BALANCING PAIRWISE COMPARISON MATRICES BY TRANSITIVE
MATRICES
András Farkas
Faculty for Business and Economics Óbuda University
1084 Budapest, Tavaszmez˝o út 17, Hungary e-mail: farkas.andras@kgk.uni-obuda.hu
Abstract
We discuss the development and use of a recursive rank-one residue iter- ation (triple R-I) to balancing pairwise comparison matrices (PCMs). This class of positive matrices is in the centre of interest of a widely used multi- criteria decision making method called analytic hierarchy process (AHP).
To find a series of the ’best’ transitive matrix approximations to the original PCM the Newton-Kantorovich (N-K) method is employed for the solution to the formulated nonlinear problem. Applying a useful choice for the up- date in the iteration, we show that the matrix balancing problem can be transformed to minimizing the Frobenius norm, and, equivalently, for cer- tain matrices thel1- and thel∞-norms. Convergence proofs for this scaling algorithm are given. A comprehensive numerical example is included.
Mathematics subject classification: 15A12, 15B99, 65F35, 90B50 Key words: numerical mathematics, matrix balancing, diagonal similarity scaling, pairwise comparison matrix
1 Introduction
We call an element-wise positive n × n matrix A = (aij), i, j = 1,2, . . . , n symmetrically reciprocal (SR) if aijaji = 1, i 6= j for all i, j = 1,2, . . . , nand aii = 1 for all i = 1,2, . . . , n. We call an element-wise positive n ×n matrix B = (bij)transitiveif bijbjk = bik, for all i, j, k = 1,2, . . . , n. As we proved in [4, p.425], a transitive matrix is necessarily in SR and has rank one, hence it may be expressed asB=uv>, whereu(resp.v>)is its first column (resp. row).
Non-transitive SR matrices are used in Saaty’s multi-criteria decision making method called the analytic hierarchy process (AHP) [18]. Such matrices also occur in the field of macroeconomics, in both the static and the dynamic input-output analyses (see [5] and [21]). In the AHP, an entry aij of the so called pairwise comparison matrix A (PCM) represents the strength with which an alternativei dominates an other alternative j. After a PCM has been constructed it is required to derive implicitweights,w1, . . . , wn, associated with thendecision alternatives A1, . . . , An, respectively. Saaty proposed theui elements of the principal (Perron) eigenvector of A to give the priority weights of the alternatives with respect to a given criterion. This solution for the weights is unique up to a multiplicative constant. The entries of the weight vectorw = (wi),wi > 0, i= 1,2, . . . , n, are then usually normalized so that the sum of its elements is unity.
The nonnegative matrix balancing problem has attracted immense interest over the past decades. Osborne [16] was the first who presented remarkable results in pre-conditioning matrices and showed that a matrix balanced in thel2 -norm has minimal Frobenius norm. For balancing in the l1 -norm, a relevant result related to similarity scalings was established in [9]. There, it was proven that if Ais ir- reducible, then a diagonal balancing matrix exists and is unique up to scalar mul- tiples. Explicit characterizations of nonnegative matrices for which such scalings do exist were obtained by [1]. Necessary and sufficient conditions that a matrix is a similarity-scaling of another matrix were presented in [19]. Then, Eaves et al.
[3] provided characterization theorems on nonnegative balanceable matrices.
Iterative scaling algorithms as well as optimization algorithms for different matrix balancing problems are well-known in the related literature (see the Os- borne, the Parlett-Reinsch, the Krylov-based, cycle based and weighted balancing algorithms). For a detailed discussion and comparisons of these procedures, we re- fer to the paper [20] and the work of [2]. Recently, Genma et al. [8] have proposed an algorithm for a fractional minimization problem equivalent to minimizing the sum of linear ratios over the positive orthant as a matrix balancing problem. This approach was then applied to the so-called binary AHP method (see in [10] and
[22]) which is an oversimplified version of the traditional AHP having any com- parison valueaij ∈[α1, α].
In this paper we present a new scaling algorithm, termed arecursive rank-one residue iteration(triple R-I) to balancing such SR matrices. For this purpose, we employ a system of inhomogeneous nonlinear equations and discuss the related least-squares optimization problem. We show that a sequence of transitive approx- imations to the original PCM will monotonically improve the objective function value, and, simultaneously, minimize the Frobenius-norm of the balanceable ma- trix. In practice, this means that repeated iteration of the proposed update rule guarantees convergence of a locally optimal matrix balancing. The balanced ma- trices have some useful properties for the application of the AHP. The algorithm seems to be efficient in computational time and easy usage.
This article is organized as follows. In Section 2, we introduce some notation and necessary definitions. In Section 3, we outline our earlier results related to the Newton-Kantorovich (N-K) procedure. The iterative scaling algorithm is pre- sented in Section 4. We give proofs for the main results in Section 5, concerning convergence of the algorithm, existence of the similarity scalings and obtaining a limit point which comprises a stationary vector. Finally, in Section 6, a com- prehensive numerical example provides a means of clear understanding for the readers underlying the use of our findings in the AHP.
2 Notations and definitions
Two mutually connected notations will be used for the weights:w= (wi),wi >0, i = 1,2, . . . , nis the weight (column) vector from Rn, whereas W = diag[wi], denotes a diagonal matrix with the diagonal entries w1, w2, . . . , wn. Thus, W is a positive definite diagonal matrix if and only if w is an element-wise positive column vector.
