MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS-1963-A

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1 AD-R149 B17 AXIOMATIC CHARACTERIZATIONS OF CONTINUUM STRUCTURE i/l FUNCTIONS(U) STATE UNIV OF NEW YORK AT STONY BROOK DEPT OF APPLIED MRTHEMA.. C KIM ET AL. NOV 84 UNCLRSSIFIED AFOSR-TR-84-i282 AFOSR F/G 20/11 NL Elllllllllll I.

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3 q *11 'o..0 MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS-1963-A I I

4 S S -TR - y - -V AXIOMATIC CHARACTERIZATIONS OF CONTINUUM STRUCTURE FUNCTIONS* _Chul Kim and Laurence A. Baxter 00 Department of Applied Mathematics and Statistics State University of New York at Stony Brook Brook, NY 11794, USA 0Stony DTIC SELECTE J2A N '2 1 Approve for publ 5..re., ee- 3 L *Research supported by the National Science Foundation under grant ECS and by the Air Force Office of Scientific Research, AFSC, USAF, under grant AFOSR The US Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. k.- q C"i I.." t. ".- ":, ". " ",. " "- ' ' - ' "" I l'..". ".,- ",

5 r r. - < r ~nl ~ ~ 9 - S.. fft' * -.. _.. *% -~ - - -= UNCLk'SSIFIED SIEC ITv CASS:IICA'iON OF THIS PAGE is) REPcRT SECURITY CLASSIFICATION UNCLASSIFIED REPORT DOCUMENTATION PAGE 1b. RESTRICTIVE MARKINGS 2& SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION/AVAILABILITY OF REPORT 2b OECLASSIFICATION/DOWNGRADING SCHEDULE Approved for public release; distribution unlimited. 4 PERFORMING ORGANIZATION REPORT NUMBER(S) S. MONITORING ORGANIZATION REPORT NUMBERIS) XFOSR.-Tit $ s, NAME OF PERFORMING ORGANIZATION b. OFFICE SYMBOL 7s. NAME OF MONITORING ORGANIZATION State University of New York S c. ADDRESS (City. State and ZIP Code) fir plcalj Air Force Office of Scientific Research 7b. ADDRESS (City, State and ZIP Cod. Department of Applied Mathematics and Directorate of Mathematical & Information Statistics, Stony Brook NY Sciences, Bolling AFB DC a. NAME OF FUNOINGSPONSORING Bb OFFICE SYMBOL 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ORGANIZATION (if appicable) AFOSR NM AFOSR c ADDRESS icity, Stat. and ZIP Code) 10 SOURCE OF FUNDING NOS. 4 PROGRAM PROJECT TASK WORK UNIT ELEMENT NO. NO. NO. NO. Boiling AFB DC F 2304 A5 11. TITLE (include Security Cla" sfication) AXIOMATIC CHARACTERIZATIONS OF CONTINUUM STRUCTURE FUNCTIONS 12. PERSONAL AUTHOR(S) Chul Kim and Laurence A. Baxter 13a. T'vPE OF REPORT 13b. TIME COVERED 14 DATE OF REPORT (Yr. Mo.. Day) 15. PAGE COUNT Technical FROM TO _NOV IS SUPPLEMENTARY NOTATION 17. COSATI CODES 18. SUBJECT TERMS icontinue ov reverse it necemry and identify by block number) FIELD GROUP SUB GR. Reliability; continuum structure function; multistate structure function. 1S ABSTRACT (Continue on ey.erse if nhcesary and identify by bloc number) A continuum s-ricture function is a nondecreasing mapping from the unit hypercube to the unit interval. Axiomatic characterizations of the continuum structure functions based on the Barlow-Wu and Natvig multistate structure functions are derived. 20. DISTRIBUTION/AVA ILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CLASSIFICATION UNCLASSIFIED/UNLIMITED M SAME AS apt. 0 OTICUSERS 0 UNCLASSIFIED 22g NAME OF RESPONSIBLE INDIVIDUAL 22b TELEPHONE NUMBER 22c. OFFICE SYMBOL (Include A me CodeD MAJ Brian W. Woodruff (202) NM I Ed DO FORM 1473, 83 APR EDITION OF I JAN 73 IS OBSOLETE. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGI,', % " _.',.. ' _..' ' :, f ' '

