New insights on multi-state component criticality and importance Jose E. Ramirez-Marquez a, * , Claudio M. Rocco b , Bethel A. Gebre a , David W. Coit c , Michael Tortorella c a Department of System Engineering and Engineering Management, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030, USA b Facultad de Ingenierı ´a, Universidad Central de Venezuela, Caracas, Venezuela c Department of Industrial and Systems Engineering, Rutgers University, Piscataway, NJ 08854, USA Received 4 May 2005; accepted 30 August 2005 Available online 2 November 2005 Abstract In this paper, new importance measures for multi-state systems with multi-state components are introduced and evaluated. These new measures complement and enhance current work done in the area of multi-state reliability. In general, importance measures are used to evaluate and rank the criticality of component or component states with respect to system reliability. The focus of the study is to provide intuitive and clear importance measures that can be used to enhance system reliability from two perspectives: (1) how a specific component affects multi-state system reliability and (2) how a particular component state or set of states affects multi-state system reliability. The first measure unsatisfied demand index, provides insight regarding a component or component state contribution to unsatisfied demand. The second measure multi-state failure frequency index, elaborates on an approach that quantifies the contribution of a particular component or component state to system failure. Finally, the multi-state redundancy importance identifies where to allocate component redundancy as to improve system reliability. The findings of this study indicate that both perspectives can be used to complement each other and as an effective tool to assess component criticality. Examples illustrate and compare the proposed measures with previous multi-state importance measures. q 2005 Elsevier Ltd. All rights reserved. Keywords: Reliability; Multi-state system; Component criticality; Importance measures 1. Introduction Traditionally, system reliability has been analyzed from a binary perspective assuming the system and its components can be in either of two states: completely functioning or failed. However, many systems that provide basic services, such as telecommunications, gas and oil production, transportation and electric power distribution, operate at various levels of performance as opposed to the binary perspective. These types of systems may provide a service or function at degraded component performance levels. Therefore, it is essential to model and analyze them accordingly. For these systems, multi- state system reliability methods have been proposed as a more appropriate modeling and computational approach. Currently, most reliability work on multi-state systems has focused on two cases: (1) multi-state systems with binary capacitated components [1–4] where in general, the system has to fulfill a number of different demands during a specified time interval assuming the components can either work at a nominal capacity level or not work at all, and (2) multi-state systems with multi-state components (MSMC) where in general, the system has to fulfill a known demand based on the different component performance states [5–13]. This paper deals with systems following the second case. In reliability theory, importance measures (IM) have been recognized to provide critical information regarding the impact components have in system reliability [14–22]. IM are essential in determining and explaining the effects of components on the overall reliability of a system. For systems exhibiting binary behavior, IM have enabled engineers to determine system configuration improvements and ultimately cost effective methods to maintain high levels of system reliability. For the binary case, a variety of IM have been proposed and are in existence today. Amongst these, Vasseur and Llory [21] recognize reliability achieve- ment worth (RAW), reliability reduction worth (RRW), Fussell–Veseley (FV) and Birnbaum as the most widely used in industry. For the binary case, system components can be ranked with respect to the impact they have on system reliability based on a given importance measure. Reliability Engineering and System Safety 91 (2006) 894–904 www.elsevier.com/locate/ress 0951-8320/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ress.2005.08.009 * Corresponding author. Tel.: C1 201 216 8003. E-mail address: jmarquez@stevens.edu (J.E. Ramirez-Marquez).