Reliability analysis of systems and lattice polynomial description

Transcription

1 Reliability analysis of systems and lattice polynomial description Jean-Luc Marichal University of Luxembourg Luxembourg

2 A basic reference Selected references R. E. Barlow and F. Proschan. Statistical theory of reliability and life testing. Holt Rinehart and Wilson, New York, Personal contributions J.-L. Marichal, Weighted lattice polynomials of independent random variables, Discrete Appl. Math. 156: , A. Dukhovny and J.-L. Marichal, System reliability and weighted lattice polynomials, Proba. Eng. Inform. Sc. 22: , J.-L. Marichal and P. Mathonet, Extensions of system signatures to dependent lifetimes, J. Multivar. Anal. 102: , J.-L. Marichal, P. Mathonet, and T. Waldhauser, On signature-based expressions of system reliability, J. Multivar. Anal. 102: , A. Dukhovny and J.-L. Marichal, Reliability of systems with dependent components based on lattice polynomial description, Stoch. Model. 28: , J.-L. Marichal and P. Mathonet, On the extensions of Barlow-Proschan importance index and system signature to dependent lifetimes, J. Multivar. Anal. 115: 48-56, J.-L. Marichal and P. Mathonet, Computing system signatures through reliability functions, Stat. Proba. Lett. 83: , J.-L. Marichal, Subsignatures of systems, J. Multivar. Anal. 124: , 2014.

3 Sketch of the presentation Part I : Semicoherent systems Part II : Reliability analysis Part III : Lattice polynomial language Part IV : Signature and importance indexes Part V : Additional results in the i.i.d. case

4 Part I : Semicoherent systems

5 System Definition. A system is a set of interconnected components Example. Home video system 1. Blu-ray player 2. PlayStation 3 3. LED television 4. Sound amplifier 5. Speaker A 6. Speaker B Assumptions C = {1,..., n} = [n] The system and the components are of the crisply on/off kind The components are nonrepairable

6 Structure function State of a component j C = [n] Boolean variable x j = 1 if component j is functioning 0 if component j is in a failed state State of the system Boolean function φ {0, 1} n {0, 1} φ(x 1,..., x n ) = 1 if the system is functioning 0 if the system is in a failed state This function is called the structure function of the system S = (C, φ)

7 Representations of Boolean functions Boolean function set function φ {0, 1} n {0, 1} φ 2 [n] {0, 1} φ(1 A ) = φ(a) A [n] Polynomial representation of a Boolean function φ(x) = A [n] φ(a) x j j A (1 x j ) j [n] A

8 Representations of Boolean functions Simple form φ(x) = A [n] m(a) x j j A where m(a) = ( 1) A B φ(b) B A φ(a) = m(b) B A (Hammer and Rudeanu 1968)

9 Coherent and semicoherent systems The system is said to be semicoherent if φ is nondecreasing : x x φ(x) φ(x ) φ(0) = 0, φ(1) = 1 The system is said to be coherent if, in addition every component is relevant to φ x {0, 1} n φ(1 j, x) φ(0 j, x) where (1 j, x) = (x 1,..., (j) 1,..., x n ) (0 j, x) = (x 1,..., (j) 0,..., x n )

10 Representations of Boolean functions x 1 x 2 = min(x 1, x 2 ) = x 1 x 2 x 1 x 2 = max(x 1, x 2 ) = 1 (1 x 1 )(1 x 2 ) Since φ is nondecreasing and nonconstant φ(x) = A [n] φ(a)=1 x j j A φ(x) = A [n] φ([n] A)=0 x j j A (Hammer and Rudeanu 1968)

11 Block diagrams A serially connected segment of components is functioning if and only if every single component is functioning A system of parallel components is functioning if and only at least one component is functioning 1 2 3

12 Series structure Block diagrams Parallel structure φ(x) = x 1 x 2 x 3 = 1 3 x i i=1 2 3 φ(x) = 1 (1 x 1 )(1 x 2 )(1 x 3 ) = 3 x i i=1

13 Example. Home video system 1. Blu-ray player 2. PlayStation 3 3. LED television 4. Sound amplifier 5. Speaker A 6. Speaker B Block diagrams φ(x) = (x 1 x 2 ) x 3 x 4 (x 5 x 6 )

