New hybrid AHP-QFD-PROMETHEE decision-making support method in the hesitant fuzzy environment: an application in packaging design selection

Abstract

This paper presents a new hybrid decision-making support method (New Hesitant Fuzzy AHP-QFD-PROMETHEE II Method), which jointly uses the Analytic Hierarchy Process (AHP), the Quality Function Deployment (QFD) and the Preference Ranking Method for Enrichment Evaluation (PROMETHEE II), as well as the Hesitant Fuzzy Linguistic Term Sets (HFLTS) to capture hesitation and aggregate divergent opinions from different experts. A real application of the new method to a packaging design selection problem for an automotive company is described, finding that AHP assisted in determining the importance of QFD’s customer requirements (CRs) and PROMETHEE II was used to select the best packaging design. With this same problem, for the purpose of validating the proposed method, a comparative analysis was made with the use of the Hesitant Fuzzy AHP-QFD-TOPSIS method and also with the traditional AHP-QFD-PROMETHEE method, which makes it impossible to capture the hesitation of decision makers. The result showed similarity in the rankings of design alternatives found in the three methods application. The proposed method proved advantageous for solving problems that can generally be solved with the QFD House of Quality but have serious difficulties when decision makers have divergent opinions and hesitate in evaluating criteria and alternatives.

Keywords

Decision making problem hesitant fuzzy linguistic term sets hesitant fuzzy house of quality AHP PROMETHEE

1 Introduction

Quality Function Deployment –QFD is a customer-oriented tool widely used in improvements and new product development because it efficiently translates customer requirements (CRs) into de-sign requirements (DRs) [39, 56]. QFD has been widely applied in quality planning and continuous product improvement in many sectors, such as manufacturing, electronics [14] and in the service field.

Multiple-criteria decision-making methods –MCDM are widely used to improve the quality of group decisions involving several criteria, making the process more explicit, rational, and efficient [1, 26]. The main objective of MCDM is to find the best of a set of viable alternatives regarding several criteria [6].

A growing proposition of hybrid MCDM methods has been found with the combined use of QFD and, for instance, the Analytic Hierarchy Process –AHP, Analytic Network Process –ANP and Technique for Order Preference by Similarity to Ideal Solution –TOPSIS.

In addition, fuzzy MCDM methods have been adopted to address the subjectivity involved in the analysis of the House of Quality in the first matrix of the QFD [27 , 66].

In this regard, recently, hesitant fuzzy sets and fuzzy intuitionistic sets have been applied to MCDM methods and integrated into QFD [35 , 56].

In fact, systems with criteria and inaccurate information are inherent to several decision problems in the real world, where the occurrence of uncertainties has a relevant role. Additionally, in practice, there is usually a group of experts (decision makers) who are consulted about their preferences, and it is necessary to turn individual preferences, often conflicting and divergent ones, into a single collective preference [1].

In several decision making problems, when adopting a fuzzy MCDM, experts must estimate the degree of relevance, or association of an element to a set, using different values, and considering their personal interest, time constraints and lack of information [8].

A generalization of fuzzy sets, the Hesitant Fuzzy Set (HFS), developed by Torra [62], has proven to be one of the most effective tools to deal with this type of uncertainty associated with human nature.

HFSs have been proposed to deal with decision making problems where experts hesitate at the time of their assessment. In addition, HFSs are interesting techniques to aggregate divergent opinions in group decision-making [35].

The main characteristic of HFSs is that the degree of relevance of an element to a given set is described by different values between 0 and 1, allowing an expert to better express his or her assessment [53].

The Hesitant Fuzzy Linguistic Term Sets (HFLTS) are extensions of HFSs and have the advantage that they can be used in situations where there is a lack of information, or uncertainties, as the decision maker can use more than one linguistic term to model his or her hesitation in judgements.

Despite the numerous advantages mentioned above, there are still few publications of QFD applications in the hesitant fuzzy environment. In research in Scopus (https://www.scopus.com) and Web of Science (https://webofknowledge.com) databases, it was found that only the AHP, DEMATEL (Decision making trial and evaluation laboratory), TOPSIS and VIKOR (vlsekriterijumska optimizacija i kompromisno resenje in serbian) methods were used together with the QFD for the proposition of hybrid hesitant fuzzy MCDM methods.

It was also found that, in the literature consulted, there are few publications on the application of the Preference Ranking Organization Method for Enrichment Evaluation (PROMETHEE) family in group decision-making and, in particular, in conjunction with the QFD [38, 55].

The application of the AHP method together with the QFD is more popular, although PROMETHEE [29], considered an important outranking method, has the advantage of allowing decision makers to specify their preferred functions.

It was also found that, to address qualitative problems of MCDM, there are few articles relating the PROMETHEE method with HFLTSs and concomitantly with QFD.

In this context, the main contributions of this article are (1) to propose a new, more versatile and flexible hybrid MCDM hesitant method composed of the already known AHP, QFD and PROMETHEE II methods, (2) to present a multicriteria decision problem application where there are divergent opinions and expert hesitations, (3) to propose a new hesitant fuzzy PROMETHEE II method using the OWA operator by Liu & Rodriguez [23], to aggregate expert opinions.

Additionally, to illustrate the advantages of this new method, and to make a comparative analysis with other traditionally adopted methods, a real application is described, in an automotive company, related to the problem of packaging design selection.

The article is organized in four other Sections. Section 2 presents a literature review on HFS and HFLTS, Hesitant Fuzzy QFD, Hesitant Fuzzy AHP and Hesitant Fuzzy PROMETHEE. Section 3 details the proposed new approach. Section 4 describes the application of the new method to a real problem and compares the results if the traditional AHP-QFD-PROMETHEE method, without considering the occurrence of uncertainties, and the Hesitant Fuzzy AHP-QFD-TOPSIS method were used. Finally, in Section 5 are the conclusions and suggestions for future research, followed by the references consulted.

2 Preliminaries

2.1 Hesitant fuzzy sets and HFLTS

Fuzzy set theory and fuzzy logic allow managing inaccurate and vague information. This inaccuracy is reflected by the degree of relevance of objects belonging to a concept [69].

The theory of fuzzy set has been widely applied successfully to deal with this type of uncertainty in decision problems in different areas. However, it has limitations in dealing with inaccurate and vague information when different sources of inaccuracy appear simultaneously [52].

Hesitant fuzzy sets (HFS) are extensions of fuzzy sets that seek to address the difficulty of determining the degree of relevance of an element to a set, as they represent human hesitation when there are several possible values for the degree of relevance and it is not clear what would be the best choice for that value [62].

HFS are used to deal with circumstances where decision experts hesitate between many values to evaluate an alternative, or a variable etc. Recently, the hesitant fuzzy multi-criteria decision making (MCDM) has received a lot of attention among researchers from many different fields and methods have been proposed to deal with this problem [6].

Definition 1. A hesitant fuzzy set (HFS) on X, where X is a fixed set, can be defined as follows: $E = {〈 x, h_{E} (x) 〉 | x ɛ X}$ (1) where h_E (x) denotes possible membership degrees of the element xɛX to the set E and is a set of values in [0, 1]. Where h = h_E (x) is a hesitant fuzzy element (HFE) [70].

According to Rodríguez et al. [52], HFS can be classified as: dual hesitant fuzzy sets, interval valued hesitant fuzzy sets, generalized hesitant fuzzy sets, triangular fuzzy hesitant sets and hesitant fuzzy linguistic term sets.

When experts face decision situations with a high degree of uncertainty, they often hesitate between different linguistic terms and would like to use more complex linguistic expressions that could not be expressed using classical linguistic approaches.

This limitation is due to the use of a priori defined linguistic terms and because most linguistic approaches model information using only one linguistic term [52].

To overcome this limitation, different proposals have been made to provide more flexible and richer expressions, i.e., that may contain more than one linguistic term [28 , 67]. To meet the needs and requirements of experts in hesitation situations, Rodríguez et al. [51] proposed the concept of Hesitant fuzzy linguistic term set (HFLTS), which maintains the basis of Zadeh’s [36] fuzzy linguistic approach and extends the idea of HFS to linguistic contexts.

Experts often hesitate to select the appropriate linguistic expression. In classical fuzzy linguistic approaches, a single expression must be selected, limiting the experts in expressing their opinion [53]. HFLTS are used when experts hesitate among different language expressions.

The basic definitions of HFLTS are described below [51 –53]:

Definition 2. An HFLTS, H_s, is an ordered finite subset of consecutive linguistic terms of a linguistic term set S which can be shown as S ={ s₀, s₁, …, s_g }.

