# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Stability analysis of an optimal balance for an assembly line with fixed cycle time. (English) Zbl 1102.90321
We address the simple assembly line balancing problem: minimize the number of stations $m$ for processing $n$ partially ordered operations $V=1, 2,\dots, n$ within the cycle time $c$. The processing time $t_i$ of operation $i\in V$ and cycle time $c$ are given. However, during the life cycle of the assembly line the values $t_i$ are definitely fixed only for the subset of automated operations $V\setminus \widetilde V$. Another subset $\widetilde V\subseteq V$ includes manual operations, for which it is impossible to fix the exact processing times during the whole life cycle of the assembly line. If $j\in\widetilde V$, then operation time $t_j$ can be different for different cycles of production process. For the optimal line balance $\bold b$ of a paced assembly line with vector $t = (t_1, t_2,\dots, t_n)$ of the operation times, we investigate stability of its optimality with respect to possible variations of the processing times $t_j$ of the manual operations $j\in\widetilde V$. In particular, we derive necessary and sufficient conditions when optimality of the line balance $\bold b$ is stable with respect to sufficiently small variations of the operation times $t_j$, $j\in\widetilde V$. We show how to calculate the maximal value of independent variations of the processing times of all the manual operations, which definitely keep the feasibility and optimality of the line balance $\bold b$.

##### MSC:
 90B30 Production models 90C31 Sensitivity, stability, parametric optimization
Full Text:
##### References:
 [1] Baybars, I.: A survey of exact algorithms for the simple assembly line balancing problem. Management science 32, No. 8, 909-932 (1986) · Zbl 0601.90081 [2] Bukchin, J.; Tzur, M.: Design of flexible assembly line to minimize equipment cost. IIE transactions 32, 585-598 (2000) [3] Dolgui, A.; Guschinski, N.; Harrath, Y.; Levin, G.: Une approche de programmation linéaire pour la conception des lignes de transfert. European journal of automated systems (APII-JESA) 36, No. 1, 11-33 (2002) [4] A. Dolgui, N. Guschinsky, G. Levin, On problem of optimal design of transfer lines with parallel and sequential operations, in: J.M. Fuertes (Ed.), Proceedings of the 7th IEEE International Conference on Emerging Technologies and Factory Automation (ETFA’99), Barcelona, vol. 1, 1999, pp. 329-334. [5] Erel, E.; Sarin, S. C.: A survey of the assembly line balancing procedures. Production planning and control 9, No. 5, 414-434 (1998) [6] Gen, M.; Tsujimura, Y.; Li, Y.: Fuzzy assembly line balancing using genetic algorithms. Computers and industrial engineering 31, No. 3/4, 631-634 (1996) [7] Kravchenko, S. A.; Sotskov, Yu.N.; Werner, F.: Optimal schedules with infinitely large stability radius. Optimization 33, 271-280 (1995) · Zbl 0821.90067 [8] Lee, H. F.; Johnson, R. V.: A line-balancing strategy for designing flexible assembly systems. The international journal of flexible manufacturing systems 3, 91-120 (1991) [9] Pinto, P. A.; Dannenbring, D. G.; Khumawala, B. M.: Assembly line balancing with processing alternatives: an application. Management science 29, 817-830 (1983) [10] Sarin, S. C.; Erel, E.; Dar-El, E. M.: A methodology for solving single-model, stochastic assembly line balancing problem. OMEGA--international journal of management science 27, 525-535 (1999) [11] Scholl, A.: Balancing and sequencing of assembly lines. (1999) · Zbl 0949.90034 [12] Scholl, A.; Klein, R.: Balancing assembly lines effectively: A computational comparison. European journal of operational research 114, 51-60 (1998) · Zbl 0949.90034 [13] Sotskov, Yu.N.: Stability of an optimal schedule. European journal of operational research 55, 91-102 (1991) · Zbl 0755.90048 [14] Yu.N. Sotskov, A. Dolgui, Stability radius of the optimal assembly line balance with fixed cycle time, in: Proceedings of the IEEE Conference ETFA’2001, 2001, pp. 623-628. [15] Sotskov, Yu.N.; Tanaev, V. S.; Werner, F.: Stability radius of an optimal schedule: A survey and recent developments. Industrial applications of combinatorial optimization 16, 72-108 (1998) · Zbl 0936.90030 [16] Tsujimura, Y.; Gen, M.; Kubota, E.: Solving fuzzy assembly-line balancing problem with genetic algorithms. Computers and industrial engineering 29, No. 1-4, 543-547 (1995)