Prioritization of Factors Impacting Lecturer Research Productivity Using an Improved Fuzzy Analytic Hierarchy Process Approach

: Improving the scientiﬁc research productivity of lecturers is an important strategy contribut-ing to improving the reputation of universities, attracting external funding sources, and improving the credibility of both domestic and international students. This study was carried out with the aim of determining the priority of the university’s governance factors that affect lecturers’ scientiﬁc research productivity. Six university governance factors were considered, including (i) research objectives and strategies, (ii) decentralization, (iii) leadership, (iv) support for research activities, (v) policy towards lecturers, and (vi) resources for research activities. In this study, an improved analytic hierarchy process method using generalized triangular fuzzy numbers and a centroid index was proposed. The research data were collected via in-depth interviews with experts and administrators at Vietnam National University, Hanoi (VNU). The results indicate that “resources for research activities” consti-tute the most important factor affecting the research productivity of lecturers at VNU, followed by research objectives and strategies and leadership.


Introduction
In recent years, administrators and universities all over the world have been increasingly interested in improving lecturers' research productivity, which is a key criterion for university rankings and one of the most important factors to consider when evaluating a university's research performance [1,2]. Many scholars have conducted research on individual and organizational factors that influence university lecturers' research productivity and research outcomes [3][4][5]. According to Nafukho et al. [3], the research productivity of lecturers at top Kenyan universities varies with gender, subject major, number of years of experience, terminal degree, number of students enrolled, proportion of students pursuing a doctoral degree, and research funding. These authors also point out that at top Kenyan universities, lecturers' experience is not a determining factor in their research productivity. Abramo et al. [4] investigated how individual and organizational factors interact to influence lecturer collaboration and research productivity in Italy. According to their findings, f T (x) = 0, for all x ∈ (a 4 , ∞], where a 1 , a 2 , a 3 and a 4 are real numbers.

Definition 2.
The fuzzy number T = (a 1 , a 2 , a 3 , a 4 ; w) is a trapezoidal fuzzy number if its membership function is given by: where f L T (x) and f R T (x) are the left and right membership functions of T, respectively. If a 1 < a 2 = a 3 < a 4 then T becomes a generalized triangular fuzzy number and can be denoted by T = (a 1 , a 2 , a 4 ; w). If w = 1, then T is a normal fuzzy number.

Chang's Extent Analysis Method on Fuzzy AHP
Chang [21] proposed an extent analysis method for fuzzy AHP to obtain a crisp priority vector from a triangular fuzzy comparison matrix. Chang's [21] approach is briefly discussed as follows: Let T = {t 1 , t 2 , . . . , t n } be an object set and G = {g 1 , g 2 , . . . , g m } be a goal set. According to Chang's [21] approach, each object is taken, and an extent analysis for each goal (g i ) is performed. Therefore, m extent analysis values for each object can be obtained as  Assuming that M j g i = (l ij , m ij , u ij ) are the values of the extent analysis of the ith object for m goals, the value of the fuzzy synthetic extent S i is defined as: Letting S 1 = (l 1 , m 1 , u 1 ) and S 2 = (l 2 , m 2 , u 2 ) be two TFNs, the degree of possibility of S 1 ≥ S 2 is defined as follows: The membership degree of possibility can be expressed as in Equation (4) V where d is the ordinate of the highest intersection point of two membership functions µ S 1 (x 1 ) and µ S 2 (x 2 ), as shown in Figure 1. The degree of possibility for a convex fuzzy number to be greater than k convex fuzzy numbers ( 1,2,..., ) can be defined as: The weight vector is given by: Via normalization, the weight vectors can be obtained as: where W is a non-fuzzy number.

Shortcomings of Chang's Approach and the Proposed Improved Generalized Fuzzy AHP Method
Wang et al. [32] indicated that using Chang's [21] approach may assign an irrational zero weight to some useful decision criteria and alternatives, leading to them not being The degree of possibility for a convex fuzzy number to be greater than k convex fuzzy numbers S I (i = 1, 2, . . . , k) can be defined as: The weight vector is given by: where A i (i = 1, 2, . . . , n) are n elements, Via normalization, the weight vectors can be obtained as: where W is a non-fuzzy number.

