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* 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*

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Statement of dual problems …

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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)

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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

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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.

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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.

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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

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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

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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.

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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

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