\r\n\tThe LED can be lingering further into three major categories are (i) Traditional inorganic LEDs, (ii) Organic LEDs (Small Molecule OLED, Polymer LED, Passive Matrix OLED Active Matrix OLED), (iii) High brightness LEDs, (iv) Deep-UV LEDs, (v) Active Matrix Organic Light-Emitting Diodes (AMOLED).
",isbn:"978-1-83968-886-7",printIsbn:"978-1-83968-885-0",pdfIsbn:"978-1-83968-887-4",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"97e861d1556a639f0e5cc6ee8bdb0a0f",bookSignature:"Prof. Jagannathan Thirumalai",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10559.jpg",keywords:"Aluminum Gallium Arsenide, Gallium Arsenide Phosphide, Indium Phosphide, Thin-Film-Display, Organic Rare-Earth Complexes, Colour Rendering Index, High Brightness Leds, Luminous Control, Air Purification, Skin Therapy, Organic Compounds Form the Electroluminescent Material, Specific Type of Thin-Film-Display",numberOfDownloads:14,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"September 21st 2020",dateEndSecondStepPublish:"October 19th 2020",dateEndThirdStepPublish:"December 18th 2020",dateEndFourthStepPublish:"March 8th 2021",dateEndFifthStepPublish:"May 7th 2021",remainingDaysToSecondStep:"3 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,biosketch:"As an expert in the optoelectronics and nanotechnology area, Dr.Thirumalai has been invited to examine several MSc and Ph.D. theses, invited to give a talk in various forums, and to review papers for international and national journals.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"99242",title:"Prof.",name:"Jagannathan",middleName:null,surname:"Thirumalai",slug:"jagannathan-thirumalai",fullName:"Jagannathan Thirumalai",profilePictureURL:"https://mts.intechopen.com/storage/users/99242/images/system/99242.png",biography:"Dr. J. Thirumalai received his Ph.D. from Alagappa University, Karaikudi in 2010. \n\nHe was awarded the Post-doctoral Fellowship from Pohang University of Science and Technology (POSTECH), Republic of Korea, in 2013.\nHe worked as an Assistant Professor of Physics, B.S. Abdur Rahman University, Chennai, India (2011 to 2016). \nCurrently, he is working as an Assistant Professor & Head of the Department of Physics, SASTRA Deemed to be University, Kumbakonam (T.N.), India. \n\nHis research interests focus on luminescence, self-assembled nanomaterials, thin-film optoelectronic devices & Supercapacitors. \n\nHe has published more than 60 SCOPUS/ISI indexed papers, 11 book chapters, and he edited 5 books. He is serving as a member in various national and international societies. Currently, he is acting as a principal investigator for a funded project towards the application of luminescence-based thin-film optoelectronic devices, funded by the Science and Engineering Research Board (SERB), India. \nAs an expert in optoelectronics and nanotechnology area, he has been invited to examine several MSc and Ph.D. theses, invited to give a talk in various forums and to review papers for international and national journals.",institutionString:"SASTRA University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"10",totalChapterViews:"0",totalEditedBooks:"6",institution:{name:"SASTRA University",institutionURL:null,country:{name:"India"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"11",title:"Engineering",slug:"engineering"}],chapters:[{id:"74673",title:"Economic Applications for LED Lights in Industrial Sectors",slug:"economic-applications-for-led-lights-in-industrial-sectors",totalDownloads:14,totalCrossrefCites:0,authors:[{id:"150046",title:"Prof.",name:"Muhammad M.A.S.",surname:"Mahmoud",slug:"muhammad-m.a.s.-mahmoud",fullName:"Muhammad M.A.S. Mahmoud"}]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"297737",firstName:"Mateo",lastName:"Pulko",middleName:null,title:"Mr.",imageUrl:"https://mts.intechopen.com/storage/users/297737/images/8492_n.png",email:"mateo.p@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"5348",title:"Luminescence",subtitle:"An Outlook on the Phenomena and their Applications",isOpenForSubmission:!1,hash:"d982c49fed4423a0ea7367af4f917b82",slug:"luminescence-an-outlook-on-the-phenomena-and-their-applications",bookSignature:"Jagannathan Thirumalai",coverURL:"https://cdn.intechopen.com/books/images_new/5348.jpg",editedByType:"Edited by",editors:[{id:"99242",title:"Prof.",name:"Jagannathan",surname:"Thirumalai",slug:"jagannathan-thirumalai",fullName:"Jagannathan Thirumalai"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"6489",title:"Light-Emitting Diode",subtitle:"An Outlook On the Empirical Features and Its Recent Technological 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by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"73530",title:"Polynomial