 1046
Makespan found: 1127
Machine order:
3 4 6 10 5 7 1 9 2 8 4 3 1 2 10 9 7 6 5 8 2 1 4 5 7 10 9 6 3 8 5 3 9 6 4 8 2 7 10 1 9 10 3 5 4 1 8 7 2 6 9 8 7 10 3 2 6 5 1 4 5 6 4 10 1 9 7 8 3 2 6 5 3 7 2 8 1 4 10 9 2 6 1 4 3 8 9 7 10 5 3 6 7 10 2 4 9 1 8 5 2 5 1 3 10 9 6 4 8 7 6 10 1 5 7 4 3 2 9 8 6 10 9 8 5 7 4 1 2 3 2 9 1 3 10 4 6 7 5 8 5 4 7 6 3 9 2 10 8 1
Process time:
34 55 95 16 21 71 53 52 21 26 39 31 12 42 79 77 77 98 55 66 19 83 34 92 54 79 62 37 64 43 60 87 24 77 69 38 87 41 83 93 79 77 98 96 17 44 43 75 49 25 35 95 9 10 35 7 28 61 95 76 28 59 16 43 46 50 52 27 59 91 9 20 39 54 45 71 87 41 43 14 28 33 78 26 37 8 66 89 42 33 94 84 78 81 74 27 69 69 45 96 31 24 20 17 25 81 76 87 32 18 28 97 58 45 76 99 23 72 90 86 27 48 27 62 98 67 48 42 46 17 12 50 80 50 80 19 28 63 94 98 61 55 37 14 50 79 41 72 18 75
59
Hybrid Genetic Algorithm for
Jobshop Production Problem
Chart No. 23
Sequence: 15 1 2 9 3 13 9 13 7 11 2 5 2 5 2 14 12 2 12 6 4 10 9 2
15 3 8 7 9 5 12 6 7 8 7 13 7 1 5 4 15 10 11 5 8 3 1 13
6 15 4 10 11 1 14 7 4 9 1 2 12 13 10 8 8 14 6 8 2 15 4
12 15 10 11 2 12 3 6 15 5 4 11 8 10 11 14 7 14 9 14 10 1 13
8 7 12 6 12 1 15 3 4 3 13 12 14 7 6 11 5 14 6 8 2 4 10
15 7 10 9 12 13 3 6 13 5 11 4 9 8 14 11 6 10 9 15 1 9 3
13 5 5 14 1 3 11 4 3 1
60
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 22 No. of Jobs: 15
No. of machines: 10
Optimum Makespan: 927
Makespan found: 948
Machine order:
10 6 5 3 8 4 2 1 9 7 4 3 5 2 10 1 7 6 8 9 9 8 3 1 10 6 7 4 2 5 4 3 7 5 8 9 6 10 1 2 5 7 2 3 8 1 9 6 4 10 7 1 5 4 8 9 2 6 3 10 4 10 7 6 1 9 5 3 8 2 5 2 9 1 8 7 6 4 10 3 10 2 5 4 9 3 7 1 8 6 4 3 7 10 8 1 5 6 2 9 2 5 1 3 10 7 8 9 6 4 2 4 1 3 10 8 9 5 7 6 6 4 7 2 1 8 9 10 3 5 2 1 8 5 4 6 10 9 7 3 5 9 3 4 2 7 8 10 6 1
Process time:
66 91 87 94 21 92 7 12 11 19 13 20 7 14 66 75 77 16 95 7 77 20 34 15 88 89 53 6 45 76 27 74 88 62 52 69 9 98 52 88 88 15 52 61 54 62 59 9 90 5 71 41 38 53 91 68 50 78 23 72 95 36 66 52 45 30 23 25 17 6 65 8 85 71 65 28 88 76 27 95 37 37 28 51 86 9 55 73 51 90 39 15 83 44 53 16 46 24 25 82 72 48 87 66 5 54 39 35 95 60 46 20 97 21 46 37 19 59 34 55 23 25 78 24 28 83 28 5 73 45 37 53 87 38 71 29 12 33 55 12 90 17 49 83 40 23 65 27 7 48
61
Hybrid Genetic Algorithm for
Jobshop Production Problem
Chart No. 24
Sequence: 2 6 10 5 8 1 7 6 9 4 5 1 1 9 10 5 7 3 9 1 3 10 3 10
7 8 4 8 9 8 4 1 7 9 3 1 3 2 6 10 7 8 9 5 10 2 4 10
9 4 4 5 7 6 7 6 6 9 1 6 5 4 9 1 2 10 2 8 8 5 8 2 7
3 7 2 10 3 3 5 3 4 8 2 4 2 6 1 8 1 7 6 10 5 6 9 4 2
3 5
62
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 23 No. of Jobs: 15
No. of machines: 10
