2.1. Background and Literature Review
Analytic hierarchy process (AHP) is a commonly used MCDM technique originally proposed by Saaty [
1]. However, it has been subject to criticism since it employs an unbalanced scale of judgments and it is unable to handle imprecision and uncertainty in the pairwise comparison process [
9]. In order to address these shortcomings, FAHP was developed to solve the hierarchical problems arising from the fact that decision-makers usually find that giving interval judgments is more accurate than giving fixed value judgments. As a result, FAHP uses both, fuzzy set theory and fuzzy numbers in order to express the uncertain comparison of opinions and it enables the incorporation of the incomplete, unquantifiable and non-obtainable information into the decision-making process.
Several authors have proposed fuzzy analytic hierarchy process (FAHP) applications [
6,
10,
11,
12,
13,
14], since it represents a systematic approach to the selection of alternatives and the resolution of problems by applying fuzzy set theory, which helps to express the uncertain comparison of opinions through the use of fuzzy numbers and AHP. The methods employed by van Laarhoven and Pedrcyz [
10], Buckley [
11], Enea and Piazza [
13] and Krejčí et al. [
14] derive fuzzy priorities represented as fuzzy numbers or fuzzy sets. On the other hand, Chang [
12] and Mikhailov [
6] obtain crisp priorities from fuzzy comparisons.
FAHP is frequently applied along with other tools, namely, goal programming (GP), fuzzy linear programming (FLP), fuzzy DEMATEL (FD), MOORA and fuzzy MOORA (FMOORA), TOPSIS and fuzzy TOPSIS (FTOPSIS), VIKOR, strengths, weaknesses, opportunities, threats (SWOT) analysis, grey relational analysis (GRA), fuzzy comprehensive evaluation method (FCEM), particle swarm optimization (PSO) and DEA. In
Table 1, we display some relevant FAHP applications in which fuzzy numbers with linear membership functions, that is, triangular numbers (TN) are the main membership function used, followed by trapezoidal number (TrN).
As
Table 1 shows, a great number of contributions only apply FAHP. In other cases, however, when applying FAHP combined with other methodologies, a first step is to determine weights for each criterion using FAHP, while a second step entails establishing a ranking using some of the aforementioned methods. These techniques are primarily MCDM methodologies which complement FAHP and have been applied to many fields such as economics, finance, environment or engineering.
2.2. Mikhailov’s Model: Fuzzy Preference Programming (FPP)
FAHP models operate basically using triangular or trapezoidal fuzzy numbers with linear membership functions, which involves the subsequent limitation for the decision-makers when their opinions must be represented.
The main steps in FAHP are the following:
Just like in classical AHP, obtain a hierarchical structure from a decision-making problem.
The next step is to develop pairwise fuzzy comparison matrices. Take a prioritization problem with
n components, where fuzzy numbers denote pairwise fuzzy comparisons. As in classical AHP, every set of comparisons for each level needs
n (
n−1)/2 judgments, these being used to build a positive fuzzy reciprocal comparison matrix
that:
Saaty [
1] has set out a pairwise comparison scale ranging from 1 to 9, where a value of 9 represents “extremely preferred”, and a value of 1 is “equally preferred”. Due to the complexity and uncertainty of many real-world decision-making problems, researchers acknowledge exact judgments are often unrealistic. When the information provided by the decision-makers is vague and imprecise, even more, when it is formulated in linguistic terms, FAHP becomes an appropriate tool. Therefore, different approaches of the fuzzy Saaty’s fundamental scale emerge in the literature [
19].
A fuzzy judgment matrix
that is constructed as in (1) where the components of the pairwise fuzzy comparison matrix are expressed by triangular numbers
, where
Besides,
Thus, when the proposed scale is , the fuzzy fundamental scale is known as a multiplicatively reciprocal pairwise comparison (multiplicative PCM).
In other cases, an alternative fuzzy scale is proposed for pairwise comparisons where fuzzy judgment matrix
is constructed as in (1) using elements from interval [0, 1], where 0.1 is “extremely not preferred” and 1 “extremely preferred”. The components are expressed by triangular numbers
, where
Besides,
In this case, the scale is known as the Additive reciprocal pairwise comparison (Additive PCM).
Multiplicative and additive PCM’s are equivalent, and moreover a multiplicative PCM can be transformed to an additive PCM (see Krejčí [
54]). The substantial difference between the multiplicative PCM and the additive PCM are the scales decision-makers used [
55]. In our proposal, we have opted for the multiplicative scale PCM as it is the most preferred by most researches [
14], as well as by Mikhailov’s model.
