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Operations Research Letters 31 (2003) 268–272

OperationsResearchLetters

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On the characterization of continuum structure functions�

Seung Min LeeDepartment of Statistics, Hallym University, Chunchon 200-702, South Korea

Received 10 May 2002; received in revised form 19 September 2002; accepted 13 December 2002

Abstract

A continuum structure function is a non-decreasing mapping from the unit hypercube to the unit interval. Within the classof continuum structure functions, new axiomatic characterizations of the Natvig and the Barlow–Wu subclass are obtained.c© 2003 Elsevier Science B.V. All rights reserved.

Keywords: Reliability; Continuum structure function

1. Introduction

In reliability theory, structure functions model theoperation of a complex system by relating the statesx=(x1; x2; : : : ; xn) of the components C={1; 2; : : : ; n}of a system to that of the system itself. A binary struc-ture function is a mapping � : {0; 1}n → {0; 1}whichis non-decreasing in each augument. A binary struc-ture function � is said to be coherent if

min16i6n maxx∈{0;1}n [�(1i ; x)− �(0i ; x)] = 1;

where (�i; x) denotes (x1; : : : ; xi−1; �; xi+1; : : : ; xn), i.e.if each component is relevant to the system. Extend-ing the domain and range from {0; 1} to {0; 1; : : : ; M},Barlow and Wu [1] propose a class of multistate struc-ture functions having a one-to-one correspondence be-tween the multistate structure functions and their un-derlying binary structures, and Natvig [11] suggests ageneralization of this class by permitting the underly-ing binary structure to vary. Characterizations of the

� Research supported by KOSEF research project No.R05-2001-000-00074-0.

E-mail address: smlee1@hallym.ac.kr (S.M. Lee).

Natvig and the Barlow–Wu subclass within the classof all multistate structure functions have been made byBorges and Rodrigues [7]. See also Block and Savits[5].

A continuum structure function (CSF) is a map-ping :� → [0; 1] which is non-decreasing in eachargument and which satisBes (0) = 0 and (1) = 1,where � denotes the unit hypercube [0; 1]n and �denotes (�; �; : : : ; �)∈�. In the spirit of Natvig’s sug-gestion, Baxter [3] proposes the following class ofCSFs.

De�nition 3 (Baxter [3]). Let {��; 0¡�6 1} be aclass of binary coherent structure functions such that��(I�(x)) is a left-continuous and non-increasingfunction of � for Bxed x. If

(x)¿ � iD ��(I�(x)) = 1; x∈�; �∈ (0; 1];

is said to be a Natvig CSF, where I�(x) =(I�(x1); : : : ; I�(xn)) and I�(xi) is the indicator of{xi¿ �}, i = 1; : : : ; n.

A Natvig CSF reduces to a Barlow–Wu CSF [2],if the underlying binary structures ��; 0¡�6 1, are

0167-6377/03/$ - see front matter c© 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0167-6377(02)00240-7

S.M. Lee /Operations Research Letters 31 (2003) 268–272 269

identical. Within the class of CSFs, Kim and Baxter[9] obtain axiomatic characterizations of the Natvigand the Barlow–Wu subclass. See also Block and Sav-its [6], GriGth [8], and Mak [10] for various charac-terizations of the Barlow–Wu subclass.

In this paper, we present new axomatic character-izations of the Natvig and the Barlow–Wu subclasswithin the class of CSFs. The meaning of each ax-iom is clariBed in the characterization theorems. InSection 2, we present a characterization of the Natvigsubclass and show that Natvig CSFs are continuous.A numerical example illustrates that our characteriza-tion facilitates determination of whether a CSF is ofthe Natvig type. An analogous characterization of theBarlow–Wu subclass is also presented. A discussionfollows in Section 3.

2. The charaterization theorems

In this section, we present four conditions and ad-equate combinations of these conditions would char-acterize the corresponding subclasses. Conditions B1and B2 reveal the essential features of the underlyingstructures, and conditions R1 and R2 describe compo-nent relevancy of diDerent levels.

