An integrated group decision-making method for brand packaging design effect evaluation based on the 2-tuple linguistic Pythagorean fuzzy sets

Abstract

In the current era of economic and cultural globalization, the demand for packaging design is increasing, and the packaging design requirements for brands and products in the entire consumer market are becoming increasingly strict and refined. Designers are facing one greater challenge after another. Packaging design, as a comprehensive discipline that integrates commerce, culture, and art, diverse cultural forms and elements will run through the entire process of brand packaging design. Good products need to be sold through good channels in order to achieve greater value. As a bridge connecting products and the market, the role of packaging design is even more evident. It can not only improve the recognition of products, but also help enterprises stand out among many similar products, thereby increasing product value. The brand packaging design effect evaluation is a classical multiple attribute group decision making (MAGDM) problem. This paper extends the EDAS method to the 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs). On the basis of the original EDAS method, 2-tuple linguistic Pythagorean fuzzy EDAS (2TLPF-EDAS) method is built for MAGDM. Finally, a case study for brand packaging design effect evaluation show that the new method proposed in this paper is reasonable.

Keywords

Multiple attribute group decision making (MAGDM)2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs)EDAS method brand packaging design effect evaluation

1 Introduction

Decision making is a process of action from a multitude of alternatives. Multi attribute decision-making (MADM), also known as finite scheme multi-objective decision-making, refers to the decision-making problem of selecting the optimal alternative solution or ranking solutions considering multiple attributes [1 –3]. It is an important component of modern decision-making science. Its theory and methods are widely applied in many fields such as engineering, technology, economy, management, and military. Then, Atanassov [4] defined the intuitionistic fuzzy sets(IFSs). Yager [5] defined Pythagorean fuzzy sets (PHFSs). Then, the methods related to the MAGDM have been fully applied to PHFSs, for example, EDAS method [6 –8], PRETHEE method [9], GRA method [10], MULTIMOORA method [11]. Furthermore, Deng et al. [12] proposed the 2-tuple linguistic Pythagorean fuzzy sets (2TLPFSs) based on Hamy Mean operators [13 –18]. Deng et al. [19] proposed the weighted BM (WBM) operator, generalized WBM (GWBM) operator and dual GWBM operator under 2TLPFSs based on the classical BM operator [20 –25]. Deng and Gao [26] proposed the TODIM method under 2TLPFSs for MADM. He et al. [27] proposed the CODAS method under 2TLPFSs for MAGDM. Deng et al. [28] proposed some Muirhead Mean (MM) operators [29 –36] under 2TLPFSs.

Whether it is packaging design or brand image, the ultimate attraction is always consumers. To shape a successful brand image, it is necessary to achieve a transformation from quantity to quality. The success of a brand comes from the decisions of consumers or service recipients. The formation of a brand is achieved through continuous innovation in stable development [37, 38]. At a certain stage, it is necessary to obtain innovative and functional packaging design based on the original basic functions of the packaging [39, 40]. This has gradually become an important tool for shaping a good brand image, and is an important part of establishing a good and deeply rooted brand image and identity. Package design is an important channel that connects brands and end users [41, 42]. It is not only the main sales method for enterprise products, but also the necessary support for establishing a unified visual image of the brand through good packaging design, reflecting the standard colors, text, graphics, and other characteristics of the enterprise. Promoting a unified visual image of enterprise products is beneficial for distinguishing between brands, enabling consumers to have a deeper understanding of the brand and corporate image, and to a certain extent, to have a sense of identity [43, 44]. This promotes enterprises to highlight their own characteristics in competitive products, and forms scale on this basis. Secondly, packaging design has its own unique expression form and contains unique creative concepts, which can effectively strengthen the brand personality of the product and provide support for the brand to achieve accurate market positioning. A successful packaging design should not only have a visually appealing appearance, but also have a memorable brand impression for consumers [45, 46]. The meaning of the images on the packaging and the brand name need to complement each other, and the uniqueness of packaging design can effectively provide an important foundation for the success of brand personality highlighting. Brands can still be recognized and representative by society to form a culture, which requires recognizing that the visual image of every successful product brand always relies on the characteristics of creative overall packaging visual design [47, 48]. While strengthening the strength of brand personality cultivation, it meets the needs of brand image shaping, making the brand image deeply rooted in people’s hearts. Once again, packaging design deepens the brand image into people’s hearts. As people’s living standards continue to improve, the requirements for natural quality also need to be significantly elevated [49, 50]. Therefore, packaging design plays an important role in the development process of homogenization of this product. In different uses and grades, and when observing different objects, it is also necessary to strengthen the serialization and matching design of packaging design, provide support to avoid being eliminated by brands in the same industry [51, 52]. Finally, with the global warming and frequent occurrence of natural disasters, China needs to fully reflect the characteristics of green packaging design when carrying out brand design, helping people realize the importance of protecting nature and coexisting harmoniously with nature. This has also become a successful brand trend with directional characteristics. On this basis, in order to be able to design a more mature corporate brand, it is necessary to permeate the meaning and values of brand culture into different aspects of company behavior, which are usually reflected in corporate responsibility and the moral and ethical values it represents. In the information society, the development of high-tech has a greater impact on the current market order. The load capacity of the natural environment is constantly increasing, and environmental protection is also an important issue of global concern [38, 53]. The important role of packaging design in shaping the visual brand image of enterprises has also received attention and attention. During this period, brand images with new environmental concepts will also be widely concerned by consumers, Enterprises should also actively face new challenges in packaging design, lay an important foundation for brand building, gain consumer favor, attract consumer attention, effectively enhance market competitiveness, and enable sustainable development of the enterprise [37, 54]. In short, on the basis of ensuring the effectiveness of visual brand image, it is necessary to deeply highlight the role of brand packaging design. In packaging design, design works that have more brand characteristics to ensure that the brand image is more deeply rooted in people’s hearts [55, 56].

The brand packaging design effect evaluation is a classical MAGDM problem. This paper extends the EDAS method to the 2TLPFSs. On the basis of the classical EDAS method [6], 2-tuple linguistic Pythagorean fuzzy EDAS (2TLPF-EDAS) method is proposed for MAGDM. Finally, a case study for brand packaging design effect evaluation show that the new method proposed in this paper is reasonable. In order to realize the research purpose, the research framework of this paper is given. In the Sect. 2, we introduce the 2TLPFSs. In the Sect. 3, the new 2TLPF-EDAS method in built for MAGDM. In the Sect. 4, by a numerical case about brand packaging design effect evaluation is given. In Sect. 5, we get some conclusion.

2 Preliminary knowledge

Deng et al. [12] defined the 2TLPFSs.

