A review of the literature on GMS problems was provided in this chapter. In §2.1, a general description of the GMS problem was given in some detail, explaining the notions of a scheduling window and of scheduling resolution, as well as referring the typical scheduling criteria pertaining to GMS model formulations in the literature. Constraints applicable to GMS model formula- tions and other related energy problems found in the literature were also described. This was followed, in§2.2, by a discussion on the most popular GMS model formulations in the literature, citing typical objective functions for and constraints included in model incarnations of the GMS problem. GMS model solution approaches that have been adopted in the literature were de- scribed in§2.3, including mathematical programming techniques, expert systems, fuzzy solution approaches, heuristics and finally metaheuristics. The approximate solution methodology that is employed later in this dissertation, namely the method of SA, was finally described in more detail in §2.4.
CHAPTER 3
Reliability theory
Contents
3.1 General considerations . . . 41 3.2 Basic mathematical notions . . . 42 3.3 Lifetime distribution models for non-repairable systems . . . 44 3.4 Repairable systems . . . 50 3.5 Life data classification . . . 53 3.6 Trend tests . . . 55 3.7 Model parameter estimation methods . . . 58 3.8 Acceleration models . . . 60 3.9 Chapter summary . . . 62
This chapter contains a literature review on the research area of reliability theory. A number of general considerations in respect of reliability theory are presented, with a focus on the mathematical formulation of the notation of reliability. This is followed by a description of the most popular trend tests employed in practice to determine whether a system exhibits non- repairable or repairable characteristics. The chapter also includes mathematical treatments of the notion of reliability in the contexts of repairable systems and non-repairable systems, as well as popular models for describing the reliabilities of both types of systems.
3.1 General considerations
Survival theory is a subfield of general statistics concerned with the analysis of the durations between the occurrences of successive events in a system. The nature of these events may vary according to the type of system being analysed, and may include mechanical failures of machinery or illnesses of biological organisms [159]. The intersection between survival theory and engineering is known as reliability theory or reliability analysis, which is a general theory about system failures. Its constituent components are ideas, mathematical models and methods that can be used to estimate, predict, understand and optimise the lifespans of components as well as of systems as a whole [19]. The objective in reliability theory is usually to quantify a suitable trade-off between wasting a system’s residual life and running the risk of unexpected failure of the system. The two main branches of reliability theory are theories that have been developed for repairable systems and for non-repairable systems [181]. A brief overview is given in this section of these two subtheories.
A system is classified as a non-repairable system when the population of items contained as components in the system is one for which individual items that fail are permanently removed from the population and the system is repaired by replacing these items with items from either the same or from a different population. A repairable system, on the other hand, may be defined as a system that can be restored to fully operational performance by any maintenance action other than replacing the entire system after failing to perform some of its intended functions [11]. The effective reliability of a system over a certain time frame during the system’s lifetime can be determined based on historical failure data of the system. Historical failure data may exhibit an increasing failure rate (IFR), a constant failure rate (CFR) or a decreasing failure rate (DFR). A system exhibiting an IFR, also called an ageing system, generally comprises components that wear out over the lifetime of the system, which causes the time between consecutive failures to decrease. Systems with IFRs generally require the most attention with respect to planned maintenance, since failures in the system are observed more frequently towards the end of the system’s lifetime [85]. Some systems exhibit CFRs, where the failures in the system are observed to be random with inter-failure rates that are exponentially distributed. These systems require less attention when planning maintenance due to the randomness between consecutive failures [23]. In the case where a system exhibits a DFR, the time between consecutive failures is increasing — hence fewer failures are observed towards the end of the system’s lifetime — and so the system may be described as an improving system [3]. For improving systems, it might sometimes be harmful to conduct planned maintenance as the system naturally increases in reliability over time.
The well-known bathtub curve, presented in Figure 3.1, may be used to represent the failure rate of a system graphically as a function of the lifetime of the system. Typically, from the start of the lifetime of a system up to a certain time t1, the system exhibits an DFR. Between times t1
and t2 > t1, the system is in its useful stage and exhibits a CFR. The final part of the graph,
from t2 onwards, represents the final part of the system’s lifetime, which exhibits an IFR.
Time
Early life Useful Life Wear-out
Failure rate
Figure 3.1: The failure rate of a system as a function of time, represented by the well-known bathtub curve [214].