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Abstract

Since the reform and opening up, China’s foreign trade has developed rapidly. The import and export trade volume has doubled, and the team of foreign trade enterprises is constantly expanding. After China’s accession to the WTO and the liberalization of foreign trade management rights, a group of small and medium-sized foreign trade enterprises have grown up in recent years and are becoming a new force in the development of China’s foreign trade industry. So, how to make these enterprises grow and grow in the international market competition, and focus on cultivating the core competitiveness of small and medium-sized foreign trade enterprises, has become a strategic issue related to the overall development of foreign trade. Recently, COmbinative Distance-based ASsessment (CODAS) method computes the Euclidean distances as the primary measure and Hamming distances as the secondary measure to assess alternatives based on predetermined attribute. Moreover, the probabilistic linguistic term sets (PLTSs) are effective tool for depicting uncertainty of the multiple attribute group decision making (MAGDM) problems. The core competitiveness evaluation of small and medium-sized foreign trade enterprises is a classical MAGDM. In this paper, the CODAS method is extended to the MAGDM with PLTSs. Firstly, the definition of PLSs is introduced. Then, on the basis of the classical CODAS method, the PL-CODAS method is proposed to cope with MAGDM under PLTSs and its significant characteristic is that it can fully consider PLED and PLHD. Finally, a practical example for core competitiveness evaluation of small and medium-sized foreign trade enterprises is given to verify the developed approach and some comparative analysis was also given to verify the PL-CODAS approach.

Keywords

Multiple attribute group decision making (MAGDM)probabilistic linguistic term sets (PLTSs)CODAS method core competitiveness evaluation

1. Introduction

According to data from the National Bureau of Economic Statistics, in 2013, China’s total import and export volume of goods reached 258 168 A historical high of 8.9 billion yuan, with small and medium-sized foreign trade enterprises contributing up to 60% [1]. However, small and medium-sized foreign trade enterprises have always been difficult to form a scale. They are inherently weak and cannot reach the basic scale, that is, diseconomies of scale are very prominent [2, 3, 4]. In addition, small and medium-sized foreign trade enterprises lack independent brands, and their products are often directly labeled as “Made in China”. Enterprises can only earn a small processing fee, put in a lot, and receive little return, always in a state of “working” for the world. China’s small and medium-sized foreign trade enterprises only emerged on the international economic stage in the 1990s after the country opened up its foreign trade management rights, with a history of only about 20 years [5, 6, 7]. Due to their youth, small and medium-sized foreign trade enterprises inevitably lack experience in business management and foreign trade practices. Another prominent issue faced by small and medium-sized foreign trade enterprises is “short lifespan”, which means a shorter life cycle. According to data from the “China Private Enterprise Development Report”, the average lifespan of small and medium-sized foreign trade enterprises is less than 3 years; The average annual growth of small and medium-sized foreign trade enterprises is about 150000, but nearly 50000 have gone bankrupt; 60% go bankrupt within 5 years, and 80% disappear within 10 years. Small and medium-sized foreign trade enterprises in China mainly rely on simple processing or reprocessing to earn meager profits, and their innovation ability is not strong [8, 9]. The vast majority of exported products belong to low value-added goods with low technological content. The key reason is that one cannot master the core technology; In addition, the government also lacks corresponding incentive policies for small and medium-sized foreign trade enterprises in terms of policy support [10, 11]. Both internal and external factors lead to weak competitiveness of small and medium-sized foreign trade enterprises, resulting in poor risk resistance and low survival rate. Many small and medium-sized foreign trade enterprises lack management experience due to their short time in conducting foreign trade business. Managers tend to be conservative in their thinking and have a fixed mindset, pursuing short-term profit maximization without considering long-term development. In the international market competition, we adhere to the business strategy of good quality and low price, and long-term use low-priced competition as the main means of expanding the market [12, 13, 14]. We do what we do to make money, lack sufficient research on the market, and cannot reasonably determine the marketing methods that are suitable for our own characteristics. We also lack the ability to cultivate and enhance the core competitiveness of the enterprise from aspects such as product research and development, brand building, and after-sales service.

There are many small and medium-sized foreign trade enterprises in a state of extensive management. On the other hand, for a long time, small and medium-sized foreign trade enterprises have not received strong policy support, and the quality and efficiency of government services are difficult to meet the needs of production, operation, and export-oriented development of small and medium-sized foreign trade enterprises, putting enormous pressure on their survival and development [15, 16]. For example, regarding the “export tax refund” policy, many small and medium-sized foreign trade enterprises need to enjoy it after the tax refund of large enterprises, and the approval process is complex and strict. It is generally believed that one of the important reasons why small and medium-sized enterprises can maintain their longevity in fierce market competition is because they possess a certain core competitiveness [17, 18, 19]. Therefore, the research on the evaluation of core competitiveness of small and medium-sized enterprises naturally has important theoretical value and practical significance. Small and medium-sized foreign trade enterprises refer to small and medium-sized enterprises engaged in foreign trade activities, which have two basic characteristics: “small and medium-sized” and “foreign trade”. In fact, different countries have different definitions of small and medium-sized foreign trade enterprises [20, 21, 22]. The definition of small and medium-sized foreign trade enterprises in China is based on sales revenue. Any enterprise with import and export business rights with sales revenue below 300 million yuan is called a small and medium-sized foreign trade enterprise. At the beginning of 2014, there were approximately 15 million small and medium-sized foreign trade enterprises in China. Small and medium-sized foreign trade enterprises are an indispensable component of a country’s economic development and foreign trade, and are also the backbone of improving international competitiveness and influence. Its development also plays a crucial role in increasing employment, stabilizing social structure, and promoting technological development [23, 24]. Especially in China, small and medium-sized foreign trade enterprises are relatively active and important economic units, with export growth of over 40% for five consecutive years. Their important role in the economy is self-evident. However, behind the impressive achievements, small and medium-sized foreign trade enterprises face many unavoidable and urgent problems in terms of their core competitiveness, which is related to their survival and development destiny [25, 26].

