Disjoint sum forms in reliability theory

The structure function f of a binary monotone system is assumed to be known and given in a disjunctive normal form, i.e. as the logical union of products of the indicator variables of the states of its subsystems. Based on this representation of f, an improved Abraham algorithm is proposed for generating the disjoint sum form of f. This form is the base for subsequent numerical reliability calculations. The approach is generalized to multivalued systems. Examples are discussed.


INTRODUCTION
We deal with a basic problem of reliability theory, namely with finding and analyzing algorithms for calculating reliability criteria of systems based on reliability criteria of its elements (subsystems).Even in the simple case of binary monotone systems with independent elements, these algorithms generally have running times which increase exponentially fast with the complexity of the system [5].Within that limitation it is, however, imperative to develop algorithms, which are relatively fast and applicable to a broad class of problems.The first step is to determine the structure function of the system.This paper requires the structure function to be known and given in a disjunctive normal form.

BINARY SYSTEMS
Let S be the system under consideration and e e e n 1 2 , ,..., its n elements.Let furthermore z s be indicator variable of the system state and z i the indicator variable of element e i .Then, in a binary system, http://orion.journals.ac.za/ z S s = R S T The structure function ϕ of S is a Boolean function, which describes the mutual relationship between the states of the system and its elements: = ϕ( , ,..., ) Let Z be the set of all 2 n state vectors of the system.The system is monotone if ϕ is nondecreasing in each variable z i and each z i is relevant with respect to ϕ, i.e. z i influences the value of ϕ.A vector r z ∈ Z is called a path vector of ϕ if ϕ( Obviously, the concepts of (minimal) path vectors and (minimal) path sets are equivalent.For a thorough discussion of these concepts see, for instance, [7] and [15].It is easy to see that the structure function of any binary monotone system with minimal path sets with A z j w Since there are computer-aided methods for determining the minimal path sets of any binary monotone system, the disjunctive normal form of ϕ given by ( 2) is most widely used for generating disjoint sum forms of structure functions.A disjoint sum form of ϕ has structure ϕ( ) where the D k d k ; , ,..., = 1 2 ; are products of some z i and z i j j ; .
≠ Hence, ϕ given by ( 3) is a sum of mutually disjoint terms D i .
Let us now take into account that the z i are random variables with ϕ r be the probability that the system is operating.p s is called the availability of the system and p i is the availability of element e i .Since z z s = ϕ( ) This relationship shows the practical importance of a disjoint sum form (3): The system availability is simply given by If the z i are independent random variables, i.e. the elements operate independently from each other, then E D i ( ) is obtained from D i simply by replacing there the z i and z j with the corresponding p i and p j , respectively.
Structure functions, in particular disjoint sum forms, are not unique.For the sake of computational efficiency, a disjoint sum form should be of low complexity, i.e. both d and the total number of factors in the D i should be small.Numerous algorithms transforming the disjunctive normal form (2) into a disjoint sum form of low complexity have been developed.
Most popular is the algorithm of Abraham [2].It has formed the basis for substantially improved versions yielding disjoint sum forms of lower complexity than the original version of Abraham [7], [8], [9], [14].The probably most efficient algorithm not based on Abraham's one, is due to Torrey [22].For surveys, see [21], [23]. http://orion.journals.ac.za/ The following version of Abraham's algorithm is based on an algorithm given in [7].To describe the algorithm, some further notation is needed.Let M be the set of all possible products of some z i and z j .The principle of the algorithm consists in replacing each A k in (2) with a sum of disjoint products L consists of mutual disjoint products from M .Thus, the set M ϕ can be identified with the structure function ϕ.The sums L k are successively generated from sums where The process starts for each k = 2, 3, …, w at j = 1 and stops at . The transition from L j k −1, to L j k , or, equivalently, from M j k −1, to M j k , , depends on which of the following three cases occurs.To characterize these cases, let A be the product of some z i and ∈ , the set of all those z i , which are factors in A , but not in . In case c), M j k , contains the products , is obtained if this procedure is repeated for all B M j k ∈ −1, and starts with M A k k 0, = l q.Note that the sets M k are generated independently from each other.Hence, they can be determined in an arbitrary order.

Algorithm 1 1
Order the A j according to the number of their factors.
For all B M j k ∈ −1, : 5.1 If A j and B are disjoint, B becomes element of M j k , .Select another B .

5.3
If C A B j ( , ) = ∅ , delete B and select another B .

If
5.5 Select another B.