Definition 3. Suppose E_{G
_H} is a function that converts language expressions into HFLTS, H_S. Suppose that G_H is a context-free grammar that uses the linguistic term set, and S and S_ll is the domain of expression generated by G_H. This relationship can be expressed as E_{G
_H} : S_ll → H_S.

Using the following transformations, comparative linguistic expressions are converted into HFLTSs: $E_{G_{H}} (s_{i}) = {s_{i} | s_{i} \in S}$ (2) $E_{G_{H}} ({atmosts}_{i}) = {s_{j} | s_{j} \in S and s_{j} ⩽ s_{i}}$ (3) $E_{G_{H}} ({lowerthans}_{i}) = {s_{j} | s_{j} \in S and s_{j} < s_{i}}$ (4) $E_{G_{H}} ({atleasts}_{i}) = {s_{j} | s_{j} \in S and s_{j} ⩾ s_{i}}$ (5) $E_{G_{H}} ({greaterthans}_{i}) = {s_{j} | s_{j} \in S and s_{j} 〉 s_{i}}$ (6)

$E_{G_{H}} ({betweens}_{i} {ands}_{j}) = {s_{k} | s_{k} \in S and s_{i} ⩽ s_{k} ⩽ s_{j}}$ (7)

Definition 4. The envelope of an HFLTS, env (H_S), is a linguistic interval which is limited by its maximum value and minimum value, $env (H_{S}) = [H_{S^{-}}, H_{S^{+}}], H_{S^{-}} ⩽ H_{S^{+}}$ (8) where $H_{S^{-}} = min (s_{i}) = s_{j}, s_{i} \in H_{S} and s_{i} ⩾ s_{j} \forall i$ $H_{S^{+}} = max (s_{i}) = s_{j}, s_{i} \in H_{S} and s_{i} ⩽ s_{j} \forall i$

Definition 5. Let S ={ s₀, s₁, …, s_g } be a linguistic term set. A HFLTS, is defined as an ordered finite subset of consecutive linguistic terms of S:

$H_{S} = {s_{i}, s_{i + 1}, \dots s_{j}} such that s_{k} \in S, k \in {i, \dots, j}$ (9)

Definition 6. An ordered weighted average (OWA) operator of dimension n is a mapping OWA: Rⁿ → R, so that $OWA (a_{1}, a_{2}, \dots, a_{n}) = \sum_{j = 1}^{n} w_{j} b_{j}$ (10)

Where b_j is the j^th largest of the aggregated arguments a₁, a₂, …, a_n, and W = (w₁, w₂, …, w_n) ^T is the associated weighting vector satisfying w_i ∈ [0, 1] , i = 1, 2, …, n and $\sum_{i = 1}^{n} w_{i} = 1$ .

Definition 7. A triangular fuzzy membership function $\tilde{A} = (a, b, c)$ is utilised as the representation of the comparative linguistic expressions based on HFLTS H_S, the definition domain of $\tilde{A}$ should be the same as the linguistic terms {s_i, s_i+1, …, s_j } ∈ H_S. The min and the max operators are used to compute a and c $a = min {a_{L}^{i}, a_{M}^{i}, a_{M}^{i + 1}, \dots, a_{M}^{j}, a_{R}^{j}} = a_{L}^{i}$ (11) $c = max {a_{L}^{i}, a_{M}^{i}, a_{M}^{i + 1}, \dots, a_{M}^{j}, a_{R}^{j}} = a_{R}^{j}$ (12)

The remaining elements $a_{M}^{i}, a_{M}^{i + 1}, \dots, a_{M}^{j} \in T$ should contribute to the computation of the parameter b. The OWA aggregation operator will be utilised to aggregate them: $b = {OWA}_{W^{S}} (a_{M}^{i}, a_{M}^{i + 1}, \dots, a_{M}^{j})$ (13)

2.2 Hesitant fuzzy QFD, AHP and PROMETHEE II

Fuzzy systems have been successfully applied to deal with imperfect, vague and inaccurate information often found when using the QFD method. One of the first publications Kraslawski et al. [5], was related to the development of a robust design method, where fuzzy systems were applied in the House of Quality. The main idea was to minimize product quality variability under inaccurate technological and economic constraints. Also in the early 1990 s, linear fuzzy regression equations were applied to the QFD to estimate the relationships between variables.

Fuzzy systems were also applied to determine the weights of customer requirements [11 , 48]. In addition, new MCDM methods were created using QFD along with other methods such as AHP and TOPSIS [27 , 66].

The HFLTS has been used in several works in literature [21 , 53]. The use of HFLTS in QFD is recent. Onar et al. [53] determined the importance of customer requirements (CRs), the relationships between CRs and design requirements (DRs), and the correlations between DRs by means of hesitant fuzzy linguistic term sets (HFLTS). In addition, AHP, QFD and TOPSIS methods based on HFLTS were used to select the best computer workstation. Other methods developed can be consulted in Liu et al. [24]; Osiro et al. [35] and Dinçer et al. [20].

The Analytic Hierarchy Process (AHP) proposed by Saaty [61] is a multicriteria decision method used for complex decision problems. In recent decades, the AHP and related research topics have attracted much attention from researchers [9 , 57].

Although the AHP is the most popular method of pairwise comparison, due to the uncertainties inherent in human judgements it becomes very subjective. Thus, the use of linguistic expressions, instead of numerical values, is more appropriate [19]. Hesitation is a common phenomenon in the decision-making process. In view of this, Zhu et al. [10] developed the Hesitant analytic hierarchy process. Algumas dessas publicações podem ser consultadas em Dogan and Öztaysi [43]; Zhu and Xu [8]; Öztaysi et al. [7]; Yavuz et al. [40]; Zhu et al. [10]; Li, Dong and Wang [49] Ren et al. [45]; Büyükozkan et al. [18], entre outros.

PROMETHEE [29] is a family of methods that is characterized by allowing the construction of outranking relationships between the alternatives. It is considered an outranking method due to the use of a preference function [38, 56], and was developed in the early 1980 s by Brans et al. [30]. It has been applied in several areas due to its flexibility and ease of use [30, 38].

Specifically, PROMETHEE II is a method that results in an ordering of the alternatives based on a preferred value relationship between the alternatives, based on the concept of net flow. Thus, the Promethée II method proves to be more effective in the evaluation of designs, as it is able to prioritize the alternatives through complete ordering [46].

Among the family methods, particularly PROMETHEE I and PROMETHEE II play a critical role in ad-dressing MCDM problems involving uncertainty [50]. However, there are still very few publications on the use of the PROMETHEE method to deal with multicriteria decision problems in the hesitant fuzzy environment.

Liang et al. [50] proposed two extensions of the PROMETHEE method for multicriteria decision making using HFLTSs. Wu et al. [68], proposed the PROMETHEE method, using HFLTSs and ANP to assess the social sustainability of small hydroelectric power plants.

There are some publications that include hybrid fuzzy MCDM methods composed by PROMETHEE and QFD separately and together. In the bibliographic research carried out, no studies were found involving the application of these two methods together in the hesitant fuzzy environment. Additionally, the OWA operator developed by Liu and Rodriguez [23] has not yet been used in the PROMETHEE hesitant fuzzy to aggregate the opinions of experts.

Guo et al. [47] used the fuzzy House of Quality integrated to fuzzy PROMETHEE in a design selection problem. Motlagh et al. [55] used fuzzy PROMETHEE fuzzy in a Group Decision Support System (GDSS) approach to classify the technical requirements of a product in the House of Quality.

Ighravwe [15] used fuzzy QFD in conjunction with PROMETHEE and TOPSIS method to select the best energy storage technology for small-scale renewable energy systems. Roghanian & Alipour [17] used the AHP, QFD and Promethee methods in the fuzzy environment to prioritize tools to be applied in lean production. Yadav & Gangele [32] developed an approach integrating QFD with other MCDM methods (i.e. AHP, TOPSIS, PROMETHEE, etc.) to select the optimal product design and manufacturing solution.

3 Methodology

The steps of the proposed Hesitant Fuzzy-AHP-QFD-PROMETHEE method can be seen in Fig. 1. The structure applied to classify designs has three main parts, as detailed below.

Fig. 1

Schematic representation of the algorithm.

In this article, the hesitant fuzzy linguistic ordered weighted average (HFLOWA), proposed by Liu & Rodriguez [23], was used to aggregate hesitant linguistic terms sets.

3.1 Stage 01: Determine customer requirements (CRs) and weights using Fuzzy Hesitant AHP method

Step 1: Identify and build the hierarchy of CRs;

Step 2: Calculate the weight of the CRs as described below [53]

Step 2.1: Build the pairwise comparison matrix and get the experts’ opinion using the Hesitant Fuzzy Linguistic Term Sets (HFLTS), according to Table 1 and context-free grammar; such as between, greater than, less than, at most, at least etc.

