ARAS-H: A ranking-based decision aiding method for hierarchically structured criteria

In most of real world problems, alternatives are evaluated according to a large set of criteria which confuses the Decision Maker (DM) in terms of allotting alternatives’ assessments. So, to reduce the complexity of the presented problem, it is recommended to organize the criteria into a hierarchy tree to decompose the main problem into sub-ones. Therefore, the DM gains detailed insight on each level of the hierarchy instead of focusing only on one level. In this context, we propose an extension of the Additive Ratio ASsessment (ARAS) method to the case of hierarchically structured criteria. The proposed approach is called Hierarchical Additive Ratio ASsessment (ARAS-H) method. A major advantage of the ARAS-H method is that it enables the DM to analyze the partial pre-orders (the rankings of the alternatives) at each node of the criteria tree i.e. according to each sub-criterion. The partial pre-orders present solutions of the problem with respect to each subset of criteria. In view of determining the criteria weights at each level of the hierarchy tree, we apply the AHP method. Finally, we apply the ARAS-H method on a case study related to tourism which aims to rank tourist destination websites brands in accordance with a four levels criteria hierarchy.


Introduction
MCDA can be defined as a general framework for supporting complex decision-making situations with multiple and often conflicting objectives ( [1,2]). The International Society on Multiple Criteria Decision Making (MCDM) defines it as "The study of methods and procedures by which multiple and conflicting criteria can be incorporated into the decision process". [3] defines MCDA as "an umbrella term to describe a collection of formal approaches which seek to take explicit account of multiple criteria in helping individuals or groups explore decisions that matter". As a matter of fact, most of Multiple Criteria Decision Aiding (MCDA) methods deal with a flat structure of criteria. However, it is often the case that a practical application imposes a hierarchical structure of criteria. Actually, the hierarchy aims to decompose a complex decision making problem into smaller and manageable sub-ones to facilitate the task for the decision makers (DMs). In fact, for ranking problems, very few methods in the MCDA literature used a hierarchy of criteria for the decomposition of decision problems. The ARAS method [4] aims to rank the alternatives from the best to the worst one evaluated according to one level of criteria. Nonetheless, most of case studies require a hierarchical structure of criteria namely in the field of water resource management, public healthcare management, business management, road safety problem and more. Indeed, the hierarchical structure decomposes the primary objective into separate components analyzed on their turn into sub-dimensions toward the lowest level of the hierarchy tree. In view of implementing the hierarchical ARAS (ARAS-H) method [6], we used the terminology defined by the Multiple Criteria Hierarchy Process (MCHP) methodology since the MCHP can be applied to any MCDA method [5]. Thus, the ARAS-H is considered as an extension of the ARAS method in the case of hierarchically structured criteria. The aim is to construct partial pre-orders at each node of the hierarchy tree so that the DM gains insight on all partial dimensions of the problem, instead of focusing uniquely on the comprehensive level. Indeed, structuring decision problems is needed in circumstances requiring a large set of criteria depicting the examined problem. In this way, employing a hierarchical decomposition facilitates the interpretation of the results as it permits DMs to explore feasible elementary dimensions of the whole problem. We adopted a bottom up approach to analyze criteria in different levels of the hierarchy tree. In effect, basing on partial pre-orders obtained at the level of elementary criteria, we proceed to construct partial preorders at the upper level of the hierarchy. As a matter of fact, most of multi-criteria methods require fixing criteria weights in order to be implemented. Henceforth, we opted for the AHP [7] being a popular MCDM method to elicit ARAS-H criteria weights at each level of the hierarchy tree by conducting pairwise comparisons of criteria with respect to their importance.
The paper is divided into six sections. Section 1 will give a brief state of the art survey on hierarchical MCDA methods. In section 2, we will present the AHP method. In section 3, we will state the steps of the ARAS method. In section 4, we will explain the different steps of the ARAS-H method. In section 5, a case study will be presented to discuss the feasibility of the proposed model.
In section 6, we will conclude and we will present our perspectives.