Definition 1 [57, 58]. Let pλ ={ pλ₁, pλ₂, …, pλ_T } be a LTSs and pλ is defined:

$p λ = {\begin{matrix} p λ_{0} = extremely poor, p λ_{1} = very poor, p λ_{2} = poor, p λ_{3} = medium, \\ p λ_{4} = good, p λ_{5} = very good, p λ_{6} = extremely good . \end{matrix}}$ (1)

Definition 2 [12]. Let pδ_j (j = 1, 2, …, n) be a 2TLSs. If pδ =〈 (pl_t, pξ) , (pl_f, pζ) 〉 is defined for pl_t, , pl_f ∈ pλ, pξ, pζ ∈ [- 0.5, 0.5) , (pl_t, pξ) , (pl_f, pζ) depict the truth-membership and falsity-membership through 2TLSs, then the 2TLPFSs are defined: $p δ_{j} = 〈 ({pl}_{t_{j}}, p ξ_{j}), ({pl}_{f_{j}}, p ζ_{j}) 〉$ (2) where 0 ⩽ Δ^-1 (pl_{t
_j}, pξ_j) ⩽ T, 0 ⩽ Δ^-1 (pl_{f
_j}, pζ_j) ⩽ T, and $0 ⩽ {(\frac{Δ^{- 1} ({pl}_{t_{j}}, p ξ_{j})}{T})}^{2} + {(\frac{Δ^{- 1} ({pl}_{f_{j}}, p ζ_{j})}{T})}^{2} ⩽ 1 .$

Definition 3 [12]. Let pδ =〈 (pl_t, pξ) , (pl_f, pζ) 〉, the score function SF (pδ) and accuracy function AF (pδ) are defined:

$\begin{matrix} SF (p δ) \\ = Δ (\frac{T (1 + {(\frac{Δ^{- 1} ({pl}_{t_{j}}, p ξ_{j})}{T})}^{2} - {(\frac{Δ^{- 1} ({pl}_{f_{j}}, p ζ_{j})}{T})}^{2})}{2}) \end{matrix}$ (3)

$\begin{matrix} AF (p δ) \\ = Δ (T ({(\frac{Δ^{- 1} ({pl}_{t_{j}}, p ξ_{j})}{T})}^{2} + {(\frac{Δ^{- 1} ({pl}_{f_{j}}, p ζ_{j})}{T})}^{2})) \end{matrix}$ (4)

Definition 4 [12]. Let pδ₁ =〈 (pl_{t
₁}, pξ₁) , (pl_{f
₁}, pζ₁) 〉 and pδ₂ =〈 (pl_{t
₂}, pξ₂) , (pl_{f
₂}, pζ₂) 〉, then (1) If SF (pδ₁) < SF (pδ₂), pδ₁ < pδ₂; (2) If SF (pδ₁) = SF (pδ₂) , AF (pδ₁) < AF (pδ₂), pδ₁< pδ₂ ; (3) If SF (pδ₁) = SF (pδ₂) , AF (pδ₁) = AF (pδ₂), pδ₁ = pδ₂ .

Definition 5 [12]. Let pδ₁ =〈 (pl_{t
₁}, pξ₁) , (pl_{f
₁}, pζ₁) 〉, pδ₂ =〈 (pl_{t
₂}, pξ₂) , (pl_{f
₂}, pζ₂) 〉, pδ =〈 (pl_t, pξ) , (pl_f, pζ) 〉, then, the operational laws are defined:

$p δ_{1} \oplus p δ_{2} = {\begin{matrix} Δ (T \sqrt{1 - (1 - {(\frac{Δ^{- 1} ({pl}_{t_{1}}, p ξ_{1})}{T})}^{2}) (1 - {(\frac{Δ^{- 1} ({pl}_{t_{2}}, p ξ_{2})}{T})}^{2})}) \\ Δ (T (\frac{Δ^{- 1} ({pl}_{f_{1}}, p ζ_{1})}{T}) (\frac{Δ^{- 1} ({pl}_{f_{2}}, p ζ_{2})}{T})) \end{matrix}};$ (5)

$p δ_{1} \otimes p δ_{2} = {\begin{matrix} Δ (T (\frac{Δ^{- 1} ({pl}_{t_{1}}, p ξ_{1})}{T}) (\frac{Δ^{- 1} ({pl}_{t_{2}}, p ξ_{2})}{T})) \\ Δ (T \sqrt{1 - (1 - {(\frac{Δ^{- 1} ({pl}_{f_{1}}, p ζ_{1})}{T})}^{2}) (1 - {(\frac{Δ^{- 1} ({pl}_{f_{2}}, p ζ_{2})}{T})}^{2})}) \end{matrix}}$ (6)

$α p δ = {Δ (T \sqrt{1 - {(1 - {(\frac{Δ^{- 1} ({pl}_{t}, p ξ)}{T})}^{2})}^{a}}) Δ (T {(\frac{Δ^{- 1} ({pl}_{f}, p ζ)}{T})}^{a})}, α > 0;$ (7)

$p δ^{α} = {Δ (T {(\frac{Δ^{- 1} ({pl}_{t}, p ξ)}{T})}^{a}), Δ (T \sqrt{1 - {(1 - {(\frac{Δ^{- 1} ({pl}_{f}, p ζ)}{T})}^{2})}^{a}})}, α > 0 .$ (8)

Definition 6 [26]. If pδ₁ =〈 (pl_{t
₁}, pξ₁) , (pl_{f
₁}, pζ₁) 〉 and pδ₂ =〈 (pl_{t
₂}, pξ₂) , (pl_{f
₂}, pζ₂) 〉, the Hamming distance between pδ₁ =〈 (pl_{t
₁}, pξ₁) , (pl_{f
₁}, pζ₁) 〉 and pδ₂ =〈 (pl_{t
₂}, pξ₂) , (pl_{f
₂}, pζ₂) 〉 is defined:

$\begin{matrix} HD (p δ, p δ_{2}) \\ = (\frac{1}{2} | \frac{Δ^{- 1} ({pl}_{t_{1}}, p ξ_{1})}{T} - \frac{Δ^{- 1} ({pl}_{f_{1}}, p ζ_{1})}{T} |) \end{matrix}$ (9)

Definition 7 [59]. Let pδ_j =〈 (pl_{t
_j}, pξ_j) , (pl_{f
_j}, pζ_j) 〉, the 2TLPFWG operator is defined:

$\begin{matrix} 2 TLPFWG (p δ_{1}, p δ_{2}, \dots, p δ_{n}) \\ = {(p δ_{1})}^{{wp}_{1}} \otimes {(p δ_{2})}^{{wp}_{2}} \otimes \dots \otimes {(p δ_{n})}^{{wp}_{n}} = \otimes_{j = 1}^{n} {(p δ_{j})}^{{wp}_{j}} \\ = {Δ (T \prod_{j = 1}^{n} {(\frac{Δ^{- 1} ({pl}_{t_{j}}, p ξ_{j})}{T})}^{{wp}_{j}}), Δ (T \sqrt{1 - \prod_{j = 1}^{n} {(1 - {(\frac{Δ^{- 1} ({pl}_{f_{j}}, p ζ_{j})}{T})}^{2})}^{{wp}_{j}}})} \end{matrix}$ (10)

where wp_j is weight of pδ_j, $0 ⩽ {wp}_{j} ⩽ 1, \sum_{j = 1}^{n} {wp}_{j} = 1 .$

