An effective Markov based approach for calculating the Limit Matrix in the analytic network process

European Journal of Operational Research 214 (2011) 85–90

Contents lists available at ScienceDirect

European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Stochastics and Statistics

An effective Markov based approach for calculating the Limit Matrix in the analytic network process Konstantinos Kirytopoulos a, Dimitra Voulgaridou b, Agapios Platis b,⇑, Vrassidas Leopoulos a a b

National Technical University of Athens, School of Mechanical Engineering, 9 Heroon Polytechniou Str., 15780 Athens, Greece University of the Aegean, Financial and Management Engineering Dept., 41 Kountouriotou, 82100 Chios, Greece

a r t i c l e

i n f o

Article history: Received 12 December 2009 Accepted 23 March 2011 Available online 29 March 2011 Keywords: Markov processes Multiple criteria decision analysis Analytic network process ‘Power’ matrix method

a b s t r a c t Analytic network process is a multiple criteria decision analysis (MCDA) method that aids decision makers to choose among a number of possible alternatives or prioritize the criteria for making a decision in terms of importance. It handles both qualitative and quantitative criteria, that are compared in pairs, in order to forge a best compromise answer according to the different criteria and influences involved. The method has been widely applied and the literature review reveals a rising trend of ANP-related articles. The ‘power’ matrix method, a procedure necessary for the stability of the decision system, is one of the critical calculations in the mathematical part of the method. The present study proposes an alternative mathematical approach that is based on Markov chain processes and the well-known Gauss–Jordan elimination. The new approach obtains practically the same results as the power matrix method, requires slightly less time and number of calculations and handles effectively cyclic supermatrices, optimizing thus the whole procedure. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Analytic network process (ANP) is an MCDA method that takes simultaneously, several criteria, both qualitative and quantitative, into consideration, allowing dependence and feedback and making numerical tradeoffs to arrive at a synthetic conclusion indicating the best solution out of a set of possible alternatives. ANP was officially introduced by Saaty (1996) as a generalization of the analytic hierarchy process (Saaty, 1980). Since then it has gained the attention of many academics and practitioners. The first journal publication of the initial concept of the method appeared in 1986 (Hamalainen and Seppalainen, 1986) followed by another one in 1993 (Azhar and Leung, 1993) where the method was still immature and poorly documented. The method appeared again during 1998, with two articles (Ashayeri et al., 1998; Meade and Sarkis, 1998). After that, the rate of publications kept low although the interest of the researchers in this new method was gradually growing. As a result, the number of ANP publications increased rapidly between 2005 and 2009. The trend for the ANP-related articles shows that publications will still rise thus, more articles are expected in the near future. This phenomenon is largely attributed to the fact that ANP has been proven valid for many different fields. Its applications vary from strategy selection, energy-related deci⇑ Corresponding author. Tel.: +30 2271035457; fax: +30 2271035499. E-mail addresses: [email protected] (K. Kirytopoulos), [email protected] (D. Voulgaridou), [email protected] (A. Platis), [email protected] (V. Leopoulos). 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.03.043

sions, information technology and conflict resolution problems to medical, financial and forecasting issues. Despite its many advantages, ANP, as most of the sophisticated decision making methods, entails some limitation regarding the easiness of calculations. Its implementation requires a significant amount of calculating effort by the analyst that is usually compromised with the use of relevant software (Kirytopoulos et al., 2009). One of the calculating difficulties is that ANP engages the utilization of the ‘power’ matrix method, where an N  N size matrix has to be raised to an arbitrarily large power until it converges. This procedure is indispensable in order for the results to be stable and reliable. For more information about this ‘power’ matrix method the reader may refer to Saaty (2005). Based on the above, the authors propose here an alternative approach for the ANP analysts to achieve the stable state (steady state) of the decision. This approach has never been used in the past, as far as ANP is concerned. Thus, the academic contribution of this paper consists in offering an alternative mathematical approach for solving a well-known problem that may spark further research and lead to ANP optimization. Apart from introducing a different way to achieve practically the same results (which has always been regarded as an interesting academic endeavour), it is shown here that the proposed approach is also less complex and slightly quicker than the one currently used. It uses the concept of the Markov chain processes as well as the Gauss–Jordan elimination method. It should be noted that by the time the ANP method was firstly introduced, Saaty (1996) had noticed that the

