Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm

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Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm Q1

Bosko Blagojevic∗, Bojan Srdjevic, Zorica Srdjevic, Tihomir Zoranovic

Q2

University of Novi Sad, Faculty of Agriculture, Department of Water Management, Trg D. Obradovica 8, 21000 Novi Sad, Serbia

a r t i c l e

i n f o

Article history: Received 20 November 2014 Revised 13 October 2015 Accepted 17 October 2015 Available online xxx Keywords: Analytic hierarchy process Group decision making Consensus Heuristic aggregation

a b s t r a c t There are various aggregation procedures for obtaining a group priority vector within Analytic Hierarchy Process (AHP) supported decision making processes. This paper will introduce a heuristic aggregation procedure based on simulated annealing (SA) algorithm to be used for the purposes of obtaining a group priority vector at any node of an AHP hierarchy. The proposed procedure performs its aggregation process by minimizing the group Euclidean distance (GED) (consensus measure) across group weights and judgments, and the group vector obtained in this way is invariant to any prioritization method. In other words, there is no need to have individual priority vectors as is required by some other aggregation procedures. Along with SA minimization of the GED, the group rank reversal (minimum violation) criterion is implemented as a control mechanism, as well as the consensus measure based on the ranks of alternatives. The presented procedure is compared with several reported combinations of different prioritization methods and group aggregation procedures. Five examples from literature are used to show that the proposed procedure performs better or at least equally to several other well known combinations of prioritization and aggregation in AHP group decision making frameworks. © 2015 Published by Elsevier Inc.

1

1. Introduction

2

The Analytic Hierarchy Process (AHP) [28] is a multi criteria decision-making method that has been used in many applications related to decision-making problems [17], and is applicable to both individual and group decision making situations. The two most used procedures to obtain group priorities in the AHP are the aggregation of individual judgments (AIJ) and the aggregation of individual priorities (AIP) [1,13,26]. In situations which require the AIJ, the group matrix can be generated using consensus, voting or the mathematical aggregation of individual judgments, after which the group’s priorities can be calculated using any prioritization method. However, because in many group settings it is difficult to achieve consensus and obtain an accurate and successful round of voting, the usual choice for generating the group matrix ends up being mathematical aggregation. Geometric mean aggregation is suggested for the AIJ in order to preserve the reciprocal property of the generated group matrix and to account for possible significant differences in individual decision makers’ judgments [13]. In the second case (AIP), the priorities of decision makers (DMs) are first calculated using the selected prioritization method, after which individual priorities are aggregated using either the weighted geometric mean or weighted arithmetic mean method, though Ishizaka and Labib [18] encourage using the weighted arithmetic mean method.

3 4 5 6 7 8 9 10 11 12 13

∗

Corresponding author. Tel.: +381 21 4853 293; fax: +381 21 455 713. E-mail addresses: [email protected], [email protected] (B. Blagojevic), [email protected] (B. Srdjevic), [email protected] (Z. Srdjevic), [email protected] (T. Zoranovic). http://dx.doi.org/10.1016/j.ins.2015.10.033 0020-0255/© 2015 Published by Elsevier Inc.

Please cite this article as: B. Blagojevic et al., Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.10.033

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There is also a third procedure that should be considered for use in the context of group aggregation: the consensus convergence model (CCM) [21], where by repeatable mathematical procedure and through mutual respect the decision makers not only achieve consensus on the issue under consideration but also agree on the overall relative weight of each decision maker [32]. The original model assumes that DMs have opinions about the expertise and rationality of other DMs. The model proposed by Regan et al. [27] uses a weight of respect based on the strength of the difference in the alternative weights assigned by all DMs, and when used in combination with the AHP, priority vectors obtained from the individual judgment matrices of all DMs represent the input data to this model [32]. Group priorities can also be obtained by combining the AHP with compatible ’soft’ consensus models [e.g., 7,11,25,35–37]. Consensus is usually defined as the unanimous agreement of involved DMs after possible alternatives have been considered, but because more often than not unanimous consensus is impossible to achieve in real life [24], success of these procedures depends on correctly measuring the consensus degree, i.e. to measure the consistency of individual decisions against the group decision. The main idea is to generate advice on how the DMs with the highest inconsistencies with the group decision should change their preferences and contribute to the group consistency. In this study, the geometric cardinal consensus model (GCCM) [11], one of the most cited models, is used. This model requires the use of the logarithmic least squares (LLS) prioritization method and the geometric cardinal consensus index (GCCI) to measure consensus; an index is based on the geometric consistency index (GCI) given by Crawford and Williams [9]. The GCCM automatically changes individual judgments of DMs with the highest GCCI values according to the group priority vector, and this model has two desired features: firstly, in reaching a consensus the adjusted judgment matrix has a better individual consistency index (i.e., GCI) than the corresponding original judgment matrix, and secondly it is preserved that the Pareto principle of social choice theory is satisfied. We also find that this model is correct in the aforementioned sense although we are aware that the problem with the implementation of the model could be that it automatically modifies DMs’ opinions to reach consensus while in real life situations sometimes the DM may not want to change his/her opinion even if he/she demonstrates the highest inconsistency with the group decision. Determining the effectiveness of the AIJ, AIP or CCM as group aggregation procedures requires acknowledging that group priorities will also depend on the selected prioritization method. This is because in AHP-supported group decision processes it is quite often impossible to expect full consistency from both the individual and group pairwise comparison matrices, especially if the order of matrices is higher than three. Srdjevic [30] compared six prioritization methods against two consistency measures (Euclidean distance and minimum violation), and concluded that, though being quite simple, the additive normalization method (AN) is highly competitive with other more sophisticated methods such as, e.g., eigenvector method (EV) [28] or logarithmic least square method (LLS) [9]. Athough comparison analyses of prioritization methods are carried out by many authors [6,9,10,14,20,22], so far there is no clear agreement on which method is the best; rather, an open discussion is maintained on the effectiveness of different prioritization methods. In our study we used prioritization methods EV, AN and LLS. It is important to note here that when using EV and AN, the group priority vector obtained by the AIP and AIJ can be different, while in case of LLS, the AIJ and AIP produce the same group priority vector [3]. In this paper we propose a heuristic stochastic approach to group decision making based on the use of the simulated annealing (SA) algorithm, an aggregation procedure which searches for the best group priority vector for a given node in an AHP hierarchy. The precision of the group priority vector is then further refined by minimizing the group Euclidean distance (GED) of all individual judgments from group priority vectors, which are collected during the iterative SA run. Following the logic that the commitment of the group to the implementation of the outcomes depends upon the level of consensus achieved by the group [24], we believe that minimization of GED can be considered as an objective search for maximum consensus between individuals within the group. The group priority vector obtained in this way is invariant to any prioritization method; that is, there is no need to have individual priority vectors as is required by some other aggregation procedures. In order to check the validity of our approach, five examples are used to compare our results with results obtained by various combinations of aggregations (AIJ and AIP), consensus models (CCM and GCCM) and prioritization methods (EV, AN and LLS), and hereafter these combinations will be referred to as ’aggregation schemes’. For comparison purposes, all results obtained by our and other aggregation schemes are ’controlled’ by two well known criteria, group minimum violation (GMV) measure, which we also purposely introduce in our approach, and the consensus measure (CM) proposed by Herrera-Viedma et al. [16]. The structure of the paper is as follows. In Section 2 we present preliminary knowledge about the most commonly used prioritization methods and group aggregation procedures within the AHP. Section 3 introduces the proposed simulated annealing aggregation procedure (SAAP). Section 4 describes criteria used for comparison of the results obtained by SAAP and other aggregation schemes. In Section 5, a Wilcoxon matched-pairs signed-ranks nonparametric test is described as it was carried out to ascertain whether the application of SAAP and other aggregation schemes produce significant differences in the values of tested criteria. Five numerical examples are given in Section 6 followed by discussion in Section 7 on the philosophical implications and benefits of applying SAAP compared to the other tested consensus models. Section 8 closes the paper with concluding remarks.