3 EDAS method for MAGDM with 2TLPFSs

The 2TLPF-EDAS method is used to solve the MAGDM with 2TLPFSs. Suppose that {PA₁, PA₂, … , PA_m } is the alternatives, {PO₁, PO₂, … , PO_n } is the attributes, {PD₁, PD₂, … , PD_t } the experts. Let call (wp₁, wp₂, ⋯ , wp_n) for the weight of attributes and {dζ₁, dζ₂, ⋯ , dζ_t } for the weight of experts. Then, the evaluation matrix $PR = {[p φ_{ij}^{t}]}_{m \times n} = {({pl}_{t_{ij}}^{t}, p ξ_{ij}^{t}), ({pl}_{f_{ij}}^{t}, p ζ_{ij}^{t})}_{m \times n}$ are constructed. Based on the 2TLPF-EDAS, the calculating steps for MAGDM are built as follows.

Step 1. According to the experts’ weight, the overall matrix $PR = {[p φ_{ij}]}_{m \times n}$ is derived based on the 2TLPFWG operator.

$\begin{matrix} p φ_{ij} = {({pl}_{t ij}, p ξ_{ij}), ({pl}_{f_{ij}}, p ζ_{ij})} \\ = 2 TLPFWG (p δ_{ij}^{1}, p δ_{ij}^{2}, \dots, p δ_{ij}^{t}) \\ = {(p δ_{ij}^{1})}^{d ζ_{1}} \otimes {(p δ_{ij}^{2})}^{d ζ_{2}} \otimes \dots \otimes {(p δ_{ij}^{t})}^{d ζ_{t}} = \otimes_{d = 1}^{t} {(p φ_{ij}^{d})}^{d ζ_{d}} \\ = {\begin{matrix} Δ (T \prod_{d = 1}^{t} {(\frac{Δ^{- 1} ({pl}_{t_{ij}}, p ξ_{ij})}{T})}^{d ζ_{d}}) \\ Δ (T \sqrt{1 - \prod_{d = 1}^{t} {(1 - {(\frac{Δ^{- 1} ({pl}_{f_{ij}}, p ζ_{ij})}{T})}^{2})}^{d ζ_{d}}}) \end{matrix}} \end{matrix}$ (11)

Step 2. Normalize the overall matrix from $PR = {[p φ_{ij}]}_{m \times n} = {({pl}_{t_{ij}}, p ξ_{ij}), ({pl}_{f_{ij}}, p ζ_{ij})}_{m \times n}$ to $NR = {[np φ_{ij}]}_{m \times n} = {({npl}_{t_{ij}}, np ξ_{ij}), {({npl}_{f_{ij}}, np ζ_{ij})}}_{m \times n}$ . For benefit decision attribute: ${[np φ_{ij}]}_{m \times n} = {[p φ_{ij}]}_{m \times n} = {({pl}_{t_{ij}}, p ξ_{ij}), ({pl}_{f_{ij}}, p ζ_{ij})}_{m \times n}$ . For cost decision attribute: ${[np φ_{ij}]}_{m \times n} = {({pl}_{f_{ij}}, p ζ_{ij}), ({pl}_{t_{ij}}, p ξ_{ij})}_{m \times n}$ .

Step 3. Obtain the weight information through the entropy weight method.

Firstly, we normalize decision matrix. ${NAF}_{ij} = \frac{AF {({npl}_{t_{ij}}, np ξ_{ij}), ({npl}_{f_{ij}}, np ζ_{ij})}}{\sum_{i = 1}^{m} AF {({npl}_{t_{ij}}, np ξ_{ij}), ({npl}_{f_{ij}}, np ζ_{ij})}}$ (12)

Then, compute the degree of entropy. ${SE}_{j} = - \frac{1}{ln m} \sum_{i = 1}^{m} {NAF}_{ij} ln {NAF}_{ij}$ (13)

where, if NAF_ij = 0, let ln NAF_ij = 0.

Finally, we can get the entropy weight. ${wp}_{j} = \frac{(1 - {SE}_{j})}{\sum_{j = 1}^{n} (1 - {SE}_{j})}$ (14)

Step 4. Calculate the average solution of each attribute.

$\begin{matrix} {AV}_{j} = {({apl}_{t_{ij}}, ap ξ_{ij}), ({apl}_{f_{ij}}, ap ζ_{ij})} \\ = \frac{\oplus_{i = 1}^{m} {({npl}_{t_{ij}}, np ξ_{ij}), ({npl}_{f_{ij}}, np ζ_{ij})}}{m} \\ = {Δ (T \sqrt{1 - \prod_{i = 1}^{m} {(1 - {(\frac{Δ^{- 1} ({npl}_{t_{ij}}, np ξ_{ij})}{T})}^{2})}^{\frac{1}{m}}}) \\ Δ (T \prod_{i = 1}^{m} {(\frac{Δ^{- 1} ({npl}_{f_{ij}}, np ζ_{ij})}{T})}^{\frac{1}{m}})} \end{matrix}$ (15)

Step 5. Obtain the negative ideal solution (NIS) with 2TLPFSs: ${NIS}_{j} = {({pl}_{t_{j}}^{-}, p ξ_{j}^{-}), ({pl}_{f_{j}}^{-}, p ζ_{j}^{-})}$ (16) $SF ({NIS}_{j}) = min_{i} SF {({pl}_{t_{ij}}, p ξ_{ij}), ({pl}_{f_{ij}}, p ζ_{ij})}$ (17)

Step 6. Calculate the PDA and NDA under 2TLPFSs.

For the positive attributes,

${PDA}_{ij} = \frac{max (0, HD (np φ_{ij}, {NIS}_{j}) - HD ({AV}_{j}, {NIS}_{j}))}{HD (np φ_{ij}, {NIS}_{j})}$ (18) ${NDA}_{ij} = \frac{max (0, HD ({AV}_{j}, {NIS}_{j}) - HD (np φ_{ij}, {NIS}_{j}))}{HD (np φ_{ij}, {NIS}_{j})}$ (19)

For the negative attributes,

${PDA}_{ij} = \frac{max (0, HD ({AV}_{j}, {NIS}_{j}) - HD (np φ_{ij}, {NIS}_{j}))}{HD (np φ_{ij}, {NIS}_{j})},$ (20) ${NDA}_{ij} = \frac{max (0, HD (np φ_{ij}, {NIS}_{j}) - HD ({AV}_{j}, {NIS}_{j}))}{HD (np φ_{ij}, {NIS}_{j})},$ (21)

Step 7. Based on the weight, the weighted PDA and NDA is obtained.

