The heuristic optimization techniques were commonly used in solving several optimization
problems. The present work aims to develop a hybrid algorithm to solve the scheduling optimization problem of
JSSP. There are different variants of these algorithms that were addressed in several previous works. The
impacts of these two kinds (Genetic Algorithm (GA) and Simulated Annealing (SA) based optimization model)
of initial condition on the performance of these two algorithms were studied using the convergence curve and
the achieved makespan. Even though genetic algorithm performed better than other evolutionary algorithms, it
has some weakness. During running GA, sometimes, it will produce same result without any improvement. SA
has a mechanism to overcome from that situation. During SA, if same result will be repeated, then it is rapidly
changing the change in temperature variable and re-initiates another random search. By using this feature of
SA, it has been implemented a hybrid based evolutionary model for solving JSSP by improving GA.
Comparison has been made with the performance of the proposed SA-GA-Hybrid model with GA as well as SA.
Job-shop manufacturing environment requires planning of schedules for the systems of low-volume having numerous variations. For a job-shop scheduling, ‘k’ number of operations and ‘n’ number of jobs on ‘m’ number of machines processed through an assured objective function to be minimized (makespan). This paper presents a capable genetic algorithm for the job-shop scheduling problems among operating parameters such as random population generation with a population size of 50, operation based chromosome structure, tournament selection as selection scheme, 2-point random crossover with probability 80%, 2-point mutation with probability 20%, elitism, repairing of chromosomes and no. of iteration is 1000. An algorithm is programmed for job shop scheduling problem using MATLAB 2009 a 7.8. The proposed genetic algorithm with certain operating parameters is applied to the two case studies taken from literature. The results also show that genetic algorithm is the best optimization technique for solving the scheduling problems of job shop manufacturing systems evolving shortest processing time and transportation time due to its implications to more practical and integrated problems.
A case study on Machine scheduling and sequencing using Meta heuristicsIJERA Editor
Modern manufacturing systems are constantly increasing in complexity and become more agile in nature such
system has become more crucial to check feasibility of machine scheduling and sequencing because effective
scheduling and sequencing can yield increase in productivity due to maximum utilization of available resources
but when number of machine increases traditional scheduling methods e.g. Johnson‟s ,rule is becomes in
effective Due to the limitations involved in exhaustive enumeration, for such problems meta-heuristics has
become greater choice for solving NP hard problems because of their multi solution and strong neighbourhood
search capability in a reasonable time.
AN EFFICIENT HEURISTIC ALGORITHM FOR FLEXIBLE JOB SHOP SCHEDULING WITH MAINTE...mathsjournal
This document summarizes a heuristic algorithm for solving the flexible job shop scheduling problem with preventive maintenance constraints. The objectives are to minimize makespan, total workload, and maximum machine workload.
The algorithm uses a constructive procedure to sequentially assign each operation (job or maintenance) to machines based on a calculated total cost (TC) value. TC considers factors like processing time, workload balance, and maintenance window compliance. Multiple parameter settings are tested to generate initial solutions. Computational results on benchmark problems show the heuristic finds good quality solutions very quickly, making it suitable for practical applications.
This paper considers a flow shop scheduling problem. The flow shop scheduling is
characterized “n” jobs and “m” machines in series with unidirectional flow of work with
variety of jobs being processed sequentially in a single pass manner. Most real world
problems are NP-hard in nature. The essential complexity of the problem necessitates the
use of meta-heuristics for solving flow shop scheduling problem. The paper addresses the
flow shop scheduling problem to minimize the makespan time with specific batch size
using Particle Swarm Optimization (PSO) and comparing the results with an real time
company production sequence.
The document summarizes research in the field of stochastic scheduling. It discusses three broad categories of stochastic scheduling models: 1) models for scheduling a batch of stochastic jobs, 2) multi-armed bandit models, and 3) models for scheduling queueing systems. For many models in these categories, relatively simple priority-index rules have been shown to optimize important performance objectives like expected flowtime. However, more complex models generally lack tractable optimal solutions. Recent research thus focuses on designing near-optimal heuristic policies.
A review on non traditional algorithms for job shop schedulingiaemedu
The document provides a review of non-traditional algorithms that have been used for job shop scheduling problems. It discusses how job shop scheduling is an NP-hard problem and researchers have focused on hybrid methods and metaheuristics. The review covers various techniques including tabu search, genetic algorithms, simulated annealing, ant colony optimization, and iterative local search methods. It also includes tables summarizing different approximation algorithms and literature on job shop scheduling using techniques like priority dispatch rules, insertion algorithms, artificial intelligence methods, and local search methods.
This document discusses quantitative methods for decision making, also known as operations research. It defines decision making as the process of choosing among alternatives. The document then discusses different types of decisions, such as sequential, conscious/unconscious, and managerial decisions. It also discusses various farm management decisions including organizational, administrative, and marketing decisions. Finally, it provides an example of using linear programming to solve a problem involving maximizing profit from toy production given resource constraints.
The document summarizes job shop scheduling by:
1) Describing the job shop environment with m machines and n jobs that each follow predetermined routes on the machines that may differ between jobs and allow recirculation.
2) Formulating the problem as a network with nodes representing job operations and arcs representing precedence constraints, with the objective to minimize makespan.
3) Explaining how to construct feasible schedules by selecting disjunctive arcs from the network to create an acyclic graph and correspond to a machine sequence.
Job-shop manufacturing environment requires planning of schedules for the systems of low-volume having numerous variations. For a job-shop scheduling, ‘k’ number of operations and ‘n’ number of jobs on ‘m’ number of machines processed through an assured objective function to be minimized (makespan). This paper presents a capable genetic algorithm for the job-shop scheduling problems among operating parameters such as random population generation with a population size of 50, operation based chromosome structure, tournament selection as selection scheme, 2-point random crossover with probability 80%, 2-point mutation with probability 20%, elitism, repairing of chromosomes and no. of iteration is 1000. An algorithm is programmed for job shop scheduling problem using MATLAB 2009 a 7.8. The proposed genetic algorithm with certain operating parameters is applied to the two case studies taken from literature. The results also show that genetic algorithm is the best optimization technique for solving the scheduling problems of job shop manufacturing systems evolving shortest processing time and transportation time due to its implications to more practical and integrated problems.
A case study on Machine scheduling and sequencing using Meta heuristicsIJERA Editor
Modern manufacturing systems are constantly increasing in complexity and become more agile in nature such
system has become more crucial to check feasibility of machine scheduling and sequencing because effective
scheduling and sequencing can yield increase in productivity due to maximum utilization of available resources
but when number of machine increases traditional scheduling methods e.g. Johnson‟s ,rule is becomes in
effective Due to the limitations involved in exhaustive enumeration, for such problems meta-heuristics has
become greater choice for solving NP hard problems because of their multi solution and strong neighbourhood
search capability in a reasonable time.
AN EFFICIENT HEURISTIC ALGORITHM FOR FLEXIBLE JOB SHOP SCHEDULING WITH MAINTE...mathsjournal
This document summarizes a heuristic algorithm for solving the flexible job shop scheduling problem with preventive maintenance constraints. The objectives are to minimize makespan, total workload, and maximum machine workload.
The algorithm uses a constructive procedure to sequentially assign each operation (job or maintenance) to machines based on a calculated total cost (TC) value. TC considers factors like processing time, workload balance, and maintenance window compliance. Multiple parameter settings are tested to generate initial solutions. Computational results on benchmark problems show the heuristic finds good quality solutions very quickly, making it suitable for practical applications.
