* United Institute of Informatics Problem, Minsk, Belarus ** Otto-von-Guericke-Universität, Magdeburg, Germany Calculation of the stability radius of an.

Slides:

Advertisements

Similar presentations

Agenda of Week X. Layout Capacity planning Process selection Linebalancing Review of week 9 13 Approaches Purposes : Finishing the capacity planning Understanding.

Advertisements

Yuri N. Sotskov 1, Omid Gholami 2, Frank Werner 3 1. United Institute of Informatics Problems, Minsk, Belarus, 2. Islamic.

C&O 355 Lecture 8 N. Harvey TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A.

Spatial Embedding of Pseudo-Triangulations Peter Braß Institut für Informatik Freie Universität Berlin Berlin, Germany Franz Aurenhammer Hannes Krasser.

Incremental Linear Programming Linear programming involves finding a solution to the constraints, one that maximizes the given linear function of variables.

1 Graphs with Maximal Induced Matchings of the Same Size Ph. Baptiste 1, M. Kovalyov 2, Yu. Orlovich 3, F. Werner 4, I. Zverovich 3 1 Ecole Polytechnique,

Partial Derivatives Definitions : Function of n Independent Variables: Suppose D is a set of n-tuples of real numbers (x 1, x 2, x 3, …, x n ). A real-valued.

1 Material to Cover  relationship between different types of models  incorrect to round real to integer variables  logical relationship: site selection.

Solving a job-shop scheduling problem by an adaptive algorithm based on learning Yuri N. Sotskov 1, Omid Gholami 2, Frank Werner 3 1. United Institute.

Transportation Problem (TP) and Assignment Problem (AP)

MATHEMATICS 3 Operational Analysis Štefan Berežný Applied informatics Košice

Optimization of thermal processes2007/2008 Optimization of thermal processes Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery.

Optimization of thermal processes2007/2008 Optimization of thermal processes Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery.

Parallel Scheduling of Complex DAGs under Uncertainty Grzegorz Malewicz.

Optimization of thermal processes2007/2008 Optimization of thermal processes Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery.

Algorithmic and Economic Aspects of Networks Nicole Immorlica.

Proximal Support Vector Machine Classifiers KDD 2001 San Francisco August 26-29, 2001 Glenn Fung & Olvi Mangasarian Data Mining Institute University of.

Approximation Algorithms

Support Vector Machines Formulation  Solve the quadratic program for some : min s. t.,, denotes where or membership.  Different error functions and measures.

Contents Introduction Related problems Constructions –Welch construction –Lempel construction –Golomb construction Special properties –Periodicity –Nonattacking.

EE 685 presentation Optimization Flow Control, I: Basic Algorithm and Convergence By Steven Low and David Lapsley Asynchronous Distributed Algorithm Proof.

Computability and Complexity 24-1 Computability and Complexity Andrei Bulatov Approximation.

Multihop Paths and Key Predistribution in Sensor Networks Guy Rozen.

Lot sizing and scheduling

Mathematics1 Mathematics 1 Applied Informatics Štefan BEREŽNÝ.

Linear programming. Linear programming… …is a quantitative management tool to obtain optimal solutions to problems that involve restrictions and limitations.

Optimal pricing of a heterogeneous portfolio for a given risk level Yaniv Zaks 1,2, Esther Frostig 2 and Benny Levikson 2 1 Department of Mathematics,

LP-based Algorithms for Capacitated Facility Location Chaitanya Swamy Joint work with Retsef Levi and David Shmoys Cornell University.

Steady and Fair Rate Allocation for Rechargeable Sensors in Perpetual Sensor Networks Zizhan Zheng Authors: Kai-Wei Fan, Zizhan Zheng and Prasun Sinha.

THE STABILITY BOX IN INTERVAL DATA FOR MINIMIZING THE SUM OF WEIGHTED COMPLETION TIMES Yuri N. Sotskov Natalja G. Egorova United Institute of Informatics.

1 On single machine scheduling with processing time deterioration and precedence constraints V. Gordon 1, C. Potts 2, V.Strusevich 3, J.D. Whitehead 2.

Markov Decision Processes1 Definitions; Stationary policies; Value improvement algorithm, Policy improvement algorithm, and linear programming for discounted.

Approximation Algorithms Department of Mathematics and Computer Science Drexel University.

A Graphical Approach for Solving Single Machine Scheduling Problems Approximately Evgeny R. Gafarov Alexandre Dolgui Alexander A. Lazarev Frank Werner.

Flexible Assembly line Minimum part set(MPS) Suppose there are l product types. Let N l denote the number of jobs for each product type l. If z is the.

TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A A A A A A A Image:

Course: Logic Programming and Constraints

1 Combinatorial Algorithms Local Search. A local search algorithm starts with an arbitrary feasible solution to the problem, and then check if some small,

Optimization of Continuous Models

12.6 Surface Area and Volume of a Sphere

EE 685 presentation Optimization Flow Control, I: Basic Algorithm and Convergence By Steven Low and David Lapsley.

Part 3 Linear Programming 3.3 Theoretical Analysis.

