An intelligent approach for the evaluation of transformers in a power distribution project

Abstract

In this study, a hybrid approach is presented for the evaluation and selection of transformers in a power distribution project. Ranking transformers and selecting the best among alternatives is a complex multiple criteria decision making (MCDM) problem with various possibly conflicting quantitative and qualitative criteria. In this research, two hesitant fuzzy MCDM methods; hesitant fuzzy Analytic Hierarchy Process (hesitant F-AHP) and hesitant fuzzy Preference Ranking Organization Method for Enriching Evaluations II (hesitant F-PROMETHEE II) are combined to evaluate and rank transformers. In the hesitant fuzzy AHP-PROMETHEE II, hesitant F-AHP is implemented to determine criteria weights and hesitant F-PROMETHEE II is applied to rank transformer alternatives, utilizing obtained criteria weights. An illustrative example is presented to demonstrate the effectiveness and applicability of the proposed approach. In the example, five transformers are evaluated based on twelve criteria by three decision makers (DMs) and best alternative is selected. For comparison analysis, integration of hesitant F-AHP and hesitant fuzzy Technique for Order Preference by Similarity to Ideal Solution (hesitant F-TOPSIS) is used and results are compared.

Keywords

Transformer selection hesitant fuzzy multiple-criteria decision making AHP-PROMETHEE II AHP-TOPSIS

1 Introduction

A Transformer is the hub equipment of a power system and it is a significant equipment of electrical power transmission and distribution system since the stability of a power network system is directly related to the safety and reliability of the transformer [27, 42]. The transformer manufacturing industry is overcapacity and the quality and type of transformers vary, therefore it is really significant to evaluate and select the best transformer in order to ensure the safety and reliability of the power grid [10]. In power distribution industry, long lasting and reliable transformers are desired which is related to the health condition and degradation rate of transformer’s insulation [42]. Evaluation and selection of a transformer for a power distribution project is naturally a MCDM problem since there are various possibly conflicting qualitative and quantitative criteria that need to be taken into consideration. Therefore, a MCDM method is needed to manage the decision making process efficiently.

In the literature, as the MCDM method, AHP or its fuzzy extension F-AHP is frequently used for the evaluation of conditions of transformers, detection methods and design of the components. Song et al. [43] applied failure mode and effects analysis to understand severity and occurrence of failures and detection methods of transformer faults and then implemented AHP to prioritize these detection methods. Sun et al. [44] used F-AHP and modified weighted averaging to assess conditions of power transformers. Daneshmand et al. [16] implemented AHP to evaluate winding schemes and select the best geometry for the high-temperature superconducting transformers. Suwanasri et al. [45] implemented AFP to determine criteria weights of the components that effect the conditions of power transformers for risk-based maintenance. Tee et al. [46] applied principal component analysis and AHP to evaluate and rank in-service transformers based on their insulation conditions. Wan et al. [54] developed a transformer health state evaluation method, where they applied association rules for the transformer faults and AHP for the calculation of weights of comprehensive states. Wang et al. [56] first applied Kano model to collect and classify user requirements for oil-immersed distribution transformer product service system and then applied AHP to sort these requirements and recombine.

In this research, as the MCDM method, hesitant F-AHP is incorporated with hesitant F-PROMETHEE II to have both methods’ advantages in the transformer evaluation and ranking process. The key advantage of PROMETHEE methods is that they are easy to apply and they deliver stable results [31 , 53]. However, this integration is needed, since just like TOPSIS, ELECTRE and VIKOR methods, PROMETHEE methods also do not provide precise guidelines for assigning criteria weights, assuming DM(s) are able to determine those, properly [29 , 53]. Moreover, PROMETHEE methods are weaker in assessing qualitative criteria then AHP, and when the number of criteria is more than seven, with PROMETHEE methods, it may be difficult for DMs to obtain a clear view and evaluate criteria and alternatives [29 , 37]. On the other hand, with (hesitant fuzzy) AHP, utilizing pairwise comparisons of both quantitative and qualitative criteria and consistency checks of these comparisons, dependable criteria weights can be determined. However, when there are many alternatives and criteria in the problem, implementing (hesitant fuzzy) AHP both to the evaluation of criteria and to the ranking of alternatives (without the integration) can take a lot of time and effort and be burdensome since a large number of pairwise comparisons are needed [38 –40]. Therefore, integration of both methods is essential. In the hesitant fuzzy AHP-PROMETHEE II, reliable results are obtained without large number of pairwise comparisons, especially when there are many alternatives. Also, with the utilization of hesitant fuzzy linguistic term sets (HFLTS) and hesitant fuzzy set concepts in the hesitant fuzzy AHP-PROMETHEE II, uncertainty, ambiguity and hesitations DMs might have in their preferences can be reflected in the decision making process and this makes the process more flexible and real.

AHP [36] has a multi-level, hierarchical structure and alternatives are evaluated with respect to each criterion utilizing pairwise comparisons and then ranked based on a total weighted score. To capture the ambiguity and fuzziness of DMs’ judgements, concepts of fuzzy set theory [61, 63] was integrated to AHP and F-AHP was developed. PROMETHEE, developed by Brans [6], is a family of outranking methods. PROMETHEE, different versions of PROMETHEE and F-PROMETHEE, and their real life application areas are given in details in the authors previous work [38].

Traditional (fuzzy) AHP-PROMETHEE integration has been used to benefit from both methods’ advantages in several applications in the literature such as selection of milling machines to be purchased in a company [15], ranking the tasks of the assignments [3], evaluation of energy supply configuration of autonomous desalination units [19], evaluation of location alternatives of solar power plants [38]. In the traditional F-AHP and F-PROMETHEE II, DMs give a single linguistic expression to evaluate criteria and alternatives and this does not reflect the hesitation DM might have in reality. Instead, in hesitant F-AHP and hesitant F-PROMETHEE II, DM might use “hesitant” terminologies and HFLTS. For example, while doing pairwise comparisons of criteria in hesitant F-AHP, DM might asses as “criterion 3 is between absolutely weak and fairly weak in comparison to criterion 9”. Also, while evaluating alternatives with respect to criteria in hesitant F-PROMETHEE II, DM might assess as “alternative 1 is between medium good and poor in terms of criterion 1”. In this manner, DMs might express their opinions in a more flexible, practical and accurate manner with multi-linguistic expressions [23].

