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On the characterization of continuum structure functions
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Operations Research Letters 31 (2003) 268 – 272
Operations Research Letters www.elsevier.com/locate/dsw
On the characterization of continuum structure functions Seung Min Lee Department 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 class of 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 the operation of a complex system by relating the states x = (x1 ; x2 ; : : : ; x n ) of the components C = {1; 2; : : : ; n} of a system to that of the system itself. A binary structure function is a mapping : {0; 1}n → {0; 1} which is non-decreasing in each augument. A binary structure 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 ; : : : ; x n ), i.e. if each component is relevant to the system. Extending the domain and range from {0; 1} to {0; 1; : : : ; M }, Barlow and Wu [1] propose a class of multistate structure functions having a one-to-one correspondence between the multistate structure functions and their underlying binary structures, and Natvig [11] suggests a generalization of this class by permitting the underlying binary structure to vary. Characterizations of the Research supported by KOSEF research project No. R05-2001-000-00074-0. E-mail address: [email protected] (S.M. Lee).
Natvig and the Barlow–Wu subclass within the class of all multistate structure functions have been made by Borges and Rodrigues [7]. See also Block and Savits [5]. A continuum structure function (CSF) is a mapping : → [0; 1] which is non-decreasing in each argument and which satisBes (0) = 0 and (1) = 1, where denotes the unit hypercube [0; 1]n and denotes (; ; : : : ; ) ∈ . In the spirit of Natvig’s suggestion, Baxter [3] proposes the following class of CSFs. Denition 3 (Baxter [3]). Let { ; 0 ¡ 6 1} be a class of binary coherent structure functions such that (I (x)) is a left-continuous and non-increasing function 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 (x n )) 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
c 2003 Elsevier Science B.V. All rights reserved. 0167-6377/03/$ - see front matter 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 Natvig and the Barlow–Wu subclass. See also Block and Savits [6], GriGth [8], and Mak [10] for various characterizations of the Barlow–Wu subclass. In this paper, we present new axomatic characterizations of the Natvig and the Barlow–Wu subclass within the class of CSFs. The meaning of each axiom is clariBed in the characterization theorems. In Section 2, we present a characterization of the Natvig subclass and show that Natvig CSFs are continuous. A numerical example illustrates that our characterization facilitates determination of whether a CSF is of the Natvig type. An analogous characterization of the Barlow–Wu subclass is also presented. A discussion follows in Section 3.
observe that
2. The charaterization theorems
Theorem 1. A CSF is of the Natvig type if and only if it satis3es conditions B1 and R1.
In this section, we present four conditions and adequate combinations of these conditions would characterize the corresponding subclasses. Conditions B1 and B2 reveal the essential features of the underlying structures, and conditions R1 and R2 describe component 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 an x ∈ 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 binary structure functions, not necessarily coherent, such that (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 so that (z ) = 1. Hence, is non-decreasing. Further,
(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 the proof. Let { ; 0 ¡ 6 1} be a class of underlying binary structures of CSF as in Lemma 1. Then, left-continuity and monotonicity of directly follow from that is well deBned and non-decreasing as a CSF.
Proof. It is easily veriBed that a Natvig CSF satisBes B1 and R1. To prove the converse, for ∈ (0; 1] Bxed, let be the underlying binary structure as in Lemma 1 under B1. Then, for each i ∈ C, there exists an x ∈ U such that (0i ; x) ¡ by R1. For the binary vector I (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 coherent. Since is chosen arbitrarily, this completes the proof. The Natvig subclass is characterized in Kim and Baxter [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 where y ¡ x means y 6 x but y = x. K3. For each i ∈ C and all ∈ (0; 1], there exists an x ∈ such that (i ; x) ¿ whereas (i ; x) ¡ for all ¡ . Baxter [3] proves that Natvig CSFs are rightcontinuous, and the Kim and Baxter characterization
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S.M. Lee / Operations Research Letters 31 (2003) 268 – 272
includes right-continuity (K1) to ensure that all P (in K2) are non-empty. In our charaterization, right-continuity is eliminated and the relevancy condition R1 is weaker than their K3. Here, we show that CSFs which satisfy condition B1 are in fact continuous and hence so are Natvig CSFs, which is a stronger result than Theorem 3.5 of Baxter [3]. Theorem 2. A CSF is continuous if it satis3es condition 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, if x ∈ U , then, since x ¿ I (x), it follows that x ∈ V and hence U ⊂ V . Thus, U = V and, as V is obviously closed (in the relative topology on ), it follows that U is also closed. Since is arbitrary, we have shown that every U is closed. Hence, by Proposition 1 of Kim and Baxter [9], is right-continuous. Further, we show that is also left-continuous. Choose x ∈ and suppose that (x)=. Given ∈ (0; ), since
is non-decreasing, it suGces to show that ((x − ”)∗ ) ¿ − , where y∗ is the vector whose ith element is 0 ∨ yi , i = 1; 2; : : : ; n. Let ; 0 ¡ 6 1, be the underlying binary structures under B1. Then, − (I− ((x − ”)∗ )) ¿ − (I (x)); ¿ (I (x));
since 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 also left-continuous, it follows that is continuous, completing the proof.