14 Block diagrams Example. Bridge structure φ(x) = x 3 φ(1 3, x) + (1 x 3 )φ(0 3, x) φ(1 3, x) = (x 1 x 2 )(x 4 x 5 ) φ(0 3, x) = (x 1 x 4 ) (x 2 x 5 ) Pivotal decomposition of the structure function φ(x) = x j φ(1 j, x) + (1 x j )φ(0 j, x)

15 Example. k-out-of-n structure Block diagrams The system fails upon the kth component failure i.e., the system is functioning if and only if at least n k + 1 of the n components are functioning φ(x) = n 1 if x j n k + 1 j=1 0 otherwise φ(x) = x k n = A =n k+1 j A x j = A =k j A x j

16 Block diagrams Example. 2-out-of-3 structure φ(x) = x 2 3 = A =2 j A x j = x 1 x 2 x 1 x 3 x 2 x

17 Path and cut sets Definition. A subset A C of components is a path set of φ if φ(a) = 1 a cut set of φ if φ(c A) = 0 A path (cut) set is minimal if it does not strictly contain a path (cut) set. Bridge structure Minimal path sets : {1, 4}, {2, 5}, {1, 3, 5}, {2, 3, 4} Minimal cut sets : {1, 2}, {4, 5}, {1, 3, 5}, {2, 3, 4}

18 Path and cut sets If P 1,..., P r denote the minimal path sets φ(x) = r j=1 i P j x i If K 1,..., K s denote the minimal cut sets φ(x) = s j=1 i K j x i Bridge structure φ(x) = (x 1 x 4 ) (x 2 x 5 ) (x 1 x 3 x 5 ) (x 2 x 3 x 4 ) = (x 1 x 2 )(x 4 x 5 )(x 1 x 3 x 5 )(x 2 x 3 x 4 )

19 Correspondence Reliability/Game Theory Reliability Component Semicoherent structure Structure function Irrelevant component Path set Cut set Minimal path set Minimal cut set Series structure Paralell structure Module Modular set Game Theory Player Simple game Characteristic function Null player Winning coalition Blocking coalition Minimal winning coalition Minimal blocking coalition Unanimity game Decisive game Committee Committee set (Ramamurthy 1990)

20 State variable Random variable x j X j (t) X j (t) = 1 if j is functioning at time t 0 if j is in a failed state at time t X j (t) Failure 0 T j t T j = random lifetime of component j C X j (t) = Ind(T j > t) = random state of j at time t 0

21 System lifetime and component lifetimes T S = system lifetime X S (t) = Ind(T S > t) = random state of the system at time t 0 X S (t) = φ(x 1 (t),..., X n (t)) t 0 Expression of T S in terms of T 1,..., T n?

22 System lifetime and component lifetimes Series structure φ(x) = x 1 x 2 x 3 T S = T 1 T 2 T 3 Parallel structure φ(x) = x 1 x 2 x 3 T S = T 1 T 2 T 3

23 System lifetime and component lifetimes General structure (Dukhovny & M. 2012) φ(x) = A [n] φ(a)=1 x j T S = j A A [n] φ(a)=1 T j j A Life function p φ (t 1,..., t n ) = A [n] φ(a)=1 t j t j 0 j A lattice polynomial (lattice term) T S = p φ (T 1,..., T n )

24 System How to describe T 1,..., T n? Cumulative distribution function (c.d.f.) of the component lifetimes F (t 1,..., t n ) = Pr(T 1 t 1,..., T n t n ) t 1,..., t n 0 S = (C, φ, F ) Classical assumptions F absolutely continuous + i.i.d. lifetimes F absolutely continuous + exchangeable lifetimes F has no ties Pr(T i = T j ) = 0 i j

25 Part II : Reliability analysis

26 Reliability analysis Reliability function of component j C R j (t) = Pr(T j > t) t 0 = probability that component j does not fail in the interval [0, t] X j (t) = Ind(T j > t) R j (t) = Pr(X j (t) = 1) = E[X j (t)] System reliability function R S (t) = Pr(T S > t) t 0 = probability that the system does not fail in the interval [0, t] R S (t) = Pr(X S (t) = 1) = E[X S (t)]

27 We have Reliability analysis R S (t) = E[X S (t)] = E[φ(X 1 (t),..., X n (t))] = A C φ(a) E[ X j (t) j A (1 X j (t))] j C A Pr ( j C X j (t)=1 j A) Theorem (Dukhovny 2007) R S (t) = φ(a) Pr(X(t) = 1 A ) t 0 A C All the needed information is the distribution of X(t) (the knowledge of the joint distribution F of the component lifetimes is not necessary)