Due to the complexity and uncertainty of society, it is difficult for an expert or decision-maker to consider all aspects of a MAGDM problem [27, 28, 29, 30, 31]. Therefore, in order to consider more fully and make decisions more reasonable in a complex environment, it is necessary for multiple decision-makers to jointly provide their professional knowledge and experience [32, 33, 34, 35, 36]. Considering that decision-makers are often influenced by their knowledge structure, personal preferences, and familiarity with the solution when making decisions, different decision-makers often have different levels of understanding and conclusions about the decision-making problem, sometimes with significant differences [37, 38, 39, 40, 41, 42]. Similar to MAGDM problem, the Multicriteria Decision Analysis (MCDA) is an important research domain in modern management science [43, 44, 45]. There are some MCDA methods are designed to manage the decision issues, such as, Data vARIability Assessment TOPSIS (DARIA-TOPSIS method) [46], Characteristic Objects Method (COMET) [47, 48], Stable Preference Ordering Towards Ideal Solution (SPOTIS) method [49], Sequential Interactive Modelling for Urban Systems (SIMUS) method [43, 44, 50, 51] and RANking COMparison (RANCOM) method [45]. In order to depict the uncertain information, Herrera and Martinez [52] built the 2-tuple fuzzy linguistic representation model. Rodriguez, Martinez and Herrera [53] built the hesitant fuzzy linguistic term sets s (HFLTSs). Pang, Wang and Xu [54] built the probabilistic linguistic term sets (PLTSs). Recently, PLTSs have become a very hot topic in the area of HFLTSs [55, 56, 57, 58, 59] and hesitant fuzzy sets (HFSs) [60, 61, 62, 63]. The core competitiveness evaluation of small and medium-sized foreign trade enterprises is a classical MAGDM. Recently, CODAS method [64] computes the Euclidean distances as the primary measure and Hamming distances as the secondary measure to assess alternatives based on predetermined attribute. Moreover, the PLTSs [54] are effective tool for depicting uncertainty during the core competitiveness evaluation of small and medium-sized foreign trade enterprises. But there is no study on the CODAS method [65] for MAGDM under PLTSs in the existing literature. Therefore, it is necessary to pay attention to this issue. The aim of this paper is to extend the CODAS method to solve the MAGDM with the PLTSs. The main originality and research motivation of the paper are summarized: (1) the CODAS method is extended under PLTSs; (2) the probabilistic linguistic CODAS (PL-CODAS) method is proposed to cope with the MAGDM under PLTSs; (3) a numerical example for core competitiveness evaluation of small and medium-sized foreign trade enterprises is proposed to verify the developed approach; (4) some comparative studies are given to verify the rationality of PL-CODAS method.

The remainder of this paper is set out as follows. Section 2 presents the concepts related to PLTSs. In Section 3 the PL-CODAS method is proposed for MAGDM under PLTSs. In Section 4, an illustrative example for core competitiveness evaluation of small and medium-sized foreign trade enterprises is pointed out and some comparative analysis is conducted. The study ends with some conclusions in Section 5.

2. Preliminaries

The concept of PLTS [54] is introduced.

Definition 1 [66]. Let $ZL=\{{zl_{z\alpha}|z\alpha=-z\theta,\ldots,-2,-1,0,1,2,\ldots z\theta}\}$ be an LTS, the linguistic terms $l_{\alpha}$ express the equivalent information to $\beta$ with the function $z g$ :

$\displaystyle zg:[{zl_{-z\theta},zl_{-z\theta}}]\to[{0,1}],zg({zl_{z\alpha}})=% \frac{\alpha+z\theta}{2z\theta}=\beta$ (1)

$\beta$ expressed the equivalent information to $zl_{z\alpha}$ with the function $zg^{-1}$ :

$\displaystyle zg^{-1}:[{0,1}]\to[{zl_{-z\theta},zl_{-z\theta}}],zg^{-1}(\beta)% =zl_{({2\beta-1})z\theta}=zl_{z\alpha}$ (2)

Definition 2 [54]. Given an LTS $ZL=\{{zl_{z\alpha}|z\alpha=-z\theta,\ldots,-2,-1,0,1,2,\ldots z\theta}\}$ , a PLTS based on $Z L$ is defined as:

$\displaystyle ZL(p)=\left\{{zl^{({z\phi})}({p^{({z\phi})}})\left|{zl^{({z\phi}% )}\in ZL,p^{({z\phi})}\geqslant 0,z\phi=1,2,\ldots,\#ZL(p),\sum\limits_{z\phi=% 1}^{\#ZL(p)}{p^{({z\phi})}\leqslant 1}}\right.}\right\}$ (3)

where $zl^{({z\phi})}({p^{({z\phi})}})$ is the $z\phi$ th linguistic term $zl^{({z\phi})}$ with the probability $p^{({z\phi})}$ , and $\#ZL(p)$ is the number $ZL(p)$ . The linguistic term $zl^{({z\phi})}$ in $ZL(p)$ are listed with ascending order.

Pang, Wang and Xu [54] normalized the PLTS $ZL(p)$ as:

$\displaystyle ZL({zp})$ $\displaystyle=\left\{{zl^{({z\phi})}({zp^{({z\phi})}})\left|{zl^{({z\phi})}\in ZL% ,zp^{({z\phi})}\geqslant 0,\;z\phi=1,2,\ldots,\#ZL({zp}),\sum\limits_{z\phi=1}% ^{\#ZL({zp})}{zp^{({z\phi})}=1}}\right.}\right\},$

where $zp^{({z\phi})}={p^{({z\phi})}}\left/{\sum\limits_{z\phi=1}^{\#ZL(p)}{p^{({z% \phi})}}}\right.$ for all $z\phi=1,2,\ldots,\#ZL({zp})$ .