in {0; }n for some ∈ (0; 1]. Note that x ∈ {0; }n implies 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 and Baxter characterization determines whether is of the Natvig type or not, our characterization immediately concludes the same without further investigation. The following example illustrates that the converse is not true. Example 1. Given a CSF on [0; 1]2 , suppose that the 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 = 13 , x = (1; 0) ∈ U but I (x) = ( 31 ; 0) ∈ U and hence (ii) also contradicts B1. Thus, in each case of (i) or (ii), our characterization determines that is not of the Natvig type whereas the Kim and Baxter characterization needs further investigation. Observe that each of (i) or (ii) holds on both 1 and 2 , where x1 ∧ x2 if x1 ∨ x2 ¡ (6) 12 ;
1 ( 2 )(x1 ; x2 ) = x1 ∨ x2 otherwise and notice that 1 satisBes K1 but not K2 while 2 satisBes K2 but not K1. As a specialization of the Natvig subclass, we characterize the Barlow–Wu subclass by adding condition B2. Under B1 and B2, the relevancy conditions R1 and R2 are equivalent and hence, R1 can be replaced with a weaker condition R2 in the characterization.
Corollary 1. Natvig CSFs are continuous.
Lemma 2. If is a CSF which satis3es condition B2, in addition to B1, then the underlying binary structures , under B1, are identical.
The key conditions which prescribe underlying structures are K1 and K2 in the Kim and Baxter characterization, and B1 in our characterization. Let a CSF
be given. If satisBes K1 and K2, then it satisBes B1. If does not satisfy K1, then it does not satisfy B1 either by Theorem 2. Suppose that does not satisfy K2, i.e. there exists an x which is in P but not
Proof. Let ; 0 ¡ 6 1, be the underlying binary structures under B1, and suppose contrapositively that = for 0 ¡ ¡ 6 1. Then, there exists a binary vector x such that (x) = 1 and (x) = 0 so 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
(x) = ( x), in contradiction to B2, completing the proof. Each of the following conditions is possibly used, instead of B2, to ensure that the underlying binary structures 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 weaker than B2 and neither of B2 or B2 implies the other. In practice, 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 of such CSFs satisfying () = . Theorem 3. A CSF is of the Barlow–Wu type if and only if it satis3es conditions B1, B2 and R2. Proof. It is easily veriBed that a Barlow–Wu CSF satisBes B1, B2 and R2. Conversely, for i ∈ C Bxed, there exists an x ∈ such that (1i ; x) ¿ (0i ; x) by R2. 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 as shown in the proof of Theorem 1. Since, under B2, the underlying 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 binary structure is coherent, completing the proof.
