Combining reference point based composite indicators with data envelopment analysis: application to the assessment of universities

In our knowledge society, where universities are key players, the assessment of higher education institutions should meet the new demands of the present complex environment. This calls for the use of techniques that are able to manage this complexity. In this paper, we propose a novel combination of methodologies, jointly using a multi-criteria reference point scheme and the data envelopment analysis (DEA) for the assessment of universities. This combination allows us to take into account all the aspects regarded as relevant to assess university performance, and use them as outputs in the efficiency analysis. Our findings highlight the convenience to assess the university performance by using both compensatory and non-compensatory schemes. This way, the information provided allows to detect the actions needed to improve the performances of the universities, rather than just giving an overall performance measure. Furthermore, combining the use of composite indicators with the DEA analysis provides a more complete picture of the institutions assessed, allowing universities to check their efficiency and to detect their weaknesses and strengths accordingly. The approach is illustrated using data of 47 Spanish public universities for the academic year, 2016–2017.


Introduction
The evaluation of universities' performance has become indispensable, due to their vital role in the economic growth and social development of our society. Higher education institutions are key drivers of growth of performance, prosperity and competitiveness, at both the national and international levels and as a result, university rankings, an outcome of the competitiveness boosted by globalization, have put pressure on the universities to adapt to market forces without compromising quality (Heitor & Horta, 2013). However, most of the existing rankings operate under the concept of "the higher the value for all the indicators, the higher the performance achieved" (González-Garay et al., 2019). Hence, they assign certain performance measures to universities, regardless of whether they make an efficient use of their resources or not.
An increasing pressure on public finances is giving rise to the need to operate public universities with a higher degree of efficiency (Abbott & Doucouliagos, 2003). Since the pioneering works (Charnes et al., 1978(Charnes et al., , 1981, the comparative analysis of the efficiency in education has been brought into the focus of public and policy interest (De Witte & López-Torres, 2017). There is no doubt that this topic will continue to dominate headlines in the field of higher education, where a university is regarded as efficient if it produces the highest output levels by using the lowest input levels (Visbal-Cadavid et al., 2017). Specifically, following the economic recession in 2008, the general trend in reducing public funds for Spanish universities has led to define the allocation of resources in an efficient way as a priority for the Spanish public higher education sector (de la Torre et al., 2017a;Martínez-Campillo & Fernandez-Santos, 2020).
Data envelopment analysis (DEA) and its variations have been the most frequently used methodologies for measuring the efficiency in the context of higher education. Empirical research on the higher education efficiency using DEA models is becoming increasingly important at the international level (e.g. Abbott & Doucouliagos, 2003;Agasisti et al., 2019;Johnes, 2006). In Spain, according to Martínez-Campillo & Fernandez-Santos (2020), there has been limited research on the technical efficiency of the public universities (Berbegal-Mirabent et al., 2013;Berbegal-Mirabent, 2018;de la Torre et al., 2017ade la Torre et al., , 2017bde Jorge Moreno et al., 2019;Martínez-Campillo & Fernandez-Santos, 2020;Salas-Velasco, 2020a, 2020b. This approach is suitable in this context for several reasons: (a) it can handle multiple inputs and multiple outputs in a simple manner, (b) it does not assume any underlying functional relationships between inputs and outputs and (c) it determines the weights of inputs and outputs in an endogenous manner, looking for those that evaluate each unit in the best possible way, as compared to all the units (Abbott & Doucouliagos, 2003).However, the suitable number of inputs and outputs in relation to the number of units included in the analysis is still debatable (Akbari et al., 2020). Some authors claim that the number of units has to be at least twice the number of inputs and outputs to be used (Dyson et al., 2001), while others suggest that the number of units has to be three times higher. In fact, problems of discrimination between efficient and inefficient units often arise when there is a relatively large number of variables, as compared to the number of units (Charles et al., 2019) and thus, the greater the number of inputs and outputs to be used in relation to the number of units, the lower the discrimination in assessing relative efficiency (de Jorge Moreno et al., 2019). It is worth highlighting some existing approaches on variable (indicator) aggregation before DEA, such us principal component analysis (Adler & Yazhemsky, 2010), where the main applying components replace the original variables. It has also been suggested to set weights to the original variables, using information about prices (Nguyen & Zelenyuk, 2021), or value judgments via analytic hierarchical process (Meng et al., 2008), or applying a variant of DEA which offers the summarized variable from variable subsets defined by experts (Charles et al., 2019). In our case, the output variables in the DEA model will be composite indicators that summarize information on multiple aspects of the units' performance, in order to analyze whether better performance is jointed with an efficient use of available resources.
In the particular case of universities, most of the papers that use DEA combine labor and capital inputs to produce two main outputs: human capital (teaching) and scientific knowledge (research). Few studies, such as those by Berbegal-Mirabent et al. (2013) and de la Torre et al. (2017a) consider the technology transfer as a separated mission from the traditional ones of the university (teaching and research). We follow this focus in this paper. Interestingly, the number of outputs considered in the previously referenced papers ranges from two to four, being the number of graduate students and the number of publications the most frequently used ones. Nevertheless, the higher education system is quite diverse and multi-faceted, and therefore, it is difficult to condense the diversified activity going on within universities into few variables as outputs proxies (González-Garay et al., 2019). For this reason, it is necessary to introduce new metrics that allow us to combine a greater variety of aspects and avoid, at the same time, the use of an excessive number of variables, which could make the DEA model reduce its discriminatory power. Besides, in today's globalized and highly competitive environment, the higher education system should have a powerful tool to detect improvement areas, rather than just ranking the universities. Consequently, the evaluation of the university system requires the use of techniques that are able to manage all this complexity (Attardi et al., 2017).
In recent years, the measurement of such multidimensional frameworks has received special attention. Within this context, the use of composite indicators is highly suitable. They enable aggregation of many single indicators into one measure, making it possible to compare many units (Molinos-Senante, 2018). This way, they integrate large amounts of information in a clear and understandable way that is easy for decision makers to interpret (Szuwarzynski, 2018). The use of composite indicators greatly simplifies the comparison among higher education institutions, helping them to benchmark their performance against each other in a straightforward manner (González-Garay et al., 2019).
Few initiatives have tackled the evaluation of university performance without using composite indicators, such as the U-Multirank, which considers the scores of universities on individual indicators and place these in five performance groups ("very good" through to "weak"). Oppositely, a general widespread trend towards the construction of composite indicators has been established in the field. Most of the existing rankings have developed composite indicators to provide rankings of higher education institutions or countries (the ARWU ; the Times Higher Education World University Ranking (THE); the QS World University Ranking (QS); the U-Ranking project of the BBVA Foundation and the IVIE 1 (Aguillo et al., 2008;Alasehir et al., 2014;Giannoulis & Ishizaka, 2010;Torres-Salinas et al., 2011).
The construction of composite indicators has been tackled from different angles. One of the most extended approaches consists of using multiple criteria decision making (MCDM) methods (see, e.g., El Gibari et al., 2019). Murias et al. (2008) present a composite indicator for quality assessment in the Spanish public university system using DEA, while Giannoulis and Ishizaka (2010) adopt ELECTRE III to evaluate the performance of British universities, reflecting personal preferences. El Gibari et al. (2018) develop composite indicators for different compensation degrees for each of the three main missions of universities (research, teaching and technology transfer). To this end, they evaluate the performance of public universities using a multi-criteria analysis technique, based on the double reference point method (Ruiz et al., 2011;Wierzbicki, 1980). The use of this technique in the evaluation of university performance has two main advantages. First, the use of reference levels for each indicator makes the results obtained easily interpretable in terms of the performance of the university with respect to these levels. Second, aggregations for different compensation degrees are provided, and thus, apart from giving an overall performance measure of the universities, they also provide warning signals that assists the user in strategic decision making for policy purposes. This method was generalized to the multiple reference point weak-strong composite indicator (MRP-WSCI) approach, where any number of reference levels and any scale can be used (Ruiz et al., 2020).
According to Nardo et al. (2005), weighting is a key step in constructing composite indicators. Weights can have a significant effect on the overall composite indicator and the results obtained. The literature provides a wide range of methodological approaches. Generally, weights are usually adopted on the basis of expert's opinions and decisions makers, but the equal weighting approach is also widely used (Antanasijevic et al., 2017;Blancas et al., 2010;Langhans et al., 2014;Zhou et al., 2010). Anyway, weighting implies introducing controversial and subjective elements on which there is no general consensus (Huang, 2011;Marginson & Van der Wende, 2007).
In this paper, we discuss how the combination of the MRP-WSCI approach with the data envelopment analysis (DEA) can add value to traditional composite indicators, not only in terms of securing high performance, but also in how to make an appropriate use of resources. Thus, the contribution of this paper to the current literature is twofold. First, we analyze the added information provided by the joint consideration of the MRP-WSCI composite indicators and the DEA analysis. Unlike the approach followed in previous studies, we use different composite indicators for each mission (teaching, research and technology transfer), which allow us to take into account a greater variety of aspects regarding the activities of universities. Besides, we consider two scenarios, depending on whether the compensability among certain indicators is or not possible, in order to test its influence on the evaluation of the efficiency of universities. Second, the DEA efficiency scores for the Spanish public universities is carried out in a robust environment, with respect to changes of the weights when constructing the MRP-WSCI composite indicators. The suggested methodology is applied to the case of the Spanish public universities, but it can be used as a supporting tool in other university systems, allowing decision makers to detect and improve important factors of the educational efficiency and effectiveness of universities. In fact, the ultimate goal of the research proposed in this article is to provide the decision makers of the higher education system with a tool that allows them to evaluate the performance and efficiency of the universities and make decisions accordingly, taking into account their preferences.
Following this introduction, "Available system of indicators" gives a short overview of the data used for the evaluation of the Spanish public universities. "Research methodologies" describes the MRP-WSCI procedure and its combination with the DEA analysis. The empirical results for the compensatory and non-compensatory scenarios are presented in "Compensatory scenario" and "Non-compensatory scenario", while "Discussion" discusses some remarkable aspects related with the results of this paper, and finally, "Conclusions" presents some conclusions.

