Enhanced EDAS technique for quality evaluation of online games interface design based on three types of probabilistic linguistic similarity measures

Abstract

The highest level of interface design for online games is to achieve invisibility, which means that the interface is very natural and easy to use. Players are proficient in using it and fully immerse themselves in the world of online games. It seems that the interface is transparent and they no longer feel its existence. Excellent online game interface design is the requirement of users that runs through the entire design process. But this is not to say that the usability of online game software can surpass other factors. All great designs seek a balance and harmony between artistic beauty, reliability, security, usability, cost, and performance. The quality evaluation of online games interface design is a multiple-attribute group decision-making (MAGDM) problem. Recently, the Evaluation Based on Distance from Average Solution (EDAS) technique, cosine similarity measure (CSM), Dice similarity measure (DSM), Jaccard similarity measure (JSM) and entropy technique has been separately employed to cope with MAGDM issues. The probabilistic linguistic term sets (PLTSs) are employed as a tool for conveying uncertain information during the quality evaluation of online games interface design. In this paper, the EDAS technique is expanded to the PLTSs and the probabilistic linguistic EDAS (PL-EDAS) technique based on cosine similarity measure (CSM), Dice similarity measure (DSM) and Jaccard similarity measure (JSM) is constructed to manage MAGDM issue. The information entropy technique is employed to implement the weight values based on CSM technique, DSM technique and JSM technique under PLTSs. Finally, the quality evaluation of online games interface design is employed to demonstrate the PL-EDAS technique and some comparative analysis is employed to demonstrate the PL-EDAS technique. Thus, the main research contribution of this work is constructed: (1) the information entropy technique is constructed to implement the attribute weight values based on CSM technique, DSM technique and JSM technique; (2) the PL-EDAS technique is constructed under PLTSs based on CSM technique, DSM technique and JSM technique; (3) an example for quality evaluation of online games interface design is employed to verify PL-EDAS technique and several decision comparative analysis is employed to verify the PL-EDAS technique.

Keywords

Multiple attribute group decision making (MAGDM)probabilistic linguistic term sets (PLTSs)EDAS technique information entropy online games interface design

1. Introduction

The comprehensive coverage of the Internet and the rapid development of mobile terminal devices have also led to the rapid development and growth of online games, becoming one of the mainstream leisure and entertainment methods in contemporary times [1]. Especially in recent years, more and more people have become members of online game users, and various esports game leagues have also gained higher attention [2, 3]. This development trend has brought huge profits to game developers. The diverse games in the market are prone to similar game styles, but each game should have a unique game interface [4, 5]. At present, the main game development companies in China are Tencent and NetEase. Most of the games produced by these two companies have excellent graphics, novel and interesting content gameplay, and meet the needs of most market users. Just like in real life, the first impression of people comes from external observation, online games also apply this principle. Users’ first impression of online games comes from the most intuitive game interface. There are many game products in the market, so the design of the game interface directly affects users’ click through volume [6, 7]. The online game development market in China has become increasingly mature, but there is still a lot of room for improvement in interface design. A beautiful and elegant game interface can add impression points to the game. There are many factors that influence the success of a game, with first impressions occupying an important position. Therefore, game interface designers should be good at observing the interface design of various online games, summarizing and categorizing the complex game interface design styles, and discovering the game styles that users like, laying a solid foundation for future game interface design [8, 9]. Although the purpose of players playing online games is for leisure and entertainment, and to obtain fun from the game, a comfortable game interface can provide players with a better visual experience, so it has high practical value [10]. Firstly, there are a wide variety of online games of all sizes, and the styles between different games are also ever-changing [11]. The design and production of basic online games are not very difficult in terms of technical difficulty, but once the popularity of a certain online game style gradually increases, a large number of online games of the same type and design style will also begin to emerge. Currently, the interface design of online games in China mostly draws inspiration from the Japanese and Korean game interface design styles, which mostly use cartoon and cute elements. Although this style is favored by female players, uniformity inevitably leads to aesthetic fatigue. At present, the online game styles of shooting and sports are mostly based on the interface design of European and American games, and through some realistic art styles, players have a stronger sense of immersion [12, 13]. Although this design style is more appealing to male players, it still lacks innovation. Secondly, the design of online game interfaces directly affects the sales of this game in many cases. The internal operations of the game are important, but first of all, players need to generate interest through the game interface to appreciate whether the game’s operations are excellent. This requires the game interface’s painting style and design elements to be aesthetically pleasing. And there are usually differences between the promotional interface and the internal operation interface of online games, because in order to ensure the smooth operation of the game process, the graphics often cannot be described in detail and displayed delicately [14, 15]. However, in the initial promotional interface of the game, each online game strives for excellence. Therefore, the complex graphics and magnificent colors are the common characteristics of current online games. Moreover, due to the unique nature of online games, there are significant differences in the interface style of the same game across different clients. Finally, as the online gaming industry continues to mature, more and more new styles have emerged in the interface design of online games [16]. It has also won many awards in international online game interface design. At present, various online games are popular worldwide, and online games of all sizes have developed very mature in terms of game operation content, overall visual quality, content, and gameplay, allowing players to fully immerse themselves in interesting games [17]. In the future, online game interface designers should continuously improve their creative thinking and abilities, seek a balance and harmony between the artistic beauty, ease of operation, and cost-effectiveness of game interfaces, pay more attention to the psychological needs of players, extract more localized design elements, and design more local and popular online game interfaces [18].

When decision information is uncertain or insufficient, uncertain variables and fuzzy variables are commonly used to express the decision-maker’s judgment. Among them, linguistic variables are often considered an effective way to describe decision-makers’ judgments [19, 20, 21]. Due to the complexity of unexpected events, decision-makers may need multiple linguistic variables to express their judgments in a richer way. To address this issue, Rodriguez, Martinez and Herrera [22] constructed the concept of Hesitant Fuzzy Linguistic Term Sets (HFLTS). Subsequently, decision-making research based on HFLTS gradually increased. Liang, Tu, Ju and Shen [23] extended the TODIM technique to a multi-granularity proportional hesitation fuzzy linguistic environment and developed a multi granularity proportional hesitation language TODIM technique for mine emergency decision-making. In addition, Wu, Xu, Jiang and Zhong [24] constructed a MAGDM based on HFLTS, combined with VIKOR and TOPSIS techniques. In reality, decision-makers not only use multiple linguistic variables to express their judgments, but also exhibit hesitation and varying degrees of preference. For this reason, Pang, Wang and Xu [25] constructed a new linguistic variable called Probabilistic Linguistic Term Sets (PLTSs), which can reflect both decision-makers’ hesitation and their preference for language terms. Given that PLTSs can comprehensively express decision-makers’ judgments, they have been widely adopted in decision analysis [26, 27, 28, 29, 30, 31, 32]. In addition, many scholars have conducted research on the information processing techniques of PLTSs [33, 34]. Among them, the techniques for determining the similarity between two PLTSs mainly include distance measurement, correlation measurement, and entropy measurement [35]. Pang, Wang and Xu [25] constructed for the first time a deviation measurement technique between PLTSs similar to the hesitant fuzzy Euclidean distance. Lin et al. [36] constructed a correlation coefficient measurement technique between two PLTSs. Liu et al. [37] introduced the fuzzy entropy, hesitant entropy, and total entropy measurement techniques of PLTSs, and then constructed a multi standard decision model based on these measurement techniques. The quality evaluation of online games interface design is a MAGDM problem. Recently, the EDAS technique [38, 39, 40, 41, 42], cosine similarity measure (CSM) [43, 44, 45], Dice similarity measure (DSM) [46, 47], Jaccard similarity measure (JSM) [48, 49] and entropy technique [50] has been separately employed to cope with MAGDM issues. The probabilistic linguistic term sets (PLTSs) [51] are employed as a tool for conveying uncertain information during the quality evaluation of online games interface design. Thus, it has significant significance to put up with the quality evaluation of online games interface design. Unfortunately, few valuable works were unable to manage the decision-making issues through employing the EDAS technique [38, 39] based on CSM technique, DSM technique and JSM technique under PLTSs [51] during existing research literatures. Therefore, it is valuable to go deeply into EDAS technique with PLTSs based on CSM technique, DSM technique and JSM technique. In this paper, the EDAS technique is expanded to the PLTSs and the probabilistic linguistic EDAS (PL-EDAS) technique based on cosine similarity measure (CSM), Dice similarity measure (DSM) and Jaccard similarity measure (JSM) is constructed to manage MAGDM issue. The information entropy technique is employed to implement the weight values based on CSM technique, DSM technique and JSM technique under PLTSs. Finally, the quality evaluation of online games interface design is employed to demonstrate the PL-EDAS technique and some comparative analysis is employed to demonstrate the PL-EDAS technique. Thus, the main research goal and motivation of this work is constructed: (1) the information entropy technique is constructed to implement the attribute weight values based on CSM technique, DSM technique and JSM technique; (2) the PL-EDAS technique is constructed under PLTSs based on CSM technique, DSM technique and JSM technique; (3) an example for quality evaluation of online games interface design is employed to verify PL-EDAS technique and several decision comparative analysis is employed to verify the PL-EDAS technique.