3. Discussion (1) Other subclasses: The relevancy condition R1 states that each component is relevant to the system at all ∈ (0; 1], which ensures that, with B1, all underlying binary structures of a Natvig CSF are coherent. When R1 is replaced with a weaker condition R2, each component is relevant to the system, but not all need be coherent. Such CSFs, characterized by conditions B1 and R2, would constitute a new subclass which includes the Natvig subclass. Baxter and Lee [4] have considered a more generalized model, called F-type, of which the underlying structures are
271
not binary but multistate structures, and characterized the F-type subclass within the class of CSFs by right-continuity and Bniteness of P . (2) Characterization of multistate structures: A multistate structure function (MSF) is a mapping : S n → S which is non-decreasing in each argument and which satisBes (0) = 0 and (m) = m, where S = {0; 1; : : : ; m} and k denotes (k; k; : : : ; k) ∈ S n . The same techniques of Section 2 can be used to obtain the following characterizations of the Natvig and the Barlow–Wu subclass within the class of MSFs: An MSF is of the Natvig type if and only if it satisBes conditions M1 and M3, and is of the Barlow–Wu type if and only if it satisBes conditions M1, M2 and M4, where M1. for all k ¿ 0, if x ∈ Uk , then kIk (x) ∈ Uk , where Uk = {x ∈ S n | (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 an x ∈ Uk such that (0i ; x) ¡ k, M4. for each i ∈ C, there exists an x ∈ S n such that (0i ; x) ¡ (mi ; x). In the characterizations given in Borges and Rodrigues [7], the key conditions, which prescribe underlying structures, are: C1. for all k ¿ 0, if x ∈ Uk , then there exists y ∈ {0; k}n such that y 6 x and y ∈ Uk , C2. ({0; m}n ) = {0; m}. In our characterizations, condition M1 is simpler than C1 which essentially says that Pk ⊂ {0; k}n , k ¿ 0, where Pk = {x ∈ S n | (x) ¿ k whereas (y) ¡ k for all y ¡ x}, and condition M2 is weaker than C2 within the class of such MSFs satisfying (k) = k, k ∈ S.
Acknowledgements The author is grateful to the Associate Editor for valuable comments and suggestions on earlier versions of this paper, which substantially improved the presentation.
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References [1] R.E. Barlow, A.S. Wu, Coherent systems with multi-state components, Math. Oper. Res. 3 (4) (1978) 275–281. [2] L.A. Baxter, Continuum structures I, J. Appl. Probab. 21 (1984) 802–815. [3] L.A. Baxter, Continuum structures II, Math. Proc. Camb. Phil. Soc. 99 (1986) 331–338. [4] L.A. Baxter, S.M. Lee, Structure functions with Bnite minimal vector sets, J. Appl. Probab. 26 (1989) 196–201. [5] H.W. Block, T.H. Savits, A decomposition for multistatemonotone systems, J. Appl. Probab. 19 (1982) 392–402. [6] H.W. Block, T.H. Savits, Continuous multistate structure functions, Oper. Res. 32 (1984) 703–714.
[7] W.de.S. Borges, F.W. Rodrigues, An axiomatic characterization of multistate coherent structures, Math. Oper. Res. 8 (3) (1983) 435–438. [8] W.S. GriGth, A note on the characterization of the Barlow and Wu continuum structure functions, Oper. Res. Lett. 21 (1997) 65–67. [9] C. Kim, L.A. Baxter, Axiomatic characterization of continuum structure functions, Oper. Res. Lett. 6 (1987) 297–300. [10] K. Mak, A note on Barlow–Wu structure functions, Oper. Res. Lett. 8 (1989) 43–44. [11] B. Natvig, Two suggestions of how to deBne a multistate coherent system, Adv. Appl. Probab. 14 (1982) 434–455.
Operations Research Letters 31 (2003) 268 – 272
Operations Research Letters www.elsevier.com/locate/dsw
On the characterization of continuum structure functions Seung Min Lee Department 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 class of 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 the operation of a complex system by relating the states x = (x1 ; x2 ; : : : ; x n ) of the components C = {1; 2; : : : ; n} of a system to that of the system itself. A binary structure function is a mapping : {0; 1}n → {0; 1} which is non-decreasing in each augument. A binary structure 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 ; : : : ; x n ), i.e. if each component is relevant to the system. Extending the domain and range from {0; 1} to {0; 1; : : : ; M }, Barlow and Wu [1] propose a class of multistate structure functions having a one-to-one correspondence between the multistate structure functions and their underlying binary structures, and Natvig [11] suggests a generalization of this class by permitting the underlying binary structure to vary. Characterizations of the Research supported by KOSEF research project No. R05-2001-000-00074-0. E-mail address: [email protected] (S.M. Lee).
