Tropical Optimization Framework for Analytical Hierarchy Process

Transcription

1 Tropical Optimization Framework for Analytical Hierarchy Process Nikolai Krivulin 1 Sergeĭ Sergeev 2 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

2 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

3 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

4 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

5 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

6 Idempotent Algebra Definitions and Notation Vector and Matrix Algebra Over R max, The scalar idempotent semifield R max, is routinely extended to idempotent systems of vectors in R n + and of matrices in R m 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, {xa} 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, k=1 {xa} ij = xa ij N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 6 / 31

7 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 i if x i 0, 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 ), where a ij = a 1 ji if a ji 0, 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

8 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 = λx 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

9 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 Rn + 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

10 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 max u>0 1 T Bu(Bu) 1 min u>0 1 T Bu(Bu) 1 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 lkb)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

11 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: { 0, if b ij < 1 1 T b j ; bij = 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 u = (I 1 B 1 11T B)v, v > 0, B 1 B N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 11 / 31

12 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 1 i equally important as j 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

13 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

14 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 x = m w k x k k=1 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 14 / 31

15 Analytical Hierarchy Process Minimax approximation based AHP Minimax approximation based AHP Log-Chebyshev Approximation of Comparison Matrices Consider the problem to approximate a pairwise comparison matrix A = (a ij ) by a consistent matrix X = (x ij ), where a ij = 1/a ji, x ij = x i /x j The log-chebyshev distance between A and X is defined as ( max log a ij log x ij = log max max aij, x ) ij 1 i,j n 1 i,j n x ij a ij Minimizing the log-chebyshev distance is equivalent to minimizing ( max max aij, x ) ( ij = max 1 i,j n x ij a max aij x j, a ) jix i a ij x j = max ij 1 i,j n x i x j 1 i,j n x i N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 15 / 31

16 Analytical Hierarchy Process Minimax approximation based AHP Approximation as Tropical Optimization Problem Tropical representation of the objective function in terms of R max, a ij x j max = x 1 1 i,j n i a ij x j = x Ax x i 1 i,j n In the framework of the idempotent semifield R max,, the minimax approximation problem takes the form min x>0 x Ax Theorem Let A be a pairwise comparison matrix with tropical spectral radius λ, and B = (λ 1 A). Then, all priority vectors are given by x = Bu, u > 0 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 16 / 31

17 Analytical Hierarchy Process Minimax approximation based AHP Weighted Approximation Under Several Criteria Simultaneous minimax approximation of the matrices A k = (a (k) ij ) with weights w k > 0 by a consistent matrix involves minimizing max 1 k m w k ( max 1 i,j n (a(k) ij x j)/x i ) = max 1 i,j n max (w ka (k) ij )x j/x i. 1 k m In terms of R max,, the approximation problem takes the form min x>0 x (w 1 A 1 w m A m )x Theorem Let A 1,..., A m be comparison matrices, w 1,..., w m be weights, C = w 1 A 1 w m A m be a matrix with tropical spectral radius µ, and B = (µ 1 C). Then, all priority vectors are given by x = Bu, u > 0 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 17 / 31

18 Analytical Hierarchy Process Minimax approximation based AHP Most and Least Differentiating Priority Vectors The priority vectors x = Bu obtained by the minimax approximation may be not unique up to a positive factor Further analysis is then needed to reduce to a few representative solutions, such as some best and worst priority vectors One can take two vectors that most and least differentiate between the choices with the highest and lowest priorities The most and least differentiating priority vectors are obtained by maximizing and minimizing the contrast ratio max x i/ min x i = max x i max 1 i n 1 i n 1 i n 1 i n x 1 i In terms of the semifield R max,, the contrast ratio is written as 1 T xx 1 = 1 T Bu(Bu) 1 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 18 / 31

19 Analytical Hierarchy Process Minimax approximation based AHP Most Differentiating Priority Vector In terms of the semifield R max,, the problem to maximize the contrast ratio is written as max u>0 1 T Bu(Bu) 1 If u is a solution, then x = Bu is the most differentiating vector N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 19 / 31

20 Analytical Hierarchy Process Minimax approximation based AHP Most Differentiating Priority Vector Theorem Let B be a matrix defining a set of priority vectors x = Bu, u > 0, and B lk be the matrix obtained from B = (b j ) 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 The most differentiating priority vectors are given by x = B(I B lkb)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 20 / 31

21 Analytical Hierarchy Process Minimax approximation based AHP Least Differentiating Priority Vector In terms of the semifield R max,, the problem to minimize the contrast ratio is written as min u>0 1 T Bu(Bu) 1 If u is a solution, then x = Bu is the least differentiating vector N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 21 / 31

22 Analytical Hierarchy Process Minimax approximation based AHP Least Differentiating Priority Vector Theorem Let B be a matrix defining a set of priority vectors x = Bu, u > 0. 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: { 0, if b ij < 1 1 T b j ; bij = 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 the least differentiating priority vectors are given by u = (I 1 B 1 11T B)v, v > 0, B 1 B N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 22 / 31

