# Identification of Homogeneous Groups of Actors in a Local AHP-Multiactor Context with a High Number of Decision-Makers: A Bayesian Stochastic Search

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. The Analytic Hierarchy Process

_{(nxn)}= (r

_{ij}) is consistent [1] if it verifies the cardinal transitivity of the judgments, that is, r

_{ij}r

_{jk}= r

_{ik}∀ i, j, k = 1, …, n. Otherwise, the matrix is said to be inconsistent. The two most used indicators for measuring the level of inconsistency are the Saaty Consistency Ratio (CR) [1] and the Geometric Consistency Index (GCI) [14,15]. The values of CR < 5% for n = 3; CR < 8% for n = 4, and CR < 10% for n > 4 are considered as acceptable levels of inconsistency. The associated thresholds for GCI are [15]: GCI < 0.31 for n = 3, GCI < 0.35 for n = 4, and GCI < 0.37 for n > 4.

#### 2.2. Multi-Actor Decision Making

## 3. Methodology

#### 3.1. Problem Formulation

_{p}(

**µ**,

**∑**) denotes the p-variant normal distribution of mean vector

**µ**and the matrix of variances and covariances

**∑**; T

_{p}(

**µ**,

**∑**, υ) denotes the p-variant Student t distribution with mean vector

**µ**, scale matrix

**∑**and degrees of freedom υ; Gam(p, a) denotes the gamma distribution with shape parameter p and scale parameter 1/a; ${\chi}_{\nu}^{2}$ denotes the chi-squared distribution with υ degrees of freedom; I

_{A}denotes the indicator function of set A; ∝ indicates proportional to; and [Y|X] denotes the density function of the conditional distribution of Y given X.

**D**= $\left\{{\mathrm{D}}^{[1]},\dots ,{\mathrm{D}}^{[\mathrm{K}]}\right\}$ be the set of decision-makers,

**A**= $\left\{{\mathrm{A}}_{1},\dots ,{\mathrm{A}}_{\mathrm{n}}\right\}$ be the set of n alternatives and

**R**

^{(k)}= $\left({\mathrm{r}}_{\mathrm{i}\mathrm{j}}^{(\mathrm{k})}\right)$; k = 1, …, K be the nxn pairwise comparison matrices (PCMs) elicited by each decision-maker. Let ℘(

**D**) the set of partitions of

**D**.

**D**, with

**G**

_{g}= $\left\{{\mathrm{D}}_{}^{\left[{\mathrm{i}}_{\mathrm{g},\mathrm{1}}\right]},\dots ,{\mathrm{D}}_{}^{\left[{\mathrm{i}}_{\mathrm{g},{\mathrm{n}}_{\mathrm{g}}}\right]}\right\}$ ⊆ D; n

_{g}the size of group

**G**

_{g}for g = 1,…, m,

**G**

_{g}∩

**G**

_{g′}= ∅ if g ≠ g′, $\underset{\mathrm{g}=1}{\overset{\mathrm{m}}{\cup}}{\mathbf{G}}_{\mathrm{g}}$ =

**D**; [i

_{g,j}] the j-th (j = 1, …,n

_{g}) decision-maker of group

**G**

_{g}.

_{g,1}< i

_{g′,1}if g < g′.

**M**(

**$\mathcal{G}$**) with log-normal errors is used [3,16]. The model

**M**(

**$\mathcal{G}$**) assumes that the decision-makers who belong to a group

**G**

_{g}of the partition

**$\mathcal{G}$**have homogeneous preferences regarding the priorities of the alternatives of

**A**so that:

- (a)
- D
^{[k]}∈**G**_{g(k)}with g(k) ∈ {1, …, m} being the index of the group of $\mathcal{G}$ which contains D^{[k]} - (b)
- ${\mathsf{\mu}}_{\mathrm{i}}^{\left(\mathrm{g}(\mathrm{k})\right)}=\mathrm{log}\left({\mathrm{v}}_{\mathrm{i}}^{\left(\mathrm{g}(\mathrm{k})\right)}\right)$; i = 1, …, n being ${\mathrm{v}}_{\mathrm{i}}^{\left(\mathrm{g}(\mathrm{k})\right)}$ the priority (without normalising) given to the alternative A
_{i}by the members of the group**G**_{g(k)} - (c)
- ${\mathrm{v}}_{\mathrm{n}}^{\left(\mathrm{g}(\mathrm{k})\right)}=1$ (that is to say, ${\mathsf{\mu}}_{\mathrm{n}}^{\left(\mathrm{g}(\mathrm{k})\right)}=0$) to avoid identifiability problems
- (d)
- ${\mathsf{\epsilon}}_{\mathrm{i}\mathrm{j}}^{(\mathrm{k})}~\mathrm{N}\left(0,{\mathsf{\sigma}}^{(\mathrm{g}(\mathrm{k}))}\right)$; k = 1, …, K; 1 ≤ i < j ≤ n independent.

#### 3.2. Analysis of the Priorities and the Homogeneity of the Groups of $\mathcal{G}$

**G**

_{g}(g = 1,…, m) will be given by the vector:

^{(g)}; g = 1, …, m} which quantifies the level of compatibility of each decision-maker with the priorities vector

**w**

^{(g)}of his/her group. The estimation of (

**w**(g), σ

^{(g)}) is carried out using a Bayesian approach which let us obtain exact inferences about them. To do this, we use the standard conjugate normal-gamma prior distributions given by:

**µ**

^{(G)}is not significant. The hyper-parameters ${\mathrm{n}}_{0}$ and ${\mathrm{s}}_{0}^{2}$ are determined from the maximum levels of incompatibility ${\mathsf{\sigma}}_{\mathrm{max}}^{2}$ allowed for each decision-maker so that:

#### 3.2.1. Posterior Distribution

**µ**

^{(g)},τ

^{(g)}) is given by:

**y**

^{(k)}= ${\left({\mathrm{y}}_{\mathrm{i}\mathrm{j}}^{(\mathrm{k})};1\le \mathrm{i}<\mathrm{j}\le \mathrm{n}\right)}^{\prime}$ for k = 1, …, K and

**X**= (x

_{ij}) (J × (n − 1)) with J = $\frac{\mathrm{n}(\mathrm{n}-1)}{2}$ is the regression matrix of model (1) so that:

- -
- x
_{ij}= 1 if the i-th judgement is y_{jk}with k ≠ j; - -
- x
_{ij}= −1 if the i-th judgement is y_{kj}with k ≠ j; - -
- x
_{ij}= 0 in any other case.

