Decision Support Biomedical Application Based on Consistent ...

Decision Support Biomedical Application Based on Consistent Optimization of Preference Matrices Richard Cimler1,2(B) , Martin Gavalec1 , Karel Mls1 , and Daniela Ponce1 1

Faculty of Informatics and Management, University of Hradec Kr´ alov´e, Hradec Kr´ alov´e, Czech Republic [email protected] 2 Center for Basic and Applied Research (CZAV), University of Hradec Kr´ alov´e, Hradec Kr´ alov´e, Czech Republic http://www.uhk.cz

Abstract. In designing a medical application, big emphasis must be given on correct understanding of experts’ opinions. This article deals with the decision making support based on Analytic Hierarchy Process (AHP). Inconsistency measures of preference matrices and the problem of consistent optimization are studied. A new method for computing the optimal consistent approximation of a given preference matrix is described. Furthermore, algorithms for finding a consensus of several experts are discussed. The proposed methods are presented and explained on numerical examples.

Keywords: Analytic Hierarchy Process consistent optimization

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Inconsistency

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Orthogonal

Introduction

Smart technologies are nowadays an inherent part of human life. In the last decades, the increase in computational power and miniaturization enabled to create various devices simplifying human life. Health care is one of branches where modern devices are widely used. There are various technologies for collecting data [3,8,27], data processing and, based on the results, evaluating the health status of a person. Complex algorithms are used to process the data and built-in devices make decisions about actions. The devices may only be used as an information source and as a support for decision making, or they can also perform autonomous actions and operate without human interaction. During creation of every application, there must be experts who share their knowledge with the programmers of the application. The aim of every expert system is to create an autonomous application which substitutes the expert’s decisions, and is able to make decisions based on expert’s knowledge and experience, even without his/her personal presence. c Springer International Publishing Switzerland 2016  N.T. Nguyen et al. (Eds.): ICCCI 2016, Part II, LNAI 9876, pp. 292–302, 2016. DOI: 10.1007/978-3-319-45246-3 28

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In designing a medical application, big emphasis must be given on correct understanding and programming of experts opinions. Nobody is perfect, and even an expert on a particular domain might be sometimes not 100 % sure. The decision can be influenced by different perturbations. Creation of medical applications requires that experts from different domains discuss the problem and try to find consensus during the creation of the rules for an expert system. Finding the right way to the optimal solution might be sometimes very difficult. There might by different views on the same thing by various experts. This article deals with the decision making based on Analytic Hierarchy Process (AHP). Matrices used in this process are created by many experts. Algorithms for finding a consensus of several experts are introduced. The article is divided into the following sections. The Analytic Hierarchy Process is introduced in the next section. Inconsistency of preference matrices and ways to measure it are discussed in Sect. 3. Our proposed method for finding the optimal consistent approximation of a given preference matrix is described in Sect. 4. The problem of aggregation of multiple experts’ opinions is discussed in Sect. 5. Proposed methods are presented on implementation examples in Sect. 6. The last section of the article summarizes the results.

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Analytic Hierarchy Process

In the AHP approach to modeling decision situations, priority scales are derived formally after pair-wise subjective judgments are made. According to Saaty [26], there are at least three reasons to make comparisons instead of direct measurements: 1. the lack of instrument or scale to make a measurement, 2. our belief that the outcome of comparisons using our judgments would be better than using some general scale of measurement, 3. a way to measure something at present - happiness, popularity or aesthetic appeal.

