679

A publication of

CHEMICAL ENGINEERING TRANSACTIONS

VOL. 81, 2020 The Italian Association
of Chemical Engineering
Online at www.cetjournal.it

Guest Editors: Petar S. Varbanov, Qiuwang Wang, Min Zeng, Panos Seferlis, Ting Ma, Jiří J. Klemeš
Copyright © 2020, AIDIC Servizi S.r.l.
DOI: 10.3303/CET2081114
ISBN 978-88-95608-79-2; ISSN 2283-9216

Predicting Higher Education Outcomes with Hyperbox

Machine Learning: What Factors Influence Graduate
Employability?
Kathleen B. Avisoa, Jose Isagani B. Janairob, Rochelle Irene G. Lucasc, Michael
Angelo B. Promentillaa, Derrick Ethelbhert C. Yud, Raymond R. Tana,*
a
Chemical Engineering Department, De La Salle University, 2401 Taft Avenue, 0922 Manila, Philippines
b
Biology Department Department, De La Salle University, 2401 Taft Avenue, 0922 Manila, Philippines
c
Department of English and Applied Linguistics, De La Salle University, 2401 Taft Avenue, 0922 Manila, Philippines
d
Chemistry Department, De La Salle University, 2401 Taft Avenue, 0922 Manila, Philippines
raymond.tan@dlsu.edu.ph

A machine learning approach to predict university attributes that influence graduate employability is presented
in this work. The machine learning technique used here is the hyperbox model, which is based on the principle
of generating if / then rules to predict outcomes. The rule-based hyperbox model can be generated from
empirical data using a mixed integer linear programming model. This machine learning approach is applied to
the problem of predicting employability of chemical engineering graduates based on institutional attributes. The
analysis shows that research intensity and quality do not necessarily result in high employability.

1. Introduction
Artificial intelligence (AI) and machine learning (ML) have become commonplace tools in the modern world.
Different AI/ML techniques have been developed for use in many practical applications (Jordan and Mitchell,
2015). These techniques offer the prospect of improved decision-making in industry (Makridakis, 2017) and
government (Sharma et al., 2020). AI/ML tools such as artificial neural networks (ANN), support vector machines
(SVM), and Bayesian networks (BN) are powerful alternatives to classical statistical techniques. There is an
emerging body of literature on the use of AI/ML for the analysis of education data. Recent work has been
reported for both basic education (Abad and Chaparro Caso López, 2017) and higher education (Alyahyan and
Düştegör, 2020). Institutional attributes such as research intensity and research quality may indirectly influence
graduate employability, which is an important measure of higher education effectiveness (Gyenes, 2019).
However, a search of the Scopus database indicates that research on the use of AI/ML to analyse the
effectiveness of chemical engineering education is rare. Application of AI/ML can lead to new insights to improve
educational outcomes and career prospects.
In ML, prediction models are calibrated to fit a set of training data. The training procedure uses a second model
to optimize fit. This process is analogous to using least squares optimization to generate a regression equation.
Mathematical Programming (MP) models can be developed for training in ML. Mixed-integer linear programming
(MILP) models are a useful class of models for this purpose. The use of continuous and integer variables allows
MILP models to explore optimal and near-optimal solutions during supervised training (Voll et al., 2015). Notable
examples of the use of MILP in ML have been reported. Iannarilli and Rubin (2003) developed an MILP-based
feature selection technique for multiclass discrimination. Kim and Ryoo (2007) proposed an MILP model for
non-linear data separation, while Bal and Örkcü (2011) developed a multi-class classification approach based
on MILP. Xu et al. (2011) proposed a 0-1 programming model for attribute reduction. Yan and Ryoo (2017) also
proposed a 0-1 multi-linear programming approach for pattern recognition. Rudin and Ertekin (2018) used an
MILP approach for optimal rule generation, while Corrêa et al. (2019) proposed an approach for binary single-
group classification. MILP models have also been combined with other ML techniques such as rough sets
(Chang et al., 2019) and SVM (Labbé et al., 2019).