Then vectore> = [1,1, . . . ,1]is defined to be the row vector ofRn, and the n×nmatrixE= (eij)=ee>to be the square matrix ofRnwith all entries equal to one. The n ×n matrix In of Rn denotes the identity matrix with ones on the main diagonal and zeros elsewhere.
Ann×nmatrixAwith nonnegative entries is said to bebalancedif for each i= 1,2, . . . , n, the sum of the elements in theith row ofAequals the sum of the elements in theith column ofA, i.e., ifAisline-sum-symmetricso that
Ae=A>e. (2.1)
A matrixA is said to bebalanceable via diagonal similarity-scalingif there ex- ists a nonsingular diagonal matrix W with positive diagonal elements such that WAW−1is balanced, i.e., if
WAW−1e=W−1A>We. (2.2)
For a real numberp ≥ 1 the lp -norm of a vectorw is defined by kwkp = (|w1|p +|w2|p +· · ·+|wn|p)1p. If kAk denotes the norm of matrix A, then the l1 -norm is: kAk1 = max
j
Pn
i=1|aij|, the l∞ -norm is:kAk∞ = max
i
Pn j=1|aij|.
Using the p norms, in the special case p = 2, the Frobenius norm is: kAkF = qPn
i=1
Pn
j=1|aij|2.
Kalantari et al. [12] have defined the matrix balancing problem in more gener- ality. According to their definition ann×nmatrixQwith arbitrary real entries is said to bebalanced in thelpnorm(p >0)if for eachi= 1, . . . , n, theith row and column ofQhave the samelp-norm . An invertible (nonsingular) diagonal matrix X= diag [x1, x2, . . . , xn]balancesQin thelp-normif for eachi= 1, . . . , n, the lp-norm of theith row and column ofXQX−1are identical, i.e.
n
X
j=1
qij
xi xj
p
=
n
X
j=1
qji
xj xi
p
, i= 1, . . . , n. (2.3) Clearly, an invertible diagonal matrixX= diag [x1, x2, . . . , xn]balancesQin the lp -norm if and only if the positive diagonal matrix W = diag [w1, w2, . . . , wn] balances the nonnegative matrix A = (|qij|p) in l1 -norm. The general matrix balancing problem in lp -norm can thus be reduced to the case of nonnegative matrix balancing via a positive diagonal matrix.
Ann×nmatrixAis said to bereducibleif and only if for some permutation matrix A, the matrix P>AP=B C
0 D
is block upper triangular, where B and D are both square. OtherwiseAis said to be anirreduciblematrix. The graphG(A) of Ais defined to be thedirected graph ofN nodes in which there is a directed arc leading from nodei to nodej if and only ifaij > 0.G(A)is calledstrongly connected if for each pair of nodes (i, k) there is a sequence of directed arcs leading from node ito nodek. A is an irreducible matrix if and only ifG(A)is strongly connected, i.e., if it contains exactly one strongly connected component which includes all of the nodes.
Ann×n nonnegative matrixA is calledcompletely reducible ifi ∈ N has access to j ∈ N if and only if j has access toi. In particular, every irreducible matrix is completely reducible.
3 Preliminaries
An approach to producing a ‘best’ transitive rank-one matrix approximationBto a SR matrix A (to a general PCM) was developed and presented in [6]. There, the ‘best’ is assessed in a least-squares (LS) sense. Thereby, in order to extract a vector of the weights w from A the following expression, i.e. the Euclidean- distance between matricesAandBshould be minimized:
S2(w) :=kA−Bk2F =
n
X
i=1 n
X
j=1
aij −wj wi
2
, (3.1)
where the subscript F denotes the Frobenius norm; the square root of the sum of the squares of the elements. As we have shown in [6, p.694] a stationary valuew of the error functionalS2(w), denoted byw∗, satisfies thehomogeneousnonlinear equation
R(w)w= 0, (3.2) where the variable dependent coefficient matrix, R(w)=(rij), i, j = 1,2, . . . , n, fromRn,has the form:
R(w) =W−2(A−W−1ee>W)−(A−W−1ee>W)>W−2. with the off-diagonal elements
rij =
−(wj−aijwi) 1
wi3 + 1 aijw3j
for i < j, (wi−ajiwj)
1
wj3 + 1 ajiw3i
for i > j, and the diagonal elements
rii= 0, i= 1,2, . . . , n.
SinceAis an SR matrix, obviously,R(w)is askew-symmetricmatrix.
However, the solution of equation (3.2) cannot be unique, since any constant multiple of this solution would produce an other solution. To overcome these shortcomings a nonzero vector c ∈ Rn is introduced. Moreover, let (3.2) have a positive solutionwnormalized so thatc>w= 1. (A convenient choice isj = 1 and thusc> = [1,0, . . . ,0], i.e. this condition is thenw1 = 1.) This way, for any
j, 1 ≤ j ≤ n, apparently, a stationary vectorw∗ is a solution to the following inhomogeneoussystem ofnnonlinear equations
c>w= 1, f(w) = Rk∗(w)w= 0, k 6=j, 1≤k ≤n, (3.3) where a conventional notation for thejth row of matrixMbyMj∗is used. Observe in (3.3) thatf(w)is at least twice differentiable.