6 ABSTRACT A continuum structure function is a nondecreasing mapping from the unit hypercube to the unit interval. Axiomatic characterizations of the continuum structure functions based on the Barlow-Wu and Natvig multistate structure functions are derived..",, Dis I L I I..J 4t function. * h AMS 1980 Subject Classification: Primary 90B25 OR/MS Index 1978 Subject Classification: Primary 721 Reliability *,. NO IC

7 *~XW -I O* W'* -'~ 1. INTRODUCTION n Let C - {1,2,...,n} denote a set of components and let A = (0,] n. A nondecreasing mapping y: A " [0,1] with y(o) = 0 and y(l) - 1 is said to be a continuum structure function (CSF). If sup [(lix)-y(oix)] > 0 XEA for each i E C, where (6i,X) denotes (Xi,...,Xi 1,6,X+ 1,...,X), y is said to be weakly coherent. Definition Let PI...,Pr denote the r minimal path sets of a binary coherent structure function. If Y)= max min X i 15<j<r iep. y is said to be a Barlow-Wu CSF [2]. Definition Let {4, O<q<l} be a class of binary coherent structure functions such that (Y ) is a left-continuous and non-increasing function of a for fixed X where Y is the indicator of {X >M}u, i-1,2,...,n. If Y(X) > a iff a(y) 1 (XEA,0<a<l), y is said to be a Natvig CSF [3]. In this paper, we present axiomatic characterizations of the Barlow-Wu and Natvig CSFs. In particular, we show that y is a Barlow-Wu CSF if and only if it satisfies the following conditions: 6.

8 . 2 'A Cl y is continuous C2 P 0 and P c {O,a}n, O < a < a a - C3 There is no nonempty open set A c A such that y is constant on A C4 y is weakly coherent where P = {XEAIy(X) > a whereas y(y) < a for all Y<X} and where Y < X means that Y < X but that Y # X. Some consequences of these axioms are deduced in Section 2, and in Section 3 we present our main results: an axiomatic characterization of the Barlow-Wu CSF and an analogous characterization of the Natvig CSF. Our approach was suggested by the Borges-Rodrigues characterizations of the Barlow-Wu and Natvig multistate structure functions [51 though, as we show in Section 4, their characterizations are incorrect. 2. SOME DEDUCTIONS FROM THE AXIOMS Let U. = {XEAIY(X) >a} and L = {XEAIy(X) <a}, 0 < a < 1. Further, define K a = {XEAIy(X) <a _ whereas y(y) > a for all Y - > Z1, 0 < a < 1. Proposition 2.1 Let y be a CSF. () y is right (left)-continuous if and only if each U (L ) is closed. a a (ii) If y is right (left)-continuous, then each P a (K ) is nonempty and X E U (L ) if and only if X > (<) Y E P (K ). - a a (iii) If y is continuous, then y(p ) {a}, 0 < a < 1, and y(k ) {a}, a a 0

9 3 Proof: The proofs of (i) and (iii) are straightforward; see [4] for the proof of (ii). Proposition 2.2 If y is a continuous CSF, conditions 02 and C2' K j#0and K c{,l}n, 0< a <1 are equivalent. )k'! Proof: Since y is continuous, each K is nonempty. We show that, if C2 holds, then 1~C{,l}n for all a E [0,1). a Suppose, conversely, that for some a E [0,1) there exists a vector Y E K asuch that Y f {a,l}n. Then there exists at least one component, k say, such that Yk {a,l1. Either 0 < Y < a < 1 or 0 < a < Y < 1; we k-k k consider these two cases separately. Suppose, firstly, that 0 < Y k < a < 1. By Proposition 2.1, ryq) = a and yr(6ky > at if Y < 6 < a. Let y( 6 k'q=~ hnckx ~ ic U is closed there exists, by Proposition 2.1, an X < ( 6 k~ such that, X EP. Now Y ju and n so Y <Xk< 6. Thus 0< Y < X.k< 6< :{,~ a < and C2 K a 1 so X J {0F~, in contradiction to 02. Suppose, now, that 0 < a < Y <1. AganyY.LtylY _ k git()=a Le lk9~) 6 > a. Since Y(xkY) is a continuous, nondecreasing function of x for fixed Y),~~ it follows from the intermediate value theorem that, for given with a < &< YoA6, there exists a w E (y" such that Y(wkY)= Thus (wty) E Uo and hence there exists an X < (war; such that XE P NowY ( U and so X <. It follows thato ac Y X w so k < k <w <a< <Yk <k-w and hence X { 0,n, in contradiction to C2.