Available system of indicators
The first step of our study consists of analyzing the performance of the Spanish public universities. To this end, we have used an existing set of indicators that is frequently used by several Spanish institutions. More precisely, a set of 23 performance indicators is selected, taking into account both their relevance to measure the Spanish public universities performance and their availability (for further details about the performance indicators considered in each mission, see Table 1). All of them are outcome indicators, and some have been relativized in order to avoid biases due to the size of the university (Alasehir et al., 2014). We would like to point out that the aim of this research is showing the validity of the methodology proposed in this field. We do not discuss the suitability of the particular system of indicators used, which is the system used by Spanish higher education managers. Our methodology can be applied to any other system of available indicators, should it be regarded as more suitable by the decision makers.
The research and teaching missions are further broken down into three and two subblocks, respectively, while the technology transfer missions has no sub-blocks. We have considered that indicators within a sub-block are substitutable, that is, poor performance in some ones can be offset by good behaviors in other ones. However, the different sub-blocks belonging to the same mission may be substitutable or not. This issue will lead to two scenarios depending on whether the compensability is assumed or not.
Second, we measure the efficiency of the Spanish public universities, where the outputs considered are the composite indicators provided by the MRP-WSCI approach. According to (Salas-Velasco, 2020b), the "perfect set of inputs" to evaluate university efficiency does not exist. In fact, the literature of higher education efficiency provides a wide range of inputs. According to Berbegal-Mirabent et al. (2013) and Anderson et al. (2007), in the higher education field, the labour force and capital inputs are vital. In particular, it is important to consider faculty members, as they transmit knowledge to students and carry out most of the research, while the R &D income provides financial resources necessary to develop new research and technology transfer outputs. Moreover, certain financial expenditures are key inputs to the development of the activities of each university (Zhong et al., 2011). In this paper, four inputs are considered: Ratio of faculty members with PhD, R &D funds per faculty member, Staff expenses per students and Expenses on good and services per student (similar to, Agasisti & Pérez-Esparrells, 2010;Expósito-García & Velasco-Morente, 2018;Yang et al., 2018, these expenses refer mainly to expenditure on academic staff, expenditure on non-academic staff and running expenses in relation to goods and services). The input and output specifications of each DEA analysis performed are displayed in Table 2.
In summary, the performance indicators and the inputs considered in this paper have been used by the Spanish Education authorities to evaluate the Spanish universities in different areas, such as access to funding, quality and excellence, internationalization and results. Our database contains data on research, teaching and technology transfer for 47 Spanish public universities for the academic year 2016-2017 ( Table 5 in the Appendix records the names of such universities and their abbreviations). The data were collected from two sources: the Spanish University Rectors' Conference 2 (CRUE) and the IUNE Observatory 3 .

