This document summarizes a research paper that examines optimal scheduling for a two-stage flow shop problem using the branch and bound method. It presents the mathematical model and provides a numerical example to illustrate the algorithm. The paper studies a flow shop problem where equivalent jobs are defined for group jobs. The objective is to minimize total elapsed time. The branch and bound method is applied to obtain the optimal job sequence. A 5-job, 2-machine problem is used to demonstrate the step-by-step approach to determining the optimal schedule with minimum completion time of 138 units.
Fuzzy Sequencing Problem Using Generalized Triangular Fuzzy NumbersIJERA Editor
In this paper, we present the different methods to solve fuzzy sequencing problem using fuzzy technological
values like generalized triangular fuzzy numbers. The procedure adopted was the fuzzy sequencing problems
are defuzzified using ranking functions and hence solving the crisp sequencing problem by standard sequencing
algorithm for obtaining the optimal sequence and minimum completion time in terms of fuzzy values which is
illustrated with numerical examples and solutions
Fuzzy Sequencing Problem Using Generalized Triangular Fuzzy NumbersIJERA Editor
In this paper, we present the different methods to solve fuzzy sequencing problem using fuzzy technological
values like generalized triangular fuzzy numbers. The procedure adopted was the fuzzy sequencing problems
are defuzzified using ranking functions and hence solving the crisp sequencing problem by standard sequencing
algorithm for obtaining the optimal sequence and minimum completion time in terms of fuzzy values which is
illustrated with numerical examples and solutions
International Journal of Engineering Research and Development (IJERD)IJERD Editor
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Radix-3 Algorithm for Realization of Type-II Discrete Sine TransformIJERA Editor
In this paper, radix-3 algorithm for computation of type-II discrete sine transform (DST-II) of length N =
3푚 (푚 = 1,2, … . ) is presented. The DST-II of length N can be realized from three DST-II sequences, each of
length N/3. A block diagram of for computation of the radix-3 DST-II algorithm is given. Signal flow graph for
DST-II of length 푁 = 32 is shown to clarify the proposed algorithm.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Moment Preserving Approximation of Independent Components for the Reconstruct...rahulmonikasharma
The application of Independent Component Analysis (ICA) has found considerable success in problems where sets of observed time series may be considered as results of linearly mixed instantaneous source signals. The Independent Components (IC’s) or features can be used in the reconstruction of observed multivariate time seriesfollowing an optimal ordering process. For trend discovery and forecasting, the generated IC’s can be approximated for the purpose of noise removal and for the lossy compression of the signals.We propose a moment-preserving (MP) methodology for approximating IC’s for the reconstruction of multivariate time series.The methodologyis based on deriving the approximation in the signal domain while preserving a finite number of geometric moments in its Fourier domain.Experimental results are presented onthe approximation of both artificial time series and actual time series of currency exchange rates. Our results show that the moment-preserving (MP) approximations of time series are superior to other usual interpolation approximation methods, particularly when the signals contain significant noise components. The results also indicate that the present MP approximations have significantly higher reconstruction accuracy and can be used successfully for signal denoising while achieving in the same time high packing ratios. Moreover, we find that quite acceptable reconstructions of observed multivariate time series can be obtained with only the first few MP approximated IC’s.
In today's world developers are faced with the problem of writing high-performing algorithms that scale efficiently across a range of multi-core processors. Traditional blocked algorithms need to be tuned to each processor, but the discovery of cache-oblivious algorithms give developers new tools to tackle this emerging challenge. In this talk you will learn about the external memory model, the cache-oblivious model, and how to use these tools to create faster, scalable algorithms.
Justification of Montgomery Modular Reductionacijjournal
one of the most known and widely used methods in Cryptography is the method suggested by Peter
Montgomery; this method is based on the changing of the original reduction modulo by some other
convenient modulo, the original Montgomery paper gives the algorithm without any considerations
leading to that algorithm.
Effect Estimation Method of Parallel Computing Based on Dynamic Generation of...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
In an earlier paper in 2017, Rastogi and Bajpai[1] defined and studied a special vector field of the first kind in a Finsler space as follows:
Definition 1: A vector field Xi(x), in a Finsler space, is said to be a special vector field of the first kind, if (i) Xi/j = - δij and (ii) Xi hij = Ɵj, where Ɵj is a non-zero vector field in the given Finsler space.