To search for a vector rootw∗> = [w1∗, w2∗, . . . , w∗n]of equation (3.3) we used the Newton-Kantorovich (N-K) method, which employs the recurrent procedure:
w(p+1) =w(p)− ∇2f(w)(p)−1
∇f(w)(p), p= 0,1,2, . . . , (3.4) where ∇2f denotes the second Fr`echet derivative (i.e. the Hessian matrix in the finite-dimensional case). The main convergence result for iteration (3.4) origi- nated with Kantorovich [14]. As it is well-known, an appropriately chosen initial approximation, say w(0), is critical for the convergence of the procedure. This means that the norm of the vector function, kf(w(0))kF should be small enough, i.e., w(0) must be close to a solution. We always have chosen the solution of the
’best’ linear approximation to problem (3.1), denoted by φ0 in [6, p.693], as an initial value for running the procedure. Applying this strategy to the minimiza- tion problem (3.3) subject to an equality constraint on the weights, these N-K se- quences were alwaysconvergentand producedlocalminima. Furthermore, com- putations with different choices forcandjhave always led to scalar multiples of the same solutionw∗, giving some confidence the conjecture, that in these cases, this stationary vector w∗ is associated with a global minimum of S2. We men- tion here that the interested reader may find a relatively new approach to the N-K method, where global convergence can be achieved for functions that are not nec- essarily convex and the iteration converges globally for an arbitrary initial point [15]. For further problem background we refer to the excellent paper of Polyak [17].
It is apparent that by the weight vectorw(and thus by the matrixW) the ’best’
approximating transitive matrixBto a matrixAin a LS sense can be obtained as B =W−1EW =W−1ee>W=
wj wi
, i, j = 1,2, . . . , n. (3.5) From (3.5), it is easy to see thatBW−1e = nW−1e, i.e. the only nonzero (prin- cipal) eigenvalue of B is n and its associated Perron-eigenvector is W−1e, i.e.
a vector whose elements are the reciprocals of the weights. For the nontransitive cases, however,λmax > n(see a proof for this relationship e.g. in [5, p.407]).
4 A recursive rank-one residue iteration
In this section we discuss the development of a particular scaling method to bal- ancing SR matrices. For this purpose, we formulate a least-squares (LS) optimiza- tion algorithm called arecursive rank-one residue iteration(triple R-I).
Let the setΩdenote the feasible region for problem (3.1):
Ω =n
w∈Rn
c>w= 1,w>0o .
The triple R-I starts by using the N-K method for solving equation (3.3) to find a stationary vectorw∗(0)(and thus the diagonal matrixW0∗) at the initial step, k = 0. The normalization conditionc>w= 1is imposed in order to hold{w∗(k)}, k = 0,1,2, . . ., in a bounded set throughout the entire process. ByW∗0 and with the expression (3.5), the ‘best’ transitive matrix approximationB0 to the original SR matrixAin a LS sense can thus be determined.
A strategy to design an iterative procedure by establishing a successively ad- justed sequence of rank-one matrices is the following. It is clear that the ‘best’
approximation of an entryaij of matrixAiswj∗(0)/wi∗(0) i, j = 1,2, . . . , n. Since we may reasonably expect that(w∗(0)i /w∗(0)j )aij produces a ‘good’ approximation of 1, it is readily apparent that
"
wi∗(0) wj∗(0)aij
#
=W∗0AW∗(−1)0 ≈E, i, j = 1,2, . . . , n. (4.1) The main idea is to achieve continuous improvement in further approximating E. For this purpose, let a positive n × n matrix Hk = (h(k)ij ), k = 0,1,2, . . ., called a residue be defined. It is convenient to set H0 = A, at k = 0. Hence, necessarily,Hkis also in SR. Next, at the consecutive steps of the iteration process each entries of Hk will simultaneously be updated by performing a similarity transformation (diagonal similarity scaling) of the previous updateHk−1 with the generating diagonal matricesWk−1andWk−1−1 . This yields theupdating rule:
Hk =Wk−1Hk−1Wk−1−1 =
"
w(k−1)i w(k−1)j
#
◦h
h(k−1)ij i
, k = 1,2, . . . . (4.2) Note here that (4.2) can also be written in the form of a Hadamard product. For updating matrixHk−1,formula (4.2) is referred to as thestep-operator, Sk(Hk),
in order to the rank-one matrix
∼
Bk be recursively adjusted to the original matrix Aat the consecutive iteration steps, so that
∼
Bk=
∼
W
−1 k E
∼
Wk, k = 0,1, . . . , where
∼
Wk=
k−1
Y
i=0
Wi and
∼
W
−1
k =
k−1
Y
i=0
Wi−1. (4.3) It can be readily seen that each of the adjustment errors, S(w∼(k))=kA−B∼kkF, will be greater fork = 1,2, . . . ,than that of fork = 0. An other transitive matrix,
BPk=Wk−1EWk,=
"
w(k)j w(k)i
#
◦h e(k)ij i
, k = 1,2, . . . , (4.4) called apatternwill represent the ’best’ transitive matrix approximation to its cor- responding residueHk. Its approximation error is:S(w(k))=kHk−BPkkF. Obvi- ously,B0=BP0=
∼
B0. It is evident that the updating rule (4.2) will force all entries of
∼
Bkto be set to 1, while the elements of the Perron-eigenvectors of the pattern BPk will successively approach to those of matrixA.
The process is repeated until some convergence criterion is met. The stopping rule is to halt the algorithm at iteration stepk = q once the numerical error falls below a predefined tolerance (a reasonably small positive number,ε >0) yielding the "stabilized" matrices
∼
Wq,Hq,
∼
Bq andBPq.
The formal description of the algorithm is presented below:
Triple R-I Algorithm
Input module.Enter the SR matrixA. Calculate its Perron-eigenvalue, λmax(A), and its normalized right and left Perron-eigenvectors,umax(A)andvmax(A).