Research methodologies
In this paper, we propose a combination of the MRP-WSCI approach (Ruiz et al., 2020) with the DEA analysis (Charnes et al., 1978) for the evaluation of the Spanish public universities. This section describes both methodologies. Then, we describe the stages for combining them.

MRP-WSCI approach
The MRP-WSCI approach, defined in Ruiz et al. (2020), is an adaptation of the original reference point method proposed by Wierzbicki (1980) to the construction of composite indicators, which generalizes the double reference point method (Ruiz et al., 2011;Wierzbicki et al., 2000). In order to construct the MRP-WSCI composite indicators, different aspects are taken into account: • Reference levels. The performance of the units (J), in our case, 47 Spanish public universities, is assessed in terms of how far is each indicator (i) from given reference levels. The decision-maker can give any number of reference levels ( q t i ) for each indicator. These reference levels can be "absolute" or "relative". In the first case, the reference levels are given by one or a group of experts, if they have enough knowledge about the problem and they wish to do so. In this case, the final composite indicator gives us an absolute measure of performance, with respect to these values. Alternatively, they can be set statistically, given a data set. In this case, the composite indicator measures the relative position of the units with respect to those belonging to the data set. • Achievement functions. Once the reference levels are established, a so-called achievement function ( s i (x ij , q i ) ) measures the deviation between the values of the objective functions and the reference levels, and at the same time, brings all the indicators down to a common scale. The achievement function for the "the more, the better" type of indicators is calculated as follows: are the reference levels of indicator i and each t (t = 0, … , n + 1) is the value in the common scale. • Weights. Unavoidably, the different indicators used to measure university performance do have different relative importance, and this has to be taken into account when building an overall measure. The methodology must be designed in order to clearly identify these subjective elements, and to allow potential users to include their own weights ( i ) in the process. • Compensation. The MRP-WSCI method allows to construct two different composite indicators, depending on the compensation degree among the indicators.
First, the weak indicator (WCI) allows for full compensation among the individual indicators. (1) where w i are the normalized weights, which add up to 1. Second, the strong indicator (SCI) does not allow for any compensation. Therefore, it reflects the worst values achieved by an unit, relativized by the weight of the indicator. To this end, the original achievement functions have to be modified as follows: where s i are the corrected normalized weights. When constructing the SCI, a different normalization of the weights is carried out, where the greatest weight takes value 1. This way, SCI takes a worse value if the corresponding indicator has a higher weight, and in particular, it takes the worst possible value if unit j has the worst possible value in the highest weighted indicator.
Therefore, the SCI takes the form: Summing up, the WCI gives an overall performance measure of the units (universities), while the SCI provides warning signals that let the user detect improvement areas.

Data envelopment analysis
Data envelopment analysis (DEA) is a non-parametric technique, originally introduced by Farrell (1957) and extended by Charnes et al. (1978Charnes et al. ( , 1981. It was proposed to analyse the efficiency problem of complex organizations that produce many outputs ( o = 1, … , O ) using several inputs ( m = 1, … , M ) at a time. In a DEA model, technical efficiency is defined as the relative ability of each decision making unit (DMU, j = 1, … , J ) in producing outputs given a certain set of inputs. An important aspect to emphasize in the DEA model is the choice of a set of weights which combines several outputs and several inputs. This issue is left to DEA through a linear programming technique which chooses the best set of weights for each DMU to maximize the efficiency ratio (outputs/inputs).
In this paper, we employ an output-oriented model, because our aim with the DEA model is to analyze whether the performance of a university, measured by MRP-WSCI indicators (outputs), is the best possible one, given its available resources (inputs). Besides, we use the BCC model (Banker et al., 1984), which assumes variable returns to scale (VRS), so that each university is compared to others of the same "relative" size. Both hypotheses are usual in this context (Agasisti & Pérez-Esparrells, 2010;Visbal-Cadavid et al., 2017). Furthemore, in our case, the BCC model allows us to guarantee that the projected output is within the common scale used for the composite indicators obtained in the first step.
Given these considerations, the linear program used to obtain the efficiency level of each DMU (university j) is: where 1 ≤ j < +∞ indicates the score efficiency for university j. When j = 1 , university j attains an efficient (best-practice) performance, in the sense that each of its outputs could not be improved, remaining the inputs at the same level. Contrariwise, when j > 1 , university j is failing to attain best-practice performance. k is the intensity vector and is the non-Archimedean infinitesimal constant. s + o is the output slack vector, while s − m is the input slack vector. y oj is the amount of output o from university j and x mj is the amount of input m from university j.