10 4 Thus, a continuous CSF satisfying C2 also satisfies C2'. A similar argument verifies the converse. Proposition 2.3 If y is a CSF which satisfies Cl, C2 and C3, then y({o,a}n) = {0,} for all a E [0,1]. Proof: If a = 0 there is nothing to prove, so suppose that, for some a E (0,1], there exists a vector X E 1 0,a}n such that = y() {O,a}. It is easily seen that 0 < B < a and that X 0 0 or a, and hence we can write 0 for j=l,2,...,k Xi. = 3 I a for j=k+l,...,n for some k with I < k < n-1. Since X E U n L, and both are closed, it follows from Proposition 2.1 that there exist a Z E P and a W E K such that Z < X < W. This ordering will only hold if Z E {0, 0 }n - if W E 81} {0} satisfies Zi. = 0 for j=l,2,...,k and {L} satisfies W i. = 1 for J=k+l,...,n and so A - (ZIWI)X... X 3 (Zn,W ) c A is open. Further, since Z E P and W E K, it follows that n9 n ' * y(x) = 1 for all X E A, in contradiction to C3. Thus y(x) E {0,a} as claimed. * Proposition 2.4 a E (0,1]. If y is a CSF which satisfies Cl, C2 and C3, then P. = Ol for all, ,, "......" _ ".....'.. ", ",. -.

11 * 5 Proof: Suppose that a < 1, otherwise there is nothing to prove, and let./~ XE P a so that y(x) = a. Then X < -xand so y(x) -< y(1x). y-) Since ic 1 1~ -X E {,Ol}n' it follows from Proposition 2.3 that y(-x) 1. We claim that -!X E P 0-1' Suppose, conversely, that it P. Since U 1 is closed, it follows from Proposition 2.1 that there exists a W < -X such that W E P the vector aw E {O,a}n; it is easily seen that y(aw) = a and thus there - Consider exists a vector aw < X such that aw E U. This contradicts the assumption that X E P and hence -X E P as claimed. This holds for all X E P and a G- 1- a so P C P V Similarly, it can be shown that ap C Pa" 3. THE CHARACTERIZATION THEOREMS Theorem 3.1 A CSF y is of the Barlow-Wu type if and only if it satisfies conditions Cl, C2, C3 and C4. *Proof: It is easily verified that the Barlow-Wu CSF satisfies Cl, C2, C3 and C4. To prove the converse, observe that y(x) > a - X > Y E P Smin X > a for some Y E P {i1y 1 =a} 1 a1 "max min X > a YEP {i I Yi=a} 0 ;....j.; , 2 : - i '. < " -. : : , '..' '. - -

12 ~~~- T-* ' -. -, r -b r.,.,% 9 -%.-I *V , ,. 6 o-* max min X. > a by Proposition 2.4 YEaP I tilyi=a} - *-- max min X, > a where Z =Y. ZEP {iizi=l} ~ This holds for all XE A and a E (0,1] and so y(x) = max min X 1. ZEP 1 {iiz.=l} Write P = X ( 1 ) '. X X(N) and let T. = {iecx 11. By the definition 4 of Pit it is clear that each T. is nonempty and that T. G Tk for all j,k=l,2,...,n with j # k. Thus y(x) max min X l<j_<n ietj N where each T. c C. Condition C4 ensures that UT. = j=l 3 proof. C, completing the Theorem 3.2 A CSF y is of the Natvig type if and only if it satisfies C2 and - Cl' y is right-continuous C4' For each i E C and all a E (0,1], there exists an X E A such *" that y(ai,) > a whereas y(oi,x) < a for all 1 < a. Proof: Baxter (3] proves that Natvig CSFs are right-continuous, and it is readily seen that such functions satisfy C2 and C4'. Conversely, from the preceding proof,