This paper considers a flow shop scheduling problem. The flow shop scheduling is
characterized “n” jobs and “m” machines in series with unidirectional flow of work with
variety of jobs being processed sequentially in a single pass manner. Most real world
problems are NP-hard in nature. The essential complexity of the problem necessitates the
use of meta-heuristics for solving flow shop scheduling problem. The paper addresses the
flow shop scheduling problem to minimize the makespan time with specific batch size
using Particle Swarm Optimization (PSO) and comparing the results with an real time
company production sequence.
The document summarizes research in the field of stochastic scheduling. It discusses three broad categories of stochastic scheduling models: 1) models for scheduling a batch of stochastic jobs, 2) multi-armed bandit models, and 3) models for scheduling queueing systems. For many models in these categories, relatively simple priority-index rules have been shown to optimize important performance objectives like expected flowtime. However, more complex models generally lack tractable optimal solutions. Recent research thus focuses on designing near-optimal heuristic policies.
A review on non traditional algorithms for job shop schedulingiaemedu
The document provides a review of non-traditional algorithms that have been used for job shop scheduling problems. It discusses how job shop scheduling is an NP-hard problem and researchers have focused on hybrid methods and metaheuristics. The review covers various techniques including tabu search, genetic algorithms, simulated annealing, ant colony optimization, and iterative local search methods. It also includes tables summarizing different approximation algorithms and literature on job shop scheduling using techniques like priority dispatch rules, insertion algorithms, artificial intelligence methods, and local search methods.
This document discusses quantitative methods for decision making, also known as operations research. It defines decision making as the process of choosing among alternatives. The document then discusses different types of decisions, such as sequential, conscious/unconscious, and managerial decisions. It also discusses various farm management decisions including organizational, administrative, and marketing decisions. Finally, it provides an example of using linear programming to solve a problem involving maximizing profit from toy production given resource constraints.
The document summarizes job shop scheduling by:
1) Describing the job shop environment with m machines and n jobs that each follow predetermined routes on the machines that may differ between jobs and allow recirculation.
2) Formulating the problem as a network with nodes representing job operations and arcs representing precedence constraints, with the objective to minimize makespan.
3) Explaining how to construct feasible schedules by selecting disjunctive arcs from the network to create an acyclic graph and correspond to a machine sequence.
Let x1 = Number of units of M1 produced
x2 = Number of units of M2 produced
ii) Write the constraints:
4x1 + 2x2 ≤ 80 (Grinding constraint)
2x1 + 5x2 ≤ 180 (Polishing constraint)
x1, x2 ≥ 0
iii) Write the objective function:
Maximize Z = 3x1 + 4x2
iv) Solve the LP problem graphically or by simplex method to find the optimal solution.
The document describes an Operations Research course. It includes 8 units covering topics like linear programming, transportation problems, queuing theory, PERT-CPM techniques, game theory, and integer programming. It provides details of each unit including the number of lecture hours and the topics to be covered. It also lists the textbooks and reference books for the course. The course aims to introduce students to various operations research techniques and their applications in decision making.
This document provides an introduction to linear programming. It defines linear programming as a mathematical modeling technique used to optimize resource allocation. The key requirements are a well-defined objective function, constraints on available resources, and alternative courses of action represented by decision variables. The assumptions of linear programming include proportionality, additivity, continuity, certainty, and finite choices. Formulating a problem as a linear program involves defining the objective function and constraints mathematically. Graphical and analytical solutions can then be used to find the optimal solution. Linear programming has many applications in fields like industrial production, transportation, and facility location.
Linear Programming - Meaning, Example and Application in BusinessSundar B N
This document discusses linear programming and its applications. It begins with an introduction and definition of linear programming as a quantitative technique to solve allocation problems. It then provides an example problem formulation to maximize revenue from two products under time and capacity constraints. The document outlines several application areas of linear programming in business, including production management, personnel management, inventory management, marketing management, financial management, and blending problems. It concludes that linear programming is an important mathematical technique for managers to make optimal decisions.
This document discusses linear programming techniques for managerial decision making. Linear programming can determine the optimal allocation of scarce resources among competing demands. It consists of linear objectives and constraints where variables have a proportionate relationship. Essential elements of a linear programming model include limited resources, objectives to maximize or minimize, linear relationships between variables, homogeneity of products/resources, and divisibility of resources/products. The linear programming problem is formulated by defining variables and constraints, with the objective of optimizing a linear function subject to the constraints. It is then solved using graphical or simplex methods through an iterative process to find the optimal solution.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Liner programming on Management ScienceAbdul Motaleb
The document discusses management science and linear programming. It provides details on:
1) Management science uses various scientific principles and analytical methods to help organizations make rational decisions to maximize profit or minimize expenses.
2) Management science research can be done on fundamental, modeling, and application levels.
3) Linear programming is a method to achieve the optimal outcome given linear constraints and can be used to solve production planning, marketing mix, product distribution, and staff scheduling problems in business.
4) The key characteristics of linear programming problems are that they involve optimization with an objective function and constraints, and have linear relationships between variables.
Linear programming is an optimization technique used to maximize or minimize a linear objective function subject to linear constraints. It involves defining variables, constraints, and an objective function in terms of those variables. The optimal solution is found by systematically considering all extreme points of the feasible region defined by the constraints. Linear programming has wide applications in fields like production, transportation, finance, and resource allocation.
Linear Programming Problems {Operation Research}FellowBuddy.com
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
- Linear programming (LP) is a form of constrained optimization where the objective function and constraints are linear. It is widely used to model and solve problems in areas like transportation and production scheduling.
- The Acme Bicycle Company example problem involves determining production levels to maximize profit, given constraints like production capacity. It is formulated as an LP with variables for production rates, an objective to maximize profit, and constraints representing limits.
- Optimal solutions in LP problems will always occur at corner points of the feasible region. Checking all corner points is more efficient than randomly sampling interior points, and is the basis for the simplex method for solving LPs.
The document provides an overview of linear programming, including its applications, assumptions, and mathematical formulation. Some key points:
- Linear programming is a tool for maximizing or minimizing quantities like profit or cost, subject to constraints. 50-90% of business decisions and computations involve linear programming.
- Applications in business include production, personnel, inventory, marketing, financial, and blending problems. The objective is to optimize variables like costs, profits, or resources while meeting constraints.
- Assumptions of linear programming include certainty, linearity/proportionality, additivity, divisibility, non-negativity, finiteness, and optimality at corner points.
- A linear programming problem is modeled mathemat
The document discusses adapting a genetic algorithm to schedule variants of manufacturing shop models. It presents two examples - a flow shop and a shop with alternative machine routing - to demonstrate how genetic algorithms can be applied to schedule jobs for more complex shop models. The genetic algorithm is implemented using standard spreadsheet software, showing how simple it is to apply genetic algorithms to production scheduling problems.
Master thesis job shop generic time lag max plusSiddhartha Verma
The document discusses solving the job shop scheduling problem with time lags using max-plus algebra. It begins with motivations and an outline. It then provides background on max-plus algebra, the job shop problem with time lags, and previous related works. The document models the job shop problem with time lags constraints using max-plus algebra, representing the constraints as potential inequalities and disjunctive constraints. It presents case studies, including a naval industry example of a 3x3 job shop problem with time lags, modeled with max-plus matrices. The document analyzes feasibility of schedules using spectral theory and max-plus algebra properties.
This document discusses resource optimization and linear programming. It defines optimization as finding the best solution to a problem given constraints. Linear programming is introduced as a mathematical technique to optimize allocation of scarce resources. The key components of a linear programming model are described as decision variables, an objective function, and constraints. Graphical and algebraic methods for solving linear programming problems are also summarized.