Shortcomings of Chang's Approach and the Proposed Improved Generalized Fuzzy AHP Method
Wang et al. [32] indicated that using Chang's [21] approach may assign an irrational zero weight to some useful decision criteria and alternatives, leading to them not being considered in decision analysis. This problem is shown in Example 1.

Example 1.
Assume that a university wishes to evaluate its lecturers' performance. Three decision makers, D 1 , D 2 , and D 3 , are responsible for the evaluation of three candidates, A 1 , A 2 , and A 3 . Five criteria are chosen for evaluating the lecturers: publication (C 1 ), classroom teaching (C 2 ), student advising (C 3 ), personality factors (C 4 ), and fluency in a foreign language (C 5 ). Table 1 shows the fuzzy comparison matrix of the five criteria based on three decision makers. Accordingly, the priority vector of the five criteria is estimated by Chang's approach as W = (1, 0, 0, 0.008, 0), which means that the criteria C 2 , C 3 , and C 5 are given a zero weight and are not considered in decision analysis. In addition, Liu et al. [34] found that Chang's [21] approach is inappropriate for determining the relative importance in some cases. Consider two TFNs, i.e., A = (l a , m a , u a ) and B = (l b , m b , u b ). In the case that m a = m b but l b ≺ l a and u b ≺ u a , one can logically infer that A should have priority over B (Figure 2a). However, according to Equation (4), when m a = m b , V( A ≥ B) = V( B ≥ A) = 1, the priorities of A and B are the same. In the case that m b = m a + δ, u b = m b + δ, l a = m a − δ, l b = m b + γ, u a = m a + γ, where δ is a very small positive number close to 0 and γ is a large positive number (as shown in Figure 2b), intuitively, the TFN A should be preferred over B. However, using Equation (4), we have V( B ≥ A) = 1 > V( A ≥ B), which indicates that the TFN B has a higher priority. Chang's approach, therefore, fails to correctly determine the relative importance of fuzzy numbers.
Furthermore, Chang's [21] approach only can apply to normal fuzzy numbers. However, in the real world, it is impossible to restrict the membership function to the normal form in some cases. In order to overcome the shortcomings of Chang's [21] approach, this paper proposes a revised fuzzy AHP based on the centroid index ranking approach as follows.
The first step defines the generalized triangular fuzzy comparison matrix. The matrix is expressed by:  Figure   2b), intuitively, the TFN A  should be preferred over B  . However, using Equation (4),  , which indicates that the TFN B  has a higher priority. Chang's approach, therefore, fails to correctly determine the relative importance of fuzzy numbers. Furthermore, Chang's [21] approach only can apply to normal fuzzy numbers. However, in the real world, it is impossible to restrict the membership function to the normal form in some cases. In order to overcome the shortcomings of Chang's [21] approach, this paper proposes a revised fuzzy AHP based on the centroid index ranking approach as follows.
The first step defines the generalized triangular fuzzy comparison matrix. The matrix is expressed by: The second step determines the values of the fuzzy synthetic extents. In this paper, the values of fuzzy synthetic extents S i are defined using the correct normalization formula presented by Wang et al. [32] in the following equation: The third step is to calculate the centroid indices of the fuzzy synthetic extent, S i by using Dat et al.'s [35] approach.
Suppose S 1 , S 2 , . . . , S n are the values of the fuzzy synthetic extents. The centroid point of all fuzzy numbers C i = (x S i , y S i ), i = 1, 2, . . . , n can be calculated by: The distance between the centroid point C i = (x S i , y S i ), i = 1, 2, . . . , n and the minimum point G = (x min , y min ), is determined by Equation (12) and Figure 3: Example 2. This re-considers the data in Example 1. Using Equations (9)- (11), the new fuzzy synthetic extents and the centroid point of five criteria are obtained, respectively (as shown in Table  2). The minimum point of the five criteria is min min By using Equation (12), the distances between the centroid point and minimize point of five criteria are calculated. Finally, the priority vector of five criteria is obtained using Equation (13). Evidently, the priority vector obtained by the proposed fuzzy AHP approach is more reasonable than the outcome obtained using Chang's [21] approach.   0.375, 0.500, and 0.875 and 0.125, 0.500, and 0.625, respectively. Then, the centroid points of fuzzy numbers A  and B  are obtained by using Equations (10) and (11)  and B  are 0.611 and 0.389, respectively, and   The fourth step defines the weight vector W = (w 1 , . . . , w n ) T of the fuzzy comparison matrix as: Example 2. This re-considers the data in Example 1. Using Equations (9)-(11), the new fuzzy synthetic extents and the centroid point of five criteria are obtained, respectively (as shown in Table 2). The minimum point of the five criteria is (x min , y min ) = (0.07, 1.0). By using Equation (12), the distances between the centroid point and minimize point of five criteria are calculated. Finally, the priority vector of five criteria is obtained using Equation (13). Evidently, the priority vector obtained by the proposed fuzzy AHP approach is more reasonable than the outcome obtained using Chang's [21] approach.  (10) and (11), i.e., C A = (0.583, 0.333) and C B = (0.417, 0.333). By using Equation (12), the relative importance measures of A and B are 0.611 and 0.389, respectively, and thus A B. Note that the priority A ∼ B obtained using Chang's approach is seen as unreasonable and not consistent with human intuition. Clearly, the revised approach can overcome the shortcomings of the inconsistency of Chang's [21] approach. Figure 2b, i.e., A = (6, 6.2, 11.6) and B = (1, 6.4, 6.6). Using Equation (9), the fuzzy synthetic extent values of fuzzy numbers A and B are 0.476, 0.492, and 0.921 and 0.079, 0.508, and 0.524, respectively. Then, the centroid points of fuzzy numbers A and B are obtained by using Equations (10) and (11), i.e., C A = (0.63, 0.333) and C B = (0.37, 0.333). By using Equation (12), the relative importance of A and B is 0.654 and 0.346, respectively, and thus A B. Note that the priority B A obtained using Chang's [21] approach is not consistent with human intuition. Again, this example shows that the proposed approach can overcome the shortcomings of Chang's [21] approach.