Table 1
Linguistic scale for hesitant fuzzy AHP

Linguistic term s _i Abb. Triangular fuzzy number

Absolutely high importance s ₁₀ (AHI) (7, 9, 9)

Very high importance s ₉ (VHI) (5, 7, 9)

Essentially high importance s ₈ (ESHI) (3, 5, 7)

Weakly high importance s ₇ (WHI) (1, 3, 5)

Equally high importance s ₆ (EHI) (1, 1, 3)

Exactly low importance s ₅ (EE) (1, 1, 1)

Equally low importance s ₄ (ELI) (0.33, 1, 1)

Weakly low importance s ₃ (WLI) (0.2, 0.33, 1)

Essentially low importance s ₂ (ESLI) (0.14,0.2,0.33)

Very low importance s ₁ (VLI) (0.11,0.14,0.2)

Absolutely low importance s ₀ (ALI) (0.11,0.11,0.14)

Linguistic term	s _i	Abb.	Triangular fuzzy number
Absolutely high importance	s ₁₀	(AHI)	(7, 9, 9)
Very high importance	s ₉	(VHI)	(5, 7, 9)
Essentially high importance	s ₈	(ESHI)	(3, 5, 7)
Weakly high importance	s ₇	(WHI)	(1, 3, 5)
Equally high importance	s ₆	(EHI)	(1, 1, 3)
Exactly low importance	s ₅	(EE)	(1, 1, 1)
Equally low importance	s ₄	(ELI)	(0.33, 1, 1)
Weakly low importance	s ₃	(WLI)	(0.2, 0.33, 1)
Essentially low importance	s ₂	(ESLI)	(0.14,0.2,0.33)
Very low importance	s ₁	(VLI)	(0.11,0.14,0.2)
Absolutely low importance	s ₀	(ALI)	(0.11,0.11,0.14)

Step 2.2: Check the consistency of each expert’s responses by calculating the Consistency Ratio (CR). Details of the method used in this work can be found in Ho et al. [63]

Step 2.3: Aggregate and build fuzzy envelope for HFLTS by using the OWA operator, as proposed by Liu and Rodriguez [23]. In this approach, the result of aggregation yields a trapezoidal fuzzy number.

Initially, the scale presented in Table 1 is sorted from the lowest (S₀) to the highest (S_g). Assume the experts evaluations vary between two terms i.e. S_i and S_j. Then S₀ ⩽ S_i < S_j ⩽ S_g. The parameters of trapezoidal fuzzy membership function $\tilde{A} = (α, β, γ, δ)$ are computed as follows: [53] $α = min {a_{L}^{i}, a_{M}^{i}, a_{M}^{i + 1}, \dots, a_{M}^{j}, a_{R}^{j}} = a_{L}^{i}$ (14) $δ = max {a_{L}^{i}, a_{M}^{i}, a_{M}^{i + 1}, \dots, a_{M}^{j}, a_{R}^{j}} = a_{R}^{j}$ (15) $β = {\begin{matrix} a_{M}^{i}, if i + 1 = j \\ {OWA}_{W^{2}} (a_{M}^{i}, \dots, a_{M}^{\frac{i + j}{2}}), if i + j is even \\ {OWA}_{W^{2}} (a_{M}^{i}, \dots, a_{M}^{\frac{i + j - 1}{2}}), if i + j is odd \end{matrix}$ (16)

$γ = {\begin{matrix} a_{M}^{i + 1}, if i + 1 = j \\ {OWA}_{W^{1}} (a_{M}^{j}, a_{M}^{j - 1} \dots, a_{M}^{\frac{i + j}{2}}), if i + j is even \\ {OWA}_{W^{1}} (a_{M}^{j}, a_{M}^{j - 1} \dots, a_{M}^{\frac{i + j + 1}{2}}) if i + j is odd \end{matrix}$ (17)

OWA operation presented in Definition 6 requires a weight vector. The first and second types of weights can be obtained through α parameter which belongs to the unit range [0, 1] [53]. First type of weights $W^{1} = (W_{1}^{1}, W_{2}^{1}, \dots, W_{n}^{1})$ is defined as:

$W_{1}^{1} = α_{2}, W_{2}^{1} = α_{2} (1 - α_{2}), \dots, W_{n}^{1} = α_{2} {(1 - α_{2})}^{n - 2}$ (18)

The second type of weights $W^{2} = (W_{1}^{2}, W_{2}^{2}, \dots, W_{n}^{2})$ is defined as:

$W_{1}^{2} = α_{1}^{n - 1}, W_{2}^{2} = (1 - α_{1}) α_{1}^{n - 2}, \dots, W_{n}^{2} = 1 - α_{1}$ (19)

Where $α_{1} = \frac{g - (j - i)}{g - 1}$ , $α_{2} = \frac{(j - i) - 1}{g - 1}$ and g is the number of terms in the evaluation scale, j is the rank of highest evaluation and i is the rank of lowest evaluation value of the given interval.

Step 2.4: Obtain pairwise comparison matrix $(\tilde{C})$ composed of aggregated fuzzy number in Step. 2.3. Since the fuzzy envelopes, obtained in previous step are trapezoidal fuzzy numbers, reciprocal values are calculated as follows [53] ${\tilde{c}}_{ji} = (\frac{1}{c_{iju}}, \frac{1}{c_{ijm 2}}, \frac{1}{c_{ijm 1}}, \frac{1}{c_{ijl}})$ (20)

Step 2.5: Compute fuzzy geometric mean for each row $({\tilde{r}}_{i})$ of the matrix $\tilde{C}$ using Equation (2). ${\tilde{r}}_{i} = {({\tilde{c}}_{i 1} \otimes {\tilde{c}}_{i 2}, \dots, \otimes {\tilde{c}}_{in})}^{1 / n}$ (21)

Step 2.6: The fuzzy weight $({\tilde{w}}_{i}^{CR})$ of each main customer requirement is calculated using $({\tilde{r}}_{i})$ values as follows: ${\tilde{w}}_{i}^{CR} = {\tilde{r}}_{i} \otimes {({\tilde{r}}_{1} \oplus {\tilde{r}}_{2} \dots \oplus {\tilde{r}}_{n})}^{- 1}$ (22)

Step 2.7: Defuzzify the trapezoidal fuzzy number ${\tilde{w}}_{ij}^{CR} = (α, β, γ, δ)$ by Center of Gravity (COG) using Equation 23 [57] and normalize the defuzzified values using Equation 24.

$\begin{matrix} defuzz (w_{ij}^{CR}) \\ = \frac{- α \times β + γ \times δ + (\frac{1}{3}) \times {(δ - γ)}^{2} - (\frac{1}{3}) \times {(β - α)}^{2}}{- α - β + γ + δ} \end{matrix}$ (23)

$w_{ij}^{N} = \frac{w_{ij}^{G}}{\sum_{i} \sum_{j} w_{ij}^{G}}$ (24)

3.2 Stage 02: Determine the effect of CRs on design requirements (DRs) and determine the weight of DRs using hesitant fuzzy-QFD

Step 1: Collect the data for the relations between DRs and CRs by using HFLTS from expert. The HFLS of relations are obtained by utilizing the linguistic terms in Table 2 and context-free grammar.

