A fuzzy TOPSIS for assessing higher vocational education development levels in uncertainty environments

Abstract

It has an important practical significance to assess the higher vocational education development level in one specific region, but few effective methods are reported in the literature. Based on the practical challenges, we propose a fuzzy TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) for assessing higher vocational education development levels. Firstly, in order to deal with decision-makers’ preferences on interval numbers, we develop a preference-based fuzzy number comparison method and then integrate it with the classic TOPSIS method to formulate a fuzzy TOPSIS method. An application example shows the effectiveness of the work and observes the impact of decision-makers’ preferences on the assessment results. Several insights are also found to improve the assessment process in the real world.

Keywords

Fuzzy number comparison fuzzy TOPSIS higher vocational education development level assessment optimism degrees

1 Introduction

Education is one of the most important sources of human capital accumulation, and higher vocational education is an important part of the whole education system [13]. The main objective of higher vocational education is to train skilled technicians for specific production and service positions by equipping the students with theoretical knowledge and technical skills. Thus, higher vocational education is with characteristics of professionalism and specialization, playing an extremely important part in the formation and accumulation of human capital.

The differences of regional education, personnel policy and social environment often cause regional differences in the level and structure of human capital, which in turn lead to regional economic development imbalance [14]. Thus, it has an important practical significance to assess the higher vocational education development level. However, the assessment is challenging due to the following characteristics:

There is no a standard as the reference for assessing the higher vocational education development level. This is because the vocational education in one country should be in accord with the economic and social development of the country [8]. Improper vocational education cannot provide adequate labors for the development of one country.

The assessment often involves multiple factors in different dimensionalities such as population background, education structure, educational resource and investment, etc [2]. To our knowledge, few assessment index systems are well recognized in the literature.

The information on assessment factors is not always precise due to the uncertainty in the real-world [13]. When it is difficult or impossible to obtain precise information, decision-makers have to estimate the factors using uncertain numbers.

In the literature, few quantitative assessment methods have been reported. A few studies are contributed to assess the teaching and training performance [1 , 23], but none of them is on how to assess the higher vocational education development level in one region.

Motivated by these practical observations, we are focusing on how to properly assess the higher vocational education development level in accordance with the development of local economics and society. The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is a multi-criteria decision analysis method, which has been used in wide areas such as financing [9, 18], risk assessment [7, 20], medical treatment [27], and control decision [19, 21]. The classical TOPSIS is well recognized as one effective method for the MCDA problems when all the data are crisp values. Unfortunately, in real world situations, the factor values of higher vocational education development levels often include uncertain information which can be represented by fuzzy numbers. Meanwhile, different kinds of decision-makers often have different preferences on fuzzy numbers, which may further impact the assessmentresults.

Thus, in this work, we present a fuzzy number comparison method and integrate it with the classical TOPSIS method to assess higher vocational education development levels with uncertain information. The contributions of this work include: (i) A preference-based index is presented for comparing interval numbers with the consideration of decision-makers’ preferences; (ii) A fuzzy number comparison method is formulated by introducing the α-cut technique into the preference-based index; (iii) By integrating the presented fuzzy number comparison method with classical TOPSIS, a fuzzy TOPSIS is proposed for assessing higher vocational education development levels with uncertain information; (vi) An application example is given to show the effectiveness of our work.

The remainder of this paper is organized as follows. In Section 2, we present the fuzzy TOPSIS for assessing higher vocational education development levels with uncertain information, with the consideration of decision-makers’ preferences. In Section 3, we give an application example to show the effectiveness of our work. Conclusions are finally drawn in Section 4.

2 The proposed approach

2.1 The classical TOPSIS based method

The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) was originally proposed by Hwang and Yoon [6]. This method is based on two concepts: the positive ideal solution and the negative ideal solution. The positive ideal solution is formulated by the best values among all the assessment alternatives, and the negative ideal solution is formulated by the worst ones. The closer one alternative is to the positive ideal solution, the better the alternative is; on the contrary, the closer one alternative is to the negative ideal solution, the worse the alternative is.