67

2. Preliminary knowledge

68

2.1. The problem statement

69 70

The group prioritization problem of n elements E1 , E2 , … , En at a given level of the AHP hierarchy can be formalized in the following way. The decision maker k semantically compares any two elements, Ei and Ej , and indirectly (verbally) or di-

71

rectly (numerically) assigns a value ai j which represents his judgment of the relative importance of decision element Ei over Ej

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(k)

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respecting the element in the upper level of the hierarchy. This is done through the use of Saaty’s 9-point scale [28] and can be

73 74

mathematically explained as deriving the group priority vector w(g) = (w1 , . . . , wn ) from m matrices, where m is the number of decision makers.

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2.2. Prioritization methods

76

Matrix A = (aij )n×n is a set of pairwise judgments where aij > 0 and aij × a ji = 1. If Saaty’s 9-point scale is used, then each aij can be assigned a value from the discrete interval [1/9, 9]. The prioritization method refers to the process of deriving a priority  vector w = (w1 ,...,wn )T from matrix A, where wi > 0 and ni=1 wi = 1. Below, we briefly describe three prioritization methods that are of interest to our study: EV, AN and LLS. The EV method Saaty [28] has proposed the principal eigenvector of A as the desired priority vector w. This vector can be obtained by solving the linear system:

77 78 79 80 81 82

(g)

Aw = λw, eT w = 1, 83 84 85

87

(1)

where λ is the principal eigenvalue of A. As a consistency check, a consistency index (CI) has also been proposed and associated as relevant for both EV and AN method. It is defined by relation:

λmax − n

CI = 86

n−1

,

(2)

where n is the dimension of the matrix and λmax is the maximal eigenvalue. By introducing the random index (RI) (the average CI of 500 randomly filled matrices), the consistency ratio (CR) is computed as:

CR = CI/RI. 88 89 90 91

ai j

i=1

92

n n

,

i, j = 1, 2, . . . , n (calculations by column),

(4)

,

i = 1, 2, . . . , n (calculations by rows).

(5)

The LLS method The following optimization problem is solved:

min w

95

ai j

j=1

wi = 93

(3)

Saaty has proposed that if the value of CR is less than 0.1, then the matrix can be considered as having an acceptable consistency. The AN method The method is based on simple calculations along columns and rows as given by Eqs. (4) and (5):

ai j ai j = n

94

(g) T

n  n 

( ln ai j − ( ln wi − ln w j ))2 ,

(6)

i=1 i< j

subject to: n 

wi > 0,

wi = 1,

i = 1, . . . , n.

(7)

i=1

96 97

Crawford and Williams [9] have shown that the solution to the above problem is unique and represented by normalized geometric means of the rows of matrix A:

 n n

w i = n i=1

98

j=1

ai j

 n n

j=1

ai j

.

(8)

For this method, the geometric consistency index (GCI) [2,9] is developed as: 2 GCI(A) = (n−1)( n−2)



( ln (ai j ) − ln (wi ) + ln (w j ))2 .

(9)

i< j

99 100

For GCI(A) = 0, A is considered fully consistent, while the threshold values (GCIt ) are defined to indicate acceptable consistency of matrices of different sizes: GCIt = 0.31 (for n = 3), GCIt = 0.35 (for n = 4), and GCIt = 0.37 (for n > 4) [2]. Please cite this article as: B. Blagojevic et al., Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.10.033

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2.3. Group preference aggregation procedures employed in AHP

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Aggregation of individual judgments (AIJ) For AIJ, the weighted geometric mean procedure aggregates judgments at all corresponding positions in individual matrices (g) to obtain a group judgment matrix A(g) = (ai j ) . Aggregation is performed by:

103 104

n×n

a(i jg) =

m 

αk

(a(i jk) ) ,

(10)

k=1

105 106 107 108 109 110 111

 where m is the number of individuals, α k is the weight of individual k and m k=1 αk = 1. In group settings, prioritization methods EV, AN or LLS should be directly applied at matrix A(g) to derive the group priority vector. Aggregation of individual priorities (AIP) (k) (k) Let w(k) = (w1 , . . . , wn )T be the individual priority vector for decision maker k derived from individual judgment matrix (g)

(g)

A(k) . Then, the group priority vector w(g) = (w1 , . . . , wn )T can be obtained through use of the weighted arithmetic mean aggregation of priorities:

w(i g) =

m 

αk w(i k) , i = 1, . . . , n.

(11)

k=1

112 113 114 115 116

Consensus convergence model (CCM) The core of CCM [27] is calculating the weights of respect based on the strength of differences in weights assigned to alterna(k) tives by decision makers. In this model, value wi j indirectly describes the respect of the decision maker i regarding the expertise of decision maker j in evaluating the alternative k. If the initial weights for alternative k given by m decision makers are P(k) = (p01 , p02 , … , p0m )T , then:



(k)

w i j = m

j=1

117 118

119 120 121 122 123 124



1 − p0i − p0j



,

(12)

1 − p0i − p0j

and the matrix W(k) of size m is created. The consensual vector of the alternative weights is obtained by the iterative procedure: (k) Pc(k) = W (k) Pc−1 ,

(13) (k)

which is repeated until all the values in the vector Pc are identical, and it represents the weight of alternative k. The complete description of the CCM is provided in [21]. Geometric cardinal consensus model (GCCM) Inspired by the idea that consensus indices within a group decision making context can be determined as a distance between individual preference values and group preference values [7,16], Dong et al. [11] defined the geometric cardinal consensus index (GCCI) for AHP group decision making: 2 GCCI(A(k) ) = (n−1)( n−2)

n 

2

( ln (a(i jk) ) − ln (w(i g) ) + ln (w(jg) )) .

(14)

i< j i, j=1

125 126 127 128 129 130 131

By assumption, Eq. (14) is valid if the LLS method is used for prioritization. If GCCI(A(k) ) = 0, then the decision maker k is completely in cardinal consensus with the group priority vector. Otherwise, the smaller the value of GCCI(A(k) ), the larger the cardinal consensus. This model is based on the iterative adjustment of judgments within individual matrices until acceptable consensus (GCCImax ) is reached. Inspired by the work of Aguarón and Moreno-Jiménez [2], in [11] the following thresholds for the geometric cardinal consensus index (GCCI) are defined: (1) GCCI∗ max = 0.31 (for n = 3), GCCI∗ max = 0.35 (for n = 4), and GCCI∗ max = 0.37 (for n > 4), where n is matrix size; and (2) GCCI∗∗ max = 0.001 (full consensus).

133

When these threshold values are met, the procedure stops and it is assumed that an acceptable and total consensus has been reached among the decision makers.

134

3. Simulated annealing aggregation procedure (SAAP)

135

3.1. Consensus consistency

132

(k)

(k)

(k)

(k)

136

If decision maker k is perfectly consistent, then all elements ai j in the judgment matrix A(k) have exact values ai j = wi /w j .