Fig. 1. AHP model with criteria Ci and alternatives Aj

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Moreover, thinking about values in economics is mostly based on utility theory, implicitly subsuming benefits, opportunities, costs and risks. The construction and evaluation procedure of the AHP model can be summarized as follows [24], see Fig. 1: – Model the problem as a hierarchy containing the decision goal, the alternatives for reaching it, and the criteria for the alternatives evaluation. – Establish priorities among the elements of the hierarchy by making a series of judgments based on pairwise comparisons of the elements. – Synthesize these judgments to yield a set of overall priorities for the hierarchy. – Check the consistency of the judgments. – Reach the final decision based on the results of this process. Given alternatives A1 , A2 , .., An will be considered till the end of the paper. N will denote the set {1, 2, . . . , n} and R the set of all real numbers. For every pair Ai , Aj , the real number aij is interpreted as an evaluation of the relative preference of Ai with respect to Aj , in the additive sense. The matrix A = (aij ), i, j ∈ N is called additive preference matrix (for short: preference matrix) of the alternatives A1 , A2 , .., An . The basic properties of preference matrices are defined as follows A is antisymmetric if aij = −aji for every i, j ∈ N , A is consistent if aij + ajk = aik for every i, j, k ∈ N . Clearly, if A is consistent, ⎛ then A is⎞antisymmetric, but the converse implication 0 1 1 is not true. E.g., A = ⎝ −1 0 1 ⎠ is antisymmetric, but it is not consistent, −1 −1 0 because a12 + a23 = 1 + 1 = 2 = a13 . Another frequently used form for the relative preferences of alternatives are multiplicative preference matrices. If M ∈ R+ (n, n) is a multiplicative preference matrix, then every entry mij with i, j ∈ N is considered as an multiplicative evaluation of the relative preference. The multiplicative preference matrices have analogous properties as the additive preference matrices. In fact, they can be equivalently transferred to each other by the logarithmic and exponential transformations. The reason for which the additive form is used here for expressing the relative preferences is that we work with the notions of linear spaces and with linear methods, which are based on linear combinations of variables.

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Inconsistency Measures

The fundamental question in the AHP decision making process is how to find an appropriate preference matrix for a set of alternatives. The preferences given by human experts are often inconsistent and do not reflect the deep relations between the processed notions, see [6,16,20–22].

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One way of solving the inconsistency problem for a preference matrix is to define the consistency index of A and the consistency ratio of A CI(A) =

λmax − n , n−1

CR(A) =

CI(A) , ARI(n)

(1)

where λmax is the principal eigenvalue of A, n is its size and ARI(A) is the average consistency index of randomly generated reciprocal matrices of size n. Then the preference matrix A is considered to be acceptable if CR does not exceeds the empirical value 0.1, see [23,25]. Many further inconsistency measures were defined like: simple normalized column sum, preference weighted geometric mean, least square deviation and/or least absolute error deviation sum (see [2], for a discussion on these and other inconsistency measures). Based on a numerical study of several inconsistency measures, the authors in [4] point out that there is no agreement in the research community on which measure of inconsistency should be used, and moreover, little attention is paid to similarities and differences of various measures. When the preference matrix is not consistent, the decision maker faces a dilemma: either to reject the preference matrix and replace it by a new preference matrix or to modify the preference matrix in such a way that it becomes consistent. This decision can be supported by the inconsistency measure of the matrix, that is, when the inconsistency measure of the preference matrix is acceptably low for the decision maker, then the consistent approximation of the preference matrix can be reasonable. Put this way, the measure of inconsistency of a matrix A can be interpreted as a measure of acceptability of the approximating matrix X. The smaller the measure of inconsistency of A is, the more acceptable the consistent approximation X to matrix A is for the decision maker, because it is the nearest consistent matrix to A from all consistent matrices. The search for consistent approximation of an inconsistent preference matrix has been investigated in some previous works, see e.g. [10,13,17]. Inconsistency measure may be based on distance of preference matrices (see e.g. [15]), but not necessarily. For example, error-free correctness methods [7] give 0 for consistent matrices and positive values otherwise, using some formula with the inconsistent matrix as input value. In their work [5], Cao et al. propose an algorithm that gradually modifies the given inconsistent matrix till the inconsistency measure of the matrix drops below an acceptable value. Another approach to the inconsistency problem is to take the values in the expert’s preference matrix A as a starting point for computing a good consistent approximation of A. Such computations have been suggested e.g. in [12,18,19]. Various distance metrics are used and compared in [18]. The additive form of expressing the relative preferences is applied. The additive form is more convenient for the optimization purposes than the multiplicative form, as the notions and methods from linear algebra can directly be applied. A new method for measuring the inconsistency of preference matrices is described in this paper. The method is based on the orthogonal projection of

