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.
Job Shop Scheduling Using Mixed Integer ProgrammingIJMERJOURNAL
ABSTRACT: In this study, four different models in terms of mixed integer programming (MIP) are formulated for fourdifferent objectives. The first model objective is to minimizethemaximum finishing time (Makespan) without considering the products’ due dates, while the second model is formulated to minimize the makespan considering the due dates for all the products, the third model is to minimize the total earliness time, and the fourth one is to minimize the total lateness time. The proposed models are solved, and their computational performance levels are compared based on parameters such as makespan, machine utilization, and time efficiency. The results are discussed to determine the best suitable formulation
Job Shop Scheduling Using Mixed Integer ProgrammingIJMERJOURNAL
ABSTRACT: In this study, four different models in terms of mixed integer programming (MIP) are formulated for fourdifferent objectives. The first model objective is to minimizethemaximum finishing time (Makespan) without considering the products’ due dates, while the second model is formulated to minimize the makespan considering the due dates for all the products, the third model is to minimize the total earliness time, and the fourth one is to minimize the total lateness time. The proposed models are solved, and their computational performance levels are compared based on parameters such as makespan, machine utilization, and time efficiency. The results are discussed to determine the best suitable formulation
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.
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.
A Hybrid Evolutionary Optimization Model for Solving Job Shop Scheduling Prob...iosrjce
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.
Hybridizing guided genetic algorithm and single-based metaheuristics to solve...IAESIJAI
This paper focuses on solving unrelated parallel machine scheduling with
resource constraints (UPMR). There are j jobs, and each job needs to be
processed on one of the machines aim at minimizing the makespan. Besides
the dependence of the machine, the processing time of any job depends on the
usage of a rare renewable resource. A certain number of those resources (Rmax)
can be disseminated to jobs for the purpose of processing them at any time,
and each job j needs units of resources (rjm) when processing in machine m.
When more resources are assigned to a job, the job processing time minimizes.
However, the number of resources available is limited, and this makes the
problem difficult to solve for a good quality solution. Genetic algorithm shows
promising results in solving UPMR. However, genetic algorithm suffers from
premature convergence, which could hinder the resulting quality. Therefore,
the work hybridizes guided genetic algorithm (GGA) with a single-based
metaheuristics (SBHs) to handle the premature convergence in the genetic
algorithm with the aim to escape from the local optima and improve the
solution quality further. The single-based metaheuristics replaces the mutation
in the genetic algorithm. The evaluation of the algorithm performance was
conducted through extensive experiments.
An efficient simulated annealing algorithm for a No-Wait Two Stage Flexible F...ijait
In this paper, no wait two stage flexible flow shop scheduling problem (FFSSP) is solved using two metaheuristic algorithms. This problem with minimum makespan performance measure is NP-Hard. The proposed algorithms are Simulated Annealing and Genetic Algorithm. The results are analyzed in terms of Relative Percentage Deviation of Makespan. The performance of the proposed algorithms are studied and compared with that of MDA algorithm. For this propose a number of problems in different sizes are solved. The results of the studies proposes the effective algorithm. This is followed by describing the
outline of the study, concluding remarks and suggesting potential areas for further researches
Artificial Neural Networks can achieve high degree of computation rates by
employing a massive number of simple processing elements with a high degree of
connectivity between elements. In this paper an attempt is made to present a Constraint
Satisfaction Adaptive Neural Network (CSANN) to solve the generalized job-shop
scheduling problem and it shows how to map a difficult constraint satisfaction job-shop
scheduling problem onto a simple neural net, where the number of neural processors equals
the number of operations, and the number of interconnections grows linearly with the total
number of operations. The proposed neural network can be easily constructed and can adjust
its weights of connections based on the sequence and resource constraints of the job-shop
scheduling problem during its processing. Simulation studies have shown that the proposed
neural network produces better solutions to job-shop scheduling problem.
An efficient simulated annealing algorithm for a No-Wait Two Stage Flexible F...ijait
In this paper, no wait two stage flexible flow shop scheduling problem (FFSSP) is solved using two metaheuristic algorithms. This problem with minimum makespan performance measure is NP-Hard. The proposed algorithms are Simulated Annealing and Genetic Algorithm. The results are analyzed in terms of Relative Percentage Deviation of Makespan. The performance of the proposed algorithms are studied and compared with that of MDA algorithm. For this propose a number of problems in different sizes are solved. The results of the studies proposes the effective algorithm. This is followed by describing the outline of the study, concluding remarks and suggesting potential areas for further researches
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.
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.
It is well-known that SRPT is optimal for minimizing
ow time on machines that run one job at a time.