^{(g)}using the posterior median and the corresponding posterior quantiles can be obtained. In addition, and using Monte Carlo, for each group {

**G**

_{g}; g = 1, …, m} the posterior distribution of their priorities vectors {

**w**

^{(g)}; g = 1, …, m} could be calculated (see reference [3] for details).

_{g}could also be calculated using the alpha distributions ${\mathbf{P}}_{\mathsf{\alpha}}^{{\mathbf{G}}_{\mathrm{g}}}=\left({\mathrm{P}}_{\mathsf{\alpha},1}^{{\mathbf{G}}_{\mathrm{g}}},\dots ,{\mathrm{P}}_{\mathsf{\alpha},\mathrm{n}}^{{\mathbf{G}}_{\mathrm{g}}}\right)$ with:

**γ**

_{h}= (γ

_{h,1}, …, γ

_{h,n}) is the h-th permutation of the elements of

**A**sorted according to the lexicographical order (see references [3,16] for more details).

#### 3.2.2. Analysis of the Representativeness of a Partition $\mathcal{G}$

**Y**| $\mathcal{G}$] where

**Y**= {

**y**

^{(k)}; k ∈ {1, …, K}}. This density evaluates the goodness of fit of the model (1)–(3) with respect to the judgments issued by the decision-makers of

**D**so that, the higher its value, the greater the degree of representativeness of G. This density is given by:

**Q**

^{(g)}=${\mathrm{n}}_{0}{\mathrm{s}}_{0}^{2}+{\displaystyle \sum _{\mathrm{k}:\mathrm{g}\left(\mathrm{k}\right)=\mathrm{g}}{\mathbf{y}}^{(\mathrm{k}){}^{\prime}}{\mathbf{y}}^{(\mathrm{k})}}-{\mathbf{m}}^{\left(\mathrm{g}\right){}^{\prime}}\left({\mathrm{n}}_{\mathrm{g}}\left(\mathbf{X}{}^{\prime}\mathbf{X}\right)+{\mathrm{c}}_{0}{\mathrm{I}}_{\mathrm{n}-1}\right){\mathbf{m}}^{\left(\mathrm{g}\right)}$(see reference [3]).

#### 3.2.3. Selection of the Best Partitions $\mathcal{G}$. Stochastic Search Algorithm

**D**). Let 0 < β < 1 be a threshold to determine if there exist significant differences in the fitness of the data of $\mathsf{\mathcal{G}}$ and $\mathsf{\mathcal{G}}{}^{\prime}$ in such a way that if $\frac{\left[\mathbf{Y}|\mathsf{\mathcal{G}}\right]}{\left[\mathbf{Y}|\mathsf{\mathcal{G}}{}^{\prime}\right]}<\mathsf{\beta}$ the goodness of fit of ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ is better than $\mathsf{\mathcal{G}}$ and, therefore, partition ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ is more representative than $\mathsf{\mathcal{G}}$. In our case, and following to [13], we have taken β = 0.05.

**D**) is based on the determination of the Occam’s window given by:

**D**) is equal to the Bell number:

**Y**|$\mathsf{\mathcal{G}}$] ∀ $\mathsf{\mathcal{G}}$ ∈ ℘(

**D**), determining ${\mathsf{\mathcal{G}}}_{\mathrm{max}}$ and the partitions $\mathsf{\mathcal{G}}$ which verify (9). This is the approach followed in reference [3]. However, the larger the K the more computationally infeasible becomes the problem. For instance, if K = 22, which is the number of decision-makers of our illustrative example, then B

_{22}= 4.507 × 10

^{15}. For this reason, it is necessary to use searching algorithms that determine the Occam’s window in a computationally acceptable time. To that aim, we propose a stochastic search algorithm that we will now describe.

#### Stochastic Search Algorithm

**D**) whose models

**M**($\mathcal{G}$) have an adequate goodness of fit to data

**Y**. Figure 1 shows the main steps of the algorithm for determining the groups with homogeneous opinions.

#### Algorithm

**Step 0: Input**

**Step 1: Start**

^{(0)}is calculated.

**M**($\mathcal{G}$): $\mathsf{\mathcal{G}}$ ∈ ℘(

**D**)} with better goodness of fit properties for data Y measured by L(

**Y**|$\mathcal{G}$) = log([

**Y**|$\mathcal{G}$]). Given that the initial solutions are calculated from the observed individual pairwise comparison matrices, which are part of the dataset

**Y**, we think that this way of starting the algorithm is noticeably better than starting from randomly selected partitions.

**Step 2: Selection of the movement**

**D**, $\mathcal{G}$′, with higher values of L(Y|$\mathcal{G}$′). Movements 1 and 2 are divisive algorithms that seek to dismember groups G to get some new partitions with adequate goodness of fit properties for data. Movement 1 locates groups G with high discrepancy between their decision-makers, which is revealed by low values of L(

**Y**|{

_{G}**G**}), and then try to divide them into two subgroups

**C**and

_{1}**C**of

_{2}**G**such that L(

**Y**|{

_{G}**G**}) < L(

**Y**|{

_{G}**C**,

_{1}**C**}). Movement 2 tries to divide a random selected group

_{2}**G**in several subgroups considering the individual gamma distributions of its members. Movement 3 is based on an agglomerative algorithm that seeks to unite groups

**G′**and

**G″**with high values of L(

**Y**|{

_{G′∪G″}**G′∪G″**}). Finally, the fourth movement seeks to explore, in a random way, new partitions by combining an agglomerative step with a divisive step.