B1. For all �∈ (0; 1], if x∈U�, then �I�(x)∈U�,where U� = {x∈�|(x)¿ �}.

B2. For all �∈ (0; 1), if x∈U� ∩ {0; 1}n, then(�x)¡(x).

R1. For each i∈C and all �∈ (0; 1], there exists anx∈U� such that (0i ; x)¡�.

R2. For each i∈C, there exists an x∈� such that(0i ; x)¡(1i ; x).

Lemma 1. If is a CSF which satis3es condition B1,then there exists a class {��; 0¡�6 1} of binarystructure functions, not necessarily coherent, suchthat (x)¿ � i5 ��(I�(x)) = 1 (x∈�, �∈ (0; 1]).

Proof. For �∈ (0; 1] Bxed, deBne a binary function�� : {0; 1}n → {0; 1} by ��(z) = 1 iD �z∈U�. Let z′

and z be binary vectors such that z′¿ z and ��(z)=1.Then, since z′¿ z, it follows that �z′¿ �z and then,since �z∈U� and is non-decreasing, �z′ ∈U� sothat ��(z′)=1. Hence, �� is non-decreasing. Further,

observe that

(x)¿ �⇔ x∈U�;

⇔ �I�(x)∈U�;

by B1 and since x¿ �I�(x);

⇔��(I�(x)) = 1; by deBnition of ��:

Thus, �� is the desired binary structure function at�. Since � is chosen arbitrarily, this completes theproof.

Let {��; 0¡�6 1} be a class of underlying bi-nary structures of CSF as in Lemma 1. Then,left-continuity and monotonicity of �� directly followfrom that is well deBned and non-decreasing as aCSF.

Theorem 1. A CSF is of the Natvig type if andonly if it satis3es conditions B1 and R1.

Proof. It is easily veriBed that a Natvig CSF satisBesB1 and R1. To prove the converse, for �∈ (0; 1] Bxed,let �� be the underlying binary structure as in Lemma1 under B1. Then, for each i∈C, there exists an x∈U�

such that (0i ; x)¡� by R1. For the binary vectorI�(x), we show that ��(1i ; I�(x))−��(0i ; I�(x)) = 1.Now, since x∈U� and (1i ; x)¿ x, it follows that(1i ; x)∈U� so that ��(I�(1i ; x)) = ��(1i ; I�(x)) = 1.Similarly, (0i ; x)¡� implies that ��(0i ; I�(x)) = 0.Hence, the binary structure function �� is coher-ent. Since � is chosen arbitrarily, this completes theproof.

The Natvig subclass is characterized in Kim andBaxter [9] by the following three conditions:

K1. is right-continuous.K2. P� ⊂ {0; �}n, 0¡�6 1, where P�={x∈�|(x)

¿ � whereas (y)¡� for all y¡ x} and wherey¡ x means y6 x but y = x.

K3. For each i∈C and all �∈ (0; 1], there exists anx∈� such that (�i; x)¿ � whereas (�i; x)¡�for all �¡�.

Baxter [3] proves that Natvig CSFs are right-continuous, and the Kim and Baxter characterization

270 S.M. Lee /Operations Research Letters 31 (2003) 268–272

includes right-continuity (K1) to ensure that allP� (in K2) are non-empty. In our charaterization,right-continuity is eliminated and the relevancy con-dition R1 is weaker than their K3. Here, we showthat CSFs which satisfy condition B1 are in fact con-tinuous and hence so are Natvig CSFs, which is astronger result than Theorem 3.5 of Baxter [3].

Theorem 2. A CSF is continuous if it satis3es con-dition B1.