Hesitant fuzzy set concepts and HFLTS, introduced by Torra and Narukawa [49] and Rodrigez et al. [34] are integrated with several MCDM in the literature. Some of these integrations with application areas are: hesitant F-AHP for evaluation of performance of a cargo company [50], CNC router selection in woodwork manufacturing [9] and evaluation of sustainability of hydrogen production methods [1]; hesitant F-TOPSIS and F-VIKOR for quality evaluation of teachers [52]; hesitant F-AHP and TOPSIS for evaluation of summer sport schools [4]; hesitant F-AHP and then F-VIKOR for ranking power generation enterprises [23]; interval-valued intuitionistic hesitant fuzzy entropy based VIKOR for selection of industrial robots [32]; Superiority and Inferiority Ranking (SIR) method with FFLTS for evaluation of sustainable energy technologies such as natural fossil fuel, hydrogen and renewable energy technologies [62]; hesitant fuzzy AHP-VIKOR for evaluation of innovation projects [39]; proportional hesitant fuzzy linguistic term set (PHFLTS) TOPSIS for evaluation of hazardous materials transportation alternatives [11]; proportional interval type-2 hesitant fuzzy TOPSIS for evaluation of data science and machine learning platform vendors [13]; hesitant fuzzy AHP, VIKOR and TOPSIS for evaluation of energy technologies [14], hesitant F-AHP and GRA for performance evaluation of bank regions [51] and HFLTS ARAS for selecting ecofriendly cities [8].

PROMETHEE methods are applied with hesitant fuzzy set concepts and HFLTS in various applications in the literature. Wu et al. [57] first applied ANP to determine the weights of indicators and then used hesitant fuzzy PROMETHEE to rank social sustainability of small hydropower stations in China. Mahmoudi et al. [30] presented an extension of PROMETHEE with typical hesitant fuzzy sets and gave two examples about ranking overseas outstanding teachers and strategy initiatives. Liu et al. [26] applied double hierarchy hesitant fuzzy linguistic PROMETHEE with subjective and objective information to evaluate public-private-partnership advancement from stakeholder perspective. Tian et al. [47] implemented hesitant F-PROMETHEE to rank start-up firms for venture capitalists. Feng et al. [17] presented a new hesitant fuzzy linguistic PROMETHEE method where they proposed to use possibility theory instead of preference functions. Liang et al. [17] developed a projection-based hesitant F-PROMETHEE and applied it to a green supply chain management problem to evaluate green practices for the domestic organizations in China. Li and Wang [22] used hesitant probabilistic fuzzy sets with QUALIFLEX and PROMETHEE II and gave an illustrative example about evaluation of candidates in doctoral entrance interviews. Liao et al. [24] implemented hesitant F-PROMETHEE method to evaluate green logistic providers.

At present, to the best of the authors knowledge, integration of hesitant F-AHP and hesitant F-PROMETHEE II has never been studied, especially for the evaluation of transformers in a power distribution project. With the proposed hesitant fuzzy AHP-PROMETHEE II, transformers are evaluated and ranked without too many tiresome pairwise comparisons and complex calculations. Application steps of hesitant fuzzy AHP-PROMETHEE II are explained in Section 2. For comparison analysis, hesitant fuzzy AHP-TOPSIS is applied and its steps are presented in Section 3. An illustrative example and comparison analysis are given in Section 4 along with the conclusions in Section 5.

2 Hesitant fuzzy AHP-PROMETHEE II

Fuzzy set theory includes classes with unsharp boundaries [21, 28] and sets in crisp theories can be fuzzified by applying the concept of fuzzy sets [61]. In the proposed hesitant fuzzy AHP-PROMETHEE II approach, and in hesitant fuzzy AHP-TOPSIS, triangular fuzzy numbers (TFNs) are utilized due to its uncomplicatedness. A fuzzy number is a special fuzzy set F ={ (x, μ_F (x)) , x ∈ R } where x is a real number, R : - ∞ < x < + ∞ and μ_F (x) is from R to [0, 1]. A TFN, $\tilde{M} = (l, m, u)$ l ⩽ m ⩽ u has the triangular type membership function; $μ_{F} (x) = {\begin{matrix} 0 \\ x - l / m - l \\ u - x / u - m \\ 0 \end{matrix} \begin{matrix} x < l \\ l ⩽ x ⩽ m \\ m ⩽ x ⩽ u \\ x > u \end{matrix}$ (1)

Arithmetic operations between two positive TFNs $\tilde{C} = (l_{1}, m_{1}, u_{1})$ , $\tilde{D} = (l_{2}, m_{2}, u_{2})$ l₁ ⩽ m₁ ⩽ u_1,l₂ ⩽ m₂ ⩽ u₂ and a crisp number E are explained as [2 , 58]: $\tilde{C} + \tilde{D} = (l_{1} + l_{2}, m_{1} + m_{2}, u_{1} + u_{2})$ (2)

$\tilde{C} + E = (l_{1} + E, m_{1} + E, u_{1} + E)$ (3)

$\tilde{C} - \tilde{D} = (l_{1} - u_{2}, m_{1} - m_{2}, u_{1} - l_{2})$ (4)

$\tilde{C} - E = (l_{1} - E, m_{1} - E, u_{1} - E)$ (5)

$\tilde{C} * \tilde{D} = (l_{1} * l_{2}, m_{1} * m_{2}, u_{1} * u_{2})$ (6)

$\begin{matrix} \tilde{C} * E = (l_{1} * E, m_{1} * E, u_{1} * E) \\ for E ⩾ 0 \end{matrix}$ (7)

$\frac{\tilde{C}}{\tilde{D}} = (\frac{l_{1}}{u_{2}}, \frac{m_{1}}{m_{2}}, \frac{u_{1}}{l_{2}})$ (8)