**Step 3: Partition refinement**

_{max}iterations, we stop the random search process carried out in Step 2 and we continue with the local exploration process of Step 4. Otherwise, we go to Step 2.

**Step 4: Local exploration**

**Y**|$\mathcal{G}$) for each partition G of the current estimation of the Occam’s window by relocating individual decision-makers in other groups of $\mathcal{G}$. If an improvement of L(

**Y**|$\mathcal{G}$) is obtained, we update the Occam’s window estimation by incorporating the information provided by the new partition and we restart the random search process of Step 2. Otherwise, we go to Step 5.

**Step 5: Output**

#### 3.2.4. Solution Post-Processing

**V**,

**E**). The vertices or nodes would be the set of decision-makers (

**V**=

**D**) and the edges or links (

**E**) between two nodes D and D′ ∈

**D**would represent the existence of a partition $\mathcal{G}$ ∈ ${\mathsf{\mathcal{G}}}^{(\mathrm{O}\mathrm{c}\mathrm{c}\mathrm{a}\mathrm{m}\mathrm{Window})}$ that contains a group

**G**∈ $\mathcal{G}$ such that D, D’ ∈

**G**. In the graph the decision-makers D, D’ ∈

**D**are closer (further away) if the probability to be classified in the same group is higher (lower). This probability is given by:

## 4. Case Study

- A1:
- Build a new tramline
- A2:
- Use a tram and bus combination called Tran bus
- A3:
- Use a tram combination with commuter lines
- A4:
- Do nothing

#### 4.1. Simulation Study

#### 4.2. Empirical Study

**igraph**package of the statistical program

**R**. The different groups are indicated by surrounding the corresponding decision-makers with coloured lines. The preferred rankings of each group are enclosed in light blue rectangles. The preference structures of each group are constructed from their gamma distributions and a distinction is made between strict preference (>) and non-strict preference (≥) depending on the appearance of one or more preference structures with a posteriori probability greater than or equal to 0.2.

_{1}= {D

_{1}, D

_{6}, D

_{7}, D

_{9}, D

_{10}, D

_{11}, D

_{15}, D

_{19}, D

_{20}}, G

_{2}= {D

_{2}, D

_{8}, D

_{13}}, G

_{3}= {D

_{3}}, G

_{4}= {D

_{4}, D

_{5}, D

_{12}, D

_{14}, D

_{18}, D

_{21}, D

_{22}} and G

_{5}= {D

_{16}, D

_{17}}. The posterior median priorities of G

_{1}are ${\mathrm{w}}_{\mathrm{E}\mathrm{C}\mathrm{O}}^{\left(1\right)}$= 0.6420, ${\mathrm{w}}_{\mathrm{S}\mathrm{O}\mathrm{C}}^{\left(1\right)}$ = 0.2481, ${\mathrm{w}}_{\mathrm{E}\mathrm{N}\mathrm{V}}^{\left(1\right)}$ = 0.1094 (Table 2) and the most probable preference structure is ECO > SOC > ENV (Figure 3). This structure reflects, on the one hand, that said group places the Economic criterion as the most preferred, followed by the Social criterion and the Environmental criterion as the least preferred. Some of these preferences are not strict. Therefore, for instance, group G

_{2}has ENV ≥ ECO > SOC as the most probable preference structure, because the most preferred structures are ENV > ECO > SOC and ECO > ENV > SOC with posterior probabilities 0.5938 and 0.4059, respectively. This reflects that the group priorities ${\mathrm{w}}_{\mathrm{E}\mathrm{C}\mathrm{O}}^{\left(2\right)}$ = 0.4212 and ${\mathrm{w}}_{\mathrm{E}\mathrm{N}\mathrm{V}}^{\left(2\right)}$= 0.4623 are similar (see Table 2).

_{1}on the one hand, and G

_{4}∪ G

_{5}on the other hand, that place the economic or social criteria as the most important ones, respectively. In addition, the graph of the Goal highlights that decision-makers who support economic criterion as the most preferred are more homogeneous in their opinions about the rankings of compared criteria than those who support social criterion, and these differ in their opinion about the importance of the economic and environmental criteria (see Figure 3).

_{4}(17 decision-makers in the investment criterion, 18 decision-makers in the maintenance and 12 decision-makers in other economic aspects. For the rest of the alternatives, the support shown for alternative A

_{2}in the investment sub-criterion stands out and tends to occupy the first or second place in the opinion of 17 decision-makers.

_{1}(14 decision-makers in population, 12 decision-makers in comfort and 9 decision-makers in other social aspects). For the rest of the alternatives, the support shown for alternative A

_{2}to occupy the first or second place (19 decision-makers in population, 17 in comfort and 13 in other social aspects) stands out.

_{4}is the most supported alternative by the 3 sub-criteria (15 decision-makers in environmental impact, 16 decision-makers in reversibility and 15 decision-makers in other environmental aspects). For the rest of the alternatives, it highlights the support shown for alternative A

_{2}to occupy the first or second place (16 decision-makers in impact, 16 in reversibility and 14 in other environmental aspects).

