# A Recursive Optimization Approach for Buffer Allocation in Large Production Lines
## 2 Literature Review  
In the literature, there is a substantial amount of research works on BAPs. For a comprehensive review, the reader is referred to [PERSON] et al. [2], [PERSON] et al. [3], [PERSON] and [PERSON] [8], [PERSON] et al. [9], and [PERSON] and [PERSON] [10]. According to [PERSON] et al. [2], the solution methods to BAPs can be classified into explicit solutions, integrated optimization methods, and iterative optimization methods. The three categories differ in how they incorporate the performance evaluation of a given buffer allocation. The iterative optimization method is the most widely used method; it consists of an evaluative method and a generative method. The two methods are combined in a closed-loop configuration: the evaluative method calculates the performance measure of a buffer allocation candidate, the generative method uses the evaluation result to select a new candidate, and this candidate is further evaluated. In the following, we review the most relevant works on the evaluative and generative methods. Typical evaluative methods include simulation, exact analytical methods, and approximate analytical methods. Comprehensive surveys of performance evaluation techniques were provided by [PERSON] and [PERSON] [11], [PERSON] and [PERSON] [10], [PERSON] et al. [12], [PERSON], [PERSON], and [PERSON] [1], among others. Simulation is a flexible tool for evaluating complex realistic manufacturing systems [13], but it is very time consuming, especially when many alternative system configurations have to be evaluated [14; 15]. Exact analytical methods are based on Markovian analysis and are only applicable to small systems due to the state-space explosion problem. Therefore, considerable research effort was devoted to approximate analytical methods, such as decomposition methods [16; 17] and aggregation methods [4; 18]. Our work is based on decomposition methods. [PERSON] [16] proposed a well-known decomposition method for the approximate analysis of synchronous production lines with geometrically distributed machine failure and repair times. The core idea of this method was to decompose a long production line into a set of equivalent two-machine systems with unknown parameters. A set of decomposition equations were derived to determine the unknown parameters and the performance measures of the original line was obtained by evaluating the two-machine systems. [PERSON], [PERSON], and [PERSON] [17] proposed a similar method for the analysis of continuous flow lines with exponentially distributed failure and repair times. Moreover, they developed an efficient computational algorithm (referred to as the DDX algorithm) to solve the decomposition equations. [PERSON] [19] extended the decomposition method for evaluating inhomogeneous continuous flow lines and developed a more efficient algorithm. In general, the above decomposition methods provide accurate performance estimates. However, the accuracy of these methods may be poor when the machine reliability parameters have different orders of magnitude. To solve this problem, [PERSON] and [PERSON] [20] proposed an improved method by using generalized exponential distributions instead of exponential distributions to model the repair times of the virtual machines in the decomposed systems. [PERSON] and [PERSON] [21] developed a more accurate method by using hyper-exponential distributions to model the repair times. Other extensions of the decomposition methods include [PERSON] et al. [22], [PERSON] et al. [23] and [PERSON] and [PERSON] [24], among others. As far as the generative methods are concerned, the existing methods can be roughly classified into the enumeration method, metaheuristics, dynamic programming, and search algorithms [2]. The enumeration method can guarantee the optimality of allocation results, but is only applicable to small systems [25]. In the literature, it is typically used as a benchmark for other generative methods [26].