## 4 Assembly Line Balancing Problems with Stochastic or Fuzzy Parameters
### Stochastic Assembly and Production Line Balancing Problems
The performance of the developed Pareto-based artificial bee colony algorithm was compared with a multi-objective algorithm, NSGA II. Computational results showed that the proposed algorithm outperforms the NSGA II algorithm in both Pareto solution quality and CPU time. In [84], a workforce assignment is studied in the assembly line with several workstations and executing final parts of different types. The objective is to minimize the number of human operators over the assembly conveyer. Due to the complex market environment, possible changes in product demands have an influence on production balancing and scheduling. The uncertain demands were investigated in [84]. An ambiguity set was applied to portray the demand uncertainty. A chance-constrained programming was developed. Two probability-distribution-free algorithms were chosen: approximations based on Markov inequality and a mixed-integer second-order conic program, to approximate the chance constraints of the tested problems. Computational experiments were conducted to compare the performances of the two proposed algorithms. The model assembly conveyer is an industrial arrangement of the available workstations, needed equipment and assembly operators for continuous flow of workpieces in mass production operations. The reliability of the assembly production has been investigated by taking into account operation duration uncertainties. The paper [85] provides a reliability metric which encompasses two types of operation duration uncertainties. A multi-objective mathematical model was developed to maximize the reliability and efficiency of the conveyers. Neighborhood search methods with two restart mechanisms were devised to solve the problem, and then they were compared. The computational results showed some managerial implications for the production planners. The methodology proposed in [85] can be applied to many assembly industries when some historical data of uncertain inputs are available, while some others are not. In [86], chance-constrained binary programming for the stochastic straight- and U-shaped line balancing problems were proposed and investigated. The proposed algorithms were used for solving several instances, which are available from the OR literature. The obtained computational results were described and compared. A goal programming algorithm was also developed for increasing the assembly line reliability, which is needed to investigate the stochastic ALBP. In the real assembly lines, production planning and inventory control are often subject to different types of uncertainty. The paper [87] is devoted to a control problem for a single-level multi-component inventory, which arises in the assembly line replenishment under stochastic component procurement lead times. In order to follow the common assumption of the MRP software tool, the discrete distributions of random component lead times were investigated. The latter was expressed as the number of the tested time periods. Since the finished product was assembled using several component types simultaneously, the assembly process was stopped if a single type of component was delayed. The assembly stoppage forced by a component delay or stock-out was penalized by a backlogging cost. The considered objective aimed to minimize the total cost composed of holding and backlogging costs. To solve this problem, a joint chance-constrained algorithm was developed based on an equivalent linear reformulation. The practical advantages of the developed approach were estimated in its release from backlogging costs, which are difficult to quantify in the real-world industrial assembly productions. Summarizing the results of [68; 69; 87], one can conclude that the stochastic SALBP turned out to be more complex than the deterministic SALBP. However, stochastic problems do not quite correspond to some real assembly conveyer productions, where it is not possible to obtain sufficient information to determine the probability distribution of a random duration, \(x_{i}\), of each assembly operation, \(i\in V\).