137

Now let’s suppose that all m decision makers are perfectly consistent and have identical judgments ai j which can be understood

(k)

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as their full consensus. In that case, all decision makers are in full agreement with the group priority vector w(g) and all elements (k) (k) (g) (g) ai j have exact values ai j = wi /w j . However, complete agreement is often impossible to achieve in real life examples of AHP group applications, which has led to the use of consensus degree (consensus consistency). In general, the computation of the consensus degree among the decision makers is done by measuring the distance between their preference values (individual judgments) and the group decision (group priority vector). Here, the group Euclidean distance (GED) is used as a group consistency measure representing the total distance (k) between all judgment elements ai j in all corresponding comparison matrices at a given node of an AHP hierarchy and related ratios of the weights contained within group priority vector w(g) . GED is given by Eq. (15)

⎡ GED = ⎣

n  m  n 

a(i jk) −

k=1 i=1 j=1

(g)

wi

2 ⎤ 12 ⎦ ,

w(jg)

(15)

152

where m is the number of decision makers and n is the number of compared elements, i.e. the size of the pair-wise comparison matrix. Obviously, if GED = 0, then all decision makers are fully in consensus with the group priority vector. Otherwise, the smaller the value of the GED, the larger the consensus. GED is chosen because it is a universal cardinal error measure and is not related to any prioritization method. Also, there are other universal cardinal error measures (Manhattan distance, for example) that can be also used for this purpose, but many authors [10,14,22,30,31] chose Euclidean distance as the universal criterion for comparing different prioritization methods in the AHP.

153

3.2. SAAP optimization model

154

The relationship between the annealing of solids and optimization problems establishes a meta-heuristic procedure, called simulated annealing (SA) [12]. With the work of Kirkpatrick et al. [19], research regarding SA applicability gained noticeable interest and SA is now used for solving a wide range of optimization problems (see, for instance, [15]). Here we propose to use the SA to minimize a single objective function defined as a group Euclidean distance (GED) (described in Section 3.1). For easier referencing, we label this procedure as the simulated annealing aggregation procedure (SAAP) with the mathematical model given by Eqs. (16) and (17):

146 147 148 149 150 151

155 156 157 158 159

⎡ min f (x) = GED = ⎣

n  n m   k=1 i=1 j=1

160

s.t.

n  w(i g) = 1,

w(i g) > 0.

a(i jk) −

w(i g) w(jg)

2 ⎤ 12 ⎦ ,

(16)

(17)

i=1

164

The goal of the proposed approach is to use Simulated Annealing (SA) in order to minimize the difference (GED) between the input (individual judgments) and output (group priority vector) when deriving the group priority vector at a given node of an AHP hierarchy. In this way, inherently achieved consensus is higher and the resulting group priority vector becomes more effective the smaller the GED is. In the presented examples, SAAP is executed through the Mathematica 6 software.

165

4. Comparison criteria

166

Within the individual AHP context, for the direct validation of inputs (judgments in a decision matrix) and output (priority vector) of the applied prioritization method, Srdjevic [30], Srdjevic and Srdjevic [31] and Kou and Lin [20] suggest to use two general error measures that are applicable to all prioritization methods: Euclidean distance (ED) and rank reversal, i.e. minimum violations (MV) criterion. Here, the same criteria are used but adjusted to be applicable within a group context. Group Euclidean distance (GED) is used as an objective function in the proposed SAAP (see Section 3.2), while group minimum violation (GMV) served as a control criterion. Following the work of Herrera-Viedma et al. [16], an additional criterion known as the consensus measure (CM) is added. Namely, it was expected that SAAP will beat the other aggregation schemes on the GED as a cardinal error measure since the method tries to minimize the associated function. By adding two ordinal measures (GMV and CM) based on ranks of compared elements in a group decision, the results obtained by SAAP and other aggregation schemes could be compared in a more detail manner. The following is a brief description of the used ordinal measures. The Group minimum violation (GMV) This measure sums up all local violations associated with the group priority vector w(g) and is expressed as

161 162 163

167 168 169 170 171 172 173 174 175 176 177

GMV =

n  m  n  Ii(jk) ,

(18)

k=1 i=1 j=1

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where

Ii(jk) =

⎧ 1 ⎪ ⎪ ⎨

179

181 182 183 184 185 186 187 188 189 190 191

193 194 195

otherwise.

The conditions of violation defined by Eq. (19) penalize possible order reversals for all DMs. For example, if the alternative j is (k) (g) (g) preferred to the alternative i for the decision maker k (i.e. ai j > 1) but the derived group priorities are such that wi > w j , then there is a violation, or element preference reversal [14]. Notice that the value of GMV should be scaled to the size and number of analyzed comparison matrices (by dividing GMV by m x n2 ) in order to preserve consistent comparisons of different matrix sizes and the number of decision makers. The Consensus measure (CM) In group decision making, different individual priority weights for alternatives can represent similar (or identical) rankings of alternatives. Because of this, Herrera-Viedma et al. [16] proposed a consensus measure (CM) based on the comparison of the ranks of alternatives within the individual decision and the group decision. The consensus measure can be defined in the unit interval [0, 1], where the value of 0 signifies no consensus and value of 1 indicates a full consensus among the decision makers. Boroushaki and Malczewski [5] summarize the procedure for calculating CM in the following steps: 1. Calculating the distance for each DM and for each alternative by comparing the rank of each alternative within the group decision using Eq. (20):

(k)

whereDi

(g) (k) Ri − Ri n−1

198

(20)

(k)

(g)

(k)

(g)

(k)

(k)

(g)

(k)

the group decision. Obviously, Di = 0 if Ri = Ri and Di = 1 if the proximity between Ri and Ri is at a maximum, which is n - 1; where n is the number of alternatives. 2. Computing the consensus degrees, CDi , among the decision makers for each alternative using the following equation:

m

197

,

∈ [0, 1] and Di is the distance of the decision maker k for the alternative i and Ri is the rank of the alternative i in

CDi = 1 − 196

(19)

(g) (g) (k) ⎪ ⎪ ⎩0.5 i f wi = w j and a ji = 1,

D(i k) = 192

i f w(i g) > w(jg) and a(jik) > 1,

0.5 i f w(i g) = w(jg) and a(jik) = 1, 0

180

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k=1

D(i k)

m

,

(21)

where m is the number of DMs and i stands for an alternative. 3. Calculating the consensus measure CM (overall consensus among decision makers), over the set of alternatives by aggregating (combining) the consensus degrees as follows:

CM = (1 − β)

m

i=1 CDi

n

+ β CD f irst ,

(22)

202

where CDfirst represents the consensus degree on the first ranked alternative for the group decision, and β represents a parameter defined in the unit interval, β ∈[0, 1]. Notice that consensus measures CM for two β values are calculated in our examples in order to compare SAAP with other aggregation schemes; CM for parameter β = 1 represents the consensus degree on the only first ranked alternative, and CM for β = 0 is the average of the consensus degrees for all alternatives.

203

5. Wilcoxon matched-pairs signed-ranks statistical test

204

In this section we present a statistical analysis which we found useful to carry out in order to ascertain whether the application of SAAP and other tested aggregation schemes produce significant differences in the values of four tested criteria GED, GMV, CM (β = 0) and CM (β = 0). The analysis is based on the use of the Wilcoxon matched-pairs signed-ranks nonparametric test described in detail in [29]. Here, only brief comments are given regarding this test concerning how we used it in our study. For each pair of compared criteria values we analyzed two related samples. Note that a commonly used parametric test is the t test applied to the difference scores. However, Chiclana et al. [8] concluded that nonparametric tests are the most appropriate for an experimental study such as ours because they can achieve greater generality in conclusions. For continuous data and two related samples, the main nonparametric tests available are the sign test and the Wilcoxon signed-rank test [8,29,34]. The sign test can be used to test the hypothesis (H0 ) that the median of the differences between the two variable scores is zero, i.e. if the null hypothesis is true we would expect about half the differences to be negative and half to be positive. The null hypothesis is rejected if ‘too few’ differences of one sign occur. Chiclana et al. [8] suggested the Wilcoxon signed-rank test as being more powerful, which incorporates more information about the data. This test takes into account information from the sign of the differences and also their magnitude so that they are appropriately ranked in order of absolute magnitude. Below, in short, we describe the Wilcoxon matched-pairs signed-ranks statistical test procedure:

199 200 201

205 206 207 208 209 210 211 212 213 214 215 216 217 218 219

1 For i = 1, …, n, xi and yi are the two variable values and n is the sample size (number of pairs); 2 H0 : The difference between the variable values of each pair (xi , yi ) has median value zero; Please cite this article as: B. Blagojevic et al., Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.10.033

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Table 1 Combination of group aggregation procedures and prioritization methods. Aggregation schemes

Description

AIJ-EV AIJ-AN AIJ-LLS

Individual judgment matrices are aggregated into a group matrix using the weighted geometric mean procedure. Group priority vectors are obtained from the group matrix using EV, AN and LLS prioritization methods.