The vector of exact priorities is
The third step is control of coherence and resulting priorities, which evaluates consistency and also obtains priorities from the pairwise fuzzy matrices.
One last step is aggregation of priorities and classification of alternatives. By applying a simple weighted sum, we aggregate the local priorities computed in the distinct levels of the hierarchy of decisions. The global priorities thus obtained provide the final ranking and the selection of the best alternative.
The reason why Mikhailov’s methodology [
56] has been selected is because it helps us evaluate consistency of the decision-makers’ opinions by using the so-called
or “index of consistency” [
52]. According to this methodology, fuzzy preference programming (FPP) is proposed to obtain priorities from the fuzzy comparison judgments, which removes some of the drawbacks of the fuzzy prioritization methods currently employed. This proposed approach does not involve the building up of complete fuzzy comparison matrices, and besides it allows us to derive priorities from an incomplete set of fuzzy judgments. Moreover, the approach remains invariant to the precise shape of the fuzzy sets that have been employed in the representation of judgments [
52].
By employing α-cuts, initial fuzzy judgments are converted into a series of interval judgments. The method is used to transform the FPP priority allocation problem into a fuzzy program. This allows us to derive clear priorities from interval judgments, which correspond to each α-level cut. Therefore, the need for another fuzzy classification procedure disappears.
The FPP priority allocation problem consists in solving the following program [
6]:
Mikhailov denotes as “consistency index”, which is used to evaluate the satisfaction level of the optimal priority vector . When is positive, all the solution coefficients entirely satisfy fuzzy opinions. This means that the initial set of fuzzy judgments is significantly consistent. Conversely, a negative value of shows that the fuzzy judgments are highly inconsistent, that is to say, we can employ the optimal value of as a consistency measure of the initial set of fuzzy judgments.
2.3. Extended FAHP (E-FAHP) with (m,n)-Trapezoidal Numbers
An extension of FAHP Mikhailov’s model for its application with (m,n)-trapezoidal numbers called Extended FHP (E-FAHP) is proposed. Before establishing the E-FAHP model, let us begin with a basic definition for (m,n)-trapezoidal number.
Definition 1. (m,n)-trapezoidal number.Let us now define a type of fuzzy number called (m,n)-trapezoidal number,where. Its membership function is provided by Appadoo [7]: The representation of
, from the α-cuts is:
From
, we can obtain a trapezoidal number
, when m = n = 1, that is:
In a similar way we could obtain a triangular number
, from
, if m = n = 1, and from
, and we rewrite
for
, that is:
The membership function of
where m,n
[0, ∞], is displayed in
Figure 1:
Figure 1 depicts different types of graphic representations for the
, which is symmetric when
and m = n, and asymmetric when
or m ≠ n or both. On the other hand, when m = n = 1 and
we have the representation of the trapezoidal number
, while if m = n = 1 and
we can derive the triangular number
.
Next, we state the main operations (see Appadoo [
57]), with
and
being two (m,n)-trapezoidal numbers,
The aggregation of
and
, will be given by:
The difference between
and
, will be given by:
The multiplication of
and
, will be given by:
The division between
and
, will be given by:
In our case, and unlike Mikhailov’s model, let us suppose a fuzzy judgment matrix
, built as in (1). We represent the components of the pairwise fuzzy comparison matrix by
, where
Also,
As a result, an exact priority vector
which derives from
should satisfy fuzzy inequations:
where
and symbol
represent “fuzzy less than or equal to”.
In order to measure the satisfaction degree of different crisp relationships
as regards double side inequality in Equation (4), we can define a new membership function from (3):
The solution to the prioritization problem through FPP relies on two main assumptions [
56].
Assumption 1. This assumption requires the existence of non-empty fuzzy feasible areaon the–dimensional simplex Being defined as an intersection of the membership functions, similar to (5) and the simplex hyperplane (6), the membership function of the fuzzy feasible areais given by: Once membership functions (5) are defined as L-fuzzy sets, we can relax the assumption of non-emptiness ofon the simplex. If fuzzy judgments are significantly inconsistent, thencould take negative values for all normalized priority vectors.
Assumption 2. The second assumption incorporates a selection rule determining a priority vector which has the maximum degree of membership in aggregate membership function (7). It can be easily proven thatis a convex set and therefore priority vectoralways has the highest degree of membership. Let us represent the maximin of prioritization problem (8) as follows:
Taking into account the particular form of membership function (5), problem (9) can be converted into the E-FAHP preference programming:
If the elements of the pairwise fuzzy comparison matrix were represented by trapezoidal numbers
, where
, that is m = n = 1, then the problem would become: