Stability analysis of an optimal balance for an assembly line with fixed cycle time

We address the simple assembly line balancing problem: minimize the number of stations m for processing n partially ordered operations V = {1, 2, …, n} within the cycle time c. The processing time ti of operation i ∈ V and cycle time c are given. However, during the life cycle of the assembly line the values ti are definitely fixed only for the subset of automated operations V⧹V∼. Another subset V∼⊆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∈V∼, then operation time tj can be different for different cycles of production process. For the optimal line balance b of a paced assembly line with vector t = (t1, t2, …, tn) of the operation times, we investigate stability of its optimality with respect to possible variations of the processing times tj of the manual operations j∈V∼. In particular, we derive necessary and sufficient conditions when optimality of the line balance b is stable with respect to sufficiently small variations of the operation times tj, j∈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 b.