Mathematical Optimisation - Fundamentals and ApplicationsGokul Alex
My Session on Mathematical Optimisation Fundamentals and Industry applications for the Academic Knowledge Refresher Program organised by Kerala Technology University and College of Engineering Trivandrum, Department of Interdisciplinary Studies.
ON Fuzzy Linear Programming Technique Applicationijsrd.com
A solution approach based on fuzzy linear programming is proposed and applied to optimal machine scheduling problem. In this solution approach errors in the demand of various products during the next production period are considered to be fuzzy in nature. In conventional linear programming approach it is assumed that there is no error in the expected demand of various products. A fuzzy linear programming approach is proposed to obtain an optimal solution under fuzzy conditions. In the proposed method expected demand of products & profit are expressed by fuzzy set notations. The proposed fuzzy linear programming formulation is then transformed to an equivalent conventional linear programming problem and solutions obtained by solving this transformed linear programming problem. For illustration purpose the proposed method is applied to a profit maximization related machine scheduling problem.
Linear programming deals with optimizing a linear objective function subject to linear constraints. It involves determining the values of decision variables to maximize or minimize the objective function. The general linear programming model involves maximizing or minimizing a linear combination of n decision variables subject to m linear constraints, along with non-negativity restrictions on the decision variables. Formulating a linear programming problem involves identifying decision variables, expressing constraints and the objective function linearly in terms of the variables, and adding non-negativity restrictions.
This document discusses using a genetic algorithm to solve the job shop scheduling problem. It begins with an abstract that introduces using genetic algorithms for job shop scheduling. It then provides more details on the problem and discusses using genetic algorithms with modifications like generating the initial population using priority rules and incorporating critical block and disjunctive graph distance in the crossover and mutation operations. The document outlines the genetic algorithm approach with sections on chromosome representation and decoding, data structures, generating the initial population, crossover and mutation operations. The goal is to minimize the makespan value for job shop scheduling.
A case study on Machine scheduling and sequencing using Meta heuristicsIJERA Editor
Modern manufacturing systems are constantly increasing in complexity and become more agile in nature such
system has become more crucial to check feasibility of machine scheduling and sequencing because effective
scheduling and sequencing can yield increase in productivity due to maximum utilization of available resources
but when number of machine increases traditional scheduling methods e.g. Johnson‟s ,rule is becomes in
effective Due to the limitations involved in exhaustive enumeration, for such problems meta-heuristics has
become greater choice for solving NP hard problems because of their multi solution and strong neighbourhood
search capability in a reasonable time.
Design and Implementation of a Multi-Agent System for the Job Shop Scheduling...CSCJournals
This document proposes an agent-based parallel genetic algorithm for solving the job shop scheduling problem. The approach divides the genetic algorithm population into multiple subpopulations that are evolved independently in parallel on different hosts. Agents are used to create the initial populations in parallel and execute the genetic algorithm operations. The genetic algorithm runs in execution phases where subpopulations evolve independently, and migration phases where the subpopulations exchange solutions. Experimental results showed this parallel approach improves the performance over a non-parallel genetic algorithm.
Modified heuristic time deviation Technique for job sequencing and Computatio...ijcsit
The document discusses job sequencing and scheduling techniques. It begins by defining job sequencing as the arrangement of tasks to be performed in a specified order. It then discusses several algorithms and methods for solving job sequencing problems, including tabu search, fuzzy TOPSIS, genetic algorithms, Johnson's rule, and mathematical modeling approaches. It also covers classification of jobs, performance metrics, and discusses modified time deviation as a heuristic technique to arrange job sequences to minimize total elapsed time.
Let x1 = Number of units of M1 produced
x2 = Number of units of M2 produced
ii) Write the constraints:
4x1 + 2x2 ≤ 80 (Grinding constraint)
2x1 + 5x2 ≤ 180 (Polishing constraint)
x1, x2 ≥ 0
iii) Write the objective function:
Maximize Z = 3x1 + 4x2
iv) Solve the LP problem graphically or by simplex method to find the optimal solution.
The document describes an Operations Research course. It includes 8 units covering topics like linear programming, transportation problems, queuing theory, PERT-CPM techniques, game theory, and integer programming. It provides details of each unit including the number of lecture hours and the topics to be covered. It also lists the textbooks and reference books for the course. The course aims to introduce students to various operations research techniques and their applications in decision making.
This document provides an introduction to linear programming. It defines linear programming as a mathematical modeling technique used to optimize resource allocation. The key requirements are a well-defined objective function, constraints on available resources, and alternative courses of action represented by decision variables. The assumptions of linear programming include proportionality, additivity, continuity, certainty, and finite choices. Formulating a problem as a linear program involves defining the objective function and constraints mathematically. Graphical and analytical solutions can then be used to find the optimal solution. Linear programming has many applications in fields like industrial production, transportation, and facility location.
Linear Programming - Meaning, Example and Application in BusinessSundar B N
This document discusses linear programming and its applications. It begins with an introduction and definition of linear programming as a quantitative technique to solve allocation problems. It then provides an example problem formulation to maximize revenue from two products under time and capacity constraints. The document outlines several application areas of linear programming in business, including production management, personnel management, inventory management, marketing management, financial management, and blending problems. It concludes that linear programming is an important mathematical technique for managers to make optimal decisions.
This document discusses linear programming techniques for managerial decision making. Linear programming can determine the optimal allocation of scarce resources among competing demands. It consists of linear objectives and constraints where variables have a proportionate relationship. Essential elements of a linear programming model include limited resources, objectives to maximize or minimize, linear relationships between variables, homogeneity of products/resources, and divisibility of resources/products. The linear programming problem is formulated by defining variables and constraints, with the objective of optimizing a linear function subject to the constraints. It is then solved using graphical or simplex methods through an iterative process to find the optimal solution.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Liner programming on Management ScienceAbdul Motaleb
The document discusses management science and linear programming. It provides details on:
1) Management science uses various scientific principles and analytical methods to help organizations make rational decisions to maximize profit or minimize expenses.
2) Management science research can be done on fundamental, modeling, and application levels.
3) Linear programming is a method to achieve the optimal outcome given linear constraints and can be used to solve production planning, marketing mix, product distribution, and staff scheduling problems in business.
4) The key characteristics of linear programming problems are that they involve optimization with an objective function and constraints, and have linear relationships between variables.
Linear programming is an optimization technique used to maximize or minimize a linear objective function subject to linear constraints. It involves defining variables, constraints, and an objective function in terms of those variables. The optimal solution is found by systematically considering all extreme points of the feasible region defined by the constraints. Linear programming has wide applications in fields like production, transportation, finance, and resource allocation.
Linear Programming Problems {Operation Research}FellowBuddy.com
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
- Linear programming (LP) is a form of constrained optimization where the objective function and constraints are linear. It is widely used to model and solve problems in areas like transportation and production scheduling.
- The Acme Bicycle Company example problem involves determining production levels to maximize profit, given constraints like production capacity. It is formulated as an LP with variables for production rates, an objective to maximize profit, and constraints representing limits.
- Optimal solutions in LP problems will always occur at corner points of the feasible region. Checking all corner points is more efficient than randomly sampling interior points, and is the basis for the simplex method for solving LPs.
The document provides an overview of linear programming, including its applications, assumptions, and mathematical formulation. Some key points:
- Linear programming is a tool for maximizing or minimizing quantities like profit or cost, subject to constraints. 50-90% of business decisions and computations involve linear programming.
- Applications in business include production, personnel, inventory, marketing, financial, and blending problems. The objective is to optimize variables like costs, profits, or resources while meeting constraints.