Table 2
Linguistic scale for correlations

Linguistic term Abb. Triangular fuzzy number

Absolutely low AL (1,2,3)

Very low VL (2,3,4)

Low L (3,4,5)

Medium M (4,5,6)

High H (5,6,7)

Very high VH (6,7,8)

Absolutely high AH (7,8,9)

Linguistic term	Abb.	Triangular fuzzy number
Absolutely low	AL	(1,2,3)
Very low	VL	(2,3,4)
Low	L	(3,4,5)
Medium	M	(4,5,6)
High	H	(5,6,7)
Very high	VH	(6,7,8)
Absolutely high	AH	(7,8,9)

Step 2: Aggregate HFLTS relations by using the aggregation operator defined in Step 2.3 and obtain relation matrix $\tilde{R}$ with trapezoidal fuzzy numbers as given by Equation (14). $\tilde{R} = \begin{matrix} \begin{matrix} {DR}_{1} & {DR}_{2} & \dots & {DR}_{z} \end{matrix} \\ \begin{matrix} {CR}_{11} \\ {CR}_{12} \\ ⋮ \\ {CR}_{nv} \end{matrix} & | \begin{matrix} {\tilde{R}}_{111} & {\tilde{R}}_{112} & \dots & {\tilde{R}}_{11 z} \\ {\tilde{R}}_{121} & {\tilde{R}}_{122} & \dots & {\tilde{R}}_{12 z} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{R}}_{nv 1} & {\tilde{R}}_{nv 2} & \dots & {\tilde{R}}_{nvz} \end{matrix} | \end{matrix}$ (25)

Step 3: Obtain weighted relation matrix $({\tilde{R}}^{w})$ whose elements are obtained by using Equation (15). ${\tilde{R}}_{ijk}^{w} = w_{ij}^{N} \times {\tilde{R}}_{ijk}$ (26)

Step 4: Obtain fuzzy importance value of DRs by summing the elements in each column of ${\tilde{R}}^{w}$ as shown in Equation (16). ${\tilde{DR}}_{k}^{Imp} = \sum_{i} \sum_{j} {\tilde{R}}_{ijk}^{w}$ (27) where ${\tilde{DR}}_{k}^{Imp}$ denotes the fuzzy importance of design requirement k

Step 5: Using Equations (23) and (28), obtain the crisp importance weights of DRs by defuzzifying ${\tilde{DR}}_{k}^{Imp}$ and normalizing them. ${DR}_{k}^{N} = \frac{{DR}_{k}^{Imp}}{{max}_{k = 1, \dots, z} {DR}_{k}^{Imp}}$ (28) where ${DR}_{k}^{Imp}$ and ${DR}_{k}^{N}$ denotes the defuzzified and normalized importance values of design requirement k, respectively.

3.3 Stage 03: Apply the hesitant fuzzy PROMETHEE II method to prioritize the alternatives

Step 1: Determine evaluation criteria and evaluation committee, and then define the context-free grammar and the linguistic term set. In this work, the scale presented in Table 3 will be used [23].

Table 3
Set of seven terms with corresponding semantics

Linguistic term s _i Abb. Triangular fuzzy number

Perfect s ₆ (P) (0.83, 1, 1)

Very good s ₅ (VG) (0.67,0.83, 1)

Good s ₄ (G) (0.5,0.67,0.83)

Medium s ₃ (M) (0.33,0.5,0.67)

Bad s ₂ (B) (0.17,0.33,0.5)

Very bad s ₁ (VB) (0,0.17,0.33)

Nothing s ₀ (N) (0,0,0.17)

Linguistic term	s _i	Abb.	Triangular fuzzy number
Perfect	s ₆	(P)	(0.83, 1, 1)
Very good	s ₅	(VG)	(0.67,0.83, 1)
Good	s ₄	(G)	(0.5,0.67,0.83)
Medium	s ₃	(M)	(0.33,0.5,0.67)
Bad	s ₂	(B)	(0.17,0.33,0.5)
Very bad	s ₁	(VB)	(0,0.17,0.33)
Nothing	s ₀	(N)	(0,0,0.17)

Step 2: Construct a fuzzy decision matrix $(\tilde{V})$ by aggregating fuzzy rating of all experts by using the aggregation operator defined in Step 2.3 (Stage 01).

Step 3: Construct the preference function (Equation 18). In this paper, the Hamming distance method is used for comparing two alternatives g and f on each criterion. At first the maximum between two fuzzy numbers is computed. So their least upper bound is determined to find $max ({\tilde{v}}_{gj}, {\tilde{v}}_{fj})$ . Then the Hamming distance $d (max ({\tilde{v}}_{gj}, {\tilde{v}}_{fj}), {\tilde{v}}_{gj})$ and $d (max ({\tilde{v}}_{gj}, {\tilde{v}}_{fj}), {\tilde{v}}_{fj})$ are calculated. [1 , 16]. $P_{j} (g, f) = {\begin{matrix} d (max ({\tilde{v}}_{gj}, {\tilde{v}}_{fj}), {\tilde{v}}_{gj} {\tilde{v}}_{gj} < {\tilde{v}}_{fj} \\ d (max ({\tilde{v}}_{gj}, {\tilde{v}}_{gj}), {\tilde{v}}_{fj} {\tilde{v}}_{gj} ⩾ {\tilde{v}}_{fj} \end{matrix}$ (29)

Step 4: Fuzzy preference index is calculated to determine the value of outranking relation (j = 1,2,...,n)(Eq.19) $\tilde{π} (g, f) = \sum_{j = 1}^{n} {\tilde{w}}_{j} P_{j} (g, f)$ (30)

Step 5: The leaving and entering flows are calculated for ranking of alternatives (Equation 20 and Eq.21). ${\tilde{ϕ}}^{+} (g) = \sum_{\begin{matrix} f \neq g \\ f = 1 \end{matrix}}^{m} \tilde{π} (g, f)$ (31) ${\tilde{ϕ}}^{-} (g) = \sum_{\begin{matrix} f \neq g \\ f = 1 \end{matrix}}^{m} \tilde{π} (f, g)$ (32)

Step 6: Calculation of net outranking flows and the PROMETHEE II complete ranking. Calculate the fuzzy net outranking for each alternative (Equation 22). $ϕ (g) = ϕ^{+} (g) - ϕ^{-} (g)$ (33)

The results of leaving and entering flows are presented in the form of fuzzy numbers. Defuzzification is used for ranking of fuzzy flows that converts fuzzy numbers to appropriate crisp values.

4 Results

Stage 01: Determine customer requirements (CRs) and weights using Fuzzy Hesitant AHP

The internal customers who approve the packaging are: Planning and Production Control (PPC), Manufacturing, Quality and Ergonomics and Safety. This way, the quality matrix construction began from the definition of its quality requirements.

Through a closed questionnaire, the requirements were listed and scores were assigned. The expert group identified 8 CRs, as shown in Table 4.

Table 4
Customer Requirements (CRs)

CRs Description

CR1 Not to hurt the operator’s hands or back

CR2 Not to accumulate water

CR3 Be demountable

CR4 Pack as many parts as possible.

CR5 Preserve the integrity of the product

CR6 Be able to be stacked

CR7 Easy and fast loading/unloading

CR8 Correct identification

CRs	Description
CR1	Not to hurt the operator’s hands or back
CR2	Not to accumulate water
CR3	Be demountable
CR4	Pack as many parts as possible.
CR5	Preserve the integrity of the product
CR6	Be able to be stacked
CR7	Easy and fast loading/unloading
CR8	Correct identification

By adopting the codes in Table 1, each of the four experts attributed the degree of importance of the CRs, as shown in Tables 5 –8.

Table 5

Pairwise comparisons of the customer requirement –PPC

	CR1	CR2	CR3	CR4	CR5	CR6	CR7	CR8
CR1	EE	EE	ESHI	ESHI	EE	EE	EE	AHI
CR2		EE	ESHI	ESHI	EE	EE	EE	VHI
CR3			EE	EHI	ESLI	ESLI	ESLI	ESHI
CR4				EE	ESLI	ESLI	ESLI	ESHI
CR5					EE	EE	EE	VHI
CR6						EE	EE	AHI
CR7							EE	AHI
CR8								EE

Table 6

Pairwise comparisons of the customer requirement –Manufacturing

	CR1	CR2	CR3	CR4	CR5	CR6	CR7	CR8
CR1	EE	ESHI	AHI	ESHI	ESHI	VHI	VHI	ESHI
CR2		EE	ESHI	EE	EE	ESHI	ESHI	EE
CR3			EE	ESLI	ESLI	EHI	EHI	ESLI
CR4				EE	EHI	ESHI	ESHI	EE
CR5					EE	ESHI	ESHI	EE
CR6						EE	EE	ESLI
CR7							EE	ESLI
CR8								EE

Table 7

Pairwise comparisons of the customer requirement –quality

	CR1	CR2	CR3	CR4	CR5	CR6	CR7	CR8
CR1	EE	ESHI	ESHI	ESHI	EE	EE	EE	AHI
CR2		EE	EE	EE	ESLI	ESLI	ESLI	ESHI
CR3			EE	EHI	ESLI	ESLI	ESLI	ESHI
CR4				EE	ESLI	ESLI	ESLI	ESHI
CR5					EE	EE	EE	VHI
CR6						EE	EE	AHI
CR7							EE	AHI
CR8								EE

Table 8

Pairwise comparisons of the customer requirement –Ergonomics and Safety

	CR1	CR2	CR3	CR4	CR5	CR6	CR7	CR8
CR1	EE	EE	AHI	AHI	ESHI	VHI	VHI	AHI
CR2		EE	VHI	VHI	ESHI	VHI	VHI	VHI
CR3			EE	EHI	ESLI	EHI	EHI	EE
CR4				EE	ESLI	EE	EE	EE
CR5					EE	ESHI	ESHI	ESHI
CR6						EE	EE	EE
CR7							EE	EE
CR8								EE

Consistency was checked in each of the Tables (5 to 8), with all the CR values obtained below 0.10. Table 9 shows the global comparison between the CRs aggregating the opinion of the four experts. For example, by comparing CR1 with CR3, the opinions expressed by the four experts were: ESHI, ESHI, AHI and AHI. Thus, the global opinion used was “At least ESHI”, in bold in Table 9.

