Spanning trees and logarithmic least squares optimality for complete and incomplete pairwise comparison matrices Sándor Bozóki Institute for Computer Science and Control Hungarian Academy of Sciences (MTA SZTAKI); Corvinus University of Budapest Vitaliy Tsyganok Laboratory for Decision Support Systems, The Institute for Information Recording of National Academy of Sciences of Ukraine; Department of System Analysis, State University of Telecommunications May 24, 2017 p. 1/46
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Multi Criteria Decision Making Analytic Hierarchy Process (Saaty, 1977) Criterion tree Pairwise comparison matrix p. 3/46
The aim of multiple criteria decision analysis The aim is to select the overall best one from a finite set of alternatives, with respect to a finite set of attributes (criteria), or, to rank the alternatives, or, to classify the alternatives. p. 4/46
Properties of multiple criteria decision problems criteria contradict each other there is not a single best solution, that is optimal with respect to each criterion subjective factors influence the decision contradictive individual opinions have to be aggregated p. 5/46
Properties of multiple criteria decision problems criteria contradict each other there is not a single best solution, that is optimal with respect to each criterion subjective factors influence the decision contradictive individual opinions have to be aggregated Examples of multi criteria decision problems tenders, public procurements, privatizations evaluation of applications environmental studies ranking, classification p. 6/46
Main tasks in multi criteria decision problems to assign weights of importance to the criteria to evaluate the alternatives to aggregate the evaluations with the weights of criteria sensitivity analysis p. 7/46
Decomposition of the goal: tree of criteria main criterion 1 criterion 1.1 criterion 1.2 criterion 1.3 criterion 1.4 criterion 1.5 main criterion 2 criterion 2.1 criterion 2.2 main criterion 3 criterion 3.1 subcriterion 3.1.1 subcriterion 3.1.2 criterion 3.2 p. 8/46
Estimating weights from pairwise comparisons How many times criterion 1 is more important than criterion 2? A = 1 a 12 a 13... a 1n a 21 1 a 23... a 2n a 31 a 32 1... a 3n,....... a n1 a n2 a n3... 1 is given, where for any i,j = 1,...,n indices a ij > 0, a ij = 1 a ji. The aim is to find the w = (w 1,w 2,...,w n ) R n + weight vector such that ratios w i w j are close enough to a ij s. p. 9/46
Evaluation of the alternatives Alternatives are evaluated directly, or by using a function, or by pairwise comparisons as before. How many times alternative 1 is better than alternative 2 with respect to criterion 1.1? B = 1 b 12 b 13... b 1m b 21 1 b 23... b 2m b 31 b 32 1... b 3m....... b m1 b m2 b m3... 1 p. 10/46
Aggregation of the evaluations total scores are calculated as a weighted sum of the evaluations with respect to leaf nodes of the criteria tree (bottom up); partial sums are informative p. 11/46
Weighting methods Eigenvector Method (Saaty): Aw = λ max w. Logarithmic Least Squares Method (LLSM): min n i=1 n j=1 ( log a ij log w ) 2 i w j n w i = 1, w i > 0, i = 1, 2,...,n. i=1 p. 12/46
incomplete pairwise comparison matrix A = 1 a 12 a 14 a 15 a 16 a 21 1 a 23 a 32 1 a 34 a 41 a 43 1 a 45 a 51 a 54 1 a 61 1 p. 13/46
incomplete pairwise comparison matrix and its graph A = 1 a 12 a 14 a 15 a 16 a 21 1 a 23 a 32 1 a 34 a 41 a 43 1 a 45 a 51 a 54 1 a 61 1 p. 14/46
The Logarithmic Least Squares (LLS) problem min i, j : a ij is known [ log a ij log w i > 0, ( wi w j )] 2 i = 1, 2,...,n. The most common normalizations are n and w 1 = 1. i=1 w i = 1, n i=1 w i = 1 p. 15/46