$\frac{\tilde{C}}{\tilde{D}} = (min (\frac{l_{1}}{l_{2}}, \frac{l_{1}}{u_{2}}, \frac{u_{1}}{l_{2}}, \frac{u_{1}}{u_{2}}), \frac{m_{1}}{m_{2}}, max (\frac{l_{1}}{l_{2}}, \frac{l_{1}}{u_{2}}, \frac{u_{1}}{l_{2}}, \frac{u_{1}}{u_{2}}))$ (9) if $\tilde{C}$ and $\tilde{D}$ are two TFNs (not necessarily positive TFNs). $\frac{\tilde{C}}{E} = (\frac{l_{1}}{E}, \frac{m_{1}}{E}, \frac{u_{1}}{E}) for E > 0$ (10)

${\tilde{C}}^{- 1} = (\frac{1}{u_{1}}, \frac{1}{m_{1}}, \frac{1}{l_{1}})$ (11)

$MAX (\tilde{C} + \tilde{D}) = (max (l_{1}, l_{2}), max (m_{1}, m_{2}), max (u_{1}, u_{2}))$ (12) $MIN (\tilde{C} + \tilde{D}) = (min (l_{1}, l_{2}), min (m_{1}, m_{2}), min (u_{1}, u_{2}))$ (13)

The graded mean integration approach [59] is applied for defuzzification of TFNs as seen below: $Crisp (\tilde{C}) = (4 m_{1} + l_{1} + u_{1}) / 6$ (14)

While making judgements, DMs might have some hesitations in deciding the membership degree of an element since there may be more than one likely value. To reflect these hesitations, the concept of hesitant fuzzy sets (HFS) was introduced [48, 49]. In Triangular Fuzzy Hesitant Fuzzy Sets (TFHFS), the membership degree of an element to a given set is expressed by several possible TFNs.

If X is a fixed set, the HFS on X returns a subset of [0, 1] by: $G = {〈 x, h_{G} (x) 〉 | x \in X}$ (15) where h_G (x) is the possible membership degrees of element x ∈ X to set G with values in [0, 1]. The lower and upper bounds are calculated as: $h_{(x)}^{-} = min h (x), h_{(x)}^{+} = max h (x)$ (16)

Basic operations for 3 HFS h, h1, h2 are given as: $h^{ℷ} = \cup_{γ \in h} {γ^{ℷ}}$ (17)

$ℷ h = \cup_{γ \in h} {1 - {(1 - γ)}^{ℷ}}$ (18)

$h_{1} \pm h_{2} = \cup_{γ 1 \in h 1, γ 2 \in h 2} {γ^{1} + γ^{2} - γ^{1} γ^{2}}$ (19)

$h_{1} \cap h_{2} = \cup_{γ 1 \in h 1, γ 2 \in h 2} \min {γ^{1}, γ^{2}}$ (20)

$h_{1} \cup h_{2} = \cup_{γ 1 \in h 1, γ 2 \in h 2} \max {γ^{1}, γ^{2}}$ (21)

$h_{1} \otimes h_{2} = \cup_{γ 1 \in h 1, γ 2 \in h 2} {γ^{1} γ^{2}}$ (22)

An Ordered Weighting Averaging (OWA) operator that can be implemented is given as: $OWA (a_{1}, a_{2}, \dots, a_{n}) = \sum_{j = 1}^{n} w_{j} b_{j}$ (23)

where, b_j is the jth largest of a₁, a₂, …, a_n, $w_{j} \in [0, 1] \forall j and \sum_{j = 1}^{n} w_{j} = 1 .$ [5, 39]

2.1 Combining DM Evaluations in Hesitant Fuzzy AHP-PROMETHEE II and Hesitant Fuzzy AHP-TOPSIS

Fuzzy envelope approach [25] is implemented to combine DM evaluations in hesitant F-AHP, hesitant F-PROMETHEE II and hesitant F-TOPSIS steps. Scales given for DM evaluations are sorted from the lowest s_o to the highest s_g so if the DM’s evaluations are between s_i and s_j, then s_o ⩽ s_i ⩽ s_j ⩽ s_g.

Based on the HFLTS, linguistic expressions can be represented by a triangular fuzzy membership function $\tilde{A} = (a, b, c)$ , where a, b and c are calculated as: $a = \min {a_{L}^{i}, a_{M}^{i}, a_{M}^{i + 1}, \dots, a_{M}^{j}, a_{R}^{j}} = a_{L}^{i}$ (24)

$b = {\begin{matrix} a_{M}^{i} ifi + 1 = j \\ {OWA}_{W} {a_{M}^{i}, a_{M}^{i + 1}, \dots, a_{M}^{j}} otherwise \end{matrix}$ (25)

$c = \max {a_{L}^{i}, a_{M}^{i}, a_{M}^{i + 1}, \dots, a_{M}^{j}, a_{R}^{j}} = a_{R}^{j}$ (26)

Weight vector in OWA operator [18] is defined as: $w_{1} = α^{n - 1}, w_{2} = (1 - α) α^{n - 2}, \dots, w_{n} = (1 - α),$ (27) $where α = (l - j + i) / (l - 1)$

Here, l depends on the number of terms in DM’s evaluation scale (in Table 1 or Table 2), j is the rank of the highest, and i is the rank of the lowest evaluation value. i and j can take ranks starting from 0 to l and n = j - I [5, 39].

Table 1

Scale for the evaluation of criteria in hesitant F-AHP [39, 41]

Linguistic terms	Triangular fuzzy number (TFN)
Absolutely strong (AS)	(2, 5/2, 3)
Very strong (VS)	(3/2, 2, 5/2)
Fairly strong (FS)	(1, 3/2, 2)
Slightly strong (SS)	(1, 1, 3/2)
Equal (E)	(1, 1, 1)
Slightly weak (SW)	(2/3, 1, 1)
Fairly weak (FW)	(1/2, 2/3, 1)
Very weak (VW)	(2/5, 1/2, 2/3)
Absolutely weak (AW)	(1/3, 2/5, 1/2)

Table 2

Scale for the evaluation of alternatives in hesitant F-PROMETHEE II and hesitant F-TOPSIS [41]

Linguistic terms	TFN
Very poor (VP)	(0,0,1)
Poor (P)	(0,1,3)
Medium poor (MP)	(1,3,5)
Fair (F)	(3,5,7)
Medium good (MG)	(5,7,9)
Good (G)	(7,9,10)
Very good (VG)	(9,10,10)

2.2 Determination of weights of criteria with hesitant F-AHP

In hesitant F-AHP, the DMs make pairwise comparisons of criteria using the linguistic terms given in Table 1.