Table 9

Pairwise comparisons of the customer requirement –global

	CR1	CR2	CR3	CR4	CR5	CR6	CR7	CR8
CR1	EE	Between EE and ESHI	At least ESHI	At least ESHI	Between EE and ESHI	At least EE	At least EE	At least ESHI
CR2		EE	At least EE	At least EE	Between ESLI and ESHI	Between ESLI and VHI	Between ESLI and VHI	At least EE
CR3			EE	Between ESLI and EE	ESLI	Between ESLI and EE	Between ESLI and EE	Between ESLI and ESHI
CR4				EE	Between ESLI and EE	Between ESLI and ESHI	Between ESLI and ESHI	Between EE and ESHI
CR5					EE	Between EE and ESHI	Between EE and ESHI	At least EE
CR6						EE	EE	Between ESLI and VHI
CR7							EE	Between ESLI and VHI
CR8								EE

In this approach, the result of aggregation produces a trapezoidal fuzzy number. To facilitate the approach understanding, a sample calculation was shown for each table. The bold numbers in the tables indicate how these examples were obtained.

Table 10 shows that the pairwise comparison value in CR1 and CR3 whose result was the trapezoidal fuzzy number (3, 7.88, 9, 9). The experts’ language assessments for the comparison are “at least s8 = essentially high importance”. The triangular fuzzy numbers associated with the linguistic terms mentioned are (3,5,7), (5,7,9) and (7,9,9), respectively.

Table 10

Pairwise comparisons values of the main CRs with respect to the goal

	CR1	CR2	CR3	CR4	CR5	CR6	CR7	CR8
CR1	(1, 1, 1, 1)	(1, 1, 3, 7)	(3, 7.88, 9, 9)	(3, 7.88, 9, 9)	(1, 1, 3, 7)	(1, 1.89, 4.11, 9)	(1, 1.89, 4.11, 9)	(3, 7.88, 9, 9)
CR2	(0.14, 0.33, 1, 1)	(1, 1, 1, 1)	(1, 1.89, 4.11, 9)	(1, 1.89, 4.11, 9)	(0.14, 0.5, 1.5, 7)	(0.14, 0.32, 1.68, 9)	(0.14, 0.32, 1.69, 9)	(1, 1.89, 4.11, 9)
CR3	(0.11, 0.11, 0.13, 0.33)	(0.11, 0.24, 0.53, 1)	(1, 1, 1, 1)	(0.14, 0.58, 1.42, 3)	(0.14, 0.2, 0.2, 0.33)	(0.14, 0.58, 1.42, 3)	(0.14, 0.58, 1.42, 3)	(0.14, 0.5, 1.5, 7)
CR4	(0.11, 0.11, 0.13, 0.33)	(0.11, 0.24, 0.53, 1)	(0.33, 0.70, 1.72, 7.14)	(1, 1, 1, 1)	(0.14, 0.58, 1.42, 3)	(0.14, 0.5, 1.5, 7)	(0.14, 0.5, 1.5, 7)	(1, 1, 3.7)
CR5	(0, 14, 0.33, 1, 1)	(0, 14, 0.67, 2, 7, 14)	(3, 5, 5.7)	(0.33, 0.70, 1.72, 7.14)	(1, 1, 1, 1)	(1, 1, 3, 7)	(1, 1, 3, 7)	(1, 1.89, 4.11, 9)
CR6	(0.11, 0.24, 0.53, 1)	(0.11, 0.59, 3.15, 7.14)	(0.33, 0.70, 1.72, 7.14)	(0.14, 0.67, 2, 7.14)	(0.14, 0.33, 1, 1)	(1, 1, 1, 1)	(1, 1, 1, 1)	(0.14, 0.32, 1.68, 9)
CR7	(0.11, 0.24, 0.53, 1)	(0.11, 0.59, 3.15, 7.14)	(0.33, 0.70, 1.72, 7.14)	(0.14, 0.67, 2, 7.14)	(0.14, 0.33, 1, 1)	(1, 1, 1, 1)	(1, 1, 1, 1)	(0.14, 0.32, 1.68, 9)
CR8	(0.11, 0.11, 0.13, 0.33)	(0.11, 0.24, 0.53, 1)	(0.14, 0.67, 2, 7.14)	(0.14, 0.33, 1, 1)	(0.11, 0.24, 0.53, 1)	(0.11, 0.11, 0.13, 0.33)	(0.11, 0.11, 0.127, 0.33)	(1, 1, 1, 1)

Using the OWA operator proposed by Liu and Rodriguez [23], the trapezoidal fuzzy association function representing the linguistic evaluation is calculated as: $\begin{matrix} α & = min {a_{l}^{8}, a_{M}^{8}, a_{L}^{9}, a_{R}^{8}, a_{M}^{9}, a_{L}^{10}, a_{R}^{9}, a_{M}^{10}, a_{R}^{10}} \\ = min {3, 5, 5, 7, 7, 7, 9, 9, 9} = 3 \end{matrix}$ $\begin{matrix} δ & = max {a_{l}^{8}, a_{M}^{8}, a_{L}^{9}, a_{R}^{8}, a_{M}^{9}, a_{L}^{10}, a_{R}^{9}, a_{M}^{10}, a_{R}^{10}} \\ = max {3, 5, 5, 7, 7, 7, 9, 9, 9} = 9 \end{matrix}$

Since i = 8, g = 10, it obtains α = 8/10 and the associated OWA weighting vector: $W^{2} = ({(\frac{8}{10})}^{2}, (1 - \frac{8}{10}) \times \frac{8}{10}, (1 - \frac{8}{10}))$

The OWA operator was used to obtain the element β:

$\begin{matrix} β = & {(\frac{8}{10})}^{2} \times 9 + (1 - \frac{8}{10}) \times \frac{8}{10} \times 7 \\ + (1 - \frac{8}{10}) \times 5 = 7.88 \end{matrix}$

And the parameter γ is $a_{M}^{g} = a_{M}^{10} = 9$ . Therefore, the fuzzy envelope is (3,7.88,9,9).

After determining the values of the comparison matrix, the geometric mean of each line is calculated and the standard fuzzy weight of each criterion is obtained. For example, the value (1.51, 2.54,4.27, 6.42) in Table 11 is calculated as:

Table 11

Weight of the customer requirements

CRs	Geometric Means	Normalized Weight	De-fuzzified weight	Normalized weight
CR1	(1.51,2.54,4.27,6.42)	(0.05,0.18,0.64,1.81)	0.72	0.25
CR2	(0.37,0.76,2.04,5.03)	(0.01,0.05,0.30,1.42)	0.51	0.18
CR3	(0.17,0.39,0.7,1.46)	(0.01,0.03,0.10,0.41)	0.15	0.05
CR4	(0.24,0.48,1.01,2.65)	(0.01,0.03,0.15,0.75)	0.27	0.09
CR5	(0.61,1.05,2.24,4.46)	(0.02,0.08,0.33,1.26)	0.47	0.16
CR6	(1.24,0.54,1.33, 2.75)	(0.01,0.04,0.20,0.78)	0.28	0.10
CR7	(1.24,0.54,1.33,2.75)	(0.01,0.04,0.20,0.77)	0.28	0.10
CR8	(1.16,0.39,0.96,1.82)	(0.01,0.03,0.14,0.51)	0.19	0.07
TOTAL			2.89	1

(1×1×3×3×1×1×1×3)^1/5 = 1.51; (1×1×7.88×7.88×1×1.89×1.89×7.88)^1/5 = 2.54; (1×3×9×9×3×4.11×4.11×9)^1/5 = 4.27; (1×7×9×9×7×9×9×9)^1/5 = 6.42

Then, the geometric means are summarized. The sum of the geometric means given in Table 11 is (3.55, 6.69, 13.88, 27.36), so the standardized weight for the criterion performance is obtained as follows:

$\frac{1.51}{27.36} = 0.05; \frac{2.54}{13.88} = 0.18; \frac{4.27}{6.69} = 0.64;$ $\frac{6.42}{3.55} = 1.81$

When we defuzzify the trapezoidal fuzzy number using Equation 23, we obtained the following result: $\begin{matrix} W_{11}^{G} & = \frac{- 0.01 + 1.16 + \frac{1}{3} \times {(1, 17)}^{2} - \frac{1}{3} \times (0.13)}{- 0.05 - 0.18 + 0.64 + 1.81} \\ = 0.72 \end{matrix}$

To calculate the normalized value 0.25, we first sum the defuzzified values and then the defuzzified value 0.72 is divided by this sum: $W_{11}^{N} = \frac{0.72}{2.89} = 0.25$

Stage 02: Determine the effect of CRs on design requirements (DRs) and determine the weight of DRs using hesitant fuzzy-QFD.