8
If k w < , then k k ← +1 and go to 3. If k w = , STOP.From Figure 1, W 1 1 4 = , l q, W 2 2 5 = , l q, W 3 1 3 5 = , , l q, W 4 2 3 4 = , , l q.Hence, A z z A z z A z z z A z z z To apply algorithm 1, firstly, initialize M A = l q, M M z z z z z z z = l q.Therefore, M z z z z = l q.Thus, M M z z z z z = l q.This gives M z z z z = l q.Therefore, M z z z z z 2 4 1 2 3 4 5 , = l q.
Lastly, with B z z z z z = 1 2 3 4 5 , A 3 and B are disjoint (case a).Hence, B is also element of Algorithm 1 is not applicable to generating disjoint sum forms of non-monotone structure functions.However, there are technical systems, whose reliability behaviour can only be described by non-monotone structure functions.Examples are given in [7].Moreover, the problem of generating disjoint sum forms from disjunctive normal forms also arises in probabilistic model-based reasoning and in the Dempster-Shafer theory of evidence.Here the Boolean functions of interest ("structure functions") are usually non-monotone.Abraham's approach to generating disjoint sum forms from disjunctive normal forms of Boolean functions has been generalized to non-monotone Boolean and even multi-valued functions in [3,4], 10,12,16,17,18,20].

MULTIVALUED SYSTEMS
To assume that the system S and its elements e e e n and the state space Z has r r r n ( , ,..., ) , r r Z is commonly defined as follows: 1) ϕ is nondecreasing in each argument and Usually, the elements of the Z i are real numbers.Otherwise, a total order in the set Z Z Z ∪ ∪ ∪ L n must be given (see, for instance, [1,11,19]).In these papers it is generally assumed that the state spaces Z s and Z i are identical or r s = 2 and r i > 2 .Multivalued systems with nondenumerable state spaces are, for instance, considered in [6].Here a partial generalization of these models is dealt with: The state spaces Z Z Z Z s n , , ,.
need not be identical (although this can be assumed without loss of generality) and the structure function ϕ need not be nondecreasing.However, as in the previous section, the state z s of the system can only assume values 0 or 1, i.e.Z s = { , } 0 1 (system is not available, is available).
To be able to present an algorithm for constructing a disjoint sum form of the structure function, concepts introduced in section 2 have to be generalized.Let Z i ⊆ Z .A set constraint (SC) over z i with respect to Z is denoted as z Z i ∈ and defined by Thus, Z is that subset of states, in which element e i operates satisfactorily.From the point of view of logic, z Z i ∈ is a predicate, which is true iff z i assumes a value from Z .Obviously, if Z i = 0 1 , l q, then the Boolean variable z i is equivalent to z i i ∈Z .Hence, SC's are generalizations of Boolean variables with , ' , '' ⊆ ⊆ Z Z .Disjunction, conjunction and negation of SC's are defined as In particular, for SC's over the same variable, An SC-clause (SC-term) is a disjunction (conjunction) of proper SC's with every variable z i occurring at most once.

L
As in chapter 2, it will be assumed that the z z z n 1 2 , ,..., are independent.Then, since SC's are random (0-1)-variables, e j e j e j Let the structure function of a possibly noncoherent multivalued system be given in the form ϕ( where the f k are proper terms with SC's over all or some of the z z z n 1 2 , ,..., .The usefulness of transforming structure function of type (5) into disjoint sum forms is motivated as in chapter 2. The following algorithm is an adaptation of algorithm 1 to noncoherent system functions with multi-valued arguments [3,20].It is based on a version firstly presented in [18].An alternative approach using the information that every element is in exactly one mode is presented in [4].
Algorithm 2 (order: f j contains not more SC's than f j+1 ) ; output: sets M j r i j , ,..., l q , then add the following formulas to M i j , : L Z \ http://orion.journals.ac.za/

1 Furthermore
Boolean, binary or, more specific, (0-1)-variable.For any two (0-1)variable x and y, disjunction, conjunction and complement are defined as follows: , x and y are called disjoint if x y = 0. Hence,

Figure 1 Example 1
Figure 1 Bridge structure

,
for all D in M i j −1,if D and f i are disjoint, then add D to M i j ) .
w be the set of all minimal path vectors.The minimal path set Consider, for example, systems (elements) with different operating and/or failure modes.Hence, it makes sense to assume that the indicator variables z s and z i of the states of S and e i can assume values from sets , ,..., can only be in either state "available" or "not available" is frequently an inadmissible oversimplification of the real situation.
called the frame of the indicator s .A multivalued coherent system with finite state sets Z Z