The framework structure of this paper is conveyed below. In Section 2, the PLTSs is conveyed. In Section 3, PL-EDAS technique is constructed under PLTSs based on CSM technique, DSM technique and JSM technique with entropy. Section 4 gives the numerical example for quality evaluation of online games interface design and some comparative analysis. Some remarks are conveyed in Section 5.

2. Preliminaries

Pang, Wang and Xu [52] conveyed the PLTSs.

Definition 1 [52]. Let $K L = {k l_{ϕ} | ϕ = 0, 1, 2, \dots, 2 ξ}$ be the LTSs, the linguistic terms $k l_{ϕ}$ depict the information to $φ$ along with mathematical function $k f$ :

$k f : [k l_{0}, k l_{2 ξ}] \to [0, 1], k f (k l_{ϕ}) = \frac{ϕ}{2 ξ} = φ$ (1)

$φ$ is the equivalent information to the $k l_{ϕ}$ with help of function $k f^{- 1}$ :

$k f^{- 1} : [0, 1] \to [k l_{0}, k l_{2 ξ}], k f^{- 1} (φ) = k l_{2 φ ξ} = k l_{ϕ}$ (2)

Definition 2 [52]. Let $K L = {k l_{ϕ} | ϕ = 0, 1, 2, \dots, 2 ξ}$ , the PLTS is created:

$K L (k p) = {k l^{(k γ)} (k p^{(k γ)}) | (\begin{matrix} k l^{(k γ)} \in K L, k p^{(k γ)} ⩾ 0, \\ k γ = 1, 2, \dots, # K L (k p), \sum_{k γ = 1}^{# K L (k p)} k p^{(k γ)} ⩽ 1 \end{matrix})}$ (3)

where $k l^{(k γ)} (k p^{(k γ)})$ is the $k γ$ th $k l^{(k γ)}$ with the probability $k l^{(k γ)}$ , and $# K L (k p)$ is the length of $K L (k p)$ . The $k l^{(k γ)}$ in $K L = {k l_{ϕ} | ϕ = 0, 1, 2, \dots, 2 ξ}$ are created through employing the ascending order.

Definition 3 [53]. Let $K L = {k l_{ϕ} | ϕ = 0, 1, 2, \dots, 2 ξ}$ be LTSs,

$K L_{1} (k p) = {k l_{1}^{(k γ)} (k p_{1}^{(k γ)}) | (\begin{matrix} k l_{1}^{(k γ)} \in K L, k p_{1}^{(k γ)} ⩾ 0, \\ k γ = 1, 2, \dots, # K L_{1} (k p), \sum_{k γ = 1}^{# K L_{1} (k p)} k p_{1}^{(k γ)} ⩽ 1 \end{matrix})}$

and

$K L_{2} (k p) = {k l_{2}^{(k γ)} (k p_{2}^{(k γ)}) | (\begin{matrix} k l_{2}^{(k γ)} \in K L, k p_{2}^{(k γ)} ⩾ 0, \\ k γ = 1, 2, \dots, # K L_{2} (k p), \sum_{k γ = 1}^{# K L_{2} (k p)} k p_{2}^{(k γ)} ⩽ 1 \end{matrix})}$

be two PLTSs, $# K L_{1} (k p)$ and $# K L_{2} (k p)$ are the length of PLTS $K L_{1} (k p)$ and $K L_{2} (k p)$ . If $# K L_{1} (k p) \neq # K L_{1} (k p)$ , The PLTS could be normalized through employing the following technique:

$K L_{1} (k p) = {k l_{1}^{(k γ)} (k p_{1}^{(k γ)}) | (\begin{matrix} k l_{1}^{(k γ)} \in K L, k p_{1}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} k p_{1}^{(k γ)} ⩽ 1 \end{matrix})}$ (4) $K L_{2} (k p) = {k l_{2}^{(k γ)} (k p_{2}^{(k γ)}) | (\begin{matrix} k l_{2}^{(k γ)} \in K L, k p_{2}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} k p_{2}^{(k γ)} ⩽ 1 \end{matrix})}$ (5)

For the $K L_{1} (k p)$ and $K L_{2} (k p)$ , the added LTS along with corresponding probability zero.

Furthermore, if $\sum_{k γ = 1}^{2 ξ} k p_{1}^{(k γ)} ⩽ 1$ , $\sum_{k γ = 1}^{2 ξ} k p_{2}^{(k γ)} ⩽ 1$ , then, the PLTSs could be normalized through employing the following technique [53]:

$K L_{1} (k p) = {k l_{1}^{(k γ)} (k p_{1}^{(k γ)}) | (\begin{matrix} k l_{1}^{(k γ)} \in K L, k p_{1}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} k p_{1} {}^{(k γ)}= 1 \end{matrix})}$ (6) $K L_{2} (k p) = {k l_{2}^{(k γ)} (k p_{2}^{(k γ)}) | (\begin{matrix} k l_{2}^{(k γ)} \in K L, k p_{2}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} k p_{2}^{(k γ)} = 1 \end{matrix})}$ (7)

where $k p_{1}^{(k γ)} = k p_{1}^{(k γ)} / \sum_{k γ = 1}^{2 ξ} k p_{1}^{(k γ)}$ , $k p_{2}^{(k γ)} = k p_{2}^{(k γ)} / \sum_{k γ = 1}^{2 ξ} k p_{2}^{(k γ)}$ , for all $k γ = 1, 2, \dots, 2 ξ$ .

