This document presents a branch and bound technique to solve a three-stage flow shop scheduling problem that considers breakdown intervals and transportation time. The algorithm aims to minimize the total elapsed time. It provides notation for the problem, describes the mathematical model and development, provides a numerical example, and references other related work. The algorithm calculates lower bounds at each step of the branch and bound tree to find the optimal sequence that minimizes the total elapsed time, accounting for any impacts of breakdown intervals.
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
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
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
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 Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
An Efficient Elliptic Curve Cryptography Arithmetic Using Nikhilam Multiplica...theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
An Efficient Elliptic Curve Cryptography Arithmetic Using Nikhilam Multiplica...theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
SchedulingPart 1. Question 1. How many of the statements are tru.docxanhlodge
Scheduling
Part 1. Question 1. How many of the statements are true?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
Statement 1. The Johnson’s rule is a sequencing rule for an ‘nx1’ system.
Statement 2. The total flow time is minimized in an ‘nx2’ system when the schedule is based on the MPT rule.
Statement 3. The flow time for a job is equal to the processing time minus the queue time.
Statement 4. The start date for a job is equal to the due date minus the processing time.
Part 2. Questions 2-3
Five jobs arrived to be processed.
Present Date=
180
Work Order
A
B
C
D
E
Processing Time (Days)
20
8
25
4
15
Due Date
211
200
224
210
205
Question 2. The FIFO schedule is “ABCDE” and the LIFO schedule is “EDCBA”.
Consider the schedules based on the sequencing rules: Minimum Processing Time (MPT); Earliest Due Date (EDD); Minimum Slack Time (MST); and Minimum Critical Ratio (MCR). How many of the schedules are correct?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
The schedule based on the MPT sequencing rule is E,A,B,C,D
The schedule based on the EDD sequencing rule is B,E,D,A,C
The schedule based on the MST sequencing rule is D,B,E,A,C
The schedule based on the MCR sequencing rule is A,E,C,B,D
Question 3. What is the flow of the MPT schedule?
(A) 162 (B) 230 (C) 72 (D) 233 (E) none of the above
Part 3. Questions 4-5
The jobs A,B,C,D,E, arrived in that order to be processed on two machines.
Job
A
B
C
D
E
Time on Machine 1
35
41
49
28
53
Time on Machine 2
32
56
36
48
55
Question 4. Which is the Johnson’s Rule schedule?
(A) ‘CEBAD’ (B) ‘DBECA’ (C) ‘DCABE’ (D) ‘DABEC’
Question 5. How many of the statements are correct?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 4
Statement 1. In the FIFO schedule, the total time of completion is 271.
Statement 2. In the FIFO schedule, the flow time of job D is 216.
Statement 3. In the FIFO schedule, the total queue time of job C is 76.
Statement 4. In the FIFO schedule, the total idle time of machine 2 is 35.
Part 4. Questions 6-7
An organization that continuously processes proposals is addressing a large backlog due to an historical FIFO sequencing rule. You have been asked to determine a schedule that minimizes the processing time of project proposals through the editing department and revision department. The editing department identifies changes and the revision department incorporates the changes and produces a final proposal document for release. A proposal must pass through each department. Each department has personnel to service only one document at a time. The editing time is dependent on the type of proposal and the revision time is dependent on the size of the proposal. The time estimates below have been provided to you for scheduling.
Question 6. What is the minimum processing time in minutes?
(A) 525 (B) 1025 (C) 522 (D) 502 (E) none of the above
Question 7. What is the minimum average jobs per minute that can be achieved.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
Let's dive deeper into the world of ODC! Ricardo Alves (OutSystems) will join us to tell all about the new Data Fabric. After that, Sezen de Bruijn (OutSystems) will get into the details on how to best design a sturdy architecture within ODC.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Search and Society: Reimagining Information Access for Radical FuturesBhaskar Mitra
The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Branch and bound technique for three stage flow shop scheduling problem including breakdown interval and transportation time
1. Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5782 (print) ISSN 2225-0506 (online)
Vol 2, No.1, 2012
Branch and Bound Technique for three stage Flow Shop
Scheduling Problem Including Breakdown Interval and
Transportation Time
Deepak Gupta*
Department of Mathematics Maharishi Markandeshwar University Mullana, Ambala (India)
* E-mail of the corresponding author: guptadeepak2003@yahoo.co.in
Abstract
This paper deals with minimization of the total elapsed time for nx3 flow shop
scheduling problem in which the effect of breakdown interval and the transportation
time are considered. A Branch and Bound technique is given to optimize the
objective of minimize the total elapsed time. The algorithm is very simple and easy to
understand and, also provide an important tool for decision makers to design a
schedule. A numerical illustration is given to clarify the algorithm.