13 7 y(x) > a~ max min 7 1 YEP {iliy a) x where Zi is the indicator of {XI>} (0<a<l, XEA). Write P = {X(,l),X ( ' N( a )) } and let T a = {ieci4 'J) a}, j=l,2,..,n(c). Then y(x) > a if and only if 4)(Z ) - iwhere ax --a a (Za)= max min Z. IS<j<N (a) ieta ai 3 We claim that the binary functions {' O<a<l} satisfy the conditions of the " definition of the Natvig CSF. It is clear that 4 is nondecreasing in each argument for all a E (0,1] and that 4)a(Za) is nonincreasing in a for fixed X. To verify left-continuity, it is sufficient to consider the point at which the function decreases. Thus, suppose that y(x) = a (O<a<1); then there exists an X' < X such that X' E P. Clearly, y(x') = a and hence ' ) c(z'c i= 1 whereas, if a > a, y(x') < a and so 4 " (Z') = 0. Thus 4)(Z) is left-continuous as claimed. Lastly, observe that, by C4', for each i E C and all a E (0,1], there exists an X E A such that (iiz) = 1 whereas a(c 0 iza) = 0 and so each 4) is coherent. This completes the proof. El t- I, S 4: 'S

14 8 4. SOME REMARKS ON THE BORGES-RODRICUES CHARACTERIZATION Let S = {0,i.. M}, M > 1. A nondecreasing mapping (P: Sn 1+ S with '(O) = 0 and D(M) = N is said to be a multistate structure function (MSF). It is weakly coherent if max [((Mi, ) - v(oix)] > 1 for each i E C. If '(P) max min X. (XESn) l<j<r iep. i I where PI... P are the r minimal path sets of a binary coherent structure function, then 4 is said to be a Barlow-Wu MSF [1]. If 4)(X) > j if and only p if p.(yj) = 1 (XES n, j=l,2,...,m) where {M...,qM} is a collection of binary coherent structure functions such that.(y.) is nonincreasing in j for fixed X, and where Y.. is the indicator of {X.>j}, then (P is said 31 to be a Natvig MSF 16]. Borges and Rodrigues [5] present axiomatic characterizations of the Barlow-Wu and Natvig MSFs in terms of the following conditions: B1 For every X E S n with 4'(X) > k > 1, there exists a Y E {O,k}n such that Y < X and O(Y) > k B2 B3 4,(O,M} n ) = (O,M} 4' is weakly coherent. Borges and Rodrigues [5] claim (1) 4) is a Barlow-Wu MSF if and only if it satisfies Bl, B2 and B3 (2) 4 is a Natvig MSF if and only if it satisfies B1 and B3. Both claims are false as the following examples attest. '" '"-"',' %*,"' "'""~~... "%' "'". ".. '-.. % *. " " i - " '.

15 I... 9 Example 4.1 Consider the MSF [i: {0,1,2}2,, {0,1,2} defined as follows: S(0,0) = 0 )i(0, ) = 0 D (0,2) = 2 D (1,0) = 0 DI(l,1) = 1 =1(1,2) 2 (D (2,0) = 2 (i(2,1) = 2 ( (2,2) = This satisfies Bi, B2 and B3 and yet is clearly not of the Barlow-Wu type since the only Barlow-Wu MSFs of size two are X 1 AX 2 and X 1 vx 2. Notice in particular that 4i provides a counter-example to Lemma 4 of [5]. Example 4.2 Let ) =(YiIY2 11 and 2(Y21'Y22= Y21AY22 and define the MSF " 2: {0,1,212 b- {0,1,2} as the function which satisfies D2(XlX 2 > j if and-'only if (YjIYj2) = I where Yji is the indicator of {Xi>j} (i,j,=l,2). This is clearly not a Natvig MSF since the binary function is not coherent, but it is easily verified that T2 satisfies BI and B3.! ' "II 7 F a 'I k X 'i-,..*..

16 REFERENCES [11] Barlow, R. E. and Wu, A. S. (1978). "Coherent Systems with Multi- Components", Math. Operat. Res., 3, [21 Baxter, L. A. (1984). "Continuum Structures I", J. Appl. Prob., (to appear). [3] Baxter, L. A. (1984). "Continuum Structures II", submitted for pt [4] Block, H. W. and Savits, T. H. (1984). "Continuous Multistate Structure Functions", Operat. Res., 32, [5 Borges, W. de S. and Rodrigues, F. W. (1983). "An Axiomatic Characterization of Multistate Coherent Structures", Math. Operat. Res., 8, [6] Natvig, B. (1982). "Two Suggestions of How to Define a Multistatn Coherent System", Adv. Appl. Prob., 14, r I" I..::. -:... :? :-.:: :2 ===== == = ==== == === ====== - :::.-: :.:., :: : :.:.:" :

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