Definition 4 [53]. Let $K L = {k l_{ϕ} | ϕ = 0, 1, 2, \dots, 2 ξ}$ be LTSs,

$K L_{1} (k p) = {k l_{1}^{(k γ)} (k p_{1}^{(k γ)}) | (\begin{matrix} k l_{1}^{(k γ)} \in K L, k p_{1}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} k p_{1}^{(k γ)} = 1 \end{matrix})}$

and

$K L_{2} (k p) = {k l_{2}^{(k γ)} (k p_{2}^{(k γ)}) | (\begin{matrix} k l_{2}^{(k γ)} \in K L, k p_{2}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} k p_{2}^{(k γ)} = 1 \end{matrix})}$

be two PLTSs, some basic operational laws are constructed:

$(1) K L_{1} (k p) \oplus K L_{2} (k p) = \underset{k l_{1}^{(k γ)} (k p_{1}^{(k γ)}) \in K L_{1} (k p), k l_{2}^{(k γ)} (k p_{2}^{(k γ)}) \in K L_{2} (k p)}{\cup} {k l_{1}^{(k γ)} (k p_{1}^{(k γ)} + k p_{2}^{(k γ)})}; (2) K L_{1} (k p) \otimes K L_{2} (k p) = \underset{k l_{1}^{(k γ)} (k p_{1}^{(k γ)}) \in K L_{1} (k p), k l_{2}^{(k γ)} (k p_{2}^{(k γ)}) \in K L_{2} (k p)}{\cup} {k l_{1}^{(k γ)} (k p_{1}^{(k γ)} \times k p_{2}^{(k γ)})}; (3) λ K L_{1} (k p) = \underset{k l_{1}^{(k γ)} (k p_{1}^{(k γ)}) \in K L_{1} (k p)}{\cup} {k l_{1}^{(k γ)} (λ k p_{1}^{(k γ)})}, 0 ⩽ λ ⩽ 1; (4) {(K L_{1} (k p))}^{λ} = \underset{k l_{1}^{(k γ)} (k p_{1}^{(k γ)}) \in K L_{1} (k p)}{\cup} {k l_{1}^{(k γ)} ({(k p_{1}^{(k γ)})}^{λ})}, 0 ⩽ λ ⩽ 1 .$

Definition 5 [52, 53]. Let $K L = {k l_{ϕ} | ϕ = 0, 1, 2, \dots, 2 ξ}$ be LTSs,

and

be two PLTSs, The expected values $E V (K L_{1} (k p))$ and $E V (K L_{2} (k p))$ of $K L_{1} (k p)$ and $K L_{2} (k p)$ are created as follows:

$E V (K L_{1} (k p)) = \sum_{k γ = 0}^{2 ξ} (k f (k l_{1}^{(k γ)}) k p_{1}^{(k γ)})$ (8) $E V (K L_{2} (k p)) = \sum_{k γ = 0}^{2 ξ} (k f (k l_{2}^{(k γ)}) k p_{2}^{(k γ)})$ (9)

The deviation degree $D D (K L_{1} (k p))$ and $D D (K L_{2} (k p))$ of $K L_{1} (k p)$ and $K L_{2} (k p)$ are created as follows:

$D D (K L_{1} (k p)) = \sqrt{\sum_{k γ = 0}^{2 ξ} {(k f (k l_{1}^{(k γ)}) k p_{1}^{(k γ)} - E V (K L_{1} (k p)))}^{2}}$ (10) $D D (K L_{2} (k p)) = \sqrt{\sum_{k γ = 0}^{2 ξ} {(k f (k l_{2}^{(k γ)}) k p_{2}^{(k γ)} - E V (K L_{2} (k p)))}^{2}}$ (11)

In line with Eqs (4) and (5), the order is constructed between

$K L_{1} (k p) = {k l_{1}^{(k γ)} (k p_{1} {}^{(k γ)}) | (\begin{matrix} k l_{1}^{(k γ)} \in K L, k p_{1}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} k p_{1}^{(k γ)} = 1 \end{matrix})}$

and

(1) if $E V (K L_{1} (k p)) > E V (K L_{2} (k p))$ , $K L_{1} (k p) > K L_{2} (k p)$ ;

(2) if $E V (K L_{1} (k p)) = E V (K L_{2} (k p))$ , if $D D (K L_{1} (k p)) = D D (K L_{2} (k p))$ , then $K L_{1} (k p) = K L_{2} (k p)$ ; if $D D (K L_{1} (k p)) < D D (K L_{2} (k p))$ , then, $K L_{1} (k p) > K L_{2} (k p)$ .

Definition 6 [53, 54]. Let $K L = {k l_{ϕ} | ϕ = 0, 1, 2, \dots, 2 ξ}$ be LTSs,

and

be two PLTSs, then the probabilistic linguistic Hamming distance $𝑃𝐿𝐻𝐷 (K L_{1} (k p), K L_{2} (k p))$ and probabilistic linguistic Euclidean distance $𝑃𝐿𝐸𝐷 (K L_{1} (k p), K L_{2} (k p))$ are constructed:

$𝑃𝐿𝐻𝐷 (K L_{1} (k p), K L_{2} (k p)) = \frac{\sum_{k γ = 0}^{2 ξ} | k p_{1}^{(k γ)} k f (k l_{1}^{(k γ)}) - k p_{2}^{(k γ)} k f (k l_{2}^{(k γ)}) |}{2 ξ}$ (12) $𝑃𝐿𝐸𝐷 (K L_{1} (k p), K L_{2} (k p)) = \sqrt{\frac{\sum_{k γ = 0}^{2 ξ} {(k p_{1}^{(k γ)} k f (k l_{1}^{(k γ)}) - k p_{2}^{(k γ)} k f (k l_{2}^{(k γ)}))}^{2}}{2 ξ}}$ (13)

Definition 7 [55]. Let $K L = {k l_{ϕ} | ϕ = 0, 1, 2, \dots, 2 ξ}$ be LTSs,

and

be two PLTSs, probabilistic linguistic cosine similarity measure (PLCSM) between $K L_{1} (k p)$ and $K L_{2} (k p)$ is constructed:

$𝑃𝐿𝐶𝑆𝑀 (K L_{1} (k p), K L_{2} (k p))$ $= \frac{(\sum_{k γ = 0}^{2 ξ} ((k f (k l_{1}^{(k γ)}) k p_{1}^{(k γ)}) \times (k f (k l_{2}^{(k γ)}) k p_{2}^{(k γ)})))}{\sqrt{\sum_{k γ = 0}^{2 ξ} {(k f (k l_{1}^{(k γ)}) k p_{1}^{(k γ)})}^{2}} \sqrt{\sum_{k γ = 1}^{2 ξ} {(k f (k l_{2}^{(k γ)}) k p_{2}^{(k γ)})}^{2}}},$ (14)

Definition 8 [56]. Let $K L = {k l_{ϕ} | ϕ = 0, 1, 2, \dots, 2 ξ}$ be LTSs,

and

be two PLTSs, probabilistic linguistic Dice similarity measure (PLDSM) between $K L_{1} (k p)$ and $K L_{2} (k p)$ is constructed:

$𝑃𝐿𝐷𝑆𝑀 (K L_{1} (k p), K L_{2} (k p))$ $= \frac{2 (\sum_{k γ = 0}^{2 ξ} ((k f (k l_{1}^{(k γ)}) k p_{1}^{(k γ)}) \times (k f (k l_{2}^{(k γ)}) k p_{2}^{(k γ)})))}{(\sum_{k γ = 0}^{2 ξ} {(k f (k l_{1}^{(k γ)}) k p_{1}^{(k γ)})}^{2} + \sum_{k γ = 1}^{2 ξ} {(k f (k l_{2}^{(k γ)}) k p_{2}^{(k γ)})}^{2})},$ (15)

