Reliability modelling of performance functions containing correlated basic variables, with application to construction project risk management

Ker-Fox, Gregory Mark (2002-12)

Thesis (PhD)--Stellenbosch University, 2002.

Thesis

ENGLISH ABSTRACT: Correlation mechanisms describing systematic variations and common sensitivities are critical contributors to uncertainty in quantitative functions modelling project performance in terms of probabilistic or basic variables. Current reliability methods transform dependent vectors to an equivalent set of independent standard normal variates. A simple method is developed for dealing with correlation in the original variable space. An algebraic description of the direction cosine (or alpha) for performance functions under conditions of dependence is formally derived and numerically validated. The resultant General First Order Second Moment (GFOSM) method for correlated basic variables is shown to be equivalent to the orthogonal transformation method. Geometric and physical interpretations of the general direction cosine are developed, with alpha found to be equivalent to the correlation between a basic variable and performance function. Corresponding inequalities and normalizing conditions are also developed for alpha. Expressions for a number of applications utilising the general dependent form for the direction cosine are derived and demonstrated. The current definition of the direction cosine as an importance factor is validated for dependent conditions, and conditions established under which this descriptor is no longer adequate. Expressions are derived to measure the significance of a variable in terms of stochastic importance and function sensitivity, to establish reliability index sensitivity to the omission of non-critical items, quantifying variable elasticity and an elasticity index. The general FOSM method for correlated basic variables is applied to system analysis to generate modal correlation coefficients between failure modes. The general direction cosine is stable for multivariate linear functions and functions of limited curvature across a range of reliabilities and correlation levels. This characteristic further simplifies the process by providing for deterministic reliability modelling of performance functions containing dependent variables, avoiding the solution of the more complex joint density function. The extension of the current theory and the treatment of performance functions in the original vector space develop invaluable insight into the correlation mechanisms driving risk and reliability. This will assist project managers to better understand areas that can affect project performance, to focus management attention, develop mitigation strategies and to allocate resources for the optimal management of project risk.

AFRIKAANSE OPSOMMING: Korrelasie meganismes wat sistematiese afwykings en gemeenskaplike sensitiwiteite veroorsaak, is kritieke bydraers tot onsekerheid in kwantitatiewe funksies wat projek prestasie modelleer m terme van probabilistiese of basiese veranderlikes. Huidige betroubaarheidsmetodes transformeer afhanklike vektore tot 'n ekwivalente stel van standaard normaalonafhanklike veranderlikes. '0 Eenvoudige metode is ontwikkelom die effekte van korrelasie in die oorspronklike vektorspasie te hanteer. 'n Algebraise beskrywing van die rigtingseosines (genoem alfa) vir prestasiefunksies onder omstandighede van afhanklikheid is formeel afgelei en numeries gevalideer. Dit is bewys dat die resulterende Algemene Eerste Orde Tweede Moment metode vir gekorreleerde basiese veranderlikes ekwivalent is aan die tradisionele Ortogonale Transformasie metode. Geometriese en fisiese interpretasies vir die algemene rigtingscosinus is ontwikkel, met bewys dat alfa ekwivalent is aan die korrelasie tussen 'n basiese veranderlike en die prestasiefunksie. Ooreenstemmende ongelykhede en normaliserings-kondisies is ook vir alfa ontwikkel. Uitdrukkings vir 'n aantal toepassings wat gebruik maak van die algemene afhanklike vorm van die rigtingscosinus is afgelei en gedemonstreer. Die huidige definisie van die rigtingscosinus as 'n belangrikheidsfaktor is gevalideer vir kondisies van afhanklikheid en omstandighede is uitgewys wanneer dit onvoldoende is. Uitdrukkings is afgelei om stochastiese belangrikheid te meet asook funksie sensitiwiteit, die sensitiwiteit van die betroubaarheidsindeks tot die weglating van nie kritiese veranderlikes, sowel as die kwantifisering van elastisiteit en die elastisiteitsindeks. Die Algemene Eerste Orde Tweede Moment metode vir gekorreleerde' veranderlikes is toegepas op sisteem analise om die korrelasie tussen falingsmodes te genereer. Die algemene rigtingscosinus is stabiel vir liniêre funksies en funksies met 'n beperkte kromming oor 'n reeks betroubaarheidswaardes en korrelasie vlakke. Hierdie kenmerk vereenvoudig die metode verder deur voorsiening te maak vir deterministiese betroubaarheidsmodellering van prestasie funksies met afhanklike veranderlikes, deur die oplossing van die meer komplekse gesamentlike-digtheidsfunksies te vermy. Die uitbreiding van die huidige teorie en die hantering van prestasie funksies in die oorspronklike vektor spasie ontwikkel waardevolle insig in die korrelasie meganismes wat risiko en betroubaarheid oorheers. Hierdie insig sal projekbestuurders in staat stelom kritieke gebiede wat projek prestasie kan affekteer beter te verstaan, om hulle aandag daarop te fokus, om teenmaatreël-strategieë te ontwikkel en hulpbronne toe te ken vir die optimale bestuur van projek risiko.

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