Paper Received: 29/04/2020; Revised: 13/06/2020; Accepted: 15/06/2020

Please cite this article as: Aviso K.B., Janairo J.I.B., Lucas R.I.G., Promentilla M.A.B., Yu D.E.C., Tan R.R., 2020, Predicting Higher Education
Outcomes with Hyperbox Machine Learning: What Factors Influence Graduate Employability?, Chemical Engineering Transactions, 81, 679-
684 DOI:10.3303/CET2081114
680

Xu and Papageorgiou (2009) developed a hyperbox-based ML approach for classification problems. Compared
to many black-box ML techniques, this approach has the advantage of generating transparent results (Yang et
al., 2015a), since the hyperboxes can be interpreted as if/then rules (Tan et al., 2020). The original algorithm
required repeated re-optimization of MILP models for a satisfactory fit. Maskooki (2013) proposed an improved
training algorithm with a reduced number of steps. An improved version of the hyperbox-based ML approach
was also developed to account for Type I (false positive) and Type II (false negative) prediction errors (Yang et
al., 2015a). Applications of hyperbox-based ML include business performance prediction (Xu and Papageorgiou,
2009), disease diagnosis (Yang et al., 2015b), and geological reservoir classification for CO2 storage (Tan et
al., 2020). The latter work improved the hyperbox-based ML approach for binary classification by (a) using
concentric hyperboxes to separate positive and negative samples, and (b) enabling both rule simplification and
attribute reduction.
In this paper, the hyperbox-based ML technique developed by Tan et al. (2020) is applied to the problem of
predicting graduate employability based on high-level university attributes. The rest of this paper is organized
as follows. Section 2 discusses the hyperbox concept. Section 3 gives the formulation of the MILP model for
generating the hyperbox decision model. Section 4 applies the methodology to predicting employability of
chemical engineering graduates in the UK. Section 5 gives the conclusions and prospects for future work.

2. Problem statement
The formal problem can be stated as follows:
● Given an information system with a set of criteria (I), which can be further divided into a set of condition
attributes (A) and a binary decision set (D);
● Given a set of samples (J) which have known performance levels relative to the set of condition attributes
(A) and final classification in the decision set (D);
● Given a predefined limit on the rate of false negatives;
● Given the minimum margin of separation between positive and negative samples for each given criterion;
The objective is to determine the boundaries of the hyperbox which will minimize the number of false positive
results based on the training data set. The concept of hyperbox-based ML for binary classification is illustrated
in Figure 1 for a problem with two criteria. The hyperboxes should enclose the positive samples while excluding
the negative samples. Each hyperbox has an error margin; any sample that falls within this margin is considered
as misclassified. It can be seen that one positive sample is misclassified, because it does not fall within any of
the hyperboxes. One of the negative samples is also misclassified, since it is enclosed in the error margin of
hyperbox 2.
The hyperboxes can also be interpreted as if/then rules (Tan et al., 2020). Hyperbox 1 can be specified using
two-sided inequalities in both dimensions. Hyperbox 2 can be specified using one-sided inequalities (upper
bounds only), and is said to be semi-infinite in both dimensions. Dimension reduction is illustrated by hyperbox
3, which can be projected to a one-dimensional space along x to make y irrelevant. The next section describes
the MILP model (Tan et al., 2020) to generate the hyperboxes.