Operational Research & ManagementOperations Scheduling Economic Lot Scheduling 1.Summary Machine Scheduling 2.ELSP (one item, multiple items) 3.Arbitrary.

1.2 Guidelines for strong formulations  Running time for LP usually depends on m and n ( number of iterations O(m), O(log n)). Not critically depend on.

STABILITY ANALYSIS IN MULTICRITERIA OPTIMIZATION Tallinn 2010 Karelkina Olga University of Turku Department of Mathematics.

Introduction to Optimization

What is a matroid? A matroid M is a finite set E, with a set I of subsets of E satisfying: 1.The empty set is in I 2.If X is in I, then every subset of.

10-3 Areas of Regular Polygons. Parts of a Regular Polygon center radius apothem.

Part 3 Linear Programming 3.3 Theoretical Analysis.

Tuesday, March 19 The Network Simplex Method for Solving the Minimum Cost Flow Problem Handouts: Lecture Notes Warning: there is a lot to the network.

Evgeny R. Gafarov Alexander A. Lazarev Frank Werner

Irina V. Gribkovskaia Molde University College, Norway

Single Machine Scheduling with a Non-renewable Financial Resource

The minimum cost flow problem

ENGM 535 Optimization Networks

James B. Orlin Presented by Tal Kaminker

k-center Clustering under Perturbation Resilience

دانشگاه شهیدرجایی تهران

Part 3 Linear Programming

تعهدات مشتری در کنوانسیون بیع بین المللی

PTAS for Bin-Packing.

Max Flow Application: Precedence Relations

I.4 Polyhedral Theory (NW)

Flow Feasibility Problems

I.4 Polyhedral Theory.

Performance Optimization

Chapter 1. Formulations.

1.2 Guidelines for strong formulations

1.2 Guidelines for strong formulations

Presentation transcript:

* United Institute of Informatics Problem, Minsk, Belarus ** Otto-von-Guericke-Universität, Magdeburg, Germany Calculation of the stability radius of an optimal line balance Yuri N. Sotskov*, Frank Werner**, Aksana Zatsiupa*

Statement of dual problems …

SALBP-1 is to find an optimal balance of the assembly line for the given cycle time c, i.e., to find a feasible assignment of all operations V to a minimal possible number m of stations. Condition 1: Inclusion implies that operation i is assigned to some station and operation j is assigned to some station such that. (Set B) Condition 2: The cycle time c is not violated for each station. (Set B(t)) Condition 3: The line balance b uses a minimal number of m stations. (Set B opt (t)) 3 FORMULATION OF PROBLEM SALBP-1 B B(t)B(t) B opt (t)

4 SALBP-2 is to find an optimal line balance for a given set of m stations, i.e., to find a feasible assignment of the operations V to the m stations in such a way that the cycle time c reaches its minimal value. Condition 1: Inclusion implies that operation i is assigned to some station and operation j is assigned to some station such that. (Set B) Condition 2: The line balance b has to use all m stations. (Set B(t)) Condition 3: The line balance b provides the minimal cycle time с. (Set ) B B(t)B(t) B opt (t) FORMULATION OF PROBLEM SALBP-2

5 DEFINITION OF THE STABILITY RADIUS t1t1 t2t2 0 The closed (open) ball in the space with the radius and the center is called a stability ball of the line balance, if for each vector of the processing times with the line balance remains optimal. The maximal value of the radius of a stability ball is called the stability radius of the line balance.

6 CRITERIA OF STABILITY FOR OPTIMUM LINE BALANCES Theorem 1 The line balance is not stable for the problem SALBP-1 if and only if there exists a subset such that and Let be the set of subsets for which equality holds. Theorem 2 The line balance is not stable for the problem SALBP-2 if and only if there exists a line balance such that condition does not hold.

7 EXACT VALUE OF THE STABILITY RADIUS FOR THE PROBLEM SALBP-1 Theorem 3 If and then where and are defined by where where and

8 EXACT VALUE OF THE STABILITY RADIUS FOR THE PROBLEM SALBP-2 Theorem 4 If and then where and are defined by Let denote a non-decreasing sequence of the processing times of the operations from the set

9 REFERENCES 1. Sotskov, Y.N.; Sotskova, N. (2004): Теория расписаний. Системы с неопределенными числовыми параметрами; Мн.: ОИПИ НАН Беларуси, 290 p. 2. Sotskov, Y.N.; Dolgui, A.; Portmann, M.-C. (2006): Stability analysis of optimal balance for assembly line with fixed cycle time. European Journal of Operational Research. vol p. 783− Sotskov, Y.N.; Dolgui, A.; Sotskova, N.; Werner, F. (2004): Stability of optimal line balance with given station set. Supply Chain Optimization. Product/Process Design, Facility Location and Flow Control. p. 135−149.

Thanks for your attention! 10 Yuri N. Sotskov*, Frank Werner**, Aksana Zatsiupa* *United Institute of Informatics Problem, Minsk, Belarus **Otto-von-Guericke-Universität, Magdeburg, Germany