AIP-EV AIP-AN

Individual priority vectors obtained through EV and AN prioritization methods are aggregated into a group priority vector using the weighted arithmetic mean procedure.

CCM-EV CCM-AN CCM-LLS

Individual priority vectors obtained through EV, AN and LLS prioritization methods are aggregated into a group priority vector using CCM.

GCCM∗ GCCM∗∗

A group priority vector is obtained by GCCM. In the first case (GCCM∗ ), the model stops when GCCI∗ max is reached while in the second case (GCCM∗∗ ) it stops when GCCI∗∗ max is reached; z is the number of iterations.

Table 2 Group priority vectors and corresponding ranks for different aggregation schemes (Example A). Aggregation schemes

AIJ-EV AIJ-AN AIJ-LLS AIP-EV AIP-AN CCM-EV CCM-AN CCM- LLS GCCM ∗ GCCM ∗∗ SAAP

220 221 222 223 224 225 226 227

Alternatives А1

А2

А3

А4

0.532 (1) 0.529 (1) 0.531 (1) 0.533 (1) 0.524 (1) 0.504 (1) 0.495 (1) 0.502 (1) 0.579 (1) 0.583 (1) 0.526 (1)

0.263 (2) 0.264 (2) 0.264 (2) 0.249 (2) 0.253 (2) 0.257 (2) 0.261 (2) 0.258 (2) 0.248 (2) 0.246 (2) 0.285 (2)

0.133 (3) 0.135 (3) 0.134 (3) 0.151 (3) 0.155 (3) 0.176 (3) 0.179 (3) 0.177 (3) 0.109 (3) 0.107 (3) 0.113 (3)

0.071 (4) 0.072 (4) 0.071 (4) 0.067 (4) 0.068 (4) 0.073 (4) 0.074 (4) 0.076 (4) 0.064 (4) 0.064 (4) 0.076 (4)

3 Calculate the absolute difference Di =| xi -yi | for each pair; 4 Exclude pairs with Di = 0. Let nr be the reduced sample size; 5 Consider the remaining absolute differences Di and rank them from 1 to nr . Equal absolute differences get the same rank–all are ranked with the mean of the rank numbers that would have been assigned if they would have been different; 6 Assign to each rank a “+” sign when xi -yi > 0 and a “-” sign when xi -yi < 0; 7 Calculate T+ and T- , where T+ is the sum of ranks assigned to those Di ’s that are positive and T- is the sum of ranks assigned to those Di ’s that are negative; 8 Compare T values with the corresponding critical region values (for the level of significance α = 0.05) and accept or reject H0 .

228

6. Numerical examples

229

Five examples are taken from literature (A: Dong et al. [11]; B: Moreno-Jimenez and Polasek [23]; C: Xu et al. [36]; D: Srdjevic et al. [33]; E: Blagojevic et al. [4]) and used to compare the proposed SAAP with the other ten aggregation schemes made from combinations of group aggregation procedures and prioritization methods (Table 1). In the first example (A), most of the DMs created identical rankings for all alternatives, while in the second example (B) all DMs ranked a different alternative as the optimal choice. Not surpisingly, the decision problems are not easy to solve in practice since the order of the decision matrices in the AHP is usually different. Therefore, in the first two examples (A and B) we applied SAAP to lower order (4 × 4) matrices to provide the first insight into the proposed approach. In the following three examples (C, D and E) matrices of a higher order (5 × 5, 7 × 7 and 9 × 9) are used to validate the effectiveness of the SAAP and better reflect its performance. Notice that in Table 1 there is no AIP-LLS. This is because results from AIP-LLS are identical with AIJ-LLS. Example A The five judgment matrices (4 × 4) of acceptable consistency are provided in [11]. This example is selected because four of the five DMs identically ranked all alternatives. Group priority vectors computed by 11 aggregation schemes are presented in Table 2. The computed values of four comparison criteria (GED, GMV, CM for β = 0 and CM for β = 1) are given in Table 3. Notice that aggregation schemes with the lowest GED and GMV are considered better than others, while for CM (for all β values) the opposite is true, i.e. the aggregation scheme with the highest value of CM is better than others. Results in Table 3 show that the application of SAAP outperformed other aggregation schemes, that is, a group priority vector computed by SAAP had a minimal group Euclidean distance of GED = 10.202. All aggregation schemes gave identical results for all other criteria.

230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247

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B. Blagojevic et al. / Information Sciences xxx (2015) xxx–xxx Table 3 Values of comparison criteria for different aggregation schemes (Example A). Aggregation schemes

GED

GMV

CM β = 0

CM β = 1

AIJ-EV AIJ-AN AIJ-LLS AIP-EV AIP-AN CCM-EV CCM-AN CCM-LLS GCCM ∗ GCCM ∗∗ SAAP

10.486 10.486 10.505 11.024 10.972 11.280 11.367 11.310 11.490 11.592 10.202∗

0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04

0.933 0.933 0.933 0.933 0.933 0.933 0.933 0.933 0.933 0.933 0.933

0.867 0.867 0.867 0.867 0.867 0.867 0.867 0.867 0.867 0.867 0.867

Table 4 Group priority vectors and corresponding ranks for different aggregation schemes (Example B). Aggregation schemes

AIJ-EV AIJ-AN AIJ-LLS AIP-EV AIP-AN CCM-EV CCM-AN CCM- LLS GCCM ∗ GCCM ∗∗ SAAP

Alternatives А1

А2

А3

А4

0.308 (2) 0.308 (2) 0.308 (2) 0.319 (1) 0.320 (1) 0.295 (2) 0.296 (2) 0.296 (2) 0.316 (2) 0.312 (2) 0.338 (1)

0.246 (3) 0.246 (3) 0.246 (3) 0.257 (3) 0.256 (3) 0.269 (3) 0.269 (3) 0.268 (3) 0.237 (3) 0.239 (3) 0.242 (3)

0.330 (1) 0.330 (1) 0.331 (1) 0.314 (2) 0.314 (2) 0.324 (1) 0.324 (1) 0.323 (1) 0.329 (1) 0.330 (1) 0.312 (2)

0.115 (4) 0.115 (4) 0.115 (4) 0.109 (4) 0.110 (4) 0.112 (4) 0.113 (4) 0.113 (4) 0.117 (4) 0.118 (4) 0.108 (4)

Table 5 Values of comparison criteria for different aggregation schemes (Example B).