Combining the MRP-WSCI approach and the DEA analysis
In order to evaluate the Spanish public universities by combining the MRP-WSCI approach and the DEA analysis, as can be seen in Fig. 1, the empirical analysis carried out in this paper follows these steps: 1. Construction of the MRP-WSCI composite indicators for each mission, in a robust environment with respect to the indicator weights.
• Normalization. In this paper, in order to compare the Spanish public universities among themselves, the statistical option for establishing the reference levels has been chosen. This way, we will obtain a measure of the relative performance of each university with respect to all the Spanish public universities. Namely, for each indicator i, apart from the minimum ( Min i ) and maximum ( Max i ) values, we have used two intermediate reference levels: the reservation level ( r i , which corresponds to the average value between the mean and the Min i value) and the aspiration level ( a i , which corresponds to the average value between the mean and the Max i value). Once the reference levels are given, all the performance indicators are translated to a common default scale from 0 to 3, that is, we set 0 = 0 , 1 = 1 , 2 = 2 and 3 = 3 , which define the performance levels of each indicator i, e.g. poor (0 to 1, the university performs worse than the corresponding r i ), fair (1 to 2, the university performs better than the r i level, but worse than the a i level), and good (2 to 3, the university performs better than the corresponding a i ). It should be mentioned that an outlier detection, using the interquartile range ( IQR = Q 3 − Q 1 ) method was carried out (Tukey, 1977). This test allows to distinguish among mild outliers ( Q 1 − 1.5 × IQR , Q 3 + 1.5 × IQR ) or extreme outliers ( Q 1 − 3 × IQR , Q 3 + 3 × IQR ). In our case study, no extreme outliers were detected, while the mild outliers detected were assigned, on the corresponding 1 3 achievement function, the maximum or minimum value (depending on its relative position) of the scale. • Weighting. As previously mentioned, the developed methodology aims to provide decision makers with a useful tool for decision making. Therefore, the evaluation of the performance of the universities has to take into account their preferences, that is, the relative importance they give to the different elements of the system of indicators used. For this reason, in this study, the weights of the performance individual indicators and sub-blocks were assessed by a group of experts in the field of the Spanish universities, integrated by researchers and professionals from different fields of research and different Spanish public universities (the weights assigned to each indicator and sub-block are displayed in Table 1 in "Available system of indicators"). These experts have been chosen according to the criteria of people with great decision-making capacity in the field of the Spanish public university system. In any case, given the impact of these weights on the final results obtained, in "Discussion" a comparison is made with the results that would be obtained if equal weights were used. • Robustness analysis. In order to construct the MRP-WSCI composite indicators in a robust environment with respect to the indicator weights, we have implemented 500 computation rounds, changing randomly ±10% the weights of all the indicators (and corresponding sub-blocks) considered for the construction of the MRP-WSCI approach for each mission. The values of the weights are sampled from a uniform distribution centered in its actual value, and censored at 0 and 1. • Aggregation. Finally, the MRP-WSCI composite indicators are calculated in a robust environment with respect to the indicator weights. In our case, as mentioned in "Available system of indicators", the research and teaching missions are further broken down into three and two sub-blocks, respectively, while the technology transfer mission has no sub-blocks (for further details, see Table 1 in "Available system of indicators"). This way, the aggregation is carried out in two steps: -In the first step, since the information provided by the individual indicators belonging to each sub-block is homogeneous, a full compensation among them is allowed. That is, bad performances in certain single indicators can be offset by good performances in other. This way, we get the WCI of each sub-block of the research and the teaching missions. -In the second aggregation step, different compensation degrees among the subblocks are provided. First, we construct the WWCI, which allows for a full compensation among the different sub-blocks. This way, the WWCI provides a measure of overall performance of each Spanish public university in each mission. Second, the SWCI does not allow for any compensation among the sub-blocks. Therefore, the use of SWCI is useful to point out the worst sub-block of each mission, and, to detect possible improvement areas of the Spanish public universities. In the case of the technology transfer mission, which has no sub-blocks, the WCI and SCI are obtained directly in the first step (by aggregating the individual indicators).
As mentioned before, the construction of the MRP-WSCI composite indicators is made in a robust environment with respect to the indicator weights. This way, we obtain 500 MRP-WSCI composite indicators for each compensatory scenario (and for each mission), that is, we construct 500 WWCI and 500 SWCI for each mission, changing randomly ±10% the corresponding weights considered. 2. The MRP-WSCI composite indicators (used later on as outputs in the DEA analysis) are represented through box-plots for the 500 instances for each Spanish public university (and for each mission). This way, we can graphically see the variability of the corresponding weak and strong composite indicators across the 500 instances, and analyze their reliability. 3. Once the performance of the Spanish public universities is analyzed using the MRP-WSCI approach, the next step consists of calculating the efficiency scores of the Spanish public universities, considering the composite indicators obtained as outputs. To this end, we perform two DEA analyses with different outputs. First, the WWCI of each mission are considered as outputs. Similarly, for each mission, we consider as outputs the SWCI. As mentioned in "Available system of indicators", four different inputs are considered (Ratio of faculty members with PhD, R &D funds per faculty member, Staff expenses per students and Expenses on good and services per student). As previously 1 3 mentioned, all the composite indicators obtained are on a scale from 0 to 3, and since we use the BCC model (5), we have that is, the projected output is in the same scale as the composite indicators. It must be noticed that a DEA analysis is carried out for each one of the 500 instances, and the robustness of the results is also analyzed. 4. Finally, the input and output levels and the efficiency scores are represented through bar charts for each Spanish public university.

Application to the Spanish public higher education system
In this section, we show and analyze the results obtained. To this end, the combination of the MRP-WSCI approach and DEA analysis developed in "Combining the MRP-WSCI approach and the DEA analysis" is applied to the data of the 47 Spanish public universities. As previously mentioned, different DEA analyses with different outputs are performed (for further details about the input and output specifications, see Table 2 in "Available system of indicators"). First, we analyze the performance of the Spanish public universities in the compensatory scenario (WWCI of each mission), and we measure the efficiency of the universities in this scenario. Similarly, we analyze the performance and efficiency of the Spanish public universities in the non-compensatory scenario (SWCI of each mission). As mentioned in "Available system of indicators", the inputs are the same in both analyses. Note that, as mentioned before, our analysis is made in a robust environment with respect to the performance indicator (and sub-blocks) weights.