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a given (possibly inconsistent) preference matrix to the linear subspace of all consistent matrices. In biomedical applications, usually more than one expert participates in creating the preference matrix. Every expert creates his/her personal version of preferences and the individual judgments are then merged to make the common preference matrix. In practical applications, the preference matrix is created by a human expert in the given field. While the antisymmetricity is easy to verify, the consistency of a preference is not directly seen from the data. As a consequence, the preference matrices given by experts are often inconsistent. The following approximation problem is investigated in this section: given a (possibly inconsistent) preference matrix A, find a consistent matrix X which will be as close to A as possible. Matrix X is then called the optimal consistent approximation of A. Clearly, every approximation depends on the distance measure that is used in the optimization. A general family of distances lp with 1 ≤ p ≤ ∞ is known in the literature. For chosen value p, the distance of vectors x, y ∈ R(n) is  lp (x, y) =



1/p |xi − yi |

p

(2)

i∈N

For p = 1, p = 2, or p = ∞, (2) gives the so-called Manhattan distance l1 (x, y) =

2 2 i∈N |xi − yi |, Euclidean distance l (x, y) = i∈N |xi − yi | , or Chebyshev ∞ distance l (x, y) = maxi∈N |xi − yi |. The optimal consistent approximation according to the above three distance functions have been compared in [18]. In this contribution, the Euclidean distance will only be considered, and denoted simply by l.

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Orthogonal Consistent Optimization

In the AHP theory, the consistent preference matrices are closely related with preference vectors showing the importance for every of alternatives. In the additive notation, vector w ∈ R(n) is called a balanced preference vector (for short: a balanced vector) if i∈N wi = 0. When alternatives A1 , A2 , . . . , An are considered, then wi is interpreted as the preference degree of Ai for every i ∈ N . The differences of preferences are the entries of the corresponding matrix of relative preferences A(w) with aij (w) = wi − wj for i, j ∈ N . We say that A(w) is induced by w. The relation between preference matrices and balanced vectors has been described in [19]. Theorem 1. [19] Let A ∈ R(n, n) be a preference matrix. (i) If w ∈ R(n), then the induced preference matrix A(w) is consistent. (ii) If w, w ∈ R(n) and A(w) = A(w ), then w = w + δ for some δ ∈ R.

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(iii) If A is consistent, then there is a unique balanced vector w such that A = A(w). We are interested in solving the following optimization problem: OCA (optimal consistent approximation) given A ∈ R(n, n), minimize z = l(A, X) −→ min

(3)

such that X ∈ C. Suppose n alternatives A1 , A2 , . . . , An are considered. Denote by B the set of all balanced vectors w ∈ R(n), and by C the set of all consistent matrices A ∈ R(n, n). C is sometimes called the consistency subspace. According to Theorem 1(iii), any w ∈ B induces a uniquely determined consistent matrix A(w). Moreover, any linear combination i∈K ci wi with ci ∈ R and wi ∈ B, for every i ∈ K = {1, 2, . . . , k} induces the linear combination      (4) A ci wi = ci A wi i∈K

i∈K

of consistent matrices in C with the same coefficients. Using Theorem 1 (iii) and (4), we get the following result. Theorem 2. Suppose n alternatives are considered. Then B is a linear subspace of R(n), C is a linear subspace of R(n, n), (w1 , w2 , . . . , wk ) is a base in B if and only if A(w1 ), A(w2 ), . . . , A(wk ) is a base in C, (iv) dim(B) = dim(C) = k = n − 1.