Table 12 presents the relationships between design requirements and customer requirements using the Hesitant Fuzzy Linguistic Term Sets (HFLTS) in Table 2. Table 13 presents the aggregate correlation matrix $\tilde{R}$ between CRs and Drs.

Table 12

Identified design requirements and their relations with the customer requirements

	Design requirements (DRs)
	Decrease the number of accessories	Prevent contact between parts	Mechanical resistance	Free of burrs, sharp edges or any similar element	Ergonomic Design	The machined and sensitive parts of the pieces should not be supported on the packaging	Higher number of parts per package	Add holes for water evacuation	Clear and readable identification
CR1	Between L and M			At least VH	At least VH		Between L and M	Between L and M
CR2	At most L			Between M and H	Between M and H			At least VH
CR3	Between L and M		Between L and M	At most L	Between L and M	Between L and M	Between L and M
CR4	At most L	At most L	Between L and M			Between M and VH	At least VH
CR5	Between L and M	At least VH	Between M and H	Between M and H		At least VH	Between L and M	Between M and H
CR6	Between L and M	At least VH	At least VH	Between L and M		Between VL and L	Between VL and L	Between VL and L
CR7	At least VH			Between L and M	Between M and H	Between VL and L	Between VL and L		Between L and M
CR8									At least VH

Table 13

Aggregated relation matrix $\tilde{R}$ between CRs and DRs.

	Weight	DR1	DR2	DR3	DR4	DR5	DR6	DR7	DR8	DR9
CR1	0.25	(3,4,5,6)	(6,7,8,9)	(4,5,6,7)	(4,5,6,7)		(6,7,8,9)	(3,4,5,6)	(4,5,6,7)
CR2	0.18	(3,4,5,6)			(6,7,8,9)	(6,7,8,9)		(3,4,5,6)	(3,4,5,6)
CR3	0.05	(3,4,5,6)		(3,4,5,6)	(1,2,2.9,5)	(3,4,5,6)	(3,4,5,6)	(3,4,5,6)
CR4	0.09	(1,2,3,4)	(1,2,2.9,5)	(3,4,5,6)			(4,5.8,6.3,8)	(6,7,8,9)
CR5	0.16	(3,4,5,6)	(6,7,8,9)	(6,7,8,9)	(3,4,5,6)		(2,3,4,5)	(2,3,4,5)	(2,3,4,5)
CR6	0.10	(6,7,8,9)			(3,4,5,6)	(4,5,6,7)	(2,3,4,5)	(2,3,4,5)		(3,4,5,6)
CR7	0.10									(6,7,8,9)
CR8	0.07	(1,2,2.9,5)			(4,5,6,7)	(4,5,6,7)			(6,7,8,9)

Using the OWA operator, the aggregated value for CR1 and “DR1 = Decrease number of accessories corresponding to “between L and M” is calculated as (3, 4, 5, 6).

Table 14 shows the weights relation matrix $({\tilde{R}}^{w})$ and the fuzzy importance value of DRs. The weighed correlation for CR1 and DR1 relation is calculated as follows:

$0.250 \times (3, 4, 5, 6) = (0.75, 1, 1.25, 1.5)$

Table 14

Weighted correlation matrix

	DR1	DR2	DR3	DR4	DR5	DR6	DR7	DR8	DR9
CR1	(0.75, 1, 1.25, 1.5)	(1.5, 1.75, 2, 2.25)	(1, 1.25, 1.5, 1.75)	(1, 1.25, 1.5, 1.75)		(1.5, 1.75, 2, 2.25)	(0.75, 1, 1.25, 1.5)	(1, 1.25, 1.5.1.75)
CR2	(0.53, 0.71, 0.88, 1.06)			(1.06, 1.24, 1.41, 1.59)	(1.06, 1.24, 1.41, 1.59)		(0.53, 0.71, 0.88, 1.06)	(0.53, 0.71, 0.83, 1.06)
CR3	(0.16, 0.2, 0.27, 0.32)		(0.16, 0.21, 0.27, 0.32)	(0.05, 0.11, 0.15, 0.27)	(0.16, 0.21, 0.27, 0.32)	(0.16, 0.21, 0.27, 0.32)	(0.16, 0.21, 0.27, 0.32)
CR4	(0.09, 0.19, 0.28, 0.37)	(0.09, 0.19, 0.27, 0.47)	(0.28, 0.37, 0.47, 0.56)			(0.37, 0.54, 0.58, 0.75)	(0.56, 0.65, 0.75, 0.84)
CR5	(0.49, 0.65, 0.81, 0.98)	(0.98, 1.14, 1.3, 1.46)	(0.98, 1.14, 1.3, 1.46)	(0.49, 0.65, 0.81, 0.98)		(0.33, 0.49, 0.65, 0.81)	(0.33, 0.49, 0.65, 0.81)	(0.33, 0.49, 0.65, 0.81)
CR6	(0.59, 0.69, 0.79, 0.89)			(0.3, 0.39, 0.49, 0.59)	(0.39, 0.49, 0.59, 0.69)	(0.2, 0.3, 0.39, 0.49)	(0.2, 0.3, 0.39, 0.49)		(0.3, 0.39, 0.49, 0.59)
CR7									(0.59, 0.69, 0.79, 0.89)
CR8	(0.07, 0.13, 0.19, 0.33)			(0.26, 0.33, 0.4, 0.46)	(0.26, 0.33, 0.4, 0.46)			(0.4, 0.46, 0.53, 0.59)
Total	(2.68, 3.584.48, 5.45)	(2.57, 3.08, 3.57, 4.18)	(2.42, 2.98, 3.54, 4.1)	(3.16, 3.97, 4.77, 5.64)	(1.81, 2.27, 2.67, 3.06)	(2.56, 3.29, 3.89, 4.63)	(2.52, 3.36, 4.2, 5.03)	(2.25, 2.91, 3.56, 4.22)	(0.89, 1.09, 1.28, 1.48)
Total Normaliz.	(0.07, 0.11, 0.17, 0.26)	(0.07, 0.10, 0.13, 0.20)	(0.06, 0.09, 0.13, 0.20)	0.08, 0.12, 0.18, 0.27)	(0.05, 0.07, 0.10, 0.15)	(0.07, 0.10, 0.15, 0.22)	(0.07, 0.10, 0.16, 0.24)	(0.06, 0.09, 0.13, 0.20)	(0.02, 0.03, 0.05, 0.07)

The crisp importance weights of DRs are given in Table 15. The defuzzified score for DR1 is calculated as follows: First, the weighted correlations in the DR1 column in Table 14 are summed. It is found to be ${\tilde{DR}}_{1}^{Imp} = (2.68, 3.58, 4.48, 5.45)$ . Then this sum is defuzzified using Equation 23 to calculate ${\tilde{DR}}_{1}^{N}$ (4.04). The normalized score is calculated by using Equation 28:

${\tilde{DR}}_{1}^{N} = \frac{4.04}{29.28} = 0.14$

Table 15

Weights of design requirements

	DRs	Defuzzified score	Normalized score
DR1	Decrease the number of accessories	4.05	0.14
DR2	Prevent contact between parts	3.36	0.11
DR3	Mechanical resistance	3.26	0.11
DR4	Free of burrs, sharp edges or any similar element	4.39	0.15
DR5	Ergonomic Design	2.47	0.08
DR6	The machined and sensitive parts of the pieces cannot be supported on the packaging	3.59	0.12
DR7	Higher number of parts per package	3.78	0.13
DR8	Add holes for water evacuation	3.24	0.11
DR9	Clear and readable identification	1.18	0.04

Stage 03. Apply the hesitant fuzzy PROMETHEE II method to prioritize the alternatives.

Once the DR scoring process has been completed, the next step is to apply the proposed Hesitant Fuzzy PROMETHEE to select the packaging design. Thus, adopting the note made in Table 3, the evaluation committee provides the linguistic values of the alternatives considering each indicator (DRs) through the set of linguistic terms. The result is shown in Table 16.