Figure 1: Illustration of hyperbox concept (adapted from Tan et al., 2020)

 681

3. MILP for generating Hyperbox Decision Model

The objective of the optimization model is to minimize , as shown in Eq(1), where  is the ratio of the total
number of false positives (∑𝑗 𝐹𝑃𝑗 ) to the total number of negative samples in the training data. Eq(2) indicates
that the ratio of false negatives (∑𝑗 𝐹𝑁𝑗 ) to the total number of positive samples in the training data () should
be less than a predefined threshold . Eq(3) and Eq(4) define  and . A sample j is counted as a false positive
if the hyperbox model classifies it as positive (𝑐𝑗 = 1) when its true classification is negative (𝐶𝑗∗ = 0), as
indicated in Eq(5). The reverse is true for false negatives, as shown in Eq(6).
𝛼 (1)

𝛽≤𝜀 (2)
∑𝑗 𝐹𝑃𝑗
𝛼= (3)
𝑇𝑁
∑𝑗 𝐹𝑁𝑗
𝛽= (4)
𝑇𝑃

𝐹𝑃𝑗 ≥ 𝑐𝑗 − 𝐶𝑗∗ ∀𝑗 (5)

𝐹𝑁𝑗 ≥ 𝐶𝑗∗ − 𝑐𝑗 ∀𝑗 (6)

Eq(7) and Eq(8) are meant to determine the outer boundaries of the hyperboxes, while Eq(9) and Eq(10) are
𝐿 𝑈
meant to determine the inner boundaries. 𝑋𝑗𝑖 is the performance of training data point j in criterion I; 𝑥𝑖𝑘 and 𝑥𝑖𝑘
are the lower and upper boundaries of hyperbox k for criterion I;  is the distance between the inner and outer
limits of the hyperbox; 𝑏𝑗𝑘 is a binary variable which indicates if sample j is enclosed within hyperbox k; and M
is an arbitrary large number.
𝐿
𝑋𝑗𝑖 > 𝑥𝑖𝑘 − ∆ − 𝑀(1 − 𝑏𝑗𝑘 ) ∀ 𝑗, 𝑖 (7)

𝑈
𝑋𝑗𝑖 < 𝑥𝑖𝑘 + ∆ + 𝑀(1 − 𝑏𝑗𝑘 ) ∀ 𝑗, 𝑖 (8)

𝐿
𝑋𝑗𝑖 > 𝑥𝑖𝑘 − 𝑀(1 − 𝑏𝑗𝑘 ) ∀ 𝑗, 𝑖 (9)

𝑈
𝑋𝑗𝑖 < 𝑥𝑖𝑘 + 𝑀(1 − 𝑏𝑗𝑘 ) ∀ 𝑗, 𝑖 (10)
𝐿
Eq(11) and Eq(12) account for the possibility that boundaries may not exist for criterion i in hyperbox k. 𝑍𝑖𝑘 and
𝑈 𝐿 𝑈
𝑍𝑖𝑘 approximate −∞ and +∞ (or very large negative and positive values); 𝑏𝑖𝑘 and 𝑏𝑖𝑘 are binary variables which
𝐿 𝑈
indicate whether the lower boundary (𝑏𝑖𝑘 = 1) or upper boundary (𝑏𝑖𝑘 = 1) for criterion i in hyperbox k
disappears. Eq(13) and Eq(14) account for samples that lie outside of the outer boundaries of the hyperbox k;
𝐿 𝑈
𝑞𝑖𝑗𝑘 and 𝑞𝑖𝑗𝑘 are binary variables indicating that the performance of sample i in criterion j is less than the lower
𝐿 𝑈
limit (𝑞𝑖𝑗𝑘 = 1) or more than the upper limit (𝑞𝑖𝑗𝑘 = 1) set for hyperbox k. Eq(15) and Eq(16) define when a
sample is enclosed in hyperbox k; 𝑏𝑗𝑘 is a binary variable which takes a value of 1 when sample j is enclosed
within hyperbox k and 0 otherwise. Eq(17) considers a sample to belong in the positive decision set (𝑐𝑗 = 1) if
it is enclosed by at least one hyperbox. Eq(18) defines all the binary variables in the model.
𝐿 𝐿 𝐿 𝐿 𝐿
𝑍𝑖𝑘 − 𝑀(1 − 𝑏𝑖𝑘 ) ≤ 𝑥𝑖𝑘 ≤ 𝑍𝑖𝑘 + 𝑀𝑏𝑖𝑘 ∀ 𝑖, 𝑘 (11)