- Assumptions of linear programming include certainty, linearity/proportionality, additivity, divisibility, non-negativity, finiteness, and optimality at corner points.
- A linear programming problem is modeled mathemat
The document discusses adapting a genetic algorithm to schedule variants of manufacturing shop models. It presents two examples - a flow shop and a shop with alternative machine routing - to demonstrate how genetic algorithms can be applied to schedule jobs for more complex shop models. The genetic algorithm is implemented using standard spreadsheet software, showing how simple it is to apply genetic algorithms to production scheduling problems.
Master thesis job shop generic time lag max plusSiddhartha Verma
The document discusses solving the job shop scheduling problem with time lags using max-plus algebra. It begins with motivations and an outline. It then provides background on max-plus algebra, the job shop problem with time lags, and previous related works. The document models the job shop problem with time lags constraints using max-plus algebra, representing the constraints as potential inequalities and disjunctive constraints. It presents case studies, including a naval industry example of a 3x3 job shop problem with time lags, modeled with max-plus matrices. The document analyzes feasibility of schedules using spectral theory and max-plus algebra properties.
This document discusses resource optimization and linear programming. It defines optimization as finding the best solution to a problem given constraints. Linear programming is introduced as a mathematical technique to optimize allocation of scarce resources. The key components of a linear programming model are described as decision variables, an objective function, and constraints. Graphical and algebraic methods for solving linear programming problems are also summarized.
Mathematical Optimisation - Fundamentals and ApplicationsGokul Alex
My Session on Mathematical Optimisation Fundamentals and Industry applications for the Academic Knowledge Refresher Program organised by Kerala Technology University and College of Engineering Trivandrum, Department of Interdisciplinary Studies.
ON Fuzzy Linear Programming Technique Applicationijsrd.com
A solution approach based on fuzzy linear programming is proposed and applied to optimal machine scheduling problem. In this solution approach errors in the demand of various products during the next production period are considered to be fuzzy in nature. In conventional linear programming approach it is assumed that there is no error in the expected demand of various products. A fuzzy linear programming approach is proposed to obtain an optimal solution under fuzzy conditions. In the proposed method expected demand of products & profit are expressed by fuzzy set notations. The proposed fuzzy linear programming formulation is then transformed to an equivalent conventional linear programming problem and solutions obtained by solving this transformed linear programming problem. For illustration purpose the proposed method is applied to a profit maximization related machine scheduling problem.
Linear programming deals with optimizing a linear objective function subject to linear constraints. It involves determining the values of decision variables to maximize or minimize the objective function. The general linear programming model involves maximizing or minimizing a linear combination of n decision variables subject to m linear constraints, along with non-negativity restrictions on the decision variables. Formulating a linear programming problem involves identifying decision variables, expressing constraints and the objective function linearly in terms of the variables, and adding non-negativity restrictions.
This document discusses using a genetic algorithm to solve the job shop scheduling problem. It begins with an abstract that introduces using genetic algorithms for job shop scheduling. It then provides more details on the problem and discusses using genetic algorithms with modifications like generating the initial population using priority rules and incorporating critical block and disjunctive graph distance in the crossover and mutation operations. The document outlines the genetic algorithm approach with sections on chromosome representation and decoding, data structures, generating the initial population, crossover and mutation operations. The goal is to minimize the makespan value for job shop scheduling.
A case study on Machine scheduling and sequencing using Meta heuristicsIJERA Editor
Modern manufacturing systems are constantly increasing in complexity and become more agile in nature such
system has become more crucial to check feasibility of machine scheduling and sequencing because effective
scheduling and sequencing can yield increase in productivity due to maximum utilization of available resources
but when number of machine increases traditional scheduling methods e.g. Johnson‟s ,rule is becomes in
effective Due to the limitations involved in exhaustive enumeration, for such problems meta-heuristics has
become greater choice for solving NP hard problems because of their multi solution and strong neighbourhood
search capability in a reasonable time.
Design and Implementation of a Multi-Agent System for the Job Shop Scheduling...CSCJournals
This document proposes an agent-based parallel genetic algorithm for solving the job shop scheduling problem. The approach divides the genetic algorithm population into multiple subpopulations that are evolved independently in parallel on different hosts. Agents are used to create the initial populations in parallel and execute the genetic algorithm operations. The genetic algorithm runs in execution phases where subpopulations evolve independently, and migration phases where the subpopulations exchange solutions. Experimental results showed this parallel approach improves the performance over a non-parallel genetic algorithm.
Modified heuristic time deviation Technique for job sequencing and Computatio...ijcsit
The document discusses job sequencing and scheduling techniques. It begins by defining job sequencing as the arrangement of tasks to be performed in a specified order. It then discusses several algorithms and methods for solving job sequencing problems, including tabu search, fuzzy TOPSIS, genetic algorithms, Johnson's rule, and mathematical modeling approaches. It also covers classification of jobs, performance metrics, and discusses modified time deviation as a heuristic technique to arrange job sequences to minimize total elapsed time.
An Efficient Heuristic Algorithm for Flexible Job Shop Scheduling with Mainte...mathsjournal
This paper deals with the flexible job shop scheduling problem with the preventive maintenance constraints where the objectives are to minimize the overall completion time (makespan), the total workload of machines and the workload of the most loaded machine. A fast heuristic algorithm based on a constructive procedure is developed to solve the problem in very short time. The algorithm is tested on the benchmark instances from the literature in order to evaluate its performance. Computational results show that, the proposed heuristic method is computationally efficient and promising for practical problems.
Parallel Line and Machine Job Scheduling Using Genetic AlgorithmIRJET Journal
This document summarizes research on using a genetic algorithm to solve a parallel line and machine job scheduling problem. The objective is to minimize the makespan (completion time) of all jobs. A genetic algorithm is applied that represents possible job schedules as chromosomes. The fitness function evaluates schedules based on makespan. Genetic operators like crossover and mutation create new schedules from existing ones. The algorithm was tested on sample job data using a MATLAB program. Results showed the genetic algorithm approach can find optimal job allocations and schedules that minimize makespan for parallel line scheduling problems.
This paper examines using a shifting bottleneck heuristic algorithm to minimize the makespan of a job shop production system at Dejena Aviation Industry in Ethiopia. The authors collected secondary data on five machines processing five jobs. Applying the shifting bottleneck algorithm resulted in an 8.33% reduction in total makespan. Machine one (41%) and three (36%) were the least utilized, while machine three (64%) and five (59%) were the busiest. The paper reviews literature on job shop scheduling and shifting bottleneck approaches, provides background on modeling job shops, and describes the shifting bottleneck algorithm used in the study.
This document summarizes a research paper that investigates minimizing the makespan of a job shop production system at Dejena Aviation Industry using a shifting bottleneck heuristic algorithm. The paper analyzes production data from 5 machines and 5 jobs. It finds that applying the shifting bottleneck algorithm reduces the total makespan by 8.33%. Machine 1 and 3 are the least utilized at 41% and 36% respectively, while machine 3 and 5 are the busiest at 64% and 59%. The algorithm iteratively identifies the bottleneck machine and reschedules jobs to minimize the maximum lateness time until an optimal solution is reached.
This document summarizes a research paper that investigates minimizing the makespan of a job shop production system at Dejena Aviation Industry using a shifting bottleneck heuristic algorithm. The paper analyzes production data from 5 machines and 5 jobs. It finds that applying the shifting bottleneck algorithm reduces the total makespan by 8.33%. Machine 1 and 3 are the least utilized at 41% and 36% respectively, while machine 3 and 5 are the busiest at 64% and 59%. The algorithm iteratively identifies bottleneck machines and reschedules jobs to minimize the maximum lateness time and reduce the makespan.