_{4}(do nothing) is supported mostly by decision-makers in the economic and environmental sub-criteria, while alternative A

_{1}(build a new tram line) is supported by the social sub-criteria. Therefore, the negotiation should be considered at the level of economic and environmental criteria, on one hand, and social criteria, on the other, deciding which of them is given more priority. Based on this priority, the chosen alternative would be to build the new tram line if social criteria are given higher priority or to do nothing, if economic and environmental criteria are given higher priority. On the other hand, analysing the graphs of the sub criteria, another alternative negotiation line can be appreciated. This line would propose the alternative A

_{2}(use a tram and bus combination called Tran bus) as a final solution since a large number of decision-makers place it in the first or second place of their preferences in all the sub-criteria.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Stochastic Search Algorithm

**Y**|$\mathsf{\mathcal{G}}$) = log([

**Y**|$\mathsf{\mathcal{G}}$]) the logarithm of the marginal prior density of a partition $\mathsf{\mathcal{G}}$ ∈ ℘(D). Noticing that (9) can be alternatively written as L(Y|$\mathrm{G}$) ≥ L(

**Y**| ${\mathsf{\mathcal{G}}}_{\mathrm{max}}$) + log(β).

**D**.

**G**⊆

**D**.

**γ**= ${\mathrm{A}}_{{\mathrm{i}}_{1}}>{\mathrm{A}}_{{\mathrm{i}}_{2}}>\dots >{\mathrm{A}}_{{\mathrm{i}}_{\mathrm{n}}}$ with i

_{1}≠ i

_{2}≠ … ≠ i

_{n}and 1 ≤ i

_{j}≤ n; j = 1,…, n a preference ranking of alternatives in A where > means “more preferred than”.

**℘**(

**D**); l = 1,…, L

_{s}; m

_{s,ℓ}is the number of groups of ${\mathsf{\mathcal{G}}}^{(\mathrm{s},\ell )}$.

**D**explored by the algorithm.

**Step 0: Input**

**Step 1: Start**

^{[1]}, …, D

^{[K]}the individual P.α and P.γ distributions:

**D**}, ${\mathsf{\mathcal{G}}}^{(0,2)}$ = {{D

^{[1]}}, …, {D

^{[K]}}}, ${\mathsf{\mathcal{G}}}^{(0,3)}$ = $\left\{{\mathbf{G}}_{\mathsf{\alpha}}^{(1)},\dots ,{\mathbf{G}}_{\mathsf{\alpha}}^{(\mathrm{n})}\right\}$, ${\mathsf{\mathcal{G}}}^{(0,4)}$ = $\left\{{\mathbf{G}}_{\mathsf{\gamma}}^{(1)},\dots ,{\mathbf{G}}_{\mathsf{\gamma}}^{(\mathrm{n}!)}\right\}$ where ${\mathbf{G}}_{\mathsf{\alpha}}^{(\mathrm{i})}=\left\{{\mathrm{D}}^{[\mathrm{k}]}:{\mathrm{P}}_{\mathsf{\alpha},\mathrm{i}}^{[\mathrm{k}]}={\mathrm{max}}_{1\le \mathrm{j}\le \mathrm{n}}\left\{{\mathrm{P}}_{\mathsf{\alpha},\mathrm{j}}^{[\mathrm{k}]}\right\}\right\}$, ${\mathbf{G}}_{\mathsf{\gamma}}^{(\mathrm{i})}=\left\{{\mathrm{D}}^{[\mathrm{k}]}:{\mathrm{P}}_{\mathsf{\gamma},{\mathsf{\gamma}}_{\mathrm{i}}}^{[\mathrm{k}]}={\mathrm{max}}_{1\le \mathrm{h}\le \mathrm{n}!}\left\{{\mathrm{P}}_{\mathsf{\gamma},{\mathsf{\gamma}}_{\mathrm{h}}}^{[\mathrm{k}]}\right\}\right\}$ where ${\mathrm{P}}_{\mathsf{\alpha},\mathrm{j}}^{[\mathrm{k}]}$ and ${\mathrm{P}}_{\mathsf{\gamma},{\mathsf{\gamma}}_{\mathrm{h}}}^{[\mathrm{k}]}$ are given in (6) and (7) with

**G**= $\left\{{\mathrm{D}}_{\mathrm{k}}\right\}$.

**D**the non-hierarchical divisive movement describe in Step 2 b

_{2}) during ${\mathrm{L}}_{0}$ iterations. Then, we eliminate the repeated partitions ${\mathsf{\mathcal{G}}}^{(0,\mathrm{s})};\mathrm{s}=1,\dots ,{\mathrm{L}}_{0}$ and include the partitions non eliminated in ${\mathsf{\mathcal{G}}}^{\mathrm{explored}}$. Finally, we put

**Y**|${\mathsf{\mathcal{G}}}^{(0,\ell )}$) ≥ max

_{ℓ}{L(

**Y**|${\mathsf{\mathcal{G}}}^{(0,\ell )}$)} + log(β)}.

**Step 2: Selection of the movement**

- a)
- Let $\mathsf{\mathcal{G}}$ = $\left\{{\mathbf{G}}_{1},\dots ,{\mathbf{G}}_{\mathrm{m}}\right\}$ be the partition drawn from ${\mathsf{\mathcal{G}}}^{(\mathrm{s})}$ = $\left\{{\mathsf{\mathcal{G}}}^{(\mathrm{s},\ell )}\in \wp \left(\mathbf{D}\right);\ell =1,\dots ,{\mathrm{L}}_{\mathrm{s}}\right\}$.
- b)
- Draw one of the four movements described below with the same probabilityb
_{1})**Movement 1**- i)
- Draw
**G**from $\mathsf{\mathcal{G}}$ with a probability proportional to $\mathrm{exp}\left[-\frac{\mathrm{L}\left({\mathbf{Y}}_{\mathbf{G}}|\left\{\mathbf{G}\right\}\right)}{\left|\mathbf{G}\right|}\right]$. Put**C**_{1}= ∅ and**C**_{2}=**G**the clusters into which**G**is to be subdivided. - ii)
- Determining D ∈
**C**_{2}such that $\mathrm{L}\left({\mathbf{Y}}_{{\mathbf{C}}_{1}\cup \left\{\mathrm{D}\right\}}|{\mathbf{C}}_{1}\cup \left\{\mathrm{D}\right\}\right)$ + $\mathrm{L}\left({\mathbf{Y}}_{{\mathbf{C}}_{2}\backslash \left\{\mathrm{D}\right\}}|{\mathbf{C}}_{2}\backslash \left\{\mathrm{D}\right\}\right)$ is maximum. - iii)
- Checking if $\mathrm{L}\left({\mathbf{Y}}_{{\mathbf{C}}_{1}}|{\mathbf{C}}_{1}\right)+\mathrm{L}\left({\mathbf{Y}}_{{\mathbf{C}}_{2}}|{\mathbf{C}}_{2}\right)$ ≤ $\mathrm{L}\left({\mathbf{Y}}_{{\mathbf{C}}_{1}\cup \left\{\mathrm{D}\right\}}|{\mathbf{C}}_{1}\cup \left\{\mathrm{D}\right\}\right)$ + $\mathrm{L}\left({\mathbf{Y}}_{{\mathbf{C}}_{2}\backslash \left\{\mathrm{D}\right\}}|{\mathbf{C}}_{2}\backslash \left\{\mathrm{D}\right\}\right)$If this condition is verified, put ${\mathrm{C}}_{1}$ = ${\mathbf{C}}_{1}\cup \left\{\mathrm{D}\right\}$ and ${\mathbf{C}}_{2}={\mathbf{C}}_{2}\backslash \left\{\mathrm{D}\right\}$ and go to ii). Otherwise, put ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ = $\mathsf{\mathcal{G}}$ ∪ $\left\{{\mathbf{C}}_{1},{\mathbf{C}}_{2}\right\}\backslash \left\{\mathbf{G}\right\}$ and go to Step 3).