Literature review
It is a fact that in the literature, very few authors dealt with the hierarchical structure of criteria in MCDA.
To start with, Del Vasto-Terrientes et al [8] proposed a method for ranking the alternatives using multiple and conflicting criteria organized in a hierarchical structure. Therefore, an extension of the ELECTRE-III method [9], called ELECTRE-III-H, was presented. Thus, using a bottom-up approach, the authors constructed outranking relations by propagating the partial pre-orders upwards in the hierarchy. In [10], the authors integrated the indicators of the Web Quality Index (WQI) into the ELECTRE-III-H method. As a matter of fact, MCHP methodology [5] has emerged in few works. In this context, Corrente et al. [11] applied the MCHP to the ELECTRE III ranking method. Also, they generalized the SRF (Simos-Roy-Figueira) method [12] and applied Stochastic Multiobjective Acceptability Analysis (SMAA) [13] so that the rankings and preference relations can be constructed at each level of the hierarchy tree. Moreover, Corrente et al. [14] applied MCHP to the ELECTRE-Tri methods [15,16] as extensions of ELECTRE-Tri-B [17], ELECTRE-Tri-C [17] and ELECTRE-Tri-nC [18]. Actually, they considered the case of interaction between criteria (it could be a strengthening, weakening or antagonistic effect) and they extended the SRF method to determine criteria weights in a hierarchical structure of criteria. Nevertheless, they adopted a top-down procedure to infer criteria weights. Additionally, Corrente et al. [19] proposed an extension of ELECTRE [20], [21] and PROMETHEE [22] methods to the case of the hierarchy of criteria and adapted Robust Ordinal Regression (ROR) [23] to the hierarchical versions of ELECTRE and PROMETHEE methods. Indeed, ROR takes into account the preference information provided by the DM but its main specificity is that it gives recommendations taking into account not only one but the plurality of instances of the assumed preference model compatible with such preferences. Likewise, Corrente et al. [5] applied MCHP on ROR which permits to consider necessary and possible preference information at each node of the hierarchy tree. As a matter of fact, Podinovskaya and Podinovski [24] developed a new model with a hierarchical structure of criteria. They considered the qualitative and quantitative importance of criteria which enables the comparison of decision alternatives for different types of criteria scale. Moreover, Corrente et al. [25] proposed an extension of the MCHP methodology to sorting problems within a hierarchical structure of criteria. They formulated a model allowing the inference of preference relations through preference disaggregation techniques based on an additive value function model (namely the UTADIS [26,27] and UTADIS GMS [28] methods). The combination between MCHP and UTADIS, UTADIS GMS allows the consideration of global and partial preference judgments to add more flexibility to the specification of the input preference information required in the decision aiding process. Apart from this mentioned approaches, Saaty developed two well-known hierarchical methods: the AHP and the ANP. The application of Analytic Hierarchy Process (AHP) begins with a problem being decomposed into a hierarchy of criteria so as to be more easily analyzed and compared in an independent manner. After this logical hierarchy is constructed, the DM can systematically assess the alternatives by making pair-wise comparisons according to each of the chosen criteria. The Analytic Network Process (ANP) [29] is considered as a generalization of the AHP. The ANP can model complex decision problems, where a hierarchical model is not sufficient. However, the ANP allows for feedback connections and loops.

The AHP method
The AHP method decomposes the main problem into a hierarchy of criteria to be easily analyzed by the DM. After the construction of the hierarchy tree, the DMs can systematically assess the alternatives by making pair-wise comparisons for each criterion. In fact, this comparison may use concrete data from the alternatives or human judgments. The steps of the AHP method are as follow: Step 1: The first step consists in decomposing the problem into a hierarchy of goal, criteria, sub-criteria and alternatives.
Step 2: The second step of AHP method consists of transforming the empirical comparisons into numerical values to be compared for each criterion ( Step 3: The pairwise comparisons of criteria are organized into a square matrix. The diagonal elements of the matrix are equal to 1. The criterion in the i th row is more important than criterion in the j th column if the value of element (i, j) is greater than 1. The (j, i) element of the matrix corresponds to the reciprocal of the (i, j) element.
Step 4: The contribution of each criterion to the organizational goal is determined by calculating the priority vector (or Eigenvector). The Eigenvector shows the relative weights between each criterion. It is obtained by calculating the mathematical average according to all criteria. The sum of all values from the vector is always equal to 1. The elements of the normalized eigenvector are known as weights for the criteria and ratings for the alternatives.
Step 5: The next step is to look for any data inconsistencies. The objective is to capture enough information to determine whether the DMs have been consistent in their choices or not. Thus, the consistency condition should verify this equality: aij = aik × akj for all i; k; j. It expresses the transitivity of preferences. The consistency index (CI) is calculated as: CI = ʎ − −1 where ʎmax is the maximum eigenvalue of the judgment matrix. This consistency index (CI) can be compared with an average consistency index (RI) ( Table 2). The derived ratio, CI RI is called the consistency ratio (CR). Saaty states that the value of CR should be less than 0,1.  Step 6: The last step of the AHP method is to produce the alternatives' priority values based on the judged importance of one alternative over another with respect to a common criterion.