This document summarizes a research paper that investigates minimizing the makespan of a job shop production system at Dejena Aviation Industry using a shifting bottleneck heuristic algorithm. The paper analyzes production data from 5 machines and 5 jobs. It finds that applying the shifting bottleneck algorithm reduces the total makespan by 8.33%. Machine 1 and 3 are the least utilized at 41% and 36% respectively, while machine 3 and 5 are the busiest at 64% and 59%. The algorithm iteratively identifies bottleneck machines and reschedules jobs to minimize the maximum lateness time and reduce the makespan.
Smooth-and-Dive Accelerator: A Pre-MILP Primal Heuristic applied to SchedulingAlkis Vazacopoulos
This article describes an effective and simple primal heuristic to greedily encourage a reduction in the number of binary or 0-1 logic variables before an implicit enumerative-type search heuristic is deployed to find integer-feasible solutions to “hard” production scheduling problems. The basis of the technique is to employ well-known smoothing functions used to solve complementarity problems to the local optimization problem of minimizing the weighted sum over all binary variables the product of themselves multiplied by their complement. The basic algorithm of the “smooth-and-dive accelerator” (SDA) is to solve successive linear programming (LP) relaxations with the smoothing functions added to the existing problem’s objective function and to use, if required, a sequence of binary variable fixings known as “diving”. If the smoothing function term is not driven to zero as part of the recursion then a branch-and-bound or branch-and-cut search heuristic is called to close the procedure finding at least integer-feasible primal infeasible solutions. The heuristic’s effectiveness is illustrated by its application to an oil-refinery’s crude-oil blendshop scheduling problem, which has commonality to many other production scheduling problems in the continuous and semi-continuous (CSC) process domains.
In industries, the completion time of job problems in the manufacturing unit has risen significantly. In several types of current study, the job's completion time, or makespan, is reduced by taking straight paths, which is time-consuming. In this paper, we used an Improved Ant Colony Optimization and Tabu Search (ACOTS) algorithm to solve this problem by precisely defining the fault occurrence location in order to rollback. We have used a short-term memory-based rollback recovery strategy to minimise the job's completion time by rolling back to its own short-term memory. The recent movements in Tabu quest are visited using short term memory. As compared to the ACO algorithm, our proposed ACOTS-Cmax solution is more efficient and takes less time to complete.
AN IMPROVE OBJECT-ORIENTED APPROACH FOR MULTI-OBJECTIVE FLEXIBLE JOB-SHOP SCH...ijcsit
Flexible manufacturing systems are not easy to control and it is difficult to generate controlling systems for this problem domain. Flexible job-shop scheduling problem (FJSP) is one of the instances in this domain. It is a problem which acquires the job-shop scheduling problems (JSP). FJSP has additional routing subproblem in addition to JSP. In routing sub-problem each task is assigned to a machine out of a set of capable machines. In scheduling sub-problem, the sequence of assigned operations is obtained while optimizing the objective function(s). In this work an object-oriented (OO) approach with simulated annealing algorithm is used to simulate multi-objective FJSP. Solution approaches provided in the literature generally use two-string encoding scheme to represent this problem. However, OO analysis, design and programming methodology helps to present this problem on a single encoding scheme effectively which result in a practical integration of the problem solution to manufacturing control systems where OO paradigm is frequently used. Three parameters are considered in this paper: maximum completion time, workload of the most loaded machine and total workload of all machines which are the benchmark used to show the propose system achieve effective result.
Flexible manufacturing systems are not easy to control and it is difficult to generate controlling systems for this problem domain. Flexible job-shop scheduling problem (FJSP) is one of the instances in this domain. It is a problem which acquires the job-shop scheduling problems (JSP). FJSP has additional routing subproblem in addition to JSP. In routing sub-problem each task is assigned to a machine out of a set of capable machines. In scheduling sub-problem, the sequence of assigned operations is obtained while optimizing the objective function(s). In this work an object-oriented (OO) approach with simulated annealing algorithm is used to simulate multi-objective FJSP. Solution approaches provided in the literature generally use two-string encoding scheme to represent this problem. However, OO analysis, design and programming methodology helps to present this problem on a single encoding scheme effectively which result in a practical integration of the problem solution to manufacturing control systems where OO paradigm is frequently used. Three parameters are considered in this paper: maximum completion time, workload of the most loaded machine and total workload of all machines which are the benchmark used to show the propose system achieve effective result.
A Discrete Firefly Algorithm for the Multi-Objective Hybrid Flowshop Scheduli...Xin-She Yang
The document describes a discrete firefly algorithm proposed to solve hybrid flowshop scheduling problems with two objectives: minimizing makespan and mean flow time. Hybrid flowshop scheduling problems involve scheduling jobs through multiple stages with parallel machines in some stages, and are known to be NP-hard. The proposed discrete firefly algorithm adapts the continuous firefly algorithm to the discrete problem by using a smallest position value rule to map continuous firefly positions to discrete job permutations. Computational experiments show the proposed algorithm outperforms other metaheuristics for hybrid flowshop scheduling problems.
A Model for Optimum Scheduling Of Non-Resumable Jobs on Parallel Production M...IOSR Journals
This document presents a mathematical model for optimizing the scheduling of non-resumable jobs on parallel production machines that are subject to multiple fixed periods of unavailability. The problem is formulated as an integer linear programming model to minimize the makespan. Two algorithms are proposed to solve the problem - the first uses the longest processing time rule to generate an initial solution and upper bound, and the second is a greedy algorithm that iteratively achieves an optimal solution by backtracking. The algorithms are implemented in software and validated using numerical examples and industrial case studies, showing makespan reductions of 49% and 40% compared to actual industry scheduling.
This document presents a mathematical model for optimizing the scheduling of non-resumable jobs on parallel production machines that are subject to multiple fixed periods of unavailability. The problem is formulated as an integer linear programming model to minimize the makespan. Two algorithms are proposed to solve the problem - the first uses the longest processing time rule to generate an initial solution and upper bound, and the second is a greedy algorithm that iteratively achieves an optimal solution by backtracking. The algorithms are implemented in software and validated using numerical examples and industrial case studies, showing makespan reductions of 49% and 40% compared to actual industry scheduling.
IMPROVED MUSIC BASED HARMONY SEARCH (IMBHS) FOR SOLVING JOB SHOP SCHEDULING P...ijpla
The document describes an Improved Music Based Harmony Search (IMBHS) algorithm for solving Job Shop Scheduling Problems (JSSPs). The traditional Music Based Harmony Search (MBHS) algorithm uses fixed parameters that can limit its performance. The IMBHS algorithm improves on MBHS by adjusting the Pitch Adjustment Rate (PAR) and bandwidth (bw) parameters during the search process - using larger PAR and smaller bw in later iterations to fine tune solutions. The authors apply both MBHS and IMBHS to benchmark JSSP instances and find that IMBHS often finds better quality solutions than MBHS and benchmark solutions. The results suggest IMBHS is an effective new approach for solving JSSPs.
EFFICIENT DISPATCHING RULES BASED ON DATA MINING FOR THE SINGLE MACHINE SCHED...cscpconf
This document discusses using data mining techniques to generate new dispatching rules for single machine scheduling problems based on solutions obtained from a genetic algorithm. Specifically, it proposes using decision trees to extract dispatching rules from a genetic algorithm's scheduling of jobs on a single machine. Two heuristics are introduced to address potential issues with inversions in job order that could arise from the decision tree's pairwise job comparisons. An experiment is conducted using 125 benchmark problems to compare the performance of the generated dispatching rules to other rules from literature. The genetic algorithm finds the best known solution for 32 of the problems.