In 2019, some more special vector fields in a Finsler space of two and three dimensions have been defined and studied by the authors Dwivedi et al.[2] and Dwivedi et al.[3] In Dwivedi et al.[3], the authors defined and studied six kinds of special vector fields in a Finsler space of three dimensions and, respectively, called them special vector fields of the second, third, fourth, fifth, sixth, and seventh kind. In the present paper, we shall study some curvature properties of special vector fields of the first and seventh kind in a Finsler space of three dimensions.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Radix-3 Algorithm for Realization of Type-II Discrete Sine TransformIJERA Editor
In this paper, radix-3 algorithm for computation of type-II discrete sine transform (DST-II) of length N =
3푚 (푚 = 1,2, … . ) is presented. The DST-II of length N can be realized from three DST-II sequences, each of
length N/3. A block diagram of for computation of the radix-3 DST-II algorithm is given. Signal flow graph for
DST-II of length 푁 = 32 is shown to clarify the proposed algorithm.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Moment Preserving Approximation of Independent Components for the Reconstruct...rahulmonikasharma
The application of Independent Component Analysis (ICA) has found considerable success in problems where sets of observed time series may be considered as results of linearly mixed instantaneous source signals. The Independent Components (IC’s) or features can be used in the reconstruction of observed multivariate time seriesfollowing an optimal ordering process. For trend discovery and forecasting, the generated IC’s can be approximated for the purpose of noise removal and for the lossy compression of the signals.We propose a moment-preserving (MP) methodology for approximating IC’s for the reconstruction of multivariate time series.The methodologyis based on deriving the approximation in the signal domain while preserving a finite number of geometric moments in its Fourier domain.Experimental results are presented onthe approximation of both artificial time series and actual time series of currency exchange rates. Our results show that the moment-preserving (MP) approximations of time series are superior to other usual interpolation approximation methods, particularly when the signals contain significant noise components. The results also indicate that the present MP approximations have significantly higher reconstruction accuracy and can be used successfully for signal denoising while achieving in the same time high packing ratios. Moreover, we find that quite acceptable reconstructions of observed multivariate time series can be obtained with only the first few MP approximated IC’s.
In today's world developers are faced with the problem of writing high-performing algorithms that scale efficiently across a range of multi-core processors. Traditional blocked algorithms need to be tuned to each processor, but the discovery of cache-oblivious algorithms give developers new tools to tackle this emerging challenge. In this talk you will learn about the external memory model, the cache-oblivious model, and how to use these tools to create faster, scalable algorithms.
Justification of Montgomery Modular Reductionacijjournal
one of the most known and widely used methods in Cryptography is the method suggested by Peter
Montgomery; this method is based on the changing of the original reduction modulo by some other
convenient modulo, the original Montgomery paper gives the algorithm without any considerations
leading to that algorithm.
Effect Estimation Method of Parallel Computing Based on Dynamic Generation of...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
In an earlier paper in 2017, Rastogi and Bajpai[1] defined and studied a special vector field of the first kind in a Finsler space as follows:
Definition 1: A vector field Xi(x), in a Finsler space, is said to be a special vector field of the first kind, if (i) Xi/j = - δij and (ii) Xi hij = Ɵj, where Ɵj is a non-zero vector field in the given Finsler space.
In 2019, some more special vector fields in a Finsler space of two and three dimensions have been defined and studied by the authors Dwivedi et al.[2] and Dwivedi et al.[3] In Dwivedi et al.[3], the authors defined and studied six kinds of special vector fields in a Finsler space of three dimensions and, respectively, called them special vector fields of the second, third, fourth, fifth, sixth, and seventh kind. In the present paper, we shall study some curvature properties of special vector fields of the first and seventh kind in a Finsler space of three dimensions.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
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Essentials of Automations: Optimizing FME Workflows with Parameters
11.application of branch and bound method for optimal two stage flow shop scheduling problem with group job restrictions
1. Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5758 (print) ISSN 2224-896X (online)
Vol 1, No.5, 2011
Application of Branch and Bound Method for Optimal Two
Stage Flow Shop Scheduling Problem with Group Job-
Restrictions
Deepak Gupta* , Payal Singla, Harminder Singh
Department of Mathematics Maharishi Markandeshwar University Mullana, Ambala (India)
* E-mail of the corresponding author: guptadeepak2003@yahoo.co.in
Abstract:
This paper studies two stage flow shop scheduling problem in which equivalent job for group job,
processing times on each machine are given. The objective of the study is to get optimal sequence of
jobs in order to minimize the total elapsed time. The given problem is solved with branch and bound
method.The method is illustrated by numerical example.
Keywords: Equivalent job, processing time, elapsed time, Branch and Bound.
1. Introduction:
A flowshop scheduling problem has been one of classical problems in production scheduling
since Johnson [6] proposed the well known Johnson’s rule in the two-stage flowshop makespan
scheduling problem. Yoshida and Hitomi [11] further considered the problem with setup times. Yang
and Chern [10] extended the problem to a two-machine flowshop group scheduling problem. Maggu
and Dass[9] introduced the concept of equivalent job for a job-block when the situations of giving
priority of one job over another arise. Kim, et al.[7] considered a batch scheduling problem for a two-
stage flowshop with identical parallel machines at each stage. Brah and Loo [1] studied a flow shop
scheduling problem with multiple processors. Futatsuishi, et al. [4] further studied a multi-stage
flowshop scheduling problem with alternative operation assignments.