Theorem (Bozóki, Fülöp, Rónyai, 2010): Let A be an incomplete or complete pairwise comparison matrix such that its associated graph G is connected. Then the optimal solution w = expy of the logarithmic least squares problem is the unique solution of the following system of linear equations: (Ly) i = log a ik for all i = 1, 2,...,n, y 1 = 0 k:e(i,k) E(G) where L denotes the Laplacian matrix of G (l ii is the degree of node i and l ij = 1 if nodes i and j are adjacent). p. 16/46
example 1 a 12 a 14 a 15 a 16 a 21 1 a 23 a 32 1 a 34 a 41 a 43 1 a 45 a 51 a 54 1 a 61 1 4 1 0 1 1 1 1 2 1 0 0 0 0 1 2 1 0 0 1 0 1 3 1 0 1 0 0 1 2 0 1 0 0 0 0 1 y 1 (= 0) y 2 y 3 y 4 y 5 y 6 = log(a 12 a 14 a 15 a 16 ) log(a 21 a 23 ) log(a 32 a 34 ) log(a 41 a 43 a 45 ) log(a 51 a 54 ) log a 61 p. 17/46
The spanning tree approach (Tsyganok, 2000, 2010) 1 a 12 a 14 a 15 a 16 a 21 1 a 23 a 32 1 a 34 a 41 a 43 1 a 45 a 51 a 54 1 a 61 1 p. 18/46
The spanning tree approach (Tsyganok, 2000, 2010) 1 a 12 a 14 a 15 a 16 a 21 1 a 23 a 32 1 a 34 a 41 a 43 1 a 45 a 51 a 54 1 a 61 1 1 a 12 a 14 a 15 a 16 a 21 1 a 23 a 32 1 a 41 1 a 51 1 a 61 1 p. 19/46
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The spanning tree approach Every spanning tree induces a weight vector. Natural ways of aggregation: arithmetic mean, geometric mean etc. p. 21/46
Theorem (Lundy, Siraj, Greco, 2017): The geometric mean of weight vectors calculated from all spanning trees is logarithmic least squares optimal in case of complete pairwise comparison matrices. p. 22/46
Theorem (Lundy, Siraj, Greco, 2017): The geometric mean of weight vectors calculated from all spanning trees is logarithmic least squares optimal in case of complete pairwise comparison matrices. Theorem (Bozóki, Tsyganok): Let A be an incomplete or complete pairwise comparison matrix such that its associated graph is connected. Then the optimal solution of the logarithmic least squares problem is equal, up to a scalar multiplier, to the geometric mean of weight vectors calculated from all spanning trees. p. 23/46
proof Let G be the connected graph associated to the (in)complete pairwise comparison matrix A and let E(G) denote the set of edges. The edge between nodes i and j is denoted by e(i,j). The Laplacian matrix of graph G is denoted by L. Let T 1,T 2,...,T s,...,t S denote the spanning trees of G, where S denotes the number of spanning trees. E(T s ) denotes the set of edges in T s. Let w s,s = 1, 2,...,S, denote the weight vector calculated from spanning tree T s. Weight vector w s is unique up to a scalar multiplication. Assume without loss of generality that w s 1 = 1. Let y s := logw s, s = 1, 2,...,S, where the logarithm is taken element-wise. p. 24/46
proof Let w LLS denote the optimal solution to the incomplete Logarithmic Least Squares problem (normalized by w1 LLS = 1) and y LLS := logw LLS, then (Ly LLS) = for all i = 1, 2,...,n, i k:e(i,k) E(G) b ik where b ik = log a ik for all (i,k) E(G). b ik = b ki for all (i,k) E(G). In order to prove the theorem, it is sufficient to show that ( L 1 S ) S y s s=1 i = k:e(i,k) E(G) b ik for all i = 1, 2,...,n. p. 25/46
proof Challenge: the Laplacian matrices of the spanning trees are different from the Laplacian of G. Consider an arbitrary spanning tree T s. Then ws i w = a j s ij for all e(i,j) E(T s ). Introduce the incomplete pairwise comparison matrix A s by a s ij := a ij for all e(i,j) E(T s ) and a s ij := ws i w for all j s e(i,j) E(G)\E(T s ). Again, b s ij := log as ij (= ys i ys j ). Note that the Laplacian matrices of A and A s are the same (L). p. 26/46
proof 1 a 12 a 14 a 15 a 16 a 21 1 a 23 a 32 1 a 32 a 21 a 14 a 41 a 41 a 12 a 23 1 a 41 a 15 a 51 a 51 a 14 1 a 61 1 1 a 12 a 14 a 15 a 16 a 21 1 a 23 a 32 1 a 32 a 21 a 14 a 41 a 41 a 12 a 23 1 a 41 a 15 a 51 a 51 a 14 1 a 61 1 p. 27/46
