Hybrid Genetic Algorithm For JobShop Scheduling

1

Hybrid Genetic Algorithm for

Jobshop Production Problem

CHAPTER No. 01

INTRODUCTION

2



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

3



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.

4



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

5



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).

6



CHAPTER No. 02

LITERATURE REVIEW

7



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

8



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

9



(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.

10



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.

11



CHAPTER No. 03

METHODOLOGY

12



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

13



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.

14



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.

15



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

16



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.

17



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.

http://en.wikipedia.org/wiki/Computer_science

http://en.wikipedia.org/wiki/Algorithm

http://en.wikipedia.org/wiki/A_priori_and_a_posteriori

http://en.wikipedia.org/wiki/A_priori_and_a_posteriori

18



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.

19



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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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

21



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

22



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

23



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

24



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.

25



Roulette wheel Selection

Fig No. 1

26



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

27



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.

28



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

29



CHAPTER No. 04

RESULTS

30



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



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



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



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




No. of machines: 5


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




No. of machines: 5


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


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




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


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



LA 5

No. of Jobs: 10

No. of machines: 5


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



LA 6

No. of Jobs: 15

No. of machines: 5


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



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




No. of machines: 5


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



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




No. of machines: 5


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




No. of machines: 5


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



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




No. of machines: 5


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




No. of machines: 5


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




No. of machines: 5



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




No. of machines: 5



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




No. of machines: 5



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




No. of machines: 5



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




No. of machines: 10


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




No. of machines: 10


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




No. of machines: 10


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




No. of machines: 10


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




No. of machines: 10


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



FT 10 No. of Jobs: 10

No. of machines: 10


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



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




No. of machines: 10



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



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




No. of machines: 10


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



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




No. of machines: 10



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



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



LA 24

No. of Jobs: 15

No. of machines: 10



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



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



LA 25

No. of Jobs: 15

No. of machines: 10



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



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



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



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Problems.

Journal of the Operations Research Society of America 3 (1955) 429-442

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70



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Hybrid Genetic Algorithm For JobShop Scheduling

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