L-superadditive structure functions

Abstract

Structure functions relate the level of operations of a system as a function of the level of the operation of its components. In this paper structure functions are studied which have an intuitive property, called L-superadditive (L-subadditive). Such functions describe whether a system is more series-like or more parallel-like. L-superadditive functions are also known under the names supermodular, quasi-monotone and superadditive and have been studied by many authors. Basic properties of both discrete and continuous (i.e., taking a continuum of values) L-superadditive structure functions are studied. For binary structure functions of binary values, El-Neweihi (1980) showed that L-superadditive structure functions must be series. This continues to hold for binary-valued structure functions even if the component values are continuous (see Proposition 3.1). In the case of non-binary-valued structure functions this is no longer the case. We consider structure functions taking discrete values and obtain results in various cases. A conjecture concerning the general case is made.

The number of critical connection vectors of L-superadditive structure functions

“…Such assumptions generalize familiar ones in the binary case. However, no such conditions have been assumed for LSP (LSB) structure functions by Block et al (1989). The following, unexpected, result shows that the upper bound in Theorem 2.1 (which is sharp in general) is quite crude when such additional relevance and other conditions are imposed on LSP structure functions.…”

“…Various classes of discrete multistate structures have been introduced and studied by several researchers (see EI-Neweihi and Prosch an (1984) for a survey). Block et al (1989) study the class of L-superadditive structures by imposing the following condition on the structure function ep:…”

“…Such functions have the interesting interpretation of describing whether a system is more series-like than parallel-like or vice versa (see Block et al (1989». A vector x is said to be a critical connection vector to level k > 0 for a multistate structure ep if ep(x)= k and y < x implies ep(y) < k, where y < x means Yi~Xi for each i and strict inequality holds for at least one i. Critical connection vectors to the various performance levels of a multistate structure ep playa central role similar to the one played by minimal path vectors in the binary case.…”

“…Critical connection vectors to the various performance levels of a multistate structure ep playa central role similar to the one played by minimal path vectors in the binary case. So naturally, Block et al (1989) examine critical connection vectors to levels of performance of an LSP structure function. They conjecture an upper bound for the number of such vectors and prove it for some special cases.…”

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AbstractA conjecture due to Block et al (1989), concerning the number of critical connection vectors to the various performance levels of a discrete L-superadditive structure function, is proved. When the components of the discrete L-superadditive structure function are further assumed to satisfy a certain relevance condition due to Griffith (1980), it is shown that there is exactly one critical connection vector to each performance level.

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