(i) (ii) (iii)

In view of Theorem 2, the solution to the optimization problem OCA can be found as the orthogonal projection X = P (A) of the input matrix A into C. Assume B1 , B2 , . . . , Bk is a base in C ⊂ R(n, n). Then the solution can be expressed as linear combination X = j∈K cj Bj with cj ∈ R. The coefficients must satisfy the orthogonality conditions ⎞ ⎛  Bi · ⎝A − (5) cj Bj ⎠ = 0 for every i ∈ K j∈K

where the dots in (5) indicate the sumproduct operation in R(n, n). By easy modification we get  cj (Bi · Bj ) = Bi · A for every i ∈ K (6) j∈K

If c1 , c2 , . . . , ck satisfy conditions (6), then Y = A − X is orthogonal to every Bi from the base of C. That is, Y is orthogonal to subspace C and X is the

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orthogonal projection of A to C. Therefore, X is determined uniquely, and does not depend on the base B1 , B2 , . . . , Bk . On the other hand, for a fixed base, the coefficients c1 , c2 , . . . , ck are determined uniquely, and (6) has exactly one solution. The above considerations are formulated in the following theorem. Theorem 3. Assume A ∈ R(n, n) is a preference matrix. Then (i) the optimization problem OCA with input A has unique solution X, (ii) X is the orthogonal projection of A to the consistency subspace C, (iii) if B1 , B2 , . . . , Bk is a base in C, then X can be computed as a linear combination with coefficients satisfying (6). The orthogonal consistent optimization method described in Theorem 3 is called OCO. Remark 1. It is clear that if the input matrix A is consistent, then X = A. Remark 2. Along with other optimization problems, OCA has been studied in [18]. Solution has been found by the least squares method, under the assumption that A is antisymmetric.

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Methods of Aggregation

There are various approaches how to reach consensus from several (usually inconsistent) experts judgments in group decision making. Dong in [11] proved that both consistency and consensus may not be perfect in general and so there is a need of some measure or degree for them. The consistency measure is then used to quantify the difference among individual decision makers and to help in finding an acceptable solution of the consensus model. Considering AHP as a platform for decision making support, opinions of individual experts can be merged in several ways. Aggregation of individual opinions can be done at the initial stage of the decision making process by searching consensus on the given judgments (aggregation of individual judgments AIJ) or at the last stage by individually induced preferences with importance weights of the group members (aggregation of individual priorities - AIP) [11]. Between these stages, other approaches, e.g. voting, can be applied to the decomposed intermediate judgments aggregation [14]. We use the Synergy Aggregation Method (SAM) that takes preference matrices (i.e. individual judgments of experts involved), approximates them by the nearest consistent matrices if needed and produces an aggregated consistent preference matrix (i.e. collective judgment). We furthermore suppose that each expert excels in a particular part of the problem and therefore for every expert we assign weights to pairwise alternative comparisons formulated by that expert. These weights are then used in the calculation of the weighted arithmetic mean of preference matrices. Because of the specific features of the problem, we can suppose that experts work in synergy and they agree on the rank of alternatives.

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As a consequence, the preference matrices of all experts can be all transformed into matrices such that they contain non-negative elements only in the upper triangle matrices. In the case of discordance, other aggregation methods are applied, see [1].