Definition 9. Let $K L = {k l_{ϕ} | ϕ = 0, 1, 2, \dots, 2 ξ}$ be LTSs,

and

be two PLTSs, probabilistic linguistic Jaccard similarity measure (PLJSM) between $K L_{1} (k p)$ and $K L_{2} (k p)$ is constructed:

$𝑃𝐿𝐽𝑆𝑀 (K L_{1} (k p), K L_{2} (k p))$ $= \frac{(\sum_{k γ = 0}^{2 ξ} ((k f (k l_{1}^{(k γ)}) k p_{1}^{(k γ)}) \times (k f (k l_{2}^{(k γ)}) k p_{2}^{(k γ)})))}{(\begin{matrix} (\sum_{k γ = 0}^{2 ξ} {(k f (k l_{1}^{(k γ)}) k p_{1}^{(k γ)})}^{2} + \sum_{k γ = 1}^{2 ξ} {(k f (k l_{2}^{(k γ)}) k p_{2}^{(k γ)})}^{2}) \\ - \sum_{k γ = 0}^{2 ξ} ((k f (k l_{1}^{(k γ)}) k p_{1}^{(k γ)}) \times ((k f (k l_{2}^{(k γ)}) k p_{2}^{(k γ)}))) \end{matrix})},$ (16)

3. PL-EDAS technique for MAGDM based on CSM, DSM and JSM

In this section, PL-EDAS technique is constructed for MAGDM. Let $K G = {K G_{1}, K G_{2}, \dots, K G_{m}}$ be alternatives and $K P = {K P_{1}, K P_{2}, \dots, K P_{n}}$ be attributes with weight information $k w$ , where $k w_{j} \in [0, 1]$ , $\sum_{j = 1}^{n} k w_{j} = 1$ . Suppose that the $K G_{i}$ under $K P_{j}$ are assessed with PLTSs through integrating different experts’ information.

Then, PL-EDAS technique is employed to solve the PL-MAGDM. The calculating steps are depicted:

Figure 1.

The steps of PL-EDAS method for MAGDM based on CSM, DSM and JSM with information entropy.

Step 1. Implement the normalized probabilistic linguistic matrix $K L = {(K L_{i j} (k p))}_{m \times n}$ ,

$K L_{i j} (k p) = {k l_{i j}^{(k γ)} (k p_{i j}^{(k γ)}) | (\begin{matrix} k l_{i j}^{(k γ)} \in K L, k p_{i j}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} k p_{i j}^{(k γ)} = 1 \end{matrix})}$

through integrating different experts’ information. Thus, probabilistic linguistic information for $K G_{i}$ is depicted as:

$j = 1, 2, \dots, n$ .

Step 2. Implement the probabilistic linguistic average decision alternative (PLADA).

$𝑃𝐿𝐴𝐷𝐴 = {({𝑃𝐿𝐴𝐷𝐴}_{j})}_{1 \times n}$ (17) ${𝑃𝐿𝐴𝐷𝐴}_{j} = {k l_{j}^{(k γ)} (k p_{j}^{(k γ)}) | (\begin{matrix} k l_{j}^{(k γ)} \in K L, k p_{j}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} k p_{j}^{(k γ)} = 1 \end{matrix})}$ (18) $k l_{j}^{(k γ)} (k p_{j}^{(k γ)}) = \frac{\sum_{i = 1}^{m} k l_{j}^{(k γ)}}{m} (\frac{\sum_{i = 1}^{m} k p_{j}^{(k γ)}}{m})$ (19)

Step 3. Implement the probabilistic linguistic positive ideal alternative (PLPIA):

${𝑃𝐿𝑃𝐼𝐴}_{j} = {{𝑝𝑘𝑙}_{j}^{(k γ)} ({𝑝𝑘𝑝}_{j}^{(k γ)}) | (\begin{matrix} {𝑝𝑘𝑙}_{j}^{(k γ)} \in K L, {𝑝𝑘𝑝}_{j}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} {𝑝𝑘𝑝}_{j}^{(k γ)} = 1 \end{matrix})}$ (20) $E V ({𝑃𝐿𝑃𝐼𝐴}_{j}) = E V ({{𝑝𝑘𝑙}_{j}^{(k γ)} ({𝑝𝑘𝑝}_{j}^{(k γ)}) | (\begin{matrix} {𝑝𝑘𝑙}_{j}^{(k γ)} \in K L, {𝑝𝑘𝑝}_{j}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} {𝑝𝑘𝑝}_{j}^{(k γ)} = 1 \end{matrix})})$ $= max_{i} E V (K L_{i j} (k p))$ $= max_{i} E V ({k l_{i j}^{(k γ)} (k p_{i j}^{(k γ)}) | (\begin{matrix} k l_{i j}^{(k γ)} \in K L, k p_{i j}^{(k γ)} ⩾ 0, \\ k γ = 0, 1, 2, \dots, 2 ξ, \sum_{k γ = 0}^{2 ξ} k p_{i j}^{(k γ)} = 1 \end{matrix})})$ (21)

Step 4. Implement the PLCSM between

and

$𝑃𝐿𝐶𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j})$ $= \frac{(\sum_{k γ = 0}^{2 ξ} ((k f (k l_{i j}^{(k γ)}) k p_{i j}^{(k γ)}) \times (k f ({𝑝𝑘𝑙}_{j}^{(k γ)}) {𝑝𝑘𝑝}_{j}^{(k γ)})))}{\sqrt{\sum_{k γ = 0}^{2 ξ} {((k f (k l_{i j}^{(k γ)}) k p_{i j}^{(k γ)}))}^{2}} \sqrt{\sum_{k γ = 1}^{2 ξ} {((k f ({𝑝𝑘𝑙}_{j}^{(k γ)}) {𝑝𝑘𝑝}_{j}^{(k γ)}))}^{2}}},$ (22)

Step 5. Implement the PLDSM between

and

$𝑃𝐿𝐷𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j})$ $= \frac{2 (\sum_{k γ = 0}^{2 ξ} ((k f (k l_{i j}^{(k γ)}) k p_{i j}^{(k γ)}) \times (k f ({𝑝𝑘𝑙}_{j}^{(k γ)}) {𝑝𝑘𝑝}_{j}^{(k γ)})))}{(\sum_{k γ = 0}^{2 ξ} {((k f (k l_{i j}^{(k γ)}) k p_{i j}^{(k γ)}))}^{2} + \sum_{k γ = 0}^{2 ξ} {((k f ({𝑝𝑘𝑙}_{j}^{(k γ)}) {𝑝𝑘𝑝}_{j}^{(k γ)}))}^{2})}$ (23)

Step 6. Implement the PLDSM between

and

$𝑃𝐿𝐽𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j})$ $= \frac{(\sum_{k γ = 0}^{2 ξ} ((k f (k l_{i j}^{(k γ)}) k p_{i j}^{(k γ)}) \times (k f ({𝑝𝑘𝑙}_{j}^{(k γ)}) {𝑝𝑘𝑝}_{j}^{(k γ)})))}{(\begin{matrix} \sum_{k γ = 0}^{2 ξ} {((k f (k l_{i j}^{(k γ)}) k p_{i j}^{(k γ)}))}^{2} + \sum_{k γ = 0}^{2 ξ} {((k f ({𝑝𝑘𝑙}_{j}^{(k γ)}) {𝑝𝑘𝑝}_{j}^{(k γ)}))}^{2} \\ - \sum_{k γ = 0}^{2 ξ} ((k f (k l_{i j}^{(k γ)}) k p_{i j}^{(k γ)}) \times (k f ({𝑝𝑘𝑙}_{j}^{(k γ)}) {𝑝𝑘𝑝}_{j}^{(k γ)})) \end{matrix})}$ (24)

Step 7. Implement the attributes weight values through employing information entropy.