248 249 250 251 252 253 254 255 256 257 258 259 260

Aggregation schemes

GED

GMV

CM β = 0

CM β = 1

AIJ-EV AIJ-AN AIJ-LLS AIP-EV AIP-AN CCM-EV CCM-AN CCM-LLS GCCM ∗ GCCM ∗∗ SAAP

4.967 4.968 4.973 4.912 4.914 5.010 5.012 5.010 4.970 4.989 4.883∗

0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13

0.833 0.833 0.833 0.833 0.833 0.833 0.833 0.833 0.833 0.833 0.833

0.778∗ 0.778∗ 0.778∗ 0.667 0.667 0.778∗ 0.778∗ 0.778∗ 0.778∗ 0.778∗ 0.667

Example B The three judgment matrices (4 × 4) provided in [23] also have acceptable consistencies. This example is chosen because all DMs ranked alternatives differently, identifying different alternatives to be ranked first which is opposite to Example A where the majority of five DMs ranked all alternatives identically. Group priority vectors computed by 11 aggregation schemes are given in Table 4. The computed values of four comparison criteria are given in Table 5. Again, SAAP obtained a group priority vector with a minimal group Euclidean distance (GED = 4.883). Also, SAAP was equal with other aggregation schemes for GMV and CM (β = 0) criterion. For the CM (β = 1) SAAP was not ranked first, but was equal with AIP-EV and AIP-AN. Example C The three judgment matrices (5 × 5) are provided in Xu et al. [36]. Group priority vectors computed by 11 aggregation schemes are given in Table 6. The computed values of the comparison criteria are given in Table 7. For Example C, SAAP again obtained a group priority vector with a minimal group Euclidean distance (GED = 13.738). Also, SAAP was equal with all other aggregation schemes for Please cite this article as: B. Blagojevic et al., Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.10.033

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Table 6 Group priority vectors and corresponding ranks for different aggregation schemes (Example C). Aggregation schemes

AIJ-EV AIJ-AN AIJ-LLS AIP-EV AIP-AN CCM-EV CCM-AN CCM-LLS GCCM∗ GCCM ∗∗ SAAP

Alternatives A1

A2

A3

A4

A5

0.359 (1) 0.357 (1) 0.357 (1) 0.333 (1) 0.326 (1) 0.359 (1) 0.351 (1) 0.348 (1) 0.379 (1) 0.382 (1) 0.371 (1)

0.120 (4) 0.121 (4) 0.120 (4) 0.150 (4) 0.149 (4) 0.142 (4) 0.142 (4) 0.144 (4) 0.085 (4) 0.084 (4) 0.088 (4)

0.072 (5) 0.072 (5) 0.072 (5) 0.067 (5) 0.072 (5) 0.075 (5) 0.081 (5) 0.074 (5) 0.063 (5) 0.064 (5) 0.057 (5)

0.246 (2) 0.246 (2) 0.247 (2) 0.210 (3) 0.215 (3) 0.212 (2) 0.215 (2) 0.220 (2) 0.214 (3) 0.214 (3) 0.237 (3)

0.203 (3) 0.204 (3) 0.204 (3) 0.240 (2) 0.237 (2) 0.209 (3) 0.208 (3) 0.210 (3) 0.260 (2) 0.257 (2) 0.248 (2)

Table 7 Values of comparison criteria for different aggregation schemes (Example C). Aggregation schemes

GED

GMV

CM β = 0

CM β = 1

AIJ-EV AIJ-AN AIJ-LLS AIP-EV AIP-AN CCM-EV CCM-AN CCM-LLS GCCM ∗ GCCM ∗∗ SAAP

14.594 14.625 14.599 14.900 15.135 15.068 15.337 15.082 13.859 13.887 13.738∗

0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

0.750 0.750 0.750 0.750 0.750 0.750 0.750 0.750 0.750 0.750 0.750

0.733 0.733 0.733 0.767∗ 0.767∗ 0.733 0.733 0.733 0.767∗ 0.767∗ 0.767∗

Table 8 Group priority vectors and corresponding ranks for different aggregation schemes (Example D). Aggregation schemes

AIJ-EV AIJ-AN AIJ-LLS AIP-EV AIP-AN CCM-EV CCM-AN CCM-LLS GCCM∗ GCCM ∗∗ SAAP

261 262 263 264 265 266 267 268 269 270 271

Alternatives A1

A2

A3

A4

A5

A6

A7

0.164 (3) 0.164 (3) 0.162 (3) 0.157 (3) 0.159 (3) 0.149 (3) 0.152 (3) 0.140 (3) 0.165 (3) 0.166 (3) 0.177 (3)

0.247 (1) 0.243 (1) 0.242 (1) 0.251 (1) 0.240 (1) 0.238 (1) 0.225 (1) 0.225 (1) 0.259 (1) 0.261 (1) 0.247 (1)

0.075 (7) 0.076 (7) 0.076 (7) 0.071 (7) 0.077 (7) 0.079 (7) 0.086 (7) 0.080 (7) 0.071 (7) 0.070 (7) 0.060 (7)

0.149 (4) 0.149 (4) 0.152 (4) 0.148 (4) 0.147 (4) 0.123 (5) 0.122 (5) 0.127 (5) 0.161 (4) 0.163 (4) 0.152 (4)

0.203 (2) 0.205 (2) 0.204 (2) 0.185 (2) 0.186 (2) 0.190 (2) 0.191 (2) 0.197 (2) 0.195 (2) 0.194 (2) 0.224 (2)

0.086 (5) 0.087 (5) 0.087 (5) 0.109 (5) 0.109 (5) 0.134 (4) 0.133 (4) 0.139 (4) 0.073 (6) 0.072 (6) 0.064 (6)

0.077 (6) 0.077 (6) 0.078 (6) 0.079 (6) 0.083 (6) 0.083 (6) 0.087 (6) 0.087 (6) 0.076 (5) 0.075 (5) 0.076 (5)

GMV and CM (β = 0). For the CM (β = 1) SAAP was better than AIJ-EV, AIJ-AN, AIJ-LLS, CCM-EV, CCM-AN, CCM-LLS and was equal with AIP-EV, AIP-AN, GCCM∗ and GCCM∗∗ . Example D Three large judgment matrices (7 × 7) are provided in Srdjevic et al. [33] from a real-life decision making problem. In this particular example one DM demonstrated a slightly higher inconsistency (CR = 0.182) than what is recommended as the tolerant limit (CR = 0.10). Group priority vectors computed by 11 aggregation schemes are given in Table 8. The computed values of four comparison criteria are given in Table 9. Notice that in the last three rows there are different SAAP identifications related to three prioritization methods, even though the group priority vector is identical in all three cases and was marked as SAAP before (in Table 8). This is because input data for CM criteria (for all β values) calculations are the individual ranks for alternatives, and these ranks can differ between different prioritization methods. This allows for CM values to differ as well. Please cite this article as: B. Blagojevic et al., Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.10.033

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B. Blagojevic et al. / Information Sciences xxx (2015) xxx–xxx Table 9 Values of comparison criteria for different aggregation schemes (Example D). Aggregation schemes

GED

GMV

CM β = 0

CM β = 1

AIJ-EV AIJ-AN AIJ-LLS AIP-EV AIP-AN CCM-EV CCM-AN CCM-LLS GCCM ∗ GCCM ∗∗ SAAP-EV SAAP-AN SAAP-LLS

16.610 16.649 16.662 17.069 17.242 17.790 18.024 17.991 16.339 16.313 16.077∗ 16.077∗ 16.077∗

0.177∗ 0.177∗ 0.177∗ 0.177∗ 0.177∗ 0.184 0.184 0.184 0.184 0.184 0.184 0.184 0.184

0.794 0.794 0.810∗ 0.794 0.794 0.794 0.794 0.794 0.810∗ 0.810∗ 0.794 0.794 0.810∗

0.889∗ 0.833 0.833 0.889∗ 0.833 0.889∗ 0.833 0.833 0.833 0.833 0.889∗ 0.833 0.833

Table 10 Group priority vectors and corresponding ranks for different aggregation schemes (Example E). Aggregation schemes

AIJ-EV AIJ-AN AIJ-LLS AIP-EV AIP-AN CCM-EV CCM-AN CCM-LLS GCCM∗ GCCM ∗∗ SAAP

Alternatives A1

A2

A3

A4

A5

A6

A7

A8

A9

0.204 (2) 0.201 (2) 0.202 (2) 0.215 (2) 0.206 (2) 0.250 (1) 0.240 (1) 0.252 (1) 0.195 (2) 0.192 (2) 0.211 (3)

0.166 (3) 0.166 (3) 0.171 (3) 0.181 (3) 0.174 (3) 0.201 (2) 0.191 (2) 0.200 (2) 0.169 (3) 0.172 (3) 0.226 (2)