${NSP}_{i} = \sum_{j = 1}^{n} {PDA}_{ij} \cdot {wp}_{j}, i = 1, 2, \dots, m$ (22) ${NSN}_{i} = \sum_{j = 1}^{n} {NDA}_{ij} \cdot {wp}_{j}, i = 1, 2, \dots, m$ (23)

Step 8. Compute the value of the weighted normalized PDA and NDA. ${NNSP}_{i} = \frac{{NSP}_{i}}{max_{i} ({NSP}_{i})}, i = 1, 2, \dots, m$ (24) ${NNSN}_{i} = 1 - \frac{{NSN}_{i}}{max_{i} ({NSN}_{i})}, i = 1, 2, \dots, m$ (25)

Step 9. On the basics of ${NNSP}_{i} = \frac{{NSP}_{i}}{max_{i} ({NSP}_{i})}$ and ${NNSN}_{i} = 1 - \frac{{NSN}_{i}}{max_{i} ({NSN}_{i})}$ , the scores of AS_i is obtained. ${AS}_{i} = \frac{1}{2} ({NNSP}_{i} + {NNSN}_{i}), i = 1, 2, \dots, m$ (26)

Step 10. Based on the values of AS_i, rank the final scores and choose the best alternative.

4 Numerical example and comparative analysis

4.1 Numerical example

Appropriate product packaging design can effectively enhance product display opportunities, increase product sales, and bring more economic benefits to enterprises. It is precisely because of this that businesses are currently paying more attention to product packaging design. But is your product packaging design effective? In order to achieve a more significant effect in packaging design, the following principles should be followed: (1) The simple and concise packaging design style has been popular for many years, and it is still very popular in the packaging market today. Simple packaging design can better convey important information about products, allowing consumers to easily find the products they need through product design among numerous products. If the packaging design content of a product is too complex, consumers may have looked at it for a long time without knowing what the packaging design is intended to convey, and they are likely to give up choosing the product. (2) When designing product packaging, it is important to pay attention to the selection of colors, as those with low saturation will give people a sincere and warm feeling, and visually give consumers a pleasant impression. Therefore, in packaging design, it is necessary to avoid using overly bright colors. (3) The main purpose of packaging products with movie posters is to make them stand out among similar products. You can try designing product packaging in the form of movie posters, which is also a way to attract consumers’ attention. The form of movie posters can effectively highlight the theme of the product and convey the main information of the product to consumers. (4) Is the core message you want to convey highlighted in large font in your packaging design? It should be noted that the large font on the packaging can effectively attract consumers’ attention and allow them to see the characteristics of the product in the first place. So the important information in packaging design can be bolded or paired with different colors, and I believe your product will stand out among similar products. (5) When designing packaging for products, it is best to choose a creative design. For example, if we want to package a certain food, we can place the food in a tree hole or fruit juice in a bamboo joint on the packaging. This is a very correct design, and the unique packaging design will become a major selling point of the product. The brand packaging design effect evaluation is a MAGDM. In this section, we provide a numerical example for brand packaging design effect evaluation by using 2TLPF-EDAS method. Assume that five possible brand packaging design schemes PA_i (i = 1, 2, 3, 4, 5) to be selected and four attributes PO_j (j = 1, 2, 3, 4) to assess the brand packaging design schemes:

PO₁ conveys the visual attraction effect. Visual attraction effect refers to the effect of packaging on attracting customers’ attention. As long as customers notice the existence of the product, they can further observe the product and make purchasing decisions based on the information obtained. Visual attractiveness is the primary condition for evaluating packaging design. Only with good attractiveness can a product stand out on the shelf, achieve further sales goals, and convey brand information.

PO₂ conveys the information communication effect. The information conveyed in packaging design can generally be divided into two types. One type is to clarify information, such as practical information such as product quantity, specifications, and application explanations. This type of information is mainly used to help users use the product correctly. The second type is promotional information, such as product features, advertising slogans, brand positioning, and so on. The function of this type of information is the function of shopping malls, such as stimulating users to purchase, strengthening brand information, etc. The information on a good packaging should be graded, meaning there are differences in importance.

PO₃ conveys the artistic and aesthetic effects. Artistic and aesthetic effects refer to the appearance of packaging that should bring visual pleasure to users and have a certain aesthetic feeling. In some packaging plans, visual elements such as text, graphics, and logos are often inappropriately highlighted in order to emphasize promotion or brand information. Perhaps all product information that can be placed is piled up on the packaging, leading to a decrease in packaging quality and leaving users with a vulgar and cheap image.

PO₄ conveys the functional effect. The functional effect mainly refers to the examination of the practical function of packaging. The most basic function of product packaging is to protect and use the product. In general, packaging should meet the following application functions, including product protection, easy transportation and storage, convenient carrying and use, safety, and environmental protection.

The brand packaging design effect evaluation is depicted with 2TLPFSs by DMs by using four attributes PO_j (j = 1, 2, 3, 4), The five brand packaging design schemes PA_i (i = 1, 2, 3, 4, 5) will be evaluated with 2TLPFSs with the four criteria by three experts PD_t (t = 1, 2, 3) (Suppose expert’s weight is (0.27, 0.34, 0.39). The evaluation matrix is depicted in Tables 1 –3.

Table 1
The evaluation matrix PD¹

PO ₁ PO ₂ PO ₃ PO ₄

PA ₁ { (pλ₁, 0) , (pλ₃, 0)} { (pλ₄, 0) , (pλ₃, 0)} { (pλ₂, 0) , (pλ₁, 0)} { (pλ₃, 0) , (pλ₄, 0)}

PA ₂ { (pλ₃, 0) , (pλ₁, 0)} { (pλ₂, 0) , (pλ₄, 0)} { (pλ₁, 0) , (pλ₃, 0)} { (pλ₄, 0) , (pλ₂, 0)}

PA ₃ { (pλ₄, 0) , (pλ₂, 0)} { (pλ₁, 0) , (pλ₃, 0)} { (pλ₂, 0) , (pλ₄, 0)} { (pλ₃, 0) , (pλ₁, 0)}

PA ₄ { (pλ₂, 0) , (pλ₄, 0)} { (pλ₁, 0) , (pλ₂, 0)} { (pλ₄, 0) , (pλ₃, 0)} { (pλ₂, 0) , (pλ₄, 0)}

PA ₅ { (pλ₄, 0) , (pλ₃, 0)} { (pλ₂, 0) , (pλ₄, 0)} { (pλ₃, 0) , (pλ₁, 0)} { (pλ₃, 0) , (pλ₂, 0)}