Application of the Proposed Fuzzy AHP Approach
In this section, the improved fuzzy AHP approach is applied to define the priority of the university's governance factors that affect lecturers' scientific research productivity at Vietnam National University, Hanoi, Vietnam (VNU). VNU is one of two leading multidisciplinary and multi-sectoral national universities in Vietnam. VNU is entrusted with the task of producing highly qualified human resources for the industrialization and modernization of the country. VNU holds a special position in Vietnam's higher education system, operating according to a special regulation promulgated by the Prime Minister. Currently, this university has 35 members (including 8 affiliated universities, 4 affiliated schools, 7 research institutes, 2 training and research centers, and 13 support/service units), 33 research groups, 216 laboratories, and 4.326 staff members. As of the end of 2020, VNU had 488 training programs (including 185 undergraduate programs, 187 master's programs, and 116 doctoral programs), with 40.038 undergraduate students, 7500 graduate students, and 819 international students.
In this study, data were collected by conducting semi-structured interviews with the director of VNU's Organization-Personnel Department, University of economics and Business -VNU's Office of Human resources and department head, University of economics and Business-VNU's faculty head. A committee of three decision makers (D 1 , D 2 , and D 3 ) conducted the evaluation of the university's governance factors.
The entire the evaluation of university's governance factors was characterized by the following steps: Step 1: Defining the university's governance factors.
Step 2: Determining fuzzy judgment matrices of factors and sub-factors.
Step 3: Aggregating decisions from decision makers based on geometric means.
Step 4: Calculating fuzzy synthetic extent values of factors and sub-factors.
Step 5: Determining the weight vector of factors and sub-factors impacting lecturer research productivity.
Steps 1 and 2 were performed by the committee (three decision makers, i.e., D 1 , D 2 , and D 3 ) without any intervention from the authors. Steps 3-5 were calculated using the improved fuzzy AHP approach on a spreadsheet.

