a b s t r a c tThe utility of the Weibull distribution has been traditionally justified with the belief that it is the mathematical expression of the weakest-link concept in the case of flaws locally initiating failure in a stressed volume. This paper challenges the Weibull distribution as a mathematical formulation of the weakest-link concept and its suitability for predicting probability of failure locally initiated by flaws. The paper shows that the Weibull distribution predicts correctly the probability of failure locally initiated by flaws if and only if the probability that a flaw will be critical is a power law or can be approximated by a power law of the applied stress.Contrary to the common belief, on the basis of a theoretical analysis and Monte Carlo simulations we show that in general, for non-interacting flaws randomly located in a stressed volume, the distribution of the minimum failure stress is not necessarily a Weibull distribution. For the simple cases of a single group of identical flaws or two flaw size groups each of which contains identical flaws, for example, the Weibull distribution fails to predict correctly the probability of failure. Furthermore, if in a particular load range, no new critical flaws are created by increasing the applied stress, the Weibull distribution also fails to predict correctly the probability of failure of the component. In all these cases however, the probability of failure is correctly predicted by the suggested alternative equation. This equation is the correct mathematical formulation of the weakest-link concept related to random flaws in a stressed volume. The equation does not require any assumption concerning the physical nature of the flaws and the physical mechanism of failure and can be applied in cases of locally initiated failure by non-interacting entities.

The paper demonstrates for the first time uncertainty reduction and superior performance through segmentation based on algebraic inequalities. Meaningful interpretation of algebraic inequalities has been used for generating new knowledge in unrelated application domains. Thus, the method of segmentation through an abstract inequality led to a new fundamental theorem related to electrical circuits. The power output from a source with particular voltage, on elements connected in series, is smaller than the total power output from the segmented sources applied to the individual elements. Segmentation attained through the same abstract inequality led to another fundamental theorem related to electrical capacitors. The energy stored by a charge of given size on a single capacitor is smaller than the total energy stored in multiple capacitors with the same equivalent capacity, by segmenting the initial charge over the separate capacitors.Finally, inequalities based on sub-additive and super-additive functions have been introduced for reducing uncertainty and obtaining superior performance by a segmentation or aggregation of controlling factors. By a meaningful interpretation of sub-additive and superadditive inequalities, superior performance and important properties have been established for processes described by a power-law dependence.

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The paper discuses new, very efficient augmentation algorithms and theorems related to maximising the flow in singlecommodity and multi-commodity networks. For the first time, efficient algorithms with linear average running time ) (m O in the size m of the network, are proposed for restoring the maximum flow in single-commodity and multi-commodity networks after a component failure. The proposed algorithms are particularly suitable for discrete-event simulators of repairable production networks whose analysis requires generating thousands of simulation histories, each including hundreds of component failures. In this respect, a new, very efficient augmentation method with linear running time has been proposed for restoring the maximum output flow of oil in oil and gas production networks, after a component failure. Another important application of the proposed algorithms is in networks controlled in real time, where upon failure, the network flows need to be redirected quickly in order to maintain a maximum output flow.

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