Steps of hesitant F-AHP:

Step 1: Identify K DMs, n criteria; and linguistic terms and scale for the pairwise comparison of criteria. Based on the scale used in Table 1 and Eqs. (23)-(27), combine DM’s evaluations with fuzzy envelope approach and obtain TFNs corresponding to the evaluation of each DM. Calculate ${\tilde{x}}_{ij} = \frac{1}{K} ({\tilde{x}}_{ij}^{1} (+) {\tilde{x}}_{ij}^{2} (+) . . . (+) {\tilde{x}}_{ij}^{K})$ where ${\tilde{x}}_{ij}^{K} = (a_{ij}^{K}, b_{ij}^{K}, c_{ij}^{K}) \forall i, j, k$ is the TFN corresponding to the evaluation of the Kth DM.

Step 2: $\tilde{X} = [\begin{matrix} (1, 1, 1) & \tilde{x_{12}} & . . & . . & \tilde{x_{1 n}} \\ \tilde{x_{21}} & (1, 1, 1) & . . & . . & \tilde{x_{2 n}} \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ \tilde{x_{n 1}} & \tilde{x_{n 2}} & . . & . . & (1, 1, 1) \end{matrix}]$

with elements ${\tilde{x}}_{ij} = (a_{ij}, b_{ij}, c_{ij})$ is defuzzified with Equation (14) and w = (w₁, w₂, . . . , w_n) (approximate crisp criteria weights) is determined by averaging the entries on each row of normalized X. So the normalized principal eigen vector is w. The largest eigen value, called the principal eigen value (λ_max) is determined with Eq. (28). ${Xw}^{T} = λ_{\max} w^{T}$ (28)

The consistency index (CI) is determined as: $CI = (λ_{\max} - n) / (n - 1)$ (29)

The consistency ratio (CR) in Eq. (30) is used to estimate the consistency of pairwise comparisons. $CR = CI / RI$ (30)

RI is the average index for randomly generated weights [36]. If CR < 0.10, the comparisons are acceptable, otherwise they are not. [36] If comparisons are not acceptable (inconsistent), then DMs are asked to re-evaluate and adjust their judgements. Note that, also, there are several studies [12 , 55] in the literature about adjusting inconsistencies and improving consistency.

2.3 Ranking alternatives with hesitant F-PROMETHEE II

w = (w₁, w₂, . . . , w_n) obtained with hesitant F-AHP is used in hesitant F-PROMETHEE II to rank the alternatives. In hesitant F-PROMETHEE II, the alternatives are evaluated using the linguistic terms given in Table 2.

Steps of hesitant F-PROMETHEE II: [38, 60]

Step 1: Specify m alternatives; and linguistic terms and scale for the evaluation of alternatives with respect to each criterion. Based on the scale used in Table 2 and Eqs. (23)–(27), combine DM’s evaluations with fuzzy envelope approach and obtain TFNs corresponding to the evaluation of each DM. Calculate ${\tilde{r}}_{ij} = \frac{1}{K} ({\tilde{r}}_{ij}^{1} (+) {\tilde{r}}_{ij}^{2} (+) . . . (+) {\tilde{r}}_{ij}^{K})$ , where ${\tilde{r}}_{ij}^{K} = (a_{ij}^{K}, b_{ij}^{K}, c_{ij}^{k})$ is the TFN corresponding to the evaluation of the Kth DM. After this aggregation, the fuzzy matrix with evaluations of m alternatives with respect to n criteria is given as: $\tilde{D} = [\begin{matrix} \tilde{r_{11}} & \tilde{r_{12}} & . . & . . & \tilde{r_{1 n}} \\ \tilde{r_{21}} & \tilde{r_{22}} & . . & . . & \tilde{r_{2 n}} \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ \tilde{r_{n 1}} & \tilde{r_{n 2}} & . . & . . & \tilde{r_{mn}} \end{matrix}],$ where ${\tilde{r}}_{ij} = (a_{ij}, b_{ij}, c_{ij}) \forall i, j$ are positive TFNs.

Step 2: $\tilde{D}$ is defuzzified with Eq. (14) and $D = [\begin{matrix} r_{11} & r_{12} & . . & . . & r_{1 n} \\ r_{21} & r_{22} & . . & . . & r_{2 n} \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ r_{m 1} & r_{m 2} & . . & . . & r_{mn} \end{matrix}]$ is obtained where r_ij are crisp values.

Step 3: Calculate aggregated preference indices: P_j (A_i, A_k) = P_j (d (A_i, A_k)) = P_j (r_ij - r_kj) is a preference function showing how much DM prefers A_i to A_k with respect to c_j. From the generalized functions in Table 3, Gaussian criterion was selected. The Gaussian criterion is generally preferred especially when there is continuing data in practical applications and the major criteria in this study contain continuing data. Here, the preference function is:

Table 3
Generalized criteria [7]

$P (d) = {\begin{matrix} 0 & d ⩽ 0 \\ 1 - e^{- \frac{d^{2}}{2 s^{2}}} & d > 0 \end{matrix}},$ (31)

for maximum criterion and $P (d) = {\begin{matrix} 0 & d ⩾ 0 \\ 1 - e^{- \frac{d^{2}}{2 s^{2}}} & d < 0 \end{matrix}},$ (32)

for minimum criterion.