Initial module. For k = 0. Given a positive initial value φ0 and a reasonably smallε >0. Using the N-K method find the stationary vectorw∗(0) (and thus the diagonal matrixW∗(0))by solving the following system of nonlinear equations:
{W0−2(A−W0−1EW0)−(A−W−10 EW0)>W−20 }W0e= 0, (4.5) where in (4.5), W0e = w(0) = [w(0)i ], i = 1,2, . . . , n, is normalized so that c>w(0) = 1.
a) Ifw(0) is stationary, compute
∼
B0 =BP0 =W∗(−1)0 EW∗0 =
"
wj∗(0) wi∗(0)
#
, i, j = 1,2, . . . , n. (4.6) b) Else choose an other promising positive initial value and repeat the N-K procedure untilw(0)is stationary, then compute
∼
B0 =BP0 according to (4.6).
c) Calculate the error of the ’best’ transitive matrix approximation
∼
B0 to the original matrixAas:S(w(0)) =S(w∼(0))=kA−B∼0kF.
SetH0 =A.
Recursion module. For k = 1,2, . . .. Using the N-K method find the stationary vectorw∗(k)(andW∗(k)) by solving the following system of nonlinear equations:
{Wk−2(Hk−W−1k EWk)−(Hk−W−1k EWk)>Wk−2}Wke= 0, (4.7) where in (4.7), the vectorWke=w(k) = [wi(k)],i= 1,2, . . . , n, is normalized so thatc>w(k) = 1and the residue (updating rule) is given by the formula (4.2).
a) IfkWk−Ink< ε,fork > N(ε), setk=q, then compute
∼
Wq,Hq,
∼
Bq andBPq,then calculate the Perron-eigenvalue,λmax(Hq), and its normalized right and left Perron-eigenvectors,umax(Hq)andvmax(Hq)and stop.
b) Else compute
∼
Wk,Hk,
∼
Bk andBPk.
c) Calculate the adjustment error of the rank-one matrix
∼
Bkto the original matrixAas:S(w∼(k))=kA−B∼kkF.
d) Continue the iteration process fork+ 1.
5 Diagonal similarity scaling of pairwise compari- son matrices
This section discusses the matrix balancing problem done through successive ad- justments of the residue and the pattern matrices. We will show that matrices A and B0 are balanceable in the sense of (2.2) and can be balanced by virtue of (2.1). The balanced matrices have useful properties which provide some novel contributions to the theory of the AHP as well. In particular, we will give proofs that our triple R-I algorithm with a user specified termination criterion results in the similarity scalings,BPq andHq.
We start by focusing on the sequence of the pattern{BPk},k = 1,2, . . . ,gen- erated via line-sum-symmetric diagonal similarity scaling (DSS). We will assume that matrixBPk is irreducible by permutation matrices. Since BPk > 0, according to the Perron-theorem this condition holds. The reason for this restriction is that in the triple R-I it is desirable to have the entries of the sequence of diagonal matrices bounded. The iterative algorithm looks for a sequence of {Wk}, so that at each stepk, the patternBPk=W−1k EWkis yielded. It is easy to recognize here that this is equivalent with the DSS:BPk>=WkEW−1k . Accordingly, the following results will be related to the transpose of the pattern. Thus, the first row (andnotits resp.
column as forBPk) of this matrix will contain the reciprocals of the weights,wi−1, i= 1,2, . . . , n, i.e. the priorities of the decision alternatives in the AHP.
We first present an important result to characterize existence of line-sum- symmetric similarity scalings of balanceable matrices [3]:
Corollary 1. (Eaves et al. [3, p.133]). LetAbe ann×nnonnegative matrix and letf be the real valued function defined onΩ{w∈Rn :w0}by
f(w) =
n
X
i=1 n
X
j=1
wiaijw−1j . (5.1)
Then the following are equivalent:
(a) Ahas a line-sum-symmetric similarity-scaling, (b) Ais completely reducible, and
(c) f attains a minimum overΩ.
An other famous result originated with the same authors relates to the charac- terization of the set of diagonal matrices yielding line-sum-symmetric similarity scalings of balanceable matrices [3]:
Theorem 1. (Eaves et al. [3, p.134]). LetAbe ann×nnonnegative matrix and let f be the real valued function defined on Ω {w ∈ Rn : w 0} by (5.1).
Consider the following properties of a vectorw∗ ∈Ω:
(a) w∗ minimizes the function f overΩ, and
(b) the matrixW(w∗)AW(w∗)−1 is line-sum-symmetric.
Thew∗satisfies(a)if and only ifw∗ satisfies(b).
Proofs of the aforesaid results are given in [3, pp.133-134].
Theorem 1 shows that the problem of searching for a line-sum-symmetric sim- ilarity scaling of a given balanceable matrix is equivalent to the problem of min- imizing a (nonlinear) function over the positive orthant. Corollary 1 shows that these problems have a solution if and only if the underlying matrix is completely reducible. Theorem 1 states also that a stationary vector,w∗ minimizes the func- tionf overΩwhich is a unique optimal solution up to a positive scalar multiple, if and only if the matrixWkAW−1k is in line-sum-symmetry.
We now define the following vector functions of the row and column sums for a positive vector wof the diagonal similarity scaling BPk>=WkEW−1k . (To simplify notation, hereafter we will omit the iteration step indexk):
r(w)≡
n
X
j=1
e1jwj
w1 2
, . . . ,
n
X
j=1
enjwj
wn 2!