Optimum Makespan: 1032
Makespan found: 1032
Process time:
84 58 77 44 97 89 5 58 96 9 21 87 15 39 81 85 31 57 73 77 40 71 34 82 70 22 10 80 48 49 75 17 7 72 11 62 47 35 91 55 20 12 71 67 64 94 15 50 75 90 93 93 57 70 77 58 52 29 7 68 56 95 48 26 82 63 36 27 87 6 76 15 78 8 41 36 30 84 36 76 75 13 81 29 54 82 88 78 40 13 6 26 32 64 54 52 82 6 88 54 62 67 32 62 69 61 35 72 5 93 78 90 85 72 64 63 11 82 88 7 28 11 50 88 44 31 27 66 49 35 14 39 56 62 97 66 69 7 47 76 18 93 58 47 69 57 41 53 79 64
Machine order:
8 6 9 3 5 7 4 2 10 1 7 2 5 6 3 4 8 9 10 1 1 6 9 10 4 7 5 8 3 2 6 3 4 7 5 8 9 10 2 1 10 5 7 8 1 3 9 6 4 2 7 6 2 8 9 5 1 3 10 4 8 1 9 5 3 2 10 4 7 6 4 6 10 2 9 3 5 7 1 8 1 8 3 9 5 7 6 2 10 4 3 2 8 7 5 1 6 4 10 9 9 3 6 1 8 4 2 5 10 7 3 10 1 2 9 7 4 8 6 5 5 10 8 7 1 6 3 2 9 4 3 6 7 5 4 10 8 2 9 1 2 9 8 7 4 10 3 6 5 1
63
Hybrid Genetic Algorithm for
Jobshop Production Problem
Chart No. 25
Sequence: 5 3 13 7 10 12 1 4 4 9 2 15 15 14 2 4 3 11 12 9 3 15 6 7
10 4 14 5 8 14 4 9 11 15 1 7 8 13 12 4 1 15 2 3 13 1 5 3
6 14 2 10 9 13 10 7 8 11 5 11 13 15 4 9 1 7 2 14 12 6 4 3
14 1 6 13 8 6 10 11 8 12 2 5 3 10 6 15 4 7 9 1 1 12 15
11 7 6 13 3 2 1 10 14 7 8 12 5 8 6 2 13 6 10 14 9 15 5 2
5 8 4 7 15 11 11 11 1 12 3 6 12 5 10 9 14 9 14 13 3 8 2 5
13 9 10 7 11 12 8
64
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 24
No. of Jobs: 15
No. of machines: 10
Optimum Makespan: 935
Makespan found: 1053
Machine order: 8 10 1 7 5 9 3 6 2 4
7 9 4 1 2 5 6 10 3 8
2 4 6 5 1 3 7 9 10 8
2 8 5 7 6 1 9 4 10 3
8 3 9 6 2 7 4 1 10 5
9 1 5 6 10 2 8 7 4 3
7 3 9 2 10 5 8 1 6 4
9 8 6 4 3 5 10 2 1 7
5 1 10 6 8 4 3 9 7 2
10 1 4 9 2 7 3 6 5 8
8 4 5 6 3 7 1 10 2 9
1 4 3 8 9 6 10 2 7 5
10 2 4 7 3 9 8 1 6 5
5 3 6 7 9 8 4 2 1 10
3 6 10 9 1 7 4 8 2 5
Process times: 8 75 72 74 30 43 38 98 26 19
19 73 43 23 85 39 13 26 67 9
50 93 80 7 55 61 57 72 42 46
68 43 99 60 68 91 11 96 11 72
84 34 40 7 70 74 12 43 69 30
60 49 59 72 63 69 99 45 27 9
71 91 65 90 98 8 50 75 37 17
62 90 98 31 91 38 72 9 72 49
35 39 74 25 47 52 63 21 35 80
58 5 50 52 88 20 68 24 53 57
99 91 33 19 18 38 24 35 49 9
68 60 77 10 60 15 72 18 90 18
79 60 56 91 40 86 72 80 89 51
10 92 23 46 40 72 6 23 95 34
24 29 49 55 47 77 77 8 28 48
65
Hybrid Genetic Algorithm for
Jobshop Production Problem
Sequence:
6 15 4 13 9 2 2 7 13 13 14 1 14 6 5 3 15 15 7 6 1 6 9 8
10 2 15 4 11 12 13 13 15 10 7 14 11 3 15 5 4 8 13 12 4 7 6
9 2 6 11 8 9 5 11 5 1 12 7 5 1 8 7 3 10 13 9 14 10 9
1 2 14 7 11 13 10 15 4 13 11 3 8 5 3 6 12 11 11 8 15 9 2
8 7 12 9 14 10 7 12 11 9 3 2 4 5 12 10 8 1 14 4 13 1 2
