Optimization with Max-Min Fuzzy Relational Equations Shu-Cherng - PowerPoint PPT Presentation

Optimization with Max-Min Fuzzy Relational Equations Shu-Cherng Fang Industrial Engineering and Operations Research North Carolina State University Raleigh, NC 27695-7906, U.S.A www.ie.ncsu.edu/fangroup October 31, 2008 ISORA’08 at Lijiang, China Co-author: Pingke Li NC STATE UNIVERSITY

Problem Facing • Problem(P) Minimize f ( x ) s.t. A ∘ x = b x ∈ [0,1] n where f : R n → R is a function, [ ] = ∈ mn A ( a ) 0,1 , × ij m n [ ] = ∈ m b ( ) b 0,1 , × i m 1 “ ∘ ” is a matrix operation replacing “product” by “minimum” and “addition” by “maximum”, i.e., = = K max min( a , x ) b , for i 1, , m . ij j i ≤ ≤ 1 j n NC STATE UNIVERSITY 2

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

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

Triangular Norms [ Schweizer B. and Sklar A. (1961), “Associative functions and statistical triangle inequalities”, Mathematical Debrecen 8, 169-186.] t-norm: × → t :[0,1] [0,1] [0,1] such that = 1 ( t x,y ) t y,x ( ) ) (commutative) = 2 ( ( , )) ( ( , ), ) t x,t y z t t x y z ) (associative) ≤ ≤ 3 t x,y ( ) t x z ( , ), y z ) if (monotonically nondecreasing) = = 4 and dition) . ( 0) t x, 0 ( 1) t x, x ) (boundary con s-norm (t co-norm): × → : s [ 0 , 1 ] [ 0 , 1 ] [ 0 , 1 ] such that = − − − ∀ ∈ s ( x,y ) 1 t ( 1 x, 1 y ) x , y [ 0 , 1 ] NC STATE UNIVERSITY 5

Triangular Norms μ μ μ μ = ⎧ min{ ( ), x ( )} if max{ x ( ), x ( )} x 1 μ μ = ⎨ % % % % A B A B t ( ( ), x ( )) x % % w B A ⎩ 0, otherwise (drastic product) μ μ μ μ = ⎧ max{ ( ), x ( )} if min{ x ( ), x ( )} x 0 μ μ = % % % % ⎨ A B A B s ( ( ), x ( )) x % % w A B ⎩ 1, otherwise (drastic sum) μ μ = μ + μ − t ( ( ), x ( )) x max{0, ( ) x ( ) 1} x bounded difference % % % % 1 B B A A μ μ = μ + μ ( ( ), ( )) min{1, ( ) ( )} bounded sum s x x x x % % % % 1 A B A B μ ⋅ μ ( ) x ( ) x μ μ = % % A B t ( ( ), x ( )) x Einst ein product % % − μ + μ − μ ⋅ μ 1.5 B A 2 [ ( ) ( ) ( ) ( )] x x x x % % % % A B A B μ + μ ( ) x ( ) x μ μ = + % % A B s ( ( ), x ( )) x Einstein sum % % μ ⋅ μ 1.5 B A 1 ( ) x ( ) x % % A B NC STATE UNIVERSITY 6

Triangular Norms μ μ = μ ⋅ μ t ( ( ), x ( )) x ( ) x ( ) x algebraic product % % % % 2 A B A B μ μ = μ + μ − μ ⋅ μ s ( ( ), x ( )) x ( ) x ( ) x ( ) x ( ) x algebraic sum % % % % % % 2 B B B A A A μ ⋅ μ ( ) x ( ) x μ μ = % % A B t ( ( ), x ( )) x Hamacher product % % μ + μ − μ ⋅ μ 2.5 A B ( ) x ( ) x ( ) x ( ) x % % % % B B A A μ + μ − μ ⋅ μ ( ) x ( ) x 2 ( ) x ( ) x μ μ = % % % % A B A B s ( ( ), x ( ) x ) Hamacher sum % % − μ ⋅ μ 2.5 A B 1 ( ) ( ) x x % % B A μ μ = μ μ t ( ( ), x ( )) x min{ ( ), x ( )} x minimum % % % % 3 A B A B μ μ = μ μ s ( ( ), x ( )) x max{ ( ), x ( )} x maximum % % % % 3 B B A A ≤ ≤ ≤ ≤ = ≤ = ≤ ≤ ≤ L L L L L L t t t t min s max s s s w 1 2 3 3 2 1 w NC STATE UNIVERSITY 7

