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Calculation of the stability radius of an optimal line balance

Calculation of the stability radius of an optimal line balance. Yuri N. Sotskov* , Frank Werner**, Aksana Zatsiupa*. * United Institute of Informatics Problem, Minsk, Belarus ** Otto-von-Guericke-Universität, Magdeburg, Germany. Statement of dual problems. …. 5. 6. 1. 3. 4. 2.

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Calculation of the stability radius of an optimal line balance

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

  2. Statement of dual problems … 5 6 1 3 4 2

  3. B(t) Bopt(t) B FORMULATION OF PROBLEMSALBP-1 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. (SetB) Condition 2: The cycle time c is not violated for each station . (SetB(t)) Condition 3:The line balanceb uses a minimal number of m stations. (SetBopt(t))

  4. B(t) Bopt(t) B FORMULATION OF PROBLEMSALBP-2 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. (SetB) Condition 2: The line balance b has to use all m stations. (SetB(t)) Condition 3:The line balance b provides the minimal cycle timeс.(Set )

  5. DEFINITION OF THE STABILITY RADIUS 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 . t2 t1 0

  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 PROBLEMSALBP-1 Theorem 3 If and then where and are defined by where where and

  8. EXACT VALUE OF THE STABILITY RADIUS FOR THE PROBLEMSALBP-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. 168. p. 783−797. 3. 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.

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

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