Optimization with Max-Min Fuzzy Relational Equations

Transcription

1 Optimization with Max-Min Fuzzy Relational Equations Shu-Cherng Fang Industrial Engineering and Operations Research North Carolina State University Raleigh, NC , U.S.A October 31, 2008 ISORA 08 at Liiang, China Co-author: Pingke Li

2 Problem(P) Minimize f(x) Problem Facing s.t. A x = b x [0,1] n 2 where f: R n R is a function, [ ] [ ] mn A= ( a ) 0,1, i m n b= ( b i) m 1 0,1, is a matrix operation replacing product by minimum and addition by maximum, i.e., max min( a, x ) = b, for i = 1, K, m. 1 n m i i

3 Examples 1. Capacity Planning Multimedia server Regional server 1 End Users 1 i n m a i : bandwidth in field from server to user i b i : bandwidth required by user i x : capacity of server Consider max min ( a, x ) = b, for i = 1, K, m. 1 n i i

4 Examples 2. Fuzzy control / diagnosis / knowledge system Input Output... System... i a i : degree of input relating to output i b i : degree of output at state i (symptom) x : degree of input at state (cause) A fuzzy system is usually characterized by max ta (, x) = b, i, or 1 n min sa (, x) = b, i, 1 n i i i i where " t" and " s" are triangular norms. 4

5 Triangular Norms [ Schweizer B. and Sklar A. (1961), Associative functions and statistical triangle inequalities, Mathematical Debrecen 8, ] t-norm: t :[0,1] [0,1] [0,1] such that 1) tx,y ( ) = ty,x ( ) (commutative) 2) tx,tyz ( (, )) = ttxy ( (, ), z) (associative) tx,y ( ) txz (, ), y z 3) if (monotonically nondecreasing) 4) tx, ( 0) = 0 and tx, ( 1) = x (boundary condition). s-norm (t co-norm): :[0,1] s [0,1] [0,1] such that s( x,y) = 1 t(1 x, 1 y) x, y [0,1] 5

6 Triangular Norms min{ μ ( ), ( )} if max{ ( ), ( )} 1 A% x μb% x μa% x μb% x = tw( μ ( ), ( )) A% x μb% x = 0, otherwise (drastic product) max{ μ ( x), ( x)} if min{ ( x), ( x)} 0 A% μb% μ μ A% B% = sw( μ ( x), μ ( x)) = A% B% 1, otherwise (drastic sum) t ( μ ( x), μ ( x)) = max{0, μ ( x) + μ ( x) 1} bounded difference 1 s ( μ ( x), μ ( x)) = min{1, μ ( x) + μ ( x)} bounded sum 1 A% B% A% B% A% B% A% B% μ ( ) ( ) A% x μb% x t1.5( μ ( ), ( )) Einst A% x μb% x = ein product 2 [ μ ( x) + μ ( x) μ ( x) μ ( x)] A% B% A% B% μ ( ) ( ) A% x + μb% x s1.5( μ ( x), ( x)) Einstein sum A% μb% = 1 + μ ( x) μ ( x) A% B% 6

7 Triangular Norms t ( μ ( x), μ ( x)) = μ ( x) μ ( x) algebraic product 2 s ( μ ( x), μ ( x)) = μ ( x) + μ ( x) μ ( x) μ ( x) algebraic sum 2 A% B% A% B% A% B% A% B% A% B% μ ( ) ( ) A% x μb% x t2.5( μ ( ), ( )) Hamacher product A% x μ x B% = μ ( x) + μ ( x) μ ( x) μ ( x) A% B% A% B% μ ( ) ( ) 2 ( ) ( ) A% x + μb% x μa% x μb% x s2.5( μ ( x), μ ( x) A% B% ) = Hamacher sum 1 μ ( x) μ ( x) A% B% t ( μ ( x), μ ( x)) = min{ μ ( x), μ ( x)} minimum 3 s ( μ ( x), μ ( x)) = max{ μ ( x), μ ( x)} maximum 3 A% B% A% B% A% B% A% B% t L t L t L t = min s = maxl s L s L s w w 7

8 Fuzzy Relational Equations 8 Given find m n A ( ) [0,1], m b (,, ) [0,1], x= ( x, K, x ) 1 1 n = a 1 i = b K b n [0,1] such that ( max-t-norm composition ) max t(a, x ) = b, i. 1 n m i i n Ao x ( min-s-norm composition Ao x= b) min s(a, x ) = b, i. i i The solution set is denoted by ( A, b). = b

9 Difficulties in Solving Problem (P) 1. Algebraically, neither maximum nor minimum operations has an inverse operation x+ 0.3 = 0.5 x= = max 0.3, min 0.2, = 0.5 x=? ( x) ( ) 2. Geometrically, the solution set (A, b) is a combinatorially generated non-convex set. 9