28 Reliability analysis When T 1,..., T n are independent, we have Corollary R S (t) = A C = A C φ(a) E[X j (t)] j A φ(a) R j (t) j A If T 1,..., T n are independent, then (1 E[X j (t)]) j C A (1 R j (t)) j C A R S (t) = φ(r 1 (t),..., R n (t)) t 0 Multilinear extension of φ φ [0, 1] n [0, 1] φ(x) = A C φ(a) x j j A (1 x j ) j C A

29 Reliability analysis Simple form of φ φ(x) = A C m(a) x j j A Corollary (Dukhovny and M. 2008) We have R S (t) = m(a) Pr(T j > t j A) t 0 A C In case of independence R S (t) = A C m(a) R j (t) t 0 j A

30 Mean time-to-failure of the system Mean time-to-failure of the system MTTF S = E[T S ] = 0 t dr S (t) MTTF S = 0 R S (t) dt In case of independence MTTF S = A C MTTF S = A C φ(a) m(a) 0 j A 0 j A R j (t) (1 R j (t)) dt j C A R j (t) dt

31 Mean time-to-failure of the system Example. Assume R j (t) = e λ j t, j C MTTF S = A C = A C m(a) m(a) e λ j t dt 0 j A 0 = m(a) 1 A C λ A A e λ At dt (λ A = λ j ) j A Series structure: MTTF S = 1 λ C Parallel structure: MTTF S = A C A ( 1) A 1 1 λ A

32 Part III : Lattice polynomial language

33 Life function p φ (t 1,..., t n ) = A [n] φ(a)=1 t j t j 0 j A lattice polynomial (lattice term) T S = p φ (T 1,..., T n )

34 Advantage of the lattice polynomial language Suppose there is (i) an upper bound on lifetimes of a subset A of components (imposed by the physical properties of the assembly) A T c subset lifetime = T c (ii) a lower bound (imposed by a back-up block with a constant lifetime) A T c subset lifetime = T c

35 Advantage of the lattice polynomial language The lifetime of a general system with upper and/or lower bounds can be described through a lattice polynomial function T S = p(t 1,..., T n ) Example. 1 2 Suppose that the lifetime of component #2 must lie in the time interval [c, d] T S = T 1 median(c, T 2, d) = T 1 (c (T 2 d)) = (c T 1 ) (d T 1 T 2 )

36 Lattice polynomial functions Representations of a l.p. function (Goodstein 1967) p(t 1,..., t n ) = A [n] (α(a) t j ) t 1,..., t n 0 j A α(a) = p(e A ) (e A ) j = if j A 0 otherwise

37 Lattice polynomial functions p(t 1,..., t n ) = A [n] (α(a) t j ) t 1,..., t n 0 j A Theorem (Dukhovny & M. 2008) If T S = p(t 1,..., T n ) then X S (t) = φ t (X 1 (t),..., X n (t)) t 0 where φ t (x) = A [n] Ind(α(A) > t) x j j A (1 x j ) j [n] A This extends the classical formula X S (t) = φ(x 1 (t),..., X n (t)) t 0

38 Lattice polynomial functions Example (cont d) 1 2 T S = (c T 1 ) (d T 1 T 2 ) p(t 1, t 2 ) = (c t 1 ) (d t 1 t 2 ) Then we have X S (t) = (Ind(c > t) X 1 (t)) (Ind(d > t) X 1 (t) X 2 (t))

39 Reliability analysis Exact reliability formulas (Dukhovny & M. 2008) In case of independence R S (t) = φ t (A) Pr(X(t) = 1 A ) A C R S (t) = m t (A) Pr(T j > t j A) A C R S (t) = A C R S (t) = A C φ t (A) R j (t) (1 R j (t)) j A j C A m t (A) R j (t) j A

40 Mean time-to-failure of the system MTTF S = 0 = A C = A C = A C = A C R S (t) dt 0 0 B A B A m t (A) R j (t) dt j A ( ( 1) A B φ t (B)) R j (t) dt B A j A ( 1) A B ( 1) A B 0 0 α(b) Ind(α(B) > t) R j (t) dt j A j A R j (t) dt

41 Mean time-to-failure of the system Example. Assume R j (t) = e λ j t, j C MTTF S = A C = A C B A B A ( 1) A B ( 1) A B 0 0 α(b) α(b) j A e λ j t dt e λ At dt = α( ) + ( 1) A B 1 e λ A α(b) A [n] B A λ A A

42 Part IV : Signature and importance indexes

43 Simple game Let N = {1,..., n} be the set of players Characteristic function of the game = set function v 2 N R which assigns to each coalition S N of players a real number v(s) which represents the worth of S The game is said to be simple if v takes on its values in {0, 1} The set function v can be regarded as a Boolean function v {0, 1} n {0, 1}

44 Power indexes Let v 2 N {0, 1} be a simple game on a set N of n players Let j N be a player Banzhaf power index (Banzhaf 1965) ψ B (v, j) = 1 2 n 1 S N {j} (v(s {j}) v(s)) Shapley power index (Shapley 1953) ψ Sh (v, j) = S N {j} 1 n ( n 1 (v(s {j}) v(s)) ) S

45 Cardinality index Cardinality index (Yager 2002) C k = 1 (n k)( n k ) S =k j N S (v(s {j}) v(s)) (k = 0,..., n 1) C k = 1 ( n k+1 ) S =k+1 v(s) 1 ( n k ) S =k v(s) Interpretation: C k is the average gain that we obtain by adding an arbitrary player to an arbitrary k-player coalition

46 Barlow-Proschan importance index System S = (C, φ, F ) Assume that the components have independent lifetimes Importance index (Barlow-Proschan 1975) I (j) BP = Pr(T S = T j ) j C I BP = (I (1) BP,..., I (n) BP ) j I (j) BP = 1 I (j) BP is an measure of importance of component j

47 In the i.i.d. case: Barlow-Proschan importance index I BP = (I (1) BP,..., I (n) BP ) b = (b 1,..., b n ) b j = A C {j} 1 n ( n 1 (φ(a {j}) φ(a)) ) A b j = ψ Sh (φ, j) b j is independent of F! b defines a structure importance index

48 System signature Assume that F is absolutely continuous and the components have i.i.d. lifetimes Order statistics T 1,..., T n T 1 n T n n System signature (Samaniego 1985) s k = Pr(T S = T k n ) k = 1,..., n s = (s 1,..., s n ) k s k = 1

49 System signature Explicit expression (Boland 2001) s k = 1 ( n n k+1 ) φ(a) A C A =n k+1 1 ( n n k ) φ(a) A C A =n k C k = 1 ( n k+1 ) S =k+1 v(s) 1 ( n k ) S =k v(s) s k = C n k s k is independent of F! s defines the structure signature

50 Barlow-Proschan importance index and system signature Series structure I BP = ( 1 3, 1 3, 1 ) s = (1, 0, 0) 3

51 Barlow-Proschan importance index and system signature Bridge structure I BP = ( 7 30, 7 30, 2 30, 7 30, 7 30 ) s = (0, 1 5, 3 5, 1 5, 0)

52 Barlow-Proschan importance index and system signature Home video system I BP = ( 2 30, 2 30, 11 30, 11 30, 2 30, 2 30 ) s = ( 5 15, 6 15, 4, 0, 0, 0) 15

53 Correspondence Reliability/Game Theory Reliability Component Importance of a component Barlow-Proschan importance index Birnbaum importance index Signature Game Theory Player Power of a player Shapley power index Banzhaf power index Cardinality index

54 Extension of signature to dependent lifetimes General dependent case : we only assume that F has no ties Probability signature (Navarro-Spizzichino-Balakrishnan 2010) p k = Pr(T S = T k n ) k = 1,..., n p = (p 1,..., p n ) k p k = 1 Can we provide an explicit expression for p k in terms of φ and F? S = (C, φ, F )

55 Extension of signature to dependent lifetimes Relative quality function q 2 C [0, 1] q(a) = Pr (T i < T j i A, j A) = Pr ( max i A T i < min j A T j) (M. & Mathonet 2011) q(a) = probability that the best A components (those having the longest lifetimes) are exactly A q(a) measures the overall quality of the components A when compared with the components C A Remark: q is independent of φ (q depends only on C and F )

56 Extension of signature to dependent lifetimes Theorem (M. & Mathonet 2011) p k = q(a) φ(a) A C A =n k+1 q(a) φ(a) A C A =n k extends Boland s formula s k = 1 ( n n k+1 ) φ(a) A C A =n k+1 1 ( n n k ) φ(a) A C A =n k Open problem Find necessary and sufficient conditions under which a set function on C is the relative quality function of a system S = (C, φ, F )

57 Extension of signature to dependent lifetimes Proposition If T 1,..., T n are exchangeable, then q is symmetric q(a) = 1 ( n A ) p k = s k = 1 ( n n k+1 ) φ(a) A C A =n k+1 1 ( n n k ) φ(a) A C A =n k p = s

58 Extension of signature to dependent lifetimes Theorem (M. & Mathonet & Waldhauser 2011) The identity p = s holds for every n-component semicoherent system if and only if q is symmetric

59 Extension of BP index to dependent lifetimes Relative quality function of component j q j 2 C {j} [0, 1] q j (A) = Pr ( max i C A T i = T j < min i A T i) (M. & Mathonet 2013) q j (A) = probability that the components that are better than component j are precisely A.