Keywords: Flow shop scheduling, Processing time, Transportation time, Branch and Bound
Technique, Optimal sequence.
1. Introduction:
Scheduling problems are common occurrence in our daily life e.g. ordering of
jobs for processing in a manufacturing plant, programs to be run in a sequence at a
computer center etc. Such problems exist whenever there is an alternative choice in
which a number of jobs can be done. Now-a-days, the decision makers for the
manufacturing plant have interest to find a way to successfully manage resources in
order to produce products in the most efficient way. They need to design a production
schedule to minimize the flow time of a product. The number of possible schedules in
a flow shop scheduling problem involving n-jobs and m-machines is ( n !)m . The
optimal solution for the problem is to find the optimal or near optimal sequence of
jobs on each machine in order to minimize the total elapsed time.Johnson (1954) first
of all gave a method to minimise the makespan for n-job, two-machine scheduling
problems. The scheduling problem practically depends upon the important factors
namely, Transportation time, break down effect, Relative importance of a job over
another job etc. These concepts were separately studied by Ignall and Scharge (1965),
Maggu and Dass (1981), Temiz and Erol(2004),Yoshida and Hitomi (1979), Lomnicki
(1965), Palmer (1965) , Bestwick and Hastings (1976), Nawaz et al. (1983) , Sarin and
Lefoka (1993) , Koulamas (1998) , Dannenbring (1977) , etc.
Singh T.P. and Gupta Deepak (2005)studied the optimal two stage production
schedule in which processing time and set up time both were associated with
probabilities including job block criteria. Heydari (2003)dealt with a flow shop
scheduling problem where n jobs are processed in two disjoint job blocks in a string
consists of one job block in which order of jobs is fixed and other job block in which
order of jobs is arbitrary. Lomnicki (1965) introduced the concept of flow shop
scheduling with the help of branch and bound method. Further the work was developed
by Ignall and Scharge (1965), Chandrasekharan (1992), Brown and Lomnicki(1966) ,
with the branch and bound technique to the machine scheduling problem by introducing
different parameters. The concept of transportation time is very important in scheduling
when the machines are distantly situated. The break down of the machines have
significant role in the production concern. The effect of break down interval is
important as there are feasible situations where machine during process may get sudden
24
2. Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5782 (print) ISSN 2225-0506 (online)
Vol 2, No.1, 2012
break down due to either failure of any component of machine or the machines are
supposed to stop their working for a certain interval of time due to some external
imposed policy such as electric cut/shortage due to government policy. The working of
machine no longer remains continuous and is subject to break-down for a certain
interval of time. This paper extends the study made by Ignall and Scharge (1965) by
introducing the concept of transportation time and break down interval. Hence the
problem discussed here is wider and has significant use of theoretical results in process
industries.
2 Practical Situation
Many applied and experimental situations exist in our day-to-day working in factories
and industrial production concerns etc. In many manufacturing companies different
jobs are processed on various machines. These jobs are required to process in a
machine shop A, B, C, ---- in a specified order. When the machines on which jobs are
to be processed are planted at different places, the transportation time (which includes
loading time, moving time and unloading time etc.) has a significant role in
production concern. The break down of the machines (due to delay in material,
changes in release and tails date, tool unavailability, failure of electric current, the
shift pattern of the facility, fluctuation in processing times, some technical
interruption etc.) have significant role in the production concern.
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 i on machine A
Bi : Processing time for job i on machine B
Ci : Processing time for job i on machine C
Cij : Completion time for job i on machines A, B and C
ti : Transportation time of ith job from machine A to machine B.
gi : Transportation time of ith job from machine B to machine C.