b_{2})**Movement 2**Draw**G**from $\mathsf{\mathcal{G}}$ such that |**G**| > 1. Draw, for each D ∈**G**, a preference ranking γ^{D}using his/her individual P.γ distribution ${\mathbf{P}}_{\mathsf{\gamma}}^{(\mathrm{D})}$. Determine the groups G_{γ,i}= {D ∈**G**: γ^{D}= γ_{i}}; i = 1, …, n! where γ_{i}is the i-th permutation of $\left\{{\mathrm{A}}_{1},\dots ,{\mathrm{A}}_{\mathrm{n}}\right\}$ ordered according to the lexicographic order. Notice that**G**_{γ,i}contains the decision-makers of**G**who have the same preference ranking γ_{i}. Let**C**_{1}, …,**C**_{m}the non-empty groups of {**G**_{γ,i}; 1 ≤ i ≤ n!}.Put ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ = $\mathsf{\mathcal{G}}$ ∪ $\left\{{\mathbf{C}}_{1},\dots ,{\mathbf{C}}_{\mathrm{m}}\right\}\backslash \left\{\mathrm{G}\right\}$ and go to Step 3.b_{3})**Movement 3**Draw**C**_{1}≠**C**_{2}from $\mathsf{\mathcal{G}}$ with probability proportional to $\mathrm{exp}\left[-\frac{\mathrm{L}\left({\mathbf{Y}}_{{\mathbf{C}}_{1}\cup {\mathbf{C}}_{2}}|\left\{{\mathbf{C}}_{1}\cup {\mathbf{C}}_{2}\right\}\right)}{\left|{\mathbf{C}}_{1}\cup {\mathbf{C}}_{2}\right|}\right]$. Calculate**G**= ${\mathbf{C}}_{1}\cup {\mathbf{C}}_{2}$ and put ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ = $\mathsf{\mathcal{G}}$ ∪ $\left\{\mathbf{G}\right\}\backslash \left\{{\mathbf{C}}_{1},{\mathbf{C}}_{2}\right\}$. Go to Step 3.b_{4})**Movement 4**Draw**C**_{1}≠**C**_{2}from $\mathsf{\mathcal{G}}$ without replacement. Put**G**= ${\mathbf{C}}_{1}\cup {\mathbf{C}}_{2}$ and applying to**G**the Movement 2.

**Step 3: Partition refinement**

- a)
- If ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ ∉ ${\mathsf{\mathcal{G}}}^{\mathrm{explored}}$ put ${\mathsf{\mathcal{G}}}^{\mathrm{explored}}$ = ${\mathsf{\mathcal{G}}}^{\mathrm{explored}}$ ∪ {${\mathsf{\mathcal{G}}}^{{}^{\prime}}$} and go to Step 3 b).If ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ ∈ ${\mathsf{\mathcal{G}}}^{\mathrm{explored}}$ put counter = counter + 1. If counter = ${\mathrm{C}}_{\mathrm{max}}$ go to Step 4; otherwise go to Step 2.
- b)
- Calculate L
_{max}= $\mathrm{M}\mathrm{a}{\mathrm{x}}_{\mathsf{\mathcal{G}}\in {\mathsf{\mathcal{G}}}^{(\mathrm{s})}\cup \left\{\mathsf{\mathcal{G}}{}^{\prime}\right\}}\mathrm{L}\left(\mathbf{Y}|\mathsf{\mathcal{G}}\right)$ = L(Y|${\mathsf{\mathcal{G}}}_{\mathrm{max}}$). If there are ties in the maximum, the partitions with minimum number of groups are selected. - c)
- Calculate ${\mathsf{\mathcal{G}}}^{(\mathrm{s}+1)}$ = {$\mathsf{\mathcal{G}}$ ∈ ${\mathsf{\mathcal{G}}}^{(\mathrm{s})}$ ∪ {${\mathsf{\mathcal{G}}}^{{}^{\prime}}$}: L(
**Y**|$\mathrm{G}$) ≥ L_{max}+ log(α) and |$\mathsf{\mathcal{G}}$| ≤ |${\mathsf{\mathcal{G}}}_{\mathrm{max}}$|} - d)
- If ${\mathsf{\mathcal{G}}}^{(\mathrm{s}+1)}$ ≠ ${\mathsf{\mathcal{G}}}^{(\mathrm{s})}$ put s = s + 1 and counter = 0 and go to Step 2.If ${\mathsf{\mathcal{G}}}^{(\mathrm{s}+1)}$ = ${\mathsf{\mathcal{G}}}^{(\mathrm{s})}$ put counter = counter + 1. If counter = ${\mathrm{C}}_{\mathrm{max}}$ go to Step 4; otherwise, go to Step 2.