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Supermatrix concept is similar to the Markov chain processes, however, the method is still based on the power method for the calculation of the Limit Matrix (Saaty, 2010). The proposed research falls into the category of optimization of MCDA methods, frequently published in EJOR. Specifically, it belongs to the same category as the articles of Yu and Cheng (2007), Leung and Cao (2001), Raharjo et al. (2009), Saaty and Shih (2009), Saaty (2006), Zahir (1999) and Saaty (2003) that foster the use and optimization of the AHP–ANP methods. The rest of the paper is organized as follows: Section 2 provides a general description of the ANP, its applications as well as the rationale of the method. Section 3 describes the algorithm proposed here to replace the traditional method and offers its verification, complexity and validation. Finally, Section 4 concludes the paper.

2. The analytic network process Fig. 1. Applications of ANP.

2.1. General description The analytic network process is the generalization of the analytical hierarchy process (AHP) as it incorporates feedback and interdependent relationships among decision criteria and alternatives (Jharkharia and Shankar, 2007; Voulgaridou et al., 2008; Voulgaridou et al., 2009). Technically, the model consists of clusters and elements. The dominance or relative importance of influence is the central concept. The ANP provides a general framework to deal with decisions without making assumptions about the independence of higher-level elements from lower-level elements and about the independence of the elements within a level as in hierarchal decision making methods. In fact, the ANP uses a network without the need to specify levels. As in the AHP, the decision maker provides judgments using the fundamental scale of the AHP by answering the question (Saaty, 2005): Which of the two elements Y, Z influences a third element X more with respect to the control criterion? In order that all such influences are considered with respect to the same criterion so they would be meaningful to synthesize, it is essential that the same criterion is used to make all the comparisons. Such a criterion is called control criterion. The control criterion is directly connected to the structure of the problem and it usually represents the ultimate goal (e.g. to select a vendor, to choose the most important project, etc.). The following section presents the applications of the ANP found in related literature. 2.2. Applications of ANP

decision criterion. During this step, the decision maker and the analyst determine the criteria, the alternatives (which in ANP are treated as criteria) and the interdependencies among them. Step 2 – Clusters’ and elements’ pairwise comparisons. During this step, the decision makers perform a number of pairwise comparisons with respect to a control criterion. Clusters and elements are compared with respect to their influence on another cluster or element, accordingly, by applying the fundamental scale of AHP (Saaty, 1996). The consistency of the judgments should be taken into account and the inconsistency ratio is an indicator that identifies possible errors. For example, if cluster A is more important than B and B is more important than C, C should be probably less important than A. In general, the inconsistency ratio should be less than 0.1 (Demirtas and Ustun, 2008). Then, the eigenvector method is employed to obtain local priority vectors for each pairwise comparison matrix. The matrix containing the weights of all clusters is called Clusters’ Priority Matrix. Step 3 – The elements are arranged both vertically and horizontally by clusters. This matrix is known as the Supermatrix. The priorities of the elements are read from the vertical columns. The Supermatrix of an ANP network and detail of the matrices in it are presented in the matrices (1) and (2), respectively.