Definition 3 [54]. Let $ZL=\{{zl_{z\alpha}|z\alpha=-z\theta,\ldots,-2,-1,0,1,2,\ldots z\theta}\}$ be an LTS, $ZL_{1}({zp})=\{zl_{1}^{({z\phi})}\linebreak({zp_{1}^{({z\phi})}})|{z\phi=1,2,% \ldots,\#ZL_{1}({zp})}\}$ and $ZL_{2}({zp})=\{{zl_{2}^{({z\phi})}({zp_{2}^{({z\phi})}})|{z\phi=1,2,\ldots,\#% ZL_{2}({zp})}}\}$ , where $\#ZL_{1}({zp})$ and $\#ZL_{2}({zp})$ are the numbers of $ZL_{1}({zp})$ and $ZL_{2}({zp})$ . If $\#ZL_{1}({zp})>\#ZL_{2}({zp})$ , then add $\#ZL_{1}({zp})-\#ZL_{2}({zp})$ linguistic terms to $ZL_{2}({zp})$ . The added linguistic terms should be the smallest one in $ZL_{2}({zp})$ and the probabilities values should be zero.

Definition 4 [54]. For a PLTS $ZL({zp})=\{{zl^{({z\phi})}({zp^{({z\phi})}})|{z\phi=1,2,\ldots,\#ZL({zp})}}\}$ , the score function $SF({ZL({zp})})$ and deviation degree $DD({ZL({zp})})$ is defined:

$\displaystyle SF({ZL({zp})})={\sum\limits_{z\phi=1}^{\#ZL({zp})}{zg({ZL({zp})}% )zp^{({z\phi})}}}\left/{\sum\limits_{z\phi=1}^{\#ZL_{(}{zp})}{zp^{({z\phi})}}}\right.$ (4) $\displaystyle DD({ZL({zp})})=\left.{\sqrt{\sum\limits_{z\phi=1}^{\#zL({zp})}{(% {zg({ZL({zp})})zp^{({z\phi})}-SF({ZL({zp})})})^{2}}}}\right/{\sum\limits_{z% \phi=1}^{\#ZL_{(}{zp})}{zp^{({z\phi})}}}$ (5)

According to the Eqs (4) and (5), the order is defined as: (1) if $SF({ZL_{1}({zp})})>SF({ZL_{2}({zp})})$ , then $ZL_{1}({zp})>ZL_{2}({zp})$ ; (2) if $SF({ZL_{1}({zp})})=SF({ZL_{2}({zp})})$ , then if $DD({ZL_{1}({zp})})=DD({ZL_{2}({zp})})$ , $ZL_{1}({zp})=ZL_{2}({zp})$ ; if $DD({ZL_{1}({zp})})<DD({ZL_{2}({zp})})$ , $ZL_{1}({zp})>ZL_{2}({zp})$ .

Definition 5 [67]. Let $ZL=\{{zl_{z\alpha}|{z\alpha=-z\theta,\ldots,-2,-1,0,1,2,\ldots z\theta}}\}$ be an LTS. And let $ZL_{1}({zp})=\{{zl_{1}^{({z\phi})}({zp_{1}^{({z\phi})}})|{z\phi=1,2,\ldots,\#% ZL_{1}({zp})}}\}$ and $ZL_{2}({zp})=\{{zl_{2}^{({z\phi})}({zp_{2}^{({z\phi})}})|{z\phi=1,2,\ldots,\#% ZL_{2}({zp})}}\}$ be two PLTSs with $\#ZL_{1}({zp})=\#ZL_{2}({zp})=ZL({zp})$ , then the Euclidean distance $\textit{PLED}(ZL_{1}({zp}),\linebreak ZL_{2}({zp}))$ and Hamming distance $\textit{PLHD}({ZL_{1}({zp}),ZL_{2}({zp})})$ is defined:

$\displaystyle\textit{PLED}({ZL_{1}({zp}),ZL_{2}({zp})})=\sqrt{\left.{\sum% \limits_{z\phi=1}^{\#ZL({zp})}{({zp_{1}^{(\phi)}zg({zl_{1}^{({z\phi})}})-zp_{2% }^{({z\phi})}zg({zl_{2}^{({z\phi})}})})^{2}}}\right/{\#ZL({zp})}}$ (6) $\displaystyle\textit{PLHD}({ZL_{1}({zp}),ZL_{2}({zp})})=\left.{\sum\limits_{z% \phi=1}^{\#ZL({zp})}{\left|{zp_{1}^{(\phi)}zg({zl_{1}^{({z\phi})}})-zp_{2}^{({% z\phi})}zg({zl_{2}^{({z\phi})}})}\right|}}\right/{\#ZL({zp})}$ (7)

3. CODAS method for probabilistic linguistic MAGDM problems

In this section, the probabilistic linguistic CODAS (PL-CODAS) method is proposed for MAGDM.

3.1 MAGDM problems description

The following notations are employed to express the probabilistic linguistic MAGDM Let $ZA=\{{ZA_{1},ZA_{2},\ldots,ZA_{m}}\}$ be alternatives, and $ZG=\{{ZG_{1},ZG_{2},\ldots,ZG_{n}}\}$ with weight information $zw=({zw_{1},zw_{2},\ldots,zw_{n}})$ , where $zw_{j}\in[{0,1}]$ , $\sum\limits_{j=1}^{n}{zw_{j}}=1$ , and a set of experts $ZE=\{{ZE_{1},ZE_{2},\ldots,ZE_{q}}\}$ . Suppose that the values are assessed as linguistic expressions $zl_{ij}^{k}(i=1,2,\ldots,m,j=1,2,\ldots,n,k=1,2,\ldots,q)$ .