	PO ₁	PO ₂	PO ₃	PO ₄
PA ₁	{ (pλ₁, 0) , (pλ₃, 0)}	{ (pλ₄, 0) , (pλ₃, 0)}	{ (pλ₂, 0) , (pλ₁, 0)}	{ (pλ₃, 0) , (pλ₄, 0)}
PA ₂	{ (pλ₃, 0) , (pλ₁, 0)}	{ (pλ₂, 0) , (pλ₄, 0)}	{ (pλ₁, 0) , (pλ₃, 0)}	{ (pλ₄, 0) , (pλ₂, 0)}
PA ₃	{ (pλ₄, 0) , (pλ₂, 0)}	{ (pλ₁, 0) , (pλ₃, 0)}	{ (pλ₂, 0) , (pλ₄, 0)}	{ (pλ₃, 0) , (pλ₁, 0)}
PA ₄	{ (pλ₂, 0) , (pλ₄, 0)}	{ (pλ₁, 0) , (pλ₂, 0)}	{ (pλ₄, 0) , (pλ₃, 0)}	{ (pλ₂, 0) , (pλ₄, 0)}
PA ₅	{ (pλ₄, 0) , (pλ₃, 0)}	{ (pλ₂, 0) , (pλ₄, 0)}	{ (pλ₃, 0) , (pλ₁, 0)}	{ (pλ₃, 0) , (pλ₂, 0)}

Table 2

The evaluation matrix PD²

	PO ₁	PO ₂	PO ₃	PO ₄
PA ₁	{ (pλ₄, 0) , (pλ₃, 0)}	{ (pλ₁, 0) , (pλ₃, 0)}	{ (pλ₂, 0) , (pλ₄, 0)}	{ (pλ₂, 0) , (pλ₃, 0)}
PA ₂	{ (pλ₃, 0) , (pλ₁, 0)}	{ (pλ₃, 0) , (pλ₄, 0)}	{ (pλ₂, 0) , (pλ₁, 0)}	{ (pλ₄, 0) , (pλ₂, 0)}
PA ₃	{ (pλ₁, 0) , (pλ₂, 0)}	{ (pλ₂, 0) , (pλ₄, 0)}	{ (pλ₃, 0) , (pλ₁, 0)}	{ (pλ₄, 0) , (pλ₃, 0)}
PA ₄	{ (pλ₃, 0) , (pλ₁, 0)}	{ (pλ₃, 0) , (pλ₂, 0)}	{ (pλ₄, 0) , (pλ₂, 0)}	{ (pλ₁, 0) , (pλ₂, 0)}
PA ₅	{ (pλ₂, 0) , (pλ₄, 0)}	{ (pλ₄, 0) , (pλ₁, 0)}	{ (pλ₂, 0) , (pλ₃, 0)}	{ (pλ₃, 0) , (pλ₄, 0)}

Table 3

The evaluation matrix PD³

	PO ₁	PO ₂	PO ₃	PO ₄
PA ₁	{ (pλ₂, 0) , (pλ₄, 0)}	{ (pλ₃, 0) , (pλ₄, 0)}	{ (pλ₂, 0) , (pλ₁, 0)}	{ (pλ₄, 0) , (pλ₁, 0)}
PA ₂	{ (pλ₁, 0) , (pλ₂, 0)}	{ (pλ₂, 0) , (pλ₄, 0)}	{ (pλ₃, 0) , (pλ₄, 0)}	{ (pλ₂, 0) , (pλ₁, 0)}
PA ₃	{ (pλ₂, 0) , (pλ₃, 0)}	{ (pλ₄, 0) , (pλ₁, 0)}	{ (pλ₂, 0) , (pλ₁, 0)}	{ (pλ₄, 0) , (pλ₃, 0)}
PA ₄	{ (pλ₄, 0) , (pλ₂, 0)}	{ (pλ₃, 0) , (pλ₂, 0)}	{ (pλ₂, 0) , (pλ₁, 0)}	{ (pλ₃, 0) , (pλ₄, 0)}
PA ₅	{ (pλ₂, 0) , (pλ₃, 0)}	{ (pλ₁, 0) , (pλ₂, 0)}	{ (pλ₃, 0) , (pλ₄, 0)}	{ (pλ₁, 0) , (pλ₃, 0)}

The 2TLPF-EDAS method is used to solve the brand packaging design effect evaluation.

Step 1. The overall matrix is derived through 2TLPFWG operator (See Table 4).

Table 4

The overall evaluation matrix

	PO ₁	PO ₂
PA ₁	${\begin{matrix} (p λ_{2}, 0.1228), \\ (p λ_{1}, 0.2143) \end{matrix}}$	${\begin{matrix} (p λ_{3}, 0.3158), \\ (p λ_{2}, 0.5124) \end{matrix}}$
PA ₂	${\begin{matrix} (p λ_{1}, 0.4124), \\ (p λ_{1}, 0.4276) \end{matrix}}$	${\begin{matrix} (p λ_{3}, 0.3372), \\ (p λ_{2}, 0.4175) \end{matrix}}$
PA ₃	${\begin{matrix} (p λ_{3}, 0.3435), \\ (p λ_{3}, 0.0121) \end{matrix}}$	${\begin{matrix} (p λ_{1}, 0.3963), \\ (p λ_{1}, 0.4306) \end{matrix}}$
PA ₄	${\begin{matrix} (p λ_{2}, 0.4621), \\ (p λ_{2}, 0.3124) \end{matrix}}$	${\begin{matrix} (p λ_{3}, 0.1437), \\ (p λ_{2}, 0.4214) \end{matrix}}$
PA ₅	${\begin{matrix} (p λ_{3}, 0.3115), \\ (p λ_{2}, 0.4103) \end{matrix}}$	${\begin{matrix} (p λ_{2}, 0.1136), \\ (p λ_{1}, 0.4021) \end{matrix}}$
	PO ₃	PO ₄
PA ₁	${\begin{matrix} (p λ_{2}, 0.3901), \\ (p λ_{1}, 0.2032) \end{matrix}}$	${\begin{matrix} (p λ_{3}, 0.3142), \\ (p λ_{1}, 0.4108) \end{matrix}}$
PA ₂	${\begin{matrix} (p λ_{1}, 0.2078), \\ (p λ_{2}, 0.0215) \end{matrix}}$	${\begin{matrix} (p λ_{2}, 0.3471), \\ (p λ_{2}, 0.3025) \end{matrix}}$
PA ₃	${\begin{matrix} (p λ_{2}, 0.1036), \\ (p λ_{2}, 0.3371) \end{matrix}}$	${\begin{matrix} (p λ_{2}, 0.1102), \\ (p λ_{1}, 0.4326) \end{matrix}}$
PA ₄	${\begin{matrix} (p λ_{3}, 0.2014), \\ (p λ_{2}, 0.3015) \end{matrix}}$	${\begin{matrix} (p λ_{3}, 0.2015), \\ (p λ_{1}, 0.3017) \end{matrix}}$
PA ₅	${\begin{matrix} (p λ_{2}, 0.3106), \\ (p λ_{2}, 0.2215) \end{matrix}}$	${\begin{matrix} (p λ_{1}, 0.3502), \\ (p λ_{2}, 0.4045) \end{matrix}}$

Step 2. Normalize the overall matrix. All attributes are beneficial, thus, $NR = {[np φ_{ij}]}_{5 \times 4}$ is same to $PR = {[p φ_{ij}]}_{5 \times 4}$ (See Table 5).