ANP is a well-known method that since its introduction has been recognized by scholars and widely used in various decision analysis processes (Kirytopoulos et al., 2008). The authors performed an extended literature review concerning the applications of ANP and found 133 articles published by eight publishers in 51 different journals, between years 1986 and 2009 (July). The variety of these journals covering areas such as operations research, engineering, environmental issues, medical decisions, indicates the broad-spectrum applications of the ANP. Fig. 1 depicts the applications of ANP per topic. 2.3. Implementation steps and the mathematical part of ANP Implementation of ANP requires that the analyst along with the decision maker have to fulfil five steps: Step 1 – Model construction: The problem is decomposed into a network where nodes correspond to clusters. Each cluster is a set of elements and each element corresponds to a

ð1Þ

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where:

2

ðj Þ

W 1 6 i1 6 ðj Þ 6W 1 6 i2 W ij ¼ 6 6 .. 6 . 4 ðj Þ

W in1 i

ðjn Þ

ðj Þ

   W i1 j

ðj Þ

   W i2 j .. .. . .

W i12 W i22 .. .

ðj Þ

W in2 i

3

7 7 7 7 7 7 7 5 Þ

ðjn Þ

ðjn

   W in j

ð2Þ

i

andci: cluster i, eij: element j of cluster i, ni the number of elements of cluster i, jk the kth element of cluster j (where k = 1, . . . , nj). Step 4 – Formation of the Weighted Supermatrix. The weighted priorities at the Clusters Priority Matrix are used to weight all the elements in the block of column priorities of the Supermatrix corresponding to the impact of the elements of that cluster on another cluster (Saaty, 2010). This process is repeated for all the clusters and results in the Weighted Supermatrix. Thus, the elements of each column of the Weighted Supermatrix sums to one and the Matrix becomes column stochastic. This ‘column stochastic’ feature of the Weighted Supermatrix allows its convergence to a Limit Matrix. Step 5 – Formation of the Limit Matrix. Finally, the Weighted Supermatrix is transformed into the Limit Matrix by raising itself to powers (Saaty, 1996, 2005). The reason for multiplying the Weighted Supermatrix is because we wish to capture the transmission of influence along all possible paths of the Supermatrix. The entries of the Weighted Supermatrix represent only the direct influence of any element on any other element, but an element can influence a second element indirectly through its influence on a third element that has the direct influence on the second element. Such one-step indirect influences are captured by squaring the Weighted Supermatrix, and two-step indirect influences are obtained from the cubic power of the matrix, and so on. Raising the Weighted Supermatrix to the power 2k + 1, where k is an arbitrarily large number, allows convergence of the matrix, which means that all columns are the same (take the same value for each specific row of the matrix). The stopping criterion for iterations is satisfied when all the elements of W2k+1 are equal to all the elements of W 2k ðw2kþ1 ¼ w2k ij Þ with a cerij tain precision. The precision usually used ranges from 2 to 6 decimal places. In this research the fourth decimal place is used. The resulting matrix is called the Limit Matrix, which yields limit priorities capturing all the indirect influences of each element on every other element. From this Limit Matrix the decision maker can derive the final solution (final weights for alternatives and criteria). For more details on Supermatrix characteristics and theory, the reader may refer to Saaty (2005, 2010). This paper is focusing on the last step of the method and hereafter proposes an alternative approach for calculating the Limit Matrix. 3. Proposing an alternative approach for calculating Limit Matrix 3.1. Mathematical formulation of the proposed algorithm The Weighted Supermatrix W has interesting properties such as that all the coefficient matrix elements are between 0 and 1 and the sum of the elements of each column equals one. Let us define P = WT, this matrix has the sum of all its rows equal to one and all its elements between 0 and 1. As a result, this

matrix can be considered stochastic and furthermore, the transition probability matrix of a Markov chain X. The computation of the Limit Matrix W1 is therefore equivalent to the computation of the asymptotic steady state probability distribution of the chain X. The existence and uniqueness of the steady state probability distribution is guaranteed if the chain X is irreducible and acyclic, which are similar necessary properties for the existence and uniqueness of the Limit Matrix W1. Various papers and books treat the ergodicity of Markov chains (Revuz, 1984; Trivedi, 2001; Platis and Drosakis, 2009), semiMarkov processes (Limnios and Oprisan, 2001), cyclic nonhomogeneous Markov chains (Platis et al., 1997; Platis, 2006; Koutras et al., 2009) and general nonhomogeneous Markov chains (Platis et al., 1998). If the Markov chain is irreducible and acyclic, then P raised to an arbitrary large power tends to an ergodic matrix E. This matrix has similar rows (p), i.e.,

lim Pn ¼ E ¼ 1:p;