Then, the PL-CODAS is proposed to solve MAGDM. The calculating steps are involved as follows:

Step 1. Transform the linguistic variables $zl_{ij}^{k}$ into probabilistic linguistic matrix $ZL=({ZL_{ij}({zp})})_{m\times n}$ , $ZL_{ij}(p)=\{{zl_{ij}^{({z\phi})}({p_{ij}^{({z\phi})}})|{z\phi=1,2,\ldots,\#ZL% _{ij}(p)}}\}({i=1,2,\ldots,m,j=1,2,\ldots,n})$ .

Step 2. Derive the normalized probabilistic linguistic matrix $ZL_{ij}({zp})=\{zl_{ij}^{({z\phi})}({zp_{ij}^{({z\phi})}})|z\phi=1,2,% \linebreak\ldots,\#ZL_{ij}({zp})\}$ .

Step 4. Compute the weight with entropy.

Entropy [68] is a conventional tool which is also used to determine weight of attributes. Firstly, the normalized matrix $\textit{NZL}_{ij}({zp})$ is derived [69]:

$\displaystyle\textit{NZL}_{ij}({zp})=\frac{\sum\limits_{z\phi=1}^{\#Z\tilde{L}% ({zp})}{({zg({zl_{ij}^{({z\phi})}})})}}{\sum\limits_{i=1}^{m}{\sum\limits_{z% \phi=1}^{\#ZL({zp})}{({zg({zl_{ij}^{({z\phi})}})})}}},j=1,2,\ldots,n,$ (8)

Then, the information of Shannon entropy $\textit{ISE}=({\textit{ISE}_{1},\textit{ISE}_{2},\ldots,\textit{ISE}_{n}})$ is calculated:

$\displaystyle\textit{ISE}_{j}=-\frac{1}{\ln m}\sum\limits_{i=1}^{m}{\textit{% NZL}_{ij}({zp})\ln\textit{NZL}_{ij}({zp})}$ (9)

and $\textit{NZL}_{ij}({zp})\ln\textit{NZL}_{ij}({zp})=0$ , if $\textit{NZL}_{ij}({zp})=0$ .

Finally, the attribute weights $zw=({zw_{1},zw_{2},\ldots,zw_{n}})$ is computed:

$\displaystyle zw_{j}=\frac{1-\textit{ISE}_{j}}{\sum\limits_{j=1}^{n}{({1-% \textit{ISE}_{j}})}},j=1,2,\ldots,n.$ (10)

Step 3. Obtain the probabilistic linguistic negative ideal solution (PLNIS):

$\displaystyle\textit{PLNIS}=[{\textit{PLNIS}_{j}}]_{1\times n},$ (11) $\displaystyle\textit{PLNIS}_{j}=\{{zl_{j}^{({z\phi})}({zp_{j}^{({z\phi})}})|{z% \phi=1,2,\ldots,\#ZL_{ij}({zp})}}\}$ (12) $\displaystyle SF({\textit{PLNIS}_{j}})=\mathop{\min}\limits_{i}SF({ZL_{ij}({zp% })})$ (13)

Step 4. Determine the probabilistic linguistic Euclidean distance (PLED) and probabilistic linguistic Hamming distance (PLHD):

$\displaystyle\textit{PLED}({ZL_{i}({zp}),\textit{PLNIS}})=\sum\limits_{j=1}^{n% }{zw_{j}\textit{PLED}({ZL_{ij}({zp}),\textit{PLNIS}_{j}})}$ (14) $\displaystyle\textit{PLHD}({ZL_{i}({zp}),\textit{PLNIS}})=\sum\limits_{j=1}^{n% }{zw_{j}\textit{PLHD}({ZL_{ij}({zp}),\textit{PLNIS}_{j}})}$ (15)

Step 5. Construct the probabilistic linguistic relative assessment matrix (PLRAM):

$\displaystyle\textit{PLRAM}=[{\textit{PLH}_{ik}}]_{m\times m}$ (16) $\displaystyle\textit{PLH}_{ik}=({\textit{PLED}({ZL_{i}({zp}),\textit{PLNIS}_{j% }})-\textit{PLED}({ZL_{k}^{(}{zp}),\textit{PLNIS}_{j}})})$ (17) $\displaystyle\quad{}+\left({\begin{array}[]{l}zf({\textit{PLED}({ZL_{i}({zp}),% \textit{PLNIS}_{j}})-\textit{PLED}({ZL_{k}({zp}),\textit{PLNIS}_{j}})})\\ {}\times({\textit{PLHD}({ZL_{i}({zp}),\textit{PLNIS}_{j}})-\textit{PLHD}({ZL_{% k}({zp}),\textit{PLNIS}_{j}})})\\ \end{array}}\right),$

$z f$ denotes a threshold function which is defined:

$\displaystyle zf({zx})=\left\{{\begin{array}[]{ll}1&\text{if }|{zx}|\geqslant z% \tau\\ 0&\text{if }|{zx}|<z\tau\\ \end{array}}\right.,$ (18)

where $z\tau$ is a threshold value and $0.01\leqslant z\tau\leqslant 0.05$ . In this paper, $z\tau=0.02$ .

Step 6. Calculate the probabilistic linguistic assessment score $\textit{PLAS}_{i}$ .