As stated in the Introduction, the classical TOPSIS is effective to deal with MCDA problems with no assessment standards and multiple assessment factors. Thus, the TOPSIS is suitable for the assessment of higher vocational education development levels. A brief introduction of applying the classical TOPSIS into the assessment of higher vocational education development levels is as below.

let m represent the number of alternative areas for attending the assessment of higher vocational education development levels and n represent the number of assessment factors. The intersection of each alternative area and criterion is given as x_ij, i = 1, 2, …, m, j = 1, 2, …, n. Then an assessment matrix can be obtained: $X = {[\begin{matrix} x_{11} & \dots & x_{1 n} \\ ⋮ & \dots & ⋮ \\ x_{m 1} & \dots & x_{mn} \end{matrix}]}_{m \times n}$ (1) where x_ij is the assessment value of the jth factor for the ith alternative area. In order to reduce the impact of different dimensionalities, the above assessment matrix is normalized into the normalization matrix:

$R = {[\begin{matrix} r_{11} & \dots & r_{1 n} \\ ⋮ & \dots & ⋮ \\ r_{m 1} & \dots & r_{mn} \end{matrix}]}_{m \times n}$ (2) where $r_{ij} = \frac{x_{ij}}{\sqrt{\sum_{i = 1}^{m} x_{ij}^{2}}}, i = 1, 2, \dots, m, j = 1, 2, \dots, n$ (3)

After getting the normalization matrix, the positive ideal solution and the negative ideal solution can be defined as follows.

Definition 1. The positive ideal solution A_b is made up of the best factor values among all the alternative areas, that is, $\begin{matrix} A_{b} = {〈 max (w_{j} r_{ij} | - i = 1, 2, \dots, m) | j \in J_{+} 〉, \\ 〈 min (w_{j} r_{ij} | - i = 1, 2, \dots, m) | j \in J_{-} 〉} \equiv {t_{bj}} \end{matrix}$ (4) where w_j is the weight of the jth factor, J₊ denotes the set of all the positive factors, and J_- denotes the set of all the negative factors.

Definition 2. The negative ideal solution A_w is made up of the worst factor values among all the alternative areas, that is, $\begin{matrix} A_{w} = {〈 min (w_{j} r_{ij} | - i = 1, 2, \dots, m) | j \in J_{+} 〉, \\ 〈 max (w_{j} r_{ij} | - i = 1, 2, \dots, m) | j \in J_{-} 〉} \equiv {t_{wj}} \end{matrix}$ (5)

With A_b and A_w, the assessment result of the higher vocational education development level in the ith alternative area can be obtained by: $s_{i} = \frac{d_{iw}}{d_{iw} + d_{ib}}$ (6) where d_ib and d_iw denotes the distance of the ith alternative area from the positive ideal solution A_b and the negative ideal solution A_w, respectively, that is, $d_{ib} = \sqrt{\sum_{j = 1}^{n} (r_{ij} - t_{bj})^{2}}, i = 1, 2, \dots, m$ (7) $d_{iw} = \sqrt{\sum_{j = 1}^{n} (r_{ij} - t_{wj})^{2}}, i = 1, 2, \dots, m$ (8)

According to the assessment result {s_i, i = 1,2, …, m}, we can rank all the alternative areasaccording to their higher vocational education development levels.

If all the values of assessment factors are crisp, it is easy to determine the assessment results using the above classical TOPSIS based method. However, uncertainties can arise from various sources in the real world, and many MCDA methods have been reported in order to deal with the uncertain MCDA problems [4 , 15–17]. Similarly, the values of the factors for assessing higher vocational education development levels often include incomplete or uncertain information. In addition, different kinds of decision-makers often have different preferences when they face uncertain information, which may have impacts on the assessment results.

Motivated by these observations, in this work we extend the classical TOPSIS based method to one fuzzy TOPSIS based method. A preference-based index is presented for comparing interval numbers, and then the α-cut technique is integrated with the preference-based index to formulate a fuzzy TOPSIS based method, which can assess higher vocational education development levels with fuzzy information.