Sk : Sequence using johnson’s algorithm
L : Length of break down interval.
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 three machines A, B & C in the
order ABC. A job i (i=1,2,3…n) has processing time Ai , Bi & Ci on each machine
respectively, assuming their respective probabilities pi , qi & ri such that 0 ≤ pi ≤ 1,
Σpi = 1, 0 ≤ qi ≤ 1, Σqi = 1, 0≤ ri ≤ 1, Σri = 1. Let ti and gi be the transportation time of
machine A to machine B and machine B to machine C respectively. The mathematical model
of the problem in matrix form can be stated as :
25
3. Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5782 (print) ISSN 2225-0506 (online)
Vol 2, No.1, 2012
Jobs Machine A Machine B Machine C
ti gi
i Ai Bi Ci
1 A1 t1 B1 g1 C1
2 A2 t2 B2 g2 C2
3 A3 t3 B3 g3 C3
4 A4 t4 B4 g4 C4
- ---- --- --- --- ----
- -- -- -- --- --
n An tn Bn gn Cn
Tableau – 1
Our objective is to obtain the optimal schedule of all jobs which minimize the total
elapsed time whenever the effect of break down interval (a, b) is given, using branch
and bound technique.
5 Algorithm:
Step1: Calculate
(i) g1 = t ( J r ,1) + ∑ Ai + min( Bi + Ci )
′
i∈J r
′
i∈J r
(ii) g2 = t ( J r , 2) + ∑ Bi + min(Ci )
′
i∈J r
′
i∈ jr
(iii) g3= t ( J r ,3) + ∑ Ci
′
i∈ jr
Step 2: Calculate g = max [g1, g2, g3] We evaluate g 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 3: Now explore the vertex with lowest label. Evaluate g 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 4: Prepare in-out table for the optimal sequence obtained in step 4 and read the
effect of break down interval (a, b) on different jobs.
Step 5: Form a modified problem with processing times p′ , p′ & p′ on machines A,
i1 i2 i3
B & C respectively. If the break down interval (a, b) has effect on job i then p′ =pi1 + L
i1
, p′ = pi2 + L and p′ = pi3 + L
i2 i3 where L = b – a, the length of the break down
interval.
If the break down interval (a, b) has no effect on job i then p′ =pi1 , p′ = pi2 and p′ = pi3.
i1 i2 i3
26
4. Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5782 (print) ISSN 2225-0506 (online)
Vol 2, No.1, 2012
Step 6: Repeat the procedure to get the optimal sequence for the modified scheduling
problem using step1 to step 3. Compute the in-out table and get the minimum total
elapsed time.
6 Numerical Example:
Consider 5 jobs 3 machine flow shop problem. processing time of the jobs on each
machine is given. Our objective is to obtain the optimal schedule of all jobs which
minimize the total elapsed time whenever the effect of break down interval (25, 35) is
given.
Jobs Machine A Machine B Machine C
I Ai ti Bi gi Ci
1 15 4 20 5 16
2 25 7 10 8 5
3 10 6 12 3 12
4 18 9 15 7 18
5 16 2 25 6 3
Tableau – 2
Solution:Step1: Calculate
(i) g1 = t ( J r ,1) + ∑ Ai + min( Bi + Ci )
′
i∈J r
′
i∈J r
(ii) g2 = t ( J r , 2) + ∑ Bi + min(Ci )
′
i∈J r
′
i∈ jr
(iii) g3= t ( J r ,3) + ∑ Ci
′
i∈ jr
For J1 = (1).Then J′(1) = {2,3,4}, we get g1 = 43 , g2 = 37 & g3 = 43
g = max(g1, g2, g3) = 43 similarly, we have LB(2)= 51 , LB(3)= 52 and LB(4)= 58
Step 2 & 3: Now branch from J1 = (1). Take J2 =(12). Then J′2={3,4} and LB(12) = 51
Proceeding in this way, we obtain lower bound values on the completion time on
machine C as shown in the tableau- 3
Step 4 :Therefore the sequence S1 is 1-3-4-2 and the corresponding in-out table and
checking the effect of break down interval (25, 35) on sequence S1 is as in tableau4:
Step 5: The modified problem after the effect of break down interval (25,35) with
processing times A′i, B′i and C′i on machines A, B & C respectively is as in tableau-5:
Step 6: Now, on repeating the procedure to get the optimal sequence for the modified
scheduling problem using step 1 to step 3, we obtain lower bound values on the
27
5. Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5782 (print) ISSN 2225-0506 (online)
Vol 2, No.1, 2012
completion time on machine C as shown in the tableau- 6 we have get the sequence
S2 : 3-1-4-5-2. Compute the in-out table for S2 and get the minimum total elapsed time
as in tableau-7.