Algorithm for Constructing Pareto-Optimal Schedules for Problem 1∣rj∣Lmax,Cmax",doi:"10.5772/intechopen.93677",slug:"polynomial-algorithm-for-constructing-pareto-optimal-schedules-for-problem-1-em-r-em-sub-j-sub-em-l-",body:'\nWe consider a classical scheduling problem on a single machine. A release time of each job is predefined and represents the minimum possible start time of the job. When constructing schedules, we consider two objective functions. The first one is to minimize the maximum lateness, that is, maximum difference between each job due date and its actual completion time. The second one is to minimize the maximum completion time, that is, to complete all the jobs as soon as possible. The problem is NP-hard in the strong sense [1]. We provide a polynomial time algorithm for constructing a Pareto-optimal set of schedules on criteria of maximum lateness and maximum completion time, that is, problem \n
We consider single machine scheduling problem, where a set of \n
The schedule is represented by a set \n
We denote the completion time of all jobs of the set \n
The problem is to find the optimal schedule \n
For any arbitrary set of jobs \n
In the standard notation of Graham et al. [2], this problem is denoted as \n
Potts [3] introduced an iterative version of extended Jackson rule (IJ) [4] and proved that \n
A number of polynomially solvable cases of the problem were found, starting with Jackson’s early result [4] for the case \n
Schedule is constructed according to the extended Jackson rule (Schrage schedule): on the next place in the schedule we select a released non-ordered job with the minimum due date; if there are no such jobs, then we select the job with the minimum release time among the unscheduled jobs.
\nIf process times of all jobs are equal, the complexity can be reduced to \n
A case when job processing times are mutually divisible is considered in [9]. Author suggest a polynomial-time algorithm with a complexity of \n
Special cases \n
We denote by \n
This paper deals with a problem with two objective functions \n
Definition 1.1 For any instance \n
Early schedules play an important role in our construction, since it is sufficient to check all early schedules to find the optimal schedule of any problem instance.
\nBy \n
We denote by \n
This section deals with the problem of constructing a Pareto-optimal set by criteria \n
There is no schedule \n
We denote the precedence of the jobs \n
In cases when its obvious how many jobs we mean, we write \n
We assume that the job parameters satisfy the following constraints:
\nFor example, these constraints correspond to the case when \n
We assume that \n
where \n
Lemma 1.1 If the jobs of the set \n
is true, and for all \n
where \n
Proof: For each job \n
therefore (12) is valid.
\nIf for job \n
Since \n
The inequality (13) can be proved in a similar way.
\nFor each job \n
Let for the job \n
Since \n
Since \n
Hence, \n
Theorem 1.1 If conditions (9) are true for jobs in the subset \n
and one of the jobs \n
Proof: Let \n
From the lemma 1.1 we have
\nObviously, the following inequality is true for job \n
From (20)–(23) we get \n
Let \n
Let us assume \n
For each job \n
Job \n
Since \n
From the lemma 1.1
\nMoreover, it is obvious that
\nFrom inequalities (24)–(29) it follows that \n
We call a schedule \n
Thus, when constraints (9) are satisfied for jobs from the set \n
We define the set of schedules \n
Example 1.1
\nThe set \n
Theorem 1.2 If for the jobs of the subset \n
Proof: Let \n
According to the theorem 1.1 for the jobs of the set \n
Let us denote \n
After no more than \n
We form the following partial schedule \n
Algorithm 1.1 for constructing schedule \n
1: Initial step. Let \n
2: Main step. Find the jobs \n
3: if\n
4: \n
5: else
\n6: algorithm stops;
\n7: end if
\n8: Let \n
Lemma 1.2 The complexity of the algorithm 1.1 for finding the schedule \n
Proof: At each iteration of the algorithm 1.1 we find two jobs: \n
The main step of algorithm 1.1 is finding the jobs \n
Lemma 1.3 If for jobs of the set \n
Proof: If \n
Let \n
Let us use the following notation \n
Consequence 1.1 from Lemma 1.3. If the jobs of the set \n
Theorem 1.3 If the jobs of the set \n
Proof: In the case \n
If \n
Consider the case of \n
Let us choose any arbitrary schedules \n
For case \n
The other case is \n
Therefore, our assumption is not true and \n
Therefore, jobs of the set \n
The problem \n
Lemma 1.4 The complexity of algorithm 1.2 does not exceed \n
Proof: At each iteration of the main step of the algorithm 1.2 we find the schedules \n
Algorithm 1.2 for solving the problem \n
1: Initial step. Let \n
2: Main step.