2.2 A preference-based interval comparison method

A lot of interval comparison methods have been reported in the literature. In this work, we integrate the Giove’s method [25] with the classical TOPSIS to develop a fuzzy TOPSIS for assessing higher vocational education development levels with uncertain information. To this end, we first present a preference-based interval comparison method by introducing decision-makers’ preferences into the Giove’s interval comparison method.

For the incomplete information of one assessment factor, we use one interval number to represent the factor value: $\begin{matrix} {\bar{x}}_{ij} = [x_{ij}^{low}, x_{ij}^{up}] \\ = {x_{ij} | x_{ij}^{low} \leq x_{ij} \leq x_{ij}^{up}, x_{ij} \in R} \end{matrix}$ (9) where $x_{ij}^{low}$ and $x_{ij}^{up}$ are the lower and upper bounds of ${\bar{x}}_{ij}$ , respectively, $x_{ij}^{low} \leq x_{ij}^{up}$ . Sengupta and Pal [3] presented an index to compare two interval values ${\bar{x}}_{ij}$ and ${\bar{x}}_{jl}$ (denoted by $α ({\bar{x}}_{ij} ≻ {\bar{x}}_{jl})$ ):

$α ({\bar{x}}_{ij} ≻ {\bar{x}}_{jl}) = \frac{m ({\bar{x}}_{ij}) - m ({\bar{x}}_{jl})}{w ({\bar{x}}_{ij}) + w ({\bar{x}}_{jl})}$ (10) where $m ({\bar{x}}_{ij})$ and $m ({\bar{x}}_{jl})$ respectively denote the mid-points of ${\bar{x}}_{ij}$ and ${\bar{x}}_{jl}$ , that is, $m ({\bar{x}}_{ij}) = \frac{1}{2} (x_{ij}^{low} + x_{ij}^{up})$ (11) $m ({\bar{x}}_{jl}) = \frac{1}{2} (x_{jl}^{low} + x_{jl}^{up})$ (12) and $w ({\bar{x}}_{ij})$ and $w ({\bar{x}}_{jl})$ respectively denote the half-widths of ${\bar{x}}_{ij}$ and ${\bar{x}}_{jl}$ , that is, $w ({\bar{x}}_{ij}) = \frac{1}{2} (x_{ij}^{up} - x_{ij}^{low})$ (13) $w ({\bar{x}}_{jl}) = \frac{1}{2} (x_{jl}^{up} - x_{jl}^{low})$ (14)

The above Sengupta and Pal’s method is effective to compare two real interval numbers. However, the method cannot compare crisp numbers, because the half-widths of crisp numbers are 0 which makes the denominator of Equation (10) invalid. In order to fix this issue, Giove [25] modified the index (10) into: $β ({\bar{x}}_{ij} ≻ {\bar{x}}_{jl}) = \frac{m ({\bar{x}}_{ij}) - m ({\bar{x}}_{jl})}{w ({\bar{x}}_{ij}) + w ({\bar{x}}_{jl}) + 1}$ (15)

In the real world, decision-makers often have preferences on the uncertainty, as Fig. 1 shows. These different preferences may produce different assessment results of higher vocational education development levels with uncertain information.

In order to reflect decision-makers’ preferences on interval numbers, we introduce the optimism degree γ into (15):

$β ({\bar{x}}_{ij} ≻ {\bar{x}}_{jl}) = \frac{o ({\bar{x}}_{ij}) - o ({\bar{x}}_{jl})}{w ({\bar{x}}_{ij}) + w ({\bar{x}}_{jl}) + 1}$ (16) $o ({\bar{x}}_{ij}) = (γ x_{ij}^{low} + (1 - γ) x_{ij}^{up})$ (17) $o ({\bar{x}}_{jl}) = (γ x_{jl}^{low} + (1 - γ) x_{jl}^{up})$ (18) where γ denotes the optimism degree of decision-makers, 0 ≤ γ ≤ 1, and $o ({\bar{x}}_{ij})$ and $o ({\bar{x}}_{jl})$ respectively denote the preferred mid-points of ${\bar{x}}_{ij}$ and ${\bar{x}}_{jl}$ .