Natvig and the Barlow–Wu subclass within the class of all multistate structure functions have been made by Borges and Rodrigues [7]. See also Block and Savits [5]. A continuum structure function (CSF) is a mapping : → [0; 1] which is non-decreasing in each argument and which satisBes (0) = 0 and (1) = 1, where denotes the unit hypercube [0; 1]n and denotes (; ; : : : ; ) ∈ . In the spirit of Natvig’s suggestion, Baxter [3] proposes the following class of CSFs. Denition 3 (Baxter [3]). Let { ; 0 ¡ 6 1} be a class of binary coherent structure functions such that (I (x)) is a left-continuous and non-increasing function 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 (x n )) 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
c 2003 Elsevier Science B.V. All rights reserved. 0167-6377/03/$ - see front matter 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 Natvig and the Barlow–Wu subclass. See also Block and Savits [6], GriGth [8], and Mak [10] for various characterizations of the Barlow–Wu subclass. In this paper, we present new axomatic characterizations of the Natvig and the Barlow–Wu subclass within the class of CSFs. The meaning of each axiom is clariBed in the characterization theorems. In Section 2, we present a characterization of the Natvig subclass and show that Natvig CSFs are continuous. A numerical example illustrates that our characterization facilitates determination of whether a CSF is of the Natvig type. An analogous characterization of the Barlow–Wu subclass is also presented. A discussion follows in Section 3.
observe that
2. The charaterization theorems
Theorem 1. A CSF is of the Natvig type if and only if it satis3es conditions B1 and R1.
In this section, we present four conditions and adequate combinations of these conditions would characterize the corresponding subclasses. Conditions B1 and B2 reveal the essential features of the underlying structures, and conditions R1 and R2 describe component 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 an x ∈ 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 binary structure functions, not necessarily coherent, such that (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 so that (z ) = 1. Hence, is non-decreasing. Further,
(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 the proof. Let { ; 0 ¡ 6 1} be a class of underlying binary structures of CSF as in Lemma 1. Then, left-continuity and monotonicity of directly follow from that is well deBned and non-decreasing as a CSF.
Proof. It is easily veriBed that a Natvig CSF satisBes B1 and R1. To prove the converse, for ∈ (0; 1] Bxed, let be the underlying binary structure as in Lemma 1 under B1. Then, for each i ∈ C, there exists an x ∈ U such that (0i ; x) ¡ by R1. For the binary vector I (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 coherent. Since is chosen arbitrarily, this completes the proof. The Natvig subclass is characterized in Kim and Baxter [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 where y ¡ x means y 6 x but y = x. K3. For each i ∈ C and all ∈ (0; 1], there exists an x ∈ such that (i ; x) ¿ whereas (i ; x) ¡ for all ¡ . Baxter [3] proves that Natvig CSFs are rightcontinuous, and the Kim and Baxter characterization
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S.M. Lee / Operations Research Letters 31 (2003) 268 – 272
includes right-continuity (K1) to ensure that all P (in K2) are non-empty. In our charaterization, right-continuity is eliminated and the relevancy condition R1 is weaker than their K3. Here, we show that CSFs which satisfy condition B1 are in fact continuous and hence so are Natvig CSFs, which is a stronger result than Theorem 3.5 of Baxter [3]. Theorem 2. A CSF is continuous if it satis3es condition 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, if x ∈ U , then, since x ¿ I (x), it follows that x ∈ V and hence U ⊂ V . Thus, U = V and, as V is obviously closed (in the relative topology on ), it follows that U is also closed. Since is arbitrary, we have shown that every U is closed. Hence, by Proposition 1 of Kim and Baxter [9], is right-continuous. Further, we show that is also left-continuous. Choose x ∈ and suppose that (x)=. Given ∈ (0; ), since
is non-decreasing, it suGces to show that ((x − ”)∗ ) ¿ − , where y∗ is the vector whose ith element is 0 ∨ yi , i = 1; 2; : : : ; n. Let ; 0 ¡ 6 1, be the underlying binary structures under B1. Then, − (I− ((x − ”)∗ )) ¿ − (I (x)); ¿ (I (x));
since 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 also left-continuous, it follows that is continuous, completing the proof.
in {0; }n for some ∈ (0; 1]. Note that x ∈ {0; }n implies 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 and Baxter characterization determines whether is of the Natvig type or not, our characterization immediately concludes the same without further investigation. The following example illustrates that the converse is not true. Example 1. Given a CSF on [0; 1]2 , suppose that the 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 = 13 , x = (1; 0) ∈ U but I (x) = ( 31 ; 0) ∈ U and hence (ii) also contradicts B1. Thus, in each case of (i) or (ii), our characterization determines that is not of the Natvig type whereas the Kim and Baxter characterization needs further investigation. Observe that each of (i) or (ii) holds on both 1 and 2 , where x1 ∧ x2 if x1 ∨ x2 ¡ (6) 12 ;
1 ( 2 )(x1 ; x2 ) = x1 ∨ x2 otherwise and notice that 1 satisBes K1 but not K2 while 2 satisBes K2 but not K1. As a specialization of the Natvig subclass, we characterize the Barlow–Wu subclass by adding condition B2. Under B1 and B2, the relevancy conditions R1 and R2 are equivalent and hence, R1 can be replaced with a weaker condition R2 in the characterization.
