## 4 Assembly Line Balancing Problems with Stochastic or Fuzzy Parameters
### Stochastic Assembly and Production Line Balancing Problems
This interpretation of the non-productive times in available workstations was presented by introducing the concept of the fractal buffer time to manage the effect of manufacturing uncertainties. To overcome the problem of excessive robustness, a robust algorithm with conservatism-level flexibility was used, focusing on the cycle time in a bottleneck workstation. The effect of the uncertainties and conservatism levels on the cycle time was analyzed through several numerical instances. Computational results of the study can be used for improving a manufacturing system in which uncertainties in assembly operations and inter-operation times may significantly degrade its productivity. Human learning algorithms were developed in many research fields, including the ALBP. Despite the plethora of real contributions and different algorithms used for solving the optimization problems, the autonomous learning phenomenon (the time-dependent or position-dependent reduction of the assembly operation durations due to possible process repetitions) should be explored using a stochastic model, which has been disregarded. In the paper [81], a cost-based stochastic balancing property was coupled with a time-learning curve in order to investigate the role of learning in the rebalancing of the existing assembly conveyers with repetitive assembly operations. A real case study was conducted to demonstrate the applicability of the new algorithms. The paper [82] presents a mixed-model assembly operation sequencing problem with stochastic operation durations in a multi-workstation assembly line. A mixed-integer nonlinear programming was developed to minimize a weighted sum of the expected total workstation overload and workstation idieness, which was converted into a mixed-integer linear programming to optimally solve small-sized instances of the sequencing problem. Due to the proven NP-hardness of the considered sequencing problem [82], a simulated annealing algorithm was developed. This algorithm employs a learning procedure to select an appropriate heuristic through a search process. Several numerical results were presented on the tested and benchmark instances taken from the OR literature. The computational results of the statistical analysis indicated that the developed algorithm was quite competitive in comparison with the published software packages. The developed algorithm was superior to other published simulated annealing algorithms. These computational results highlight the advantages of the mixed-model sequencing in comparison with deterministic algorithms used for the mixed-model sequencing problems. Assembly lines of determining the optimal order of the available workstations in the U-shaped assembly lines with stochastic durations of the assembly operations were studied in the paper [82], as well. The ALBP is highly important for efficient and cost-effective assembly production of similar products. Different uncertain events might cause a variation in the assembly operation duration. Due to these variations, there remains a possibility that the completion time of the assembly operations might exceed the predetermined cycle time. To hedge against such an issue, a single-model ALBP with the uncertain operation durations and multiple objective functions was studied in [83]. This research aimed to minimize the cycle time in addition to maximizing the probability that completion times of the operations on the workstations will not exceed the predetermined cycle time and will minimize the smoothness index. A Pareto-based artificial bee colony algorithm was proposed to obtain a Pareto solution for the multiple objective functions. The proposed algorithm introduced extra steps, as follows: sorting of food sources, a niche technique and preserving some elitists in the traditional artificial bee colony algorithm to obtain a Pareto solution. The main parameters of the developed algorithm were tuned using the Taguchi method. Computational experiments were conducted to solve the standard ALBP, which were taken from the OR library.