𝑈 𝑈 𝑈 𝑈 𝑈
𝑍𝑖𝑘 − 𝑀𝑏𝑖𝑘 ≤ 𝑥𝑖𝑘 ≤ 𝑍𝑖𝑘 + 𝑀(1 − 𝑏𝑖𝑘 ) ∀ 𝑖, 𝑘 (12)

𝐿 𝐿
𝑋𝑗𝑖 ≤ 𝑥𝑖𝑘 − ∆ + 𝑀(1 − 𝑞𝑖𝑗𝑘 ) ∀ 𝑗, 𝑖 (13)

𝑈 𝑈
𝑋𝑗𝑖 ≥ 𝑥𝑖𝑘 + ∆ − 𝑀(1 − 𝑞𝑖𝑗𝑘 ) ∀ 𝑗, 𝑖 (14)

𝐿
∑𝑖 (𝑞𝑖𝑗𝑘 𝑈
+ 𝑞𝑖𝑗𝑘 ) ≤ 𝑀(1 − 𝑏𝑗𝑘 ) ∀ 𝑗, 𝑘 (15)

𝐿
∑𝑖 (𝑞𝑖𝑗𝑘 𝑈
+ 𝑞𝑖𝑗𝑘 ) ≥ (1 − 𝑏𝑗𝑘 ) ∀ 𝑗, 𝑘 (16)

∑𝑘 𝑏𝑗𝑘 ≤ 𝑀𝑐𝑗 ∀ 𝑗, 𝑘 (17)

682

𝑈 𝐿 𝑈 𝐿
𝑏𝑗𝑘 , 𝑏𝑖𝑘 , 𝑏𝑖𝑘 , 𝑄𝑖𝑗𝑘 , 𝑄𝑖𝑗𝑘 , 𝐶𝑗 ∈ {0,1} (18)
There are two phases in the developed methodology. The first phase is to split the data set into the training and
validation sets. The training data set is then used to generate the hyperboxes. The second phase tests the
resulting bianary classifier on the validation data set.

4. Case study: Predicting employability

This case study analyzes the employability of chemical engineering graduates of different UK universities.
Industry expections are an important measure of educational outcomes (Gyenes, 2019). The data used were
reported by Gonzalez-Garay et al. (2019). Five university attributes are considered: (1) entry standards, (2)
research intensity, (3) staff to student ratio, (4) budget per student, and (5) research quality. The decision
attribute is employability, which is transformed into binary form using a threshold value of 80 %. The total number
of samples was 25. From this data set, 15 samples were used as training data (Table 1), while the other 10
samples were used as validation data (Table 2).
The hyperbox-based ML approach was used to derive empirical if/then rules from Table 1. Supervised training
was done by solving the MILP using the optimization software LINGO. Predictive performance was gauged
using the validation data in Table 2. Different classifiers can also be generated by using integer cuts. Expert
knowledge can be used to determine if the model produces genuine insights, or merely detects spurious
patterns. These alternative classifiers are not shown here due to space constraints.

Table 1: Training data (adapted from Gonzalez-Garay et al., 2019)

Entry Research Staff/Student Budget per Research Employability
Standards Intensity student Quality
Portsmouth 134 0.580 0.052 3 2.46 1
Surrey 152 0.800 0.065 4 2.98 0
Teesside 113 0.260 0.061 4 2.63 0
Herriot Watt 165 0.860 0.065 5 3.30 1
Sheffield 157 0.880 0.045 5 3.06 1
Bath 203 0.880 0.052 4 3.08 0
Imperial College 233 0.990 0.057 9 3.34 1
Edinburgh 197 0.910 0.067 8 3.30 0
Swansea 141 0.830 0.053 4 3.29 1
Queens Belfast 154 1.000 0.063 5 2.99 1
Leeds 189 0.830 0.078 5 2.97 1
West of Scotland 161 0.450 0.056 4 2.48 0
Lancaster 157 1.000 0.082 7 3.08 1
Cambridge 235 1.000 0.089 10 3.38 1
Hull 111 0.530 0.058 2 2.82 1