Table 5

The normalized overall matrix

	PO ₁	PO ₂
PA ₁	${\begin{matrix} (p λ_{2}, 0.1228), \\ (p λ_{1}, 0.2143) \end{matrix}}$	${\begin{matrix} (p λ_{3}, 0.3158), \\ (p λ_{2}, 0.5124) \end{matrix}}$
PA ₂	${\begin{matrix} (p λ_{1}, 0.4124), \\ (p λ_{1}, 0.4276) \end{matrix}}$	${\begin{matrix} (p λ_{3}, 0.3372), \\ (p λ_{2}, 0.4175) \end{matrix}}$
PA ₃	${\begin{matrix} (p λ_{3}, 0.3435), \\ (p λ_{3}, 0.0121) \end{matrix}}$	${\begin{matrix} (p λ_{1}, 0.3963), \\ (p λ_{1}, 0.4306) \end{matrix}}$
PA ₄	${\begin{matrix} (p λ_{2}, 0.4621), \\ (p λ_{2}, 0.3124) \end{matrix}}$	${\begin{matrix} (p λ_{3}, 0.1437), \\ (p λ_{2}, 0.4214) \end{matrix}}$
PA ₅	${\begin{matrix} (p λ_{3}, 0.3115), \\ (p λ_{2}, 0.4103) \end{matrix}}$	${\begin{matrix} (p λ_{2}, 0.1136), \\ (p λ_{1}, 0.4021) \end{matrix}}$
	PO ₃	PO ₄
PA ₁	${\begin{matrix} (p λ_{2}, 0.3901), \\ (p λ_{1}, 0.2032) \end{matrix}}$	${\begin{matrix} (p λ_{3}, 0.3142), \\ (p λ_{1}, 0.4108) \end{matrix}}$
PA ₂	${\begin{matrix} (p λ_{1}, 0.2078), \\ (p λ_{2}, 0.0215) \end{matrix}}$	${\begin{matrix} (p λ_{2}, 0.3471), \\ (p λ_{2}, 0.3025) \end{matrix}}$
PA ₃	${\begin{matrix} (p λ_{2}, 0.1036), \\ (p λ_{2}, 0.3371) \end{matrix}}$	${\begin{matrix} (p λ_{2}, 0.1102), \\ (p λ_{1}, 0.4326) \end{matrix}}$
PA ₄	${\begin{matrix} (p λ_{3}, 0.2014), \\ (p λ_{2}, 0.3015) \end{matrix}}$	${\begin{matrix} (p λ_{3}, 0.2015), \\ (p λ_{1}, 0.3017) \end{matrix}}$
PA ₅	${\begin{matrix} (p λ_{2}, 0.3106), \\ (p λ_{2}, 0.2215) \end{matrix}}$	${\begin{matrix} (p λ_{1}, 0.3502), \\ (p λ_{2}, 0.4045) \end{matrix}}$

Step 3. Calculate the attributes’ weight in Table 6.

Table 6

The attributes weight

	PO ₁	PO ₂	PO ₃	PO ₄
weight	0.3005	0.2376	0.2262	0.2357

Step 4. Compute the average solution (See Table 7).

Table 7

The average solution

	The average solution
PO ₁	{ (pλ₂, 0.4634) , (pλ₂, 0.3351)}
PO ₂	{ (pλ₃, 0.1387) , (pλ₂, 0.3512)}
PO ₃	{ (pλ₂, 0.2187) , (pλ₃, 0.2516)}
PO ₄	{ (pλ₂, 0.3609) , (pλ₁, 0.4216)}

Step 5. Obtain the NIS with 2TLPFSs (See Table 8).

Table 8

The NIS with 2TLPFSs

	The NIS with 2TLPFSs
PO ₁	{ (pλ₂, 0.1228) , (pλ₁, 0.2143)}
PO ₂	{ (pλ₁, 0.3963) , (pλ₁, 0.4306)}
PO ₃	{ (pλ₁, 0.2078) , (pλ₂, 0.0215)}
PO ₄	{ (pλ₁, 0.3502) , (pλ₂, 0.4045)}

Step 6. The PDA and NDA are derived in Table 9-10.

Table 9

The PDA

	PO ₁	PO ₂	PO ₃	PO ₄
PA ₁	0.0000	0.0000	0.0000	0.0684
PA ₂	0.0000	0.0000	0.0503	0.0000
PA ₃	0.0385	0.0259	0.0000	0.0000
PA ₄	0.0000	0.0000	0.0000	0.0000
PA ₅	0.0000	0.0463	0.0000	0.0000

Table 10

The NDA

	PO ₁	PO ₂	PO ₃	PO ₄
PA ₁	0.0658	0.0443	0.0672	0.0000
PA ₂	0.1128	0.0618	0.0000	0.1215
PA ₃	0.0000	0.0000	0.0447	0.1098
PA ₄	0.0547	0.1127	0.0831	0.0709
PA ₅	0.0639	0.0000	0.2316	0.0984

Step 7. Calculate the NSP and NSN (See Table 11).

Table 11

The values of NSP and NSN

	NSP	NSN
PA ₁	0.0161	0.0455
PA ₂	0.0114	0.0772
PA ₃	0.0177	0.0360
PA ₄	0.0000	0.0787
PA ₅	0.0110	0.0948

Step 8. Calculate NNSP and NNSN in Table 12.

Table 12

The values of NNSP and NNSN

	NNSP	NNSN
PA ₁	0.9097	0.5200
PA ₂	0.6420	0.1853
PA ₃	1.0000	0.6203
PA ₄	0.0000	0.1694
PA ₅	0.6207	0.0000

Step 9. Calculate the AS (See Table 13).

Table 13

The AS

	AS	Order
PA ₁	0.7148	2
PA ₂	0.4136	3
PA ₃	0.8101	1
PA ₄	0.0847	5
PA ₅	0.3104	4

Step 10. According to the AS, the order is: PA₃ > PA₁ > PA₂ > PA₅ > PA₄ and PA₃ is the best brand packaging design scheme.