Aggregated preference index π (A_i, A_k) expresses the degree of how much DM prefers A_i to A_k with respect to all the criteria. The aggregated preference indices are given as: $\begin{matrix} π (A_{i}, A_{k}) = \frac{\sum_{j = 1}^{n} P_{j} (A_{i}, A_{k}) . w_{j}}{\sum_{j = 1}^{n} w_{j}}, \\ \forall A_{i}, A_{k} \in A and i \neq k \end{matrix}$ (33)

Step 4: Calculate the outranking flows to rank the alternatives. Each alternative A_i faces (m-1) other alternatives in A. The positive outranking flow defines howA_i outranks other and it is given as: $Ø^{+} (A_{i}) = \sum_{k = 1}^{m} π (A_{i}, A_{k})$ (34)

The negative outranking flow articulates how A_i is outranked by others and it is given as: $Ø^{-} (A_{i}) = \sum_{k = 1}^{m} π (A_{k}, A_{i})$ (35)

Step 5: Finally, the net outranking flow is determined as: $Ø (A_{i}) = Ø^{+} (A_{i}) - Ø^{-} (A_{i}), \forall i \in {1, \dots, m}$ (36)

Based on the net outranking flow alternatives are ranked; highest Ø (A_i) specifies the best alternative.

3 Hesitant fuzzy AHP-TOPSIS

For comparison analysis, hesitant fuzzy AHP-TOPSIS is applied. After the determination of weights of criteria with hesitant F-AHP, hesitant F-TOPSIS is implemented to determine the best alternative. The steps of hesitant F-TOPSIS are given below.

Step 1: After implementing Steps 1 and 2 of hesitant F-PROMETHEE II in Section 2.3, normalize D and obtain normalized decision matrix F such that elements of F are $f_{ij} = r_{ij} / (\sum_{i} r_{ij}^{2})^{1 / 2}$ .

Step 2: Construct the weighted normalized decision matrix G such that elements of G are g_ij = w_jf_ij.

Step 3: Determine the positive ideal A^* and negative ideal A^- solutions. $A^{*} = (g_{1}^{*}, \dots, g_{n}^{*})$ where $g_{j}^{*} = max_{i} (g_{ij}) for \max . criteria and g_{j}^{*} = min_{i} (g_{ij}) for \min . criteria$ .

Also $A^{-} = (g_{1}^{-}, \dots, g_{n}^{-})$ , where $g_{j}^{-} = min_{i} (g_{ij}) for \max . criteria and g_{j}^{-} = max_{i} (g_{ij}) for \min . criteria$ .

Step 4: Calculate the separation measures of ach alternative. Separation from ideal solution is $S_{i}^{*} = (\sum_{j} {(g_{j}^{*} - g_{ij})^{2})}^{1 / 2}$ and separation from negative ideal solution is $S_{i}^{-} = (\sum_{j} {(g_{j}^{-} - g_{ij})^{2})}^{1 / 2}$ .

Step 5: Calculate closeness index values C_i for each alternative as: $C_{i} = S_{i}^{-} / (S_{i}^{-} + S_{i}^{*})$ . Alternatives are ranked based on C_i values and alternative with the highest C_i value is the best.

4 Illustrative example and comparison analysis

For the evaluation of transformers in a power distribution project, 12 benefit (maximization) criteria (C1, C2, ... ,C12) are determined which are listed in Table 4 along with detailed explanations [35 , 65]. Five alternative transformers (A1, A2, A3, A4, A5) are going to be evaluated with respect to these criteria by three DMs (DM1, DM2, DM3).

Table 4
Evaluation criteria for transformers and explanations

C1 Nominal power (kVA): kVA stands for Kilovolt-Ampere and is the rating normally used to rate a transformer. The size of a transformer is determined by the kVA of the load. C7 Isolation type and class: The capacity of a transformer is limited by the temperature that isolation type and class can tolerate.

C2 Cost efficiency (for product cost and maintenance cost): Purchasing a high efficiency transformer upfront will result in significant savings over the life of a transformer. Operators will also benefit from reduced maintenance costs as well as added environmental benefits. C8 Safety level: Protect the human health from the hazards of high voltage capacity in operation and maintenance of a transformer.

C3 Efficiency level: It is the ratio of useful power output to the power input of the transformer operating at a particular load and power factor. C9 Metal enclosure and ventilation: Availability of metal enclosure of a transformer and fans to alleviate excessive heating caused by abnormal operating conditions. The enclosures against solid materials, water and dust in a working environment.

C4 Input-Output voltages and current values: Primary and secondary voltages and their values according to the duty factor. C10 Petiteness: Size of a transformer with its dimensions.

C5 Input-Output voltage steps availability: The steps of primary and secondary voltages in a transformer. C11 Noise level reduction: Availability of a system to reduce noise level depending on the load of a transformer.

C6 Input-Output connection types availability: The types of the input and output sides of a transformer, also called primary and secondary sides (delta, star, zigzag, etc.). C12 Level of service quality after sale: The service quality level for maintenance and availability of spare parts after the sale.

First, with hesitant F-AHP, w = (w₁, w₂, . . . , w_n) is determined. In hesitant F-AHP, DMs compare criteria pairwise with the help of linguistic terms in Table 1 and these are given in Table 5.

Table 5

Pairwise comparison of evaluation criteria by 3 DMs

After the combination of each DM’s evaluations with fuzzy envelope approach and aggregation of the corresponding TFNs of 3 DMs evaluations, the fuzzy evaluation matrix for the criteria weights ( $\tilde{X}$ ) in Table 6 is obtained. Then, elements of $\tilde{X}$ are defuzzified with Eq. (14) and evaluation matrix X in Table 7 is obtained. Afterwards, w = (w₁, w₂, . . . , w_n) is determined by taking the average of the entries in each row of normalized X. λ_max = 13.227 and CR of X is checked with Eqs. (28)–(30) as CI=(13.227-12)/11 = 0.1116 and CR = CI/RI=0.1116/1.48 = 0.0754. Since CR < 0.1, the pairwise comparisons are determined to be consistent. w = (w₁, w₂, . . . , w_n) obtained with hesitant F-AHP is then used in hesitant F-PROMETHEE II and hesitant F-TOPSIS to rank transformer alternatives.