Entropy [50] is an efficient technique to obtain weight values. Firstly, the normalized probabilistic linguistic matrix ${𝑁𝑃𝐿𝑀}_{i j}$ is managed:

${𝑁𝑃𝐿𝑀}_{i j} = \frac{(\begin{matrix} 𝑃𝐿𝐶𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) \\ + 𝑃𝐿𝐷𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) \\ + 𝑃𝐿𝐽𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) \end{matrix})}{\sum_{i = 1}^{m} (\begin{matrix} 𝑃𝐿𝐶𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) \\ + 𝑃𝐿𝐷𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) \\ + 𝑃𝐿𝐽𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) \end{matrix})},$ (25)

Then, the probabilistic linguistic Shannon entropy values (PLSEV) is managed by Eq. (26):

${𝑃𝐿𝑆𝐸𝑉}_{j} = - \frac{1}{\ln m} \sum_{i = 1}^{m} {𝑁𝑃𝐿𝑀}_{i j} \ln {𝑁𝑃𝐿𝑀}_{i j}$ (26)

and ${𝑁𝑃𝐿𝑀}_{i j} \ln {𝑁𝑃𝐿𝑀}_{i j} = 0$ if ${𝑁𝑃𝐿𝑀}_{i j} = 0$ .

Then, the weight information $k w = (k w_{1}, k w_{2}, \dots, k w_{n})$ is constructed:

$k w_{j} = \frac{1 - {𝑃𝐿𝑆𝐸𝑉}_{j}}{\sum_{j = 1}^{n} (1 - {𝑃𝐿𝑆𝐸𝑉}_{j})}, j = 1, 2, \dots, n .$ (27)

Step 8. Implement the probabilistic linguistic positive distance measures (PLPDM) from PLADA and probabilistic linguistic negative distance measures (PLNDM) from PLADA:

For the positive attributes:

${𝑃𝐿𝑃𝐷𝑀}_{i j}$ $= \frac{1}{3} (\begin{matrix} \frac{\max (0, 𝑃𝐿𝐶𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \\ + \frac{\max (0, 𝑃𝐿𝐷𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \\ + \frac{\max (0, 𝑃𝐿𝐽𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \end{matrix})$ (28) ${𝑃𝐿𝑁𝐷𝑀}_{i j}$ $= \frac{1}{3} (\begin{matrix} \frac{\max (0, 𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐶𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \\ + \frac{\max (0, 𝑃𝐿𝐷𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐷𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐷𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \\ + \frac{\max (0, 𝑃𝐿𝐽𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐽𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐽𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \end{matrix})$ (29)

For the negative attributes:

${𝑃𝐿𝑃𝐷𝑀}_{i j}$ $= \frac{1}{3} (\begin{matrix} \frac{\max (0, 𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐶𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \\ + \frac{\max (0, 𝑃𝐿𝐷𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐷𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐷𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \\ + \frac{\max (0, 𝑃𝐿𝐽𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐽𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐽𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \end{matrix})$ (30) ${𝑃𝐿𝑁𝐷𝑀}_{i j}$ $= \frac{1}{3} (\begin{matrix} \frac{\max (0, 𝑃𝐿𝐶𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \\ + \frac{\max (0, 𝑃𝐿𝐷𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \\ + \frac{\max (0, 𝑃𝐿𝐽𝑆𝑀 (K L_{i j} (k p), {𝑃𝐿𝑃𝐼𝐴}_{j}) - 𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j}))}{𝑃𝐿𝐶𝑆𝑀 ({𝑃𝐿𝐴𝐷𝐴}_{j}, {𝑃𝐿𝑃𝐼𝐴}_{j})} \end{matrix})$ (31)

Step 9. Implement the weighted sum of PLPDM (WSPLPDM) and weighted sum of PLNDM (WSPLNDM)

${𝑊𝑆𝑃𝐿𝑃𝐷𝑀}_{i} = \sum_{j = 1}^{n} k w_{j} \cdot {𝑃𝐿𝑃𝐷𝑀}_{i j}$ (32) ${𝑊𝑆𝑃𝐿𝑁𝐷𝑀}_{i} = \sum_{j = 1}^{n} k w_{j} \cdot {𝑃𝐿𝑁𝐷𝑀}_{i j}$ (33)

Step 10. Implement the normalized ${𝑊𝑆𝑃𝐿𝑃𝐷𝑀}_{i}$ and ${𝑊𝑆𝑃𝐿𝑁𝐷𝑀}_{i}$ :

${𝑁𝑊𝑆𝑃𝐿𝑃𝐷𝑀}_{i} = \frac{{𝑊𝑆𝑃𝐿𝑃𝐷𝑀}_{i}}{max_{i} ({𝑊𝑆𝑃𝐿𝑃𝐷𝑀}_{i})}$ (34) ${𝑁𝑊𝑆𝑃𝐿𝑁𝐷𝑀}_{i i} = 1 - \frac{{𝑊𝑆𝑃𝐿𝑁𝐷𝑀}_{i}}{max_{i} ({𝑊𝑆𝑃𝐿𝑁𝐷𝑀}_{i})}$ (35)

Step 11. Implement the probabilistic linguistic appraisal ranking values (PLARV).

${𝑃𝐿𝐴𝑅𝑉}_{i} = \frac{1}{2} ({𝑁𝑊𝑆𝑃𝐿𝑃𝐷𝑀}_{i} + {𝑁𝑊𝑆𝑃𝐿𝑁𝐷𝑀}_{i})$ (36)

Step 12. In line with PLARV, the higher PLARV is the optimal alternative.

4. Numerical example and comparative analysis

4.1 Numerical example

With the gradual increase in the influence of games, more and more people are using games as their main form of entertainment [57, 58]. According to the 2022 global gaming market report released by market research company Newzoo, in 2022, global gaming players reached nearly 3.2 billion, with a total gaming consumption of $196.8 billion. Each player has many gaming and payment options, and games are becoming truly accessible to everyone. The report mentioned that RPG games are currently the most popular form of games [59]. The first three RPG games that are most popular on PC platforms are Genshin Impact, World of Warcraft and Final Fantasy 14. Male players are the main gaming group, with players aged 10–20 accounting for up to 30% of the total. With the establishment of more comprehensive internet infrastructure and more affordable networking costs, the number of players will continue to grow [60, 61]. At present, the mainstream works in the game market have exquisite graphics, novel and interesting gameplay, and captivating storylines, which can meet users’ needs for various aspects of the game [2]. However, many games still need to be optimized in terms of game system interface design. Excellent game system interfaces can provide users with a smooth interactive experience, increase their immersion while playing, enhance their focus, and help a game gain recognition from players. The content of the game itself is the focus of the user experience, and the game system can ensure that users can smoothly experience the game content and achieve interaction between the game and people. The development of information technology has made online games popular worldwide, and the design of online game interfaces also greatly affects whether this game is loved by players [62, 63]. It can be said that online game interface design is a high-tech mass entertainment art that integrates various technologies such as computer graphics, game programming, electronic animation, music, art, etc. Software interface design mainly focuses on users’ convenient and accessible choices, while online game interfaces are more focused on fun interactions [64]. The interface design requirements for online games are stricter, and interface elements should be arranged based on various factors such as player identity, age, gender, etc. The design of game pages should fully consider the aesthetic perspective and players’ appreciation level. In summary, in the current game industry where various online games are flourishing, how to deeply explore players’ preferences and needs from multiple perspectives, and design novel and interesting online game interfaces is an urgent problem that every online game designer needs to solve [11]. The quality evaluation of online games interface design is the MAGDM issue. Therefore, the quality evaluation of online games interface design is presented to demonstrate the PL-EDAS technique developed in this paper. There is a panel with five online games $K G_{i} (i = 1, 2, 3, 4, 5)$ to choose. The experts select seven attributes (see Table 1 [65]) to assess the interface design quality of five online games.