5 Conclusion

Packaging design, as a carrier of culture, inherits the national cultural spirit and regional cultural characteristics of a country or region, such as social ideology, aesthetic tendencies, moral ethics, folk customs, etc. The physical expression of packaging that integrates regional culture can appeal to people’s visual senses through its direct and artistic image. It can effectively guide consumer choices, enhance consumer taste, promote product sales, and achieve the shaping, development, and dissemination of brand image. Brand image is the individual characteristics of a brand in the market and the public, reflecting the public’s understanding and evaluation of the brand, reflecting the essence and strength of the brand. It includes three parts: brand concept, brand behavior, and brand vision. Among them, brand concept lays the central framework of the overall brand image, brand behavior reflects the dynamic process of the brand, and brand vision displays the brand image through material media such as products, packaging, advertising, and directly connects with the public. The brand packaging design effect evaluation is a MAGDM problem.This paper extends the EDAS method to the 2TLPFSs. On the basis of the original EDAS method, the 2TLPF-EDAS is built for MAGDM. Finally, a case study for brand packaging design effect evaluation is given and some comparative analysis with the other methods show that the new method proposed in this paper is effective and accurate.

References

Agrawal

V.P.

, Kohli

and Gupta

, Computer-aided robot selection - the multiple attribute decision-making approach, International Journal of Production Research 29 (1991), 1629–1644.

Dunn

E.G.

, Keller

J.M.

and Marks

L.A.

, Fuzzy multiple attribute decision making (MADM): A tool for agricultural and resource economics, American Journal of Agricultural Economics 78 (1996), 1394–1394.

Hon

C.C.

, Guh

Y.Y.

, Wang

K.M.

and Lee

E.S.

, Fuzzy multiple attributes and multiple hierarchical decision making, Computers & Mathematics with Applications 32 (1996), 109–119.

Atanassov

K.T.

, More on intuitionistic fuzzy-sets, Fuzzy Sets and Systems 33 (1989), 37–45.

Yager

, Pythagorean fuzzy subsets., IFSA word congress and NAFIPS annual meeting (IFSA/NAFIPS), Edmonton, Canada, New York: IEEE, (2013), 57–61.

Karunanithi

, Han

, Lee

C.J.

, Shi

W.C.

, Duan

and Qian

, Identification of a hemodynamic parameter for assessing treatment outcome of EDAS in Moyamoya disease, Journal of Biomechanics 48 (2015), 304–309.

Keshavarz Ghorabaee

, Zavadskas

E.K.

, Olfat

and Turskis

, Multi-criteria inventory classification using a new method of evaluation based on distance from average solution (EDAS), Informatica 26 (2015), 435–451.

Keshavarz Ghorabaee

, Zavadskas

E.K.

, Amiri

and Turskis

, Extended EDAS Method for Fuzzy Multi-criteria Decision-making: An Application to Supplier Selection, International Journal of Computers Communications & Control 11 (2016), 358–371.

Molla

M.U.

, Giri

B.C.

and Biswas

, Extended PROMETHEE method with Pythagorean fuzzy sets for medical diagnosis problems, Soft Computing.

10.

Khan

M.S.A.

and Abdullah

, Interval-valued Pythagorean fuzzy GRA method for multiple-attribute decision making with incomplete weight information, International Journal of Intelligent Systems 33 (2018), 1689–1716.

11.

Huang

, Lin

M.W.

and Xu

Z.H.

, Pythagorean fuzzy MULTIMOORA method based on distance measure and score function: its application in multicriteria decision making process, Knowledge and Information Systems 62 (2020), 4373–4406.

12.

Deng

X.M.

, Wang

, Wei

G.W.

and Lu

, Models for Multiple Attribute Decision Making with Some 2-Tuple Linguistic Pythagorean Fuzzy Hamy Mean Operators, Mathematics 6 (2018).

13.

Guan

K.Z.

, The Hamy symmetric function and its generalization, Mathematical Inequalities & Applications 9 (2006), 797–805.

14.

Qian

W.M.

, Schur convexity for the ratios of the Hamy and generalized Hamy symmetric functions, Journal of Inequalities and Applications (2011).

15.

Liu

, Khan

and Mahmood

, Application of Interval Neutrosophic Power Hamy Mean Operators in MAGDM, Informatica 30 (2019), 293–325.

16.

Liu

P.D.

and Liu

X.X.

, Linguistic Intuitionistic Fuzzy Hamy Mean Operators and Their Application to Multiple-Attribute Group Decision Making, IEEE Access 7 (2019), 127728–127744.

17.

Wang

H.D.

, He

S.F.

, Li

C.D.

and Pan

X.H.

, Pythagorean Uncertain Linguistic Variable Hamy Mean Operator and Its Application to Multi-attribute Group Decision Making, Ieee-Caa Journal of Automatica Sinica 6 (2019), 527–539.

18.

Wang

, Wei

G.W.

, Lu

J.P.

, Alsaadi

F.E.

, Hayat

, Wei

and Zhang

, Some q-Rung Orthopair Fuzzy Hamy mean Operators in Multiple Attribute Decision Making and their application to enterprise resource planning systems selection, International Journal of Intelligent Systems 34 (2019), 2429–2458.

19.

Deng

X.M.

, Wei

G.W.

, Gao

and Wang

, Models for Safety Assessment of Construction Project With Some 2-Tuple Linguistic Pythagorean Fuzzy Bonferroni Mean Operators, IEEE Access 6 (2018), 52105–52137.

20.

Bonferroni

, Sulle medie multiple di potenze, Bolletino Matematica Italiana 5 (1950), 267–270.

21.

Beliakov

, Pradera

and Calvo

, Aggregation functions:Aguide for practitioners, Springer, Berlin; New York, 2007.

22.

Z.S.

and Yager

R.R.

, Intuitionistic Fuzzy Bonferroni Means, Ieee Transactions on Systems Man and Cybernetics Part B-Cybernetics 41 (2011), 568–578.

23.

and Wu

, Interval-valued intuitionistic fuzzy Heronian mean operators and their application in multi-criteria decision making, African Journal of Business Management 6 (2012), 4158–4168.

24.

Zhou

and He

J.M.

, Intuitionistic Fuzzy Normalized Weighted Bonferroni Mean and Its Application in Multicriteria Decision Making, Journal of Applied Mathematics (2012).

25.

Zhu

, Xu

Z.S.

and Xia

M.M.

, Hesitant fuzzy geometric Bonferroni means, Information Sciences 205 (2012), 72–85.

26.

Deng

X.M.

and Gao

, TODIM method for multiple attribute decision making with 2-tuple linguistic pythagorean fuzzy information, Journal of Intelligent & Fuzzy Systems 37 (2019), 1769–1780.

27.

T.T.

, Zhang

S.Q.

, Wei

G.W.

, Wang

, Wu

and Wei

, CODAS method for 2-tuple linguistic pythagorean fuzzy multiple attribute group decision making and its application to financial management performance assessment, Technological and Economic Development of Economy 26 (2020), 920–932.

28.

Deng

, Wang

, Wei

and Wei

, Multiple attribute decision making based on Muirhead mean operators with 2-tuple linguistic Pythagorean fuzzy information, Scientia Iranica 28 (2021), 2294–2322.