Compensatory scenario
The results of the 500 instances for the compensatory scenario are shown in Fig. 2, where each sub-figure presents the box-plot for the 500 values of the WWCI for each Spanish public university, by changing randomly ±10% the weights of all the individual indicators (and corresponding sub-blocks) considered in each mission. We can observe that the range of variation in all missions is very small. In general, we can assume that the WWCI of the Spanish public universities are fairly robust, since the solution proved to be stable for all missions.
Let us analyze the overall performance of the Spanish public university system in each mission. As seen in Fig. 2, if we consider the median value of the WWCI for the 500 instances, most universities are in the "fair" performance level for the three missions (1 < value of the WWCI < 2). That is, most of the Spanish public universities perform better than the corresponding reservation level, but worse than the aspiration level for all the university missions analyzed. Note that the Spanish public universities are compared among themselves.
Specifically, in the research mission, among the 47 Spanish public universities in the data set, there is only one university (ULPGC) in the "poor" performance level (median value of the WWCI = 0.88 < 1), while four universities (UPF, UAB, UAM and UB) are j y oj ≤ J ∑ k=1 k y ok ≤ 3, Fig. 2 Robustness analysis on weights: effect on WWCI of each mission in the "good" performance level (median value of the WWCI > 2). The case of UPF is noteworthy, getting the best WWCI value for the 500 instances (median value of the WWCI = 2.60). This is because the "intermediate" performance in two individual indicators (participation in projects and Official recognition research) is compensated by good results in the rest of the research performance indicators. In fact, UPF has the best possible value (3), that is, the best value among all the Spanish universities, in four research indicators, while the rest of indicators are over the aspiration level. Regarding the teaching mission, in general, the overall performance of all the Spanish public universities is quite similar, and only one university (UV) achieves a median value of the WWCI better than the aspiration level (median value of the WWCI = 2.08 > 2), while two universities (ULL and US) have the median value of the WWCI worse than the reservation level (median value of the WWCI < 1). The rest of Spanish public universities are in the "fair" performance level (1 < value of the WWCI < 2). With respect to the technology transfer mission, a higher number of universities perform poorly. Specifically, the median values of the WWCI for fourteen Spanish public universities are worse than the reservation level, while only UPM performs better than the aspiration level in the technology transfer mission (median value of the WWCI = 2.04 > 2).
Next, we apply DEA to each of the 500 instances. First, it should be pointed out that the efficiency scores for all the Spanish public universities are also very robust, since the solution proved to be stable for the 500 instances, specially for the efficient ones ( j = 1 for the 500 instances in all of them). Second, in order to analyze the DEA results in a more intuitive way, for each Spanish public university, Fig. 3 shows the input levels (the first four sub-figures), while the output levels (the fifth sub-figure) and the efficiency scores (the sixth sub-figure) are represented taking into account the median value of the 500 instances.
As can be seen in Fig. 3, on the one hand, 18 Spanish public universities are efficient in the compensatory scenario (colored in red in the bottom part of Fig. 3). That is, when a total compensation is allowed among the individual indicators (and the corresponding sub-blocks), the DEA analysis point towards the presence of a group of 18 Spanish universities that make an efficient use of their resources. The rest of the universities are not efficient (colored in blue). It must be taken into account that the level of inefficiency increases as the DEA inefficiency score grows. For example, a DEA score of 1.42 means that this university would have to significantly increase (42%) its level of outputs (WWCI of each mission), given its available inputs. For some of the most inefficient universities, it can be observed in Fig. 3 that, despite having high values of the inputs considered (topmost part of the figure), they do not manage to perform well in any of the three missions analyzed (middle part of the figure, research in blue, teaching in red and transfer in yellow).
Furthermore, our analysis shows that, in addition to the Spanish public universities that are always present in the most prestigious university rankings (e.g. UPF, UAB, UAM and UB), other Spanish public universities, occupying the lowest positions in some missions in the previous analysis, are among the efficient ones. Let us analyse, for example the case of ULPGC, which in Fig. 2, has the worst median value of the WWCI for the research mission (0.88) and the second worst median value of the WWCI for the technology transfer mission (0.51), while in the teaching mission, it performs better (1.48). However, the result of the DEA analysis (Fig. 3) points to an efficient use of its resources. It should be noted that ULPGC has the lowest value of R &D funds per faculty member. In other words, taking into account that the inputs available are quite modest, the performances are, comparatively, good enough. On the other hand, it is noteworthy the case of UV, which is among the efficient Spanish public universities.

3
This university, despite having one of the lowest Staff expenses per students, was the only university with a median value of the WWCI for the teaching mission better than the aspiration level, and it has one of the best performances in the "fair" level for the Therefore, these findings highlight the need to complement the use of composite indicators and university rankings with other analytic tools, such as the DEA analysis. Overall, according to González-Garay et al. (2019), DEA provides further insight into the assessment of universities, allowing institutions to better understand their weaknesses and strengths.