Lomnicki [8] introduced the concept of flow shop scheduling with the help of
branch and bound method. Further the work was developed by Ignall and Scharge [5], Chandrasekharan
[3] , Brown and Lomnicki [2], with the branch and bound technique to the machine scheduling problem
by introducing different parameters In this paper we consider a two-stage flowshop scheduling problem
with job-block with the help of branch and bound method. The given method is very simple and easy to
understand. Thus, the problem discussed here has significant use of theoretical results in process
industries.
2. Assumptions:
i. No passing is allowed.
ii. Each operation once started must performed till completion.
iii. Jobs are independent to each other.
iv. A job is entity, i.e. no job may be processed by more than one machine at a time.
3.Notations:
We are given n jobs to be processed on three stage flowshop scheduling problem and we have used the
following notations:
Ai : Processing time for job ith on machine A
Bi : Processing time for job ith on machine B
Cij : Completion time for job ith on machines A and B
S0 : Optimal sequence
1
2. Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5758 (print) ISSN 2224-896X (online)
Vol 1, No.5, 2011
Jr : Partial schedule of r scheduled jobs
Jr′ : The set of remaining (n-r) free jobs
4. Mathematical Development:
Consider n jobs say i=1, 2, 3 … n are processed on two machines A & B in the order AB. A job i
(i=1,2,3…n) has processing time Ai & Bi on each machine respectively. . Let an equivalent job β is
defined as (k, m) where k, m are any jobs among the given n jobs such that k occurs before job m in the
order of job block (k , m).The mathematical model of the problem in matrix form can be stated as :
Jobs Machine A Machine B
i Ai Bi
1 A1 B1
2 A2 B2
3 A3 B3
4 A4 B4
- ---- ---
- -- --
n An Bn
Tableau – 1
Our objective is to obtain the optimal schedule of all jobs which minimize the total elapsed time, using
branch and bound technique.
5. Algorithm:
Step 1: (i) Aβ = Ak + Am − min( Ak , Bm )
(ii) Aβ = Ak + Am − min( Ak , Bm )
Step 2: Calculate
(i) l1 = t ( J r ,1) + ∑ Ai + min( Bi )
′
i∈J r
′
i∈J r
(ii) l1 = t ( J r , 2) + ∑ Bi
′
i∈J r
Step 3: Calculate
l = max(l1 , l2 )
We evaluate l first for the n classes of permutations, i.e. for these starting with 1, 2, 3………n
respectively, having labelled the appropriate vertices of the scheduling tree by these values.
Step 4: Now explore the vertex with lowest label. Evaluate l for the (n-1) subclasses starting with this
vertex and again concentrate on the lowest label vertex. Continuing this way, until we reach at the end
of the tree represented by two single permutations, for which we evaluate the total work duration. Thus
we get the optimal schedule of the jobs.
Step 5: Prepare in-out table for the optimal sequence obtained in step 3 and get the minimum total
elapsed time.
2
3. Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5758 (print) ISSN 2224-896X (online)
Vol 1, No.5, 2011
6.Numerical Example:
Consider 5 jobs 2 machine flow shop problem whose processing time of the jobs on each machine is
given.
Job Machine A Machine B
i Ai Bi
1 11 17
2 15 21
3 22 25
4 34 19
5 27 38
Tableau – 2
Our objective is to obtain optimal schedule for above said problem in which jobs 2,4 are to be
processed as a group job (2,4)
Solution:
Step1: Calculate
(i) Aβ = A2 + A4 − min( A4 , B2 )
= 28
(ii) Bβ = B2 + B4 − min( A4 , B2 )
= 21
The problem reduced as in tableau-3.
Step2: Calculate
(i) l1 = t ( J r ,1) + ∑ Ai + min( Bi )
′
i∈J r
′
i∈J r
(ii) l1 = t ( J r , 2) + ∑ Bi
′
i∈J r
For J1 = (1).Then J′(1) = {2,3,4}, we get l 1 = 43 , l 2 = 37
LB(1) l = max(l1 , l2 ) =112 similarly,
we have LB(β)= 129, LB(3)= 123, LB(5)= 128
Step 3 & 4:
Now branch from J1 = (1). Take J2 =(1β). Then J′2={3,4} and LB(1β) = 123
Proceeding in this way, we obtain lower bound values as shown in the tableau- 4
Step 5 :
Therefore the sequence S1 is 1-3-5- β i.e. 1-3-5- 2 - 4 and the corresponding in-out table on sequence
S1 is as in tableau-5:
Hence the total elapsed time is 138 units.