29.

Muirhead

R.F.

, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proceedings of the Edinburgh Mathematical Society 21 (1902), 144–162.

30.

Wang

M.K.

, Chu

Y.M.

and Qiu

Y.F.

, Some Comparison Inequalities for Generalized Muirhead and Identric Means, Journal of Inequalities and Applications (2010), 10.

31.

Deng

Y.P.

, Wu

S.H.

, Chu

Y.M.

and He

, The Schur-Convexity of the Generalized Muirhead-Heronian Means, Abstract and Applied Analysis, (2014), 11.

32.

Qin

J.D.

and Liu

X.W.

, 2-tuple linguistic Muirhead mean operators for multiple attribute group decision making and its application to supplier selection, Kybernetes 45 (2016), 2–29.

33.

Liu

P.D.

and Li

D.F.

, Some Muirhead Mean Operators for Intuitionistic Fuzzy Numbers and Their Applications to Group Decision Making, Plos One 12 (2017), 28.

34.

, Zhang

R.T.

, Wang

, Zhu

X.M.

and Xing

Y.P.

, Pythagorean fuzzy power Muirhead mean operators with their application to multi-attribute decision making, Journal of Intelligent & Fuzzy Systems 35 (2018), 2035–2050.

35.

Liu

P.D.

, Li

, Zhang

M.C.

, Zhang

and Zhao

, Multiple-attribute decision-making method based on hesitant fuzzy linguistic Muirhead mean aggregation operators, Soft Computing 22 (2018), 5513–5524.

36.

Liu

P.D.

and Teng

, Some Muirhead mean operators for probabilistic linguistic term sets and their applications to multiple attribute decision-making, Applied Soft Computing 68 (2018), 396–431.

37.

Presenting German Packaging Museum bauhaus popular From the Bauhaus to the Department Store: The New Typography and its Influence on Brand Design, Wochenbl Papierfabr 147 (2019), 431–431.

38.

Favier

, Celhay

and Pantin-Sohier

, Is less more or a bore ? Package design simplicity and brand perception: an application to Champagne, J Retail Consum Serv 46 (2019), 11–20.

39.

Burns

, Brand protection in package design, in: Society of Plastics Engineers, Flexible Packaging Conference, FLEXPACKCON 2006, Dec 14–16 2006, Society of Plastics Engineers, Orlando, FL, United states, 2006, pp. 33–53.

40.

Bailey

C.K.

and O’Neill

G.C.

, Shaping a brand through package design, in, CRC Press, 2007, pp. 505–522.

41.

Zhang

and Xie

, Analytical Facts of Product Designing, Branding, Packaging, and Distributing through Internet of Things Applications, Wireless Communications and Mobile Computing 2022 (2022).

42.

Romeo-Arroyo

, Jensen

, Hunneman

and Velasco

, Assessing the influence of packaging design symmetry, curvature, and mark on the perception of brand premiumness, Int J Gastron Food Sci 31 (2023), 12.

43.

Qiao

and Griffin

W.G.

, Brand imitation strategy, package design and consumer response: what does it take to make a difference? J Prod Brand Manag 31 (2022), 177–188.

44.

Zhang

and Xie

, Analytical Facts of Product Designing, Branding, Packaging, and Distributing through Internet of Things Applications, Wirel Commun Mob Comput 2022 (2022), 8.

45.

Hwang

and Kim

, The effects of packaging design of private brands on consumers’ responses, Psychol Mark 39 (2022), 777–796.

46.

Min

, The Application of Computer Technology in Zen Brand Packaging Design, in: 11th International Conference on Frontier Computing, FC 2021, July 14, 2021 – July 14, 2021, Springer Science and Business Media Deutschland GmbH, Virtual, Online, 2022, pp. 1061–1065.

47.

Yan

J.L.

and Li

S.B.

, Visual Communication Design of Nano Material Packaging Based on Brand Positioning, Basic Clin Pharmacol Toxicol 126 (2020), 401–402.

48.

Wen

and Huang

M.-J.

, A color study on the packaging design of Korean brand cosmetics -Focusing on affordable brands for female consumers, in: 3rd International Conference on Energy Resources and Sustainable Development, ICERSD 2020, December 25, 2020 – December 27, 2020, EDP Sciences, Harbin, China, 2021.

49.

Yang

Y.C.

and Zhao

, Exploring the relationship of packaging design with consumers’ green trust, and brand attachment, Soc Behav Pers 47 (2019), 10.

50.

, Research on Packaging Visual Communication Design in Sportswear Brand Design, Basic Clin Pharmacol Toxicol 127 (2020), 178–178.

51.

Mitchell

, Moodie

, Critchlow

and Bauld

, Adolescents’ perceptions of standardised cigarette packaging design and brand variant name post-implementation: a focus group study in Scotland, BMC Public Health 19 (2019), 9.

52.

Orth

U.R.

, Rose

G.M.

and Merchant

, Preservation, rejuvenation, or confusion? Changing package designs for heritage brands, Psychol Mark 36 (2019), 831–843.

53.

and Si

, Research on brand packaging design of Chinese women’s cosmetics, in: 21st IAPRI World Conference on Packaging 2018 – Packaging: Driving a Sustainable Future, June 19, 2018 – June 22, 2018, DEStech Publications Inc., Zhuhai, China, 2019, pp. 303–308.

54.

Huang

Y.-C.

, Hsiao

M.-Y.

, Yi

H.-J.

and Yeh

T.-C.

, A teaching research for packaging design in brand building –Acase study of tea packaging in teama, in: 10th International Conference on Education Technology and Computers, ICETC 2018, October 26, 2018 – October 28, 2018, Association for Computing Machinery, Tokyo, Japan, 2018, pp. 212–216.

55.

Timney

and Chamberlain

, Integrated package design: an interdisciplinary approach to package design that benefits consumer experience and brand perception, Computer-Aided Design and Applications 14 (2017), 33–40.

56.

Huang

Y.C.

, Hsiao

M.Y.

, Yi

H.J.

and Yeh

T.C.

, Acm, A Teaching Research for Packaging Design in Brand Building –ACase Study of Tea Packaging in Teama, in: 10th International Conference on Education Technology and Computers (ICETC 2018), Assoc Computing Machinery, Tokyo Inst Technol, Tokyo, JAPAN, 2018, pp. 212–216.

57.

Herrera

and Martinez

, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Transactions on Fuzzy Systems 8 (2000), 746–752.

58.

Herrera

and Martinez

, An approach for combining linguistic and numerical information based on the 2-tuple fuzzy linguistic representation model in decision-making, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 8 (2000), 539–562.

59.

Wang

, Wei

G.W.

and Wei

, Models for Green Supplier Selection with Some 2-Tuple Linguistic Neutrosophic Number Bonferroni Mean Operators, Symmetry-Basel 10 (2018), 36.