Keywords: interval probability theory, linguistic variables, dependency, confidence, weighted intervals, vulnerability.

Vulnerability to bomb blast is a socio-technical or "soft" or imprecise phenomenon that requires both learning from past experience and a calculus that permits incomplete knowledge [1]. Grounded theory [2] has been used to learn from past experience and to generate a hierarchical causal tree. These members of the hierarchy are called "holons", conveying the idea that they are both wholes and parts. The generation of this hierarchy has been reported elsewhere [1] and is not the subject of this paper.

The main objective of this study was to use interval probability theory [3] to arrive at an interval value for the top level holon "vulnerability" from a number of lower level holons, each of which is characterized by an interval number and a weight, reflecting importance. Different ways of combining connectives and dependencies between lower level (child) holons were explored, in order to arrive at the interval for an upper level (parent) holon. This aspect of the work is a novel contribution. Proposing a method for converting the final two valued vulnerability assessment back to a linguistic label having a level of confidence was another objective with a novel character. The final objective was to demonstrate that the computerized hierarchy could be used as a management tool - i.e. to check the contribution to overall vulnerability from various holons, so that effective improvements and/or assessments could be considered.

The evidence for proneness to failure is first given a linguistic label (i.e. very low, low, moderate, high, very high) that is related to a fuzzy set. Next, the confidence in that assessment is treated as a horizontal line that cuts the relevant fuzzy set at a lower and upper bound, thus creating an interval number for a holon, the lower number referred to as S_n or "necessary" support and the upper one as S_p or "possible" support.

The combining of two weighted interval numbers treated the weights as influences on the upper level holon, considering two boundary conditions, namely the case when one of the weights was unity and the other zero, and the case when both weights were equal. This was extended to the combining of more than two weighted interval numbers, ensuring that the result was independent of the order of combination. In addition, the combination of evidence also depends on the dependency relationship between holons; in this study, a maximum dependence assumption was used together with the OR connective. The above approach was preferred as a result of its simplicity to the more recent developments in interval probability theory [4].

The entire system was computerized using a spreadsheet, with a reasonably simple user interface. An existing building was assessed as a test case. The interval number for vulnerability was converted back to a linguistic variable with a degree of confidence by determining the area between the vertical lines through the two numbers on the fuzzy set graphs; and then determining the overlapping of that area with the areas of the fuzzy sets corresponding to the various linguistic labels (e.g. moderate, high, very high etc.). The ratio between the individual area for each fuzzy set and the total area can be taken as a measure of confidence of the assessment that the building is vulnerable to the degree defined by that fuzzy set.

The model yielded results that were intuitively satisfactory. The interval values for the top level holon "vulnerability" were also shown to be satisfactorily and realistically sensitive to changes in the evidence (of proneness to failure) and confidence (in the assessment) of lower level holons.

1

Chandratilake, S.R. and Dias, W.P.S. Identifying Vulnerability of Buildings to Blast Events using Grounded Theory, 10th Engineering Research Unit Symposium on Research for Industry, University of Moratuwa, Sri Lanka, August 2004.

2

Glaser, B. and Strauss, A.L. The Discovery of Grounded Theory: Strategies for Qualitative Research, Weidenfeld and Nicolson, London, 1967.

3

Cui, W.C. and Blockley, D.I. Interval probability theory for evidential support, Int. Jnl. of Intelligent Systems, Vol. 5, No. 2, June, pp. 183-192, 1990. doi:10.1002/int.4550050204

4

Davis, J.P. and Hall, J.W. A software-supported process for assembling evidence for handling uncertainty in decision making, Decision Support Systems, Vol. 35, pp. 415-433, 2003. doi:10.1016/S0167-9236(02)00117-3

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