Using the training data, several scenarios were considered by varying the value of  from 1.0 to 0.0 in
increments of 0.2, with  = 0.05 and using three hyperboxes. The resulting rules were then tested on the
validation data. The resulting values for  and  both for the training and validation data are summarized in
Table 3.
The three hyperboxes shown in Table 4 can be translated into three rules as follows:
● Rule 1: IF (140.2 ≤ Entry Standards ≤ 189.0) AND (0.49 ≤ Research Intensity) AND (4.4 ≤ Budget per
Student ≤ 7.6) THEN (Employability = 1)
● Rule 2: IF (0.09 ≤ Staff/Student Ratio) AND (Research Quality ≤ 2.58) THEN (Employability = 1)
● Rule 3: IF (Entry Standards ≤ 134.0) AND (Staff/Student Ratio ≤ 0.06) AND (Research Quality ≤ 3.24)
THEN (Employability = 1)
These rules are disjunctive. Since there are three hyperboxes, a sample is classified as Employability = 1 if it
satisfies at least one of the rules indicated above. This set of rules gives good prediction performance, with 78 %
balanced accuracy, 56 % sensitivity, and 100 % specificity. One key finding is that high employability does not
necessarily result from high research intensity or quality. A possible reason for this surprising result is that the
benefits of university research accrue more to postgraduate rather than undergraduate programs. This result
needs to be examined further in future work.
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Table 2: Validation data (adapted from Gonzalez-Garay et al., 2019)

Entry Research Staff/Student Budget per Research Employability
Standards Intensity student Quality
Strathclyde 225 0.870 0.040 3 3.03 1
Nottingham 174 0.860 0.064 7 3.16 1
Loughborough 160 0.930 0.067 4 3.08 0
Birmingham 190 1.000 0.056 9 3.14 1
Newcastle 162 0.740 0.059 5 3.02 1
UCL 187 0.980 0.066 9 3.12 1
Bradford 124 0.280 0.078 2 2.69 1
London South Bank 108 0.910 0.037 2 2.59 1
Aston 127 0.610 0.053 5 2.83 1
Manchester 186 0.970 0.047 5 3.20 1

Table 3: Performance of model for training and validation data sets with varying 
 Training Data Validation Data
   
1.0 0.0 1.0 0.0 1.0
0.8 0.0 0.80 0.0 0.89
0.6 0.0 0.50 0.0 0.78
0.4 0.0 0.30 0.0 0.44
0.2 0.0 0.10 0.0 0.89
0.0 0.0 0.0 0.0 0.67

The best validation performance occurs at  = 0.4. The dimensions of the different hyperboxes are summarized
in Table 4.

Table 4: Lower and upper bounds for criteria

Box 1 Box 2 Box 3
LL UL LL UL LL UL
Entry Standards 140.2 189.0 134.0
Research Intensity 0.49
Staff/Student 0.09 0.06
Budget per student 4.4 7.6
Research Quality 2.58 3.24

5. Conclusions
This paper has applied hyperbox-based ML for predicting the employability of chemical engineering graduates
based on UK university rankings. The rules generated had satisfactory predictive ability, as characterized by
78 % balanced accuracy, 56 % sensitivity, and 100 % specificity. Results show that research intensity and
quality do not necessarily result in high employability. The application highlights the capability of this approach
to generate classification rules which offer better insights than conventional black-box ML approaches. In the
future, the geographic scope can be broadened to larger regions, or even to a global scale, using public data
on world university rankings. Future work also can focus on the use of the same approach for other applications.
This technique can be useful for classification problems where transparent, interpretable rule-based models are
needed.

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