7. Remarks:
The study may further be extended by considering various parameters such as transportation time,
mean weightage time etc.
Refrences:
[1] Brah, S.A. and Loo, L.L.,(1999), “Heuristics for Scheduling in a Flow Shop with Multiple
Processors,”European Journal of Operation Research, Vol. 113, No.1, pp.113-122.
[2] Brown, A.P.G. and Lomnicki, Z.A. (1966), “Some applications of the branch and bound
algorithm to the machine scheduling problem”, Operational Research Quarterly, Vol. 17,
pp.173-182.
[3] Chander Shekharan, K, Rajendra, Deepak Chanderi (1992), “An efficient heuristic approach
to the scheduling of jobs in a flow shop”, European Journal of Operation Research Vol. 61,pp.
318-325.
3
4. Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5758 (print) ISSN 2224-896X (online)
Vol 1, No.5, 2011
[4] Futatsuishi, Y., Watanabe, I., and Nakanishi, T. ( 2002), “A Study of the Multi-Stage
Flowshop Scheduling Problem with Alternative Operation Assignments,”Mathematics and
Computers in Simulation,Vol. 59, No. 3, pp.73-79.
[5] Ignall, E. and Schrage, L. (1965), “Application of the branch-and-bound technique to some
flowshop scheduling problem”, Operations Research, Vol. 13, pp.400-412.
[6] Johnson, S.M. (1954), “Optimal Two and Three Stage Prouction Schedules with Setup Times
Include,”Naval Research Logistics Quarterly,Vol. 1 No. 1, pp. 61-68.
[7] Kim, J.S., Kang, S.H.,and Lee, S.M.(1997), “ Transfer Batch Scheduling for a Two-Stage
Flowshop with Identical Parallel Machines at Each Stage,” Omega,Vol. 25, No. 1, pp. 547-
555
[8] Lomnicki, Z.A. (1965), “A branch-and-bound algorithm for the exact solution of the three-
machine scheduling problem”, Operational Research Quarterly, Vol. 16, pp.89-100.
[9] Maggu & Das (1981), “On n x 2 sequencing problem with transportation time of jobs”, Pure
and Applied Mathematika Sciences, 12-16.
[10] Yang, D.L. and Chern, M.S. (2000), “Two-Machine Flowshop Group Scheduling
Problem,”Computers and Operations Research, Vol.27, No.10, pp.975-985.
[11] Yoshida, T. and Hitomi, K., (1979), “Optimal Two-Stage Production Scheduling with Setup
Times Separated,”AIIE Transactions, Vol. 11, No. 1, pp. 261-273.
Tables and Figures:
Table 3: The reduced as in tableau-3:
Job Machine A Machine B
I Ai Bi
1 11 17
Β 28 21
3 22 25
5 27 38
Tableau –3
Table 4: lower bounds for respective jobs are as in tableau-4
Tableau-4
Table 5: In-out table for S1 and the minimum total elapsed time as in tableau-5.
Job Machine A Machine B Node Jr LB
I In-out In-out (Jr)
(1) 112
1 0 - 11 11 – 28
(β) 129
3 11 – 33 33 – 58 (3) 123
5 33 – 60 60 - 98 (5) 128
3 60 -75 98 - 119 Tableau- 5 (1β) 123
4 75 - 109 119 - 138 (13) 117
(15) 122
(13β) 126
(135) 119
4
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send their full paper to the following email address. More information can be found in the IISTE website : www.iiste.org
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Education, and other Social Sciences PAPER SUBMISSION EMAIL
Journal of Education and Practice JEP@iiste.org
Journal of Law, Policy and Globalization JLPG@iiste.org Global knowledge sharing:
New Media and Mass Communication NMMC@iiste.org EBSCO, Index Copernicus, Ulrich's
Journal of Energy Technologies and Policy JETP@iiste.org Periodicals Directory, JournalTOCS, PKP
Historical Research Letter HRL@iiste.org Open Archives Harvester, Bielefeld
Academic Search Engine, Elektronische
Public Policy and Administration Research PPAR@iiste.org Zeitschriftenbibliothek EZB, Open J-Gate,
International Affairs and Global Strategy IAGS@iiste.org OCLC WorldCat, Universe Digtial Library ,
Research on Humanities and Social Sciences RHSS@iiste.org NewJour, Google Scholar.
Developing Country Studies DCS@iiste.org IISTE is member of CrossRef. All journals
Arts and Design Studies ADS@iiste.org have high IC Impact Factor Values (ICV).