Non-compensatory scenario
Next, we will analyze the performance and efficiency of the Spanish public university system in the non-compensatory scenario (SWCI of each mission).
Similarly to the previous analysis, the results of the 500 instances for the non-compensatory scenario are shown in Fig. 4, where each sub-figure presents the box-plot for the 500 values of the SWCI for each Spanish public university, by changing randomly ±10% the weights of all the individual indicators (and corresponding sub-blocks) considered in each mission. First, comparing Figs. 2 and 4, we can notice that, to some extent, due to its compensatory nature, results for WWCI are more robust than results for SWCI. Anyway, the results are still robust enough to be analyzed and used in the DEA analysis.
In this case, most of the Spanish public universities are in the "fair" performance level for the research and teaching missions (1 < values of the SWCI < 2), meaning that all their sub-blocks perform better than the reservation level. However, in the technology transfer mission, most universities are in the "poor" performance level (values of the SCI < 1). That is, most Spanish public universities perform worse than the corresponding reservation level for at least one indicator (given that there are no sub-blocks in this mission).
Let us analyze the results of the SWCI for the research mission (Fig. 4, first sub-figure). Let us point out the case of UPF, which is the only Spanish public university in the "good" performance level (median value of the SWCI = 2.43 > 2, that is, with all the sub-blocks performing better than the aspiration level), and the one with a widest variability range compared to the rest of universities. First, UPF has the best possible value (3) in all the individual indicators for the publications sub-block, which has the highest weight in the research mission (0.425). This way, the median value of the WCI of the publications sub-block is 3. On the other hand, the median values of the WCI of the projects sub-block and the other research activities sub-block are 2.53 and 2.01, respectively (the sub-block weights are 0.345 and 0.23, respectively). The median value of the SWCI of the research mission is 2.43. However, the SWCI value for many experiments is worse than this median value (wider variability), which is mainly due to its worse performance in the other research activities sub-block (in the previous analysis, we have mentioned that this university has two individual indicators in the "intermediate" performance level, participation in projects and Official recognition research, which both correspond to the other research activities sub-block). As a result, when the weight of this sub-block is increased, the SWCI gets a worse value. This allows decision makers at UPF learn that a special attention should be paid to these indicators.
With respect to the technology transfer mission, which has no sub-blocks, the SCI is obtained directly by aggregating the individual indicators. Only 8 Spanish public universities are in the "fair" performance level, being the SCI value for UPC the best one for the 500 instances (median value of the SCI = 1.88). This is because, all the individual indicators of this university are better than the corresponding reservation level (values of its achievement 1 3 functions > 1). In fact, its worst indicators correspond to the PCT extensions per doctor and Spin-offs per doctor (1.85 and 1.86, respectively). These two indicators are the lowest weighted ones in the technology transfer mission (0.23 and 0.19, respectively). This way, its SCI is not so bad, since its worse performance corresponds to the lowest weighted indicator.
The rest of universities are in the "poor" performance level. Let us analyze, for instance, the case of ULPGC. This university has the worst possible value (0), that is, the worst value among all the Spanish universities, in the highest weighted indicator (National patents per doctor, 0.287). This way, for most instances, its SCI takes the worst possible value (0).
Regarding the efficiency analysis in the non-compensatory scenario, 17 Spanish public universities are efficient (colored in red in Fig. 5). Once again, the efficiency scores for all the Spanish public universities proved to be stable for the 500 instances, specially for the efficient ones. The differences in the efficiency assignments under the two scenarios are discussed in further detail in "Discussion".
Comparing Figs. 3 and 5, we can observe that four Spanish public universities (UAB, UAM, USAL and UJAEN) are efficient in the compensatory scenario (Fig. 3), while in the non-compensatory one they are inefficient (Fig. 5). First, it can be noticed that the median DEA efficiency scores of UAB, UAM, USAL and UJAEN are 1.02, 1.01, 1.01 and 1.07, respectively. This suggests that in response to a small proportional increase (only 2%, 1% and 7%, respectively) of their level of outputs (SWCI of each mission, that is, concentrating this increase in the indicators or sub-blocks with the worst behaviors). Note that additional reductions in some inputs and/or increases in some outputs may be necessary.
All these universities perform worse than the reservation level of the technology transfer mission ("poor" performance level). Specifically, in the case of UAB, our analysis shows that, despite having one of the best SWCI in the research mission, the value of its SCI for the technology transfer mission has still much room for improvement. This university performs poorly in the highest weighted technology transfer indicator (median value of its SCI in the National patents per doctor is 0.22). Similarly, UAM has one of the best SWCI in the research and teaching missions. However, its SCI for the technology transfer mission is in the "poor" performance level. Therefore, these universities should not neglect any of the higher education missions, and pay special attention to the technology transfer mission, since our results reveal that they have a good performance in the teaching and research missions, but they perform poorly the technology transfer one.
On the other hand, UNICAN is a medium size university, which is inefficient in the compensatory scenario (Fig. 3), while in Fig. 5, it manages to make an efficient use of its resources (Fig. 5). This university has one of the lowest value of Faculty members with PhD, and is among the Spanish public universities with the best SWCI performance in the research and technology transfer missions. In fact, as mentioned in the previous analysis, only 8 Spanish public universities are in the "fair" performance level, being UNICAN among them (median value of its SCI = 1.37 > reservation level = 1).
Summing up, our analysis highlights the convenience of assessing the university performance by using both compensatory scenario (WCI) and non-compensatory one (SCI), where the WCI gives an overall performance measure of the universities, while the SCI helps to detect the actions needed to improve the performances of the universities. Furthermore, combining the MRP-WSCI composite indicators with the DEA analysis gives us a more complete picture of the institutions under assessment, allowing universities to better understand their weaknesses and strengths.

Discussion
In this section, we will discuss certain relevant aspects related methodology proposed for the assessment of universities. First, we report the results of a robustness analysis on the reference levels used. Second, the impact of the weights used is discussed, and results obtained with equal weights are reported and commented. Third, we discuss and compare the results obtained for the compensatory and non-compensatory scenarios. Finally, we give some hints about how this methodology can be used to make improvement decisions for a given university.

Robustness analysis on reference levels
With respect to the MRP-WSCI composite indicators, logically, the results obtained depend on the reference levels chosen, which must be set taking into account the aim of the analysis. According to the experts opinions, in this paper, the reference levels have been set by considering the 47 Spanish public universities, and this way, they are compared among themselves. In order to test how a change of the reference levels would affect the MRP-WSCI composite indicators, a robustness analysis, with respect to changes of the reference levels has been carried out. To this end, similarly to the case of the weights, we have implemented 500 computation rounds for each option analyzed in "Application to the Spanish public higher education system", changing randomly ±10% the Spanish universities reference levels (Fig. 7 in the Appendix shows the results of the 500 instances for both MRP-WSCI composite indicators for each mission). First, we can conclude that, in general terms, a high stability can be detected in both MRP-WSCI composite indicators for each mission. Second, in general, results with respect to changes of the reference levels are even more robust than results with respect to changes of the weights.