$\displaystyle\textit{PLAS}_{i}=\sum\limits_{k=1}^{m}{\textit{PLH}_{ik}}.$ (19)

Step 7. Rank the alternatives according to $\textit{PLAS}_{i}$ . The alternative with highest $\textit{PLAS}_{i}$ is the best.

4. Numerical example and comparative analysis

4.1 Numerical example

The deepening of the world economic integration pattern will further affect the development of China’s foreign trade. With the increase in foreign trade of small and medium-sized foreign trade enterprises in China, in order to ensure the normal expansion of their foreign trade business and market expansion, they need to have their own core competitiveness in order to win in the fierce global competition. When dealing with such competition, they may face the dilemma of weak core competitiveness and insufficient strength. Moreover, many small and medium-sized foreign trade enterprises have a biased understanding, believing that as long as they ensure low prices, they can ensure a competitive advantage. What is lacking in this idea is the construction of corresponding core competitiveness, and enterprises that lack core competitiveness will inevitably not be able to develop for a long time. The competition between enterprises ultimately belongs to talent competition, but for small and medium-sized foreign trade enterprises, talent reserves have become a problem. Firstly, its own foundation is relatively weak, making it difficult to recruit excellent specialized talents in many cases. Moreover, many small and medium-sized foreign trade enterprises may experience a large number of talent loss issues in their daily operations due to the lack of various individual incentive models. Such problems will inevitably lead to a further decline in one’s own competitiveness, ultimately leading to a lack of competitiveness and a disadvantage in competition. Innovation ability is a very important ability for any enterprise, and it is also the driving force and foundation of enterprise development. In the operation of small and medium-sized foreign trade enterprises, innovation includes various types of innovation models, such as marketing innovation and technological content innovation, which belong to the overall innovation ability of the enterprise. Through effective innovation, the core products of the enterprise can be effectively produced, forming core competitiveness. In general, foreign trade enterprises need to find corresponding export trade agents, who are mainly responsible for corresponding export processing, which makes foreign trade enterprises constrained by exporters and makes their decision-making difficult, leading to various problems in their own operations. This type of problem belongs to the issue of management philosophy, which has led to a decline in the actual core competitiveness of China’s small and medium-sized foreign trade enterprises. Risk resilience is the ability of modern enterprises to cope with various external market risk factors. Many small and medium-sized foreign trade enterprises cannot properly cater to national policies and establish their own risk prevention and control systems in their development, which can lead to systemic business problems once they encounter market risks. The ability to resist risks may vary depending on the enterprise, and of course, it may also differ from the market environment. For small and medium-sized foreign trade enterprises in China, it is necessary to recognize the important role of risk resilience.

Table 1
Linguistic matrix by the first expert

	ZG₁	ZG₂	ZG₃	ZG₄	ZG₅	ZG₆
ZA₁	ZG	ZVG	ZVP	ZEP	ZVG	ZP
ZA₂	ZVP	ZVG	ZG	ZEG	ZG	ZEG
ZA₃	ZEG	ZG	ZP	ZEG	ZEG	ZEG
ZA₄	ZVG	ZVP	ZVG	ZG	ZP	ZEG
ZA₅	ZG	ZEG	ZP	ZVP	ZEG	ZEG

Table 2

Linguistic decision matrix by the second expert

	ZG₁	ZG₂	ZG₃	ZG₄	ZG₅	ZG₆
ZA₁	ZEG	ZEG	ZVP	ZVP	ZVP	ZVG
ZA₂	ZP	ZVG	ZEG	ZEG	ZEG	ZG
ZA₃	ZVG	ZG	ZP	ZEG	ZVG	ZG
ZA₄	ZVG	ZP	ZEG	ZEG	ZG	ZP
ZA₅	ZG	ZEG	ZP	ZP	ZVP	ZVG

Table 3

Linguistic decision matrix by the third expert

	ZG₁	ZG₂	ZG₃	ZG₄	ZG₅	ZG₆
ZA₁	ZEG	ZEG	ZVP	ZEP	ZEG	ZG
ZA₂	ZVP	ZVG	ZEG	ZVG	ZVG	ZVP
ZA₃	ZEG	ZG	ZM	ZEG	ZG	ZP
ZA₄	ZVG	ZP	ZVG	ZVG	ZVP	ZEG
ZA₅	ZG	ZEG	ZP	ZP	ZEG	ZP

The core competitiveness evaluation of small and medium-sized foreign trade enterprises is a MAGDM. Thus, in this section we present a numerical example for core competitiveness evaluation of small and medium-sized foreign trade enterprises to illustrate the method proposed in this paper. There is a panel with five possible small and medium-sized foreign trade enterprises $ZA_{i}({i=1,2,3,4,5})$ to select. The experts select six attribute to evaluate the five possible small and medium-sized foreign trade enterprises: ⟀ ZG₁ is the management innovation ability; ⟁ ZG₂ is the technological innovation capability; ⟂ ZG₃ is the risk resilience; ⟃ ZG₄ is the management level of the enterprise; ⟄ ZG₅ is product quality; ⟅ ZG₆ is marketing channels innovation. The five possible small and medium-sized foreign trade enterprises $ZA_{i}({i=1,2,3,4,5})$ are to be evaluated by using the linguistic term set

$\displaystyle ZL=\{zl_{-3}=\textit{extremely poor}(\textit{ZEP}),zl_{-2}=% \textit{very poor}(\textit{ZVP}),$ $\displaystyle zl_{-1}=\textit{poor}(ZP),zl_{0}=\textit{medium}(ZM),zl_{1}=% \textit{good}(ZG),$ $\displaystyle zl_{2}=\textit{very good}(\textit{ZVG}),zl_{3}=\textit{extremely% good}(\textit{ZEG})\}$

by the five DMs under four attributes, as listed in the Tables 1–5.