60 Extension of BP index to dependent lifetimes We have q j (A) = 1 (j C) A C {j} Theorem (M. & Mathonet 2013) I (j) BP = q j (A) (φ(a {j}) φ(a)) A C {j} In the i.i.d. case: I (j) BP = b j = A C {j} 1 n ( n 1 (φ(a {j}) φ(a)) ) A

61 Extension of BP index to dependent lifetimes Proposition If T 1,..., T n are exchangeable, then q j (A) = 1 n ( n 1 A ) I (j) BP = b j = A C {j} 1 n ( n 1 (φ(a {j}) φ(a)) ) A I BP = b

62 Extension of BP index to dependent lifetimes Theorem (M. & Mathonet 2013) The identity I BP = b holds for every n-component semicoherent system if and only if 1 q j (A) = n ( n 1 A )

63 Case of independent lifetimes We now assume that T 1,..., T n are independent lifetimes Every T j has a - a p.d.f. f j - a c.d.f. F j with F j (0) = 0 Theorem q(a) = j A 0 f j (t) F i (t) F i (t) dt (A ) i A i A {j} where F j (t) = 1 F j (t) provides an explicit expression for the signature in the independent case

64 Case of independent lifetimes Example: independent exponential lifetimes F j (t) = 1 e λ j t λ j > 0 Corollary q(a) = ( 1) B λ A (A ) B C A λ A B where λ A = j A λ j

65 Case of independent lifetimes The ratio λ {j} λ C = q(c {j}) is the probability that j is the worst component More generally, λ A = q(c {j}) λ C j A is the probability that the worst component is in A

66 Case of independent lifetimes Theorem q j (A) = 0 f j (t) i A {j} F i (t) F i (t) dt i A provides an explicit expression for Barlow Proschan index in the independent case Corollary For independent exponential lifetimes q j (A) = ( 1) B B C (A {j}) λ {j} λ A B {j}

67 Interpretation in game theory Is there an interpretation in game theory of the formula Pr(T S = T j ) = q j (A) (φ(a {j}) φ(a))? A C {j} Yes : based on the derivation of the Shapley power index from a bargaining procedure (Shapley 1953)

68 Interpretation in game theory The players agree to play the game v in a grand coalition The coalition adds one player at a time until everyone has been admitted The order in which the players are to join is determined by chance, with all arrangements equally probable Each player, on his admission, is promised the amount corresponding to his marginal contribution Let S N {j} be the set of players preceding j payment to j : v(s {j}) v(s) probability of that contingency is total expectation of player j ψ Sh (v, j) = S N {j} 1 n( n 1 S ) 1 n ( n 1 (v(s {j}) v(s)) ) S

69 Interpretation in game theory General case : T j = time at which player j is admitted in the coalition probability that S is the set of players preceding j total expectation of player j p j (S) = Pr ( max i S T i < T j = min i N S T i) S N {j} p j (S) (v(s {j}) v(s)) If the game is monotone and simple Pr(T N = T j ) = S N {j} p j (S) (v(s {j}) v(s)) T N = time at which the forming coalition turns from losing to winning

70 Interpretation in game theory Let S N, S = k, be the set of the first k players (k = 0,..., n 1) probability that this coalition forms is p(s) = Pr ( max i S T i < min i N S T i) average marginal contribution of an additional arbitrary player p(s) v(s) S N S =k+1 If the game is monotone and simple p(s) v(s) S N S =k Pr(T N = T k+1 n ) = p(s) v(s) S N S =k+1 p(s) v(s) S N S =k

71 Subsignature Let M C Subsignature (M. 2014) p (k) M = Pr(T S = T k M ) k = 1,..., M Explicit formula p (k) M = A C M A =k j M A q j (A) (φ(a {j}) φ(a)) + interpretation in game theory