14 5 5 3 10 2 8 11 1 6 10 4 15 12 10 15 12 3 7 4 3 9 2
14 6 6 4 1 14 5 3 1 8 12
66
Hybrid Genetic Algorithm for
Jobshop Production Problem
LA 25
No. of Jobs: 15
No. of machines: 10
Optimum Makespan: 977
Makespan found: 1090
Machine order: 9 5 4 3 1 6 10 2 8 7
6 4 3 5 7 10 1 2 8 9
10 2 1 7 5 8 4 6 9 3
3 2 1 6 5 8 10 9 4 7
7 3 4 9 5 8 2 6 10 1
9 3 8 1 6 4 5 7 10 2
1 3 4 6 5 10 9 7 8 2
4 8 10 1 3 5 6 2 9 7
3 4 5 7 2 10 9 1 6 8
8 9 5 7 1 6 3 10 4 2
9 7 8 5 6 4 1 3 10 2
2 6 9 7 5 1 4 3 8 10
3 7 8 2 5 9 1 4 10 6
7 3 6 9 2 8 10 5 4 1
5 8 9 2 4 3 7 10 6 1
Process time: 14 75 12 38 76 97 12 29 44 66
38 82 85 58 87 89 43 80 69 92
5 84 43 48 8 7 41 61 66 14
42 8 96 19 59 97 73 43 74 41
55 70 75 42 37 23 48 5 38 7
9 72 31 79 73 95 25 43 60 56
97 64 78 21 94 31 53 16 86 7
86 85 63 61 65 30 32 33 44 59
44 16 11 45 30 84 93 60 61 90
36 31 47 52 32 11 28 35 20 49
20 49 74 10 17 34 85 77 68 84
85 7 71 59 76 17 29 17 48 13
15 87 11 39 39 43 19 32 16 64
32 92 33 82 83 57 99 91 99 8
88 07 27 38 91 69 21 62 39 48
67
Hybrid Genetic Algorithm for
Jobshop Production Problem
Sequence:
15 4 12 1 4 3 15 10 3 6 7 9 5 8 9 14 9 11 13 7 2 12 4 2
14 10 15 2 9 8 1 15 4 11 3 10 8 11 12 14 8 13 4 9 9 13 5
14 13 4 11 4 7 2 12 15 7 3 6 2 2 15 10 6 13 13 11 11 15 5
12 11 5 13 8 5 13 6 12 2 1 6 14 4 14 7 5 12 3 15 1 4
10 9 12 2 3 9 13 15 3 6 8 7 12 14 10 11 2 6 10 8 7 7 7 4
11 9 8 6 8 9 5 12 11 1 1 7 10 1 5 6 15 5 3 13 2 3 14 10
14 3 1 10 5 1 6 8 1 14
68
Hybrid Genetic Algorithm for
Jobshop Production Problem
CONCLUSION:
Since JSSP falls into the class of NP-hard problems, they are among the most
difficult to formulate and solve. Research analyst and engineers have been pursuing
solutions to these problems for more than 5 decades, with varying degree of success.
They impact the ability of manufacturers to meet customer demands and make a profit.
The study on GA and job shop scheduling problem provides a rich experience for
the constrained combinatorial optimization problems. Application of genetic algorithm
gives a good result most of the time. Although GA takes time to provide a good result,
yet it provides a flexible framework for evolutionary computation and it can handle
varieties of objective function and constraint.
Our proposed algorithm was quite efficient and gave optimal solutions for most of
the benchmark problems with some exceptions leaving behind room for improvement.
69
Hybrid Genetic Algorithm for
Jobshop Production Problem
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Hybrid Genetic Algorithm for
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