## 4 Assembly Line Balancing Problems with Stochastic or Fuzzy Parameters
### Assembly and Production Line Balancing Problems with Fuzzy Parameters  
The input data for many practical ALBP are uncertain or questionable, and so only some limits can be set on the original data. The data questionability can be represented by fuzzy numbers to reduce possible errors associated with input data. The duration of assembly operations in such an ALBP can be represented as fuzzy numbers. Such problems are called fuzzy ALBP and can be formulated as follows. Let the set of assembly operations, \(V=\{1,2,\ldots,n\}\), be given. The duration of each operation is presented by a fuzzy number. The precedence digraph \(G=(V,A)\) is given, which determines the partial strict order on the set \(V\). The problem is to assign a set of operations, \(V\), to the ordered workstations, \(S_{1},S_{2},\ldots,S_{m}\), in such a way that the precedence constraints are not disturbed and the objective function would take an optimal fuzzy value [26; 33; 88; 89]. The articles [88; 89] provide approximate solutions to the ALBP with fuzzy durations of the assembly operations. A genetic algorithm was developed to minimize the maximum total duration of the assembly operations assigned to each workstation. Genetic operators suitable for solving the fuzzy ALBP were considered. The article [88] provides a heuristic solution to the ALBP with fuzzy durations of the assembly operations. In [26], a mixed-model assembly line is designed to assemble several types of the final product. The optimal sequence of product models was determined to minimize the number of assembly conveyor stops required to move from assembling one product model to assembling another model of the same product. When the operating conditions of the assembly line change or the requirements for the product model change, it is important to determine the assembly sequence of all models of the product, for which the stops of the assembly conveyor would be minimal and there would be no frequent need to rebalance the assembly line. Three objective functions conflicting with each other were considered. To solve the fuzzy ALBP, a mathematical programming method with a fuzzy goal was developed. The mixed-model assembly lines are highly adopted in the automobile industry and so the part feeding process becomes critical. In [90], the dynamic part feeding scheduling problem was studied for optimization of the throughput of the mixed-model assembly lines, along with the total delivery distance of the automatic guide vehicles. A process considering the feeding operation generation, the loading, sequencing and dispatching problems was analyzed, determining appropriate models and algorithms. To heuristically solve these problems, the research [90] contains a hybrid fuzzy-neural dynamic scheduling algorithm, which integrates the self-organizing maps with a fuzzy algorithm and a knowledge base. This algorithm was adapted to the pre-cluster status of the mixed-model assembly line in order to optimize the initial clustering centers. After that, the algorithm was availed to guide the clusters in a way to improve the clustering performance. The computational experiments were conducted in order to evaluate the scheduling performance in the dynamic manufacturing environment and verify the algorithm's superiority over the benchmark ones. The developed algorithm allows a decision-maker to select a rational scheduling scheme based on the decision impacts in the productivity, feeding costs and the real-time status of the assembly conveyers. The paper [91] presents the results of the investigation of a single-model SALBP with fuzzy assembly operation durations. This problem is referred to as a fuzzy SALBP-E, consisting of finding the best combination of the number of workstations and the cycle time as well as a respective assembly line balance, such that the determined measure of efficiency of the single-model assembly line is maximized. A fuzzy SALBP-E is an extension of the deterministic SALBP-E under fuzziness of input data.