Proof. For �∈ (0; 1] Bxed, let Q� = {�I�(x)|x∈U�}and deBne V� =

⋃y∈Q�

{x∈�|x¿ y}. Since Q� ⊂U� under B1, it follows that V� ⊂ U�. Further, ifx∈U�, then, since x¿ �I�(x), it follows that x∈V�

and hence U� ⊂ V�. Thus, U� =V� and, as V� is obvi-ously closed (in the relative topology on �), it followsthat U� is also closed. Since � is arbitrary, we haveshown that every U� is closed. Hence, by Proposition1 of Kim and Baxter [9], is right-continuous. Fur-ther, we show that is also left-continuous. Choosex∈� and suppose that (x)=�. Given �∈ (0; �), since is non-decreasing, it suGces to show that ((x −”)∗)¿ � − �, where y∗ is the vector whose ith ele-ment is 0 ∨ yi, i = 1; 2; : : : ; n. Let ��; 0¡�6 1, bethe underlying binary structures under B1. Then,

��−�(I�−�((x− ”)∗))¿��−�(I�(x)); since I�−�((x− ”)∗)¿ I�(x);

¿��(I�(x)); since ��

is non-increasing in � for Bxed x;

=1; since x∈U�:

Hence, (x − ”)∗ ∈U�−�, i.e. ((x − ”)∗)¿ � − �,as claimed. Since is right-continuous and is alsoleft-continuous, it follows that is continuous, com-pleting the proof.

Corollary 1. Natvig CSFs are continuous.

The key conditions which prescribe underlyingstructures are K1 and K2 in the Kim and Baxter char-acterization, and B1 in our characterization. Let a CSF be given. If satisBes K1 and K2, then it satisBesB1. If does not satisfy K1, then it does not satisfyB1 either by Theorem 2. Suppose that does not sat-isfy K2, i.e. there exists an x which is in P� but not

in {0; �}n for some �∈ (0; 1]. Note that x ∈ {0; �}nimplies �I�(x)¡ x and hence, by deBnition of P�,x∈U� but �I�(x) ∈ U�, which contradicts B1. Thus,in terms of the key conditions, whenever the Kim andBaxter characterization determines whether is of theNatvig type or not, our characterization immediatelyconcludes the same without further investigation. Thefollowing example illustrates that the converse is nottrue.

Example 1. Given a CSF on [0; 1]2, suppose thatthe following is known in each case:

(i) is not continuous.(ii) (1; 0) = 1 and ( 13 ; 0) = 0.

It is seen that, by Theorem 2, (i) contradicts B1.With (ii), notice that, for � = 1

3 , x = (1; 0)∈U� but�I�(x) = (13 ; 0) ∈ U� and hence (ii) also contradictsB1. Thus, in each case of (i) or (ii), our characteriza-tion determines that is not of the Natvig type whereasthe Kim and Baxter characterization needs further in-vestigation. Observe that each of (i) or (ii) holds onboth 1 and 2, where

1(2)(x1; x2) =

{x1 ∧ x2 if x1 ∨ x2 ¡ (6) 1

2 ;

x1 ∨ x2 otherwise

and notice that 1 satisBes K1 but not K2 while 2satisBes K2 but not K1.

As a specialization of the Natvig subclass, we char-acterize the Barlow–Wu subclass by adding conditionB2. Under B1 and B2, the relevancy conditions R1and R2 are equivalent and hence, R1 can be replacedwith a weaker condition R2 in the characterization.

Lemma 2. If is a CSF which satis3es conditionB2, in addition to B1, then the underlying binarystructures ��, under B1, are identical.

Proof. Let ��; 0¡�6 1, be the underlying binarystructures under B1, and suppose contrapositivelythat �� = �� for 0¡�¡�6 1. Then, there existsa binary vector x such that ��(x) = 1 and ��(x) = 0so that �6 (x)¡�. Let (x)= �′. Then, since�′x∈U�′ by B1, it follows that �′6 (�′x)6 (x)=�′. Thus, for �′ ∈ (0; 1), x∈U�′ ∩ {0; 1}n but

S.M. Lee /Operations Research Letters 31 (2003) 268–272 271

(x) = (�′x), in contradiction to B2, completing theproof.