\n3: if\n
4: \n
5: end if
\n6: Find \n
7: if\n
8: the algorithm stops.
\n9: else
\n10: if\n
11: \n
12: end if
\n13: if\n
14: \n
15: end if
\n16: if\n
17: \n
18: end if
\n19: end if
\nThe problem \n
We denote by \n
Theorem 1.4 Let the jobs of the set \n
Proof: In case if for schedule \n
According to Lemma 1.3, all schedules of the set \n
After \n
Let \n
If \n
Thus, at each main step of the algorithm, we choose the fastest continuation of the partial schedule \n
Let us assume that after the \n
Repeating our proof as many times as the main step of algorithm 1.2 (no more than \n
Let us develop an algorithm for constructing a set of Pareto schedules \n
Schedule \n
Algorithm 1.3 for constructing a set of Pareto schedules by criteria \n
1: Initial step.\n
2: if\n
3: \n
4: end if
\n5: Main step.
\n6: if\n
7: \n
8: end if
\n9: if\n
10: if\n
11: find \n
12: if\n
13: \n
14: else
\n15: \n
16: if\n
17: \n
18: else
\n19: \n
20: end if
\n21: end if
\n22: if\n
23: find \n
24: if\n
25: \n
26: else
\n27: \n
28: end if
\n29: end if
\n30: end if
\n31: end if
\nAs a result of the algorithm 1.3, a set of schedules \n
Lemma 1.5 The complexity of the algorithm 1.3 does not exceed \n
Proof: At each iteration of the main step of the algorithm 1.3 we find schedules \n
Theorem 1.5 If case if (9) is true for each job of the set \n
Proof: According to Theorem 1.2, there exists an optimal (by \n
Let \n
If \n
If \n
Repeating our proof no more than \n
The set of schedules \n
Suppose there is a schedule \n
We assume that the first \n
\nFigure 1 schematically shows the considered schedule.
\nThe set of Pareto-optimal schedules.
For the set of schedules \n
The schedule \n
Single machine scheduling problem with given release dates and two objective functions is considered in this chapter, which is \n
an algorithm for constructing a Pareto-optimal set of schedules by criteria \n
An experimental study of the algorithm showed that it can be used to construct optimal schedules (by \n
The research was supported by RFBR (project 20-58-S52006).
\nWe consider a classical scheduling problem on a single machine. A release time of each job is predefined and represents the minimum possible start time of the job. When constructing schedules, we consider two objective functions. The first one is to minimize the maximum lateness, that is, maximum difference between each job due date and its actual completion time. The second one is to minimize the maximum completion time, that is, to complete all the jobs as soon as possible. The problem is NP-hard in the strong sense [1]. We provide a polynomial time algorithm for constructing a Pareto-optimal set of schedules on criteria of maximum lateness and maximum completion time, that is, problem \n
We consider single machine scheduling problem, where a set of \n
The schedule is represented by a set \n
We denote the completion time of all jobs of the set \n
The problem is to find the optimal schedule \n
For any arbitrary set of jobs \n
In the standard notation of Graham et al. [2], this problem is denoted as \n
Potts [3] introduced an iterative version of extended Jackson rule (IJ) [4] and proved that \n
A number of polynomially solvable cases of the problem were found, starting with Jackson’s early result [4] for the case \n
Schedule is constructed according to the extended Jackson rule (Schrage schedule): on the next place in the schedule we select a released non-ordered job with the minimum due date; if there are no such jobs, then we select the job with the minimum release time among the unscheduled jobs.
\nIf process times of all jobs are equal, the complexity can be reduced to \n
A case when job processing times are mutually divisible is considered in [9]. Author suggest a polynomial-time algorithm with a complexity of \n
Special cases \n
We denote by \n
This paper deals with a problem with two objective functions \n
Definition 1.1 For any instance \n
Early schedules play an important role in our construction, since it is sufficient to check all early schedules to find the optimal schedule of any problem instance.
\nBy \n
We denote by \n
This section deals with the problem of constructing a Pareto-optimal set by criteria \n
There is no schedule \n
We denote the precedence of the jobs \n
In cases when its obvious how many jobs we mean, we write \n
We assume that the job parameters satisfy the following constraints:
\nFor example, these constraints correspond to the case when \n
We assume that \n
where \n
Lemma 1.1 If the jobs of the set \n
is true, and for all \n
where \n
Proof: For each job \n
therefore (12) is valid.
\nIf for job \n
Since \n
The inequality (13) can be proved in a similar way.