However, running one job at a time is a big under-
utilization for modern systems where sharing, simultane-
ous execution, and virtualization-enabled consolidation
are a common trend to boost utilization. Such machines,
used in modern large data centers and clouds, are
powerful enough to run multiple jobs/VMs at a time
subject to overall CPU, memory, network, and disk
capacity constraints.
Motivated by this pr
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively
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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
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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
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effective Due to the limitations involved in exhaustive enumeration, for such problems meta-heuristics has
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A Hybrid Evolutionary Optimization Model for Solving Job Shop Scheduling Prob...iosrjce
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.
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Hybridizing guided genetic algorithm and single-based metaheuristics to solve...IAESIJAI
This paper focuses on solving unrelated parallel machine scheduling with
resource constraints (UPMR). There are j jobs, and each job needs to be
processed on one of the machines aim at minimizing the makespan. Besides
the dependence of the machine, the processing time of any job depends on the
usage of a rare renewable resource. A certain number of those resources (Rmax)
can be disseminated to jobs for the purpose of processing them at any time,
and each job j needs units of resources (rjm) when processing in machine m.
When more resources are assigned to a job, the job processing time minimizes.
However, the number of resources available is limited, and this makes the
problem difficult to solve for a good quality solution. Genetic algorithm shows
promising results in solving UPMR. However, genetic algorithm suffers from
premature convergence, which could hinder the resulting quality. Therefore,
the work hybridizes guided genetic algorithm (GGA) with a single-based
metaheuristics (SBHs) to handle the premature convergence in the genetic
algorithm with the aim to escape from the local optima and improve the
solution quality further. The single-based metaheuristics replaces the mutation
in the genetic algorithm. The evaluation of the algorithm performance was
conducted through extensive experiments.
An efficient simulated annealing algorithm for a No-Wait Two Stage Flexible F...ijait
In this paper, no wait two stage flexible flow shop scheduling problem (FFSSP) is solved using two metaheuristic algorithms. This problem with minimum makespan performance measure is NP-Hard. The proposed algorithms are Simulated Annealing and Genetic Algorithm. The results are analyzed in terms of Relative Percentage Deviation of Makespan. The performance of the proposed algorithms are studied and compared with that of MDA algorithm. For this propose a number of problems in different sizes are solved. The results of the studies proposes the effective algorithm. This is followed by describing the
outline of the study, concluding remarks and suggesting potential areas for further researches
Artificial Neural Networks can achieve high degree of computation rates by
employing a massive number of simple processing elements with a high degree of
connectivity between elements. In this paper an attempt is made to present a Constraint
Satisfaction Adaptive Neural Network (CSANN) to solve the generalized job-shop
scheduling problem and it shows how to map a difficult constraint satisfaction job-shop
scheduling problem onto a simple neural net, where the number of neural processors equals
the number of operations, and the number of interconnections grows linearly with the total
number of operations. The proposed neural network can be easily constructed and can adjust
its weights of connections based on the sequence and resource constraints of the job-shop
scheduling problem during its processing. Simulation studies have shown that the proposed
neural network produces better solutions to job-shop scheduling problem.
An efficient simulated annealing algorithm for a No-Wait Two Stage Flexible F...ijait
In this paper, no wait two stage flexible flow shop scheduling problem (FFSSP) is solved using two metaheuristic algorithms. This problem with minimum makespan performance measure is NP-Hard. The proposed algorithms are Simulated Annealing and Genetic Algorithm. The results are analyzed in terms of Relative Percentage Deviation of Makespan. The performance of the proposed algorithms are studied and compared with that of MDA algorithm. For this propose a number of problems in different sizes are solved. The results of the studies proposes the effective algorithm. This is followed by describing the outline of the study, concluding remarks and suggesting potential areas for further researches
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.
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.
It is well-known that SRPT is optimal for minimizing
ow time on machines that run one job at a time.
However, running one job at a time is a big under-
utilization for modern systems where sharing, simultane-
ous execution, and virtualization-enabled consolidation
are a common trend to boost utilization. Such machines,
used in modern large data centers and clouds, are
powerful enough to run multiple jobs/VMs at a time
subject to overall CPU, memory, network, and disk
capacity constraints.
Motivated by this pr
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OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
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algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
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a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
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OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
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audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
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Mathematics subject classifications: 45H10, 54H25
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
Table of Contents - September 2022, Volume 9, Number 2/3mathsjournal
Applied Mathematics and Sciences: An International Journal (MathSJ ) aims to publish original research papers and survey articles on all areas of pure mathematics, theoretical applied mathematics, mathematical physics, theoretical mechanics, probability and mathematical statistics, and theoretical biology. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
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Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdf
An Efficient Heuristic Algorithm for Flexible Job Shop Scheduling with Maintenance Constraints
1. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
19
AN EFFICIENT HEURISTIC ALGORITHM FOR
FLEXIBLE JOB SHOP SCHEDULING WITH
MAINTENANCE CONSTRAINTS
Mohsen Ziaee∗
Department of Industrial Engineering, University of Bojnord, 94531-55111
Bojnord, Iran
Abstract
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.
Keywords
Scheduling, Multi-Objective Flexible Job Shop, Preventive Maintenance, Heuristic, Local Search.
1.Introduction
The job shop scheduling problem (JSP) is one of the most popular scheduling problems and has
attracted many researchers due to both its practical importance and its complexity [1]. In the n×m
classical JSP, a set of n jobs have to be processed on a group of m machines, where the processing
of each job i consists of Ji operations performed on these machines. Each job has a processing
order on the machines which is fixed and known in advance, i.e., each operation has to be
performed on a given machine. The processing times of all operations are also fixed and known.
Each machine can process at most one operation at a time, and the operations are processed on the
machines without interruption [2,3]. A typical performance indicator for the JSP is the makespan,
i.e., the time needed to complete all the jobs.
The flexible job shop scheduling problem (FJSP) is a generalization of the classical JSP, in which
each operation is allowed to be processed by any among set of candidate machines, instead of a
given machine; and thus, the scheduling problem is to choose for each operation, a machine and a
starting time at which the operation must be processed. The FJSP is more difficult than the
classical JSP because it contains an additional problem which is to determine the job routes, or to
assign the operations to the machines. This problem is known to be strongly NP-hard even if each
job has at most three operations and there are two machines [4].
The FJSP with PM constraints is to assign each operation to an appropriate machine out of a set
2. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
20
of machines capable of executing it, and to sequence the job operations and PM operations on
each machine, in order to minimize one or more criteria. This paper presents a solution method to
solve this problem in order to minimize the following three objectives:
(1) The maximum completion time of the machines, i.e. the makespan (Cmax).
(2) The total workload of the machines, which represents the total working time of all
machines (WT). This objective is of interest if machines differ with respect to the efficiency.
(3) The maximal machine workload, i.e., the maximum working time spent on any machine
(Wmax). This objective is used to prevent assignment of too much work to a single machine and to
keep the balance of work distribution over the machines.
In this article, the weighted sum of the above three objectives is taken as the objective function.
This approach, i.e. the weighted sum, in dealing with the multi-objective optimization, is easy for
decision makers to understand, convenient for developers to implement, and available to modify
the weights for satisfying the requirements of decision makers. In this study, the PM periods are
also included in the above three criteria, since the PM periods are mandatory and the scheduler
has to schedule them along with the jobs to be processed. The objective function of the studied
problem is therefore computed as (1):
Minimize Obj=W1×Cmax+W2×WT+W3×Wmax
(1)
such that: W1+W2+W3=1; W1, W2, W3 ≥ 0.
W1, W2, and W3 represent the weight coefficients of the three objectives.
The FJSP with PM constraints is strongly NP-hard, since the problem without PM scheduling is
already strongly NP-hard [4]. Therefore, as it can be seen in the literature review section,
approximate algorithms, mainly metaheuristics, have been used to solve the problem.
In this study, a simple and easily extendable heuristic algorithm based on a constructive
procedure is presented to solve the FJSP with PM constraints (section 3). The main purpose is to
produce reasonable and applicable schedules very quickly. It can also be used to improve the
quality of the initial feasible solution of metaheuristics applied to solve the problem, since the
choice of a good initial solution is an important aspect of the performance of algorithms in terms
of computing time and solution quality [5,6,7]. In order to evaluate the performance of the
proposed heuristic, it is implemented using several benchmark problems and the results of the
computational experiments are presented (section 4). The results show that our novel method can
obtain good solutions in very short time. Concluding remarks are given in the last section.
Assumptions considered in this paper are as follows:
(1) There is only one PM operation on each machine during the planning horizon. Each PM
operation has to be performed within a predefined time window, and its duration is determined in
advance.
(2) Jobs are independent of each other.
(3) Machines are independent of each other.
(4) Setup and transportation times are negligible.
(5) Preemption is not allowed, i.e. a started operation (either operation of a job or a PM
operation) can not be interrupted during its processing.
(6) Each machine can process at most one operation (either job operation or PM operation)
at the same time.
(7) An operation can not be performed by more than one machine at the same time.
(8) All jobs have equal priorities.
3. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
21
(9) Machines never break down and are available throughout the scheduling period.
(10) All jobs are available at time zero.
The notations used throughout the paper are as follows:
n: number of jobs,
m: number of machines,
i,z: index of jobs; i,z=1,…,n,
Ji: number of operations of job i,
maxJ: maximum number of operations per job (i.e., maxJ= i
J
i
max ),
j: index of operations; j=1,…, Ji,
k,y: index of machines; k,y=1,…,m,
tijy: processing time of operation j of job i on machine y,
Cij: completion time of operation j of job i,
WT: total workload of machines (the workload of each machine is defined by the sum of
processing times of all operations assigned to it),
Wmax: critical machine workload, i.e. the workload of the most loaded machine,
dury: duration of the preventive maintenance task of machine y,
(tEy , tLy): time window associated with the PM task on machine y, where tEy is the earliest
possible completion time, and tLy is the latest possible completion time.
2.Literature Review
The vast majority of papers dealing with scheduling problems assume that the machines are
continuously available for processing during the whole planning horizon. However, this
assumption may not be true in real industrial settings, since the machines may become
unavailable during the planning period for many reasons such as unforeseen breakdowns
(stochastic unavailability [8]), or due to the use of the equipments for planned activities such as
preventive maintenance (PM) tasks that conflict with scheduling decisions (i.e. deterministic
unavailability, in which the periods of unavailability are known in advance).
Recently, many researchers began to study the scheduling problems with PM constraints, because
PM is considered as a common reason for machine unavailability. PM is used by manufacturing
industries for following reasons: PM can reduce the probability of unforeseen breakdowns; in
manufacturing systems that use PM, if a machine breaks down, then its unavailability time
interval is expected to be significantly reduced; and therefore PM can restore the reliability of
machines and improve the machine utilization ratio. This paper deals with the FJSP with PM
constraints and assumes that the starting time of PM operations is not known in advance and must
be determined during the production scheduling process. However, each PM operation has to be
executed within a given time window. These constraints are referred to as non-fixed availability
constraints in the literature. In fixed availability constraints, each PM operation is started at a
fixed time point which is predetermined by maintenance planning system. We also suppose that
there is only one PM operation on each machine during the planning horizon. This assumption
brings the problem closer to the real manufacturing situations [9].
Scheduling problems under the machine availability constraints have recently been investigated in
numerous papers. Schmidt [10] presents a survey of existing methods for solving deterministic
scheduling problems with availability constraints, as well as complexity results; and most recent
survey of scheduling with deterministic machine availability constraints can be seen in reference
4. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
22
[11]. However, for the FJSP, there are only a few papers that deal with the FJSP under resource
constraints. Dalfard and Mohammadi [12] propose two metaheuristics (a hybrid genetic algorithm
and a simulated annealing algorithm) for the multi-objective FJSP with parallel machines,
maintenance costs, jobs due dates and jobs release times to minimize the mean tardiness, the
makespan and the mean flowtime. As there is no similar work in the literature, they compare the
solutions of these metaheuristic methods with those obtained by solving a mathematical model
using the software LINGO. Gao et al. [9] consider the FJSP with non-fixed availability
constraints where each machine is subject to an arbitrary number of PM tasks, and present a
hybrid genetic algorithm (hGA) to solve the problem with a multiobjective function including
makespan, total machine workload and the workload of the most loaded machine. Chan et al. [13]
develop a heuristic based on genetic algorithm (GA), namely iterative GA (IGA), for solving the
bicriteria FJSP under resource constraints. The objectives considered are the minimization of
makespan and machine idle cost. Zribi et al. [14] investigate the multi-purpose machine (MPM)
job shop scheduling problem with fixed machine availability constraints, and apply a GA to solve
the problem. Rajkumar et al. [15] present a greedy randomized adaptive search procedure
(GRASP) to solve the FJSP under non-fixed availability constraints and with the same objectives
as those considered by Gao et al. [9]. Moradi et al. [16] solve the FJSP with PM under two
objectives: the minimization of makespan for the production part and the minimization of system
unavailability for the maintenance part. Vilcot and Billaut [17] study a general job shop
scheduling problem with multiple constraints, coming from printing and boarding industry. They
present an algorithm based on tabu search and GA for minimizing the makespan and the
maximum lateness. Chan et al. [18] consider the distributed flexible manufacturing system (FMS)
scheduling problem subject to machine maintenance constraints, in which the maintenance time is
related to the machine age. The presented method also covers the FJSP with maintenance
activities. Wang and Yu [19] present a heuristic based on the filtered beam search (FBS)
algorithm to solve the FJSP with the non-fixed and fixed machine availability constraints due to
the PM.