72 Decomposition of reliability Recall that R S (t) = Pr(T S > t) and R k n (t) = Pr(T k n > t) Proposition (Samaniego 1985) If F is absolutely continuous with i.i.d. lifetimes, we have R S (t) = n s k R k n (t) k=1 for every t 0 and every n-component coherent system

73 Decomposition of reliability Theorem (M. & Mathonet & Waldhauser 2011) For any t 0, we have R S (t) = n s k R k n (t) k=1 for every n-component coherent system if and only if the state variables X 1 (t),..., X n (t) are exchangeable Remark. This condition is weaker than exchangeability of the component lifetimes T 1,..., T n

74 Part V : Additional results in the i.i.d. case

75 Manual computation of the Barlow-Proschan index b j = ψ Sh (φ, j) = A C {j} 1 n ( n 1 (φ(a {j}) φ(a)) ) S φ(x) = multilinear extension of φ(x) Theorem (Owen 1972) b j 1 = ψ Sh (φ, j) = ( 0 x j φ)(x,..., x) dx

76 Manual computation of the Barlow-Proschan index Example. Home video system φ(x 1,..., x 6 ) = (x 1 x 2 ) x 3 x 4 (x 5 x 6 ) φ(x 1,..., x 6 ) = x 1 x 3 x 4 x 5 + x 2 x 3 x 4 x 5 + x 1 x 3 x 4 x 6 + x 2 x 3 x 4 x 6 x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 6 x 1 x 3 x 4 x 5 x 6 x 2 x 3 x 4 x 5 x 6 +x 1 x 2 x 3 x 4 x 5 x 6 Example: b 2 =? ( x 2 φ)(x,..., x) = 2x 3 3x 4 + x 5 b 2 1 = (2x 3 3x 4 + x 5 ) dx =

77 Manual computation of the signature How can we efficiently compute the system signature s k = 1 ( n n k+1 ) φ(a) A C A =n k+1 1 ( n n k ) φ(a)? A C A =n k

78 Manual computation of the signature With any n-degree polynomial p R R we associate the reflected polynomial R n p R R defined by (R n p)(x) = x n p( 1 x ) p(x) = a 0 +a 1 x+ +a n x n (R n p)(x) = a n +a n 1 x+ +a 0 x n (M. 2014) Setting p(x) = d dx φ(x,..., x), we have x 0 (Rn 1 p)(t + 1) dt = n ( n k=1 k ) s k x k

79 Manual computation of the signature Example. Home video system φ(x 1,..., x 6 ) = x 1 x 3 x 4 x 5 + x 2 x 3 x 4 x 5 + x 1 x 3 x 4 x 6 + x 2 x 3 x 4 x 6 x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 6 x 1 x 3 x 4 x 5 x 6 x 2 x 3 x 4 x 5 x 6 +x 1 x 2 x 3 x 4 x 5 x 6 φ(x,..., x) = 4x 4 4x 5 + x 6 p(x) = d dx φ(x,..., x) = 16x 3 20x 4 + 6x 5 (R 5 p)(x) = 6 20x + 16x 2 x 0 (R5 p)(t + 1) dt = 2 x + 6 x x 3 = ( 6 1 ) s 1 x + ( 6 2 ) s 2 x ( 6 6 ) s 6 x 6

80 Barlow-Proschan importance index and system signature Home video system s = ( 5 15, 6 15, 4, 0, 0, 0) 15 C = (0, 0, 0, 4 15, 6 15, 5 15 ) I BP = ( 2 30, 2 30, 11 30, 11 30, 2 30, 2 30 )

81 Computation of the signature from the minimal path sets The multilinear extension φ(x) can be obtained from the minimal path sets P 1,..., P r simply by (i) expressing the structure function, e.g., as a coproduct over the minimal path sets (ii) expanding the coproduct φ(x) = r j=1 i P j x i (iii) simplifying the resulting algebraic expression (using x 2 j = x j ) until it becomes multilinear. Then we compute p(x) = d dx φ(x,..., x) and x 0 (Rn 1 p)(t + 1) dt = n ( n k=1 k ) s k x k

82 Open problems One can show that there is a linear bijection between the signature s and the polynomial function φ(x,..., x) Find necessary and sufficient conditions under which an n-tuple a = (a 1,..., a n ) is the signature of a semicoherent system Find necessary and sufficient conditions under which an n-degree polynomial function P(x) is the function φ(x,..., x) of a semicoherent system Enumerate all the possible semicoherent systems having a prescribed φ(x,..., x)

83 Thank you for your attention!