Table 6

The fuzzy evaluation matrix for the criteria weights ( $\tilde{X}$ )

	C1	C2	C3	C4	C5	C6
C1	(1.000,1.000,1.000)	(1.000,1.204,1.667)	(0.633,0.858,1.000)	(0.890,1.000,1.167)	(1.167,1.333,1.667)	(1.167,1.500,1.833)
C2	(0.690,0.968,1.000)	(1.000,1.000,1.000)	(1.333,1.500,1.667)	(0.890,1.371,1.667)	(1.000,1.333,1.833)	(0.833,0.890,1.000)
C3	(1.000,1.371,1.833)	(0.777,0.800,0.833)	(1.000,1.000,1.000)	(1.057,1.333,1.500)	(0.690,0.833,0.890)	(0.633,0.722,0.890)
C4	(0.890,1.000,1.167)	(0.778,0.930,1.167)	(0.800,0.833,1.057)	(1.000,1.000,1.000)	(0.890,1.167,1.333)	(1.167,1.333,1.667)
C5	(0.689,0.833,0.890)	(0.579,0.918,1.223)	(1.167,1.333,1.667)	(0.833,0.890,1.167)	(1.000,1.000,1.000)	(0.600,0.667,0.780)
C6	(0.633,0.723,0.890)	(1.000,1.167,1.333)	(1.167,1.643,2.000)	(0.690,0.833,0.890)	(1.333,1.667,2.000)	(1.000,1.000,1.000)
C7	(0.890,1.000,1.000)	(1.057,1.333,1.500)	(0.579,0.833,0.890)	(0.800,0.833,0.890)	(1.057,1.500,1.833)	(1.000,1.333,1.667)
C8	(1.167,1.390,1.667)	(0.800,0.833,0.890)	(1.167,1.390,1.667)	(0.667,0.779,1.000)	(0.800,0.833,0.890)	(0.600,0.667,0.780)
C9	(0.667,0.780,1.000)	(0.833,0.890,1.333)	(1.167,1.810,2.167)	(0.690,0.833,0.890)	(0.690,0.968,1.000)	(1.333,1.667,2.000)
C10	(0.777,0.800,0.833)	(0.777,0.800,0.833)	(0.833,0.890,1.333)	(0.833,1.057,1.333)	(0.500,0.690,0.833)	(0.890,1.000,1.167)
C11	(1.333,1.667,2.000)	(0.723,0.890,1.000)	(1.223,1.500,1.667)	(0.610,0.831,1.167)	(0.777,0.837,1.500)	(0.580,0.833,0.890)
C12	(0.667,0.800,0.833)	(1.000,1.223,1.500)	(1.110,1.300,1.500)	(0.490,0.802,0.890)	(0.888,1.201,1.500)	(1.333,1.500,1.667)
	C7	C8	C9	C10	C11	C12
C1	(1.000,1.000,1.167)	(0.777,0.967,1.167)	(1.000,1.333,1.667)	(1.333,1.500,1.667)	(0.600,0.667,0.780)	(1.333,1.500,1.833)
C2	(0.800,0.833,1.057)	(1.167,1.333,1.500)	(0.780,1.167,1.333)	(1.333,1.500,1.667)	(1.000,1.167,1.500)	(0.800,1.000,1.223)
C3	(1.167,1.333,1.833)	(0.777,0.967,1.167)	(0.578,0.714,0.890)	(0.780,1.167,1.333)	(0.777,0.800,1.000)	(1.110,1.300,1.500)
C4	(1.167,1.333,1.500)	(1.000,1.476,1.833)	(1.167,1.333,1.667)	(0.833,1.057,1.333)	(1.167,1.531,2.000)	(1.167,1.537,2.167)
C5	(0.633,0.723,1.057)	(1.167,1.333,1.500)	(1.000,1.204,1.667)	(1.333,1.667,2.167)	(1.111,1.430,1.667)	(1.110,1.481,2.167)
C6	(0.667,0.780,1.000)	(1.333,1.667,2.000)	(0.600,0.667,0.780)	(0.890,1.000,1.167)	(1.167,1.333,1.833)	(0.777,0.800,0.833)
C7	(1.000,1.000,1.000)	(1.110,1.300,1.500)	(0.890,1.167,1.333)	(0.833,1.000,1.167)	(0.890,1.167,1.333)	(1.333,1.500,1.833)
C8	(1.110,1.300,1.500)	(1.000,1.000,1.000)	(0.800,0.833,0.890)	(1.000,1.167,1.333)	(1.333,1.500,1.667)	(1.000,1.476,2.000)
C9	(0.833,0.890,1.167)	(1.167,1.333,1.500)	(1.000,1.000,1.000)	(1.000,1.167,1.500)	(1.333,1.667,2.167)	(0.889,1.167,1.333)
C10	(0.889,1.085,1.333)	(0.833,0.890,1.000)	(0.723,0.890,1.000)	(1.000,1.000,1.000)	(1.000,1.476,1.833)	(1.223,1.500,1.667)
C11	(0.833,0.890,1.167)	(0.777,0.800,0.833)	(0.490,0.667,0.780)	(0.667,0.779,1.000)	(1.000,1.000,1.000)	(1.167,1.333,1.667)
C12	(0.667,0.800,0.833)	(0.557,0.779,1.000)	(0.833,0.890,1.167)	(0.777,0.800,1.000)	(0.690,0.833,0.890)	(1.000,1.000,1.000)

Table 7

X and criteria weights (w)

