## 6 Designing, Balancing and Scheduling Assembly Lines with Uncertain Parameters
### Stability Analysis of Assembly and Production Lines  
The paper [131] deals with the optimization problem, which arises when a new simple assembly line has to be designed subject to a fixed number of the workstations, a predetermined cycle time constraint and the precedence constraints determined on a set of assembly operations. The studied ALBP consists in assigning a set of assembly operations to workstations so as to find the robust assembly line, which can withstand operation duration uncertainty in the robust sense. The assembly line robustness was measured by an indicator called a stability factor. The studied ALBP was proven to be strongly NP-hard, upper bounds were derived on the stability factor and the relation of the stability factor with the stability indicator, called a stability radius, was investigated. A mixed-integer linear programming was proposed for maximizing the stability factor. An alternative formulation was derived when uncertainty originates in the used workstations only. Computational results were reported on a collection of ALBP instances derived from benchmark data used in the OR literature for the deterministic SALBP. The paper [132] is devoted to study optimization problems arising if a transfer line has to be designed subject to a limited number of available workstations, a cycle time constraint and predetermined precedence relations on the set of assembly operations. The considered problem consists in assigning a set of operations to blocks and the determined blocks to workstations in order to construct a robust transfer line configuration under operation duration uncertainty. The robustness of a transfer line configuration is measured by the stability radius, determined as the maximal amplitude of deviations from the nominal value of the durations of uncertain operations that do not violate the solution feasibility. In order to consider different hypotheses on operation duration uncertainty, the stability radius was based on the Manhattan norm or the Chebyshev norm. The considered problem was proven to be strongly NP-hard. A mixed-integer linear programming was proposed for addressing these problems. In order to accelerate the search for an optimal transfer line, two heuristic algorithms and several reduction rules were derived for the mixed-integer linear programming. Computational results were reported on a collection of instances derived from benchmark data used in the OR literature for the deterministic transfer line balancing problems. The paper [133] is devoted to the SALBP, which arises when a paced simple assembly line has to be designed for the limited number of workstations, predetermined cycle time and precedence constraints given on a set of the assembly operations. The studied problem consists in assigning a set of assembly operations to available workstations in such a way that the constructed line configuration (a feasible solution) will be robust under the operation duration variability. The solution robustness was measured via the stability radius, which is equal to the maximal amplitude of deviations of assembly operation durations that do not violate the solution feasibility. In this paper, the concept of the stability radius was considered for the Manhattan and Chebyshev norms. For each of these norms, the SALBP was proven to be strongly NP-hard, and a mixed-integer linear programming was developed for addressing these uncertain problems. To accelerate the search for optimal solutions, an upper bound on the stability radius was proven and integrated into the corresponding mixed-integer linear programming. Computational results were reported on uncertain instances derived from benchmark data used for the deterministic SALBP. In [38], the GALBP was studied, where several workplaces were associated with each available workstation. The sets of assembly operations assigned to the workstation have to be partitioned into blocks. Each assembly operation block regroups all assembly operations to be performed at the same workplace of the workstation.