Each of the following conditions is possibly used,instead of B2, to ensure that the underlying binarystructures under B1 are identical.

B2′. There is no non-empty open set A ⊂ � on which is constant. (C3 of Kim and Baxter [9])

B2′′. ({0; 1}n) = {0; 1}. (C1 of GriGth [8])

Notice that, within the class of CSFs, B2 is weakerthan B2′ and neither of B2 or B2′′ implies the other. Inpractice, however, it seems reasonable to claim a CSF as a reliability model to satisfy (�) = �, �∈ [0; 1],and then B2 is weaker than B2′′ within the class ofsuch CSFs satisfying (�) = �.

Theorem 3. A CSF is of the Barlow–Wu type ifand only if it satis3es conditions B1, B2 and R2.

Proof. It is easily veriBed that a Barlow–Wu CSFsatisBes B1, B2 and R2. Conversely, for i∈C Bxed,there exists an x∈� such that (1i ; x)¿(0i ; x) byR2. Choose �∈ (0; 1] such that (1i ; x)¿ �¿(0i ; x),and let �� be the underlying binary structure at �.Then, we have ��(1i ; I�(x)) − ��(0i ; I�(x)) = 1 asshown in the proof of Theorem 1. Since, under B2, theunderlying binary structures are identical, i.e., �� =�say, we have shown that component i is relevant to �.This holds for each i∈C so that the underlying binarystructure � is coherent, completing the proof.

3. Discussion

(1) Other subclasses: The relevancy condition R1states that each component is relevant to the systemat all �∈ (0; 1], which ensures that, with B1, all un-derlying binary structures �� of a Natvig CSF are co-herent. When R1 is replaced with a weaker conditionR2, each component is relevant to the system, but notall �� need be coherent. Such CSFs, characterized byconditions B1 and R2, would constitute a new sub-class which includes the Natvig subclass. Baxter andLee [4] have considered a more generalized model,called F-type, of which the underlying structures are

not binary but multistate structures, and character-ized the F-type subclass within the class of CSFs byright-continuity and Bniteness of P�.

(2) Characterization of multistate structures: Amultistate structure function (MSF) is a mapping :Sn → S which is non-decreasing in each argumentand which satisBes (0) = 0 and (m) = m, whereS = {0; 1; : : : ; m} and k denotes (k; k; : : : ; k)∈ Sn. Thesame techniques of Section 2 can be used to obtainthe following characterizations of the Natvig and theBarlow–Wu subclass within the class of MSFs: AnMSF is of the Natvig type if and only if it satisBesconditions M1 and M3, and is of the Barlow–Wu typeif and only if it satisBes conditions M1, M2 and M4,where

M1. for all k ¿ 0, if x∈Uk , then kIk(x)∈Uk , whereUk = {x∈ Sn| (x)¿ k},

M2. for all k; m¿k ¿ 0, if x∈Uk ∩ {0; m}n, then (kIk(x))¡ (x),

M3. for each i∈C and all k ¿ 0, there exists anx∈Uk such that (0i ; x)¡k,

M4. for each i∈C, there exists an x∈ Sn such that (0i ; x)¡ (mi; x).

In the characterizations given in Borges and Rodrigues[7], the key conditions, which prescribe underlyingstructures, are:

C1. for all k ¿ 0, if x∈Uk , then there existsy∈{0; k}n such that y6 x and y∈Uk ,

C2. ({0; m}n) = {0; m}.

In our characterizations, condition M1 is simpler thanC1 which essentially says that Pk ⊂ {0; k}n, k ¿ 0,where Pk = {x∈ Sn| (x)¿ k whereas (y)¡k forall y¡ x}, and condition M2 is weaker than C2 withinthe class of such MSFs satisfying (k) = k, k ∈ S.

Acknowledgements

The author is grateful to the Associate Editor forvaluable comments and suggestions on earlier ver-sions of this paper, which substantially improved thepresentation.

272 S.M. Lee /Operations Research Letters 31 (2003) 268–272

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