**Step 4: Local exploration**

**Y**|$\mathsf{\mathcal{G}}$) is improved. Specifically, the elements of ${\mathsf{\mathcal{G}}}^{(\mathrm{s})}$ are randomly ordered. Then, for each

**G**∈ ${\mathsf{\mathcal{G}}}^{(\mathrm{s})}$ we carry out the following steps a) to e):

- a)
- Put ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ = $\mathsf{\mathcal{G}}$
- b)
- Determine D
_{min}∈**D**such that:$\mathrm{L}\left({\mathbf{Y}}_{\left\{{\mathrm{D}}_{\mathrm{min}}\right\}}|\left\{{\mathbf{G}}_{{\mathrm{D}}_{\mathrm{min}}}\right\}\right)$ = Min D∈**D**{L(**Y**{_{D}}|{**G**_{D}})}where ${\mathbf{G}}_{\mathrm{D}}$ ∈ ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ is the group of ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ such that D ∈ ${\mathbf{G}}_{\mathrm{D}}$. Therefore, ${\mathrm{D}}_{\mathrm{min}}$ is the worst decision-maker classified according to ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$. - c)
- Determine ${\mathbf{G}}_{\mathrm{max}}$ ∈ ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ such that: $\mathrm{L}\left({\mathbf{Y}}_{\left\{{\mathrm{D}}_{\mathrm{min}}\right\}}|\left\{{\mathbf{G}}_{\mathrm{max}}\cup \left\{{\mathrm{D}}_{\mathrm{min}}\right\}\right\}\right)={\mathrm{max}}_{\mathbf{G}\in {\mathsf{\mathcal{G}}}^{{}^{\prime}}}\mathrm{L}\left({\mathbf{Y}}_{\left\{{\mathrm{D}}_{\mathrm{min}}\right\}}|\left\{\mathbf{G}\cup \left\{{\mathrm{D}}_{\mathrm{min}}\right\}\right\}\right)$
- d)
- Calculate ${\mathsf{\mathcal{G}}}_{{\mathrm{D}}_{\mathrm{min}}}$ = ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ ∪ $\left\{{\mathbf{G}}_{{\mathrm{D}}_{\mathrm{min}}}\backslash \left\{{\mathrm{D}}_{\mathrm{min}}\right\},{\mathbf{G}}_{\mathrm{max}}\cup \left\{{\mathrm{D}}_{\mathrm{min}}\right\}\right\}\backslash \left\{{\mathbf{G}}_{{\mathrm{D}}_{\mathrm{min}}},{\mathbf{G}}_{\mathrm{max}}\right\}$If L(
**Y**|${\mathsf{\mathcal{G}}}_{{\mathrm{D}}_{\mathrm{min}}}$) > L(**Y**|${\mathsf{\mathcal{G}}}^{{}^{\prime}}$) or L(**Y**|${\mathsf{\mathcal{G}}}_{{\mathrm{D}}_{\mathrm{min}}}$) ≥ L(**Y**|${\mathsf{\mathcal{G}}}^{{}^{\prime}}$) + log(β) with |${\mathsf{\mathcal{G}}}_{{\mathrm{D}}_{\mathrm{min}}}$| < |${\mathsf{\mathcal{G}}}^{{}^{\prime}}$| put ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ = ${\mathsf{\mathcal{G}}}_{{\mathrm{D}}_{\mathrm{min}}}$ and go to Step 4 b). Otherwise, go to Step 4 e). - e)
- If ${\mathsf{\mathcal{G}}}^{{}^{\prime}}$ ≠ $\mathsf{\mathcal{G}}$ go to Step 3. Otherwise, proceed to examine another partition of ${\mathsf{\mathcal{G}}}^{(\mathrm{s})}$ by repeating Steps 4 a)–4 d) until all its elements have been examined without any change in the partitions. In this last case go to Step 5.

**Step 5: Output**

## References

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Criterion | Exhaustive Average CPU Time (Standard Deviation) (K = 11) | Stochastic Search Average CPU Time (Standard Deviation) (K = 11) | % of Simulations Where $\mathcal{G}$_{opt} = $\mathcal{G}$_{max} (K = 11) | Average Factor Bayes (Standard Deviation) (K = 11) | Stochastic Search Average CPU Time (Standard Deviation) (K = 22) |
---|---|---|---|---|---|

Environmental | 98.68 (0.55) | 7.19 (0.20) | 93.00% | 0.9735 (0.1267) | 17.93 (1.50) |

Comfort | 112.68 (2.10) | 9.20 (1.18) | 73.00% | 0.8762 (0.2583) | 80.38 (18.27) |

Economic | 101.85 (6.81) | 7.21 (0.54) | 97.00% | 1.0000 (0.0000) | 15.58 (0.19) |

Impact | 116.53 (5.73) | 8.59 (0.96) | 68.00% | 0.8271 (0.2826) | 51.65 (1.97) |

Investment | 112.23 (1.17) | 7.92 (0.60) | 81.00% | 0.9162 (0.2033) | 29.83 (1.47) |

Maintenance | 115.58 (6.11) | 9.33 (1.59) | 61.00% | 0.7716 (0.3288) | 68.16 (1.25) |

Goal | 99.59 (1.72) | 7.24 (0.25) | 100.00% | 1.0000 (0.0000) | 16.43 (0.28) |

Other Environmental Aspects | 125.25 (7.29) | 9.25 (1.53) | 52.00% | 0.6996 (0.3576) | 74.60 (3.11) |

Other Economic Aspects | 120.29 (2.00) | 8.19 (0.93) | 52.00% | 0.6772 (0.3782) | 50.54 (4.17) |

Other Social Aspects | 121.41 (4.70) | 8.71 (1.06) | 69.00% | 0.8734 (0.2428) | 70.29 (4.20) |

Population | 131.03 (11.65) | 9.01 (1.13) | 58.00% | 0.7771 (0.3130) | 56.67 (10.01) |

Reversibility | 130.84 (11.03) | 10.80 (1.94) | 73.00% | 0.8649 (0.2693) | 88.74 (4.51) |

Social | 105.16 (2.21) | 7.25 (0.37) | 93.00% | 0.9581 (0.1742) | 17.57 (1.86) |

**Table 2.**Groups (G

_{g}) and group priorities $\left({\mathrm{w}}_{\mathrm{i}}^{\left(\mathrm{g}\right)}\right)$ corresponding to the goal and criteria.