Fuzzy Relational Equations Given × = ∈ m n A ( a ) , [0,1] ij = ∈ m K b ( , , ) , b b [0,1] 1 m find = ∈ n K ( , , ) x x x [0,1] such that 1 n = o ( A x b ) max-t-norm composition = ∀ max (a , t x ) b , i . ij j i ≤ ≤ 1 j n = o ( A x b ) min-s-norm composition = ∀ min (a , s x ) b , i . ij j i ≤ ≤ 1 j n ∑ The solution set is denoted by ( , ). A b NC STATE UNIVERSITY 8

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

Solution Set of Max-t Equations ∈Σ ˆ x ( , ) A b : s a maximum solution 1. Definition i ≤ ∀ ∈Σ ˆ, ( , ). x x x A b if ∈Σ x ( , ) A b : s a minimum solution 2. i Definition ≥ ∀ ∈Σ x x , x ( , ). A b if ∈Σ ˆ x ( , ) A b : s a maximal solution 3. Definition i ≥ = ∀ ∈Σ ˆ ˆ x x x x , x ( , ). A b if implies ∈Σ ( , ) A b : is a minimal solution 4. x Definition ≤ = ∀ ∈Σ x x x x , x ( , ). A b if implies NC STATE UNIVERSITY 10

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)] ˆ x . . . x 1 x 3 x 2 [Root System] NC STATE UNIVERSITY 11

Characteristics of Solution Sets • Existence [di Nola / Sessa / Pedrycz / Sanchez (1989)] Σ ≠ φ ( , ) A b if and only if Theorem : For a continuous t - norm, = K ˆ ˆ ˆ it has a maximum solution x ( x , , x ) 1 j = ϕ ˆ with x min ( a ) b where j ij i ≤ ≤ 1 i m { } ϕ ≡ ∈ ≤ . a b sup u [0,1] ( t a,u) b NC STATE UNIVERSITY 12

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, 103-113.] • Complexity [Wang / Sessa/ di Nola/ Pedrycz (1984), “How many lower solutions does a fuzzy relation have?,” BUSEFAL 18, 67-74.] upper bound = n m NC STATE UNIVERSITY 13

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. ˆ ˆ ˆ x x x 1 2 3 x [Crown System] NC STATE UNIVERSITY 14

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

Optimization with Fuzzy Relation Equations � f ( x ) =c T x linear function [Fang / Li (1999), “Solving fuzzy relation equations with a linear objective function, Fuzzy Sets and Systems 103, 107-113.] ≤ ˆ 0 , c j x : If for all then is an Lemma1 j optimal solution. ≥ c 0 j , : If for all then one of the Lemma2 j minimal solutions is an optimal solution. NC STATE UNIVERSITY 16

Optimization with Fuzzy Relation Equations Theorem : Let ≥ ⎧ ∗ ⎧ ≥ ⎪ ⎪ c if c 0 x if c 0 j j = = j j ⎨ ⎨ c ' and * x < < j ⎪ ⎪ if c ˆ 0 0 , ⎩ x if c 0 ⎩ j j j = T where * solves the problem with ( ) x f x ( c' ) x , then * is an optimal solution. x 0-1 integer programming with a branch-and-bound solution technique. NC STATE UNIVERSITY 17

Optimization with Fuzzy Relational Equations • Extensions 1. Objective 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. NC STATE UNIVERSITY 18

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

Major 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”: ∑ ⎡ ⎤ − ˆ Minimize f ( x ) f (0) u ⎣ ⎦ j j j j j ≥ m (SCP) s. t. Qu e { } ∈ n u 0,1 . NC STATE UNIVERSITY 20