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	Weights (w)
C1	1.000	1.247	0.844	1.009	1.361	1.500	1.028	0.968	1.333	1.500	0.674	1.528	0.088
C2	0.927	1.000	1.500	1.340	1.361	0.899	0.865	1.333	1.130	1.500	1.194	1.004	0.088
C3	1.386	0.802	1.000	1.315	0.819	0.735	1.389	0.968	0.721	1.130	0.829	1.302	0.078
C4	1.009	0.944	0.865	1.000	1.148	1.361	1.333	1.456	1.361	1.066	1.548	1.580	0.092
C5	0.819	0.913	1.361	0.927	1.000	0.674	0.764	1.333	1.247	1.694	1.416	1.533	0.085
C6	0.736	1.167	1.623	0.819	1.667	1.000	0.798	1.667	0.674	1.009	1.389	0.802	0.083
C7	0.982	1.315	0.800	0.837	1.482	1.333	1.000	1.302	1.148	1.000	1.148	1.528	0.087
C8	1.399	0.837	1.399	0.797	0.837	0.674	1.302	1.000	0.837	1.167	1.500	1.484	0.082
C9	0.798	0.954	1.762	0.819	0.927	1.667	0.927	1.333	1.000	1.194	1.694	1.148	0.088
C10	0.802	0.802	0.954	1.066	0.682	1.009	1.094	0.899	0.881	1.000	1.456	1.482	0.076
C11	1.667	0.881	1.482	0.850	0.938	0.801	0.927	0.802	0.656	0.797	1.000	1.361	0.076
C12	0.783	1.232	1.302	0.764	1.199	1.500	0.783	0.779	0.927	0.829	0.819	1.000	0.075

In hesitant F-PROMETHEE II, DMs evaluate alternatives with respect to each criterion using the linguistic terms in Table 2 and these are given in Table 8. After the combination of each DM’s evaluations, the fuzzy evaluation matrix $\tilde{D}$ in Table 9 is obtained. Elements of $\tilde{D}$ is defuzzified with Eq. (14) and decision matrix D in Table 10 is attained. Gaussian criterion is selected for the criteria and aggregated preference index matrix for the alternatives is given in Table 11. Using the aggregated preference indices, the positive, negative and net outranking flows are calculated and presented in Table 12. Alternatives are ranked based on net outranking flows and the ranking is given in Table 12. As seen in Table 12, based on the hesitant fuzzy AHP-PROMETHEE II, the best alternative is A3 due to the highest net outranking flow.

Table 8

3 DMs’ evaluations of the transformers with respect to each criterion

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
A1	between MG, P	bet. MG, P	P	G	P	MG	P	VG	VP	MG	G	P
	P	F	G	G	MG	P	bet. G, F	VG	G	P	G	VP
	P	G	G	MP	F	VP	G	G	G	VP	MP	P
A2	P	VP	bet. F, P	VP	P	P	G	P	P	P	F	bet. P, VP
	VP	G	VP	F	MP	VP	VG	P	G	G	G	G
	G	G	G	F	G	MG	G	MP	bet. G, MG	G	G	G
A3	bet. MG, P	P	MG	P	bet. F, P	F	G	MG	P	VG	bet. MG, MP	VP
	VG	G	P	G	G	G	F	VG	VP	F	P	G
	MP	F	P	G	F	G	F	G	F	G	VP	F
A4	VP	P	P	G	P	P	P	F	VP	MG	VP	MP
	G	F	G	MG	P	P	bet. MP, VP	G	P	MP	F	P
	F	G	G	P	F	bet. G, F	G	F	G	VP	G	MP
A5	F	G	P	MG	G	G	MG	G	P	P	G	bet. MG, VP
	G	F	MP	P	VG	VG	P	G	MP	VP	VG	P
	G	VG	G	VP	G	P	P	MG	G	P	MP	P

Table 9

Fuzzy decision matrix $\tilde{D}$

	C1	C2	C3	C4
A1	(0.000,2.307,5.000)	(3.333,6.307,8.667)	(4.667,6.333,7.667)	(5.000,7.000,8.333)
A2	(2.333,3.333,4.667)	(4.667,6.000,7.000)	(2.333,4.533,6.000)	(2.000,3.333,5.000)
A3	(3.333,5.973,8.000)	(3.333,5.000,6.667)	(1.667,3.000,5.000)	(4.667,6.333,7.667)
A4	(3.333,4.667,6.000)	(3.333,5.000,6.667)	(4.667,6.333,7.667)	(4.000,5.667,7.333)
A5	(5.667,7.667,9.000)	(6.333,8.000,9.000)	(2.667,4.333,6.000)	(1.667,2.667,4.333)
	C5	C6	C7	C8
A1	(2.667,4.333,6.333)	(1.667,2.667,4.333)	(3.333,6.200,7.667)	(8.333,9.667,10.00)
A2	(2.667,4.333,6.000)	(1.667,2.667,4.333)	(7.667,9.333,10.00)	(0.333,1.667,3.667)
A3	(3.333,6.200,8.000)	(5.667,7.667,9.000)	(4.333,6.333,8.000)	(7.000,8.667,9.667)
A4	(1.000,2.333,4.333)	(1.000,3.533,5.333)	(2.333,4.200,6.000)	(4.333,6.333,8.000)
A5	(7.667,9.333,10.00)	(5.333,6.667,7.667)	(1.667,3.000,5.000)	(6.333,8.333,9.667)
	C5	C6	C7	C8
A1	(4.667,6.000,7.000)	(1.667,2.667,4.333)	(5.000,7.000,8.333)	(0.000,0.667,2.333)
A2	(4.000,5.667,7.667)	(4.667,6.333,7.667)	(5.667,7.667,9.000)	(4.667,6.000,7.667)
A3	(1.000,2.000,3.667)	(6.333,8.000,9.000)	(0.333,2.533,4.333)	(3.333,4.667,6.000)
A4	(2.333,3.333,4.667)	(2.000,3.333,5.000)	(3.333,4.667,6.000)	(0.667,2.333,4.333)
A5	(2.667,4.333,6.000)	(0.000,0.667,2.333)	(5.667,7.333,8.333)	(0.000,1.483,5.000)

Table 10

Decision matrix D

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12
A1	2.371	6.204	6.278	6.889	4.389	2.778	5.967	9.500	5.944	2.778	6.889	0.833
A2	3.389	5.944	4.411	3.389	4.333	2.778	9.167	1.778	5.722	6.278	7.556	6.056
A3	5.871	5.000	3.111	6.278	6.022	7.556	6.278	8.556	2.111	7.889	2.467	4.667
A4	4.667	5.000	6.278	5.667	2.444	3.411	4.189	6.278	3.389	3.389	4.667	2.389
A5	7.556	7.889	4.333	2.778	9.167	6.611	3.111	8.222	4.333	0.833	7.222	1.822

