FURTHER EXPOSITION OF THE VALUE OF RELIABILITY

As the demands placed on transport systems have increased relative to extensions in supply, problems of network unreliability have become ever more prevalent. The response of some transport users has been to accommodate expectations of unreliability in their decision-making, particularly through their trip scheduling. In the analysis of trip scheduling, Small’s (1982) approach has received considerable support. Small extends the microeconomic theory of time allocation (e.g. Becker, 1965; De Serpa, 1971), accounting for scheduling constraints in the specification of both utility and its associated constraints. Small makes operational the theory by means of the random utility model (RUM). This involves a process of converting the continuous departure time variable into discrete departure time segments, specifying the utility of each departure time segment as a function of several components (specifically journey time, schedule delay and the penalty of late arrival), and adopting particular distributional assumptions concerning the random error terms of contiguous departure time segments (whilst his 1982 paper assumes IID, Small’s 1987 paper considers a more complex pattern of covariance). A fundamental limitation of Small’s approach is that individuals make choices under certainty, an assumption that is clearly unrealistic in the context of urban travel choice. The response of microeconomic theory to such challenge is to reformulate the objective problem from the maximisation of utility, to one of maximising expected utility, with particular reference to the works of von Neumann & Morgenstern (1947) and Savage (1954). Bates et al. (2001) apply this extension to departure time choice, but specify choice as being over continuous time; the latter carries the advantage of simplifying some of the calculations of optimal departure time. Moreover Bates et al. offer account of departure time choice under uncertainty, but retain a deterministic representation. Batley & Daly (2004) develop ideas further by reconciling the analyses of Small (1982) and Bates et al. Drawing on early contributions to the RUM literature by Marschak et al. (1963), Batley and Daly propose a probabilistic model of departure time choice under uncertainty, based on an objective function of random expected utility maximisation. Despite this progression in the generality and sophistication of methods, significant challenges to the normative validity of RUM and transport network models remain. Of increasing prominence in transport research, is the conjecture that expected utility maximisation may represent an inappropriate objective of choice under uncertainty. Significant evidence for this conjecture exists, and a variety of alternative objectives proposed instead; Kahneman & Tversky (2000) offer a useful compendium of such papers. With regards to these alternatives, Kahneman & Tversky’s (1979) own Prospect Theory commands considerable support as a theoretical panacea for choice under uncertainty. This theory distinguishes between two phases in the choice process - editing and evaluation. Editing may involve several stages, so-called ‘coding’, ‘combination’, ‘cancellation’, ‘simplification’ and ‘rejection of dominated alternatives’. Evaluation involves a value function that is defined on deviations from some reference point, and is characterised by concavity for gains and convexity for losses, with the function being steeper for gains than for losses. The present paper begins by formalising the earlier ideas of Batley and Daly (2004); the paper thus presents a theoretical exposition of a random expected utility model of departure time choice. The workings of the model are then illustrated by means of numerical example. The scope of the analysis is subsequently widened to consider the possibility of divergence from the objective of expected utility maximisation. An interesting feature of this discussion is consideration of the relationship between Prospect Theory and a generalised representation of the random expected utility model. In considering this relationship, the paper draws on Batley & Daly’s (2003) investigation of the equivalence between RUM and elimination-by-aspects (Tversky, 1972); the latter representing one example of a possible ‘editing’ model within Prospect Theory. Again, the extended model is illustrated by example.

A reliability analysis method using binary decision diagrams in phased mission planning

The use of autonomous systems is becoming increasingly common in many fields. A significant example of this is the ambition to deploy unmanned aerial vehicles (UAVs) for both civil and military applications. In order for autonomous systems such as these to operate effectively, they must be capable of making decisions regarding the appropriate future course of their mission responding to changes in circumstance in as short a time as possible. The systems will typically perform phased missions and, owing to the uncertain nature of the environments in which the systems operate, the mission objectives may be subject to change at short notice. The ability to evaluate the different possible mission configurations is crucial in making the right decision about the mission tasks that should be performed in order to give the highest possible probability of mission success. Because binary decision diagrams (BDDs) may be quickly and accurately quantified to give measures of the system reliability it is anticipated that they are the most appropriate analysis tools to form the basis of a reliability-based prognostics methodology. The current paper presents a new BDD-based approach for phased mission analysis, which seeks to take advantage of the proven fast analysis characteristics of the BDD and enhance it in ways that are suited to the demands of a decision-making capability for autonomous systems. The BDD approach presented allows BDDs representing the failure causes in the different phases of a mission to be constructed quickly by treating component failures in different phases of the mission as separate variables. This allows flexibility when building mission phase failure BDDs because a global variable ordering scheme is not required. An alternative representation of component states in time intervals allows the dependencies to be efficiently dealt with during the quantification process. Nodes in the BDD can represent components with any number of failure modes or factors external to the system that could affect its behaviour, such as the weather. Path simplification rules and quantification rules are developed that allow the calculation of phase failure probabilities for this new BDD approach. The proposed method provides a phased mission analysis technique that allows the rapid construction of reliability models for phased missions and, with the use of BDDs, rapid quantification