ð3Þ

n!1

where 1 is a Nth dimensional column vector containing ones. If a is the initial distribution of the chain, the state probability vector at time n, SP(n), is given by

SPðnÞ ¼ aPn ;

ð4Þ

and the steady state probability vector, SSP is given as follows:

h i SSP ¼ lim aPn ¼ a lim Pn ¼ a:1:p ¼ p; n!1

ð5Þ

n!1

since a  1 = 1. If we assume that the chain is irreducible and acyclic, vector p, is the sole steady state probability distribution of the chain. If p is the asymptotic probability distribution at an arbitrary instant of time t, at the next unit of time, the distribution of the chain which is pP at the instant t + 1, remains stationary, i.e.:

pP ¼ p or pðP  IÞ ¼ 0;

ð6Þ

where 0 is a row vector containing zeros. Solving p(P  I) = 0 will not give a unique solution, an additional constraint p  1 = 1 (the sum of the steady state probabilities equals one) is necessary in order to guarantee the uniqueness of the solution. In order to take into account the additional constraint, the last column of the matrix P  I is replaced by a column of ones and the right hand side is replaced by a row vector containing zeros except its last element which is one. After a final transposition, the following linear system is obtained in compact matrix notation:

Au ¼ s;

ð7Þ

where u is the solution column vector (pT), sT = (0, 0, . . ., 0, 1) and A = (W  I) with its last row replaced by a row of ones, i.e.:

2

W 1;1  1

6 6 W 2;1 6 6 6 .. A¼6 . 6 6 6 W N1;1 4 1

W 1;2 W 2;2  1 .. . W N1;2 1

...

W 1;N

3

7 W 2;N 7 7 7 7 .. .. 7: . . 7 7 . . . W N1;N 7 5 ...

ð8Þ

1

It should be noted that the resulting dense linear system can be solved by exploiting the Gauss–Jordan elimination method. Supposing that

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2

Q ¼ A1

3 Q 1;N Q 2;N 7 7 7 .. 7 . 5

Q 1;1 6 Q 2;1 6 ¼6 6 .. 4 .

Q 1;2 Q 2;2 .. .

...

Q N;1

Q N;2

. . . Q N;N

..

.

ð9Þ

we get the solution column vector, u, by multiplying the Q matrix (N  N) with the column matrix s (N  1), containing N  1 zeros and 1 in the Nth element.

2

Q 1;1 6 6 Q 2;1 6 u¼Q s¼6 6 .. 6 . 4

Q 1;2

Q N;1

Q N;2

Q 2;2 .. .

2 3 0 3 2 Q 1;N . . . Q 1;N 6 7 7 6 7 607 7 6 6 Q 2;N 7 Q 2;N 7 7 7 6 7 6 7 6 7  607 ) u ¼ 6 7 7 6 .. .. 7 . 6 7 6 .. 7 .. 7 . . 7 5 6 5 4 6. 7 4 5 . . . Q N;N Q N;N 1 3