=
n
X
j=1
wj w1
2
, . . . ,
n
X
j=1
wj wn
2! , (5.2) c(w)≡
n
X
i=1
ei1w1
wi 2
, . . . ,
n
X
i=1
einwn
wi 2!
=
n
X
i=1
w1 wi
2
, . . . ,
n
X
i=1
wn wi
2! . (5.3) Observe now that any positivewsatisfiesf(w)=r(w)e=c(w)eand for a pos- itive vectorw∗,r(w∗)=c(w∗)holds, if and only ifw∗ is a stationary vector. The triple R-I iterates over all rows and columns of the balanceable matrix at a partic- ular stepkto find an appropriate scale vector, i.e. each entry ofWk and thusBPk
is updated and so the rows/columns are scaled simultaneously, making this algo- rithm very efficient. The following two lemmas and their proofs are extensions of similar results that were presented by Genma et al. [8].
Lemma 1. For a positive vector w, at each iteration step k, k = 1,2, . . . , of the triple R-I algorithm, letr=r(w)andc=c(w), defined as(5.2)and(5.3). For any particular row and column, assume that rl6=cl, l = 1,2, . . . , n. Let an index κ(l)=p
rl/clbe introduced and letwi(κ(l))=κ(l)wi fori=l. Then, min
κ(l)>0f (w(κ(l))−f(w) (5.4)
has a unique optimal solutionw∗(−1)>= p
r1/c1, . . . ,p rn/cn
and its optimal objective function value isZ∗=
n
P
l=1
−(rl−cl)2.
Proof. We first note that inκ(l)=p
rl/cl,the restriction imposed onBPk pre- ventsclfrom equaling0. After some algebraic manipulations we may obtain that
f (w(κ(l)))−f(w)=
=
n
X
j=1
wj κ(l)wl
2
+
n
X
i=1
κ(l)wl wi
2
−
n
X
j=1
wj wl
2
+
n
X
i=1
wl wi
2!
=
= 1
κ2(l)rl2+κ2(l)c2l−rl2−c2l, l= 1, . . . , n.
(5.5)
To the following steps, it is convenient to introduce the functions resulted in (5.5) as:g(κ(l))=κ21(l)r2l+κ2(l)c2l−r2l−c2l. Since, obviously, bothrl2>0andc2l>0, therefore, theg(κ(l))’s are each strictly convex. Minimizing them, we get that
dg
dκ(l)=− 2
κ3(l)r2l+2κ(l)c2l=0, l= 1, . . . , n. (5.6) Now, it is easy to see that a stationary value,κ∗(l)=p
rl/cl, satisfies (5.6). Hence, Z∗=
n
X
l=1
g(κ∗(l))=
n
X
l=1
1
rl
cl
c2l+rl cl
r2l−rl2−c2l=
n
X
l=1
−r2l−c2l+2rlcl=
n
X
l=1
−(rl−cl)2. (5.7)
With κ∗(l), the minimum that can be achieved in (5.5) yields 2rlcl.The up- dating rule (4.2) ensures the reduction of the value of the objective function Z∗. Thus, the sequence {Wk} k = 1,2, . . . , generated by the recursive process in- duces a decrease in the Frobenius norm,
BP>k
F,at each stepk,as is shown by the following lemma.
Lemma 2. At a particular iteration stepk, k = 1,2, . . . ,of the triple R-I iteration and utilizing the indexκ(l), l = 1,2, . . . , n,it yields
n
X
l=1
−
r(k)l −c(k)l 2
=f w(k+1)
−f w(k)
. (5.8)
Therefore,f w(k)
>f w(k+1)
, k = 1,2, . . ..
Proof. Sincew(k+1) is equal to a positive scalar multiple ofwq
rl
cl
at step k+ 1, it follows fromf w(k+1)
=f wq
rl
cl
and Lemma 1 that 0>
n
X
l=1
−
rl(k)−c(k)l 2
= min
κ(l)>0f w(k)(κ(l))
−f w(k)
=
=f
w rrl
cl
−f(w(k))=f w(k+1)
−f w(k) .
(5.9)
As it can be seen from (5.9) the norm is never increased. For the casesrl(k) 6= c(k)l there is a reduction in
BPk
2
F. The restriction of irreducibility placed upon BPk preventsr(k)l andc(k)l from identically vanishing. Thus,
BPk>
2 F−
BPk+1>
2 F=
n
X
l=1
r(k)l −c(k)l 2
≥0, k = 1,2, . . . . (5.10)
We remark that using similar arguments and applying the technique of proofs used for Lemmas 1 and 2 accordingly, one could easily verify that there is a strict decrease in thel1 -norm,
BPk
1,and in thel∞-norm, BPk
∞,as well (see also Corollary 1 and Theorem 1), if one defines:
rl(k)=
n
X
j=1
bPlj(k)
and c(k)l =
n
X
j=1
bPjl(k)
, l= 1,2, . . . , n. (5.11) The convergence theorem can now be stated.
Theorem 2. LetBPk> = (bPji),i, j = 1,2, . . . , n,be a transitive matrix with posi- tive entries and called a pattern. Assume thatBPk>is irreducible by permutations.
Through the triple R-I iteration, the sequence{BP>k },k = 1,2, . . ., generated by the diagonal matricesWkandW−1k defined in(4.3)converges to someBPq∗over the feasible setΩandq indicates the step of the termination of the algorithm for a prescribed reasonably small toleranceε >0. Then,
(i) lim
k→∞BPk>=BPq>exists;
(ii) BP>q =WqEW−1q ; (iii) Wq= lim
k→∞Wk= lim
k→∞W−1k =In; (iv)
BPq>
F=inf
Wψ
WψEW−1ψ F; (v) BP>q =BPq=E=BP∗;
(vi) BP∗ is in sum-symmetry;
where in(iv),Wψ ranges over the class of all nonsingular diagonal matrices.