2.3 A novel method for comparing fuzzy numbers

In this section, we use the α-cut technique and the preference-based interval comparison method to develop a novel method for comparing fuzzy numbers.

A fuzzy number ${\tilde{x}}_{ij} = (a, b, c; 1)$ is a fuzzy subset of a universal set R with the membership function: $μ_{{\tilde{x}}_{ij}} (x) = {\begin{matrix} \frac{x - a}{b - a}, a \leq x \leq b \\ \frac{c - x}{c - b}, b \leq x \leq c \\ 0, otherwise \end{matrix}$ (19) where a, b and c are crisp numbers, a < b < c.

The basic idea of the α-cut based method for comparing fuzzy numbers is firstly to transform fuzzy numbers into interval numbers, and then compare the cut interval numbers.

For a fuzzy number ${\tilde{x}}_{ij} = (a, b, c; 1)$ , the α-cut of ${\tilde{x}}_{ij}$ (denoted by ${\tilde{x}}_{ij}^{α}$ ) is a crisp set, that is, ${\tilde{x}}_{ij}^{α} = {\begin{matrix} {x | μ_{{\tilde{x}}_{ij}} (x) \geq α, x \in R}, 0 < α \leq 1 \\ [a, c], α = 0 \end{matrix}$ (20) where α denotes the cut level.

Then, for a fuzzy number ${\tilde{x}}_{ij} = (a, b, c; 1)$ , the left α-cut of ${\tilde{x}}_{ij}$ is the value whose membership degree is equal to α between a and b, that is, (b - a) α + a, and the right α-cut of ${\tilde{x}}_{ij}$ is the value whose membership degree is equal to α between b and c, that is, c - (c - b) α. Meanwhile, because the membership function of ${\tilde{x}}_{ij}$ is strictly increasing on [a, b] and is strictly decreasing on [b, c], the left and right α-cuts of ${\tilde{x}}_{ij}$ are unique values. Thus, the α-cut interval value of ${\tilde{x}}_{ij}$ (denoted by ${\bar{\tilde{x}}}_{ij}^{α}$ ) is a unique interval number, that is, ${\bar{\tilde{x}}}_{ij}^{α} = [(b - a) α + a, c - (c - b) α]$ (21)

Using (21), the fuzzy number ${\tilde{x}}_{ij}$ can be transformed into an interval number. Then, we can use the preference-based interval comparison method proposed in Section 2.2 to compare fuzzy numbers.

For two fuzzy values ${\tilde{x}}_{ij} = (a_{1}, b_{1}, c_{1}; 1)$ and ${\tilde{x}}_{kl} = (a_{2}, b_{2}, c_{2}; 1)$ , ${\tilde{x}}_{ij}$ and ${\tilde{x}}_{kl}$ can be transformed into two interval numbers, respectively: ${\bar{\tilde{x}}}_{ij}^{α} = [(b_{1} - a_{1}) α + a_{1}, c_{1} - (c_{1} - b_{1}) α]$ (22) ${\bar{\tilde{x}}}_{kl}^{α} = [(b_{2} - a_{2}) α + a_{2}, c_{2} - (c_{2} - b_{2}) α]$ (23) where 0 ≤ α ≤ 1.

Based on (16–18), we compare ${\tilde{x}}_{ij}$ and ${\tilde{x}}_{kl}$ by $δ ({\bar{\tilde{x}}}_{ij}^{α} ≻ {\bar{\tilde{x}}}_{kl}^{α}) = \frac{o ({\bar{\tilde{x}}}_{ij}^{α}) - o ({\bar{\tilde{x}}}_{kl}^{α})}{w ({\bar{\tilde{x}}}_{ij}^{α}) + w ({\bar{\tilde{x}}}_{kl}^{α}) + 1}$ (24) where $\begin{matrix} o ({\bar{\tilde{x}}}_{ij}^{α}) = γ ((b_{1} - a_{1}) α + a_{1}) + (1 - γ) \\ (c_{1} - (c_{1} - b_{1}) α) \end{matrix}$ (25) $\begin{matrix} o ({\bar{\tilde{x}}}_{kl}^{α}) = γ ((b_{2} - a_{2}) α + a_{2}) + (1 - γ) \\ (c_{2} - (c_{2} - b_{2}) α) \end{matrix}$ (26) $w ({\bar{\tilde{x}}}_{ij}^{α}) = \frac{1}{2} ((c_{1} - (c_{1} - b_{1}) α) - ((b_{1} - a_{1}) α + a_{1}))$ (27) $w ({\bar{\tilde{x}}}_{kl}^{α}) = \frac{1}{2} ((c_{2} - (c_{2} - b_{2}) α) - ((b_{2} - a_{2}) α + a_{2}))$ (28)