Table 1
The seven attributes

Attributes	Description
$K P_{1}$	Guiding satisfaction for online games interface design
$K P_{2}$	Interactive usability for online games interface design
$K P_{3}$	Satisfaction with interaction mode for online games interface design
$K P_{4}$	Functional clarity for online games interface design
$K P_{5}$	Satisfaction with the design style of screen elements for online games interface design
$K P_{6}$	Sound effect satisfaction for online games interface design
$K P_{7}$	Creative novelty for online games interface design

All attributes are beneficial one. The five potential online games $K G_{i} (i = 1, 2, 3, 4, 5)$ are to be assessed through utilizing the linguistic term set:

$K L = {k l_{0} = very poor (V P), k l_{1} = 𝑝𝑜𝑜𝑟 (P), k l_{2} = 𝑚𝑒𝑑𝑖𝑢𝑚 (M),$ $k l_{3} = 𝑔𝑜𝑜𝑑 (G), k l_{4} = very good (V G))}$

through online review of some game enthusiasts and gamers. After that, the PL-EDAS technique is implemented to manage the quality evaluation of online games interface design.

Step 1. Implement the $K L = {(K L_{i j} (k p))}_{5 \times 7}$ online review of some game enthusiasts and gamers and these integrated data are managed in Table 2.

Table 2

Probabilistic linguistic information matrix

Alternatives	$K P_{1}$
$K G_{1}$	${k l_{0} (0.0909), k l_{1} (0.0909), k l_{2} (0.2727), k l_{3} (0.4545), k l_{4} (0.0909)}$
$K G_{2}$	${k l_{0} (0.0500), k l_{1} (0.0000), k l_{2} (0.4000), k l_{3} (0.1500), k l_{4} (0.4000)}$
$K G_{3}$	${k l_{0} (0.1000), k l_{1} (0.0000), k l_{2} (0.1000), k l_{3} (0.7000), k l_{4} (0.1000)}$
$K G_{4}$	${k l_{0} (0.1250), k l_{1} (0.0000), k l_{2} (0.4375), k l_{3} (0.3125), k l_{4} (0.1250)}$
$K G_{5}$	${k l_{0} (0.0000), k l_{1} (0.1000), k l_{2} (0.5000), k l_{3} (0.3000), k l_{4} (0.1000)}$
Alternatives	$K P_{2}$
$K G_{1}$	${k l_{0} (0.0000), k l_{1} (0.0909), k l_{2} (0.3636), k l_{3} (0.5455), k l_{4} (0.0000)}$
$K G_{2}$	${k l_{0} (0.0000), k l_{1} (0.1000), k l_{2} (0.3000), k l_{3} (0.3000), k l_{4} (0.3000)}$
$K G_{3}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.2000), k l_{3} (0.6000), k l_{4} (0.2000)}$
$K G_{4}$	${k l_{0} (0.0625), k l_{1} (0.1250), k l_{2} (0.5625), k l_{3} (0.1250), k l_{4} (0.1250)}$
$K G_{5}$	${k l_{0} (0.0000), k l_{1} (0.1000), k l_{2} (0.3000), k l_{3} (0.5000), k l_{4} (0.1000)}$
Alternatives	$K P_{3}$
$K G_{1}$	${k l_{0} (0.0000), k l_{1} (0.1818), k l_{2} (0.2727), k l_{3} (0.4545), k l_{4} (0.0909)}$
$K G_{2}$	${k l_{0} (0.0000), k l_{1} (0.0500), k l_{2} (0.3000), k l_{3} (0.4000), k l_{4} (0.2500)}$
$K G_{3}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.2000), k l_{3} (0.5000), k l_{4} (0.3000)}$
$K G_{4}$	${k l_{0} (0.0625), k l_{1} (0.1250), k l_{2} (0.3125), k l_{3} (0.3125), k l_{4} (0.1875)}$
$K G_{5}$	${k l_{0} (0.1000), k l_{1} (0.0000), k l_{2} (0.1000), k l_{3} (0.7000), k l_{4} (0.1000)}$
Alternatives	$K P_{4}$
$K G_{1}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.2727), k l_{3} (0.4545), k l_{4} (0.2727)}$
$K G_{2}$	${k l_{0} (0.0000), k l_{1} (0.1000), k l_{2} (0.4000), k l_{3} (0.1500), k l_{4} (0.3500)}$
$K G_{3}$	${k l_{0} (0.0000), k l_{1} (0.1000), k l_{2} (0.2000), k l_{3} (0.5000), k l_{4} (0.2000)}$
$K G_{4}$	${k l_{0} (0.1250), k l_{1} (0.1250), k l_{2} (0.1250), k l_{3} (0.3750), k l_{4} (0.2500)}$
$K G_{5}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.2000), k l_{3} (0.6000), k l_{4} (0.2000)}$
Alternatives	$K P_{5}$
$K G_{1}$	${k l_{0} (0.0000), k l_{1} (0.1818), k l_{2} (0.0909), k l_{3} (0.4545), k l_{4} (0.2727)}$
$K G_{2}$	${k l_{0} (0.0500), k l_{1} (0.0000), k l_{2} (0.2000), k l_{3} (0.2000), k l_{4} (0.5500)}$
$K G_{3}$	${k l_{0} (0.0000), k l_{1} (0.1000), k l_{2} (0.0000), k l_{3} (0.4000), k l_{4} (0.5000)}$
$K G_{4}$	${k l_{0} (0.0625), k l_{1} (0.1250), k l_{2} (0.3125), k l_{3} (0.3125), k l_{4} (0.1875)}$
$K G_{5}$	${w l_{0} (0.0000), w l_{1} (0.0000), w l_{2} (0.2000), w l_{3} (0.6000), w l_{4} (0.2000)}$
Alternatives	$K P_{6}$
$K G_{1}$	${k l_{0} (0.0000), k l_{1} (0.0909), k l_{2} (0.0909), k l_{3} (0.4545), k l_{4} (0.3636)}$
$K G_{2}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.2000), k l_{3} (0.2000), k l_{4} (0.6000)}$
$K G_{3}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.3000), k l_{3} (0.4000), k l_{4} (0.3000)}$
$K G_{4}$	${k l_{0} (0.0000), k l_{1} (0.1250), k l_{2} (0.3125), k l_{3} (0.3125), k l_{4} (0.2500)}$
$K G_{5}$	${k l_{0} (0.0000), k l_{1} (0.1000), k l_{2} (0.2000), k l_{3} (0.4000), k l_{4} (0.3000)}$
Alternatives	$K P_{7}$
$K G_{1}$	${k l_{0} (0.1818), k l_{1} (0.3636), k l_{2} (0.1818), k l_{3} (0.1818), k l_{4} (0.0909)}$
$K G_{2}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.3500), k l_{3} (0.2500), k l_{4} (0.4000)}$
$K G_{3}$	${k l_{0} (0.2000), k l_{1} (0.2000), k l_{2} (0.2000), k l_{3} (0.2000), k l_{4} (0.2000)}$
$K G_{4}$	${k l_{0} (0.0000), k l_{1} (0.3750), k l_{2} (0.3125), k l_{3} (0.0625), k l_{4} (0.2500)}$
$K G_{5}$	${k l_{0} (0.1000), k l_{1} (0.2000), k l_{2} (0.1000), k l_{3} (0.4000), k l_{4} (0.2000)}$

Step 2. Implement the PLADA (see Table 3).