Proof. As a consequence of (5.10) in Lemma 2, the sequence{
WkEW−1k F} of similarity scalings is bounded. Therefore, sincew(k)1 =1is fixed for allk,there exists α > 0 and β > 0 such that α ≤ w(k)i (κ(l)) ≤ β for all k and i, l = 1,2, . . . , n. From this, it is clear that there exists γ > 0 such that rl > γ and cl > γ for all k. Following the line of the technique that was used in [16], from BP>1
F ≥ BP2>
F ≥. . .≥0follows the existence ofLsuch that
k→∞lim
BPk>
F=L≥0, BP>k
F ≥L for allk. (5.12) According to (5.10), this implies that
k→∞lim
n
X
l=1
(rl−cl)2= lim
k→∞
BPk+1
2 F−
BPk
2 F
=0, (5.13)
or
rl=cl+η(k), lim
k→∞
η(k)
=0, for alllandk. (5.14) But
wi(−1)2(κ(l))=rl
cl=1 +η(k)
cl and rl> γ. (5.15)
Therefore,
k→∞lim wi(−1)(κ(l))=1, i, l= 1,2, . . . , n. (5.16) The values of |rl−cl| pass through zero at least once for every k consecutive steps of the iterative process. The sums rl and cl are bounded for all k so that (5.16) implies that the changes in rl andcl over anyk successive steps approach zero askincreases without limit. This implies that
k→∞lim |rl−cl|=0, l = 1,2, . . . , n. (5.17) Another consequence of the fact that the diagonal elements of{Wk},k = 1,2, . . ., are bounded away from zero and from above is that a subsequence{Wk¯}of{Wk} can be chosen such that
¯lim
k→∞ W¯k=Wq, Wq is nonsingular. (5.18) Also, since the entries of BPk>are continuous functions of the diagonal elements ofWk,it follows that
k→∞lim BP¯k>=WqEW−1q =BPq> (5.19) exists, representing a unique limit point of the triple R-I iteration, denoted asBPq∗. The iterate Wk converges to In in a limiting sense. The algorithm stops for a stipulated arbitrarily smallε >0if a certainN can be found such that
kWq−Ink< ε, forq > N(ε).
Thus, S2(w(q+p)) = S2(w(q)), for p > 0. A direct implication of (5.16) is that BP>q =BPq=E=BP∗q . Also, it is straightforward that the matricesBP>q andBPq are in sum-symmetry, since their entries are all ones. Indeed, it is easy to see that BPq embodies the sole line-sum-symmetric transitive matrix over the class of all transitive matrices. This completes the proof.
In the sequel, we consider the residue matrixHk. We will show that the up- dating rule of the triple R-I algorithm is essentially analogous to the fixed point iteration
Hk+1 =Sk(Hk) = WkHkWk−1, k = 0,1, . . . (5.20) whereSk(Hk)is the step operator of the triple R-I. The objective is to minimize the Frobenius norm:
kHk+1−HkkF ⇒minimum, k = 0,1, . . . . (5.21)
The convergence theorem for the sequence{Hk}is stated below.
Theorem 3. Let H = (hij), i, j = 1,2, . . . , n be an SR matrix with positive entries and called a residue. The sequence{Hk}, k = 1,2, . . ., generated by the fixed point iteration(5.20)using the step operatorSkconverges to someH∗q ∈ H∗, where H∗ is the set of stationary points of problem (5.21) over the feasible set Ω, and q indicates the step of the termination of the triple R-I for a prescribed reasonably small toleranceε >0.
Proof. As follows from its construct, Sk(Hk) is non-expansive, therefore {Hk}lies in a compact set and must have a limit point, say Hˆ = limj→∞Hkj. Additionally, for anyH∗q ∈ H∗,
Hk+1−H∗q F =
(Sk(Hk)−Skj(H∗q) F≤
Hk−H∗q F,
which implies that the sequence{kHk−H∗qkF}is monotonically non-increasing under the updating rule (4.2). Hence,
k→∞lim
Hk−H∗q F=
Hˆ −H∗q F
, (5.22)
whereHˆ can be any limit point of{Hk}. Considering thatSk(Hk)is continuous, the step operator forH,ˆ
Skj(H) = limˆ
j→∞Skj(Hkj) = lim
j→∞Hkj+1, produces also a limit point of{Hk}. Therefore, we have
Skj(H)ˆ −Sq(H∗q) F
=
Skj(H)ˆ −H∗q F
=
Hˆ −H∗q F
which shows that Hˆ is a stationary point of problem (5.21). Finally, by setting H∗q = ˆH∈ H∗in (5.22), we obtain
k→∞lim
Hk−Hˆ F
= lim
j→∞
Hkj −Hˆ F
= 0,
i.e.{Hk}converges to its limit pointH. In each stepˆ kof the recursive algorithm the N-K method is used to solve the system of nonlinear equations (4.7). There- fore, at stepk =q, when the iteration has converged to any limit point H∗q in the interior of the feasible region Ω, this point is necessarily a stationary point (see Farkas et al. [6, p.695]). This completes the proof.
