Tropical Optimization Framework for Analytical Hierarchy Process Nikolai Krivulin 1 ı Sergeev 2 Serge˘ 1 Faculty of Mathematics and Mechanics Saint Petersburg State University, Russia 2 School of Mathematics University of Birmingham, UK LMS Workshop on Tropical Mathematics & its Applications University of Birmingham, November 15, 2017
Outline Introduction Tropical Optimization Idempotent Algebra Definitions and Notation Tropical Optimization Problems Solution Examples Analytical Hierarchy Process Traditional Approach Minimax approximation based AHP Illustrative Example Selecting Plan for Vacation Concluding Remarks N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 2 / 31
Introduction Tropical Optimization Introduction: Tropical Optimization ◮ Tropical (idempotent) mathematics focuses on the theory and applications of semirings with idempotent addition ◮ The tropical optimization problems are formulated and solved within the framework of tropical mathematics ◮ Many problems have objective functions defined on vectors over idempotent semifields (semirings with multiplicative inverses) ◮ The problems find applications in many areas to provide new efficient solutions to various old and novel problems in ◮ project scheduling, ◮ location analysis, ◮ transportation networks, ◮ decision making, ◮ discrete event systems N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 3 / 31
Idempotent Algebra Definitions and Notation Idempotent Algebra: Definitions and Notation Idempotent Semifield ◮ Idempotent semifield: the algebraic system � X , 0 , 1 , ⊕ , ⊗� ◮ The binary operations ⊕ and ⊗ are associative and commutative ◮ The carrier set X has neutral elements, zero 0 and identity 1 ◮ Multiplication ⊗ is distributive over addition ◮ Addition ⊕ is idempotent: x ⊕ x = x for all x ∈ X ◮ Multiplication ⊗ is invertible: for each nonzero x ∈ X , there exists an inverse x − 1 ∈ X such that x ⊗ x − 1 = 1 ◮ Algebraic completeness: the equation x p = a is solvable for any a ∈ X and integer p (there exist powers with rational exponents) ◮ Notational convention: the multiplication sign ⊗ will be omitted N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 4 / 31
Idempotent Algebra Definitions and Notation Semifield R max , × (Max-Algebra) ◮ Definition: R max , × = � R + , 0 , 1 , max , ×� with R + = { x ∈ R | x ≥ 0 } ◮ Carrier set: X = R + ; zero and identity: 0 = 0 , 1 = 1 ◮ Binary operations: ⊕ = max and ⊗ = × ◮ Idempotent addition: x ⊕ x = max( x, x ) = x for all x ∈ R + ◮ Multiplicative inverse: for each x ∈ R + \ { 0 } , there exists x − 1 ◮ Power notation: x y is routinely defined for each x, y ∈ R + ◮ Further examples of real idempotent semifields: R max , + = � R ∪ {−∞} , −∞ , 0 , max , + � , R min , + = � R ∪ { + ∞} , + ∞ , 0 , min , + � , R min , × = � R + ∪ { + ∞} , + ∞ , 1 , min , ×� N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 5 / 31
Idempotent Algebra Definitions and Notation Vector and Matrix Algebra Over R max , × ◮ The scalar idempotent semifield R max , × is routinely extended to + and of matrices in R m × n idempotent systems of vectors in R n + ◮ The matrix and vector operations follow the standard entry-wise formulas with the addition ⊕ = max and the multiplication ⊗ = × ◮ For any vectors a = ( a i ) and b = ( b i ) in R n + , and a scalar x ∈ R + , the vector operations follow the conventional rules { a ⊕ b } i = a i ⊕ b i , { x a } i = xa i ◮ For any matrices A = ( a ij ) ∈ R m × n , B = ( b ij ) ∈ R m × n and + + C = ( c ij ) ∈ R n × l , and x ∈ R + , the matrix operations are given by + n � { A ⊕ B } ij = a ij ⊕ b ij , { AC } ij = a ik c kj , { x A } ij = xa ij k =1 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 6 / 31
Idempotent Algebra Definitions and Notation Vector and Matrix Algebra Over R max , × ◮ All vectors are column vectors, unless otherwise specified ◮ The zero vector and vector of ones: 0 = (0 , . . . , 0) T , 1 = (1 , . . . , 1) T ◮ Multiplicative conjugate transposition of a nonzero column vector x = ( x i ) is the row vector x − = ( x − i ) , where x − i = x − 1 if x i � = 0 , i and x − i = 0 otherwise ◮ The zero matrix and identity matrix: 0 = (0) , I = diag(1 , . . . , 1) ◮ Multiplicative conjugate transposition of a nonzero matrix A = ( a ij ) is the matrix A − = ( a − ij = a − 1 ij ) , where a − if a ji � = 0 , ji and a − ij = 0 otherwise ◮ Integer powers of square matrices: A 0 = I , A p = A p − 1 A = AA p − 1 , p ≥ 1 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 7 / 31