Table 4

Linguistic decision matrix by the fourth expert

	ZG₁	ZG₂	ZG₃	ZG₄	ZG₅	ZG₆
ZA₁	ZG	ZVG	ZVP	ZVP	ZEG	ZG
ZA₂	ZVP	ZVG	ZG	ZVG	ZVG	ZP
ZA₃	ZEG	ZG	ZP	ZEG	ZEG	ZVG
ZA₄	ZVG	ZVP	ZEG	ZVG	ZM	ZEG
ZA₅	ZEG	ZEG	ZP	ZVP	ZVG	ZVG

Table 5

Linguistic decision matrix by the fifth expert

	ZG₁	ZG₂	ZG₃	ZG₄	ZG₅	ZG₆
ZA₁	ZEG	ZVG	ZVP	ZP	ZP	ZVG
ZA₂	ZP	ZVG	ZG	ZG	ZVG	ZP
ZA₃	ZEG	ZVG	ZP	ZVG	ZVG	ZG
ZA₄	ZVG	ZVP	ZEG	ZVG	ZVG	ZP
ZA₅	ZG	ZEG	ZP	ZVP	ZVP	ZEG

Then, the PL-CODAS method is developed for core competitiveness evaluation of small and medium-sized foreign trade enterprises.

Step 1. Transform the linguistic values into probabilistic linguistic matrix (Table 6).

Table 6

Probabilistic linguistic matrix

Alternatives	ZG₁	ZG₂
ZA₁	$\{{zl_{1}({0.4}),zl_{3}({0.6})}\}$	$\{{zl_{2}({0.4}),zl_{3}({0.6})}\}$
ZA₂	$\{{zl_{-2}({0.6}),zl_{-1}({0.4})}\}$	$\{{zl_{2}(1)}\}$
ZA₃	$\{{zl_{2}({0.2}),zl_{3}({0.8})}\}$	$\{{zl_{1}({0.8}),zl_{2}({0.2})}\}$
ZA₄	$\{{zl_{2}(1)}\}$	$\{{zl_{-2}({0.6}),zl_{-1}({0.4})}\}$
ZA₅	$\{{zl_{1}({0.8}),zl_{3}({0.2})}\}$	$\{{zl_{3}(1)}\}$
Alternatives	ZG₃	ZG₄
ZA₁	$\{{zl_{-2}(1)}\}$	$\{{zl_{-3}({0.4}),zl_{-2}({0.4}),zl_{-1}({0.2})}\}$
ZA₂	$\{{zl_{1}({0.6}),zl_{3}({0.4})}\}$	$\{{zl_{1}({0.2}),zl_{2}({0.4}),zl_{3}({0.4})}\}$
ZA₃	$\{{zl_{-1}({0.8}),zl_{0}({0.2})}\}$	$\{{zl_{3}(1)}\}$
ZA₄	$\{{zl_{2}({0.4}),zl_{3}({0.6})}\}$	$\{{zl_{1}({0.2}),zl_{2}({0.6}),zl_{3}({0.2})}\}$
ZA₅	$\{{zl_{-1}(1)}\}$	$\{{zl_{-2}({0.6}),zl_{-1}({0.4})}\}$
Alternatives	ZG₅	ZG₆
ZA₁	$\{{zl_{2}({0.7}),zl_{1}({0.3})}\}$	$\{{zl_{3}({0.4}),zl_{2}({0.4}),zl_{1}({0.2})}\}$
ZA₂	$\{{zl_{2}({0.4}),zl_{1}({0.6})}\}$	$\{{zl_{0}({0.6}),zl_{1}({0.4})}\}$
ZA₃	$\{{zl_{0}({0.4}),zl_{1}({0.6})}\}$	$\{{zl_{-1}({0.2}),zl_{-2}({0.8})}\}$
ZA₄	$\{{zl_{3}({0.2}),zl_{2}({0.8})}\}$	$\{{zl_{2}({0.4}),zl_{1}({0.4}),zl_{0}({0.2})}\}$
ZA₅	$\{{zl_{2}({0.5}),zl_{1}({0.5})}\}$	$\{{zl_{-1}({0.4}),zl_{-2}({0.6})}\}$

Step 2. Calculate the normalized probabilistic linguistic matrix (Table 7).