10 Solution Set of Max-t Equations xˆ Σ( A, b) x xˆ, x Σ( A, b). x Σ( A, b) x x, x Σ( A, b). xˆ Σ( A, b) x xˆ implies x= xˆ, x Σ( A, b). x Σ( Ab, ) x x x= x, x Σ( A, b). 1. Definition : i s a maximum solution if 2. Definition : i s a minimum solution if 3. Definition : i s a maximal solution 4. if Definition if implies : is a minimal solution 10

11 Solution Set of Max-t Equations Theorem: For a continuous t-norm, if Σ(A, b) is nonempty, then Σ(A, b) can be completely determined by one maximum and a finite number of minimal solutions. [Czogala / Drewhiak / Pedrycz (1982), Higashi / Klir (1984), di Nola (1985)] xˆ x 1 x 2... x 3 [Root System] 11

12 Characteristics of Solution Sets Existence [di Nola / Sessa / Pedrycz / Sanchez (1989)] Theorem : For a continuous t - norm, it has a maximum solution with xˆ = min ( a ϕ b ) i i 1 i m xˆ = ( xˆ, K, xˆ ) where { } a ϕ b sup u [0,1] t( a,u) b 1 Σ( Ab, ) φ if and only if. 12

13 Characteristics of Solution Sets Uniqueness [Sessa S. (1989), Finite fuzzy relation equations with a unique solution in complete Brouwerian lattices, Fuzzy Sets and Systems 29, ] Complexity [Wang / Sessa/ di Nola/ Pedrycz (1984), How many lower solutions does a fuzzy relation have?, BUSEFAL 18, ] upper bound = n m 13

14 Characteristics of Solution Sets Theorem: For a continuous s-norm, if Σ(A, b) is nonempty, then Σ(A, b) is completely determined by one minimum and a finite number of maximal solutions. ˆx 1 ˆx 2 ˆx 3 x [Crown System] 14

15 Problem(P) Minimize f(x) Problem Facing s.t. A x = b x [0,1] m A nonconvex optimization problem over a region defined by a combinatorial number of vertices. 15

16 Optimization with Fuzzy Relation Equations f (x)=c T x linear function [Fang / Li (1999), Solving fuzzy relation equations with a linear obective function, Fuzzy Sets and Systems 103, ] Lemma1 c 0, xˆ : If for all then is an optimal solution. Lemma2 c 0, : If for all then one of the minimal solutions is an optimal solution. 16

17 c' Optimization with Fuzzy Relation Equations Theorem : Let c if c 0 x if c 0 = and x* = 0 if c < 0 xˆ if c < 0, where x * solves the problem with f ( x) = ( c' ) then x * is an optimal solution. T x, 0-1 integer programming with a branch-and-bound solution technique. 17

18 Optimization with Fuzzy Relational Equations Extensions 1. Obective function f(x) - linear fractional - geometric - general nonlinear - vector-valued - max-t or min-s operated 2. Constraints - interval-valued - max-t or min-s operated 3. Latticized optimization on the unit interval. 18

19 Maor Results Theorem 1: Let A x = b be a consistent system of max-min (max-t) equations with a maximum solution ˆx. The Problem (P) can be reduced to Minimize f( x) s. t. Qu e (MIP) Gu e where Q is m-by-r, G is n-by- r, V is n-by- r, m n e, e are vectors of all ones, and r is an integer (up to m n). m n Vu x xˆ u { 0,1} r 19

20 Maor Results Theorem 2: As in Theorem 1, if f(x) is linear, fractional linear, or monotone in each variable, then the Problem (P) can be further reduced to a 0-1 integer programming problem. In particular, when the t-norm is Archimedean and f(x) is separable and monotone in each variable, then Problem (P) is equivalent to a set covering problem : Minimize f ( xˆ ) f (0) u (SCP) s. t. Qu e u m { } n 0,1. 20

21 Maor Results When the t-norm is non-archimedean, then Problem (P) is equivalent to a constrained set covering problem. Corollary: Problem (P) is in general NP-hard. Theorem 3: In case with f ( ) being continuous and monotone for each, then Problem (P) can be solved in polynomial time. f ( x) = max f ( x ) N 21

22 Challenges Remain Efficient solution procedures to generate (A, b). Efficient algorithms for optimization problems with relational equation constraints. Efficient algorithms to generate an approximate solution of (A, b), i.e., Minimize dist Ao x, b ( ) [ ] s. t. x 0,1. n 22

23 References 1. Li, P., Fang, S.-C., On the resolution and optimization of a system of fuzzy relational equations with sup-t composition, Fuzzy Optimization and Decision Making, 7 (2008) Li, P., Fang, S.-C., Minimizing a linear fractional function subect to a system of sup-t equation with a continuous Archimedean triangular norm, to appear in Journal of Systems Science and Complexity. 3. Li, P., Fang, S.-C., A survey on fuzzy relational equations, Part I: Classification and solvability, to appear in Fuzzy Optimization and Decision Making. 4. Li, P., Fang, S.-C., Latticized linear optimization on the unit interval, submitted to IEEE Transactions on Fuzzy System. 23

24 Thank you! Questions? < 24