Idempotent Algebra Definitions and Notation Square Matrices ◮ Trace: the trace of a matrix A = ( a ij ) ∈ R n × n is given by + tr A = a 11 ⊕ · · · ⊕ a nn ◮ Eigenvalue: a scalar λ such that there is a vector x � = 0 to satisfy Ax = λ x ◮ Spectral radius: the maximum eigenvalue given by ρ = tr A ⊕ · · · ⊕ tr 1 /m ( A m ) ◮ Asterate: the asterate operator (the Kleene star) is given by A ∗ = I ⊕ A ⊕ · · · ⊕ A n − 1 , ρ ≤ 1 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 8 / 31
Tropical Optimization Problems Solution Examples Tropical Optimization Problems: Solution Examples Problem with Pseudo-Quadratic Objective Given a matrix A ∈ R n × n , find positive vectors x ∈ R n + that solve the + problem x − Ax min x > 0 Theorem Let A be a matrix with tropical spectral radius λ > 0 , and denote B = ( λ − 1 A ) ∗ . Then, ◮ the minimum of x − Ax is equal to λ ; ◮ all positive solutions are given by x = Bu , u > 0 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 9 / 31
Tropical Optimization Problems Solution Examples Maximization Problem with Hilbert (Range, Span) Seminorm Given a matrix B ∈ R n × m , find positive vectors u ∈ R l + that solve the + problems 1 T Bu ( Bu ) − 1 1 T Bu ( Bu ) − 1 max min u > 0 u > 0 Lemma Let B be a positive matrix, and B lk be the matrix derived from B = ( b k ) m k =1 by fixing the entry b lk and replacing the others by 0 . ◮ The maximum of 1 T Bu ( Bu ) − 1 is equal to ∆ = 1 T BB − 1 ◮ All positive solutions are given by u = ( I ⊕ B − lk B ) v , v > 0 , where the indices k and l satisfy the condition 1 T b k b − 1 lk = ∆ N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 10 / 31
Tropical Optimization Problems Solution Examples Minimization Problem with Hilbert (Range, Span) Seminorm Lemma Let B be a matrix without zero rows and columns. ◮ The minimum of 1 T Bu ( Bu ) − 1 is equal to ∆ = ( B ( 1 T B ) − ) − 1 . ◮ Denote by � B be the sparsified matrix with entries: � if b ij < ∆ − 1 1 T b j ; 0 , � b ij = b ij , otherwise . Let B be the set of matrices obtained from � B by fixing one nonzero entry in each row and setting the others to 0 . Then, all positive solutions are given by 1 11 T B ) v , u = ( I ⊕ ∆ − 1 B − v > 0 , B 1 ∈ B N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 11 / 31
Analytical Hierarchy Process Traditional Approach Analytical Hierarchy Process: Traditional Approach Pairwise Comparison ◮ Given m criteria and n choices, the problem is to find priorities of choices from pairwise comparisons of criteria and of choices ◮ Outcome of comparison is given by a matrix A = ( a ij ) , where a ij shows the relative priority of choice i over j ◮ Note that a ij = 1 /a ji > 0 ◮ Scale (Saaty, 2005): a ij Meaning i equally important as j 1 3 i moderately more important than j 5 i strongly more important than j 7 i very strongly more important than j 9 i extremely more important than j N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 12 / 31
Analytical Hierarchy Process Traditional Approach Consistency ◮ A pairwise comparison matrix A is consistent if its entries are transitive to satisfy the condition a ij = a ik a kj for all i, j, k ◮ Each consistent matrix A has unit rank and is given by A = xx T , where x is a positive vector that entirely specifies A ◮ If a comparison matrix A is consistent, the vector x represents, up to a positive factor, the individual priorities of choices ◮ Since the comparison matrices are usually inconsistent, a problem arises to approximate these matrices by consistent matrices N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 13 / 31
Analytical Hierarchy Process Traditional Approach Principal Eigenvector Method and Weighted Sum Solution ◮ The traditional AHP uses approximation of pairwise comparison matrices by consistent matrices with the principal eigenvectors ◮ Let A 0 be a matrix of pairwise comparison of criteria, and A k be a matrix of pairwise comparison of choices for criterion k ◮ Let w = ( w k ) m k =1 be the principal eigenvector of A 0 : the vector of priorities (weights) for criteria ◮ Let x k be the principal eigenvector of A k : the vector of priorities of choices with respect to criterion k ◮ The resulting vector x of priorities of choices is calculated as m � x = w k x k k =1 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 14 / 31
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