0.029 (9) 0.029 (9) 0.030 (9) 0.028 (9) 0.030 (9) 0.026 (9) 0.028 (9) 0.024 (9) 0.028 (9) 0.028 (9) 0.030 (9)

0.077 (6) 0.079 (6) 0.080 (5) 0.078 (5) 0.079 (6) 0.082 (4) 0.083(5) 0.079 (5) 0.075 (6) 0.075 (6) 0.074 (4)

0.061 (7) 0.062 (8) 0.063 (7) 0.057 (8) 0.062 (8) 0.048 (8) 0.053 (8) 0.046 (8) 0.061 (8) 0.061 (8) 0.050 (7)

0.086 (4) 0.087 (4) 0.088 (4) 0.080 (4) 0.085 (4) 0.072 (6) 0.078 (6) 0.075 (6) 0.082 (5) 0.081 (5) 0.069 (5)

0.058 (8) 0.060 (7) 0.059 (8) 0.059 (7) 0.062 (7) 0.050 (7) 0.054 (7) 0.051 (7) 0.063 (7) 0.063 (7) 0.048 (8)

0.079 (5) 0.082 (5) 0.079 (6) 0.075 (6) 0.080 (5) 0.079 (5) 0.085 (4) 0.081 (4) 0.086 (4) 0.087 (4) 0.051(6)

0.239 (1) 0.234 (1) 0.229 (1) 0.228 (1) 0.224 (1) 0.194 (3) 0.189 (3) 0.195 (3) 0.241 (1) 0.242 (1) 0.240 (1)

Table 11 Values of comparison criteria for different aggregation schemes (Example E).

272 273 274 275 276 277 278 279 280 281

Aggregation schemes

GED

GMV

CM β = 0

CM β = 1

AIJ-EV AIJ-AN AIJ-LLS AIP-EV AIP-AN CCM-EV CCM-AN CCM-LLS GCCM ∗ GCCM ∗∗ SAAP-EV SAAP-AN SAAP-LLS

42.287 42.458 42.341 41.920 42.363 41.971 42.107 42.331 42.455 42.437 41.023∗ 41.023∗ 41.023∗

0.14 0.14 0.15 0.15 0.15 0.14 0.13∗ 0.13∗ 0.14 0.14 0.15 0.15 0.15

0.802 0.795 0.788 0.795 0.795 0.802 0.795 0.809∗ 0.802 0.802 0.788 0.774 0.781

0.813∗ 0.781 0.813∗ 0.813∗ 0.781 0.750 0.750 0.750 0.813∗ 0.813∗ 0.813∗ 0.781 0.813∗

In this example, SAAP again obtained a group priority vector with a minimal group Euclidean distance (GED = 16.077). Also, SAAP-LLS was ranked first for CM (β = 0) and SAAP-EV for CM (β = 1). For the GMV, SAAP was not ranked first, but was equal with CCM-EV, CCM-AN, CCM-LLS, GCCM∗ and GCCM∗∗ . Example E The four largest judgment matrices (9 × 9) are taken from Blagojevic et al. [4]. This example is also from a real-life decision process where the inconsistency level of all DMs is higher than allowed (CR > 0.10). It is chosen because the number of compared elements is 9 (maximum according to Saaty [28]) and this could very well be the main reason for the DMs’ inconsistencies. Group priority vectors computed by 11 aggregation schemes are given in Table 10. Values of four comparison criteria for all aggregation schemes (Table 11) show that the group priority vector found by SAAP was better than others regarding the GED. All aggregation schemes have similar values for GMV criterion (0.13, 0.14 and 0.15) and Please cite this article as: B. Blagojevic et al., Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.10.033

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Table 12 Summary of results for all five examples (A–E). Ranks of aggregation schemes Example GED

GMV

CM β = 0

CM β = 1

Sum of first ranks

A B C D E A B C D E A B C D E A B C D E 11

AIJ-EV

AIJ-AN

AIJ-LLS

AIP-EV

AIP-AN

CCM-EV

CCM-AN

CCM-LLS

GCCM ∗

GCCM ∗∗

SAAP

2-3 4 4 4 5 1-11 1-11 1-11 1-5 3-7 1-11 1-11 1-11 5-11 2-5 1-11 1-8 6-11 1-4 1-6 9

2-3 5 6 5 10-11 1-11 1-11 1-11 1-5 3-7 1-11 1-11 1-11 5-11 6-9 1-11 1-8 6-11 5-11 7-8 11

4 7 5 6 7 1-11 1-11 1-11 1-5 8-11 1-11 1-11 1-11 1-4 10-11 1-11 1-8 6-11 5-11 1-6 11

6 2 7 7 2 1-11 1-11 1-11 1-5 8-11 1-11 1-11 1-11 5-11 6-9 1-11 9-11 1-5 1-4 1-6 9

5 3 10 8 8 1-11 1-11 1-11 1-5 8-11 1-11 1-11 1-11 5-11 6-9 1-11 9-11 1-5 5-11 7-8 9

7 9-10 8 9 3 1-11 1-11 1-11 6-11 3-7 1-11 1-11 1-11 5-11 2-5 1-11 1-8 6-11 1-4 9-11 9

9 11 11 11 4 1-11 1-11 1-11 6-11 1-2 1-11 1-11 1-11 5-11 6-9 1-11 1-8 6-11 5-11 9-11 10

8 9-10 9 10 6 1-11 1-11 1-11 6-11 1-2 1-11 1-11 1-11 5-11 1 1-11 1-8 6-11 5-11 9-11 11

10 6 2 3 10-11 1-11 1-11 1-11 6-11 3-7 1-11 1-11 1-11 1-4 2-5 1-11 1-8 1-5 5-11 1-6 11

11 8 3 2 9 1-11 1-11 1-11 6-11 3-7 1-11 1-11 1-11 1-4 2-5 1-11 1-8 1-5 5-11 1-6 16

1∗ 1∗ 1∗ 1∗ 1∗ 1-11 1-11 1-11 6-11 8-11 1-11 1-11 1-11 1-4 10-11 1-11 9-11 1-5 1-4 1-6

283

CCM-AN and CCM-LLS were first ranked with GMV = 0.13. Analysis of CM (β = 1) values indicates that SAAP-LLS and SAAP-EV tied the first place with AIJ-EV, AIJ-LLS, AIP-EV, GCCM ∗ and GCCM∗∗ . For CM (β = 0), CCM-LLS gave the best result.

284

6.1. Summary of results for examples

285

292

Table 12 summarizes the results for five example applications of SAAP and ten other aggregation schemes. The results are presented as ranks of aggregation schemes per criteria and then again per examples. Our intent was to see how many times each aggregation scheme is ranked first (maximum is 20, because four criteria are used in five examples). At a first glance, it comes out that the SAAP is ranked first 16 times and outperformed other aggregation schemes. The data was then analyzed further to determine whether any aggregation outranked the others based on the individual criterion. All cases where one aggregation scheme held a first place position without sharing it with other aggregation schemes were marked, and our results showed that SAAP was the only aggregation scheme that did not share a first ranked position with another aggregation scheme within the GED criterion. In all other cases, first place was shared with other aggregation schemes.