3.3. Complexity comparison and validation of the proposed algorithm

ð10Þ In general, for solving linear systems there is a choice between direct methods such as the Gauss–Jordan elimination and wellknown iterative methods such as the power method, Jacobi, Gauss–Seidel and Successive Over-Relaxation (SOR) methods. 3.2. Verification of the proposed algorithm The proposed algorithm was verified by comparing the results between the traditional algorithm (power method) and the proposed one in 22 cases that appear in the bibliography. In order to find those cases the authors searched the Scopus database by using the keywords ANP and analytic network process in the article title, abstract and keywords fields. Of the results returned, the papers that provided the Weighted Supermatrices were selected as inputs in both the traditional and the proposed algorithm. Thus, the set of the cases/papers represent an impartial sample and the result can be considered objective. The bibliographic references of the examined cases/papers are given in Appendix A. The paper describing each case offers the Weighted Supermatrix and the Limit Matrix, which, as stated in the relevant cases/papers, were calculated with the traditional ‘‘power method’’. The authors, taking into account the Weighted Supermatrix offered in each case/paper, calculated the Limit Matrix by using the proposed algorithm. The outcomes of the calculation with the proposed algorithm of this paper were practically the same as those offered in the papers describing each of the examined cases thus, the algorithm was verified. For the calculations with the proposed algorithm of this paper, the authors used ‘C’ programming language. An interesting finding was the capability of the method to handle cyclic matrices as well, without having to modify the algorithm described in Section 3.1. A well known consideration for the power method is that under specific initial matrix formation (in our case weighted supermatrix formation) the process of raising it to an arbitrarily large power yields to a cyclic matrix of the following form (example for a matrix with two periods cycle):

W n1 ¼

Wn ¼





A

0

0

B

0

C

D

0

 ;

When this is the case, the power method should be stopped by the time when for each row of the matrix the non-zero elements are the same (stabilise) in two successive raisings (we reach Wn1 and Wn with the properties described above). The limit cycles in blocks and the different limits are summed and averaged and normalized to give the final importance of each element of the ANP method. In such a case, the proposed approach can be used unmodified and offer exactly the same results. Typical examples of such instances can be found in cases/papers 10 and 13 (refer to Appendix A).

 ð11Þ

ð12Þ

where the size of each sub-matrix is A(n⁄m), B(k⁄l), C(n⁄l) and D(k⁄m) and all elements of the nth row of sub-matrix A equal all elements of the nth row of sub-matrix Cand respectively, all elements of the kth row of sub-matrix D equal all elements of the kth row of sub-matrix B.

The same cases/papers (as in verification) were exploited for the validation of the model which comprises in the optimization of the calculation. The two approaches were compared in terms of complexity. At first sight, the two approaches seem to be equal as the power method entails matrices multiplication and the proposed one is based on matrix inversion with Gauss–Jordan elimination. Thus, complexity of N  N-matrices multiplication and inversion of a N  N matrix is O(n3) for both (for more on Big O notation the reader may refer to Knuth (1976)). However, there is still a slight advantage for the number of calculations in the proposed approach than in the commonly used power method. The number of calculations in the proposed exact method for the inversion of an N  N matrix are (3/2)⁄N3 + (1/2)⁄N  1 multiplications and (3/ 2)⁄N3  2⁄N2 (1/2)⁄N additions. On the other hand, the power method is an iterative one thus, the number of calculations needed is proportional to the amount of data and the exact number of matrices multiplications. Specifically, for a N  N matrix, it entails K⁄(N3) multiplications and K⁄(N3  N2) additions, where K is the number of iterations in order to reach the steady state. In the typical case for exploring complexity, we assume that n -> 1 thus, the complexity of both approaches is the same. However, in the ANP this is not the case at all. Since ‘N’ represents the number of criteria, it will always range from (let us suppose) four to a relatively small number. In all the examples examined in this research, ‘N’ was ranging from 5 to 29 where K which is the number of iterations for the power method was ranging from 6 to 262. That is, K is at the same (if not higher) order of magnitude with N which in turn means that the number of calculations lies on the level of N4(=K⁄(N3)). This finding was also proven empirically by measuring, for each case/paper of Appendix A, with the Power method and the Gauss– Jordan elimination method the time needed for calculating the Limit Matrix given the Weighted Supermatrix. The time for both algorithms was measured on a PC with processor Intel Core (2)Duo, Clock Speed 2.40 GHz and RAM 4.00 GB. For each case/paper the time needed for both methods was measured 50 times in order to avoid skewed results due to any random reason during time measurement and the mean values for each case/paper and method (Power method – Gauss–Jordan method) were calculated. The standard deviation for all cases/papers and methods were small. Fig. 2 depicts the time needed to calculate the Limit Matrix with the Power Method (dark coloured bars) and the Gauss–Jordan elimination (light coloured bars), the acceleration achieved with the Gauss–Jordan elimination (stars inside the graph), while the data table in Fig. 2 includes also the number of iterations needed for convergence with the power method, as well as the dimensions of the Weighted Supermatrix for each case explored (x-axis shows each case number). It is indicative of the performance of the Gauss–Jordan elimination method that in all of the cases/papers examined the increase in calculation speed was over 80%. Although both methods are executed instantly by the use of typical personal computers, the proposed algorithm proved to be quicker in pure mathematical terms.