3.Proposed Heuristic Approach
In this section, we present a heuristic method to solve the problem. This approach is motivated by
the idea of developing a constructive heuristic that considers simultaneously several factors
affecting the solution quality and intelligently balances their effects in the process of schedule
generation, and the observation that it could lead to good results in some preliminary
computational experiments on a wide range of difficult scheduling problems. This algorithm has a
simple structure, is easy to implement, and requires very little computational effort which makes
it preferable over other more complex and time-consuming approaches. Some notations that will
be used in the algorithm are defined as follows:
Aij: set of machines which are capable to execute operation j of job i,
Nij: number of members of the set Aij,
s′ij: mean processing time of operation j of job i over the machines belonging to the set Aij
(i.e., ij
s′ = ij
A
y
ijy N
t
ij
/
)
( ∑
∈
),
sji: total mean processing time of job i (i.e., sji = ∑
=
′
i
J
j
ij
s
1
),
5. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
23
sky: total weighted processing time on machine y which is calculated as: sky
= ∑ ∑
=
∈
=
′
n
i
J
A
y
if
j ij
ij
i
ij
N
s
1 1
,
M: a large number.
An outline of the proposed heuristic algorithm is given in Fig. 1 and the pseudocode of the
algorithm is shown in Fig. 2. Some other notations used in these two figures will be defined later.
until all operations of all jobs are scheduled, repeat
{
• For all i, j, k (such that: 1. j ≤ Ji, 2. j=1 or (j-1)th operation of job i is already
scheduled, and 3. jth operation of job i is an unscheduled operation and machine k is
capable of processing this operation), calculate the value of TC.
• For all unscheduled PM operations, calculate the value of TC.
• Select the operation (either job operation or PM operation) with minimum TC and
schedule it on the last position of current partial sequence on the corresponding
machine.
}
Fig. 1. General outline of the proposed heuristic algorithm
Initialization:
• Sort the jobs in increasing order of their sji and call the
resulting set: i_sort. Let i_sortz be zth job of the list
i_sort.
• Sort the machines in increasing order of their sky and call the
resulting set: k_sort. Let k_sorty be yth machine of the list
k_sort.
Constructive Algorithm:
for x1:=L_x1 to U_x1 do
for x2:=L_x2 to U_x2 do
for x6:=L_x6 to U_x6 do
for x7:=0 to 1 do
{
% Beginning of a schedule generation
until all job operations and all PM operations are
scheduled, repeat the following steps:
{
for j:= 1 to maxJ do
{
Set TC*
:=M
6. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
24
for i’:=1 to n do
{
Set i:=x7.(i_sorti’)+(1-x7).(i_sort(n-i’+1)),
Set b:=0,
if ( 1. j ≤ Ji, and
2. j=1 or (j-1)th operation of job i
is already scheduled, and
3. jth operation of job i is an
unscheduled operation) then
{
for k’:=1 to m do
{
Set k:= k_sort(m-k’+1),
if (machine k is capable of
processing jth operation of job
i) then
{
if ( 1. PM of machine
k is already
scheduled; or
2.PM task of machine
k is not already
scheduled, and: Cij <= tLk -
durk, (such that,
Cij=max (Cmaxk, Ci,j-
1)+tijk )
) then
{
Set TC:=
5
1
r
r
r
r C
.
x
.
w
if TC<TC*
then
{
Set TC*
:= TC
Set z:=i
Set y:=k
Set b:=0
}
}
}
if ( PM operation of machine k is an unscheduled operation) then
{
Set TC:=
5
1
r
r
r
r
6 C
.
x
.
w
.
x
if TC<TC*
then
{
Set TC*
:= TC
7. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
25
Set b:=1
Set y:=k
}
}
}
}
}
if TC*<M then
{
if b:=1 then schedule the PM task of
machine y on the last position of the
current partial sequence on this machine,
and set its completion time as max (Cmaxy+
dury, tEy).
else schedule jth operation of job z on the
last position of the current partial
sequence on machine y to finish at time czj.
}
}
}
% End of a schedule generation
If the objective value of the obtained sequence (Obj) is
less than the best objective value obtained so far (Obj*),
then set Obj*:=Obj and xr
*
=xr (r=1,2,…, 7) corresponding to
Obj*.