The following lemma refers to the limit of the sequence of the entries of ma- trices{Hk},k = 1,2, . . ., and verifies that this is also a convergent sequence.
Lemma 3. For the convergence of a sequence of matrices{Hk},k = 0,1,2, . . . it is necessary and sufficient that the generalized Cauchy test hold, namely for any ε >0there must be a numberN =N(ε)such that fork > N,p >0
kHk+p−Hkk< ε, (5.23)
where the matrix norm can be any canonical norm.
Proof. Indeed, since according to Theorem 3 the sequence {kHkkF} is de- creasing and therefore, inequality (5.23) is valid. Thus, for every elementh(k)ij of the matrices of the sequence{Hk}the Cauchy test (see e.g. in [13]) will hold, and hence, there exists
k→∞lim Hk= [ lim
k→∞h(k)ij ]=Hq. (5.24) The matrixHq is stabilized at stepq, and repeats itself in the succeeding steps, if we would continue the iteration. Therefore,Hq=Hq+p,forp >0.
Next, we show that matrixH∗q is in line-sum-symmetry.
Corollary 2. For the limit matrix H∗q, of the triple R-I the right and the left eigenvectors associated with the zero eigenvalue of the skew-symmetric matrix (H∗q−H∗>q ), are the vectorseande>, respectively.
Proof. Using the diagonal matrix
∼
Wdefined as (4.3) we can write the product of the diagonal matricesWk in a limiting sense as
k→∞lim(Wk−1Wk−2. . .W2W1W0) =
∼
W. (5.25)
By taking the limit of (4.2) we have
k→∞lim Hk =
∼
WqA
∼
Wq−1. (5.26)
Applying (5.26) fork > N, the system of nonlinear equations (4.3) leads to the following equation:
∼ WqA
∼
Wq−1−
∼
Wq−1A>
∼
Wq
e= (H∗q−H∗q>)e= 0, (5.27) where it is apparent that the right eigenvector associated with the zero eigenvalue of the skew-symmetric matrix (H∗q −H∗>q )ise, while the left eigenvector is e>.
As it is readily seen from equation (5.27) the matrixH∗q is balanced, since it is in line-sum-symmetry in a sense of (2.2).
Writing the recursion formula (4.2) in an element-wise form we have
h(k)ij =h(k−1)ij w(k−1)j w(k−1)i
!
for i < j, h(k)ji =h(k−1)ji w(k−1)i
w(k−1)j
!
for i > j, and for the diagonal elements
hii= 1, i= 0,1, . . . , n.
Making use of Lemma 1 and from Theorems 1 and 2 it can be seen that the se- quences {h(k)ij } and{h(k)ji }, k = 0,1,2, . . ., for all i, j = 1,2, . . . , n,as well as {w(k)i },k = 0,1,2, . . ., for alli, j = 1,2, . . . , n,converge to their limits. We now show that the quotients
w(k)j w(k)i
and their products with{h(k)ij }also converge to a limit point. For this purpose, we apply the following well-known theorem.
Theorem 4. Suppose that{w(k)i }and{w(k)j }are two convergent sequences gen- erated by the triple R-I over the feasible set Ω. Let these sequences simply be denoted by ({an}) and({bn}), respectively, with limits A and B. Then, the fol- lowing rules apply:
(i) Product Rule.The product({an} {bn})is convergent and lim
n→∞({an}{bn})=AB;
(ii) Quotient Rule.IfB 6= 0,then {a
n} {bn}
is also convergent,and lim
n→∞
{a
n} {bn}
=BA. Proof. To see that(i)is valid, observe that the following relation holds:
n→∞lim({an}) =A ⇐⇒ lim
n→∞({an} −A) = 0. (5.28) Applying (5.28) to our problem, we have to admit that
n→∞lim({an}{bn} −AB) = 0. (5.29)
Using the triangular inequality:
|{an}{bn} −AB|=|{an}{bn} −A{bn}+A{bn} −AB|
=|{bn}({an} −A) +A({bn} −B)|
≤ |{bn}| |{an} −A|+|A| |{bn} −B|.
(5.30)
Since |{bn}|is a bounded sequence (as it is convergent), whereas |{an} −A| is a null sequence, their product yields a null sequence. The constant sequence |A|
is, obviously bounded, while the sequence |{bn} −B| is a null sequence, thus their product is also a null sequence. Whence, the sum is a null sequence as well.
Therefore, we can write
(|{an}{bn} −AB|) nullsequence =⇒ lim
n→∞({an}{bn} −AB) = 0, (5.31) since a null sequence is invariant to the absolute value. To see that(ii) holds, we make the following rearrangements:
{an} {bn} − A
B
= |{an}B −A{bn}|
|{bn}B| = |{an}B −AB+BA−A{bn}|
|{bn}B|
=
|B({an} −A) +A(B − {bn})|
|{bn}B| ≤ |B| |{an} −A|
|{bn}| |B| + |A| |B− {bn}|
|{bn}| |B|
= 1
|{bn}|
| {z }
bounded
· |{an} −A|
| {z }
nullsequence
+ |A|
|B|
|{z}
bounded
· 1
|{bn}|
| {z }
bounded
· |{bn} −B|
| {z }
nullsequence
| {z }
nullsequence
(5.32)
From the last expression in (5.32) it is seen that
{an} {bn} − AB
is a null sequence.
By the Product rule (i), then, lim
n→∞
{an}
{bn} = AB. This completes the proof.