23 Illustrative Example Selecting Plan for Vacation Illustrative Example: Selecting Plan for Vacation Problem: Select a Place to Spend a Week (Saaty, 1977) Five criteria: (1) cost of the trip from Philadelphia, (2) sight-seeing opportunities, (3) entertainment (doing things), (4) way of travel, (5) eating places; with the criteria comparison matrix 1 1/5 1/5 1 1/ /5 1/5 1 A 0 = / /5 1 Four places: (1) short trips from Philadelphia (i.e., New York, Washington, Atlantic City, New Hope, etc.), (2) Quebec, (3) Denver, (4) California N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 23 / 31

24 Illustrative Example Selecting Plan for Vacation Problem (cont.) Pairwise comparison matrices of places with respect to criteria /5 1/6 1/4 A 1 = 1/ /7 1/6 1 3, A 2 = /2 1 6, 1/9 1/7 1/ /4 1/ / /4 1/3 A 3 = 1/ /7 1/ /7, A 4 = 1/4 1 1/ , /3 1/ A 5 = 1/7 1/6 1 1/4 1/4 1/3 4 1 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 24 / 31

25 Illustrative Example Selecting Plan for Vacation Solution: Evaluating Priority Vector (Weights) for Criteria The tropical spectral radius of the comparison matrix A 0 λ = (a 14 a 43 a 32 a 21 ) 1/4 = 5 3/4 The Kleene star of the matrix λ 1 A 0 (λ 1 A 0 ) = I λ 1 A 0 λ 2 A 2 0 λ 3 A 3 0 λ 4 A /4 5 1/2 5 3/4 5 1/2 5 1/ /4 5 1/2 5 1/4 = 5 1/2 5 1/ / /4 5 1/2 5 1/ / / / / /4 The priority (weight) vector for criteria (pseudo-quadratic problem) w = (1, 5 1/4, 5 1/2, 5 3/4, 3 5 3/4 ) N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 25 / 31

26 Illustrative Example Selecting Plan for Vacation Derivation of All Priority Vectors for Places The weighted combination of comparison matrices of places C = A 1 5 1/4 A 2 5 1/2 A 3 5 3/4 A 4 (3 5 3/4 )A 5 5 3/ / /2 9 = 5 5/4 5 3/ / / /4 5 3/ / / / /2 5 3/4 The tropical spectral radius of matrix C µ = (c 13 c 31 ) 1/2 = 2 5 5/8 7 1/2 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 26 / 31

27 Illustrative Example Selecting Plan for Vacation Derivation of All Priority Vectors for Places (cont.) The Kleene star of matrix µ 1 C (µ 1 C) = I µ 1 C µ 2 C 2 µ 3 C 3 1 r/4 r/4 3/4 = 3/r 1 3/4 3/r 4/r 1 1 3/r, r = 2 71/2 5 1/ r/4 r/4 1 All solution vectors (pseudo-quadratic problem) 1 r/4 3/4 x = Bu, B = 3/r 1 3/r 4/r 1 3/r, u > 0 1 r/4 1 N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 27 / 31

28 Illustrative Example Selecting Plan for Vacation Evaluation of Most Differentiating Solutions The vectors with maximum differentiation between choices of lowest and highest priorities (maximization of Hilbert seminorm) 1 3/4 3/r 3/r x 1 = 4/r 1 u, x 2 = 3/r 1 Examples of vectors with u = v = 1 v, u, v > 0, r = 2 71/2 5 1/ x 1 (1.00, 0.69, 0.92, 1.00) T, x 2 (0.75, 0.69, 0.69, 1.00) T The priority order of places according to the vectors (4) (1) (3) (2), (4) (1) (3) (2) N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 28 / 31

29 Illustrative Example Selecting Plan for Vacation Evaluation of Least Differentiating Solutions The vectors with minimum differentiation between choices of lowest and highest priorities (minimization of Hilbert seminorm) 1 x 1 = 4/r 4/r u > 0, r = 2 71/2 5 1/ Example of vectors with u = 1 and related priority order x 1 (1.00, 0.92, 0.92, 1.00) T, (4) (1) (3) (2) Combined new orders versus order by (Saaty, 1977) NEW: (4) (1) (3) (2) OLD: (1) (3) (4) (2) N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 29 / 31

30 Concluding Remarks Concluding Remarks We have proposed a new implementation of the AHP method, based on minimax approximation and tropical optimization The new AHP implementation uses log-chebyshev matrix approximation instead of the principal eigenvector method The weights of criteria are incorporated into the evaluation of the priorities of choices rather then used to form the result directly N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 30 / 31

31 Concluding Remarks Concluding Remarks Since the solution obtained is usually non-unique, a technique has been proposed to find two representative priority vectors As such solutions, those vectors are taken which most and least differentiate between choices with the highest and lowest priorities The above problems have been formulated in the framework of tropical mathematics, and solved as tropical optimization problems Exact solutions to the problems have been given in a compact vector form ready for further analysis and practical implementation N. Krivulin and S. Sergeev Tropical Optimization for AHP Tropical Workshop 31 / 31