}
Fig. 2. Pseudocode of the proposed heuristic method
In the above algorithm, each unscheduled job operation (i, j) (operation j of job i) to be scheduled
on machine y is evaluated by the following criterion (2):
TC =
5
r r r
r 1
w .x .C ,
=
∑ (2)
Also, unscheduled PM operation of machine y is evaluated by the following criterion (3):
TC =
5
6 r r r
r 1
x . w .x .C ,
=
′
∑ (3)
and the unscheduled operation (either job operation or maintenance operation) with minimum TC
is selected for scheduling.
C1 to C5 and C′1 to C′5 are calculated as (4) to (13):
8. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
26
C1 = max (Cmaxy, Ci,j-1 )+ tijy (4)
C2 = max (0,( Ci,j-1 -Cmaxy)) (5)
C3 = tijy (6)
C4 = w′T + tijy (7)
C5 = w′y + tijy (8)
C′1 = max (Cmaxy+ dury, tEy) (9)
C′2 = max (0,(tEy-dury-Cmaxy)) (10)
C′3 = dury (11)
C′4 = w′T + dury (12)
C′5 = w′y + dury (13)
TC is weighted sum of some criteria which are established based on the factors affecting the
objective function value. Minimization of TC in the process of schedule generation leads to
improvement in solution quality. wr (r=1,2,…,5) are constants and xr (r=1,2,…,6) are integer
variables used to increase the flexibility and effectiveness of criterion TC and have a significant
impact on the performance of the algorithm. The constant weights (wr) are preliminary estimated
weights assigned to criteria according to their importance, and the coefficients xr are variables
bounded in a given range and used to refine the TC. Cmaxy is the maximum completion time
across all the operations scheduled on machine y; i.e., the completion time of last operation
(either job operation or PM operation) scheduled on machine y. w′T is the total workload of
machines for the partial schedule. w′y is the workload of machine y for the partial schedule. C1
and C′1 are applied to decrease Cmaxy; C2 and C′2 are used to decrease idle times; clearly, both
these objectives (Cmaxy and idle times) affect the main objective function, i.e. Cmax. The values of
other objective functions, i.e. WT and Wmax, are directly affected by (C4 and C′4) and (C5 and C′5),
respectively. For assigning operations to a machine, their processing time are also taken into
account by C3 and C′3.
Other notations used in the pseudocode of the proposed heuristic are as follows:
TC*: denotes the best value of TC. After each operation is scheduled, TC* is reset to M.
L_xr (r=1,2,…,6): lower limit of xr.
U_xr (r=1,2,…,6): upper limit of xr.
b: a binary variable taking value 1 if PM task of a machine is selected for scheduling, and
0 if an operation of a job is selected for scheduling.
As it can be seen in Fig. 2, the algorithm first sorts the jobs in increasing (decreasing) order of
their sji and then uses this order for evaluating their operations. Therefore, if two unscheduled
operations belonging to two different jobs have the same value of TC, then according to this
sorting of the jobs, the operation of job with smaller (greater) sji is selected for scheduling sooner
than the other operation. Binary variable x7 is applied for setting the order of the sorting (i.e.
either increasing order or decreasing order), it takes a value of 1 for increasing order and 0 for
decreasing one. Similarly, the algorithm first sorts the machines in decreasing order of their sky
and then uses this order to evaluate assigning the operations to each of them. In our preliminary
computational experiments, we used these sortings of the jobs and machines instead of randomly
selecting them, and interestingly observed that these sortings can lead to better solutions.
Specially, the results showed that in most cases, the sorting of the machines in decreasing order of
their sky leads to better solutions in comparison with increasing order. It is because the machines
with larger sky which are firstly selected for scheduling have more sensibility and effect on the
objective value. In other words, the schedule of these machines determines the performance of
overall schedule of the problem. Therefore, we have used only decreasing order of them in the
computational experiments. xr
*
(r=1,2,…,7) are the best values of variables xr (i.e. the values
9. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
27
corresponding to the best solutions). Indeed, for various values of xr (r=1,2,…,7), the algorithm of
Fig. 1 is run and a complete schedule is generated. Among all these schedules, the one with
minimum Obj is reported as the final solution. The values of variables xr for this best solution are
also reported and denoted by xr
*
(see Table 2). This best schedule obtained from the heuristic is
next improved by a shift neighborhood based local search procedure. The pseudocode of this
local search is shown in Appendix.
As mentioned earlier, the evaluation of the operations for scheduling them is done using the
criterion TC, i.e. the unscheduled operation with minimum TC is selected for scheduling.
4.Computational Results
This section describes the computational experiments conducted in order to evaluate the
performance of the proposed heuristic method. First, some preliminary experiments have been
conducted for the parameter settings. Regarding the test on various values for the parameters of
the algorithm and considering the computational results, we used the settings of Table 1 for
benchmarking the presented algorithm.