Table 16

Evaluation of alternatives with respect to design requirements

	Design A1	Design A2	Design A3
DR1	Between M and G	Between M and VG	Between M and G
DR2	Between G and VG	Between M and G	At least G
DR3	Between B and M	Between M and G	Between M and G
DR4	At least G	Between M and VG	Between M and G
DR5	At least G	Between M and VG	Between M and VG
DR6	Between B and M	Between B and M	Between M and G
DR7	Between M and G	At least G	Between M and G
DR8	At least G	At least G	At least G
DR9	Between G and VG	Between G and VG	Between M and G

After that, linguistic values can be transformed into HFLTS decision matrix. The opinions are grouped using the same OWA operator defined in step 2.3 (Stage 01), as shown in Table 17.

Table 17

The aggregation HFLTS for the indicators of Three alternatives

	Design A1	Design A2	Design A3
DR1	(0.33,0.5,0.67,0.83)	(0.33,0.64,0.7,1)	(0.33,0.5,0.67,0.83)
DR2	(0.5,0.67,0.83,1)	(0.33,0.5,0.67,0.83)	(0.5,0.85,1,1)
DR3	(0.17,0.33,0.5,0.67)	(0.33,0.5,0.67,0.83)	(0.33,0.5,0.67,0.83)
DR4	(0.5,0.85,1,1)	(0.33,0.64,0.7,1)	(0.33,0.5,0.67,0.83)
DR5	(0.5,0.85,1,1)	(0.33,0.64,0.7,1)	(0.33,0.64,0.7,1)
DR6	(0.17,0.33,0.5,0.67)	(0.17,0.33,0.5,0.67)	(0.33,0.5,0.67,0.83)
DR7	(0.33,0.5,0.67,0.83)	(0.5,0.85,1,1)	(0.33,0.5,0.67,0.83)
DR8	(0.5,0.85,1,1)	(0.5,0.85,1,1)	(0.5,0.85,1,1)
DR9	(0.5,0.67,0.83,1)	(0.5,0.67,0.83,1)	(0.33,0.5,0.67,0.83)

Hamming distance was used to compare two fuzzy numbers f and g, as shown in Table 18. The first and second number in each cell of Table 18 represent the $d (max ({\tilde{v}}_{gj}, {\tilde{v}}_{fj}), {\tilde{v}}_{gj})$ and $d (max ({\tilde{v}}_{gj}, {\tilde{v}}_{fj}), {\tilde{v}}_{fj})$ , respectively. For example, considering the performance rating of the fuzzy numbers A1 and A2 against design requirement 1 (DR1) presented in Table 17 as ${\tilde{v}}_{A 1 DR 1} = (0.33, 0.5, 0.67, 0.83)$ e ${\tilde{v}}_{A 2 DR 1} = (0.33, 0.64, 0.7, 1)$ can be represented by the following association functions:

$μ_{{\tilde{v}}_{A 1 DR 1}} (x) = {\begin{matrix} 0, x < 0.33, \\ \frac{x - 0.33}{0.17}, 0.33 ⩽ x < 0.5, \\ 1, 0.5 ⩽ x < 0.67, \\ \frac{x - 0.83}{- 0.16}, 0.67 ⩽ x < 0.83 \\ 0, x ⩾ 0.83 \end{matrix}$ $μ_{{\tilde{v}}_{A 2 DR 1}} (x) = {\begin{matrix} 0, x < 0.33, \\ \frac{x - 0.33}{0.31}, 0.33 ⩽ x < 0.64, \\ 1, 0.64 ⩽ x < 0.7, \\ \frac{x - 1}{- 0.3}, 0.7 ⩽ x < 1 \\ 0, x ⩾ 1 \end{matrix}$

The Hamming distance between ${\tilde{v}}_{A 1 DR 1}$ and ${\tilde{v}}_{A 2 DR 1}$ , $\int_{R} | μ_{{\tilde{v}}_{A 2 DR 1}} (x) - μ_{{\tilde{v}}_{A 1 DR 1}} (x) |$ , is 0 for interval [– ∞, 0.33], |– 0.04| for interval [0.33, 0.5], |– 0.03| for the interval [0.5, 0.64], 0 for the interval 0.64, 0.67], 0 for the interval [0.67, 0.7], 0.05 for the interval [0.7, 0.83], 0.05 for the interval [0.83, 1] and 0 for the interval [1, ∞. The sum module of the integrals is:

|0 + 0.04 + 0.03 + 0+0 + 0.05 + 0.05 + 0|=0.17.

The distances shown in Table 18 are expressed as a preferred function for each pair of alternatives using Equation 29 and shown in Table 19. The fuzzy preference indexes shown in Table 20 are calculated using the preference function value and criteria weights (Equation 30), which in this case are the design requirements weights. Then fuzzy flows are determined for each alternative through Equation 31 and Equation 32 and shown in Table 21.

Table 18

The distance between two alternatives g and f with respect to each criterion

	X11	X21	X31		X12	X22	X32		X13	X23	X33
X11	–	(0.17;0)	(0;0)	X12	–	(0; 0.33)	(0.17;0)	X13	–	(0.33;0)	(0.33;0)
X21		–	(0; 0.17)	X22		–	(0.42;0)	X23		–	(0;0)
X31			–	X32			–	X33			–
	X14	X24	X34		X15	X25	X35		X16	X26	X36
X14	–	(0;0.28)	(0;0.42)	X15	–	(0;0.28)	(0;0.28)	X16	–	(0;0)	(0,33;0)
X24		–	(0; 0,17)	X25		–	(0;0)	X26		–	(0.33;0)
X34			–	X35			–	X36			–
	X17	X27	X37		X18	X28	X38		X19	X29	X39
X17	–	(0.42;0)	(0;0)	X18	–	(0;0)	(0;0)	X19	–	(0;0)	(0; 0.33)
X27		–	(0;0.42)	X28		–	(0;0)	X29		–	(0; 0.33)
X37			–	X38			–	X39			–

Table 19

The preference function in terms of distances

	P_j (a₁, a₂)	P_j (a₁, a₃)	P_j (a₂, a₁)	P_j (a₂, a₃)	P_j (a₃, a₁)	P_j (a₃, a₂)
DR1	0	0	0.17	0.17	0	0
DR2	0.33	0	0	0	0.17	0.42
DR3	0	0	0.33	0	0.33	0
DR4	0.28	0.42	0	0.17	0	0
DR5	0.28	0.28	0	0	0	0
DR6	0	0	0	0	0.33	0.33
DR7	0	0	0.42	0.42	0	0
DR8	0	0	0	0	0	0
DR9	0	0.33	0	0.33	0	0

Table 20

The fuzzy preference index

	A1	A2	A3
A1	–	(0.06,0.09,0.12,0.18)	(0.06,0.08,0.12,0.18)
A2	(0.06,0.09,0.14,0.21)	–	(0.06,0.09,0.14,0.21)
A3	(0.05,0.08,0.12,0.17)	(0.05,0.07,0.10,0.16)	–

Table 21

Fuzzy flows for each alternative

	Fuzzy leaving flows	Fuzzy entering flows
A1	(0.12,0.17,0.24,0.36)	(0.12,0.17,0.25,0.38)
A2	(0.12,0.19,0.28,0.42)	(0.11,0.16,0.23,0.34)
A3	(0.11,0.16,0.22,0.33)	(0.12,0.18,0.26,0.39)

The leaving and entering fuzzy flows are defuzzied by Equation (23) and the absolute values are shown in Table 22. To determine the classification of design alternatives, the net flows are calculated by using Equation (31) and the ranking result is shown in the last column of Table 22.

Table 22

Crisp flows for each alternative

	Leaving flows	Entering flows	Net flows	Ranking
A1	0.22	0.23	– 0.01	2nd
A2	0.26	0.21	0.04	1st
A3	0.20	0.24	– 0.03	3rd

4.3 Comparison with Traditional AHP-QFD-PROMETHEE and AHP-QFD-TOPSIS Hesitant Fuzzy methods

To check the robustness of the proposed method, the same problem data were applied to the traditional AHP-QFD-PROMETHEE method, which uses absolute values and the recent method proposed by Onar et al. [53], AHP-QFD-TOPSIS in the fuzzy hesitant environment. Table 23 shows the ranking result obtained by the traditional AHP-QFD-PROMETHEE method, i.e., using absolute values in all stages of the method.