2.4 The fuzzy TOPSIS based method

In this section, we integrate the fuzzy number comparison method in Section 2.3 with the classical TOPSIS based method in 2.1 to develop a fuzzy TOPSIS based method for assessing higher vocational education development levels with uncertain information.

Under the uncertain environment, the assessment matrix is a fuzzy matrix and is represented by $\tilde{X} = {[\begin{matrix} {\tilde{x}}_{11} & \dots & {\tilde{x}}_{1 n} \\ ⋮ & \dots & ⋮ \\ {\tilde{x}}_{m 1} & \dots & {\tilde{x}}_{mn} \end{matrix}]}_{m \times n}$ (29) where ${\tilde{x}}_{ij} = (x_{ij}^{low}, x_{ij}^{mid}, x_{ij}^{up}), i = 1, 2, \dots$ , m, j = 1, 2, …, n.

In order to deal with the fuzzy values in (29), we use (24–28) to develop a relative value: $\begin{matrix} RV ({\tilde{x}}_{ij}) = {\sum_{k = 1}}_{k \neq i}^{m} δ ({\bar{\tilde{x}}}_{ij}^{α} ≻ {\bar{\tilde{x}}}_{kj}^{α}) \\ = {\sum_{k = 1}}_{k \neq i}^{m} \frac{o ({\bar{\tilde{x}}}_{ij}^{α}) - o ({\bar{\tilde{x}}}_{kj}^{α})}{w ({\bar{\tilde{x}}}_{ij}^{α}) + w ({\bar{\tilde{x}}}_{kj}^{α}) + 1} \end{matrix}$ (30)

Then, we normalize the relative values into: $\hat{\tilde{R}} = {[\begin{matrix} \frac{RV ({\tilde{x}}_{11})}{\sqrt{\sum_{i = 1}^{m} RV ({\tilde{x}}_{i 1})^{2}}} & \dots & \frac{RV ({\tilde{x}}_{1 n})}{\sqrt{\sum_{i = 1}^{m} RV ({\tilde{x}}_{in})^{2}}} \\ ⋮ & \dots & ⋮ \\ \frac{RV ({\tilde{x}}_{m 1})}{\sqrt{\sum_{i = 1}^{m} RV ({\tilde{x}}_{i 1})^{2}}} & \dots & \frac{RV ({\tilde{x}}_{mn})}{\sqrt{\sum_{i = 1}^{m} RV ({\tilde{x}}_{in})^{2}}} \end{matrix}]}_{m \times n}$ (31)