Table 1. Parameter settings for the heuristic
Parameter Value Parameter Value Parameter Value
L_x1 0 U_x1 2 w1 1
L_x2 0 U_x2 2 w2 1
L_x3 0 U_x3 2 w3 1
L_x4 0 U_x4 2 w4 0.5
L_x5 0 U_x5 2 w5 0.2
L_x6 0 U_x6 2
The algorithm was coded in C language and run on a Pentium IV, 2.2 GHz and 2.0 GB RAM PC.
The benchmark problems used were the set of 4 instances presented by Kacem et al. [20,21,22]
and extended by Gao et al. [9] and Rajkumar et al. [15] to problems involving maintenance
constraints. All these instances have exactly one PM activity on each machine in the planning
horizon. Table 2 shows a comparison of the results of our algorithm with those of two recently
published algorithms: the hybrid genetic algorithm (hGA) presented by Gao et al. [9] and the
greedy randomized adaptive search procedure (GRASP) developed by Rajkumar et al. [15]. The
results are obtained under two objective functions: Obj1=0.3×Cmax+0.5×WT+0.2×Wmax, and
Obj2=0.2×Cmax+0.5×WT+0.3×Wmax. Weights denotes the weights of the objectives. Name and
Size refer to the name of each instance and its size in terms of the number of jobs, machines and
operations, respectively. Cmax, WT and Wmax stand for the makespan, total workload and
maximum workload, respectively. RPD is the relative percentage deviation and calculated as (14):
100
×
−
= *
*
lg
a
Obj
Obj
Obj
RPD , (14)
where Objalg is the objective function (Obj) value generated by the algorithm and Obj*
is the best
value of Obj obtained from the three algorithms. Sol.1 and Sol.2 show the results obtained by the
heuristic algorithm and local search procedure, respectively. Time(s) indicates the computational
time to solve each instance by the heuristic (including the time spent on the local search) in
seconds. The best values of variables xr (i.e. xr
*
), r=1,2,…,7; are also reported in Table 2. Symbol
10. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
28
‘−’ denotes that the result was not presented in the given reference. As it can be seen in the table,
for the results corresponding to Obj2, i.e. cases with larger weights for Wmax, the value of x7 is
equal to 0.25, i.e. near to zero. It means that the sorting of the jobs in decreasing order of their sji
leads to better solutions in comparison with increasing order. It is because the jobs with larger sji
which are firstly selected for scheduling have more impact on Wmax. The results of Table 2 also
show that the value of x6 is equal to zero in almost all instances, meaning that TC for PM
operations is set to zero, i.e. PM operations are started at earliest possible time. This intuitively
gives more opportunities to job operations to be inserted in good positions of the overall schedule.
The average value of each variable xr, r=1,2,…,5 can be considered as the relative effect of the
corresponding criterion on the quality of solutions. Of course, as it can be seen in the table, the
values of each variable xr, r=1,2,…,5 have relatively high variance, meaning that they are
strongly dependent on the specifications of problem instance under consideration and on the
values of other variables xr. The proposed algorithm selects for each instance, best combination of
xr values leading to best result.
In Table 2, an interesting observation is that the proposed heuristic is better than the other two
metaheuristic algorithms in terms of the average RPD, considering that it is very fast and needs
only 0.5 sec. on average. Figures 3 and 4 show a graphical comparison of the RPD of the three
methods for Obj1 and Obj2, respectively.