293

6.2. Results of statistical comparative study

294

The four hypotheses that we are testing in this paper can be stated as follows: H1 : The application of the SAAP and other tested aggregation schemes (AS) do not produce significant differences in the values of GED; H2 : The application of the SAAP and other tested AS do not produce significant differences in the values of GMV; H3 : The application of the SAAP and other tested AS do not produce significant differences in the values of CM (β = 0); H4 : The application of the SAAP and other tested AS do not produce significant differences in the values of CM (β = 1). To test the above four hypothesis, for each of five examples (A, B, ..., E) and for each of four criteria (GED, GMV, CM (β = 0) and CM (β = 1)), ten (10) pairs of tested criteria values are created: CSAAP vs. CASj . CSAAP represents the value of SAAP for a corresponding criterion, while CASj represents the values of other AS for the same criterion. j represents identification of other AS and j=1, ..., 10. For example, CAS1 is the value of the corresponding criterion for AIJ-EV (first aggregation scheme) while CAS10 is the value of the corresponding criterion for GCCM ∗∗ (last aggregation scheme). Additionally, we tested three hypotheses (H2 -H4 ) for the sum of five examples (A-E). In this case all criteria values were tested in 50 pairs: CSAAPi vs. CASij , where i stands for an example (i=A, ..., E); and j represents identification of AS (j=1, ..., 10). It was possible to summarize examples (although they have different matrix sizes and different numbers of decision makers) because values of GMV, CM (β = 0) and CM (β = 1) are scaled to the size and number of analyzed comparison matrices in order to preserve consistent comparisons of different matrix sizes and the number of decision makers (see Section 4). Also, there was no need to test H1 (the application of the SAAP and other tested AS do not produce significant differences in the values of GED), because it was possible to reach a conclusion based on tests of H1 in individual examples. Results in the last column of Table 13 (sum of all five examples) show that pairs of SAAP and other tested AS produce significantly different results for GED and GMV criteria. For GED criterion SAAP performed better than other AS because it produced the group priority vector with a minimal GED in all examples (for each example T+ = 0 and T- = 55). For GMV criterion, other

282

286 287 288 289 290 291

295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314

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Table 13 Results of Wilcoxon matched-pairs signed-ranks statistical test. A

B

C

D

E

A+B+C+D+E

T+ = 0 T- = 55 nr = 10 H1 rejected, SAAP is better

T+ = 0 T- = 55 nr = 10 H1 rejected, SAAP is better

T+ = 0 T- = 55 nr = 10 H1 rejected, SAAP is better

T+ = 0 T- = 55 nr = 10 H1 rejected, SAAP is better

T+ = 0 T- = 55 nr = 10 H1 rejected, SAAP is better

H1 rejected, SAAP is better

GMV

nr = 0

nr = 0

nr = 0

CM β = 0

nr = 0

nr = 0

nr = 0

T+ = 15 T- = 0 nr = 5∗ H2 rejected, SAAP is worse T+ = 28 T- = 0 nr = 7 H3 rejected, SAAP is better

T+ = 28 T- = 0 nr = 7 H2 rejected, SAAP is worse T+ = 0 T- = 45 nr = 9 H3 rejected, SAAP is worse

CM β = 1

nr = 0

T+ = 0 T- = 36 nr = 8 H4 rejected, SAAP is worse

T+ = 21 T- = 0 nr = 6 H4 rejected, SAAP is better

T+ = 28 T- = 0 nr = 7 H4 rejected, SAAP is better

T+ = 15 T- = 0 nr = 5∗ H4 rejected, SAAP is better

GED

∗

T+ = 78 T- = 0 nr = 12 H2 rejected, SAAP is worse T+ = 84 T- = 52 nr = 16 H3 accepted, no significant differences T+ = 171 T- = 180 nr = 26 H4 accepted, no significant differences

when nr = 5 we can only test H0 for α = 0.10

316

AS outperformed SAAP (T+ = 78 and T- = 0). On the other hand, pairs of SAAP and other tested AS do not produce significantly different results for CM (β = 0) (T+ = 84 and T- = 52) and CM (β = 1) criterion (T+ = 171 and T- = 180).

317

7. Discussion

318

Consensus building is not only related to mathematical models, but also to philosophical issues [21]. Therefore, it would be interesting to comment on some psychological aspects of SAAP and the other two tested consensus models, CCM and GCCM. One important question to consider is how cooperation among decision makers is treated between the two models, and in both CCM and GCCM it was found that cooperation is rewarded. In the case of CCM, non-cooperation yields the risk of being excluded from the process or having very little impact on the outcome as a result of being assigned low respect weightings [27]. In the case of GCCM, non-cooperation results in frequent preference modifications for outliers within the group. Conversely, in the case of SAAP non-cooperation is not penalized, because outliers will have an impact on the group decision equal to that of the other group members. This is a key difference between SAAP and the other two tested consensus models. As a result, an important question is what causes someone to become an outlier in a group. According to Regan et al. [27] extreme views might be caused by three potential situations:

315

319 320 321 322 323 324 325 326 327 328 329 330

1. outliers know more than the other group members, 2. outliers know considerably less, or 3. outliers are deliberately misrepresenting their views to push the central tendency towards their own more moderate position.

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CCM and GCCM are able to penalize the second and third situations, but they are inadequate for handling the first one. On the other hand, SAAP is appropriate for instances of the first situation but inadequate for the second situation. SAAP is also able to penalize the third situation, since each decision maker can fill in their own AHP decision matrix separately, without knowing the preferences of other decision makers. According to what has been previously stated, it is obvious that consensus models CCM, GCCM and SAAP have differing philosophical bases. It is justified to use CCM and GCCM when decision makers are willing to change their preferences and when outliers know considerably less than the other decision makers, but in cases when decision makers do not want to change their initial individual judgments or outliers know more than the other group members, use of SAAP should be preferable.

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8. Conclusion

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Whenever using the AHP within a group decision making context, it optimal outcome is decision makers achieving the highest degree of consensus when deriving the group priority vector. Our experience in real life applications of AHP shows that in many cases some decision makers do not want to change their individual judgments in the hopes of obtaining stronger consensus for the group decision, therefore we focused our research on evidencing the practicality and usefulness of establishing a group aggregation procedure which would return a more concise group priority vector corresponding with the consensus degree (consensus level) principle between decision makers. As a result of this focus we developed the procedure described in this paper

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and tested it along with several other procedures presented in literature. The core of our approach holds that the group Euclidean distance should be used to measure the consensus level among decision makers while the simulated annealing (SA) algorithm is used to maximize consensus level. One important reason for such an approach is that the group Euclidean distance is a universal cardinal error measure which in many cases perfectly follows the purpose of the AHP to calculate cardinal information (weights) and not only ranks of alternatives like many other multi-criteria methods do. The proposed method labeled SAAP (simulated annealing aggregation procedure) was compared with other aggregation schemes such as the aggregation of individual judgments, aggregation of individual priorities and two consensus models in combination with several of the most commonly used prioritization methods. For comparison purposes, the results obtained with SAAP are controlled by two other criteria known as group minimum violation and the consensus measure. Five examples are taken from literature, and in all cases the group priority vector generated by the SAAP appears to be superior to the others if the group Euclidean distance is used as a measure of group consistency. Consensus measures (for both β values) obtained by SAAP were found to be equal in a comparison against results obtained from computations made with the same measures for other aggregation schemes. However, in regards to group minimum violation values SAAP was not better than other aggregation schemes, but in three of five cases it was equal with the others. We believe that the main advantage of the SAAP is that it is independent of the used prioritization method while other tested group aggregation procedures are not. With SAAP, minimum values of the group Euclidean distance are computed and the highest degree of consensus is achieved without changing any of the individual judgments of decision makers participating in the group. Based on the results presented in this paper, we think that the proposed approach within the AHP group decision making framework could be extended to situations when decision makers do not have equal weights, unlike in the examples we used in this paper, and also for cases when other cardinal error measures (e.g. Manhattan distance) are used.

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Acknowledgments

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This work was supported in part by Ministry of Education, Science and Technological Development of Serbia under the Grant 174003 (2011-2014) - Theory and application of Analytic hierarchy process (AHP) in multi-criteria decision making under conditions of risk and uncertainty (individual and group context).