proof Consider an arbitrary spanning tree T s. Then ws i w = a j s ij for all e(i,j) E(T s ). Introduce the incomplete pairwise comparison matrix A s by a s ij := a ij for all e(i,j) E(T s ) and a s ij := ws i w for all e(i,j) E(G)\E(T s ). Again, j s b s ij := log as ij (= ys i ys j ). Note that the Laplacian matrices of A and A s are the same (L). Since weight vector w s is generated by the matrix elements belonging to spanning tree T s, it is the optimal solution of the LLS problem regarding A s, too. Equivalently, the following system of linear equations holds. (Ly s ) i = k:e(i,k) E(T s ) b ik + k:e(i,k) E(G)\E(T s ) b s ik for all i = 1,...,n p. 28/46
proof Lemma S b ik + b s ik = S b ik s=1 k:e(i,k) E(T s ) k:e(i,k) E(G)\E(T s ) k:e(i,k) E(G) p. 29/46
proof of the lemma p. 30/46
proof of the lemma p. 31/46
proof of the lemma p. 32/46
proof of the lemma b 1 12 = b 15 + b 54 + b 43 + b 32 p. 33/46
proof of the lemma b 1 12 = b 15 + b 54 + b 43 + b 32 p. 34/46
proof of the lemma b 1 12 = b 15 + b 54 + b 43 + b 32 b 4 15 = b 12 + b 23 + b 34 + b 45 p. 35/46
proof of the lemma b 1 12 = b 15 + b 54 + b 43 + b 32 b 4 15 = b 12 + b 23 + b 34 + b 45 p. 36/46
proof of the lemma b 1 12 = b 15 + b 54 + b 43 + b 32 b 4 15 = b 12 + b 23 + b 34 + b 45 b 1 12 + b4 15 = b 12 + b 15 p. 37/46
proof of the lemma b 1 12 = b 15 + b 54 + b 43 + b 32 b 4 15 = b 12 + b 23 + b 34 + b 45 b 1 12 + b4 15 = b 12 + b 15 p. 38/46
proof of the lemma b 1 12 = b 15 + b 54 + b 43 + b 32 b 4 15 = b 12 + b 23 + b 34 + b 45 b 1 12 + b4 15 = b 12 + b 15 p. 39/46
proof Finally, to complete the proof, take the sum of equations (Ly s ) i = b ik + for all i = 1,...,n k:e(i,k) E(T s ) k:e(i,k) E(G)\E(T s ) for all s = 1, 2,...,S and apply the lemma S b ik + = S s=1 k:e(i,k) E(T s ) k:e(i,k) E(G)\E(T s ) b s ik b s ik k:e(i,k) E(G) b ik to conclude that y LLS = 1 S S s=1 y s. p. 40/46
Remark. Complete pairwise comparison matrices (S = n n 2 ) are included in our theorem as a special case, and our proof can also be considered as a second, and shorter proof of the theorem of Lundy, Siraj and Greco (2017). p. 41/46
Conclusions The equivalence of two fundamental weighting methods has been shown. The advantages of two approaches have been united. p. 42/46
Main references in historical order 1/2 Tsyganok, V. (2000): Combinatorial method of pairwise comparisons with feedback, Data Recording, Storage & Processing 2:92 102 (in Ukrainian). Tsyganok, V. (2010): Investigation of the aggregation effectiveness of expert estimates obtained by the pairwise comparison method, Mathematical and Computer Modelling, 52(3-4) 538 54 Siraj, S., Mikhailov, L., Keane, J.A. (2012): Enumerating all spanning trees for pairwise comparisons, Computers & Operations Research, 39(2) 191 199 Siraj, S., Mikhailov, L., Keane, J.A. (2012): Corrigendum to Enumerating all spanning trees for pairwise comparisons [Comput. Oper. Res. 39(2012) 191-199], Computers & Operations Research, 39(9) page 2265 p. 43/46
Main references in historical order 2/2 Lundy, M., Siraj, S., Greco, S. (2017): The mathematical equivalence of the spanning tree and row geometric mean preference vectors and its implications for preference analysis, European Journal of Operational Research 257(1) 197 208 Bozóki, S., Tsyganok, V. ( 2017) The logarithmic least squares optimality of the geometric mean of weight vectors calculated from all spanning trees for (in)complete pairwise comparison matrices. Under review, https://arxiv.org/abs/1701.04265 p. 44/46
Thank you for attention. bozoki.sandor@sztaki.mta.hu http://www.sztaki.mta.hu/ bozoki p. 45/46
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