## 4 Assembly Line Balancing Problems with Stochastic or Fuzzy Parameters
### Stochastic Assembly and Production Line Balancing Problems
A Benders' decomposition algorithm was developed along with different inequalities and a preprocessing stage as a solution algorithm. Three datasets were proposed and used for testing the developed algorithm and treating uncertainty in the mixed-model assembly line. The integration of the strategic ALBP with the operational sequencing one was used in robust assembly lines. Most of the research papers related to different assembly lines are concentrated on the ALBP, provided that the precedence relations among the given assembly operations are not violated and the objective function is optimized in the desired line balance. The multi-objective ALBP with stochastic assembly operation durations is an important practical topic of the traditional ALBP involving conflicting criteria, such as minimizing the cycle time, variation of workload or the processing cost under uncertain manufacturing conditions. The paper [78] proposes a hybrid multi-objective evolutionary algorithm for heuristically solving such an ALBP, with stochastic assembly operation durations to minimize the cycle time and the processing cost with the given fixed set of the workstations. The special fitness function was adopted, and a hybrid selection was designed to improve the convergence of the solution process. The computational experiments with tested instances showed that the developed multi-objective evolutionary algorithm could display a better convergence distribution performance than other published algorithms. The paper [79] includes a multi-objective genetic algorithm for heuristically solving a mixed-model ALBP, simultaneously considering the cycle time and number of available workstations. A mixed-model assembly line is capable of producing different types of products to respond to uncertain market demands, while minimizing capital costs of designing a multiple assembly line. According to the stochastic environment of the considered assembly productions, a mixed-model assembly line was put forth in the make-to-order environment. A multi-objective genetic algorithm was developed for solving the corresponding ALBP and a decision-maker was provided with the subsequent replies to pick one of them based on the specific situation. A computational comparison on the computer was carried out between six multi-objective evolutionary algorithms in order to determine the best algorithm to heuristically solve the specified ALBP. Possible variations of the assembly operation durations in the manufacturing assembly line can result in a longer processing time to complete assembly operations than a given cycle time. This may lead to the assembly line stoppage and to loss of the production time. In practical assembly production, a portion of the cycle time is often allocated as a predefined fixed-size buffer time, which is determined based on experience for accounting uncertain variations of the durations of assembly operations for a paced assembly line without storage-buffers between workstations. The size of the required buffer time in each available workstation depends on the variation levels of the durations of the assembly operations and the desired conservatism level for preventing a cycle time violation. There are uncertainties in other added activity times in available workstations, which are called inter-operation times. Although many studies on designing a stochastic manufacturing assembly line focused on minimizing the cost incurred when the cycle time is exceeded due to assembly operation duration variations, they mostly disregarded the inter-operation times. Therefore, it is worth studying the simultaneous effect of the manufacturing time uncertainty and that of the conservatism level on the cycle time. The paper [80] proposes the algorithm for a robust manufacturing assembly line design that incorporates the conservatism level and uncertainties in the assembly operation and inter-operation times.