Table 2. Computational results for benchmark instances
Weights Name Size Cm ax Wt Wm ax Obj1 x1 x2 x3 x4 x5 x6 x7 Cm ax Wt Wm ax Obj1 RPD Time(s)
4-5-nfa 4,5,12 15 40 9 26.3 1 1 1 2 1 0 0 13 40 9 25.7 0 0.05
8-8-nfa 8,8,27 31 103 16 64 0 0 0 1 1 0 0 20 103 16 60.7 0.998 0.22
10-10-nfa 10,10,30 11 60 8 34.9 2 1 1 0 2 0 1 9 60 8 34.3 0.587 0.38
15-10-nfa 15,10,56 20 108 13 62.6 2 1 2 0 1 1 1 15 104 14 59.3 0 1.45
1.25 0.75 1 0.75 1.25 0.25 0.5 0.396 0.52
Weights Name Size Cm ax Wt Wm ax Obj2 x1 x2 x3 x4 x5 x6 x7 Cm ax Wt Wm ax Obj2 RPD Time(s)
4-5-nfa 4,5,12 15 40 9 25.7 1 1 1 2 1 0 0 13 40 9 25.3 0 0.05
8-8-nfa 8,8,27 31 103 16 62.5 0 0 0 1 1 0 0 20 103 16 60.3 0.668 0.22
10-10-nfa 10,10,30 11 60 8 34.6 2 1 1 0 2 0 1 9 60 8 34.2 0.885 0.36
15-10-nfa 15,10,56 27 104 14 61.6 0 0 0 2 2 0 0 14 105 12 58.9 0 1.09
0.75 0.5 0.5 1.25 1.5 0 0.25 0.388 0.43
Weights Name Size Cm ax Wt Wm ax Obj1 RPD Cm ax Wt Wm ax Obj1 RPD
4-5-nfa 4,5,12 − − − − − 16 40 9 26.6 3.502
8-8-nfa 8,8,27 17 105 15 60.6 0.832 18 103 16 60.1 0
10-10-nfa 10,10,30 8 61 7 34.3 0.587 9 60 7 34.1 0
15-10-nfa 15,10,56 12 109 12 60.5 2.024 16 107 13 60.9 2.698
1.147 1.55
Weights Name Size Cm ax Wt Wm ax Obj2 RPD Cm ax Wt Wm ax Obj2 RPD
4-5-nfa 4,5,12 − − − − − 16 40 9 25.9 2.372
8-8-nfa 8,8,27 17 105 15 60.4 0.835 18 103 16 59.9 0
10-10-nfa 10,10,30 8 61 7 34.2 0.885 9 60 7 33.9 0
15-10-nfa 15,10,56 12 109 12 60.5 2.716 16 107 13 60.3 2.377
1.479 1.187
w
1
=0.3
w
2
=0.5
w
3
=0.2
w
1
=0.2
w
2
=0.5
w
3
=0.3
Average
Average
GRASP
Heuristic
Sol.1 Sol.2
hGA
Average
Average
w
1
=0.3
w
2
=0.5
w
3
=0.2
w
1
=0.2
w
2
=0.5
w
3
=0.3
11. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
29
0
0.5
1
1.5
2
2.5
3
3.5
4
4-5-nfa 8-8-nfa 10-10-nfa 15-10-nfa
Instance name
RPD
hGA
GRASP
Heuristic
Fig. 3 Comparison of the three methods for Obj1 (i.e. W1=0.3, W2=0.5, W3=0.2)
0
0.5
1
1.5
2
2.5
3
4-5-nfa 8-8-nfa 10-10-nfa 15-10-nfa
Instance name
RPD
hGA
GRASP
Heuristic
Fig. 4 Comparison of the three methods for Obj2 (i.e. W1=0.2, W2=0.5, W3=0.3)
5.Conclusion
This paper investigates the flexible job shop scheduling problem with preventive maintenance
constraints. The objective is to minimize the makespan, the total workload of machines and the
workload of most loaded machine. The main purpose is to produce reasonable schedules very
quickly. A simple and easily extendable heuristic based on a constructive procedure is presented.
The proposed approach uses an accurate, relatively comprehensive and flexible criterion for
scheduling job operations and PM operations and constructing a feasible high-quality solution. In
this criterion, several factors affecting the quality of solutions are used and to each of these
factors, a variable weight is assigned. By setting different values to these variable weights,
different solutions are generated and evaluated. The algorithm is tested on benchmark instances
from the literature in order to evaluate its performance. The computational results show that the
proposed approach can yield very good solutions with very little computational time. Since the
presented method is a heuristic, its results cannot be compared in a meaningful way with those of
the methods evaluated as they are metaheuristic based algorithms. However, the computational
results show that the proposed heuristic outperforms the other metaheuristic methods evaluated,
12. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 1, No. 1, May 2014
30
in terms of both the average RPD and the computational time. Further research needs to be
conducted in applying other criteria in the TC in order to improve the solution quality and to
adapt the approach to other objectives and process constraints.
Appendix. Shift neighborhood based local search procedure(l and l′ are indices of position in the
machine path. L, L′ denote the number of operations assigned to machines y and y′ in the current
solution, respectively.)
Repeat:
for y:=1 to m do
for l:=1 to L do
for y′:=1 to m do
for l’:=1 to L’ do
if (y′≠y or (y′=y and l’≠l)) then
{
• Remove the operation placed in position l
on machine y and insert it into position l’
on machine y′, leaving all other relative
operation orders unchanged.
• If the objective value of the obtained
sequence (Obj) is less than the best
objective value obtained so far (Obj*), then
set Obj*:=Obj; Otherwise, insert the
operation into its previous position.
}
until (no improvement occurs over the best solution)
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