Table 3

The PLADA

Attribute	PLADA
$K P_{1}$	${k l_{0} (0.0732), k l_{1} (0.0382), k l_{2} (0.3420), k l_{3} (0.3834), k l_{4} (0.1632)}$
$K P_{2}$	${k l_{0} (0.0125), k l_{1} (0.0832), k l_{2} (0.3452), k l_{3} (0.4141), k l_{4} (0.1450)}$
$K P_{3}$	${k l_{0} (0.0325), k l_{1} (0.0714), k l_{2} (0.2370), k l_{3} (0.4734), k l_{4} (0.1857)}$
$K P_{4}$	${k l_{0} (0.0250), k l_{1} (0.0650), k l_{2} (0.2395), k l_{3} (0.4159), k l_{4} (0.2545)}$
$K P_{5}$	${k l_{0} (0.0225), k l_{1} (0.0814), k l_{2} (0.1607), k l_{3} (0.3934), k l_{4} (0.3420)}$
$K P_{6}$	${k l_{0} (0.0000), k l_{1} (0.0632), k l_{2} (0.2207), k l_{3} (0.3534), k l_{4} (0.3627)}$
$K P_{7}$	${k l_{0} (0.0964), k l_{1} (0.2277), k l_{2} (0.2289), k l_{3} (0.2189), k l_{4} (0.2282)}$

Step 3. Implement the PLPIA (see Table 4).

Table 4

The PLPIA

Attribute	PLPIA
$K P_{1}$	${k l_{0} (0.0500), k l_{1} (0.0000), k l_{2} (0.4000), k l_{3} (0.1500), k l_{4} (0.4000)}$
$K P_{2}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.2000), k l_{3} (0.6000), k l_{4} (0.2000)}$
$K P_{3}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.2000), k l_{3} (0.5000), k l_{4} (0.3000)}$
$K P_{4}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.2000), k l_{3} (0.6000), k l_{4} (0.2000)}$
$K P_{5}$	${k l_{0} (0.0000), k l_{1} (0.1000), k l_{2} (0.0000), k l_{3} (0.4000), k l_{4} (0.5000)}$
$K P_{6}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.2000), k l_{3} (0.2000), w l_{4} (0.6000)}$
$K P_{7}$	${k l_{0} (0.0000), k l_{1} (0.0000), k l_{2} (0.3500), k l_{3} (0.2500), k l_{4} (0.4000)}$

Step 4. Implement the PLCSM between $K L_{i j} (k p)$ and ${𝑃𝐿𝑃𝐼𝐴}_{j}$ (see Table 5).

Table 5

The PLCSM

	$K P_{1}$	$K P_{2}$	$K P_{3}$	$K P_{4}$
$K G_{1}$	0.5836	0.8980	0.9030	0.9642
$K G_{2}$	1.0000	0.8668	0.9864	0.6675
$K G_{3}$	0.4406	1.0000	1.0000	0.9954
$K G_{4}$	0.7570	0.5868	0.9584	0.9506
$K G_{5}$	0.7108	0.9728	0.8806	1.0000
	$K P_{5}$	$K P_{6}$	$K P_{7}$
$K G_{1}$	0.9310	0.8725	0.7865
$K G_{2}$	0.9473	1.0000	1.0000
$K G_{3}$	1.0000	0.8513	0.9654
$K G_{4}$	0.8312	0.8480	0.9156
$K G_{5}$	0.8013	0.8591	0.8305

Table 6

The PLDSM

	$K P_{1}$	$K P_{2}$	$K P_{3}$	$K P_{4}$
$K G_{1}$	0.5726	0.8922	0.8749	0.9599
$K G_{2}$	1.0000	0.8468	0.9741	0.6567
$K G_{3}$	0.4356	1.0000	1.0000	0.9859
$K G_{4}$	0.7256	0.5340	0.8973	0.9164
$K G_{5}$	0.6856	0.9560	0.8771	1.0000
	$K P_{5}$	$K P_{6}$	$K P_{7}$
$K G_{1}$	0.8957	0.8511	0.5783
$K G_{2}$	0.9473	1.0000	1.0000
$K G_{3}$	1.0000	0.8067	0.8353
$K G_{4}$	0.7228	0.7501	0.8412
$K G_{5}$	0.7924	0.8060	0.8038

Table 7

The PLJSM

	$K P_{1}$	$K P_{2}$	$K P_{3}$	$K P_{4}$
$K G_{1}$	0.4011	0.8054	0.7775	0.9229
$K G_{2}$	1.0000	0.7344	0.9495	0.4889
$K G_{3}$	0.2784	1.0000	1.0000	0.9722
$K G_{4}$	0.5694	0.3642	0.8137	0.8457
$K G_{5}$	0.5216	0.9157	0.7811	1.0000
	$K P_{5}$	$K P_{6}$	$K P_{7}$
$K G_{1}$	0.8111	0.7408	0.4068
$K G_{2}$	0.8998	1.0000	1.0000
$K G_{3}$	1.0000	0.6761	0.7172
$K G_{4}$	0.5660	0.6002	0.7259
$K G_{5}$	0.6562	0.6750	0.6720

Step 5. Implement the PLDSM between $K L_{i j} (k p)$ and ${𝑃𝐿𝑃𝐼𝐴}_{j}$ (see Table 6).

Step 6. Implement the PLJSM between $K L_{i j} (k p)$ and ${𝑃𝐿𝑃𝐼𝐴}_{j}$ (see Table 7).

Step 7. Implement the attributes weight through employing information entropy (see Table 8).

Table 8

The weight information

	$K P_{1}$	$K P_{2}$	$K P_{3}$	$K P_{4}$
Weight information	0.4116	0.2109	0.0195	0.1223
	$K P_{5}$	$K P_{6}$	$K P_{7}$
Weight information	0.0698	0.0514	0.1145

Step 8. Implement the PLPDM from PLADA and PLNDM from PLADA (see Tables 9 and 10).

Table 9

The PLPDM from PLADA

	$K P_{1}$	$K P_{2}$	$K P_{3}$	$K P_{4}$
$K G_{1}$	0.0000	0.0000	0.0000	0.0100
$K G_{2}$	0.4234	0.0000	0.0109	0.0000
$K G_{3}$	0.0000	0.0759	0.0382	0.0499
$K G_{4}$	0.0000	0.0000	0.0000	0.0000
$K G_{5}$	0.0000	0.0247	0.0000	0.0634
	$K P_{5}$	$K P_{6}$	$K P_{7}$
$K G_{1}$	0.0000	0.0000	0.0000
$K G_{2}$	0.0029	0.1403	0.1086
$K G_{3}$	0.0608	0.0000	0.0000
$K G_{4}$	0.0000	0.0000	0.0000
$K G_{5}$	0.0000	0.0000	0.0000

Table 10

The PLNDM from PLADA

	$K P_{1}$	$K P_{2}$	$K P_{3}$	$K P_{4}$
$K G_{1}$	0.1907	0.0468	0.0816	0.0045
$K G_{2}$	0.1108	0.0848	0.0028	0.2543
$K G_{3}$	0.3208	0.0252	0.0117	0.0137
$K G_{4}$	0.0220	0.3238	0.0498	0.0240
$K G_{5}$	0.0686	0.0095	0.0879	0.0186
	$K P_{5}$	$K P_{6}$	$K P_{7}$
$K G_{1}$	0.0437	0.0413	0.2250
$K G_{2}$	0.0094	0.0434	0.0434
$K G_{3}$	0.0202	0.0761	0.0331
$K G_{4}$	0.1696	0.1091	0.0465
$K G_{5}$	0.1462	0.0737	0.0985

Step 9. Implement the WSPLPDM and WSPLNDM (see Table 11).