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Implementation Examples

The same procedures as in [9] are used in the examples below describing a health-monitoring application. We assume that experts, in our case doctors, have prepared a preference matrix A. The submitted matrix may be neither consistent nor antisymmetric. Example 1 In the first example there is only one expert who created an inconsistent and non-antisymmetric matrix A. For implementation into the application we need to find a consistent and antisymmetric matrix which is as close as possible to A. The method OCO described in Theorem 3, is used for computing such a matrix. The alternatives in this example are the levels of alert: Normal, Warning level 1, Warning level 2, and Alarm. The entries of the matrix have been suggested by the expert to express which level of alert is the most appropriate in the given situation (such as: blood pressure = increased, heart rate = slightly increased, body temperature = high). ⎛ ⎞ 0 −2 −3 1 ⎜ 2 0 −1 2 ⎟ ⎟ A=⎜ ⎝3 1 0 2⎠ 3 −2 −2 0 The computed matrix X is consistent and antisymmetric. The Euclidean distance from the original matrix A is 3. ⎛ ⎞ 0 −2.25 −3.0 −0.75 ⎜ 2.25 0 −0.75 1.5 ⎟ ⎟ X=⎜ ⎝ 3.0 0.75 0 2.25 ⎠ 0.75 −1.5 −2.25 0 Example 2 In the second example we assume that several experts are creating preference matrices related to the same problem. There might be different opinions on the same problem thus preference matrices can be different. Using the methods OCO and SAM described in Sects. 4 and 5, we can get one consistent antisymetric matrix which will be the closest one to all matrices created by the individual experts. According to the experts’ competences there are different weights q of experts’ suggestions. In this example: q (1) = 4, q (2) = 3, q (3) = 1. ⎛

A(1)

⎛ ⎛ ⎞ ⎞ ⎞ 0 −2 −3 1 0 −2 −3 −2 0 −2 −4 −3 ⎜ 2 0 −1 2 ⎟ (2) ⎜ 2 0 ⎜ ⎟ 1 4⎟ ⎟A =⎜ ⎟ A(3) = ⎜ 3 0 −3 −2 ⎟ =⎜ ⎝3 1 ⎝ 3 −1 ⎝4 3 0 2⎠ 0 4⎠ 0 4⎠ 3 −2 −2 0 2 −4 −3 0 2 3 −4 0

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In the first phase of the solution we compute the approximations by consistent and antisymmetric matrices using the ⎛ ⎛ described in Sect. 4. ⎞ methods ⎞ X

(1)

0 −2.25 −3.0 −0.75 ⎜ 2.25 0 −0.75 1.5 ⎟ ⎟ X (2) =⎜ ⎝ 3.0 0.75 0 2.25 ⎠ 0.75 −1.5 −2.25 0 ⎛ 0 −1.5 ⎜ 1.5 0 (3) X =⎜ ⎝ 5.0 3.5 2.5 1.0

0 −3.5 −3.125 −0.375 ⎜ 3.5 0 0.375 3.125 ⎟ ⎟ =⎜ ⎝ 3.125 −0.375 0 2.75 ⎠ 0.375 −3.125 −2.75 0 ⎞ −5.0 −2.5 −3.5 −1.0 ⎟ ⎟ 0 2.5 ⎠ −2.5 0

By the aggregation method SAM, we then get from several consistent and antisymetric matrices one merged matrix X, which can be used in the biomedical application as a criterion matrix. The weighted average of experts’ individual preference matrices based on the experts’ particular competences has been used in our case. ⎛ ⎞ 0 −2.63 −3.30 −0.83 ⎜ 2.63 0 −0.67 1.8 ⎟ ⎟ X (final) ≈ ⎜ ⎝ 3.3 0.67 0 2.47 ⎠ 0.83 −1.80 −2.47 0

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Conclusions

The decision making in a medical application requires big emphasis on experts’ opinions and experience. Solving the inconsistency problem for preference matrices is therefore of vital importance. The euclidean distance of a given preference matrix to the linear subspace of all consistent matrices has been suggested in the paper as the inconsistency measure for preference matrices. Thus, the nearest consistent matrix can be computed as the orthogonal projection of a given preference matrix submitted by an expert. For a group of experts, the individual consistent approximations are merged into one final result, taking in view the experts’ competence levels. The proposed methods have been illustrated by numerical examples. ˇ #14-02424S and the Acknowledgment. The support of the grant project GACR specific research project SPEV at FIM UHK is gratefully acknowledged.

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