The ARAS method
Step 1 Constructing the decision matrix X for m alternatives and n criteria.
Step 2 Normalizing the decision making matrix to unify the incommensurable measures of the criteria so all the performances can be compared. The authors suggest these two normalization formula.
The criteria to be maximized, are normalized as follows.
The criteria to be minimized, are normalized as follows.
Where is the performance value of the alternative i with respect to the criterion j; ̅ ij denotes the normalized values of the normalized decision-making matrix ̅ and * is the original value of minimized criteria.
Step 3 Forming the weighted-normalized matrix ̂ through this formula: ̂= ̅ ij ; i = 1,…,m and j = 1,…, n Where ̅ ij is the normalized value of the criterion j; wj is the weight of the criterion j such that ∑ =1 = 1 and wj > 0 for all gj.
Step 4 Determining the values of optimality function denoted by such that: Step 5 In ARAS method, the utility degree value Ki determines the relative efficiency of a feasible alternative .
That is to say, Ki = Where 0 is the optimal value (i.e. the maximum value of S ) and the calculated values Ki are in the interval [0,1].

Step 6
Ranking in an increasing order the values of the utility degrees Ki to rank the alternatives from the best one to the least important.
Nonetheless, the ARAS method deals with a flat structure of criteria. However, in most real world applications, criteria are organized naturally according to a hierarchical structure. For that, in the next section, we will define the MCHP methodology and present the ARAS-H algorithm.

The Multiple Criteria Hierarchy Process
In this section, we define the different concepts and notation with regard to a hierarchical structure of criteria. Therefore, we will distinguish between three types of criteria ( Figure 1). G is the set of all criteria such that G = R ∪ E ∪ I Taking figure 1 as an example of a hierarchy of criteria, then: In what follows, we present the basic notations of the hierarchy of criteria introduced by [5].

Let:
A is the finite set of alternatives; l is the number of levels in the hierarchy of criteria; G is the set of all criteria at all considered levels; IG is the set of indices presenting position of the specific criteria in the hierarchy tree; wj: be the criteria weight such that ∑ ∈ I G = 1 and wj > 0.
We adopt a bottom-up approach for the construction of partial pre-orders i.e. we start by ranking the alternatives from the lowest level of the hierarchy tree upward to the root criterion.

The ARAS-H algorithm
The For h = l Step 1: Fixing the hierarchical structure of the set of criteria using the MCHP, distinguishing the subsets of elementary criteria, and higher level criteria, up to the root criterion.
Step 2: Asking the DM to provide the performance matrix in which the alternatives are evaluated according to elementary criteria.
Step 3: Normalizing the performance matrix with respect to the min-max normalization technique. Step 7: Constructing the partial pre-orders of sub-criteria from ranking in a decreasing order the values of utility degrees .
For h = l-1 Step 8: Calculating the utility degrees Ki for each alternative i ∈ A according to each first level Criteria weights are normalized such that ∑ ∈ = 1.
Step 9: Calculating the fraction ′ = ∑ ∈ Step 10: Ranking in a decreasing order the values of ′ Step 11: Constructing the partial pre-orders at the upper level of the criteria's hierarchy tree.
For h = 1 (i.e. at the first level of the hierarchy tree) Go to step 8 Go to step 9 Go to step 10 Step 12: Constructing the complete pre order at the first level of the hierarchy tree (i.e. ranking the alternatives according to the root criterion).
The proposed ARAS-H algorithm is illustrated in the below flow chart.