Table 7

Normalized probabilistic linguistic matrix

Alternatives	ZG₁	ZG₂
ZA₁	$\{{zl_{3}({0.6}),zl_{1}({0.4}),zl_{1}(0)}\}$	$\{{zl_{3}({0.6}),zl_{2}({0.4}),zl_{2}(0)}\}$
ZA₂	$\{{zl_{-1}({0.4}),zl_{-2}({0.6}),zl_{-2}(0)}\}$	$\{{zl_{2}(1),zl_{2}(0),zl_{2}(0)}\}$
ZA₃	$\{{zl_{3}({0.8}),zl_{2}({0.2}),zl_{2}(0)}\}$	$\{{zl_{2}({0.2}),zl_{1}({0.8}),zl_{1}(0)}\}$
ZA₄	$\{{zl_{2}(1),zl_{2}(0),zl_{2}(0)}\}$	$\{{zl_{-1}({0.4}),zl_{-2}({0.6}),zl_{-2}(0)}\}$
ZA₅	$\{{zl_{3}({0.2}),zl_{1}({0.8}),zl_{1}(0)}\}$	$\{{zl_{3}(1),zl_{3}(0),zl_{3}(0)}\}$
Alternatives	ZG₃	ZG₄
ZA₁	$\{{zl_{-2}(1),zl_{-2}(0),zl_{-2}(0)}\}$	$\{{zl_{-1}({0.2}),zl_{-2}({0.4}),zl_{-3}({0.4})}\}$
ZA₂	$\{{zl_{3}({0.4}),zl_{1}({0.6}),zl_{1}(0)}\}$	$\{{zl_{3}({0.4}),zl_{2}({0.4}),zl_{1}({0.2})}\}$
ZA₃	$\{{zl_{0}({0.2}),zl_{-1}({0.8}),zl_{-1}(0)}\}$	$\{{zl_{3}(1),zl_{3}(0),zl_{3}(0)}\}$
ZA₄	$\{{zl_{3}({0.6}),zl_{2}({0.4}),zl_{2}(0)}\}$	$\{{zl_{3}({0.2}),zl_{2}({0.6}),zl_{1}({0.2})}\}$
ZA₅	$\{{zl_{-1}(1),zl_{-1}(0),zl_{-1}(0)}\}$	$\{{zl_{-1}({0.4}),zl_{-2}({0.6}),zl_{-2}(0)}\}$
Alternatives	ZG₅	ZG₆
ZA₁	$\{{zl_{2}({0.7}),zl_{1}({0.3}),zl_{1}({0.0})}\}$	$\{{zl_{3}({0.4}),zl_{2}({0.4}),zl_{1}({0.2})}\}$
ZA₂	$\{{zl_{2}({0.4}),zl_{1}({0.6}),zl_{1}({0.0})}\}$	$\{{zl_{1}({0.4}),zl_{0}({0.6}),zl_{0}({0.0})}\}$
ZA₃	$\{{zl_{1}({0.6}),zl_{0}({0.4}),zl_{0}({0.0})}\}$	$\{{zl_{-1}({0.2}),zl_{-2}({0.8}),zl_{-2}({0.0})}\}$
ZA₄	$\{{zl_{3}({0.2}),zl_{2}({0.8}),zl_{2}({0.0})}\}$	$\{{zl_{2}({0.4}),zl_{1}({0.4}),zl_{0}({0.2})}\}$
ZA₅	$\{{zl_{2}({0.5}),zl_{1}({0.5}),zl_{1}({0.0})}\}$	$\{{zl_{-1}({0.4}),zl_{-2}({0.6}),zl_{-2}({0.0})}\}$

Step 3. Compute the attributes weight (See Table 8).

Table 8

The attribute weight information

	ZG₁	ZG₂	ZG₃	ZG₄	ZG₅	ZG₆
$z w$	0.2207	0.1826	0.2194	0.1464	0.1072	0.1237

Step 4. Obtain the PLNIS (Table 9).

Table 9

PLNIS

	ZG₁	ZG₂
PLNIS	$\{{zl_{-1}({0.4}),zl_{-2}({0.6}),zl_{-2}(0)}\}$	$\{{zl_{-1}({0.4}),zl_{-2}({0.6}),zl_{-2}(0)}\}$
	G₃	G₄
PLNIS	$\{{zl_{-1}(1),zl_{-1}(0),zl_{-1}(0)}\}$	$\{{zl_{-3}({0.4}),zl_{-2}({0.4}),zl_{-1}({0.2})}\}$
	G₅	G₆
PLNIS	$\{{zl_{1}({0.6}),zl_{0}({0.4}),zl_{0}({0.0})}\}$	$\{{zl_{-1}({0.2}),zl_{-2}({0.8}),zl_{-2}({0.0})}\}$

Step 5. Determine the $\textit{PLED}({ZL_{i}({zp}),\textit{PLNIS}})$ and $\textit{PLHD}({ZL_{i}({zp}),\textit{PLNIS}})$ (Table 10).

Table 10

Weighted Euclidean distance and Hamming distance

Alternatives	$\textit{PLED}({ZL_{i}({zp}),\textit{PLNIS}})$	$\textit{PLHD}({ZL_{i}({zp}),\textit{PLNIS}})$
ZA₁	0.4614	0.1495
ZA₂	0.4927	0.1791
ZA₃	0.6106	0.2370
ZA₄	0.5254	0.2037
ZA₅	0.5417	0.1951

Step 5. Compute the PLRAM (Table 11).

Table 11

The PLRAM

	ZA₁	ZA₂	ZA₃	ZA₄	ZA₅
ZA₁	0.0000	$-$ 0.0590	$-$ 0.2346	$-$ 0.1163	$-$ 0.1240
ZA₂	0.0628	0.0000	$-$ 0.1738	$-$ 0.0554	$-$ 0.0631
ZA₃	0.2384	0.1775	0.0000	0.1202	0.1126
ZA₄	0.1201	0.0592	$-$ 0.1164	0.0000	$-$ 0.0144
ZA₅	0.1277	0.0669	$-$ 0.1088	0.0182	0.0000

Step 6. Calculate the $\textit{PLAS}_{i}({i=1,2,3,4,5})$ (Table 12).

Table 12

The PLAS

Alternatives	ZA₁	ZA₂	ZA₃	ZA₄	ZA₅
PLAS	$-$ 0.5339	$-$ 0.2295	0.6487	0.0485	0.1040

Step 7. According to $\textit{PLAS}_{i}({i=1,2,3,4,5})$ , the order is: $ZA_{3}>ZA_{5}>ZA_{4}>ZA_{2}>ZA_{1}$ and the best alternative is $ZA_{3}$ .

4.2 Comparative analysis

Then, PL-CODAS method is compared with PLWA operator [54], PL-TOPSIS method [54], PL-GRA method [70], probabilistic linguistic ELECTRE II (PL-ELECTRE II) method [71], probabilistic linguistic ELECTRE III (PL-ELECTRE III) method [72], Probabilistic linguistic MULTIMOORA (PL-MULTIMOORA) method [73], probabilistic linguistic PROMTHEE II (PL-PROMTHEE II) method [74] and Probabilistic linguistic QUALIFLEX(PL-QUALIFLEX) method [75]. The comparative decision results are shown in Table 13.