GOAL | |||||||||||

Groups | Priorities | ||||||||||

ECO | SOC | ENV | |||||||||

1, 6, 7, 9, 10, 11, 15, 19, 20 | 0.6420 | 0.2481 | 0.1094 | ||||||||

2, 8, 13 | 0.4212 | 0.1145 | 0.4623 | ||||||||

3 | 0.0826 | 0.3497 | 0.5650 | ||||||||

4, 5, 12, 14, 18, 21, 22 | 0.2515 | 0.6673 | 0.0806 | ||||||||

16, 17 | 0.1171 | 0.5791 | 0.3031 | ||||||||

ECONOMIC | SOCIAL | ENVIRONMENTAL | |||||||||

Groups | Priorities | Groups | Priorities | Groups | Priorities | ||||||

INV | MAN | OEC | POP | COM | OSA | IMP | REV | OEN | |||

1, 8, 11, 22 | 0.5652 | 0.1150 | 0.3182 | 1, 2, 3, 5, 6, 7, 10, 16, 17 | 0.6503 | 0.2274 | 0.1217 | 1,11 | 0.5423 | 0.2278 | 0.2292 |

2,7, 10, 13, 15, 19, 20, 21 | 0.6296 | 0.2435 | 0.1267 | 4, 14 | 0.0858 | 0.6437 | 0.2666 | 2, 3, 6, 13, 15, 16, 18, 20, 22 | 0.6568 | 0.2266 | 0.1160 |

3, 4, 5, 6, 9, 12, 14, 16, 17 | 0.2626 | 0.6441 | 0.0929 | 8, 11, 13, 18, 20, 21 | 0.6344 | 0.1239 | 0.2410 | 4, 5 | 0.1347 | 0.3111 | 0.5533 |

18 | 0.1164 | 0.2764 | 0.6039 | 9, 12, 15, 19 | 0.2237 | 0.6460 | 0.1290 | 7, 12, 19, 21 | 0.2578 | 0.6333 | 0.1078 |

22 | 0.2419 | 0.1095 | 0.6438 | 8, 9, 10, 14, 17 | 0.4649 | 0.1137 | 0.4186 |

**Table 3.**Groups (G

_{g}) and group priorities $\left({\mathrm{w}}_{\mathrm{i}}^{\left(\mathrm{g}\right)}\right)$ corresponding to the economic sub-criteria.

INVESTMENT | MAINTENANCE | OTHER ECONOMIC ASPECTS | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Groups | Priorities | Groups | Priorities | Groups | Priorities | |||||||||

A_{1} | A_{2} | A_{3} | A_{4} | A_{1} | A_{2} | A_{3} | A_{4} | A_{1} | A_{2} | A_{3} | A_{4} | |||

1, 2, 3, 7, 12, 15, 16, 17, 19, 20, 21, 22 | 0.0626 | 0.2573 | 0.1197 | 0.5597 | 1, 16, 19 | 0.1898 | 0.054 | 0.1615 | 0.5891 | 1, 11 | 0.1562 | 0.5718 | 0.1398 | 0.127 |

4, 10 | 0.6129 | 0.1602 | 0.0925 | 0.1318 | 2, 6, 7, 12, 17, 20 | 0.2295 | 0.1238 | 0.0788 | 0.5657 | 2, 7, 16 | 0.2280 | 0.1829 | 0.1607 | 0.4255 |

5, 6 | 0.2979 | 0.1300 | 0.0639 | 0.5032 | 3 | 0.1247 | 0.5702 | 0.0655 | 0.2339 | 3 | 0.1163 | 0.5631 | 0.0658 | 0.2479 |

8, 18 | 0.1132 | 0.2716 | 0.0512 | 0.5609 | 4, 10 | 0.5738 | 0.2082 | 0.1151 | 0.0985 | 4, 10, 20 | 0.5589 | 0.1828 | 0.1517 | 0.1042 |

9, 14 | 0.1427 | 0.1128 | 0.6566 | 0.078 | 5, 8, 15, 18, 22 | 0.0878 | 0.2337 | 0.0753 | 0.6014 | 5, 6, 8, 17, 19 | 0.1366 | 0.1800 | 0.0532 | 0.6287 |

11 | 0.1225 | 0.4685 | 0.2744 | 0.1289 | 9, 14 | 0.2169 | 0.1261 | 0.5848 | 0.0701 | 9, 14, 18, 22 | 0.1560 | 0.1212 | 0.6302 | 0.0882 |

13 | 0.1014 | 0.1672 | 0.2855 | 0.4422 | 11 | 0.2613 | 0.446 | 0.1424 | 0.1483 | 12, 13, 15, 21 | 0.0663 | 0.1773 | 0.1811 | 0.5734 |

13, 21 | 0.1145 | 0.1233 | 0.2621 | 0.4969 |

**Table 4.**Groups (G

_{g}) and group priorities $\left({\mathrm{w}}_{\mathrm{i}}^{\left(\mathrm{g}\right)}\right)$ corresponding to the social sub-criteria.