Phased mission analysis using the cause–consequence diagram method

Most reliability analysis techniques and tools assume that a system used for a mission consists of a single phase. However, multiple phases are natural in many missions. A system that can be modelled as a mission consisting of a sequence of phases is called a phased mission system. In this case, for successful completion of each phase the system may have to meet different requirements. System failure during any phase will result in mission failure. Fault tree analysis, binary decision diagrams and Markov techniques have been used to model phased missions. The cause–consequence diagram method is an alternative technique capable of modelling all system outcomes (success and failure) in one logic diagram. [Continues.

A binary decision diagram method for phased mission analysis of non-repairable systems

Phased mission analysis is carried out to predict the reliability of systems which undergo a series of phases, each with differing requirements for success, with the mission objective being achieved only on the successful completion of all phases. Many systems from a range of industries experience such missions. The methods used for phased mission analysis are dependent upon the repairability of the system during the phases. If the system is non-repairable, fault-tree-based methods offer an efficient solution. For repairable systems, Markov approaches can be used. This paper is concerned with the analysis of non-repairable systems. When the phased mission failure causes are represented using fault trees, it is shown that the binary decision diagram (BDD) method of analysis offers advantages in the solution process. A new way in which BDD models can be efficiently developed for phased mission analysis is proposed. The paper presents a methodology by which the phased mission models can be developed and analysed to produce the phase failure modes and the phase failure likelihoods

Systems reliability for phased missions

The concept of a phased mission has been introduced as a sequential set of objectives that operate over different time intervals. During each phase of the mission, the system may alter such that the logic model, system configuration, or system failure characteristics may change to accomplish a required objective. A new fault tree method has been proposed to enable the probability of failure in each phase to be determined in addition to the whole mission unreliability. Phase changes are assumed to be instantaneous, and component failure rates are assumed to be constant through the mission. For any phase, the method combines the causes of success of previous phases with the causes of failure for the phase being considered to allow both qualitative and quantitative analysis of both phase and mission failure. A new set of Boolean laws is introduced to combine component success and failure events through multiple phases so that the expression for each phase failure can be reduced into minimal form. [Continues.

A reliability analysis method using binary decision diagrams in phased mission planning

The use of autonomous systems is becoming increasingly common in many fields. A significant example of this is the ambition to deploy UAVs (unmanned aerial vehicles) for both civil and military applications. In order for autonomous systems such as these to operate effectively they must be capable of making decisions regarding the appropriate future course of their mission responding to changes in circumstance in as short a time as possible. The systems will typically perform phased missions and, due to the uncertain nature of the environments in which the systems operate, the mission objectives may be subject to change at short notice. The ability to evaluate the different possible mission configurations is crucial in making the right decision about the mission tasks that should be performed in order to give the highest possible probability of mission success. Since Binary Decision Diagrams (BDD) may be quickly and accurately quantified to give measures of the system reliability it is anticipated that they are the most appropriate analysis tools to form the basis of a reliability-based prognostics methodology. This paper presents a new Binary Decision Diagram based approach for phased mission analysis, which seeks to take advantage of the proven fast analysis characteristics of the BDD and enhance it in ways which are suited to the demands of a decision making capability for autonomous systems. The BDD approach presented allows BDDs representing the failure causes in the different phases of a mission to be constructed quickly by treating component failures in different phases of the mission as separate variables. This allows flexibility when building mission phase failure BDDs since a global variable ordering scheme is not required. An alternative representation of component states in time intervals allows the dependencies to be efficiently dealt with during the quantification process. Nodes in the BDD can represent components with any number of failure modes or factors external to the system that could affect its behaviour, such as the weather. Path simplification rules and quantification rules are developed that allow the calculation of phase failure probabilities for this new BDD approach. The proposed method provides a phased mission analysis technique that allows the rapid construction of reliability models for phased missions and, with the use of BDDs, rapid quantification

Diversification Preferences in the Theory of Choice

Diversification represents the idea of choosing variety over uniformity. Within the theory of choice, desirability of diversification is axiomatized as preference for a convex combination of choices that are equivalently ranked. This corresponds to the notion of risk aversion when one assumes the von-Neumann-Morgenstern expected utility model, but the equivalence fails to hold in other models. This paper studies axiomatizations of the concept of diversification and their relationship to the related notions of risk aversion and convex preferences within different choice theoretic models. Implications of these notions on portfolio choice are discussed. We cover model-independent diversification preferences, preferences within models of choice under risk, including expected utility theory and the more general rank-dependent expected utility theory, as well as models of choice under uncertainty axiomatized via Choquet expected utility theory. Remarks on interpretations of diversification preferences within models of behavioral choice are given in the conclusion