\nFor each job \n
Let for the job \n
Since \n
Since \n
Hence, \n
Theorem 1.1 If conditions (9) are true for jobs in the subset \n
and one of the jobs \n
Proof: Let \n
From the lemma 1.1 we have
\nObviously, the following inequality is true for job \n
From (20)–(23) we get \n
Let \n
Let us assume \n
For each job \n
Job \n
Since \n
From the lemma 1.1
\nMoreover, it is obvious that
\nFrom inequalities (24)–(29) it follows that \n
We call a schedule \n
Thus, when constraints (9) are satisfied for jobs from the set \n
We define the set of schedules \n
Example 1.1
\nThe set \n
Theorem 1.2 If for the jobs of the subset \n
Proof: Let \n
According to the theorem 1.1 for the jobs of the set \n
Let us denote \n
After no more than \n
We form the following partial schedule \n
Algorithm 1.1 for constructing schedule \n
1: Initial step. Let \n
2: Main step. Find the jobs \n
3: if\n
4: \n
5: else
\n6: algorithm stops;
\n7: end if
\n8: Let \n
Lemma 1.2 The complexity of the algorithm 1.1 for finding the schedule \n
Proof: At each iteration of the algorithm 1.1 we find two jobs: \n
The main step of algorithm 1.1 is finding the jobs \n
Lemma 1.3 If for jobs of the set \n
Proof: If \n
Let \n
Let us use the following notation \n
Consequence 1.1 from Lemma 1.3. If the jobs of the set \n
Theorem 1.3 If the jobs of the set \n
Proof: In the case \n
If \n
Consider the case of \n
Let us choose any arbitrary schedules \n
For case \n
The other case is \n
Therefore, our assumption is not true and \n
Therefore, jobs of the set \n
The problem \n
Lemma 1.4 The complexity of algorithm 1.2 does not exceed \n
Proof: At each iteration of the main step of the algorithm 1.2 we find the schedules \n
Algorithm 1.2 for solving the problem \n
1: Initial step. Let \n
2: Main step.
\n3: if\n
4: \n
5: end if
\n6: Find \n
7: if\n
8: the algorithm stops.
\n9: else
\n10: if\n
11: \n
12: end if
\n13: if\n
14: \n
15: end if
\n16: if\n
17: \n
18: end if
\n19: end if
\nThe problem \n
We denote by \n
Theorem 1.4 Let the jobs of the set \n
Proof: In case if for schedule \n
According to Lemma 1.3, all schedules of the set \n
After \n
Let \n
If \n
Thus, at each main step of the algorithm, we choose the fastest continuation of the partial schedule \n
Let us assume that after the \n
Repeating our proof as many times as the main step of algorithm 1.2 (no more than \n
Let us develop an algorithm for constructing a set of Pareto schedules \n
Schedule \n
Algorithm 1.3 for constructing a set of Pareto schedules by criteria \n
1: Initial step.\n
2: if\n
3: \n
4: end if
\n5: Main step.
\n6: if\n
7: \n
8: end if
\n9: if\n
10: if\n
11: find \n
12: if\n
13: \n
14: else
\n15: \n
16: if\n
17: \n
18: else
\n19: \n
20: end if
\n21: end if
\n22: if\n
23: find \n
24: if\n
25: \n
26: else
\n27: \n
28: end if
\n29: end if
\n30: end if
\n31: end if
\nAs a result of the algorithm 1.3, a set of schedules \n
Lemma 1.5 The complexity of the algorithm 1.3 does not exceed \n
Proof: At each iteration of the main step of the algorithm 1.3 we find schedules \n
Theorem 1.5 If case if (9) is true for each job of the set \n
Proof: According to Theorem 1.2, there exists an optimal (by \n
Let \n
If \n
If \n
Repeating our proof no more than \n
The set of schedules \n
Suppose there is a schedule \n
We assume that the first \n
\nFigure 1 schematically shows the considered schedule.
\nThe set of Pareto-optimal schedules.
For the set of schedules \n
The schedule \n
Single machine scheduling problem with given release dates and two objective functions is considered in this chapter, which is \n
an algorithm for constructing a Pareto-optimal set of schedules by criteria \n
An experimental study of the algorithm showed that it can be used to construct optimal schedules (by \n
The research was supported by RFBR (project 20-58-S52006).
\nIntechOpen’s team of Scientific Advisors supports the publishing team by providing editorial and academic input and ensuring the highest quality output of free peer-reviewed articles. The Boards consist of independent external collaborators who assist us on a voluntary basis. Their input includes advising on new topics within their field, proposing potential expert collaborators and reviewing book publishing proposals if required. Board members are experts who cover major STEM and HSS fields. All are trusted IntechOpen collaborators and Academic Editors, ensuring that the needs of the scientific community are met.
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