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A Boolean Algebra and K-Map Approach Proceedings of the International Conference on Industrial Engineering and Operations Management Washington DC, USA, September 27-29, 2018 Reliability Assessment of Bufferless Production System: A Boolean Algebra and K-Map Approach Firas Sallumi ([email protected]) and Walid Abdul-Kader ([email protected]) Industrial and Manufacturing Systems Engineering University of Windsor, Windsor (ON) Canada Abstract In this paper, system reliability is determined based on a K-map (Karnaugh map) and the Boolean algebra theory. The proposed methodology incorporates the K-map technique for system level reliability, and Boolean analysis for interactions. It considers not only the configuration of the production line, but also effectively incorporates the interactions between the machines composing the line. Through the K-map, a binary argument is used for considering the interactions between the machines and a probability expression is found to assess the reliability of a bufferless production system. The paper covers three main system design configurations: series, parallel and hybrid (mix of series and parallel). In the parallel production system section, three strategies or methods are presented to find the system reliability. These are the Split, the Overlap, and the Zeros methods. A real-world case study from the automotive industry is presented to demonstrate the applicability of the approach used in this research work. Keywords: Series, Series-Parallel, Hybrid, Bufferless, Production Line, Reliability, K-Map, Boolean algebra 1. Introduction Presently, the global economy is forcing companies to have their production systems maintain low inventory and provide short delivery times. Therefore, production systems are required to be reliable and productive to make products at rates which are changing with the demand. Moreover, many products have a very short life cycle that results in reducing the time available to develop production systems. So, the designers of production systems need systematic models to predict system reliability and performance under diverse production scenarios. Productivity is one of the most important performance measures for a certain machine or system. Usually, productivity is estimated about both the reliability and the production rate of the production system. The parameters selected for a machining process at a certain production rate affect the reliability of the machine and consequently its productivity. The productivity of a system is not only dependent on its reliability, but is also dependent on its configuration. Thus, the productivity of a production system can be estimated by considering both system design and process planning simultaneously. To determine production system productivity, the individual machine reliability should be measured. At the system level, the reliability depends on the system configuration of the production process. Production system configurations could be serial, such as a traditional transfer line, or parallel, and usually a duplicate of serial lines in parallel (parallel-series) as required for a slow production processes. Another configuration is the hybrid line, which consists of a mix of serial and parallel (series-parallel) connected machines that could take any configuration. Reliability is a major attribute in the design and operation of today’s large and complex production systems. The integrity of modern production systems is strongly dependent on the availability and reliability of their machines. However, reliability theory depends heavily on an understanding of failure physics modeling and on techniques of probability and statistics. Therefore, reliability models play a significant role in system analysis. Typically, reliability predictions for complex production systems begin with predictions of the probabilities of mission success for the machines in a system. Predictions are then combined in accordance with a logic model that describes how the machines interact in the system. The result is a predicted mission success probability for the production system. Therefore, it is of great importance to have practical models which efficiently predict the reliability of complex systems, and also give useful design information with respect to individual machines. There are two main benefits of using a K-map regarding system design. First, the number of machines in the system will present all possible states the system can have and which of those states are producing. This can be easily found from the system truth table. Second, by constructing a K-map of the production system, it will make it easier to implement changes on the K-map and re-project them on the original production system. Great attention should be given to keep the integrity of the production system intact. © IEOM Society International 1030 Proceedings of the International Conference on Industrial Engineering and Operations Management Washington DC, USA, September 27-29, 2018 Evaluating system reliability is not new using Boolean algebra minimization. In contrast, based on the review of related literature presented below and to the best of our knowledge, there has been no detailed study found on using K-map technique to simplify system configuration and find its reliability. This paper exploits a clear-cut method to find system reliability. A K-map is used by considering all possible cases that can occur while solving to find the final probability expression. Based on the production system configuration, a Boolean equation can be generated to model the system. A Boolean variable is assigned to each machine in the system since any machine can be in one of two states: either working (logic 1, or X) or not working (logic 0, or X’). Depending on the production system configuration, the product will move from one machine to another. Usually the flow of a product is in a series configuration of machines, thus the system equation that relates the Boolean variables is represented by AND operation ((.) or (*) operator). Sometimes there are more than one way for a product to flow; in this case, the system equation relates the Boolean variables as OR operation represented by a (+) sign. Without using a K-map and Boolean algebra, any complex hybrid system can be reduced to a single machine by applying the following two rules successively (Bazovsky, 2004): Rule 1: When n machines are all in series, the reliability of the system is equivalent to a single machine with n reliability = Pi )( i1 Rule 2: When n machines are all in parallel, the reliability of the system is equivalent to a single machine with reliability = n Pi )1(1 i1 However, the solution would be simplifying mathematical equations only. The K-map technique provides a complete visualization of the production system since every (+) means machines in parallel and every (.) means machines in series. Consequently, any production system can be easily constructed from its final Boolean expression. On the other hand, buffers are not considered in the system equation because they couple machines and there are no Boolean variables for them as stated in Freiheit et al. (2004). Accordingly, this research work is concerned with bufferless production systems. For example, the production system shown in Figure 1 has a Boolean system equation that is a function of the three Boolean variables that represent its three machines: F (A, B, C) = A + (B.C) = A + BC A F B C Figure 1: Three-machine production system From the Boolean expression, F (A, B, C), and without looking at Figure 1, it can be noticed that machine B and machine C are in series (B.C) and both of them are in parallel with machine A. The system states (minterms in Boolean algebra) can be determined from the number of machines (in this case; 23 = 8 states). The production system is producing if the Boolean system equation equals one, i.e., F = 1. This condition will be achieved if either machine (A) is working, or both machines (B and C) are working, or all three machines are working. Since machines (B) and (C) are in series, then if either one is not working, the whole branch will not be producing. The corresponding truth table is shown below. From Table 1, A is a variable denoting the state of machine A when it is operating, and A’ is the variable when it is not operating. The same applies for the remaining machines B and C. The minterms m0 –m7 present the possibility of “Anding” all variables (A, B, and C) in a function, which also represents all possible system states of a production system. In this research work, a K-map method is presented to calculate the reliability of the production system. Originally, the K-map is a graphical representation used to simplify Boolean expressions of digital circuits. The K-map technique is simple and can be easily applied to production lines regardless of their configurations (series, parallel or hybrid). The construction of a K-map from a Boolean expression will be explained later in this paper to emphasize the logic behind using a K-map. © IEOM Society International 1031 Proceedings of the International Conference on Industrial Engineering and Operations Management Washington DC, USA, September 27-29, 2018 Table 1: Truth table for production system shown in Figure 1 Decimal Minterms Minterms or A B C Output (F) (state #) ID System States 0 m0 A’B’C’ 0 0 0 0 1 m1 A’B’C 0 0 1 0 2 m2 A’BC’ 0 1 0 0 3 m3 A’BC 0 1 1 1 4 m4 AB’C’ 1 0 0 1 5 m5 AB’C 1 0 1 1 6 m6 ABC’ 1 1 0 1 7 m7 ABC 1 1 1 1 The remainder of this paper is divided in 5 more sections where section 2 covers the literature review, followed by section 3 on Boolean algebra and K-map techniques. Section 4 presents the design methodology. It also shows the applicability of this approach on a real-world case study. Sections 5 and 6 cover discussions of the approach and conclude the paper, respectively.
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