Corollary 1. Natvig CSFs are continuous.
Lemma 2. If is a CSF which satis3es condition B2, in addition to B1, then the underlying binary structures , under B1, are identical.
The key conditions which prescribe underlying structures are K1 and K2 in the Kim and Baxter characterization, and B1 in our characterization. Let a CSF
be given. If satisBes K1 and K2, then it satisBes B1. If does not satisfy K1, then it does not satisfy B1 either by Theorem 2. Suppose that does not satisfy K2, i.e. there exists an x which is in P but not
Proof. Let ; 0 ¡ 6 1, be the underlying binary structures under B1, and suppose contrapositively that = for 0 ¡ ¡ 6 1. Then, there exists a binary vector x such that (x) = 1 and (x) = 0 so 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
(x) = ( x), in contradiction to B2, completing the proof. Each of the following conditions is possibly used, instead of B2, to ensure that the underlying binary structures 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 weaker than B2 and neither of B2 or B2 implies the other. In practice, 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 of such CSFs satisfying () = . Theorem 3. A CSF is of the Barlow–Wu type if and only if it satis3es conditions B1, B2 and R2. Proof. It is easily veriBed that a Barlow–Wu CSF satisBes B1, B2 and R2. Conversely, for i ∈ C Bxed, there exists an x ∈ such that (1i ; x) ¿ (0i ; x) by R2. 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 as shown in the proof of Theorem 1. Since, under B2, the underlying 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 binary structure is coherent, completing the proof.
3. Discussion (1) Other subclasses: The relevancy condition R1 states that each component is relevant to the system at all ∈ (0; 1], which ensures that, with B1, all underlying binary structures of a Natvig CSF are coherent. When R1 is replaced with a weaker condition R2, each component is relevant to the system, but not all need be coherent. Such CSFs, characterized by conditions B1 and R2, would constitute a new subclass which includes the Natvig subclass. Baxter and Lee [4] have considered a more generalized model, called F-type, of which the underlying structures are
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not binary but multistate structures, and characterized the F-type subclass within the class of CSFs by right-continuity and Bniteness of P . (2) Characterization of multistate structures: A multistate structure function (MSF) is a mapping : S n → S which is non-decreasing in each argument and which satisBes (0) = 0 and (m) = m, where S = {0; 1; : : : ; m} and k denotes (k; k; : : : ; k) ∈ S n . The same techniques of Section 2 can be used to obtain the following characterizations of the Natvig and the Barlow–Wu subclass within the class of MSFs: An MSF is of the Natvig type if and only if it satisBes conditions M1 and M3, and is of the Barlow–Wu type if and only if it satisBes conditions M1, M2 and M4, where M1. for all k ¿ 0, if x ∈ Uk , then kIk (x) ∈ Uk , where Uk = {x ∈ S n | (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 an x ∈ Uk such that (0i ; x) ¡ k, M4. for each i ∈ C, there exists an x ∈ S n such that (0i ; x) ¡ (mi ; x). In the characterizations given in Borges and Rodrigues [7], the key conditions, which prescribe underlying structures, are: C1. for all k ¿ 0, if x ∈ Uk , then there exists y ∈ {0; k}n such that y 6 x and y ∈ Uk , C2. ({0; m}n ) = {0; m}. In our characterizations, condition M1 is simpler than C1 which essentially says that Pk ⊂ {0; k}n , k ¿ 0, where Pk = {x ∈ S n | (x) ¿ k whereas (y) ¡ k for all y ¡ x}, and condition M2 is weaker than C2 within the class of such MSFs satisfying (k) = k, k ∈ S.
Acknowledgements The author is grateful to the Associate Editor for valuable comments and suggestions on earlier versions of this paper, which substantially improved the presentation.
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