## 4 Assembly Line Balancing Problems with Stochastic or Fuzzy Parameters
### Stochastic Assembly and Production Line Balancing Problems  
In the stochastic ALBP, the durations of assembly operations are random variables with a known law of distribution of their probabilities (usually, the normal law of distribution of a random variable with known mathematical expectation and variance is used). The stochastic ALBP can be formulated as follows. The set, \(V=\{1,2,\ldots,n\}\), of assembly operations is given, and the duration of each assembly operation is a random variable for which the probability distribution law is specified before solving the ALBP. The precedence digraph \(G=(V,A)\) is given, which determines the partial strict order on the set \(V\) of the given assembly operations. The problem is to assign assembly operations, \(V\), to the ordered workstations, \(S_{1},S_{2},\ldots,S_{m}\), in such a way that the precedence relations determined by the digraph \(G=(V,A)\) are not disturbed, and the mathematical expectation of the objective function would take the optimal (respectively, minimum or maximum) value for the desired balance, \(b=(V_{1}^{b},V_{2}^{b},\ldots,V_{m}^{b})\), of the assembly line. As for the deterministic version of the assembly line balancing problem, for the stochastic ALBP, there is a finite but sufficiently large number of feasible solutions (line balances) at large values of \(m\) and \(n\). In the article [3], the algorithms for solving stochastic problems of balancing the assembly line were divided into the following three classes: (1) modifications of algorithms developed for the deterministic ALBP [68; 72], (2) study of the specific properties of the stochastic ALBP on the basis of computer modeling, with a subsequent comparison of the obtained results of solving the stochastic version and the deterministic version of the ALBP [69], and (3) algorithms developed specifically for the stochastic ALBP [70; 71]. Articles [68; 69; 70; 72] are devoted to the generalizations of the SALBP as a result of restriction of the condition (A-1). It was assumed that the durations of assembly operations are random variables with laws of probability distributions, which are known before solving the problem. Instead of a deterministic criterion, a corresponding stochastic criterion has to be optimized. Namely, in the stochastic SALBP-1, the mathematical expectation, \(Em\), of the number of the used workstations, \(m\), has to be minimized for the fixed cycle time, \(c\). In the stochastic SALBP-2, the expected value of the cycle time, \(c\), has to be minimized for the given number, \(m\), of the used workstations. The Industry 4.0 concept aims to bring more flexibility and agility to the assembly and production shop floor. The sequencing and scheduling problems are important issues of Industry 4.0. In fact, a good (or optimal, which is better) schedule has to guarantee a high performance level, which allows a scheduler to take into consideration possible changes and machine perturbations occurring in the production workshop. In [73], the different approaches aim to find stochastic, fuzzy, robust or stable schedules capable of optimizing corresponding single or several criteria, considering possible machine perturbations and variations of assembly operation durations. With the new requirements of the modern production workshop and the high importance of a decision-making process when implementing the constructed schedule, it is essential to extend the scheduling problem with inaccurate data to be adaptable to the needs of a decision-maker in evaluating with respect to properties of the stochastic, fuzzy, robust or stable schedules. In the paper [74], it is considered a robust scheduling problem. Based on a decision-making framework, a robust specification was developed to evaluate the possible schedule perturbations. The robust measure was based on the service level with a robustness metric. It is defined as a framework gathering several relevant tasks of robustness.