Finally, we show that using the triple R-I the sequence of matrices {Hk}, k = 1,2, . . ., achieves a minimum, represented by the stabilized matrixH∗q for a prescribed reasonably small tolerance ε >0. The following theorem utilizes the element-wise form of the recursion rule:h(k+1)ij =wi(k)h(k)ij wj−1(k).
Theorem 5. Let the elements h(k)ij > 0 and wi(k) > 0, i, j = 1,2, . . . , n, k = 0,1,2, . . . ,be generated by the triple R-I algorithm. The sum of products
n2
X
l=1
h(k)ij
l
w(k)i w(k)j
!
l
i, j = 1,2, . . . , n, k = 0,1,2, . . . , l = 1,2, . . . , n2, (5.33) attains a minimum if
h(k)ij
1
≤ h(k)ij
2
≤. . .≤ h(k)ij
n2
and w(k)i
w(k)j
!
1
≥ wi(k) wj(k)
!
2
≥. . .≥ w(k)i w(k)j
!
n2
(5.34)
hold.
Proof. Suppose that the statement of Theorem 5 is not true. This would mean that the minimum of (5.33) is generated by such a sum of products for which the (5.34) orders are not held, i.e. (thereafter we use a simplified notation for the members of the products):
h1w´1+h2w´2+. . .+hjw´j+1+hj+1w´j+. . .+hlw´l (5.35) If the statement of the theorem is false, then the sum (5.35) is less than that of (5.33), i.e.
h1w´1+. . .+hjw´j+1+hj+1w´j +. . .+hlw´l <
n2
X
l=1
hlw´l (5.36) Since the right hand and the left hand sides of the inequality (5.36) differ in two members only, therefore
hjw´j+1+hj+1w´j < hjw´j+hj+1w´j+1 (5.37) must hold, i.e.
0< hj( ´wj −w´j+1) +hj+1( ´wj+1−w´j) (5.38) and
0<(hj−hj+1)( ´wj −w´j+1). (5.39)
Since by (5.34) hj −hj+1 ≤ 0andw´j −w´j+1 ≥ 0, inequality (5.39) is certain not to happen. Thus, an opposite statement than the statement of the theorem leads to a contradiction. Therefore, by Lemma 3 and considering the fact that at iteration step q, w(q)i =1 for all i, the sum of products (5.33) attains a minimum.
This completes the proof.
Finally, it should be noted that depending upon a certain degree of perturbation of a PCM (termed a level of inconsistency of matrix A in the field of decision sciences) the algorithm may produce more than one limit point. We refer to [7] for some details of this phenomenon, however, this issue is subject to further research.
6 Numerical illustration
The following example demonstrates the results discussed in the previous sec- tions. The prescribed accuracy is: ε = 10−6. Numerical results are presented to four digits. For comparison, weights are normalized so that the sum of their ele- ments is one.
Example. We consider a 5×5 SR matrixA. The objective is to prioritize five given alternatives. Saaty’s nine-point scale [1/9, . . . ,1/2,1,2, . . . ,9] is used for the entries ofA.
A =
1 3 5 4 2
1/3 1 4 4 6
1/5 1/4 1 2 2 1/4 1/4 1/2 1 2 1/2 1/6 1/2 1/2 1
.
In the input module of the algorithm the spectral properties of A are calculated.
The principal eigenvalue of A is: λmax = 5.5737. The right and left Perron- eigenvectors of A, where the right one represents the weights (priority scores) of the alternatives, are, respectively (observe here that the eigenvectors are not element-wise reciprocal, see [11] for more detail):
u>(A) = [0.4254,0.3030,0.1071,0.0859,0.0786], v>(A) = [0.0695,0.0859,0.2190,0.2701,0.3561].
The output of the initial module of the algorithm yields the stationary vector w∗(0) which appears in the first row of the ‘best’ transitive matrix approximation
B0to matrixAin an LS sense given in a non-normalized form:
w∗>(0) = [1.0000,0.8574,3.7688,3.6920,4.0897].
Its inverse, displayed in a normalized form, represents the ’best’ estimate of the weights in a LS sense:
w(−1)∗>(0) = [0.3393,0.3958,0.0903,0.0919,0.0830],
which is apparently not a good adjustment to the right Perron-eigenvector,u(A), of the original matrixA.
The triple R-I terminates at stepq= 66, producing the stationary vector:
w∗>(q) = [1.0000,1.0000,1.0000,1.0000,1.0000],
which would repeat itself if one continues the iteration. Therefore,BPq=E=BP∗, are clearly in line-sum-symmetry, thus the pattern matrixBP has been balanced.
The stabilized SR matrixH∗qyields:
H∗q =
1 2.2972 1.4109 0.8946 0.3690 0.4353 1 1.4740 1.1683 1.4457 0.7088 0.6784 1 1.5853 1.3078 1.1178 0.8559 0.6308 1 1.6499 2.7099 0.6917 0.7647 0.6061 1
.
The principal eigenvalue ofH∗q is: λmax = 5.5737. It is easy to check that H∗q is in line-sum symmetry, since it has been balanced. The right and the left Perron- eigenvectors of H∗q as function of the selected ε are very close to e. They are, respectively,
u>q(H∗q) = [0.2045,0.2014,0.1983,0.1997,0.1961], vq>(H∗q) = [0.2034,0.2027,0.1988,0.1995,0.1956].
Up to the termination step of the iteration,q = 66,the inverse matrix
∼
W−1q of the product of the diagonal matricesW−1k ,k = 0,1,2, . . . , q, is obtained as
∼
W−1q =
1 0
0.7658
0.2822
0.2237
0 0.1845
.