Table 13
Order of the different methods

Different methods	Order
PLWA operator [54]	$ZA_{3}>ZA_{5}>ZA_{4}>ZA_{2}>ZA_{1}$
PL-TOPSIS method [54]	$ZA_{3}>ZA_{5}>ZA_{4}>ZA_{2}>ZA_{1}$
PL-GRA method [70]	$ZA_{3}>ZA_{5}>ZA_{4}>ZA_{2}>ZA_{1}$
PL-ELECTRE II method [71]	$ZA_{3}>ZA_{5}>ZA_{4}>ZA_{2}>ZA_{1}$
PL-ELECTRE III method [72]	$ZA_{3}>ZA_{5}>ZA_{4}>ZA_{2}>ZA_{1}$
PL-MULTIMOORA method [73]	$ZA_{3}>ZA_{5}>ZA_{4}>ZA_{2}>ZA_{1}$
PL-PROMTHEE II method [74]	$ZA_{3}>ZA_{5}>ZA_{2}>ZA_{4}>ZA_{1}$
PL-QUALIFLEX method [75]	$ZA_{3}>ZA_{5}>ZA_{2}>ZA_{4}>ZA_{1}$

In line with WS coefficients [76, 77], the WS coefficient calculation between PLWA operator [54], PL-TOPSIS method [54], PL-GRA method [70], probabilistic linguistic ELECTRE II (PL-ELECTRE II) method [71], probabilistic linguistic ELECTRE III (PL-ELECTRE III) method [72], probabilistic linguistic MULTIMOORA (PL-MULTIMOORA) method [73], probabilistic linguistic PROMTHEE II (PL-PROMTHEE II) method [74] and probabilistic linguistic QUALIFLEX (PL-QUALIFLEX) method [75] and the proposed PL-CODAS method is 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 1.0000, 0.7917, 0.7917, respectively. The WS coefficient shows the ranking results of the proposed PL-CODAS method are same to the ranking results of PLWA operator [54], PL-TOPSIS method [54], PL-GRA method [70], probabilistic linguistic ELECTRE II (PL-ELECTRE II) method [71], probabilistic linguistic ELECTRE III (PL-ELECTRE III) method [72] and probabilistic linguistic MULTIMOORA (PL-MULTIMOORA) method [73]; the WS coefficient shows the ranking results of the proposed PL-CODAS method are slightly different to the ranking results of probabilistic linguistic PROMTHEE II (PL-PROMTHEE II) method [74] and probabilistic linguistic QUALIFLEX (PL-QUALIFLEX) method [75]. Furthermore, the reason for this subtle difference is that probabilistic linguistic PROMTHEE II (PL-PROMTHEE II) method [74] and probabilistic linguistic QUALIFLEX (PL-QUALIFLEX) method [75] lost some information in pairwise comparison. This verifies PL-CODAS method we proposed is reasonable and effective in this paper. The proposed PL-CODAS method combines the PLED as the primary measure and PLHD as the secondary measure to assess alternatives only from the PLNIS.

5. Conclusion

At present, there is a common problem of insufficient understanding of core competitiveness among small and medium-sized foreign trade enterprises in China. Therefore, small and medium-sized foreign trade enterprises need to search for factors that do not have strong core competitiveness and continuously improve their internal strength. At the same time, the government should also help and guide small and medium-sized foreign trade enterprises to cultivate the concept of core competitiveness and accumulate competitiveness at the macro policy environment level, and continuously improve the scale and strength of China’s small and medium-sized foreign trade enterprises by enhancing their core competitiveness, Thus, making small and medium-sized foreign trade enterprises an important engine for achieving China’s economic takeoff. The core competitiveness evaluation of small and medium-sized foreign trade enterprises is a classical MAGDM. In this paper, the CODAS method is extended to the MAGDM with PLTSs. Firstly, the definition of PLSs is introduced. Then, on the basis of the classical CODAS method, the PL-CODAS method is proposed to cope with MAGDM under PLTSs and its significant characteristic is that it can fully consider PLED and PLHD. Finally, a practical example for core competitiveness evaluation of small and medium-sized foreign trade enterprises is given to verify the developed approach and some comparative analysis was also given to verify the PL-CODAS approach. The main contribution of this paper is constructed: (1) the CODAS method is extended under PLTSs; (2) the probabilistic linguistic CODAS (PL-CODAS) method is proposed to cope with the MAGDM under PLTSs; (3) a numerical example for core competitiveness evaluation of small and medium-sized foreign trade enterprises is proposed to verify the developed approach; (4) some comparative studies are given to verify the rationality of PL-CODAS method. In the future, the application of the proposed models and methods under PLTSs needs to be explored in the other uncertain decision making and other uncertain environment. Simultaneously, more and more methods for determining weights could be also designed in our future works [78, 79, 80, 81, 82].

Footnotes

Acknowledgments

The work was supported by the Guangdong Provincial Department of Education 2021 Education Science Planning Project (Higher Education Special Project) (Project No.2021GXJK465); Guangdong Institute of Higher Education “14th Five-Year Plan” 2022 Higher Education Research Project (Project No.22GQN52); Guangzhou Huali College 2022 Higher Education Teaching Research and Reform Project.

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A group decision framework for core competitiveness evaluation of small and medium-sized foreign trade enterprises under probabilistic linguistic term sets

Abstract

Keywords

1. Introduction

2. Preliminaries

3.1 MAGDM problems description

4.1 Numerical example

Table 1 Linguistic matrix by the first expert

Table 13 Order of the different methods

Footnotes

Acknowledgments

References

Table 1
Linguistic matrix by the first expert

Table 13
Order of the different methods