Hence the total elapsed time is 131 units.
References :
[1] 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.
[2] Bestwick, P.F. and Hastings, N.A.J. (1976), “A new bound for machine
scheduling”, Operational Research Quarterly, Vol. 27, pp.479-490.
[3] Cormen, T.H., Leiserson, C.E. and Rivest, R.L. (1990), “Introduction to
Algorithms”, Cambridge, MA: MIT Press.
[4] Chandramouli, A.B.(2005), “Heuristic approach for N job 3 machine flow
shop scheduling problem involving transportation time, break-down time and
weights of jobs”, Mathematical and Computational Application, Vol.10
(No.2), pp 301-305.
[5] 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 61, 318-325.
[6] Dannenbring, D.G. (1977) ,“An evaluation of flowshop sequencing
heuristics”, Management Science, Vol. 23, No. 11, pp.1174-1182.
[7] Heydari (2003), “On flow shop scheduling problem with processing of jobs in
a string of disjoint job blocks: fixed order jobs and arbitrary order jobs”,
JISSOR , Vol. XXIV , pp 1- 4.
[8] Ignall, E. and Schrage, L. (1965), “Application of the branch-and-bound
technique to some flowshop scheduling problems”, Operations Research, Vol.
13, pp.400-412.
[9] Johnson S. M. (1954), “Optimal two and three stage production schedule with
set up times included”. Nay Res Log Quart Vol 1, pp 61-68
[10] Koulamas, C. (1998), “A new constructive heuristic for the flowshop
scheduling problem”, European Journal of Operations Research’, Vol. 105,
pp.66-71.
[11] 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.
[12] Maggu & Das (1981), “On n x 2 sequencing problem with transportation
time of jobs”, Pure and Applied Mathematika Sciences, 12-16.
[13] Nawaz M., Enscore Jr., E.E. and Ham, I. (1983) , “A heuristic algorithm for
the m-machine n-job flowshop sequencing problem”, OMEGA International
Journal of Management Science, Vol. 11, pp.91-95.
[14] Palmer, D.S.(1965), “Sequencing jobs through a multi-stage process in the
minimum total time - a quick method of obtaining a near-optimum”,
Operational Research Quarterly, Vol. 16,No. 1, pp.101-107.
28
6. Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5782 (print) ISSN 2225-0506 (online)
Vol 2, No.1, 2012
[15] Park, Y.B. (1981), “A simulation study and an analysis for evaluation of
performance-effectiveness of flowshop sequencing heuristics: a static and
dynamic flowshop model”, Master’s Thesis, Pennsylvania State University.
[16] Singh, T.P., K, Rajindra & Gupta Deepak (2005), “Optimal three stage
production schedule the processing time and set up times associated with
probabilities including job block criteria”, Proceeding of National Conference
FACM- (2005), pp 463-470.
[17] Sarin, S. and Lefoka, M. (1993), “Scheduling heuristics for the n-job, m-
machine flowshop”, OMEGA, Vol. 21, pp.229-234.
[18] Turner S. and Booth D. (1987), “Comparison of heuristics for flowshop
sequencing”, OMEGA,Vol.15, pp.75-78.
[19] Temiz Izzettin and Serpil Erol(2004), “Fuzzy branch and bound algorithm
for flow shop scheduling”, Journal of Intelligent Manufacturing, Vol.15,
pp.449-454.
[20] Yoshida and Hitomi (1979), “Optimal two stage production scheduling with
set up times separated”,AIIETransactions. Vol. II.pp 261-263.
29