Efficient dispatching rules based on data mining for the single machine sched...csandit
In manufacturing the solutions found for scheduling
problems and the human expert’s
experience are very important. They can be transfor
med using Artificial Intelligence techniques
into knowledge and this knowledge could be used to
solve new scheduling problems. In this
paper we use Decision Trees for the generation of n
ew Dispatching Rules for a Single Machine
shop solved using a Genetic Algorithm. Two heuristi
cs are proposed to use the new Dispatching
Rules and a comparative study with other Dispatchin
g Rules from the literature is presented.
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A Hybrid Evolutionary Optimization Model for Solving Job Shop Scheduling Problem using GA and SA
1. IOSR Journal of Computer Engineering (IOSR-JCE)
e-ISSN: 2278-0661,p-ISSN: 2278-8727, Volume 17, Issue 6, Ver. I (Nov – Dec. 2015), PP 16-24
www.iosrjournals.org
DOI: 10.9790/0661-17611624 www.iosrjournals.org 16 | Page
A Hybrid Evolutionary Optimization Model for Solving Job
Shop Scheduling Problem using GA and SA
Dr. S. Jayasankari
Assistant Professor, Department of Computer Science,
P.K.R. Arts College for Women, Gobichettipalayam, Tamilnadu, India
Email id: jai.nivi.ravi@gmail.com
Abstract: The heuristic optimization techniques were commonly used in solving several optimization
problems. The present work aims to develop a hybrid algorithm to solve the scheduling optimization problem of
JSSP. There are different variants of these algorithms that were addressed in several previous works. The
impacts of these two kinds (Genetic Algorithm (GA) and Simulated Annealing (SA) based optimization model)
of initial condition on the performance of these two algorithms were studied using the convergence curve and
the achieved makespan. Even though genetic algorithm performed better than other evolutionary algorithms, it
has some weakness. During running GA, sometimes, it will produce same result without any improvement. SA
has a mechanism to overcome from that situation. During SA, if same result will be repeated, then it is rapidly
changing the change in temperature variable and re-initiates another random search. By using this feature of
SA, it has been implemented a hybrid based evolutionary model for solving JSSP by improving GA.
Comparison has been made with the performance of the proposed SA-GA-Hybrid model with GA as well as SA.
Keywords: Job Shop Scheduling Problem, Optimization, Genetic Algorithm, Simulated Annealing.
I. Introduction
Scheduling is an optimization method to make the best possible way to use the limited resources by
making a suitable allocation of the constraint resources over a period of time. Those problems are combinatorial
optimization problems, where a set of possible solutions is distinct or can be reduced to a discrete one, and the
goal is to find the best possible solution. Scheduling falls under the NP-hard class of combinatorial optimization
problems category. Developing of job processing schedules in a scheduling environment is a very difficult task.
The number of possible schedules increases with the increase in the number of jobs and related operations. This
growth makes it practically impossible to use mathematical programming or thorough search-based approaches
to find the global optimum schedules.
In the recent competitive atmosphere in manufacturing and service industries, the operational
sequencing and scheduling has become a crucial for survival in the marketplace. Companies have to produce
their product advance as opposed to due date. Otherwise, it will affect upon reputation of a business. At the same
time, the activities and operations need to be scheduled with the intention that the available resources will be
used in an efficient manner. As a result, there is a great good scheduling algorithm and heuristics are invented.
Most of the principal practical scheduling problems exist in stochastic and dynamic environment.
Stochastic is a problem in that some of the variables are undefined while dynamic problem is when jobs
arrive randomly. In another way, the problems with ready time is known and fixed are known problems such as
static and for problem where all the parameter such as processing times are known and fixed is called
deterministic problems. Regardless of this, it is impossible to predict exactly when jobs will become available for
processing.
The core objective in solving the job shop scheduling problem is to find the sequence for each operation
on each machine that optimizes the objective function. The objective function that has been used in scheduling
the job shop problem is minimization of makespan value or the time to complete all jobs.
1.1 Problem Definition
The Job Shop Scheduling Problem (JSSP) may be described as follows: Given „n‟ jobs, each composed
of several operations that must be processed on „m‟ machines. Each operation uses one of the „m‟ machines for a
fixed duration. Each machine can process at most one operation at a time and once an operation initiates
processing on a given machine it must complete processing on that machine without interruption. The
operations of a given job have to be processed in a given order. The problem consists in finding a
schedule of the operations on the machines, taking into account the precedence constraints, that
minimizes the makespan (Cmax), that is, the finish time of the last operation completed in the schedule.
The focus on job-shop scheduling problems composed of the following elements:
2. A Hybrid Evolutionary Optimization model for Solving Job Shop Scheduling Problem using GA…
DOI: 10.9790/0661-17611624 www.iosrjournals.org 17 | Page
Jobs: J = {J1 , · · · , Jn } is a set of n jobs to be scheduled. Each job Ji consists of a predetermined sequence
of operations. Oi,j is the operation j of Ji. All jobs are released at time 0.
Machines: M = {M1 , · · · , Mm } is a set of m machines. Each machine can process only one operation at a
time. And each operation can be processed without interruption during its performance on one of the set of
machines. All machines are available at time 0.
Constraints: The constraints are rules that limit the possible assignments of the operations. They can be
divided mainly into following situations:
- Each operation can be processed by only one machine at a time (disjunctive constraint).
- Each operation, which has started, runs to completion (non-preemption condition).
- Each machine performs operations one after another (capacity constraint).
- Although there are no precedence constraints among operations of different jobs, the predetermined
sequence of operation for each job forces each operation to be scheduled after all predecessor operations
(precedence/conjunctive constraint).
- The machine constraints emphasize the operations can be processed only by the machine from the given
set (resource constraint).
Objective: The objective is to find a schedule that has minimum time required to complete all operations
(minimum makespan).
1.2 Different Approaches for Solving Scheduling Problems
Job shop scheduling problem is a NP-hard problem with no easy solution. Branch-and-bound, Tabu
search, and biologically inspired approaches such as GA, Swarm Intelligence and other stochastic model such as
Simulated Annealing algorithm were proposed for achieving possible solutions to complex problems such as job
shop scheduling.
SA is a stochastic heuristic algorithm which is used to resolve combinatorial optimization problems.
Simulated Annealing optimization is analogous to the annealing of metals. This Simulated Annealing is different
from other algorithms, such that SA uses a probability mechanism to have power over the process of jumping out
of the local minimum. In the process of searching, SA is not only accepting better solutions, but also accepting
worse solutions randomly with a certain probability. At high temperatures, the probability of accepting better
solutions is comparatively big. With the decrease of the temperature, the chance of accepting worse solutions also
move downs, and when the temperature closes in upon zero, SA no longer accepts any worse solution. These
make SA have more chance to avoid getting trapped in a local minimum and avoid the limitation of other local
search algorithms and the gradient algorithms. Because of its qualities above, SA has become an enormously
accepted method for solving large-sized and practical problems like job shop scheduling, timetabling, traveling
salesman and packing problem. However, like many other search algorithms, SA may get trapped in a local
minimum or take a long time to find a reasonable solution. For these reasons, SA is often used as a part of a
hybrid method.
II. The JSSP And The Hybrid Evolutionary Model For Solving JSSP
Let J = {0, 1, …, n, n+1} denote the set of operations to be scheduled and M = {1,..., m} the set of
machines. The operations 0 and n+1 are dummy, have no duration and represent the initial and final
operations. The operations are interrelated by two kinds of constraints. First, the precedence constraints, which
force each operation j to be scheduled after all predecessor operations, Pj, are completed. Second,
operation j can only be scheduled if the machine it requires is idle. Further, let dj denote the (fixed)
duration (processing time) of operation j. Let Fj represent the finish time of operation j. A schedule can be
represented by a vector of finish times (F1, , Fm, ... , Fn+1). Let A(t) be the set of operations being processed at
time t, and let rj,m = 1 if operation j requires machine m to be processed and rj,m = 0 otherwise.