Based on Definition 1 and Definition 2, the positive ideal solution and negative ideal solution with fuzzy information (respectively denoted by ${\hat{\tilde{A}}}_{b}$ and ${\hat{\tilde{A}}}_{w}$ ) can be obtained by: $\begin{matrix} {\hat{\tilde{A}}}_{b} = \\ {〈 max (w_{j} \frac{RV ({\tilde{x}}_{ij})}{\sqrt{\sum_{i = 1}^{m} RV ({\tilde{x}}_{ij})^{2}}} | - i = 1, 2, \dots, m) | j \in J_{+} 〉, \\ 〈 min (w_{j} \frac{RV ({\tilde{x}}_{ij})}{\sqrt{\sum_{i = 1}^{m} RV ({\tilde{x}}_{ij})^{2}}} | - i = 1, 2, \dots, m) | j \in J_{-} 〉} \\ \equiv {{\hat{\tilde{t}}}_{bj}} \end{matrix}$ (32) $\begin{matrix} {\hat{\tilde{A}}}_{w} = \\ {〈 min (w_{j} \frac{RV ({\tilde{x}}_{ij})}{\sqrt{\sum_{i = 1}^{m} RV ({\tilde{x}}_{ij})^{2}}} | - i = 1, 2, \dots, m) | j \in J_{+} 〉, \\ 〈 max (w_{j} \frac{RV ({\tilde{x}}_{ij})}{\sqrt{\sum_{i = 1}^{m} RV ({\tilde{x}}_{ij})^{2}}} | - i = 1, 2, \dots, m) | j \in J_{-} 〉} \\ \equiv {{\hat{\tilde{t}}}_{wj}} \end{matrix}$ (33)

With ${\hat{\tilde{A}}}_{b}$ and ${\hat{\tilde{A}}}_{w}$ , the assessment result of the higher vocational education development level in the ith alternative area with fuzzy information can be obtained by: $\frac{{\hat{\tilde{s}}}_{i} = \sqrt{\sum_{j = 1}^{n} {(w_{j} {\frac{R V ({\tilde{x}}_{i j})}{\sqrt{\sum_{i = 1}^{m} R V {({\tilde{x}}_{i j})}^{2}}}}_{i j} - {\hat{\tilde{t}}}_{w j})}^{2}}}{\sqrt{\sum_{j = 1}^{n} {(w_{j} \frac{R V ({\tilde{x}}_{i j})}{\sqrt{\sum_{i = 1}^{m} R V {({\tilde{x}}_{i j})}^{2}}} - {\hat{\tilde{t}}}_{w j})}^{2}} + \sqrt{\sum_{j = 1}^{n} {(w_{j} \frac{R V ({\tilde{x}}_{i j})}{\sqrt{\sum_{i = 1}^{m} R V {({\tilde{x}}_{i j})}^{2}}} - {\hat{\tilde{t}}}_{b j})}^{2}}}$ (34)

According to the assessment results ${{\hat{\tilde{s}}}_{i}, i = 1, 2, \dots, m}$ , we can rank all the alternative areas according to their higher vocational education development levels with fuzzy information.

3 Application example

In this section, we use an application example to show the effectiveness and advantage of applying our fuzzy TOPSIS based method into the assessment of higher vocational education development levels with uncertain information.

3.1 Assessment data

A province in China wants to assess higher vocational education development levels in its 11 cities C₁, C₂, …, C₁₁. Four assessment factors are recognized as the main indicators: the average number of higher vocational graduates (denoted as F₁), average education years (denoted as F₂), the average number of higher vocational teachers (denoted as F₃) and average educational funds per year (denoted as F₄). Based on the situations in these 11 cities, the original assessment data (the fuzzy assessment matrix $\tilde{X}$ ) are obtained, as Table 1 shows.

As we can see, all the factor values are represented with fuzzy numbers, so we cannot recognize which city is better than other cities in terms of higher vocational education development. We also cannot directly use the classical TOPSIS based method, since the classical method is only effective when all the assessment values are precise. In order to use our fuzzy TOPSIS based method, we first get the normalized values.

3.2 Results with α= 0.5 and γ= 0.5

As analyzed in Section 2.3, the α-cut level has impact on fuzzy comparison results, which may further impact the assessment results. Here we first give the results with α= 0.5. According to Equation (30) in Section 2.4, we can get the α-cut values of the original assessment data, as Table 2 shows. Given γ= 0.5, we can get the relative values, as Table 3 shows.

The relative values in Table 3 can verify the effectiveness of our preference-based fuzzy comparison method. For example, city C₄ with the fuzzyF₁ value [35000, 37000, 38000] in Table 1 is thebest, and its relative value 6.627 in Table 3 isalso the best in terms of F₁; city C₆ with the fuzzy F₁ value [5000, 5600, 6000] in Table 1 is the worst,and its relative value – 3.444 is also the worst interms of F₁.