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Fig. 2. Mean time for calculating the Limit Matrix.

4. Discussion ANP is a well known and widely applied technique that has gained the attention of many authors and decision makers since its development. The main advantages of the method are considered to be the simultaneous inclusion of tangible and intangible factors, the ability to capture the interactions among them and its accordance with the human way of thinking. The ‘power’ matrix method, a procedure necessary for the stability of the decision system, is one of the critical calculations in the mathematical part of the method. The aim of this paper is to propose an alternative approach for the calculation of the Limit Matrix. The proposed approach is based on Markov chain processes and the well-known Gauss–Jordan elimination. It obtains practically the same results as the power matrix method, requires slightly less time and number of calculations and handles effectively cyclic supermatrices, optimizing thus the whole procedure. If one would like to compare the two approaches based on the findings from this research, the following conclusions could be drawn. Initially, it appears that the power method requires saving the intermediate results of each step in order to be able to check different halting conditions, which is a more complex process, in contrast to the proposed approach where there is no need to do so, as there are not intermediate results or multiple conditions to check. This is caused by the fact that the power method uses specific processes for each type of initial matrix. As an example, the process followed to compute final priorities of a primitive irreducible matrix is different of the process for cyclic matrices, something that is not the case in the proposed method that provides a unique process for computing final priorities regardless of the initial matrix form, provided that it is invertible. On the other hand, the power method provides a way to compute final priorities in any case, despite the type of initial stochastic matrix given as input, whereas the proposed algorithm cannot return results in case the initial matrix is singular (not invertible) which is an interesting point for further research. Another remark is that the number of needed iterations, K, of the power method is sensitive to the precision used to make computations, as the time for convergence is analogous to the requested precision of the inputs. This iteration process is also leading to another differentiation between the two methods. Due to the need to raise a matrix to a large power until it converges

for the power method, it is essential to use specialised software. However, the proposed approach, as an exact method, can be executed with a common spread sheet thus, its use is more appropriate in cases where the analyst has to raise the weighted matrix to a very large power for achieving convergence. The proposed method is expected to facilitate the diffusion of an already promising method, by attracting more scholars and practitioners and moreover, give new opportunities for further research. Acknowledgements The authors thank the editor and the anonymous reviewers for their constructive remarks that helped significantly the improvement of this paper. Appendix A. Cases/papers examined Case/paper ID

Reference

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Niemira and Saaty (2004) Huang et al. (2005) Neaupane and Piantanakulchai (2006) Promentilla et al. (2006) Chang et al. (2007a) Chang et al. (2007b) Whitaker (2007) Demirtas and Ustun (2008) Raharjo et al. (2008) García-Melon et al. (2008) case 1 García-Melon et al. (2008) case 2 Lin and Tsai (2010) Aragonés-Beltrán et al. (2008) case 1 Aragonés-Beltrán et al. (2008) case 2 Aragonés-Beltrán et al. (2008) case 3 Ayag and Ozdemir (2007) Lee et al. (2009) Lin et al. (2009) Lin and Tsai (2009) Liao and Chang (2009) Cortés-Aldana et al. (2009) Kirytopoulos et al. (2009)

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