Analytical Approach to Estimate Efficiency of Series Machines in Production Lines

Series machines, i.e., machines (which are usually unreliable) arranged in series with no buffering, are pervasive in production systems. In the analysis, design, and optimization of the series-machine system, the efficiency analysis is one of the most fundamental issues. There are not a lot of researches analyzing the efficiency of the series-machine system, and almost all of them assume that the system operates under type-I failure mechanisms (i.e., the breakdown of a machine could make all other series machines forced down) rather than under type-II mechanisms (i.e., the breakdown of a machine does not make any other series machines forced down). The reason that the type-I failure mechanisms are usually assumed in the literature is that the analysis of the series-machine system under type-II mechanisms is much more complex than under type-I mechanisms, although type-II mechanisms are more common in practice. To thoroughly and systematically estimate the efficiency of the series-machine system, in this paper, we propose a unified analytical approach to investigate the efficiency under both type-I and type-II failure mechanisms. Both cases of deterministic and random cycle times are considered. Different from under type-I failure mechanisms, analytical expressions of the efficiency of series-machine systems under type-II failure mechanisms are extremely hard to obtain, and thus, limit bounds of the efficiency are derived and algorithms are developed to calculate its exact value. Results show that the series-machine system under type-II failure mechanisms is more efficient than under type-I mechanisms, which, intuitively making sense, is the reason that type-II mechanisms are more common in the industry. Note to Practitioners —The series-machine system is pervasive in industry and it usually operates under type-II failure mechanisms (i.e., the breakdown of a machine does not make any other series machines forced down). Nevertheless, the existing research on type-II failure mechanisms is very limited. Thus, the efficiency and other performance measures of such systems are, in general, approximately evaluated based on type-I mechanisms (i.e., the breakdown of a machine could make all other series machines forced down). However, the error of this approximation may be large and unacceptable. In this paper, a unified approach is proposed to analyze the efficiency of the series-machine system. Due to the difficulties in deriving closed-form expressions under type-II failure mechanisms, algorithms are developed to calculate the efficiency of the series-machine system. Theoretical results ensure the effectiveness of the algorithms.