Table 23
Crisp flows for each alternative (Traditional AHP-QFD-PROMETHEE)

Leaving flows Entering flows Net flows Ranking

A1 0.11649 0.11016 0.00632 2nd

A2 0.12181 0.10893 0.01288 1st

A3 0.10028 0.11948 – 0.01920 3rd

	Leaving flows	Entering flows	Net flows	Ranking
A1	0.11649	0.11016	0.00632	2nd
A2	0.12181	0.10893	0.01288	1st
A3	0.10028	0.11948	– 0.01920	3rd

Table 24 shows the ranking results obtained by the Hesitant fuzzy AHP-QFD-TOPSIS method. The application of Hesitant fuzzy AHP-QFD has already been performed in the case study (Stage 01 and 02), except for the application of Hesitant TOPSIS to rank the packaging design alternatives.

Table 24

Classification of alternatives using the Hesitant Fuzzy AHP-QFD-TOPSIS method

DRs	Weights	Design A1	Design A2	Design A3
DR1	(0.071,0.112,0.169,0.260)	(0.33,0.5,0.67,0.83)	(0.33,0.64,0.70,1)	(0.33,0.5,0.67,0.83)
DR2	(0.068, 0.096,0.135,0.200)	(0.5,0.67,0.83,1)	(0.33,0.5,0.67,0.83)	(0.5,0.85,1,1)
DR3	(0;064,0.093,0.133,0.196)	(0.17,0.33,0.5,0.67)	(0.33,0.5,0.67,0.83)	(0.33,0.5,0.67,0.83)
DR4	(0.084,0.124,0.180,0.269)	(0.5,0.85,1,1)	(0.33,0.64,0.70,1)	(0.33,0.5,0.67,0.83)
DR5	(0.050,0.071,0.101,0.146)	(0.5,0.85,1,1)	(0.33,0.64,0.70,1)	(0.33,0.64,0.70,1)
DR6	(0.068,0.103,0.147,0.221)	(0.17,0.33,0.5,0.67)	(0.17,0.33,0.5,0.67)	(0.33,0.5,0.67,0.83)
DR7	(0.066,0.105,0.158,0.240)	(0.33,0.5,0.67,0.83)	(0.5,0.85,1,1)	(0.33,0.5,0.67,0.83)
DR8	(0.060,0.091,0.134,0.201)	(0.5,0.85,1,1)	(0.5,0.85,1,1)	(0.5,0.85,1,1)
DR9	(0.023,0.034,0.048,0.071)	(0.5,0.67,0.83,1)	(0.5,0.67,0.83,1)	(0.33,0.5,0.67,0.83)
$d_{i}^{+}$		8.194	8.181	8.204
$d_{i}^{-}$		0.806	0.819	0.796
CC _i		0.090	0.091	0.088

The TOPSIS method considers that a preferred alternative is the one with the closest distance to the positive optimal solution and the furthest distance from the negative optimal solution.

A positive optimal solution consists of the best performance values of each criterion, while the negative optimal solution consists of the worst performance values.

The closeness coefficient determines the ranking order of the alternatives, as it calculates the distances of the alternatives to the positive and negative optimal solutions [55]. The distance of each alternative was calculated in relation to the positive optimal solution, ${\tilde{A}}^{+} = ({\tilde{v}}_{1}^{+}, \dots, {\tilde{v}}_{p}^{+})$ , and the negativa optimal solution, ${\tilde{A}}^{-} = ({\tilde{v}}_{1}^{-}, \dots, {\tilde{v}}_{p}^{-})$ , defined as ${\tilde{v}}_{i}^{+} = (1, 1, 1, 1)$ and ${\tilde{v}}_{i}^{+} = (0, 0, 0, 0)$ .

According to the closeness coefficient (CC) of alternatives, the ranking order of eight alternatives is determined as A2 >A1 >A3, as shown in Table 24.

It can be observed, that in this problem, there is no difference between the final solution produced by the three MCDA methods, Hesitant Fuzzy AHP-QFD-PROMETHEE, traditional AHP-QFD-PROMETHEE and Hesitant Fuzzy AHP-QFD-TOPSIS. This indicates that the classifications obtained by the proposed method are highly reliable for the purposes of classifying A1, A2 and A3 design alternatives.

Despite this, some comments can be made regarding these comparisons:

Comparison of the proposed hesitant fuzzy AHP-QFD-PROMETHEE method to the traditional AHP-QFD-PROMETHEE method.

In the application of the traditional AHP-QFD-PROMETHEE method absolute values were used. Each expert judged the alternatives and a simple average was obtained from that score. Although there was no difference between the rankings of the alternatives, it was possible to observe that in the application of the method in the hesitant fuzzy environment the difference of Net flows between the first place project (A2) and the second place (A1) was higher using HFLTS (0,056) than when using absolute values (0,005). Hence, it could be observed that in the hesitant fuzzy environment the superiority of A2 packing in relation to A1 was better evidenced.

Comparison of the proposed hesitant fuzzy AHP-QFD-PROMETHEE method to the hesitant fuzzy AHP-QFD-PROMETHEE method from Onar et al. [53].

The difference between these two methods is that in the proposed method a new type of proposal of hesitant fuzzy PROMETHEE method was used. Both in the proposed method and in method of Onar et al. [53] hesitant fuzzy linguistic ordered weighted average (HFLOWA) was used, proposed by Liu & Rodriguez [23], to aggregate hesitant linguistic terms sets.

Fuzzy TOPSIS is one of the most used MCDM fuzzy methods and it is less complex. Thus, it was of paramount importance to compare the proposed method to the hesitant fuzzy method of Onar et al. [53]. The ranking of the projects using both methods was the same, showing the effectiveness of the proposed method.

5 Conclusion

This article proposed a hybrid fuzzy MCDM method composed of AHP, QFD and PROMETHEE II methods that helps to reflect more objectively the hesitation and aggregate divergent opinions of evaluators at the time of the decision, when compared to other classical extensions of the fuzzy set. Additionally, the decision-making process has become more flexible, as decision makers can choose the most appropriate preferred function at the time of evaluating alternatives, with the inclusion of the hesitant fuzzy PROMETHEE method proposed in the article.

To illustrate its use, it was applied to a packaging design selection problem. Design selection is a multicriteria problem in a fuzzy environment, as experts often express their assessments using linguistic terms.

The solution of this selection process was supported by a QFD approach, considering the DRs associated to CRs. and the best alternative was determined through the PROMETHEE method. The proposed method allowed the analysis of vague and inaccurate relationships between CRs and DRs. The CR weights, obtained from the fuzzy hesitant AHP technique, led to selecting the best packaging design option.

The proposed hybrid method has been validated by comparing its results with those obtained by the traditional AHP-QFD-PROMETHEE, which uses absolute values, and the AHP-QFD-TOPSIS fuzzy hesitant. It is noted that the rankings of design alternatives were similar for all three methods, with the advantage of the proposed method allowing the decision-maker to choose one more linguistic term when expressing his or her opinion.

Future research may explore the allocation of weights for expert opinion, as well as the possibility of applications in other decision problems. Other preferred functions may also be used in the application step of the PROMETHEE II method.

Another interesting possibility for future work might be using other aggregation operators instead of the OWA operator, such as Triangular Fuzzy Hesitant Weighted Averaging (TFHFWA) or Triangular Fuzzy Hesitant Weighted Geometric (TFHFWG). Finally, the use of intuitionistic fuzzy sets is suggested in the proposed hybrid MCDM method.

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, Distance and similarity measures for hesitant fuzzy sets, Information Sciences 181(11) (2011), 2128–2138.

New hybrid AHP-QFD-PROMETHEE decision-making support method in the hesitant fuzzy environment: an application in packaging design selection

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Hesitant fuzzy sets and HFLTS

3 Methodology

Table 2 Linguistic scale for correlations Linguistic term Abb. Triangular fuzzy number Absolutely low AL (1,2,3) Very low VL (2,3,4) Low L (3,4,5) Medium M (4,5,6) High H (5,6,7) Very high VH (6,7,8) Absolutely high AH (7,8,9)

Table 23 Crisp flows for each alternative (Traditional AHP-QFD-PROMETHEE) Leaving flows Entering flows Net flows Ranking A1 0.11649 0.11016 0.00632 2nd A2 0.12181 0.10893 0.01288 1st A3 0.10028 0.11948 – 0.01920 3rd

References

Table 2
Linguistic scale for correlations

Linguistic term Abb. Triangular fuzzy number

Absolutely low AL (1,2,3)

Very low VL (2,3,4)

Low L (3,4,5)

Medium M (4,5,6)

High H (5,6,7)

Very high VH (6,7,8)

Absolutely high AH (7,8,9)

Table 23
Crisp flows for each alternative (Traditional AHP-QFD-PROMETHEE)

Leaving flows Entering flows Net flows Ranking

A1 0.11649 0.11016 0.00632 2nd

A2 0.12181 0.10893 0.01288 1st

A3 0.10028 0.11948 – 0.01920 3rd