With the relative values in Table 3, we can determine the positive ideal solution and negative ideal solution (The weight of each assessment factor isrecognized as equivalent): ${\hat{\tilde{A}}}_{b}$ = {6.627, 3.849, 6.624, 5.467} and ${\hat{\tilde{A}}}_{w}$ = {–3.444, –3.616, –3.162,–3.795}.

Finally, we can obtain the assessment results, as Table 4 shows. As we can see, when α= 0.5 and γ= 0.5, the higher vocational education development of city C₄ is in the highest level in the province, and city C₆ is in the lowest level.

In order to verify the effectiveness and advantage of our proposed method, we compare our results with those by the centroid-based method, as Table 4 shows. The centroid-based method is one of extant methods for dealing with fuzzy information [10 , 28]. As we can see, the ranking by our method is fully consistent with that by the centroid-based method. However, the centroid-based method cannot reflect decision-makers’ preference on fuzzy information. As observed in the following, both optimism degree and α-cut level have the impact on the assessment results, so our proposed method is more flexible.

3.3 Results with different optimism degrees

Using the similar process in Section 3.2, we can obtain the assessment values with different optimism degrees, as Table 5 shows (Here we set α= 0.5). Figure 2 shows the variation analysis of the assessment results with different optimismdegrees.

From the results in Table 5 and Fig. 2, we can get the following observations:

(1) Based on the assessment results, the higher vocational education development levels of the 11 cities can be divided into three categories: The assessment values of cities C₁, C₄, C₇ and C₁₁ are bigger than 0.6, so the higher vocational education development of these four cities, especially C₄, is relatively in good level; The assessment values of cities C₂, C₃, C₈ and C₁₀ are between 0.3 and 0.6, so these four cities are in average level; The assessment values of cities C₅, C₆ and C₉ are below 0.3, so these three cities are in low level.

(2) The decision-makers’ optimism degree indeed has impact on the assessment values, and the impact has a relationship with the uncertainty degree in fuzzy numbers. The bigger the uncertainty degree in the assessment data is, the bigger the impact of decision-makers’ optimism degree on the assessment results is. As Fig. 2(a) shows, the assessment values of city C₁₁ have bigger impact by the decision-makers’ optimism degree than cities C₁, C₄ and C₇. This observation verifies the impact of decision-makers’ preference and shows the reasonability of integrating the preference-based fuzzy comparison method with the classical TOPSIS. This observation also gives us one insight: Decision-makers should try their best to reduce the uncertainty in the assessment to minimize the impact of their preferences.

3.4 Results with different α-cuts

Using the similar process in Section 3.2, we can obtain the assessment values with different α-cuts, as Table 6 shows (Here we set γ= 0.5). Figure 3 shows the variation analysis of the assessment results with different α-cuts.

From the results in Table 6 and Fig. 3, we can find: With different α-cuts, the higher vocational education development levels of the 11 cities can also be divided into three categories, and the assessment orders are consistent with those in Section 3.3. Similarly, the α-cut level has different impacts on the assessment values due to the different uncertainty degrees in assessment data.

4 Conclusions

In this work, we propose a fuzzy TOPSIS method by integrating the fuzzy comparison method with the classical TOPSIS method. In the integrated method, a preference-based interval comparison method is presented in order to consider decision-makers’ preferences into the comparison process. Then, the α-cut technique is introduced into the preference-based index to formulate a fuzzy number comparison method. Finally, the classical TOPSIS method is extended to the fuzzy TOPSIS based method using the preference-based fuzzy comparison method. Based on application results, the reasonability of considering decision-makers’ preferences into the assessment of higher vocational education development levels is verified.

Footnotes

Acknowledgments

This research is supported by Hebei High-Level Personnel Funded Project (No. A201500112), Hebei Province Outstanding Experts Overseas Training Project, National Natural Science Foundation of China (No. 71503199), Natural Science Basic Research Project in Shaanxi Province (No. 2016JQ7005), and China Ministry of Education Social Sciences and Humanities Research Youth Fund Project (No. 16YJC630102).

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