The conceptual model of the JSP can be described the following way (Goncalvesetal 2005):
Minimize Fn+1 (Cmax) ……….(1)
Subject to:
Fk ≤ Fj – dj j-1,…,n+2 ; k ϵ Pj ……….(2)
( ) ,jA t j m
r ≤ 1 m ϵ M ; t ≥ 0 ……….(3)
3. A Hybrid Evolutionary Optimization model for Solving Job Shop Scheduling Problem using GA…
DOI: 10.9790/0661-17611624 www.iosrjournals.org 18 | Page
Fj ≥ 0 j=1,….,n+1 ……….(4)
The objective function (1) minimizes the finish time of operation n+1 (the last operation), and
therefore minimizes the makespan. Constraints (2) impose the precedence relations between operations
and constraints (3) state that one machine can only process one operation at a time. Finally (4) forces the finish
times to be non- negative. The JSP is amongst the hardest combinatorial optimization problems. The JSP is NP-
hard (Lenstra and Rinnooy Kan, 1979), and has also proven to be computationally challenging.
2.1 Simulated Annealing
Simulated Annealing (SA) is motivated by an analogy to annealing in solids. The idea of SA comes
from a paper published by Metropolis etc al in 1953 [Metropolis, 1953]. The algorithm in this paper simulated
the cooling of material in a heat bath. This is a process known as annealing. If a solid is heated up to the melting
point of that metal and then cool it, the structural properties of the solid depend on the rate of cooling. If the liquid
metal is cooled very slowly, large crystals will be formed. However, if the liquid is cooled quickly the crystals
will contain imperfections. Metropolis's algorithm simulated the material as a system of particles. This algorithm
simulates the cooling process by slowly lowering the temperature of the system until it converges to a steady,
frozen state. In 1982, Kirkpatrick et al (Kirkpatrick, 1983) took the idea of the Metropolis algorithm and applied
it to optimization problems. The idea is to use simulated annealing to search for feasible solutions and converge
to an optimal solution.
Simulated annealing refers to the annealing process done on a computer by simulation. In this model, a
parameter T, equivalent to temperature in annealing, is reduced slowly. The law of thermodynamics state that at
temperature, t, the probability of an increase in energy of magnitude, δE, is given by (Kirkpatrick et al 1983):
P(δE) = exp(-δE /kt)
Where k is a constant known Boltzmann's constant.
The simulation in the Metropolis algorithm calculates the new energy of the system. If the energy has
decreased then the systems move to this state. If the energy has increased then the new state is established using
the probability returned by the above formula. A certain number of iterations are carried out at each temperature
and then the temperature is decreased. This is repeated until the system freezes into a steady state.
The following equation is directly used in simulated annealing, although it is usual to drop the Boltzmann
constant as this was only introduced into the equation to cope with different materials. Therefore, the probability
of accepting a worse state is given by the equation (Kirkpatrick et al 1983):
P = exp(-c/t) > r
Where
c = the change in the evaluation function
t = the current temperature
r = a random number between 0 and 1
The probability of accepting a worse move is a function of both the temperature of the system and of the
change in the cost function. It can be valued that as the temperature of the system decreases the probability of
accepting a worse move is decreased. This is the same as slowly moving to a frozen state in physical annealing.
Also note, that if the temperature is zero then only better moves will be accepted which effectively makes
simulated annealing act like hill climbing. The following algorithm is mentioned by Russell, 1995.
Function SIMULATED-ANNEALING(Problem, Schedule) returns a solution state
Inputs :Problem, a problem
Schedule, a mapping from time to temperature
Local Variables :Current, a node
Next, a node
T, a “temperature” controlling the probability of downward steps
Current = MAKE-NODE(INITIAL-STATE[Problem])
For t = 1 to do
T = Schedule[t]
If Termination Condition then
return Current
Next = a randomly selected successor of Current
E = fittness[Next] – fittness[Current]
if E > 0 then
Current = Next
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DOI: 10.9790/0661-17611624 www.iosrjournals.org 19 | Page
else if (exp(-E/T) > probability then
Current = Next
2.2 Genetic Algorithm (GA)
Genetic algorithms are adaptive methods, which may be used to solve search and optimization
problems (Beasley et al. (1993)). They are based on the genetic process of biological organisms. Over many
generations, natural populations evolve according to the principles of natural selection, i.e. Survival of the fittest,
first clearly stated by Charles Darwin in The Origin of Species. By simulating this process, genetic algorithms
are able to evolve solutions to real world problems.
Before a genetic algorithm can be run, an appropriate encoding (or representation) for the problem must
be developed. A fitness function is also required, which assigns a figure of merit to each encoded solution.
During the run, parents must be selected for reproduction, and recombined to generate offspring. It is
assumed that a solution to a problem may be represented as a set of parameters. These parameters
(known as genes) are coupled together to form a string of values (chromosome). In genetic terminology, the set
of parameters represented by a particular chromosome is referred to as an individual. The fitness of an
individual depends on its chromosome and is evaluated by the fitness function.
The individuals, during the reproductive phase, are selected from the population and recombined,
producing offspring, which involve the next generation. Parents are randomly selected from the population
using a scheme, which favors fitter individuals. Having selected two parents, their chromosomes are
recombined, typically using mechanisms of crossover and mutation. Mutation is usually applied to some
individuals, to guarantee population diversity. The basic genetic pseudo code is as follows.
Genetic Algorithm
{
Generate initial population Pt
Evaluate population Pt
While stopping criteria not satisfied Repeat
{
Select elements from Pt to copy into Pt+l
Crossover elements of Pt and put into Pt+l
Mutation elements of Pt and put into Pt+l
Evaluate new population Pt+l
Pt = Pt+l
}
}
2.3 Proposed Hybrid Evolutionary Optimization Model
The following steps explains the Proposed SA-GA-Hybrid Model for solving JSSP
1. Initialize the solution X of size [p x (rxc)] (where r x c is the size of the JSSP and p is the population size)
2. For Each Generation Repeat 3 to 6
3. Xmin X1 ; Fmin Fittness(X1) ; Current X1
4. Do GA based perturbation on X and produce Xnew
T = Schedule[t]
If SA_Termination_Condition then
Go to 8
Xnew= GeneticOperationsOn(X)
5. Calculate Fitness of Xnew
Fnew= Fittness(Xnew)
Where
Fnew be the p x 1 matrix which will contain the
individual fitness values.
6. Main SA based Check
[FSorted, Index] sort(Fnew) ;
Next FSorted (top)
E = fittness[Next] – fittness[Current]
if E > 0 then
Current Next
else if (exp(-E/T) > probability then
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Current Next
7. X(Index (top)) Current
go to step 4
8. Finally return the results
Makespan fittness[Current]
BestSolution Current
Stop.
The following is an example for a 3 x 3 JSSP.
Table 1 : A 3x3 JSSP
Job Operations routing (processing time)
1 1(3) 2(3) 3(3)
2 1(2) 3(3) 2(4)
3 2(3) 1(2) 3(1)
The one of the optimum schedule of the above problem is [ J1,1, J3,1, J2,1, J1,2, J3,2, J2,2, J2,3, J1,3, J3,3 ]. Here, for
example, J1,2 represents the operation 2 of the Job 1. Figure1shows one of such a optimum solution for the
problem represented by “Gantt-Chart".