Table 11

Aggregated preference index matrix

A1	A1	A2	A3	A4	A5
A1	0	0.083	0.066	0.050	0.058
A2	0.067	0	0.071	0.092	0.106
A3	0.104	0.113	0	0.095	0.096
A4	0.014	0.045	0.024	0	0.032
A5	0.096	0.132	0.072	0.110	0

Table 12

Outranking flow indices and rank

Alter.	A1	A2	A3	A4	A5
Ø⁺	0.257	0.336	0.408	0.115	0.409
Ø^-	0.280	0.373	0.234	0.346	0.292
Ø	–0.023	–0.037	0.175	–0.232	0.117
Rank	3	4	1	5	2

For comparison analysis, hesitant fuzzy AHP-TOPSIS is applied to the same example. Decision matrix D in Table 10 is normalized and after multiplication with weight matrix, weighted normalized decision matrix G is obtained as seen in Table 13. Positive and negative ideal solutions (A^*andA^-), separation measures (S^*, S^-), and closeness index of each alternative (C_i) are also presented in Table 13. Alternatives are ranked based on closeness indices and the ranking is given in Table 13. As seen in Table 13, based on the hesitant fuzzy AHP-TOPSIS, the best alternative is also A3 (same as hesitant fuzzy AHP-PROMETHEE II) due to the highest closeness index.

Table 13

Weighted normalized decision matrix G, A^*andA^-, separation measures (S^*, S^-), closeness indices (C_i) and ranking of alternatives

	C1	C2	C3	C4	C5	C6	C7	C8	C9	C10	C11	C12	S ^*	S ^-	C _i	Rank
A1	0.018	0.040	0.044	0.054	0.029	0.020	0.038	0.047	0.052	0.019	0.039	0.008	0.089	0.074	0.455	4
A2	0.026	0.038	0.031	0.027	0.029	0.020	0.059	0.009	0.050	0.043	0.042	0.055	0.077	0.085	0.525	2
A3	0.045	0.032	0.022	0.049	0.040	0.056	0.040	0.043	0.018	0.054	0.014	0.042	0.063	0.092	0.594	1
A4	0.036	0.032	0.044	0.044	0.016	0.025	0.027	0.031	0.030	0.023	0.026	0.022	0.089	0.052	0.367	5
A5	0.058	0.051	0.030	0.022	0.061	0.049	0.020	0.041	0.038	0.006	0.040	0.017	0.083	0.084	0.503	3
A ^*	0.058	0.051	0.044	0.054	0.061	0.056	0.059	0.047	0.052	0.054	0.042	0.055
A ^-	0.018	0.032	0.022	0.022	0.016	0.020	0.020	0.009	0.018	0.006	0.014	0.008

5 Discussion and conclusion

In this paper, an integrated hesitant fuzzy AHP-PROMETHEE II method is proposed to evaluate transformer alternatives in a power distribution project. In this integration, hesitant F-AHP is applied to determine the weights of criteria, and hesitant F-PROMETHEE II method is implemented to evaluate and rank alternatives utilizing the obtained weights.

At present, there does not appear to be a study in the literature that integrates hesitant F-AHP and hesitant F-PROMETHEE II for transformer evaluation problem. Adoption of hesitant fuzzy linguistic terms in F-AHP and F-PROMETHEE II captures the ambiguity and fuzziness in decisions of DMS, provides flexibility in decision making and reflects the degree of hesitation DMs might have in reality. With hesitant fuzzy AHP-PROMETHEE II, transformers are evaluated and ranked without large number of monotonous pairwise comparisons and dense calculations. For comparison analysis, hesitant fuzzy AHP-TOPSIS is applied to the same example. The aim of this study is to determine the best transformer alternative among all alternatives. Proposed hesitant fuzzy AHP-PROMETHEE II method is determined to be credible in selecting the best alternative since it chose the same best alternative (A3) as the compared hesitant fuzzy AHP-TOPSIS method.

In this research the DMs are thought to be at the same hierarchical level in the decision making process, however in reality this might not be the case. For future research, different hierarchical levels of DMs might be taken into consideration in the group decision making process. Also, correlations between criteria, outer and inner dependence and feedback relations between criteria can be studied with hesitant correlated F-AHP and hesitant F-ANP and these methods can be incorporated with hesitant F-PROMETHEE II and hesitant F-TOPSIS for evaluation and selection problems.

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C1	Nominal power (kVA): kVA stands for Kilovolt-Ampere and is the rating normally used to rate a transformer. The size of a transformer is determined by the kVA of the load.	C7	Isolation type and class: The capacity of a transformer is limited by the temperature that isolation type and class can tolerate.
C2	Cost efficiency (for product cost and maintenance cost): Purchasing a high efficiency transformer upfront will result in significant savings over the life of a transformer. Operators will also benefit from reduced maintenance costs as well as added environmental benefits.	C8	Safety level: Protect the human health from the hazards of high voltage capacity in operation and maintenance of a transformer.
C3	Efficiency level: It is the ratio of useful power output to the power input of the transformer operating at a particular load and power factor.	C9	Metal enclosure and ventilation: Availability of metal enclosure of a transformer and fans to alleviate excessive heating caused by abnormal operating conditions. The enclosures against solid materials, water and dust in a working environment.
C4	Input-Output voltages and current values: Primary and secondary voltages and their values according to the duty factor.	C10	Petiteness: Size of a transformer with its dimensions.
C5	Input-Output voltage steps availability: The steps of primary and secondary voltages in a transformer.	C11	Noise level reduction: Availability of a system to reduce noise level depending on the load of a transformer.
C6	Input-Output connection types availability: The types of the input and output sides of a transformer, also called primary and secondary sides (delta, star, zigzag, etc.).	C12	Level of service quality after sale: The service quality level for maintenance and availability of spare parts after the sale.

An intelligent approach for the evaluation of transformers in a power distribution project

Abstract

Keywords

1 Introduction

2 Hesitant fuzzy AHP-PROMETHEE II

Table 3 Generalized criteria [7]

4 Illustrative example and comparison analysis

References

Table 3
Generalized criteria [7]