Defining the University's Governance Factors
Following a survey of the literature and discussions with the committee, six factors and twenty-two sub-factors affecting lecturers' scientific research productivity were chosen. Table 3 shows the factors and sub-factors used in this study. Table 3. Factors affecting lecturers' scientific research productivity.

No. Factors Sub-Factors and Description References
1 Research objectives and strategies (RO&S) Development orientation towards research university (RO&S1) [2,5,6] Establishing a set of plans and strategies to boost research activities (RO&S2) [9,11,36] Specific and widely communicated objectives (RO&S3) [7] Research strategies and objectives are built up in multiple dimensions (from top to bottom, from bottom to top, there is coordination between units and peer groups) (RO&S4) [1,7,17] 2 Decentralization (DC) Governance decisions are made on the basis of extensive stakeholder engagement, with emphasis on feedback systems and collaboration (DC1) [7,8] Governance activities promote academic independence, equality, communication, and connection among members (DC2) [1,17] The autonomy of institutions and lecturers (DC3) [9] Degree of lecturers' participation in the decision-making process about scientific research activities in their faculty/university (DC4) [17] 3

Leadership (LD)
Professional competence and research experience (LD1) [9] Ability to connect research groups, attract funding, and create an environment that ensures research activities and innovation (LD2) [20] Having a high reputation and clear communication skills (LD3) [6,10] Leaders' support for scientific research work and ability to create a departmental research atmosphere (LD4) [15] 4 Support for research activities (SR) Specific policies to support scientific research activities (SR1) [19] Administrative support of employees, coordinators in departments, and lecturers' efforts in promoting research (SR2) [6,11,12,17] Support for the transfer and commercialization of research products (SR3) [16,37] 5

Determining Fuzzy Judgment Matrices of Factors and Sub-Factors
In this study, the committee was requested to separately proceed to their own fuzzy judgment matrices for the evaluation of the university's governance factors. This study adopts an intensity scale for fuzzy numbers to transform the linguistic values into TFNs, as shown in Table 4.

Calculating Fuzzy Synthetic Extent Values of Factors and Sub-Factors
Using Equation (9) and Tables 5-11, the fuzzy synthetic extent values of factors and sub-factors are calculated (as shown in Table 12).

Determining the Weight Vector of Factors and Sub-Factors Impacting on Lecturer Research Productivity
Using Equations (10)-(13), the weight vectors of the factors and sub-factors affecting lecturer research productivity are shown in Table 13. The result indicates that "resources for research activities" is the most important factor affecting the research productivity of lecturers at VNU. This result is similar to other studies in which it has been found that the quality of human resources, investment in facilities, equipment, and investment funds for research activities have a strong impact on the research productivity of lecturers. VNU is a multi-disciplinary university with a higher percentage of lecturers with doctoral degrees than many other universities in Vietnam. This has been used by VNU in the establishment of interdisciplinary research groups and has resulted in many important achievements through the implementation of national and international research missions. However, at present, the facilities of many universities under VNU are quite limited, scattered, and not synchronized. The financial resources invested in science and technology activities are not commensurate with the potential or the need to improve the research capacity of the teaching staff to meet international standards.

Conclusions
Increasing lecturers' research productivity plays an important role in improving teaching quality, as well as universities' global status and prestige. This study indicated the shortcomings of Chang's extent analysis method for fuzzy AHP and proposed an innovative revised fuzzy AHP approach. The proposed AHP approach considers both normal triangular fuzzy numbers and non-normal triangular fuzzy numbers. Comparative examples were presented to demonstrate the validity and advantages of the proposed fuzzy AHP approach. This study shows that the results obtained using the proposed fuzzy AHP approach are more consistent with human intuition than Chang's approach. The proposed fuzzy AHP approach was further applied to determine the priority of the university's governance factors that affect lecturers' scientific research productivity. The result indicated that "Resources for research activities" is the most important factor affecting the research productivity of lecturers at VNU, followed by research objectives and strategies and leadership.