Table 11

The WSPLPDM and WSPLNDM

	WSPLPDM	WSPLNDM
$K G_{1}$	0.0012	0.1214
$K G_{2}$	0.1943	0.1025
$K G_{3}$	0.0271	0.1484
$K G_{4}$	0.0000	0.1040
$K G_{5}$	0.0130	0.0595

Step 10. Calculate the NWSPLPDM and NWSPLNDM (see Table 12).

Table 12

The NWSPLPDM and NWSPLNDM

	NWSPLPDM	NWSPLNDM
$K G_{1}$	0.0063	0.1817
$K G_{2}$	1.0000	0.3091
$K G_{3}$	0.1395	0.0000
$K G_{4}$	0.0000	0.2989
$K G_{5}$	0.0667	0.5991

Step 11. Implement the PLARV (see Table 13).

Table 13

The PLARV

	PLARV	Order
$K G_{1}$	0.0940	4
$K G_{2}$	0.6546	1
$K G_{3}$	0.0698	5
$K G_{4}$	0.1495	3
$K G_{5}$	0.3329	2

Step 12. In line with the PLARV, the order is: $K G_{2} > K G_{5} > K G_{4} > K G_{1} > K G_{3}$ and $K G_{2}$ is the best online game.

4.2 Comparative analysis

Then, the PL-EDAS technique is compared with PL-TOPSIS technique [66], PL-CODAS technique [67], PL-GRA technique [68], probabilistic linguistic TODIM (PL-TODIM) technique [69], probabilistic linguistic MABAC (PL-MABAC) technique [70], probabilistic linguistic TODIM-VIKOR (PL-TODIM-VIKOR) technique [71]. The comparative results are dealt with Table 14 and Fig. 2.

Table 14
Order for different techniques

Different techniques	Order
PL-TOPSIS technique [66]	$K G_{2} > K G_{5} > K G_{4} > K G_{1} > K G_{3}$
PL-CODAS technique [67]	$K G_{2} > K G_{5} > K G_{4} > K G_{1} > K G_{3}$
PL-GRA technique [68]	$K G_{2} > K G_{5} > K G_{4} > K G_{1} > K G_{3}$
PL-TODIM technique [69]	$K G_{2} > K G_{4} > K G_{5} > K G_{1} > K G_{3}$
PL-MABAC technique [70]	$K G_{2} > K G_{5} > K G_{4} > K G_{1} > K G_{3}$
PL-TODIM-VIKOR technique [71]	$K G_{2} > K G_{4} > K G_{5} > K G_{1} > K G_{3}$
The proposed PL-EDAS technique	$K G_{2} > K G_{5} > K G_{4} > K G_{1} > K G_{3}$

Figure 2.

Order for different techniques.

From the above analysis, it could be known that the order of these techniques is same to the order of the PL-TOPSIS technique [66], PL-CODAS technique [67], PL-GRA technique [68] and PL-MABAC technique [70]; the order of these techniques is slightly different from the order of PL-TODIM technique [69] and PL-TODIM-VIKOR technique [71]. However, these techniques have the same optimal online game and worst online game. This verifies the proposed PL-EDAS technique is reasonable and effective. Furthermore, the reason for the slight difference is that PL-TODIM technique [69] emphasized the psychological behavior of decision-makers, while PL-TODIM-VIKOR technique [71] emphasized both psychological behavior of decision-makers and group maximum utility & individual regret value. The differences of proposed PL-EDAS technique from existing PL-EDAS technique [72] are outlined: (1) the weight of proposed PL-EDAS technique is constructed by entropy technique based on the based on CSM technique, DSM technique and JSM technique; while the weight of existing PL-EDAS technique [72] is constructed by entropy technique based on the expected values; (2) The key steps of PLPDM and PLNDM of proposed PL-EDAS technique is constructed based on the based on CSM technique, DSM technique and JSM technique; while The key steps of PLPDM and PLNDM of existing PL-EDAS technique [72] is constructed by based on expected values and Hamming distance.

5. Conclusion

The development of information technology has made online games popular worldwide, and the design of online game interfaces also greatly affects whether this game is loved by players. It can be said that online game interface design is a high-tech mass entertainment art that integrates various technologies such as computer graphics, game programming, electronic animation, music, art, etc. Especially in recent years, various online game competitions have become increasingly frequent and specialized, and designers have subverted the design principles of traditional single player games by designing online game interfaces. They are constantly developing in the process of exploring new interactive and humanized approaches. The quality evaluation of online games interface design is MAGDM problem. Recently, the EDAS technique, CSM technique, DSM technique, JSM technique and entropy technique has been separately employed to cope with MAGDM issues. The PLTSs are employed as a tool for conveying uncertain information during the quality evaluation of online games interface design. In this paper, the EDAS technique is expanded to the PLTSs and the PL-EDAS technique based on CSM technique, DSM technique and JSM technique is constructed to manage MAGDM issue. The information entropy technique is employed to implement the weight values based on CSM technique, DSM technique and JSM technique under PLTSs. Finally, the quality evaluation of online games interface design is employed to demonstrate the PL-EDAS technique and some comparative analysis is employed to demonstrate the PL-EDAS technique. Thus, the main research contribution of this work is constructed: (1) the information entropy technique is constructed to implement the attribute weight values based on CSM technique, DSM technique and JSM technique; (2) the PL-EDAS technique is constructed under PLTSs based on CSM technique, DSM technique and JSM technique; (3) an example for quality evaluation of online games interface design is employed to verify PL-EDAS technique and several decision comparative analysis is employed to verify the PL-EDAS technique.

There may be some possible limitations for quality evaluation of online games interface design, which could be further studied in our future research for quality evaluation of online games interface design: (1) It is a worthwhile research work to manage consensus issues [73, 74, 75, 76] for quality evaluation of online games interface design under PLTSs and manage regret theory [77, 78, 79, 80] under PLTSs; (2) At present, there is a contradiction in the interface design of online games, which is that for the development of online games, designers hope to present all the content in the game to players, so that players can quickly grasp the attraction points and strengthen their attractiveness to players. However, at the same time, players are unable to make choices when facing complex information, and the surface of the game screen is also occupied by a large number of menus and charts, resulting in a reduced operating space for players and a sense of fatigue. In response to the above issues, the interface design process should follow the principle of simplification, and designers should fully consider the player’s feelings, scientifically analyze people’s memory patterns, as well as attention and other issues. (3) With the continuous development of science and technology, online games have gradually become an emerging industry that can earn more profits for businesses. As a popular entertainment project, it has gradually occupied people’s daily lives. Online games have gradually become a new economic growth point in the new era, and the competitive pressure in the field is also increasing. interface design is the first impression that players have of games, which plays an important role in the development of games. Therefore, relevant research should be strengthened.

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Enhanced EDAS technique for quality evaluation of online games interface design based on three types of probabilistic linguistic similarity measures

Abstract

Keywords

1. Introduction

2. Preliminaries

4.1 Numerical example

Table 1 The seven attributes

Table 14 Order for different techniques

References

Table 1
The seven attributes

Table 14
Order for different techniques