POPULATION | COMFORT | OTHER SOCIAL ASPECTS | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Groups | Priorities | Groups | Priorities | Groups | Priorities | |||||||||

A_{1} | A_{2} | A_{3} | A_{4} | A_{1} | A_{2} | A_{3} | A_{4} | A_{1} | A_{2} | A_{3} | A_{4} | |||

1, 7, 18, 21 | 0.1432 | 0.3243 | 0.466 | 0.0615 | 1, 11, 19 | 0.2409 | 0.4903 | 0.1865 | 0.0758 | 1, 11 | 0.1139 | 0.6195 | 0.1401 | 0.1182 |

2, 14 | 0.4726 | 0.1551 | 0.2581 | 0.1094 | 2, 21 | 0.4476 | 0.1967 | 0.2676 | 0.0858 | 2, 4, 10, 13, 15, 16, 17, 19, 20 | 0.4938 | 0.2627 | 0.1703 | 0.0715 |

3, 6, 8, 9 | 0.1269 | 0.3035 | 0.0716 | 0.4931 | 3, 9 | 0.0938 | 0.4797 | 0.0864 | 0.3337 | 3, 5, 6, 8, 9 | 0.1428 | 0.3025 | 0.0820 | 0.4691 |

4, 5, 10, 12, 15, 16 | 0.5502 | 0.2368 | 0.1074 | 0.1041 | 4, 5, 10 | 0.5373 | 0.2482 | 0.0884 | 0.1247 | 7, 21, 22 | 0.2144 | 0.2916 | 0.4194 | 0.0722 |

11 | 0.2248 | 0.4525 | 0.1446 | 0.1754 | 6, 8 | 0.1856 | 0.1924 | 0.0669 | 0.5498 | 12, 18 | 0.2998 | 0.0972 | 0.5464 | 0.0531 |

13, 17, 19, 20, 22 | 0.4269 | 0.3742 | 0.1479 | 0.0486 | 7, 13, 15, 16, 17, 20, 22 | 0.5012 | 0.2781 | 0.1607 | 0.0594 | 14 | 0.1141 | 0.0867 | 0.5519 | 0.2400 |

12, 14 | 0.2418 | 0.116 | 0.5813 | 0.0582 | ||||||||||

18 | 0.2500 | 0.2500 | 0.2500 | 0.2500 |

**Table 5.**Groups (G

_{g}) and group priorities $\left({\mathrm{w}}_{\mathrm{i}}^{\left(\mathrm{g}\right)}\right)$ corresponding to the environmental sub-criteria.

IMPACT | REVERSIBILITY | OTHER ENVIRONMENTAL ASPECTS | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Groups | Priorities | Groups | Priorities | Groups | Priorities | |||||||||

A_{1} | A_{2} | A_{3} | A_{4} | A_{1} | A_{2} | A_{3} | A_{4} | A_{1} | A_{2} | A_{3} | A_{4} | |||

1 | 0.0773 | 0.5945 | 0.1991 | 0.1274 | 1 | 0.0579 | 0.6617 | 0.2061 | 0.0679 | 1 | 0.0680 | 0.6584 | 0.1869 | 0.0807 |

2, 7 | 0.1802 | 0.0951 | 0.2849 | 0.4369 | 2, 3, 7, 9, 12, 17, 21, 22 | 0.0722 | 0.3139 | 0.1298 | 0.4831 | 2, 6, 16, 17 | 0.2396 | 0.1144 | 0.0735 | 0.5705 |

3, 5, 8, 18, 20 | 0.1137 | 0.3355 | 0.0772 | 0.4728 | 4, 11 | 0.2668 | 0.4765 | 0.1374 | 0.1173 | 3, 13, 15, 19, 21, 22 | 0.0764 | 0.1791 | 0.1584 | 0.584 |

4, 9 | 0.3354 | 0.0967 | 0.4601 | 0.1027 | 5 | 0.3090 | 0.2674 | 0.1559 | 0.2656 | 4, 11 | 0.2549 | 0.4865 | 0.1370 | 0.1195 |

6, 17 | 0.2783 | 0.1141 | 0.0799 | 0.5253 | 6, 8, 16, 18 | 0.1275 | 0.2291 | 0.0632 | 0.5767 | 5, 8 | 0.1192 | 0.2793 | 0.0711 | 0.5278 |

10, 16 | 0.5395 | 0.2163 | 0.1176 | 0.1221 | 10, 20 | 0.5753 | 0.2007 | 0.1253 | 0.0948 | 7, 12, 18 | 0.2329 | 0.0666 | 0.1686 | 0.5282 |

11 | 0.2387 | 0.5230 | 0.1150 | 0.1208 | 13, 14, 15, 19 | 0.0854 | 0.1145 | 0.2095 | 0.5874 | 9, 10, 20 | 0.4846 | 0.2108 | 0.1770 | 0.1180 |

12, 13, 15, 19, 21, 22 | 0.0572 | 0.2056 | 0.1167 | 0.6190 | 14 | 0.1006 | 0.1461 | 0.5129 | 0.2358 | |||||

14 | 0.0945 | 0.1985 | 0.5457 | 0.1565 |

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Altuzarra, A.; Gargallo, P.; Moreno-Jiménez, J.M.; Salvador, M.
Identification of Homogeneous Groups of Actors in a Local AHP-Multiactor Context with a High Number of Decision-Makers: A Bayesian Stochastic Search. *Mathematics* **2022**, *10*, 519.
https://doi.org/10.3390/math10030519

**AMA Style**

Altuzarra A, Gargallo P, Moreno-Jiménez JM, Salvador M.
Identification of Homogeneous Groups of Actors in a Local AHP-Multiactor Context with a High Number of Decision-Makers: A Bayesian Stochastic Search. *Mathematics*. 2022; 10(3):519.
https://doi.org/10.3390/math10030519

**Chicago/Turabian Style**

Altuzarra, Alfredo, Pilar Gargallo, José María Moreno-Jiménez, and Manuel Salvador.
2022. "Identification of Homogeneous Groups of Actors in a Local AHP-Multiactor Context with a High Number of Decision-Makers: A Bayesian Stochastic Search" *Mathematics* 10, no. 3: 519.
https://doi.org/10.3390/math10030519