An analysis of the overall results changing the weights
In this paper, given that the purpose is to provide university managers with tools to support their decisions, it has been decided to use weights assigned by experts recognized as such by the decision-making centers. Anyway, as mentioned in "Introduction", weights do have an important impact on the overall composite indicator and the results obtained. For this reason, an analysis of the performance and efficiency of the Spanish public university system has been carried out, considering slight modifications (random changes ±10%) of the weights given by the experts. In any case, in this section we want to go a step further and discuss the results that would be obtained using a totally different weighting scheme: equal weights for both simple indicators and sub-blocks. In this way, we can clarify the impact of the weights on the solutions obtained.
The comparison of the results obtained can be seen in Fig. 8 of the Appendix. Note that, in the case of equal weights, only one problem has been solved, so the statistical values refer to the values obtained for the different universities. On the other hand, the statistical values of the experts' weights case refer to the values obtained in the 500 instances, for the different universities. In the compensatory scenario (WWCI), it can be seen that the median and mean values of the research and teaching missions improve when considering equal weights, while there is a decrease in the range of values due to a decrease in the maximum values. On the other hand, the results worsen in general in the technology transfer mission, which suggests that the universities present worse behaviors in indicators to which 1 3 the experts assigned greater weights. These variations cause greater homogeneity among the universities studied, so the efficiency scores improve. As a result, in the case of equal weights, one more efficient university appear, apart from the 18 that were efficient with the experts' weights.
In the non-compensatory scenario (SWCI), the worsening of the performance measures when using equal weights is generalized in the three missions and especially significant in the technology transfer one. This is because a bad behavior in any indicator is regarded as equally bad when building the composite indicator. Once again, the result of this general decline is that efficiency scores improve. As a result, in the case of equal weights, up to 6 more efficient universities appear, apart from the 17 that were efficient with the weights of the experts. Therefore, it can be concluded that the use of equal weights is less demanding when establishing efficient universities. The experts' weights place more demand on certain indicators and sub-blocks, which translates into the existence of less efficient universities.

A comparison of the compensatory and non-compensatory scenarios
As mentioned in "Combining the MRP-WSCI approach and the DEA analysis", in the first aggregation step, a full compensation among the individual indicators is allowed, while in the second step, depending on the compensation degree allowed among the different subblocks, two different composite indicators are provided (WWCI and SWCI). On the one hand, the WWCI allows to analyse the overall performance of the universities, and on the other hand, the SWCI offers alert signs which let the decision-maker detect improvement areas. This way, due to the philosophy behind their construction, there are obvious differences among the results obtained in both scenarios. In order to allow a more direct interpretation of the results obtained, in this section the weights assigned by the experts are used directly, without taking into account the previously described robustness analysis. Figure 6 shows the levels of efficiency for each university in both scenarios. Each university is represented by a dot, whose horizontal component is its DEA score in the compensatory scenario, and whose vertical component is its DEA score in the non-compensatory scenario. Among the 47 Spanish public universities in the data set, 18 are efficient in both scenarios, and thus, they share the dot placed in (1,1). Their names are colored in red in the figure. There are two other universities placed on the vertical line x = 1 , which means that they are efficient in the compensatory scenario, but not in the non-compensatory one (colored in green). This means that they perform efficiently in an overall fashion, but some sub-block (or indicator for the technology transfer mission) is performing inefficiently, given the inputs available. Conversely, there are two universities in the horizontal line y = 1 , colored in blue, which are efficient under the non-compensatory scenario, but they are inefficient in the compensatory scenario. This means that no single sub-block performs in a too poor way given the available inputs, but the overall performance is not good enough to reach efficiency. The rest of the universities are inefficient in both scenarios, with the inefficiency growing as they are placed more to the left and the bottom of the figure. The results show that, on average, the scores among the Spanish public universities are quite similar in both analyses (1.11 and 1.12, respectively). This indicates that, given their available inputs, Spanish universities should expand, on average, their outputs by 11% in the compensatory scenario and 12% in the non-compensatory one, in order to operate efficiently. If we look at the bisector line drawn in Fig. 6, we can see that most of these inefficient universities (17) are below the line, meaning that their inefficiency in the 1 3 non-compensatory scenario is greater than that of the compensatory scenario. This most common case happens when at least one sub-block is performing significantly worse than the rest, which gives hints to decision makers about fields of improvement. Finally, there are 8 universities above the line, which means that many sub-blocks or indicators are performing in an inefficient way, although none of them is significantly worse that the rest. In our opinion, these findings show the usefulness of simultaneously considering both scenarios for the DEA efficiency analysis.