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References

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371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

E. Abel, L. Mikhailov, J. Keane, Group aggregation of pairwise comparisons using multi-objective optimization, Inf. Sci. 322 (2015) 257–275. J. Aguarón, J.M. Moreno-Jiménez, The geometric consistency index: Approximated thresholds, Eur. J. Oper. Res. 147 (2003) 137–145. J. Barzilai, B. Golany, AHP rank reversal normalization and aggregation rules, INFOR 32 (1994) 57–64. B. Blagojevic, B. Srdjevic, Z. Srdjevic, M. Lakicevic, Allocation of budget funds on agricultural loan programs: group consensus decision making in the Provincial Fund for Agricultural Development of Vojvodina Province in Serbia, Ind. (J. Econ. Inst.) 3 (2012) 57–70. S. Boroushaki, J. Malczewski, Measuring consensus for collaborative decision-making: A GIS-based approach, Comput. Environ. Urban Syst. 34 (2010) 322– 332. L. Chen, Z. Xu, A new fuzzy programming method to derive the priority vector from an interval reciprocal comparison matrix, Inf. Sci. 316 (2015) 148–162. F. Chiclana, F. Mata, L. Martínez, E. Herrera-Viedma, S. Alonso, Integration of a consistency control module within a consensus decision making model, Int. J. Uncertain. Fuzziness Knowl. Based Syst. 16 (2008) 35–53. F. Chiclana, J.T. Garcia, M.J. del Moral, E. Herrera-Viedma, A statistical comparative study of different similarity measures of consensus in group decision making, Inf. Sci. 221 (2013) 110–123. G. Crawford, C. Williams, A note on the analysis of subjective judgment matrices, J. Math. Psychol. 29 (1985) 387–405. Y. Dong, Y. Xu, H. Li, M. Dai, A comparative study of the numerical scales and the prioritization methods in AHP, Eur. J. Oper. Res. 186 (2008) 229–242. Y.C. Dong, G.Q. Zhang, W.C. Hong, Y.F. Xu, Consensus models for AHP group decision making under row geometric mean prioritization method, Dec. Support Syst. 49 (2010) 281–289. K.A. Dowsland, Simulated annealing, in: C.R. Reeves (Ed.), Modern Heuristic Techniques for Combinatorial Problems, McGraw-Hill, 1995. E. Forman, K. Peniwati, Aggregating individual judgments and priorities with the analytic hierarchy process, Eur. J. Oper. Res. 108 (1998) 165–169. B. Golany, M. Kress, A multicriteria evaluation of methods for obtaining weights from ratio-scale matrices, Eur. J. Oper. Res. 69 (1993) 210–220. K. Haridass, J. Valenzuela, A.D. Yucekaya, T. McDonald, Scheduling a log transport system using simulated annealing, Inf. Sci. 264 (2014) 302–316. E. Herrera-Viedma, F. Herrera, F. Chiclana, A consensus model for multiperson decision making with different preference structures, IEEE Trans. Syst. Man Cybern. Part A: Syst. Humans 32 (3) (May 2002) 394–402, doi:10.1109/TSMCA.2002.802821. W. Ho, Integrated analytic hierarchy process and its applications – A literature review, Eur. J. Oper. Res. 186 (2008) 211–228. A. Ishizaka, A. Labib, Review of the main developments in the analytic hierarchy process, Expert Syst. Appl. 38 (2011) 14336–14345. S. Kirkpatrick, C.D. Gelatt, M.P Vecchi, Optimization by simulated annealing, Science 220 (1983) 671–680. G. Kou, C. Lin, A cosine maximization method for the priority vector derivation in AHP, Eur. J. Oper. Res. 235 (2014) 225–232. K. Lehrer, C. Wagner, Rational Consensus in Science and Society, Reidel, Dordrecht, 1981. L. Mikhailov, M.G. Singh, Comparison analysis of methods for deriving priorities in the analytic hierarchy process, in: Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, 1999, pp. 1037–1042. J.M. Moreno-Jimenez, W. Polasek, e-Democracy and Knowledge, A Multicriteria Framework for the New Democratic Era, J. Multi-Criteria Dec. Anal. 12 (2003) 163–176. W. Pedrycz, P. Ekel, R. Parreiras, Fuzzy Multicriteria Decision-Making: Models, Methods And Applications, John Wiley & Sons, 2011. W. Pedrycz, M. Song, A granulation of linguistic information in AHP decision-making problems, Inf. Fusion 17 (2014) 93–101. R. Ramanathan, L.S. Ganesh, Group preference aggregation methods employed in AHP: An evaluation and an intrinsic process for deriving members’ weightages, Eur. J. Oper. Res. 79 (1994) 249–265. H.M. Regan, M. Colyvan, L. Markovchick-Nicholls, A formal model for consensus and negotiation in environmental management, J. Environ. Manag. 80 (2006) 167–176. T.L. Saaty, The Analytical Hierarchy Process, McGraw Hill, New York, 1980. S. Siegel, Nonparametric Statistics for the Behavioral Sciences, McGraw-Hill, New York, 1956. B. Srdjevic, Combining different prioritization methods in AHP synthesis, Comput. Oper. Res. 25 (2005) 1897–1919. B. Srdjevic, Z. Srdjevic, Bi-criteria evolution strategy in estimating weights from the AHP ratio-scale matrices, Appl. Math. Comput. 218 (2011) 1254–1266. B. Srdjevic, Z. Srdjevic, B. Blagojevic, K. Suvocarev, A two-phase algorithm for consensus building in AHP-group decision making, Appl. Math. Model. 37 (2013) 6670–6682.

Please cite this article as: B. Blagojevic et al., Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.10.033

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[33] B. Srdjevic, Z. Srdjevic, B. Blagojevic, O. Cukaliev, Multi-Criteria Evaluation of Groundwater Ponds as Suppliers to Urban Water Distribution Systems, Sustain. Cities Mil. Install. (2014) 163–175. [34] F. Wilcoxon, Individual comparisons by ranking methods, Biom. Bull. 1 (1945) 80–83. [35] Z. Xu, X. Cai, Group consensus algorithms based on preference relations, Inf. Sci. 181 (2011) 150–162. [36] Y. Xu, K.W. Li, H. Wang, Distance-based consensus models for fuzzy and multiplicative preference relations, Inf. Sci. 253 (2013) 56–73. [37] Z. Zhang, C. Wu, A decision support model for group decision making with hesitant multiplicative preference relations, Inf. Sci. 282 (2014) 136–166.

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Bosko Blagojevic, Ph.D., working as a Teaching Assistant in Water Resources Systems Analysis and Informatics. His research focus is on the development and application of decision-making methods in agriculture and water resources. He is the author of more than 30 papers in international and national journals and conference proceedings and a referee for the European Journal of Operational Research.

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Bojan Srdjevic is a Full Professor in Water Resources Systems Analysis and Informatics. He is lecturing on various graduate and postgraduate courses at the Universities of Novi Sad (Serbia), Salvador (Brazil) and Stuttgart (Germany). His research interests include decision-making methodologies and supporting tools and water resources planning and management. He is the author of more than 200 papers in international and national journals and conference proceedings, and a referee for IEEE, Elsevier, Springer, ASCE and IWA international journals, beside others.

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Zorica Srdjevic is an Associate Professor at the University of Novi Sad, Serbia. She is lecturing on various undergraduate, master and doctoral courses. Her research interests include multi-criteria analysis and decision-making methodologies and supporting tools. She is the author of more than 100 articles and other publications, mainly in peer-reviewed international and national journals. She is a referee for several international journals, including Elsevier and Springer.

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Tihomir Zoranovic is an Assistant Professor at the University of Novi Sad, Serbia. He is lecturing on graduate and postgraduate courses. His research area is the development and application of decision making methods, tools and software. He is the author of more than 100 papers in international and national journals and several computer programs.

Please cite this article as: B. Blagojevic et al., Heuristic aggregation of individual judgments in AHP group decision making using simulated annealing algorithm, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.10.033