Figure 1 : The Gantt-Chart Representation of the Solution of the above 3x3 Problem
If we denote the operations of the job as follows,
Job1: Op1, Op2, Op3
Job2: Op4, Op5, Op6
Job3: Op7, Op8, Op9
Or simply
1 2 3
4 5 6
7 8 9
then, the schedule [ J1,1, J1,2, J1,3, J2,1, J2,2, J2,3, J3,1, J3,2, J3,3] or simply [1, 2, 3, 4, 5, 6, 7, 8, 9] will be
the one of the known worst case schedule which will satisfy all the conditions of the JSSP. But in this case, the
makespan will not be optimum. The schedule [ J1,1, J3,1, J2,1, J1,2, J3,2, J2,2, J2,3, J1,3, J3,3 ] or simply [1, 7, 4, 2, 8, 5,
6, 3, 9] will be the one of the best know optimum solution. Here the strings “1, 2, 3, 4, 5, 6, 7, 8, 9” and “1, 7, 4,
2, 8, 5, 6, 3, 9” represents solutions and known as valid strings.
In GA, a legal string or a illegal string (of numbers) which represent the order of the schedule can be
represented by a chromosome. For example, the known worst case solution can be represented as a chromosome
of GA by a string “1, 2, 3, 4, 5, 6, 7, 8, 9”. Similarly, the chromosome of GA “1, 7, 4, 2, 8, 5, 6, 3, 9” will
represent a legal string which is an optimal solution of JSSP. And for example, the chromosome “3, 9, 4, 2, 1, 5,
6, 7, 8” will be a invalid string which correspond to a illegal operation or schedule since this schedule will not
satisfy the conditions of JSSP.
So, if the initial selection of chromosomes of GA with random values, then there will be lot of invalid
strings in the initial guess. The scope of the evolutionary algorithm is to permute the most optimal string to better
most optimal string which will hopefully make that string as a legal string in proceeding generations/steps and
finally it will end up with a string belongs to a better solution or optimal schedule with minimum makespan.
This assumption will be good and can produce meaningful solutions for lower order scheduling
problems such as 3x3 JSSP or 4x4 JSSP. But, it may produce illegal solutions even after very long runs in the
case of higher order scheduling problems like 15x15 JSSP. Because, if we randomly chose initial population then
there will be much chance for getting all illegal strings in the initial set which belongs to no nearby solution. So
the fitness calculation methods will lead to meaningless fitness values and the selection method will also be
incapable of selecting a better solution in each generation or step. So in each step of the evolutionary process
there will not be any guaranty of getting progressive solution.
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So we believe that the random selection of initial solution/seed in an evolutionary algorithm will not lead
to a better result in higher order scheduling problems such as 15x15 JSSP. So in this work, we have evaluated the
performance of two evolutionary optimization techniques SA and GA with different initial conditions.
III. Results And Analysis
In the experiments, we have given the known worst case solution as initial “seed” for the evolutionary
process. We expect that, the system will be capable of producing at latest one meaningful legal string in every
generation/step/iteration and hence there will be a much good probability of achieving a better solution in the
succeeding generations or steps.
3.1 Analysis of Performance with Different Problem Size
The GA was run for 100 generations with population size of 100. The SA was run for 3000 iterations
since this simple SA will only handle one solution at a time.
The following figure shows a result (makespan=68) found by the GA based method for a 6x6 problem.
Figure 2 : Solution found by GA for a 6x6 Problem.
The following figure show a s result (makespan=77) found by the SA based method for a 6x6 problem.
Figure 3 : Solution found by SA for a 6x6 Problem
The following figure shows a result (makespan=62) found by the SA-GA-Hybrid based method for the same
6x6 problem.
Figure 4 : A Solutions found by SA-GA-Hybrid for a 6x6 Problem .
The following figure shows another result (makespan=61) found by the SA-GA-Hybrid based method for the
same 6x6 problem.
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Figure 5: Another Solutions found by SA-GA-Hybrid for a 6x6 Problem
The Following fugures show the average fitness of the two methods over generations. If we see these plots, we
can realize that both the algorithms behaves in a similar fashion.
Figure 6 : The Average Fitness Plot of the Solutions found by GA and SA-GA-Hybrid algorithm for a 6x6
Problem .
The Following fugures show the best fitness of the two methods over generations. If we see these plots, we can
realize that proposed SA-GA-Hybrid algorithm performes little bit better than the normal GA .
Figure 7 : The Best Fitness Plot of the Solutions found by GA and SA-GA-Hybrid algorithm for a 6x6 Problem
3.2 Performance in terms of speed
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To measure the performance in terms of speed, the model was run with problems of different sizes and the results
were tabulated as follows:
Table 2 : The time taken for different JSSP size
Sl.No JSSP Size
Time Taken(sec)
GA SA
SA-GA
Hybrid
1 3x3 2.87 8.44 2.86
2 4x4 3.61 9.14 3.47
3 6x6 5.70 10.51 5.61
4 10x10 16.75 12.72 16.85
5 15x15 58.28 19.82 55.73
The following figure shows the performance in terms of time with respect to different JSSP problem
size. If we carefully see the lines of GA and SA-GA Hybrid, then we can say that the performance of SA-GA
Hybrid is little bit better than GA in some cases and almost equal in some cases.
Figure 8 : JSSP Size vs Time Chart
3.3 Convergence Capability of the Algorithms
The convergence was measured in terms of makespan for different size of the problems was tabulated
below. We only considered the convergence up to the first 100 iterations in the case of GA and SA-GA-Hybrid
model during its run.
To measure the performance in terms of convergence, the model was tested with problems of different
sizes and the results were tabulated as follows:
Table 3 : Startup with known worst case solution
Sl.
No
JSSP
Achieved Optimal Solution
(Makespan)
Size
Known
optimum
value
GA SA
SA-GA
Hybrid
1 3x3 12 12 12 12
2 4x4 272 272 286 272
3 6x6 55 68 72 61
4 10x10 902 2214 2899 2011
5 15x15 1268 6771 8241 6637
The following figure shows the performance in terms of Makespan with respect to different JSSP problem size.
J S S P S ize vs C o n ve rg e n c e in te rm s o f M a k e s p a n
0
1 0 0 0
2 0 0 0
3 0 0 0
4 0 0 0
5 0 0 0
6 0 0 0
7 0 0 0
8 0 0 0
9 0 0 0
1 0 0 0 0
3 x3 4 x4 6 x6 1 0 x1 0 1 5 x1 5
J S S P S ize
Makespan.
G A S A S A -G A H yb rid
Figure 9: JSSP Size Vs Makespan Chart.
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DOI: 10.9790/0661-17611624 www.iosrjournals.org 24 | Page
Even though they arrived solution is far from the optimal solution, the better performance in the case of
SA-GA-Hybrid is very obvious.
IV. Conclusion And Future Work
The algorithms have successfully implemented two basic evolutionary models for solving JSSP using
GA & SA, and also the proposed SA-GA-Hybrid. They arrived results showing that the models produced optimal
or near-optimal solutions medium level job shop scheduling problems in a shot duration. While initializing with
known, worst case solution, the evolutionary process was capable of converging into meaningful and more
optimum solutions. Further, as shown in the convergence curves in the previous section, The SA-GA-Hybrid
performed in a very better way than GA. Future works may address the issued involved in improving the
performance of the SA-GA-Hybrid for high dimensional problems.
Acknowledgement
My sincere thanks to the management and principal of P.K.R. Arts College for Women,
Gobichettipalayam and friends, who have motivated & helped me towards the topic and development of this
research paper.
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