Analyzing particular universities: how to move towards efficiency
In what follows, we will use the results obtained by two universities (UDC and UAB), to illustrate the information provided by combining the MRP-WSCI approach and DEA analysis and its usefulness for decision making purposes. As mentioned in "Available system of indicators", in this paper, two DEA analyses with different outputs are performed, while the inputs are the same in both analyses (Ratio of faculty members with PhD, R &D funds per faculty member, Staff expenses per students and Expenses on good and services per student). Our analysis shows that UAB is efficient in the compensatory scenario, while UDC is efficient in the non-compensatory one.
The achievement functions of the individual indicators, the WCI of each sub-block and the WWCI and SWCI of each mission are provided in Table 3. First, using the MRP-WSCI approach, the performance of both universities is analyzed in Table 3a. The scores below 1 appear in red (values worse than the reservation, "poor" performance level); the scores between 1 and 2 are in yellow (values better than the reservation level, but worse than the aspiration level, "fair" performance level); and the scores over 2 are highlighted in green (values better than the aspiration level, "good" performance level). Then, the inputs and  Table 3b.
As can be observed in Table 3a, UAB is one of the Spanish public universities with the lowest value of the SCI in the technology transfer mission. Note that this mission has no sub-blocks. Then, the construction of the MRP-WSCI is made in one step, where the SCI reflects the worst value achieved by the Spanish universities in the individual indicators of the technology transfer mission, relativized by the weight of the indicator. UAB performs poorly in the highest weighted technology transfer indicator (value of its SCI in the National patents per doctor is 0.22).
On the other hand, UDC also has its SCI value worse than the reservation level in the technology transfer mission. However, its SCI value is much better than this of UAB (0.84 and 0.22, respectively). Specifically, only 13 Spanish public universities manage to have a better SCI value in the above mentioned mission. Besides, UDC is performing worse in the indicator PCT extensions per doctor, whose weight is less than the National patents per doctor.
Furthermore, the worst values achieved by UDC in the research and teaching missions are better than the corresponding reservation level but worse than the aspiration level ("fair" performance level). With respect to the research mission, UAB has a better SCI value than UDC (1.93 and 1.18, respectively). This is because, the worst performance of UAB takes place in the other research activities sub-block, which is the lowest weighted sub-block of the research mission, while in the publications and projects sub-blocks, UAB has a good performance (among all the Spanish public universities, it has the second best performance, just worse than UPF).
Regarding the compensatory scenario, UAB compensates its unfavorable performance in the National patents per doctor with a better value in the rest of technology transfer indicators, and its WCI in the technology transfer is in the "fair" performance level (WCI = 1.04), although this of UDC is still better (WCI = 1.15). This is because, as commented before, while UAB performs poorly in the highest weighted indicator, UDC performs poorly in a less weighted indicator, being its value in the PCT extensions per doctor indicator one of the best among all the Spanish public universites. Therefore, UDC manages to compensate better than UAB in the technology transfer mission. However, UAB performs much  Table 3b, despite being the inputs levels of UAB higher than this of UDC, with the exception of the Staff expenses per students, UDC makes an efficient use of its resources in the non-compensatory scenario, while in the compensatory one, UDC is inefficient, being UAB efficient. Summing up, when the overall performance (WWCI) is analyzed, UAB is in the "fair" performance level for the teaching and technology transfer missions (1 < WWCI < 2), and in the "good" performance level for the research mission (WWCI > 2), while UDC is in the "fair" performance level for all the three missions. Specifically, the non-compensatory analysis shows that UAB has still much room for improvement in the technology transfer mission. Thus, these findings should encourage UAB to pay greater attention to the technology transfer mission. Moreover, UDC does not perform poorly in any sub-block, since its worst performance is not that bad, but given its input levels, UDC should improve its overall performance.
Based on the information obtained in the previous analysis, decision makers can be provided with a tool to simulate improvements in certain indicators and see their impact on the values of the composite indicators and on the efficiency scores. As an example, we will now make a proposal for UDC. As previously mentioned, this university is inefficient for the compensatory indicator, with an efficiency score of 1.131. This implies that it must experience an improvement of about 13.1% in the WWCI of each of the missions to achieve efficiency (if there is no slack in any of the inputs). Table 4 shows a sensible proposal on how this improvement could be achieved. In the table, the indicators that have been improved in the simulation have been highlighted and for them the improved value is shown next to the current one. In addition, the impact (absolute and in percentage) of these improvements in the composite indicators of the sub-blocks and missions can be seen. The strategy followed has been to improve indicators with worse behavior and those with good behavior and with high weights. Among them, those that are estimated to be most immediately improvable have been chosen. As a result of the improvements in the chosen indicators, a fairly homogeneous improvement has been achieved in all the sub-blocks, and just the desired one in each of the three missions. Obviously, this is a hypothetical proposal and the real one would have to be implemented by the university authorities, but this simulation shows the potential usefulness of the tool provided by this methodology. For the case of the UAB, which is inefficient in the non-compensatory scenario, the simulation should be based more on improving the indicators and sub-blocks with the worst performance. Finally, a university that is inefficient in both scenarios should combine both strategies to improve its scores. As a conclusion, we can say that the combination of the MRP-WSCI composite indicators and DEA analysis provides a rich amount of information. First, the joint consideration of the compensatory and non-compensatory scenarios is highly suitable, where the former gives an overall measure of the universities, while the latter helps to detect the actions needed to improve the performances of the universities. Second, combining both scenarios with the DEA analysis allows universities to better understand their weaknesses and strengths and provides a valuable tool for decision making purposes.

Conclusions
The evaluation of multidimensional frameworks, such as the higher education field, is a complex decision making problem which involves multiple criteria. Within this context, multiple criteria decision making provides a set of tools and techniques that can support such complex decision making processes. In this paper, we propose a combination, in a robust environment, of the MRP-WSCI approach with the DEA analysis, where all the aspects regarded as relevant to assess university performance are included as outputs.
Specifically, in order to illustrate the information provided by combining both approaches in such complex decision-making process, we have used the data of 47 Spanish public universities for the academic year 2016-2017. It is important to point out that the main emphasis of this paper is on the methodological contribution, while the evaluation of the Spanish public university system serves as an illustration of the combination proposed. Anyway, in order to detect and improve important factors of the educational efficiency and effectiveness of universities, the approach proposed can be used as a supporting tool for decision making, allowing the decision makers to select appropriate solutions and identify weaknesses and strengths of the universities analyzed, using multiple criteria.
From the practical point of view, our results highlight the need to assess the university performance by using both compensatory and non-compensatory scenarios, where the overall performance measure is given by the former, while the latter allows the decision maker to detect improvement areas. Furthermore, combining the use of composite indicators with the DEA analysis gives us a more complete picture of the universities under assessment, allowing us to consider the inherently multi-faceted academic quality in higher education.
As further research lines, we would like to extend this analysis to a dynamic scheme, in order to study the evolution of the universities along a given period, either by using fixed reference levels for the whole period, or even changing these levels according to new requirements established by the academic authorities. Besides, it would also be interesting to make a wider comparison among universities from different countries, where the results will give an idea of the relative position of each Spanish university with respect to all international universities.