An illustrative example
The aim of this example is to present websites designed to promote tourist destination brands. In this section, we address the analysis of this dataset with the proposed ARAS-H method. Indeed, websites have become important tools for the communication of destination brands and the sale of a range of tourism services and related items [30]. •Constructing the partial pre-orders at the upper level of the criteria's hierarchy tree.
For h=1 •Repeat step 8 •Repeat step 9 •Repeat step 10 step 12 •Constructing the complete pre order at the first level of the hierarchy tree Organizing criteria into a hierarchical structure will permits the DM to gain insight on rankings not only on the comprehensive level but at each node of the hierarchy. This will help him to identify the strengths and weaknesses of each website and therefore to develop appropriate strategies to deal with the discovered weaknesses. Due to the complexity of the problem, experts organized criteria into a hierarchical structure (figure 2). They are asked to identify the possible criteria to evaluate 10 tourist     After constructing the weighted-normalized decision matrix, the ARAS method is applied for each MAX g 1,1,1,1 g 1,1,1,2 g 1,1,1,3 g 1,1,2,1 g 1,1,2,2 g 1,1,2,3 g 1,2,1 g 1,2,2 g 1,3,1,1 g 1,3,1,2 g 1,3,2,1 g 1,3,2, 2  Andalusia  100  100  60  25  62,5  16  72,2 75,53  100  58  100  60  Catalonia  74  30  0  75  87,5 38,67  33,3  59  0  78  70  30  Barcelona  100  60  80  75  62,5  100  74,1 85,93  0  74  100  70  Madrid  88  100  60  50  100 61,33  75,9 72,87  0  38  50  60  Santiago  100  80  20  50  50  32  100  68,6  100  48  80  60  Rias Baixas  50  30  0  75  62,5  16  46,3  59  100  38  50  30  Stockholm  100  100  100  100  62,5      The final ranking of the alternatives according to the root criterion as presented in figure 4 is: As can be seen, Barcelona outranks all the other alternatives according to sub-criterion "brand treatment". Also, it is considered as the best alternative for the root criterion. Together with Rias Baixas, which is considered to be the worst according to sub-criterion "usability and accessibility", is also the worst alternative for the root criterion. The founded results are robust and therefore recommended. structure. Among them, we cite the Simos method [31] which is considered to be a very simple and easy technique for the DMs. The procedure consists of ranking the subsets of cards from the least important to the most important while assigning a position to each card including the white ones. The procedure ends with the determination of the normalized weights. Another criteria weight elicitation method is the Entropy [32]. In this method, the DM interferes indirectly in the weight elicitation process. The criteria weights values are determined through the alternative performances from the decision making matrix.
As a matter of fact, [33], extended the classical Entropy method to the case of hierarchically structured criteria. Besides these mentioned techniques, we used the ANP method. Therefore,

Conclusion
In this paper, we proposed a ranking method which deals with a hierarchical structure of criteria called ARAS-H method. This method adopts a bottom-up approach. It constructs partial pre-orders from the lowest level of the hierarchy tree and propagates them to the upward level until reaching the root criterion to construct the complete pre-order. Also, we presented a sensitivity analysis which aims to study the effect of some criteria weight elicitation methods on the complete pre-order of the ARAS-H method. The advantage of the proposed approach is that the DM gains insight on the ranking of the alternatives at each node of the hierarchy tree. Therefore, he will be able to detect the inconsistencies and analyzing them in a deepest way. Nevertheless, the proposed method deals with a small problem within a small set of alternatives as well as criteria. As a perspective, we intend to elicit ARAS-H criteria weights through mathematical programming at each level of the hierarchy tree. Moreover, we aim to deal with fuzzy data to allow the model to be applicable in the case of lack of certainty. Consequently, we will develop the fuzzy ARAS-H (F-ARAS-H) method.   Table 9: Weighted normalized decision-making matrix g 1,1,1,1 g 1,1,1,2 g 1,1,1,3 g 1,1,2,1 g 1,1,2,2 g 1,1,2,3 g 1,2,1 g 1,2,2 g 1,3,1,1 g 1,3,1,2 g 1,3,2,1 g 1,3,2,2 Andalusia    1,1,1, g1,1,1,2, g1,1,1,3 criteria (g1,1,2,1, g1,1,2,2, g1,1,2 Switzerland 0,64 6 Figure 9: The partial pre-order generated from the aggregation of "usability" and "accesibility" criteria